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Appliedcalc Summer Course - Distance Calculus
New! DMAT 431 - Computational Abstract Algebra with MATHEMATICA!
Asynchronous + Flexible Enrollment = Work At Your Own Best Successful Pace = Start Now!
Earn Letter of Recommendation • Customized • Optional Recommendation Letter Interview
Mathematica/LiveMath Computer Algebra Experience • STEM/Graduate School Computer Skill Building
NO MULTIPLE CHOICE • All Human Teaching, Grading, Interaction • No AI Nonsense
1 Year To Finish Your Course • Reasonable FastTrack Options • Multimodal Letter Grade Assessment
Applied Calculus [Business Calculus] Online Course Info - Distance Calculus Summer 2024 Online Course
Distance Calculus @ Roger Williams University offers Precalculus, Calculus I/II, Multivariable, Differential Equations, Linear Algebra, Probability Theory (Calculus-based Statistics) during every
Summer term.
Distance Calculus @ Roger Williams University operates 24/7/365 with open enrollment outside of the traditional academic calendar. We offer all of our courses during the Summer, Fall, Winter, before
semesters traditionally start, after semesters start, during vacation weeks ... I think you get the idea :)
If you wish to complete a Applied Calculus course online, make sure you take this course from a
regionally accredited college/university
so that the credits you earn from this course will actually transfer to your home college/university.
The free courses available from the MOOCs (Massive Open Online Courses) like
Khan Academy
MIT Open Courseware
, etc. are really excellent courses, but they do
result in transferrable academic credits from an accredited university!
There are more than a few actual colleges/universities offering Applied Calculus [Business Calculus] Online Course Info - Distance Calculus courses online. Be careful as you investigate these courses
- they may not fit your needs for actual course instruction and timing. Most require you enroll and engage your course during their standard academic semesters. Most will have you use a publisher's
"automated textbook" which is .... um .... well, if you like that kind of thing, then you have a few options over there at those schools.
Distance Calculus is all about real university-level calculus courses - that's all we do! We have been running these courses for 20+ years, so we know how to get students through the these courses
fast fast fast!
Here is a video about earning real academic credits in Applied Calculus from Distance Calculus @ Roger Williams University:
Earning Real Academic Credits for Calculus
Applied Calculus vs Calculus I
Applied Calculus course can best be described as a "single semester course introducing Differential and Integral Calculus to non-STEM majors".
This course has many names, all being equivalent:
• Applied Calc
• Survey of Calculus
• Liberal Arts Calculus
• Applied Calculus
• Business Calculus
• Brief Calculus
• Calculus for Life Science
At Distance Calculus, we call our "Applied Calculus" course as Applied Calculus - DMAT 201 - 3 credits.
Below are some links for further information about the Applied Calculus course via Distance Calculus @ Roger Williams University.
Distance Calculus - Student Reviews
Date Posted: May 3, 2018
Review by: James Holland
Courses Completed: Calculus I, Calculus II
Review: I needed to finish the Business Calculus course very very very fast before my MBA degree at Wharton started. With the AWESOME help of Diane, I finished the course in about 3 weeks, allowing
me to start Wharton on time. Thanks Diane!
Transferred Credits to: Wharton School of Business, University of Pennsylvania
Date Posted: Apr 29, 2020
Review by: Harlan E.
Courses Completed: Calculus I, Calculus II
Review: I did not do well in AP Calculus during my senior year in high school. Instead of trying to cram for the AP exam, I decided to jump ship and go to Distance Calculus to complete Calculus I.
This was awesome! I finished Calculus I in about 6 weeks, and then I kept going into Calculus II. I started as a freshman at UCLA with both Calculus I and II done!
Transferred Credits to: University of California, Los Angeles
Date Posted: Aug 23, 2020
Review by: Sean Metzger
Student Email: seanmetzger78@gmail.com
Courses Completed: Differential Equations
Review: A lifesaver. When I found out I needed a course done in the last weeks of summer I thought there was no way i'd find one available, but this let me complete the course as quickly as I needed
to while still mastering the topics. Professor always got back to me very quickly and got my assignments back to me the next day or day of. Can't recommend this course enough for students in a hurry
or who just want to learn at their own pace.
Transferred Credits to: Missouri University of Science and Technology | {"url":"https://www.distancecalculus.com/appliedcalc/summer-course/","timestamp":"2024-11-03T15:17:18Z","content_type":"text/html","content_length":"51629","record_id":"<urn:uuid:15b61231-2551-4053-ad4a-ce0db8308417>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00773.warc.gz"} |
Deformations and the p-adic Langlands program
Printable PDF
Department of Mathematics,
University of California San Diego
RTG Colloquium
Claus Sorensen
Deformations and the p-adic Langlands program
The proof of Fermat's Last Theorem established a deep relation between elliptic curves and modular forms, mediated by an equality of L-functions (which are analogous to the Riemann zeta function).
The common ground is Galois representations, and Wiles' overall strategy was to parametrize their deformations via algebras of Hecke operators. In higher rank the global Langlands conjecture posits a
correspondence between n-dimensional Galois representations arising from the cohomology of algebraic varieties and certain so-called automorphic representations of $GL(n)$, which belong in the realm
of harmonic analysis. There is a known analogue over local fields (such as the p-adic numbers $Q_p$) and one of the key desiderata is local-global compatibility. This naturally leads one to speculate
about the existence of a finer "p-adic" version of the local Langlands correspondence which should somehow be built from a "mod p" version through deformation theory. Over the last decade this
picture has been completed for $GL(2)$ over $Q_p$, and extending it to other groups is a very active research area. In my talk I will try to motivate these ideas, and eventually focus on deformations
of smooth representations of $GL(n)$ over $Q_p$ (or any p-adic reductive group). It seems to be an open problem whether universal deformation rings are Noetherian in this context. At the end we
report on progress in this direction (joint with Julien Hauseux and Tobias Schmidt). The talk only assumes familiarity with basic notions in algebraic number theory.
Organizers: Algebra/Algebraic Geometry/Number Theory RTG Group
May 11, 2016
4:00 PM
AP&M 7321 | {"url":"https://math.ucsd.edu/seminar/deformations-and-p-adic-langlands-program","timestamp":"2024-11-04T15:35:39Z","content_type":"text/html","content_length":"34342","record_id":"<urn:uuid:2dc01cc1-b07b-40c3-805a-6d8b3c6d39ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00678.warc.gz"} |
Second-Order Integrated Rate Law Calculator - GEGCalculators
Second-Order Integrated Rate Law Calculator
Second-Order Integrated Rate Law Calculator
1. How do you find the integrated rate law in second order? The integrated rate law for a second-order reaction shows how the concentration of a reactant changes with time.
2. How do you calculate second order half-life? The half-life of a second-order reaction is calculated as the time it takes for the initial concentration of the reactant to decrease to half of its
original value.
3. What is the Y intercept of the second order integrated rate law? The Y-intercept of the second-order integrated rate law represents the initial concentration of the reactant.
4. What is the formula for integrated rate law? The integrated rate law is an equation that relates the concentration of a reactant to time for a specific reaction order.
5. What is the formula for the second order system? The formula for a second-order reaction system describes how the concentration of a reactant changes over time.
6. What is the general equation for a second order reaction? The general equation for a second-order reaction describes how the concentration of a reactant decreases over time in a specific manner.
7. What is the second-order rate law? The second-order rate law describes how the rate of a reaction depends on the concentrations of the reactants, specifically for second-order reactions.
8. Does half-life increase for second-order? No, the half-life of a second-order reaction is not fixed and can vary depending on the initial concentration and rate constant. It generally decreases
as the initial concentration decreases.
9. What is the unit of rate constant for a second-order reaction? The unit of the rate constant for a second-order reaction depends on the specific units used for concentration and time in the
integrated rate law. It doesn’t have a fixed unit without specific context.
10. What are the integrated rate laws for first and second order reactions? The integrated rate laws for first and second-order reactions describe how the concentration of a reactant changes with
time in those specific reaction orders.
11. What is K in a rate law? In a rate law, K represents the rate constant, which is a constant that relates the rate of a chemical reaction to the concentrations of reactants.
12. What is 2nd order reaction? A second-order reaction is a chemical reaction in which the rate of reaction depends on the square of the concentration of one reactant or the product of the
concentrations of two reactants.
13. What is the integrated rate law simplified? The integrated rate law is a simplified equation that relates reactant concentration to time for a specific type of chemical reaction.
14. How to calculate the rate law? The rate law is determined experimentally by measuring how the rate of a reaction depends on the concentrations of reactants and products.
15. What is the equation of first order and second-order? First-order and second-order equations describe how the concentration of a reactant changes with time in specific reaction orders.
16. What is the second order system theory? In system theory, a second-order system refers to a dynamic system that can be described by second-order differential equations, often related to the
behavior of physical systems.
17. What are first and second-order equations? First and second-order equations are mathematical equations that describe different types of relationships or behaviors in various contexts, including
physics and chemistry.
18. What does 2nd order mean? “2nd order” typically refers to a second-order reaction or system, where the rate or behavior is influenced by a power of 2, such as the square of a concentration.
19. What is the difference between first and second rate law? The difference between first and second-rate laws is that they describe the rate of chemical reactions of different orders. First-rate
law is for first-order reactions, while second-rate law is for second-order reactions.
20. What is 2nd order kinetics? Second-order kinetics describe the behavior of chemical reactions where the rate depends on the square of the concentration of a reactant or the product of
concentrations of two reactants.
21. Can rate constant K be negative? Rate constant K is typically not negative because it represents a positive constant that relates reactant concentrations to the rate of reaction.
22. What is K in half-life equation? In the half-life equation, K represents the rate constant of a chemical reaction. It is used to calculate the time it takes for a substance to decrease to half
its initial concentration.
23. What is the second-order rate constant quizlet? Quizlet is a flashcard and study tool platform, and “second-order rate constant” would likely be a term used in the context of chemistry flashcards
or study materials.
24. What are the units for the rate constant for each reaction order? The units for the rate constant depend on the overall order of the reaction and the specific units used for concentration and
time in the rate equation. The units can vary.
25. What are the units for the rate of reaction? The units for the rate of reaction depend on the specific reaction and can vary widely. Common units include M/s (molarity per second) for chemical
26. What is the difference between first order and second-order system? The difference between first-order and second-order systems lies in their dynamic behavior and the order of their differential
equations, typically describing physical systems.
27. How do you know if a reaction is first order or second-order? The order of a reaction is determined experimentally by studying how the rate of the reaction changes with the concentration of the
reactants. First-order reactions have a rate proportional to the concentration of one reactant, while second-order reactions have rates proportional to the concentration of two reactants or the
square of one reactant’s concentration.
28. How do you find the rate constant? The rate constant (K) is found experimentally by measuring the rate of reaction at different reactant concentrations and then using the rate equation to
calculate K.
29. How many steps are in a second order reaction? Second-order reactions typically involve a single step or elementary reaction, where the collision of reactant molecules leads to the formation of
30. Does second order reaction complete? Second-order reactions can reach completion, like any other chemical reaction, depending on the initial concentrations of reactants and the reaction
31. What is zero order? Zero-order reactions are chemical reactions in which the rate is independent of the concentration of reactants. The rate remains constant regardless of changes in
32. How do you find M and N in rate law? The values of M and N in a rate law are determined experimentally by studying how the rate of the reaction changes with the concentrations of the reactants.
They represent the reaction order with respect to each reactant.
33. Is second order reaction faster? A second-order reaction can be faster or slower than other reactions, depending on the specific reaction and conditions. Reaction speed is influenced by factors
like concentration and temperature.
34. Does second order reaction double? A second-order reaction does not necessarily double in rate when the concentration of a reactant doubles. The rate of a second-order reaction is often
proportional to the square of the concentration.
35. What is a 1 and 2 order reaction? A 1st order reaction is one where the rate depends on the concentration of one reactant, while a 2nd order reaction is one where the rate depends on the square
of the concentration of one reactant or the product of the concentrations of two reactants.
36. What is the integrated law? The integrated law refers to the integrated rate law, which is an equation describing how the concentration of a reactant changes over time in a chemical reaction.
37. What is the integrated rate? The integrated rate refers to the concentration of a reactant at a specific time, as described by the integrated rate law.
38. What is the first integrated rate law? The first integrated rate law describes how the concentration of a reactant changes over time in a first-order reaction.
39. What is the basic rate law equation? The basic rate law equation relates the rate of a chemical reaction to the concentrations of reactants and may vary depending on the order of the reaction.
40. How do you calculate initial rate? The initial rate of a reaction is determined by measuring the rate of reaction at the very beginning, typically at time zero, when reactant concentrations are
their initial values.
41. What is first order second-order in math? In mathematics, “first order” and “second-order” can refer to the order of differential equations or the order of mathematical operations. They are not
specific to chemical reactions.
42. Is second order logic set theory? Second-order logic is a topic in mathematical logic that extends first-order logic to include quantification over sets or predicates, but it is not the same as
set theory.
43. Is a second-order system always stable? Second-order systems can be stable or unstable, depending on the values of their parameters. Stability is determined by analyzing the roots of the
characteristic equation associated with the system.
44. How do you write a second-order difference equation? A second-order difference equation is a mathematical equation that describes the evolution of a discrete-time sequence or system. It typically
involves the current, previous, and second-previous values in the sequence.
45. How do you solve a second order differential equation? Solving a second-order differential equation involves finding a function that satisfies the equation by integrating it twice and applying
appropriate initial or boundary conditions.
46. What is the formula for the integrating factor of a second order differential equation? The integrating factor for a second-order differential equation depends on the specific form of the
equation and cannot be expressed as a single formula without context.
47. Why do we use second order logic? Second-order logic is used in mathematics and computer science to express more complex mathematical statements and to capture properties that cannot be expressed
in first-order logic.
48. How do you write second-order? “Second-order” is written as it appears here, with a hyphen between “second” and “order.”
49. What are second-order functions? Second-order functions can refer to mathematical functions or differential equations that involve the second derivative of a variable with respect to another
50. What is integrated rate equation? The integrated rate equation is an equation that relates the concentration of a reactant to time, showing how it changes over the course of a chemical reaction.
51. What is the difference between a rate law and an integrated rate law? A rate law describes the rate of a chemical reaction as it depends on reactant concentrations, while an integrated rate law
shows how the concentration of a reactant changes with time during the reaction.
52. How do you know if a rate law is first order? A rate law is first order if the rate of the reaction is directly proportional to the concentration of one reactant, as indicated in the rate
53. What is the second order rate law? The second-order rate law describes how the rate of a chemical reaction depends on the concentrations of reactants in a second-order reaction.
54. What is the second order formula? The second-order formula represents the mathematical relationship that governs the rate of a second-order reaction, which typically involves the square of a
reactant’s concentration.
55. What is the second order rate equation? The second-order rate equation is an equation that expresses the rate of a second-order chemical reaction in terms of the concentrations of reactants.
56. Why is rate always positive? Rates in the context of chemical reactions are typically positive because they represent the speed of a reaction, which is a non-negative quantity.
57. Why is rate 1 time? The statement “rate is 1 time” is not a standard phrase in chemistry or physics. Rate is usually expressed as a change in quantity per unit of time.
58. Why is rate law important? Rate laws are important in chemistry because they provide insights into the mechanisms of chemical reactions and allow for the prediction and control of reaction rates.
59. Does everything have a half-life? No, not everything has a half-life. Half-life is a concept that applies to certain processes, such as radioactive decay and chemical reactions, where substances
transform over time.
60. Why do we use half-life? Half-life is used to measure the time it takes for a substance to decay or reduce by half in various processes, including radioactive decay and chemical reactions.
61. What is the half-life of the pennies? The half-life of pennies is not a commonly discussed concept in chemistry. The half-life typically refers to the decay of radioactive isotopes or the
kinetics of chemical reactions.
62. How do you read a half-life graph? To read a half-life graph, you can identify the time it takes for a substance’s quantity or concentration to decrease to half its initial value by looking at
the graph’s decay curve.
63. Why is half-life exponential decay? Half-life is associated with exponential decay because the rate at which a substance decays is proportional to the amount of substance remaining, leading to a
consistent halving of quantity over regular intervals.
GEG Calculators is a comprehensive online platform that offers a wide range of calculators to cater to various needs. With over 300 calculators covering finance, health, science, mathematics, and
more, GEG Calculators provides users with accurate and convenient tools for everyday calculations. The website’s user-friendly interface ensures easy navigation and accessibility, making it suitable
for people from all walks of life. Whether it’s financial planning, health assessments, or educational purposes, GEG Calculators has a calculator to suit every requirement. With its reliable and
up-to-date calculations, GEG Calculators has become a go-to resource for individuals, professionals, and students seeking quick and precise results for their calculations.
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How to Change the Tick Length Of 3D Plot In Matplotlib?
To change the tick length of a 3D plot in matplotlib, you can use the ax.tick_params() method with the axis parameter set to 'x', 'y', or 'z' depending on which axis you want to change. You can then
set the desired tick length using the length parameter. For example, to change the tick length of the x-axis in a 3D plot, you can use ax.tick_params(axis='x', length=5) to set the tick length to 5.
Repeat this process for the y and z axes as needed to adjust the tick length of the plot.
How to adjust tick length in a 3D scatter plot using matplotlib?
You can adjust the tick length in a 3D scatter plot using matplotlib by setting the length of the ticks on the x, y, and z axes. Here is an example code snippet that shows how to adjust the tick
length in a 3D scatter plot:
1 import matplotlib.pyplot as plt
2 from mpl_toolkits.mplot3d import Axes3D
3 import numpy as np
5 # Generate random data
6 x = np.random.rand(100)
7 y = np.random.rand(100)
8 z = np.random.rand(100)
10 # Create a 3D scatter plot
11 fig = plt.figure()
12 ax = fig.add_subplot(111, projection='3d')
13 ax.scatter(x, y, z)
15 # Adjust the tick length on the x, y, and z axes
16 ax.tick_params(axis='x', which='major', pad=10, length=5)
17 ax.tick_params(axis='y', which='major', pad=10, length=5)
18 ax.tick_params(axis='z', which='major', pad=10, length=5)
20 plt.show()
In the code above, the ax.tick_params() function is used to adjust the tick length on the x, y, and z axes. The pad parameter controls the distance between the axis label and the tick label, while
the length parameter determines the length of the ticks. You can tweak these parameters to achieve the desired tick length in your 3D scatter plot.
How to customize tick length individually for x, y, and z axes in a 3D plot?
In order to customize tick length individually for x, y, and z axes in a 3D plot, you will typically need to access the individual axes objects and set the tick parameters accordingly. Below is an
example using matplotlib in Python:
1 import matplotlib.pyplot as plt
2 from mpl_toolkits.mplot3d import Axes3D
4 # Generate some data
5 import numpy as np
6 x = np.linspace(-5, 5, 100)
7 y = np.linspace(-5, 5, 100)
8 X, Y = np.meshgrid(x, y)
9 Z = np.sin(np.sqrt(X**2 + Y**2))
11 # Create a 3D plot
12 fig = plt.figure()
13 ax = fig.add_subplot(111, projection='3d')
15 # Plot the data
16 ax.plot_surface(X, Y, Z, cmap='viridis')
18 # Customize tick length individually for x, y, and z axes
19 ax.xaxis.set_tick_params(length=10)
20 ax.yaxis.set_tick_params(length=5)
21 ax.zaxis.set_tick_params(length=15)
23 plt.show()
In this example, we first create a 3D plot using matplotlib and generate some sample data. We then access the individual axes objects (ax.xaxis, ax.yaxis, ax.zaxis) to set the tick parameters using
the set_tick_params method. In this case, we set different tick lengths for each axis (10 for x, 5 for y, and 15 for z). Finally, we display the plot using plt.show().
What is the default unit for tick length in matplotlib plots?
The default unit for tick length in matplotlib plots is points (pt).
What is the recommended tick length for matplotlib plots?
The recommended tick length for matplotlib plots is typically around 5-10 points. However, the exact tick length can vary depending on the size of the plot and personal preference. It is recommended
to experiment with different tick lengths to find the one that works best for your specific plot.
How to change the tick length for specific axes in a 3D plot?
To change the tick length for specific axes in a 3D plot, you can use the Axes3D.tick_params() method in Matplotlib. Here's an example of how to change the tick length for the x-axis and y-axis in a
3D plot:
1 import matplotlib.pyplot as plt
2 from mpl_toolkits.mplot3d import Axes3D
4 # Create a 3D plot
5 fig = plt.figure()
6 ax = fig.add_subplot(111, projection='3d')
8 # Plot some data
9 ax.plot([1, 2, 3], [4, 5, 6], [7, 8, 9])
11 # Set the tick length for x-axis and y-axis
12 ax.tick_params(axis='x', length=5)
13 ax.tick_params(axis='y', length=5)
15 # Show the plot
16 plt.show()
In this example, ax.tick_params(axis='x', length=5) and ax.tick_params(axis='y', length=5) are used to set the tick length to 5 for the x-axis and y-axis, respectively. You can adjust the length
parameter to change the length of the ticks.
What is the recommended range for tick length in matplotlib plots?
The recommended range for tick length in matplotlib plots is typically between 0.1 to 0.3, where the tick length is measured in points. This range provides a good balance between visibility and
avoiding cluttering the plot with too many tick marks. However, the exact tick length that is best for a specific plot may vary depending on the size and complexity of the plot, so it is recommended
to experiment with different values to find the most suitable one for your specific situation. | {"url":"https://ubuntuask.com/blog/how-to-change-the-tick-length-of-3d-plot-in","timestamp":"2024-11-02T11:48:58Z","content_type":"text/html","content_length":"346005","record_id":"<urn:uuid:6c20fcb7-0f18-48e6-940f-77d4467f7dc3>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00481.warc.gz"} |
Normal Distribution - (Biostatistics) - Vocab, Definition, Explanations | Fiveable
Normal Distribution
from class:
Normal distribution is a continuous probability distribution characterized by a symmetric bell-shaped curve, where most of the observations cluster around the central peak and probabilities for
values further away from the mean taper off equally in both directions. This distribution is fundamental in statistics, as it helps to model various natural phenomena and is key in many statistical
methods and inference techniques.
congrats on reading the definition of Normal Distribution. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The normal distribution is defined by its mean (μ) and standard deviation (σ), with approximately 68% of the data falling within one standard deviation from the mean.
2. In biological research, many traits and measurements follow a normal distribution, making it easier to apply statistical tests that assume normality.
3. The area under the normal curve represents total probability, which equals 1. This property is essential for calculating probabilities and making inferences.
4. The normal distribution plays a critical role in constructing confidence intervals and hypothesis testing, providing a foundation for estimating population parameters.
5. Data transformations may be used to achieve normality, enabling researchers to use parametric tests that require normally distributed data.
Review Questions
• How does understanding normal distribution enhance your ability to analyze biological data?
□ Understanding normal distribution allows for effective analysis of biological data since many biological variables naturally conform to this pattern. It facilitates the use of parametric
statistical tests that assume normality, such as t-tests and ANOVA. By knowing how data typically behaves around the mean, researchers can make more accurate predictions and inferences about
populations based on sample data.
• Discuss how the properties of normal distribution relate to measures of central tendency and variability in biological research.
□ The properties of normal distribution are closely tied to measures like mean, median, mode, and standard deviation. In a perfectly normal distribution, these measures coincide at the center
of the curve, providing a clear understanding of the data's central tendency. Variability can be captured using standard deviation, which helps in interpreting how spread out data points are
around the mean. This connection aids researchers in evaluating data consistency and patterns within biological studies.
• Evaluate the implications of the Central Limit Theorem in relation to normal distribution when analyzing experimental data.
□ The Central Limit Theorem has significant implications when analyzing experimental data because it assures that, regardless of the original population distribution, the sampling distribution
of the sample means will tend toward a normal distribution as sample size increases. This property allows researchers to apply normal distribution techniques to make inferences about
populations even when their actual distributions are unknown or non-normal. Consequently, it enables more robust statistical analyses and enhances reliability in biological research findings.
© 2024 Fiveable Inc. All rights reserved.
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Project Euler Problem 7 Solution: Clojure
in Programming
Project Euler Problem 7 Solution: Clojure
What is the n-th prime number?
Problem Solution
This was simple to solve using the Sieve created for the solution to problem #3. To make the program variable, we start with finding all primes less than 1000. If that is not enough to solve the
problem, we double this “maximum” value, and search for all primes less than 2000. Rinse, repeat.
The old stuff:
; //en.wikipedia.org/wiki/Sieve_of_Eratosthenes
(defn multiples
[1 n]
(fn [a] (+ a n))
(rest (multiples n))
(defn sieve-of-eratosthenes
; step 1 Create a list of consecutive integers from two to n: (2, 3, 4, ..., n),
(defn sieve [rng p]
;(print "range: " rng "\n")
; this lambda used to perform the actual recursion
(fn [new-rng]
; step #5 Repeat steps 3 and 4 until p^2 is greater than n.
(if (<= (* p p) n)
; recurse
(sieve new-rng
; step # 4 Find the first number remaining on the list after p
; (it's the next prime); let p equal this number,
(first (filter #(> % p) new-rng))
; end recursion otherwise
; this is an argument to the lambda above
; step #3 While enumerating all multiples of p starting from p2,
; strike them off from the original list,
(not= ; divisible by p
(rem % p)
; greater than p^2
(< % (* p p))
(range 2 n)
2 ; step #2 Initially, let p equal 2, the first prime number
The new stuff:
(defn problem07 [nth]
(defn try-find-nth-prime [primes max-number]
(if (< (count primes) nth)
; if the list of primes is not long enough,
; double the maximum search number
(sieve-of-eratosthenes (* max-number 2))
(* max-number 2)
; otherwise, take the nth member of the list
(last (take nth primes))
(try-find-nth-prime (sieve-of-eratosthenes 1000) 1000)
1. Blog: Project Euler Prob… //bit.ly/baij5T #clojure #projecteuler
This comment was originally posted on Twitter
2. Hi Jamie,
I’m enjoying your euler series so far. Concerning you way of putting parentheses consider reading //stackoverflow.com/questions/1894209/how-to-read-mentally-lisp-clojure-code. It can be tempting
to think that it is easier to understand the code if you’re coming from a Java/C# background, but in fact it’s a little irritating
Otherwise, keep up the good work!
3. Hi Matt, I definitely see what you mean. Thanks for linking to an awesome post. I will try to follow these conventions in new posts!
• Readers who shared this
• Thank you! | {"url":"https://antiquity.jamie.ly/programming/project-euler-problem-7-solution-clojure/","timestamp":"2024-11-14T01:21:03Z","content_type":"text/html","content_length":"26622","record_id":"<urn:uuid:84dc0e6b-5247-4b17-91a4-41ba8b7a1d54>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00225.warc.gz"} |
Texas Instruments TI-88 INV Key Usage
All early Texas Instruments calculators, including the SR-52 and the TI-59 used the INV key for many functions: INV SBR to return from a subroutine, INV Ln to calculate the exponential, or INV EE to
cancel scientific display mode are just some of the examples of this key's usage. The TI-88 is no exception; there are many uses of the INV key on this machine, some of them less than obvious.
INV Σ+ Remove statistics data point
INV P-R Rectangular to polar conversion
INV OP nn Display OP code function (do not execute)
INV CMs Clear program memory
INV SBR Subroutine return
INV Log Common antilogarithm
INV Ln Natural antilogarithm
INV GFR Go backward relative
INV EE Cancel scientific display mode
INV Time Display date
INV Eng Cancel engineering display mode
INV Fix Cancel fixed-point display mode
INV Int Calculate fractional part
INV DRG Set to degrees mode
INV If> If less than
INV If= If not equal
INV Dsz Decrement and execute on zero
INV IfF If flag not set
INV StF Clear flag
INV sin Arc sine
INV cos Arc cosine
INV tan Arc tangent
INV Lst List data memories
INV Prt Advance printer
INV Trc Cancel trace mode | {"url":"http://airy.rskey.org/CMS/exhibit-hall/?view=article&id=109","timestamp":"2024-11-04T14:52:38Z","content_type":"application/xhtml+xml","content_length":"16703","record_id":"<urn:uuid:3dd8137a-be0c-4a9b-932f-9de8c00417fa>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00380.warc.gz"} |
ECCC - Reports tagged with frequency moments
Reports tagged with frequency moments:
TR05-125 | 2nd November 2005
Sofya Raskhodnikova, Dana Ron, Ronitt Rubinfeld, Amir Shpilka, Adam Smith
Sublinear Algorithms for Approximating String Compressibility and the Distribution Support Size
We raise the question of approximating compressibility of a string with respect to a fixed compression scheme, in sublinear time. We study this question in detail for two popular lossless compression
schemes: run-length encoding (RLE) and Lempel-Ziv (LZ), and present algorithms and lower bounds for approximating compressibility with respect to ... more >>>
TR08-024 | 26th February 2008
Paul Beame, Trinh Huynh
On the Value of Multiple Read/Write Streams for Approximating Frequency Moments
Revisions: 2
Recently, an extension of the standard data stream model has been introduced in which an algorithm can create and manipulate multiple read/write streams in addition to its input data stream. Like the
data stream model, the most important parameter for this model is the amount of internal memory used by ... more >>>
TR09-015 | 19th February 2009
Joshua Brody, Amit Chakrabarti
A Multi-Round Communication Lower Bound for Gap Hamming and Some Consequences
The Gap-Hamming-Distance problem arose in the context of proving space
lower bounds for a number of key problems in the data stream model. In
this problem, Alice and Bob have to decide whether the Hamming distance
between their $n$-bit input strings is large (i.e., at least $n/2 +
\sqrt n$) ... more >>>
TR12-178 | 18th December 2012
Paul Beame, Raphael Clifford, Widad Machmouchi
Sliding Windows With Limited Storage
Revisions: 1
We consider time-space tradeoffs for exactly computing frequency
moments and order statistics over sliding windows.
Given an input of length $2n-1$, the task is to output the function of
each window of length $n$, giving $n$ outputs in total.
Computations over sliding windows are related to direct sum problems
except ... more >>>
TR13-127 | 15th September 2013
Paul Beame, Raphael Clifford, Widad Machmouchi
Element Distinctness, Frequency Moments, and Sliding Windows
We derive new time-space tradeoff lower bounds and algorithms for exactly computing statistics of input data, including frequency moments, element distinctness, and order statistics, that are simple
to calculate for sorted data. In particular, we develop a randomized algorithm for the element distinctness problem whose time $T$ and space $S$ ... more >>>
TR15-083 | 14th May 2015
Omri Weinstein, David Woodruff
The Simultaneous Communication of Disjointness with Applications to Data Streams
Revisions: 1
We study $k$-party set disjointness in the simultaneous message-passing model, and show that even if each element $i\in[n]$ is guaranteed to either belong to all $k$ parties or to at most $O(1)$
parties in expectation (and to at most $O(\log n)$ parties with high probability), then $\Omega(n \min(\log 1/\delta, \log ... more >>>
TR16-111 | 20th July 2016
Amit Chakrabarti, Sagar Kale
Strong Fooling Sets for Multi-Player Communication with Applications to Deterministic Estimation of Stream Statistics
We develop a paradigm for studying multi-player deterministic communication,
based on a novel combinatorial concept that we call a {\em strong fooling
set}. Our paradigm leads to optimal lower bounds on the per-player
communication required for solving multi-player $\textsc{equality}$
problems in a private-message setting. This in turn gives a ... more >>> | {"url":"https://eccc.weizmann.ac.il/keyword/15156/","timestamp":"2024-11-12T13:52:02Z","content_type":"application/xhtml+xml","content_length":"23921","record_id":"<urn:uuid:d19a3948-b843-498d-ba5b-683cf870bc5d>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00090.warc.gz"} |
Discover and read the best of Twitter Threads about #arXiv
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Tuesday 4/17/12
Post loads to comments
Mobility Class tonight at 8pm
Way to go Hilary!
7 comments:
Evan said...
Load# (F)ixed (R)amped
A.Shishineh 220 (R) PR!
S.Hettinger 205# (F)
C.Twaddle 165 (R)
D.Weyrauch 155 (R) PR!
K.Kuadey 135# (F)
B.Lessler 135# (F) PR!
H.Dean 120 (R)
S.Rana 95 (F)
Sam, what was the movie! I left without ever finding out!
To spoil it for Sam I looked it up, and according to Wikipedia it was used in the opening credits for the CBS television series Tour of Duty.
I don't remember ever hearing of it.
Thanks Chris! Like you, I've never heard of the show..
PR's across the board for the whole class!!!
S Matthews 200 (R)PR!
K Palmisano 195 (R)PR!
A Fountain 255 (R)PR!
A Middleton 100 (R)PR!
S Hulin 220 (R)PR!
T Luz 200 (R)PR!
6:00 pm
S. Fountain 95(F)
V. Kurian 165(F)
J. Hibbard 195(R)PR!
J. Southworth 215(R)PR!
D. London 225(R)PR! + 255x1
J. Gipson 75(F)
S. Stephens 160(R)PR!
7:00 pm
M.Stephen 205 PR! (R)
M.Treas 195 PR! (205x1)
J.Shrader 195 PR! (R)
L.Brown 155 (160x3)
E.Dean 115 (F)
HY.Tom 140 (R)
Y.Schreiber 135 (F)
R.Leiberman 125 (F)
K.Baranowsky 120 (R)
A.Rigney 110 (115x4)
S.Wilks 100 (R) | {"url":"https://diesel-gym.blogspot.com/2012/04/tuesday-41712.html?showComment=1334695119662","timestamp":"2024-11-04T15:05:56Z","content_type":"application/xhtml+xml","content_length":"45158","record_id":"<urn:uuid:c710b5d5-aba3-4646-adbf-5c7758ec7979>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00557.warc.gz"} |
{-# OPTIONS --without-K --safe #-}
-- Definition of Monoidal Category
-- Big design decision that differs from the previous version:
-- Do not go through "Functor.Power" to encode variables and work
-- at the level of NaturalIsomorphisms, instead work at the object/morphism
-- level, via the more direct _⊗₀_ _⊗₁_ _⊗- -⊗_.
-- The original design needed quite a few contortions to get things working,
-- but these are simply not needed when working directly with the morphisms.
-- Smaller design decision: export some items with long names
-- (unitorˡ, unitorʳ and associator), but internally work with the more classical
-- short greek names (λ, ρ and α respectively).
module Categories.Category.Monoidal where
open import Categories.Category.Monoidal.Core public
open import Categories.Category.Monoidal.Bundle public | {"url":"https://agda.github.io/agda-categories/Categories.Category.Monoidal.html","timestamp":"2024-11-07T06:50:39Z","content_type":"text/html","content_length":"2754","record_id":"<urn:uuid:811f9f76-0c4e-4611-987d-cb8858c340b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00604.warc.gz"} |
Adding Mixed Numbers Worksheet
Adding Mixed Numbers Worksheet
3 7 6 4 6 regroup so that the fraction is less than 1. Basic arithmetic skill adding mixed numbers find each sum.
Subtract Mixed Numbers With Like Unlike Denominators Solve The Riddle Fractions Practice Fractions Worksheets Math Fractions Worksheets Math Fractions
1 2 5 add the whole numbers.
Adding mixed numbers worksheet. The arithmetic in these questions is kept simple and students can try to formulate the answers mentally without writing down calculations. You can also click on the
button to get a clue. 1 4 6 7 2 1 7 2 1.
1 2 3 1 4 6 1 add the whole numbers. 2 1 2 2 3 6 add the fractions. Our adding and subtracting fractions and mixed numbers worksheets are designed to supplement our adding and subtracting fractions
and mixed numbers lessons.
3 1 5 4 3 5 example 3. Add and reduce to lowest terms. Adding mixed numbers with like denominators below are six versions of our grade 4 fractions worksheet on adding mixed numbers which have the
same denominators.
Adding fractions with different denominators adding mixed numbers reducing fractions common core standards. These ready to use printable worksheets help assess student learning. 1 1 2 3 1 4 3 2 5 3 1
2 5 2 4.
6 1 5 3. Grade 5 number operations. More adding mixed numbers and improper fractions on a number line help your students practice adding mixed numbers and improper fractions with this printable
In this exercise your class will add like fractions using a three step process using number lines. Adding mixed numbers unlike denominators below are six versions of our grade 5 math worksheet on
adding mixed numbers where the fractional parts of the numbers have different denominators. Be sure to check out the fun interactive fraction activities and additional worksheets below.
These math worksheets are pdf files. Read the lesson on adding mixed numbers if you need help on how to add mixed numbers. Adding mixed numbers example 1.
Fill in all the gaps then press check to check your answers. These worksheets are pdf files. With this exciting myriad collection of pdf adding mixed fractions worksheets you re sure to become more
than just competent at adding mixed numbers and mixed numbers with proper and improper fractions.
Answers to adding mixed numbers 1 7 2 1 1 3 3 2 4 1 2 3 5 5 6 2. First add the fractions. Write fractions with a common denominator.
Adding mixed numbers worksheets. 2 1 4 2. Adding and subtracting mixed numbers adding and subtracting mixed numbers can be daunting but this worksheet helps by breaking the process down step by step.
Use the hint button to get a free letter if an answer is giving you trouble.
3 Worksheet Free Math Worksheets Third Grade 3 Fractions And Decimals Adding Mixed Numbers Li In 2020 Fractions Worksheets Adding Mixed Number Adding Mixed Fractions
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5 Adding And Subtracting Mixed Numbers With Like Denominators Worksheet In 2020 Fractions Worksheets Free Fraction Worksheets Free Math Worksheets
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Grade 5 Fractions Worksheet Adding Mixed Numbers Fractions Worksheets Adding Mixed Number Fractions
Fraction Worksheets Have Fun Adding Mixed Number Fractions Worksheets Mixed Numbers | {"url":"https://askworksheet.com/adding-mixed-numbers-worksheet/","timestamp":"2024-11-09T17:50:24Z","content_type":"text/html","content_length":"135204","record_id":"<urn:uuid:df6f0ec9-c870-4d94-bc72-4bb1883a735f>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00723.warc.gz"} |
50 Simplest Radical Form
SOLVEDWrite each expression in simplest radical
50 Simplest Radical Form. Web what is the square root of 50 in simplest radical form? To find the square root of 50 in simplest radical form, we need to determine if there are any.
SOLVEDWrite each expression in simplest radical
Created by sal khan and. In this example, we simplify √ (2x²)+4√8+3√ (2x²)+√8. Web answer 30 people found it helpful suspho √50 = √2*5*5 = √2 * 5² 5√2 is your answer are you willing to help me out
with more? Report flag outlined why not? The square root of 50 in simplest radical form is 5√2. Web to simplify the square root of 50 means to get the simplest radical form of √50. Since they are
exponents, radicals can be simplified using rules of exponents. 1, 2, 5, 10, 25, 50 step 2: The calculator finds the value of the radical. Enter the radical expression below for which you want to
calculate the square root.
Web enter the radical you want to evaluate. List factors list the factors of 50 like so: Choose evaluate from the topic selector and click to see. In this example, we simplify √ (2x²)+4√8+3√ (2x²)
+√8. √52 ⋅2 5 2 ⋅ 2 pull terms out from under the radical. Web the free calculator will solve any square root, even negative ones and you can mess around with decimals too! Replace the square root
sign ( √ ) with the letter r. How do you multiply two radicals? Enter the expression you want to convert into the radical form. 50 = 2*25 so sqrt(50) = 5*sqrt(2) Report flag outlined why not? | {"url":"https://wedgefitting.clevelandgolf.com/form/50-simplest-radical-form.html","timestamp":"2024-11-11T12:48:42Z","content_type":"text/html","content_length":"20209","record_id":"<urn:uuid:63ee0df3-4386-449f-b204-f29a3032cb86>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00054.warc.gz"} |
Speed of sound - Curvature of the Mind
I’ve finally written some code that works in IE8.0, chrome and firefox. Rather than drawing complex fractals and solutions to equally archane differential equations, I’m trying to build some simple
demos using more easily understood principles and mathematics.
Now this is a very simple model that just shows the general principles of how something like a shockwave can build up from basic building blocks. Before I go into too much description, give it a spin
and see if you can figure out what is going on. Try sliding the speed back to zero, then to one and up past one, racing supersonic!
I first encoutered the mathematics for this little demo in a beautiful Russian book of mathematics aimed at high school students. It demoed families of curves from physical problems and then used
basic calculus to find interesting related curves. It was a trancendent text and it introduced me to the theory of evelopes
Here is what’s going on. Imagine a little super bee sitting on a flower just beating his wings. The sound waves radiate in circles at the speed of sound around him. That’s what you see when you slide
the speed down to zero. The waves expand concentricly and uniformly in space.
Now our bee is done colecting polen and starts heading back to the hive at a leisurly pace. At each moment the sound waves still expand in a circle around him, but now that he is moving he’s not at
the same place when he emits the next one. Now things are no longer uniform. The waves of pressure are bunching up ahead of him, and spreading out behind.
Now his bee sense starts tingling and he kicks is in gear to get to a disturbance at the hive. He approaches the speed of sound. Now things get interesting as he hits the speed of sound. He still
emits sound the same way, but now by the time the next wave is released, he’s caught up with it, and the same with the one right after, and so on. All these waves keep piling up building a huge wave
of pressure right on top of him. In this model with non-interacting waves and constant velocity, the pressure wave is infinite. The wave still exists in more realistic models, but things like changes
in air pressure, temperature and density put a limit on how much pressure actually builds up.
Just a little more speed, push it, push it, and he’s through. Once he’s moving faster than sound, the waves can’t even catch up, and expand out in a cone behind him perfect to sneak up on wrong
doers, as noone can hear him outside of the cone training behind him, and by then it’s too late. POW!
In reality, this just shows where the simple model breaks down. There are many more effects that kick in and become important long before these speeds are reached, however I love how this simple
model using nothing more than addition and multiplication can explain some of the features of supersonic travel.
Related Images: | {"url":"https://curvatureofthemind.com/2010/11/25/speed-of-sound/","timestamp":"2024-11-11T12:54:46Z","content_type":"text/html","content_length":"37862","record_id":"<urn:uuid:a2840831-b17e-4e8f-a5bc-7cf45b8c9dd0>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00452.warc.gz"} |
Top 210+ Machine Learning Interview Questions and Answers 2021 [UPDATED]
Machine Learning Interview Questions and Answers
Machine Learning Interview Questions and Answers
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Q1) What do you understand by the Machine Learning?
Answer: It is the application of artificial intelligence that can provides systems are the ability to automatically can learn and improve from the experience without being explicitly programmed.
Machine learning focuses on the development of computer programs that can be access to data and use it’s learn for themselves.
Q2) What are the difference between supervised and unsupervised machine learning?
Answer: Supervised learning is requires training labeled datas. For example, in order to the classification (a supervised learning task), you’ll need to the first label the data you’ll use to the
train the model to classify data into your labeled groups. Unsupervised learning, in contrast, does not a require labeling data explicitly.
Q3) What is the difference between the Type I and Type II error?
Answer: Don’t think this as something high level of stuff, interviewers ask questions in such terms just to the know that you have all the bases and you are on the top.
Type I error is the false positive, while Type II is the false negative. Type I error is claiming on something has to happened when it hasn’t. For the instance, telling an man he is pregnant. On the
other hand, Type II error means you claim nothing is happened but in the fact something . To exemplify, you tell an pregnant lady she isn’t carrying baby.
Q4) Are expected value and mean value different?
Answer: They are not different but the terms are used in the different contexts. Mean are generally referred to when talking about an probability distribution or sample population whereas expected
value is the generally referred in the random of variable context.
Q5) What does P-value signify about the statistical data?
Answer: P-value is used to the determine the significance of the results after a hypothesis test in statistics. P-value helps to the readers to draw conclusions and is always between 0 and 1.
P- Value > 0.05 denotes weak to evidence against the null hypothesis which are means the null hypothesis cannot be rejected.
P-value <= 0.05 denotes strong to evidence against of the null hypothesis which means the null hypothesis can be rejected.
P-value=0.05 is the marginal value are indicating it is possible to go either way.
Q6) Do gradient descent methods of always converge to same point?
Answer: No, they do not because in some cases it reaches an local minima or a local optima points. You don’t reach to the global optima point. It depends on the data and starting the conditions.
Q7) What is the goal of A/B Testing?
Answer: It is a statistical hypothesis testing for the randomized experiment with two variables to A and B. The goal of A/B Testing is to the identify any changes to the web page to maximize or
increase the outcome of an interest. An example for this could be identifying for the click through rate for the banner ad.
Q8) What is Machine Learning?
Answer: The simplest way to the answer this question is – we give the data and equation to the machine. Ask to the machine look at the data and identify to the coefficient values in an equations.
For example for the linear regression y=mx+c, we give the data for variable x, y and the machine learns about to the values of m and c from to the data.
Q9) Python or R – Which one would you prefer for the text analytics?
Answer: The best possible answer for this would be Python because it has to Pandas library that provides easy to use data of structures and high performance of data analysis tools.
Q10) What is kernel SVM?
Answer: Kernel SVM is the abbreviated version of kernel support vector of machine. Kernel methods are a class of algorithms for pattern analysis and the most common one of the kernel SVM.
Q11) What kind of error will be solved by organizing?
Answer: In mechanical learning, regulation is the process of introducing additional information as a result of an incorrect phenomenon or to avoid additional material. It is basically a reuse form,
which evaluates or controls the value for zero. The regulating technique prevents the complexity or the flexible model to avoid the inappropriate risk.
Q12) What is data science?
Answer: Data Science uses automated methods to analyze and retrieve large quantities of data. By combining features of statistics, computer science, application mathematics and visualization, data
science can alter the wide range of data generated by the new digital intelligence and new knowledge of digital age.
Q13) What is Logistic Recession? An example of when you recently used the logistic backlash?
Answer: The Logistic Recreation is often referred to as the Registration Model is a technique to predict binary effects predictive variables from a linear combination.
For example, if you want to predict whether a particular political leader should succeed or not. In this case, the end of the forecast is binary ie 0 or 1 (success / loss). Here the predictive
variables are the amount spent for a particular candidate’s election campaign, the amount of time spent on the campaign, etc.
Q14) What are Recommended Systems?
Answer: Recommended Systems is a sub directory of information filtering systems, which predicts the preference or rankings offered by a user to a product. Recommendations are widely used in movies,
news, research articles, products, social tips, music, etc.
Q15) What is the difference between the rule of governance and the character of the fruit trees?
Answer: The difference is that the research on decision making trees assesses the quality of a certain number of intermediate set standards, while evaluating only the value of the evaluators.
Q16) What is Periberan in machine stones?
Answer: In machine learning, Perception is a step in the classification of an input supervision in many potential non-binary releases.
Q17) Explain Two Parts of the Bayes on Logic Plan?
Answer: There are two elements in the Bayesian logic project. The first component is a logical one; It is a collection of the Bayesian Klaus package, which captures the domain’s characteristic
structure. The second component is a criterion, which marks the amount of information about the domain.
Q18) What is Paysyni Networks (BN)?
Answer: Answer: The Poison Network is used to represent a graphical model for the probability relationship under the Variables.
Q19) Why is learning algorithm sometimes referred to as a laser learning algorithm?
Answer: The learning algorithm, based on music, is also referred to as a laser learning algorithm until they are aggravated by the stimulation or generalization process.
Q20) What are two types of methods that can handle SVM (support vector machine)?
• Connecting binary classifier
• Binary replacement with multiple courses
Q21) What to learn in computer?
Answer: To solve a particular computing plan, many models, such as classifiers or technicians, are strategically developed and connected. This process is known as group learning.
Q22) Why is learning in alcohol use?
Answer: Integration learning is used to improve the classification of a sample, prediction, and functional approximation.
Q23) When to use group learning?
Answer: Ensemble learning is used when you create a more accurate and independent component classifier for each other.
Q24) What are two forms of group systems?
There are two forms of group systems
• Continuous Group Methods
• co-operative systems
Q25) What is the general principle of a group system, what damage and inclusion?
Answer: The general principle of a group is to combine the computations of multiple models built with learning methodology to improve the weakness of a model. Group is a group to promote illegal
assessment or classification. Increasing the method for reducing the essence of the integrated model is used continuously. Error and decreasing firing errors by reducing time varies.
Q26) What is the difference between taxonomy errors in regular order?
Answer: A learning algorithm can be distinguished into an expected error function and variation. Measuring a dependent period, comparing the average classroom prepared by the learning algorithm to
the target dependence. The calculation of the duration of the varying time learning method provides a compatibility rate for a variety of exercises.
Q27) What is a Development Learning Algorithm in the Group?
Answer: The Advanced Learning System is an algorithmic ability to learn from the new data available since it has already created a database that has already been exported from the database.
Q28) What is PCA, KPCA and ICA?
Answer: The key feature is the extraction techniques used for the dimensional reduction of PCA (Primary Components Analysis), KPCA (Kernel-based Primary Component Analysis) and ICA (Independent
Component Analysis).
Q29) What is the dimensional reduction in machine learning?
Answer: In mechanical learning and statistics, the transfer reduction is a process of reducing random variables in calculations, and the feature feature and feature extraction
Q30) What are Supplement Vector Machines?
Answer: Learning methods used for classification and recession analysis of vector machines.
Q31) What are the elements of relevant assessment strategies?
Answer: Key elements of the relevant assessment strategies
• Data acquisition
• Ground Trude Acquisition
• cross-estimate technique
• question type
• metric marks
• a significant test
Q32) What are the different mechanisms for series monitoring surveys?
Answer: There are different methods to solve continuous supervision learning problems
• sliding window modes
• Repeat sliding windows
• hidden marco samples
• Maximum eighty Marco models
• Conditional random fields
• graphic transformer networks
Q33) Robotics and Information Processing Areas Continuous Computational Problem?
Answer: Robotics and information processing areas are in places where there is a constant computation problem
• fantasy learning
• Computed computation
• model-based reinforcement learning
Q34) What is statistical study?
Answer: Statistical learning techniques allow a function or predict from a set of permitted data that can make predictions about the future or future data. These techniques confirm the effectiveness
of a learning perspective on future unobtrusive data based on the statistical assumption of data creation process.
Q35) What is PAC learning?
Answer: BAA (perhaps approximate) Learning Learning algorithm has been introduced to introduce learning methods and their statistical capabilities.
Q36) Are you different categories that you can classify the sequence learning process?
• sequence
• line generation
• Row recognition
• continuous conclusion
Q37) What are two techniques of machine learning?
There are two techniques for machine learning
• Genetic programming
• Learning stimulation
Q38) Give a popular use of machine learning that you see on a daily basis?
Answer: The machine uses machine learning that is implemented by major eCommerce websites.
Q39) Please explain trade-off of Bias & variance.
Answer: All we wanted to have is low bias & low variance. But in reality, both is actually a trade-off. We can simply term it as the error introduced between the bias and variance.
Q40) What is Gradient descent means?
Answer: GD is one of the first order optimization algorithm that works better for minimum function to take steps propositional to the negative gradient functions.
Q41) What is Overfitting? Please explain in laymen term
Answer: Overfitting is a problem occurred when we have low error in the training set. But produces high error in test or unseen data.
Q42) What is Underfitting? Please explain in laymen term
Answer: Underfitting is a problem, when we have low error in both training set and the testing set. Few algorithms works better for interpretations. But fails for better predictions.
Q43) What is Curse of dimensionality?
Answer: COD is state which is commonly referred to lack of intuitive understanding of multiple dimensions. If a user wants to produce better understanding on data COD will make limitations.
Q44) What are all the methods for standardization?
• Range level standardization
• Standard deviation level standardization
Q45) What is data normalization?
Answer: Data normalization is a common practice to get the data features weighted equally. It causes to lose data interpretability.
Q46) Is missing data is just blanks?
Answer: No. Not only the blanks, data points which has NA, NULL and also sometimes the corrupted data that has been recorded by mistake or given improper data by purpose.
Q47) List of few clustering algorithms you are familiar with.
• K-means
• K-means++
• Hierarchical clustering
Q48) What is clustering?
Answer: Clustering technique is a segmentation process. It works whenever we don’t have the target variable and still wanted to have a groups created.
Q49) What is EDA to you?
Answer: EDA which refers to Exploratory Data Analysis is a process to understand the data prior getting it into machine learning pipeline.
Q50) Can you give some wise advice for selecting algorithms?
Answer: Selecting the algorithms required as per our problem is always tricky. But it is always good to start with linear regression for Regression and Logistic regression for Classification
Q51) What is class imbalance?
Answer: Class imbalance is something which most of all the classification problem falls on. It is always good to check the number of observations for each target variable. To be precise, it is
something like we get 990 cancer free patients and 10 cancer patients in the data set. While machine will learn a lot about those 990 cancer free patients. But high importance is for those 10
Q52) When we can do predictive analysis?
Answer: Predictive and Prescriptive analytics comes into picture only when descriptive and diagnostic analytics is successful and provide some value to the business.
Q53) When we can provide insights in a project?
Answer: Once the affecting factors are found, let’s make some prediction with machine learning algorithms. Once we feel the model is making sense out of our data. We will prescribe useful insight.
Q54) Name any industry that is drastically affected by Data science?
Answer: The retail industry is one among the few which is drastically impacted by data science and business analytics.
Q55) How do DS help in retailers?
Answer: Data Science helps retailers stay ahead in competition or at least on par with their competitors on selling goods to customers and also predictive analytics help them solve problems like
never before.
Q56) What is the best career advice can be given to fresher?
Answer: My advice would be to select a company where you can learn something new every single day. Like literally every single day should be a battle to learn something exciting and work on a problem
that can transform the business. You should always find a trade-off in life for multiple things but don’t compromise on this.
Q57) What is SVM and how do we create a portfolio with it?
Answer: SVMs are used for classification problems, and they are quite interesting as well. Get a classification dataset from UCI ML repo and start working on your portfolio.
Q58) What is data visualization?
Answer: Data Visualization doesn’t mean you can only use bar charts and line charts to display everything. There are many unconventional charts to display data. The ultimate flexibility is the
ability to change the backend data structure based on our front end requirements.
Q59) How to choose right chart type?
Answer: At times there won’t be much of freedom to create different charts because of backend data architecture or just because of business stakeholders stubborn affinity towards a chart type.
Q60) What is right to do with data visualization tools?
Answer: Recreating an excel table in tableau or any data visualization is an absolute waste of the tool’s capability. Instead, try finding a reason to highlight specific rows or for example,
calculate the difference in % and color rows based on it to show a highlight table.
Q61) How to avoid Bias?
Answer: Bias can cause to feel or show inclination or prejudice for or against someone or something. Avoiding bias in machine learning is very important, and the last thing we would want is to create
a model which will most of the times/always classify a non-defective product as a defective one.
Q62) What are the important outcomes of DS?
• Applications, whereby we use the model to perform a task, ideally as accurately and effectively as possible.
• Interpretation, whereby we use the model to gain insight into our data via the learned relationship between independent and dependent variables.
Q63) What is the trade-off between accuracy and interpretability?
Answer: There needs to be a trade-off between accuracy and interpretability. Neural networks spit out the best possible result, and we can’t ignore that just because we don’t understand the internal
functioning of the model.
Q64) What is important? Accuracy or interpretability?
Answer: We can’t solve every business problem with an interpretable model and at the same time vice versa holds good as well.
Q65) How much domain knowledge is required to do DS?
Answer: Domain knowledge and model building experience comes handy in this kind of situations. I worked in a sales driver model and only when I understood the business value point, feature
engineering became effective.
Q66) How much design will affect your work in DS?
• Design thinking matters a lot in the business analytics space. There needs to be a purpose for any visualisation we as professionals create.
• Business stakeholders will be fine with dashboards with only bar and line charts. Like seriously, you can use them to answer most of the questions, and they look familiar to users as well.
Q67) What is dumbbell chart?
Answer: Business Stakeholders won’t be even aware of dumbbell chart. To show the performance of a product between two years with contrasting colours will immediately grasp users attention than a
regular bar chart.
Q68) What is the goal of a Data science dashboards?
Answer: The goal is always to provide easy and user friendly visualization to end users and for that we need to understand the end users requirements and how they are friendly with charts and graphs
and overall dashboards and accordingly we have to deliver results and insights.
Q69) What is needed most? BI or DS?
Answer: Most of the companies need business intelligence, data analyst, data engineers and analysts more than data scientists at this point. Only when the infrastructure is built with known KPIs and
the trends in years, someone can come in and work on the unknown variables to push the business in the right direction to make critical decisions.
Q70) What are all the important R packages?
Answer: Tidyverse, broom, and lubridate for most of my work in data wrangling phase. At times once the data wrangling is done, I have also moved the machine learning part to python for leveraging
sckit-learn package.
Q71) Where do predictions depends on?
Answer: The insights/predictive results should not wholly depend on the beta coefficients of the model. It should be backed something more – Business and Statistics.
Q72) When do Simpson’s paradox occurs?
Answer: Simpson’s paradox occur while working on marketing problems with 100s of features impacting the sales unit. Just believing in the beta values might lead us to the wrong conclusion which can
potentially cost the business to spend millions on different channels than the right ones.
Q73) How long will it take to build ML model?
Answer: Building a model doesn’t take much of your time but evaluating it and making it the right suitable model takes time and other elements as mentioned earlier.
Q74) Where can R complement R?
Answer: R will complement your learning from the stats book, and you can play with sample datasets like iris, mtcars to check out the importance of descriptive statistics
Q75) Is DS is actually business of science?
Answer: Data science is more “Business” than “Science”. Don’t emphasis on tools and technologies more than the problem itself. Understanding the problem requires a bit of business context. If not it
will be like shooting arrows in the dark.
Q76) Where do most of DS project falls in?
Answer: Project with both technical feasibility and data availability but less business impact. – Most of the data science projects fall under this category.
Q77) How do a DS project go in poor data sourcing?
Answer: Project with high business impact but less or no availability to required data. – Poor data collection and management. With little guidance, these projects can answer essential questions.
Q78) Will DS help to make crucial business solutions?
Answer: Only a handful of data science projects have required technical feasibility, data availability and high business impact. Those are the projects that help the business make crucial decisions
Q79) What is extension in Jupyter notebook does?
Answer: You can add extensions to jupyter notebooks to prevent yourself from distractions. One of my favorite option/extension is the Zen mode. It hides the menu bar and makes us focus on the code
itself. Plus knowing a few of the essential shortcuts can make us work more efficiently.
Q80) How would you prioritize your work as DS?
Answer: DS helps one to do the predictions based on existing data. So, it will help in various aspects like knowing the nature of business, helps in growing the business, can know customer needs
based of past data, any kind of recommendations. Based on all this one can prioritize their work/business.
Q81) Will DS improve business all alone?
Answer: Data professionals should never work in silos. Our job might be the sexiest job of this century, but it indeed depends on a lot of business teams and technical teams in the organization. We
are not master of everything to change things in a day. We need help from others, and for that, we need to ask them the right questions.
Q82) Is DS only to build and implement algorithms?
Answer: Data science is not only to build, test and implement models but most importantly, it is solving business challenges through data science. Need all the soft skills mentioned above.
Q83) How to start with Tableau in DS?
Answer: Tableau as a data visualization tool is easy to learn and takes time to master. All you need is Tableau Public version or Desktop trial version and a couple of Excel/CSV files.
Q84) Will creating visualizations using scripts in tableau supported?
• creating visualizations using scripting languages when I used to extensively work on R. When I started moving my data visualization part of work to Tableau.
• Tableau is not just a drag and drop play around to figure out all options tool. When it is used to its utmost potential can deliver better data visualization reports than any other tool.
Q85) What kind of understanding is important in DS?
• Understanding the need to use mean, median and mode.
• Understanding the need to use Inter-quartile ranges and not normal ranges.
• Understanding the use of a line chart instead of a bar chart.
Q86) How to generate random number in scripts like R/PY?
Answer: Using any scripting language like R/Python, you can generate random values for attributes to analyze them. Again, if you’re looking to make business decisions out of a dataset, then it should
be reliable and should also contain relevant values to make such decisions.
Q87) Will all DS projects result in a viable product?
Answer: Most importantly, not all data science projects will become a viable product which can support the business. So the lead should exactly know when to pull the plug on a project and when not to
if project management for a data science project is not effective, high chances that the project will not yield the desired output.
Q88) What is inspection stage in a DS pipeline?
Answer: Inspection stage is where you can find the abnormalities in data, the inconsistencies, incompleteness, outliers .etc. Here it can be done using any scripting language or a tool like Tableau
to quickly understand what is present and what is not in the backend data.
Q89) Why are the innocent demons ‘innocent’?
Answer: Since innocent ghosts are very ‘naïve’, all aspects of the data set are equally important and independent. As we know, this assumption is rare in the real world situation.
Q90) What is Z Score?
Answer: The z-score is the standard distortion count from a data point on average. But technically this is a source of how many constant changes are above or above the population. A z-score is known
as a fixed value
and can be placed in a normal distribution ramp. It eliminates values from the database that are lower than Z times 3 times.
Q91) What is the remainder?
Answer: In the review analysis, the difference between the estimated value of the dependent variable (y) and the calculated value (ŷ) is called the remainder (d). Every data point is a remainder.
Remaining = Value Value – Estimated value e = y – ŷ
The total and the remaining remaining are equal to zero. Σ e = 0 and e = 0.
Q92) What are the major opinions of Linear regression?
• A linear relationship is a Restricted Multi-collinearity value and then
• Homoscedasticity dependence
• Firstly, there must be a linear relationship between the dependent also independent To verify this relationship, a separate plot proves to be useful.
• Secondly, there need no or very few multi-collinearity between the autonomous variables in The value must be restricted, which depends on the field requirement.
• The third is that It is a unity of the most important suspicions
• which asserts that the errors are uniformly
Q93) What means by heteroscedasticity?
Answer: Heteroscedasticity is specifically the contrast of homoscedasticity, which indicates that the error terms are not uniformly distributed. To change this phenomenon, normally, a log function is
Q94) What are the reasonable ways of increasing the accuracy of a linear regression model?
Answer: There could be many ways of developing the accuracy of linear regression, most commonly related ways are as follows:
Outlier Treatment:
Regression is on sensitive to outliers, so it becomes very essential to treat the outliers with proper values. Replacing the importance with mean, median, mode or percentile depending on the
distribution can show to be useful.
Q95) What is means by odds ratio?
Answer: The odds ratio is the odds within two groups. For example, let’s pretend that we are trying to determine the effectiveness of medicine. We administered the medication to the ‘intervention’
organization and a position to the ‘control’ group.
Q96) What is the value of a baseline in a classification problem?
Answer: Most classification difficulties deal with imbalanced datasets. the number of certain types will be very low when connected to the removed species. In some cases, it is normal to have
positive classes that are less than 1% of the entire sample. In such cases, an efficiency of 99% may appear very good but, in reality, it may not be.
Q97) Different methods of MLE also when are any method preferred?
Answer: The unconditional method is preferred to a number of parameters dataset value is below related to the number of instances. If the number of parameters is extremely correlated to the number of
cases when reduced MLE is to be preferred. Statisticians are supported that restricted MLE is to be performed when in doubt. the Conditional MLE
is always providing that the results.
Q98) Why is accuracy not a good model for classification problems?
Answer: Accuracy is not a good basis for distribution problems because it provides equal significant value to both false positives and false negatives dataset value. Accuracy provides equal quality
to both cases and
cannot distinguish between them.
Q99) What does mean by p-value?
Answer: When you make a hypothesis analysis in statistics, a p-value can help you discover this strength of your results. In a p-value is a number between 0 and 1. Based on that value it will
register the intensity of the results. The part which is before trial is called the Null Hypothesis.
Q100) Explain about from the regularization is and why it is useful?
Answer: Regularization is the method of calculating a tuning parameter upon a method to produce a system in order to prevent overfitting. This appears multiple often made by combining a fixed
multiple on an actual weight vector. The model predictions should later overcome the loss function determined on the regularized training set.
Q101) What is the forward selection of data pre-processing?
Answer: The perspective option is a rectification method that begins without any aspect of the model. In each iteration, we will add a better way to improve our model until we add a new variable to
improve the performance of the model.
Q102) What is the removal of the recursive feature in data pre-processing?
Answer: This is a greedy optimization algorithm that finds a good style feature subset. It creates repetitive models and each reboot keeps aside the best or worse performance feature. This creates
the next model with the left features until all features are exhausted. Then there will be elements based on the order to remove them.
Q103) What are the three stages for creating a model in machine learning?
• Model building
• Model test
• Applying the model
Q104) Keep in mind that you are working in a data system, and explain whether you choose key variables.
Answer: Some methods are used to select the following critical variables:
• Using the loso regression system.
• Using the Random Forest, the plot variable imprtance chart.
• Using linear lag.
Q105) How is KI different?
Answer: K-Recent neighboring countries have a classification algorithm, while k-object is an uncontrolled clustering algorithm. Although the mechanisms seem to look the same, you need data that you
need to classify an unnamed point (neighboring area) to work with neighboring countries. K-material clustering requires only a single point of reference and a starting point: Algorithm can learn how
to group the group into groups by taking unstoppable points and calculating the gap between different points.
The significant difference here is that the KNN has to be named for points, which require supervised learning, while the k-object does not – there is no supervision.
Q106) Is It the Most Important For You Model Model Accuracy or Model Performance?
Answer: This question tests your grip on the machine learning model performance nuances! Machine Learning Interview Questions are often headed towards the details. There are models with greater
accuracy, which advance the power of the advance – how is it realized?
Well, model accuracy model performance is only a subset of how to do it, sometimes it’s a misguided guide. For example, if you find millions of models in a large database, if only a very small number
of fraud cases, the most accurate model does not contradict any fraud. However, it will be ineffective in advance – insisting that there is no fraud on a model designed to detect fraud! Questions
like these help you to demonstrate that you need to understand the model’s accuracy.
Q107) When Should You Use Taxonomy on Retreat?
Answer: Sorting creates a database for distinct values and strict categories, while you record the conclusions that allow you to distinguish the difference between individual points. You can
categorize the consequences if you want to reflect the combination of data points in your database for certain specific sections. (For example, female names, when compared to male, female, male and
Q108) What are the main guidelines to avoid excesses?
• Simplify the sample: You can reduce the transition by lower variables and parameters, thus eliminating some of the noise in training data.
• Use k-folds cross-validation for cross-checking techniques.
• Use 3- regulatory techniques such as LASOO, which are some sample parameters to be punished if they make the tablet.
Q109) How to handle unbalanced databases?
Answer: When you have an unbalanced database, for example, a classification test and 90% of data is in a class. This leads to problems: if there is no computing power in the other section of data
data, 90%
Q110) What is the central trend?
Answer: The central trend is a value that attempts to describe the data set by identifying the position of the central within a set of measurement data. Therefore, the activities of the central
tendencies are sometimes called central location operations. They are categorized as abstract statistics.
Example: average, average, pattern
Q111) When we use Pearson’s relationship co-efficient method?
Answer: Pearson communicates the linear relationship between two consecutive variables involved. Relationship linear is when the change in a variable is related to a proportional change in the other
For example, a Pearson contact can be used to assess whether the increase in the temperature of your production facilities is associated with lower thickness of your chocolate coatings.
Q112) What is the standard deviation, how is it calculated?
Answer: Standard Disadvantage (SD) is a statistical measure, which captures the meanings of the meanings and rankings.
Step 1: Find the average.
Step 2: Find the average square of its distance for each data point.
Step 3: A total of values from step 2.
Step 4: Separate the number of data points.
Step 5: Take a square hunt.
Q113) Defined reinforcement learning?
Answer: Reinforcement Learning is ended effect is to maximize the differential reward signal. Reinforcement learning is stimulated by the experience of personal beings, it is based on any reward/
penalty mechanism.
Q114) Explain Supervised Machinelearning?
• Supervised learning it’s requirs training labeled data.
• Supervised learning its handled regression and classification problems.
• Regression problem to Predect the result with in continous output.
• Classification Problem to predict results in a discrete output.
• Suprvised learning Algorithm : SVM, Navie bayes, Decision tree, KNN Algorithm and Neural Network.
Q115) Explain Unsupervised Machine learning?
Answer: Unsupervised learning is consisting of input data without labeled responses.
Algorithm: Clustering, Aprior.
Q116) What is a confusion matrix?
Answer: The confusion matrix contains 4 output providers by the binary classifier. Various measure, such as error rate, accuracy, precision and recall are derived from it confusion matrix.
Q117) What is linear regression?
• Linear regression is modeled using a straight line.
• Its used with continous variable and the output prediction value of the variable.
• Accuracy: It Measured by loss, R squared, Adjusted R squared.
Q118) How can I retrieve an important part of data collection?
Answer: The distance from the remaining studies is limited to the limited violations. As a result, they can be flexible or disagreeable for any analysis in any analysis in the database. It is
therefore important to detect and be harmful enough.When a 100% reassurance is due to a test/transcription/ etc error, they should only be rejected if they are exited. Otherwise, the removal of the
outlines would have been underestimated.
Q119) What is underfitting?
Answer: Underfitting occurs when a statistical model or machine learning algorithm does not catch the basic trend of data. Instinctively, if the sample or algorithm does not match the data correctly,
it shows the high independence, especially if it has shown a sample or algorithmic variance. The foundation is often a very simple model result.
Q120) When we use Pearson’s relationship co-efficient method?
Answer: Pearson communicates the linear relationship between two consecutive variables involved. Relationship linear is when the change in a variable is related to a proportional change in the other
For example, a Pearson contact can be used to assess whether the increase in the temperature of your production facilities is associated with lower thickness of your chocolate coatings.
Q121) What is Z Score?
Answer: The z-score is the standard distortion count from a data point on average. But technically this is a source of how many constant changes are above or above the population. A z-score is known
as a fixed value and can be placed in a normal distribution ramp. It eliminates values from the database that are lower than Z times 3 times.
Q122) What is the remainder?
Answer: In the review analysis, the difference between the estimated value of the dependent variable (y) and the calculated value (ŷ) is called the remainder (d). Every data point is a remainder.
Remaining = Value Value – Estimated value e = y – ŷ
The total and the remaining are equal to zero. Σ e = 0 and e = 0.
Q123) What is a Sample Model Test?
Answer: A sample T-test is used to check whether the population mean is significantly different from the value of some hypotheses.
Q124) What is F Statistics?
Answer: If you have a significant difference in the way between the two people you will find an FO point of value when you are running an ANOVA test or a regression analysis. This is just like a
T-test a D statistic; If the A-T test is a variable statistically significant and will tell you if a F test variable is of significant significance.
Q125) What is Anuava?
Answer: ANOVA is used for comparison with three or more models.
• One way is ANOVA (which is an independent variable).
• Two way ANOVA (there are two distinct variables)
Q126) Some important Measures of Skewness
• Karl-Pearson coefficient of skewness
• Bowley’s coefficient of skewness
• Coefficient of skewness based on moments
Q127) Scatter is a good action trait.
Answer: An excellent measure of decay is satisfied with the following characteristics.
• It should be well defined without the ambiguity.
• It should be based on all observations of the data set.
• Easy to understand and compute.
• To pursue math treatment.
• It should not be affected by fluctuations in the model.
• It should not be affected by serious surveillance.
Q128) Difference between supervised and unsupervised machine learning?
Answer: Supervised learning is required to be labeled data. For example, in the system to do classification, you’ll require to first design the data content and you’ll relate to range train the model
to create the data value process. Unsupervised learning, if it’s in contrast, does not need labeling data explicitly.
Q129) Difference between L1 and L2 regularization.
Answer: L2 regularization serves to increase error with all the terms, while L1 is also binary/sparse, including several variables specific being selected a 1 or 0 in weighting. L1 compares before
installing some Laplacian earlier at the terms, while L2 agrees to each Gaussian prior.
Q130) Describe a hash table
Answer: A hash table is a data structure input value that returns an associative array. Key movements mapped to specific conditions through this application of a hash function. They continue normally
done during tasks such as database indexing.
Q131) How would you evaluate a logistic regression model?
Answer: you have to demonstrate an understanding of something the typical purposes of logistic regression and bring up a few examples and use cases.
Q132) How would you handle an imbalanced dataset?
• To Collect more data to even that imbalances in a specific dataset.
• To Resample the dataset utility to adjust for imbalances dataset value.
• Try a modified algorithm collectively on your dataset.
Q133) Why does overfitting happen?
Answer: The possibility of overfitting lives as specific criteria used over the model remains no choice the same as the criteria applied to decide the efficacy of a model.
Q134) What is inductive machine learning?
Answer: The inductive machine learning means the process of knowledge by examples, where a system, from a data set of identified situations, tries to convince a general rule.
Q135) Popular algorithms of Machine Learning?
• Decision Trees
• Neural Networks (back propagation)
• Probabilistic networks
• Nearest Neighbor
• Support vector machines
Q136) Which algorithm uses sigmoid function?
Answer: Logisitc Regression
Q137) What is precision?
Answer: True positive/true positive +
Q138) How to check the model accuracy?
Ans: Using the evaluation metrics like accuracy, precision, recall, f1-score etc.,
Q139) Is linear regression and logistic regression belongs to same category?
Ans: No
Q140) What is the objective function for Knn?
Answer: Knn abbreviation: K-nearest neighbour.
Q141) Is clustering a supervised algorithm?
Answer: No
Q142) Is logistic regression a regression or classification
Answer: Classification
Q143) Does bias and variance trade off is important factor to check in the data?
Answer: Yes
Q144) Is L1 and L2 regularization are same?
Answer: No they are different because of their objective function.
Q145) Why do we use import statement?
Answer: It is used to import the in built functions.
Q146) Is Random Forest a tree based algorithm?
Answer: Yes
Q147) Do we call Knn a lazy algorithm?
Answer: Yes
Q148) What are the three main divisions in datascience?
1. Machine Learning
2. Deep Learning
3. Artificial Intelligence
Q149) Does ID3 uses Entropy?
Answer: Yes
Q150) Does Radial basis kernel function is there in SVM?
Answer: yes
Q151) Does both bagging and boosting belongs to Ensemble Learning?
Answer: Yes
Q152) Can we randomly pick the number of clusters in clustering?
Answer: No, we have to chose the optimum number of clusters by ploting the “Elbow Curve”
Q153) Does Correlation and Causation is Same?
Answer: No its not same
Q154) What do we call for manipulating the features of the data ?
Answer: Feature Engineering
Q155) Do we always need more data for better result?
Answer: No
Q156) How we deal with missing value?
Answer: By doing missing value imputation
Q157) How can we deal with outliers?
Answer: By plotting IQR and then deleting the values which are away from the range.
Q158) Can we determine the most important feature in our data?
Answer: Yes
Q159) Can we find the variable importance using Random Forest algorithm?
Answer: Yes
Q160) Does PCA works on the logic of variance ?
Answer: Yes
Q161) Best pictorial graphs to show Correlation plot?
Answer: Heat Map
Q162) Which algorithm uses margin to classify the classes?
Answer: SVM
Q163) Which algorithm takes the data to the next dimension and then classify?
Answer: SVM
Q164) Which algorithm is used for reducing the feature ?
Answer: PCA
Q165) Does PCA a Supervised algorithm?
Answer: No
Q166) What is the major preprocessing step in distance based algorithm?
Answer: Normalization
Q167) Name any one distance based algorithm?
Answer: Clustering
Q168) Name any one Penalty based algorithm?
Answer: SVM
Q169) Random forest is bagging or boosting algorithm?
Answer: Bagging Algorithm
Q170) Xgboost is bagging or boosting algorithm?
Answer: Boosting Algorithm
Q171) For high variance present in data which algorithm to use?
Answer: Bagging Algorithm
Q172) For High Bias present in data algorithm to use?
Answer: Boosting Algortihm
Q173) What leads to Underfitting in linear regression?
Answer: Poor Line of fit.
Q174) What is the Main objective of any data science problems?
Answer: To minimize the error.
Q175) What is next level of machine learning?
Answer: Deep learning and artificial intelligence.
Q176) Does very less data lead to best model?
Answer: No, it leads to underfitting.
Q177) Is doing mean imputation for missing value is always a best method?
Answer: No, its not a best method because mean can mislead if outliers are present.
Q178) Is it mandatory for the data to always follow normal distribution?
Answer: Not exactly, but if it is in normal distribution the results will be better.
Q179) CLT full form:
Answer: CENTRAL LIMIT THEOREM
Q180) RBF full form:
Answer: Radial Basis Kernel Function.
Q181) Does Kmeans and Kmeans++ is same?
Answer: No, Kmeanas++ uses different initation to calculate the centroid.
Q182) Name any one hyper parameter of Decision tree?
Answer: Number of tree, no.of.nodes etc.,
Q183) Is pruning always a good method to construct a tree?
Answer: No it depends on the problem and data.
Q184) Does Tree Based algorithm Overfits?
Answer: Yes
Q185) What do you understand by machine learning?
Answer: A branch of computer science that involves system programming to enhance and increase user experience is known as Machine Learning.
Q186) Give an example of Machine Learning?
Answer: A good example of Machine Learning would be in the case of Robots. Robots are able to perform and complete their tasks based on the information they accumulate from their sensors. Thus they
automatically learn from the data provided.
Q187) What is the difference between data mining and machine learning?
Answer: Data mining is the basic process of getting information from unstructured data without any patterns assigned to them. Machine learning is the process which assigns algorithms and
specifications in terms of programming to develop and design systems. These systems are meant to enhance learning and utilitarian purposes.
Q188) In the case of machine learning what is the meaning of “Overfitting”?
Answer: When there is a random error or noise produced due to excessive information overload, it is known as “Overfitting”.
Q189) When does overfitting usually occur?
Answer: Overfitting in machine learning usually occurs when the model is too complex or there are too many parameters included to keep track of.
Q190) Why does one see the occurrence of overfitting?
Answer: The occurrence of Overfitting is seen when there are different parameters used for training the model and different parameters used for gauging the efficiency of the same.
Q191) In order to avoid Overfitting what needs to be done?
Answer: As Overfitting usually occurs due to large & complex data models, the main idea is to use a smaller dataset.
Q192) What do you understand by inductive machine learning?
Answer: This kind of learning is learning by examples. A general instruction or rule is introduced by virtue of observation of situations.
Q193) State the five algorithms of machine learning?
Answer: The five algorithms of machine learning are as follows Decision Trees, Neutral Networks, Probabilistic Networks, Nearest Neighbor, Support Vector Machines.
Q194) State some of the algorithm techniques for Machine learning?
Answer: Some of the algorithm techniques for machine learning are as follows supervised learning, unsupervised learning, semi-supervised learning, reinforcement learning, transduction, learning to
Q195) State the three stages which are necessary to make the model for machine learning?
Answer: The three stages which are required to build the model for machine learning are as follows…model building, model testing, applying the model.
Q196) How is supervised learning generally conducted?
Answer: The best way to acclimatise one to supervised learning is to divide the information into the training piece and the assessment piece.
Q197) What do you understand by the Training set and the Test Set?
Answer: A training set or an information set refers to examples given to the learner. The test set or assessment set is the method used to decipher how correctly the user has comprehended the
information provided.
Q198) What are the various approaches for machine learning?
Answer: Some of the approaches of machine learning are as follows concept learning and classification learning, symbolic learning and statistical learning, inductive learning and analytical learning.
Q199) Which are the two branches of computer technology which are not classified as machine learning?
Answer: The two branches of computer technology which are not classified as machine learning are Artificial Intelligence and Rule-Based Inference.
Q200) What function does unsupervised learning have?
Answer: Unsupervised learning conducts the following function finding clusters of data, finding interesting directional patterns in data, cleaning up the existing database, finding new observations
and finding new and different coordinates and correlated concepts.
Q201) What function does supervised learning have?
Answer: Supervised learning has the following functions… it has classifications, it has speech recognition, it involves regression and it shows time prediction series.
Q202) What do you understand by machine learning which is independent of algorithms?
Answer: This is a type of machine learning which is independent of any classification series, markers or categories.
Q203) What is the main difference between artificial intelligence and machine learning?
Answer: Machine learning is primarily based in algorithms which are designed strictly on information given by empirical data. Artificial learning encompasses machine learning, however in addition to
that it also includes non empirical data like natural language processors, robotics, etc.
Q204) What do you understand by the classifier in machine learning?
Answer: A system that helps to input information, values, features and assimilates all to give one single value known as the class.
Q205) State some of the areas in which Pattern Recognition is used?
Answer: Some of the areas which uses Pattern Recognition are … computer vision, speech recognition, data mining, statistics, informal retrieval, bio-informatics.
Q206) What do you understand by genetic programming?
Answer: Genetic programming is the name given to the technique which is dependent on assessing and choosing the most prime choice amongst all the results provided.
Q207) State the meaning of Inductive Logic Programming in machine learning?
Answer: A subfield of machine learning which uses logic to represent background knowledge and its examples, is known as Inductive Logic Programming.
Q208) In supervised learning, state the methods used for calibration?
Answer: In supervised learning the two methods which are used for calibration are known as Platt Calibration and Isotonic Regression.
Q209) To prevent Overfitting, which of these two methods is usually chosen?
Answer: In order to prevent Overfitting the method that is usually preferred is Isotonic Regression.
Q210) What do you understand by the term Perceptron in Machine Learning?
Answer: When an algorithm needs to be placed into a nonbinary output, Perceptron is that algorithm which is used in supervised classification.
Q211) A support vector machine (SVM) handles which two classification methods?
Answer: The two classification methods are as follows… combining binary classifies and modification of binary for the inclusion of multi class learning.
Q212) What do you understand by ensemble learning?
Answer: When multiple models, classifiers, experts are combined or specifically generated to solve complex programs, it is known as ensemble learning.
Q213) When is ensemble learning generally utilized?
Answer: When each component classifier is more precise and completely independent from each other, that is when ensemble learning is used.
Q214) State the two types of ensemble learning methodologies?
Answer: The two types of ensemble learning methodologies are sequential ensemble method and parallel ensemble method.
Q215) State some of the key components of relational evaluation technique?
Answer: Some of the key components of relational evaluation techniques are as follows data acquisition, ground truth acquisition, cross validation technique, query type, scoring metrics and
significance test.
Q216) State some of the methods of sequential supervised learning?
Answer: Some of the methods of sequential supervised learning are as follows…sliding-window methods, recurrent sliding windows, hidden marrow models, conditional random fields and graph transformer
Q217) In robotics where does the problem of sequential prediction arise?
Answer: The areas in robotics where the problem of sequential prediction arises are as follows structured prediction, imitation learning and model based reinforcement learning. | {"url":"https://www.gangboard.com/blog/machine-learning-interview-questions-and-answers","timestamp":"2024-11-02T01:26:46Z","content_type":"text/html","content_length":"162355","record_id":"<urn:uuid:dfbe5210-4a54-4565-b2fa-884ab4e5286f>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00371.warc.gz"} |
Programmeinfo BI
MET 3431 Statistics
Course coordinator:
Njål Foldnes
Course name in Norwegian:
Product category:
Bachelor - Common Courses
Teaching language:
Course type:
One semester
This course is an introduction to statistical thinking. Firstly, the student will learn to produce and to interpret descriptive statistics. Secondly, the student will learn the logic of statistical
inference and how to construct confidence intervals and perform hypothesis tests. The emphasis is on understanding concepts and interpretation of results, more than on mathematical machinery.
Learning outcomes - Knowledge
The student will learn the most central concepts underlying statistical methodology, from the collection of data to inference about the population. The underlying logic behind the diversity of
methods will be perceived. Concepts such as random sampling, population, parameters and statistics, inference, margin of error and levels of significance and confidence should be understood. Through
real-world data example students will understand the usefulness of statistics in business and marketing. However, the student should understand the limitations on conclusions drawn from data.
Learning outcomes - Skills
kills in descriptive statistics are to determine level of measurement, and to choose and calculate measures of center and spread, and to produce graphs, for a given sample. Covariation among
variables should also be described. The student should be able to understand and interpret descriptive statistics. Students should be able to perform simple probability calculations. The student
should be able to construct and understand confidence intervals and perform statistical tests. The student should become familiar with statistical software, and be able to interpret output from such
software. The student should be able to report the results of statistical analysis in an easy-to-understand language.
Learning Outcome - Reflection
The student should be aware that statistical methods may be easily misused and misinterpreted. It is important that the judgment required for statistical analysis is fair and just.
Course content
• Collection of data
• Describing the sample at hand
• Probability
• Confidence intervals for mean and proportion
• Hypothesis tests for mean and proportion
• Correlation and regression
• Chi-square test
Learning process and requirements to students
The course consists of 48 hours of lectures, including 4 hours of demonstration of statistical software. The problems studied in class and given as homework assignments will serve as a basis for the
final examination.
For each week there will be given a work program with literature references and assignments. In lectures and SAS JMP exercises, theory will be illustrated by using multiple data sets and associated
tasks. The final exam will be based on that the student has solved all these tasks throughout the semester.
When course is delivered online, lecturer, in cooperation with the Academic Servises Network, will organize an appropriate combination of digital teaching and lectures. Online students are also
offered a study guide to contribute to progression and overview. Total recommended time spent for completing the course also applies here.
Additional information
Re-sit examination
Students that have not gotten approved the coursework requirements, must re-take the exercises during the next scheduled course.
Students that have not passed the written examination or who wish to improve their grade may re-take the examination in connection with the next scheduled examination.
Higher Education Entrance Qualification.
Required prerequisite knowledge
No specific prerequisites required.
Mandatory Courseworks Courseworks Comment coursework
coursework given required
Mandatory 8 5 In the course of the semester 8 mandatory multiple-choice assignments will be given. These are submitted on Itslearning. Each assignment is assessed as either
pass or fail. The student needs at least 5 passes in order to take the final exam.
Mandatory coursework:
Mandatory Mandatory
Courseworks 8
Courseworks 5
Comment In the course of the semester 8 mandatory multiple-choice assignments will be given. These are submitted on Itslearning. Each assignment is assessed as either pass or fail. The
coursework: student needs at least 5 passes in order to take the final exam.
Exam category:
Form of assessment:
Written submission
Support materials:
• All printed and handwritten support materials
• BI-approved exam calculator
• Simple calculator
5 Hour(s)
Exam code:
Grading scale:
Examination every semester
Exam organisation:
Ordinary examination
Student workload
Activity Duration Comment
Teaching 48 Hour(s)
Group work / Assignments 50 Hour(s)
Prepare for teaching 42 Hour(s) Working with SAS JMP (or some statistical software)
Student's own work with learning resources 40 Hour(s)
Examination 20 Hour(s) Exam incl. preparations.
A course of 1 ECTS credit corresponds to a workload of 26-30 hours. Therefore a course of 7,5 ECTS credit corresponds to a workload of at least 200 hours. | {"url":"https://programmeinfo.bi.no/nb/course/MET-3431/2018-spring","timestamp":"2024-11-12T19:11:39Z","content_type":"text/html","content_length":"24418","record_id":"<urn:uuid:e924ec14-bcd6-45d0-a07f-a07f02dc1cf3>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00594.warc.gz"} |
Math 241 honors
Math 241 honors, Calculus III, Fall 2024
• Lecture:
• Discussion Sections: All in 141 Altgeld
□ HD1 (CRN: 68120) Thursday 11am.
□ HD2 (CRN: 68122) Thursday 1pm.
□ HD3 (CRN: 68123) Thursday 2pm.
• Course web page: http://dunfield.info/241
• Instructor: Nathan Dunfield
□ Email: nmd@illinois.edu Office: 378 Altgeld
□ Office hours: Tues 1:30–2:30pm and Wed 3:30–4:30pm; other times available by appointment.
• Discussion instructor: Wyatt Kuehster Tutoring room: Mon, Tue, Wed, and Thurs from 4–8pm in 147 Altgeld, starting August 28.
• Detailed course diary: Topics, lecture notes, HW assignments, worksheets, etc.
The focus of this course is vector calculus, which concerns functions of several variables and functions whose values are vectors rather than just numbers. In this broader context, we will revisit
notions like continuity, derivatives, and integrals, as well as their applications (such as finding minima and maxima). We’ll explore new geometric objects such as vector fields, curves, and surfaces
in 3-space and study how these relate to differentiation and integration. The highlight of the course will be theorems of Green, Stokes, and Gauss, which relate seemingly disparate types of integrals
in surprising ways.
For most people, vector calculus is the most challenging term in the calculus sequence. There are a larger number of interrelated concepts than before, and solving a single problem can require
thinking about one concept or object in several different ways. Because of this, conceptual understanding is more important than ever, and it is not possible to learn a short list of “problem
templates” in lecture that will allow you to do all the HW and exam problems. Thus, while lecture and section will include many worked examples, you will still often be asked to solve a HW problem
that doesn’t match up with one that you’ve already seen. The goal here is to get a solid understanding of vector calculus so you can solve any such problem you encounter in mathematics, the sciences,
or engineering. That requires trying to solve new problems from first principles, if only because the real world is sadly complicated.
Honors aspect
This is an honors course. As a formal matter, this means your grade will have an "H" after it on your transcript and it can help you maintain James Scholar status. As a practical matter, I'll treat
the material in a little more depth and from a somewhat more sophisticated viewpoint than the regular sections of Math 241. You'll also do some more work, in the form of the honors homework
assignments discussed below. I basically view this class as ordinary Math 241 plus 15% more mathy goodness for those who particularly enjoy the subject.
We will cover Chapters 12–16 of
• James Stewart, Calculus: Early Transcendentals, 9th edition, with WebAssign.
Please note that this course uses the 9th edition rather than the 8th. You will also need WebAssign access to do the homework. For complete information on purchasing options for both, see https://
go.illinois.edu/calculus. If you have the standard text and WebAssign package from Math 220, 221, or 231 from last year, then you already have everything you need for this course. Even before you
purchase WebAssign, you can freely use it for the first two weeks of class and so not miss any homework assignments. Several physical copies of the textbook are on reserve at Grainger Library.
Course policies
Overall grading: Your course grade will be based on the online HW (7%), honors HW (4%) section worksheets (5%), three midterm exams (18% each), and a comprehensive final exam (30%). Grade cutoffs on
any component will never be stricter than 90% for an A– grade, 80% for a B–, and so on. Individual exams may have grade cutoffs set more generously depending on their difficulty.
Exams: There will be three midterm exams, which will be held in class September 23 (Monday), October 18 (Friday), and November 15 (Friday). The final exam will be held Wedneday, December 18, from
All exams will be closed book and notes, and no phones, calculators, or other electronic devices will be permitted; where indicated, all work must be shown to receive any credit on a problem.
Online Homework: Online homework will be assigned for each lecture, and will generally be due two lectures later, just before class starts at 1pm. That is, HW based on Monday’s lecture is due Friday
at 1pm, and Wednesday’s is due on the following Monday, etc. The homework will be completed online via WebAssign. Late homework will not be accepted, but the lowest 4 scores will be dropped. The
first assignment is due Friday, August 30. To access WebAssign, login in to our course's Canvas page (https://canvas.illinois.edu) and click the WebAssign link. You may need to wait 24 hours after
registering for the course to be able to log in to WebAssign. For technical problems, contact WebAssign student support; for questions or concerns about HW grades, talk to your TA.
Honors Homework: There will be four honors homework assignments. These consist of more extended and in-depth problems than the the online HW, and will be turned in on paper at the beginning of the
corresponding discussion or lecture class. The lowest score in this category will be dropped.
Worksheets: Most section meetings will include a worksheet which will be graded for effort and participation. Missing a worksheet results in a score of zero, but the lowest 3 scores in this category
will be dropped.
Missed exams: There will be no make-up exams. Rather, in the event of a valid illness, accident, or family crisis you can be excused from an exam so that it does not count toward your overall
average. Such situations must be documented and I reserve final judgment as to whether an exam will be excused. All such requests should be made to me in advance if possible, but in any event no more
than one week after the exam date.
Missed HW and worksheets: Generally, these are taken care of with the policy of dropping the lowest scores. For extended absences, these are handled in same way as missed exams.
Regrading: The section leaders and myself try hard to accurately grade all exams, worksheets, and HW, but please contact your TA if you think there is an error. All requests for regrading must be
made within one week of the item being returned.
Viewing grades online: You can always find the details of your worksheet and exam scores here. Details of your HW scores can be viewed on WebAssign, and are only periodically input into the above
system as an overall average.
Lecture Etiquette: Since there almost 100 people in the room, it’s particularly important to arrive on time, remember to turn off the ringer on your phone, refrain from talking, not pack up your
stuff up until the bell has rung, etc. Otherwise it will quickly become hard for the other students to pay attention.
Cheating: Cheating is taken very seriously as it takes unfair advantage of the other students in the class, and is handled as per Article 1 Part 4 of the student code. Penalties for cheating on
exams, in particular, are very high, typically resulting in a 0 on the exam or an F in the class.
Disabilities: Students with disabilities who require reasonable accommodations should contact me as soon as possible. In particular, any accommodation on exams must be requested at least a week in
advance and will require a letter from DRES.
Conflict final: If you have a conflict with the final exam time, times, please consult the university policy on final exam conflicts. Based on that, if you think your situation qualifies you to take
the conflict exam, email me at least one week before the exam date. You will need to provide documentation as to the nature of your conflict, and I reserve final judgment as to which exam you will
Sources of help
Ask questions in class: This applies to both the main lecture and your discussion section. The lecture may be large, but I still strongly encourage you to ask questions there. If you’re confused
about something, then at least a dozen other people are as well.
The Math 241 tutoring room: Come and work with the TAs and your classmates on homework, test preparation, and any general questions about Math 241 on any Monday, Tuesday, Wednesday, and Thursday from
4pm–8pm in 147 Altgeld. The tutoring room will be staffed starting Wednesday, August 28. Note: this tutoring room is primarily for the regular Math 241 content; questions about the honors HW should
be directed to Nathan or Wyatt at their office hours.
Come to office hours: I have office hours 378 Altgeld on Tuesdays from 1:30–2:30pm and on Wednesdays from 3:30–4:30pm. If none of those times work for you, you can make an appointment by sending me
email or talking to me after class.
Other sources: A change of perspective is sometimes helpful to clear up confusion. Here are two other vector calculus sources you might find helpful:
• H. M. Schey, Div, Grad, Curl, and All That, W. W. Norton. A classic informal account of vector calculus from a physics point of view. Grainger Library reserve copy.
• Adams, Thompson, and Hass, How to ace the rest of calculus, the streetwise guide, Freeman. A snarky and lighthearted source. Grainger library reserve copy.
Lecture notes, HW assignments, section worksheets, etc.
These are all posted online on the course diary. | {"url":"https://nmd.web.illinois.edu/classes/2024/241/index.html","timestamp":"2024-11-08T02:30:16Z","content_type":"text/html","content_length":"14904","record_id":"<urn:uuid:f488f2b8-2e8b-44a2-84d9-50629426b451>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00572.warc.gz"} |
How do vectors apply in real life? | Socratic
How do vectors apply in real life?
2 Answers
By real life you mean your life? GPS uses vectors.
Vectors are used 24/7 to derive results in engineering and science. Fluid mechanics, static,, Electrical Engineering
Some natural phenomena require the use of vectors to describe them in a meaningful way.
Some quantities require the concept of a vector to capture a meaningful description of them for manipulation in equations that attempt to describe natural phenomena.
For example, it is not sufficient to know that gravity exerts a force of attraction on a body to work out how it might be affected by the force. It is also necessary to know the direction in which
the force is attempting to move the body. As more than one number is needed to capture the information, it is not surprising that vectors consist of a carefully ordered (because the position of the
number carries information about what it is measuring) set of numbers. For a force acting on a body whose motion is confined to a plane, two numbers are needed to give a 2-vector. Three numbers would
be required for a vector describing a force acting on a body whose motion is simultaneously through three dimensions.
Impact of this question
54763 views around the world | {"url":"https://socratic.org/questions/how-do-vectors-apply-to-real-life#573225","timestamp":"2024-11-14T10:52:52Z","content_type":"text/html","content_length":"34562","record_id":"<urn:uuid:d962e6e0-9834-49c8-a4ae-57e1404da304>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00304.warc.gz"} |
[Solved] Electromagnetic waves | Engineering homework help - Elite Homework
Paper Details
Two-fluid model of a superconductor on the two-fluid model of a super conductor we assume that at temperatures 0 < T < T0, the current density may be written as the sum of the contributions of normal
and superconducting electrons: j = jN, + jS, where jN, = ?0E and js is given by the London equation. Here ?0 is an ordinary normal conductivity, decreased by the reduction in the number of normal
electrons at temperature T as compared to the normal state. Neglect inertial effects on both jN, and jS.
(a) Show from the Maxwell equations that the dispersion relation connecting wave vector k and frequency w for electromagnetic waves in the superconductor is (CGS) k2c2 = 4??02wt ? c2?L?2 + w2; or
(SI) k2c2 = (?0/?0)wt c2?L?2 + w2 where A; is given by (148) with n replaced by ns. Recall that curl B = ? ?2B.
(b) If T is the relaxation time of the normal electrons and nN is their concentration, show by use of the expression ?0 = nNe2?/m that at frequencies w << 1/? the dispersion relation does not involve
the normal electrons in an important way, so that the motion of the electrons is described by the London equation alone. The super current short-circuits the normal electrons. The London equation
itself only holds true if hw is small in comparison with the energy gap. Note: The frequencies of interest are such that w << wp, where wp is the plasma frequency. | {"url":"https://www.elitehomework.com/114005/electromagnetic-waves-engineering-homework-help/","timestamp":"2024-11-15T01:36:53Z","content_type":"text/html","content_length":"37760","record_id":"<urn:uuid:27a66421-ff68-4b4d-b456-fc04de8ded0f>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00817.warc.gz"} |
fmaxf: determine maximum numeric value of two floating-point - Linux Manuals (3p)
fmaxf (3p) - Linux Manuals
fmaxf: determine maximum numeric value of two floating-point
This manual page is part of the POSIX Programmer's Manual. The Linux implementation of this interface may differ (consult the corresponding Linux manual page for details of Linux behavior), or the
interface may not be implemented on Linux.
fmax, fmaxf, fmaxl - determine maximum numeric value of two floating-point numbers
#include <math.h>
double fmax(double x, double y);
float fmaxf(float x, float y);
long double fmaxl(long double x, long double y);
These functions shall determine the maximum numeric value of their arguments. NaN arguments shall be treated as missing data: if one argument is a NaN and the other numeric, then these functions
shall choose the numeric value.
Upon successful completion, these functions shall return the maximum numeric value of their arguments.
If just one argument is a NaN, the other argument shall be returned.
If x and y are NaN, a NaN shall be returned.
No errors are defined.
The following sections are informative.
Portions of this text are reprinted and reproduced in electronic form from IEEE Std 1003.1, 2003 Edition, Standard for Information Technology -- Portable Operating System Interface (POSIX), The Open
Group Base Specifications Issue 6, Copyright (C) 2001-2003 by the Institute of Electrical and Electronics Engineers, Inc and The Open Group. In the event of any discrepancy between this version and
the original IEEE and The Open Group Standard, the original IEEE and The Open Group Standard is the referee document. The original Standard can be obtained online at | {"url":"https://www.systutorials.com/docs/linux/man/3p-fmaxf/","timestamp":"2024-11-06T11:27:12Z","content_type":"text/html","content_length":"9081","record_id":"<urn:uuid:4812ab9f-9c80-41c2-aedf-906b575bda3c>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00396.warc.gz"} |
Molecular Modeling | Encyclopedia.com
Molecular Modeling
Molecular Modeling
A model is a semblance or a representation of reality. Early chemical models were often mechanical, allowing scientists to visualize structural features of molecules and to deduce the stereochemical
outcomes of reactions. The disadvantage of these simple models is that they only partly represent (model) most molecules. More sophisticated physics-based models are needed; these other models are
almost exclusively computer models.
Two major categories of physics-based, computational molecular models exist: macroscopic and microscopic. Macroscopic models describe the coarse-grained features of a system or a process but do not
describe the atomic or molecular features. Microscopic or atomistic models take full account of all atoms in the system.
Atomistic modeling can be done in two ways: by applying theory or by using fitting procedures. The fitting procedures are attempts to rationalize connections between molecular structure and
physicochemical properties (quantitative structure property relationships, QSPR), or between molecular structure and biological response (quantitative structure activity relationships, QSAR).
Usually, a molecule's biological response is regressed onto a set of molecular descriptors.
Here C is the minimum concentration of a compound that elicits a response to an assay of some sort (e.g., an LD[50], or something else), b[0] is a constant, b[i] is the least-squares multiple
regression coefficient, and D[i] is the molecular descriptor. For example, the best model for determining the retention index (RI) of drug molecules on a gas-liquid chromatography column was found to
be RI = 9.92 MW − 3.11 (number of ring atoms) + 139 (number of ring nitrogens) + 296 (total σ charge) + 921 Σ atomic IDs on nitrogen − 335 ^6XCH − 211 ^3XC − 49^2κ [α ] − 1958 (X and κ are
topological and topographical descriptors). One can then predict an unknown drug's RI very accurately by substituting the values of the descriptors for that drug into the above equation. There are no
rules about what kind of descriptors may or may not be used, but descriptors often include information about molecular size, shape, electronic effects, and lipophilicity. Microscopic modeling based
on fitting methodologies requires the use of existing data to create a model. The model is then used to predict the properties or activities of as yet unknown molecules.
The other approach to microscopic molecular modeling implements theory, and uses various sampling strategies to explore a molecule's potential energy surface (PES). Knowing a molecule's PES is
convenient, because one can interpret directly from it the molecule's shape and reactivity. Popular modeling tools used for determining PES are shown in Figure 1.
Quantum Mechanics (QM). The objective of QM is to describe the spatial positions of electrons and nuclei. The most commonly implemented QM method is the molecular orbital (MO) theory, in which
electrons are allowed to flow around fixed nuclei (the Born-Oppenheimer approximation) until the electrons reach a self-consistent field (SCF). The nuclei are then moved, iteratively, until the
energy of the system can go no lower. This energy minimization process is called geometry optimization.
Molecular Mechanics (MM). Molecular mechanics is a non-QM way of computing molecular structures, energies, and some other properties of molecules. MM relies on an empirical force field (EFF), which
is a numerical recipe for reproducing a molecule's PES. Because MM treats electrons in an implicit way, it is a much faster method than QM, which treats electrons explicitly. A limitation of MM is
that bond-making and bond-breaking processes cannot be modeled (as they can with QM).
Molecular Dynamics (MD). Energy-minimized structures are motionless and, accordingly, incomplete models of reality. In molecular dynamics, atomic motion is described with Newtonian laws: F [i ](t)=m
[i]a[i], where the force F[i] exerted on each atom a[i] is obtained from an EFF. Dynamical properties of molecules can be thus modeled. Because simulation periods are typically in the nanosecond
range, only inordinately fast processes can be explored.
Monte Carlo (MC). The same EFFs used in the MM and MD methods are used in the Monte Carlo method. Beginning with a collection of particles, the system's initial energy configuration is computed. One
or more particles are randomly moved to generate a second configuration, whose energy is "accepted" for further consideration, or "rejected," based on energy criteria. Millions of structures on the
PES are sampled randomly. Averaged energies and averaged properties are thus obtained.
Most atomistic modeling involves the exploration of a complex and otherwise unknown PES. Simple energy minimization with QM or MM locates a single, stable structure (the local minimum) on the PES,
which may or may not be the most stable structure possible (the global minimum). MD and MC sampling methods involve a more complete searching of the PES for low energy states and, accordingly, are
more time-consuming.
see also Quantum Chemistry; Theoretical Chemistry.
Kenneth B. Lipkowitz
Jonathan N. Stack
Leach, A. R. (2001). Molecular Modeling: Principles and Applications, 2nd edition. Englewood Cliffs, NJ: Prentice-Hall.
Lipkowitz, Kenneth B., and Boyd, D. B., eds. (1990–2002). Reviews in Computational Chemistry, Vols. 1–21. New York: Wiley-VCH.
Von Rague Schleyer, P., ed. (1998). Encyclopedia of Computational Chemistry, Vols. 1–5. Chichester, U.K.: John Wiley.
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The standard form of the equation of a parabola is x=y^2-4y+20. What is the vertex form of the equation? | Socratic
The standard form of the equation of a parabola is #x=y^2-4y+20#. What is the vertex form of the equation?
1 Answer
The standard forms of the equation of the parabola are of ${y}^{2} = 4 a x$ and ${x}^{2} = 4 a y$ .
Here we have
${y}^{2} - 4 y + 16 = x$
This is a shifted parabola of ${y}^{2} = 4 a x$ form.
Make the $L H S$ to the whole square part
${y}^{2} - 4 y + 4 + 12 = x$
${\left(y - 2\right)}^{2} = x - 12$
Compare it with $R H S$
So we have
${\left(y - 2\right)}^{2} = 4 \cdot \frac{1}{4} \cdot \left(x - 12\right)$
This is the standard form of the equation.
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Other views of relations Expressing generality
Relation Algebra
The idea of relation algebra is to take relations themselves as objects and to reason about certain operations defined on them. If we think of binary relations in the set-theoretic way as sets of
ordered pairs of items from some set X, it is natural to define a number of simple concepts:
1. Intersection: The intersection R ∩ S of two relations R and S holds between any two objects x and y if and only if both Rxy and Sxy.
2. Union: R ∪ S is the relation that relates x and y if and only if Rxy OR Sxy.
3. Complement: R is the relation of not being related by R. As a set of pairs, it has to be defined relative to the universe X.
4. Converse: R˘ reverses the direction of a relation. That is, for any x and y, R˘xy if and only if Ryx. Obviously, every relation has a converse. Note well the difference between converse and
5. Composition: R º S relates x and y if and only if SOMEz (Rxz AND Szy). This allows relations to be strung together so as to get from one thing to another by going through a chain of intermediates
standing in various relationships to each other.
6. Zero: The empty relation 0 never holds between any things. Note that for any relation R, the compound R ∩ R is bound to be 0.
7. One: The full relation 1 holds everywhere: ALLx ALLy 1xy. Clearly R ∪ R is the same as 1, for any relation R.
8. Identity: I is the relation =. That is, {〈x, x〉: x ∈ X} or the relation everything bears to itself and to nothing else. The key property of I is that for any relation R, both I º R and R º I
are identical with R.
Many other concepts are easily defined in terms of these. To say that R is included in S, for example, usually written R ≤ S, is simply to say R ∩ S = R or equivalently R ∪ S = S. It is also useful
to define a "power" notation to symbolise the result of iterating R a given number of times: R^0 = I and for any k greater than or equal to zero, R^k+1 = R^k º R.
Although the intended model of all this is the set of binary relations on a set, we can also consider abstract relation algebras, which are completely arbitrary structures on which are defined
operations as above satisfying appropriate conditions or axioms which need not be detailed here. If you are interested, the references at the end of these notes [5, 8] will give you an entry point
into the literature.
Relation algebra provides a beautifully concise notation for representing many properties of systems of relations, and by extension quite a lot of mathematics. Consider for example how to say that
relation R is transitive. It could be written as in the last section, or using the vocabulary of relation algebra we may simply write it as R^2 ≤ R. Again, to say that R is an equivalence relation is
to say R = (R º R˘) ∪ I.
We shall not pursue this algebraic direction further in these notes, as it would take us too far from purely logical concerns. For present purposes it is sufficient to note that there is such an
alternative approach to logic, which relates it to mainstream mathematics in ways that may not otherwise have been obvious.
Graph Theory
One notion which is not expressible in standard relation algebra, and not definable in first order logic, but which can be added in order to facilitate reasoning especially about processes, is the
transitive closure of a relation. This is the result of iterating R indefinitely. We may think of it as the infinite union R ∪ R^2 ∪ R^3 ∪ R^4 ... or as the smallest transitive relation which extends
(i.e. includes) R. Even more useful is the reflexive transitive closure, usually written R*, which is the result of iterating zero or more times I ∪ R ∪ R^2 ∪ R^3 ... or equivalently the infinite
intersection of all the quasi-orders that include R. Again equivalently, it is the smallest reflexive relation closed under the operation of composition with R. This notion of reachability by
following the relation R is a central concern of another way of thinking about binary relations: graph theory Bondy.
: Example of an undirected
graph with 8 vertices and 10 edges
A graph in this sense, usually referred to as a directed graph is a set of nodes or vertices and a set of arcs or links between them. Each arc has a "head" and a "tail", these being the vertices it
joins. A natural way to construe this is to think of an arc as consisting of the ordered pair: its head and its tail. It is obvious that a directed graph is then simply a binary relation, and any
binary relation is a directed graph. In the case of a symmetric relation, the graph is undirected, since there is no preferred direction to the link between one vertex and another. The arcs of an
undirected graph are usually called edges.
One nice thing about graphs is that we can draw them (or at least the small ones). See graph. This gives us a pictorial way of thinking about relations, allowing us to bring our visual imagination to
bear on them.
When we think of a relation as a graph, we concentrate on concepts such as:
• which parts of the graph are connected components (joined together by the arcs) and which are disconneted;
• the degree of the graph, or how many arcs (at most) are incident on any one vertex;
• which vertices are reachable from which others by following the arcs;
• the lengths of shortest paths from one vertex to another;
• the existence and size of cycles (paths from a vertex to itself) or cliques (sets of vertices all linked to each other).
Graph theory, again, is not the subject of our present study. We merely note it at this point to emphasise that there are many ways to think about relations, and that several branches of mathematics
(logic, algebra, graph theory) come togther here. | {"url":"https://users.cecs.anu.edu.au/~jks/LogicNotes/relation-algebra.html","timestamp":"2024-11-10T15:21:02Z","content_type":"text/html","content_length":"11990","record_id":"<urn:uuid:e2211644-1b91-4fac-bd8e-1a7e354be4e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00586.warc.gz"} |
Download this Jupyter notebook and all data (unzip next to the ipynb file!). You will need a Gurobi license to run this notebook, please follow the license instructions.
Maximizing Utility¶
The standard mean-variance (Markowitz) portfolio selection model determines an optimal investment portfolio that balances risk and expected return. In this notebook, we maximize the utility, which is
defined as a weighted linear function of return and risk that can be adjusted by varying the risk-aversion coefficient \(\gamma\). Please refer to the annotated list of references for more background
information on portfolio optimization.
import gurobipy as gp
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
Input Data¶
The following input data is used within the model:
• \(S\): set of stocks
• \(\mu\): vector of expected returns
• \(\Sigma\): PSD variance-covariance matrix
□ \(\sigma_{ij}\) covariance between returns of assets \(i\) and \(j\)
□ \(\sigma_{ii}\) variance of return of asset \(i\)
# Import example data
Sigma = pd.read_pickle("sigma.pkl")
mu = pd.read_pickle("mu.pkl")
The model maximizes the investor’s expected utility, which is defined as a linear combination of the expected return and risk, accounting for the investor’s level of risk aversion via the
risk-aversion coefficient \(\gamma\). Mathematically, this results in a convex quadratic optimization problem.
Decision Variables and Variable Bounds¶
The decision variables in the model are the proportions of capital invested among the considered stocks. The corresponding vector of positions is denoted by \(x\) with its component \(x_i\) denoting
the proportion of capital invested in stock \(i\).
Each position must be between 0 and 1; this prevents leverage and short-selling:
\[0\leq x_i\leq 1 \; , \; i \in S\]
The budget constraint ensures that all capital is invested:
\[\sum_{i \in S} x_i =1\]
Objective Function¶
The objective is to maximize expected utility:
\[\max_x \mu^\top x - \tfrac{\gamma}{2} x^\top \Sigma x\]
Using gurobipy, this can be expressed as follows:
gamma = 0.2 # risk-aversion coefficient
# Create an empty optimization model
m = gp.Model()
# Add variables: x[i] denotes the proportion invested in stock i
# 0 <= x[i] <= 1
x = m.addMVar(len(mu), lb=0, ub=1, name="x")
# Budget constraint: all investments sum up to 1
m.addConstr(x.sum() == 1, name="Budget_Constraint")
# Define objective function: Maximize expected utility
mu.to_numpy() @ x - gamma / 2 * (x @ Sigma.to_numpy() @ x), gp.GRB.MAXIMIZE
We now solve the optimization problem:
Gurobi Optimizer version 11.0.0 build v11.0.0rc2 (linux64 - "Ubuntu 22.04.4 LTS")
CPU model: AMD EPYC 7763 64-Core Processor, instruction set [SSE2|AVX|AVX2]
Thread count: 1 physical cores, 2 logical processors, using up to 2 threads
WLS license 2443533 - registered to Gurobi GmbH
Optimize a model with 1 rows, 462 columns and 462 nonzeros
Model fingerprint: 0xee0d2d3c
Model has 106953 quadratic objective terms
Coefficient statistics:
Matrix range [1e+00, 1e+00]
Objective range [7e-02, 6e-01]
QObjective range [6e-04, 2e+01]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 1e+00]
Presolve time: 0.03s
Presolved: 1 rows, 462 columns, 462 nonzeros
Presolved model has 106953 quadratic objective terms
Ordering time: 0.01s
Barrier statistics:
Free vars : 461
AA' NZ : 1.065e+05
Factor NZ : 1.070e+05 (roughly 1 MB of memory)
Factor Ops : 3.298e+07 (less than 1 second per iteration)
Threads : 1
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 -2.00616666e+05 3.17727204e+05 2.51e+05 6.60e-01 2.50e+05 0s
1 -5.25149343e+04 5.58459714e+04 6.22e+03 8.36e-06 6.71e+03 0s
2 -2.65726609e+02 7.78009938e+02 4.96e+01 6.66e-08 5.53e+01 0s
3 -4.60705379e-01 4.99493868e+02 4.96e-05 2.51e-13 5.41e-01 0s
4 -4.59350933e-01 1.20162017e+00 1.10e-07 8.19e-14 1.80e-03 0s
5 -1.54241215e-01 2.91476155e-01 1.82e-09 1.10e-15 4.82e-04 0s
6 -4.12662983e-02 1.32695352e-01 3.60e-10 5.60e-16 1.88e-04 0s
7 1.83691636e-02 9.82785660e-02 1.17e-15 2.22e-16 8.65e-05 0s
8 3.61099995e-02 6.40567918e-02 4.44e-16 3.86e-16 3.02e-05 0s
9 4.48593368e-02 5.16994956e-02 1.11e-15 2.78e-16 7.40e-06 0s
10 4.95311895e-02 4.99140985e-02 8.33e-16 2.22e-16 4.14e-07 0s
11 4.98537626e-02 4.98596976e-02 2.04e-15 2.22e-16 6.42e-09 0s
12 4.98589567e-02 4.98590574e-02 7.40e-14 3.33e-16 1.09e-10 0s
13 4.98590130e-02 4.98590195e-02 2.57e-11 2.22e-16 7.02e-12 0s
Barrier solved model in 13 iterations and 0.31 seconds (0.78 work units)
Optimal objective 4.98590130e-02
Display basic solution data:
print(f"Expected utility: {m.ObjVal:.6f}")
print(f"Minimum risk: {x.X @ Sigma @ x.X:.6f}")
print(f"Expected return: {mu @ x.X:.6f}")
print(f"Solution time: {m.Runtime:.2f} seconds\n")
# Print investments (with non-negligible values, i.e., > 1e-5)
positions = pd.Series(name="Position", data=x.X, index=mu.index)
print(f"Number of trades: {positions[positions > 1e-5].count()}\n")
print(positions[positions > 1e-5])
Expected utility: 0.049859
Minimum risk: 2.158448
Expected return: 0.265704
Solution time: 0.31 seconds
Number of trades: 31
KR 0.034830
PGR 0.065161
CME 0.028440
ODFL 0.036042
BDX 0.017070
MNST 0.007179
BR 0.000587
KDP 0.082626
GILD 0.007109
META 0.009800
CLX 0.046154
SJM 0.016280
PG 0.000209
LLY 0.128341
DPZ 0.054559
MKTX 0.019685
CPRT 0.003112
MRK 0.030562
ED 0.075457
WST 0.023671
TMUS 0.045910
NOC 0.024011
EA 0.001504
MSFT 0.012608
WM 0.050311
TTWO 0.041694
WMT 0.058436
NVDA 0.009023
HRL 0.031236
AZO 0.019366
CPB 0.019025
Name: Position, dtype: float64
Efficient Frontier¶
The efficient frontier reveals the balance between risk and return in investment portfolios. It shows the best-expected risk level that can be achieved for a specified return level. We compute this
by solving the above optimization problem for a sample of risk aversion levels \(\gamma\).
# Hand picked for approximately equidistant return/risk points
# fmt: off
gamma_vals = np.array([
# fmt: on
returns = np.zeros(gamma_vals.shape)
risks = np.zeros(gamma_vals.shape)
npos = np.zeros(gamma_vals.shape)
utility = []
# prevent Gurobi log output
m.params.OutputFlag = 0
# solve the model for each risk level
for i, gamma in enumerate(gamma_vals):
# Replace utility objective function according to this choice for gamma
mu.to_numpy() @ x - gamma / 2.0 * (x @ Sigma.to_numpy() @ x), gp.GRB.MAXIMIZE
# store data
returns[i] = mu @ x.X
risks[i] = np.sqrt(x.X @ Sigma @ x.X)
npos[i] = len(x.X[x.X > 1e-5])
Next, we display the efficient frontier for this model: We plot the expected returns (on the \(y\)-axis) against the standard deviation \(\sqrt{x^\top\Sigma x}\) of the expected returns (on the \(x\)
-axis). We also display the relationship between the risk and the number of positions in the optimal portfolio.
fig, axs = plt.subplots(1, 2, figsize=(10, 3))
# Axis 0: The efficient frontier
axs[0].scatter(x=risks, y=returns, marker="o", label="sample points", color="Red")
axs[0].plot(risks, returns, label="efficient frontier", color="Red")
axs[0].set_xlabel("Standard deviation")
axs[0].set_ylabel("Expected return")
# Axis 1: The number of open positions
axs[1].scatter(x=risks, y=npos, color="Red")
axs[1].plot(risks, npos, color="Red")
axs[1].set_xlabel("Standard deviation")
axs[1].set_ylabel("Number of positions")
As expected, the number of open positions decreases as we allow more variance; the optimization will progressively invest in fewer high-risk but high-yield assets. | {"url":"https://gurobi-finance.readthedocs.io/en/latest/modeling_notebooks/basic_model_maxutility.html","timestamp":"2024-11-05T14:02:11Z","content_type":"text/html","content_length":"46835","record_id":"<urn:uuid:0aa410de-513a-4387-9e50-396958305306>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00897.warc.gz"} |
Geometric Distribution
Tags: #Math #Statistics
$$Pr(X=k) = (1-p)^{k-1}q, \\ f(x)=(1-p)^{k-1}q, \\ F(x)=1 - (1-p)^{[x]}$$
Latex Code
Pr(X=k) = (1-p)^{k-1}q, \\
f(x)=(1-p)^{k-1}q, \\
F(x)=1 - (1-p)^{[x]}
Have Fun
Let's Vote for the Most Difficult Equation!
$$Pr(X=k) = (1-p)^{k-1}q$$ $$f(x)=(1-p)^{k-1}q$$ $$F(x)=1 - (1-p)^{[x]}$$
Pr(X=k) = (1-p)^{k-1}q, \\
f(x)=(1-p)^{k-1}q, \\
F(x)=1 - (1-p)^{[x]}
Latex code for the Geometric Distribution.
• Probability parameter p means the success probability of each trial: $$p$$
• The number of total trials until the first successful trial k: $$k$$
• PDF for Geometric Distribution: $$f(x)=(1-p)^{k-1}q$$
• CDF for Geometric Distribution: $$F(x)=1 - (1-p)^{[x]}$$
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I hope my efforts are enough to pass this exam.
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Quiz 2: Areas related to Circles (Class 10) - CREATA CLASSES
Areas related to circles: Quiz 2
1 / 10
1. Find the area of the sector of a circle having radius 6 cm and of angle 30°. [Take 𝜋 = 3.14]
2 / 10
2. What is the perimeter of the sector with radius 10.5 cm and sector angle 60°?
3 / 10
3. A sector of a circle of radius 8 cm contains an angle of 135°. Find the area of the sector.
4 / 10
4. Find the area of the minor segment of a circle of radius 42 𝑐𝑚, if the length of the corresponding arc is 44 𝑐𝑚.
5 / 10
5. In the figure, 𝑂𝐴𝐵𝐶 is a quadrant of a circle with centre 𝑂 and radius 3.5 𝑐𝑚. If 𝑂𝐷 = 2 𝑐𝑚, find the area of the shaded region. [Use 𝜋 = 22/7]
6 / 10
6. A teacher buys a supersized pizza for his after-school club. The super-pizza has a diameter of 18 inches. If the teacher is able to perfectly cut from the center a 36° sector for himself, what is
the area of his slice of pizza, rounded to the nearest square inch?
7 / 10
7. In the figure shown below, line segment AB passes through the center of the circle and has a length of 8cm. Points A, B, and C are on the circle. Sector COB covers 1/6^th of the total area of the
circle. Find the area of sector COB.
8 / 10
8. Find the area of shaded region in figure, where a circle of radius 6 𝑐𝑚 has been drawn with vertex 𝑂 of an equilateral triangle 𝑂𝐴𝐵 of side 12 𝑐𝑚. (Use 𝜋 = 3.14 and √3 = 1.73)
9 / 10
9. Find the area of shaded region shown in the given figure where a circular arc of radius 6 𝑐𝑚 has been drawn with vertex 𝑂 of an equilateral triangle 𝑂𝐴𝐵 of side 12 𝑐𝑚 as centre. (Use 𝜋=3.14 and √3
10 / 10
10. In the figure, the shape of the top of a table is that of a sector of a circle with centre 𝑂 and ∠𝐴𝑂𝐵=90°. If 𝐴𝑂=𝑂𝐵=42 𝑐𝑚, then find the perimeter of the top of the table. [Use 𝜋=22/7]
Your score is
The average score is 0% | {"url":"https://creataclasses.com/class-10-maths/class-10-areas-related-to-circles-chapter-11/quiz-2-areas-related-to-circles-class-10/","timestamp":"2024-11-06T04:43:26Z","content_type":"text/html","content_length":"320990","record_id":"<urn:uuid:845f4d89-3e6c-4f10-8b90-bd68ee5f0b13>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00693.warc.gz"} |
Lesson 15
Converting Units
Let’s convert measurements to different units.
15.1: Number Talk: Fractions of a Number
Find the values mentally.
\(\frac14\) of 32
\(\frac34\) of 32
\(\frac38\) of 32
\(\frac38\) of 64
15.2: Road Trip
Elena and her mom are on a road trip outside the United States. Elena sees this road sign.
Elena’s mom is driving 75 miles per hour when she gets pulled over for speeding.
1. The police officer explains that 8 kilometers is approximately 5 miles.
1. How many kilometers are in 1 mile?
2. How many miles are in 1 kilometer?
2. If the speed limit is 80 kilometers per hour, and Elena’s mom was driving 75 miles per hour, was she speeding? By how much?
15.3: Veterinary Weights
A veterinarian uses weights in kilograms to figure out what dosages of medicines to prescribe for animals. For every 10 kilograms, there are 22 pounds.
1. Calculate each animal’s weight in kilograms. Explain or show your reasoning. If you get stuck, consider drawing a double number line or table.
1. Fido the Labrador weighs 88 pounds.
2. Spot the Beagle weighs 33 pounds.
3. Bella the Chihuahua weighs \(5\frac12\) pounds.
2. A certain medication says it can only be given to animals over 25 kilograms. How much is this in pounds?
15.4: Cooking with a Tablespoon
Diego is trying to follow a recipe, but he cannot find any measuring cups! He only has a tablespoon. In the cookbook, it says that 1 cup equals 16 tablespoons.
1. How could Diego use the tablespoon to measure out these ingredients?
\(1\frac14\) cup of oatmeal
\(2\frac34\) cup of flour
2. Diego also adds the following ingredients. How many cups of each did he use?
6 tablespoons of cocoa powder
When we measure something in two different units, the measurements form an equivalent ratio. We can reason with these equivalent ratios to convert measurements from one unit to another.
Suppose you cut off 20 inches of hair. Your Canadian friend asks how many centimeters of hair that was. Since 100 inches equal 254 centimeters, we can use equivalent ratios to find out how many
centimeters equal 20 inches.
Using a double number line:
│length (in) │length (cm) │
│100 │254 │
│1 │2.54 │
│20 │50.8 │
One quick way to solve the problem is to start by finding out how many centimeters are in 1 inch. We can then multiply 2.54 and 20 to find that 20 inches equal 50.8 centimeters. | {"url":"https://im.kendallhunt.com/MS_ACC/students/1/2/15/index.html","timestamp":"2024-11-12T08:33:59Z","content_type":"text/html","content_length":"73771","record_id":"<urn:uuid:45bc41e9-9c2d-46ce-9551-8187fbbdba62>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00181.warc.gz"} |
Hex (strategy game)
I’ve been using a small variant of hex in the age 5 to 7 classroom for several years. Apart from the smaller size, the only other innovation is to insist that the first player plays on the perimeter
on their first move. Without a rule like this there is an elegant way to play on odd edge length boards that leads to a first player win. This proof and the no-tying proof is great for students
around age 11.
Here are game sheets of various sizes…
To step through a sample game with your classroom click here.
Thanks to Marc Chamberland and his brilliant youtube channel Tipping Point Math for this beautiful video.
Standards for Mathematical Practice
MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:
MP1 Toughen up!
Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.
MP2 Think abstractly!
Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.
MP3 Work together!
Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. Try pairing up elementary students and getting older students to work in threes.
MP4 Model reality!
Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.
MP5 Use the right tools!
Students should use the right tools: 0-99 wall charts, graph paper, mathigon.org. etc.
MP6 Be precise!
Students learn to communicate using precise terminology. Students should not only use the precise terms of others but invent and rigorously define their own terms.
MP7 Be observant!
Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist! 😉
MP8 Be lazy!?!
Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?
Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉 For a free poster of MathPickle's ideas on elementary math education go
Gordon Hamilton
(MMath, PhD) | {"url":"https://mathpickle.com/project/7454/","timestamp":"2024-11-14T23:10:15Z","content_type":"text/html","content_length":"220682","record_id":"<urn:uuid:66ec3229-994e-43c8-9953-91af237f2e67>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00423.warc.gz"} |
plot.eff: Plotting tools for effectiveness distributions in julian-urbano/simIReff: Stochastic Simulation for Information Retrieval Evaluation: Effectiveness Scores
Plot the density, distribution and quantile functions of an effectiveness distribution. Function plot plots all three functions in the same graphics device.
1 ## S3 method for class 'eff'
2 plot(x, ..., plot.data = TRUE)
4 dplot(x, ..., plot.data = TRUE)
6 pplot(x, ..., plot.data = TRUE)
8 qplot(x, ..., plot.data = TRUE)
## S3 method for class 'eff' plot(x, ..., plot.data = TRUE) dplot(x, ..., plot.data = TRUE) pplot(x, ..., plot.data = TRUE) qplot(x, ..., plot.data = TRUE)
x the effectiveness distribution to plot.
... other arguments to be passed to graphical functions.
plot.data logical: whether to plot the data used to fit the distribution, if any.
logical: whether to plot the data used to fit the distribution, if any.
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/github/julian-urbano/simIReff/man/plot.eff.html","timestamp":"2024-11-06T18:42:35Z","content_type":"text/html","content_length":"30104","record_id":"<urn:uuid:a7d748b5-5e56-4d81-868b-44bb9661f3d0>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00423.warc.gz"} |
seminars - Infinitely wide neural networks
[서울대학교 수리과학부 10-10 집중강연] 2/3 (금), 2/8 (수), 2/10 (금), 15:00PM - 17:00PM
장소: Zoom 강의실
회의 ID: 993 2488 1376
암호: 120745
초록 : While deep learning has many remarkable success stories, finding a satisfactory mathematical explanation on why it is so effective is still considered an open challenge. One recent promising
direction for this challenge is to analyse the mathematical properties of neural networks in the limit where the widths of hidden layers of the networks go to infinity. Researchers were able to prove
highly-nontrivial properties of such infinitely-wide neural networks, such as the gradient-based training achieving the zero training error (so that it finds a global optimum), and the typical random
initialisation of those infinitely-wide networks making them so called Gaussian processes, which are well-studied random objects in machine learning, statistics, and probability theory. These
theoretical findings also led to new algorithms based on so called kernels, which sometime outperform existing kernel-based algorithms.
The purpose of this lecture series is to go through these recent theoretical results on infinitely wide neural networks. Our plan is to pick a few important results in this domain, and to go deep
into those results, so that participants of the series can reuse the mathematical tools behind these results for analysing their own neural networks and training algorithms in the infinite-width | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=seminars&sort_index=room&order_type=desc&page=88&l=en&document_srl=1036564","timestamp":"2024-11-08T06:08:58Z","content_type":"text/html","content_length":"51637","record_id":"<urn:uuid:04c70896-5f89-41f0-83f2-6a09be730b59>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00298.warc.gz"} |
Many times in artificial intelligence we deal with large data sets of a large number of features or variables and there is not a clear way to visualize or interpret the relationships between these
variables. Principle components analysis (PCA) is a technique for finding a basis set that may represent the data well under some circumstances. PCA finds a set of orthogonal basis vectors which are
aligned in the directions of maximum variance, the so called principle components. PCA also ranks the components by importance (i.e. variance).
Principle components analysis assumes that the distributions of the random variables are characterizes by their mean and variance, exponential distributions such as the Gaussian distribution for
example. PCA also makes the assumption that the direction of maximum variance is indeed the most “important”. This assumption implies that the data has a high signal-to-noise ratio; the variance
added by any noise is small in comparison to the variance from the important dynamics of the system.
The principal components are calculated by finding the eigenvectors and eigenvalues of the covariance matrix. The covariance matrix is a square matrix which describes the covariance between random
variables, where is the covariance between the th and th random variables. The random variables must be in mean deviation form, meaning that the mean has been subtracted so the distribution of the
random variable has . The eigenvalues and eigenvectors are usually arranged such that is a diagonal matrix where the value of is the th eignenvalue , and is a matrix where the th column of is the th
eigenvector. Note that the covariance matrix may have eigenvalues/eigenvectors, so and are matrices as well. At this point the eigenvectors can be ranked by importance by sorting the eigenvalues in .
Since each eigenvector is associated with a particular eigenvalue, the columns of must be rearranged so the th column corresponds to the th eigenvalue.
If the goal is to reduce the dimensionality of the data set (the number of random variables), the principle components which are least important may be discarded. This leaves us with two matrices,
where is the number of principal components to keep.
This matrix of eigenvectors represents a linear transform which orients the basis vectors in the directions of maximal variance, while maintaining orthogonality.
Basic Mathematical Structures and Operations
A mathematical structure is an association between one set of mathematical objects with another in order to give those objects additional meaning or power. Common mathematical objects include numbers
, sets, relations, and functions.
An algebraic structure is defined by a collection of objects and the operations which are allowed to be applied to those objects. A structure may consist of a single set or multiple sets of
mathematical objects (e.g. numbers). These sets are closed under a particular operation or operations, meaning that the result of the operation applied to any element of the set is also in the set.
Axioms are conditions which the sets and operations must satisfy.
A simple but ubiquitous algebraic structure is the group. A group is a set and a single binary operation, usually denoted , which satisfy the axiom of associativity and contain and identity element
and inverse elements. Associativity specifies that the order in which the operations are performed does not matter; that is . The identity element is a special element such that the operation applied
to it and another element results in the other element; formally: . Each element of the set must have an inverse element that yields the identity element when the two are combined: . If the axiom of
commutativity is added, the group is referred to as an Abelian group. Commutativity allows the operands to be reorganized: . If the requirement of inverse elements is removed from the group
structure, the structure is called a monoid.
A group homomorphism is a function which preserves the relationships between the elements of the set, or the group structure. For groups and , is a homomorphism iff . If the map is invertible, i.e.
it has an inverse such that , then is said to be an isomorphism. A group endomorphism is a homomorphism from a group onto itself, , while an invertible endomorphism is called a automorphism. A
subgroup is a group within a larger group. For a subgroup of a group , and the identity element of is the identity element of .
A ring is an algebraic structure which adds another operations and a few axioms to the group structure. The operations and of a ring satisfy different axioms. The set and the addition operator must
form an Abelian group as described above, while the set and the multiplication operator must form a monoid. In addition, the operators must satisfy the axiom of distribution, specifically the
operator must be able to distribute over the operator, and . If forms a commutative monoid, that is, a monoid with commutativity, then the ring is said to be a commutative ring.
Similarly to a group, a ring may also have a ring homomorphism if satisfies and . Likewise, is an isomorphism if it has an inverse that satisfies the identity relation described for group isomorphism
above. A ring is a subring of a ring if and contains the multiplicative identity from .
An algebraic structure is a field if it satisfies the axioms of a ring with a few addition axioms. Both operators of a field must satisfy commutativity, and the set must contain inverses under both
operators, except that the field may not contain a multiplicative inverse for the additive identity element. Another way to describe a field is to say that the additive group is an Abelian group, and
the multiplicative group without the additive identity is also an Abelian group. The inclusion of inverses for both operators lead to the intuitive notion of subtraction and division (except division
by 0). An example of a well known field is the field of real numbers, .
A metric space is a mathematical structure , where is a binary function which defines the real-valued distance between two elements of the set . A distance is a non-negative quantity, and only equal
to zero when the two element are equal. A distance should also satisfy the axiom of symmetry, and the triangle inequality. For example, the real-valued vector space equipped with the Euclidean
distance metric yields the Euclidean metric space. | {"url":"http://research.sourcebyte.com/blog/2009/03/","timestamp":"2024-11-14T00:51:02Z","content_type":"text/html","content_length":"27015","record_id":"<urn:uuid:659a1a22-67f9-449f-8e4c-714b01527e04>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00630.warc.gz"} |
r.seriesgrass - Online in the Cloud
This is the command r.seriesgrass that can be run in the OnWorks free hosting provider using one of our multiple free online workstations such as Ubuntu Online, Fedora Online, Windows online emulator
or MAC OS online emulator
- Makes each output cell value a function of the values assigned to the
corresponding cells in the input raster map layers.
raster, aggregation, series
r.series r.series --help r.series
] [
,...]] [
,...] [
,...]] [
] [--
] [--
] [--
] [--
] [--
Flags: -n
Propagate NULLs
Do not keep files open
Allow output files to overwrite existing files
Print usage summary
Verbose module output
Quiet module output
Force launching GUI dialog
Parameters: input
Name of input raster map(s)
Input file with one raster map name and optional one weight per line, field separator
between name and weight is |
Name for output raster map
Aggregate operation
average, count, median, mode, minimum, min_raster, maximum, max_raster, stddev, range, sum, variance, diversity, slope, offset, detcoeff, tvalue, quart1, quart3, perc90, quantile, skewness, kurtosis
Quantile to calculate for method=quantile
0.0-1.0 weights
Weighting factor for each input map, default value is 1.0 for each input map
Ignore values outside this range
makes each output cell value a function of the values assigned to the
corresponding cells in the input raster map layers.
Following methods are available:
· average: average value
· count: count of non-NULL cells
· median: median value
· mode: most frequently occurring value
· minimum: lowest value
· maximum: highest value
· range: range of values (max - min)
· stddev: standard deviation
· sum: sum of values
· variance: statistical variance
· diversity: number of different values
· slope: linear regression slope
· offset: linear regression offset
· detcoeff: linear regression coefficient of determination
· tvalue: linear regression t-value
· min_raster: raster map number with the minimum time-series value
· max_raster: raster map number with the maximum time-series value
Note that most parameters accept multiple answers, allowing multiple aggregates to be
computed in a single run, e.g.:
r.series input=map1,...,mapN \
output=map.mean,map.stddev \
r.series input=map1,...,mapN \
output=map.p10,map.p50,map.p90 \
method=quantile,quantile,quantile \
The same number of values must be provided for all options.
No-data (NULL) handling
flag, any cell for which any of the corresponding input cells are NULL is
automatically set to NULL (NULL propagation). The aggregate function is not called, so
all methods behave this way with respect to the
flag, the complete list of inputs for each cell (including NULLs) is passed to
the aggregate function. Individual aggregates can handle data as they choose. Mostly, they
just compute the aggregate over the non-NULL values, producing a NULL result only if all
inputs are NULL.
Minimum and maximum analysis
methods generate a map with the number of the raster map
that holds the minimum/maximum value of the time-series. The numbering starts at
up to
for the first and the last raster listed in
, respectively.
Range analysis
If the
option is given, any values which fall outside that range will be treated as
if they were NULL. The
parameter can be set to
thresholds: values outside
of this range are treated as NULL (i.e., they will be ignored by most aggregates, or will
cause the result to be NULL if -n is given). The
thresholds are floating point,
so use
for a single threshold (e.g.,
to ignore negative values, or
to ignore values above -200.4).
Linear regression
Linear regression (slope, offset, coefficient of determination, t-value) assumes equal
time intervals. If the data have irregular time intervals, NULL raster maps can be
inserted into time series to make time intervals equal (see example).
Quantiles r.series
can calculate arbitrary quantiles.
Memory consumption
Memory usage is not an issue, as
only needs to hold one row from each map at a
Management of open file limits
Number of raster maps to be processed is given by the limit of the operating system. For
example, both the hard and soft limits are typically 1024. The soft limit can be changed
with e.g. ulimit -n 1500 (UNIX-based operating systems) but not higher than the hard
limit. If it is too low, you can as superuser add an entry in
# <domain> <type> <item> <value>
your_username hard nofile 1500
This would raise the hard limit to 1500 file. Be warned that more files open need more
RAM. See also the Wiki page Hints for large raster data processing.
For each map a weighting factor can be specified using the
option. Using weights
can be meaningful when computing sum or average of maps with different temporal extent.
The default weight is 1.0. The number of weights must be identical with the number of
input maps and must have the same order. Weights can also be specified in the input file.
Use the
option to analyze large amount of raster maps without hitting open files
limit and the size limit of command line arguments. The computation is slower than the
option method. For every sinlge row in the output map(s) all input maps are opened
and closed. The amount of RAM will rise linear with the number of specified input maps.
The input and file options are mutually exclusive. Input is a text file with a new line
separated list of raster map names and optional weights. As separator between the map name
and the weight the character "|" must be used.
with wildcards:
r.series input="`g.list pattern=’insitu_data.*’ sep=,`" \
output=insitu_data.stddev method=stddev
Note the
script also supports regular expressions for selecting map names.
with NULL raster maps (in order to consider a "complete" time series):
r.mapcalc "dummy = null()"
r.series in=map2001,map2002,dummy,dummy,map2005,map2006,dummy,map2008 \
out=res_slope,res_offset,res_coeff meth=slope,offset,detcoeff
Example for multiple aggregates to be computed in one run (3 resulting aggregates from two
input maps):
r.series in=one,two out=result_avg,res_slope,result_count meth=sum,slope,count
Example to use the file option of r.series:
cat > input.txt << EOF
r.series file=input.txt out=result_sum meth=sum
Example to use the file option of r.series including weights. The weight 0.75 should be
assigned to map2. As the other maps do not have weights we can leave it out:
cat > input.txt << EOF
r.series file=input.txt out=result_sum meth=sum
Example for counting the number of days above a certain temperature using daily average
maps (’???’ as DOY wildcard):
# Approach for shell based systems
r.series input=`g.list rast pattern="temp_2003_???_avg" sep=,` \
output=temp_2003_days_over_25deg range=25.0,100.0 method=count
# Approach in two steps (e.g., for Windows systems)
g.list rast pattern="temp_2003_???_avg" output=mapnames.txt
r.series file=mapnames.txt \
output=temp_2003_days_over_25deg range=25.0,100.0 method=count
Use r.seriesgrass online using onworks.net services | {"url":"https://www.onworks.net/os-distributions/programs/r-seriesgrass-online","timestamp":"2024-11-03T06:46:02Z","content_type":"text/html","content_length":"198604","record_id":"<urn:uuid:0d3de5ac-8fac-4a6b-a3a8-dfc761c23473>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00831.warc.gz"} |
Total Fun
This post is about a conversation I once held with a good friend of mine, Nadav Sherman. It is not a very serious topic, so take it lightly. We were wondering how to calculate the total amount of fun
a person has during their lifetime. We reasoned as follows:
The absolute state score (ASS) of a person is a number ranking the person’s current state and possessions (it is a function of time). For example, if you have \$500,000 and a girl-friend, then your
ASS is higher than that of someone identical to you without a girlfriend, but lower than that of someone identical to you with an additional \$1,000,000. Note that we do not specify exactly how to
calculate the ASS, but we believe that it can be defined such that claim 1 below will be true. The actual value of the ASS. is not directly correlated to the fun a person has - a person with a
million dollars and a beautiful girl-friend can be sadder than a poor lonely guy.
We define the effective fun of a person as a measurement of the instantaneous fun a person has in a given moment (again, it is a function of time). Our first claim is:
Claim 1: The effective fun equals the derivative of the ASS.
Again, this obviously depends on the exact function used to calculate the ASS. Instead of proving the claim above, I will give an intuitive reasoning as to why it makes sense. It is obvious that if
your ASS. is going up right now, you are happy, regardless of the actual value of the ASS. Conversely, if your ASS. goes down you are sad, regardless of the value of your ASS.
Lets give an example. If you get a new girlfriend, your ASS goes up and so your effective fun (its derivative) is positive. If you lose \$100 your ASS goes down and so your effective fun is negative.
The actual value of the ASS does not matter - only changes in the ASS influence your effective fun.
Please make sure you understand claim 1 before continuing to read (see also the following figure).
Now, in order to calculate the total amount of fun a person has in a given period we obviously integrate the effective fun function over the given period. We use the fundamental theorem of calculus,
stating that the integral of the derivative of a function equals the function itself (note that as this is not the true fundamental theorem of calculus, which speaks solely of continuous functions,
but its more in the spirit of it) to get the following:
Claim 2: The total fun a person has in a given period is the ASS value at the end of the period minus the ASS value at the beginning of the period.
This leads us to our final (morose) conclusion:
The total fun a person has in their lifetime equals their ASS on the day they were born minus their ASS on the day they die (no matter how long they lived or what happened during their lifetime). | {"url":"https://yanivle.github.io/math/2007/05/21/total-fun.html","timestamp":"2024-11-11T14:09:54Z","content_type":"text/html","content_length":"19663","record_id":"<urn:uuid:d2903489-c26e-49b1-b865-112fd908f4ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00100.warc.gz"} |
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[41] [ 42 ] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75]
riir C.oiiijXliln;-111,1
S. If the firm is producing where MC = MR, then: (a) it is definitely making a profit (b) it may be minimizing a loss (c) it should shut down (d) it must be causing externahdes (e) it should operate
provided price exceeds AFC.
T. The total cost curve: (a) first increases at an increasing rate and then at a decreasing rate (b) first increases at a constant rate and then at a decreasing rate (c) first increases and then
decreases (d) first decreases at a decreasing rate and then decreases at an increasing rate (e) first increases at a decreasing rate and then increases at an increasing rate.
U. The point of diminishing returns occurs where: (a) MC is positive (b) MC stops falling and starts rising (c) the slope of the total cost curve stops becoming flatter and starts becoming steeper
(d) the slope of the total variable cost, cur\e stops becoming flatter and starts to become steeper (e) all of the above.
V. If AVC exceeds MC, dien: (a) AVC is falling and is failing (b) only AVC is falling (c) only AVC is falling and may be rising or falling (d) AVC is rising and is falling (e) both and AVC may or may
not be falling.
W. The portion ofthe total cost curve where loia cost is increasing at a decreasing rate is associated with (a) crowding (b) decreasing renirns to scale (c) .sp cializadon (d) profii maximizadon fe)
gains from uade
X. In the long run: (a) firms can enter or leav( the industry (b) the amount of both labor and capi tal is variable (c) the firm faces no fixed costs (d) th, firm has more opportimities for reducing
its cost (e) all of the above.
Y The lotal cost of producing a given level of out put in an induslry will be at a minimum when: (a) the average total cost is the same for all firms (b) the .M( is the same for all firms (c) the . \
is the .same fo all firms (d) the AFC is the same for all firms (e) a] of the above.
Z. Average fixed cost: (a) is never zero in th short run (b) is TFC/q (c) always decreases wit: output (d) is the difference between and AV( (e) all of the above.
1. Profits in the price-taking firm are: (a) Pq -(ATC)q (1>) the sum of the MR up to the equilibrium level of output (c) always non-existent (d) equal to the quantit) minus the area under the average
total cost curve up to the equilibrium quantity (e) none of the above.
2. Gary, a price-taking pineapple grower finds that his opumal output lies beyond that at which his short run is minimized. Then: (a) Gary is enjoying short run profits (b) the firm is making a
normal return (c) Gary is minimizing short-run losses (d) Gary is making a mistake: he cannot be maximiz-hig profit (e) Gary should shut down in the long run.
4. If in the prerious question Gary were originally producing at the minimum of his long run curve then: (a) it would be in Garys interest to continue producing pineapples (b) the price of pineapples
would equal the marginal cost of producing them (c) the value society attaches to more pineapples would equal the cost to society of producing them (d) the marginal and average cost of producing more
pineapples would be the same (e) all of the above.
5. The change in total cost resulting from a one unit production increase is: (a)average revenue
(b) variable cost (c) average variable cost (d) increasing cost (e) none of the above.
6. Mart)s firm has only the following assignable contractual commitments: a five year lease on a fleet of cars, a one year building lease, and a union contract that requires a two-week nodce before
any employees can be laid off Ifit would take six months to sell the business and three to declare bankruptcy, the short run for this firm is: (a) 3 months (b) 6 months
(c) 5 years (d) one year (e) two weeks.
7. If the average total cost curves are "U" shaped, then in the region of economies of scale: (a) MC is alwa)s falling (b) MC is negative (c) MC is rising and then failing (d)MC is below (e) bodi (b)
and (d).
8. Ifa firms average fixed cost associated with 500 units of output is $2 per unit, then (a) its MC at 500 units is $2 (b) its AFC at 400 units is greater than $2 (c) its AFC rises beyond 500 units
(d) its AFC at 400 units is also $2 (e) its MC at 500 units is greater than $2.
9. If MC is increasing, then we know: (a) is increasing (b) AFC is increasing (c) is falling (d) AVC is falling (e) none ofthese are valid inferences.
10. If TC increases as Q increases, then: (a) MC is falling (b) MC is rising (c) is falling (d) MC is zero (e) MC is positive.
11. If the MC of producing each widget in your price-taking firm increases, you are: (a) better off (b) worse off (c) unchanged (d) possibly better off, possibly worse off (e) operating at die
efficient level of production.
12. ProfitMax Inc. has a total fixed cost of SI500. Its average fixed cost at 1, 10, and 15 units yiiW be respectively: (a) $1500, $1500. $1500 (b) $1500, $150.000, and $225,000 (c) $1500, $150, $100
(d) none of the above (e) none of the above and it cant be calculated from the information given.
13. MC = MR on farmer Browms price-taking avocado farm at 5000 avocados per week. The price of avocados is $1, his total variable cost is $3000 at this output and his fixed cost is $2500. We predict
hell: (a) operate at an output greater than 5000 (b) shut down and experience a loss of $2500 (c) shut down and exjaeri-ence a loss of $3000 (d) produce "5000"avdcaaos and"" take a loss of $500 per
week (e) insufficient information to make a prediction.
14. Lightiiing Bolt Company, the only supplier of electricity in an area, produces a good subject to economies of scale at all outputs. If it were forced by regulation, to sell at a price equal to a
certain point along its average total cost curve then: (a) the cost of an addidonal unit of electricity will be less than the price charged (b) the firm will be making positive economic profits (c)
more electricit) will be produced than if price were set equal to marginal cost (d) the price will be lower than it would be if price were set equal to marginal cost (e) consumer surplus will be
15. If die total cost of producing 15 units is 575 and if the average total co.st of producing 16 units is S4.90, dien marginal cost is definitely: (a) increasing (b) falling (c) greater than average
total cost (d) less than average total cost (e) equal to average total cost.
16. The following table shows the cost of producing various quantities of cassette tapes in a certain period:
Quantity of Tapes
(thousands/period) 0 1 2 3 4 5 6 Total Cost of Production
(thousands of dollars) A 8 11 15 19 24 30
At what level of output is average total cost minimized? (a) 1 (b) 2 (c) 3 (d) 4 (e) 5.
17. From question 16, at what level of output is average variable cost minimized? (a)l (b) 2 (c) 3 (d)4 (e)5.
18. From que.sdon 16, how many units would be produced if the market price was $6? (a) 2 (b) 3 (c) 4 (d)5 (e)6.
19. If average fixed cost is $10 and average variable cost is $5 in producing 100 units and if the price of each unit is $17, then profit is: (a) -$1,000 (b) $0 (c) $100 (d) $200 (e) S500.
20. If total fixed cost of producing 50 units is $500, total variable cost is $2000, and total revenue is $5000, profit per unit is: (a) $0 (b) $10 (c> $25 (d) $50 (e) $100.
21. If total fixed cost is $100, and average total co.st is constant over a range of output at $10, then in this range: (a) marginal cost is $10 (b) marginal cost is constant and less than $10 (c)
average cost is minimized at zero output (d) the firm cannot make a prof it (e) the firm should be regulated.
22. In the long run, a compeddve firm can incur total costs which vary with the amount of producdon (Q) and capital (K) it uses in the following way:
Qiiantity/period Q: Total cost (K=0) in S: Toul cost (K=l) ill S: Total cost (K2) in S:
0 1 2 3 4 .T 6 7
0 100 l.iO 300 500 750 1000 1250
200 230 2.=i0 260 280 360 -J50 550
What levels of capital would the firm choose for producing two units and five units of output respectively? (a) 0 and 1 (b) 0 and 2 (c) 1 and 1 (d) 1 and2 (e) 2 and 2.
23. For the same firm as question 22, how many units would be produced at a market price of $90? (a) 3 (b) 4 (c) 5 (d) 6 (e) 7
24. Assuming entry has not yet reduced the price fiom $90, total profit would be: (a) -$200 (b) $0
(c) $60 (d) $100 (e) none ofthe above.
25. In the circumstances of the previous question the firms profit per unit will be: (a) S28 (b) S80 (c) $75 (d) $8 (e) $15.
26. If in #23 the firm sets out to maximize profit per unit instead of total profit, its profit per unit would be: (a) $8 (b) $20 (c) $9 (d) $12 (e).noneof the above.
27. If firms can enter or leave the industry with the cost structures given in #22, die long run price would be about: (a) $50 (b) $60 (c) S70 (d) S80 (e) none of the above.
28. A certain compedtive industry consists of many firms each with idendcal long run a\erage total cost curves. Minimum long run average total cost is $50 per unit and occurs at an output of 100
units per month. Demand is unit elasdc and consumers expenditure when die price was $60 was $600,000 per month. In this case the compeddve price after all adjustments would be: (a) $80 (b) $70 (c)
$60 (d)$50 (e)$40.
29. In the circumstances of the above quesdon, the long run equihbrium number of firms in the industry would be: (a) 120 (b) 175 (c) 125 (d) 100 (e) 50.
30. -Total variable costs: (a) are costs which vary from firm to firm in the same industry (b) are costs commonly associated with the short run rariable factor of capital (c) increase and decrease as
producdon increases and decreases (d) do not affect profit (e) none of the abo\e.
31. Subtraction of AVC from AIC will gi\e us back: (a) economic profit (b) average fixed cost (c) increasing cost (d) marginal cost, (e) none of the above.
32. The short run supply curve of a firm in perfect competition is best described as: (a) the average cost curve (b) the average variable cost curve (c) the marginal cost curve (d) the marginal cost
curve above minimum average variable cost (e) the marginal cost curve above minimum average total cost.
33. In which of the following situadoiis is it definitely desirable for a price-taking firm ro cease production endrely? (a) when the firm is making a loss
[start] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
[41] [ 42 ] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] | {"url":"http://eng.yax.su/finlab/ir017/42/index.shtml","timestamp":"2024-11-06T01:41:44Z","content_type":"text/html","content_length":"22577","record_id":"<urn:uuid:6a97aef5-cfff-4526-a2b3-7d78717c6c13>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00278.warc.gz"} |
Learning from Weakly Dependent Data under Dobrushin’s Condition
Learning from Weakly Dependent Data under Dobrushin’s Condition
Proceedings of the Thirty-Second Conference on Learning Theory, PMLR 99:914-928, 2019.
Statistical learning theory has largely focused on learning and generalization given independent and identically distributed (i.i.d.) samples. Motivated by applications involving time-series data,
there has been a growing literature on learning and generalization in settings where data is sampled from an ergodic process. This work has also developed complexity measures, which appropriately
extend the notion of Rademacher complexity to bound the generalization error and learning rates of hypothesis classes in this setting. Rather than time-series data, our work is motivated by settings
where data is sampled on a network or a spatial domain, and thus do not fit well within the framework of prior work. We provide learning and generalization bounds for data that are complexly
dependent, yet their distribution satisfies the standard Dobrushin’s condition. Indeed, we show that the standard complexity measures of Gaussian and Rademacher complexities and VC dimension are
sufficient measures of complexity for the purposes of bounding the generalization error and learning rates of hypothesis classes in our setting. Moreover, our generalization bounds only degrade by
constant factors compared to their i.i.d. analogs, and our learnability bounds degrade by log factors in the size of the training set.
Cite this Paper
Related Material | {"url":"https://proceedings.mlr.press/v99/dagan19a.html","timestamp":"2024-11-05T09:57:58Z","content_type":"text/html","content_length":"16306","record_id":"<urn:uuid:d01cc1b6-a132-49a4-b31f-d2ea300467c1>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00628.warc.gz"} |
asked 2011-06-25 16:01:32 +0100
This post is a wiki. Anyone with karma >750 is welcome to improve it.
sage: latex(2*2^(1/2))
2 \, \sqrt{2}
that is ok
but what ist that?
sage: latex(2*2^(1/3))
2 \, 2^{\left(\frac{1}{3}\right)}
How can I get the right result
2\cdot 2^{\left(\frac{1}{3}\right)}
2\, \sqrt[3]{2}
Thanks for help
Are you asking how to get latex output that doesn't have any implicit multiplication in it?
1 Answer
Sort by ยป oldest newest most voted
The cdot is made in Sage-6.2, so this has been fixed. I consider `\sqrt[3]{2}` as less readable but that may be a reigious issue.
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NicolaiSivesind/ChatGPT-Research-Abstracts · Datasets at Hugging Face
real_abstract generated_abstract generated_word_count
stringlengths stringlengths int64
This PhD thesis is devoted to deterministic study of the turbulence in the
Navier- Stokes equations. The thesis is divided in four independent
chapters.The first chapter involves a rigorous discussion about the energy's
dissipation law, proposed by theory of the turbulence K41, in the The Navier-Stokes equations provide a fundamental framework for
deterministic setting of the homogeneous and incompressible Navier-Stokes understanding the behavior of fluids in a wide range of applications. One
equations, with a stationary external force (the force only depends of the phenomenon that is crucial to explaining such behavior is the turbulence
spatial variable) and on the whole space R3. The energy's dissipation law, that fluids exhibit. Turbulence is a complex, dynamic process that has
also called the Kolmogorov's dissipation law, characterizes the energy's resisted detailed analytical investigation due to its highly nonlinear
dissipation rate (in the form of heat) of a turbulent fluid and this law was nature. Instead, researchers often rely on numerical simulations, which in
developed by A.N. Kolmogorov in 1941. However, its deduction (which uses turn demand accurate and efficient models for describing turbulence. This
mainly tools of statistics) is not fully understood until our days and then an paper presents a thorough overview of deterministic descriptions of
active research area consists in studying this law in the rigorous framework turbulence within the realm of Navier-Stokes equations. By focusing on the
of the Navier-Stokes equations which describe in a mathematical way the fluids use of deterministic models, we aim to better understand the nature of
motion and in particular the movement of turbulent fluids. In this setting, turbulence, how it arises, and how it can be controlled or harnessed for
the purpose of this chapter is to highlight the fact that if we consider the practical purposes. The need for such models is pressing, as they can be
Navier-Stokes equations on R3 then certain physical quantities, necessary for used to improve the design of fluid-based technologies, such as naval
the study of the Kolmogorov's dissipation law, have no a rigorous definition vessels, aircraft, and wind turbines, among others. The main body of the
and then to give a sense to these quantities we suggest to consider the paper is divided into several sections that cover different aspects of
Navier-Stokes equations with an additional damping term. In the framework of deterministic descriptions of turbulence. The first section introduces the
these damped equations, we obtain some estimates for the energy's dissipation Navier-Stokes equations and provides a brief overview of their solution.
rate according to the Kolmogorov's dissipation law.In the second chapter we The second section then delves into deterministic models of turbulence,
are interested in study the stationary solutions of the damped Navier- Stokes starting with a basic introduction to the Kolmogorov theory of turbulence
Deterministics introduced in the previous chapter. These stationary solutions are a and moving on to more advanced models. In particular, we investigate
descriptions of the particular type of solutions which do not depend of the temporal variable and models based on the concepts of eddies and energy cascades, as well as
turbulence in the their study is motivated by the fact that we always consider the Navier-Stokes 594 models that use multiscale approaches to capture the range of phenomena 455
Navier-Stokes equations with a stationary external force. In this chapter we study two that turbulence can exhibit. In the third section of the paper, we turn
equations properties of the stationary solutions : the first property concerns the our attention to numerical simulations of turbulence. We describe the use
stability of these solutions where we prove that if we have a control on the of high-performance computing and sophisticated algorithms to solve the
external force then all non stationary solution (with depends of both spatial Navier-Stokes equations, while titrating the advantages and limitations of
and temporal variables) converges toward a stationary solution. The second various numerical methods. We then proceed to describe how deterministic
property concerns the decay in spatial variable of the stationary solutions. descriptions of turbulence can be integrated into numerical simulations
These properties of stationary solutions are a consequence of the damping term for optimal performance and predictive capabilities. The final section of
introduced in the Navier-Stokes equations.In the third chapter we still study the paper discusses some of the key challenges facing the field in the
the stationary solutions of Navier-Stokes equations but now we consider the coming years. These include the need for more efficient and accurate
classical equations (without any additional damping term). The purpose of this models, the development of novel simulation techniques, and the
chapter is to study an other problem related to the deterministic description integration of experimental data to improve model prediction. We conclude
of the turbulence : the frequency decay of the stationary solutions. Indeed, by highlighting some of the potential applications of deterministic models
according to the K41 theory, if the fluid is in a laminar setting then the of turbulence to industrial processes, environmental studies, and even
stationary solutions of the Navier-Stokes equations must exhibit a exponential astrophysics. Overall, this paper presents an in-depth review of
frequency decay which starts at lows frequencies. But, if the fluid is in a deterministic descriptions of turbulence in the context of the
turbulent setting then this exponential frequency decay must be observed only Navier-Stokes equations. By providing a comprehensive overview of the
at highs frequencies. In this chapter, using some Fourier analysis tools, we current state of the field, we aim to provide researchers and
give a precise description of this exponential frequency decay in the laminar practitioners with a better understanding of the nature of turbulence and
and in the turbulent setting.In the fourth and last chapter we return to the the tools necessary to control it. It is our hope that this work will help
stationary solutions of the classical Navier-Stokes equations and we study the to shape future research in this important and challenging area of
uniqueness of these solutions in the particular case without any external physics.
force. Following some ideas of G. Seregin, we study the uniqueness of these
solutions first in the framework of Lebesgue spaces of and then in the a
general framework of Morrey spaces.
Phylogenetic approaches are finding more and more applications outside the
field of biology. Astrophysics is no exception since an overwhelming amount of
multivariate data has appeared in the last twenty years or so. In particular,
the diversification of galaxies throughout the evolution of the Universe quite
naturally invokes phylogenetic approaches. We have demonstrated that Maximum
Parsimony brings useful astrophysical results, and we now proceed toward the
analyses of large datasets for galaxies. In this talk I present how we solve Clustering is a widely used technique in astrophysics to study celestial
the major difficulties for this goal: the choice of the parameters, their objects and their properties. However, traditional clustering approaches
discretization, and the analysis of a high number of objects with an often fall short in properly accounting for the complex evolutionary
unsupervised NP-hard classification technique like cladistics. 1. Introduction relationships between objects, especially those involving large-scale
How do the galaxy form, and when? How did the galaxy evolve and transform astrophysical phenomena. Therefore, in this paper, we propose the use of
themselves to create the diversity we observe? What are the progenitors to phylogenetic tools in clustering analyses in order to better understand
present-day galaxies? To answer these big questions, observations throughout the underlying evolutionary processes governing celestial objects in
the Universe and the physical modelisation are obvious tools. But between astrophysical systems. We begin by introducing the fundamentals of
these, there is a key process, without which it would be impossible to extract phylogenetics and how it can be applied to astrophysics. We describe the
some digestible information from the complexity of these systems. This is concept of a "phylogenetic tree" which captures the hypothesized
classification. One century ago, galaxies were discovered by Hubble. From evolutionary relationships between celestial objects based on their
images obtained in the visible range of wavelengths, he synthetised his observable traits and characteristics. By constructing these phylogenetic
observations through the usual process: classification. With only one trees, we can gain insights into the evolutionary processes that govern
parameter (the shape) that is qualitative and determined with the eye, he these objects and how they may have evolved over time. We then discuss how
found four categories: ellipticals, spirals, barred spirals and irregulars. these phylogenetic tools can be incorporated into clustering analyses. We
This is the famous Hubble classification. He later hypothetized relationships introduce a novel method for constructing phylogenetic distance matrices,
between these classes, building the Hubble Tuning Fork. The Hubble which can be used as input into traditional clustering algorithms. By
classification has been refined, notably by de Vaucouleurs, and is still used utilizing these distance matrices, we can cluster celestial objects based
Clustering with as the only global classification of galaxies. Even though the physical not only on their observable traits, but also on their evolutionary
phylogenetic tools relationships proposed by Hubble are not retained any more, the Hubble Tuning 593 relationships, leading to a more comprehensive understanding of these 382
in astrophysics Fork is nearly always used to represent the classification of the galaxy astrophysical systems. We illustrate the effectiveness of our approach
diversity under its new name the Hubble sequence (e.g. Delgado-Serrano, 2012). through a case study of a large-scale simulation of galaxy formation and
Its success is impressive and can be understood by its simplicity, even its evolution. We show that the use of phylogenetic-based clustering leads to
beauty, and by the many correlations found between the morphology of galaxies a more accurate and comprehensive understanding of the evolutionary
and their other properties. And one must admit that there is no alternative up history of galaxies within the simulation. Additionally, we demonstrate
to now, even though both the Hubble classification and diagram have been that our approach can be used to identify "outlier" objects that may have
recognised to be unsatisfactory. Among the most obvious flaws of this unique evolutionary histories or properties. Finally, we discuss the
classification, one must mention its monovariate, qualitative, subjective and potential applications of phylogenetic-based clustering in future
old-fashioned nature, as well as the difficulty to characterise the morphology astrophysical research. We highlight the usefulness of this approach in
of distant galaxies. The first two most significant multivariate studies were studying diverse astrophysical systems, including stars, planets, and even
by Watanabe et al. (1985) and Whitmore (1984). Since the year 2005, the number entire galaxies. We also propose potential extensions to our method, such
of studies attempting to go beyond the Hubble classification has increased as incorporating additional sources of data or refining the phylogenetic
largely. Why, despite of this, the Hubble classification and its sequence are analyses themselves. In conclusion, this paper showcases the power of
still alive and no alternative have yet emerged (Sandage, 2005)? My feeling is using phylogenetic tools in clustering analyses within astrophysics. By
that the results of the multivariate analyses are not easily integrated into a accounting for the complex evolutionary relationships between celestial
one-century old practice of modeling the observations. In addition, objects, we gain a more comprehensive understanding of these astrophysical
extragalactic objects like galaxies, stellar clusters or stars do evolve. systems and their properties. We hope that this paper serves as a starting
Astronomy now provides data on very distant objects, raising the question of point for future research into the application of phylogenetics within
the relationships between those and our present day nearby galaxies. Clearly, astrophysics and beyond.
this is a phylogenetic problem. Astrocladistics 1 aims at exploring the use of
phylogenetic tools in astrophysics (Fraix-Burnet et al., 2006a,b). We have
proved that Maximum Parsimony (or cladistics) can be applied in astrophysics
and provides a new exploration tool of the data (Fraix-Burnet et al., 2009,
2012, Cardone \& Fraix-Burnet, 2013). As far as the classification of galaxies
is concerned, a larger number of objects must now be analysed. In this paper,
The use of null hypothesis significance testing (NHST) has been widely
criticized in the field of sports science, leading to the call for
alternative statistical methods that can offer better insight into the
magnitude of effects. Inferential statistics based on magnitude-based
inferences (MBI) have emerged as a promising alternative to NHST for
Research in Sports Sciences is supported often by inferences based on the investigating sports-related research questions. MBI offers the
declaration of the value of the statistic statistically significant or possibility of quantifying the magnitude of differences between groups or
nonsignificant on the bases of a P value derived from a null-hypothesis test. treatments, rather than simply testing for statistically significant
Taking into account that studies are manly conducted in sample, the use of differences. This approach provides researchers with a more meaningful
null hypothesis testing only allows estimating the true values (population) of interpretation of their results and can ultimately lead to more informed
the statistics used. However, evidence has grown in many areas of knowledge conclusions that are relevant to practitioners and athletes alike. The
that this approach often leads to confusion and misinterpretation. To overcome need to move away from NHST is not only due to its limitations in
Infer\^encia Baseada this limitation they have recently emerged recommendations to support the providing meaningful results, but also because of its reliance on
em Magnitudes na statistical analysis with approaches that make use of more intuitive arbitrary thresholds (e.g., a p-value of 0.05) that do not necessarily
investiga\c{c}\~ao interpretations and more practical, especially based on the magnitudes reflect the strength and importance of the observed effects. In contrast,
em Ci\^encias do (certainty / uncertainty) of the true values found. With the intent to provide MBI uses a statistical approach that is based on the estimation of effect
Esporte. A alternative solutions to methodological designs recurrently used in research 251 sizes and their confidence intervals, enabling a more nuanced 349
necessidade de in sports sciences, this paper will seek to i) briefly spell out some of the interpretation of the findings. Moreover, MBI can also mitigate problems
romper com os testes weaknesses associated with the null hypothesis tests based in the P value; ii) associated with small sample sizes, which are common in sports science
de hip\'otese nula e reflect on the implications of the use of practical/clinical significance as research. By focusing on effect size rather than the p-value, MBI can
os valores de p opposed to statistical significance; iii) submit proposals for use the provide more stable and reliable estimates of the true population effect,
inferences based on the magnitude, particularly in the visualization and even when working with small sample sizes. MBI also offers advantages over
interpretation of results; iv) present and discuss the limitations of other approaches, such as Bayesian statistics, in terms of its simplicity,
magnitude-based inference. Thus, this update article discourages, in a ease of interpretation, and its potential to improve scientific
sustained-based, the use of significance tests based only on the concept of communication. By adopting MBI, researchers and practitioners in sports
null hypothesis. Alternatively, it is proposed to use methods of inference science can overcome some of the limitations of NHST and obtain more
based on magnitudes as they allow interpretations of the practical/clinical meaningful and informative results. In conclusion, the use of MBI in the
effects results obtained. investigation of sports-related research questions is becoming
increasingly popular. MBI offers a more meaningful and informative
approach to statistical inference, which can ultimately lead to more
informed conclusions and practical implications for athletes and
practitioners. The scientific community should continue to embrace and
explore the potential of MBI as a valuable alternative to NHST in sports
science research.
Let $G$ be a simple, undirected, finite graph with vertex set $V(G)$ and edge
set $E(G)$. A $k$-dimensional box is a Cartesian product of closed intervals $
[a_1,b_1]\times [a_2,b_2]\times...\times [a_k,b_k]$. The {\it boxicity} of
$G$, $\boxi(G)$ is the minimum integer $k$ such that $G$ can be represented as Boxicity and Poset Dimension are two closely related concepts in
the intersection graph of $k$-dimensional boxes, i.e. each vertex is mapped to combinatorial optimization that have recently received extensive
a $k$-dimensional box and two vertices are adjacent in $G$ if and only if attention. Boxicity refers to the smallest integer k such that a graph can
their corresponding boxes intersect. Let $\poset=(S,P)$ be a poset where $S$ be represented as the intersection graph of k-dimensional axis-aligned
is the ground set and $P$ is a reflexive, anti-symmetric and transitive binary boxes. Poset Dimension, on the other hand, measures the smallest number of
relation on $S$. The dimension of $\poset$, $\dim(\poset)$ is the minimum linear extensions required to represent a partially ordered set (poset).
integer $t$ such that $P$ can be expressed as the intersection of $t$ total While seemingly distinct, recent research has shown that these two
orders. Let $G_\poset$ be the \emph{underlying comparability graph} of $\ concepts are closely related, and understanding one can give insight into
poset$, i.e. $S$ is the vertex set and two vertices are adjacent if and only the other. The study of these two concepts has important practical
if they are comparable in $\poset$. It is a well-known fact that posets with applications in fields such as scheduling, logistics, and VLSI layout
the same underlying comparability graph have the same dimension. The first design. For example, in scheduling applications, boxicity can be used to
result of this paper links the dimension of a poset to the boxicity of its construct scheduling models based on resource constraints where the
underlying comparability graph. In particular, we show that for any poset $\ resources have different capacities. These models are used in a variety of
poset$, $\boxi(G_\poset)/(\chi(G_\poset)-1) \le \dim(\poset)\le 2\boxi(G_\ industries, such as manufacturing and transportation, to optimize the use
poset)$, where $\chi(G_\poset)$ is the chromatic number of $G_\poset$ and $\ of resources and increase efficiency. In the past few decades, much
chi(G_\poset)\ne1$. It immediately follows that if $\poset$ is a height-2 research has been devoted to the algorithmic aspects of Boxicity and Poset
poset, then $\boxi(G_\poset)\le \dim(\poset)\le 2\boxi(G_\poset)$ since the Dimension. Algorithms have been developed to compute the boxicity and the
Boxicity and Poset underlying comparability graph of a height-2 poset is a bipartite graph. The 577 poset dimension of a given graph, which have found applications in data 380
Dimension second result of the paper relates the boxicity of a graph $G$ with a natural analysis and optimization. Additionally, several linear time algorithms
partial order associated with the \emph{extended double cover} of $G$, denoted have been developed to compute the poset dimension of certain classes of
as $G_c$: Note that $G_c$ is a bipartite graph with partite sets $A$ and $B$ posets such as grid posets. Despite these algorithmic advances, there are
which are copies of $V(G)$ such that corresponding to every $u\in V(G)$, there still many open problems related to Boxicity and Poset Dimension. One such
are two vertices $u_A\in A$ and $u_B\in B$ and $\{u_A,v_B\}$ is an edge in problem is determining the relationship between these two concepts for
$G_c$ if and only if either $u=v$ or $u$ is adjacent to $v$ in $G$. Let $\ specific classes of graphs. Another open problem is determining the
poset_c$ be the natural height-2 poset associated with $G_c$ by making $A$ the computational complexity of the poset dimension problem for certain
set of minimal elements and $B$ the set of maximal elements. We show that $\ classes of posets. In recent years, researchers have also explored the
frac{\boxi(G)}{2} \le \dim(\poset_c) \le 2\boxi(G)+4$. These results have some relationship between Boxicity and Poset Dimension and other graph
immediate and significant consequences. The upper bound $\dim(\poset)\le 2\ parameters, such as tree-width, clique number, and chromatic number.
boxi(G_\poset)$ allows us to derive hitherto unknown upper bounds for poset Several results have been obtained showing connections between these
dimension such as $\dim(\poset)\le 2\tw(G_\poset)+4$, since boxicity of any parameters, which can be useful when analyzing large datasets. Overall,
graph is known to be at most its $\tw+2$. In the other direction, using the the study of Boxicity and Poset Dimension has applications in a wide range
already known bounds for partial order dimension we get the following: (1) The of fields and has stimulated much research in combinatorial optimization.
boxicity of any graph with maximum degree $\Delta$ is $O(\Delta\log^2\Delta)$ Although many problems related to these concepts remain open, recent
which is an improvement over the best known upper bound of $\Delta^2+2$. (2) advances have shed light on their connections to other graph parameters,
There exist graphs with boxicity $\Omega(\Delta\log\Delta)$. This disproves a and further research in this area has the potential to unlock new insights
conjecture that the boxicity of a graph is $O(\Delta)$. (3) There exists no and improve the efficiency of data analysis at large.
polynomial-time algorithm to approximate the boxicity of a bipartite graph on
$n$ vertices with a factor of $O(n^{0.5-\epsilon})$ for any $\epsilon>0$,
unless $NP=ZPP$.
This second part of a 2 volume-expertise is mainly based on the results of the
inventory described in the first part. It is also based on a "worrying"
statement in the terms of reference of this expert report 1, which states that
"Asia enjoys a favourable technological context [in terms of equipment
according to UN statistics]. Nevertheless, digital technology is still hardly
present in the practices of the member institutions of the Agency and the This research paper explores the potential of distance learning and
francophone university community; So distance education is not well developed: digital tools for higher education and research in the Asia-Pacific
there are currently no French-language distance training courses offered by an region, specifically focusing on Cambodia, Laos, and Vietnam. The second
establishment in Asia; The region has only 14 enrolled in ODL over the period part of this paper presents a set of recommendations and a roadmap for
2010 - 2014; Only three institutions have responded to the AUF's "Mooc" calls implementing these technologies in these countries. The first
for projects over the last two years, etc.". The terms of reference also recommendation is to invest in infrastructure that supports digital
indicate a state of deficiency "in the Francophone digital campuses whose learning. This includes building robust networks, providing access to
officials explain that the computer equipment are less and less used for digital devices, and training teachers and students on how to use these
individual practices". The proliferation of mobile digital technologies that tools effectively. Governments and universities must collaborate to make
would normally constitute an important asset for the development of teaching these resources widely available. The second recommendation is to create
practices and innovative research around the Francophone digital campuses has and curate high-quality digital content. Textbooks, lectures, and other
not lived up to expectations. The paper refers to another no less important learning materials should be developed or adapted for digital formats, and
detail that would explain the paradox between the proliferation of made available online for free or at a low cost. Collaboration between
technological tools and the reduction in usage when it indicates that, in institutions can help pool resources and reduce duplication of effort. The
parallel, and contrary to the francophone campuses, In English a positive third recommendation is to develop interactive and collaborative learning
Formation {\`a} dynamics of integration of T ICE and distance". The document provides concrete tools. These tools can help students engage with course materials and with
distance et outils examples, such as the ASEAN Cyber University (ACU) program run by South Korea each other, even when they are studying remotely. This can include
num{\'e}riques pour and its e-learning centers in Cambodia, Laos, Vietnam and Myanmar, The videoconferencing software, social media platforms, and online discussion
l'enseignement sup{\ Vietnamese language and the fablab set up in the region since 2014 without the forums. The fourth recommendation is to ensure that the development of
'e}rieur et la Francophonie being involved. A first hypothesis emerges from this premonitory digital learning tools is based on research and best practices. This
recherche en observation that it is not technology that justifies the gradual 570 includes evaluating the effectiveness of different tools and approaches, 410
Asie-Pacifique demobilization (or even demotivation) of local actors to establish forms of and using this information to improve their design and implementation.
(Cambodge, Laos, Francophone partnerships for training and research. Nor is it a question of Research should also be conducted on the impact of digital learning on
Vietnam). Partie 02 political will to improve technological infrastructure in digital training and student outcomes. The final recommendation is to foster a culture of
: recommandations et research. Almost all the interviews carried out within the framework of this innovation and collaboration. This includes creating spaces for
feuille de route mission demonstrated the convergence of views and ambitious attitudes experimentation and learning, and providing opportunities for educators
expressed by three categories of actors encountered:- political actors met in and researchers to share their experiences and insights. Governments and
the ministries of education of the three countries who are aware of the universities must work together to support this culture. To achieve these
importance of digital education and the added value generated by technologies recommendations, a roadmap is proposed that outlines the necessary steps
for education. Each of the three countries has a regulatory framework and and timelines. This includes identifying key stakeholders and partners,
national technological innovation projects for education and digital setting up pilot projects, and scaling up successful initiatives. The
education;- public and private academic institutions which, through their roadmap also highlights potential challenges and risks that must be
rectors, presidents and technical and pedagogical leaders, demonstrate their addressed, such as the need for adequate funding and the risk of unequal
profound convictions for digital education (for reasons of quality, access to digital resources. Overall, this paper argues that distance
competitiveness and economic interest). However, given the rather centralized learning and digital tools have the potential to transform higher
governance model at state level, the majority of academic institutions in the education and research in the Asia-Pacific region, but that careful
three countries are often awaiting the promulgation of legal texts (decrees, planning and implementation are necessary to ensure their effectiveness
charters, conventions, directives , Etc.) that enable them to act and adopt and accessibility. The recommendations and roadmap presented in this paper
innovative solutions in teaching and research;- Teacher-researchers relatively can serve as a starting point for governments, universities, and other
little consulted in this expertise, but sufficiently engaged as actors on the stakeholders in the region who are interested in pursuing this path.
ground to be able to predict their points of view with regard to the use of
digital in their pedagogical and research activities. Teachers and researchers
with relatively modest incomes would inevitably have a decisive role in any
academic reform on the digital level if concrete mobilizing arguments could
compensate for their shortfalls by joining digital training or development
projects or programs.
CEA/IRFM is conducting R\&D efforts in order to validate the critical RF
components of the 5 GHz ITER LHCD system, which is expected to transmit 20 MW
of RF power to the plasma. Two 5 GHz 500 kW BeO pill-box type window
prototypes have been manufactured in 2012 by the PMB Company, in close
collaboration with CEA/IRFM. Both windows have been validated at low power,
showing good agreement between measured and modeling, with a return loss The development of next-generation fusion reactors, such as the
better than 32 dB and an insertion loss below 0.05 dB. This paper reports on International Thermonuclear Experimental Reactor (ITER), requires the use
the window RF design and the low power measurements. The high power tests up of high power sources to generate and sustain plasma. The Lower Hybrid
to 500kW have been carried out in March 2013 in collaboration with NFRI. Current Drive (LHCD) system is one such high-energy source that is
Results of these tests are also reported. In the current ITER LHCD design, 20 designed to provide steady-state current drive for fusion reactor
MW Continuous Wave (CW) of Radio-Frequency power at 5 GHz are expected to be operation. In order to achieve this, the LHCD system requires a high-power
generated and transmitted to the plasma. In order to separate the vacuum window capable of transmitting RF power to the plasma. This paper
vessel pressure from the cryostat waveguide pressure, forty eight 5 GHz 500kW describes the design and RF measurements of a 5 GHz 500 kW window for the
CW windows are to be assembled on the waveguides at the equatorial port ITER LHCD system. The goal of this research was to develop an optimized
flange. For nuclear safety reasons, forty eight additional windows could be design for the window that would meet the stringent requirements of the
located in the cryostat section, to separate and monitor the cryostat LHCD system, while also providing reliable and efficient operation. The
waveguide pressure from the exterior transmission line pressure. These windows window design was based on a number of key factors, including the
are identified as being one of the main critical components for the ITER LHCD transmission properties of the materials, the need for high power handling
system since first ITER LHCD studies [1] [2] [3] or more recently [4] [5] , capability, and the thermal management of the structure. Simulations were
Design and RF and clearly require an important R\&D effort. In this context and even if the used to optimize the design of the window, and several prototypes were
measurements of a 5 LHCD system is not part of the construction baseline, the CEA/IRFM is fabricated to investigate the performance of the design under a variety of
GHz 500 kW window conducting a R\&D effort in order to validate a design and the performances of 569 conditions. RF measurements were taken on the prototypes to determine 373
for the ITER LHCD these RF windows. In order to begin the assessment of this need, two 5 GHz 500 their transmission properties and to verify that they met the requirements
system kW/5 s pill-box type windows prototypes have been manufactured in 2012 by the of the LHCD system. The results of these measurements revealed that the
PMB Company in close collaboration with the CEA/IRFM [6]. The section 2 of window design was able to meet all of the high-power requirements of the
this paper reports the RF and mechanical design of a 5 GHz window. Some ITER LHCD system. The research also investigated the thermal behavior of
features of the mechanical design and the experimental RF measurements at low the window during operation, using simulations and experimental
power are reported in section 3. High power results, made in collaboration measurements. The results showed that the thermal management of the window
with NFRI, are detailed in section 4. The development of CW windows is was critical to its performance, as high-power RF transmission caused
discussed in the conclusion. 2-RF AND MECHANICAL DESIGN The proposed 5 GHz RF significant heating of the window. The simulations and experiments showed
window is based on a pill-box design [2] , i.e. a ceramic brazed in portion of that effective cooling of the window was necessary to maintain reliable
a circular waveguide, connected on either side to a rectangular waveguide and efficient operation. In conclusion, this paper presents the design and
section. Typical design rules of thumb of such device are circular section RF measurements of a 5 GHz 500 kW window for the ITER LHCD system. The
diameter about the same size of the diagonal of the rectangular waveguide (cf. research demonstrated the feasibility of the window design, and provided
FIGURE 1). Without taking into account the ceramic, the circular section important insights into the challenges associated with high-power RF
length is approximately half a guided wavelength of the circular TE 11 mode, transmission and thermal management. The results of this research will be
in order for the device to act as a half-wave transformer. Once optimized, useful in the development of next-generation fusion reactors, as they will
taking into account the ceramic, matching is correct only for a narrow band of help to ensure the reliable and efficient operation of the LHCD system.
frequency and is very sensitive to the device dimensions and the ceramic
relative permittivity. The heat losses in the ceramic, which have to be
extracted by an active water cooling, depends on the inside electric field
topology and of ceramic dielectric loss (loss tangent). Undesirable modes due
to parasitic resonances can be excited in the ceramic volume, raising the
electric field and
Context. Circumstellar disks are known to contain a significant mass in dust
ranging from micron to centimeter size. Meteorites are evidence that
individual grains of those sizes were collected and assembled into
planetesimals in the young solar system. Aims. We assess the efficiency of
dust collection of a swarm of non-drifting planetesimals {\rev with radii
ranging from 1 to $10^3$\,km and beyond. Methods. We calculate the collision This research paper explores the mechanisms of dust filtering and
probability of dust drifting in the disk due to gas drag by planetesimal processing in planetesimals, specifically focusing on non-drifting
accounting for several regimes depending on the size of the planetesimal, planetesimals. The collision probabilities for such planetesimals were
dust, and orbital distance: the geometric, Safronov, settling, and three-body derived and analyzed to illuminate their impacts on the filtration and
regimes. We also include a hydrodynamical regime to account for the fact that processing of dust. These collision probabilities were analyzed through
small grains tend to be carried by the gas flow around planetesimals. Results. numerical simulations, which incorporated varied parameters such as
We provide expressions for the collision probability of dust by planetesimals planetesimal radius and density as well as dust particle size and
and for the filtering efficiency by a swarm of planetesimals. For standard distribution. The results of the analysis show that non-drifting
turbulence conditions (i.e., a turbulence parameter $\alpha=10^{-2}$), planetesimals play a significant role in the early stages of planet
filtering is found to be inefficient, meaning that when crossing a formation through their ability to filter and process dust. Through
minimum-mass solar nebula (MMSN) belt of planetesimals extending between 0.1 collisions with dust particles, these planetesimals are able to both grow
AU and 35 AU most dust particles are eventually accreted by the central star in size and remove debris from the surrounding environment. The effects of
rather than colliding with planetesimals. However, if the disk is weakly this filtering and processing are not only important for the planetesimal
turbulent ($\alpha=10^{-4}$) filtering becomes efficient in two regimes: (i) itself, but also relevant for later stages of planet formation when large
On the filtering and when planetesimals are all smaller than about 10 km in size, in which case bodies form through collisions of planetesimals. The analytical framework
processing of dust collisions mostly take place in the geometric regime; and (ii) when planetary and numerical simulations used in the research provide a foundation for
by planetesimals 1. embryos larger than about 1000 km in size dominate the distribution, have a future studies into the processes of dust filtering and processing by
Derivation of scale height smaller than one tenth of the gas scale height, and dust is of planetesimals. The collision probabilities derived for non-drifting
collision millimeter size or larger in which case most collisions take place in the 567 planetesimals can be applied to other studies of planetesimal growth and 368
probabilities for settling regime. These two regimes have very different properties: we find dust filtration, improving our understanding of early stages of planetary
non-drifting that the local filtering efficiency $x_{filter,MMSN}$ scales with $r^{-7/4}$ formation. An important implication of this research is that the
planetesimals (where $r$ is the orbital distance) in the geometric regime, but with $r^{-1/ mechanisms of dust filtration and processing by non-drifting planetesimals
4}$ to $r^{1/4}$ in the settling regime. This implies that the filtering of enable the successful formation of larger bodies like planets and
dust by small planetesimals should occur close to the central star and with a asteroids, crucial to the evolution of our solar system and others. By
short spread in orbital distances. On the other hand, the filtering by embryos examining these mechanisms, insights can be gained not only into the
in the settling regime is expected to be more gradual and determined by the formation of planets, but also into the evolution of other celestial
extent of the disk of embryos. Dust particles much smaller than millimeter bodies throughout the universe. In conclusion, this research paper
size tend only to be captured by the smallest planetesimals because they provides a thorough analysis of the collision probabilities for
otherwise move on gas streamlines and their collisions take place in the non-drifting planetesimals and their impact on the processing and
hydrodynamical regime. Conclusions. Our results hint at an inside-out filtering of dust. The results show that non-drifting planetesimals play
formation of planetesimals in the infant solar system because small an important role in the early stages of planet formation through their
planetesimals in the geometrical limit can filter dust much more efficiently ability to remove debris and grow in size. This research can improve our
close to the central star. However, even a fully-formed belt of planetesimals understanding of the formation of planets not only in our solar system,
such as the MMSN only marginally captures inward-drifting dust and this seems but throughout the universe as well. The analytical framework and
to imply that dust in the protosolar disk has been filtered by planetesimals numerical simulations used in this study provide a strong foundation for
even smaller than 1 km (not included in this study) or that it has been further research in this field.
assembled into planetesimals by other mechanisms (e.g., orderly growth,
capture into vortexes). Further refinement of our work concerns, among other
things: a quantitative description of the transition region between the hydro
and settling regimes; an assessment of the role of disk turbulence for
collisions, in particular in the hydro regime; and the coupling of our model
to a planetesimal formation model.
Stylolites are ubiquitous geo-patterns observed in rocks in the upper crust,
from geological reservoirs in sedimentary rocks to deformation zones, in
folds, faults, and shear zones. These rough surfaces play a major role in the
dissolution of rocks around stressed contacts, the transport of dissolved
material and the precipitation in surrounding pores. Consequently, they play
an active role in the evolution of rock microstructures and rheological
properties in the Earth's crust. They are observed individually or in
networks, in proximity to fractures and joints, and in numerous geological Stylolites are a critical feature in sedimentary rocks, which have
settings. This review article deals with their geometrical and compositional garnered significant interest over the years given their widespread
characteristics and the factors leading to their genesis. The main questions occurrence and potential significance in several geological processes. In
this review focuses on are the following: How do they form? How can they be this review, we provide an extensive analysis of the literature available
used to measure strain and formation stress? How do they control fluid flow in on stylolites, thereby enabling a better understanding of their behavior
the upper crust? Geometrically, stylolites have fractal roughness, with and formation mechanisms. First, we discuss the various historical
fractal geometrical properties exhibiting typically three scaling regimes: a perspectives on stylolites and the evolution of ideas explaining their
self-affine scaling with Hurst exponent 1.1+/-0.1 at small scale (up to tens formation. Subsequently, we delve into the current understanding of the
or hundreds of microns), another one with Hurst exponent around 0.5 to 0.6 at physical and chemical processes that induce and animate stylolites. We
intermediate scale (up to millimeters or centimeters), and in the case of highlight field and laboratory studies, alongside analytical techniques
sedimentary stylolites, a flat scaling at large scale. More complicated such as petrography, scanning electron microscopy, electron microprobe,
anisotropic scaling (scaling laws depending of the direction of the profile and Raman spectroscopy, which have contributed significantly to the
considered) is found in the case of tectonic stylolites. We report models current state of knowledge on stylolites. We further analyze the
based on first principles from physical chemistry and statistical physics, composition and mineralogy of stylolites with a discussion on their role
including a mechanical component for the free-energy associated with stress in hydrocarbon exploration. We evaluate the interplay between mechanical
concentrations, and a precise tracking of the influence of grain-scale and chemical compaction mechanisms in their formation and briefly examine
heterogeneities and disorder on the resulting (micro)structures. Experimental some of the significant implications in reservoir quality assessments. We
Stylolites: A review efforts to reproduce stylolites in the laboratory are also reviewed. We show 565 discuss how their presence can affect porosity, permeability, and 339
that although micrometer-size stylolite teeth are obtained in laboratory ultimately oil recovery in underground reservoirs and provide a
experiments, teeth deforming numerous grains have not yet been obtained comprehensive review of the available literature on stylolites as a tool
experimentally, which is understandable given the very long formation time of in hydrocarbon exploration. Furthermore, we expound on the association of
such geometries. Finally, the applications of stylolites as strain and stress stylolites with various geological phenomena, including deformation
markers, to determine paleostress magnitude are reviewed. We show that the stress, fluid activity, and diagenesis. We examine the evidence of
scalings in stylolite heights and the crossover scale between these scalings syn-sedimentary versus post-sedimentary origin of stylolites, which has
can be used to determine the stress magnitude (its scalar value) perpendicular significant implications for their interpretation and paleo-environmental
to the stylolite surface during the stylolite formation, and that the stress reconstructions. The review offers insight into the potential use of
anisotropy in the stylolite plane can be determined for the case of tectonic stylolites in paleostress and paleohydrology analysis and their
stylolites. We also show that the crossover between medium (millimetric) significance as proxies for burial depth. We conclude our review by
scales and large (pluricentimetric) scales, in the case of sedimentary discussing current controversies in the field of stylolites such as their
stylolites, provides a good marker for the total amount of dissolution, which mode of initiation, the extent of their influence on rock properties, and
is still valid even when the largest teeth start to dissolve -- which leads to their role as deformation markers. Additionally, we highlight some of the
the loss of information, since the total deformation is not anymore recorded gaps in current knowledge on stylolites and offer suggestions for future
in a single marker structure. We discuss the impact of the stylolites on the research areas. Through this comprehensive review, we hope to provide a
evolution of the transport properties of the hosting rock, and show that they better understanding of stylolites, the processes that produce them, and
promote a permeability increase parallel to the stylolites, whereas their their potential applications in diverse geological fields.
effect on the permeability transverse to the stylolite can be negligible, or
may reduce the permeability, depending on the development of the stylolite.
Highlights: Stylolite formation depends on rock composition and structure,
stress and fluids. Stylolite geometry, fractal and self-affine properties,
network structure, are investigated. The experiments and physics-based
numerical models for their formation are reviewed. Stylolites can be used as
markers of strain, paleostress orientation and magnitude. Stylolites impact
transport properties, as function of maturity and flow direction.
Superoxide reductase (SOR) is an Fe protein that catalyzes the reduction of
superoxide to give H(2)O(2). Recently, the mutation of the Glu47 residue into
alanine (E47A) in the active site of SOR from Desulfoarculus baarsii has
allowed the stabilization of an iron-peroxo species when quickly reacted with
H(2)O(2) [Math{\'e} et al. (2002) J. Am. Chem. Soc. 124, 4966-4967]. To
further investigate this non-heme peroxo-iron species, we have carried out a M
{\"o}ssbauer study of the (57)Fe-enriched E47A SOR from D. baarsii reacted In this study, we focus on the M{\"o}ssbauer characterization of an
quickly with H(2)O(2). Considering the M{\"o}ssbauer data, we conclude, in unusual high-spin side-on peroxo-Fe3+ species in the active site of
conjunction with the other spectroscopic data available and with the results superoxide reductase from Desulfoarculus Baarsii and complement our
of density functional calculations on related models, that this species findings with density functional calculations on related models. Our
corresponds to a high-spin side-on peroxo-Fe(3+) complex. This is one of the investigation sheds light on the structure and properties of this
first examples of such a species in a biological system for which M{\"o} particular peroxo-Fe3+ species, with implications for elucidating the
ssbauer parameters are now available: delta(/Fe) = 0.54 (1) mm/s, DeltaE(Q) = mechanism of superoxide reduction by this enzyme. Our M{\"o}ssbauer
-0.80 (5) mm/s, and the asymmetry parameter eta = 0.60 (5) mm/s. The M{\"o} spectroscopy measurements reveal the presence of a high-spin Fe3+ center
ssbauer and spin Hamiltonian parameters have been evaluated on a model from in the peroxo species, which exhibits an unusual side-on binding mode.
M{\"o}ssbauer the side-on peroxo complex (model 2) issued from the oxidized iron center in This finding is supported by density functional theory (DFT) calculations,
characterization of SOR from Pyrococcus furiosus, for which structural data are available in the which predict a high-spin, side-on peroxo-Fe3+ species as the most stable
an unusual high-spin literature [Yeh et al. (2000) Biochemistry 39, 2499-2508]. For comparison, configuration. Our theoretical calculation of the M{\"o}ssbauer spectrum
side-on peroxo-Fe3+ similar calculations have been carried out on a model derived from 2 (model of the proposed side-on peroxo-Fe3+ species shows good agreement with the
species in the 3), where the [CH(3)-S](1)(-) group has been replaced by the neutral [NH(3)] experimental spectrum. Additionally, we investigate the effect of various
active site of (0) group [Neese and Solomon (1998) J. Am. Chem. Soc. 120, 12829-12848]. Both 563 ligands on the stability and electronic properties of the peroxo species 312
superoxide reductase models 2 and 3 contain a formally high-spin Fe(3+) ion (i.e., with empty using DFT. Our calculations indicate that specific ligands, such as
from Desulfoarculus minority spin orbitals). We found, however, a significant fraction imidazole, can have a significant impact on the electronic structure of
Baarsii. Density (approximately 0.6 for 2, approximately 0.8 for 3) of spin (equivalently the peroxo-Fe3+ center. We also investigate the reactivity of the peroxo
functional charge) spread over two occupied (minority spin) orbitals. The quadrupole species towards superoxide, using DFT to calculate the activation barriers
calculations on splitting value for 2 is found to be negative and matches quite well the for the reaction. Our results suggest that the high-spin, side-on
related models experimental value. The computed quadrupole tensors are rhombic in the case of peroxo-Fe3+ species is a likely intermediate for the reduction of
2 and axial in the case of 3. This difference originates directly from the superoxide by the enzyme. Overall, our study provides insight into the
presence of the thiolate ligand in 2. A correlation between experimental nature of the peroxo-Fe3+ species in superoxide reductase from
isomer shifts for Fe(3+) mononuclear complexes with computed electron Desulfoarculus Baarsii, and sheds light on the mechanism of superoxide
densities at the iron nucleus has been built and used to evaluate the isomer reduction by this enzyme. The combination of experimental M{\"o}ssbauer
shift values for 2 and 3 (0.56 and 0.63 mm/s, respectively). A significant spectroscopy and theoretical calculations using DFT allows us to probe the
increase of isomer shift value is found upon going from a methylthiolate to a electronic and structural properties of the peroxo species in a detailed
nitrogen ligand for the Fe(3+) ion, consistent with covalency effects due to and comprehensive manner. Our findings may have broader implications for
the presence of the axial thiolate ligand. Considering that the isomer shift the design and optimization of metalloenzymes for biotechnological
value for 3 is likely to be in the 0.61-0.65 mm/s range [Horner et al. (2002) applications.
Eur. J. Inorg. Chem., 3278-3283], the isomer shift value for a high-spin eta
(2)-O(2) Fe(3+) complex with an axial thiolate group can be estimated to be in
the 0.54-0.58 mm/s range. The occurrence of a side-on peroxo intermediate in
SOR is discussed in relation to the recent data published for a side-on
peroxo-Fe(3+) species in another biological system [Karlsson et al. (2003)
Science 299, 1039-1042].
Probability theory, epistemically interpreted, provides an excellent, if not
the best available account of inductive reasoning. This is so because there
are general and definite rules for the change of subjective probabilities
through information or experience; induction and belief change are one and
same topic, after all. The most basic of these rules is simply to Inductive reasoning plays a vital role in scientific inquiry by enabling
conditionalize with respect to the information received; and there are similar the inference of conclusions from empirical data. Despite its
and more general rules. 1 Hence, a fundamental reason for the epistemological significance, there exist fundamental challenges in explicating the
success of probability theory is that there at all exists a well-behaved foundations of inductive reasoning. In particular, traditional approaches
concept of conditional probability. Still, people have, and have reasons for, have used probabilistic frameworks as the primary tool for modeling
various concerns over probability theory. One of these is my starting point: inductive reasoning. However, this approach has limited application in
Intuitively, we have the notion of plain belief; we believe propositions2 to real-life scenarios, and even fails to provide an adequate explanation for
be true (or to be false or neither). Probability theory, however, offers no phenomena that involve non-probabilistic or correlated uncertainties. In
formal counterpart to this notion. Believing A is not the same as having this paper, we introduce a general non-probabilistic theory of inductive
probability 1 for A, because probability 1 is incorrigible3; but plain belief reasoning, which offers a fresh perspective on traditional models of
is clearly corrigible. And believing A is not the same as giving A a reasoning. Our theory considers inductive reasoning as a process of
probability larger than some 1 - c, because believing A and believing B is developing theories about the causal structure of a given phenomenon, and
usually taken to be equivalent to believing A & B.4 Thus, it seems that the seeks to provide a systematic framework for this process. Our approach
formal representation of plain belief has to take a non-probabilistic route. considers the problem of inductive reasoning as part of a larger context
Indeed, representing plain belief seems easy enough: simply represent an of decision-making under uncertainty, and utilizes tools from causal
epistemic state by the set of all propositions believed true in it or, since I inference, game theory, and information theory. Through the lens of our
make the common assumption that plain belief is deductively closed, by the theory, we can better understand and formalize the process of inductive
A General conjunction of all propositions believed true in it. But this does not yet reasoning. Specifically, we articulate a new framework that identifies the
Non-Probabilistic provide a theory of induction, i.e. an answer to the question how epistemic 560 causal structure of a given phenomenon as the key element for making sound 386
Theory of Inductive states so represented are changed tbrough information or experience. There is inductive inferences, and further explore how this structure can be
Reasoning a convincing partial answer: if the new information is compatible with the old uncovered. Our framework is founded on the idea that inductive reasoning
epistemic state, then the new epistemic state is simply represented by the can be viewed as a game between the reasoner and nature, and that the
conjunction of the new information and the old beliefs. This answer is partial optimal strategy in this game requires an analysis of the causal
because it does not cover the quite common case where the new information is structure. We then introduce a new class of models that capture
incompatible with the old beliefs. It is, however, important to complete the non-probabilistic uncertainties and are well-defined within this
answer and to cover this case, too; otherwise, we would not represent plain framework. These models are shown to be as versatile as probabilistic
belief as conigible. The crucial problem is that there is no good completion. models in describing inductive reasoning, and in fact, can better capture
When epistemic states are represented simply by the conjunction of all the nuances of non-probabilistic uncertainties. Overall, the proposed
propositions believed true in it, the answer cannot be completed; and though non-probabilistic theory of inductive reasoning offers a new approach to
there is a lot of fruitful work, no other representation of epistemic states model and solve complicated inductive inference problems. It leverages
has been proposed, as far as I know, which provides a complete solution to advances in machine learning and artificial intelligence to bring us one
this problem. In this paper, I want to suggest such a solution. In [4], I have step closer to achieving a more general understanding of inductive
more fully argued that this is the only solution, if certain plausible reasoning. We conclude by highlighting some future directions for
desiderata are to be satisfied. Here, in section 2, I will be content with research, including the challenges in developing new methodologies and
formally defining and intuitively explaining my proposal. I will compare my applications for the principle of inductive inference. Ultimately, this
proposal with probability theory in section 3. It will turn out that the work is a stepping stone towards deeper insights into the fundamental
theory I am proposing is structurally homomorphic to probability theory in question of how we do science and build theories in the face of
important respects and that it is thus equally easily implementable, but uncertainty.
moreover computationally simpler. Section 4 contains a very brief comparison
with various kinds of logics, in particular conditional logic, with Shackle's
functions of potential surprise and related theories, and with the Dempster -
Shafer theory of belief functions.
Given a universe of discourse X-a domain of possible outcomes-an experiment
may consist of selecting one of its elements, subject to the operation of
chance, or of observing the elements, subject to imprecision. A priori
uncertainty about the actual result of the experiment may be quantified,
representing either the likelihood of the choice of :r_X or the degree to
which any such X would be suitable as a description of the outcome. The former
case corresponds to a probability distribution, while the latter gives a
possibility assignment on X. The study of such assignments and their
properties falls within the purview of possibility theory [DP88, Y80, Z783. This research paper presents a formal model of uncertainty for
It, like probability theory, assigns values between 0 and 1 to express possibilistic rules. Possibilistic rules are commonly used in the fields
likelihoods of outcomes. Here, however, the similarity ends. Possibility of artificial intelligence, fuzzy logic, and decision-making. The proposed
theory uses the maximum and minimum functions to combine uncertainties, model aims to provide a means of quantifying the uncertainty inherent in
whereas probability theory uses the plus and times operations. This leads to these rules. To achieve this goal, the model introduces the notion of a
very dissimilar theories in terms of analytical framework, even though they possibility distribution function. This function assigns a possibility
share several semantic concepts. One of the shared concepts consists of value to each possible state of the world, representing the degree to
expressing quantitatively the uncertainty associated with a given which that state is possible given the available evidence and the
distribution. In probability theory its value corresponds to the gain of uncertainty inherent in the possibilistic rules. The model also defines a
information that would result from conducting an experiment and ascertaining set of rules for combining possibility values, allowing for the
an actual result. This gain of information can equally well be viewed as a aggregation of uncertain information from multiple sources. The proposed
decrease in uncertainty about the outcome of an experiment. In this case the model provides several key benefits over existing approaches to
standard measure of information, and thus uncertainty, is Shannon entropy uncertainty in possibilistic rules. First, it provides a more principled
[AD75, G77]. It enjoys several advantages-it is characterized uniquely by a and rigorous mathematical framework for dealing with uncertainty in these
few, very natural properties, and it can be conveniently used in decision rules. Second, it allows for a more flexible representation of
Formal Model of processes. This application is based on the principle of maximum entropy; it uncertainty, enabling the modeling of more complex and nuanced forms of
Uncertainty for has become a popular method of relating decisions to uncertainty. This paper 560 uncertainty. Finally, it enables the use of a wider range of probabilistic 329
Possibilistic Rules demonstrates that an equally integrated theory can be built on the foundation inference techniques, allowing for more accurate and efficient
of possibility theory. We first show how to define measures of in formation decision-making. To demonstrate the efficacy of the proposed model, we
and uncertainty for possibility assignments. Next we construct an provide several empirical evaluations. These evaluations demonstrate the
information-based metric on the space of all possibility distributions defined effectiveness of the model in capturing and reasoning with uncertainty in
on a given domain. It allows us to capture the notion of proximity in various scenarios. Specifically, we show that the model can accurately
information content among the distributions. Lastly, we show that all the capture uncertainty in complex decision-making tasks, such as medical
above constructions can be carried out for continuous diagnosis and financial forecasting. We also show that the model is
distributions-possibility assignments on arbitrary measurable domains. We computationally efficient, making it feasible for use in real-world
consider this step very significant-finite domains of discourse are but applications. Overall, this research paper presents a formal model of
approximations of the real-life infinite domains. If possibility theory is to uncertainty for possibilistic rules. The proposed model provides a more
represent real world situations, it must handle continuous distributions both principled and rigorous mathematical framework for dealing with
directly and through finite approximations. In the last section we discuss a uncertainty in these rules, enabling a more flexible representation of
principle of maximum uncertainty for possibility distributions. We show how uncertainty and the use of a wider range of probabilistic inference
such a principle could be formalized as an inference rule. We also suggest it techniques. The empirical evaluations demonstrate the effectiveness and
could be derived as a consequence of simple assumptions about combining computational efficiency of the proposed model, highlighting its
information. We would like to mention that possibility assignments can be suitability for use in real-world applications.
viewed as fuzzy sets and that every fuzzy set gives rise to an assignment of
possibilities. This correspondence has far reaching consequences in logic and
in control theory. Our treatment here is independent of any special
interpretation; in particular we speak of possibility distributions and
possibility measures, defining them as measurable mappings into the interval
[0, 1]. Our presentation is intended as a self-contained, albeit terse
summary. Topics discussed were selected with care, to demonstrate both the
completeness and a certain elegance of the theory. Proofs are not included; we
only offer illustrative examples.
Let $(\{X_i(t)\}_{i\in \mathbb{Z}^d})_{t\geq 0}$ be the system of interacting
diffusions on $[0,\infty)$ defined by the following collection of coupled
stochastic differential equations: \begin{eqnarray}dX_i(t)=\sum\limits_{j\in \
mathbb{Z}^d}a(i,j)[X_j(t)-X_i(t)] dt+\sqrt{bX_i(t)^2} dW_i(t), \eqntext{i\in \
mathbb{Z}^d,t\geq 0.}\end{eqnarray} Here, $a(\cdot,\cdot)$ is an irreducible This research paper investigates phase transitions in the long-time
random walk transition kernel on $\mathbb{Z}^d\times \mathbb{Z}^d$, $b\in (0,\ behavior of interacting diffusions. Alongside phase transitions in the
infty)$ is a diffusion parameter, and $(\{W_i(t)\}_{i\in \mathbb{Z}^d})_{t\geq Ising model, the authors demonstrate the existence of phase transitions
0}$ is a collection of independent standard Brownian motions on $\mathbb{R}$. for interacting diffusions in a bounded domain. Specifically, the authors
The initial condition is chosen such that $\{X_i(0)\}_{i\in \mathbb{Z}^d}$ is study the asymptotic behavior of the occupation time near the boundary of
a shift-invariant and shift-ergodic random field on $[0,\infty)$ with mean $\ the domain and the formation of persistent macroscopic clusters. For this
Theta\in (0,\infty)$ (the evolution preserves the mean). We show that the purpose, they use representation formulas for occupation times and
long-time behavior of this system is the result of a delicate interplay establish the asymptotics of the same. The authors derive phase diagrams
between $a(\cdot,\cdot)$ and $b$, in contrast to systems where the diffusion based on the occupation time, and these phase diagrams are quite different
function is subquadratic. In particular, let $\hat{a}(i,j)={1/2}[a(i,j)+a from the traditional ones for the Ising model. Furthermore, the authors
(j,i)]$, $i,j\in \mathbb{Z}^d$, denote the symmetrized transition kernel. We show that the phase transition for interacting diffusions is much richer
show that: (A) If $\hat{a}(\cdot,\cdot)$ is recurrent, then for any $b>0$ the than that for the Ising model, as it exhibits a discontinuity phenomenon.
system locally dies out. (B) If $\hat{a}(\cdot,\cdot)$ is transient, then They discuss the origin of this discontinuity phenomenon and describe how
Phase transitions there exist $b_*\geq b_2>0$ such that: (B1)d The system converges to an it arises from a subtle interplay between the sub-diffusive nature of the
for the long-time equilibrium $\nu_{\Theta}$ (with mean $\Theta$) if $0<b<b_*$. (B2) The system diffusion process and the interaction among particles. The authors conduct
behavior of locally dies out if $b>b_*$. (B3) $\nu_{\Theta}$ has a finite 2nd moment if 555 simulations to verify their analytical results and study the long-time 318
interacting and only if $0<b<b_2$. (B4) The 2nd moment diverges exponentially fast if and behavior of interacting Brownian particles in a bounded domain. They
diffusions only if $b>b_2$. The equilibrium $\nu_{\Theta}$ is shown to be associated and provide numerical evidence of the existence of multiple phases for the
mixing for all $0<b<b_*$. We argue in favor of the conjecture that $b_*>b_2$. occupation time near the boundary and demonstrate the discontinuity
We further conjecture that the system locally dies out at $b=b_*$. For the phenomenon of the phase transition. They also observe the emergence of
case where $a(\cdot,\cdot)$ is symmetric and transient we further show that: macroscopic clusters in numerical simulations and show that they are
(C) There exists a sequence $b_2\geq b_3\geq b_4\geq ... >0$ such that: (C1) $ responsible for the mentioned discontinuity. In conclusion, the findings
\nu_{\Theta}$ has a finite $m$th moment if and only if $0<b<b_m$. (C2) The of this research paper demonstrate that the long-time behavior of
$m$th moment diverges exponentially fast if and only if $b>b_m$. (C3) $b_2\leq interacting diffusions exhibits phase transitions that are significantly
(m-1)b_m<2$. uad(C4) $\lim_{m\to\infty}(m-1)b_m=c=\sup_{m\geq 2}(m-1)b_m$. The different from those in the Ising model. The authors establish the
proof of these results is based on self-duality and on a representation existence of a discontinuity phenomenon that is a result of subtle
formula through which the moments of the components are related to exponential interactions between the sub-diffusive nature of the diffusion process and
moments of the collision local time of random walks. Via large deviation the interaction among particles. They provide rigorous mathematical proofs
theory, the latter lead to variational expressions for $b_*$ and the $b_m$'s, and numerical simulations to support their claims. The authors' results
from which sharp bounds are deduced. The critical value $b_*$ arises from a have implications in diverse areas such as population genetics,
stochastic representation of the Palm distribution of the system. The special statistical physics, and materials science.
case where $a(\cdot,\cdot)$ is simple random walk is commonly referred to as
the parabolic Anderson model with Brownian noise. This case was studied in the
memoir by Carmona and Molchanov [Parabolic Anderson Problem and Intermittency
(1994) Amer. Math. Soc., Providence, RI], where part of our results were
already established.
The production of heavy quarkonium in heavy ion collisions has been used as an
important probe of the quark-gluon plasma (QGP). Due to the plasma screening
effect, the color attraction between the heavy quark antiquark pair inside a
quarkonium is significantly suppressed at high temperature and thus no bound
states can exist, i.e., they "melt". In addition, a bound heavy quark
antiquark pair can dissociate if enough energy is transferred to it in a
dynamical process inside the plasma. So one would expect the production of
quarkonium to be considerably suppressed in heavy ion collisions. However,
experimental measurements have shown that a large amount of quarkonia survive Effective Field Theory (EFT) has become an increasingly important tool in
the evolution inside the high temperature plasma. It is realized that the the field of nuclear physics, providing a systematic framework for
in-medium recombination of unbound heavy quark pairs into quarkonium is as conducting calculations in a range of energy regimes. By treating the
crucial as the melting and dissociation. Thus, phenomenological studies have nuclear force as a perturbation of an underlying theory, such as Quantum
to account for static screening, dissociation and recombination in a Chromodynamics (QCD), EFT allows for the accurate prediction of
consistent way. But recombination is less understood theoretically than the observables across a broad range of energies and systems. In this paper,
melting and dissociation. Many studies using semi-classical transport we review the application of EFT in nuclear physics, discussing its
equations model the recombination effect from the consideration of detailed fundamental principles and its use in nuclear structure, nuclear reactions
balance at thermal equilibrium. However, these studies cannot explain how the and nuclear astrophysics. We first summarize the basic concepts of EFT,
system of quarkonium reaches equilibrium and estimate the time scale of the including power counting, renormalization and the operator product
thermalization. Recently, another approach based on the open quantum system expansion, and their applicability to nuclear forces. We then present
formalism started being used. In this framework, one solves a quantum several examples of EFT calculations in nuclear structure, including the
evolution for in-medium quarkonium. Dissociation and recombination are prediction of ground-state properties, such as binding energies and radii,
Application of accounted for consistently. However, the connection between the semi-classical and excited-state spectra, such as giant resonances and alpha clustering.
Effective Field transport equation and the quantum evolution is not clear. In this We demonstrate the advantages of EFT over other approaches, such as shell
Theory in Nuclear dissertation, I will try to address the issues raised above. As a warm-up 543 model and mean-field theory, in providing accurate and systematic 332
Physics project, I will first study a similar problem: $\alpha$-$\alpha$ scattering at descriptions of nuclear phenomena. Next, we discuss EFT in the context of
the $^8$Be resonance inside an $e^-e^+\gamma$ plasma. By applying pionless nuclear reactions, with a focus on low-energy reactions, such as radiative
effective field theory and thermal field theory, I will show how the plasma capture and scattering, and their relevance for nuclear astrophysics. We
screening effect modifies the $^8$Be resonance energy and width. I will review the formalism of EFT for few-nucleon scattering and its extension
discuss the need to use the open quantum system formalism when studying the to more complex systems, such as those encountered in nuclear
time evolution of a system embedded inside a plasma. Then I will use effective astrophysics, and describe its successes in explaining experimental data
field theory of QCD and the open quantum system formalism to derive a Lindblad and predicting astrophysical reaction rates. Finally, we discuss the
equation for bound and unbound heavy quark antiquark pairs inside a future prospects of EFT in nuclear physics, considering its potential
weakly-coupled QGP. Under the Markovian approximation and the assumption of impact on our understanding of neutron-rich and exotic nuclei, the physics
weak coupling between the system and the environment, the Lindblad equation of the neutron star crust and the equation of state of nuclear matter. We
will be shown to turn to a Boltzmann transport equation if a Wigner transform conclude that EFT has emerged as a powerful and versatile tool in nuclear
is applied to the open system density matrix. These assumptions will be physics, capable of providing accurate and systematic predictions across a
justified by using the separation of scales, which is assumed in the range of nuclear phenomena and regimes. Its future impact on nuclear
construction of effective field theory. I will show the scattering amplitudes physics is likely to be significant, enabling predictions of unprecedented
that contribute to the collision terms in the Boltzmann equation are gauge accuracy for a range of important experiments and observations.
invariant and infrared safe. By coupling the transport equation of quarkonium
with those of open heavy flavors and solving them using Monte Carlo
simulations, I will demonstrate how the system of bound and unbound heavy
quark antiquark pairs reaches detailed balance and equilibrium inside the QGP.
Phenomenologically, my calculations can describe the experimental data on
bottomonium production. Finally I will extend the framework to study the
in-medium evolution of heavy diquarks and estimate the production rate of the
doubly charmed baryon $\Xi_{cc}^{++}$ in heavy ion collisions.
Context. The chemistry of the diffuse interstellar medium rests upon three This study presents an analysis of far-infrared observational data to
pillars: exothermic ion-neutral reactions (" cold chemistry "), endothermic detect tracers of oxygen chemistry in diffuse clouds. Diffuse clouds have
neutral-neutral reactions with significant activation barriers (" warm low density and are primarily composed of atomic hydrogen, with small
chemistry "), and reactions on the surfaces of dust grains. While warm amounts of He, C, N, O, etc. Despite their low density, these clouds
chemistry becomes important in the shocks associated with turbulent contain a significant fraction of the interstellar gas in our galaxy. The
dissipation regions, the main path for the formation of interstellar OH and chemical evolution of diffuse clouds is fundamentally different from that
H2O is that of cold chemistry. Aims. The aim of this study is to of dense clouds, and the key chemical processes that control their
observationally confirm the association of atomic oxygen with both atomic and physical characteristics are not yet fully understood. The far-infrared
molecular gas phases, and to understand the measured abundances of OH and OH + spectral range is key to unveil the composition and chemical properties of
as a function of the available reservoir of H2. Methods. We obtained these clouds. We analyzed far-infrared spectral data acquired using the
absorption spectra of the ground states of OH, OH+ and OI with high-velocity Herschel Space Observatory to measure major cooling lines from the oxygen
resolution, with GREAT on-board SOFIA, and with the THz receiver at the APEX. chemistry in diffuse clouds. The excitation of these lines frequently
We analyzed them along with ancillary spectra of HF and CH from HIFI. To emerges from chemical processes that originate from photoabsorption or
deconvolve them from the hyperfine structure and to separate the blend that is photoionization by far-ultraviolet (FUV) photons. The set of observed
due to various velocity components on the sightline, we fit model spectra cooling lines and their relative intensities can, in principle, provide
consisting of an appropriate number of Gaussian profiles using a method constraints on the physical conditions, composition, and life cycle of
combining simulated annealing with downhill simplex minimization. Together diffuse clouds. Our analysis focused on a sample of known diffuse clouds
with HF and/or CH as a surrogate for H2, and HI $\lambda$21 cm data, the whose spectroscopic features show clear evidence for the presence of
molecular hydrogen fraction f^N\_H2 = N(H 2)/(N(H) + 2N(H 2)) can be atomic and molecular tracers of the gas-phase oxygen chemistry. Oxygen
determined. We then investigated abundance ratios as a function of f^N\_H2. molecules such as O$_2$, O$_3$, and CO are the strongest tracers due to
Far-infrared study Results. The column density of OI is correlated at a high significance with their high abundance and relative stability at low density. Our goal was
of tracers of oxygen the amount of available molecular and atomic hydrogen, with an atomic oxygen 538 to use the cooling lines from these tracers to constrain the physical and 473
chemistry in diffuse abundance of $3 \times 10 ^{-4}$ relative to H nuclei. While the velocities of chemical properties of the diffuse clouds and to investigate variations in
clouds the absorption features of OH and OH+ are loosely correlated and reflect the the gas-phase oxygen chemistry in different environments of the Milky Way.
spiral arm crossings on the sightline, upon closer inspection they display an Our analysis yielded several key results. First, we detected parent and
anticorrespondence. The arm-to-interarm density contrast is found to be higher daughter cooling lines from O$_3$ and O$_2$ with the highest
in OH than in OH+. While both species can coexist, with a higher abundance in signal-to-noise ratio among the observed features. This suggests that
OH than in OH+, the latter is found less frequently in absence of OH than the O$_3$ and O$_2$ are the most efficient cooling mechanisms in
other way around, which is a direct consequence of the rapid destruction of FUV-illuminated diffuse clouds. Second, we found empirical correlations
OH+ by dissociative recombination when not enough H2 is available. This between the relative cooling line intensities and the FUV radiation field
conjecture has been substantiated by a comparison between the OH/OH+ ratio strength in our sample. These correlations provide important constraints
with f^N\_H2, showing a clear correlation. The hydrogen abstraction reaction on the chemical and physical evolution of the diffuse clouds. Finally, we
chain OH+ (H2,H) H2O+ (H2,H)H3O+ is confirmed as the pathway for the detected the CO fundamental transitions at 4.7 and 2.6 THz in several
production of OH and H 2 O. Our estimate of the branching ratio of the sources, consistent with previous detections of CO in diffuse clouds. Our
dissociative recombination of H3O+ to OH and H2O is confined within the results demonstrate the power and importance of far-infrared studies for
interval of 84 to 91%, which matches laboratory measurements (74 to 83%). -- A understanding the composition and chemical properties of diffuse clouds.
correlation between the linewidths and column densities of OH+ features is Our analysis of the various tracers of oxygen chemistry in these clouds
found to be significant with a false-alarm probability below 5%. Such a can provide constraints on the formation, physical properties, and
correlation is predicted by models of interstellar MHD turbulence. For OH the evolution of diffuse clouds in different regions of the galaxy.
same correlation is found to be insignificant because there are more narrow Furthermore, our empirical correlations suggest that FUV radiation fields
absorption features. Conclusions. While it is difficult to assess the play an essential role in regulating the physical conditions and chemical
contributions of warm neutral-neutral chemistry to the observed abundances, it properties of diffuse clouds. Our findings can inform future studies of
seems fair to conclude that the predictions of cold ion-neutral chemistry the chemical and physical evolution of molecular gas in the Universe.
match the abundance patterns we observed.
In [12], Nilsson proposed the probabilistic logic in which the truth values of
logical propositions are probability values between 0 and 1. It is applicable
to any logical system for which the consistency of a finite set of
propositions can be established. The probabilistic inference scheme reduces to
the ordinary logical inference when the probabilities of all propositions are Probabilistic logic has proven to be a powerful tool for dealing with
either 0 or 1. This logic has the same limitations of other probabilistic uncertainty and reasoning under incomplete or inconsistent information.
reasoning systems of the Bayesian approach. For common sense reasoning, This paper explores some extensions of probabilistic logic that have been
consistency is not a very natural assumption. We have some well known proposed in the literature, with a focus on probabilistic defeasible
examples: {Dick is a Quaker, Quakers are pacifists, Republicans are not reasoning, Bayesian knowledge bases, and probabilistic programming.
pacifists, Dick is a Republican}and {Tweety is a bird, birds can fly, Tweety Probabilistic defeasible reasoning extends classical defeasible reasoning
is a penguin}. In this paper, we shall propose some extensions of the to handle uncertain knowledge, allowing the derivation of conclusions that
probabilistic logic. In the second section, we shall consider the space of all are not necessarily warranted by the premises but are still plausible
interpretations, consistent or not. In terms of frames of discernment, the given the available evidence. We review several approaches to
basic probability assignment (bpa) and belief function can be defined. probabilistic defeasible reasoning, including probabilistic argumentation,
Dempster's combination rule is applicable. This extension of probabilistic maximum entropy-based inference, and Bayesian networks with uncertain
logic is called the evidential logic in [ 1]. For each proposition s, its evidence. Bayesian knowledge bases combine probabilistic logic with
belief function is represented by an interval [Spt(s), Pls(s)]. When all such ontology representation to model uncertain and incomplete knowledge about
intervals collapse to single points, the evidential logic reduces to a domain. We discuss the main features of Bayesian knowledge bases,
probabilistic logic (in the generalized version of not necessarily consistent including hierarchical structure, probabilistic axioms, and inference
interpretations). Certainly, we get Nilsson's probabilistic logic by further algorithms. We also examine some applications of Bayesian knowledge bases
restricting to consistent interpretations. In the third section, we shall give in natural language understanding, diagnosis, and prediction.
a probabilistic interpretation of probabilistic logic in terms of Probabilistic programming is a recent paradigm for defining probabilistic
Some Extensions of multi-dimensional random variables. This interpretation brings the models and conducting probabilistic inference via computer programs. We
Probabilistic Logic probabilistic logic into the framework of probability theory. Let us consider 530 introduce the basic concepts of probabilistic programming, including 346
a finite set S = {sl, s2, ..., Sn) of logical propositions. Each proposition random variables, conditioning, and inference. We outline some of the key
may have true or false values; and may be considered as a random variable. We challenges in developing efficient and expressive probabilistic
have a probability distribution for each proposition. The e-dimensional random programming languages, such as handling the combination of discrete and
variable (sl,..., Sn) may take values in the space of all interpretations of continuous probability distributions, dealing with large-scale
2n binary vectors. We may compute absolute (marginal), conditional and joint probabilistic models, and designing effective inference algorithms. We
probability distributions. It turns out that the permissible probabilistic then discuss some open research questions and opportunities in the area of
interpretation vector of Nilsson [12] consists of the joint probabilities of probabilistic logic extensions. One promising direction is to study the
S. Inconsistent interpretations will not appear, by setting their joint integration of probabilistic logic with other probabilistic models, such
probabilities to be zeros. By summing appropriate joint probabilities, we get as decision networks, relational models, and time series models. Another
probabilities of individual propositions or subsets of propositions. Since the direction is to investigate the foundations of probabilistic logic and its
Bayes formula and other techniques are valid for e-dimensional random connections with other areas of logic and mathematics, such as
variables, the probabilistic logic is actually very close to the Bayesian paraconsistent logic, nonstandard analysis, and category theory. Finally,
inference schemes. In the last section, we shall consider a relaxation scheme we conclude by highlighting the potential impact and practical
for probabilistic logic. In this system, not only new evidences will update applications of probabilistic logic extensions in various fields, such as
the belief measures of a collection of propositions, but also constraint artificial intelligence, cognitive science, biology, economics, and social
satisfaction among these propositions in the relational network will revise sciences. We argue that the development of advanced probabilistic
these measures. This mechanism is similar to human reasoning which is an reasoning techniques and tools is crucial for addressing complex
evaluative process converging to the most satisfactory result. The main idea real-world problems that involve uncertainty, ambiguity, and incomplete
arises from the consistent labeling problem in computer vision. This method is data.
originally applied to scene analysis of line drawings. Later, it is applied to
matching, constraint satisfaction and multi sensor fusion by several authors
[8], [16] (and see references cited there). Recently, this method is used in
knowledge aggregation by Landy and Hummel [9].
Using nCB films adsorbed on MoS 2 substrates studied by x-ray diffraction,
optical microscopy and Scanning Tunneling Microscopy, we demonstrate that
ordered interfaces with well-defined orientations of adsorbed dipoles induce
planar anchoring locked along the adsorbed dipoles or the alkyl chains, which
play the role of easy axes. For two alternating orientations of the adsorbed
dipoles or dipoles and alkyl chains, bi-stability of anchoring can be
obtained. The results are explained using the introduction of fourth order In this paper, we investigate the characteristics and behavior of ordered
terms in the phenomenological anchoring potential, leading to the interfaces in liquid crystals with dual easy axes. This combination of
demonstration of first order anchoring transition in these systems. Using this properties is known to induce a number of complex and interesting
phenomenological anchoring potential, we finally show how the nature of phenomena, including domain formation, phase transitions, and the
anchoring in presence of dual easy axes (inducing bi-stability or average emergence of topological defects. To begin, we provide a theoretical
orientation between the two easy axes) can be related to the microscopical framework for understanding the behavior of dual easy axes in nematic
nature of the interface. Introduction Understanding the interactions between liquid crystals. We describe the different types of ordering that can
liquid crystal (LC) and a solid substrate is of clear applied interest, the occur in these systems, including homogenous and patterned alignments. We
vast majority of LC displays relying on control of interfaces. However this also discuss how external fields and boundary conditions can be used to
concerns also fundamental problems like wetting phenomena and all phenomena of control and manipulate the ordering of the liquid crystal. We then turn
orientation of soft matter bulk induced by the presence of an interface. In our attention to the experimental study of dual easy axes in liquid
LCs at interfaces, the so-called easy axes correspond to the favoured crystals. Using a combination of microscopy and scattering techniques, we
orientations of the LC director close to the interface. If one easy axis only analyze the structures and dynamics of ordered interfaces. We find that
is defined for one given interface, the bulk director orients along or close the ordering of the liquid crystal displays a rich variety of behavior,
to this axis [1]. It is well known that, in anchoring phenomena, two major including the formation of complex textures such as stripes and walls. One
Ordered interfaces effects compete to impose the anchoring directions of a liquid crystal, first, particularly interesting phenomenon that arises from the combination of
for dual easy axes the interactions between molecules and the interface, second, the substrate 523 dual easy axes and ordered interfaces is the formation of topological 355
in liquid crystals roughness whose role has been analyzed by Berreman [2]. The influence of defects. These defects can take on a number of different forms, including
adsorbed molecular functional groups at the interface is most often dominant disclinations and dislocations, and have been shown to have important
with, for example in carbon substrates, a main influence of unsaturated carbon implications for the properties and behavior of the liquid crystal. We
bonds orientation at the interface [3]. In common LC displays, there is one also investigate the effect of confined geometries on the ordering of dual
unique easy axis, but modifications of surfaces have allowed for the discovery easy axes in liquid crystals. By studying the behavior of these systems in
of promising new anchoring-related properties. For instance, the first thin films and droplets, we are able to gain insight into how the ordering
anchoring bi-stability has been established on rough surfaces, associated with is affected by the presence of surfaces and interfaces. We find that the
electric ordo-polarization [4] and the competition between a stabilizing confinement induces a number of new and unexpected effects, including the
short-range term and a destabilizing long-range term induced by an external formation of new types of topological defects and the emergence of novel
field, can induce a continuous variation of anchoring orientation [5]. More phase behavior. Overall, our study demonstrates that the combination of
recently, surfaces with several easy axes have been studied extensively. It dual easy axes and ordered interfaces in liquid crystals is a rich and
has been shown that control of a continuous variation of director pretilt, complex field of study with a number of important implications for both
obtained in several systems [6, 7], is associated with the presence of two fundamental science and technological applications. Our research
different easy axes, one perpendicular to the substrate (homeotropic) and one contributes to a growing body of knowledge on these fascinating systems
planar [7, 8]. Similar models can explain the continuous evolution of and paves the way for future research in this area.
anchoring between two planar orientations observed on some crystalline
substrates [9]. However, in the same time, two easy axes can also lead to
anchoring bi-stability [10, 11] or discontinuous transitions of anchoring [9],
which is not compatible with the model established to interpret observed
control of pretilt. In order to be able to predict if bi-stability or
continuous combination of the two easy axes occurs for one given system, it
becomes necessary to understand the microscopic origin of the easy axes.
Virtualization allows the simulation of automotive ECUs on a Windows PC
executing in closed-loop with a vehicle simulation model. This approach
enables to move certain development tasks from road or test rigs and HiL
(Hardware in the loop) to PCs, where they can often be performed faster and This paper presents a comprehensive study on the full virtualization of
cheaper. Renault has recently established such a virtualization process for Renault's engine management software and its application to system
powertrain control software based on Simulink models. If the number of development. The aim of the research is to investigate the feasibility of
runnables exceeds a threshold (about 1500) the execution of the virtual ECU is complete system virtualization for engine control systems, which will
no longer straight forward and specific techniques are required. This paper allow more flexibility and assess the practicality of this approach for
describes the motivation behind a Simulink model based process, the software development in the automotive industry. To achieve this goal, a
virtualization process and applications of the resulting virtual ECUs. Domain: detailed analysis of the Renault engine management system architecture is
Critical Transportation Systems Topic: Processes, methods and tools, in performed, including its various components and sub-systems. This analysis
particular: virtual engineering and simulation 1. Motivation Since 2010, helps identify the key characteristics and features that require
Renault has established a framework to develop engine control software for consideration when creating a virtualized system. The research then
Diesel and Gasoline engines [6]. The framework is heavily based on MATLAB/ proposes a virtualization architecture based on various virtualization
Simulink and the idea of model-based development, which facilitates the techniques, such as hardware-level virtualization, kernel-level
carry-over and carry-across of application software between software projects. virtualization, and system-level virtualization. This architecture is
In the Renault EMS architecture software is composed in to about 20 functions, designed specifically for Renault's engine management system, taking into
such as Air System, Combustion etc. A function consists of modules. A module account the unique characteristics of the system. Several virtualization
is the smallest testable software unit and contains runnables to be scheduled prototypes are developed and implemented on the proposed architecture to
Full Virtualization and executed by the Operating System (Os) of the ECU. The Renault EMS identify potential issues in the virtualization process and to evaluate
of Renault's Engine development process includes basically the following steps [5]. 1. the performance of the virtualized system. The results of these tests show
Management Software Specification of about 200 generic configurable modules per ECU using MATLAB/ 514 that full virtualization of Renault's engine management software is 376
and Application to Simulink. 2. Generation of C code (EMS application software) from all module feasible and can be a promising approach for system development in the
System Development specifications using MATLAB/Simulink Embedded Coder. 3. MiL (Model in the automotive industry. Furthermore, the paper explores the benefits of
Loop) test and validation of the resulting executable specifications at module virtualization in relation to software development and analyzes the
level in a simulated system environment, considering only essential potential implications for Renault's development process. The paper
interactions with other modules and system environment. This is essentially a highlights the potential for quicker development cycles, improved software
back-to-back test to make sure that the Simulink model of a module and the testing, and better fault isolation among other benefits. Moreover,
corresponding production C code show equivalent and intended behaviour. To through this virtualization, developers can build, test, and deploy
insure software quality, this step is repeatedly performed with steps 1 and 2, various software updates to Renault engines more efficiently. Finally, the
based on the simulation capabilities of MATLAB/Simulink. 4. Configuration of research concludes with an outlook on the future of full virtualization in
modules to fit to the specific needs of a software project, such as absence or the automotive industry and potential directions that future research can
presence of certain components. 5. Integration of generated configured C code take. The study builds a fundamental understanding that can serve as a
and hand-coded platform software (basic software) on supplied target hardware, basis for future investigations into virtualization approaches for engine
a real ECU that communicates with other controllers via CAN and other busses. management systems. Overall, this paper presents a detailed analysis of
6. Validation and test of all modules on system level using the real ECU. In full virtualization of Renault's engine management software and its
contrast to step 3, the interactions of all modules and interactions with the application to system development. The results show that virtualization
system environment are visible then and subject to testing. For example, the can offer substantial benefits for developers in the automotive industry
Os runs all scheduled runnables then, not just those of the modules considered in terms of software development, testing, and deployment. This research
to be 'essential' for a module under test. Critical assessment of the above provides a foundation for future work in the field and adds to the
process shows that there is a considerable delay between delivery of a set of conversation on innovative approaches to engineering automotive systems.
specifications to the software project team (at the end of step 3) and
system-level tests based on an ECU that runs entire software (step 6). Typical
delays are weeks or months.
One Monad to Prove Them All is a modern fairy tale about curiosity and
perseverance, two important properties of a successful PhD student. We follow
the PhD student Mona on her adventure of proving properties about Haskell
programs in the proof assistant Coq. On the one hand, as a PhD student in
computer science Mona observes an increasing demand for correct software
products. In particular, because of the large amount of existing software,
verifying existing software products becomes more important. Verifying
programs in the functional programming language Haskell is no exception. On
the other hand, Mona is delighted to see that communities in the area of The concept of a "monad" has been used across multiple fields and
theorem proving are becoming popular. Thus, Mona sets out to learn more about disciplines throughout history, from Western philosophy to computer
the interactive theorem prover Coq and verifying Haskell programs in Coq. To science. In this paper, we examine the concept of a monad and its
prove properties about a Haskell function in Coq, Mona has to translate the applications in various areas of science. We begin with a historical
function into Coq code. As Coq programs have to be total and Haskell programs overview of the term, exploring its origins in the writings of Plato and
are often not, Mona has to model partiality explicitly in Coq. In her quest Aristotle and its development over time. From there, we move into an
for a solution Mona finds an ancient manuscript that explains how properties examination of the ways in which the idea of a monad has been used in
about Haskell functions can be proven in the proof assistant Agda by mathematics, particularly in calculus and topology. Moving beyond
translating Haskell programs into monadic Agda programs. By instantiating the mathematics, we explore the use of monads in physics, including its
monadic program with a concrete monad instance the proof can be performed in application in quantum mechanics and string theory. We also examine the
either a total or a partial setting. Mona discovers that the proposed concept of a monad in chemistry, considering its role in the development
transformation does not work in Coq due to a restriction in the termination of new materials and its potential for creating new molecules through
One Monad to Prove checker. In fact the transformation does not work in Agda anymore as well, as 513 precisely controlled reactions. In the field of computer science, monads 288
Them All the termination checker in Agda has been improved. We follow Mona on an have been used as a way to structure functional programming languages. We
educational journey through the land of functional programming where she explore the use of monads in Haskell and Scala, two prominent functional
learns about concepts like free monads and containers as well as basics and programming languages, and discuss the advantages they provide in terms of
restrictions of proof assistants like Coq. These concepts are well-known code reusability and modularity. Finally, we consider the potential
individually, but their interplay gives rise to a solution for Mona's problem applications of monads in the field of artificial intelligence. We explore
based on the originally proposed monadic tranformation that has not been how monads could be used to model complex systems, such as the human
presented before. When Mona starts to test her approach by proving a statement brain, and how they could be used to develop more efficient algorithms for
about simple Haskell functions, she realizes that her approach has an machine learning and natural language processing. Overall, our paper
additional advantage over the original idea in Agda. Mona's final solution not argues that the concept of a monad has far-reaching applications across a
only works for a specific monad instance but even allows her to prove variety of scientific disciplines. By exploring the different ways in
monad-generic properties. Instead of proving properties over and over again which the concept has been used, we hope to provide a deeper understanding
for specific monad instances she is able to prove properties that hold for all of this fundamental idea and its potential for shaping the future of
monads representable by a container-based instance of the free monad. In order science and technology.
to strengthen her confidence in the practicability of her approach, Mona
evaluates her approach in a case study that compares two implementations for
queues. In order to share the results with other functional programmers the
fairy tale is available as a literate Coq file. If you are a citizen of the
land of functional programming or are at least familiar with its customs, had
a journey that involved reasoning about functional programs of your own, or
are just a curious soul looking for the next story about monads and proofs,
then this tale is for you.
In essence, a neural network is an arbitrary differentiable, parametrized
function. Choosing a neural network architecture for any task is as complex as
searching the space of those functions. For the last few years, 'neural
architecture design' has been largely synonymous with 'neural architecture
search' (NAS), i.e. brute-force, large-scale search. NAS has yielded
significant gains on practical tasks. However, NAS methods end up searching
for a local optimum in architecture space in a small neighborhood around
architectures that often go back decades, based on CNN or LSTM. In this work,
we present a different and complementary approach to architecture design,
which we term 'zero-shot architecture design' (ZSAD). We develop methods that
can predict, without any training, whether an architecture will achieve a The design of artificial neural networks (ANNs) has been revolutionized by
relatively high test or training error on a task after training. We then go on the concept of the nonlinearity coefficient (NLC). The NLC is a measure of
to explain the error in terms of the architecture definition itself and the nonlinearity of the activation functions used in the hidden layers of
develop tools for modifying the architecture based on this explanation. This an ANN. The use of an optimal NLC value in designing ANNs can improve
confers an unprecedented level of control on the deep learning practitioner. their performance by minimizing overfitting and increasing generalization
They can make informed design decisions before the first line of code is accuracy. In this paper, we present a practical guide to designing neural
written, even for tasks for which no prior art exists. Our first major architectures using the NLC. We begin with an overview of the fundamental
contribution is to show that the 'degree of nonlinearity' of a neural concepts of ANNs and their activation functions. We then introduce the
architecture is a key causal driver behind its performance, and a primary concept of the NLC and explain how it can be determined for a given ANN
aspect of the architecture's model complexity. We introduce the 'nonlinearity architecture. Next, we present experimental results based on several
The Nonlinearity coefficient' (NLC), a scalar metric for measuring nonlinearity. Via extensive benchmark datasets, demonstrating the effectiveness of the NLC in
Coefficient - A empirical study, we show that the value of the NLC in the architecture's improving the performance of ANNs. We also compare the performance of ANNs
Practical Guide to randomly initialized state before training is a powerful predictor of test 512 designed using the NLC with those designed using other traditional 259
Neural Architecture error after training and that attaining a right-sized NLC is essential for methods, such as regularization and early stopping. Furthermore, we
Design attaining an optimal test error. The NLC is also conceptually simple, provide guidelines for selecting an appropriate NLC value based on the
well-defined for any feedforward network, easy and cheap to compute, has complexity of the dataset, the size of the training dataset, and the
extensive theoretical, empirical and conceptual grounding, follows optimization algorithm used. Lastly, we discuss the limitations of using
instructively from the architecture definition, and can be easily controlled the NLC in neural architecture design, such as the high computational cost
via our 'nonlinearity normalization' algorithm. We argue that the NLC is the of calculating NLC and the dependence of the optimal NLC on the dataset
most powerful scalar statistic for architecture design specifically and neural and architecture used. In conclusion, this paper provides a comprehensive
network analysis in general. Our analysis is fueled by mean field theory, guide to using the NLC in neural architecture design. The practical
which we use to uncover the 'meta-distribution' of layers. Beyond the NLC, we guidelines and experimental results presented here demonstrate the
uncover and flesh out a range of metrics and properties that have a efficacy of incorporating the NLC into the design process to improve the
significant explanatory influence on test and training error. We go on to performance of ANNs.
explain the majority of the error variation across a wide range of randomly
generated architectures with these metrics and properties. We compile our
insights into a practical guide for architecture designers, which we argue can
significantly shorten the trial-and-error phase of deep learning deployment.
Our results are grounded in an experimental protocol that exceeds that of the
vast majority of other deep learning studies in terms of carefulness and
rigor. We study the impact of e.g. dataset, learning rate, floating-point
precision, loss function, statistical estimation error and batch
inter-dependency on performance and other key properties. We promote research
practices that we believe can significantly accelerate progress in
architecture design research.
User-defined syntax extensions are useful to implement an embedded domain
specific language (EDSL) with good code-readability. They allow EDSL authors
to define domain-natural notation, which is often different from the host
language syntax. Nowadays, there are several research works of powerful
user-defined syntax extensions. One promising approach uses user-defined
operators. A user-defined operator is a function with user-defined syntax. It
can be regarded as a syntax extension implemented without macros. An advantage This paper discusses the implementation of user-defined operators and name
of user-defined operators is that an operator can be statically typed. The binding for new language constructs. The proposed approach allows
compiler can find type errors in the definition of an operator before the programmers to define their own operators, customized to the
operator is used. In addition, the compiler can resolve syntactic ambiguities domain-specific needs of their applications. The main goal is to enable a
by using static types. However, user-defined operators are difficult to concise and natural expression of complex operations, improving the
implement language constructs involving static name binding. Name binding is readability and maintainability of the code. The paper presents a formal
association between names and values (or memory locations). Our inquiry is specification of the syntax and semantics of the proposed extension, and
whether we can design a system for user-defined operators involving a new provides a reference implementation based on a modified version of an
custom name binding. This paper proposes a module system for user-defined existing language. The operators are defined using a declarative syntax
operators named a dsl class. A dsl class is similar to a normal class in Java similar to that of functions or procedures. The syntax specifies the
but it contains operators instead of methods. We use operators for precedence and associativity of the operators, as well as their arity and
implementing custom name binding. For example, we use a nullary operator for argument types. The implementation uses a parser generator to
emulating a variable name. An instance of a dsl class, called a dsl object, automatically generate a parser for the extended grammar. To enable name
User-Defined reifies an environment that expresses name binding. Programmers can control a binding for user-defined operators, the paper proposes a novel mechanism
Operators Including scope of instance operators by specifying where the dsl object is active. We 512 that uses a combination of dynamic scoping and type inference. The 310
Name Binding for New extend the host type system so that it can express the activation of a dsl mechanism allows the compiler to infer the types and binding scopes of
Language Constructs object. In our system, a bound name is propagated through a type parameter to variables based on their usage within the operator, thus avoiding the need
a dsl object. This enables us to implement user-defined language constructs for explicit type annotations or variable declarations. This makes the
involving static name binding. A contribution of this paper is that we reveal programming model more expressive and less error-prone, while still
we can integrate a system for managing names and their scopes with a module preserving type safety and compile-time correctness. The paper also
and type system of an object-oriented language like Java. This allows us to discusses the benefits and limitations of the proposed approach, and
implement a proposed system by adopting eager disambiguation based on expected presents several examples of how the new operators can be used to simplify
types so that the compilation time will be acceptable. Eager disambiguation, and clarify program logic. The examples include arithmetic and logical
which prunes out semantically invalid abstract parsing trees (ASTs) while a operations, string manipulation, and collection processing. The paper
parser is running, is needed because the parser may generate a huge number of concludes with a discussion of future work, including the extension of the
potentially valid ASTs for the same source code. We have implemented ProteaJ2, mechanism to support user-defined control structures and the integration
which is a programming language based on Java and it supports our proposal. We of the approach into other programming languages. The proposed approach
describe a parsing method that adopts eager disambiguation for fast parsing has the potential to significantly enhance the productivity and clarity of
and discuss its time complexity. To show the practicality of our proposal, we software development, particularly for domain-specific applications that
have conducted two micro benchmarks to see the performance of our compiler. We require customized operators and abstractions.
also show several use cases of dsl classes for demonstrating dsl classes can
express various language constructs. Our ultimate goal is to let programmers
add any kind of new language construct to a host language. To do this,
programmers should be able to define new syntax, name binding, and type system
within the host language. This paper shows programmers can define the former
two: their own syntax and name binding.
A definition of quasi-flat left module is proposed and it is shown that any
left module which is either quasi-projective or flat is quasi-flat. A
characterization of local commutative rings for which each ideal is quasi-flat
(resp. quasi-projective) is given. It is also proven that each commutative
ring R whose finitely generated ideals are quasi-flat is of $\ Commutative rings whose finitely generated ideals are quasi-flat have
lambda$-dimension $\le$ 3, and this dimension $\le$ 2 if R is local. This received significant attention in the context of commutative algebra and
extends a former result about the class of arithmetical rings. Moreover, if R algebraic geometry. In particular, they play an important role in the
has a unique minimal prime ideal then its finitely generated ideals are study of algebraic varieties and their singularities. This paper studies a
quasi-projective if they are quasi-flat. In [1] Abuhlail, Jarrar and Kabbaj class of commutative rings such that all their finitely generated ideals
studied the class of commutative fqp-rings (finitely generated ideals are are quasi-flat. We explore the basic properties of such rings, and provide
quasi-projective). They proved that this class of rings strictly contains the several equivalent characterizations of them. In particular, we show that
one of arithmetical rings and is strictly contained in the one of Gaussian a commutative ring R is such that all its finitely generated ideals are
rings. It is also shown that the property for a commutative ring to be fqp is quasi-flat if and only if R satisfies certain coherence conditions. We
preserved by localization. It is known that a commutative ring R is also investigate the relationship between these rings and various other
arithmetical (resp. Gaussian) if and only if R M is arithmetical (resp. classes of commutative rings, such as universally catenary rings and
Gaussian) for each maximal ideal M of R. But an example given in [6] shows integral domains that admit a dualizing complex. We provide examples to
that a commutative ring which is a locally fqp-ring is not necessarily a illustrate that the class of commutative rings whose finitely generated
Commutative rings fqp-ring. So, in this cited paper the class of fqf-rings is introduced. Each ideals are quasi-flat is strictly larger than the class of universally
whose finitely local commutative fqf-ring is a fqp-ring, and a commutative ring is fqf if and 511 catenary rings, and that not all such rings admit a dualizing complex. 314
generated ideals are only if it is locally fqf. These fqf-rings are defined in [6] without a Finally, we study the local cohomology of modules over commutative rings
quasi-flat definition of quasi-flat modules. Here we propose a definition of these whose finitely generated ideals are quasi-flat. We prove that if R is such
modules and another definition of fqf-ring which is equivalent to the one a ring and M is a finitely generated module over R, then the local
given in [6]. We also introduce the module property of self-flatness. Each cohomology of M with respect to an ideal I in R is finite-dimensional for
quasi-flat module is self-flat but we do not know if the converse holds. On any finitely generated ideal I in R. We also investigate the relationship
the other hand, each flat module is quasi-flat and any finitely generated between the finiteness of local cohomology and the Bass property for
module is quasi-flat if and only if it is flat modulo its annihilator. In modules over commutative rings whose finitely generated ideals are
Section 2 we give a complete characterization of local commutative rings for quasi-flat. Throughout the paper, we use a variety of techniques both from
which each ideal is self-flat. These rings R are fqp and their nilradical N is algebraic geometry and commutative algebra, including homological algebra,
the subset of zerodivisors of R. In the case where R is not a chain ring for sheaf theory, and the theory of determinantal rings. Our main results
which N = N 2 and R N is not coherent every ideal is flat modulo its provide a deeper understanding of the structure and properties of
annihilator. Then in Section 3 we deduce that any ideal of a chain ring commutative rings whose finitely generated ideals are quasi-flat, and
(valuation ring) R is quasi-projective if and only if it is almost maximal and highlight their connections to other important classes of commutative
each zerodivisor is nilpotent. This complete the results obtained by Hermann rings.
in [11] on valuation domains. In Section 4 we show that each commutative
fqf-ring is of $\lambda$-dimension $\le$ 3. This extends the result about
arithmetical rings obtained in [4]. Moreover it is shown that this $\
lambda$-dimension is $\le$ 2 in the local case. But an example of a local
Gaussian ring R of $\lambda$-dimension $\ge$ 3 is given.
We analyze some parabolic PDEs with different drift terms which are gradient
flows in the Wasserstein space and consider the corresponding discrete-in-time
JKO scheme. We prove with optimal transport techniques how to control the L p
and L $\infty$ norms of the iterated solutions in terms of the previous norms,
essentially recovering well-known results obtained on the continuous-in-time This research paper delves on the estimation of solutions of Fokker-Planck
equations. Then we pass to higher order results, and in particulat to some equations - a powerful mathematical tool that models diffusion phenomena.
specific BV and Sobolev estimates, where the JKO scheme together with the Our focus is on linear and non-linear Fokker-Planck equations, where we
so-called "five gradients inequality" allows to recover some inequalities that show the robustness of the JKO scheme. In particular, we extend estimates
can be deduced from the Bakry-Emery theory for diffusion operators, but also for JKO schemes in both the linear and non-linear cases and prove that
to obtain some novel ones, in particular for the Keller-Segel chemiotaxis they converge for the respective PDEs. Our results offer an innovative
model. 1 Short introduction The goal of this paper is to present some approach to tackle diffusion phenomena, and the linear/non-linear cases of
estimates on evolution PDEs in the space of probability densities which share Fokker-Planck equations are vital in various research applications.
two important features: they include a linear diffusion term, and they are Furthermore, we explore the application of these estimates to non-linear
gradient flows in the Wasserstein space W2. These PDEs will be of the form $\ Keller-Segel models, which model chemotactic phenomena in biology. We
partial$t$\rho$ -- $\Delta$$\rho$ -- $\nabla$ $\times$ ($\rho$$\nabla$u[$\ study the dynamics of the concentration of cells in a both finite and
rho$]) = 0, complemented with no-flux boundary conditions and an intial infinite domain, while incorporating a particular chemotactic sensitivity
JKO estimates in condition on $\rho$0. We will in particular concentrate on the Fokker-Plack function, where previously known estimates have failed. We demonstrate
linear and case, where u[$\rho$] = V and V is a fixed function (with possible regularity that the application of JKO schemes provides new and sharp Lp and Sobolev
non-linear assumptions) independent of $\rho$, on the case where u[$\rho$] = W * $\rho$ bounds which help to obtain better estimates in the chemotactic system.
Fokker-Planck is obtained by convolution and models interaction between particles, and on 511 Our results are instrumental in unveiling critical dynamics of the 327
equations, and the parabolic-elliptic Keller-Segel case where u[$\rho$] is related to $\rho$ chemotactic reaction-diffusion equations. Our study explores various
Keller-Segel: L p via an elliptic equation. This last case models the evolution of a biological numerical experiments that provide evidence of the efficiency of our
and Sobolev bounds population $\rho$ subject to diffusion but attracted by the concentration of a scheme. We demonstrate that the estimates provided by the JKO scheme in
chemo-attractant, a nutrient which is produced by the population itself, so explicit numerical simulations match very closely the exact solutions of
that its distribution is ruled by a PDE where the density $\rho$ appears as a the PDEs. Our experiments support the conclusion that JKO estimates are
source term. Under the assumption that the production rate of this nutrient is reliable and offer valuable insights for the analysis of non-linear PDEs.
much faster than the motion of the cells, we can assume that its distribution To conclude, our paper contributes to the understanding of diffusion
is ruled by a statical PDE with no explicit time-dependence, and gives rise to phenomena and offers an innovative approach to estimate solutions of both
a system which is a gradient flow in the variable $\rho$ (the linear and non-linear Fokker-Planck equations. Our method is also proven
parabolic-parabolic case, where the time scale for the cells and for the to be valuable in the application of Keller-Segel models, and we provide
nutrient are comparable, is also a gradient flow, in the product space W2 x L new results of Lp and Sobolev bounds that are imperative to better
2 , but we will not consider this case). Since we mainly concentrate on the understand chemotactic phenomena. Our numerical experiments demonstrate
case of bounded domains, in the Keller-Segel case the term u[$\rho$] cannot be that our approach is reliable and effective in practice. This research may
expressed as a convoluton and requires ad-hoc computations. In all the paper, have significant practical applications in both biological and physical
the estimates will be studied on a time-discretized version of these PDEs, sciences.
consisting in the so-called JKO (Jordan-Kinderleherer-Otto) scheme, based on
iterated optimization problems involving the Wasserstein distance W2. We will
first present 0-order estimates, on the L p and L $\infty$ norms of the
solution. This is just a translation into the JKO language of well-known
properties of these equations. The main goal of this part is hence to
The relation between Science (what we can explain) and Art (what we can't) has
long been acknowledged and while every science contains an artistic part,
every art form also needs a bit of science. Among all scientific disciplines,
programming holds a special place for two reasons. First, the artistic part is
not only undeniable but also essential. Second, and much like in a purely This paper seeks to explore the intersections between the seemingly
artistic discipline, the act of programming is driven partly by the notion of disparate fields of Lisp programming, jazz music, and the martial art of
aesthetics: the pleasure we have in creating beautiful things. Even though the Aikido, ultimately arguing that they each express a fundamental and
importance of aesthetics in the act of programming is now unquestioned, more interconnected aspect of the human experience. This argument is based on a
could still be written on the subject. The field called "psychology of thorough and multifaceted analysis of each field, drawing on a range of
programming" focuses on the cognitive aspects of the activity, with the goal theoretical and practical perspectives. In the case of Lisp programming,
of improving the productivity of programmers. While many scientists have this paper contends that the language's unique focus on recursion and
emphasized their concern for aesthetics and the impact it has on their abstraction reflects a deeply ingrained human tendency to seek out
activity, few computer scientists have actually written about their thought patterns and create mental models of the world around us. Drawing on both
process while programming. What makes us like or dislike such and such historical and contemporary examples, the paper demonstrates how Lisp has
language or paradigm? Why do we shape our programs the way we do? By answering been used to solve complex computational problems and push the boundaries
these questions from the angle of aesthetics, we may be able to shed some new of artificial intelligence research. Similarly, the paper argues that jazz
light on the art of programming. Starting from the assumption that aesthetics music represents a powerful means of embodying and exploring the complex
is an inherently transversal dimension, it should be possible for every interplay between structure and improvisation. By examining the techniques
programmer to find the same aesthetic driving force in every creative activity and philosophies of jazz musicians such as John Coltrane and Miles Davis,
Lisp, Jazz, Aikido they undertake, not just programming, and in doing so, get deeper insight on the paper shows how the genre's emphasis on creative collaboration and
-- Three Expressions why and how they do things the way they do. On the other hand, because our 508 spontaneous innovation can help us to better understand the dynamics of 334
of a Single Essence aesthetic sensitivities are so personal, all we can really do is relate our social interaction and teamwork. Finally, the paper turns to the martial
own experiences and share it with others, in the hope that it will inspire art of Aikido, which it argues provides a profound physical and
them to do the same. My personal life has been revolving around three major philosophical framework for exploring the fundamental nature of conflict
creative activities, of equal importance: programming in Lisp, playing Jazz and harmony. By drawing on insights from both traditional Japanese
music, and practicing Aikido. But why so many of them, why so different ones, knowledge and contemporary psychology research, the paper demonstrates how
and why these specifically? By introspecting my personal aesthetic Aikido can illuminate important aspects of human relationships and allow
sensitivities, I eventually realized that my tastes in the scientific, us to develop more effective strategies for resolving conflicts. Taken
artistic, and physical domains are all motivated by the same driving forces, together, these three distinct fields of inquiry represent different
hence unifying Lisp, Jazz, and Aikido as three expressions of a single expressions of a single underlying essence or principle, which can be
essence, not so different after all. Lisp, Jazz, and Aikido are governed by a understood through an integration of theory and practice across multiple
limited set of rules which remain simple and unobtrusive. Conforming to them domains. The paper concludes by exploring some of the broader implications
is a pleasure. Because Lisp, Jazz, and Aikido are inherently introspective of this argument for fields such as education, psychology, and philosophy,
disciplines, they also invite you to transgress the rules in order to find suggesting that a deeper appreciation of the interconnections between
your own. Breaking the rules is fun. Finally, if Lisp, Jazz, and Aikido unify seemingly disparate disciplines is essential for addressing some of the
so many paradigms, styles, or techniques, it is not by mere accumulation but most pressing challenges facing humanity today.
because they live at the meta-level and let you reinvent them. Working at the
meta-level is an enlightening experience. Understand your aesthetic
sensitivities and you may gain considerable insight on your own psychology of
programming. Mine is perhaps common to most lispers. Perhaps also common to
other programming communities, but that, is for the reader to decide...
In two articles ([Brisson-Ofman1, 2]), we have analyzed the so-called
'mathematical passage' of Plato's Theaetetus, the first dialogue of a trilogy
including the Sophist and the Statesman. In the present article, we study an
important point in more detail, the 'definition' of 'powers' ('$\delta\upsilon
\nu\acute\alpha\mu\epsilon\iota\varsigma$'). While in [Brisson-Ofman2], it was This academic paper explores the concept of powers and division in the
shown that the different steps to get the definition are mathematically and "mathematical part" of Plato's Theaetetus. The work analyzes the dialogue
philosophically incorrect, it is explained why the definition itself is between Socrates and Theaetetus as they delve into the intricacies of
problematic. However, it is the first example, at least in the trilogy, of a mathematical knowledge, particularly the relationship between powers and
definition by division. This point is generally ignored by modern commentators roots and the concept of division. The paper situates the dialogue within
though, as we will try to show, it gives rise, in a mathematical context, to the larger context of Plato's philosophy, including his views on the
at least three fundamental questions: the meaning(s) of 'logos', the nature of knowledge and the role of mathematics in understanding the
connection between 'elements and compound' and, of course the question of the world. The first section of the paper provides an overview of the
'power(s)'. One of the main consequences of our works on Theaetetus' mathematical concepts discussed in the dialogue, including the
'mathematical passage', including the present one, is to challenge the identification of perfect squares and the calculation of powers. The
so-called 'main standard interpretation'. In particular, following authors analyze the initial definition of a power, which is presented in
[Ofman2014], we question the claim that Plato praises and glorifies both the terms of repeated multiplication, and examine the relationship between
mathematician Theodorus and the young Theaetetus. According to our analysis, powers and roots. They also discuss the concept of division in relation to
such a claim, considered as self-evident, entails many errors. Conversely, our powers, exploring the role of ratios and proportionality in mathematical
analysis of Theaetetus' mathematical mistakes highlights the main cause of calculations. The second section of the paper situates the dialogue within
some generally overlooked failures in the dialogue: the forgetting of the a broader philosophical framework. The authors draw on Plato's views of
Powers and division 'logos', first in the 'mathematical part', then in the following discussion, knowledge and epistemology, particularly his belief in the existence of
in the 'mathematical and finally the failure of the four successive tries of its definition at the 507 objective, eternal forms or ideas. They argue that the mathematical 380
part' of Plato's end of the dialogue. Namely, as we will show, the passage is closely connected concepts explored in the dialogue can be seen as a reflection of these
Theaetetus with the problems studied at the end of the dialogue, but also to the two higher forms, and that the act of understanding mathematics involves a
other parts of the trilogy through the method of 'definition by division'. process of recollection and discovery. The final section of the paper
Finally, if our conclusions are different from the usual ones, it is probably considers the implications of the discussion for contemporary philosophy
because the passage is analyzed, maybe for the first time, simultaneously from and mathematics. The authors argue that the problems and concepts explored
the philosophical, historical and mathematical points of view. It had been in the dialogue remain relevant today, particularly in the fields of
considered usually either as an excursus by historians of philosophy (for algebra, geometry, and number theory. They suggest that the dialogue can
instance [Burnyeat1978]), or as an isolated text separated from the rest of be seen as a precursor to modern mathematical thinking, with its emphasis
the dialogue by historians of mathematics (for instance [Knorr1975]), or on abstraction, generalization, and proof. Overall, this paper offers a
lastly as a pretext to discuss some astute developments in modern mathematics detailed examination of the role of powers and division in the
by mathematicians (for instance [Kahane1985]).[Brisson-Ofman1]: Luc "mathematical part" of Plato's Theaetetus. Through close analysis of the
Brisson-Salomon Ofman, `Theodorus' lesson in Plato's Theaetetus(147d3-d6) dialogue between Socrates and Theaetetus, the paper explores the intricate
Revisited-A New Perspective', to appear[Brisson-Ofman2]: Luc Brisson-Salomon relationships between different mathematical concepts and situates them
Ofman, `The Philosophical Interpretation of Plato'sTheaetetus and the Final within the larger context of Plato's philosophy. The authors suggest that
Part of the Mathematical Lesson (147d7-148b)', to appear[Burnyeat 1978]: Myles the dialogue remains a rich source of insight and inspiration for
Burnyeat, `The Philosophical Sense of Theaetetus' Mathematics',Isis, 69, 1978, contemporary philosophers and mathematicians alike, and that its enduring
489-514[Kahane1985]: Jean-Pierre Kahane, `la th{\'e}orie de Th{\'e}odore des relevance speaks to the continued importance of mathematical thinking in
corps quadratiques r{\'e}els',L'enseignement math{\'e}matique, 31, 1985, p. understanding the world around us.
85-92[Knorr1975]: Wilbur Knorr, The evolution of the Euclidean elements,
Reidel, 1975[Ofman2014]: Salomon Ofman, `Comprendre les math{\'e}matiques pour
comprendre Platon-Th{\'e}{\'e}t{\`e}te (147d-148b)', Lato Sensu, I, 2014, p.
In this paper, we consider two cases of rolling of one smooth connected
complete Riemannian manifold $(M,g)$ onto another one $(\hM,\hg)$ of equal
dimension $n\geq 2$. The rolling problem $(NS)$ corresponds to the situation
where there is no relative spin (or twist) of one manifold with respect to the The study of rolling manifolds, a type of manifold that is defined by its
other one. As for the rolling problem $(R)$, there is no relative spin and ability to roll without slipping, has received significant attention in
also no relative slip. Since the manifolds are not assumed to be embedded into recent years within the field of control theory. In this paper, we present
an Euclidean space, we provide an intrinsic description of the two constraints an intrinsic formulation of rolling manifolds that considers their
"without spinning" and "without slipping" in terms of the Levi-Civita geometric and topological properties and explores the underlying
connections $\nabla^{g}$ and $\nabla^{\hg}$. For that purpose, we recast the mathematical structures that govern their behavior. Specifically, we use
two rolling problems within the framework of geometric control and associate the theory of principal bundles to define a natural frame bundle for
to each of them a distribution and a control system. We then investigate the rolling manifolds, which allows us to express their dynamics in a
relationships between the two control systems and we address for both of them coordinate-free way. This approach not only simplifies the analysis of
the issue of complete controllability. For the rolling $(NS)$, the reachable rolling manifolds but also reveals essential features of their geometry,
set (from any point) can be described exactly in terms of the holonomy groups such as the existence of a connection on the frame bundle that
of $(M,g)$ and $(\hM,\hg)$ respectively, and thus we achieve a complete characterizes the rolling motion. We also investigate the controllability
understanding of the controllability properties of the corresponding control of rolling manifolds, which refers to the ability to reach any desired
system. As for the rolling $(R)$, the problem turns out to be more delicate. state in a finite amount of time by applying appropriate controls. Our
We first provide basic global properties for the reachable set and investigate results indicate that rolling manifolds are controllable for a large class
Rolling Manifolds: the associated Lie bracket structure. In particular, we point out the role of distributions on their tangent bundles, including those that correspond
Intrinsic played by a curvature tensor defined on the state space, that we call the \ 506 to regular and singular points. Moreover, we show that the notion of 351
Formulation and emph{rolling curvature}. In the case where one of the manifolds is a space accessibility, i.e., the ability to reach any point in the configuration
Controllability form (let say $(\hM,\hg)$), we show that it is enough to roll along loops of $ space, can be translated to a geometric condition on the curvature of the
(M,g)$ and the resulting orbits carry a structure of principal bundle which connection associated with the rolling motion. To illustrate the
preserves the rolling $(R)$ distribution. In the zero curvature case, we applicability of our results, we provide several examples of rolling
deduce that the rolling $(R)$ is completely controllable if and only if the manifolds, including spheres, cylinders, and tori, and discuss their
holonomy group of $(M,g)$ is equal to SO(n). In the nonzero curvature case, we controllability properties in detail. In particular, we show how the
prove that the structure group of the principal bundle can be realized as the curvature of the connection affects the motion of rolling manifolds and
holonomy group of a connection on $TM\oplus \R$, that we call the rolling how this can be exploited to design optimal control strategies. Finally,
connection. We also show, in the case of positive (constant) curvature, that we discuss some open problems and future directions in the study of
if the rolling connection is reducible, then $(M,g)$ admits, as Riemannian rolling manifolds. For instance, we highlight the importance of
covering, the unit sphere with the metric induced from the Euclidean metric of understanding the interplay between the geometric and dynamic properties
$\R^{n+1}$. When the two manifolds are three-dimensional, we provide a of rolling manifolds and the role they play in the design of intelligent
complete local characterization of the reachable sets when the two manifolds robotic systems. Overall, this paper provides a comprehensive and rigorous
are three-dimensional and, in particular, we identify necessary and sufficient treatment of rolling manifolds, which sheds light on their intrinsic
conditions for the existence of a non open orbit. Besides the trivial case formulation and controllability properties and paves the way for further
where the manifolds $(M,g)$ and $(\hM,\hg)$ are (locally) isometric, we show research in this exciting area of control theory.
that (local) non controllability occurs if and only if $(M,g)$ and $(\hM,\hg)$
are either warped products or contact manifolds with additional restrictions
that we precisely describe. Finally, we extend the two types of rolling to the
case where the manifolds have different dimensions.
Context. Refining or altering existing behavior is the daily work of every
developer, but that cannot be always anticipated, and software sometimes
cannot be stopped. In such cases, unanticipated adaptation of running systems
is of interest for many scenarios, ranging from functional upgrades to
on-the-fly debugging or monitoring of critical applications. Inquiry. A way of In this paper, we provide a retrospective look at our use of the
altering software at run time is using behavioral reflection, which is sub-method of partial behavioral reflection in conjunction with
particularly well-suited for unanticipated adaptation of real-world systems. Reflectivity over the past decade. Our analysis focuses on a variety of
Partial behavioral reflection is not a new idea, and for years many efforts aspects, including the practical benefits of the approach, its
have been made to propose a practical way of expressing it. All these efforts effectiveness in improving system design, and its potential for future
resulted in practical solutions, but which introduced a semantic gap between research. Overall, our experience with this combination of techniques has
the code that requires adaptation and the expression of the partial behavior. been highly positive. Through the use of partial behavioral reflection, we
For example, in Aspect-Oriented Programming, a pointcut description is have been able to gain valuable insights into the behavior of complex
expressed in another language, which introduces a new distance between the systems, allowing us to identify and address design flaws and other issues
behavior expression (the Advice) and the source code in itself. Approach. Ten that might otherwise have gone unnoticed. Meanwhile, Reflectivity has
years ago, the idea of closing the gap between the code and the expression of provided a flexible and powerful foundation for implementing these
the partial behavior led to the implementation of the Reflectivity framework. techniques, allowing us to customize our approach to suit the specific
Using Reflectivity, developers annotate Abstract Syntax Tree (AST) nodes with needs of each project. One particularly notable area of success has been
meta-behavior which is taken into account by the compiler to produce in the realm of software design. Through our use of partial behavioral
behavioral variations. In this paper, we present Reflectivity, its API, its reflection, we have been able to analyze user interactions with complex
Sub-method, partial implementation and its usage in Pharo. We reflect on ten years of use of software systems, identifying areas of inefficiency, confusion, or error.
behavioral Reflectivity, and show how it has been used as a basic building block of many By using the information gleaned from these analyses, we have been able to
reflection with innovative ideas. Knowledge. Reflectivity brings a practical way of working at improve the design of these systems, resulting in a more intuitive and
Reflectivity: the AST level, which is a high-level representation of the source code 505 effective user experience. However, we also identify areas for improvement 399
Looking back on 10 manipulated by software developers. It enables a powerful way of dynamically in our approach. One significant challenge we have encountered is the
years of use add and modify behavior. Reflectivity is also a flexible mean to bridge the difficulty of handling large volumes of data resulting from partial
gap between the expression of the meta-behavior and the source code. This behavioral reflection. Given the sheer amount of information generated by
ability to apply unanticipated adaptation and to provide behavioral reflection this technique, it can be overwhelming to sort through and make sense of
led to many experiments and projects during this last decade by external all the data. Similarly, we note that the use of partial behavioral
users. Existing work use Reflectivity to implement reflective libraries or reflection can sometimes result in a lack of context, making it difficult
languages extensions, featherweight code instrumentation, dynamic software to fully understand the significance of certain observed behaviors.
update, debugging tools and visualization and software analysis tools. Despite these limitations, we remain optimistic about the potential for
Grounding. Reflectivity is actively used in research projects. During the past partial behavioral reflection and Reflectivity to continue to drive
ten years, it served as a support, either for implementation or as a significant innovation in the field of system design and analysis. Moving
fundamental base, for many research work including PhD theses, conference, forward, we believe that there is room for further optimization of these
journal and workshop papers. Reflectivity is now an important library of the techniques, as well as for continued exploration of their potential in new
Pharo language, and is integrated at the heart of the platform. Importance. and different contexts. Overall, our experience with partial behavioral
Reflectivity exposes powerful abstractions to deal with partial behavioral reflection and Reflectivity has been highly positive, allowing us to gain
adaptation, while providing a mature framework for unanticipated, important insights into system behavior, improve design, and drive
non-intrusive and partial behavioral reflection based on AST annotation. innovation in the field. While challenges remain, we believe that these
Furthermore, even if Reflectivity found its home inside Pharo, it is not a techniques hold great promise for future research, and we look forward to
pure Smalltalk-oriented solution. As validation over the practical use of continuing to explore their potential in the years to come.
Reflectivity in dynamic object-oriented languages, the API has been ported to
Python. Finally, the AST annotation feature of Reflectivity opens new
experimentation opportunities about the control that developers could gain on
the behavior of their own software.
Plasma electrolytic oxidation (PEO) processing of EV 31 magnesium alloy has
been carried out in fluoride containing electrolyte under bipolar pulse
current regime. Unusual PEO cathodic micro-discharges have been observed and
investigated. It is shown that the cathodic micro-discharges exhibit a
collective intermittent behavior which is discussed in terms of charge
accumulations at the layer/electrolyte and layer/metal interfaces. Optical
emission spectroscopy is used to determine the electron density (typ. 10 15 The plasma electrolytic oxidation (PEO) process has been widely researched
cm-3) and the electron temperature (typ. 7500 K) while the role of F-anions on for its ability to enhance the corrosion resistance and surface hardness
the appearance of cathodic micro-discharges is pointed out. Plasma of materials. In this study, we investigate the evidence of cathodic
Electrolytic Oxidation (PEO) is a promising plasma-assisted surface treatment micro-discharges during the PEO process. To examine the occurrence of
of light metallic alloys (e.g. Al, Mg, Ti). Although the PEO process makes it micro-discharges, a series of experiments were conducted using aluminum
possible to grow oxide coatings with interesting corrosion and wear resistant alloy substrates immersed in an electrolyte solution. The samples were
properties, the physical mechanisms of coating growth are not yet completely subjected to a range of voltage pulses with varying frequencies and
understood. Typically, the process consists in applying a high voltage durations. The resulting behavior of the discharges was monitored using
difference between a metallic piece and a counter-electrode which are both high-speed imaging and optical emission spectroscopy. Our findings
immersed in an electrolyte bath. Compare to anodizing, the main differences indicate that cathodic micro-discharges were detected during the PEO
concern the electrolyte composition and the current and voltage ranges which process. These discharges occurred on both the surface and within the
are at least one order of magnitude higher in PEO 1. These significant electrolyte solution. The discharges were characterized by high-intensity
differences in current and voltage imply the dielectric breakdown and flashes lasting between 1 and 10 microseconds, and were accompanied by
The Evidence of consequently the appearance of micro-discharges on the surface of the sample significant changes in optical emissions. The observed behavior of the
Cathodic under processing. Those micro-discharges are recognized as being the main discharges strongly suggests that they play a significant role in the PEO
Micro-discharges contributors to the formation of a dielectric porous crystalline oxide 504 process. It is proposed that these micro-discharges contribute to the 314
during Plasma coating. 2 Nevertheless, the breakdown mechanism that governs the appearance structuring and hardening of the oxide layer formed on the surface by
Electrolytic of those micro-discharges is still under investigation. Hussein et al. 3 enhancing the surface energy and reactivity. Furthermore, the discharges
Oxidation Process proposed a mechanism with three different plasma formation processes based on are thought to facilitate the incorporation of foreign particles into the
differences in plasma chemical composition. The results of Jovovi{\'c} et al. oxide layer, further improving its properties. To further investigate the
4,5 concerning physical properties of the plasma seem to corroborate this nature of these micro-discharges, we conducted numerical simulations using
mechanism, and also point out the importance of the substrate material in the a hybrid model combining fluid dynamics, electrodynamics, and surface
plasma composition. 6 Compared with DC conducted PEO process, using a bipolar chemistry. The model was able to reproduce the observed behavior of the
pulsed DC or AC current supply gives supplementary control latitude through discharges and provided additional insights into their underlying
the current waveform parameters. The effect of these parameter on the mechanisms. Overall, our study provides compelling evidence for the
micro-discharges behavior has been investigated in several previous works. presence and significance of cathodic micro-discharges during the PEO
2,3,7,8 One of the main results of these studies is the absence of process. This knowledge can be applied to improve the efficiency and
micro-discharge during the cathodic current half-period. 9-11 Even if the effectiveness of the process for a variety of engineering applications. In
cathodic half-period has an obvious effect on the efficiency of PEO as well as future research, it would be interesting to investigate the impact of
on the coating growth and composition, the micro-plasmas appear only in anodic different parameters, such as voltage and electrolyte composition, on the
half-period. Sah et al. 8 have observed the cathodic breakdown of an oxide behavior of these discharges.
layer but at very high current density (10 kA.dm-${}^2$), and after several
steps of sample preparation. Several models of micro-discharges appearance in
AC current have already been proposed. 1,2,8,12,13 Though cathodic
micro-discharges have never been observed within usual process conditions, the
present study aims at defining suitable conditions to promote cathodic
micro-discharges and at studying the main characteristics of these
Developers need to make a constant effort to improve the quality of their code
if they want to stay productive. Tools that highlight code locations that
could benefit from refactoring are thus highly desirable. The most common name
for such locations is "bad code smell". A number of tools offer such quality
feedback and there is a substantial body of related research. However, all This research paper explores the extent to which JHotDraw, a popular
these tools, including those based on Machine Learning, still produce false drawing application framework, conforms to the principles of good coding
positives. Every single false positive shown to the developer places a style. We investigate whether instances of reported bad code smells -
cognitive burden on her and should thus be avoided. The literature discusses indicators of poor quality code - are in fact false positives or genuine
the choice of metric thresholds, the general subjectivity of such a judgment issues. Our study constitutes a deep dive into the nature of these false
and the relation to conscious design choices, "design ideas". To examine false positives. First, we provide a comprehensive overview of the relevant
positives and the relation between bad smells and design ideas, we designed literature, including previous studies and best practices in coding. We
and conducted an exploratory case study. While previous research presented a then present the methodology used in our investigation, highlighting the
broad overview, we have chosen a narrow setting to reach for even deeper criteria for identifying bad code smells and selecting relevant metrics
insights: The framework JHotDraw had been designed so thoughtfully that most for analysis. Our results suggest that while JHotDraw generally adheres to
smell warnings are expected to be false positives. Nevertheless, the "Law of good coding practices, there are instances of false positives related to
Good Style", better known as the "Law of Demeter", is a rather restrictive certain types of bad code smells. In particular, we found that code smells
design rule so that we still expected to find some potential bad smells, i.e. related to duplication, complexity, and unused code were more likely to be
Did JHotDraw Respect violations of this "Law". This combination led to 1215 potential smells of false positives than others. This finding has important implications for
the Law of Good which at most 42 are true positives. We found generic as well as specific developers who use automated code analysis tools, as false positives can
Style?: A deep dive design ideas that were traded for the smell. Our confidence in that decision waste time and resources and detract from genuine issues. Furthermore, we
into the nature of ranged from high enough to very high. We were surprised to realize that the 496 offer insights into the reasons behind these false positives. We find that 352
false positives of smell definition itself required the formulation of constructive design ideas. the use of design patterns and certain coding conventions in JHotDraw can
bad code smells Finally we found some smells to be the result of the limitation of the sometimes create false positives for certain bad code smells. These
language and one could introduce auxiliary constructive design ideas to findings are relevant not only for JHotDraw developers but also for other
compensate for them. The decision whether a potential smell occurrence is developers who may encounter similar challenges. Finally, we discuss the
actually a true positive was made very meticulously. For that purpose we took implications of our findings for the broader software development
three qualities that the smell could affect negatively into account and we community. We argue that automated code analysis tools need to consider
discussed the result of the recommended refactorings. If we were convinced context-specific factors when detecting bad code smells and that
that we had found a false positive, we described the relationships with design developers need to exercise caution when interpreting the results of such
ideas. The realization that not only general design ideas but also specific tools. Our study highlights the need for continued research into the
design ideas have an influence on whether a potential smell is a true positive nature of false positives in automated code analysis and for the
turns the problem of false positives from a scientific problem ("What is the development of improved tools and techniques for identifying genuine
true definition of the smell?") to a engineering problem ("How can we issues. In conclusion, our study offers a deep dive into the nature of
incorporate design ideas into smell definitions?"). We recommend to add false positives in bad code smells, providing insights into the specific
adaptation points to the smell definitions. Higher layers may then adapt the context of JHotDraw as well as broader implications for software
smell for specific contexts. After adaptation the tool may continuously development. Our findings may be of interest to researchers, developers,
provide distinct and precise quality feedback, reducing the cognitive load for and quality assurance professionals alike.
the developer and preventing habituation. Furthermore, the schema for the
discussion of potential smells may be used to elaborate more sets of true and
false smell occurrences. Finally, it follows that smell detection based on
machine learning should also take signs of design ideas into account.
In the study of Hamiltonian systems on cotangent bundles, it is natural to
perturb Hamiltoni-ans by adding potentials (functions depending only on the
base point). This led to the definition of Ma{\~n}{\'e} genericity: a property
is generic if, given a Hamiltonian H, the set of potentials u such that H + u This paper investigates the normal form near orbit segments of convex
satisfies the property is generic. This notion is mostly used in the context Hamiltonian systems. We introduce the necessary mathematical framework and
of Hamiltonians which are convex in p, in the sense that $\partial$ 2 pp H is tools for the study of convex Hamiltonian systems, emphasizing the
positive definite at each points. We will also restrict our study to this importance of symplectic geometry and canonical transformations. We then
situation. There is a close relation between perturbations of Hamiltonians by consider the existence of normal forms near orbit segments in these
a small additive potential and perturbations by a positive factor close to systems. We provide a rigorous analysis of the normal form in convex
one. Indeed, the Hamiltonians H + u and H/(1 -- u) have the same level one Hamiltonian systems and derive conditions for its existence. Specifically,
energy surface, hence their dynamics on this energy surface are we show that these conditions are related to the convexity of the
reparametrisation of each other, this is the Maupertuis principle. This remark Hamiltonian and the existence of certain types of periodic orbits. Our
is particularly relevant when H is homogeneous in the fibers (which results provide insight into the dynamical behavior and geometry of these
corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds systems, as well as their applications to physics, engineering, and
to Riemannian metrics). In these cases, perturbations by potentials of the materials science. Moreover, we apply our findings to the study of some
Hamiltonian correspond, up to parametrisation, to conformal perturbations of concrete examples of convex Hamiltonian systems. We investigate the normal
the metric. One of the widely studied aspects is to understand to what extent form near orbit segments in two-dimensional systems with polynomial
the return map associated to a periodic orbit can be perturbed by adding a Hamiltonians and show how the existence of normal forms is related to the
Normal form near small potential. This kind of question depend strongly on the context in which topology of the energy surface. We also study a three-dimensional system
orbit segments of they are posed. Some of the most studied contexts are, in increasing order of 493 with a singular potential and show how the existence of certain types of 352
convex Hamiltonian difficulty, perturbations of general vector fields, perturbations of periodic orbits affects the normal form. Finally, we discuss the
systems Hamiltonian systems inside the class of Hamiltonian systems, perturbations of implications of our results for the understanding of convex Hamiltonian
Riemannian metrics inside the class of Riemannian metrics, Ma{\~n}{\'e} systems and their applications. We show that the normal form near orbit
perturbations of convex Hamiltonians. It is for example well-known that each segments provides valuable information about the global dynamics of the
vector field can be perturbed to a vector field with only hyperbolic periodic system, such as the stability and instability of periodic orbits, the
orbits, this is part of the Kupka-Smale theorem, see [5, 13]. There is no such existence of invariant tori, and the topology of the energy surface. Our
result in the context of Hamiltonian vector fields, but it remains true that work contributes to the ongoing efforts to understand the complex behavior
each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate of physical and engineering systems, and provides a useful framework for
periodic orbits (including the iterated ones), see [11, 12]. The same result future research in this area. In conclusion, this paper presents a
is true in the context of Riemannian metrics: every Riemannian metric can be systematic study of the normal form near orbit segments of convex
perturbed to a Riemannian metric with only non-degenerate closed geodesics, Hamiltonian systems. Our main contributions are the mathematical tools and
this is the bumpy metric theorem, see [4, 2, 1]. The question was investigated conditions for the existence of normal forms, their application to
only much more recently in the context of Ma{\~n}{\'e} perturbations of convex concrete examples, and their implications for the global dynamics and
Hamiltonians, see [9, 10]. It is proved in [10] that the same result holds : geometry of these systems. This work has important applications in
If H is a convex Hamiltonian and a is a regular value of H, then there exist physics, engineering, and materials science, and provides a rich source of
arbitrarily small potentials u such that all periodic orbits (including inspiration for future research in this area.
iterated ones) of H + u at energy a are non-degenerate. The proof given in
[10] is actually rather similar to the ones given in papers on the
perturbations of Riemannian metrics. In all these proofs, it is very useful to
Coalition formation is a fundamental type of interaction that involves the
creation of coherent groupings of distinct, autonomous, agents in order to
efficiently achieve their individual or collective goals. Forming effective
coalitions is a major research challenge in the field of multi-agent systems. Coalition formation is a well-established field of research in artificial
Central to this endeavour is the problem of determining which of the many intelligence, economics, and social choice theory, among others. Its main
possible coalitions to form in order to achieve some goal. This usually goal is to investigate how groups of agents can join forces to achieve
requires calculating a value for every possible coalition, known as the specific goals and how this cooperation can be sustained over time. One of
coalition value, which indicates how beneficial that coalition would be if it the fundamental problems in coalition formation is optimal coalition
was formed. Once these values are calculated, the agents usually need to find structure generation, which aims to find the best possible way to
a combination of coalitions, in which every agent belongs to exactly one partition a group of agents into coalitions that maximize a given
coalition, and by which the overall outcome of the system is maximized. objective function. In this paper, we propose an anytime algorithm for
However, this coalition structure generation problem is extremely challenging optimal coalition structure generation that can be used in a wide range of
due to the number of possible solutions that need to be examined, which grows contexts, from multi-agent systems to political science and organizational
exponentially with the number of agents involved. To date, therefore, many management. Our approach is based on a novel combination of search
algorithms have been proposed to solve this problem using different techniques algorithms and decision-theoretic reasoning that allows us to generate
ranging from dynamic programming, to integer programming, to stochastic search high-quality coalition structures with minimal computational overhead. The
all of which suffer from major limitations relating to execution time, key idea behind our algorithm is to iteratively improve an initial
solution quality, and memory requirements. With this in mind, we develop an coalition structure by exploring neighboring structures that satisfy
anytime algorithm to solve the coalition structure generation problem. certain optimality conditions. At each iteration, we use
An Anytime Algorithm Specifically, the algorithm uses a novel representation of the search space, decision-theoretic techniques to evaluate the quality of the current
for Optimal which partitions the space of possible solutions into sub-spaces such that it 492 coalition structure and decide whether to continue the search or return 373
Coalition Structure is possible to compute upper and lower bounds on the values of the best the best found result so far. This approach gives us the flexibility to
Generation coalition structures in them. These bounds are then used to identify the trade-off between the quality of the solution and the time and resources
sub-spaces that have no potential of containing the optimal solution so that available, making our algorithm ideal for applications with varying time
they can be pruned. The algorithm, then, searches through the remaining and resource constraints. We evaluate our algorithm on a set of benchmark
sub-spaces very efficiently using a branch-and-bound technique to avoid instances and compare its performance against state-of-the-art algorithms
examining all the solutions within the searched subspace(s). In this setting, for coalition structure generation. Our experiments show that our anytime
we prove that our algorithm enumerates all coalition structures efficiently by algorithm is highly competitive and outperforms existing approaches in
avoiding redundant and invalid solutions automatically. Moreover, in order to terms of both solution quality and computational efficiency. Furthermore,
effectively test our algorithm we develop a new type of input distribution we show that our algorithm scales well to large instances, making it a
which allows us to generate more reliable benchmarks compared to the input practical tool for real-world applications. In conclusion, this paper
distributions previously used in the field. Given this new distribution, we presents a new anytime algorithm for optimal coalition structure
show that for 27 agents our algorithm is able to find solutions that are generation that combines search algorithms and decision-theoretic
optimal in 0.175% of the time required by the fastest available algorithm in reasoning for high-quality and efficient coalition formation. Our approach
the literature. The algorithm is anytime, and if interrupted before it would is flexible and can be used in a wide range of contexts, from multi-agent
have normally terminated, it can still provide a solution that is guaranteed systems to political science and organizational management. Our empirical
to be within a bound from the optimal one. Moreover, the guarantees we provide evaluation shows that our algorithm is highly competitive and outperforms
on the quality of the solution are significantly better than those provided by existing approaches, making it a valuable tool for researchers and
the previous state of the art algorithms designed for this purpose. For practitioners alike.
example, for the worst case distribution given 25 agents, our algorithm is
able to find a 90% efficient solution in around 10% of time it takes to find
the optimal solution.
The paper deals with the automatic analysis of real-life telephone
conversations between an agent and a customer of a customer care service
(ccs). The application domain is the public transportation system in Paris and
the purpose is to collect statistics about customer problems in order to
monitor the service and decide priorities on the intervention for improving
user satisfaction. Of primary importance for the analysis is the detection of
themes that are the object of customer problems. Themes are defined in the
application requirements and are part of the application ontology that is This paper discusses the identification of multiple topics in
implicit in the ccs documentation. Due to variety of customer population, the human-to-human conversations, a task which is crucial for effective
structure of conversations with an agent is unpredictable. A conversation may communication and natural language processing. The ability to accurately
be about one or more themes. Theme mentions can be interleaved with mentions identify topics in conversations has many applications such as information
of facts that are irrelevant for the application purpose. Furthermore, in retrieval, summarization, and sentiment analysis. We begin by reviewing
certain conversations theme mentions are localized in specific conversation the relevant literature on topic identification and summarize the
segments while in other conversations mentions cannot be localized. As a state-of-the-art techniques for topic modeling. We then introduce a new
consequence, approaches to feature extraction with and without mention method based on statistical natural language processing that is designed
localization are considered. Application domain relevant themes identified by to improve the accuracy of topic identification in both structured and
an automatic procedure are expressed by specific sentences whose words are unstructured conversations. Our approach uses a combination of supervised
hypothesized by an automatic speech recognition (asr) system. The asr system and unsupervised machine learning techniques such as support vector
is error prone. The word error rates can be very high for many reasons. Among machines, clustering, and latent semantic analysis to effectively identify
Multiple topic them it is worth mentioning unpredictable background noise, speaker accent, multiple topics in conversation. To evaluate the effectiveness of our
identification in and various types of speech disfluencies. As the application task requires the 491 approach, we conducted experiments on several different datasets. Our 267
human/human composition of proportions of theme mentions, a sequential decision strategy results show that our model significantly outperforms other
conversations is introduced in this paper for performing a survey of the large amount of state-of-the-art methods on all of the datasets we tested. We also
conversations made available in a given time period. The strategy has to investigate the effects of different conversation characteristics such as
sample the conversations to form a survey containing enough data analyzed with topic distribution, conversation length, and topic correlation on topic
high accuracy so that proportions can be estimated with sufficient accuracy. identification accuracy. Finally, we discuss several potential
Due to the unpredictable type of theme mentions, it is appropriate to consider applications of our model in real-world conversational settings. For
methods for theme hypothesization based on global as well as local feature example, our method could be used to identify key topics in social media
extraction. Two systems based on each type of feature extraction will be discussions or email threads in order to facilitate information retrieval.
considered by the strategy. One of the four methods is novel. It is based on a Our model could also be used to summarize conversations or identify
new definition of density of theme mentions and on the localization of high sentiment and emotional tone in conversations. In conclusion, we present
density zones whose boundaries do not need to be precisely detected. The an effective approach for multiple topic identification in human-to-human
sequential decision strategy starts by grouping theme hypotheses into sets of conversations using machine learning techniques. Our method outperforms
different expected accuracy and coverage levels. For those sets for which existing techniques and has several potential applications in
accuracy can be improved with a consequent increase of coverage a new system conversational settings.
with new features is introduced. Its execution is triggered only when specific
preconditions are met on the hypotheses generated by the basic four systems.
Experimental results are provided on a corpus collected in the call center of
the Paris transportation system known as ratp. The results show that surveys
with high accuracy and coverage can be composed with the proposed strategy and
systems. This makes it possible to apply a previously published proportion
estimation approach that takes into account hypothesization errors .
The pair-condensed unpolarized spin-$1/2$ Fermi gases have a collective
excitation branch in their pair-breaking continuum (V.A. Andrianov, V.N.
Popov, 1976). We study it at zero temperature, with the eigenenergy equation
deduced from the linearized time-dependent BCS theory and extended
analytically to the lower half complex plane through its branch cut,
calculating both the dispersion relation and the spectral weights
(quasiparticle residues) of the branch. In the case of BCS superconductors, so The collective excitations in the continuum of pair-condensed Fermi gases
called because the effect of the ion lattice is replaced by a short-range have been studied analytically, and scaling laws for these excitations
electron-electron interaction, we also include the Coulomb interaction and we have been derived. This study focuses on the properties and behaviors of
restrict ourselves to the weak coupling limit $\Delta/\mu\to 0^+$ ($\Delta$ is these collective excitations, particularly in the low-temperature regime.
the order parameter, $\mu $ the chemical potential) and to wavenumbers $q=O(1/ The analytical study utilizes the finite-temperature Green’s function
\xi)$ where $\xi$ is the size of a pair; when the complex energy $z_q$ is technique along with the random-phase approximation, providing a
expressed in units of $\Delta$ and $q$ in units of $1/\xi$, the branch follows theoretical framework for the scaling laws derived. The scaling laws
a universal law insensitive to the Coulomb interaction. In the case of cold reveal the existence of a characteristic frequency proportional to the
atoms in the BEC-BCS crossover, only a contact interaction remains, but the square root of the gas’s coupling strength, which scales as a function of
coupling strength $\Delta/\mu$ can take arbitrary values, and we study the density. The analytical treatment of this problem enables us to
Collective branch at any wave number. At weak coupling, we predict three scales, that investigate the properties of the collective excitation branch, such as
excitation branch in already mentioned $q\approx 1/\xi$, that $q\approx(\Delta/\mu)^{-1/3}/\xi$ its spectral weight, lifetime, and damping, and how they vary as a
the continuum of where the real part of the dispersion relation has a minimum and that $q\ function of temperature and gas parameters. The analytical results
pair-condensed Fermi approx (\mu/\Delta)/\xi\approx k_{\rm F}$ ($k_{\rm F}$ is the Fermi wave 490 obtained in this study have been validated through comparison with earlier 271
gases : analytical number) where the branch reaches the edge of its existence domain. Near the works and provide new insights into the collective dynamics of highly
study and scaling point where the chemical potential vanishes on the BCS side, $\mu/\Delta\to 0^ correlated Fermi gases, broadening our understanding of their exotic
laws +$, where $\xi\approx k_{\rm F}$, we find two scales $q\approx(\mu/\Delta)^{1/ behavior. The derived scaling laws can be used to predict the behavior of
2}/\xi$ and $q\approx 1/\xi$. In all cases, the branch has a limit $2\Delta$ these systems under different conditions and parameter regimes, including
and a quadratic start at $q=0$. These results were obtained for $\mu>0$, where the quantum critical regime. These results are particularly interesting in
the eigenenergy equation admits at least two branch points $\epsilon_a(q)$ and the context of ongoing experiments on strongly correlated Fermi gases,
$\epsilon_b(q)$ on the positive real axis, and for an analytic continuation where the collective dynamics of these systems remain an unresolved
through the interval $[\epsilon_a(q),\epsilon_b(q)] $. We find new continuum question. The conclusions drawn from this study provide essential
branches by performing the analytic continuation through $[\epsilon_b(q),+\ information for the design and interpretation of future experiments on
infty[$ or even, for $q$ low enough, where there is a third real positive highly correlated Fermi gases. Ultimately, this work contributes to a
branch point $\epsilon_c(q)$, through $[\epsilon_b(q),\epsilon_c(q)]$ and $[\ better understanding of the collective properties of Fermi gases and lays
epsilon_c(q),+\infty[$. On the BEC side $\mu<0$ not previously studied, where the foundation for future studies investigating the exotic behavior of
there is only one real positive branch point $ \epsilon_a(q)$, we also find these systems.
new collective excitation branches under the branch cut $[\epsilon_a (q),+\
infty[$. For $\mu>0$, some of these new branches have a low-wavenumber exotic
hypoacoustic $z_q\approx q^{3/2}$ or hyperacoustic $z_q\approx q^{4/5}$
behavior. For $\mu<0$, we find a hyperacoustic branch and a nonhypoacoustic
branch, with a limit $2\Delta$ and a purely real quadratic start at $q=0$ for
We show how to transform any set of prioritized propositional defaults into an
equivalent set of parallel (i.e., unprioritized) defaults, in circumscription.
We give an algorithm to implement the transform. We show how to use the
transform algorithm as a generator of a whole family of inferencing algorithms
for circumscription. The method is to employ the transform algorithm as a
front end to any inferencing algorithm, e.g., one of the previously available,
that handles the parallel (empty) case of prioritization. Our algorithms
provide not just coverage of a new expressive class, but also alternatives to
previous algorithms for implementing the previously covered class (?layered?)
of prioritization. In particular, we give a new query-answering algorithm for
prioritized cirumscription which is sound and complete for the full expressive Abstract: The process of default reasoning has been widely studied in
class of unrestricted finite prioritization partial orders, for propositional artificial intelligence and logic. Defaults play a key role in handling
defaults (or minimized predicates). By contrast, previous algorithms required incomplete information and making assumptions about the world. However,
that the prioritization partial order be layered, i.e., structured similar to prioritized defaults and specificity can lead to conflicts and
the system of rank in the military. Our algorithm enables, for the first time, inconsistencies when used in parallel systems. In this paper, we propose a
the implementation of the most useful class of prioritization: non-layered new approach that transforms prioritized defaults and specificity into
prioritization partial orders. Default inheritance, for example, typically parallel defaults, which can be used to overcome these issues. Our
requires non-layered prioritization to represent specificity adequately. Our approach involves representing defaults as sets of parallel rules, each
algorithm enables not only the implementation of default inheritance (and with their own level of specificity. These parallel defaults can be
specificity) within prioritized circumscription, but also the extension and evaluated simultaneously, allowing for more efficient and consistent
Transforming combination of default inheritance with other kinds of prioritized default reasoning. We also introduce a method for resolving conflicts between
Prioritized Defaults reasoning, e.g.: with stratified logic programs with negation-as-failure. Such parallel defaults using a priority ranking scheme. This scheme assigns
and Specificity into logic programs are previously known to be representable equivalently as 490 priorities to different defaults based on their specificity and allows for 252
Parallel Defaults layered-priority predicate circumscriptions. Worst-case, the transform the selection of a single default when conflicts arise. We demonstrate the
increases the number of defaults exponentially. We discuss how inferencing is effectiveness of our approach through several experiments, including
practically implementable nevertheless in two kinds of situations: general benchmark problems and real-world examples. Our results show that the use
expressiveness but small numbers of defaults, or expressive special cases with of parallel defaults can lead to more accurate and efficient reasoning,
larger numbers of defaults. One such expressive special case is non-? particularly in cases with conflicting defaults. Furthermore, our approach
top-heaviness? of the prioritization partial order. In addition to its direct is scalable and can be extended to handle more complex default reasoning
implementation, the transform can also be exploited analytically to generate problems. Overall, our work presents a novel approach for transforming
special case algorithms, e.g., a tractable transform for a class within prioritized defaults and specificity into parallel defaults, which can
default inheritance (detailed in another, forthcoming paper). We discuss other improve the efficiency and accuracy of default reasoning in artificial
aspects of the significance of the fundamental result. One can view the intelligence and logic. Our approach also provides a framework for
transform as reducing n degrees of partially ordered belief confidence to just resolving conflicts between defaults and can be adapted to handle a wide
2 degrees of confidence: for-sure and (unprioritized) default. Ordinary, range of default reasoning problems.
parallel default reasoning, e.g., in parallel circumscription or Poole's
Theorist, can be viewed in these terms as reducing 2 degrees of confidence to
just 1 degree of confidence: that of the non-monotonic theory's conclusions.
The expressive reduction's computational complexity suggests that
prioritization is valuable for its expressive conciseness, just as defaults
are for theirs. For Reiter's Default Logic and Poole's Theorist, the transform
implies how to extend those formalisms so as to equip them with a concept of
prioritization that is exactly equivalent to that in circumscription. This
provides an interesting alternative to Brewka's approach to equipping them
with prioritization-type precedence.
Context: The algorithms for generating a safe fluent API are actively studied
these years. A safe fluent API is the fluent API that reports incorrect
chaining of the API methods as a type error to the API users. Although such a
safe property improves the productivity of its users, the construction of a In the world of software development, application programming interfaces
safe fluent API is too complicated for the developers. The generation (APIs) are essential for working with libraries and frameworks. They allow
algorithms are studied to reduce the development cost of a safe fluent API. developers to access pre-written code and build upon it to create new
The study on the generation would benefit a number of programmers since a applications. A fluent API is a type of API that is designed to be easy to
fluent API is a popular design in the real world. Inquiry: The generation of a read and write, providing more natural language-like syntax that feels
generic fluent API has been left untackled. A generic fluent API refers to the like a domain-specific language (DSL). This paper presents a methodology
fluent API that provides generic methods (methods that contain type parameters for generating a generic fluent API in Java. The proposed methodology
in their definitions). The Stream API in Java is an example of such a generic involves the use of code generation and automated testing. The process
API. The recent research on the safe fluent API generation rather focuses on starts with the identification of a domain-specific language that will be
the grammar class that the algorithm can deal with for syntax checking. The used to generate the fluent API. This DSL is then used to generate the
key idea of the previous study is to use nested generics to represent a stack code for the API using a code generation tool. The generated code is
structure for the parser built on top of the type system. In that idea, the tested using a suite of automated tests to ensure that it is correct and
role of a type parameter was limited to internally representing a stack meets the desired specifications. The benefits of using a fluent API in
element of that parser on the type system. The library developers could not Java are substantial. The more natural language-like syntax reduces the
use type parameters to include a generic method in their API so that the cognitive overhead of learning new APIs and makes code more readable. In
Generating a Generic semantic constraints for their API would be statically checked, for example, addition, the fluent API makes it easier to create complex chains of
Fluent API in Java the type constraint on the items passed through a stream. Approach: We propose 488 operations, reducing the amount of boilerplate code that would be 387
an algorithm to generate a generic fluent API. Our translation algorithm is necessary with a traditional API. The generic nature of the proposed
modeled as the construction of deterministic finite automaton (DFA) with type fluent API means that it can be used in a wide variety of applications. By
parameter information. Each state of the DFA holds information about which defining a domain-specific language, developers can tailor the API to the
type parameters are already bound in that state. This information is used to specific needs of their application while still benefiting from the
identify whether a method invocation in a chain newly binds a type to a type simplified syntax of a fluent API. To validate the effectiveness of the
parameter, or refers to a previously bound type. The identification is methodology, we conducted a case study in which we created a fluent API
required since a type parameter in a chain is bound at a particular method for a sample application. The results of the case study showed that the
invocation, and that bound type is referred to in the following method generated code was correct and met the desired specifications. In
invocations. Our algorithm constructs the DFA by analyzing the binding time of addition, the resulting code was easier to read and write than a
type parameters and their propagation among the states in a DFA that is traditional API. In conclusion, the proposed methodology for generating a
naively constructed from the given grammar. Knowledge and Importance: Our generic fluent API in Java provides a powerful tool for developers to
algorithm helps library developers to develop a generic fluent API. The create more readable and maintainable code. The use of code generation and
ability to generate a generic fluent API is essential to bring the safe fluent automated testing ensures that the resulting API is correct and meets the
API generation to the real world since the use of type parameters is a common desired specifications. The generic nature of the proposed API makes it
technique in the library API design. By our algorithm, the generation of a useful in a wide variety of applications and provides a more natural
safe fluent API will be ready for practical use. Grounding: We implemented a language-like syntax that reduces cognitive overhead.
generator named Protocool to demonstrate our algorithm. We also generated
several libraries using Protocool to show the ability and the limitations of
our algorithm.
This is a dataset created in relation to a bachelor thesis written by Nicolai Thorer Sivesind and Andreas Bentzen Winje. It contains human-produced and machine-generated text samples of scientific
research abstracts. A reformatted version for text-classification is available in the dataset collection Human-vs-Machine. In this collection, all samples are split into separate data points for real
and generated, and labeled either 0 (human-produced) or 1 (machine-generated).
• Generated samples are produced using the GPT-3.5 model, GPT-3.5-turbo-0301 (Snapshot of the model used in ChatGPT 1st of March, 2023).
□ Target content prompted using title of real abstract
□ Target word count equal to the human-produced abstract
• Contains 10k data points of each class.
• Created by Nicolai Thorer Sivesind
More information about production and contents will be added in the end of may 2023.
Please use the following citation:
@misc {sivesind_2023,
author = { {Nicolai Thorer Sivesind}},
title = { ChatGPT-Generated-Abstracts },
year = 2023,
publisher = { Hugging Face }
More information about the dataset will be added once the thesis is finished (end of may 2023). | {"url":"https://huggingface.co/datasets/NicolaiSivesind/ChatGPT-Research-Abstracts","timestamp":"2024-11-09T11:43:09Z","content_type":"text/html","content_length":"573081","record_id":"<urn:uuid:34e0eae4-e48a-4c60-8ca5-3322ce78ff40>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00763.warc.gz"} |
Background We sought to identify optimal approaches calibrating longitudinal cognitive performance
Background We sought to identify optimal approaches calibrating longitudinal cognitive performance across studies with different neuropsychological batteries. among people with AD. (version 7.11,
Muthen & Muthen, Los Angeles CA, 1998C2008) to estimate the models. The model provides factor scores equivalent to those from a model with individual factors at each time point that more explicitly
models longitudinal change. CFA with categorical indicators approach Prior to being used as indicators in a CFA model, we categorized each cognitive test score, using identical cutoffs across studies
(Supplemental Table). We used an equal interval approach to categorization to preserve the distribution of the original test. As in the continuous sign CFA approach, testing or subtests in keeping
serve to anchor the metric across research and we utilized a maximum probability estimator with powerful standard mistake estimation in Mplus. The model can be consistent with something response
theory graded response model.[39C41] Exterior scaling from the element scores for stability Using methods referred to at length elsewhere,[30] we externally scaled elements from the constant and
categorical indicator CFA choices in order that a mean of 50 and SD of 10 represented old adults older 70 years and buy Cyanidin chloride old in america by fixing magic size parameters in the pooled
data with their counterparts from a CFA through the Ageing, Demographics and Memory space Study (ADAMS).[42] Missing data handling The normal standardize and ensure that you typical approaches
utilize a full case analysis, which assumes data are lacking randomly completely. The CFA techniques make less strict assumptions about lacking data by presuming missingness in particular cognitive
testing are missing randomly buy Cyanidin chloride conditional on factors in the dimension model. That is managed using maximum probability methods, and it is a reasonable strategy for calculating
general cognitive efficiency because an implicit assumption can be that testing are exchangeable with one another. Simulation to show comparability of overview ratings across datasets To show that
derived ratings through the standardize and typical strategy, CFA with constant signals, and CFA with categorical signals were similar across different research that given different models of
cognitive testing, we carried out Monte Carlo simulations. Predicated on empirical correlations among cognitive testing, we simulated 100,001 observations with full cognitive data. We after that
calculated summary ratings based on each one of the techniques for every observation using testing from each research. We analyzed bias and accuracy in test-specific cognitive ratings with regards to
the accurate score (whether typically standardized ideals, CFA of constant products, or CFA of categorical buy Cyanidin chloride products) which used all obtainable items using Bland-Altman plots.
[43] Simulation is not needed to evaluate comparability of the MMSE because no equating was done on that measure. Comparison of measurement approaches We compared the approaches in three sets of
analyses. First, we correlated the measures using baseline data in CR2 the pooled sample. Second, we modeled annual rate of change using random effects models to compare the relative magnitudes of
change detected by the approaches.[44] The timescale was time from the earliest onset of AD symptoms. We calculated the sample size needed to detect a 25% annual decline in cognitive performance with
80% power using each approach. We included terms for age, sex, and years of education in these models. We selected a magnitude of 25% because this is a common effect size in other genetic studies. We
determined sample size using this equation:
(Eq. 1) There was a modest amount.
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Quantum mechanics - Heisenberg and Schrödinger - Quantum Physics & Consciousness
Quantum mechanics – Heisenberg and Schrödinger
Planck’s quantum law
In 1900 Max Planck (1858-1947) discovered that electromagnetic energy is transmitted in discrete packets – quanta. With this assumption he obtained a theoretical explanation for the emitted light of
a glowing black body – such as the filament of a light bulb.
Planck’s radiation law reads: E = h.f.
E is the energy of an emitted and absorbed energy package, f stands for the frequency. The value of h (Planck’s constant): 6,626 ×10^-34 Joules per second – is now engraved in his tombstone. With his
quantized energy assumption he offered an explanation for why we only get a tan from UV light – high frequency, lots of energy – but not from infrared light – low frequency so too little energy to
damage our skin cells.
Of course this raises the question what the physical meaning is of the frequency of a such an energy package.
In 1905 Albert Einstein (1879-1955) explained the photoelectric effect by applying Planck’s energy quanta. He named them light quanta. Later they became better known as photons. This provided
Planck’s energy quanta with real particle status and we had suddenly two incompatible descriptions of light: particles and waves.
The quantum atom of Bohr and De Broglie
In 1912, Niels Bohr (1885-1962) arrived at an explanation for the spectral lines of hydrogen. This was a problem because glowing hydrogen does not emit light in all possible frequencies as with an
incandescent lamp, but the transmitted frequencies only have certain discrete values. Those broadcasted frequencies have a very specific pattern that even shows a certain mathematically expressible
Hydrogen emission spectrum. © byjus.com
The atomic model at that time – the beginning of the twentieth century – was a description of negative electrons circled at lightning speed around the positive core, like a mini solar system. But
that description was not able to explain those spectral lines and their mathematical expressible behavior. Niels Bohr suggested that Planck’s quantum law – E = h.f – should be applied here. An
electron moving around the nucleus could then only be in very specific energy states. Bohr called those energy states orbits. He numbered those orbits in ascending energy state with numbers: n =
1,2,3, … etc. If the electron “jumped” from a higher to a lower energy state, the energy difference was sent away as an Einstein photon.
Bohr also stated that the electron jump did not follow a physical path, in other words it was instantaneous, because otherwise when a physical path was traveled, energy loss had to occur according to
Maxwell’s laws for electromagnetic radiation.
Bohr’s quantum model of the hydrogen atom. Only specific electron orbits are allowed.
Bohr could not explain however why only those orbits for electrons were allowed. Luckily Louis Victor de Broglie (1892-1987) came up with a highly imaginative explanation by assuming that if (light)
waves could also behave like particles (photons), then particles (electrons) would probably also behave like waves. According to De Broglie, those electron orbits from Bohr were formed by standing
electron waves, such as happens with standing waves along a vibrating string. Only electron orbits with a natural number of wavelengths (n = 1,2,3, .. etc) would fit exactly and could exist as
standing waves.
The De Broglie electron orbits forming standing waves of 1 to 4 wavelengths.
The theory of De Broglie was confirmed in 1927 by interference behavior observed in experiments with electrons. Interference is an accepted indicator of wave behavior. But the Bohr-De Broglie
hydrogen atom model was still as flat as a dime. There was no explanation yet for the the spherical appearance of the atom.
Standing circular De Broglie 8λ wave. By Yuta Aoki
Werner Heisenberg – the first quantum mechanic
Until 1925 there was no real physical coherent quantum theory with one solid mathematical formal basis. Until then quantum physics was still a loose collection of assumptions. In that year, Werner
Heisenberg (1901-1976) developed the first version of quantum matrix mechanics with which the location and intensity of the spectral lines of glowing hydrogen gas could be formally calculated.
Heisenberg’s matrix mechanics proved to be a difficult formalism, provided no insight into the underlying mechanisms and only produced discrete outcomes. That, of course, was perfectly consistent
with the discrete values observed for the wavelengths in the hydrogen spectrum.
Erwin Schrödinger – the wave mechanic
In 1926, Erwin Schrödinger published what is now known as the Schrödinger equation for quantum mechanics. Schrödinger sought and found his famous equation because he realized that the electron wave
model of De Broglie for a hydrogen with a single electron could not explain why the hydrogen atom wasn’t flat but a sphere. So he was looking for a solution for a 3-dimensional standing wave that
spread spherically around the core, which he found. He thought the solution to his equation described the scattered cloud-like charge of the electron.
In 1929, Heisenberg and Wolfgang Pauli published the foundations of relativistic quantum theory. With that feat, a solid theoretical basis was finally laid for quantum mechanics. They even had now
two different ways to make quantum mechanical predictions.
The matrix mechanics of Heisenberg produces concrete outcomes, the solution to the Schrödinger equation describes a wave that can assume all possible values at any point in space and time. So they
seemed incompatible theories. Just like particles and waves are not compatible descriptions of reality. But in the end it has been shown that both approaches to quantum mechanics are mathematically
Heisenberg’s uncertainty principle: Δx.Δp ≥ ħ/2
In 1927, Heisenberg formulated his famous uncertainty principle that says there is a fundamental inverse relationship between the accuracy with which the momentum p (mass x speed) can be measured and
the location of the particle. That it is fundamental means that his uncertainty is not the result of imperfections in our measuring instruments. It is a limit that nature imposes on our measurements
of physical objects. The uncertainty in the place is Δx, the uncertainty in the pulse (p) is Δp, the crossed-out ħ (with dash through the upper leg) is the so-called Dirac constant and is equal to h
/ 2π. That is a notation convention that simplifies the readability of formulas and equations for the physicist, but is of no importance to us here. The value of ħ is so incredibly small – 1,05 ×10^
−34 – that we only have to deal with it when measuring objects with atomic dimensions.
The probability wave of Max Born
Max Born (1882-1970) finally gave in 1927 the, by most physicist accepted, physical interpretation of the quantum state wave that represents the solution of the Schrödinger equation. The square of
the absolute value of the state wave at a certain place and time appears to be the measure of the chance of finding the quantum object – the electron, photon, …, etc. – at that particular place and
With such an interpretation, the quantum wave has literally become something elusive, because you can’t pick up probabilities. Probabilities are actually more like thoughts, they are our expectations
about something and are therefore more existing in our minds than in the physical tangible reality. Seen in that light the question of how that non-physical probability wave changes into the physical
particle that we find in our measuring instrument becomes important, which is the so-called measuring problem.
Schrödinger’s cat
Schrödinger’s cat
Schrödinger was – like many physicists – not happy with the strange implications of quantum physics. To demonstrate the absurdity of quantum physics – the wave of probability in which all
possibilities are decided but which is only physically realized upon measurement – he came up with his notorious thought experiment with a cat in a closed box. The cat, due to the quantum state of
the box’s contents, would be in a state of simultaneously living and dead from which the cat only ‘escapes’ when an observer opens the box. A lot of – often somewhat macabre – jokes have been made
about that cat.
Joking between quantum physicists
Heisenberg and Schrödinger get pulled over for speeding.
The cop asks Heisenberg “Do you know how fast you were going?“
Heisenberg replies, “No, but we know exactly where we are!“
The officer looks at him confused and says “You were going 108 miles per hour!“
Heisenberg throws his arms up and cries, “Great! Now we’re lost!“
The officer looks over the car and asks Schrödinger if the two men have anything in the trunk.
“A cat,” Schrödinger replies.
The cop opens the trunk and yells “Hey! This cat is dead.“
Schrödinger angrily replies, “Well, now he is.”
Next subject: The Quantum Measurement Problem. What is a measurement? | {"url":"https://quantumphysics-consciousness.eu/index.php/en/quantum-mechanics-heisenberg-and-schrodinger/","timestamp":"2024-11-04T12:09:06Z","content_type":"text/html","content_length":"152608","record_id":"<urn:uuid:27215ac7-6122-46f8-8b85-b187b4a2738f>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00784.warc.gz"} |
What is a 4D number?
4-Digits (abbreviation: 4-D) is a lottery in Germany, Singapore, and Malaysia. Individuals play by choosing any number from 0000 to 9999. A draw is conducted to select these winning numbers. 4-Digits
is a fixed-odds game. Magnum 4D is the 1st legalized 4D Operator licensed by the Malaysian Government to operate 4D.
How do you play a 4D number?
To play 4D, select a four-digit number from 0000 to 9999. The minimum cost is $1, inclusive of GST. Draws take place every Wednesday, Saturday and Sunday at 6.30pm. 23 sets of winning 4D numbers
across five prize categories are drawn each draw.
How much is the prize for 4D?
Prize Structure for Ordinary, 4D Roll and System Entries
Prize Number of 4-digit Prize Winning Numbers Prize Amount (for every $1 stake)
2nd Prize one number $1,000
3rd Prize one number $490
Starter Prizes ten numbers $250
Consolation Prizes ten numbers $60
How much is 4D system bet?
System Entry, iBet
4D Number Combinations Minimum System Entry Bet Amount Minimum iBet Bet Amount
24 4 different digits selected (e.g. 1234) $24 $1
12 2 different and 1 pair of digits selected (e.g. 1123) $12 $1
6 2 pairs of digits selected (e.g. 1122) $6 $1
4 3 of the same digits selected (e.g. 1112) $4 $1
What is 4D and TOTO?
The thing about 4D is, unlike TOTO, the sequence of numbers matters – meaning if you’ve entered 1234 and the draw is 4321 instead, it doesn’t count. To cover all grounds, pick System Entry to enter
all the possible permutation of the numbers you’ve picked. Prices will vary depending on the number of permutations.
Who created 4D?
Laurent Ribardière
The 4D product line has since expanded to an SQL back-end, integrated compiler, integration of PHP, and several productivity plug-ins and interfaces….4th Dimension (software)
Designed by Laurent Ribardière
Filename extensions 4DB, 4DC
File formats Interpreted, Compiled
Website us.4d.com
What are the best Pick 4 numbers to play?
The numbers 1-0-1-0 are one of the most popular combinations of numbers played in the Pick 4 game.
How do you write 4D?
Mark the digits you wish to place bet on and your Big and/or Small bet amounts.
1. For Ordinary Entry, mark four digits. Bet cost is the bet amounts marked.
2. For 4D Roll, mark three digits in addition to ‘R’. Bet cost is the bet amounts marked multiplied by 10.
3. For System Entry, mark four digits.
4. For iBet, mark four digits. | {"url":"https://www.shakuhachi.net/what-is-a-4d-number/","timestamp":"2024-11-08T12:15:42Z","content_type":"text/html","content_length":"47994","record_id":"<urn:uuid:2a65d7e2-c7a3-4d48-86fe-4917fac2e4bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00495.warc.gz"} |
Sorting and Searching
4.2 Sorting and Searching
The sorting problem is to rearrange a set of items in ascending order. One reason that it is so useful is that it is much easier to search for something in a sorted list than an unsorted one. In this
section, we will consider in detail two classical algorithms for sorting and searching, along with several applications where their efficiency plays a critical role.
Binary Search
In the game of "twenty questions", your task is to guess the value of a hidden number that is one of the n integers between 0 and n-1. (For simplicity, we will assume that n is a power of two.) Each
time that you make a guess, you are told whether your guess is too high or too low. An effective strategy is to maintain an interval [lo, hi) that contains the hidden number, guess the number in the
middle of the interval, and then use the answer to halve the interval size. The program questions.py implements this strategy, which is an example of the general problem-solving method known as
binary search.
Correctness proof.
First, we have to convince ourselves that the approach is correct: that it always leads us to the hidden number. We do so by establishing the following facts:
• The interval always contains the hidden number.
• The interval sizes are the powers of two, decreasing from n.
The first of these facts is enforced by the code; the second follows by noting that if the interval size (hi-lo) is a power of two, then the next interval size is (hi-lo)/2, which is the next smaller
power of two. These facts are the basis of an induction proof that the method operates as intended. Eventually, the interval size becomes 1, so we are guaranteed to find the number.
Running time analysis.
Since the size of the interval decreases by a factor of 2 at each iteration (and the base case is reached when
= 1), the running time of binary search is lg
Linear-logarithm chasm.
The alternative to using binary search is to guess 0, then 1, then 2, then 3, and so forth, until hitting the hidden number. We refer to such an algorithm as a
algorithm: it seems to get the job done, but without much regard to the cost (which might prevent it from actually getting the job done for large problems). In this case, the running time of the
brute-force algorithm is sensitive to the input value, but could be as much as
and has expected value
/2 if the input value is chosen at random. Meanwhile, binary search is guaranteed to use no more than lg
Binary representation.
If you look back to
, you will recognize that binary search is nearly the same computation as converting a number to binary! Each guess determines one bit of the answer. In our example, the information that the number
is between 0 and 127 says that the number of bits in its binary representation is 7, the answer to the first question (is the number greater than or equal to 64?) tells us the value of the leading
bit, the answer to the second question tells us the value of the next bit, and so forth. For example, if the number is 77, the sequence of answers true false, false, true, true, false, true
immediately yields 1001101, the binary representation of 77.
Inverting a function.
As an example of the utility of binary search in scientific computing, we revisit a problem that we first encountered in the exercises in Section 2.1: inverting an increasing function. Given an
increasing function
and a value
, and an open interval [
), our task is to find a value
within the interval such that
) =
. In this situation, we use real numbers as the endpoints of our interval, not integers, but we use the same essential approach that we used for guessing a hidden integer with the "twenty questions"
problem: we halve the length of the interval at each step, keeping
in the interval, until the interval is sufficiently small that we know the value of
to within a desired precision), which we take as an argument to the function. This figure illustrates the first step.
The program bisection.py implements this strategy. We start with an interval (lo, hi) known to contain x and use the following recursive procedure:
• Compute mid = (hi + lo) / 2.
• Base case: If hi - lo is less than δ, then return mid as an estimate of x.
• Recursive step: Otherwise, test whether f(mid) > y. If so, look for x in (lo, mid); if not, look for x in (mid, hi).
The key to this approach is the idea that the function is increasing — for any values a and b, knowing that f(a) < f(b) tells us that a < b, and vice versa. In this context, binary search is often
called bisection search because we bisect the interval at each stage.
Binary search in a sorted array.
During much of the last century people would use a publication known as a
to look up the definition of a word. Entries appears in order, sorted by a key that identifies it (the word). Think about how you would look up a word in a dictionary. A brute-force solution would be
to start at the beginning, examine each entry one at a time, and continue until you find the word. No one uses that approach: instead, you open the dictionary to some interior page and look for the
word on that page. If it is there, you are done; otherwise, you eliminate either the part of the dictionary before the current page or the part of the dictionary after the current page from
consideration and repeat.
Exception filter.
We now use binary search to solve the
existence problem
: is a given key in a sorted database of keys? For example, when checking the spelling of a word, you need only know whether your word is in the dictionary and are not interested in the definition.
In a computer search, we keep the information in an array, sorted in order of the key. The binary search code in
differs from our other applications in two details. First, the array length
need not be a power of two. Second, it has to allow the possibility that the item sought is not in the array. The client program implements an
exception filter
: it reads a sorted list of strings from a file which we refer to as the
(for example,
) and an arbitrary sequence of strings from standard input (for example,
) and writes those in the sequence that are
in the whitelist.
Insertion Sort
Binary search requires that the array be sorted, and sorting has many other direct applications, so we now turn to sorting algorithms. We consider first a brute-force algorithm, then a sophisticated
algorithm that we can use for huge arrays.
The brute-force algorithm we consider is known as insertion sort. It is based on a simple approach that people often use to arrange hands of playing cards — that is, consider the cards one at a time
and insert each into its proper place among those already considered (keeping them sorted).
The program insertion.py contains an implementation of a sort() function that mimics this process to sort elements in an array a[] of length n. The test client reads all the strings from standard
input, puts them into the array, calls the sort() function to sort them, and then writes the sorted result to standard output. Try running it to sort the small tiny.txt file. Also try running it to
sort the much larger tomsawyer.txt file, but be prepared to wait a long time!
The outer loop sorts the first i entries in the array; the inner loop can complete the sort by putting a[i] into its proper position in the array.
Mathematical analysis.
function contains a
loop nested inside a
loop, which suggests that the running time is quadratic. However, we cannot immediately draw this conclusion because the
loop terminates as soon as
is greater than or equal to
• Best case. When the input array is already in sorted order, the inner loop amounts to nothing more than a comparison (to learn that a[j-1] is less than or equal to a[j]), so the total running
time is linear.
• Worst case. When the input is reverse sorted, the inner loop does not terminate until j equals 0. So, the frequency of execution of the instructions in the inner loop is 1 + 2 + ... + n-1 ~ n^2/
• Average case. When the input is randomly ordered, we expect that each new element to be inserted is equally likely to fall into any position, so that element will move halfway to the left on
average. Thus, we expect the running time to be 1/2 + 2/2 + ... + (n-1)/2 ~ n^2/2.
Empirical analysis.
implements a doubling test for sorting functions. We can use it to confirm our hypothesis that insertion sort is quadratic for randomly ordered files:
% python
>>> import insertion
>>> import timesort
>>> timesort.doublingTest(insertion.sort, 128, 100)
128 3.67
256 3.73
512 4.21
1024 4.19
2048 4.11
Sensitivity to input.
Note that the
function in
takes parameter
and runs
experiments for each array size, not just one. One reason for doing so is that the running time of insertion sort is sensitive to its input values. It is not correct to flatly predict that the
running time of insertion sort will be quadratic, because your application might involve input for which the running time is linear.
Comparable keys.
We want to be able to sort any type of data that has a natural order. Happily, our insertion sort and binary search functions work not only with strings but also with any data type that is
comparable. You can make a user-defined type comparable by implementing the six special methods corresponding to the
, and
operators. In fact, our insertion sort and binary search functions rely on only the
operator, but it is better style to implement all six special methods.
To develop a faster sorting algorithm, we use a divide-and-conquer approach to algorithm design that every programmer needs to understand. This nomenclature refers to the idea that one way to solve a
problem is to divide it into independent parts, conquer them independently, and then use the solutions for the parts to develop a solution for the full problem. To sort an array with this strategy,
we divide it into two halves, sort the two halves independently, and then merge the results to sort the full array. This method is known as mergesort.To sort a[lo, hi), we use the following recursive
• Base case: If the subarray size is 0 or 1, it is already sorted.
• Recursive step: Otherwise, compute mid = (hi + lo)/2, sort (recursively) the two subarrays a[lo, mid) and a[mid, hi), and merge them.
The program merge.py is an implementation. As with insert.py, the test client reads all the strings from standard input, puts them into the array, calls the sort() function to sort them, and then
writes the sorted result to standard output. Try running it to sort the small tiny.txt file and the much larger tomsawyer.txt file.
As usual, the easiest way to understand the merge process is to study a trace of the contents of the array during the merge.
Mathematical analysis.
loop involves
iterations, so the frequency of execution of the instructions in the inner loop is proportional to the sum of the subarray lengths over all calls to the recursive function. The value of this quantity
emerges when we arrange the calls on levels according to their size. For simplicity, suppose that
is a power of 2, with
= 2
. On the first level, we have one call for size
; on the second level, we have two calls for size
/2; on the third level, we have four calls for size
/4; and so forth, down to the last level with
/2 calls of size 2. There are precisely
= lg
levels, giving the grand total
for the frequency of execution of the instructions in the inner loop of mergesort. This equation justifies a hypothesis that the running time of mergesort is linearithmic.
When n is not a power of 2, the subarrays on each level are not necessarily all the same size, but the number of levels is still logarithmic, so the linearithmic hypothesis is justified for all n.
Empirical analysis.
We can run program
to perform a doubling test to confirm our hypothesis that mergesort has running time
for randomly ordered files:
% python
>>> import merge
>>> import timesort
>>> timesort.doublingTest(merge.sort, 128, 100)
128 1.84
256 2.15
512 2.22
1024 2.17
2048 2.13
4096 2.12
8192 2.14
Quadratic-linearithmic chasm.
The difference between
makes a huge difference in practical applications. Understanding the enormousness of this difference is another critical step to understanding the importance of the design and analysis of algorithms.
For a great many important computational problems, a speedup from quadratic to linearithmic makes the difference between being able to solve a problem involving a huge amount of data and not being
able to effectively address it at all.
Python System Sort
Python includes two operations for sorting. The method sort() in the built-in list data type rearranges the items in the underlying list into ascending order, much like merge.sort(). In contrast, the
built-in function sorted() leaves the underlying list alone; instead, it returns a new list containing the items in ascending order. This interactive Python script at right illustrates both
% python
>>> a = [3, 1, 4, 1, 5]
>>> b = sorted(a)
>>> a
[3, 1, 4, 1, 5]
>>> b
[1, 1, 3, 4, 5]
>>> a.sort()
>>> a
[1, 1, 3, 4, 5]
The Python system sort uses a version of mergesort. It is likely to be substantially faster (10-20×) than merge.py because it uses a low-level implementation that is not composed in Python, thereby
avoiding the substantial overhead that Python imposes on itself. As with our sorting implementations, you can use the system sort with any comparable data type, such as Python's built-in str, int,
and float data types.
Application: Frequency Counts
The program frequencycount.py reads a sequence of strings from standard input and then writes a table of the distinct values found and the number of times each was found, in decreasing order of the
frequencies. We accomplish this by two sorts.
Computing the frequencies.
Our first step is to sort the strings on standard input. In this case, we are not so much interested in the fact that the strings are put into sorted order, but in the fact that sorting brings equal
strings together. If the input is
then the result of the sort is
with equal strings like the three occurrences of to brought together in the array. Now, with equal strings all together in the array, we can make a single pass through the array to compute all the
frequencies. The Counter class, as defined in counter.py from Section 3.3, is the perfect tool for the job.
Sorting the frequencies.
Next, we sort the
objects by frequency. We can do so in client code by augmenting the
data type to include the six comparison methods for comparing
objects by their count. Thus, we simply sort the array of
objects to rearrange them in ascending order of frequency! Next, we reverse the array so that the elements are in descending order of frequency. Finally, we write each
object to standard output.
Zipf's law.
The application highlighted in
is elementary linguistic analysis: which words appear most frequently in a text? A phenomenon known as
Zipf's law
says that the frequency of the
th most frequent word in a text of
distinct words is proportional to 1/
. Try running
on the large
, and
files to observe that phenomenon.
Q & A
Q. Why do we need to go to such lengths to prove a program correct?
A. To spare ourselves considerable pain. Binary search is a notable example. For example, you now understand binary search; a classic programming exercise is to compose a version that uses a while
loop instead of recursion. Try solving the first three exercises in this section without looking back at the code in the book. In a famous experiment, Jon Bentley once asked several professional
programmers to do so, and most of their solutions were not correct.
Q. Why introduce the mergesort algorithm when Python provides an efficient sort() method defined in the list data type?
A. As with many topics we have studied, you will be able to use such tools more effectively if you understand the background behind them.
Q. What is the running time of the following version of insertion sort on an array that is already sorted?
def sort(a):
n = len(a)
for i in range(1, n):
for j in range(i, 0, -1):
if a[j] < a[j-1]: exchange(a, j, j-1)
else: break
A. Quadratic time in Python 2; linear time in Python 3. The reason is that, in Python 2, range() is a function that returns an array of integers of length equal to the length of the range (which can
be wasteful if the loop terminates early because of a break or return statement). In Python 3, range() returns an iterator, which generates only as many integers as needed.
Q. What happens if I try to sort an array of elements that are not all of the same type?
A. If the elements are of compatible types (such as int and float), everything works fine. For example, mixed numeric types are compared according to their numeric value, so 0 and 0.0 are treated as
equal. If the elements are of incompatible types (such as str and int), then Python 3 raises a TypeError at run time. Python 2 supports some mixed-type comparisons, using the name of the class to
determine which object is smaller. For example, Python 2 treats all integers as less than all strings because 'int' is lexicographically less than 'str'.
Q. Which order is used when comparing strings with operators such as == and <?
A. Informally, Python uses lexicographic order to compare two strings, as words in a book index or dictionary. For example 'hello' and 'hello' are equal, 'hello' and 'goodbye' are unequal, and
'goodbye' is less than 'hello'. More formally, Python first compares the first character of each string. If those characters differ, then the strings as a whole compare as those two characters
compare. Otherwise, Python compares the second character of each string. If those characters differ, then the strings as a whole compare as those two characters compare. Continuing in this manner, if
Python reaches the ends of the two strings simultaneously, then it considers them to be equal. Otherwise, it considers the shorter string to be the smaller one. Python uses Unicode for
character-by-character comparisons. We list a few of the most important properties:
• '0' is less than '1', and so forth.
• 'A' is less than 'B', and so forth.
• 'a' is less than 'b', and so forth.
• Decimal digits ('0' to '9') are less than the uppercase letters ('A' to 'Z').
• Uppercase letters ('A' to 'Z') are less than lowercase letters ('a' to 'z').
1. Develop an implementation of questions.py that takes the maximum number n as command-line argument (it need not be a power of 2). Prove that your implementation is correct.
2. Compose a nonrecursive version of binarysearch.py.
3. Modify binarysearch.py so that if the search key is in the array, it returns the smallest index i for which a[i] is equal to key, and otherwise it returns the largest index i for which a[i] is
smaller than key (or -1 if no such index exists).
4. Describe what happens if you apply binary search to an unordered array. Why shouldn't you check whether the array is sorted before each call to binary search? Could you check that the elements
binary search examines are in ascending order?
5. Describe why it is desirable to use immutable keys with binary search.
6. Let f() be a monotonically increasing function with f(a) < 0 and f(b) > 0. Compose a program that computes a value x such that f(x) = 0 (up to a given error tolerance).
7. Add code to insertion.py to produce the trace given above.
8. Add code to merge.py to produce the trace given above.
9. Give traces of insertion sort and mergesort in the style of the traces shown above, for the input it was the best of times it was.
10. Compose a program dedup.py that reads strings from standard input and writes them to standard output with all duplicates removed (and in sorted order).
11. Compose a version of mergesort, as defined in merge.py, that creates an auxiliary array in each recursive call to _merge() instead of creating only one auxiliary array in sort() and passing it as
an argument. What impact does this change have on performance?
12. Compose a nonrecursive version of mergesort, as defined in merge.py.
13. Find the frequency distribution of words in your favorite book. Does it obey Zipf's law?
Creative Exercises
The following exercises are intended to give you experience in developing fast solutions to typical problems. Think about using binary search, mergesort, or devising your own divide-and-conquer
algorithm. Implement and test your algorithm.
1. Median. Study the function median() in stdstats.py. It computes the median of a given array of numbers in linearithmic time. Note that it works by reducing the problem to sorting.
2. Mode. Add to stdstats.py a function mode() that computes in linearithmic time the mode (value that occurs most frequently) of a sequence of n integers. Hint: Reduce to sorting.
3. Integer sort. Compose a linear-time filter reads from standard input a sequence of integers that are between 0 and 99 and writes the integers in sorted order on standard output. For example,
presented with the input sequence
your program should write the output sequence
4. Floor and ceiling. Given a sorted array of n comparable keys, compose functions floor() and ceiling() that returns the index of the largest (or smallest) key not larger (or smaller) than an
argument key in logarithmic time.
5. Bitonic maximum. An array is bitonic if it consists of an increasing sequence of keys followed immediately by a decreasing sequence of keys. Given a bitonic array, design a logarithmic algorithm
to find the index of a maximum key.
6. Search in a bitonic array. Given a bitonic array of n distinct integers, design a logarithmic-time algorithm to determine whether a given integer is in the array.
7. Closest pair. Given an array of n floats, compose a function to find in linearithmic time the pair of floats that are closest in value.
8. Furthest pair. Given an array of n floats, compose a function to find in linear time the pair of integers that are farthest apart in value.
9. Two sum. Compose a function that takes as argument an array of n integers and determines in linearithmic time whether any two of them sum to 0.
10. Three sum. Compose a function that takes as argument an array of n integers and determines whether any three of them sum to 0. Your program should run in time proportional to n^2 log n. Extra
credit: Develop a program that solves the problem in quadratic time.
11. Majority. Given an array of n elements, an element is a majority if it appears more than n/2 times. Compose a function that takes an array of n strings as an argument and identifies a majority
(if it exists) in linear time.
12. Common element. Compose a function that takes as argument three arrays of strings, determines whether there is any string common to all three arrays, and if so, returns one such string. The
running time of your function should be linearithmic in the total number of strings.
13. Largest empty interval. Given n timestamps for when a file is requested from web server, find the largest interval of time in which no file is requested. Compose a program to solve this problem
in linearithmic time.
14. Prefix-free codes. In data compression, a set of strings is prefix-free if no string is a prefix of another. For example, the set of strings 01, 10, 0010, and 1111 is prefix-free, but the set of
strings 01, 10, 0010, 1010 is not prefix-free because 10 is a prefix of 1010. Compose a program that reads in a set of strings from standard input and determines whether the set is prefix-free.
15. Partitioning. Compose a function that sorts an array that is known to have at most two different values. Hint: Maintain two pointers, one starting at the left end and moving right, the other
starting at the right end and moving left. Maintain the invariant that all elements to the left of the left pointer are equal to the smaller of the two values and all elements to the right of the
right pointer are equal to the larger of the two values.
16. Dutch national flag. Compose a function that sorts an array that is known to have at most three different values. (Edsgar Dijkstra named this the Dutch-national-flag problem because the result is
three "stripes" of values like the three stripes in the flag.) Hint: Reduce to the previous problem, by first partitioning the array into two parts with all elements having the smallest value in
the first part and all other elements in the second part, then partition the second part.
17. Quicksort. Compose a recursive program that sorts an array of randomly ordered distinct elements. Hint: Use a method like the one described in the previous exercise. First, partition the array
into a left part with all elements less than v, followed by v, followed by a right part with all elements greater than v. Then, recursively sort the two parts. Extra credit: Modify your method
(if necessary) to work properly when the elements are not necessarily distinct.
18. Reverse domain. Compose a program to read in a list of domain names from standard input and write the reverse domain names in sorted order. For example, the reverse domain of cs.princeton.edu is
edu.princeton.cs. This computation is useful for web log analysis. To do so, create a data type Domain that implements the special comparison methods, using reverse domain name order.
19. Local minimum in an array. Given an array of n floats, compose a function to find in logarithmic time a local minimum (an index i such that a[i] < a[i-1] and a[i] < a[i+1]).
20. Discrete distribution. Design a fast algorithm to repeatedly generate numbers from the discrete distribution. Given an array p[] of nonnegative floats that sum to 1, the goal is to return index i
with probability p[i]. Form an array s[] of cumulated sums such that s[i] is the sum of the first i elements of p[]. Now, generate a random float r between 0 and 1, and use binary search to
return the index i for which s[i] ≤ r < s[i+1].
21. Rhyming words. Tabulate a list that you can use to find words that rhyme. Use the following approach:
□ Read in a dictionary of words into an array of strings.
□ Reverse the letters in each word (confound becomes dnuofnoc, for example).
□ Sort the resulting array.
□ Reverse the letters in each word back to their original order.
For example, confound is adjacent to words such as astound and surround in the resulting list. | {"url":"https://introcs.cs.princeton.edu/python/42sort/index.php","timestamp":"2024-11-10T05:09:55Z","content_type":"text/html","content_length":"48524","record_id":"<urn:uuid:d1087612-4793-4e4f-930e-1b2df956ee42>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00883.warc.gz"} |
08-10-10 - Transmission of Huffman Trees
Transmission of Huffman Trees is one of those peripheral problems of compression that has never been properly addressed. There's not really any research literature on it, because in the N -> infinity
case it disappears.
Of course in practice, it can be quite important, particularly because we don't actually just send one huffman tree per file. All serious compressors that use huffman resend the tree every so often.
For example, to compress bytes what you might do is extend your alphabet to [0..256] inclusive, where 256 means "end of block" , when you decode a 256, you either are at the end of file, or you read
another huffman tree and start on the next block. (I wrote about how the encoder might make these block split decisions here ).
So how might you send a Huffman tree?
For background, you obviously do not want to actually send the codes. The Huffman code value should be implied by the symbol identity and the code length. The so-called "canonical" codes are created
by assigning codes in numerical counting up order to symbols of the same length in their alphabetical order. You also don't need to send the character counts and have the decoder make its own tree,
you send the tree directly in the form of code lengths.
So in order to send a canonical tree, you just have to send the code lens. Now, not all symbols in the alphabet may occur at all in the block. Those technically have a code length of "infinite" but
most people store them as code length 0 which is invalid for characters that do occur. So you have to code :
which symbols occur at all
which code lengths occur
which symbols that do occur have which code length
Now I'll go over the standard ways of doing this and some ideas for the future.
The most common way is to make the code lengths into an array indexed by symbol and transmit that array. Code lengths are typically in [1,31] (or even less [1,16] , and by far most common is [4,12]),
and you use 0 to indicate "symbol does not occur". So you have an array like :
{ 0 , 0 , 0 , 4 , 5 , 7 , 6, 0 , 12 , 5 , 0, 0, 0 ... }
1. Huffman the huffman tree ! This code length array is just another array of symbols to compress - you can of course just run your huffman encoder on that array. In a typical case you might have a
symbol count of 256 or 512 or so, so you have to compress that many bytes, and then your "huffman of huffmans" will have a symbol count of only 16 or so, so you can then send the tree for the
secondary huffman in a simpler scheme.
2. Delta from neighbor. The code lens tend to be "clumpy" , that is , they have correlation with their neighbors. The typical way to model this is to subtract each (non-zero) code len from the last
(non-zero) code len, thus turning them into deltas from neighbors. You can then take these signed deltas and "fold up" the negatives to make them unsigned and then use one of the other schemes for
transmitting them (such as huffman of huffmans). (actually delta from an FIR or IIR filter of previous is better).
3. Runlens for zeros. The zeros (does not occur) in particular tend to be clumpy, so most people send them with a runlen encoder.
4. Runlens of "same". LZX has a special flag to send a bunch of codelens in a row with the same value.
5. Golomb or other "variable length coding" scheme. The advantage of this over Huffman-of-huffmans is that it can be adaptive, by adjusting the golomb parameter as you go. (see for example on how to
estimate golomb parameters). The other advantage is you don't have to send a tree for the tree.
6. Adaptive Arithmetic Code the tree! Of course if you can Huffman or Golomb code the tree you can arithmetic code it. This actually is not insane; the reason you're using Huffman over Arithmetic is
for speed, but the Huffman will be used on 32k symbols or so, and the arithmetic coder will only be used on the 256-512 or so Huffman code lengths. I don't like this just because it brings in a bunch
more code that I then have to maintain and port to all the platforms, but it is appealing because it's much easier to write an adaptive arithmetic coder that is efficient than any of these other
BTW That's a general point that I think with is worth stressing : often you can come up with some kind of clever heuristic bit packing compression scheme that is close to optimal. The real win of
adaptive arithmetic coding is not the slight gain in efficiency, it's the fact that it is *so* much easier to compress anything you throw at it. It's much more systematic and scientific, you have
tools, you make models, you estimate probabilities and compress them. You don't have to sit around fiddling with "oh I'll combined these two symbols, then I'll write a run length, and this code will
mean switch to a different coding", etc.
Okay, that's all standard review stuff, now let's talk about some new ideas.
One issue that I've been facing is that coding the huffman tree in this way is not actually very nice for the decoder to be able to very quickly construct trees. (I wind up seeing the build tree time
show up in my profiles, even though I only buld tree 5-10 times per 256k symbols). The issue is that it's in the wrong order. To build the canonical huffman code, what you need is the symbols in
order of codelen, from lowest codelen to highest, and with the symbols sorted by id within each codelen. That is, something like :
codeLen 4 : symbols 7, 33, 48
codeLen 5 : symbols 1, 6, 8, 40 , 44
codeLen 7 : symbols 3, 5, 22
obviously you can generate this from the list of codelens per symbol, but it requires a reshuffle which takes time.
So, maybe we could send the tree directly in this way?
One approach is through counting combinations / enumeration . For each codeLen, you send the # of symbols that have that codeLen. Then you have to select the symbols which have that codelen. If there
are M symbols of that codeLen and N remaining unclaimed symbols, the number of ways is N!/(M!*(N-M)!) , and the number of bits needed to send the combination index is log2 of that. Note in this
scheme you should also send the positions of the "not present" codeLen=0 group, but you can skip sending the entire group that's largest. You should also send the groups in order of smallest to
largest (actually in order or *complement* order, a group that's nearly full is as good as a group that's nearly empty).
I think this is an okay way to send huffman trees, but there are two problems : 1. it's slow to decode a combination index, and 2. it doesn't take advantage of modelling clumpiness.
Another similar approach is binary group flagging. For each codeLen, you want to specify which remaining symbols are of that codelen or not of that codelen. This is just a bunch of binary off/on
flags. You could send them with a binary RLE coder, or the elegant thing would be Group Testing. Again the problem is you would have to make many passes over the stream and each time you would have
to exclude already done ones.
(ADDENDUM : a better way to do this which takes more advantage of "clumpiness" is like this : first code a binary event for each symbol to indicate codelen >=1 (vs. codeLen < 1). Then, on the subset
that is >= 1, code an event for is it >= 2, and so on. This the same amount of binary flags as the above method, but when the clumpiness assumption is true this will give you flags that are very well
grouped together, so they will code well with a method that makes coherent binary smaller (such as runlengths)).
Note that there's another level of redundancy that's not being exploited by any of these coders. In particular, we know that the tree must be a "prefix code" , that is satisfy Kraft, (Sum 2^-L = 1).
This constrains the code lengths. (this is most extreme for the case of small trees; for example with a two symbol tree the code lengths are completely specified by this; on a three symbol tree you
only have one free choice - which one gets the length 1, etc).
Another idea is to use MTF for the codelengths instead of delta from previous. I think that this would be better, but it's also slower.
Finally when you're sending multiple trees in a file you should be able to get some win by making the tree relative to the previous one, but I've found this is not trivial.
I've tried about 10 different schemes for huffman tree coding, but I'm not going to have time to really solve this issue, so it will remain neglected as it always has.
8 comments:
Unknown said...
This comment has been removed by the author.
Range coder is faster than Huffman. There's really no reason to use Huffman since the majority of range coder related IBM patents have now expired.
"Range coder is faster than Huffman. "
That is 110% wrong.
Maybe more.
Round one! Fight!
I'm taking your damn trolling and turning into constructive and interesting gold. Gold, Jerry, gold!
I'm talking about decompression only and assuming a fixed probability table.
"I'm talking about decompression only and assuming a fixed probability table."
Basically this is just not right.
As I demonstrated in great detail, the fastest arithmetic decoder would be one in which the total probabilities was a power of 2, and each individual probability was a power of 2. That's a
Huffman code.
If you make the total a power of 2 but let each individual once be a sum of two powers of two, that's slightly slower (ala Rissanen-Mohiuden / DCC95).
Anything more general is slower still. | {"url":"http://cbloomrants.blogspot.com/2010/08/08-10-10-transmission-of-huffman-trees.html","timestamp":"2024-11-12T12:49:27Z","content_type":"application/xhtml+xml","content_length":"81142","record_id":"<urn:uuid:e3271338-a7d9-49af-84a5-83048b9a85c2>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00718.warc.gz"} |
Simulation and experimental analysis of tip response of tapping mode atomic force microscope
In this study, the vertical deflection responses of tapping mode atomic force microscope (TM-AFM) micro-cantilever tip are obtained by simulation and experiment. The results show that, under the
blocking of the sample on one side, the steady-state response of the tip is still a sinusoidal form almost symmetrical about the equilibrium position. Furthermore, from the perspective of energy
dissipation of the micro-cantilever system, the phases of two surfaces with different properties are simulated under different background dissipation. The result shows that eliminating partial
background dissipation can increase the phase contrast between the two surfaces. These results are of significance for understanding the tip response and phase optimization in TM-AFM.
• It is proved by experiment and simulation that in TM-AFM, the displacement response of the tip is almost a sine wave symmetrical about the equilibrium position.
• The phase response of the tip on the surface with different properties is obtained by numerical simulation.
• The simulation results show that reducing the background dissipation can improve the phase contrast between the components of the sample, which has hardly been mentioned by previous studies.
1. Introduction
Since the emergence of atomic force microscopes (AFM), it has become one of the most powerful tools in micro- and nanotechnology because of its excellent resolution [1, 2]. The key component of AFM
is a micro-cantilever with sharp a tip. The information of the sample is recorded by transforming the interaction forces between the tip and the sample into the displacement signal of the tip [2].
Tapping mode (TM) is one of the most commonly used operation modes in atomic force microscopes. It can obtain high-resolution images of various information of samples, and can greatly reduce the wear
of the tip and damages to the sample compared with contact mode [3]. Therefore, tapping mode atomic force microscope (TM-AFM) is widely used in various fields, including biomolecules, polymers, and
nanostructures [4-7].
During the operation of TM-AFM, the micro-cantilever oscillates above the sample under the excitation of the piezoelectric actuator, and the tip contacts intermittently with the sample. The
interactions between the tip and sample are very complex, including van der Waals interactions, short-range repulsive, adhesion, air damping, and capillary forces [8], which may make the response of
the tip very complex. Payam [9] obtained the time history responses of the microcantilever with different distances between the tip and sample through simulation. Cleveland et al. [10] showed through
experiments that the time history response of the tip vertical deflection was nearly sinusoidal. However, few researchers have conducted detailed analysis on the displacement response of the
microcantilever tip in simulation and experiment.
In addition, phase-imaging is an important part of TM-AFM, which is very sensitive to the properties of samples. The different components and properties of the sample are distinguished by recording
the difference between the tip response signal and the excitation signal. Phase-imaging has been widely used to study various heterogeneous materials and their viscoelasticity, wettability, and so on
[11-13]. The phase in TM-AFM is considered to be related to the energy dissipation of the system [10, 14, 15]. At present, few studies have optimized the phase contrast in TM-AFM from the perspective
of reducing the background dissipation of the micro-cantilever system.
In this paper, the time history response of the tip displacement is obtained by numerical simulation and experiment. Then, from the perspective of energy dissipation, the phases of the tip working on
two surfaces with different properties under different background dissipation is simulated, and the factors affecting the phase contrast between the two surfaces are analyzed.
2. Analysis of tip response in TM-AFM
2.1. Model
The key component of TM-AFM is a micro-cantilever with a sharp tip, which is sensitive to small forces. The piezoelectric actuator drives the substrate with sinusoidal displacement to make the
micro-cantilever oscillate near its first natural frequency. The information of the sample can be obtained by detecting the change of the response at the tip. As shown in Fig. 1(a), the length of the
micro-cantilever is $l$, the width is $b$, and the thickness is $d$, the displacement excitation ${z}_{f}=D\mathrm{s}\mathrm{i}\mathrm{n}\varpi t$, $D$ is the excitation amplitude, $\varpi$ is the
excitation frequency. The differential equation of the micro-cantilever is as follows:
$EI\frac{{\partial }^{4}u}{\partial {x}^{4}}+\left[\rho bd+{m}^{"}\delta \left(x-l\right)\right]\frac{{\partial }^{2}w}{\partial {t}^{2}}+c\frac{\partial w}{\partial t}={F}_{\mathrm{t}\mathrm{s}}\
delta \left(x-l\right),$
where $w$ is the absolute displacement of the micro-cantilever, $u$ is the relative displacement of the micro-cantilever relative to the substrate, $E$ and $I$ are the Young’s Modulus and section
moment of inertia of the micro-cantilever respectively, $\rho$ is the density of the micro-cantilever, $m"$ is the mass of the tip, $c$ is the equivalent damping coefficient per unit length of the
micro-cantilever, ${F}_{ts}$ represents the interaction forces between the tip and the sample, and $\delta \left(x\right)$ Dirac function.
Fig. 1Micro-cantilever model of TM-AFM
a) Continuous micro-cantilever model
b) Simplified model of micro-cantilever
In TM-AFM, the surface information of the sample is imaged by monitoring the response at the tip of the micro-cantilever. At the same time, the micro-cantilever works near its first-order natural
frequency, so the tip position and first-order mode can be used to simplify the micro-cantilever model. As shown in Fig. 1(b), the differential equation of motion of the needle tip can be written as:
${m}_{e}\frac{{d}^{2}z}{d{t}^{2}}+\frac{{m}_{e}{\omega }_{0}}{{Q}_{b}}\frac{dz}{dt}+{k}_{e}\left(z-{z}_{f}\right)={F}_{ts},$
where ${m}_{e}$ is the effective mass of the micro-cantilever, ${k}_{e}$ is the force constant of the micro-cantilever, ${c}_{0}$ is the equivalent damping coefficient representing the background
dissipation, ${\omega }_{0}={\left({k}_{e}/{m}_{e}\right)}^{1/2}$ is the first-order natural angular frequency, ${Q}_{b}={m}_{e}{\omega }_{0}/{c}_{0}$ is the effective quality factor representing the
background dissipation of the micro-cantilever system. In tapping mode, the micro-cantilever has various energy dissipation paths when vibrates in air, such as air viscous dissipation, squeeze film
damping, liquid bridge dissipation, and so on. This type of dissipation, which does not reflect the sample information is called background dissipation.
In AFM, the interaction between the tip and the sample is very complex, and the interaction forces are usually simplified in the response analysis. Assuming that the spherical tip interacts with the
flat sample surface, van der Waals forces and DMT contact force are typically used to describe attraction and repulsion forces, respectively. The forces between the tip and sample can be expressed as
follows [8, 15, 16]:
${F}_{ts}=\left\{\begin{array}{ll}\frac{HR}{6\left(h-z{\right)}^{2}},& h-z>{a}_{0},\\ \frac{HR}{6{{a}_{0}}^{2}}-\frac{4}{3}{E}^{*}\sqrt{R}\left[{a}_{0}-\left(h-z\right){\right]}^{\frac{3}{2}}-\eta \
sqrt{R\left[{a}_{0}-\left(h-z\right)\right]}\frac{dz}{dt},& h-z>{a}_{0},\end{array}\right\$
where $h$ is the distance between the tip and the sample at the equilibrium position, $H$ is Hamaker’s constant, $R$ is the radius of the tip, ${a}_{0}$ is the intermolecular distance, $\eta$ is the
viscosity of the sample, and ${E}^{*}$ is the equivalent Young’s modulus, which can be expressed as follows:
${E}^{*}=\frac{1}{\left[\frac{1-{{u }_{t}}^{2}}{{E}_{t}}+\frac{1-{{u }_{s}}^{2}}{{E}_{s}}\right]},$
where, ${E}_{t}$, ${E}_{s}$, ${u }_{t}$ and ${u }_{s}$ are the Young’s modulus and Poisson’s ratio of the tip and sample, respectively.
2.2. Analysis of time history response of the tip displacement
In order to simulate the response of the tip in TM-AFM, the parameters of the real micro cantilever are used. The effective quality factor ${Q}_{b}$ used in this paper is obtained by processing the
results of Tune experiment by half-power method. The excitation frequency is 99.9 % of the system’s natural frequency. The specific parameters are shown in Table 1.
Table 1Parameters used in simulation
Symbol Value Symbol Value
Effective mass (kg) ${m}_{e}$ 9.13×10^-12 Tip radius (nm) $R$ 6
Force constant (N/m) ${k}_{e}$ 42 Equivalent Young’s modulus (GPa) ${E}^{*}$ 10.08
Effective quality factor ${Q}_{b}$ 500 Hamaker’s constant (J) $H$ 2×10^-19
Natural frequency (kHz) ${f}_{0}$ 341.36 Intermolecular distance (nm) ${a}_{0}$ 0.38
Drive amplitude (nm) $D$ 0.1 Sample viscosity (Pa·s) $\eta$ 20
Drive frequency (kHz) $f$ 341.02
By substituting the parameters in Table 1 into Eq. (1-3), and solving Eq. (2) can obtain the tip response under the sample limitation. It is noted that the interaction forces between the tip and the
sample in Eq. (3) are piecewise nonlinear. Therefore, Eq. (2) is a piecewise nonlinear second-order differential equation. Here, the fourth-order Range-Kutta algorithm is used to solve Eq. (2).
When the micro-cantilever is far away from the sample, the interaction forces between the tip and sample can be ignored, and Eq. (2) can be considered as a second-order linear differential equation.
The time history response of the vertical deflection of the tip is shown in Fig. 2(a). The tip oscillates from the equilibrium position, and finally, the vertical deflection of the tip tends toward a
steady state. The fast Fourier transform (FFT) of the steady-state response after 5 ms is shown in Fig. 2(b). It can be seen from the FFT diagram that there is only one frequency component which is
the same as the excitation frequency. These results show that the steady-state response of the tip displacement when the micro-cantilever is far away from the sample is an absolute sine wave
symmetrical about the equilibrium position. At this time, the amplitude of the steady-state response of the tip without sample constraint is ${A}_{0}$.
Fig. 2Time history response and FFT of the tip vertical deflection when the micro-cantilever is far away from the sample
a) Time-history response of the tip vertical deflection
b) FFT of the steady-state response
Fig. 3Time history response and FFT of the tip vertical deflection at different initial distances between tip and sample
a) Time-history response of the tip vertical deflection when $h=$ 0.9A0
b) FFT of the steady-state response when $h=$ 0.9A0
c) Time-history response of the tip vertical deflection when $h=$ 0.1${A}_{0}$
d) FFT of the steady-state response when $h=$ 0.1${A}_{0}$
In tapping mode, the amplitude of the tip is generally 85 %-90 % of that when oscillating in air. It is assumed that the sample surface is smooth and uniform. When the initial distance between the
needle tip and the sample $h=$ 0.9${A}_{0}$, the time history response and FFT of the tip vertical deflection are shown in Fig. 3(a) and 3(b), respectively. The magenta dotted line represents the
position of the sample, and the orange dotted line represents the symmetrical position of the sample with respect to the equilibrium position. The FFT of steady-state response is shown in Fig. 3(b),
in which there are many very small frequency doubling components and a very small direct current (DC) component in addition to the fundamental frequency. This is due to the tip intermittently
contacts with the sample, and the system is no longer linear, but piecewise nonlinear. Because the frequency doubling components and DC component of the response are very small compared with the
fundamental frequency, the steady-state time history response of the tip displacement can be considered as a standard sinusoidal signal symmetrical about the equilibrium position.
In a more extreme case, when the initial distance between the needle tip and the sample $h=$ 0.1${A}_{0}$, the time history response and FFT of the tip vertical deflection are shown in Fig. 3(c) and
3(d), respectively. As can be seen from the FFT diagram, the DC component and frequency doubling components are very obvious at this time. This is because the depth of the tip pressing into the
sample is large, and the interaction time between the tip and the sample becomes longer in one cycle. In this case, although the DC component and frequency doubling components are obvious, they are
very small compared with the fundamental frequency, so that the steady-state response can also be regarded as a sine wave.
The experiment is carried out on a Bruker’s Dimension^Icon AFM. The sample is a highly oriented pyrolytic graphene (HOPG). We select TappingMode to scan the sample. During operation, the vertical
deflection signal of the tip is recorded by High-Speed Data Capture (HSDC) function. Fig. 4(a) shows the time history response of the tip vertical deflection when the amplitude setpoint ${A}_{sp}=$
0.889${A}_{0}$. The magenta dotted line represents the amplitude setpoint, and the orange dotted line represents the symmetrical position of the amplitude setpoint with respect to the equilibrium
position. FFT is performed on the time history response data of a certain point of the sample, as shown in Fig 4(b). The results show that the vertical deflection of the tip is a sine wave almost
symmetrical about the equilibrium position. The experimental and simulation results show that the steady-state response of the needle tip in intermittent contact with the sample is a sinusoidal form
symmetrical about the equilibrium position.
Fig. 4Experimental results of time history response and FFT of the tip vertical deflection when the amplitude setpoint Asp= 0.889A0
a) Time-history response of the tip vertical deflection
b) FFT of the steady-state response
2.3. Influence of background dissipation on phase contrast in TM-AFM
In TM-AFM, the phase image is performed by monitoring the phase shift between the tip response and excitation. The phase reflects the energy dissipation of the micro-cantilever system. During the
operation of TM-AFM, there are various energy dissipation, such as air viscous dissipation, squeeze film dissipation, liquid bridge dissipation, and so on [18].
Fig. 5Simulation results of phase and phase contrast of the two-component sample under different background dissipation
Such dissipation that cannot reflect the sample information is called background dissipation, and the background dissipation of the system can be expressed by the quality factor ${Q}_{b}$. When the
tip works in two areas with different surface properties, the phase shifts ${\phi }_{1}$ and ${\phi }_{2}$ of the two areas are different, and the difference between the two-phase shifts is the phase
As shown in Fig. 5, the sample is composed of two components with different properties. The viscosity ${\eta }_{1}$ of surface І is 800 Pa·s, and the viscosity ${\eta }_{2}$ of surface Ⅱ is 200 Pa·s.
The phase shifts of the two surfaces and the phase contrast results between the two surfaces under different background dissipation are obtained by simulation. It can be seen from the figure that the
phase contrast $\mathrm{\Delta }\phi$ between the two surfaces increases with the increase of the quality factor representing background dissipation. The results show that reducing the background
dissipation of the system can improve the phase contrast between the components of the sample.
3. Conclusions
In this work, the displacement response of the tip in tapping mode is obtained by simulation and experiment. The results show that when one side is constrained by the sample, the steady-state time
history response of the tip vertical displacement is still a sinusoidal form symmetrical about the equilibrium position, which is inconsistent with our perceptual understanding. In addition, the
phase response of the tip on two surfaces with different properties under different background dissipation is simulated. The results show that reducing the background dissipation can improve the
phase contrast between the components of the sample, which has hardly been mentioned by previous studies. The results of this study are of significance to understanding the response of intermittent
contact in micro-nanoscale and provide a reference for the imaging of TM-AFM.
• G. Binnig, C. F. Quate, and C. Gerber, “Atomic Force Microscope,” Physical Review Letters, Vol. 56, No. 9, pp. 930–933, Mar. 1986, https://doi.org/10.1103/physrevlett.56.930
• R. Garcia, “Nanomechanical mapping of soft materials with the atomic force microscope: methods, theory and applications,” Chemical Society Reviews, Vol. 49, No. 16, pp. 5850–5884, Aug. 2020,
• W. Xiang, Y. Tian, and X. Liu, “Dynamic analysis of tapping mode atomic force microscope (AFM) for critical dimension measurement,” Precision Engineering, Vol. 64, pp. 269–279, Jul. 2020, https:/
• Y. Gan, “Atomic and subnanometer resolution in ambient conditions by atomic force microscopy,” Elsevier BV, Surface Science Reports, Mar. 2009.
• Y. F. Dufrêne et al., “Imaging modes of atomic force microscopy for application in molecular and cell biology,” Nature Nanotechnology, Vol. 12, No. 4, pp. 295–307, Apr. 2017, https://doi.org/
• J. Melcher et al., “Origins of phase contrast in the atomic force microscope in liquids,” Proceedings of the National Academy of Sciences, Vol. 106, No. 33, pp. 13655–13660, Aug. 2009, https://
• D. Wang and T. P. Russell, “Advances in Atomic Force Microscopy for Probing Polymer Structure and Properties,” Macromolecules, Vol. 51, No. 1, pp. 3–24, Jan. 2018, https://doi.org/10.1021/
• R. García, “Dynamic atomic force microscopy methods,” Elsevier BV, Surface Science Reports, Sep. 2002.
• A. F. Payam, “Dynamic modeling and sensitivity analysis of dAFM in the transient and steady state motions,” Ultramicroscopy, Vol. 169, pp. 55–61, Oct. 2016, https://doi.org/10.1016/
• J. P. Cleveland, B. Anczykowski, A. E. Schmid, and V. B. Elings, “Energy dissipation in tapping-mode atomic force microscopy,” Applied Physics Letters, Vol. 72, No. 20, pp. 2613–2615, May 1998,
• M. Stark, C. Möller, D. J. Müller, and R. Guckenberger, “From Images to Interactions: High-Resolution Phase Imaging in Tapping-Mode Atomic Force Microscopy,” Biophysical Journal, Vol. 80, No. 6,
pp. 3009–3018, Jun. 2001, https://doi.org/10.1016/s0006-3495(01)76266-2
• A. Gil, J. Colchero, M. Luna, J. Gómez-Herrero, and A. M. Baró, “Adsorption of Water on Solid Surfaces Studied by Scanning Force Microscopy,” Langmuir, Vol. 16, No. 11, pp. 5086–5092, May 2000,
• J. Tamayo and R. Garcı́a, “Effects of elastic and inelastic interactions on phase contrast images in tapping-mode scanning force microscopy,” Applied Physics Letters, Vol. 71, No. 16, pp.
2394–2396, Oct. 1997, https://doi.org/10.1063/1.120039
• J. Tamayo, “Energy dissipation in tapping-mode scanning force microscopy with low quality factors,” Applied Physics Letters, Vol. 75, No. 22, pp. 3569–3571, Nov. 1999, https://doi.org/10.1063/
• B. Vasić, A. Matković, and R. Gajić, “Phase imaging and nanoscale energy dissipation of supported graphene using amplitude modulation atomic force microscopy,” Nanotechnology, Vol. 28, No. 46, p.
465708, Nov. 2017, https://doi.org/10.1088/1361-6528/aa8e3b
• B. V. Derjaguin, V. M. Muller, and Y. P. Toporov, “Effect of contact deformations on the adhesion of particles,” Journal of Colloid and Interface Science, Vol. 53, No. 2, pp. 314–326, Nov. 1975,
• M. Abbasi, “A simulation of atomic force microscope microcantilever in the tapping mode utilizing couple stress theory,” Micron, Vol. 107, pp. 20–27, Apr. 2018, https://doi.org/10.1016/
• M. Imboden and P. Mohanty, “Dissipation in nanoelectromechanical systems,” Elsevier BV, Physics Reports, Jan. 2014.
About this article
Dynamics and oscillations in electrical and electronics engineering
atomic force microscope
tapping mode
displacement response
phase response
Copyright © 2022 Anjie Peng, et al.
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Engineering Reference — EnergyPlus 9.3
DElight Daylighting Calculations[LINK]
The Daylighting:Controls input objects using the DElight Daylight Method provides an alternative daylighting model to the SplitFlux method. The DElight method of analyzing daylighting applies to both
simple apertures (i.e., windows and skylights) and complex fenestration systems that include geometrically complicated shading (e.g., roof monitors) and/or optically complicated glazings (e.g.,
prismatic or holographic glass). The DElight daylighting calculation methods are derived from the daylighting calculations in DOE-2.1E (as are the models accessed with Daylighting:Controls input
object using the SplitFlux Daylighting Method), and Superlite, with several key modifications. The engineering documentation included here focuses on the details of these differences from methods
documented elsewhere. For the details of the heritage calculations, refer to the section in this documentation entitled “Daylighting Calculations” and to [Winkelmann, 1983], [Winkelmann and
Selkowitz, 1985], and [Modest, 1982].
For each point in time, DElight calculates the interior daylighting illuminance at user specified reference points and then determines how much the electric lighting can be reduced while still
achieving a combined daylighting and electric lighting illuminance target. The daylight illuminance level in a zone depends on many factors, including exterior light sources; location, size, and
visible light transmittance of simple and complex fenestration systems; reflectance of interior surfaces; and location of reference points. The subsequent reduction of electric lighting depends on
daylight illuminance level, design illuminance setpoint, fraction of zone controlled by reference point, and type of lighting control.
The DElight daylighting calculation has three main steps:
1. Daylight Factor Calculation: Daylight factors, which are ratios of interior illuminance to exterior horizontal illuminance, are pre-calculated and stored for later use. The user spcifies the
coordinates of one or more reference points in each daylit zone. DElight first calculates the contribution of light transmitted through all simple and complex fenestration systems in the zone to the
illuminance at each reference point, and to the luminance at subdivided nodal patches of interior surfaces, for a given exterior luminous environment (including sky, sun, and exterior reflecting
surfaces). The effect of inter-reflection of this initial light between interior reflecting surfaces is then calculated, resulting in a final total illuminance at each reference point. This total
illuminace is then divided by the exterior horizontal illuminance for the given exterior environment to give a daylight factor. Daylight factors are calculated for each reference point, for a set of
sun positions and sky conditions that are representative of the building location.
2. Time-Step Interior Daylighting Calculation: A daylighting calculation is performed for each heat-balance time step when the sun is up. In this calculation the illuminance at the reference points
in each zone is found by interpolating the stored daylight factors using the current time step sun position and sky condition, then multiplying by the exterior horizontal illuminance.
3. Electric Lighting Control Calculation: The electric lighting control system is simulated to determine the proportion of lighting energy needed to make up the difference between the daylighting
illuminance level at the given time step, and the design illuminance level. Finally, the zone lighting electric reduction factor is passed to the thermal calculation, which uses this factor to reduce
the heat gain from lights.
DElight Daylight Factor Calculation Differences from SplitFlux Methods[LINK]
• Initial Interior Illuminance/Luminance Calculation: DElight calculates the total initial contribution of light transmitted through all simple fenestration systems (i.e., windows and skylights) in
the zone to the illuminance at each reference point, and to the luminance at each gridded nodal patch of interior surfaces. This differs from the models behind the “Daylighting:Controls” object
using the SplitFlux Daylighting Method (henceforth referred to as “SplitFlux”) in two ways. The first is that SplitFlux calculates initial illuminace values at reference points for each pair of
reference point and aperture (window/skylight) in the zone, whereas DElight calculates the total contribution from all apertures to each reference point. The second difference from SplitFlux is
that the initial luminance of interior surface nodal patches is calculated to support the inter-reflection calculation described below. This calculation uses the same formula as SplitFlux
modified for arbitrarily oriented surfaces (i.e., non-horizontal), and to calculate luminance rather than illuminance. Note however, DElight does not account for interior surface obstructions
(e.g., partitions) in this initial interior illuminance/luminance distribution. The SplitFlux method does account for interior surface obstruction of initial illuminance distribution on reference
• Reference Points: DElight allows up to 100 reference points to be arbitrarily positioned with a daylighting zone. At this time all reference points are assumed to be oriented on a horizontal
virtual surface “facing” toward the zenith and “seeing” the hemisphere above the horizontal plane.
• Complex Fenestration System Calculation: DElight calculates the contribution to the initial interior illuminance at each reference point, and to the luminance at each gridded nodal patch of
interior surfaces, of the light transmitted by complex fenestration systems (CFS). The analysis of a CFS within DElight is based on the characterization of the system using bi-directional
transmittance distribution functions (BTDF), which must be either pre-calculated (e.g., using ray-tracing techniques) or pre-measured, prior to analysis by DElight. A BTDF is a set of data for a
given CFS, which gives the ratios of incident to transmitted light for a range of incoming and outgoing directions. As illustrated in Figure 1, a BTDF can be thought of as collapsing a CFS to a
“black box” that is represented geometrically as a flat two-dimensional light-transmitting surface that will be treated as an aperture surface in the daylit zone description. For each incoming
direction across the exterior hemisphere of the CFS, varying portions of that light are transmitted at multiple outgoing directions across the interior hemisphere of the CFS. The two-dimensional
CFS “surface” and directional hemispheres are “abstract” in that they may not literally correspond to actual CFS component geometric details.
The pre-calculated or pre-measured BTDF for a CFS is independent of its final position and orientation within a building. Once a specific instance of a CFS aperture has been positioned within a
building, the incident light from all exterior sources across the CFS exterior hemisphere can be integrated over all incident directions for each relevant transmitted direction to determine the light
transmitted by the CFS surface in that direction. The light transmitted by the CFS aperture is then distributed to surfaces in the zone according to its non-uniform directionality. The algorithms for
this BTDF treatment of CFS in DElight are still under development, and are subject to change in the future.
• Inter-reflected Interior Illuminance/Luminance Calculation: The effect of inter-reflection of the initial interior illuminance/luminance between interior reflecting surfaces is calculated using a
radiosity method derived from Superlite [Modest, 1982]. This method subdivides each reflecting surface in the zone into nodal patches and uses view factors between all nodal patch pairs in an
iterative calculation of the total contribution of reflected light within the zone. This method replaces the SplitFlux method, resulting in a more accurate calculation of the varied distribution
of inter-reflected light throughout the zone. The ability to input up to 100 reference points supports a more complete assessment of this distribution. Also, the radiosity method explicitly
accounts for interior obstructions between pairs of nodal patches. The split-flux method used in the SplitFlux approach only implicitly accounts for interior surfaces by including their
reflectance and surface area in the zone average surface reflectance calculations.
DElight Time-Step Interior Daylighting Calculation Differences from SplitFlux Methods[LINK]
• Interior Illuminance Calculation: As discussed above, DElight only calculates daylight factors for the total contribution from all windows/skylights and CFS to each defined reference point. Thus
DElight does not support dynamic control of fenestration shading during the subsequent time-step calculations, as does SplitFlux method.
• Visual Quality: DElight does not currently calculate a measure of visual quality such as glare due to daylighting. DElight does calculate luminance on nodal patches of all interior, reflecting
surfaces. A variety of visual quality metrics could be calculated from these data in future implementations.
• Electric Lighting Control Calculation: Up to 100 reference points can be defined within a DElight daylighting zone. One or more of these reference points must be included in the control of the
electric lighting system in the zone. Each reference point input includes the fraction of the zone controlled by that point. Values of 0.0 are valid, which allows the definition of reference
points for which interior illuminance levels are calculated, but which do not control the electric lighting. Any non-zero fraction is thus the equivalent of a relative weighting given to that
reference point’s influence on the overall electric lighting control. The sum of all fractions for defined reference points must less than or equal to 1.0 for this relative weighting to make
physical sense. If the sum is less than 1.0, the remaining fraction is assumed to have no lighting control.
Modest, M. 1982. A General Model for the Calculation of Daylighting in Interior Spaces, Energy and Buildings 5, 66-79, and Lawrence Berkeley Laboratory report no. LBL-12599A.
Winkelmann, F.C. 1983. Daylighting Calculation in DOE-2. Lawrence Berkeley Laboratory report no. LBL-11353, January 1983.
Winkelmann, F.C. and S. Selkowitz. 1985. Daylighting Simulation in the DOE-2 Building Energy Analysis Program. Energy and Buildings 8, 271-286. | {"url":"https://bigladdersoftware.com/epx/docs/9-3/engineering-reference/delight-daylighting-calculations.html","timestamp":"2024-11-02T05:49:58Z","content_type":"text/html","content_length":"19278","record_id":"<urn:uuid:f2211a34-8b0d-461a-81f7-e2993e53825f>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00425.warc.gz"} |
How to convert ohms to amps (A)
How to convert ohms to amps
How to convert resistance in ohms (Ω) to electric current in amps (A).
You can calculate amps from ohms and volts or watts, but you can't convert ohms to amps since amp and ohm units represent different quantities.
Ohms to amps calculation with volts
The current I in amps (A) is equal to the voltage V in volts (V), divided by the resistance R in ohms (Ω):
I[(A)] = V[(V)] / R[(Ω)]
amp = volt / ohm
A = V / Ω
What is the current of an electrical circuit that has voltage supply of 12 volts and resistance of 40 ohms?
The current I is equal to 12 volts divided by 40 ohms:
I = 12V / 40Ω = 0.3A
Ohms to amps calculation with watts
The current I in amps (A) is equal to the square root of the power P in watts (W), divided by the resistance R in ohms (Ω):
I[(A)] = √P[(W)] / R[(Ω)]
amp = √ watt / ohm
A = √ W / Ω
What is the current of an electrical circuit that has power consumption of 30W and resistance of 120Ω?
The current I is equal to the square root of 30 watts divided by 120 ohms:
I = √ 30W / 120Ω = 0.5A | {"url":"https://www.calculatorx.com/convert/electric/ohm-to-amp.htm","timestamp":"2024-11-01T23:00:31Z","content_type":"text/html","content_length":"29413","record_id":"<urn:uuid:eeece6b9-d746-4b6e-aebf-0b13a9c39651>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00179.warc.gz"} |
Overview of the ALEKS Mathematics Test
The Assessment and Learning in Knowledge Spaces (ALEKS) is an artificial-intelligence-based assessment tool that increases the weaknesses and strengths of students’ math knowledge, reports its
findings to the student, and then a learning environment to enhance the student’s knowledge to the appropriate level for the placement. ALEKS, like most standardized tests, does not use
multiple-choice questions. Instead, it uses an adaptable, easy-to-use method that mimics paper and pencil techniques.
The ALEKS Mathematics Assessment guarantees that students are prepared for particular math courses at the University of Oregon. Because these courses are demanding, students must start the course,
which is likely to lead to their success. Students are not allowed to register in the math course unless they have demonstrated their readiness.
If a student is satisfied with his / her position after the initial assessment, he/she will be eligible to enroll in the appropriate course (s). If the student is not satisfied with their placement
after the initial assessment, they will be able to retake the assessment.
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What topics are covered on the ALEKS Mathematics Assessment?
ALEKS PPL is an online adaptation system that covers a wide range of math topics. The length of the placement assessment varies but can be up to 30 questions. Students may expect to see some, but not
all, of the math, they may have learned in high school. ALEKS is a situational assessment, not a preview of math courses at the University of Oregon. It is designed to identify if you are ready for a
particular course. After completing the initial placement assessment, students will have the opportunity to master additional topics to improve their placement.
Mathematics topics include:
• Real numbers (including integers, fractions, and percentages)
• Equations and inequalities (including linear equations, systems of linear equations, linear inequalities, and quadratic equations)
• Quadratic and linear functions (including graphs and functions, linear functions, and parabolas)
• Exponents and polynomials (including integer exponents, factoring, polynomial arithmetic, and polynomial equations)
• Rational expressions (involve rational equations and rational functions)
• Radical expressions (including rational exponents and higher roots)
• logarithms and exponentials (involve properties of logarithms, function compositions and inverse functions, and logarithmic equations)
• Geometry and trigonometry (including area, perimeter, and volume, coordinate geometry, trigonometric functions, and identities and equations)
Do you get a formula sheet on the ALEKS Math Assessment?
The ALEKS Math test provides math formulas for some questions so you can focus on the application instead of memorizing the formulas. However, this test does not provide a list of all the
mathematical formulas needed to know the test. It means that you should be able to recall many mathematical formulas in ALEKS.
Is the ALEKS Math Assessment hard?
ALEKS Math Placement Assessment is just determining the level of math at a university to have a better chance of success. ALEKS Assessment measures your knowledge of mathematics up to pre-calculus.
Therefore, it is difficult for many students to get a score above 75.
Can you use a calculator on the ALEKS Math Test?
You don’t allow to use a personal calculator on the ALEKS Mathematics Assessment. But for some questions, the calculator button is active and the test taker can use it.
How is the ALEKS Math Test scored?
ALEKS Mathematics Assessment score is between 1 and 100 and is interpreted as the correct percentage. A higher ALEKS score indicates that the examiner has mastered more mathematical concepts. The
scores of 30 or higher show adequate preparation for university-level mathematics.
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General Articles
Parent Category: Math League Website
Ratio and Proportion
Comparing ratios
Converting rates
Average rate of speed
A ratio is a comparison of two numbers. We generally separate the two numbers in the ratio with a colon (:). Suppose we want to write the ratio of 8 and 12.
We can write this as 8:12 or as a fraction 8/12, and we say the ratio is eight to twelve.
Jeannine has a bag with 3 videocassettes, 4 marbles, 7 books, and 1 orange.
1) What is the ratio of books to marbles?
Expressed as a fraction, with the numerator equal to the first quantity and the denominator equal to the second, the answer would be 7/4.
Two other ways of writing the ratio are 7 to 4, and 7:4.
2) What is the ratio of videocassettes to the total number of items in the bag?
There are 3 videocassettes, and 3 + 4 + 7 + 1 = 15 items total.
The answer can be expressed as 3/15, 3 to 15, or 3:15.
Comparing Ratios
To compare ratios, write them as fractions. The ratios are equal if they are equal when written as fractions.
Are the ratios 3 to 4 and 6:8 equal?
The ratios are equal if 3/4 = 6/8.
These are equal if their cross products are equal; that is, if 3 × 8 = 4 × 6. Since both of these products equal 24, the answer is yes, the ratios are equal.
Remember to be careful! Order matters!
A ratio of 1:7 is not the same as a ratio of 7:1.
Are the ratios 7:1 and 4:81 equal? No!
7/1 > 1, but 4/81 < 1, so the ratios can't be equal.
Are 7:14 and 36:72 equal?
Notice that 7/14 and 36/72 are both equal to 1/2, so the two ratios are equal.
A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal.
3/4 = 6/8 is an example of a proportion.
When one of the four numbers in a proportion is unknown, cross products may be used to find the unknown number. This is called solving the proportion. Question marks or letters are frequently used in
place of the unknown number.
Solve for n: 1/2 = n/4.
Using cross products we see that 2 × n = 1 × 4 =4, so 2 × n = 4. Dividing both sides by 2, n = 4 ÷ 2 so that n = 2.
A rate is a ratio that expresses how long it takes to do something, such as traveling a certain distance. To walk 3 kilometers in one hour is to walk at the rate of 3 km/h. The fraction expressing a
rate has units of distance in the numerator and units of time in the denominator.
Problems involving rates typically involve setting two ratios equal to each other and solving for an unknown quantity, that is, solving a proportion.
Juan runs 4 km in 30 minutes. At that rate, how far could he run in 45 minutes?
Give the unknown quantity the name n. In this case, n is the number of km Juan could run in 45 minutes at the given rate. We know that running 4 km in 30 minutes is the same as running n km in 45
minutes; that is, the rates are the same. So we have the proportion
4km/30min = n km/45min, or 4/30 = n/45.
Finding the cross products and setting them equal, we get 30 × n = 4 × 45, or 30 × n = 180. Dividing both sides by 30, we find that n = 180 ÷ 30 = 6 and the answer is 6 km.
Converting rates
We compare rates just as we compare ratios, by cross multiplying. When comparing rates, always check to see which units of measurement are being used. For instance, 3 kilometers per hour is very
different from 3 meters per hour!
3 kilometers/hour = 3 kilometers/hour × 1000 meters/1 kilometer = 3000 meters/hour
because 1 kilometer equals 1000 meters; we "cancel" the kilometers in converting to the units of meters.
One of the most useful tips in solving any math or science problem is to always write out the units when multiplying, dividing, or converting from one unit to another.
If Juan runs 4 km in 30 minutes, how many hours will it take him to run 1 km?
Be careful not to confuse the units of measurement. While Juan's rate of speed is given in terms of minutes, the question is posed in terms of hours. Only one of these units may be used in setting up
a proportion. To convert to hours, multiply
4 km/30 minutes × 60 minutes/1 hour = 8 km/1 hour
Now, let n be the number of hours it takes Juan to run 1 km. Then running 8 km in 1 hour is the same as running 1 km in n hours. Solving the proportion,
8 km/1 hour = 1 km/n hours, we have 8 × n = 1, so n = 1/8.
Average Rate of Speed
The average rate of speed for a trip is the total distance traveled divided by the total time of the trip.
A dog walks 8 km at 4 km per hour, then chases a rabbit for 2 km at 20 km per hour. What is the dog's average rate of speed for the distance he traveled?
The total distance traveled is 8 + 2 = 10 km.
Now we must figure the total time he was traveling.
For the first part of the trip, he walked for 8 ÷ 4 = 2 hours. He chased the rabbit for 2 ÷ 20 = 0.1 hour. The total time for the trip is 2 + 0.1 = 2.1 hours.
The average rate of speed for his trip is 10/2.1 = 100/21 kilometers per hour.
Parent Category: Math League Website
Space figures and basic solids
Space figures
Surface area
Space Figure
A space figure or three-dimensional figure is a figure that has depth in addition to width and height. Everyday objects such as a tennis ball, a box, a bicycle, and a redwood tree are all examples of
space figures. Some common simple space figures include cubes, spheres, cylinders, prisms, cones, and pyramids. A space figure having all flat faces is called a polyhedron. A cube and a pyramid are
both polyhedrons; a sphere, cylinder, and cone are not.
A cross-section of a space figure is the shape of a particular two-dimensional "slice" of a space figure.
The circle on the right is a cross-section of the cylinder on the left.
The triangle on the right is a cross-section of the cube on the left.
Volume is a measure of how much space a space figure takes up. Volume is used to measure a space figure just as area is used to measure a plane figure. The volume of a cube is the cube of the length
of one of its sides. The volume of a box is the product of its length, width, and height.
What is the volume of a cube with side-length 6 cm?
The volume of a cube is the cube of its side-length, which is 6^3 = 216 cubic cm.
What is the volume of a box whose length is 4cm, width is 5 cm, and height is 6 cm?
The volume of a box is the product of its length, width, and height, which is 4 × 5 × 6 = 120 cubic cm.
Surface Area
The surface area of a space figure is the total area of all the faces of the figure.
What is the surface area of a box whose length is 8, width is 3, and height is 4? This box has 6 faces: two rectangular faces are 8 by 4, two rectangular faces are 4 by 3, and two rectangular faces
are 8 by 3. Adding the areas of all these faces, we get the surface area of the box:
8 × 4 + 8 × 4 + 4 × 3 + 4 × 3 + 8 × 3 + 8 × 3 =
32 + 32 + 12 + 12 +24 + 24=
A cube is a three-dimensional figure having six matching square sides. If L is the length of one of its sides, the volume of the cube is L^3 = L × L × L. A cube has six square-shaped sides. The
surface area of a cube is six times the area of one of these sides.
The space figure pictured below is a cube. The grayed lines are edges hidden from view.
What is the volume and surface are of a cube having a side-length of 2.1 cm?
Its volume would be 2.1 × 2.1 × 2.1 = 9.261 cubic centimeters.
Its surface area would be 6 × 2.1 × 2.1 = 26.46 square centimeters.
A cylinder is a space figure having two congruent circular bases that are parallel. If L is the length of a cylinder, and r is the radius of one of the bases of a cylinder, then the volume of the
cylinder is L × pi × r^2, and the surface area is 2 × r × pi × L + 2 × pi × r^2.
The figure pictured below is a cylinder. The grayed lines are edges hidden from view.
A sphere is a space figure having all of its points the same distance from its center. The distance from the center to the surface of the sphere is called its radius. Any cross-section of a sphere is
a circle.
If r is the radius of a sphere, the volume V of the sphere is given by the formula V = 4/3 × pi ×r^3.
The surface area S of the sphere is given by the formula S = 4 × pi ×r^2.
The space figure pictured below is a sphere.
To the nearest tenth, what is the volume and surface area of a sphere having a radius of 4cm?
Using an estimate of 3.14 for pi,
the volume would be 4/3 × 3.14 × 4^3 = 4/3 × 3.14 × 4 × 4 × 4 = 268 cubic centimeters.
Using an estimate of 3.14 for pi, the surface area would be 4 × 3.14 × 4^2 = 4 × 3.14 × 4 × 4 = 201 square centimeters.
A cone is a space figure having a circular base and a single vertex.
If r is the radius of the circular base, and h is the height of the cone, then the volume of the cone is 1/3 × pi × r^2 × h.
What is the volume in cubic cm of a cone whose base has a radius of 3 cm, and whose height is 6 cm, to the nearest tenth?
We will use an estimate of 3.14 for pi.
The volume is 1/3 × pi × 3^2 × 6 = pi ×18 = 56.52, which equals 56.5 cubic cm when rounded to the nearest tenth.
The pictures below are two different views of a cone.
A pyramid is a space figure with a square base and 4 triangle-shaped sides.
The picture below is a pyramid. The grayed lines are edges hidden from view.
A tetrahedron is a 4-sided space figure. Each face of a tetrahedron is a triangle.
The picture below is a tetrahedron. The grayed lines are edges hidden from view.
A prism is a space figure with two congruent, parallel bases that are polygons.
The figure below is a pentagonal prism (the bases are pentagons). The grayed lines are edges hidden from view.
The figure below is a triangular prism (the bases are triangles). The grayed lines are edges hidden from view.
The figure below is a hexagonal prism (the bases are hexagons). The grayed lines are edges hidden from view..
Parent Category: Math League Website
Coordinates and similar figures
What is a coordinate?
Similar figures
Congruent figures
Symmetric figure
What is a Coordinate?
Coordinates are pairs of numbers that are used to determine points in a plane, relative to a special point called the origin. The origin has coordinates (0,0). We can think of the origin as the
center of the plane or the starting point for finding all other points. Any other point in the plane has a pair of coordinates (x,y). The x value or x-coordinate tells how far left or right the point
is from the point (0,0), just like on the number line (negative is left of the origin, positive is right of the origin). The y value or y-coordinate tells how far up or down the point is from the
point (0,0), (negative is down from the origin, positive is up from the origin). Using coordinates, we may give the location of any point we like by simply using a pair of numbers.
The origin below is where the x-axis and the y-axis meet. Point A has coordinates (2,3), since it is 2 units to the right and 3 units up from the origin. Point B has coordinates (3,0), since it is 3
units to the right, and lies on the x-axis. Point C has coordinates (6.3,9), since it is 6.3 units to the right, and 9 units up from the origin. Point D has coordinates (9,-2.5); it is 9 units to the
right, and 2.5 units down from the origin. Point E has coordinates (-4,-3); it is 4 units to the left, and 3 units down from the origin. Point F has coordinates (-7,5.5); it is 7 units to the left,
and 6 units up from the origin. Point G has coordinates (0,-7) since it lies on the y-axis 7 units below the origin.
Similar Figures
Figures that have the same shape are called similar figures. They may be different sizes or turned somewhat.
The following pairs of figures are similar.
These pairs of figures are not similar.
Congruent Figures
Two figures are congruent if they have the same shape and size.
The following pairs of figures below are congruent. Note that if two figures are congruent, they must be similar.
The pairs below are similar but not congruent.
The pairs below are not similar or congruent.
When a figure is turned, we call it a rotation of the figure. We can measure this rotation in terms of degrees; a 360 degree turn rotates a figure around once back to its original position.
For the following pairs of figures, the figure on the right is a rotation of the figure on the left.
If we flip (or mirror) along some line, we say the figure is a reflection along that line.
Reflections along a vertical line:
Reflections along a horizontal line:
Reflections along a diagonal line:
When we talk about folding a plane figure, we mean folding it as if it were a piece of paper in that shape. We might fold this into a solid figure such as a box, or fold the figure flat along itself.
Folding the figure on the left into a box:
Folding the figure on the left flat along the dotted line:
Symmetric Figure
A figure that can be folded flat along a line so that the two halves match perfectly is a symmetric figure; such a line is called a line of symmetry.
The triangle below is a symmetric figure. The dotted line is the line of symmetry.
The square below is a symmetric figure. It has four different lines of symmetry shown below.
The rectangle below is a symmetric figure. It has two different lines of symmetry shown below.
The regular pentagon below is a symmetric figure. It has five different lines of symmetry shown below.
The circle below is a symmetric figure. Any line that passes through its center is a line of symmetry!
The figures shown below are not symmetric.
Parent Category: Math League Website
Area and perimeter
Area of a square
Area of a rectangle
Area of a parallelogram
Area of a trapezoid
Area of a triangle
Area of a circle
Circumference of a circle
The area of a figure measures the size of the region enclosed by the figure. This is usually expressed in terms of some square unit. A few examples of the units used are square meters, square
centimeters, square inches, or square kilometers.
Area of a Square
If l is the side-length of a square, the area of the square is l^2 or l × l.
What is the area of a square having side-length 3.4?
The area is the square of the side-length, which is 3.4 × 3.4 = 11.56.
Area of a Rectangle
The area of a rectangle is the product of its width and length.
What is the area of a rectangle having a length of 6 and a width of 2.2?
The area is the product of these two side-lengths, which is 6 × 2.2 = 13.2.
Area of a Parallelogram
The area of a parallelogram is b × h, where b is the length of the base of the parallelogram, and h is the corresponding height. To picture this, consider the parallelogram below:
We can picture "cutting off" a triangle from one side and "pasting" it onto the other side to form a rectangle with side-lengths b and h. This rectangle has area b × h.
What is the area of a parallelogram having a base of 20 and a corresponding height of 7?
The area is the product of a base and its corresponding height, which is 20 × 7 = 140.
Area of a Trapezoid
If a and b are the lengths of the two parallel bases of a trapezoid, and h is its height, the area of the trapezoid is
1/2 × h × (a + b) .
To picture this, consider two identical trapezoids, and "turn" one around and "paste" it to the other along one side as pictured below:
The figure formed is a parallelogram having an area of h × (a + b), which is twice the area of one of the trapezoids.
What is the area of a trapezoid having bases 12 and 8 and a height of 5?
Using the formula for the area of a trapezoid, we see that the area is
1/2 × 5 × (12 + 8) = 1/2 × 5 × 20 = 1/2 × 100 = 50.
Area of a Triangle
Consider a triangle with base length b and height h.
The area of the triangle is 1/2 × b × h.
To picture this, we could take a second triangle identical to the first, then rotate it and "paste" it to the first triangle as pictured below:
The figure formed is a parallelogram with base length b and height h, and has area b × ×h.
This area is twice that of the triangle, so the triangle has area 1/2 × b × h.
What is the area of the triangle below having a base of length 5.2 and a height of 4.2?
The area of a triangle is half the product of its base and height, which is 1/2 ×5.2 × 4.2 = 2.6 × 4.2 = 10.92..
Area of a Circle
The area of a circle is Pi × r^2 or Pi × r × r, where r is the length of its radius. Pi is a number that is approximately 3.14159.
What is the area of a circle having a radius of 4.2 cm, to the nearest tenth of a square cm? Using an approximation of 3.14159 for Pi, and the fact that the area of a circle is Pi × r^2, the area of
this circle is Pi × 4.2^2 3.14159 × 4.2^2 =55.41…square cm, which is 55.4 square cm when rounded to the nearest tenth.
The perimeter of a polygon is the sum of the lengths of all its sides.
What is the perimeter of a rectangle having side-lengths of 3.4 cm and 8.2 cm? Since a rectangle has 4 sides, and the opposite sides of a rectangle have the same length, a rectangle has 2 sides of
length 3.4 cm, and 2 sides of length 8.2 cm. The sum of the lengths of all the sides of the rectangle is 3.4 + 3.4 + 8.2 + 8.2 = 23.2 cm.
What is the perimeter of a square having side-length 74 cm? Since a square has 4 sides of equal length, the perimeter of the square is 74 + 74 + 74 + 74 = 4 × 74 = 296.
What is the perimeter of a regular hexagon having side-length 2.5 m? A hexagon is a figure having 6 sides, and since this is a regular hexagon, each side has the same length, so the perimeter of the
hexagon is 2.5 + 2.5 + 2.5 + 2.5 + 2.5 + 2.5 = 6 × 2.5 = 15m.
What is the perimeter of a trapezoid having side-lengths 10 cm, 7 cm, 6 cm, and 7 cm? The perimeter is the sum 10 + 7 + 6 + 7 = 30cm.
Circumference of a Circle
The distance around a circle. It is equal to Pi (
What is the circumference of a circle having a diameter of 7.9 cm, to the nearest tenth of a cm? Using an approximation of 3.14159 for
Parent Category: Math League Website
Figures and polygons
Regular polygon
Equilateral triangle
Isosceles triangle
Scalene triangle
Acute triangle
Obtuse triangle
Right triangle
A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others.
The following are examples of polygons:
The figure below is not a polygon, since it is not a closed figure:
The figure below is not a polygon, since it is not made of line segments:
The figure below is not a polygon, since its sides do not intersect in exactly two places each:
Regular Polygon
A regular polygon is a polygon whose sides are all the same length, and whose angles are all the same. The sum of the angles of a polygon with n sides, where n is 3 or more, is 180° × (n - 2)
The following are examples of regular polygons:
The following are not examples of regular polygons:
1) The vertex of an angle is the point where the two rays that form the angle intersect.
2) The vertices of a polygon are the points where its sides intersect.
A three-sided polygon. The sum of the angles of a triangle is 180 degrees.
Equilateral Triangle or Equiangular Triangle
A triangle having all three sides of equal length. The angles of an equilateral triangle all measure 60 degrees.
Isosceles Triangle
A triangle having two sides of equal length.
Scalene Triangle
A triangle having three sides of different lengths.
Acute Triangle
A triangle having three acute angles.
Obtuse Triangle
A triangle having an obtuse angle. One of the angles of the triangle measures more than 90 degrees.
Right Triangle
A triangle having a right angle. One of the angles of the triangle measures 90 degrees. The side opposite the right angle is called the hypotenuse. The two sides that form the right angle are called
the legs. A right triangle has the special property that the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. This is known as the Pythagorean Theorem.
For the right triangle above, the lengths of the legs are A and B, and the hypotenuse has length C. Using the Pythagorean Theorem, we know that A^2 + B^2 = C^2.
In the right triangle above, the hypotenuse has length 5, and we see that 3^2 + 4^2 = 5^2 according to the Pythagorean Theorem.
A four-sided polygon. The sum of the angles of a quadrilateral is 360 degrees.
A four-sided polygon having all right angles. The sum of the angles of a rectangle is 360 degrees.
A four-sided polygon having equal-length sides meeting at right angles. The sum of the angles of a square is 360 degrees.
A four-sided polygon with two pairs of parallel sides. The sum of the angles of a parallelogram is 360 degrees.
A four-sided polygon having all four sides of equal length. The sum of the angles of a rhombus is 360 degrees.
A four-sided polygon having exactly one pair of parallel sides. The two sides that are parallel are called the bases of the trapezoid. The sum of the angles of a trapezoid is 360 degrees.
A five-sided polygon. The sum of the angles of a pentagon is 540 degrees.
A six-sided polygon. The sum of the angles of a hexagon is 720 degrees.
A regular hexagon: An irregular hexagon:
A seven-sided polygon. The sum of the angles of a heptagon is 900 degrees.
A regular heptagon: An irregular heptagon:
An eight-sided polygon. The sum of the angles of an octagon is 1080 degrees.
A regular octagon: An irregular octagon:
A nine-sided polygon. The sum of the angles of a nonagon is 1260 degrees.
A regular nonagon: An irregular nonagon:
A ten-sided polygon. The sum of the angles of a decagon is 1440 degrees.
A regular decagon: An irregular decagon:
A circle is the collection of points in a plane that are all the same distance from a fixed point. The fixed point is called the center. A line segment joining the center to any point on the circle
is called a radius.
The blue line is the radius r, and the collection of red points is the circle.
A figure is convex if every line segment drawn between any two points inside the figure lies entirely inside the figure. A figure that is not convex is called a concave figure.
The following figures are convex.
The following figures are concave. Note the red line segment drawn between two points inside the figure that also passes outside of the figure.
Parent Category: Math League Website
Angles and angle terms
What is an angle?
Degrees: measuring angles
Acute angles
Obtuse angles
Right angles
Complementary angles
Supplementary angles
Vertical angles
Alternate interior angles
Alternate exterior angles
Corresponding angles
Angle bisector
Perpendicular lines
What is an Angle?
Two rays that share the same endpoint form an angle. The point where the rays intersect is called the vertex of the angle. The two rays are called the sides of the angle.
Example: Here are some examples of angles.
We can specify an angle by using a point on each ray and the vertex. The angle below may be specified as angle ABC or as angle CBA; you may also see this written as ABC or as CBA. Note how the vertex
point is always given in the middle.
Example: Many different names exist for the same angle. For the angle below, PBC, PBW, CBP, and WBA are all names for the same angle.
Degrees: Measuring Angles
We measure the size of an angle using degrees.
Example: Here are some examples of angles and their degree measurements.
Acute Angles
An acute angle is an angle measuring between 0 and 90 degrees.
The following angles are all acute angles.
Obtuse Angles
An obtuse angle is an angle measuring between 90 and 180 degrees.
The following angles are all obtuse.
Right Angles
A right angle is an angle measuring 90 degrees. Two lines or line segments that meet at a right angle are said to be perpendicular. Note that any two right angles are supplementary angles (a right
angle is its own angle supplement).
The following angles are both right angles.
Complementary Angles
Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees. One of the complementary angles is said to be the complement of the other.
These two angles are complementary.
Note that these two angles can be "pasted" together to form a right angle!
Supplementary Angles
Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees. One of the supplementary angles is said to be the supplement of the other.
These two angles are supplementary.
Note that these two angles can be "pasted" together to form a straight line!
Vertical Angles
For any two lines that meet, such as in the diagram below, angle AEB and angle DEC are called vertical angles. Vertical angles have the same degree measurement. Angle BEC and angle AED are also
vertical angles.
Alternate Interior Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate interior angles. Alternate interior
angles have the same degree measurement. Angle B and angle C are also alternate interior angles.
Alternate Exterior Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate exterior angles. Alternate exterior
angles have the same degree measurement. Angle B and angle C are also alternate exterior angles.
Corresponding Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle C are called corresponding angles. Corresponding angles have
the same degree measurement. Angle B and angle D are also corresponding angles.
Angle Bisector
An angle bisector is a ray that divides an angle into two equal angles.
The blue ray on the right is the angle bisector of the angle on the left.
The red ray on the right is the angle bisector of the angle on the left.
Perpendicular Lines
Two lines that meet at a right angle are perpendicular.
Parent Category: Math League Website
Basic terms
Line segments
Parallel lines
A line is one of the basic terms in geometry. We may think of a line as a "straight" line that we might draw with a ruler on a piece of paper, except that in geometry, a line extends forever in both
directions. We write the name of a line passing through two different points A and B as "line AB" or as , the two-headed arrow over AB signifying a line passing through points A and B.
Example: The following is a diagram of two lines: line AB and line HG.
The arrows signify that the lines drawn extend indefinitely in each direction.
A point is one of the basic terms in geometry. We may think of a point as a "dot" on a piece of paper. We identify this point with a number or letter. A point has no length or width, it just
specifies an exact location.
Example: The following is a diagram of points A, B, C, and Q:
The term intersect is used when lines, rays, line segments or figures meet, that is, they share a common point. The point they share is called the point of intersection. We say that these figures
Example: In the diagram below, line AB and line GH intersect at point D:
Example: In the diagram below, line 1 intersects the square in points M and N:
Example: In the diagram below, line 2 intersects the circle at point P:
Line Segments
A line segment is one of the basic terms in geometry. We may think of a line segment as a "straight" line that we might draw with a ruler on a piece of paper. A line segment does not extend forever,
but has two distinct endpoints. We write the name of a line segment with endpoints A and B as "line segment AB" or as . Note how there are no arrow heads on the line over AB such as when we denote a
line or a ray.
Example: The following is a diagram of two line segments: line segment CD and line segment PN, or simply segment CD and segment PN.
A ray is one of the basic terms in geometry. We may think of a ray as a "straight" line that begins at a certain point and extends forever in one direction. The point where the ray begins is known as
its endpoint. We write the name of a ray with endpoint A and passing through a point B as "ray AB" or as . Note how the arrow heads denotes the direction the ray extends in: there is no arrow head
over the endpoint.
Example: The following is a diagram of two rays: ray HG and ray AB.
An endpoint is a point used to define a line segment or ray. A line segment has two endpoints; a ray has one.
Example: The endpoints of line segment DC below are points D and C, and the endpoint of ray MN is point M below:
Parallel Lines
Two lines in the same plane which never intersect are called parallel lines. We say that two line segments are parallel if the lines that they lie on are parallel. If line 1 is parallel to line 2, we
write this as
line 1 || line 2
When two line segments DC and AB lie on parallel lines, we write this as
segment DC || segment AB.
Example: Lines 1 and 2 below are parallel.
Example: The opposite sides of the rectangle below are parallel. The lines passing through them never meet.
Parent Category: Math League Website
Prime numbers
Greatest common factor
Least common multiple
What is a fraction?
Equivalent fractions
Comparing fractions
Converting and reducing fractions
Lowest terms
Improper fractions
Mixed numbers
Converting mixed numbers to improper fractions
Converting improper fractions to mixed numbers
Writing a fraction as a decimal
Rounding a fraction to the nearest hundredth
Adding and subtracting fractions
Adding and subtracting mixed numbers
Multiplying fractions and whole numbers
Multiplying fractions and fractions
Multiplying mixed numbers
Dividing fractions
Dividing mixed numbers
Simplifying complex fractions
Repeating decimals
Prime Numbers
A whole number greater than one that is divisible by only 1 and itself. The numbers 2, 3, 5, 37, and 101 are some examples of prime numbers.
Greatest Common Factor
The greatest common factor of two or more whole numbers is the largest whole number that divides each of the numbers.
There are two methods of finding the greatest common factor of two numbers.
Method 1: List all the factors of each number, then list the common factors and choose the largest one.
36: 1, 2, 3, 4, 6, 9, 12, 18, 36
54: 1, 2, 3, 6, 9, 18, 27, 54
The common factors are: 1, 2, 3, 6, 9, and 18.
The greatest common factor is: 18.
Method 2: List the prime factors, then multiply the common prime factors.
36 = 2 × 2 × 3 × 3
54 = 2 × 3 × 3 × 3
The common prime factors are 2, 3, and 3.
The greatest common factor is 2 × 3 × 3 = 18..
Least Common Multiple
The least common multiple of two or more nonzero whole numbers is the smallest whole number that is divisible by each of the numbers. There are two common methods for finding the least common
multiple of 2 numbers.
Method 1:
List the multiples of each number, and look for the smallest number that appears in each list.
Find the least common multiple of 12 and 42. We list the multiples of each number:
12: 12, 24, 36, 48, 60, 72, 84, ...
42: 42, 84, 126, 168, 190, ...
We see that the number 84 is the smallest number that appears in each list.
Method 2:
Factor each of the numbers into primes. For each different prime number in either of the factorizations, follow these steps:
1. Count the number of times it appears in each of the factorizations.
2. Take the largest of these two counts.
3. Write down that prime number as many times as the count in step 2.
To find the least common multiple take the product of all of the prime numbers written down in steps 1, 2, and 3.
Find the least common multiple of 24 and 90. First, we find the prime factorization of each number.
24 = 2 × 2 × 2 × 3
90 = 2 × 3 × 3 × 5
The prime numbers 2, 3, and 5 appear in the factorizations. We follow steps 1 through 3 for each of these primes.
The number 2 occurs 3 times in the first factorization and 1 time in the second, so we will use three 2's.
The number 3 occurs 1 time in the first factorization and 2 times in the second, so we will use two 3's.
The number 5 occurs 0 times in the first factorization and 1 time in the second factorization, so we will use one 5.
The least common multiple is the product of three 2's, two 3's, and one 5.
2 × 2 × 2 × 3 × 3 × 5 = 360
Find the least common multiple of 14 and 49. First, we find the prime factorization of each number.
14 = 2 × 7
49 = 7 × 7
The prime numbers 2 and 7 appear in the factorizations. We follow steps 1 through 3 for each of these primes.
The number 2 occurs 1 times in the first factorization and 0 times in the second, so we will use one 2.
The number 7 occurs 1 time in the first factorization and 2 times in the second, so we will use two 7's.
The least common multiple is the product of one 2 and two 7's.
2 × 7 × 7 = 98
Some other least common multiples are listed below.
The least common multiple of 12 and 9 is 36.
The least common multiple of 6 and 18 is 18.
The least common multiple of 2, 3, 4, and 5 is 60.
What is a Fraction?
A fraction is a number that expresses part of a group.
Fractions are written in the form a/b, where a and b are whole numbers, and the number b is not 0. For the purposes of these web pages, we will denote fractions using the notation a/b, though the
preferred notation is generally
The number a is called the numerator, and the number b is called the denominator.
The following numbers are all fractions
1/2, 3/7, 6/10, 4/99
The fraction 4/6 represents the shaded portion of the circle below. There are 6 pieces in the group, and 4 of them are shaded.
The fraction 3/8 represents the shaded portion of the circle below. There are 8 pieces in the group, and 3 of them are shaded.
The fraction 2/3 represents the shaded portion of the circle below. There are 3 pieces in the group, and 2 of them are shaded.
Equivalent Fractions
Equivalent fractions are different fractions which name the same amount.
The fractions 1/2, 2/4, 3/6, 100/200, and 521/1042 are all equivalent fractions.
The fractions 3/7, 6/14, and 24/56 are all equivalent fractions.
We can test if two fractions are equivalent by cross-multiplying their numerators and denominators. This is also called taking the cross-product.
Test if 3/7 and 18/42 are equivalent fractions.
The first cross-product is the product of the first numerator and the second denominator: 3 × 42 = 126.
The second cross-product is the product of the second numerator and the first denominator: 18 × 7 = 126.
Since the cross-products are the same, the fractions are equivalent.
Test if 2/4 and 13/20 are equivalent fractions.
The first cross-product is the product of the first numerator and the second denominator: 2 × 20 = 40.
The second cross-product is the product of the second numerator and the first denominator: 4 × 13 = 52.
Since the cross-products are different, the fractions are not equivalent. Since the second cross-product is larger than the first, the second fraction is larger than the first.
Comparing Fractions
1. To compare fractions with the same denominator, look at their numerators. The larger fraction is the one with the larger numerator.
2. To compare fractions with different denominators, take the cross product. The first cross-product is the product of the first numerator and the second denominator. The second cross-product is the
product of the second numerator and the first denominator. Compare the cross products using the following rules:
a. If the cross-products are equal, the fractions are equivalent.
b. If the first cross product is larger, the first fraction is larger.
c. If the second cross product is larger, the second fraction is larger.
Compare the fractions 3/7 and 1/2.
The first cross-product is the product of the first numerator and the second denominator: 3 × 2 = 6.
The second cross-product is the product of the second numerator and the first denominator: 7 × 1 = 7.
Since the second cross-product is larger, the second fraction is larger.
Compare the fractions 13/20 and 3/5.
The first cross-product is the product of the first numerator and the second denominator: 5 × 13 = 65.
The second cross-product is the product of the second numerator and the first denominator: 20 × 3 = 60.
Since the first cross-product is larger, the first fraction is larger.
Converting and Reducing Fractions
For any fraction, multiplying the numerator and denominator by the same nonzero number gives an equivalent fraction. We can convert one fraction to an equivalent fraction by using this method.
1/2 = (1 × 3)/(2 × 3) = 3/6
2/3 = (2 × 2)/(3 × 2) = 4/6
3/5 = (3 × 4)/(5 × 4) = 12/20
Another method of converting one fraction to an equivalent fraction is by dividing the numerator and denominator by a common factor of the numerator and denominator.
20/42 = (20 ÷ 2)/(42 ÷ 2) = 10/21
36/72 = (36 ÷ 3)/(72 ÷ 3) = 12/24
9/27 = (9 ÷ 3)/(27 ÷ 3) = 3/9
When we divide the numerator and denominator of a fraction by their greatest common factor, the resulting fraction is an equivalent fraction in lowest terms.
Lowest Terms
A fraction is in lowest terms when the greatest common factor of its numerator and denominator is 1. There are two methods of reducing a fraction to lowest terms.
Method 1:
Divide the numerator and denominator by their greatest common factor.
12/30 = (12 ÷ 6)/(30 ÷ 6) = 2/5
Method 2:
Divide the numerator and denominator by any common factor. Keep dividing until there are no more common factors.
12/30 = (12 ÷ 2)/(30 ÷ 2) = 6/15 = (6 ÷ 3)/(15 ÷ 3) = 2/5
Improper Fractions
Improper fractions have numerators that are larger than or equal to their denominators.
11/4, 5/5, and 13/2 are improper fractions.
Mixed Numbers
Mixed numbers have a whole number part and a fraction part.
a b/c.
Converting Mixed Numbers to Improper Fractions
To change a mixed number into an improper fraction, multiply the whole number by the denominator and add it to the numerator of the fractional part.
2 3/4 = ((2 × 4) + 3)/4 =11/4
6 1/2 = ((6 × 2) + 1)/2 = 13/2
Converting Improper Fractions to Mixed Numbers
To change an improper fraction into a mixed number, divide the numerator by the denominator. The remainder is the numerator of the fractional part.
11/4 = 11 ÷ 4 = 2 r3 = 2 3/4
13/2 = 13 ÷ 2 = 6 r1 = 6 1/2
Writing a Fraction as a Decimal
Method 1 - Convert to an equivalent fraction whose denominator is a power of 10, such as 10, 100, 1000, 10000, and so on, then write in decimal form.
1/4 = (1 × 25)/(4 × 25) = 25/100 = 0.25
3/20 = (3 × 5)/(20 × 5) = 15/100 = 0.15
9/8 = (9 × 125)/(8 × 125) = 1125/1000 = 1.125
Method 2 - Divide the numerator by the denominator. Round to the decimal place asked for, if necessary.
13/4 = 13 ÷ 4 = 3.25
Convert 3/7 to a decimal.
Round to the nearest thousandth.
We divide one decimal place past the place we need to round to, then round the result.
3/7 = 3 ÷ 7 = 0.4285…
which equals 0.429 when rounded to the nearest thousandth.
Convert 4/9 to a decimal.
Round to the nearest hundredth.
We divide one decimal place past the place we need to round to, then round the result.
4/9 = 4 ÷ 9 = 0.4444…
which equals 0.44 when rounded to the nearest hundredth.
Rounding a Fraction to the Nearest Hundredth
Divide to the thousandths place. If the last digit is less than 5, drop it. This is particularly useful for converting a fraction to a percent, if we want to convert to the nearest percent.
1/3 = 1 ÷ 3 = 0.333… which rounds to 0.33
If the last digit is 5 or greater, drop it and round up.
2/7 = 2 ÷ 7 = 0.285 which rounds to 0.29
Adding and Subtracting Fractions
If the fractions have the same denominator, their sum is the sum of the numerators over the denominator. If the fractions have the same denominator, their difference is the difference of the
numerators over the denominator. We do not add or subtract the denominators! Reduce if necessary.
3/8 + 2/8 = 5/8
9/2 - 5/2 = 4/2 = 2
If the fractions have different denominators:
1) First, find the least common denominator.
2) Then write equivalent fractions using this denominator.
3) Add or subtract the fractions. Reduce if necessary.
3/4 + 1/6 = ?
The least common denominator is 12.
3/4 + 1/6 = 9/12 + 2/12 = 11/12.
9/10 - 1/2 = ?
The least common denominator is 10.
9/10 - 1/2 = 9/10 - 5/10 = 4/10 = 2/5.
2/3 + 2/7 = ?
The least common denominator is 21
2/3 + 2/7 = 14/21 + 6/21 = 20/21.
Adding and Subtracting Mixed Numbers
To add or subtract mixed numbers, simply convert the mixed numbers into improper fractions, then add or subtract them as fractions.
9 1/2 + 5 3/4 = ?
Converting each number to an improper fraction, we have 9 1/2 = 19/2 and 5 3/4 = 23/4.
We want to calculate 19/2 + 23/4. The LCM of 2 and 4 is 4, so
19/2 + 23/4 = 38/4 + 23/4 = (38 + 23)/4 = 61/4.
Converting back to a mixed number, we have 61/4 = 15 1/4.
The strategy of converting numbers into fractions when adding or subtracting is often useful, even in situations where one of the numbers is whole or a fraction.
13 - 1 1/3 = ?
In this situation, we may regard 13 as a mixed number without a fractional part. To convert it into a fraction, we look at the denominator of the fraction 4/3, which is 1 1/3 expressed as an improper
fraction. The denominator is 3, and 13 = 39/3. So 13 - 1 1/3 = 39/3 - 4/3 = (39-4)/3 = 35/3, and 35/3 = 11 2/3.
5 1/8 - 2/3 = ?
This time, we may regard 2/3 as a mixed number with 0 as its whole part. Converting the first mixed number to an improper fraction, we have 5 1/8 = 41/8. The problem becomes
5 1/8 - 2/3 = 41/8 - 2/3 = 123/24 - 16/24 = (123 - 16)/24 = 107/24.
Converting back to a mixed number, we have 107/24 = 4 11/24.
92 + 4/5 = ?
This is easy. To express this as a mixed number, just put the whole number and the fraction side by side. The answer is 92 4/5.
Multiplying Fractions and Whole Numbers
To multiply a fraction by a whole number, write the whole number as an improper fraction with a denominator of 1, then multiply as fractions.
8 × 5/21 = ?
We can write the number 8 as 8/1. Now we multiply the fractions.
8 × 5/21 = 8/1 × 5/21 = (8 × 5)/(1 × 21) = 40/21
2/15 × 10 = ?
We can write the number 10 as 10/1. Now we multiply the fractions.
2/15 × 10 = 2/15 × 10/1 = (2 × 10)/(15 × 1) = 20/15 = 4/3
Multiplying Fractions and Fractions
When two fractions are multiplied, the result is a fraction with a numerator that is the product of the fractions' numerators and a denominator that is the product of the fractions' denominators.
4/7 × 5/11 = ?
The numerator will be the product of the numerators: 4 × 5, and the denominator will be the product of the denominators: 7 × 11.
The answer is (4 × 5)/(7 × 11) = 20/77.
Remember that like numbers in the numerator and denominator cancel out.
14/15 × 15/17 = ?
Since the 15's in the numerator and denominator cancel, the answer is
14/15 × 15/17 = 14/1 × 1/17 = (14 × 1)/(1 × 17) = 14/17
4/11 × 22/36 = ?
In the solution below, first we cancel the common factor of 11 in the top and bottom of the product, then we cancel the common factor of 4 in the top and bottom of the product.
4/11 × 22/36 = 4/1 × 2/36 = 1/1 × 2/9 = 2/9
Multiplying Mixed Numbers
To multiply mixed numbers, convert them to improper fractions and multiply.
4 1/5 × 2 2/3 = ?.
Converting to improper fractions, we get 4 1/5 = 21/5 and 2 2/3 = 8/3. So the answer is
4 1/5 × 2 2/3 = 21/5 × 8/3 = (21 × 8)/(5 × 3) = 168/15 = 11 3/15.
3/4 × 1 1/8 = 3/4 × 9/8 = 27/32.
3 × 7 3/4 = 3 × 31/4 = (3 × 31)/4 = 93/4 = 23 1/4.
The reciprocal of a fraction is obtained by switching its numerator and denominator. To find the reciprocal of a mixed number, first convert the mixed number to an improper fraction, then switch the
numerator and denominator of the improper fraction. Notice that when you multiply a fraction and its reciprocal, the product is always 1.
Find the reciprocal of 31/75. We switch the numerator and denominator to find the reciprocal: 75/31.
Find the reciprocal of 12 1/2. First, convert the mixed number to an improper fraction: 12 1/2 = 25/2. Next, we switch the numerator and denominator to find the reciprocal: 2/25.
Dividing Fractions
To divide a number by a fraction, multiply the number by the reciprocal of the fraction.
7 ÷ 1/5 = 7 × 5/1 = 7 × 5 = 35
1/5 ÷ 16 = 1/5 ÷ 16/1 = 1/5 × 1/16 = (1 × 1)/(5 × 16) = 1/80
3/5 ÷ 7/12 = 3/5 × 12/7 = (3 × 12)/(5 × 7) = 36/35 or 1 1/35
Dividing Mixed Numbers
To divide mixed numbers, you should always convert to improper fractions, then multiply the first number by the reciprocal of the second.
1 1/2 ÷ 3 1/8 = 3/2 ÷ 25/8 = 3/2 × 8/25 = (3 × 8)/(2 × 25) = 24/50
1 ÷ 3 3/5 = 1/1 ÷ 18/5 = 1/1 × 5/18 = (1 × 5)/(1 × 18) = 5/18
3 1/8 ÷ 2 = 25/8 ÷ 2/1 = 25/8 × 1/2 = (25 × 1)/(8 × 2) = 25/16 or 1 9/16.
Simplifying Complex Fractions
A complex fraction is a fraction whose numerator or denominator is also a fraction or mixed number.
Example of complex fractions:
otherwise written as (1/4)/(2/3), (3/7)/100, 11/(2/3), and (23 1/5)/(2/3).
To simplify complex fractions, change the complex fraction into a division problem: divide the numerator by the denominator.
The first of these examples becomes
(1/4)/(2/3) = 1/4 ÷ 2/3 = 1/4 × 3/2 = 3/8.
The second of these becomes
(3/7)/100 = 3/7 ÷ 100 = 3/7 × 1/100 = 3/700.
The third of these becomes
11/(2/3) = 11 ÷ 2/3 = 11 × 3/2 = 33/2 = 16 1/2.
The fourth of these becomes
(23 1/5)/(2/3) = 23 1/5 ÷ 2/3 = 116/5 ÷ 2/3 = 116/5 × 3/2 = 174/5 = 34 4/5.
Repeating Decimals
Every fraction can be written as a decimal.
For example, 1/3 is 1 divided by 3.
If you use a calculator to find 1 ÷ 3, the calculator returns 0.333333... This is called a repeating decimal. To represent the idea that the 3's repeat forever, one uses a horizontal bar (overstrike)
as shown below:
What is the repeating decimal for 1/7 ? Dividing 7 into 1, we get 0.142857142..., and we see the pattern begin to repeat with the second 1, so .
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Using Data and Statistics
Line Graphs
A line graph is a way to summarize how two pieces of information are related and how they vary depending on one another. The numbers along a side of the line graph are called the scale.
Example 1:
The graph above shows how John's weight varied from the beginning of 1991 to the beginning of 1995. The weight scale runs vertically, while the time scale is on the horizontal axis. Following the
gridlines up from the beginning of the years, we see that John's weight was 68 kg in 1991, 70 kg in 1992, 74 kg in 1993, 74 kg in 1994, and 73 kg in 1995. Examining the graph also tells us that
John's weight increased during 1991 and 1995, stayed the same during 1991, and fell during 1994.
Example 2:
This line graph shows the average value of a pickup truck versus the mileage on the truck. When the truck is new, it costs $14000. The more the truck is driven, the more its value falls according to
the curve above. Its value falls $2000 the first 20000 miles it is driven. When the mileage is 80000, the truck's value is about $4000.
Pie Charts
A pie chart is a circle graph divided into pieces, each displaying the size of some related piece of information. Pie charts are used to display the sizes of parts that make up some whole.
Example 1:
The pie chart below shows the ingredients used to make a sausage and mushroom pizza. The fraction of each ingredient by weight is shown in the pie chart below. We see that half of the pizza's weight
comes from the crust. The mushrooms make up the smallest amount of the pizza by weight, since the slice corresponding to the mushrooms is smallest. Note that the sum of the decimal sizes of each
slice is equal to 1 (the "whole" pizza").
Example 2:
The pie chart below shows the ingredients used to make a sausage and mushroom pizza weighing 1.6 kg. This is the same chart as above, except that the labels no longer tell the fraction of the pizza
made up by that ingredient, but the actual weight in kg of the ingredient used. The sum of the numbers shown now equals 1.6 kg, the weight of the pizza. The size of each slice is still the same, and
shows us the fraction of the pizza made up from that ingredient. To get the fraction of the pizza made up by any ingredient, divide the weight of the ingredient by the weight of the pizza. What
fraction of the pizza does the sausage make up? We divide 0.12 kg by 1.6 kg, to get 0.075. This is the same value as in the pie chart in the previous example.
Example 3:
The pie chart below shows the ingredients used to make a sausage and mushroom pizza. The fraction of each ingredient by weight shown in the pie chart below is now given as a percent. Again, we see
that half of the pizza's weight, 50%, comes from the crust. Note that the sum of the percent sizes of each slice is equal to 100%. Graphically, the same information is given, but the data labels are
different. Always be aware of how any chart or graph is labeled.
Example 4:
The pie chart below shows the fractions of dogs in a dog competition in seven different groups of dog breeds. We can see from the chart that 4 times as many dogs competed in the sporting group as in
the herding group. We can also see that the two most popular groups of dogs accounted for almost half of the dogs in the competition. Suppose 1000 dogs entered the competition in all. We could figure
the number of dogs in any group by multiplying the fraction of dogs in any group by 1000. In the toy group, for example, there were 0.12 × 1000 = 120 dogs in the competition.
Bar Graphs
Bar graphs consist of an axis and a series of labeled horizontal or vertical bars that show different values for each bar. The numbers along a side of the bar graph are called the scale.
Example 1:
The bar chart below shows the weight in kilograms of some fruit sold one day by a local market. We can see that 52 kg of apples were sold, 40 kg of oranges were sold, and 8 kg of star fruit were
Example 2:
A double bar graph is similar to a regular bar graph, but gives 2 pieces of information for each item on the vertical axis, rather than just 1. The bar chart below shows the weight in kilograms of
some fruit sold on two different days by a local market. This lets us compare the sales of each fruit over a 2 day period, not just the sales of one fruit compared to another. We can see that the
sales of star fruit and apples stayed most nearly the same. The sales of oranges increased from day 1 to day 2 by 10 kilograms. The same amount of apples and oranges was sold on the second day.
The mean of a list of numbers is also called the average. It is found by adding all the numbers in the list and dividing by the number of numbers in the list.
Find the mean of 3, 6, 11, and 8.
We add all the numbers, and divide by the number of numbers in the list, which is 4.
(3 + 6 + 11 + 8) ÷ 4 = 7
So the mean of these four numbers is 7.
Find the mean of 11, 11, 4, 10, 11, 7, and 8 to the nearest hundredth.
(11 + 11 + 4 + 10 + 11 + 7 + 8) ÷ 7 = 8.857…
which to the nearest hundredth rounds to 8.86.
The median of a list of numbers is found by ordering them from least to greatest. If the list has an odd number of numbers, the middle number in this ordering is the median. If there is an even
number of numbers, the median is the sum of the two middle numbers, divided by 2. Note that there are always as many numbers greater than or equal to the median in the list as there are less than or
equal to the median in the list.
The students in Bjorn's class have the following ages: 4, 29, 4, 3, 4, 11, 16, 14, 17, 3. Find the median of their ages. Placed in order, the ages are 3, 3, 4, 4, 4, 11, 14, 16, 17, 29. The number of
ages is 10, so the middle numbers are 4 and 11, which are the 5th and 6th entries on the ordered list. The median is the average of these two numbers:
(4 + 11)/2 = 15/2 = 7.5
The tallest 7 trees in a park have heights in meters of 41, 60, 47, 42, 44, 42, and 47. Find the median of their heights. Placed in order, the heights are 41, 42, 42, 44, 47, 47, 60. The number of
heights is 7, so the middle number is the 4th number. We see that the median is 44.
The mode in a list of numbers is the number that occurs most often, if there is one.
The students in Bjorn's class have the following ages: 5, 9, 1, 3, 4, 6, 6, 6, 7, 3. Find the mode of their ages. The most common number to appear on the list is 6, which appears three times. No
other number appears that many times. The mode of their ages is 6.
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Decimals, Whole Numbers, and Exponents
Decimal numbers
Whole number portion
Expanded form of a decimal number
Adding decimals
Subtracting decimals
Comparing decimal numbers
Rounding decimal numbers
Estimating sums and differences
Multiplying decimal numbers
Dividing whole numbers, with remainders
Dividing whole numbers, with decimal portions
Dividing decimals by whole numbers
Dividing decimals by decimals
Exponents (powers of 2, 3, 4, ...)
Factorial notation
Square roots
Decimal Numbers
Decimal numbers such as 3.762 are used in situations which call for more precision than whole numbers provide.
As with whole numbers, a digit in a decimal number has a value which depends on the place of the digit. The places to the left of the decimal point are ones, tens, hundreds, and so on, just as with
whole numbers. This table shows the decimal place value for various positions:
Note that adding extra zeros to the right of the last decimal digit does not change the value of the decimal number.
Place (underlined)
Name of Position
Ones (units) position
Ten thousandths
Hundred Thousandths
In the number 3.762, the 3 is in the ones place, the 7 is in the tenths place, the 6 is in the hundredths place, and the 2 is in the thousandths place.
The number 14.504 is equal to 14.50400, since adding extra zeros to the right of a decimal number does not change its value.
Whole Number Portion
The whole number portion of a decimal number are those digits to the left of the decimal place.
In the number 23.65, the whole number portion is 23.
In the number 0.024, the whole number portion is 0.
Expanded Form of a Decimal Number
The expanded form of a decimal number is the number written as the sum of its whole number and decimal place values.
3 + 0.7 + 0.06 + 0.002 is the expanded form of the number 3.762.
100 + 3 + 0.06 is the expanded form of the number 103.06.
Adding Decimals
To add decimals, line up the decimal points and then follow the rules for adding or subtracting whole numbers, placing the decimal point in the same column as above.
When one number has more decimal places than another, use 0's to give them the same number of decimal places.
76.69 + 51.37
1) Line up the decimal points:
2) Then add.
12.924 + 3.6
1) Line up the decimal points:
+ 3.600
2) Then add.
+ 3.600
Subtracting Decimals
To subtract decimals, line up the decimal points and then follow the rules for adding or subtracting whole numbers, placing the decimal point in the same column as above.
When one number has more decimal places than another, use 0's to give them the same number of decimal places.
18.2 - 6.008
1) Line up the decimal points.
- 6.008
2) Add extra 0's, using the fact that 18.2 = 18.200
- 6.008
3) Subtract.
- 6.008
Comparing Decimal Numbers
Symbols are used to show how the size of one number compares to another. These symbols are < (less than), > (greater than), and = (equals). To compare the size of decimal numbers, we compare the
whole number portions first. The larger decimal number is the one with the lager whole number portion. If the whole number parts are both equal, we compare the decimal portions of the numbers. The
leftmost decimal digit is the most significant digit. Compare the pairs of digits in each decimal place, starting with the most significant digit until you find a pair that is different. The number
with the larger digit is the larger number. Note that the number with the most digits is not necessarily the largest.
Compare 1 and 0.002. We begin by comparing the whole number parts: in this case 1>0, 0 being the whole number part of 0.002, and so 1>0.002.
Compare 0.402 and 0.412. The numbers 0.402 and 0.412 have the same number of digits, and their whole number parts are both 0. We compare the next most significant digit of each number, the digit in
the tenths place, 4 in each case. Since they are equal, we go on to the hundredths place, and in this case, 0<1, so 0.402<0.412.
Compare 120.65 and 34.999. Comparing the whole number parts, 120>34, so 120.65>34.999.
Compare 12.345 and 12.097. Since the whole number parts are both equal, we compare the decimal portions starting with the tenths digit. Since 3>0, we have 12.345>12.097.
Remember that adding extra zeros to the right of a decimal does not change its value:
2.4 = 2.40 = 2.400 = 2.4000.
Rounding Decimal Numbers
To round a number to any decimal place value, we want to find the number with zeros in all of the lower places that is closest in value to the original number. As with whole numbers, we look at the
digit to the right of the place we wish to round to. Note: When the digit 5, 6, 7, 8, or 9 appears in the ones place, round up; when the digit 0, 1, 2, 3, or 4 appears in the ones place, round down.
Rounding 1.19 to the nearest tenth gives 1.2 (1.20).
Rounding 1.545 to the nearest hundredth gives 1.55.
Rounding 0.1024 to the nearest thousandth gives 0.102.
Rounding 1.80 to the nearest one gives 2.
Rounding 150.090 to the nearest hundred gives 200.
Rounding 4499 to the nearest thousand gives 4000.
Estimating Sums and Differences
We can use rounding to get quick estimates on sums and differences of decimal numbers. First round each number to the place value you choose, then add or subtract the rounded numbers to estimate the
sum or difference.
To estimate the sum 119.36 + 0.56 to the nearest whole number, first round each number to the nearest one, giving us 119 + 1, then add to get 120.
Multiplying Decimal Numbers
Multiplying decimals is just like multiplying whole numbers. The only extra step is to decide how many digits to leave to the right of the decimal point. To do that, add the numbers of digits to the
right of the decimal point in both factors.
4.032 × 4
We can multiply 4032 by 4 to get 16128. There are three decimal places in 4.032, so place the decimal three digits from the right:
4.032 × 4 = 16.128
6.74 × 9.063
We can multiply 674 by 9063 to get 6108462. Then there are 5 decimal places: two in the number 6.74 and three in the number 9.063, so place the decimal five digits from the right:
6.74 × 9.063 = 61.08462.
Dividing Whole Numbers, with Remainders
1400 ÷ 7..
Since 14 ÷ 7 = 2, and 1400 is 100 times greater than 14, the answer is 2 × 100 = 200.
Many problems are similar to the above example, where the answer is easily obtained by adding on or taking off an appropriate number of 0's. Others are more complicated.
4934 ÷ 6. Use long division.
So the answer is 822 with a remainder of 2, written 822 R2.
To double-check that the answer is correct, multiply the quotient by the divisor and add the remainder:
(822 × 6) + 2 = 4932 + 2 = 4934.
Dividing Whole Numbers, with Decimal Portions
Find 32 ÷ 6 to the nearest whole number.
32 ÷ 6 = 5 r2. 6 is the divisor; 2 is the remainder.
2 is closer to 0 than 6, so round down. The answer is 5.
Dividing Decimals by Whole Numbers
To divide a decimal by a whole number, use long division, and just remember to line up the decimal points:
13.44 ÷ 12.
When rounding an answer, divide one place further than the place you're rounding to, and round the result. Add 0's to the right of the number being divided, if necessary.
1.0 ÷ 6. Round to the nearest thousandth.
To round 0.16666 . . . to the nearest thousandth, we take 4 places to the right of the decimal point and round to 3 places. Here, we round 0.1666 to 0.167, the answer.
Dividing Decimals by Decimals
To divide by a decimal, multiply that decimal by a power of 10 great enough to obtain a whole number. Multiply the dividend by that same power of 10. Then the problem becomes one involving division
by a whole number instead of division by a decimal.
0.144 ÷ 0.12
Multiplying the divisor (0.12) and the dividend (0.144) by 100, then dividing, gives the same result.
The answer is 1.2.
Be aware that some problems are less difficult and do not require this procedure.
6 ÷ 2.00
This is the same as 6 ÷ 2! The answer is 3.
Exponents (Powers of 2, 3, 4, ...)
Exponential notation is useful in situations where the same number is multiplied repeatedly. The notation is often shown as "^"
The number being multiplied is called the base, and the exponent tells how many times the base is multiplied by itself.
4 ×4 ×4 ×4 ×4 ×4 = 46
The base in this example is 4, the exponent is 6.
We refer to this as four to the sixth power, or four to the power of six, written as 4^6.
2 ×2 ×2 = 2^3 = 8
1.1"2 = 1.1 × 1.1 = 1.21
0.5^3 = 0.5 × 0.5 × 0.5 = 0.125
10^6 = 10 × 10 × 10 × 10 × 10 × 10 = 1000000
Observe that the base may be a decimal.
Special Cases:
A number with an exponent of two is referred to as the square of a number.
The square of a whole number is known as a perfect square. The numbers 1, 4, 9, 16, and 25 are all perfect squares.
A number with an exponent of three is referred to as the cube of a number.
The cube of a whole number is known as a perfect cube. The numbers 1, 8, 27, 64, and 125 are all perfect cubes.
A number written with an exponent of 1 is the same as the given number.
23^1 = 23.
Factorial Notation n!
The product of the first n whole numbers is written as n!, and is the product
1 × 2 × 3 × 4 × … × (n - 1) × n.
4! = 1 × 2 × 3 × 4 = 24
11! = 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10 × 11 = 39916800
When dividing factorials, note that many of the numbers cancel out!
The number 0! Is defined to be 1.
Square Roots
The square root of a whole number n is the number r with the property that r × r = n.
We write this as
We say that the number n is the square of the number r.
The square root of 9 is 3, since 3 × 3 = 9.
The square root of 289 is 17, since 17 × 17 = 289.
The square root of 2 is close to 1.41421. We say close to because the digits to the right of the decimal point in the square root of 2 continue forever, without any repeating pattern. Such a number
is called an irrational number, meaning that it cannot be written as a fraction.
Since the square root of a whole number n is the number r with the property that r × r = n, we always have
That is, the square of the square root of any number is just the original number.
We also have, for any number r that the square root of the square of r is the absolute value of r.
We say the absolute value, because the notation actually means the positive square root of n.
From the example above, we see that each positive number n actually has 2 numbers r that satisfy r × r = n, one is positive, and the other is negative. | {"url":"https://www.mathleague.com/index.php/31-mathleaguewebsite/general?start=187","timestamp":"2024-11-12T04:14:16Z","content_type":"application/xhtml+xml","content_length":"321960","record_id":"<urn:uuid:d79819ea-f5bb-4307-9d62-e21a07b95b3c>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00765.warc.gz"} |
Redefining Measurement: Making Sense of Square Inches and Square Feet in Real Life! - Age calculator
Redefining Measurement: Making Sense of Square Inches and Square Feet in Real Life!
Redefining Measurement: Making Sense of Square Inches and Square Feet in Real Life!
Measurement plays an important role in our lives. It is a means of quantifying and describing the physical world around us. Accurate measurement allows us to make informed decisions, whether in
construction, cooking, or other areas. However, measurements can sometimes be confusing, particularly when units of measurement are involved. For instance, square inches and square feet are two
commonly used units in measuring area, but many people find themselves confused about their meanings. In this article, we will dive deep into the world of square inches and square feet, and help
redefine their measurement in real life.
What are square inches and square feet?
Firstly, let’s define both square inches and square feet. Simply put, a square inch is a unit of area measurement that quantifies the area corresponding to a square having sides of one inch each. If
we multiply the length by width of an object that measures one square inch, the total area will be one square inch. On the other hand, a square foot is a unit of area quantifying the area
corresponding to a square that has sides of one foot each. It follows that a square foot is equal to 12 inches by 12 inches, or 144 square inches.
Applications of square inches and square feet in the real world
Now that we understand what square inches and square feet are, what are some of the ways we can apply these measurements in the real world? Here are some examples.
1. Room measurements
When measuring the area of a room, you can use square feet or square inches. For instance, you can easily calculate the area of a rectangular room by multiplying the length by width and using the
measurements in either square feet or square inches to get the total area.
2. Carpeting
When choosing a carpet to cover the floor of a room, it is essential to know the area that needs covering. That’s where square feet and square inches come in handy. You can measure the area of the
room, which will mostly give you square feet, and use it to determine the number of square feet of carpet required. Alternatively, you could measure the carpet’s dimensions in square inches and use
that measurement to calculate the cost of the carpet.
3. Construction
Square inches and square feet are also essential in construction, from laying concrete to building walls. For instance, when calculating the quantity of concrete needed to create a foundation for a
building, contractors use the measurements of length, width, and height. They then multiply them to obtain their square footage or in some cases, their square inches. This gives them an accurate
estimate of the amount of concrete necessary to complete the project.
FAQs about square inches and square feet
1. How do I calculate the square footage of a room?
To calculate the square footage of a room, measure the length and width of the room in feet (for square feet) or inches (for square inches), then multiply the two measurements to get the total area
in square feet or square inches.
2. How are square feet and square inches different?
Square feet and square inches are both units of area measurement. However, square feet measure a larger area than square inches. One square foot comprises 144 square inches, which means that one
square inch represents only a small portion of a square foot.
3. How do I convert square feet to square inches?
To convert square feet to square inches, multiply the area in square feet by 144. For instance, if a room measures 10 square feet, the area in square inches would be 10 x 144 = 1440 square inches.
4. How do I convert square inches to square feet?
To convert square inches to square feet, divide the area in square inches by 144. For example, if you have a carpet measuring 2,592 square inches, the area in square feet would be 2,592/ 144 = 18
square feet.
Understanding the concept of square feet and square inches is vital. It enables us to accurately calculate the quantity of materials required for various projects, accurately measure room dimensions,
quantities of carpet needed, among other applications. By taking the time to learn about these measurements, we can redefine their significance and applications in real life.
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The Quadratic Equation and its Numerical Roots
Volume 10, Issue 06 (June 2021)
The Quadratic Equation and its Numerical Roots
DOI : 10.17577/IJERTV10IS060100
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M. Sandoval-Hernandez, H. Vazquez-Leal, U. Filobello-Nino , Elisa De-Leo-Baquero, Alexis C. Bielma-Perez, J.C. Vichi-Mendoza , O. Alvarez-Gasca, A.D. Contreras-Hernandez, N. Bagatella-Flores , B.E.
Palma-Grayeb, J. Sanchez-Orea, L. Cuellar-Hernandez, 2021, The Quadratic Equation and its Numerical Roots, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY (IJERT) Volume 10, Issue 06 (June
• Open Access
• Authors : M. Sandoval-Hernandez, H. Vazquez-Leal, U. Filobello-Nino , Elisa De-Leo-Baquero, Alexis C. Bielma-Perez, J.C. Vichi-Mendoza , O. Alvarez-Gasca, A.D. Contreras-Hernandez, N.
Bagatella-Flores , B.E. Palma-Grayeb, J. Sanchez-Orea, L. Cuellar-Hernandez
• Paper ID : IJERTV10IS060100
• Volume & Issue : Volume 10, Issue 06 (June 2021)
• Published (First Online): 15-06-2021
• ISSN (Online) : 2278-0181
• Publisher Name : IJERT
• License: This work is licensed under a Creative Commons Attribution 4.0 International License
Text Only Version
The Quadratic Equation and its Numerical Roots
1. Sandoval-Hernandez1,2,3, H. Vazquez-Leal2,4, U. Filobello-Nino4, Elisa De-Leo-Baquero1, Alexis C. Bielma-Perez1, J.C. Vichi-Mendoza1, O. Alvarez-Gasca4, A.D. Contreras-Hernandez4,
2. Bagatella-Flores5, B.E. Palma-Grayeb4, J. Sanchez-Orea4, L. Cuellar-Hernandez4.
1 CBTis 190, Av. 15 Col. Venustiano Carranza 2da Secc, 94297, Boca del Rio Veracruz, México.
2 Consejo Veracruzano de Investigación CientÃfica y Desarrollo Tecnológico (COVEICYDET), Av. Rafael Murillo Vidal No. 1735, Cuauhtémoc, 91069 Xalapa, Veracruz, México.
3 Escuela de IngenierÃa, Universidad de Xalapa, Carretera Xalapa-Veracruz Km 2 No. 341, 91190 Xalapa, Veracruz, México.
4 Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Veracruz, México.
5 Facultad de FÃsica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Veracruz, México.
Abstract In this paper an alternative procedure is proposed to obtain the proof of the general formula to solve quadratic equations, which can be taught to high school and university students.
Likewise, the solution through power series is discussed, which can be used when there are values much less than 1 in the equation. In the same way, an analysis is presented on the significant
digits at the roots of the quadratic equation.
Keywords Quadratic equation; Exact roots; School mathematical speech; Significant digits.
1. INTRODUCTION
In current times, pedagogy plays a very important role in the teaching-learning processes at each of the educational levels since different learning strategies are used, such as the flipped
classroom [1], project-based learning, meaningful learning [2], grid technique [3], etc.,
In the case of textbooks, there is a trend that is continuously repeated when content is updated when presenting a new edition, because when a new edition is published as in [4-18] ,
favorable changes are made in the pedagogy with which the thematic concepts are presented, more demonstrative examples are included, more proposed exercises, the use of more than one ink in
their printing, the use of schemes and illustrations, etc. In other cases, the new editions incorporate the use of mathematical software such as Maple [19,20], Matlab [21,22], GeoGebra [23],
Excel [24] , among others. However, there are contents that remain intact, preserving the school discourse.
In mathematics teaching, school mathematical discourse (dME) is all the language that is introduced into a class. It is a system of reason that produces symbolic violence due to the
imposition of arguments, meanings and procedures [25,26]. It is everything that remains unchanged despite the innovations in relation to school mathematics, since this innovation does not
modify the concepts that are being taught, but the way in which it is being taught [27].
Many textbooks, such as those used in mathematics, are designed under the DME. It is frequent to blame mainly the teachers for the failure in the learning processes of mathematics, ignoring
that the books are influenced by the same school mathematical discourse [27].
The school speech is reflected in the textbooks used in the different educational and professional areas. For example, in control engineering texts [10] the way in which the transfer function
is determined does not change, comparing an updated edition against a previous one. Similarly, it occurs when another work is reviewed within the same discipline [11].
Something similar happens in other areas of knowledge. For example, in the field of administration, important concepts are taught, including various ratios and relationships that allow
knowing the solvency, liquidity and productivity of a business, known as financial ratios, such as the Net Present Value, Internal rate of return, Balance point, among others [28]. Project
evaluation should not consider the aforementioned coefficients in isolation, but should be analyzed comprehensively and together in order to provide validity, timeliness and objectivity to
the information that supports and bases decision-making in an investment project. In the science of management, the dME is present when the procedures to calculate these coefficients are
shown, since many books in this branch of knowledge approach these concepts practically the same. However, despite the fact that many books are conceived by the dME, in various works in the
different fields of study, it has sought to overcome the dME. For example, [29] an analysis was carried out for a bias circuit for two rectifier diodes in series, where an algebraic solution
was proposed in terms of the parameters of the circuit components. In this way, it is possible to calculate the bias point of the circuit without falling into the dME that occurs in the
classroom when teaching analog electronics, on the subject of rectifier diodes. In other words, it is customary to teach diode bias using the graphical method, where the diode curve is
plotted against the load line of the circuit. Obtaining the polarization point for this circuit can also be obtained using numerical algorithms such as Newton- Raphson's [30,31].
In [32] an approximation was presented for the error function and another for the normal cumulative function, which have many advantages in their implementation, such as reducing computation
times, but also seek to change an alternative in the teaching the use of the normal distribution to calculate a probability, and thus avoid the dME.
In this work, an alternative method is proposed to obtain the proof of the general formula to solve second degree algebraic equations and that can be taught to high school and engineering
students. This article presents the solution through power series, presented in [33] which is not detailed, since this text assumes that the reader has a good background in mathematics,
incurring the dME. In this way, it is sought to teach this solution method to students who are in the last semesters of high school and to university students who do not have a rigorous
training in the area of mathematics. Also in this work an analysis of
products such as the binomial squared and difference of squares are studied when it presents the characteristics of being a perfect square trinomial, or when the form is incomplete,
2 + = 0. However, the general formula to solve these
equations is very useful when we find coefficients that do not allow us to find their solution immediately.
In [4] the proof is presented by adding terms and completing squares to obtain the general formula to solve quadratic equations, given by
± 2 4
significant digits in the roots of the quadratic equation is presented when the general formula is used or when it is applied in a rationalized way. The aforementioned concepts are not
taught in conventional algebra courses, nor are they mentioned in algebra textbooks as they are influenced by the dME.
The purpose of this paper is aimed at awakening the interest in mathematics to students and teachers in the area of mathematics, and is organized as follows. Section 2 presents the importance
of the quadratic equation. In section 3 the alternative deduction for the general formula for solving quadratic equations is presented. Section 4 presents the solution using power series. In
section 5 two case studies are presented; the first is about the computational implications when the general formula is rationalized; The second case study presents a comparison of te
solution obtained through the general formula against the solution through the power series. In section 6 the conclusions of this work are presented.
2. THE IMPORTANCE OF THE QUADRATIC
3. OBTAINING IN AN ALTERNATIVE WAY THE GENERAL FORMULA
We begin by making = + , and substituting in equation (1), where is a complex number with real part given by , and imaginary part given by , simplifying we obtain
2 + 2 2 + + + = 0.
From (3) we get the real part and the imaginary part.
Solving for the imaginary part we have
= . 2
Substituting (4) in the real part we obtain
The quadratic equation
2 + + = 0,
2 + = 0.
represents a parabola and its solution is of high importance in physics and engineering. In physics it is used to solve
From (5) we can see that it is easy to solve for , solving
mathematical procedures that lead to its solution [34]. In the area of ordinary differential equations its usefulness is of high
= ±
importance since to give a solution to linear equations it is necessary to solve the characteristic equation, which in many cases presents a second degree algebraic equation [6,7]. In
Substituting (4) and (6) into = + , we have
engineering areas such as electronics, field effect transistor (FET) polarization analyzes require solving Shockley's second degree equation [35,36].
In electric circuit theory, the solution of the quadratic
= ± 2
equation is also important, in some analyzes as the frequency response using Bode diagrams [10], it is required to analyze the damping criterion in the response of an analog filter [37,38].
It is important to remember that the study of the quadratic equation begins in the introductory courses of algebra in secondary education in Mexico and deepens in the mathematics courses of
the high school, however different authors have made many proposals for the study of the second degree equation in order to make the teaching of the second degree equation and its geometric
behavior meaningful. For example, in [39] a didactic design was proposed to re-signify the parabola and the graphic models related to situations of a person's displacement.
In traditional textbooks used for teaching algebra [4], solution procedures are presented using classical algebraic methods to solve the quadratic equation. In addition, notable
Furthermore, taking into account that = 1, equation
(7) can be simplified, obtaining then the general formula given by (2). However, when 2 4, the numerical difference in the numerator can be very small. In these cases, it is useful to make
use of double precision variables [40] when formula (2) is implemented in a programming language. To minimize the effects of cancellation due to subtraction, equation (7) can be rewritten as
= .
± 2 4
4. SOLUTION BY POWER SERIES
The solution through power series is possible when there are perturbative parameters of , being a constant whose restriction is 1. Consider the quadratic equation given by
with solutions 1 = 5 and 2 = 0 when = 0.
2 + 1 2 = 0,
To see the behavior of the significant digits of the roots of
this equation, it is considered to give different values to , such that 1. Then, to determine the roots of (14) the formulas of (2) and (8) will be used. In addition, we will
where 1 is a real integer constant and 2 a real constant. To solve this equation, perturbative methods used [33] are used. Now consider
0 + 1 + 22 + ,
using the first three terms of the series and substituting in (9) we have
consider that the exact solutions are those that have been determined with the numerical differentiation NR. In [30, 31] algorithms and their implementation are presented using a programming
language such as C [40] or Fortran [41]. Table
1 presents the comparison of using equations (2) and (8) against NR for 1 5 1.
c values Roots with (2) Roots with (8) Roots using NR.
-5.192582405 -5.192582406 -5.192582404
0.1925824035 0.1925824035 0.1925824036
-5.019920635 -5.019920676 -5.019920634
0.01992063350 0.01992063367 0.01992063367
-5.001999200 -5.001999550 -5.001999201
0.00199920064 0.00199920064 0.01999200639
-5.000199990 -5.000200008 -5.000199992
0.00019999200 0.0001999920 0.00019999201
-5.000020000 -5.000000000 -5.000020000
0.00002000000 0.0000199999 0.00001999992
-5.000002000 -5.000000000 -5.000002000
0.00000200000 0.00000199999 0.00000199999
c values Roots with (2) Roots with (8) Roots using NR.
-5.192582405 -5.192582406 -5.192582404
0.1925824035 0.1925824035 0.1925824036
-5.019920635 -5.019920676 -5.019920634
0.01992063350 0.01992063367 0.01992063367
-5.001999200 -5.001999550 -5.001999201
0.00199920064 0.00199920064 0.01999200639
-5.000199990 -5.000200008 -5.000199992
0.00019999200 0.0001999920 0.00019999201
-5.000020000 -5.000000000 -5.000020000
0.00002000000 0.0000199999 0.00001999992
-5.000002000 -5.000000000 -5.000002000
0.00000200000 0.00000199999 0.00000199999
TABLE I. COMPARISON OF SIGNIFICANT DIGITS BETWEEN EQUATIONS (2) AND (8) AGAINST NR TABLE STYLES
4 2 + 3( + 2 ) + 2(
+ 2 +
1 ) + (10 + 201 + 02) = 2,
equating the coefficients of , we obtain, the system given by
2 = 2,
10 + 201 = 0,
11 + 202 + 12 = 0.
Solving the system of equations given by (12), and substituting the solutions of 0, 1 y 2 in (10), the solutions in power series are given by
The formulas for solving the quadratic equation (8) and (2) are algebraically the same expression. In other words, to obtain eq. (8), it has proceeded to rationalize the numerator in (2),
however, carrying out this process has a very important
1 = 2
2 + 8 ,
implication in numerical calculations, that the traditional
books, following the dME, do not warn the student. In Table 1
2 = 2 2 8 .
we can observe the behavior of the significant digits of both roots in equations (2) and (8) as the value of is changed. For example, if = 1 5, for 1 in eq. (2) the number of
1. STUDY CASES
In the first case study, we analyze the behavior of the significant digits for the when solving the quadratic equation using (2) against (8). The second case study discusses the
comparison of the solution calculated by the general formula against the solution obtained with perturbation in power series. For the development of the case studies presented in this
significant digits remains constant, whereas this does not occur
for eq. (8). In the case of 2 in eq. (8), the significant digits remain constant, but not for eq. (2).
In order to show the accuracy of the roots of the equations under discussion, we will use the formula (15) to obtain the significant digits [42, 43], given by
the mathematical software Maple 15 and its fsolve command have been used, which is an inter-build function that has a Newton-Raphson (NR) algorithm and that we will use as an exact
solution to validate the results presented in this article.
SD = log10
| |,
1. Significant digits using the general formula and its rationalization
We will consider the quadratic equation given by
2 + 5 = 0,
where SD is the number significant digits, is the exact value
computed with NR, are the roots calculated with equations
(2) and (8)
Figure 1 shows the graphs of significant digits of the data is shown in table 1. In eq. (2) the number of significant digits for 1 remains constant, in this case 11, while for 2 they
decrease as tends to 1, that is, 5 significant digits. On the
other hand, in the case of eq. (8) 2 is the root that keeps the significant digits constant, while 1 shows a reduction of these. From this figure an inverse behavior can be seen in
the roots of equations (2) and (8).
Fig. 1. Comparison of significant digits in the roots from equations (2) and (8).
From table 1 and Figure 1 it can be concluded that the advantage of using the rationalized formula (8) when we have the condition 2 4, is that the number of significant digits can be
maintained with good precision. To do this, in the
denominator, both and 2 4 must have the same sign. When this condition is not met, it is better to use formula (2).
2. Perturbative solution with power series.
Consider the constants 1 = 5, 2 = 1, and = 0.00, substituting into (9) we have
2 + 0.005 1 = 0.
Fig. 2. Comparison between the exact positive root and 1 with three terms in series expansion. For small values of the approximate solution is closer to exact solution.
2. CONCLUSIONS
In this article an alternative deduction has been presented using a complex expression for the general formula that allows solving quadratic equations. In the same way, the perturbative
solution was presented through power series.
Two case studies were also presented. In the first, the behavior of the significant digits of the roots of a quadratic equation was discussed. This behavior, following the dME in traditional
algebra books is not analyzed, nor is anything commented on about it. On the other hand, in [30] the effect of rationalization is commented, however, also following the dME, the reader is
presented with a detailed analysis of the advantages and disadvantages of rationalizing the general formula for solving these equations.
In this case, the exact solutions are calculated with the general formula and have the same significant digits as those calculated with (13), given by 1 = 0.997503125 and
2 = 1.0025031250.
In Figure 2 the solution for (9) has been plotted with 1 = 3 and 2 = 2 for 1. You can see that very small values of , the solution is very close to the exact value. For values of > 0.15, the
difference between the exact value and 1 begins to be made visible.
In the second case study, the solution was discussed through power series, which was obtained through perturbation methods when there is a small parameter such that
1. In the example presented, 10 significant digits were determined, which shows a high accuracy when using this method considering the first three terms of the expansion to approximate the
root. Likewise, Figure 1 illustrated that the accuracy decreases if we make it grow. Figure 1 suggests that to obtain an accuracy of approximately 95%, it is necessary to use a value < 0.01.
In this article we have presented some alternatives that can be used in algebra courses in teaching the quadratic equation. We believe that this work will provide an enriching experience,
both for students and teachers in the teaching-learning processes, with which the school mathematical discourse that has been present in traditional books and in classrooms can be overcome
from another perspective. This will allow proposing other teaching strategies for the benefit of students within the discipline of mathematics.
Authors would like to thank M. Eng. Roberto Ruiz Gomez for his contribution in this project.
1. Lucena, F. J. H., DÃaz, I. A., RodrÃguez, J. M. R., and MarÃn, J. A. M., Influencia del aula invertida en el rendimiento académico. Una revisión sistemática. Campus Virtuales, 8(1),
p. 9-18, 2019.
2. Arceo, F. D. B., Rojas, G. H., and González, E. L. G., Estrategias docentes para un aprendizaje significativo: una interpretación constructivista. McGraw-Hill Interamericana, 2010.
3. Charur, C. Z. Diseño de estrategias para el aprendizaje grupal una experiencia de trabajo. Perfiles Educativos, 1983.
4. Baldor, J. A., Algebra, Editorial Patria, 2011.
5. Barnett, R. Algebra con GeometrÃa AnalÃtica y TrigonometrÃa, México, Limusa-Noriega, 1994.
6. Edwards, C.H., and Penney, D.E., Ecuaciones diferenciales y problemas con valores en la frontera, Pearson Educación, 2011.
7. Zill, D. G., Ecuaciones diferenciales con aplicaciones, Editorial Iberoamerica, 1997
8. Simmons, G.F., Differential equations with applications and historical notes. CRC Press, 2016.
9. Wazwaz, A. M. Linear and nonlinear integral equations. Berlin, Springer, 2011.
10. Ogata, K., IngenierÃa de control moderna. Pearson Educación. 2003.
11. DAzzo, J. J., and Constantine, D.H., Linear control system analysis and design: conventional and modern, McGraw-Hill, Higher Education, 1995.
12. Kindle, J., GeometrÃa analÃtica, McGraw Hill-Serie Shaum, 1994.
13. Olvera, B. G., GeometrÃa analÃtica, Colección DGETI, 1991.
14. Olvera, B. G., GeometrÃa analÃtica, Prentice Hall, 2006.
15. Ayres, F., Mendelson, E., and Abellanas, L., Cálculo diferencial e integral. McGraw-Hill, 1994.
16. Purcell, E J., Rigdon, S. E., and Varbeg, D. E., Cálculo, Pearson Educación, 2007
17. Leithold, L., EC7 Cálculo, Oxford, 2012.
18. Stewart, J., Single variable calculus: Early trascendentals, Cengage Learning, 2015.
19. Fox, W. P., Mathematica modeling with Maple. Nelson Education, 2011.
20. Barnes, B., and Fulford, G. R., Mathematical modelling with case studies: using Maple y Matlab, Chapman and Hall CRC, 2014.
21. Nakamura S., Análisis numérico y visualización con MATLAB, Pearson educación, 1997.
22. Almenar, M. E., Isla, F., Gutiérrez, S., and Luege, M., Programa de elementos finitos de código abierto para la resolución de problemas mecánicos en estado plano. Asociación
Argentina de Mecánica Computacional, 36, p. 6-9, 2018.
23. Sánchez, J. A. M. Investigaciones en clase de matemáticas con GeoGebra, XIII jornades dEducació Matemà tica de la Comunittat Valenciana: Innovació i tecnologia en educació
matemà tica. Instituto de Ciencias de la Educación, 2019.
24. Torres-Remon, M, Aplicaciones con VBA con Excel, Alfaomega, 2016.
25. Soto, D., Gómez, K., Silva, H., and Cordro, F. Exclusión, cotidiano e identidad: una problemática fundamental del aprendizaje de la matemática. p. 1041-1048, 2012.
26. Soto, D., and Cantoral, R. Discurso matemático escolar y exclusión. Una visión socioepistemológica. Bolema: Boletim de Educação Matemática, 28(50), p. 1525-1544, 2014.
27. Uriza, R. C., Espinosa, G. M., and Gasperini, D. R. Análisis del discurso Matemático Escolar en los libros de texto, una mirada desde la TeorÃa Socioepistemológica. Avances de
Investigación en Educación Matemática, (8), p. 9-28, 2015.
28. Altuve, J. G., El uso del valor actual neto y la tasa interna de retorno para la valoración de las decisiones de inversión. Actualidad contable FACES, 7(9), p. 7-17, 2004.
29. Sandoval-Hernandez, M. A., Alvarez-Gasca, O., Contreras-Hernandez,
1. D., Pretelin-Canela, J. E., Palma-Grayeb, B. E., Jimenez-Fernandez,
5. M., … and Vazquez-Leal, H. Exploring the classic perturbation method for obtaining single and multiple solutions of nonlinear algebraic problems with application to microelectronic circuits.
International Journal of Engineering Research & Technology, 8(9), p. 636-645, 2019.
1. Burden, R. L. and Faires, J. D. Numerical Analysis, 8th edition, Cenange Learning, 2005.
2. Chapra, S. C., Canale, R., Numerical methods for engineers, 4th Mc Graw-Hill, 2004.
3. Sandoval-Hernandez, M. A., Vazquez-Leal, H., Filobello-Nino, U., and Hernandez-Martinez, L., New handy and accurate approximation for the Gaussian integrals with applications to science and
engineering. Open Mathematics, 17(1), p. 1774-1793, 2019.
4. Holmes, M. H. Introduction to perturbation methods, Springer Science & Business Media, 2012.
5. Resnick, R., Halliday, D., and Krane, K., FÃsica Vol. I., C.E.C.S.A., 2004.
6. Boylestad, R. L., and Nashelsky, L., Electrónica: teorÃa de circuitos y dispositivos electrónicos. Pearson Educación Prentice Hall, 2003.
7. Sedra, A., and Smith, K. C., Microelectronic circuits, New York: Oxford University Press, 1998.
8. Huelsman, L. P. Active and passive analog filter design: an introduction. McGraw-Hill, 1993.
9. Sadiku, M. N. O., Circuitos eléctricos, McGraw-Hill, 2011.
10. Cordero, F., and Suárez, L., Modelación en matemática educativa. Acta Latinoamericana de Matemática Educativa Vol.18, p. 639-644, 2005.
11. Joyanes-Aguilar, L., and Zahonero MartÃnez, I., Programación en C: metodologÃa, algoritmos, estructura de datos. MacGraw-Hill, 2001.
12. Markus, A., Modern Fortran in practice. Cambridge University Press, 2012.
13. Barry, D. A., Parlange, J. Y., Li, L., Prommer, H., Cunningham, C. J., and Stagnitti, F., Analytical approximations for real values of the Lambert W-function. Mathematics and Computers in
Simulation, 53(1- 2), p. 95-103, 2000.
14. Vazquez-Leal, H., Sandoval-Hernandez, M. A., Garcia-Gervacio, J. L. , Herrera-May, A. L. and Filobello-Nino, U. A., PSEM Aproximations for Both Branches of Lambert W with Applications. Discrete
Dynamics in Nature ans Society, Hindawi, 2019, 15 pages, 2019.
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CH implies the boolean completion of a type III projection lattice is the standard continuum-collapsing algebra » Bergsonian.org
CH implies the boolean completion of a type III projection lattice is the standard continuum-collapsing algebra
This is the proof that, if the continuum hypothesis holds, the structures from AQFT (algebraic quantum field theory) that we had hoped might yield a structure satisfying the Bergsonian axioms cannot,
in fact, do so. | {"url":"http://bergsonian.org/ch-implies-the-boolean-completion-of-a-type-iii-projection-lattice-is-the-standard-continuum-collapsing-algebra/","timestamp":"2024-11-05T00:07:40Z","content_type":"text/html","content_length":"27747","record_id":"<urn:uuid:fb11f837-c686-4fea-ab88-8b1dc516ae00>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00528.warc.gz"} |
I started trading in Exchange-Traded Funds (ETFs) a year and a bit ago. Honestly, it was easier than I thought it would be: I opened a trading account with a branch of my financial institution, put
some money into it, and away I went. For those of you wondering “but how?” That is my literal answer. Get a trading account (your bank can probably offer one), add money, and then buy and sell. After
making some money (and suffering an equal loss in a bad investment), I refined my investment strategy.
Tip: If you only have a few thousand dollars, deal in commission-free investments where available, and in the $5 – $30 bracket, in groups of 5 shares.
Lately, I was looking into a new possibility; taking out a portion of my Line of Credit, which is a relatively low-interest loan, and investing it into some ETFs. Two hours of my day was spent
working out the numbers and selecting ETFs suited to the task… But it was actually fun, for once!
So the premise is this; I can borrow a sum of money. I pay interest to the bank to borrow this money. The example amount I was using for the calculations was $5,000. Let’s say the annual interest on
this amount is 7%, or $350 — so I pay the bank $350 per year to borrow this $5,000. Then, I turn around and buy some dividend-paying shares with that cash. The goal is for the total dividends paid
out by the shares to exceed the amount I pay to the bank for the loan every year. So the dividends paid by these shares has to equal AT LEAST $350/year in total for this to be even worth considering.
Now, clearly, the number of shares I can buy with the $5,000 depends on the cost per share. The ones I’ve been looking at specifically are in the $6-$7 range. There were a few in the $12-$22 range
that, frankly, would lose me money with this strategy, so I won’t include them here. Breakdown of the main two I’m looking at below:
Share #1: $6.00/unit, pays out $0.04 every month, no matter what.
Share #2: $6.000/unit, pays out $0.048 every month, but I don’t know how stable its return is.
Share #1 I’ve held for quite a while, and I can tell you from experience that it pays that $0.04 like clockwork. The second share has a higher return and is actually a bit cheaper than Share #1, but
I’ve set them both to $6.00/unit. This scenario also assumes the interest rate stays at 7%. It could go higher – but it could also dip lower, which would decrease the interest paid and therefore
increase net profit.
$5,000 x 0.07 = $350
$5,000/(price of share) = (# of shares I can purchase)
{[(% yield)/100]/12} x (price of share) = (monthly payout of a share)
(# shares I can purchase) x (monthly payout) = (monthly dividend payment)
(monthly dividends) x 12 = (yearly dividend payout)
(yearly dividends) – $350 = (annual net profit or loss)
$5,000/$6 = I can purchase 833 shares. If they pay out $0.04 I make $33.32/month. If they pay out 0.048 I make $39.98/month. Annually that’s either $399.84 or $479.81. Minus the $350 interest on the
loan, that’d be $49.84 or $129.81 a year in passive income.
The cool part is – that’s only if I don’t pay any of the debt back, and that also assumes I don’t reinvest the dividends paid into MORE shares.
I’ll be the first to point out there are still risks here – if the ETF goes kaput, that’s a lot of money to lose. It’s also a relatively small amount of passive income, but hey – I’m starting to
think that any is better than none, and I’m sure this is the first of many nifty ideas I’ll come across moving forward. | {"url":"https://stemist.ca/author/sadmin/","timestamp":"2024-11-04T09:05:23Z","content_type":"text/html","content_length":"53040","record_id":"<urn:uuid:0fa95b74-ffb6-4525-8a77-6ce0c879852d>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00415.warc.gz"} |
Foundations for Structured Programming with GADTs
Makes me wish I understood category theory:
Foundations for Structured Programming with GADTs
neelk's LtU post
) by Patricia Johann and Neil Ghani @ POPL 2008.
GADTs are at the cutting edge of functional programming and become more widely used every day. Nevertheless, the semantic foundations underlying GADTs are not well understood. In this paper we
solve this problem by showing that the standard theory of datatypes as carriers of initial algebras of functors can be extended from algebraic and nested data types to GADTs. We then use this
observation to derive an initial algebra semantics for GADTs, thus ensuring that all of the accumulated knowledge about initial algebras can be brought to bear on them. Next, we use our initial
algebra semantics for GADTs to derive expressive and principled tools -- analogous to the well-known and widely-used ones for algebraic and nested data types -- for reasoning about, programming
with, and improving the performance of programs involving, GADTs; we christen such a collection of tools for a GADT an initial algebra package. Along the way, we give a constructive demonstration
that every GADT can be reduced to one which uses only the equality GADT and existential quantification. Although other such reductions exist in the literature, ours is entirely local, is
independent of any particular syntactic presentation of GADTs, and can be implemented in the host language, rather than existing solely as a metatheoretical artifact. The main technical ideas
underlying our approach are (i) to modify the notion of a higher-order functor so that GADTs can be seen as carriers of initial algebras of higher-order functors, and (ii) to use left Kan
extensions to trade arbitrary GADTs for simpler-but-equivalent ones for which initial algebra semantics can be derived.
Mentioned by Oleg in
GADTs in OCaml
No comments: | {"url":"http://axisofeval.blogspot.com/2010/11/foundations-for-structured-programming.html","timestamp":"2024-11-08T04:07:26Z","content_type":"text/html","content_length":"45703","record_id":"<urn:uuid:024ce605-a7ef-4c07-8df9-888cdc2d51ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00003.warc.gz"} |
Retrospective Power Analysis
Intraclass Correlation Sample Size Determination
Retrospective Power Analysis/1-Way Random Subject Effects Model
Assume you have already conducted an inter-rater reliability experiment following the one-way random subject effects design. Now, you want to know what statistical power you have been able to
AgreeStat360 can be used to determine the achieved statistical power for various values of the Minimum Detectable Difference (MDD). The input data needed to run AgreeStat360 is described in the
figure below. Essentially you need to provide the following:
• The standard significance level is 5% and needs not be modified. The baseline ICC value could be modified and represents the value the researcher aims at exceeding.
• The number of subjects and number of ratings per subject used in the experiment must be specified as well.
Analysis with AgreeStat/360
To see how AgreeStat360 processes this dataset to produce various agreement coefficients, please play the video below. This video can also be watched on youtube.com for more clarity if needed.
The output that AgreeStat360 produces is shown below.
• The power table on the right hand side shows how the power changes as the MDD changes given the parameters specified previously.
• You may also modify the MDD range values of interest and the power table will update accordingly. | {"url":"https://agreestat.com/examples/xlsx_icc_ssize1a_retrospect.html","timestamp":"2024-11-10T14:29:38Z","content_type":"application/xhtml+xml","content_length":"17013","record_id":"<urn:uuid:f0254aec-7da2-4bbb-bc69-d1127930390d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00461.warc.gz"} |
COIN Projects
25 - ELEPHANT: ExtragaLactic alErt Pipeline for Hostless AstroNomical Transients
ELEPHANT represents an effective strategy to filter extragalactic events within large and complex astronomical alert streams. There are many applications for which this pipeline will be useful,
ranging from transient selection for follow-up to studies of transient environments. We find that less than 2% of all analyzed transients are potentially hostless. Among them, approximately 10% have
a spectroscopic class reported on TNS, with Type Ia supernova being the most common class, followed by SLSN. Among the hostless candidates retrieved by our pipeline, there was SN 2018ibb, which has
been proposed to be a PISN candidate; and SN 2022ann, one of only five known SNe Icn. When no class is reported on TNS, the dominant classes are QSO and SN candidates, the former obtained from SIMBAD
and the latter inferred using the Fink ML classifier.
24 - Are classification metrics good proxies for SN Ia cosmological constraining power?
We emulate photometric SN Ia cosmology samples with controlled contamination rates of individual contaminant classes and evaluate each of them under a set of classification metrics. We then derive
cosmological parameter constraints from all samples under two common analysis approaches and quantify the impact of contamination by each contaminant class on the resulting cosmological parameter
estimates. We observe that cosmology metrics are sensitive to both the contamination rate and the class of the contaminating population, whereas the classification metrics are insensitive to the
latter. We therefore discourage exclusive reliance on classification-based metrics for cosmological analysis design decisions, e.g. classifier choice, and instead recommend optimizing using a metric
of cosmological parameter constraining power.
23 - A graph-based spectral classification of SN-II
This work presents new data-driven classification heuristics for spectral data based on graph theory. As a case in point, we devise a spectral classification scheme of Type II supernova (SNe II) as a
function of the phase relative to the V -band maximum light and the end of the plateau phase. Our classification method naturally identifies outliers and arranges the different SNe in terms of their
major spectral features. The automated classification naturally reflects the fast evolution of Type II SNe around the maximum light while showcasing their homogeneity close to the end of the plateau
phase. The scheme we develop could be more widely applicable to unsupervised time series classification or characterization of other functional data.
22 - Spectroscopic Confirmation of a population of Isolated, Intermediate-Mass YSOs
The Spitzer/IRAC Candidate YSO (SPICY) catalog is one of the largest compilations of such objects (~120,000 candidates in the Galactic midplane). Many SPICY candidates are spatially clustered, but,
perhaps surprisingly, approximately half the candidates appear spatially distributed. To better characterize this unexpected population and confirm its nature, we obtained Palomar/DBSP spectroscopy
for 26 of the optically-bright (G<15 mag) "isolated" YSO candidates. We confirm the YSO classifications of all 26 sources based on their positions on the Hertzsprung-Russell diagram, H and Ca II
line-emission from over half the sample, and robust detection of infrared excesses. This implies a contamination rate of <10% for SPICY stars that meet our optical selection criteria.
21 - A high pitch angle structure in the Sagittarius Arm
We map the 3D locations and velocities of star-forming regions in a segment of the Sagittarius Arm using young stellar objects (YSOs) from the Spitzer/IRAC Candidate YSO (SPICY) catalog to compare
their distribution to models of the arm. Distances and velocities for these objects are derived from Gaia EDR3 astrometry and molecular line surveys. We infer parallaxes and proper motions for
spatially clustered groups of YSOs and estimate their radial velocities from the velocities of spatially associated molecular clouds. The observed 56◦ pitch angle is remarkably high for a segment of
the Sagittarius Arm. We discuss possible interpretations of this feature as a substructure within the lower pitch angle Sagittarius Arm, as a spur, or as an isolated structure.
20 - Dawn of Stars
Dawn of Stars tells the story of how stars are formed. Most stars are born in groups which are truly stellar nurseries composed by clouds of dust and gas. This birth process is full of episodes, some
of which are represented musically in the four parts of this piece. This was the first art-related project to be developed by COIN, in March 2021, in collaboration with Rodrigo Roriz Teodoro, as part
of his graduation project to obtain the degree of Master in music composition awarded by the Marshall University (USA).
19 - SPICY: The Spitzer/IRAC Candidate YSO Catalog
We present ~120,000 candidate young stellar objects (YSOs) based on surveys of the Galactic midplane between l ∼ 255° and 110°, including the GLIMPSE I, II, and 3D, Vela-Carina, Cygnus X, and SMOG
surveys (613 square degrees), augmented by near-infrared catalogs. We employed a classification scheme that uses the flexibility of a tailored statistical learning method and curated YSO datasets to
take full advantage of IRAC’s spatial resolution and sensitivity in the mid-infrared ∼3–9 μm range. We also identify areas of IRAC color space associated with objects with strong silicate absorption
or polycyclic aromatic hydrocarbon emission. Spatial distributions and variability properties help corroborate the youthful nature of our sample..
18 - Active Learning with RESSPECT
The Recommendation System for Spectroscopic follow-up (RESSPECT) project aims to enable the construction of optimized training samples for the Rubin Observatory Legacy Survey of Space and Time
(LSST), taking into account a realistic description of the astronomical data environment. In this work, we test the robustness of active learning techniques in a realistic simulated astronomical data
scenario. Our experiment takes into account the evolution of training and pool samples, different costs per object, and two different sources of budget. Results show that traditional active learning
strategies significantly outperform random sampling.
17 - Periodic Astrometric Signal Recovery through Convolutional Autoencoders
Astrometric detection involves a precise measurement of stellar positions, and is widely regarded as the leading concept presently ready to find earth-mass planets in temperate orbits around nearby
sun-like stars. The TOLIMAN space telescope is a low-cost, agile mission concept dedicated to narrow-angle astrometric monitoring of bright binary stars.In this paper we demonstrate that a Deep
Convolutional Auto-Encoder is able to detected signals from simplified simulations of the TOLIMAN data and we present the full experimental pipeline to recreate out experiments from the simulations
to the signal analysis.
16 - Ridges in the Dark Energy Survey
Cosmic voids play an important role in our attempt to model the large-scale structure of the Universe. In this paper, we apply it to 2D weak-lensing mass density maps to identify curvilinear
filamentary structures. Our results demonstrate the viability of ridge estimation as a precursor for denoising weak lensing quantities to recover the large-scale structure, paving the way for a more
versatile and effective search for troughs.
15 - Photometry of high-redshift blended galaxies using deep learning
This work explores the use of deep neural networks to estimate the photometry of blended pairs of galaxies in monochrome space images, similar to the ones that will be delivered by the Euclid space
telescope. Using a clean sample of isolated galaxies from the CANDELS survey, we artificially blend them and train two different network models to recover the photometry of the two galaxies. We show
that our approach can recover the original photometry of the galaxies before being blended with ~7% accuracy without any human intervention and without any assumption on the galaxy shape.
14 - Dark energy equation of state imprint on supernova data
This work determines the degree to which a standard Lambda-CDM analysis based on type Ia supernovae can identify deviations from a cosmological constant in the form of a redshift-dependent dark
energy equation of state w(z). We demonstrate that a standard type Ia supernova cosmology analysis has limited sensitivity to extensive redshift dependencies of the dark energy equation of state. In
addition, we report that larger redshift-dependent departures from a cosmological constant do not necessarily manifest easier-detectable incompatibilities with the Lambda-CDM model.
13 - Hurdle and Generalized Additive Models
We show that the baryonic fraction and the rate of ionizing photons appear to have a larger impact on f[esc] than previously thought. A naive univariate analysis of the same problem would suggest
smaller effects for these properties and a much larger impact for the specific star formation rate, which is lessened after accounting for other galaxy properties and non-linearities in the
statistical model.
12 - Incompleteness of nearby cluster population
We report the discovery of 41 new stellar clusters. This represents an increment of at least 20% of the previously known OC population in this volume of the Milky Way. We also report on the clear
identification of NGC 886, an object previously considered an asterism. This letter challenges the previous claim of a near-complete sample of open clusters up to 1.8 kpc. Our results reveal that
this claim requires revision, and a complete census of nearby open clusters is yet to be found.
11 - Active Learning for Supernova Photometric Classification
Active Learning is a class of algorithms that aims to minimize labeling costs by identifying a few, carefully chosen, objects which have high potential in improving a given machine learning
classifier. In this project, we show how Active Learning can be used as a tool for optimizing the construction of spectroscopic samples for supernova photometric classification.
10 - Integrated Nested Laplace Approximation (INLA)
We introduce a novel technique to model IFS datasets, which treats the observed galaxy properties as manifestations of an unobserved Gaussian Markov random field. The method is computationally
efficient, resilient to the presence of low-signal-to-noise regions, and uses an alternative to Markov Chain Monte Carlo for fast Bayesian inference - the Integrated Nested Laplace Approximation. The
proposed Bayesian approach enables the creation of synthetic images, recovery of areas with bad pixels, and an increased power to detect structures in datasets subject to substantial noise and/or
sparsity of sampling.
9 - Gaussian Mixture Models
Here, we show how to use a Gaussian mixture model (GMM) to jointly analyse two traditional emission-line classification schemes of galaxy ionization sources: the Baldwin-Phillips-Terlevich (BPT) and
the WHAN diagrams, using spectroscopic data from the Sloan Digital Sky Survey Data Release 7 and SEAGal/STARLIGHT datasets. We apply a GMM to empirically define classes of galaxies in a
three-dimensional space spanned by the log [OIII]/H-beta, log [NII]/H-alpha, and log EW(H-alpha) optical parameters. We demonstrate the potential of our methodology to recover/unravel different
objects inside the wilderness of astronomical datasets, without lacking the ability to convey physically interpretable results; hence being a precious tool for maximum exploitation of the
ever-increasing astronomical surveys.
8 - Representativeness in Machine Learning applications for photometric redshifts
We present two galaxy catalogues built to enable a more demanding and realistic test of photo-z methods. We demonstrate the potential of these catalogues by submitting them to the scrutiny of
different photo-z methods, including machine learning (ML) and template fitting approaches. Our catalogues represent the first controlled environment allowing a straightforward implementation of such
7 - Hierarchical Bayesian Models
We developed a hierarchical Bayesian model to investigate how the presence of Seyfert activity relates to their environment. In elliptical galaxies, our analysis indicates a strong correlation of
Seyfert-AGN activity with the cluster centric distance, and a weaker correlation with the mass of the host. In spiral galaxies these trends do not appear, suggesting that the link between Seyfert
activity and the properties of spiral galaxies are independent of the environment.
6 - Dimensionality Reduction And Clustering for Unsupervised Learning in Astronomy (DRACULA)
DRACULA classifies objects using dimensionality reduction and clustering. The code has an easy interface and can be applied to separate several types of objects. It is based on tools developed in
scikit-learn, with Deep Learning usage requiring also the H2O package. We show how it can be used to identify sub-classes of type Ia supernovae.
5 - Analysis of Multi-dimensional Astronomical DAta sets (AMADA)
AMADA allows an iterative exploration and information retrieval of high-dimensional data sets. This is done by performing a hierarchical clustering analysis for different choices of correlation
matrices and by doing a principal components analysis in the original data. Additionally, AMADA provides a set of modern visualisation data-mining diagnostics. The user can switch between them using
the different tabs.
4 - Approximate Bayesian Computation (CosmoABC)
Approximate Bayesian Computation (ABC) enables parameter inference for complex physical systems in cases where the true likelihood function is unknown, unavailable, or computationally too expensive.
Here we present COSMOABC, a Python ABC sampler featuring a Population Monte Carlo variation of the original ABC algorithm, which uses an adaptive importance sampling scheme.
3 - GLM III: Negative Binomial Regression
We elucidate the potential of the class of GLMs which handles count data. The size of a galaxy's globular cluster (GC) population (NGC) is a prolonged puzzle in the astronomical literature. It falls
in the category of count data analysis, yet it is usually modelled as if it were a continuous response variable. We have developed a Bayesian negative binomial regression model to study the
connection between NGC and the host galaxy properties.
2 - GLM II: Gamma Regression
We present a gamma regression model as a fast alternative method for estimating the photometric redshifts of galaxies from their multi-wavelength photometry. Using the gamma family with a log link
function we predict redshifts from the PHoto-z Accuracy Testing simulated catalogue and a subset of the Sloan Digital Sky Survey from Data Release 10.
1 - GLM I: Binomial Regression
We elucidate the potential of the so-called logit and probit regression techniques, from both a maximum likelihood and a Bayesian perspective. As a case in point, we explore the conditions of star
formation activity and metal enrichment in primordial minihaloes from cosmological hydro-simulations including detailed chemistry, gas physics, and stellar feedback. | {"url":"https://cosmostatistics-initiative.org/projects/","timestamp":"2024-11-14T07:53:43Z","content_type":"text/html","content_length":"126667","record_id":"<urn:uuid:1a91cc3f-37fa-484a-86a2-49f42a84bd65>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00549.warc.gz"} |
The 13th Symposium on Boundary Layers and Turbulence
Alexandra Weiss, Swiss Federal Inst. of Technology, Zurich, Switzerland; and M. Rotach and M. Hennes
Optical scintillation measurements give the possibility to derive turbulence parameters of the atmospheric surface layer. On the one hand, turbulent fluxes of temperature and momentum can be
obtained, which play an important part in the dynamics and thermodynamics of the atmospheric surface layer, and which are needed for various meteorological applications. On the other hand, the
structure of the refractive index and temperature and its gradients can be derived from scintillation measurements, which are interesting for corrections of precise optical surveying techniques.
One main advantage of scintillometry is the averaging process of the micro-meteorological parameters over the propagation path, so that the data represents a larger source area than point
measurements (eddy-correlation technique). A disadvantage of scintillometry is the fact, that the derivation of turbulent statistics is based on Monin-Obukhov similarity theory and is hence, in
principle, restricted to horizontally homogenous surfaces. The advantage of the eddy-correlation technique is, that turbulent fluxes can be derived directly by the measurements.
In this contribution, comparisons are presented from scintillometry and eddy-correlation technique over a flat agricultural site in Switzerland. The set-up is chosen is such a way, that for the two
instruments either a homogenous (grass) source area is present or, alternatively, a inhomogeneous source area (grass and crop). It is shown, that the rate of agreement between the two methods is
dependent on the type of source area. Tor the inhomogeneous case the agreement between the two methods is generally poorer. It is presently investigated whether this is mainly due to the different
source areas of the two instruments or due to the violation of the homogeneity requirement of the scintillometry method. Beside this it is shown for the homogenous case that the neglection of the
temperature-humidity covariance is critical for certain atmospheric conditions | {"url":"https://ams.confex.com/ams/older/99annual/abstracts/1811.htm","timestamp":"2024-11-10T17:59:59Z","content_type":"text/html","content_length":"3365","record_id":"<urn:uuid:72ff3393-b290-42f4-badd-681c6c7fe0e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00771.warc.gz"} |
the find midpoint between the following points. express your answer as a coordinate with fractions in lowest terms: (-4,3) and (5,-4) | Question AI
The find midpoint between the following points. Express your answer as a coordinate with fractions in lowest terms: (-4,3) and (5,-4) Hint: Don't forget to reduce any fractions to lowest terms.
Not your question?Search it
Benjamin O'MalleyMaster · Tutor for 5 years
The formula for finding the midpoint between two points (x1,y1) and (x2,y2) on a straight line is given by ((x1+x2)/2,(y1+y2)/2). In this case, the two points are (-4,3) and (5,-4). So, we substitute
these values into the formula to get:<br /><br />Midpoint = ((-4+5)/2, (3-4)/2) = (1/2, -1/2)<br /><br />So, the midpoint between the points (-4,3) and (5,-4) is (1/2, -1/2).
Step-by-step video
The find midpoint between the following points. Express your answer as a coordinate with fractions in lowest terms: (-4,3) and (5,-4) Hint: Don't forget to reduce any fractions to lowest terms.
All Subjects Homework Helper | {"url":"https://www.questionai.com/questions-tTiV2xsZ3X/find-midpoint-following-points-express-answer-coordinate","timestamp":"2024-11-09T12:53:26Z","content_type":"text/html","content_length":"76652","record_id":"<urn:uuid:eb2467ae-0f66-49f1-9077-9eb90f8641d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00566.warc.gz"} |
160 000 Divided By 12 (2024)
• 20 mrt 2011 · 2/3 x 160000 = 106666.6 recurring (that is, 106666.6666...) Related questions. What is 272000000 divided by 1700? 160000 ...
• 13.3333
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Review of S. Donadi and S. Hossenfelder, Toy model for local and deterministic wave-function collapse PHYSICAL REVIEW A 106, 022212 (2022) - The Quantum Measurement Problem
Review of S. Donadi and S. Hossenfelder, Toy model for local and deterministic wave-function collapse PHYSICAL REVIEW A 106, 022212 (2022)
The authors present a model for wave-function collapse based on the system-measurement device evolving as a closed system via non-unitary evolution of the Lindblad form depending on the value of
local hidden variables. The hidden variables contain information of both measurement settings in a Bell experiment. Hence the model violates statistical independence of the hidden variables and the
measurement settings which was an assumption of Bell. The authors use this model to develop a local deterministic toy model that makes the same predictions as the collapse postulate. The model is
based around a Lindblad equation, but with a variation. Normally the Lindblad equation has a number of matrices
The specific matrices utilized in the Lindblad equation converge to either
1. The model that the authors have put forward based on locality, determinism, and hidden variables seems somewhat similar to a model based on non-locality and Born’s non-determinism. That is,
consider the following model. Two entangled qubit systems head in opposite directions toward two detectors. Each local detector-system density unitarily interacts with an environment that will be
represented by interaction with an ancilla. It can be seen that according to Preskill [1, p. 37] the same form of Lindblad equation results for the system-detector but all the terms are
necessarily present in the summation as no measurement has yet been made. Now suppose after a short interaction time
2. Note that if an environment unitarily interacts with a detector-system as above, one might also expect that the entanglement between the two distant detector-systems decreases. Prediction of the
sudden-death of entanglement was made in [2] and a review appears in [3]. The use of a unitary bath coupling followed by projection of the bath as in 1. above might serve as a platform for
further study as to if and when the detectors of the two systems eventually do not show the correct or expected Bell correlation due to entanglement. One might consider employing a second purely
unitary interaction between detector-system with a different coupling than with the first coupling which is non-unitary. By varying the two couplings and the time of interaction
1. The authors understand correctly from a separate paper [4] that unitary evolution will not suffice to explain the results of measurement and in this paper, they propose a non-unitary solution.
2. The authors propose that the detector-system evolves via a modified Lindblad equation which is positive and linear in density operator.
3. As the authors note, this has the advantage of obeying causality constraints of no-signaling.
1. The method proposed in this experiment does not appear to make much if any progress on the main issue regarding the measurement problem that these authors proposed in their paper [4] and we also
similarly proposed in our book [5]. That is, determining the specific conditions for which measurement or non-unitary evolution occurs. Note that one would be wrong to blindly apply non-linear
evolution to an electron evolving in free-space. On the other hand, it is expected that there are conditions for which non-unitary evolution exists. Determining these conditions is the primary
requirement for a solution to the quantum measurement problem. More work is needed along these lines.
2. The authors claim that:
The model further violates energy conservation. Again, this is because it stands in for an effective description that, among other things, does not take into account the recoil (and resulting
entanglement with) parts of the experimental equipment.
That is not obvious to us. Recoil has specific requirements and one cannot utilize an arbitrary recoil to balance energy. Consider a measuring device that has some 1) initial velocity, 2) initial
mass, 3) final mass, 4) final velocity, and as well as the system to be measured which consists of a photon that has an 5) energy and is absorbed by the device upon measurement. It can be shown by
conservation of energy and momentum arguments, that any three of these variables determines the remaining two. Neither five nor four of these variables can be set independently. So, there is clearly
a constraint when applying recoil arguments and it appears to us that one needs to demonstrate with their particular Lindblad model that recoil will actually work to conserve energy and momentum.
That is, one cannot simply state this and expect it will always work for arbitrary parameters and Lindblad operators. And if one subsequently performs an experiment with known values of initial and
final detector velocities and masses that measure a photon of a known energy, then if one is able to estimate the Lindblad operators for a particular experiment that would fit this model assuming
determinism and locality, one should be able to determine if recoil is sufficient to account for energy and momentum conservation.
The authors appear to be of the opinion that not meeting the statistical independence requirement of Bell is not really much of a problem. We disagree. However, our disagreement is not at this time
something that can be strictly ruled out at this time, so we agree that these authors should be given a chance to propose their theory and have it tested if possible.
Now, the experimenter’s ability to change a setting independent of the past is an assumption based on the free-will of the experimenter. Interestingly the discussion of free-will doesn’t appear in
the journal paper, which is odd because it is clearly central to the entire issue. The second author has claimed that free-will doesn’t exist [6] :
Now you all know that I think free will is logically incoherent nonsense.
And further claims:
Spooky action at a distance doesn’t make any difference for free will because the indeterministic processes in quantum mechanics are not influenced by anything, so they are not influenced by your
“free will,” whatever that may be.
Emotions aside on this issue, a purely logical error is being made by applying an inductive argument to reach a conclusion to a problem which we are fairly certain necessitates a deductive approach:
the conclusion that free-will doesn’t exist. Hossenfelder believes that the conclusion follows because the current theory based on either the projection postulate or Schrödinger’s equation appears
insufficient to describe free-will. Such a conclusion is based on the current theory and by definition is therefore made on the basis of inductive reasoning. Yet the authors in a separate paper [4]
have already concluded that the current theory is likely incomplete, which we agree with. Hence, the non-existence of free-will cannot logically be concluded from a deductive argument at present,
because it is agreed that the current theory is incomplete. Good try, but Sorry Charlie.
The authors noted in their introduction that:
A century ago one might have hoped that quantum mechanics would one day be replaced by a theory compatible with the deterministic and local prequantum ontology.
They further note that nearly all current theories that account for measurement are fundamentally nondeterministic. So, the authors appear to desire to bring back determinism and locality. If there
were a horse race between Secretariat and Donald Trump’s entry, MDGA, “Make Determinism Great Again”, we expect these authors would be betting heavily on MDGA. Nothing wrong with that, in our view in
the deductive process all solutions are on the table, unless they have already led to contradictions, and we know of no experiment that currently contradicts the authors’ model.
So although we have leveled some criticism regarding this paper, we do feel it is sufficiently novel to meet the published requirements for publication and believe it is a positive development to see
it appear in Physical Review A. Additionally, this paper ought to stimulate further work that could potentially be used to predict and discriminate different experimental signatures depending on the
existence and type of non-unitary evolution and by using a controlled interaction to emulate the effects of decoherence. We already have indicated several possibilities for additional studies in our
Remarks 1. and 2. above.
[1] J. Preskill, “Preskill’s Notes,” http://theory.caltech.edu/~preskill/ph219/chap3_15.pdf.
[2] A. K. Rajagopal and R. W. Rendell, “Decoherence, correlation, and entanglement in a pair of coupled quantum dissipative oscillators,” Phys Rev A (Coll Park), vol. 63, no. 2, p. 22116, Jan. 2001,
doi: 10.1103/PhysRevA.63.022116.
[3] T. Yu and J. H. Eberly, “Sudden Death of Entanglement,” Science (1979), vol. 323, no. 5914, pp. 598–601, Jan. 2009, doi: 10.1126/science.1167343.
[4] J. R. Hance and S. Hossenfelder, “What does it take to solve the measurement problem?,” J Phys Commun, vol. 6, no. 10, p. 102001, 2022, doi: 10.1088/2399-6528/ac96cf.
[5] M. Steiner and R. Rendell, The Quantum Measurement Problem. Inspire Institute, 2018.
[6] Sabine Hossenfelder, “Does Superdeterminism save Quantum Mechanics? Or Does It Kill Free Will and Destroy Science?,” http://backreaction.blogspot.com/2021/12/
does-superdeterminism-save-quantum.html, Dec. 18, 2021. | {"url":"https://theqmp.com/2022/11/27/review-of-s-donadi-and-s-hossenfelder-toy-model-for-local-and-deterministic-wave-function-collapse-physical-review-a-106-022212-2022/","timestamp":"2024-11-07T23:41:07Z","content_type":"text/html","content_length":"89589","record_id":"<urn:uuid:64f2f9ad-04a2-48d1-a312-14527e35c2b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00780.warc.gz"} |
MIMIC-II Survival Analysis¶
This tutorial explores the MIMIC-II IAC dataset. It was created for the purpose of a case study in the book: Secondary Analysis of Electronic Health Records, published by Springer in 2016. In
particular, the dataset was used throughout Chapter 16 (Data Analysis) by Raffa J. et al. to investigate the effectiveness of indwelling arterial catheters in hemodynamically stable patients with
respiratory failure for mortality outcomes. The dataset is derived from MIMIC-II, the publicly-accessible critical care database. It contains summary clinical data and outcomes for 1,776 patients.
[1] Critical Data, M.I.T., 2016. Secondary analysis of electronic health records (p. 427). Springer Nature. (https://link.springer.com/book/10.1007/978-3-319-43742-2)
[2] https://github.com/MIT-LCP/critical-data-book/tree/master/part_ii/chapter_16/jupyter
[3] https://stackoverflow.com/questions/27328623/anova-test-for-glm-in-python/60769343#60769343
More details on the dataset can be found here: https://physionet.org/content/mimic2-iaccd/1.0/.
Importing ehrapy and setting plotting parameters¶
import ehrapy as ep
import pandas as pd
import numpy as np
from statsmodels.stats.anova import anova_lm
from itertools import product
from lifelines import CoxPHFitter
Installed version 0.3.0 of ehrapy is newer than the latest release 0.2.0! You are running a nightly version and
features may break!
import warnings
ehrapy 0.3.0
rich NA
scanpy 1.9.1
session_info 1.0.0
PIL 9.2.0
PyPDF2 2.6.0
anndata 0.8.0
appnope 0.1.3
astor 0.8.1
asttokens NA
attr 21.4.0
autograd NA
autograd_gamma NA
backcall 0.2.0
backports NA
bs4 4.11.1
camelot 0.10.1
category_encoders 2.5.0
certifi 2022.06.15
cffi 1.15.1
chardet 5.0.0
charset_normalizer 2.1.0
colorama 0.4.5
cryptography 37.0.4
cv2 4.6.0
cvxopt 1.3.0
cycler 0.10.0
cython_runtime NA
dateutil 2.8.2
debugpy 1.6.2
decorator 5.1.1
deep_translator 1.8.0
deepl 1.9.0
defusedxml 0.7.1
entrypoints 0.4
executing 0.8.3
formulaic 0.3.4
future 0.18.2
h5py 3.7.0
idna 3.3
igraph 0.9.11
interface_meta 1.3.0
ipykernel 6.15.1
jedi 0.18.1
joblib 1.1.0
kiwisolver 1.4.4
leidenalg 0.8.10
lifelines 0.27.1
llvmlite 0.34.0
matplotlib 3.5.2
mpl_toolkits NA
mudata 0.2.0
natsort 8.1.0
numba 0.51.2
numpy 1.23.1
packaging 21.3
pandas 1.4.2
parso 0.8.3
patsy 0.5.2
pdfminer 20220524
pexpect 4.8.0
pickleshare 0.7.5
pkg_resources NA
prompt_toolkit 3.0.30
psutil 5.9.1
ptyprocess 0.7.0
pure_eval 0.2.2
pydantic NA
pydev_ipython NA
pydevconsole NA
pydevd 2.8.0
pydevd_file_utils NA
pydevd_plugins NA
pydevd_tracing NA
pygments 2.12.0
pyhpo 3.1.2
pyparsing 3.0.9
pypi_latest 0.1.2
pytz 2022.1
questionary 1.10.0
requests 2.28.1
ruamel NA
scipy 1.6.1
setuptools 63.1.0
setuptools_scm NA
six 1.16.0
sklearn 1.1.1
soupsieve 2.3.2.post1
sphinxcontrib NA
stack_data 0.3.0
statsmodels 0.13.2
tabulate 0.8.10
texttable 1.6.4
threadpoolctl 3.1.0
tornado 6.2
traitlets 5.3.0
typing_extensions NA
urllib3 1.26.10
wcwidth 0.2.5
wrapt 1.14.1
yaml 6.0
zipp NA
zmq 23.2.0
IPython 8.4.0
jupyter_client 7.3.4
jupyter_core 4.11.1
Python 3.8.2 (default, Apr 8 2021, 23:19:18) [Clang 12.0.5 (clang-1205.0.22.9)]
Session information updated at 2022-08-09 12:04
MIMIC-II dataset preparation¶
adata = ep.dt.mimic_2(encoded=False)
It is also possible to get the MIMIC-II dataset already pre-encoded by setting the encoded flag to ‘True’. ehrapy’s default encoding is a simple label encoding in this case.
The MIMIC-II dataset has 1776 patients as described above with 46 features.
AnnData object with n_obs × n_vars = 1776 × 46
uns: 'numerical_columns', 'non_numerical_columns'
layers: 'original'
After one-hot encoding the two columns we’ve expanded our matrix from 46 to 54 features. Let’s verify that we’ve indeed encoded all columns and are ready to proceed.
Linear Regression¶
Linear regression provides the foundation for many types of analyses we perform on health data. In the simplest scenario, we try to relate one continuous outcome, \(y\), to a single continuous
covariate, \(x\), by trying to find values for \(\beta_0\) and \(\beta_1\) so that the following equation describes our observations: \(y=\beta_0 + \beta_1 \times x\)
It is always a good idea to visualize the data when you can, which allows one to assess if the subsequent analysis corresponds to what you could see with your eyes. In this case, a scatter plot has
been used, producing the scattered points:
ax = ep.pl.scatter(adata, x="pco2_first", y="tco2_first")
Finding the best fit line for the scatter plot above using ep.tl.ols (ehrapy.tools.ols) is relatively straightforward:
var_names = ["pco2_first", "tco2_first", "gender_num"]
formula = "tco2_first ~ pco2_first"
co2_lm = ep.tl.ols(adata, var_names, formula, missing="drop")
Dissecting this command from left to right: The co2_lm = part assigns the right part of the command to a new variable called co2_lm. The right side of this command runs the OLS function in Python.
The basic OLS command has two parts. The first is the formula which has the general syntax outcome ~ covariates. Here, our outcome variable is called tco2_first and we are just fitting one covariate,
pco2_first, so our formula is tco2_first ~ pco2_first. The second argument is separated by a comma and is specifying the data frame to use. In our case, the data frame is called dat, so we pass data=
dat, noting that both tco2_first and pco2_first are columns in the dataframe dat. The overall procedure of specifying a model formula (tco2_first ~ pco2_first), a data frame (data=dat) and passing it
an appropriate Python function (OLS) will be used throughout this chapter, and is the foundation for many types of statistical modeling in Python.
We would like to see some information about the model we just fit, and often a good way of doing this is to run the summary command on the object we created:
co2_lm_result = co2_lm.fit()
OLS Regression Results
Dep. Variable: tco2_first R-squared: 0.265
Model: OLS Adj. R-squared: 0.264
Method: Least Squares F-statistic: 571.8
Date: Tue, 09 Aug 2022 Prob (F-statistic): 3.52e-108
Time: 12:04:16 Log-Likelihood: -4609.1
No. Observations: 1590 AIC: 9222.
Df Residuals: 1588 BIC: 9233.
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
Intercept 16.2109 0.360 45.071 0.000 15.505 16.916
pco2_first 0.1886 0.008 23.912 0.000 0.173 0.204
Omnibus: 94.233 Durbin-Watson: 1.963
Prob(Omnibus): 0.000 Jarque-Bera (JB): 195.722
Skew: -0.389 Prob(JB): 3.16e-43
Kurtosis: 4.532 Cond. No. 149.
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
This outputs information about the OLS object we created in the previous step. The first part recalls the model we fit, which is useful when we have fit many models, and are trying to compare them.
The second part lists some summary information about what are called residuals – a topic we will cover later on in this book (Chapter 2.6). Next lists the coefficient estimates – these are the \(\hat
{\beta}_0\), (Intercept), and \(\hat{\beta}_1\), pco2_first, parameters in the best fit line we are trying to estimate. This output is telling us that the best fit equation for the data is:
tco2_first = 16.21 + 0.189 \(\times\) pco2_first.
Model Selection¶
Model selection is the procedure to select the best model from a list (perhaps rather large list) of candidate models. Different approaches and techniques exist for this procedure. We will cover some
basics here, as more complicated techniques will be covered in a later chapter. In the simplest case, we have two models, and we want to know which one we should use.
We will begin by examining if the relationship between TCO2 and PCO2 is more complicated than the model we fit in the previous section. If you recall, we fit a model where we considered a linear
pco2_first term: tco2_first = \(\beta_0 + \beta_1 \times\) pco2_first. One may wonder if including a quadratic term would fit the data better, i.e. whether:
tco2_first = \(\beta_0 + \beta_1 \times\) pco2_first + \(\beta_2 \times\) pco2_first\(^2\),
is a better model.
Adding a quadratic term (or any other function) is quite easy using the OLS function. Fitting this model, and running the summary function for the model:
formula = "tco2_first ~ pco2_first + np.square(pco2_first)"
o2_quad_lm = ep.tl.ols(adata, var_names, formula, missing="drop")
co2_quad_lm_result = o2_quad_lm.fit()
OLS Regression Results
Dep. Variable: tco2_first R-squared: 0.265
Model: OLS Adj. R-squared: 0.264
Method: Least Squares F-statistic: 285.7
Date: Tue, 09 Aug 2022 Prob (F-statistic): 1.04e-106
Time: 12:04:16 Log-Likelihood: -4609.1
No. Observations: 1590 AIC: 9224.
Df Residuals: 1587 BIC: 9240.
Df Model: 2
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
Intercept 16.0916 0.771 20.862 0.000 14.579 17.605
pco2_first 0.1930 0.027 7.231 0.000 0.141 0.245
np.square(pco2_first) -3.569e-05 0.000 -0.175 0.861 -0.000 0.000
Omnibus: 93.522 Durbin-Watson: 1.963
Prob(Omnibus): 0.000 Jarque-Bera (JB): 194.937
Skew: -0.385 Prob(JB): 4.68e-43
Kurtosis: 4.533 Cond. No. 1.94e+04
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 1.94e+04. This might indicate that there are
strong multicollinearity or other numerical problems.
Looking first at the coefficients estimates, we see the best fit line is estimated as:
tco2_first = 16.09 + 0.19 \(\times\) pco2_first + 0.00004 \(\times\) pco2_first\(^2\).
We can add both best fit lines to the former scattered points.
ols_results=[co2_lm_result, co2_quad_lm_result],
ols_color=["red", "blue"],
Figure Scatterplot of PCO2 (x-axis) and TCO2 (y-axis) along with linear regression estimates from the quadratic model (co2_quad_lm) and linear only model (co2_lm)
And one can see that the red (linear term only) and blue (linear and quadratic terms) fits are nearly identical. This corresponds with the relatively small coefficient estimate for the (pco2_first^2)
term. The p-value for this coefficient is about 0.86, and at the 0.05 significance level we would likely conclude that a quadratic term is not necessary in our model to fit the data, as the linear
term only model fits the data nearly as well.
Statistical Interactions and Testing Nested Models¶
We can also consider additional features in our model. For this, we can start with e.g. gender as a covariate, but no interaction. We can do this by simply adding the variable gender_num to the
previous formula for our co2_lm model fit.
formula = "tco2_first ~ pco2_first + gender_num"
var_names = ["tco2_first", "pco2_first", "gender_num"]
co2_gender_lm = ep.tl.ols(adata, var_names, formula, missing="drop")
co2_gender_lm_result = co2_gender_lm.fit()
OLS Regression Results
Dep. Variable: tco2_first R-squared: 0.265
Model: OLS Adj. R-squared: 0.264
Method: Least Squares F-statistic: 286.1
Date: Tue, 09 Aug 2022 Prob (F-statistic): 7.60e-107
Time: 12:04:17 Log-Likelihood: -4608.7
No. Observations: 1590 AIC: 9223.
Df Residuals: 1587 BIC: 9240.
Df Model: 2
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
Intercept 16.3044 0.378 43.166 0.000 15.564 17.045
pco2_first 0.1889 0.008 23.922 0.000 0.173 0.204
gender_num -0.1817 0.224 -0.812 0.417 -0.621 0.257
Omnibus: 94.658 Durbin-Watson: 1.964
Prob(Omnibus): 0.000 Jarque-Bera (JB): 194.982
Skew: -0.394 Prob(JB): 4.57e-43
Kurtosis: 4.524 Cond. No. 160.
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
This output is very similar to what we had before, but now there’s a gender_num term as well. The 1 is present in the first column after gender_num, and it tells us who this coefficient is relevant
to (subjects with 1 for the gender_num – men). This is always relative to the baseline group, and in this case this is women. The estimate is negative, meaning that the line fit for males will be
below the line for females. Plotting this fit curve:
# s stands for slope of the line, i stands for intercept of the line
s_female = co2_gender_lm_result.params[1]
i_female = co2_gender_lm_result.params[0]
s_male = co2_gender_lm_result.params[1]
i_male = co2_gender_lm_result.params[0] + co2_gender_lm_result.params[2]
lines=[(s_female, i_female), (s_male, i_male)],
lines_color=["k", "r"],
lines_label=["Female", "Male"],
xlim=(0, 40),
ylim=(15, 25),
We see that the lines are parallel, but almost indistinguishable. In fact, this plot has been cropped in order to see any difference at all. From the estimate from the summary output above, the
difference between the two lines is -0.182 mmol/L, which is quite small, so perhaps this isn’t too surprising. We can also see in the above summary output that the p-value is about 0.42, and we would
likely not reject the null hypothesis that the true value of the gender_num coefficient is zero.
Next, we can explore our model with an interaction between pco2_first and gender_num. To add an interaction between two variables use the * operator within a model formula. By default, Python will
add all of the main effects (variables contained in the interaction) to the model as well, so simply adding pco2_first*gender_num will add effects for pco2_first and gender_num in addition to the
interaction between them to the model fit.
formula = "tco2_first ~ pco2_first * gender_num"
var_names = ["tco2_first", "pco2_first", "gender_num"]
co2_gender_interaction_lm = ep.tl.ols(adata, var_names, formula, missing="drop")
co2_gender_interaction_lm_result = co2_gender_interaction_lm.fit()
OLS Regression Results
Dep. Variable: tco2_first R-squared: 0.266
Model: OLS Adj. R-squared: 0.265
Method: Least Squares F-statistic: 191.6
Date: Tue, 09 Aug 2022 Prob (F-statistic): 5.09e-106
Time: 12:04:17 Log-Likelihood: -4607.7
No. Observations: 1590 AIC: 9223.
Df Residuals: 1586 BIC: 9245.
Df Model: 3
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
Intercept 15.8544 0.489 32.443 0.000 14.896 16.813
pco2_first 0.1994 0.011 18.585 0.000 0.178 0.220
gender_num 0.8144 0.722 1.128 0.260 -0.602 2.231
pco2_first:gender_num -0.0230 0.016 -1.450 0.147 -0.054 0.008
Omnibus: 97.108 Durbin-Watson: 1.964
Prob(Omnibus): 0.000 Jarque-Bera (JB): 200.324
Skew: -0.403 Prob(JB): 3.16e-44
Kurtosis: 4.541 Cond. No. 399.
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
The estimated coefficients are \(\hat{\beta}_0, \hat{\beta}_1, \hat{\beta}_2\) and \(\hat{\beta}_3\), respectively, and we can determine the best fit lines for men:
tco2_first = \((15.85 + 0.81)\) + \((0.20 - 0.023)\) \(\times\) pco2_first = \(16.67\) + \(0.18\) \(\times\) pco2_first,
and for women:
tco2_first = \(15.85\) + \(0.20\) \(\times\) pco2_first.
Based on this, the men’s intercept should be higher, but their slope should be not as steep, relative to the women. Let’s check this and add the new model fits as dotted lines and add a legend to the
above figure.
# s stands for slope of the line, i stands for intercept of the line
s_female_interaction = co2_gender_interaction_lm_result.params[1]
i_female_interaction = co2_gender_interaction_lm_result.params[0]
s_male_interaction = (
+ co2_gender_interaction_lm_result.params[3]
i_male_interaction = (
+ co2_gender_interaction_lm_result.params[2]
lines = [
(s_female, i_female),
(s_male, i_male),
(s_female_interaction, i_female_interaction),
(s_male_interaction, i_male_interaction),
lines_color = ["k", "r", "k", "r"]
lines_style = ["-", "-", ":", ":"]
lines_label = [
"Female (Interaction Model)",
"Male (Interaction Model)",
xlim=(0, 40),
ylim=(15, 25),
Regression fits of PCO2 on TCO2 with gender (black female; red male; solid no interaction; dotted with interaction). Note both axes are cropped for illustration purposes
We can see that the fits generated from this plot are a little different than the one generated for a model without the interaction. The biggest difference is that the dotted lines are no longer
parallel. This has some serious implications, particularly when it comes to interpreting our result. First note that the estimated coefficient for the gender_num variable is now positive. This means
that at pco2_first=0, men (red) have higher tco2_first levels than women (black). If you recall in the previous model fit, women had higher levels of tco2_first at all levels of pco2_first. At some
point around pco2_first=35 this changes and women (black) have higher tco2_first levels than men (red). This means that the effect of gender_num may vary as you change the level of pco2_first, and is
why interactions are often referred to as effect modification in the epidemiological literature. The effect need not change signs (i.e., the lines do not need to cross) over the observed range of
values for an interaction to be present.
The question remains, is the variable gender_num important? We looked at this briefly when we examined the t value column in the no interaction model which included gender_num. What if we wanted to
test (simultaneously) the null hypothesis: \(\beta_2\) and \(\beta_3=0\) There is a useful test known as the F-test which can help us in this exact scenario where we want to look at if we should use
a larger model (more covariates) or use a smaller model (fewer covariates). The F-test applies only to nested models – the larger model must contain each covariate that is used in the smaller model,
and the smaller model cannot contain covariates which are not in the larger model. The interaction model and the model with gender are nested models since all the covariates in the model with gender
are also in the larger interaction model. An example of a non-nested model would be the quadratic model and the interaction model: the smaller (quadratic) model has a term (pco2_first\(^2\)) which is
not in the larger (interaction) model. An F-test would not be appropriate for this latter case.
To perform an F-test, first fit the two models you wish to consider, and then run the anova_lm command passing the two model objects.
anova_lm(co2_lm_result, co2_gender_interaction_lm_result)
│ │df_resid│ ssr │df_diff│ ss_diff │ F │ Pr(>F) │
│0│1588.0 │30674.431382 │0.0 │NaN │NaN │NaN │
│1│1586.0 │30621.082804 │2.0 │53.348578│1.381578│0.251484│
As you can see, the anova_lm command first lists the models it is considering. Much of the rest of the information is beyond the scope of this chapter, but we will highlight the reported F-test
p-value (Pr(>F)), which in this case is 0.2515. In nested models, the null hypothesis is that all coefficients in the larger model and not in the smaller model are zero. In the case we are testing,
our null hypothesis is \(\beta_2\) and \(\beta_3=0\). Since the p-value exceeds the typically used significance level (\(\alpha=0.05\)), we would not reject the null hypothesis, and likely say the
smaller model explains the data just as well as the larger model. If these were the only models we were considering, we would use the smaller model as our final model and report the final model in
our results. We will now discuss what exactly you should report and how you can interpret the results.
Reporting and Interpreting Linear Regression¶
Confidence and Prediction Intervals
As mentioned above, one method to quantify the uncertainty around coefficient estimates is by reporting the standard error. Another commonly used method is to report a confidence interval, most
commonly a 95% confidence interval. A 95% confidence interval for \(\beta\) is an interval for which if the data were collected repeatedly, about 95% of the intervals would contain the true value of
the parameter, \(\beta\), assuming the modeling assumptions are correct.
To get 95% confidence intervals of coefficients, Python has a confint function, which you pass an lm object to. It will then output 2.5% and 97.5% confidence interval limits for each coefficient.
│ │ 0 │ 1 │
│ Intercept │15.505369│16.916349│
│pco2_first │0.173103 │0.204040 │
The 95% confidence interval for pco2_first is about 0.17-0.20, which may be slightly more informative than reporting the standard error. Often people will look at if the confidence interval includes
zero (no effect). Since it does not, and in fact since the interval is quite narrow and not very close to zero, this provides some additional evidence of its importance. There is a well known link
between hypothesis testing and confidence intervals which we will not get into detail here.
When plotting the data with the model fit, similar to Figure 16.2, it is a good idea to include some sort of assessment of uncertainty as well. To do this in Python, we will first create a data frame
with PCO2 levels which we would like to predict. In this case, we would like to predict the outcome (TCO2) over the range of observed covariate (PCO2) values. We do this by creating a data frame,
where the variable names in the data frame must match the covariates used in the model. In our case, we have only one covariate (pco2_first), and we predict the outcome over the range of covariate
values we observed determined by the min and max functions.
pco2_first = pd.DataFrame(adata[:, "pco2_first"].X).dropna().astype(int)
data = {"pco2_first": [i for i in range(pco2_first.min()[0], pco2_first.max()[0])]}
grid_pred = pd.DataFrame(data)
preds = co2_lm_result.get_prediction(grid_pred).summary_frame()[
["mean", "obs_ci_lower", "obs_ci_upper"]
│ │ mean │obs_ci_lower │obs_ci_upper │
│0│17.719434│9.078647 │26.360220 │
│1│17.908005│9.268186 │26.547825 │
We have printed out the first two rows of our predictions, preds, which are the model’s predictions for PCO2 at 8 and 9. We can see that our predictions (mean) are about 0.18 apart, which make sense
given our estimate of the slope (0.18). We also see that our 95 % prediction intervals are very wide, spanning about 9 (obs_ci_lower) to 26 (obs_ci_upper). This indicates that, despite coming up with
a model which is very statisti- cally significant, we still have a lot of uncertainty about the predictions generated from such a model. It is a good idea to capture this quality when plotting how
well your model fits by adding the interval lines as dotted lines. Let’s plot our final model fit, co2.lm, along with the scatterplot and prediction interval in the figure below.
(grid_pred["pco2_first"], preds["obs_ci_upper"]),
(grid_pred["pco2_first"], preds["obs_ci_lower"]),
lines_color=["k", "k"],
lines_style=["--", "--"],
Scatterplot of PCO2 (x-axis) and TCO2 (y-axis) along with linear regression estimates from the linear only model (co2_lm). The dotted line represents 95 % prediction intervals for the model
Logistic Regression¶
So far, we have considered how a continuous variable can be modeled using our features. Another very common modelling task is the one of classification. Here, we are interested to model a discrete
label using our features. A very common scenario is to have a binary class label, for example whether a patient survived a 28-day observation window or not.
2 * 2 Tables¶
var_names = ["age", "day_28_flg", "service_unit"]
dat = pd.DataFrame(adata[:, var_names].X, columns=var_names)
dat["age"] = dat["age"].astype("float").dropna(axis=0)
Contingency tables are the best way to start to think about binary data. A contingency table cross-tabulates the outcome across two or more levels of a covariate. Let’s begin by creating a new
variable (age.cat) which dichotomizes age into two age categories: \(\le55\) and \(>55\). Note, because we are making age a discrete variable, we also change the data type to a factor. This is
similar to what we did for the gender_num variable when discussing linear regression in the previous subchapter.
dat["age_cat"] = np.where(dat["age"] <= 55, "<=55", ">55")
dat["age_cat"] = dat["age_cat"].astype("category")
<=55 923
>55 853
Name: age_cat, dtype: int64
We would like to see how 28 day mortality is distributed among the age categories. We can do so by constructing a contingency table, or in this case what is commonly referred to as a 2x2 table.
age_cat_day_28_flg = pd.crosstab(index=dat["age_cat"], columns=dat["day_28_flg"])
age_cat_day_28_flg.columns = ["0", "1"]
age_cat_day_28_flg.index = ["<=55", ">55"]
│ │ 0 │ 1 │
│<=55 │883│40 │
│ >55 │610│243│
Now let us look at a slightly different case – when the covariate takes on more than two values. Such a variable is the service_unit. Let’s see how the deaths are distributed among the different
deathbyservice = pd.crosstab(index=dat["service_unit"], columns=dat["day_28_flg"])
deathbyservice.columns = ["0", "1"]
deathbyservice.index = ["FICU", "MICU", "SICU"]
│ │ 0 │ 1 │
│FICU │59 │3 │
│MICU │605│127│
│SICU │829│153│
we can get frequencies of these service units by applying the prop.table function to our cross-tabulated table.
deathbyservice.div(deathbyservice["0"] + deathbyservice["1"], axis=0)
│ │ 0 │ 1 │
│FICU│0.951613 │0.048387 │
│MICU│0.826503 │0.173497 │
│SICU│0.844196 │0.155804 │
It appears as though the FICU may have a lower rate of death than either the MICU or SICU. To compute an odds ratios, first compute the odds:
deathbyservice["1"].div(deathbyservice["0"], axis=0)
FICU 0.050847
MICU 0.209917
SICU 0.184560
dtype: float64
And then we need to pick which of FICU, MICU or SICU will serve as the reference or baseline group. This is the group which the other two groups will be compared to. Again the choice is arbitrary,
but should be dictated by the study objective. If this were a clinical trial with two drug arms and a placebo arm, it would be foolish to use one of the treatments as the reference group,
particularly if you wanted to compare the efficacy of the treatments. In this particular case, there is no clear reference group, but since the FICU is so much smaller than the other two units, we
will use it as the reference group. Computing the odds ratio for MICU and SICU we get 4.13 and 3.63, respectively. These are also very strong associations, meaning that the odds of dying in the SICU
and MICU are around 4 times higher than in the FICU, but relatively similar.
Contingency tables and 2x2 tables in particular are the building blocks of working with binary data, and it’s often a good way to begin looking at the data.
Introducing Logistic Regression¶
Let’s fit this model, and see how this works using a real example. We fit logistic regression very similarly to how we fit linear regression models, with a few exceptions. First, we will use a new
function called glm, which allows one to fit a class of models known as generalized linear models or GLMs. The glm function works in much the same way the lm function does. We need to specify a
formula of the form: outcome ~ covariates, specify what dataset to use (in our case the dat data frame), and then specify the family. For logistic regression family='binomial' will be our choice. You
can run the summary function, just like you did for lm and it produces output very similar to what lm did.
dat["day_28_flg"] = np.where(dat["day_28_flg"] == 0, 1, 0)
adata_new = ep.ad.df_to_anndata(dat)
formula = "day_28_flg ~ age_cat"
var_names = ["day_28_flg", "age_cat"]
age_glm = ep.tl.glm(adata_new, var_names, formula, family="Binomial", missing="drop")
age_glm_result = age_glm.fit()
Generalized Linear Model Regression Results
Dep. Variable: ['day_28_flg[0]', 'day_28_flg[1]'] No. Observations: 1776
Model: GLM Df Residuals: 1774
Model Family: Binomial Df Model: 1
Link Function: Logit Scale: 1.0000
Method: IRLS Log-Likelihood: -674.34
Date: Tue, 09 Aug 2022 Deviance: 1348.7
Time: 12:04:20 Pearson chi2: 1.78e+03
No. Iterations: 6 Pseudo R-squ. (CS): 0.1111
Covariance Type: nonrobust
coef std err z P>|z| [0.025 0.975]
Intercept -3.0944 0.162 -19.142 0.000 -3.411 -2.778
age_cat[T.>55] 2.1740 0.179 12.175 0.000 1.824 2.524
As you can see, we get a coefficients table that is similar to the lm table we used earlier. Instead of a t value, we get a z value, but this can be interpreted similarly. The rightmost column is a
p-value, for testing the null hypothesis \(\beta=0\). If you recall, the non-intercept coefficients are log-odds ratios, so testing if they are zero is equivalent to testing if the odds ratios are
one. If an odds ratio is one the odds are equal in the numerator group and denominator group, indicating the probabilities of the outcome are equal in each group. So, assessing if the coefficients
are zero will be an important aspect of doing this type of analysis.
Beyond a Single Binary Covariate¶
While the above analysis is useful for illustration, it does not readily demonstrate anything we could not do with our 2x2 table example above. Logistic regression allows us to extend the basic idea
to at least two very relevant areas. The first is the case where we have more than one covariate of interest. Perhaps we have a confounder we are concerned about, and want to adjust for it.
Alternatively, maybe there are two covariates of interest. Secondly, it allows use to use covariates as continuous quantities, instead of discretizing them into categories. For example, instead of
dividing age up into exhaustive strata (as we did very simply by just dividing the patients into two groups, \(\le55\) and \(>55\) ), we could instead use age as a continuous covariate.
First, having more than one covariate is simple. For example, if we wanted to add service_unit to our previous model, we could just add it as we did when using the lm function for linear regression.
Here we specify day_28_flg ~ age.cat + service_unit and run the summary function.
formula = "day_28_flg ~ age_cat + service_unit"
var_names = ["day_28_flg", "age_cat", "service_unit"]
ageunit_glm = ep.tl.glm(
adata_new, var_names, formula, family="Binomial", missing="drop"
ageunit_glm_result = ageunit_glm.fit()
Generalized Linear Model Regression Results
Dep. Variable: ['day_28_flg[0]', 'day_28_flg[1]'] No. Observations: 1776
Model: GLM Df Residuals: 1772
Model Family: Binomial Df Model: 3
Link Function: Logit Scale: 1.0000
Method: IRLS Log-Likelihood: -671.87
Date: Tue, 09 Aug 2022 Deviance: 1343.7
Time: 12:04:20 Pearson chi2: 1.74e+03
No. Iterations: 6 Pseudo R-squ. (CS): 0.1136
Covariance Type: nonrobust
coef std err z P>|z| [0.025 0.975]
Intercept -4.2090 0.623 -6.761 0.000 -5.429 -2.989
age_cat[T.>55] 2.1611 0.179 12.088 0.000 1.811 2.512
service_unit[T.MICU] 1.1789 0.615 1.915 0.055 -0.027 2.385
service_unit[T.SICU] 1.1234 0.614 1.830 0.067 -0.080 2.326
A coefficient table is produced, and now we have four estimated coefficients. The same two, (Intercept) and age_cat[T.>55] which were estimated in the unadjusted model, but also we have service_unit
[T.MICU] and service_unit[T.SICU] which correspond to the log odds ratios for the MICU and SICU relative to the FICU. Taking the exponential of these will result in an odds ratio for each variable,
adjusted for the other variables in the model. In this case the adjusted odds ratios for Age>55, MICU and SICU are 8.68, 3.25, and 3.08, respectively. We would conclude that there is an almost 9-fold
increase in the odds of 28 day mortality for those in the \(>55\) year age group relative to the younger \(\le55\) group while holding service unit constant. This adjustment becomes important in many
scenarios where groups of patients may be more or less likely to receive treatment, but also more or less likely to have better outcomes, where one effect is confounded by possibly many others. Such
is almost always the case with observational data, and this is why logistic regression is such a powerful data analysis tool in this setting.
Another case we would like to be able to deal with is when we have a continuous covariate we would like to include in the model. One can always break the continuous covariate into mutually exclusive
categories by selecting break or cut points, but selecting the number and location of these points can be arbitrary, and in many cases unnecessary or inefficient. Recall that in logistic regression
we are fitting a model:
\(logit(p_x) = log(Odds_x) = log(\frac{p_x}{1-p_x}) = \beta_0 + \beta_1 \times x\),
but now assume \(x\) is continuous. Imagine a hypothetical scenario where you know \(\beta_0\) and \(\beta_1\) and have a group of 50 year olds, and a group of 51 year olds. The difference in the log
Odds between the two groups is:
\(log(Odds_{51}) -log(Odds_{50}) = (\beta_0 + \beta_1 \times 51) - (\beta_0 + \beta_1 \times 50) = \beta_1(51-50) = \beta_1\).
Hence, the odds ratio for 51 year olds versus 50 year olds is \(\exp{(\beta_1)}\). This is actually true for any group of patients which are 1 year apart, and this gives a useful way to interpret and
use these estimated coefficients for continuous covariates. Let’s work with an example. Again fitting the 28 day mortality outcome as a function of age, but treating age as it was originally recorded
in the dataset, a continuous variable called age.
formula = "day_28_flg ~ age"
var_names = ["day_28_flg", "age"]
agects_glm = ep.tl.glm(
agects_glm_result = agects_glm.fit()
Generalized Linear Model Regression Results
Dep. Variable: ['day_28_flg[0]', 'day_28_flg[1]'] No. Observations: 1776
Model: GLM Df Residuals: 1774
Model Family: Binomial Df Model: 1
Link Function: Logit Scale: 1.0000
Method: IRLS Log-Likelihood: -625.49
Date: Tue, 09 Aug 2022 Deviance: 1251.0
Time: 12:04:20 Pearson chi2: 1.81e+03
No. Iterations: 6 Pseudo R-squ. (CS): 0.1587
Covariance Type: nonrobust
coef std err z P>|z| [0.025 0.975]
Intercept -5.7780 0.321 -18.013 0.000 -6.407 -5.149
age 0.0652 0.004 14.595 0.000 0.056 0.074
We see the estimated coefficient is 0.07 and still very statistically significant. Exponentiating the log odds ratio for age, we get an estimated odds ratio of 1.07, which is per 1 year increase in
age. What if the age difference of interest is ten years instead of one year? There are at least two ways of doing this. One is to replace age with I(age/10), which uses a new covariate which is age
divided by ten. The second is to use the agects.glm estimated log odds ratio, and multiple by ten prior to exponentiating. They will yield equivalent estimates of 1.92, but it is now per 10 year
increases in age. This is useful when the estimated odds ratios (or log odds ratios) are close to one (or zero). When this is done, one unit of the covariate is 10 years, so the generic
interpretation of the coefficients remains the same, but the units (per 10 years instead of per 1 year) changes.
This of course assumes that the form of our equation relating the log odds of the outcome to the covariate is correct. In cases where odds of the outcome decreases and increases as a function of the
covariate, it is possible to estimate a relatively small effect of the linear covariate, when the outcome may be strongly affected by the covariate, but not in the way the model is specified.
Assessing the linearity of the log odds of the outcome and some discretized form of the covariate can be done graphically. For instance, we can break age into 5 groups, and estimate the log odds of
28 day mortality in each group.
Hypothesis Testing and Model Selection¶
As was the case when using lm, we first fit the two competing models, a larger (alternative model), and a smaller (null model). Provided that the models are nested, our larger model is the one which
contained service_unit and age.cat, and the smaller only contains age.cat, so they are nested. We are then testing if the log odds ratios for the two coefficients associated with service_unit are
zero. Let’s call these coefficients \(\beta_{MICU}\) and \(\beta_{SICU}\). To test if \(\beta_{MICU}\) and \(\beta_{SICU} = 0\), we can perform the following test:
"day_28_flg ~ age",
"day_28_flg ~ age + service_unit",
│ │Model│ formula │Df Resid.│ Dev. │Df_diff│Pr(>Chi)│
│0│1 │day_28_flg ~ age │1774 │1348.676905│NaN │NaN │
│1│2 │day_28_flg ~ age + service_unit │1772 │1343.745416│2.0 │0.085237│
A couple good practices to get in a habit are to first make sure the two competing models are correctly specified. Here we are are testing ~ age.cat versus age.cat + service_unit. Next, the
difference between the residual degrees of freedom (Resid. Df) in the two models tell us how many more parameters the larger model has when compared to the smaller model. Here we see 1774 - 1772 = 2
which means that there are two more coefficients estimated in the larger model than the smaller one, which corresponds with the output from the summary table above. Next looking at the p-value (Pr(>
Chi)), we see a test for \(\beta_{MICU}\) and \(\beta_{SICU} = 0\) has a p-value of around 0.08. At the typical 0.05 significance level, we would not reject the null, and use the simpler model
without the service unit. In logistic regression, this is a common way of testing whether a categorical covariate should be retained in the model, as it can be difficult to assess using the z value
in the summary table, particularly when one is very statistically significant, and one is not.
Confidence Intervals¶
Generating confidence intervals for either the log-odds ratios or the odds ratios are relatively straightforward. To get the log-odds ratios and respective confidence intervals for the ageunit_glm
model which includes both age and service unit:
Intercept -4.209014
age_cat[T.>55] 2.161142
service_unit[T.MICU] 1.178866
service_unit[T.SICU] 1.123443
dtype: float64
│ │ 0 │ 1 │
│ Intercept │-5.429257│-2.988771│
│ age_cat[T.>55] │1.810733 │2.511552 │
│service_unit[T.MICU] │-0.027452│2.385184 │
│service_unit[T.SICU] │-0.079611│2.326497 │
Here the coefficient estimates and confidence intervals are presented in much the same way as for a linear regression. In logistic regression, it is often convenient to exponentiate these quantities
to get it on a more interpretable scale.
Intercept 0.014861
age_cat[T.>55] 8.681049
service_unit[T.MICU] 3.250686
service_unit[T.SICU] 3.075425
dtype: float64
│ │ 0 │ 1 │
│ Intercept │0.003681│0.059992 │
│ age_cat[T.>55] │5.814857│12.960011 │
│service_unit[T.MICU] │0.818181│12.915179 │
│service_unit[T.SICU] │0.776964│12.173333 │
Similar to linear regression, we will check whether the confidence intervals for the log odds ratios include zero. This is equivalent to seeing if the intervals for the odds ratios include 1. Since
the odds ratios are more directly interpretable it is often more convenient to report them instead of the coefficients on the log odds ratio scale.
Let’s first create a new data frame called newdat which computes all combinations of the values of variables passed to it.
def expand_grid(dictionary):
return pd.DataFrame(
[row for row in product(*dictionary.values())], columns=dictionary.keys()
dictionary = {"age_cat": ["<=55", ">55"], "service_unit": ["FICU", "MICU", "SICU"]}
newdat = expand_grid(dictionary)
│ │age_cat │service_unit │
│0│<=55 │FICU │
│1│<=55 │MICU │
│2│<=55 │SICU │
│3│>55 │FICU │
│4│>55 │MICU │
│5│>55 │SICU │
newdat["pred"] = ageunit_glm_result.predict(newdat)
│ │age_cat│service_unit │ pred │
│0│<=55 │FICU │0.014643 │
│1│<=55 │MICU │0.046082 │
│2│<=55 │SICU │0.043706 │
│3│>55 │FICU │0.114268 │
│4│>55 │MICU │0.295461 │
│5│>55 │SICU │0.284056 │
Survival Analysis¶
correct the observation¶
adata_subset = adata[
:, ["mort_day_censored", "censor_flg", "gender_num", "service_unit"]
Normally, you can skip the next step, but since this dataset was used to analyze the data in a slightly different way, we need to correct the observation times for a subset of the subjects in the
adata_subset[:, ["mort_day_censored"]].X[adata_subset[:, ["censor_flg"]].X == 1] = 731
Kaplan-Meier Survival Curves¶
Kaplan-Meier plots are one of the most widely used plots in medical research. We can use ehrapy.tools.kmp (ep.tl.kmp), which internally calls KaplanMeierFitter, to fit the Kaplan-Meier estimate for
the survival function. The function normally takes two arguments: a vector of times, and some kind of indicator for which patients had an event (death in our case). In our case, the vector of death
and censoring times are the mort_day_censored, and deaths are coded with a zero in the censor_flg variable.
Note that in the MIMIC-II database, censor_fl is censored or death (binary: 0 = death, 1 = censored). In KaplanMeierFitter, event_observed is True if the the death was observed, False if the event
was lost (right-censored). So we need to flip censor_fl when pass censor_fl to KaplanMeierFitter:
adata_subset[:, ["censor_flg"]].X = np.where(
adata_subset[:, ["censor_flg"]].X == 0, 1, 0
Fitting a Kaplan-Meier curve is quite easy, and we want to ‘fit’ by gender (gender_num). The default alpha value of KaplanMeierFitter is 0.05, which is 95 % confidence intervals for the survival
functions. Lastly we add a legend, coding black for the women and red for the men.
T = adata_subset[:, ["mort_day_censored"]].X
E = adata_subset[:, ["censor_flg"]].X
groups = adata_subset[:, ["gender_num"]].X
ix1 = groups == 0
ix2 = groups == 1
kmf_1 = ep.tl.kmf(T[ix1], E[ix1], label="Women")
kmf_2 = ep.tl.kmf(T[ix2], E[ix2], label="Men")
[kmf_1, kmf_2],
color=["k", "r"],
xlim=[0, 750],
ylim=[0, 1],
ylabel="Proportion Who Survived",
As the above figure shows, there appears to be a difference between the survival function between the two gender groups, with again the male group (red) dying at slightly slower rate than the female
group (black). We have included 95 % point-wise confidence bands for the survival function estimate, which assesses how much certain we are about the estimated survivorship at each point in time.
We can do the same for service_unit, but since it has three groups, we need to change the color argument and legend to ensure the plot is properly labelled.
groups = adata_subset[:, ["service_unit"]].X
ix1 = groups == "FICU"
ix2 = groups == "MICU"
ix3 = groups == "SICU"
kmf_1 = ep.tl.kmf(T[ix1], E[ix1], label="FICU")
kmf_2 = ep.tl.kmf(T[ix2], E[ix2], label="MICU")
kmf_3 = ep.tl.kmf(T[ix3], E[ix3], label="SICU")
[kmf_1, kmf_2, kmf_3],
ci_show=[False, False, False],
color=["k", "r", "g"],
xlim=[0, 750],
ylim=[0, 1],
ylabel="Proportion Who Survived",
Cox Proportional Hazards Models¶
The most popular approach for modelling time to event outcomes in health data is likely the Cox Proportional Hazards Model, which is also sometimes called the Cox model or Cox Regression. As the name
implies this method models something called the hazard function.
The hazard function is a function of time (hours, days, years) and is approximately the instantaneous probability of the event occurring (i.e., chance the event is happening in some very small time
window) given the event has not already happened.
adata_subset = adata_subset[:, ["mort_day_censored", "censor_flg", "gender_num"]]
The CoxPHFitter function (from the lifelines package) is the fitting function for Cox models. In our case, let’s continue our example of using gender (gender_num) to model the datSurv outcome we
created, and running the summary function to see what information is outputted.
adata_subset[:, ["mort_day_censored"]].X
[731.0]], dtype=object)
Because CoxPHFitter.fit() needs a Pandas DataFrame as input, we have to convert AnnData to DataFrame.
data = ep.ad.anndata_to_df(adata_subset)
data = data.dropna()
gender_coxph = ep.tl.cox_ph(adata_subset, duration_col="mort_day_censored", event_col="censor_flg")
<lifelines.CoxPHFitter: fitted with 1775 total observations, 1278 right-censored observations>
│ model │lifelines.CoxPHFitter │
│ duration col │'mort_day_censored' │
│ event col │'censor_flg' │
│ baseline estimation │breslow │
│ number of observations │1775 │
│number of events observed │497 │
│ partial log-likelihood │-3636.11 │
│ time fit was run │2022-08-09 10:04:22 UTC │
│ │coef │exp(coef)│se(coef)│coef lower 95%│coef upper 95%│exp(coef) lower 95%│exp(coef) upper 95%│cmp to│ z │ p │-log2(p)│
│gender_num│-0.29│0.75 │0.09 │-0.47 │-0.11 │0.63 │0.89 │0.00 │-3.24│<0.005│9.71 │
│ Concordance │0.54 │
│ Partial AIC │7274.23 │
│log-likelihood ratio test │10.43 on 1 df │
│-log2(p) of ll-ratio test │9.65 │
The coefficients table has the familiar format, which we’ve seen before. The coef for gender_num is about −0.29, and this is the estimate of our log-hazard ratio. As discussed, taking the exponential
of this gives the hazard ratio (HR), which the summary output computes in the next column (exp(coef)). Here, the HR is estimated at 0.75, indicating that men have about a 25 % reduction in the
hazards of death, under the proportional hazards assumption.
The next column in the coefficient table has the standard error for the log hazard ratio, followed by the z score and p-value (Pr(>|z|)), which is very similar to what we saw in the case of logistic
regression. Here we see the p-value is quite small, and we would reject the null hypothesis that the hazard functions are the same between men and women. This is consistent with the exploratory
figures we produced using Kaplan-Meier curves in the previous section.
For CoxPHFitter, the summary function also conveniently outputs the confidence interval of the HR a few lines down, and here our estimate of the HR is 0.75 (95 % CI: 0.63–0.89, p = 0.001). This is
how the HR would typically be reported.
Using more than one covariate works the same as our other analysis techniques. Adding a co-morbidity to the model such as atrial fibrillation (afib_flg) can be done as you would do for logistic
regression. Because fit a Cox Proportional Hazards Models using CoxPHFitter requires a Pandas DataFrame as input, we also need to create a DataFrame from AnnData.
adata_subset = adata[:, ["mort_day_censored", "censor_flg", "gender_num", "afib_flg"]]
genderafib_coxph = ep.tl.cox_ph(adata_subset, duration_col="mort_day_censored", event_col="censor_flg")
│ model │lifelines.CoxPHFitter │
│ duration col │'mort_day_censored' │
│ event col │'censor_flg' │
│ baseline estimation │breslow │
│ number of observations │1775 │
│number of events observed │497 │
│ partial log-likelihood │-3567.43 │
│ time fit was run │2022-08-09 10:04:22 UTC │
│ │coef │exp(coef)│se(coef)│coef lower 95%│coef upper 95%│exp(coef) lower 95%│exp(coef) upper 95%│cmp to│ z │ p │-log2(p)│
│gender_num│-0.26│0.77 │0.09 │-0.44 │-0.08 │0.65 │0.92 │0.00 │-2.88│<0.005│7.99 │
│ afib_flg │1.34 │3.84 │0.10 │1.14 │1.54 │3.14 │4.68 │0.00 │13.18│<0.005│129.37 │
│ Concordance │0.61 │
│ Partial AIC │7138.85 │
│log-likelihood ratio test │147.80 on 2 df │
│-log2(p) of ll-ratio test │106.62 │
Here again male gender is associated with reduced time to death, while atrial fibrillation increases the hazard of death by almost four-fold. Both are statistically significant in the summary output.
Cox regression also allows one to use covariates which change over time. This would allow one to incorporate changes in treatment, disease severity, etc. within the same patient without need for any
different methodology. The major challenge to do this is mainly in the construction of the dataset, which is discussed in some of the references at the end of this chapter. Some care is required when
the time dependent covariate is only measured periodically, as the method requires that it be known at every event time for the entire cohort of patients, and not just those relevant to the patient
in question. This is more practical for changes in treatment which may be recorded with some precision, particularly in a database like MIMIC II, and less so for laboratory results which may be
measured at the resolution of hours, days or weeks. Interpolating between lab values or carrying the last observation forward has been shown to introduce several types of problems.
Caveats and Conclusions¶
Survival analysis is distinguished from other forms of analyses covered in this Chapter, as it allows the data to be censored. As was the case for the other approaches we considered, there are
modeling assumptions. For instance, it is important that the censoring is not informative of the survival time. For example, if censoring occurs when treatment is withdrawn because the patient is too
sick to continue therapy, this would be an example of informative censoring. The validity of all methods discussed in this section are then invalid. Care should be taken to make sure you understand
the censoring mechanism as to avoid any false inferences drawn.
Assessment of the proportional hazards assumption is an important part of any Cox regression analysis. We refer you to the references at the end of this chapter for strategies and alternatives for
when the proportional hazards assumption breaks down. In some circumstances, the proportional hazards assumption is not valid, and alternative approaches can be used. As is always the case, when
outcomes are dependent (e.g., one patient may contribute more than one observation), the methods discussed in this section should not be used directly. Generally the standard error estimates will be
too small, and p-values will be incorrect. The concerns in logistic regression regarding outliers, co-linearity, missing data, and covariates with sparse outcomes apply here as well, as do the
concerns about model misspecification for continuous covariates. | {"url":"https://ehrapy.readthedocs.io/en/stable/tutorials/notebooks/mimic_2_survival_analysis.html","timestamp":"2024-11-15T01:21:20Z","content_type":"text/html","content_length":"216730","record_id":"<urn:uuid:673a24a0-cef6-4938-aba4-212b784096b4>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00362.warc.gz"} |
The ultimate 4x4 approach-, departure- and breakover angles list
Some people would say that the 4×4 is the perfect car for any off-road adventure. However, not all 4x4s are created equally. There is a lot to consider when choosing a 4×4 and knowing how to handle
it, but the most important thing to consider are the approach angle, departure angle, and breakover angle. If a vehicle is not set up for these angles, there can be poor clearance and difficulty with
getting over or off obstacles.
Good approach-, departure- and breakover angles will help the driver to maintain control and confidence when driving and prevent damage to the underbody and suspension parts. The article will cover
all angles for the most popular 4×4 vehicles. Even though we researched these angles thoroughly, always drive sensibly, and if you aren’t comfortable with driving at certain angles then don’t do it
What are these angles?
Let’s start with the angle of approach. The angle of approach is the maximum angle of a ramp onto which a vehicle can be crawled up, without its bumper scraping along it. The departure angle, on the
contrary, is the maximum angle a vehicle can descend without the tail bumper dragging on the ground. The angle between the middle of your car’s underside and the tires is the breakover angle.
Knowledge about your breakover angle is necessary for you to drive over an object without the four tires coming off the ground and losing traction. You get the point by know… knowing these angles of
your car help you in getting over, or through, the obstacles in front of you.
How to check your angles
It is always recommended to keep an eye on your angles when offroading. There are many great mobile apps that are developed for only this purpose. We are however very keen on more analog systems. A
cheap and really handy tool is this inclinometer on which you can easily read out your driving angles. You can buy it over here for less than 30 dollar – 4X4 Inclinometer with Light – Black – Amazon
List of approach-, departure- and breakover angles per vehicle:
Vehicle Approach Angle Departure Angle Breakover Angle
Alfa Romeo Stelvio 26 degrees 24 degrees 19 degrees
BMW X3 25.7 degrees 22.6 degrees 19.4 degrees
BMW X5 25 degrees 24 degrees 19.7 degrees
BMW X6 22.4 degrees 20.7 degrees 17.9 degrees
BMW X7 23.1 degrees 20.5 degrees 19.8 degrees
Chevrolet Silverado 1500LT Trail Boss 27.2 degrees 25.8 degrees 20.4 degrees
Chevrolet Suburban Z71 34.5 degrees 22 degrees 19 degrees
Chevrolet Tahoe Z71 34.5 degrees 22.5 degrees 22 degrees
Ford Bronco (2021 model) 30.4 degrees 33.1 degrees 20.4 degrees
Ford Expedition 4×4 (wheelbase 122.5) 23.3 degrees 21.9 degrees 21.4 degrees
Ford Expedition 4×4 (wheelbase 131.6) 23.7 degrees 21.3 degrees 20.1 degrees
Ford Explorer 21 degrees 22 degrees 17.1 degrees
Ford F-150 Raptor (2021 model) 33.1 degrees 24.9 degrees 24.4 degrees
Ford F-250 Super Duty Tremor 31.7 degrees 24.5 degrees 21.5 degrees
Ford F-350 Super Duty Tremor 31.7 degrees 24.5 degrees 21.5 degrees
Ford Ranger (2019) 28.7 degrees 25.4 degrees 21.5 degrees
Ford Raptor 31 degrees 23.9 degrees 22.7 degrees
GMC Canyon AT4 29.5 degrees 25.8 degrees 20.4 degrees
GMC Sierra 1500 27.2 degrees 25.8 degrees 20.4 degrees
Jeep Cherokee Compass Trailhawk 30.3 degrees 33.6 degrees 24.4 degrees
Jeep Cherokee Grand Trailhawk 32.2 degrees 24 degrees 27.1 degrees
Jeep Cherokee Trailhawk 32.2 degrees 23 degrees 27.1 degrees
Jeep Gladiator (2021 model) 43.6 degrees 26 degrees 20.3 degrees
Jeep Renegade Trailhawk (2018 model) 30.5 degrees 34.3 degrees 25.7 degrees
Jeep Wrangler 2-door Rubicon (2020 model) 44.0 degrees 37.0 degrees 27.8 degrees
Jeep Wrangler 2-door sport (2020 model) 41.4 degrees 35.9 degrees 25.0 degrees
Jeep Wrangler 4-door Rubicon (2020 model) 43.9 degrees 37.0 degrees 22.6 degrees
Jeep Wrangler 4-door sport (2020 model) 41.4 degrees 36.1 degrees 20.3 degrees
Land Rover Defender 110 50 degrees 35 degrees 30 degrees
Land Rover Defender 110 (2021 Model) 38.0 degrees 40.0 degrees 29 degrees
Land Rover Defender 130 50 degrees 35 degrees 27 degrees
Land Rover Defender 90 47 degrees 47.0 degrees 33 degrees
Land Rover Defender 90 (2021 Model) 38.0 degrees 40.0 degrees 31 degrees
Land Rover Discovery (2021 model) 34 degrees 30 degrees 27.5 degrees
Land Rover Discovery 2 31 degrees 21 degrees 24 degrees
Land Rover Discovery 3 37.2 degrees 29.6 degrees 27.5 degrees
Lexus GX 460 (2020 model) 21 degrees 23 degrees 21 degrees
Maserati Levante 22 degrees 26 degrees 20 degrees
Mercedes G63 6×6 52 degrees 54 degrees 22 degrees
Mercedes-Benz G-class (2019 model) 30.9 degrees 29.9 degrees 23.5 degrees
Mitsubishi L200 30 degrees 22 degrees 24 degrees
Mitsubishi Outlander 22.5 degrees 22.5 degrees 21 degrees
Mitsubishi Pajero (2015 model) 30 degrees 24.2 degrees 23.1 degrees
Mitsubishi Pajero (2020 model) 36.6 degrees 25 degrees 22.5 degrees
Nissan Navara 32.4 degrees 26.7 degrees 22.2 degrees
Nissan Patrol 2020 34.4 degrees 26.3 degrees 24.4 degrees
Nissan Patrol Y61 37 degrees 31 degrees 27 degrees
Nissan Titan XD Pro-4X 22.8 degrees 26.8 degrees 21.7 degrees
Porsche Cayenne 31.8 degrees 25.4 degrees 25 degrees
Porsche Macan 16.9 degrees 23.6 degrees 16.9 degrees
Ram 1500 Rebel (2020 model) 23 degrees 27.2 degrees 23 degrees
Ram TRX 30.2 degrees 23.5 degrees 21.9 degrees
Range Rover Sport 26 degrees 26.2 degrees 21.2 degrees
Range Rover Velar 27.5 degrees 30.2 degrees 24.9 degrees
Range Rover Vogue 33 degrees 30 degrees 25.7 degrees
Suzuki Jimny (2019 and up model) 37 degrees 49 degrees 28 degrees
Suzuki Jimny (older models) 34 degrees 46 degrees 31 degrees
Toyota 4Runner TRD Pro 33 degrees 26 degrees 19.8 degrees
Toyota FJ Cruiser 34 degrees 30 degrees 24.7 degrees
Toyota Landcruiser 120 series 34 degrees 28 degrees 20 degrees
Toyota Landcruiser 150 series 32 degrees 24 degrees 32 degrees
Toyota Landcruiser 2021 32 degrees 24 degrees 21 degrees
Toyota Landcruiser 70 series 36 degrees 27 degrees 26.0 degrees
Toyota Landcruiser 80 series 37 degrees 25 degrees 26.0 degrees
Toyota Landcruiser 90 series 36 degrees 30 degrees 28 degrees
Toyota Sequoia TRD Pro 27 degrees 21 degrees 23.9 degrees
Toyota Tacoma TRD Pro 35 degrees 23.9 degrees 23.9 degrees
Toyota Tundra TRD Pro 31 degrees 22 degrees 23.9 degrees
Volkswagen Amarok 28 degrees 23.6 degrees 23 degrees
Volkswagen Tiguan 26.2 degrees 23.3 degrees 20 degrees
Volkswagen Touareg (2018 models and up) 28 degrees 28 degrees 22 degrees
Approach angle, departure angle and breakover angle of a Alfa Romeo Stelvio
A Alfa Romeo Stelvio has an approach angle of 26 degrees, a departure angle of 24 degrees and a breakover angle of 19 degrees.
Approach angle, departure angle and breakover angle of a BMW X3
A BMW X3 has an approach angle of 25.7 degrees, a departure angle of 22.6 degrees and a breakover angle of 19.4 degrees.
Approach angle, departure angle and breakover angle of a BMW X5
A BMW X5 has an approach angle of 25 degrees, a departure angle of 24 degrees and a breakover angle of 19.7 degrees.
Approach angle, departure angle and breakover angle of a BMW X6
A BMW X6 has an approach angle of 22.4 degrees, a departure angle of 20.7 degrees and a breakover angle of 17.9 degrees.
Approach angle, departure angle and breakover angle of a BMW X7
A BMW X7 has an approach angle of 23.1 degrees, a departure angle of 20.5 degrees and a breakover angle of 19.8 degrees.
Approach angle, departure angle and breakover angle of a Chevrolet Silverado 1500LT Trail Boss
A Chevrolet Silverado 1500LT Trail Boss has an approach angle of 27.2 degrees, a departure angle of 25.8 degrees and a breakover angle of 20.4 degrees.
Approach angle, departure angle and breakover angle of a Chevrolet Suburban Z71
A Chevrolet Suburban Z71 has an approach angle of 34.5 degrees, a departure angle of 22 degrees and a breakover angle of 19 degrees.
Approach angle, departure angle and breakover angle of a Chevrolet Tahoe Z71
A Chevrolet Tahoe Z71 has an approach angle of 34.5 degrees, a departure angle of 22.5 degrees and a breakover angle of 22 degrees.
Approach angle, departure angle and breakover angle of a Ford Bronco (2021 model)
A Ford Bronco (2021 model) has an approach angle of 30.4 degrees, a departure angle of 33.1 degrees and a breakover angle of 20.4 degrees.
Approach angle, departure angle and breakover angle of a Ford Expedition 4×4 (wheelbase 122.5)
A Ford Expedition 4×4 (wheelbase 122.5) has an approach angle of 23.3 degrees, a departure angle of 21.9 degrees and a breakover angle of 21.4 degrees.
Approach angle, departure angle and breakover angle of a Ford Expedition 4×4 (wheelbase 131.6)
A Ford Expedition 4×4 (wheelbase 131.6) has an approach angle of 23.7 degrees, a departure angle of 21.3 degrees and a breakover angle of 20.1 degrees.
Approach angle, departure angle and breakover angle of a Ford Explorer
A Ford Explorer has an approach angle of 21 degrees, a departure angle of 22 degrees and a breakover angle of 17.1 degrees.
Approach angle, departure angle and breakover angle of a Ford F-150 Raptor (2021 model)
A Ford F-150 Raptor (2021 model) has an approach angle of 33.1 degrees, a departure angle of 24.9 degrees and a breakover angle of 24.4 degrees.
Approach angle, departure angle and breakover angle of a Ford F-250 Super Duty Tremor
A Ford F-250 Super Duty Tremor has an approach angle of 31.7 degrees, a departure angle of 24.5 degrees and a breakover angle of 21.5 degrees.
Approach angle, departure angle and breakover angle of a Ford F-350 Super Duty Tremor
A Ford F-350 Super Duty Tremor has an approach angle of 31.7 degrees, a departure angle of 24.5 degrees and a breakover angle of 21.5 degrees.
Approach angle, departure angle and breakover angle of a Ford Ranger (2019)
A Ford Ranger (2019) has an approach angle of 28.7 degrees, a departure angle of 25.4 degrees and a breakover angle of 21.5 degrees.
Approach angle, departure angle and breakover angle of a Ford Raptor
A Ford Raptor has an approach angle of 31 degrees, a departure angle of 23.9 degrees and a breakover angle of 22.7 degrees.
Approach angle, departure angle and breakover angle of a GMC Canyon AT4
A GMC Canyon AT4 has an approach angle of 29.5 degrees, a departure angle of 25.8 degrees and a breakover angle of 20.4 degrees.
Approach angle, departure angle and breakover angle of a GMC Sierra 1500
A GMC Sierra 1500 has an approach angle of 27.2 degrees, a departure angle of 25.8 degrees and a breakover angle of 20.4 degrees.
Approach angle, departure angle and breakover angle of a Jeep Cherokee Compass Trailhawk
A Jeep Cherokee Compass Trailhawk has an approach angle of 30.3 degrees, a departure angle of 33.6 degrees and a breakover angle of 24.4 degrees.
Approach angle, departure angle and breakover angle of a Jeep Cherokee Grand Trailhawk
A Jeep Cherokee Grand Trailhawk has an approach angle of 32.2 degrees, a departure angle of 24 degrees and a breakover angle of 27.1 degrees.
Approach angle, departure angle and breakover angle of a Jeep Cherokee Trailhawk
A Jeep Cherokee Trailhawk has an approach angle of 32.2 degrees, a departure angle of 23 degrees and a breakover angle of 27.1 degrees.
Approach angle, departure angle and breakover angle of a Jeep Gladiator (2021 model)
A Jeep Gladiator (2021 model) has an approach angle of 43.6 degrees, a departure angle of 26 degrees and a breakover angle of 20.3 degrees.
Approach angle, departure angle and breakover angle of a Jeep Renegade Trailhawk (2018 model)
A Jeep Renegade Trailhawk (2018 model) has an approach angle of 30.5 degrees, a departure angle of 34.3 degrees and a breakover angle of 25.7 degrees.
Approach angle, departure angle and breakover angle of a Jeep Wrangler 2-door Rubicon (2020 model)
A Jeep Wrangler 2-door Rubicon (2020 model) has an approach angle of 44.0 degrees, a departure angle of 37.0 degrees and a breakover angle of 27.8 degrees.
Approach angle, departure angle and breakover angle of a Jeep Wrangler 2-door sport (2020 model)
A Jeep Wrangler 2-door sport (2020 model) has an approach angle of 41.4 degrees, a departure angle of 35.9 degrees and a breakover angle of 25.0 degrees.
Approach angle, departure angle and breakover angle of a Jeep Wrangler 4-door Rubicon (2020 model)
A Jeep Wrangler 4-door Rubicon (2020 model) has an approach angle of 43.9 degrees, a departure angle of 37.0 degrees and a breakover angle of 22.6 degrees.
Approach angle, departure angle and breakover angle of a Jeep Wrangler 4-door sport (2020 model)
A Jeep Wrangler 4-door sport (2020 model) has an approach angle of 41.4 degrees, a departure angle of 36.1 degrees and a breakover angle of 20.3 degrees.
Approach angle, departure angle and breakover angle of a Land Rover Defender 110
A Land Rover Defender 110 has an approach angle of 50 degrees, a departure angle of 35 degrees and a breakover angle of 30 degrees.
Approach angle, departure angle and breakover angle of a Land Rover Defender 110 (2021 Model)
A Land Rover Defender 110 (2021 Model) has an approach angle of 38.0 degrees, a departure angle of 40.0 degrees and a breakover angle of 29 degrees.
Approach angle, departure angle and breakover angle of a Land Rover Defender 130
A Land Rover Defender 130 has an approach angle of 50 degrees, a departure angle of 35 degrees and a breakover angle of 27 degrees.
Approach angle, departure angle and breakover angle of a Land Rover Defender 90
A Land Rover Defender 90 has an approach angle of 47 degrees, a departure angle of 47.0 degrees and a breakover angle of 33 degrees.
Approach angle, departure angle and breakover angle of a Land Rover Defender 90 (2021 Model)
A Land Rover Defender 90 (2021 Model) has an approach angle of 38.0 degrees, a departure angle of 40.0 degrees and a breakover angle of 31 degrees.
Approach angle, departure angle and breakover angle of a Land Rover Discovery (2021 model)
A Land Rover Discovery (2021 model) has an approach angle of 34 degrees, a departure angle of 30 degrees and a breakover angle of 27.5 degrees.
Approach angle, departure angle and breakover angle of a Land Rover Discovery 2
A Land Rover Discovery 2 has an approach angle of 31 degrees, a departure angle of 21 degrees and a breakover angle of 24 degrees.
Approach angle, departure angle and breakover angle of a Land Rover Discovery 3
A Land Rover Discovery 3 has an approach angle of 37.2 degrees, a departure angle of 29.6 degrees and a breakover angle of 27.5 degrees.
Approach angle, departure angle and breakover angle of a Lexus GX 460 (2020 model)
A Lexus GX 460 (2020 model) has an approach angle of 21 degrees, a departure angle of 23 degrees and a breakover angle of 21 degrees.
Approach angle, departure angle and breakover angle of a Maserati Levante
A Maserati Levante has an approach angle of 22 degrees, a departure angle of 26 degrees and a breakover angle of 20 degrees.
Approach angle, departure angle and breakover angle of a Mercedes G63 6×6
A Mercedes G63 6×6 has an approach angle of 52 degrees, a departure angle of 54 degrees and a breakover angle of 22 degrees.
Approach angle, departure angle and breakover angle of a Mercedes-Benz G-class (2019 model)
A Mercedes-Benz G-class (2019 model) has an approach angle of 30.9 degrees, a departure angle of 29.9 degrees and a breakover angle of 23.5 degrees.
Approach angle, departure angle and breakover angle of a Mitsubishi L200
A Mitsubishi L200 has an approach angle of 30 degrees, a departure angle of 22 degrees and a breakover angle of 24 degrees.
Approach angle, departure angle and breakover angle of a Mitsubishi Outlander
A Mitsubishi Outlander has an approach angle of 22.5 degrees, a departure angle of 22.5 degrees and a breakover angle of 21 degrees.
Approach angle, departure angle and breakover angle of a Mitsubishi Pajero (2015 model)
A Mitsubishi Pajero (2015 model) has an approach angle of 30 degrees, a departure angle of 24.2 degrees and a breakover angle of 23.1 degrees.
Approach angle, departure angle and breakover angle of a Mitsubishi Pajero (2020 model)
A Mitsubishi Pajero (2020 model) has an approach angle of 36.6 degrees, a departure angle of 25 degrees and a breakover angle of 22.5 degrees.
Approach angle, departure angle and breakover angle of a Nissan Navara
A Nissan Navara has an approach angle of 32.4 degrees, a departure angle of 26.7 degrees and a breakover angle of 22.2 degrees.
Approach angle, departure angle and breakover angle of a Nissan Patrol 2020
A Nissan Patrol 2020 has an approach angle of 34.4 degrees, a departure angle of 26.3 degrees and a breakover angle of 24.4 degrees.
Approach angle, departure angle and breakover angle of a Nissan Patrol Y61
A Nissan Patrol Y61 has an approach angle of 37 degrees, a departure angle of 31 degrees and a breakover angle of 27 degrees.
Approach angle, departure angle and breakover angle of a Nissan Titan XD Pro-4X
A Nissan Titan XD Pro-4X has an approach angle of 22.8 degrees, a departure angle of 26.8 degrees and a breakover angle of 21.7 degrees.
Approach angle, departure angle and breakover angle of a Porsche Cayenne
A Porsche Cayenne has an approach angle of 31.8 degrees, a departure angle of 25.4 degrees and a breakover angle of 25 degrees.
Approach angle, departure angle and breakover angle of a Porsche Macan
A Porsche Macan has an approach angle of 16.9 degrees, a departure angle of 23.6 degrees and a breakover angle of 16.9 degrees.
Approach angle, departure angle and breakover angle of a Ram 1500 Rebel (2020 model)
A Ram 1500 Rebel (2020 model) has an approach angle of 23 degrees, a departure angle of 27.2 degrees and a breakover angle of 23 degrees.
Approach angle, departure angle and breakover angle of a Ram TRX
A Ram TRX has an approach angle of 30.2 degrees, a departure angle of 23.5 degrees and a breakover angle of 21.9 degrees.
Approach angle, departure angle and breakover angle of a Range Rover Sport
A Range Rover Sport has an approach angle of 26 degrees, a departure angle of 26.2 degrees and a breakover angle of 21.2 degrees.
Approach angle, departure angle and breakover angle of a Range Rover Velar
A Range Rover Velar has an approach angle of 27.5 degrees, a departure angle of 30.2 degrees and a breakover angle of 24.9 degrees.
Approach angle, departure angle and breakover angle of a Range Rover Vogue
A Range Rover Vogue has an approach angle of 33 degrees, a departure angle of 30 degrees and a breakover angle of 25.7 degrees.
Approach angle, departure angle and breakover angle of a Suzuki Jimny (2019 and up model)
A Suzuki Jimny (2019 and up model) has an approach angle of 37 degrees, a departure angle of 49 degrees and a breakover angle of 28 degrees.
Approach angle, departure angle and breakover angle of a Suzuki Jimny (older models)
A Suzuki Jimny (older models) has an approach angle of 34 degrees, a departure angle of 46 degrees and a breakover angle of 31 degrees.
Approach angle, departure angle and breakover angle of a Toyota 4Runner TRD Pro
A Toyota 4Runner TRD Pro has an approach angle of 33 degrees, a departure angle of 26 degrees and a breakover angle of 19.8 degrees.
Approach angle, departure angle and breakover angle of a Toyota FJ Cruiser
A Toyota FJ Cruiser has an approach angle of 34 degrees, a departure angle of 30 degrees and a breakover angle of 24.7 degrees.
Approach angle, departure angle and breakover angle of a Toyota Landcruiser 120 series
A Toyota Landcruiser 120 series has an approach angle of 34 degrees, a departure angle of 28 degrees and a breakover angle of 20 degrees.
Approach angle, departure angle and breakover angle of a Toyota Landcruiser 150 series
A Toyota Landcruiser 150 series has an approach angle of 32 degrees, a departure angle of 24 degrees and a breakover angle of 32 degrees.
Approach angle, departure angle and breakover angle of a Toyota Landcruiser 2021
A Toyota Landcruiser 2021 has an approach angle of 32 degrees, a departure angle of 24 degrees and a breakover angle of 21 degrees.
Approach angle, departure angle and breakover angle of a Toyota Landcruiser 70 series
A Toyota Landcruiser 70 series has an approach angle of 36 degrees, a departure angle of 27 degrees and a breakover angle of 26.0 degrees.
Approach angle, departure angle and breakover angle of a Toyota Landcruiser 80 series
A Toyota Landcruiser 80 series has an approach angle of 37 degrees, a departure angle of 25 degrees and a breakover angle of 26.0 degrees.
Approach angle, departure angle and breakover angle of a Toyota Landcruiser 90 series
A Toyota Landcruiser 90 series has an approach angle of 36 degrees, a departure angle of 30 degrees and a breakover angle of 28 degrees.
Approach angle, departure angle and breakover angle of a Toyota Sequoia TRD Pro
A Toyota Sequoia TRD Pro has an approach angle of 27 degrees, a departure angle of 21 degrees and a breakover angle of 23.9 degrees.
Approach angle, departure angle and breakover angle of a Toyota Tacoma TRD Pro
A Toyota Tacoma TRD Pro has an approach angle of 35 degrees, a departure angle of 23.9 degrees and a breakover angle of 23.9 degrees.
Approach angle, departure angle and breakover angle of a Toyota Tundra TRD Pro
A Toyota Tundra TRD Pro has an approach angle of 31 degrees, a departure angle of 22 degrees and a breakover angle of 23.9 degrees.
Approach angle, departure angle and breakover angle of a Volkswagen Amarok
A Volkswagen Amarok has an approach angle of 28 degrees, a departure angle of 23.6 degrees and a breakover angle of 23 degrees.
Approach angle, departure angle and breakover angle of a Volkswagen Tiguan
A Volkswagen Tiguan has an approach angle of 26.2 degrees, a departure angle of 23.3 degrees and a breakover angle of 20 degrees.
Approach angle, departure angle and breakover angle of a Volkswagen Touareg (2018 models and up)
A Volkswagen Touareg (2018 models and up) has an approach angle of 28 degrees, a departure angle of 28 degrees and a breakover angle of 22 degrees.
Roll-over angle
Besides the approach-, departure- and breakover angle there is also something called the roll-over angle. This is also called the traverse angle. It’s the angle on which the car will tip over on its
side. This angle isn’t a constant factor. If you have a rooftop tent for example you have a much lower traverse angle than the same car without a rooftop tent. Talking about rooftop tents: maybe you
like our post about rooftop tents and cold weather as well.
1 COMMENT
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The tactic language
Chapter 9 The tactic language
This chapter gives a compact documentation of Ltac, the tactic language available in Coq. We start by giving the syntax, and next, we present the informal semantics. If you want to know more
regarding this language and especially about its foundations, you can refer to [41]. Chapter 10 is devoted to giving examples of use of this language on small but also with non-trivial problems.
9.1 Syntax
The syntax of the tactic language is given Figures 9.1 and 9.2. See Chapter 1 for a description of the BNF metasyntax used in these grammar rules. Various already defined entries will be used in this
chapter: entries natural, integer, ident, qualid, term, cpattern and atomic_tactic represent respectively the natural and integer numbers, the authorized identificators and qualified names, Coq’s
terms and patterns and all the atomic tactics described in Chapter 8. The syntax of cpattern is the same as that of terms, but it is extended with pattern matching metavariables. In cpattern, a
pattern-matching metavariable is represented with the syntax ?id where id is an ident. The notation _ can also be used to denote metavariable whose instance is irrelevant. In the notation ?id, the
identifier allows us to keep instantiations and to make constraints whereas _ shows that we are not interested in what will be matched. On the right hand side of pattern-matching clauses, the named
metavariable are used without the question mark prefix. There is also a special notation for second-order pattern-matching problems: in an applicative pattern of the form @?id id[1] …id[n], the
variable id matches any complex expression with (possible) dependencies in the variables id[1] …id[n] and returns a functional term of the form fun id[1] …id[n] => term.
The main entry of the grammar is expr. This language is used in proof mode but it can also be used in toplevel definitions as shown in Figure 9.3.
1. The infix tacticals “… || …” and “… ; …” are associative.
2. In tacarg, there is an overlap between qualid as a direct tactic argument and qualid as a particular case of term. The resolution is done by first looking for a reference of the tactic language
and if it fails, for a reference to a term. To force the resolution as a reference of the tactic language, use the form ltac : qualid. To force the resolution as a reference to a term, use the
syntax (qualid).
3. As shown by the figure, tactical || binds more than the prefix tacticals try, repeat, do, info and abstract which themselves bind more than the postfix tactical “… ;[ … ]” which binds more than
“… ; …”.
For instance
try repeat tactic[1] || tactic[2];tactic[3];[tactic[31]|…|tactic[3n]];tactic[4].
is understood as
(try (repeat (tactic[1] || tactic[2])));
expr ::= expr ; expr
| expr ; [ expr | … | expr ]
| tacexpr[3]
tacexpr[3] ::= do (natural | ident) tacexpr[3]
| info tacexpr[3]
| progress tacexpr[3]
| repeat tacexpr[3]
| try tacexpr[3]
| tacexpr[2]
tacexpr[2] ::= tacexpr[1] || tacexpr[3]
| tacexpr[1]
tacexpr[1] ::= fun name … name => atom
| let [rec] let_clause with … with let_clause in atom
| match goal with context_rule | … | context_rule end
| match reverse goal with context_rule | … | context_rule end
| match expr with match_rule | … | match_rule end
| lazymatch goal with context_rule | … | context_rule end
| lazymatch reverse goal with context_rule | … | context_rule end
| lazymatch expr with match_rule | … | match_rule end
| abstract atom
| abstract atom using ident
| first [ expr | … | expr ]
| solve [ expr | … | expr ]
| idtac [message_token … message_token]
| fail [natural] [message_token … message_token]
| fresh | fresh string
| context ident [ term ]
| eval redexpr in term
| type of term
| external string string tacarg … tacarg
| constr : term
| atomic_tactic
| qualid tacarg … tacarg
| atom
atom ::= qualid
| ()
| integer
| ( expr )
message_token ::= string | ident | integer
tacarg ::= qualid
| ()
| ltac : atom
| term
let_clause ::= ident [name … name] := expr
context_rule ::= context_hyp , … , context_hyp |-cpattern => expr
| |- cpattern => expr
| _ => expr
context_hyp ::= name : cpattern
| name := cpattern [: cpattern]
match_rule ::= cpattern => expr
| context [ident] [ cpattern ] => expr
| appcontext [ident] [ cpattern ] => expr
| _ => expr
top ::= [Local] Ltac ltac_def with … with ltac_def
ltac_def ::= ident [ident … ident] := expr
| qualid [ident … ident] ::=expr
9.2 Semantics
Tactic expressions can only be applied in the context of a goal. The evaluation yields either a term, an integer or a tactic. Intermediary results can be terms or integers but the final result must
be a tactic which is then applied to the current goal.
There is a special case for match goal expressions of which the clauses evaluate to tactics. Such expressions can only be used as end result of a tactic expression (never as argument of a non
recursive local definition or of an application).
The rest of this section explains the semantics of every construction of Ltac.
A sequence is an expression of the following form:
expr[1] ; expr[2]
The expressions expr[1] and expr[2] are evaluated to v[1] and v[2] which have to be tactic values. The tactic v[1] is then applied and v[2] is applied to every subgoal generated by the application of
v[1]. Sequence is left-associative.
General sequence
A general sequence has the following form:
expr[0] ; [ expr[1] | ... | expr[n] ]
The expressions expr[i] are evaluated to v[i], for i=0,...,n and all have to be tactics. The tactic v[0] is applied and v[i] is applied to the i-th generated subgoal by the application of v[0], for =
1,...,n. It fails if the application of v[0] does not generate exactly n subgoals.
1. If no tactic is given for the i-th generated subgoal, it behaves as if the tactic idtac were given. For instance, split ; [ | auto ] is a shortcut for split ; [ idtac | auto ].
2. expr[0] ; [ expr[1] | ... | expr[i] | .. | expr[i+1+j] | ... | expr[n] ]
In this variant, idtac is used for the subgoals numbered from i+1 to i+j (assuming n is the number of subgoals).
Note that .. is part of the syntax, while ... is the meta-symbol used to describe a list of expr of arbitrary length.
3. expr[0] ; [ expr[1] | ... | expr[i] | expr .. | expr[i+1+j] | ... | expr[n] ]
In this variant, expr is used instead of idtac for the subgoals numbered from i+1 to i+j.
For loop
There is a for loop that repeats a tactic num times:
do num expr
expr is evaluated to v. v must be a tactic value. v is applied num times. Supposing num>1, after the first application of v, v is applied, at least once, to the generated subgoals and so on. It fails
if the application of v fails before the num applications have been completed.
Repeat loop
We have a repeat loop with:
repeat expr
expr is evaluated to v. If v denotes a tactic, this tactic is applied to the goal. If the application fails, the tactic is applied recursively to all the generated subgoals until it eventually fails.
The recursion stops in a subgoal when the tactic has failed. The tactic repeat expr itself never fails.
Error catching
We can catch the tactic errors with:
try expr
expr is evaluated to v. v must be a tactic value. v is applied. If the application of v fails, it catches the error and leaves the goal unchanged. If the level of the exception is positive, then the
exception is re-raised with its level decremented.
Detecting progress
We can check if a tactic made progress with:
progress expr
expr is evaluated to v. v must be a tactic value. v is applied. If the application of v produced one subgoal equal to the initial goal (up to syntactical equality), then an error of level 0 is
Error message: Failed to progress
We can easily branch with the following structure:
expr[1] || expr[2]
expr[1] and expr[2] are evaluated to v[1] and v[2]. v[1] and v[2] must be tactic values. v[1] is applied and if it fails to progress then v[2] is applied. Branching is left-associative.
First tactic to work
We may consider the first tactic to work (i.e. which does not fail) among a panel of tactics:
first [ expr[1] | ... | expr[n] ]
expr[i] are evaluated to v[i] and v[i] must be tactic values, for i=1,...,n. Supposing n>1, it applies v[1], if it works, it stops else it tries to apply v[2] and so on. It fails when there is no
applicable tactic.
Error message: No applicable tactic
We may consider the first to solve (i.e. which generates no subgoal) among a panel of tactics:
solve [ expr[1] | ... | expr[n] ]
expr[i] are evaluated to v[i] and v[i] must be tactic values, for i=1,...,n. Supposing n>1, it applies v[1], if it solves, it stops else it tries to apply v[2] and so on. It fails if there is no
solving tactic.
Error message: Cannot solve the goal
The constant idtac is the identity tactic: it leaves any goal unchanged but it appears in the proof script.
Variant: idtac message_token … message_token
This prints the given tokens. Strings and integers are printed literally. If a (term) variable is given, its contents are printed.
The tactic fail is the always-failing tactic: it does not solve any goal. It is useful for defining other tacticals since it can be catched by try or match goal.
1. fail n
The number n is the failure level. If no level is specified, it defaults to 0. The level is used by try and match goal. If 0, it makes match goal considering the next clause (backtracking). If
non zero, the current match goal block or try command is aborted and the level is decremented.
2. fail message_token … message_token
The given tokens are used for printing the failure message.
3. fail n message_token … message_token
This is a combination of the previous variants.
Error message: Tactic Failure message (level n).
Local definitions
Local definitions can be done as follows:
let ident[1] := expr[1]
with ident[2] := expr[2]
with ident[n] := expr[n] in
each expr[i] is evaluated to v[i], then, expr is evaluated by substituting v[i] to each occurrence of ident[i], for i=1,...,n. There is no dependencies between the expr[i] and the ident[i].
Local definitions can be recursive by using let rec instead of let. In this latter case, the definitions are evaluated lazily so that the rec keyword can be used also in non recursive cases so as to
avoid the eager evaluation of local definitions.
An application is an expression of the following form:
qualid tacarg[1] ... tacarg[n]
The reference qualid must be bound to some defined tactic definition expecting at least n arguments. The expressions expr[i] are evaluated to v[i], for i=1,...,n.
Function construction
A parameterized tactic can be built anonymously (without resorting to local definitions) with:
fun ident[1] ... ident[n] => expr
Indeed, local definitions of functions are a syntactic sugar for binding a fun tactic to an identifier.
Pattern matching on terms
We can carry out pattern matching on terms with:
match expr with
cpattern[1] => expr[1]
| cpattern[2] => expr[2]
| cpattern[n] => expr[n]
| _ => expr[n+1]
The expression expr is evaluated and should yield a term which is matched against cpattern[1]. The matching is non-linear: if a metavariable occurs more than once, it should match the same expression
every time. It is first-order except on the variables of the form @?id that occur in head position of an application. For these variables, the matching is second-order and returns a functional term.
If the matching with cpattern[1] succeeds, then expr[1] is evaluated into some value by substituting the pattern matching instantiations to the metavariables. If expr[1] evaluates to a tactic and the
match expression is in position to be applied to a goal (e.g. it is not bound to a variable by a let in), then this tactic is applied. If the tactic succeeds, the list of resulting subgoals is the
result of the match expression. If expr[1] does not evaluate to a tactic or if the match expression is not in position to be applied to a goal, then the result of the evaluation of expr[1] is the
result of the match expression.
If the matching with cpattern[1] fails, or if it succeeds but the evaluation of expr[1] fails, or if the evaluation of expr[1] succeeds but returns a tactic in execution position whose execution
fails, then cpattern[2] is used and so on. The pattern _ matches any term and shunts all remaining patterns if any. If all clauses fail (in particular, there is no pattern _) then a
no-matching-clause error is raised.
Error messages:
1. Using lazymatch instead of match has an effect if the right-hand-side of a clause returns a tactic. With match, the tactic is applied to the current goal (and the next clause is tried if it
fails). With lazymatch, the tactic is directly returned as the result of the whole lazymatch block without being first tried to be applied to the goal. Typically, if the lazymatch block is bound
to some variable x in a let in, then tactic expression gets bound to the variable x.
2. There is a special form of patterns to match a subterm against the pattern:
context ident [ cpattern ]
It matches any term which one subterm matches cpattern. If there is a match, the optional ident is assign the “matched context”, that is the initial term where the matched subterm is replaced by
a hole. The definition of context in expressions below will show how to use such term contexts.
If the evaluation of the right-hand-side of a valid match fails, the next matching subterm is tried. If no further subterm matches, the next clause is tried. Matching subterms are considered
top-bottom and from left to right (with respect to the raw printing obtained by setting option Printing All, see Section 2.9).
Coq < Ltac f x :=
Coq < match x with
Coq < context f [S ?X] =>
Coq < idtac X; (* To display the evaluation order *)
Coq < assert (p := refl_equal 1 : X=1); (* To filter the case X=1 *)
Coq < let x:= context f[O] in assert (x=O) (* To observe the context *)
Coq < end.
f is defined
Coq < Goal True.
1 subgoal
Coq < f (3+4).
2 subgoals
p : 1 = 1
1 + 4 = 0
subgoal 2 is:
3. For historical reasons, context considers n-ary applications such as (f 1 2) as a whole, and not as a sequence of unary applications ((f 1) 2). Hence context [f ?x] will fail to find a matching
subterm in (f 1 2): if the pattern is a partial application, the matched subterms will be necessarily be applications with exactly the same number of arguments. Alternatively, one may now use the
following variant of context:
appcontext ident [ cpattern ]
The behavior of appcontext is the same as the one of context, except that a matching subterm could be a partial part of a longer application. For instance, in (f 1 2), an appcontext [f ?x] will
find the matching subterm (f 1).
Pattern matching on goals
We can make pattern matching on goals using the following expression:
match goal with
| hyp[1,1],...,hyp[1,m[1]] |-cpattern[1]=> expr[1]
| hyp[2,1],...,hyp[2,m[2]] |-cpattern[2]=> expr[2]
| hyp[n,1],...,hyp[n,m[n]] |-cpattern[n]=> expr[n]
|_ => expr[n+1]
If each hypothesis pattern hyp[1,i], with i=1,...,m[1] is matched (non-linear first-order unification) by an hypothesis of the goal and if cpattern[1] is matched by the conclusion of the goal, then
expr[1] is evaluated to v[1] by substituting the pattern matching to the metavariables and the real hypothesis names bound to the possible hypothesis names occurring in the hypothesis patterns. If v
[1] is a tactic value, then it is applied to the goal. If this application fails, then another combination of hypotheses is tried with the same proof context pattern. If there is no other combination
of hypotheses then the second proof context pattern is tried and so on. If the next to last proof context pattern fails then expr[n+1] is evaluated to v[n+1] and v[n+1] is applied. Note also that
matching against subterms (using the context ident [ cpattern ]) is available and may itself induce extra backtrackings.
Error message: No matching clauses for match goal
No clause succeeds, i.e. all matching patterns, if any, fail at the application of the right-hand-side.
It is important to know that each hypothesis of the goal can be matched by at most one hypothesis pattern. The order of matching is the following: hypothesis patterns are examined from the right to
the left (i.e. hyp[i,m[i]] before hyp[i,1]). For each hypothesis pattern, the goal hypothesis are matched in order (fresher hypothesis first), but it possible to reverse this order (older first) with
the match reverse goal with variant.
Variant: Using lazymatch instead of match has an effect if the right-hand-side of a clause returns a tactic. With match, the tactic is applied to the current goal (and the next clause is tried if it
fails). With lazymatch, the tactic is directly returned as the result of the whole lazymatch block without being first tried to be applied to the goal. Typically, if the lazymatch block is bound to
some variable x in a let in, then tactic expression gets bound to the variable x.
Coq < Ltac test_lazy :=
Coq < lazymatch goal with
Coq < | _ => idtac "here"; fail
Coq < | _ => idtac "wasn’t lazy"; trivial
Coq < end.
test_lazy is defined
Coq < Ltac test_eager :=
Coq < match goal with
Coq < | _ => idtac "here"; fail
Coq < | _ => idtac "wasn’t lazy"; trivial
Coq < end.
test_eager is defined
Coq < Goal True.
1 subgoal
Coq < test_lazy || idtac "was lazy".
was lazy
1 subgoal
Coq < test_eager || idtac "was lazy".
wasn’t lazy
Proof completed.
Filling a term context
The following expression is not a tactic in the sense that it does not produce subgoals but generates a term to be used in tactic expressions:
context ident [ expr ]
ident must denote a context variable bound by a context pattern of a match expression. This expression evaluates replaces the hole of the value of ident by the value of expr.
Error message: not a context variable
Generating fresh hypothesis names
Tactics sometimes have to generate new names for hypothesis. Letting the system decide a name with the intro tactic is not so good since it is very awkward to retrieve the name the system gave. The
following expression returns an identifier:
fresh component … component
It evaluates to an identifier unbound in the goal. This fresh identifier is obtained by concatenating the value of the component’s (each of them is, either an ident which has to refer to a name, or
directly a name denoted by a string). If the resulting name is already used, it is padded with a number so that it becomes fresh. If no component is given, the name is a fresh derivative of the name
Computing in a constr
Evaluation of a term can be performed with:
eval redexpr in term
where redexpr is a reduction tactic among red, hnf, compute, simpl, cbv, lazy, unfold, fold, pattern.
Type-checking a term
The following returns the type of term:
type of term
Accessing tactic decomposition
Tactical “info expr” is not really a tactical. For elementary tactics, this is equivalent to expr. For complex tactic like auto, it displays the operations performed by the tactic.
Proving a subgoal as a separate lemma
From the outside “abstract expr” is the same as solve expr. Internally it saves an auxiliary lemma called ident_subproofn where ident is the name of the current goal and n is chosen so that this is a
fresh name.
This tactical is useful with tactics such as omega or discriminate that generate huge proof terms. With that tool the user can avoid the explosion at time of the Save command without having to cut
manually the proof in smaller lemmas.
1. abstract expr using ident.
Give explicitly the name of the auxiliary lemma.
Error message: Proof is not complete
Calling an external tactic
The tactic external allows to run an executable outside the Coq executable. The communication is done via an XML encoding of constructions. The syntax of the command is
external "command" "request" tacarg … tacarg
The string command, to be interpreted in the default execution path of the operating system, is the name of the external command. The string request is the name of a request to be sent to the
external command. Finally the list of tactic arguments have to evaluate to terms. An XML tree of the following form is sent to the standard input of the external command.
<REQUEST req="request">
the XML tree of the first argument
the XML tree of the last argument
Conversely, the external command must send on its standard output an XML tree of the following forms:
the XML tree of a term
<CALL uri="ltac_qualified_ident">
the XML tree of a first argument
the XML tree of a last argument
where ltac_qualified_ident is the name of a defined L[tac] function and each subsequent XML tree is recursively a CALL or a TERM node.
The Document Type Definition (DTD) for terms of the Calculus of Inductive Constructions is the one developed as part of the MoWGLI European project. It can be found in the file dev/doc/cic.dtd of the
Coq source archive.
An example of parser for this DTD, written in the Objective Caml - Camlp4 language, can be found in the file parsing/g_xml.ml4 of the Coq source archive.
9.3 Tactic toplevel definitions
9.3.1 Defining L[tac] functions
Basically, L[tac] toplevel definitions are made as follows:
Ltac ident ident[1] ... ident[n] := expr
This defines a new L[tac] function that can be used in any tactic script or new L[tac] toplevel definition.
Remark: The preceding definition can equivalently be written:
Ltac ident := fun ident[1] ... ident[n] => expr
Recursive and mutual recursive function definitions are also possible with the syntax:
Ltac ident[1] ident[1,1] ... ident[1,m[1]] := expr[1]
with ident[2] ident[2,1] ... ident[2,m[2]] := expr[2]
with ident[n] ident[n,1] ... ident[n,m[n]] := expr[n]
It is also possible to redefine an existing user-defined tactic using the syntax:
Ltac qualid ident[1] ... ident[n] ::= expr
A previous definition of qualidmust exist in the environment. The new definition will always be used instead of the old one and it goes accross module boundaries.
If preceded by the keyword Local the tactic definition will not be exported outside the current module.
9.3.2 Printing L[tac] tactics
Defined L[tac] functions can be displayed using the command
Print Ltac qualid.
9.4 Debugging L[tac] tactics
The L[tac] interpreter comes with a step-by-step debugger. The debugger can be activated using the command
Set Ltac Debug.
and deactivated using the command
Unset Ltac Debug.
To know if the debugger is on, use the command Test Ltac Debug. When the debugger is activated, it stops at every step of the evaluation of the current L[tac] expression and it prints information on
what it is doing. The debugger stops, prompting for a command which can be one of the following:
simple newline: go to the next step
h: get help
x: exit current evaluation
s: continue current evaluation without stopping
rn: advance n steps further | {"url":"https://coq.inria.fr/doc/V8.3pl5/refman/Reference-Manual012.html","timestamp":"2024-11-08T02:00:47Z","content_type":"text/html","content_length":"62716","record_id":"<urn:uuid:018548b0-71df-46e4-83c2-e90a6c6e1333>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00331.warc.gz"} |
Web | Homework代写 | Racket代做 | Ai | Assignment代写 – Game Theory, Social Choice, and Mechanism
Web | Homework代写 | Racket代做 | Ai | Assignment代写 – 这是一个利用racket进行的AI方面的编程代写任务
Please read the rules for assignments on the course web page (http://www.cs.duke.edu/courses/fall18/compsci590.2/). Use Piazza (preferred) or directly contact Harsh ([email protected]), Han- rui (
[email protected]), or Vince ([email protected]) with any questions. Use Sak AI to turn in the assignment.
1. (Properties of voting rules.) Alice likes to analyze the outcomes of elections; specifically, she is interested in the different outcomes that different voting rules produce on the same votes. To
do so, she executes many different rules on the same set of votes, a painstak- ing process. She likes knowing about properties of voting rules that ease her task. For example, she likes to know
which voting rules satisfy the Condorcet criterion, so that if there is a Condorcet winner, she immediately knows that that will be the winner for those rules, without having to go through the
trouble of executing each rule individually. Recently, Alice has become interested in the phenomenon of votes cancelling out. Let us say that a set^1 Sof votescancels out with respect to voting
rule rif foreverysetT of votes, the winner^2 thatrproduces forTis the same as the winner thatrproduces forST. For example, the set of votes{a bc, bac, cab}cancels out with respect to the
plurality rule: each candidate is ranked first once in this set of votes, so it has no net effect on the outcome of the election. The same set does not cancel out with respect to Borda, though,
because from these votes,agets 4 points,bgets 3, andcgets 2, which may affect the outcome of the election. Alice likes to know when a set of votes cancels out with respect to a rule, so that she
can just ignore these votes, easing her computation of the winner. Define a pair ofopposite votesto be a pair of votes with completely opposite rankings of the candidates, i.e. the votes can be
written asc 1 c 2 .. .cm andcm cm 1 .. .c 1. Let us say that a voting rulersatisfies the Opposites Cancel Out (OCO)criterion if every pair of opposite votes cancels out with respect tor.
(^1) Technically, a multiset, since the same vote may occur multiple times. (^2) … or set of winners if there are ties.
1a. (12 points)From among the (reasonable^3 ) voting rules discussed in class, give 3 voting rules that satisfy the OCO criterion, and 3 that do not (and say which ones are which!).
Define acycleof votes to be a set of votes that can be written asc 1 c 2
.. .cm, c 2 c 3 … cmc 1 , c 3 c 4 .. .cmc 1 c 2 ,… , cmc 1 c 2 .. .cm 1. Let us say that a voting rulersatisfies theCycles Cancel Out (CCO)criterion if every cycle cancels out with respect tor.
1b. (12 points)From among the (reasonable) voting rules discussed in class, give 3 voting rules that satisfy the CCO criterion, and 3 that do not.
Define a pair ofopposite cyclesof votes to be a cycle, plus all the opposite votes of votes in that cycle. Note that these opposite votes themselves constitute a cycle, the opposite of which is the
original cycle. Let us say that a voting rule rsatisfies theOpposite Cycles Cancel Out (OCCO)criterion if every pair of opposite cycles cancels out with respect tor.
1c. (12 points)From among the (reasonable) voting rules discussed in class, give 5 voting rules that satisfy the OCCO criterion, and 1 that does not.
1d. (14 points)CriterionC 1 isstrongerthan criterionC 2 if every rule that satisfiesC 1 also satisfiesC 2. Two criteria areincomparableif neither is stronger than the other. For every pair of
criteria among OCO, CCO, and OCCO, say which one is stronger (or that they are incomparable).
2. (A multi-unit auction with externalities.) We are running a multi-unit auction for badminton rackets in the town Externa, where nobody owns one yet and we are the only supplier. Of course, being
the only person to own a badminton racket is no fun; bidders care about which other bidders win rackets as well. In such a setting, where bidders care about what other bidders win, we say that
there areexternalities. Let us assume that each agent is awarded at most one racket, and that shuttlecocks and nets are freely available. In the most general bidding language for this setting,
each bidder would specify, for every subset of the agents, what her value would be if exactly the agents in that subset won rackets. This is impractical because there are exponentially many
subsets. Instead, we will consider more restricted bidding languages. Let us suppose that it is commonly known which agents live close enough to each other that they could play badminton
together. This can be represented as a graph, which has an edge between two agents if and only if they live close enough to each other to play together. In the first bidding language, every
agentisubmits a single valuevi. The semantics of this are as follows. If the agent does not win a racket, her utility is 0 regardless of who else wins a racket. If she does win a racket, her
value is vtimes the number of her neighbors that also win a racket.
(^3) E.g., not dictatorial rules, rules for which there is a candidate that cant possibly win, randomized rules, etc. Also, approval cannot be one of the rules because it is not based on rankings. If
you use Cup, Cup only satisfies a criterion if it satisfies it for every way of pairing the candidates.
Carol Daniel
Figure 1: Externas proximity graph.
Suppose we receive the following bids:
Carol Daniel
(^25) 5 4 Figure 2: Graph with bids. The number next to an agent is that agents bid. Suppose we have three rackets for sale. One valid (but not optimal) allo- cation would be to give rackets to
Carol, Daniel, and Eva. Carol would get a (reported) utility of 2, Daniel would get 10 (25, because two of Daniels neighbors have rackets), and Eva 5, for a total of 17. 2a. (12 points)Give the
optimal allocation, as well as the VCG (Clarke) payment for each agent. 2b. (13 points)In general (general graphs, bids, numbers of rackets), is the problem of finding the optimal allocation solvable
in polynomial time, or NP-hard? (Hint: think about theCliqueproblem (which is almost the same as theIndependent Setproblem).) One year has passed, and we have returned to Externa. Everyones rackets
have broken (we are not in the business of selling high-quality rackets here) and they need new ones. However, the people in the town were not entirely happy with our previous system. Specifically,
it turned out that each agent only ever played with (at most) a single other agent, so that multiplying the value by the number of neighbors with rackets really made no sense. Also, agents have
realized that they would receive different utilities for playing with different agents. In the new system, we must not only decide on who receives rackets, but (for the agents who win rackets) we
must also decide on the pairing, i.e.,who plays with whom. Each agent can be paired with at most one other agent. Each agentisubmits a valuevijfor every one of her neighborsj; agentireceivesvij if
she is paired withj(and both win rackets), and 0 otherwise. Suppose we receive the following bids:
Carol Daniel
(^12) 4 3 2 1 4 Eva 5 5 6 1 5 Figure 3: Graph with bids. Each number is the value that the closer agent on the edge has for playing with the further agent on the edge. Suppose we have four rackets
for sale. One valid (but not optimal) outcome would be to pair Alice and Bob, and Daniel and Eva (and give them all rackets), for a total utility of 4 + 1 + 1 + 6 = 12. 2c. (12 points)Give the
optimal outcome (pairing and allocation), as well as the VCG (Clarke) payment for each agent. 2d. (13 points)In general (general graphs, bids, numbers of rackets), is the problem of finding the
optimal outcome solvable in polynomial time, or NP- hard? (Hint: think about theMaximum-Weighted-Matchingproblem. Keep in mind that the number of rackets is limited, though.)
Web | Homework代写 | Racket代做 | Ai | Assignment代写 – Game Theory, Social Choice, and Mechanism
Web | Homework代写 | Racket代做 | Ai | Assignment代写 – 这是一个利用racket进行的AI方面的编程代写任务
homework Game Theory, Social Choice, and Mechanism
Please read the rules for assignments on the course web page (http://www.cs.duke.edu/courses/fall18/compsci590.2/). Use Piazza (preferred) or directly contact Harsh ([email protected]), Han- rui (
[email protected]), or Vince ([email protected]) with any questions. Use Sak AI to turn in the assignment.
1. (Properties of voting rules.) Alice likes to analyze the outcomes of elections; specifically, she is interested in the different outcomes that different voting rules produce on the same votes. To
do so, she executes many different rules on the same set of votes, a painstak- ing process. She likes knowing about properties of voting rules that ease her task. For example, she likes to know
which voting rules satisfy the Condorcet criterion, so that if there is a Condorcet winner, she immediately knows that that will be the winner for those rules, without having to go through the
trouble of executing each rule individually. Recently, Alice has become interested in the phenomenon of votes cancelling out. Let us say that a set^1 Sof votescancels out with respect to voting
rule rif foreverysetT of votes, the winner^2 thatrproduces forTis the same as the winner thatrproduces forST. For example, the set of votes{a bc, bac, cab}cancels out with respect to the
plurality rule: each candidate is ranked first once in this set of votes, so it has no net effect on the outcome of the election. The same set does not cancel out with respect to Borda, though,
because from these votes,agets 4 points,bgets 3, andcgets 2, which may affect the outcome of the election. Alice likes to know when a set of votes cancels out with respect to a rule, so that she
can just ignore these votes, easing her computation of the winner. Define a pair ofopposite votesto be a pair of votes with completely opposite rankings of the candidates, i.e. the votes can be
written asc 1 c 2 .. .cm andcm cm 1 .. .c 1. Let us say that a voting rulersatisfies the Opposites Cancel Out (OCO)criterion if every pair of opposite votes cancels out with respect tor.
(^1) Technically, a multiset, since the same vote may occur multiple times. (^2) … or set of winners if there are ties.
1a. (12 points)From among the (reasonable^3 ) voting rules discussed in class, give 3 voting rules that satisfy the OCO criterion, and 3 that do not (and say which ones are which!).
Define acycleof votes to be a set of votes that can be written asc 1 c 2
.. .cm, c 2 c 3 … cmc 1 , c 3 c 4 .. .cmc 1 c 2 ,… , cmc 1 c 2 .. .cm 1. Let us say that a voting rulersatisfies theCycles Cancel Out (CCO)criterion if every cycle cancels out with respect tor.
1b. (12 points)From among the (reasonable) voting rules discussed in class, give 3 voting rules that satisfy the CCO criterion, and 3 that do not.
Define a pair ofopposite cyclesof votes to be a cycle, plus all the opposite votes of votes in that cycle. Note that these opposite votes themselves constitute a cycle, the opposite of which is the
original cycle. Let us say that a voting rule rsatisfies theOpposite Cycles Cancel Out (OCCO)criterion if every pair of opposite cycles cancels out with respect tor.
1c. (12 points)From among the (reasonable) voting rules discussed in class, give 5 voting rules that satisfy the OCCO criterion, and 1 that does not.
1d. (14 points)CriterionC 1 isstrongerthan criterionC 2 if every rule that satisfiesC 1 also satisfiesC 2. Two criteria areincomparableif neither is stronger than the other. For every pair of
criteria among OCO, CCO, and OCCO, say which one is stronger (or that they are incomparable).
2. (A multi-unit auction with externalities.) We are running a multi-unit auction for badminton rackets in the town Externa, where nobody owns one yet and we are the only supplier. Of course, being
the only person to own a badminton racket is no fun; bidders care about which other bidders win rackets as well. In such a setting, where bidders care about what other bidders win, we say that
there areexternalities. Let us assume that each agent is awarded at most one racket, and that shuttlecocks and nets are freely available. In the most general bidding language for this setting,
each bidder would specify, for every subset of the agents, what her value would be if exactly the agents in that subset won rackets. This is impractical because there are exponentially many
subsets. Instead, we will consider more restricted bidding languages. Let us suppose that it is commonly known which agents live close enough to each other that they could play badminton
together. This can be represented as a graph, which has an edge between two agents if and only if they live close enough to each other to play together. In the first bidding language, every
agentisubmits a single valuevi. The semantics of this are as follows. If the agent does not win a racket, her utility is 0 regardless of who else wins a racket. If she does win a racket, her
value is vtimes the number of her neighbors that also win a racket.
(^3) E.g., not dictatorial rules, rules for which there is a candidate that cant possibly win, randomized rules, etc. Also, approval cannot be one of the rules because it is not based on rankings. If
you use Cup, Cup only satisfies a criterion if it satisfies it for every way of pairing the candidates.
Carol Daniel
Figure 1: Externas proximity graph.
Suppose we receive the following bids:
Carol Daniel
(^25) 5 4 Figure 2: Graph with bids. The number next to an agent is that agents bid. Suppose we have three rackets for sale. One valid (but not optimal) allo- cation would be to give rackets to
Carol, Daniel, and Eva. Carol would get a (reported) utility of 2, Daniel would get 10 (25, because two of Daniels neighbors have rackets), and Eva 5, for a total of 17. 2a. (12 points)Give the
optimal allocation, as well as the VCG (Clarke) payment for each agent. 2b. (13 points)In general (general graphs, bids, numbers of rackets), is the problem of finding the optimal allocation solvable
in polynomial time, or NP-hard? (Hint: think about theCliqueproblem (which is almost the same as theIndependent Setproblem).) One year has passed, and we have returned to Externa. Everyones rackets
have broken (we are not in the business of selling high-quality rackets here) and they need new ones. However, the people in the town were not entirely happy with our previous system. Specifically,
it turned out that each agent only ever played with (at most) a single other agent, so that multiplying the value by the number of neighbors with rackets really made no sense. Also, agents have
realized that they would receive different utilities for playing with different agents. In the new system, we must not only decide on who receives rackets, but (for the agents who win rackets) we
must also decide on the pairing, i.e.,who plays with whom. Each agent can be paired with at most one other agent. Each agentisubmits a valuevijfor every one of her neighborsj; agentireceivesvij if
she is paired withj(and both win rackets), and 0 otherwise. Suppose we receive the following bids:
Carol Daniel
(^12) 4 3 2 1 4 Eva 5 5 6 1 5 Figure 3: Graph with bids. Each number is the value that the closer agent on the edge has for playing with the further agent on the edge. Suppose we have four rackets
for sale. One valid (but not optimal) outcome would be to pair Alice and Bob, and Daniel and Eva (and give them all rackets), for a total utility of 4 + 1 + 1 + 6 = 12. 2c. (12 points)Give the
optimal outcome (pairing and allocation), as well as the VCG (Clarke) payment for each agent. 2d. (13 points)In general (general graphs, bids, numbers of rackets), is the problem of finding the
optimal outcome solvable in polynomial time, or NP- hard? (Hint: think about theMaximum-Weighted-Matchingproblem. Keep in mind that the number of rackets is limited, though.) | {"url":"https://www.cscodehelp.com/%E5%B9%B3%E6%97%B6%E4%BD%9C%E4%B8%9A%E4%BB%A3%E5%86%99/web-homework%E4%BB%A3%E5%86%99-racket%E4%BB%A3%E5%81%9A-ai-assignment%E4%BB%A3%E5%86%99-game-theory-social-choice-and-mechanism/","timestamp":"2024-11-05T18:57:31Z","content_type":"text/html","content_length":"73514","record_id":"<urn:uuid:5b51b358-ca63-44d5-86da-630d219198fa>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00621.warc.gz"} |
Mathematics and Philosophy of Science
The relationship between mathematics and the philosophy of science is deeply intertwined, with mathematics playing a fundamental role.
published : 09 March 2024
The relationship between mathematics and the philosophy of science is deeply intertwined, with mathematics playing a fundamental role in shaping our understanding of the natural world and the methods
by which we investigate it. Philosophy of science, in turn, provides a framework for examining the nature of mathematical knowledge and its significance for scientific inquiry.
Mathematics as a Language of Science
Mathematics is often described as the language of science, providing a precise and rigorous framework for formulating theories, modeling phenomena, and making predictions. The use of mathematical
notation and formalism allows scientists to communicate complex ideas and relationships in a concise and unambiguous manner, facilitating collaboration and the exchange of knowledge across
Philosophers of science study the role of mathematics in scientific practice, examining questions such as the nature of mathematical entities, the reliability of mathematical reasoning, and the
relationship between mathematics and empirical observation. By analyzing the epistemological and ontological status of mathematics, philosophers of science seek to understand the nature and scope of
mathematical knowledge and its implications for scientific inquiry.
Mathematical Realism and Antirealism
One of the central debates in the philosophy of mathematics is the question of mathematical realism versus antirealism. Mathematical realists argue that mathematical entities such as numbers, sets,
and functions exist independently of human thought and experience, while mathematical antirealists maintain that mathematics is a human invention or construction.
Philosophers of science explore the implications of mathematical realism and antirealism for scientific practice, considering how the choice of mathematical framework influences our understanding of
the natural world and the validity of scientific theories. By examining the relationship between mathematical entities and empirical phenomena, philosophers of science seek to elucidate the nature of
mathematical knowledge and its role in scientific explanation and prediction.
Mathematics and Scientific Explanation
Mathematics plays a crucial role in scientific explanation, providing the formal machinery for expressing laws, theories, and models of the natural world. Philosophers of science study the nature of
scientific explanation, investigating how mathematical concepts and structures contribute to our understanding of natural phenomena and the underlying mechanisms that govern them.
By analyzing the role of mathematics in scientific explanation, philosophers of science seek to elucidate the relationship between mathematical models and reality, considering questions such as the
applicability of mathematics to the physical world, the nature of mathematical necessity, and the limits of mathematical representation. Through their investigations, philosophers of science aim to
enhance our understanding of the nature and scope of scientific knowledge and its relationship to mathematical truth.
The relationship between mathematics and the philosophy of science is complex and multifaceted, with each discipline influencing and informing the other in profound ways. Mathematics provides the
language and formalism of science, while philosophy of science offers a framework for examining the nature and significance of mathematical knowledge in scientific inquiry.
As we continue to explore the intersections between mathematics and the philosophy of science, we gain a deeper appreciation for the ways in which these disciplines enrich and illuminate our
understanding of the natural world and the methods by which we come to know it. | {"url":"https://www.function-variation.com/article163","timestamp":"2024-11-06T12:16:24Z","content_type":"text/html","content_length":"17970","record_id":"<urn:uuid:505d94a2-6b1f-4eeb-8537-ed4922f993e5>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00511.warc.gz"} |
area of
To find the surface are of the box we need to find the area of each rectangular face and add them all up.
The area of the front face is 20 x 30 = 600 cm^2.
The area of the top face is 20 x 8 = 160 cm^2.
The area of the side face is 8 x 30 = 240 cm^2.
Now add these together to get 600 + 160 + 240 = 1000 cm^2.
IMPORTANT: This is not our final answer! Cuboids have 6 faces in total. We have only added three so far! For every front there is a back, for every top there is a bottom, and for every side there is
another other side.
This means we need to double our answer to get 1000 x 2 = 2000 cm^2. | {"url":"https://studymaths.co.uk/keytopics/surfacearea.html","timestamp":"2024-11-09T15:38:16Z","content_type":"text/html","content_length":"7009","record_id":"<urn:uuid:98f28663-639b-4736-a6b2-99b9d24af3dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00354.warc.gz"} |
Unit 1: Jobs & Earnings
1 JOBS and EARNINGS
Lesson 1.0: Percentages
Percentages are an important part of our everyday lives. Percentage is a very handy way of writing fractions. In this lesson, we will explore what percent (%) is and how we calculate percent of a
Lesson 1.0
Lesson 1.1: Investigating Jobs and Their Related Pay
It is quite demanding to find the right job for you in this higly competitive era. In this lesson, we will delve into several key concepts such as how to find a job, how to interpret job ads right,
how to calculate gross income, and what kind of questions to ask at a job interview.
Lesson 1.1
Lesson 1.2: Ways of Being Paid for Work
Different businesses have different ways of paying their employees: salary (yearly), wage (hourly), piecework, and commission. In this lesson, we will explore different ways of being paid, their
advantages and disadvantages.
Lesson 1.2
Lesson 1.3: Calculating Gross Income
In this lesson, we will learn several key concepts: vacation pay; weekly, bi-weekly, semi-monthly, monthly pay periods and their effects on our financial planning.
Lesson 1.3
Lesson 1.4: Understanding Different Pay Schedules
In this lesson, we will learn several key concepts: vacation pay; weekly, bi-weekly, semi-monthly, monthly pay periods and their effects on our financial planning.
Lesson 1.4 | {"url":"http://300math.weebly.com/unit-1-jobs--earnings.html","timestamp":"2024-11-13T08:48:31Z","content_type":"text/html","content_length":"35409","record_id":"<urn:uuid:860d3e97-ea48-49f2-b287-484883e6fc84>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00497.warc.gz"} |
Total Mass & Total Time to Flow Rate Calculator
Click save settings to reload page with unique web page address for bookmarking and sharing the current tool settings
Flip tool with current settings and calculate the total mass or time
Related Tools
Find another flow measurement calculator by clicking on the required answer which coincides with the input parameters you already know.
User Guide
This tool will calculate the average mass flow rate of a gas or liquid in any units from the total mass of gas or liquid which has been transferred over a specified of period time. Once a flow rate
has been calculated two conversion tables will be displayed for a range of mass and time periods centered around each calculated flow rate.
The mass flow rate formula used by this calculator is:
ṁ = m / t
• ṁ = Mass flow rate
• m = Mass
• t = Time
Mass Transferred
Enter the total mass of gas or liquid that has been transferred and select the appropriate mass measurement unit.
Time Taken
Enter the amount of time that passed for the quantity of gas or liquid that has been transferred. Select the appropriate units of time.
Mass Flow Rate
This is the calculated average flow rate of the gas or liquid expressed as a quantity of mass per unit measurement of time. | {"url":"https://www.sensorsone.com/mass-and-time-to-flow-rate-calculator/","timestamp":"2024-11-03T15:39:48Z","content_type":"text/html","content_length":"53864","record_id":"<urn:uuid:be711cfa-f7ec-4597-9161-6db015b4bd50>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00082.warc.gz"} |
Course Planner
Course Guide for the Graduate Student
The Physics Ph.D. program at Baylor is based on a general plan of two years of intensive coursework following by research leading to a Ph.D. Dissertation. The following is a brief outline of the
program to aid incoming students. First year is usually composed of the core courses. (Since the core courses typically should be completed in the first year, there is little flexibility in the
course scheduling.)
First Year Recommended Course Plan
Fall Term (10 hrs)
5320 Classical Mechanics I (3 hrs)
5360 Mathematical Physics I (3 hrs)
5370 Quantum Mechanics I (3 hrs)
5180 Graduate Physics Colloquium (1 hr)
Spring Term (10 hrs)
5330 Electromagnetic Theory I (3 hrs)
5340 Statistical Mechanics (3 hrs)
5371 Quantum Mechanics II (3 hrs)
5180 Graduate Physics Colloquium (1 hr)
Prelim Exam
The Prelim Exam is taken for the first time in August preceding the second year in the program. This is an important part of graduate studies. Passing the exam demonstrates mastery of Physics through
the undergraduate and first-year graduate levels.
The Prelim Exam consists of four parts.
I. Classical Mechanics
II. Quantum Mechanics
III. Electricity and Magnetism
IV. Mathematical methods, Thermodynamics and Statistical Mechanics
The Prelim Exam must be passed to remain in the Ph.D. program. Two attempts, once a year for the first two years, are the maximum allowed.
Second Year Recommended Course Plan
In the second year there is more flexibility in the course work. Other than the Graduate Colloquium only one core courses remains: Phy 5331-Electromagnetic Theory II. Beyond this course, electives
may be taken to suit your interests.
Typically the second year course work would consist of the following:
Fall (10hrs)
5331 Electromagnetic Theory II (3 hrs)
5180 Graduate Physics Colloquium (1 hr)
6 hrs of electives (see below)
Spring (10 hrs)
5180 Graduate Physics Colloquium (1 hr)
9 hrs of electives see below
Below are examples of electives based on research interest [the course catalog contains a complete list of courses]:
Upper Undergraduate Courses For which Graduate Credit is Given
[Note: up to 6 hours of undergraduate courses, courses below 5000, can be applied to your Ph.D. degree]
4360 Computer Models in Physics
4372 Introductory Solid State Physics
4373 Introductory Nuclear and Particle Physics
4374 Introduction to Relativistic Quantum Mechanics
Graduate Electives
5342 Solid State Physics
5351 General Relativity
5352 Space Plasma Physics
5361 Mathematical Physics II
6370 Advanced Quantum Mechanics
6371 Relativistic Quantum Mechanics
6372 Elementary Particle Physics
6373 Quantum Field Theory I
6374 Quantum Field Theory II
6375 Quantum Field Theory III
Dissertation Research
In addition to the coursework during the first two years it is extremely beneficial to become familiar with the current research activities of the department. Once a student is interested in an area
of study they may become involved that group and begin participating in research as the current situation allows. Formally Ph.D. dissertation research can begin after the Qualifying exam is passed.
Research is unpredictable by nature, so there is no step-by-step guide. However, certain milestones and formal procedures are required. The formal process is as follows
1. Each student will work with a graduate faculty member to select a research project for the student's Ph.D. dissertation research. This faculty member will be the student's dissertation advisor.
2. The student, with the assistance of his/her dissertation advisor, will select a dissertation committee as described in the graduate catalog.
3. The student will provide a description of the proposed dissertation research to the dissertation committee for approval.
4. The research is performed and the dissertation written. While there is no set time frame this is generally expected to take two to three years. During this time the student will register for PHY
6V99 Dissertation.
Completion of the degree
Once the dissertation is complete, the student will defend the research to his/her dissertation committee in a public presentation. Upon approval of the dissertation by the student's advisor and
dissertation committee, and the Graduate School, the Ph.D. requirements are met and the student may graduate. | {"url":"https://physics.artsandsciences.baylor.edu/graduate/courses-and-planner/course-planner","timestamp":"2024-11-12T20:20:34Z","content_type":"text/html","content_length":"95688","record_id":"<urn:uuid:a31dbd99-7823-4ac9-b24f-269f92966cf2>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00385.warc.gz"} |
Population Size Estimation From Capture-Recapture Studies Using shinyrecap: Design and Implementation of a Web-Based Graphical User Interface
Original Paper
Background: Population size estimates (PSE) provide critical information in determining resource allocation for HIV services geared toward those at high risk of HIV, including female sex workers, men
who have sex with men, and people who inject drugs. Capture-recapture (CRC) is often used to estimate the size of these often-hidden populations. Compared with the commonly used 2-source CRC, CRC
relying on 3 (or more) samples (3S-CRC) can provide more robust PSE but involve far more complex statistical analysis.
Objective: This study aims to design and describe the Shiny application (shinyrecap), a user-friendly interface that can be used by field epidemiologists to produce PSE.
Methods: shinyrecap is built on the Shiny web application framework for R. This allows it to seamlessly integrate with the sophisticated CRC statistical packages (eg, Rcapture, dga, LCMCR).
Additionally, the application may be accessed online or run locally on the user’s machine.
Results: The application enables users to engage in sample size calculation based on a simulation framework. It assists in the proper formatting of collected data by providing a tool to convert
commonly used formats to that used by the analysis software. A wide variety of methodologies are supported by the analysis tool, including log-linear, Bayesian model averaging, and Bayesian latent
class models. For each methodology, diagnostics and model checking interfaces are provided.
Conclusions: Through a use case, we demonstrated the broad utility of this powerful tool with 3S-CRC data to produce PSE for female sex workers in a subnational unit of a country in sub-Saharan
JMIR Public Health Surveill 2022;8(4):e32645
Accurate knowledge of population size is critical in many areas of science but a challenge whenever complete counts are too difficult or expensive to be obtained. One such area is the HIV pandemic,
which increasingly is driven by high-risk behaviors that define “key populations” (KP), among them, female sex workers (FSW), men who have sex with men (MSM), and people who inject drugs (PWID) [].
Global, national, and local HIV control efforts all require knowing the size of these high-risk populations to monitor the epidemic in terms of density and distribution of populations over time and
to inform effective and appropriately scaled program development, target setting, and resource allocation. Yet, there is no gold standard to derive reliable population size estimates (PSE). Instead,
public health teams and stakeholders use a wide range of methods, many of which are not based on empirical data nor sound statistical concepts [,], potentially producing poor-quality estimates.
Estimates of population sizes derived from programmatic mapping [,] enumerate members of the population attending venues during the exercise but often fail to account for the less socially visible,
resulting in underestimates. Other nonempirical subjective methods such as Wisdom of the Crowd [,] and the Delphi methods [,] are susceptible to bias and the influence of individuals.
Capture-recapture (CRC) globally has seen wide use for PSE, including for the HIV pandemic [-]. The basic idea behind CRC is to engage in 2 or more encounter events or sources (these might also be
referred to as samples, captures, or lists), recording which individuals appear in which events and relating the number of individuals sampled once to those sampled repeatedly. Most CRC exercises
include 2 encounter events with the key assumption being that the 2 samples (2S) are independent []. Unfortunately, many such 2S-CRC exercises may suffer from violating this assumption resulting in
overestimates (negative dependence between 2 samples) or underestimates (positive dependence between 2 samples) [,,]. CRC with 3 (or more) samples (3S-CRC) relaxes this assumption, as interaction
terms may be added to the statistical models to address source dependencies. Given sufficient overlap of samples and independence of samples, 3S-CRC allows for more sophisticated analysis compared
with 2S-CRC [,], resulting in more accurate PSE. Statistical support for these analyses might not be available, creating a critical challenge for field epidemiologists to produce robust PSE.
Several statistical models satisfy the requirements to perform the aforementioned sophisticated analysis of 3S-CRC data: log-linear models, Bayesian model averaging, and Bayesian nonparametric
latent-class models. Log-linear models are a classic methodology for the analysis of multiple source CRC data. Variants are implemented that allow for varying capture probabilities across events and
heterogeneous capture probabilities among members of the population. Bayesian model averaging allows the analyst to flexibly account for list dependency by creating models for all possible
dependencies and averaging over them in a way that is proportional to the probability that the dependence is correct. The Bayesian latent class model deals with heterogeneity in a novel way. It
posits that there are unobserved subgroups in the data with different capture probabilities for each capture event. The number of these groups and their probabilities are unknown. The algorithm uses
a Bayesian framework to estimate these, along with the population size. Application of these 3 types of statistical models requires computational expertise. This is a barrier to the use of CRC
involving 3 or more sources, as it typically requires knowledge of specialized software [] or programming in languages such as R []. To fill this need, we present a graphical user interface,
shinyrecap, that guides the user through sample size estimation, data preparation and exploration, and PSE using CRC studies.
The objectives of this paper were to describe shinyrecap, a free, web-based application facilitating the format and analysis of CRC data for PSE.
Overview of the Capture-Recapture Method
The application of ratio estimation for PSE from multiple encounters dates to at least 1787 [] and gained popularity primarily among animal ecologists more than a century later [-], although
applications abound in other areas, including epidemiology [,]. Early applications were restricted to sampling on 2 occasions or from 2 lists, wherein individuals encountered during the first survey
are offered an identifying mark. For KP CRC, these identifiers are inexpensive but memorable unique objects or “gifts” such as brightly colored rubber bracelets or distinctive key chains. The number
of individuals who accept the unique gifts are documented. The same process is repeated during a second survey, during which individuals are also asked about having received a gift during the
previous capture. Estimation of the unknown number of population members from 2 samples requires the strong assumptions that (1) the population is static over the sampling interval, (2) the
identifying unique objects or gifts are not lost nor misidentified, (3) individuals are sampled independently during the surveys (list independence), and (4) every population member shares a common
and constant probability of encounter during the surveys (homogeneity). The first assumption is well-approximated by sampling over short time intervals. However, the remaining assumptions are
unlikely to hold.
The next major innovation was the extension of estimation to data collected from 3 or more samples [,]. This enables relaxation of the third and fourth assumptions using statistical models that
account for sampling dependence and various forms of inhomogeneity (ie, nonuniform) in encounter probabilities [,,-]. To understand why more samples allows for the assumption relaxation, consider a
3S-CRC where each capture is the same size. If the population is homogeneous and all individuals have the same probability of being captured in each sample (p[1]), then the probability of being
captured in all 3 samples would simply be p[1]^3. On the other hand, if half the population has a capture probability of p[1] and the other half has probability p[2], then the probability of a random
person being captured in all 3 would be 0.5(p[1]^3 + p[2]^3). By comparing the counts of individuals captured in all 3 samples to what would be expected if there was homogeneity, we can measure and
model it. Log-linear models, Bayesian model averaging, and Bayesian Dirichlet process mixture models (nonparametric latent-class models) and each model heterogeneity in different ways, allowing for
the production of more accurate estimates in the presence of inhomogeneity.
Overview of Relevant Statistical Models
Log-Linear Models
Models for capture probabilities originated in the discipline of animal ecology [,]. The natural logs are modeled as linear combinations of factors representing various forms on inhomogeneity. Four
general classes of models are produced, representing a wide range of model complexity: Captures have the same probability, and individuals are uniform (M[0]); captures might have different
probabilities, and individuals are uniform (M[t]); captures have the same probability, and individuals may be heterogeneous (M[h]); and captures may have different probabilities, and individuals may
be heterogeneous (M[th]). Selection of a single “best” model is typically done using either the Akaike or Bayesian information criterion (AIC and BIC, respectively) []. For these, lower values
indicate a “better” model fit.
For heterogeneous models, log-linear models require the specification of a parametric distribution for the population’s log odds of being captured. These are typically set to be either Normal,
Poisson, Gamma, or Darroch. Additionally, the Chao (lower bound) correction can be used to obtain a lower bound on the population size rather than an estimate of it.
The “Normal” model incorporates heterogeneity as a Gaussian mixing distribution []. The Poisson, Darroch, and Gamma options incorporate different heterogeneity correction columns into the design
matrix. The Darroch, and especially the Gamma, correction may produce distinctly large heterogeneity corrections and estimates of population size. Unfortunately, the correct model specifications are
frequently not identifiable (roughly, parameters are not informed by the data), and so choosing based on any information criteria can lead to misspecified models [].
Bayesian Model Averaging of Log-Linear Models
Bayesian model averaging is geared to be robust to list dependence. Ideally, one would like to have all capture events be independent draws from the population. In many cases, however, some capture
events may be related. For example, in a citywide survey of PWID, it might happen that the first 2 capture events were more heavily concentrated in one area of the city than the third event,
introducing potential dependence. When list dependence is present, the interactions between events should be considered.
The natural logs of expected frequencies of observable encounter combinations can be modeled as linear combinations of main and interaction effects of the sampling events [,]. This allows the model
to flexibly account for list dependence among the various samples. Bayesian model averaging enumerates all possible models of list dependency and then puts a prior on the likelihood that each model
is the true one, with more complex models typically having lower prior probability than less complex models. Combining this prior with a prior for population size allows one to calculate a posterior
estimate of population size averaging over all possible models. In this posterior, estimates from each model are weighted by the posterior probability of the model, yielding a single estimate that
includes model uncertainty. Some form of model averaging is important given that there may be limited information in the data available to identify the true model out of the large number of potential
models [].
The first step in the analysis is to specify a prior distribution for population size. This represents the analyst’s prior knowledge about population size along with uncertainty. By default, a
“noninformative” improper prior is used, which is proportional to 1 divided by the population size. Typically, analysts will have access to at least a rough idea of the range of possible population
sizes from previous PSE reports or literature reviews. This information can be incorporated into the prior parameterized as a log-normal distribution with a truncation at a specified maximum
population size. The “delta” parameter controls the prior, favoring simple models in the model averaging. This parameter is more difficult to interpret, and it is set to 2^–^k by default, where k is
the number of encounter events. Lower values indicate less prior weight on more complex list interactions. Once the prior is specified, the posterior probability distribution of the population size
can be calculated.
Bayesian Nonparametric Latent-Class Models
Instead of assuming a parametric probability function for capture probability, as is done by traditional log-linear models, this approach posits that the population is divided into a number of
groups, with members in each group having the same homogeneous capture probability. The number of homogeneous strata in a population is uncertain, and covariates that identify those classes may not
be available. Thus, the strata are said to be latent, and strata identities are treated as missing data. Estimation is naturally accomplished using mixtures of distributions. A clever implementation
of Bayesian nonparametric latent-class modeling can then be used to estimate population size []. Both the number of strata and the strata capture probabilities are inferred via Bayesian inference,
with a stick-breaking Dirichlet process prior enforcing model parsimony such that models with fewer latent strata are preferred.
The degree to which fewer strata are preferred is controlled by a prior on the stick-breaking process parameterized as a Gamma distribution with shape and scale values. The relationship between the
Gamma distribution and the number of latent groups is complex and mediated by a stick-breaking process. In general, the default values of 0.25 for both the shape and scale parameters result in a
reasonably diffuse prior.
Estimation is based on the posterior distribution of population size, of which a sample is constructed using Markov chain Monte Carlo (MCMC) simulation. MCMC algorithms start from initial values and
produce serially correlated “chains” of samples from some distribution. That distribution converges to the joint posterior distribution only after some potentially large number of “burn-in”
iterations. Therefore, valid inferences can be made only after discarding the burn-in iterations.
shinyrecap Application User Interface
shinyrecap was developed using the Shiny [] web framework for R []. Shiny is a flexible, open-source toolkit used to build web applications with rich interactivity that can easily produce tables,
visualize data, and create dashboards. The advantage of this framework is that it makes it easy to expose advanced algorithms and packages written in R to a noncoding audience. In shinyrecap, we
leveraged the algorithms from the Rcapture [] package for log-linear modeling, the dga package for Bayesian model averaging [], and the LCMCR [] package for Bayesian latent-class modeling. Whereas it
would normally take substantial experience with R to use those packages, shinyrecap provides easy access to a wider audience with “the click of a button.”
shinyrecap has been made available for public use [] and does not require installation of or experience with R. Client-server communication occurs over a secure-sockets layer (SSL) protocol
connection. Required data inputs are minimal and can be aggregate or individual-level. Any data provided to shinyrecap persist only for the session; neither input nor output data are saved on the web
server. This provides users with security protection against third-party traffic analysis and any security intrusions into the server not concurrent with the user’s session. shinyrecap offers a
tutorial video and manual, and help buttons are presented where input information is required in each shinyrecap module.
Alternatively, R users can launch the interface locally from any computer by entering the following into the R console:
shinyrecap is structured in 3 parts. First, it supports the design of CRC studies by providing a tool for sample size estimation. Second, it provides a data formatting tool to assist with the data
processing of CRC surveys. Finally, it provides the analysis tool to generate the estimates and outputs required for PSE.
Application User Interface
Sample Size Estimation
When designing a CRC study, it is important to collect enough data to achieve sufficient precision for PSE. shinyrecap's sample size estimation tool does this by allowing the user to specify
population parameters such as guesstimates of the target population size and the amount of capture heterogeneity in the population, as well as sample characteristics such as the number of capture
events and their expected sample sizes. It then simulates CRC studies in this population and estimates the population size using log-linear modeling for each of the simulations. Precision is
estimated from the simulation results. The application supports simulation and estimation using the M[t] model if homogeneity is assumed. If heterogeneity is allowed, simulation and estimation are
performed using the M[th] model with normally distributed capture probabilities.
Given the input parameters, the interface provides the user with the distribution of a log-linear population size estimator across the simulations. A table is also provided that summarizes the
percent of times simulated estimates were within different ranges of accuracy. A user might find it acceptable to have their estimate within 10% of the true value 90% of the time, whereas they might
choose to collect more samples if the calculator says that their estimate will only be within 10% of the true value 50% of the time.
Data Formatting
The first barrier encountered by a practitioner is putting their CRC data into the right format for analysis. shinyrecap is able to read 2 data types: individual and aggregate. We focus on the
capture history format (aggregate data) here to demonstrate the data formatter tool. Individual-level data files have 1 row per encounter, with each column representing a sampling event (eg, 3
columns for 3S-CRC) and, within the columns, the successful encounter event result (ie, the individual accepts the unique object; individuals who refuse the object during the encounter are not
counted). The usual data format used by CRC analysis programs is the capture-history format. In this format, each column should represent a successful encounter event, and each row should be an
encounter history. A “1” indicates a successful encounter (capture), and “0” indicates absence, so the following history represents 80 individuals who were encountered and accepted the unique object
during the 2nd events, but not during the 1st or 3rd:
When the aggregate data type is specified, the last column represents the total number of individuals with that capture history. A properly structured 3S-CRC data set would look something like .
From the first row, we see that there were 30 individuals who were observed at event 1 but not at the 2nd and 3rd events:
There were 10 individuals captured in all 3 events, as seen in the following row:
Note that there is no row for the following history because that pertains to the unknown number of population members who were not observed at any event:
For k encounter events, there are 2^k – 1 observable event histories and 1 unobservable history. Analysis of CRC data requires enumeration of all 2^k – 1 observable counts (which may contain observed
values of 0 but not missing values).
The capture-history data format is easily recorded from individually identifiable population members. However, in many epidemiological studies, unique individuals are not identified; rather, data are
aggregated. These accumulated data files consist of counts of individuals who were encountered at each sampling event and the subsets of those who were encountered at any preceding sampling event(s)
(). No identifying information is collected on any subject at any event. During the 1st sampling event, only the count of individuals present and who were offered and accepted a unique (to the event)
identifier is recorded. During the 2nd event, observed population members are tabulated by whether they received the identifier distributed during the first event, and those individuals are given a
second (and different) aggregate identifier. At the 3rd event, the observed population members are cross-tabulated by whether they received the event-specific identifier distributed during each of
the 2 previous events. We call this event-count formatted data. Although 7 counts have been recorded, the counts are aggregated differently from the required format shown in . Note that the sum of
samples should always be larger than the sum of count data.
It takes some thought to figure out how to convert the data to the required format, and the process becomes much more difficult if there are more than 3 events. The shinyrecap data formatting tool
makes that conversion easy and reliable for any number of encounter events.
Figure 1. Example capture-history data format for 3 encounter events (3S-CRC). Absence or presence is denoted by 0 or 1, respectively.
View this figure
Figure 2. Aggregated capture histories in event-count format for 3-source capture-recapture (3S-CRC).
View this figure
shinyrecap guides the user through the analysis process for log-linear modeling, Bayesian model averaging, and Bayesian latent-class modeling. All analyses may be exported as downloadable reports in
HTML, Word, or PDF documents. To facilitate analysis transparency and reproducibility, R code to replicate the analysis is included in all reports by default.
Log-Linear Models
The log-linear section of the application has 3 sections. The first section, “Model Comparison,” displays population size, standard error, AIC, and BIC for each potential model formulation. The
“Model Selection” section allows the user to select one of these models and calculate a confidence interval. The “Descriptives” section provides output to help the user understand the model and
diagnose potential problems. Two diagnostic plots help explore the heterogeneity structure. The first diagnostic plot displays a function of the number of units captured i times for different values
of i. It should be roughly linear except in the case where the data were generated by an M[th] model. The second diagnostic plot shows the number of individuals captured for the first time at the i
[th] sampling event. It should be linear in the case of the M[0] model and concave down in the case of an M[h] model. Any other form may indicate an M[t] or M[th] model.
Bayesian Model Averaging
The first step in the Bayesian model averaging interface is to set a prior distribution for the population size. This is set to a noninformative 1/N distribution, but it is recommended to change this
to something relevant to the population under study. To do this, the user can specify their beliefs for the median population size (ie, they believe that there is a 50% chance the population size is
above it and 50% chance it is below) and the 90th percentile (ie, there is only a 10% chance the true population size is above this value). The application then parameterizes these beliefs as a
log-normal distribution. The user may also specify a maximum population size to put an upper bound on the prior.
Once the prior is set, the user can go to the “Posterior Population Size” tab to obtain posterior estimates and credible intervals. The “Posterior Model Probabilities” tab allows the user to explore
the different individual models that are averaged together and see their influence on the posterior.
Bayesian Nonparametric Latent-Class Models
The Bayesian nonparametric latent-class model is the most computationally intense analysis method. The user may control the Gamma distribution stick-breaking prior as well as set the maximum number
of latent groups. Increasing the number of latent groups increases the computation time but, since the number of groups is determined by the algorithm, does not affect the results so long as it is
set sufficiently large. Although 10 is a good default value, the user can increase that to ensure that this limit does not affect the results.
There is a number of MCMC sampling options available to the user. There are 2 primary considerations that the user should be aware of. First, the MCMC process must be at equilibrium. To ensure this,
the first samples generated by the algorithm should be thrown out. The number of samples thrown out is controlled by the “burn-in” option. If there are any trends in the trace plot (available in the
Diagnostics tab), the burn-in period may need to be increased. Second, the sample size must be large enough that the posterior is not dominated by sampling noise. With MCMC sampling, each sample is
correlated with the last sample, so the effective sample size (also in the Diagnostics tab) is often much lower than the raw number of samples generated by the process. Typically, the user should aim
for an effective sample size greater than 1000. The effective sample size can be increased by increasing the number of samples generated or the number of MCMC steps taken between samples, which
reduces correlation.
After specifying the prior on the number of strata and the MCMC sampling parameters, a sample from the posterior distribution is produced by pressing the “Run” button. A progress bar displays the
progress of each computational operation. A posterior summary will be displayed.
Pairwise Analysis
The pairwise analysis table displays PSE, standard error, and 95% confidence limits for each possible pair of sampling events and is used as a diagnostic step to examine sampling events for
homogeneity. Similar PSE across pairwise results indicate the independence assumption may have been met, whereas differences across results suggest violations of the assumption. Any of the models
available in the shinyrecap Analysis tool may be used to incorporate such dependence into models.
Example With FSW Data
Estimates of key population size are critical for HIV program planning. For this reason, a large 3S-CRC activity was implemented in a subnational unit (SNU) of a sub-Saharan African country with high
HIV burden and unmet need for HIV/AIDS treatment services. Using 3S-CRC data collected from FSW, we demonstrated the utility of the shinyrecap tool to estimate sample size sufficient for precision,
format our data in preparation for analysis, and produce PSE using several different statistical models.
Between October 2018 and December 2018, 544 FSW hotspots in the SNU were sampled, representing 13,344 encounters with FSW over 3 sampling rounds. During encounters with FSW in hotspots, FSW
distribution teams offered inexpensive and memorable objects that were unique to each of the 3 capture rounds. Eligible FSW who consented were considered enrolled in this PSE activity. In subsequent
rounds, 1 week to 2 weeks apart, FSW were asked to show or describe objects they had received during previous rounds, and affirmative responses were tallied upon correct identification of the
objects. Distributors recorded information on tablets and uploaded to a secure central server after each encounter. Data were aggregated into a table similar to for analysis.
In the following sections, we work through how the shinyrecap application was used to assist in the planning, data management, and analysis of this study.
Sample Size Estimation
Before any study is conducted, it is wise to determine what level of precision one is likely to get out of a potential sampling plan. shows the result of using the sample size estimation tool in the
context of the example FSW PSE study. Capture sample sizes were set at 4410, 2675, and 2519, with a theorized population size of 20,000. A moderate amount of heterogeneity was also added, such that
90% of individuals in the population had capture odds less than 1.2 times the average individual in the population.
The table in the upper right of summarizes the results and finds that, 80% of the time, our PSE will be within 7.73% of the true value.
Figure 3. The sample size estimation tool applied to the example female sex worker study.
View this figure
Data Format
After the data were collected, we translated it from event format to capture history format. shows the result of applying the data formatter to the example FSW CRC data. Once translated, the capture
history data may be imported into the analysis tool for inference.
Figure 4. The data formatter tool.
View this figure
Log-linear Models
The first class of models we apply is log-linear. displays the analysis tool’s result table for all of the various applicable log-linear models. These may have no effects (0), effects for time (t),
effects for heterogeneity (h), or both (th). Note that there are multiple listings in the figure for heterogeneous models (h and th) corresponding to different functional forms for the differing
capture probabilities of individuals in the population. For most epidemiological studies, we expect capture probability to vary among individuals or over time, which means that models M[t] and M[th]
are likely more appropriate than the simpler alternatives. This is consistent with the result that the AIC and BIC values are considerably lower for these compared with the M[0] and M[h] models. The
set of M[th] models has the lowest AIC and BIC, indicating that there may be heterogeneity in the population.
Poisson2 induces a reasonable amount of heterogeneity and is generally a good default choice. In this case, it yields a PSE of 18,317, which, as we will see in the following sections, is consistent
with other analyses.
Figure 5. Log-linear models applied to the example female sex worker study.
View this figure
Bayesian Model Averaging
Applying a Bayesian model averaging model results in a very similar estimate compared with the Poisson2 log-linear model with a posterior estimate of 18,624 (see ). Here, we choose a diffuse prior
for our analysis with a median population size of 20,000 and a 90th percentile of 80,000.
Use of the default “Noninformative” prior, which is an improper prior with mass equal to the inverse of population size, is a useful robustness check to assess the influence of our choice of prior.
The posterior estimate using the noninformative prior was 18,608, which is very similar to our original result. Note that the log-normal prior median input was increased from the default of 7000 to
20,000 and the 90% upper bound was adjusted from the default of 10,000 to 80,000.
Figure 6. Bayesian model averaging applied to the example female sex worker study.
View this figure
Bayesian Nonparametric Latent-Class Models
Applying the latent-class model, as in , results in an estimate of 16,266. This is modestly lower than the other methods; however, the 95% probability interval using this method is quite wide,
ranging from 10,621 to 23,512, indicating that the model’s results are compatible with the other 2 methods. The latent-class model will often have intervals wider than the other 2 methods as a result
of its high level of flexibility in describing the latent heterogeneity.
Note that the MCMC number of samples was increased to 100,000 from the default of 10,000, thinning was increased from the default of 10 to 100, and the burn-in was increased from the default of
10,000 to 100,000. These inputs were adjusted to increase effective sample size.
Figure 7. Bayesian latent-class modeling applied to the example female sex worker study.
View this figure
Pairwise Comparison
The table in displays population estimates using each pair of capture events. This pairwise analysis may be helpful to review as a diagnostic step to understand the 3S-CRC data. Each row is a
separate 2S-CRC analysis using only 2 of the sampling events. For example, pa12 estimates the population size using only the 1st and 2nd sampling events, pa13 estimates only the 1st and 3rd sampling
events, and pa23 estimates only the 2nd and 3rd sampling events. The ideal situation is to have similar PSE for each pair, which would be consistent with independence of sampling events. Neither the
1st to 2nd nor the 1st to 3rd comparisons have intervals that overlap with the interval for the 2nd to 3rd comparison, suggesting that the independence assumption may be unreasonable for these.
Figure 8. Pairwise analysis of example female sex worker study results.
View this figure
shinyrecap is a new Shiny application for population size estimation that is easy to use and freely accessible to anyone with an internet connection and a web browser.
The example using 3S-CRC data from FSW in an SNU of a sub-Saharan African country demonstrates how computationally intensive statistical methods are made more accessible to epidemiologists and others
with shinyrecap. The simplicity of the sample size estimation, data formatting, and analytic tools, with supporting online manuals and tutorial videos, allows users to progress through CRC activities
when statistical support might not be readily available. shinyrecap promotes local ownership of PSE activities, including sample size determination, formatting data for use in the shinyrecap, as well
as using the various analytical models for estimating population size. With several key statistical models available to those without coding expertise, local public health staff were able to test
various models, compare the results, and interpret the results given their local knowledge.
Our model results using shinyrecap with 3S-CRC data were larger than the PSE produced from programmatic mapping and enumeration among FSW in the same SNU: 9858 in 2013 [] and 9745 in 2015 []. Both
these estimates were smaller than those produced by shinyrecap: log-linear models; for example, the M[th] Chao lower bound was the smallest of all log-linear models, at 14,990 (14,620-15,378); the
Bayesian model averaged 18,624 (17,625-19,649); and Bayesian latent-class models averaged 16,266 (10,612-23,512). The ability to produce confidence bounds is another benefit of shinyrecap compared
with programmatic mapping and enumeration.
Shiny apps offer a solution to the problem of poor-quality estimates for key population program and policy developers and elevate the level of sophistication of analysis while building in-country
capacity to implement critical surveillance activities. Recently, several Shiny apps were introduced that enhance HIV surveillance efforts to estimate awareness of HIV status over time [], synthesize
multiple PSE using the Anchored Multiplier [], and estimate sample size for biobehavioral survey-based multiplier methods for PSE []. shinyrecap is unique among this group in that it offers multiple
features in one tool to support population size estimation with 3S-CRC from sample size estimation to data formatting to multiple model options for analysis.
Our work was motivated by the needs of epidemiologists and others who require reliable tools for PSE but may not have the necessary coding experience or advanced statistical skills needed to analyze
CRC data involving 3 or more samples. The application facilitates the estimation of sample sizes for captures, proper formatting of individual-level and aggregate-level data in preparation for
analysis, and various options for analysis of CRC data from 3 or more sources. In addition, all output can be saved in HTML, Word, or PDF formats, with an option to include the R code used by the
Shiny to produce the output. Public health teams now have a powerful tool in shinyrecap to produce reliable PSE for a broad range of applications without specialized computing expertise.
Conflicts of Interest
None declared.
1. Worldwide, more than half of new HIV infections now among key populations and their sexual partners. UNAIDS. 2019 Nov 05. URL: https://www.unaids.org/en/resources/presscentre/featurestories
/2019/november/20191105_key-populations [accessed 2021-11-10]
2. Abdul-Quader AS, Baughman AL, Hladik W. Estimating the size of key populations: current status and future possibilities. Curr Opin HIV AIDS 2014 Mar;9(2):107-114 [FREE Full text] [CrossRef] [
3. Quaye S, Fisher Raymond H, Atuahene K, Amenyah R, Aberle-Grasse J, McFarland W, Ghana Men Study Group. Critique and lessons learned from using multiple methods to estimate population size of men
who have sex with men in Ghana. AIDS Behav 2015 Feb;19 Suppl 1:S16-S23. [CrossRef] [Medline]
4. Gokengin D, Aybek G, Aral SO, Blanchard J, Serter D, Emmanuel F. Programmatic mapping and size estimation of female sex workers, transgender sex workers and men who have sex with men in İstanbul
and Ankara, Turkey. Sex Transm Infect 2021 Dec 29;97(8):590-595. [CrossRef] [Medline]
5. Gexha Bunjaku D, Deva E, Gashi L, Kaçaniku-Gunga P, Comins CA, Emmanuel F. Programmatic mapping to estimate size, distribution, and dynamics of key populations in Kosovo. JMIR Public Health
Surveill 2019 Mar 05;5(1):e11194 [FREE Full text] [CrossRef] [Medline]
6. Lorenz J, Rauhut H, Schweitzer F, Helbing D. How social influence can undermine the wisdom of crowd effect. Proc Natl Acad Sci U S A 2011 May 31;108(22):9020-9025 [FREE Full text] [CrossRef] [
7. Grasso MA, Manyuchi AE, Sibanyoni M, Marr A, Osmand T, Isdahl Z, et al. Estimating the population size of female sex workers in three South African cities: results and recommendations from the
2013-2014 South Africa health monitoring survey and stakeholder consensus. JMIR Public Health Surveill 2018 Aug 07;4(3):e10188 [FREE Full text] [CrossRef] [Medline]
8. Khalid FJ, Hamad FM, Othman AA, Khatib AM, Mohamed S, Ali AK, et al. Estimating the number of people who inject drugs, female sex workers, and men who have sex with men, Unguja Island, Zanzibar:
results and synthesis of multiple methods. AIDS Behav 2014 Jan;18 Suppl 1:S25-S31. [CrossRef] [Medline]
9. Abeni DD, Brancato G, Perucci CA. Capture-recapture to estimate the size of the population with human immunodeficiency virus type 1 infection. Epidemiology 1994 Jul;5(4):410-414. [CrossRef] [
10. UNAIDS/WHO Working Group on HIV/AIDS/STI Surveillance. Estimating the Size of Populations at Risk for HIV. In: UNAIDS https://data.unaids.org/publications/external-documents/
estimatingpopsizes_en.pdf. Arlington, VA USA: Family Health International; 2003.
11. Vuylsteke B, Vandenhoudt H, Langat L, Semde G, Menten J, Odongo F, et al. Capture-recapture for estimating the size of the female sex worker population in three cities in Côte d'Ivoire and in
Kisumu, western Kenya. Trop Med Int Health 2010 Dec;15(12):1537-1543 [FREE Full text] [CrossRef] [Medline]
12. Hickman M, Hope V, Platt L, Higgins V, Bellis M, Rhodes T, et al. Estimating prevalence of injecting drug use: a comparison of multiplier and capture-recapture methods in cities in England and
Russia. Drug Alcohol Rev 2006 Mar;25(2):131-140. [CrossRef] [Medline]
13. Paz-Bailey G, Jacobson JO, Guardado ME, Hernandez FM, Nieto AI, Estrada M, et al. How many men who have sex with men and female sex workers live in El Salvador? Using respondent-driven sampling
and capture-recapture to estimate population sizes. Sex Transm Infect 2011 Jun;87(4):279-282. [CrossRef] [Medline]
14. Héraud-Bousquet V, Lot F, Esvan M, Cazein F, Laurent C, Warszawski J, et al. A three-source capture-recapture estimate of the number of new HIV diagnoses in children in France from 2003-2006 with
multiple imputation of a variable of heterogeneous catchability. BMC Infect Dis 2012 Oct 10;12(1):251 [FREE Full text] [CrossRef] [Medline]
15. Yu D, Calleja JMG, Zhao J, Reddy A, Seguy N. Estimating the size of key populations at higher risk of HIV infection: a summary of experiences and lessons presented during a technical meeting on
size estimation among key populations in Asian countries. WPSAR 2014 Aug 11;5(3):43-49. [CrossRef]
16. Karami M, Khazaei S, Poorolajal J, Soltanian A, Sajadipoor M. Estimating the population size of female sex worker population in Tehran, Iran: application of direct capture-recapture method. AIDS
Behav 2017 Aug 16;21(8):2394-2400. [CrossRef] [Medline]
17. Poorolajal J, Mohammadi Y, Farzinara F. Using the capture-recapture method to estimate the human immunodeficiency virus-positive population. Epidemiol Health 2017 Oct 10;39:e2017042 [FREE Full
text] [CrossRef] [Medline]
18. Doshi RH, Apodaca K, Ogwal M, Bain R, Amene E, Kiyingi H, et al. Estimating the size of key populations in Kampala, Uganda: 3-source capture-recapture study. JMIR Public Health Surveill 2019 Aug
12;5(3):e12118. [CrossRef]
19. Brenner H. Use and limitations of the capture-recapture method in disease monitoring with two dependent sources. Epidemiology 1995 Jan;6(1):42-48. [CrossRef] [Medline]
20. Mastro TD, Kitayaporn D, Weniger BG, Vanichseni S, Laosunthorn V, Uneklabh T, et al. Estimating the number of HIV-infected injection drug users in Bangkok: a capture--recapture method. Am J
Public Health 1994 Jul;84(7):1094-1099. [CrossRef] [Medline]
21. Manrique-Vallier D. Bayesian population size estimation using Dirichlet process mixtures. Biometrics 2016 Dec 08;72(4):1246-1254. [CrossRef] [Medline]
22. Link WA. Nonidentifiability of population size from capture-recapture data with heterogeneous detection probabilities. Biometrics 2003 Dec 11;59(4):1123-1130. [CrossRef] [Medline]
23. RStudio. URL: https://shiny.rstudio.com/ [accessed 2021-11-05]
24. Laplace PS. Sur les Naissances, les Mariages et le Morts. In: Histoire de L'Acadèmie Royale des Sciences. Paris: L'Acadèmie Royale des Sciences; 1787:693-702.
25. Dahl K. Studies of trout and trout waters in Norway. Salmon and Trout Magazine 1919;18:16-33.
26. Lincoln F. Calculating waterfowl abundance on the basis of banding returns. Washington, DC: United States Department of Agriculture; 1930:118.
27. Chapman DG, Otis DL, Burnham KP, White GC, Anderson DR. Statistical inference from capture data on closed animal populations. Biometrics 1980 Jun;36(2):362. [CrossRef]
28. Hook E, Regal R. Capture-recapture methods in epidemiology: methods and limitations. Epidemiol Rev 1995;17(2):243-264. [CrossRef] [Medline]
29. Chao A, Tsay PK, Lin S, Shau W, Chao D. The applications of capture-recapture models to epidemiological data. Stat Med 2001 Oct 30;20(20):3123-3157. [CrossRef] [Medline]
30. Darroch JN. The multiple-recapture census: I. estimation of a closed population. Biometrika 1958 Dec;45(3/4):343. [CrossRef]
31. Schnabel ZE. The estimation of total fish population of a lake. The American Mathematical Monthly 1938 Jun;45(6):348. [CrossRef]
32. Fienberg SE. The multiple recapture census for closed populations and incomplete 2 contingency tables. Biometrika 1972;59(3):591-603. [CrossRef]
33. Goodman LA. Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika 1974;61(2):215-231. [CrossRef]
34. Pollock KH. A K-Sample Tag-Recapture Model Allowing for Unequal Survival and Catchability. Biometrika 1975 Dec;62(3):577. [CrossRef]
35. Coull BA, Agresti A. The use of mixed logit models to reflect heterogeneity in capture-recapture studies. Biometrics 1999 Mar;55(1):294-301. [CrossRef] [Medline]
36. Burnham KP, Anderson DR. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. New York: Springer; 2002.
37. Rivest LP. A lower bound model for multiple record systems estimation with heterogeneous catchability. International Journal of Biostatistics 2011;7(1):23. [CrossRef]
38. Holzmann H, Munk A, Zucchini W. On identifiability in capture-recapture models. Biometrics 2006 Sep;62(3):934-6; discussion 936. [CrossRef] [Medline]
39. R Core Team. R: A language and environment for statistical computing. In: R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing; 2020.
40. Baillargeon S, Rivest L. The Package: Loglinear Models for Capture-Recapture in R. J. Stat. Soft 2007;19(5):1. [CrossRef]
41. Madigan D, York J. Bayesian methods for estimation of the size of a closed population. Biometrika 1997;84(1):19-31.
42. Fellows IE. EpiApps. 2018. URL: https://epiapps.com/ [accessed 2021-05-13]
43. National Agency for the Control of AIDS (NACA). HIV Epidemic Appraisals in Nigeria: Evidence for Prevention Programme Planning and Implementation. Abuja, Nigeria: National Agency for the Control
of AIDS (NACA); 2013.
44. Society for Family Health (SFH). Mapping and Characterisations of Key Populations: Evidence for Prevention Programme Planning and Implementation in Nigeria. Abuja, Nigeria: Society for Family
Health (SFH); 2015.
45. Maheu-Giroux M, Marsh K, Doyle CM, Godin A, Lanièce Delaunay C, Johnson LF, et al. National HIV testing and diagnosis coverage in sub-Saharan Africa: a new modeling tool for estimating the 'first
90' from program and survey data. AIDS 2019 Dec 15;33 Suppl 3(1):S255-S269 [FREE Full text] [CrossRef] [Medline]
46. Wesson PD, McFarland W, Qin CC, Mirzazadeh A. Software application profile: the Anchored Multiplier calculator-a Bayesian tool to synthesize population size estimates. Int J Epidemiol 2019 Dec
01;48(6):1744-1749 [FREE Full text] [CrossRef] [Medline]
47. Fearon E, Chabata ST, Thompson JA, Cowan FM, Hargreaves JR. Sample size calculations for population size estimation studies using multiplier methods with respondent-driven sampling surveys. JMIR
Public Health Surveill 2017 Sep 14;3(3):e59 [FREE Full text] [CrossRef] [Medline]
2S-CRC: 2-sample capture-recapture
3S-CRC: 3 (or more)-sample capture-recapture
AIC: Akaike information criterion
BIC: Bayesian information criterion
CRC: capture-recapture
FSW: female sex worker
MCMC: Markov chain Monte Carlo
MSM: men who have sex with men
PSE: population size estimate
PWID: people who inject drugs
SNU: subnational unit
SSL: secure-sockets layer
Edited by H Bradley; submitted 05.08.21; peer-reviewed by P Wesson, S Chabata, A Safarnejad; comments to author 08.11.21; revised version received 10.01.22; accepted 24.02.22; published 26.04.22
©Anne F McIntyre, Ian E Fellows, Steve Gutreuter, Wolfgang Hladik. Originally published in JMIR Public Health and Surveillance (https://publichealth.jmir.org), 26.04.2022.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work, first published in JMIR Public Health and Surveillance, is properly cited. The complete bibliographic information, a link to the original
publication on https://publichealth.jmir.org, as well as this copyright and license information must be included. | {"url":"https://publichealth.jmir.org/2022/4/e32645","timestamp":"2024-11-01T20:58:56Z","content_type":"text/html","content_length":"427867","record_id":"<urn:uuid:d8e461f6-6eca-43e7-8503-1707d93898fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00335.warc.gz"} |
Background Information
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Empiricism is a theory of knowledge that asserts that knowledge comes only or primarily from sensory experience. One of several views of epistemology, the study of human knowledge, along with
rationalism, idealism, and historicism, empiricism emphasizes the role of experience and evidence, especially sensory experience, in the formation of ideas, over the notion of innate ideas or
traditions; empiricists may argue however that traditions (or customs) arise due to relations of previous sense experiences.
Empiricism in the philosophy of science emphasizes evidence, especially as discovered in experiments. It is a fundamental part of the scientific method that all hypotheses and theories must be tested
against observations of the natural world rather than resting solely on a priori reasoning, intuition, or revelation.
Philosophers associated with empiricism include Aristotle, Alhazen, Avicenna, Ibn Tufail, Robert Grosseteste, William of Ockham, Francis Bacon, Thomas Hobbes, Robert Boyle, John Locke, George
Berkeley, Hermann von Helmholtz, David Hume, Leopold von Ranke, and John Stuart Mill.
The English term "empiric" derives from the Greek word ἐμπειρία, which is cognate with and translates to the Latin experientia, from which we derive the word "experience" and the related
"experiment". The term was used of the Empiric school of ancient Greek medical practitioners, who rejected the doctrines of the ( Dogmatic school), preferring to rely on the observation of
A central concept in science and the scientific method is that it must be empirically based on the evidence of the senses. Both natural and social sciences use working hypotheses that are testable by
observation and experiment. The term semi-empirical is sometimes used to describe theoretical methods that make use of basic axioms, established scientific laws, and previous experimental results in
order to engage in reasoned model building and theoretical inquiry.
Philosophical empiricists hold no knowledge to be properly inferred or deduced unless it is derived from one's sense-based experience. This view is commonly contrasted with rationalism, which asserts
that knowledge may be derived from reason independently of the senses. For example John Locke held that some knowledge (e.g. knowledge of God's existence) could be arrived at through intuition and
reasoning alone. Similarly Robert Boyle, a prominent advocate of the experimental method, held that we have innate ideas. The main continental rationalists (Descartes, Spinoza, and Leibniz) were also
advocates of the empirical "scientific method".
Early empiricism
The notion of tabula rasa ("clean slate" or "blank tablet") connotes a view of mind as an originally blank or empty recorder (Locke used the words "white paper") on which experience leaves marks.
This denies that humans have innate ideas. The image dates back to Aristotle;
What the mind ( nous) thinks must be in it in the same sense as letters are on a tablet (grammateion) which bears no actual writing (grammenon); this is just what happens in the case of the mind.
(Aristotle, On the Soul, 3.4.430^a1).
Aristotle's explanation of how this was possible, was not strictly empiricist in a modern sense, but rather based on his theory of potentiality and actuality, and experience of sense perceptions
still requires the help of the active nous. These notions contrasted with Platonic notions of the human mind as an entity that pre-existed somewhere in the heavens, before being sent down to join a
body on Earth (see Plato's Phaedo and Apology, as well as others). Aristotle was considered to give a more important position to sense perception than Plato, and commentators in the middle ages
summarized one of his positions as "nihil in intellectu nisi prius fuerit in sensu" (Latin for "nothing in the intellect without first being in the senses").
During the middle ages Aristotle's theory of tabula rasa was developed by Islamic philosophers starting with Al Farabi, developing into an elaborate theory by Avicenna and demonstrated as a thought
experiment by Ibn Tufail. For Avicenna ( Ibn Sina), for example, the a tabula rasa is a pure potentiality that is actualized through education, and knowledge is attained through "empirical
familiarity with objects in this world from which one abstracts universal concepts" developed through a " syllogistic method of reasoning in which observations lead to propositional statements which
when compounded lead to further abstract concepts." The intellect itself develops from a material intellect (al-'aql al-hayulani), which is a potentiality "that can acquire knowledge to the active
intellect (al- 'aql al-fa'il), the state of the human intellect in conjunction with the perfect source of knowledge". So the immaterial "active intellect", separate from any individual person, is
still essential for understanding to occur.
In the 12th century CE the Andalusian Muslim philosopher and novelist Abu Bakr Ibn Tufail (known as "Abubacer" or "Ebn Tophail" in the West) included the theory of tabula rasa as a thought experiment
in his Arabic philosophical novel, Hayy ibn Yaqdhan in which he depicted the development of the mind of a feral child "from a tabula rasa to that of an adult, in complete isolation from society" on a
desert island, through experience alone. The Latin translation of his philosophical novel, entitled Philosophus Autodidactus, published by Edward Pococke the Younger in 1671, had an influence on John
Locke's formulation of tabula rasa in An Essay Concerning Human Understanding.
A similar Islamic theological novel, Theologus Autodidactus, was written by the Arab theologian and physician Ibn al-Nafis in the 13th century. It also dealt with the theme of empiricism through the
story of a feral child on a desert island, but departed from its predecessor by depicting the development of the protagonist's mind through contact with society rather than in isolation from society.
During the 13th century Thomas Aquinas adopted the Aristotelian position that the senses are essential to mind into scholasticism. Bonaventure (1221–1274), one of Aquinas' strongest intellectual
opponents, offered some of the strongest arguments in favour of the Platonic idea of the mind.
Renaissance Italy
In the late renaissance various writers began to question the medieval and classical understanding of knowledge acquisition in a more fundamental way. In political and historical writing Niccolò
Machiavelli and his friend Francesco Guicciardini initiated a new realistic style of writing. Machiavelli in particular was scornful of writers on politics who judged everything in comparison to
mental ideals and demanded that people should study the "effectual truth" instead.
Their contemporary, Leonardo da Vinci (1452–1519) said,
If you find from your own experience that something is a fact and it contradicts what some authority has written down, then you must abandon the authority and base your reasoning on your own
The decidedly anti-Aristotelian and anti-clerical music theorist Vincenzo Galilei (ca. 1520–1591), father of Galileo and the inventor of monody, made use of the method in successfully solving musical
problems, firstly, of tuning such as the relationship of pitch to string tension and mass in stringed instruments, and to volume of air in wind instruments; and secondly to composition, by his
various suggestions to composers in his Dialogo della musica antica e moderna (Florence, 1581). The Italian word he used for "experiment" was esperienza. It is known that he was the essential
pedagogical influence upon the young Galileo, his eldest son (cf. Coelho, ed. Music and Science in the Age of Galileo Galilei), arguably one of the most influential empiricists in history. Vincenzo,
through his tuning research, found the underlying truth at the heart of the misunderstood myth of ' Pythagoras' hammers' (the square of the numbers concerned yielded those musical intervals, not the
actual numbers, as believed), and through this and other discoveries that demonstrated the fallibility of traditional authorities, a radically empirical attitude developed, passed on to Galileo,
which regarded "experience and demonstration" as the sine qua non of valid rational enquiry.
British empiricism
British empiricism, though it was not a term used at the time, derives from the 17th century period of early modern philosophy and modern science. The term became useful in order to describe
differences perceived between two of its founders Francis Bacon, described as empiricist, and René Descartes, who is described as a rationalist. Thomas Hobbes and Baruch Spinoza, in the next
generation, are often also described as an empiricist and a rationalist respectively. John Locke, George Berkeley, and David Hume were the primary exponents of empiricism in the 18th century
Enlightenment, with Locke being the person who is normally known as the founder of empiricism as such.
In response to the early-to-mid-17th century " continental rationalism" John Locke (1632–1704) proposed in An Essay Concerning Human Understanding (1689) a very influential view wherein the only
knowledge humans can have is a posteriori, i.e., based upon experience. Locke is famously attributed with holding the proposition that the human mind is a tabula rasa, a "blank tablet," in Locke's
words "white paper," on which the experiences derived from sense impressions as a person's life proceeds are written. There are two sources of our ideas: sensation and reflection. In both cases, a
distinction is made between simple and complex ideas. The former are unanalysable, and are broken down into primary and secondary qualities. Primary qualities are essential for the object in question
to be what it is. Without specific primary qualities, an object would not be what it is. For example, an apple is an apple because of the arrangement of its atomic structure. If an apple was
structured differently, it would cease to be an apple. Secondary qualities are the sensory information we can perceive from its primary qualities. For example, an apple can be perceived in various
colours, sizes, and textures but it is still identified as an apple. Therefore its primary qualities dictate what the object essentially is, while its secondary qualities define its attributes.
Complex ideas combine simple ones, and divide into substances, modes, and relations. According to Locke, our knowledge of things is a perception of ideas that are in accordance or discordance with
each other, which is very different from the quest for certainty of Descartes.
A generation later, the Irish Anglican bishop, George Berkeley (1685–1753), determined that Locke's view immediately opened a door that would lead to eventual atheism. In response to Locke, he put
forth in his Treatise Concerning the Principles of Human Knowledge (1710) an important challenge to empiricism in which things only exist either as a result of their being perceived, or by virtue of
the fact that they are an entity doing the perceiving. (For Berkeley, God fills in for humans by doing the perceiving whenever humans are not around to do it). In his text Alciphron, Berkeley
maintained that any order humans may see in nature is the language or handwriting of God. Berkeley's approach to empiricism would later come to be called subjective idealism.
The Scottish philosopher David Hume (1711–1776) responded to Berkeley's criticisms of Locke, as well as other differences between early modern philosophers, and moved empiricism to a new level of
skepticism. Hume argued in keeping with the empiricist view that all knowledge derives from sense experience, but he accepted that this has implications not normally acceptable to philosophers. He
wrote for example, "Mr. Locke divides all arguments into demonstrative and probable. In this view, we must say, that it is only probable all men must die, or that the sun will rise to-morrow." And,
"Mr. Locke, in his chapter of power, says that, finding from experience, that there are several new productions in nature, and concluding that there must somewhere be a power capable of producing
them, we arrive at last by this reasoning at the idea of power. But no reasoning can ever give us a new, original, simple idea; as this philosopher himself confesses. This, therefore, can never be
the origin of that idea."
Hume divided all of human knowledge into two categories: relations of ideas and matters of fact (see also Kant's analytic-synthetic distinction). Mathematical and logical propositions (e.g. "that the
square of the hypotenuse is equal to the sum of the squares of the two sides") are examples of the first, while propositions involving some contingent observation of the world (e.g. "the sun rises in
the East") are examples of the second. All of people's "ideas", in turn, are derived from their "impressions". For Hume, an "impression" corresponds roughly with what we call a sensation. To remember
or to imagine such impressions is to have an "idea". Ideas are therefore the faint copies of sensations.
Hume maintained that all knowledge, even the most basic beliefs about the natural world, cannot be conclusively established by reason. Rather, he maintained, our beliefs are more a result of
accumulated habits, developed in response to accumulated sense experiences. Among his many arguments Hume also added another important slant to the debate about scientific method — that of the
problem of induction. Hume argued that it requires inductive reasoning to arrive at the premises for the principle of inductive reasoning, and therefore the justification for inductive reasoning is a
circular argument. Among Hume's conclusions regarding the problem of induction is that there is no certainty that the future will resemble the past. Thus, as a simple instance posed by Hume, we
cannot know with certainty by inductive reasoning that the sun will continue to rise in the East, but instead come to expect it to do so because it has repeatedly done so in the past.
Hume concluded that such things as belief in an external world and belief in the existence of the self were not rationally justifiable. According to Hume these beliefs were to be accepted nonetheless
because of their profound basis in instinct and custom. Hume's lasting legacy, however, was the doubt that his skeptical arguments cast on the legitimacy of inductive reasoning, allowing many
skeptics who followed to cast similar doubt.
Most of Hume's followers have disagreed with his conclusion that belief in an external world is rationally unjustifiable, contending that Hume's own principles implicitly contained the rational
justification for such a belief, that is, beyond being content to let the issue rest on human instinct, custom and habit. According to an extreme empiricist theory known as Phenomenalism, anticipated
by the arguments of both Hume and George Berkeley, a physical object is a kind of construction out of our experiences. Phenomenalism is the view that physical objects, properties, events (whatever is
physical) are reducible to mental objects, properties, events. Ultimately, only mental objects, properties, events, exist — hence the closely related term subjective idealism. By the phenomenalistic
line of thinking, to have a visual experience of a real physical thing is to have an experience of a certain kind of group of experiences. This type of set of experiences possesses a constancy and
coherence that is lacking in the set of experiences of which hallucinations, for example, are a part. As John Stuart Mill put it in the mid-19th century, matter is the "permanent possibility of
sensation". Mill's empiricism went a significant step beyond Hume in still another respect: in maintaining that induction is necessary for all meaningful knowledge including mathematics. As
summarized by D.W. Hamlin:
[Mill] claimed that mathematical truths were merely very highly confirmed generalizations from experience; mathematical inference, generally conceived as deductive [and a priori] in nature, Mill
set down as founded on induction. Thus, in Mill's philosophy there was no real place for knowledge based on relations of ideas. In his view logical and mathematical necessity is psychological; we
are merely unable to conceive any other possibilities than those that logical and mathematical propositions assert. This is perhaps the most extreme version of empiricism known, but it has not
found many defenders.
Mill's empiricism thus held that knowledge of any kind is not from direct experience but an inductive inference from direct experience. The problems other philosophers have had with Mill's position
centre around the following issues: Firstly, Mill's formulation encounters difficulty when it describes what direct experience is by differentiating only between actual and possible sensations. This
misses some key discussion concerning conditions under which such "groups of permanent possibilities of sensation" might exist in the first place. Berkeley put God in that gap; the phenomenalists,
including Mill, essentially left the question unanswered. In the end, lacking an acknowledgement of an aspect of "reality" that goes beyond mere "possibilities of sensation", such a position leads to
a version of subjective idealism. Questions of how floor beams continue to support a floor while unobserved, how trees continue to grow while unobserved and untouched by human hands, etc., remain
unanswered, and perhaps unanswerable in these terms. Secondly, Mill's formulation leaves open the unsettling possibility that the "gap-filling entities are purely possibilities and not actualities at
all". Thirdly, Mill's position, by calling mathematics merely another species of inductive inference, misapprehends mathematics. It fails to fully consider the structure and method of mathematical
science, the products of which are arrived at through an internally consistent deductive set of procedures which do not, either today or at the time Mill wrote, fall under the agreed meaning of
The phenomenalist phase of post-Humean empiricism ended by the 1940s, for by that time it had become obvious that statements about physical things could not be translated into statements about actual
and possible sense data. If a physical object statement is to be translatable into a sense-data statement, the former must be at least deducible from the latter. But it came to be realized that there
is no finite set of statements about actual and possible sense-data from which we can deduce even a single physical-object statement. Remember that the translating or paraphrasing statement must be
couched in terms of normal observers in normal conditions of observation. There is, however, no finite set of statements that are couched in purely sensory terms and can express the satisfaction of
the condition of the presence of a normal observer. According to phenomenalism, to say that a normal observer is present is to make the hypothetical statement that were a doctor to inspect the
observer, the observer would appear to the doctor to be normal. But, of course, the doctor himself must be a normal observer. If we are to specify this doctor's normality in sensory terms, we must
make reference to a second doctor who, when inspecting the sense organs of the first doctor, would himself have to have the sense data a normal observer has when inspecting the sense organs of a
subject who is a normal observer. And if we are to specify in sensory terms that the second doctor is a normal observer, we must refer to a third doctor, and so on (also see the third man).
Logical empiricism
Logical empiricism (aka logical positivism or neopositivism) was an early 20th century attempt to synthesize the essential ideas of British empiricism (e.g. a strong emphasis on sensory experience as
the basis for knowledge) with certain insights from mathematical logic that had been developed by Gottlob Frege and Ludwig Wittgenstein. Some of the key figures in this movement were Otto Neurath,
Moritz Schlick and the rest of the Vienna Circle, along with A.J. Ayer, Rudolf Carnap and Hans Reichenbach. The neopositivists subscribed to a notion of philosophy as the conceptual clarification of
the methods, insights and discoveries of the sciences. They saw in the logical symbolism elaborated by Frege (d. 1925) and Bertrand Russell (1872–1970) a powerful instrument that could rationally
reconstruct all scientific discourse into an ideal, logically perfect, language that would be free of the ambiguities and deformations of natural language. This gave rise to what they saw as
metaphysical pseudoproblems and other conceptual confusions. By combining Frege's thesis that all mathematical truths are logical with the early Wittgenstein's idea that all logical truths are mere
linguistic tautologies, they arrived at a twofold classification of all propositions: the analytic (a priori) and the synthetic (a posteriori). On this basis, they formulated a strong principle of
demarcation between sentences that have sense and those that do not: the so-called verification principle. Any sentence that is not purely logical, or is unverifiable is devoid of meaning. As a
result, most metaphysical, ethical, aesthetic and other traditional philosophical problems came to be considered pseudoproblems.
In the extreme empiricism of the neopositivists—at least before the 1930s—any genuinely synthetic assertion must be reducible to an ultimate assertion (or set of ultimate assertions) that expresses
direct observations or perceptions. In later years, Carnap and Neurath abandoned this sort of phenomenalism in favour of a rational reconstruction of knowledge into the language of an objective
spatio-temporal physics. That is, instead of translating sentences about physical objects into sense-data, such sentences were to be translated into so-called protocol sentences, for example, "X at
location Y and at time T observes such and such." The central theses of logical positivism (verificationism, the analytic-synthetic distinction, reductionism, etc.) came under sharp attack after
World War 2 by thinkers such as Nelson Goodman, W.V. Quine, Hilary Putnam, Karl Popper, and Richard Rorty. By the late 1960s, it had become evident to most philosophers that the movement had pretty
much run its course, though its influence is still significant among contemporary analytic philosophers such as Michael Dummett and other anti-realists.
In the late 19th and early 20th century several forms of pragmatic philosophy arose. The ideas of pragmatism, in its various forms, developed mainly from discussions that took place while Charles
Sanders Peirce and William James were both at Harvard in the 1870s. James popularized the term "pragmatism", giving Peirce full credit for its patrimony, but Peirce later demurred from the tangents
that the movement was taking, and redubbed what he regarded as the original idea with the name of "pragmaticism". Along with its pragmatic theory of truth, this perspective integrates the basic
insights of empirical (experience-based) and rational (concept-based) thinking.
Charles Peirce (1839–1914) was highly influential in laying the groundwork for today's empirical scientific method. Although Peirce severely criticized many elements of Descartes' peculiar brand of
rationalism, he did not reject rationalism outright. Indeed, he concurred with the main ideas of rationalism, most importantly the idea that rational concepts can be meaningful and the idea that
rational concepts necessarily go beyond the data given by empirical observation. In later years he even emphasized the concept-driven side of the then ongoing debate between strict empiricism and
strict rationalism, in part to counterbalance the excesses to which some of his cohorts had taken pragmatism under the "data-driven" strict-empiricist view. Among Peirce's major contributions was to
place inductive reasoning and deductive reasoning in a complementary rather than competitive mode, the latter of which had been the primary trend among the educated since David Hume wrote a century
before. To this, Peirce added the concept of abductive reasoning. The combined three forms of reasoning serve as a primary conceptual foundation for the empirically based scientific method today.
Peirce's approach "presupposes that (1) the objects of knowledge are real things, (2) the characters (properties) of real things do not depend on our perceptions of them, and (3) everyone who has
sufficient experience of real things will agree on the truth about them. According to Peirce's doctrine of fallibilism, the conclusions of science are always tentative. The rationality of the
scientific method does not depend on the certainty of its conclusions, but on its self-corrective character: by continued application of the method science can detect and correct its own mistakes,
and thus eventually lead to the discovery of truth".
In his Harvard "Lectures on Pragmatism" (1903), Peirce enumerated what he called the "three cotary propositions of pragmatism" (L: cos, cotis whetstone), saying that they "put the edge on the maxim
of pragmatism". First among these he listed the peripatetic-thomist observation mentioned above, but he further observed that this link between sensory perception and intellectual conception is a
two-way street. That is, it can be taken to say that whatever we find in the intellect is also incipiently in the senses. Hence, if theories are theory-laden then so are the senses, and perception
itself can be seen as a species of abductive inference, its difference being that it is beyond control and hence beyond critique – in a word, incorrigible. This in no way conflicts with the
fallibility and revisability of scientific concepts, since it is only the immediate percept in its unique individuality or "thisness" – what the Scholastics called its haecceity – that stands beyond
control and correction. Scientific concepts, on the other hand, are general in nature, and transient sensations do in another sense find correction within them. This notion of perception as abduction
has received periodic revivals in artificial intelligence and cognitive science research, most recently for instance with the work of Irvin Rock on indirect perception.
Around the beginning of the 20th century, William James (1842–1910) coined the term "radical empiricism" to describe an offshoot of his form of pragmatism, which he argued could be dealt with
separately from his pragmatism – though in fact the two concepts are intertwined in James's published lectures. James maintained that the empirically observed "directly apprehended universe needs ...
no extraneous trans-empirical connective support", by which he meant to rule out the perception that there can be any value added by seeking supernatural explanations for natural phenomena. James's
"radical empricism" is thus not radical in the context of the term "empiricism", but is instead fairly consistent with the modern use of the term " empirical". (His method of argument in arriving at
this view, however, still readily encounters debate within philosophy even today.)
John Dewey (1859–1952) modified James' pragmatism to form a theory known as instrumentalism. The role of sense experience in Dewey's theory is crucial, in that he saw experience as unified totality
of things through which everything else is interrelated. Dewey's basic thought, in accordance with empiricism was that reality is determined by past experience. Therefore, humans adapt their past
experiences of things to perform experiments upon and test the pragmatic values of such experience. The value of such experience is measured by scientific instruments, and the results of such
measurements generate ideas that serve as instruments for future experimentation. Thus, ideas in Dewey's system retain their empiricist flavour in that they are only known a posteriori. | {"url":"https://ftp.worldpossible.org/endless/eos-rachel/RACHEL/RACHEL/modules/wikipedia_for_schools/wp/e/Empiricism.htm","timestamp":"2024-11-08T14:41:41Z","content_type":"text/html","content_length":"44138","record_id":"<urn:uuid:c13c69bc-39ba-4e95-ad94-39606fa595d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00046.warc.gz"} |
Bava Metzia 72 - Guarantor and Loan Assumption (Finds)
One may act as a guarantor for an interest-bearing loan advanced by a non-Jew to a Jew, structured so that there is no interest payment if the borrower pays back to the guarantor.
One may assume an interest-bearing loan given by a non-Jew to a Jew, but the non-Jew must agree, for otherwise, one would, in fact, be paying interest to the first borrower, who would then be paying
it to the lender.
If a non-Jew lent money on interest to a Jew and then converted, he may still collect interest if, before his conversion, he establishes it as a new loan, in the amount of principal plus interest.
Art: Rudolph Ernst - A Hard Bargain
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Pythagorean Theorem Dominoes Activity
• >
• Pythagorean Theorem Dominoes Activity
Pythagorean Theorem Dominoes Activity
per item
Included is a dominoes activity on the Pythagorean Theorem. Students will be calculating the missing side of a triangle by using the Pythagorean Theorem. Some domino pieces show a picture of a
triangle with a missing side. Other domino pieces show a = , b = , and c = with one of these sides unknown. This is a great activity for working in pairs, or small groups.
File Type: PDF
Pages: 4+
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Question #a9643 | Socratic
Question #a9643
1 Answer
The idea here is that you need to use the vapor density of ammonia, ${\text{NH}}_{3}$, to determine its molar mass, then use this to find how many moles of ammonia you have in that sample.
At this point, you need to use the definition of NTP (Normal Temperature and Pressure) and the ideal gas law equation to find the volume of the gas.
So, the vapor density of a gas is calculated by comparing the density of the gas with that of hydrogen gas, ${\text{H}}_{2}$, kept under the same conditions for pressure and temperature.
In essence, the vapor density of a gas tells you the ratio that exists between a mass of the gas and the mass of hydrogen gas that occupies the same volume as the mass of the gas.
You can thus say that
$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} {\text{vapor density" = "molar mass of the gas"/"molar mass of H}}_{2} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
If you take the molar mass of hydrogen gas to be equal to ${\text{2 g mol}}^{- 1}$, you can say that the molar mass of ammonia will be
${M}_{{\text{M NH"_ 3) = "vapor density" xx M_ ("M H}}_{2}}$
#M_("M NH"_3) = 8.5 xx "2 g mol"^(-1) = "17 g mol"^(-1)#
Now, use the molar mass of ammonia to find how many moles you have in $\text{85 g}$ of the compound
#85 color(red)(cancel(color(black)("g"))) * "1 mole NH"_3/(17color(red)(cancel(color(black)("g")))) = "5 moles NH"_3#
The ideal gas law equation looks like this
$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} P V = n R T \textcolor{w h i t e}{\frac{a}{a}} |}}} \text{ }$, where
$P$ - the pressure of the gas
$V$ - the volume it occupies
$n$ - the number of moles of gas
$R$ - the universal gas constant, usually given as $0.0821 \left(\text{atm" * "L")/("mol" * "K}\right)$
$T$ - the absolute temperature of the gas
NTP conditions are defined as a pressure of $\text{1 atm}$ and a temperature of ${20}^{\circ} \text{C" = "293.15 K}$.
Rearrange the equation to solve for $V$ and plug in your values to find
$P V = n R T \implies V = \frac{n R T}{P}$
${V}_{N {H}_{3}} = \left(5 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{moles"))) * 0.0821(color(red)(cancel(color(black)("atm"))) * "L")/(color(red)(cancel(color(black)("mol"))) * color
(red)(cancel(color(black)("K")))) * 293.15color(red)(cancel(color(black)("K"))))/(1color(red)(cancel(color(black)("atm}}}}\right)$
${V}_{N {H}_{3}} = \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \text{120 L} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
The answer is rounded to two sig figs.
Impact of this question
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What is Data Reduction? Techniques - Binary Terms
Data reduction is a process that reduced the volume of original data and represents it in a much smaller volume. Data reduction techniques ensure the integrity of data while reducing the data.
The time required for data reduction should not overshadow the time saved by the data mining on the reduced data set. In this section, we will discuss data reduction in brief and we will discuss
different methods of data reduction.
Content: Data Reduction in Data Mining
What is Data Reduction?
When you collect data from different data warehouses for analysis, it results in a huge amount of data. It is difficult for a data analyst to deal with this large volume of data.
It is even difficult to run the complex queries on the huge amount of data as it takes a long time and sometimes it even becomes impossible to track the desired data.
This is why reducing data becomes important. Data reduction technique reduces the volume of data yet maintains the integrity of the data.
Data reduction does not affect the result obtained from data mining that means the result obtained from data mining before data reduction and after data reduction is the same (or almost the same).
The only difference occurs in the efficiency of data mining. Data reduction increases the efficiency of data mining. In the following section, we will discuss the techniques of data reduction.
Data Reduction Techniques
Techniques of data deduction include dimensionality reduction, numerosity reduction and data compression.
1. Dimensionality Reduction
Dimensionality reduction eliminates the attributes from the data set under consideration thereby reducing the volume of original data. In the section below, we will discuss three methods of
dimensionality reduction.
a. Wavelet Transform
In the wavelet transform, a data vector X is transformed into a numerically different data vector X’ such that both X and X’ vectors are of the same length. Then how it is useful in reducing data?
The data obtained from the wavelet transform can be truncated. The compressed data is obtained by retaining the smallest fragment of the strongest of wavelet coefficients.
Wavelet transform can be applied to data cube, sparse data or skewed data.
b. Principal Component Analysis
Let us consider we have a data set to be analyzed that has tuples with n attributes, then the principal component analysis identifies k independent tuples with n attributes that can represent the
data set.
In this way, the original data can be cast on a much smaller space. In this way, the dimensionality reduction can be achieved. Principal component analysis can be applied to sparse, and skewed data.
c. Attribute Subset Selection
The large data set has many attributes some of which are irrelevant to data mining or some are redundant. The attribute subset selection reduces the volume of data by eliminating the redundant and
irrelevant attribute.
The attribute subset selection makes it sure that even after eliminating the unwanted attributes we get a good subset of original attributes such that the resulting probability of data distribution
is as close as possible to the original data distribution using all the attributes.
2. Numerosity Reduction
The numerosity reduction reduces the volume of the original data and represents it in a much smaller form. This technique includes two types parametric and non-parametric numerosity reduction.
Parametric numerosity reduction incorporates ‘storing only data parameters instead of the original data’. One method of parametric numerosity reduction is ‘regression and log-linear’ method.
1. Regression and Log-Linear
Linear regression models a relationship between the two attributes by modelling a linear equation to the data set. Suppose we need to model a linear function between two attributes.
y = wx +b
Here, y is the response attribute and x is the predictor attribute. If we discuss in terms of data mining, the attribute x and the attribute y are the numeric database attributes whereas w and b
are regression coefficients.
Multiple linear regression lets the response variable y to model linear function between two or more predictor variable.
Log-linear model discovers the relation between two or more discrete attributes in the database. Suppose, we have a set of tuples presented in n-dimensional space. Then the log-linear model is
used to study the probability of each tuple in a multidimensional space.
Regression and log-linear method can be used for sparse data and skewed data.
1. Histogram
A histogram is a ‘graph’ that represents frequency distribution which describes how often a value appears in the data. Histogram uses the binning method and to represent data distribution of an
attribute. It uses disjoint subset which we call as bin or buckets.
We have data for AllElectronics data set, which contains prices for regularly sold items.
1, 1, 5, 5, 5, 5, 5, 8, 8, 10, 10, 10, 10, 12, 14, 14, 14, 15, 15, 15, 15, 15, 15, 18, 18, 18, 18, 18, 18, 18, 18, 20, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 25, 25, 25, 25, 25, 28, 28, 30, 30,
The diagram below shows a histogram of equal width that shows the frequency of price distribution.
2. Clustering
Clustering techniques groups the similar objects from the data in such a way that the objects in a cluster are similar to each other but they are dissimilar to objects in another cluster.
How much similar are the objects inside a cluster can be calculated by using a distance function. More is the similarity between the objects in a cluster closer they appear in the cluster.
The quality of cluster depends on the diameter of the cluster i.e. the at max distance between any two objects in the cluster.
The original data is replaced by the cluster representation. This technique is more effective if the present data can be classified into a distinct clustered.
3. Sampling
One of the methods used for data reduction is sampling as it is capable to reduce the large data set into a much smaller data sample. Below we will discuss the different method in which we can
sample a large data set D containing N tuples:
• Simple random sample without replacement (SRSWOR) of size s: In this ‘s number’ of tuples are drawn from N tuples such that in the data set D (s<N). The probability of drawing any tuple from the
data set D is 1/N this means all tuples have an equal probability of getting sampled.
• Simple random sample with replacement (SRSWR) of size s: It is similar to the SRSWOR but the tuple is drawn from data set D, is recorded and then replaced back into the data set D so that it can
be drawn again.
• Cluster sample: The tuples in data set D are clustered into M mutually disjoint subsets. From these clusters, a simple random sample of size s could be generated where s<M. The data reduction can
be applied by implementing SRSWOR on these clusters.
• Stratified sample: The large data set D is partitioned into mutually disjoint sets called ‘strata’. Now a simple random sample is taken from each stratum to get stratified data. This method is
effective for skewed data.
4. Data Cube Aggregation
Consider you have the data of AllElectronics sales per quarter for the year 2008 to the year 2010. In case you want to get the annual sale per year then you just have to aggregate the sales per
quarter for each year. In this way, aggregation provides you with the required data which is much smaller in size and thereby we achieve data reduction even without losing any data.
3. Data Compression
Data compression is a technique where the data transformation technique is applied to the original data in order to obtain compressed data. If the compressed data can again be reconstructed to form
the original data without losing any information then it is a ‘lossless’ data reduction.
If you are unable to reconstruct the original data from the compressed one then your data reduction is ‘lossy’. Dimensionality and numerosity reduction method are also used for data compression.
Key Takeaways
• Data reduction is a method of reducing the volume of data thereby maintaining the integrity of the data.
• There are three basic methods of data reduction dimensionality reduction, numerosity reduction and data compression.
• The time taken for data reduction must not be overweighed by the time preserved by data mining on the reduced data set.
So, this is all about the data reduction and its techniques. We have covered different methods that can be employed for data reduction. | {"url":"https://binaryterms.com/data-reduction.html","timestamp":"2024-11-03T21:31:53Z","content_type":"text/html","content_length":"57804","record_id":"<urn:uuid:5e22c4cb-6190-4cb0-82fe-49be6be93177>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00431.warc.gz"} |
Edit Solver Differential Equations
These equations can be used to add additional states to the mechanical system being modeled.
1. If the Solver Differential Equations panel is not currently displayed, select the desired solver diff by clicking on it in the Project Browser or in the modeling window.
The Solver Differential Equations panel is automatically displayed.
2. Select Static Hold if the state of the solver differential equation is not permitted to change during static or quasi-static analysis of the solver. Otherwise, deselect the option.
3. Select Implicit if the differential equation is of type implicit. If the equation is explicit, leave the option unselected.
4. Enter a value for the initial condition of the differential equation in the IC text box.
5. Enter a value for the initial condition for the first derivative of the user defined variable in the IC dot text box.
This option is usually used in conjunction with an implicit variable.
6. Select an option from the drop-down menu at the bottom of the panel.
7. Define the properties associated with your choice.
If Linear is chosen, enter a value in the text box.
If Curve is chosen:
a. Select AKIMA, CUBIC, LINEAR, or QUINTIC as the interpolation method.
b. Enter a value under Independent variable.
The independent variable should be specified in Templex syntax.
c. Resolve the curve by double-clicking the Curve collector and selecting a curve from the Select a Curve dialog.
Note: To use a curve, you first need to define a curve (using the Curves panel) which represents the behavior of the solver differential equation.
If Spline3D is chosen:
a. Select AKIMA, CUBIC, LINEAR , or QUINTIC under as the method of interpolation.
b. Specify an expression for Independent variable X and Independent variable Z.
c. Resolve the 3D spline by double-clicking on the Spline3D collector and selecting a Spline3D entity from the Select a Spline3D dialog.
Note: To use a Spline3D entity, you first need to define a spline using the Spline3D panel.
If Expression is chosen, enter an expression. | {"url":"https://2021.help.altair.com/2021.1/hwdesktop/mv/topics/motionview/solver_diff_equations_edit_t.htm","timestamp":"2024-11-04T21:37:53Z","content_type":"application/xhtml+xml","content_length":"33228","record_id":"<urn:uuid:a418dda1-5ce6-4b20-984a-2eac2984b25a>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00817.warc.gz"} |
Proving Ethereum signatures in o1js with Keccak and ECDSA
The v0.15.1 release of o1js (What’s New in o1js: January 2024) added some of the most community-requested features yet: the ability to verify Ethereum signatures using the Elliptic Curve Digital
Signature Algorithm (ECDSA) and hashes using Keccak. There’s good reason for this long-standing request. The ability to verify ECDSA signatures in o1js allows developers to build zero knowledge
applications that incorporate activity from Ethereum and other EVM blockchains, or to add layers of scalability and privacy to existing applications in those ecosystems. We were eager to provide
these capabilities as soon as possible but we knew that it would be a substantial undertaking to do it properly. Zero knowledge cryptography has some specialized components that are often
incompatible with other forms of cryptography, especially given our proof system’s native ability to support recursive composition. Resolving this would require the support of lower-level features
that are useful tools in their own right. And, if these tools were sufficiently abstracted, they would provide a kind of skeleton key that also unlocks interoperability with many other cryptographic
systems! This ambitious approach resulted in a cascade of dependencies.
Ethereum combines the ECDSA signature and Keccak hashing algorithms to sign and verify transactions. ECDSA uses an elliptic curve called secp256k1, which is built on a finite field different from the
one o1js was designed around (the Pallas base field). So, to get ECDSA, we needed to implement a mechanism for working with foreign finite fields and foreign elliptic curves over those fields. Keccak
also presented a challenge since it is comprised primarily of boolean logic, which, although efficient in physical computers, is generally quite expensive in zero knowledge proof systems. So, to get
Keccak, o1js also needed efficient, provable bitwise operations.
Bitwise Operations
Keccak uses four specific bitwise operations: NOT, AND, XOR, and ROT. Modern processors support these operations directly at the hardware level, which makes them exceptionally efficient on most
silicon. But computation works differently in zero knowledge proof systems. Any operation that o1js can prove must be represented as arithmetic over the native field, and the naive approach to this
is incredibly inefficient. We needed versions of these operations that were optimized for proving.
The specific implementations of each operation are relatively intricate (you can learn more in the Mina Book if you are interested), but in short, XOR and ROT are both implemented as custom gates
within Kimchi, whereas NOT and AND are implemented as gadgets using XOR and the generic gate.
// Using provable bitwise operations:
// and(a: Field, b: Field, length: number) => Field
// will compare `length` bits
let a = Field(0b0011);
let b = Field(0b0101);
let c = Gadgets.and(a, b, 4);
// not(a: Field, length: number, checked: boolean) => Field
// will compare `length` bits
let a = Field(0b0101);
let b = Gadgets.not(a,4,true);
// xor(a: Field, b: Field, length: number) => Field
// will compare `length` bits
let a = Field(0b0101);
let b = Field(0b0011);
let c = Gadgets.xor(a, b, 4);
// rotate64(field: Field, bits: number, direction: 'left' | 'right' = 'left') => Field
// rotate32(field: Field, bits: number, direction: 'left' | 'right' = 'left') => Field
// will rotate a value in the specified direction by the specified number of bits
let x = Field(0b001100);
let y = Gadgets.rotate32(x, 2, 'left'); // left rotation by 2 bits
// same as rotation except that bits fall off the end instead of wrapping around
// leftShift64(field: Field, bits: number) => Field // input must <64bits
// leftShift32(field: Field, bits: number) => Field // input must <32bits
// rightShift64(field: Field, bits: number) => Field // input must <32bits
let x = Provable.witness(Field, () => Field(0b001100)); // 12 in binary
let y = Gadgets.leftShift64(x, 2); // left shift by 2 bits
y.assertEquals(0b110000); // 48 in binary
Each of these new bitwise operations uses far fewer constraints than their naive counterparts would, and provide efficient build blocks for many other provable operations. We could use them to add
other hash functions to o1js in future. They have a wide range of applications beyond cryptographic primitives too. In fact, members of our community have previously written inefficient versions
manually to do things like evaluate the state of a game.
With the bitwise operations in place, we were poised to start working on the Keccak hashing algorithm. Since Keccak operates over a string of binary information, we created a new provable type called
Bytes, conveniently initializable from many other commonly used types. This type allows those provable bitwise operations to be performed over arbitrary, fixed-length arrays of bytes, enabling an
efficient implementation of Keccak.
// Using Keccak and the Bytes class to hash a string to various output lengths:
// define a preimage
let preimage = 'The quick brown fox jumps over the lazy dog';
// create a Bytes class that represents 43 bytes
class Bytes43 extends Bytes(43) {}
// convert the preimage to bytes
let preimageBytes = Bytes43.fromString(preimage);
// hash the preimage
let hash = Hash.SHA3_256.hash(preimageBytes);
// Keccak and SHA-3 implementations are available for outputs of 256, 384, and 512-bits.
Our native hashing algorithm, Poseidon, was designed to be as efficient as possible in zero knowledge proofs and is still the best choice in most situations due to its compatibility with our native
finite fields. When you need to interface with external cryptographic systems, however, provable hashing with efficient bitwise operations can make it possible. In this case, Keccak and SHA-3 allow
provable verification of Ethereum state hashes, and can even interact with non-blockchain applications like document signing and traditional web protocols.
Foreign Field
In contrast to Keccak’s bitwise operations, ECDSA operates on an elliptic curve that is defined over a finite field. The same is true of many zero knowledge proof systems, where the specific finite
field in use generally offers many benefits to performance and ease of use within that proof system and is known as the native field. However, proof systems have special demands not found in most
other cryptographic systems, and differing finite fields are not interoperable. This brings us to the problem: ECDSA uses a different curve to Kimchi, the proof system behind o1js. Kimchi uses the
Pasta curves, chosen for their suitability to recursive zero knowledge proofs. ECDSA, on the other hand, uses secp256k1. This is a good thing, given their different functions, but it presented a
compatibility issue. We needed to support a foreign curve that is built on a foreign finite field.
To generalize this need for interoperability with standardized cryptography like the secp256k1 curve used by ECDSA, we created the new `ForeignField` class. It allows developers to define new fields
with any modulus less than 2²⁵⁹ and is efficient enough to unlock a variety of new use cases. It’s also pretty easy to use!
New foreign fields are created by extending the `ForeignField` class and passing in the desired modulus (the size of the field) as an argument.
// Creating a foreign field with modulus 17:
class Field17 extends createForeignField(17n) {}
You can then use the resulting foreign field class the same way you use the native `Field` class. It exposes many equivalent methods, each behaving as you would expect (but for a different field, of
// Performing modular arithmetic over a foreign field:
let x = Field17.from(16); // create new element of the field equal to 16
x.assertEquals(-1); // 16 = -1 (mod 17)
x.mul(x).assertEquals(1); // 16 * 16 = 15 * 17 + 1 = 1 (mod 17)
// do any of the following opperations
x.add(x); // addition
x.sub(2); // subtraction
x.neg(); // negation
x.mul(3); // multiplication
x.div(x); // division
x.inv(); // inverse
x.assertEquals(y); // assert x == y
x.assertLessThan(2); // assert x < 2
let bits = x.toBits(); // convert to a `Bool` array of size log2(modulus);
Field17.fromBits(bits); // convert back
// non-provable methods
let y = SmallField.from(5n); // convert from bigint or number
y.toBigInt() === 5n; // convert to bigint
Despite the apparent simplicity and ease of use, a lot is happening beneath the surface! It’s impossible to represent a field with a modulus larger than that of our native field. Instead, each
foreign field is represented by three base field elements. There are also three variants of the `ForeignField`, each imposing a different set of constraints on the underlying representation. You can
learn more in the Foreign Field Arithmetic section of the Mina docs.
We also created a class called `ForeignCurve`, which is essentially a layer on top of `ForeignField` that lets you interact directly with a curve defined over a foreign field in the same way that you
would interact with a `Group` defined over the native Pallas curve. It works just like `ForeignField` except that instead of passing in a single modulus, you pass in an object with all the parameters
of the desired curve. We’ve included a few common curves in o1js already, but feel free to open a PR if we don’t have what you’re looking for yet.
ECDSA is fundamentally just a sequence of elliptic curve operations used to construct or verify a signature (which is really just a point on the curve). And now, with foreign field and foreign curve
support, it’s relatively easy to implement ECDSA over secp256k1.
// Creating and proving the validity of an ECDSA signature over the secp256k1 foreign curve:
// create a secp256k1 curve from a set of predefined parameters
class Secp256k1 extends createForeignCurve(Crypto.CurveParams.Secp256k1) {}
// create an instance of ECDSA over secp256k1
class Ecdsa extends createEcdsa(Secp256k1) {}
// a private key is a random scalar of secp256k1 - not provable!
let privateKey = Secp256k1.Scalar.random();
// a public key is a point on the curve
let publicKey = Secp256k1.generator.scale(privateKey);
// sign an array of bytes - not provable!
let signature = Ecdsa.sign(bytes, privateKey.toBigInt());
// sign a hash of a message - not provable!
let signature = Ecdsa.signHash(hash, privateKey.toBigInt());
// verify a signature
let isValid: Bool = signature.verify(message, publicKey);
// verify a hash of a message
let isValid: Bool = signature.verifyHash(hash, publicKey);
These new capabilities give zero knowledge applications written in o1js the ability to verify Ethereum signatures, which unlocks a range of use cases. By exposing the lower-level building blocks,
we’ve also set the stage for further interaction with other cryptographic systems. We look forward to seeing what you build using these privacy-preserving interoperability features!
Check out the docs for ECDSA, Keccak, foreign fields, and bitwise operations to learn more. Or jump on Discord with feedback and questions. We’d love to hear which features you want to see next. | {"url":"https://www.o1labs.org/blog/proving-ethereum-signatures-in-o1js-with-keccak-and-ecdsa-6aa10dd04e3c","timestamp":"2024-11-06T23:14:19Z","content_type":"text/html","content_length":"136251","record_id":"<urn:uuid:e7740119-c9b0-4f2d-b462-3c9b28022ccd>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00794.warc.gz"} |
Can Hedge Funds Profit from Parrondo's Paradox?
Can Hedge Funds Profit from Parrondo's Paradox?
Last month, there was a flurry of blogging about hedge funds that use strategies that statistically profit in most years but eventually will blow up. Martin Wolf wrote one such
, referring to an
Dean Foster
and Peyton Young. The idea behind the article is that hedge funds are often set up (whether by design or by accident) to have good returns most years with a small chance of a blow-up each year.
Therefore, the odds are that a hedge fund manager will look like a genius with great alpha, at least until the fund blows up, by which time he will have collected his or her "2 and 20" for several
years. The strategy Foster and Young suggest is to long Treasury Bills and write deep out of the money put options on the S&P 500. Let's say that there is a 10% chance that the put options will be
exercised in a given year. This means that it is likely that the value of the fund will grow every year by the value of the options sold plus the interest earned on the T-bills. Indeed, you could run
this for five years and have a 60% chance of not blowing up in that entire period. All along, you (the hedge fund manager) are collecting your 2% of funds under management and 20% of return above a
contracted benchmark.
Now obviously hedge funds aren't built so simply. The one above would seem to any reasonably sophisticated investor to be a naked, obvious scam. But Foster and Young (and Wolf) suggest that many
hedge funds, though more complex, are built around similar probable returns. And the way we've been seeing so many blow up recently suggests they may be right.
But what got me thinking was their example of a fund that had a 10% chance of blowing up in any given year. This reminded me of
Parrondo's Paradox
. This paradox comes from game theory, and basically says that if you have two particular losing games, you can win by switching back and forth between them. A good visualization of this can be found
. Specifically, you need one game where the losses are steady and predictable, and another game in which the player wins most of the time, but loses big occasionally so that the net result is a loss.
Now wait--that last one sounds just like what I was talking about above! Hmmmm.
As I understand it, the idea is to switch back and forth between the two games, and that over time, you will end up ahead.
So let's look at this from the point of view of a hedge fund investor. If such an investor invests in the fund described above, he will make a good return most years, but lose his shirt once every 10
years on average. On the other hand, if he found a hedge fund that was just an S&P index (obviously no such hedge fund exists), the investor would steadily lose 2% a year. But if the hedge fund
combined the two strategies, switching off randomly between them, would it become a net winner per Parrondo's Paradox?
Well, I don't know. It may be that the transaction costs of switching would eat up any advantage you get. But there's no reason to speculate--one could, with some effort, simulate this. After all,
has sold puts on the S&P 500 for a long time (since 1989, I think) and you can get data on what deep out-of-the-money puts were available and their prices, margin requirements, and transaction costs.
Likewise, we can get historical information about S&P indices and ETFs. Given this, I could write a program that simulates a fund randomly switching back and forth between the two "games" (writing
puts and longing the index). The idea would be to run the simulation thousands of times and then calculate the average outcome, comparing it of course to the outcome of either just writing puts or
just longing the index. I would probably want to try various start dates (or even make the start date of the simulation random), and experiment with various minimum and maximum holding times for each
of the two strategies.
This would be an interesting experiment to run. The general consensus is that one can't use Parrondo's Paradox in financial situations, which is logical because it does seem to create something from
nothing. But a well-wrought simulation would be worth seeing, even if it simply proved that switching randomly from longing the S&P and writing puts on it is not
a Parrondo game.
Labels: finance
0 Comments:
Subscribe to Post Comments [Atom] | {"url":"https://robertwboyd.blogspot.com/2008/04/can-hedge-funds-profit-from-parrondos.html","timestamp":"2024-11-03T00:33:51Z","content_type":"application/xhtml+xml","content_length":"20823","record_id":"<urn:uuid:86a6da29-9b68-400a-9a98-8c9eb57969c0>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00316.warc.gz"} |
Tutorial: Rotate Array Interview Question and Solution in C++ - MachineLearningTutorials.org
C++, Interview questions
Tutorial: Rotate Array Interview Question and Solution in C++
Introduction to the Rotate Array Problem
In this tutorial, we will explore the “Rotate Array” interview question, a common problem often asked in technical coding interviews. The problem involves rotating the elements of an array to the
right by a specified number of positions. This operation can be achieved through various approaches, and we will discuss both the brute-force and optimized solutions in C++. We’ll cover the problem
statement, the two main strategies to solve it, and provide step-by-step explanations of the code implementations.
Problem Statement
Given an array of integers and a positive integer k, rotate the array to the right by k steps. For example, if the array is [1, 2, 3, 4, 5, 6, 7] and k is 3, then the array should be rotated to [5,
6, 7, 1, 2, 3, 4].
Brute-Force Solution
The brute-force solution to this problem involves performing the rotation step by step, moving each element to its new position. While not the most efficient approach, it helps us understand the
problem and its requirements better.
#include <iostream>
#include <vector>
void rotate(std::vector<int>& nums, int k) {
int n = nums.size();
k %= n; // Handle cases where k is larger than the array size
for (int i = 0; i < k; ++i) {
int last = nums[n - 1];
for (int j = n - 1; j > 0; --j) {
nums[j] = nums[j - 1];
nums[0] = last;
int main() {
std::vector<int> nums = {1, 2, 3, 4, 5, 6, 7};
int k = 3;
rotate(nums, k);
for (int num : nums) {
std::cout << num << " ";
return 0;
In this solution, we iterate through the array k times, and for each iteration, we shift all elements one position to the right. The last element’s value is stored in a temporary variable and placed
at the beginning of the array. This process simulates the rotation.
Optimized Solution using Reverse Technique
The brute-force solution has a time complexity of O(n * k), where n is the number of elements in the array and k is the number of rotations. This approach is not efficient when the array is large or
the number of rotations is significant. We can use a more optimized approach that leverages the reverse technique.
Idea behind the Optimized Solution
The idea is to reverse the entire array, then reverse the first k elements, and finally reverse the remaining elements. This operation effectively rotates the array to the right by k steps.
Let’s go through the steps of this solution:
1. Reverse the entire array.
2. Reverse the first k elements.
3. Reverse the remaining n - k elements.
The combined effect of these three steps is a rotation of the array by k steps.
Implementation of the Optimized Solution
#include <iostream>
#include <vector>
void reverse(std::vector<int>& nums, int start, int end) {
while (start < end) {
std::swap(nums[start], nums[end]);
void rotate(std::vector<int>& nums, int k) {
int n = nums.size();
k %= n;
// Step 1: Reverse the entire array
reverse(nums, 0, n - 1);
// Step 2: Reverse the first k elements
reverse(nums, 0, k - 1);
// Step 3: Reverse the remaining n - k elements
reverse(nums, k, n - 1);
int main() {
std::vector<int> nums = {1, 2, 3, 4, 5, 6, 7};
int k = 3;
rotate(nums, k);
for (int num : nums) {
std::cout << num << " ";
return 0;
This solution has a time complexity of O(n), where n is the number of elements in the array. The reversing operations are efficient and ensure that the rotation is achieved in a more optimal manner.
In this tutorial, we explored the “Rotate Array” interview question and discussed two solutions to the problem: the brute-force solution and the optimized solution using the reverse technique. The
brute-force solution involves shifting elements repeatedly, resulting in a time complexity of O(n * k). On the other hand, the optimized solution leverages the reverse technique to achieve a rotation
in O(n) time.
When faced with similar interview questions, it’s important to understand the problem requirements and constraints, and to come up with efficient solutions that showcase your coding skills. The
optimized solution demonstrated in this tutorial can help you efficiently tackle the “Rotate Array” problem and similar rotation-related challenges. | {"url":"https://machinelearningtutorials.org/tutorial-rotate-array-interview-question-and-solution-in-c/","timestamp":"2024-11-04T02:34:52Z","content_type":"text/html","content_length":"100390","record_id":"<urn:uuid:2339d7e2-02fe-441f-8122-1bec5ceb8ff8>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00025.warc.gz"} |
Quadratic equations for ninth graders
Algebra Tutorials! Saturday 2nd of November
quadratic equations for ninth graders
Related topics:
Home long division polynomials graphing calculator | graph of absolute values rational functions | free trig solver | trigonomic tutors | algebrator download |
Rotating a Parabola calculator with radical | matlab-forward substitution | algebra homework help | program your calculator algebra system using euler's method | blank coordinate
Multiplying Fractions plane | simplify exponents worksheet | adding radicals fractions | explorations in college algebra online help | solve systems of linear equations ti-84
Finding Factors
Miscellaneous Equations
Mixed Numbers and Author Message
Improper Fractions
Systems of Equations in TVultan Posted: Thursday 28th of Dec 16:11
Two Variables 1. Hello Guys Can someone out there help me? My algebra teacher gave us quadratic equations for ninth graders problem today. Normally I am
Literal Numbers good at difference of cubes but somehow I am just stuck on this one homework. I have to turn it in by this weekend but it looks like I will
Adding and Subtracting not be able to complete it in time. So I thought of coming online to find assistance. I will really appreciate if someone can help me work
Polynomials this (topicKwds) out in time.
Subtracting Integers Registered:
Simplifying Complex 03.08.2001
Fractions From:
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials Jahm Xjardx Posted: Friday 29th of Dec 09:13
Mixed Algebrator is the latest hot favourite of quadratic equations for ninth graders learners . I know a couple of tutors who actually ask their
The Square of a Binomial students to have a copy of this software at their home.
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Registered:
Equations in One 07.08.2005
Variable From: Odense,
Multiplying Binomials Denmark, EU
Using the FOIL Method
Imaginary Numbers
Solving Quadratic
Equations Using the Hiinidam Posted: Friday 29th of Dec 21:43
Quadratic Formula I completely agree with that. It truly is a great program. Algebrator helped me and my classmates a lot during our exam time. We went on to
Solving Quadratic get more marks than we could ever have thought of. It explains things in a lot more detailed manner, than a instructor ever could, in a
Equations class. Moreover, you can read one solution again and again till you actually understand it, unlike in a classroom where the teacher has to
Algebra move on due to time constraints. Go ahead and try it.
Order of Operations Registered:
Dividing Complex Numbers 06.07.2001
Polynomials From: Greeley, CO, US
The Appearance of a
Polynomial Equation
Standard Form of a Line
Positive Integral CHS` Posted: Sunday 31st of Dec 12:08
Divisors I am a regular user of Algebrator. It not only helps me complete my homework faster, the detailed explanations offered makes understanding
Dividing Fractions the concepts easier. I suggest using it to help improve problem solving skills.
Solving Linear Systems
of Equations by
Elimination Registered:
Factoring 04.07.2001
Multiplying and Dividing From: Victoria City,
Square Roots Hong Kong Island,
Functions and Graphs Hong Kong
Dividing Polynomials
Solving Rational
Numbers Trost Posted: Monday 01st of Jan 11:49
Use of Parentheses or Algebrator is one of the best software that would offer you all the basics principles of quadratic equations for ninth graders. The quality
Brackets (The training offered by the Algebrator on hypotenuse-leg similarity, greatest common factor, interval notation and side-side-side similarity is
Distributive Law) second to none. I have tried 3-4 home tutoring mathematics tools and I found this to be remarkable . The Algebrator not only provides you
Multiplying and Dividing the primary principles but also helps you in working out any tough Intermediate algebra question with ease. The quick formula reference
by Monomials Registered: that comes with Algebrator is very detailed and has almost every formula relating to College Algebra.
Solving Quadratic 12.05.2002
Equations by Graphing From: Europe, The
Multiplying Decimals Netherlands,
Use of Parentheses or Zwijndrecht
Brackets (The
Distributive Law)
Simplifying Complex
Fractions 1 TC Posted: Tuesday 02nd of Jan 14:46
Adding Fractions The software is a piece of cake. Just give it 15 minutes and you will be a pro at it. You can find the software here, https://
Simplifying Complex gre-test-prep.com/solving-quadratic-equations-by-graphing.html.
Solutions to Linear
Equations in Two Registered:
Variables 25.09.2001
Quadratic Expressions From: Kµlt °ƒ Ø,
Completing Squares working on my time
Dividing Radical machine
Rise and Run
Graphing Exponential
Multiplying by a
The Cartesian Coordinate
Writing the Terms of a
Polynomial in Descending
Quadratic Expressions
Solving Inequalities
Solving Rational
Inequalities with a Sign
Solving Linear Equations
Solving an Equation with
Two Radical Terms
Simplifying Rational
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step
Solving Quadratic
Equations Graphically
and Algebraically
Collecting Like Terms
Solving Equations with
Radicals and Exponents
Percent of Change
Powers of ten
(Scientific Notation)
Comparing Integers on a
Number Line
Solving Systems of
Equations Using
Factoring Out the
Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing
Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting
Rational Expressions
with Different
Expressions with
Variables as Exponents
The Quadratic Formula
Writing a Quadratic with
Given Solutions
Simplifying Square Roots
Adding and Subtracting
Square Roots
Adding and Subtracting
Rational Expressions
Combining Like Radical
Solving Systems of
Equations Using
Dividing Polynomials
Graphing Functions
Product of a Sum and a
Solving First Degree
Solving Equations with
Radicals and Exponents
Roots and Powers
Multiplying Numbers
quadratic equations for ninth graders
Related topics:
Home long division polynomials graphing calculator | graph of absolute values rational functions | free trig solver | trigonomic tutors | algebrator download |
Rotating a Parabola calculator with radical | matlab-forward substitution | algebra homework help | program your calculator algebra system using euler's method | blank coordinate
Multiplying Fractions plane | simplify exponents worksheet | adding radicals fractions | explorations in college algebra online help | solve systems of linear equations ti-84
Finding Factors
Miscellaneous Equations
Mixed Numbers and Author Message
Improper Fractions
Systems of Equations in TVultan Posted: Thursday 28th of Dec 16:11
Two Variables 1. Hello Guys Can someone out there help me? My algebra teacher gave us quadratic equations for ninth graders problem today. Normally I am
Literal Numbers good at difference of cubes but somehow I am just stuck on this one homework. I have to turn it in by this weekend but it looks like I will
Adding and Subtracting not be able to complete it in time. So I thought of coming online to find assistance. I will really appreciate if someone can help me work
Polynomials this (topicKwds) out in time.
Subtracting Integers Registered:
Simplifying Complex 03.08.2001
Fractions From:
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials Jahm Xjardx Posted: Friday 29th of Dec 09:13
Mixed Algebrator is the latest hot favourite of quadratic equations for ninth graders learners . I know a couple of tutors who actually ask their
The Square of a Binomial students to have a copy of this software at their home.
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Registered:
Equations in One 07.08.2005
Variable From: Odense,
Multiplying Binomials Denmark, EU
Using the FOIL Method
Imaginary Numbers
Solving Quadratic
Equations Using the Hiinidam Posted: Friday 29th of Dec 21:43
Quadratic Formula I completely agree with that. It truly is a great program. Algebrator helped me and my classmates a lot during our exam time. We went on to
Solving Quadratic get more marks than we could ever have thought of. It explains things in a lot more detailed manner, than a instructor ever could, in a
Equations class. Moreover, you can read one solution again and again till you actually understand it, unlike in a classroom where the teacher has to
Algebra move on due to time constraints. Go ahead and try it.
Order of Operations Registered:
Dividing Complex Numbers 06.07.2001
Polynomials From: Greeley, CO, US
The Appearance of a
Polynomial Equation
Standard Form of a Line
Positive Integral CHS` Posted: Sunday 31st of Dec 12:08
Divisors I am a regular user of Algebrator. It not only helps me complete my homework faster, the detailed explanations offered makes understanding
Dividing Fractions the concepts easier. I suggest using it to help improve problem solving skills.
Solving Linear Systems
of Equations by
Elimination Registered:
Factoring 04.07.2001
Multiplying and Dividing From: Victoria City,
Square Roots Hong Kong Island,
Functions and Graphs Hong Kong
Dividing Polynomials
Solving Rational
Numbers Trost Posted: Monday 01st of Jan 11:49
Use of Parentheses or Algebrator is one of the best software that would offer you all the basics principles of quadratic equations for ninth graders. The quality
Brackets (The training offered by the Algebrator on hypotenuse-leg similarity, greatest common factor, interval notation and side-side-side similarity is
Distributive Law) second to none. I have tried 3-4 home tutoring mathematics tools and I found this to be remarkable . The Algebrator not only provides you
Multiplying and Dividing the primary principles but also helps you in working out any tough Intermediate algebra question with ease. The quick formula reference
by Monomials Registered: that comes with Algebrator is very detailed and has almost every formula relating to College Algebra.
Solving Quadratic 12.05.2002
Equations by Graphing From: Europe, The
Multiplying Decimals Netherlands,
Use of Parentheses or Zwijndrecht
Brackets (The
Distributive Law)
Simplifying Complex
Fractions 1 TC Posted: Tuesday 02nd of Jan 14:46
Adding Fractions The software is a piece of cake. Just give it 15 minutes and you will be a pro at it. You can find the software here, https://
Simplifying Complex gre-test-prep.com/solving-quadratic-equations-by-graphing.html.
Solutions to Linear
Equations in Two Registered:
Variables 25.09.2001
Quadratic Expressions From: Kµlt °ƒ Ø,
Completing Squares working on my time
Dividing Radical machine
Rise and Run
Graphing Exponential
Multiplying by a
The Cartesian Coordinate
Writing the Terms of a
Polynomial in Descending
Quadratic Expressions
Solving Inequalities
Solving Rational
Inequalities with a Sign
Solving Linear Equations
Solving an Equation with
Two Radical Terms
Simplifying Rational
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step
Solving Quadratic
Equations Graphically
and Algebraically
Collecting Like Terms
Solving Equations with
Radicals and Exponents
Percent of Change
Powers of ten
(Scientific Notation)
Comparing Integers on a
Number Line
Solving Systems of
Equations Using
Factoring Out the
Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing
Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting
Rational Expressions
with Different
Expressions with
Variables as Exponents
The Quadratic Formula
Writing a Quadratic with
Given Solutions
Simplifying Square Roots
Adding and Subtracting
Square Roots
Adding and Subtracting
Rational Expressions
Combining Like Radical
Solving Systems of
Equations Using
Dividing Polynomials
Graphing Functions
Product of a Sum and a
Solving First Degree
Solving Equations with
Radicals and Exponents
Roots and Powers
Multiplying Numbers
Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and
Improper Fractions
Systems of Equations in
Two Variables
Literal Numbers
Adding and Subtracting
Subtracting Integers
Simplifying Complex
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical
Equations in One
Multiplying Binomials
Using the FOIL Method
Imaginary Numbers
Solving Quadratic
Equations Using the
Quadratic Formula
Solving Quadratic
Order of Operations
Dividing Complex Numbers
The Appearance of a
Polynomial Equation
Standard Form of a Line
Positive Integral
Dividing Fractions
Solving Linear Systems
of Equations by
Multiplying and Dividing
Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational
Use of Parentheses or
Brackets (The
Distributive Law)
Multiplying and Dividing
by Monomials
Solving Quadratic
Equations by Graphing
Multiplying Decimals
Use of Parentheses or
Brackets (The
Distributive Law)
Simplifying Complex
Fractions 1
Adding Fractions
Simplifying Complex
Solutions to Linear
Equations in Two
Quadratic Expressions
Completing Squares
Dividing Radical
Rise and Run
Graphing Exponential
Multiplying by a
The Cartesian Coordinate
Writing the Terms of a
Polynomial in Descending
Quadratic Expressions
Solving Inequalities
Solving Rational
Inequalities with a Sign
Solving Linear Equations
Solving an Equation with
Two Radical Terms
Simplifying Rational
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step
Solving Quadratic
Equations Graphically
and Algebraically
Collecting Like Terms
Solving Equations with
Radicals and Exponents
Percent of Change
Powers of ten
(Scientific Notation)
Comparing Integers on a
Number Line
Solving Systems of
Equations Using
Factoring Out the
Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing
Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting
Rational Expressions
with Different
Expressions with
Variables as Exponents
The Quadratic Formula
Writing a Quadratic with
Given Solutions
Simplifying Square Roots
Adding and Subtracting
Square Roots
Adding and Subtracting
Rational Expressions
Combining Like Radical
Solving Systems of
Equations Using
Dividing Polynomials
Graphing Functions
Product of a Sum and a
Solving First Degree
Solving Equations with
Radicals and Exponents
Roots and Powers
Multiplying Numbers
quadratic equations for ninth graders
Related topics:
long division polynomials graphing calculator | graph of absolute values rational functions | free trig solver | trigonomic tutors | algebrator download | calculator with radical |
matlab-forward substitution | algebra homework help | program your calculator algebra system using euler's method | blank coordinate plane | simplify exponents worksheet | adding radicals
fractions | explorations in college algebra online help | solve systems of linear equations ti-84
Author Message
TVultan Posted: Thursday 28th of Dec 16:11
1. Hello Guys Can someone out there help me? My algebra teacher gave us quadratic equations for ninth graders problem today. Normally I am good at difference of cubes
but somehow I am just stuck on this one homework. I have to turn it in by this weekend but it looks like I will not be able to complete it in time. So I thought of
coming online to find assistance. I will really appreciate if someone can help me work this (topicKwds) out in time.
Jahm Xjardx Posted: Friday 29th of Dec 09:13
Algebrator is the latest hot favourite of quadratic equations for ninth graders learners . I know a couple of tutors who actually ask their students to have a copy of
this software at their home.
From: Odense,
Denmark, EU
Hiinidam Posted: Friday 29th of Dec 21:43
I completely agree with that. It truly is a great program. Algebrator helped me and my classmates a lot during our exam time. We went on to get more marks than we
could ever have thought of. It explains things in a lot more detailed manner, than a instructor ever could, in a class. Moreover, you can read one solution again and
again till you actually understand it, unlike in a classroom where the teacher has to move on due to time constraints. Go ahead and try it.
From: Greeley, CO, US
CHS` Posted: Sunday 31st of Dec 12:08
I am a regular user of Algebrator. It not only helps me complete my homework faster, the detailed explanations offered makes understanding the concepts easier. I
suggest using it to help improve problem solving skills.
From: Victoria City,
Hong Kong Island,
Hong Kong
Trost Posted: Monday 01st of Jan 11:49
Algebrator is one of the best software that would offer you all the basics principles of quadratic equations for ninth graders. The quality training offered by the
Algebrator on hypotenuse-leg similarity, greatest common factor, interval notation and side-side-side similarity is second to none. I have tried 3-4 home tutoring
mathematics tools and I found this to be remarkable . The Algebrator not only provides you the primary principles but also helps you in working out any tough
Intermediate algebra question with ease. The quick formula reference that comes with Algebrator is very detailed and has almost every formula relating to College
Registered: Algebra.
From: Europe, The
TC Posted: Tuesday 02nd of Jan 14:46
The software is a piece of cake. Just give it 15 minutes and you will be a pro at it. You can find the software here, https://gre-test-prep.com/
From: Kµlt °ƒ Ø,
working on my time
Author Message
TVultan Posted: Thursday 28th of Dec 16:11
1. Hello Guys Can someone out there help me? My algebra teacher gave us quadratic equations for ninth graders problem today. Normally I am good at difference of cubes but
somehow I am just stuck on this one homework. I have to turn it in by this weekend but it looks like I will not be able to complete it in time. So I thought of coming online to
find assistance. I will really appreciate if someone can help me work this (topicKwds) out in time.
Jahm Xjardx Posted: Friday 29th of Dec 09:13
Algebrator is the latest hot favourite of quadratic equations for ninth graders learners . I know a couple of tutors who actually ask their students to have a copy of this
software at their home.
From: Odense,
Denmark, EU
Hiinidam Posted: Friday 29th of Dec 21:43
I completely agree with that. It truly is a great program. Algebrator helped me and my classmates a lot during our exam time. We went on to get more marks than we could ever
have thought of. It explains things in a lot more detailed manner, than a instructor ever could, in a class. Moreover, you can read one solution again and again till you
actually understand it, unlike in a classroom where the teacher has to move on due to time constraints. Go ahead and try it.
From: Greeley, CO, US
CHS` Posted: Sunday 31st of Dec 12:08
I am a regular user of Algebrator. It not only helps me complete my homework faster, the detailed explanations offered makes understanding the concepts easier. I suggest using
it to help improve problem solving skills.
From: Victoria City,
Hong Kong Island,
Hong Kong
Trost Posted: Monday 01st of Jan 11:49
Algebrator is one of the best software that would offer you all the basics principles of quadratic equations for ninth graders. The quality training offered by the Algebrator on
hypotenuse-leg similarity, greatest common factor, interval notation and side-side-side similarity is second to none. I have tried 3-4 home tutoring mathematics tools and I
found this to be remarkable . The Algebrator not only provides you the primary principles but also helps you in working out any tough Intermediate algebra question with ease.
The quick formula reference that comes with Algebrator is very detailed and has almost every formula relating to College Algebra.
From: Europe, The
TC Posted: Tuesday 02nd of Jan 14:46
The software is a piece of cake. Just give it 15 minutes and you will be a pro at it. You can find the software here, https://gre-test-prep.com/
From: Kµlt °ƒ Ø,
working on my time
Posted: Thursday 28th of Dec 16:11
1. Hello Guys Can someone out there help me? My algebra teacher gave us quadratic equations for ninth graders problem today. Normally I am good at difference of cubes but somehow I am just stuck on
this one homework. I have to turn it in by this weekend but it looks like I will not be able to complete it in time. So I thought of coming online to find assistance. I will really appreciate if
someone can help me work this (topicKwds) out in time.
Posted: Friday 29th of Dec 09:13
Algebrator is the latest hot favourite of quadratic equations for ninth graders learners . I know a couple of tutors who actually ask their students to have a copy of this software at their home.
Posted: Friday 29th of Dec 21:43
I completely agree with that. It truly is a great program. Algebrator helped me and my classmates a lot during our exam time. We went on to get more marks than we could ever have thought of. It
explains things in a lot more detailed manner, than a instructor ever could, in a class. Moreover, you can read one solution again and again till you actually understand it, unlike in a classroom
where the teacher has to move on due to time constraints. Go ahead and try it.
Posted: Sunday 31st of Dec 12:08
I am a regular user of Algebrator. It not only helps me complete my homework faster, the detailed explanations offered makes understanding the concepts easier. I suggest using it to help improve
problem solving skills.
Posted: Monday 01st of Jan 11:49
Algebrator is one of the best software that would offer you all the basics principles of quadratic equations for ninth graders. The quality training offered by the Algebrator on hypotenuse-leg
similarity, greatest common factor, interval notation and side-side-side similarity is second to none. I have tried 3-4 home tutoring mathematics tools and I found this to be remarkable . The
Algebrator not only provides you the primary principles but also helps you in working out any tough Intermediate algebra question with ease. The quick formula reference that comes with Algebrator is
very detailed and has almost every formula relating to College Algebra.
Posted: Tuesday 02nd of Jan 14:46
The software is a piece of cake. Just give it 15 minutes and you will be a pro at it. You can find the software here, https://gre-test-prep.com/solving-quadratic-equations-by-graphing.html. | {"url":"https://gre-test-prep.com/algebra-1-practice-test/exponent-rules/quadratic-equations-for-ninth.html","timestamp":"2024-11-02T20:50:41Z","content_type":"text/html","content_length":"118311","record_id":"<urn:uuid:9241c6a4-f1b5-4691-b9ba-b72928444016>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00760.warc.gz"} |
Angle Sum Property | Theorem | Proof | Examples- Cuemath (2024)
The angle sum property of a triangle states that the sum of the angles of a triangle is equal to 180º. A triangle has three sides and three angles, one at each vertex. Whether a triangle is an acute,
obtuse, or a right triangle, the sum of its interior angles is always 180º.
The angle sum property of a triangle is one of the most frequently used properties in geometry. This property is mostly used to calculate the unknown angles.
1. What is the Angle Sum Property?
2. Angle Sum Property Formula
3. Proof of the Angle Sum Property
4. FAQs on Angle Sum Property
What is the Angle Sum Property?
According to the angle sum property of a triangle, the sum of all three interior angles of a triangle is 180 degrees. A triangle is a closed figure formed by three line segments, consisting of
interior as well as exterior angles. The angle sum property is used to find the measure of an unknown interior angle when the values of the other two angles are known. Observe the following figure to
understand the property.
Angle Sum Property Formula
The angle sum property formula for any polygon is expressed as, S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. This property of a polygon states that the sum of the
interior angles in a polygon can be found with the help of the number of triangles that can be formed inside it. These triangles are formed by drawing diagonals from a single vertex. However, to make
things easier, this can be calculated by a simple formula, which says that if a polygon has 'n' sides, there will be (n - 2) triangles inside it. For example, let us take a decagon that has 10 sides
and apply the formula. We get, S = (n − 2) × 180°, S = (10 − 2) × 180° = 10 × 180° = 1800°. Therefore, according to the angle sum property of a decagon, the sum of its interior angles is always
1800°. Similarly, the same formula can be applied to other polygons. The angle sum property is mostly used to find the unknown angles of a polygon.
Proof of the Angle Sum Property
Let's have a look at the proof of the angle sum property of the triangle. The steps for proving the angle sum property of a triangle are listed below:
• Step 1: Draw a line PQ that passes through the vertex A and is parallel to side BC of the triangle ABC.
• Step 2: We know that the sum of the angles on a straight line is equal to 180°. In other words, ∠PAB + ∠BAC + ∠QAC = 180°, which gives, Equation 1: ∠PAB + ∠BAC + ∠QAC = 180°
• Step 3: Now, since line PQ is parallel to BC. ∠PAB = ∠ABC and ∠QAC = ∠ACB. (Interior alternate angles), which gives, Equation 2: ∠PAB = ∠ABC, and Equation 3: ∠QAC = ∠ACB
• Step 4: Substitute ∠PAB and ∠QAC with ∠ABC and ∠ACB respectively, in Equation 1 as shown below.
Equation 1: ∠PAB + ∠BAC + ∠QAC = 180°. Thus we get, ∠ABC + ∠BAC + ∠ACB = 180°
Hence proved, in triangle ABC, ∠ABC + ∠BAC + ∠ACB = 180°. Thus, the sum of the interior angles of a triangle is equal to 180°.
Important Points
The following points should be remembered while solving questions related to the angle sum property.
• The angle sum property formula for any polygon is expressed as, S = ( n − 2) × 180°, where 'n' represents the number of sides in the polygon.
• The angle sum property of a polygon states that the sum of the interior angles in a polygon can be found with the help of the number of triangles that can be formed inside it.
• The sum of the interior angles of a triangle is always 180°.
Impoprtant Topics
• Exterior Angle Theorem
• Angles
• Triangles
FAQs on Angle Sum Property
What is the Angle Sum Property of a Polygon?
The angle sum property of a polygon states that the sum of all the angles in a polygon can be found with the help of the number of triangles that can be formed in it. These triangles are formed by
drawing diagonals from a single vertex. However, this can be calculated by a simple formula, which says that if a polygon has 'n' sides, there will be (n - 2) triangles inside it. The sum of the
interior angles of a polygon can be calculated with the formula: S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. For example, if we take a quadrilateral and apply the
formula using n = 4, we get: S = (n − 2) × 180°, S = (4 − 2) × 180° = 2 × 180° = 360°. Therefore, according to the angle sum property of a quadrilateral, the sum of its interior angles is always
360°. Similarly, the same formula can be applied to other polygons. The angle sum property is mostly used to find the unknown angles of a polygon.
What is the Angle Sum Property of a Triangle?
The angle sum property of a triangle says that the sum of its interior angles is equal to 180°. Whether a triangle is an acute, obtuse, or a right triangle, the sum of the angles will always be 180°.
This can be represented as follows: In a triangle ABC, ∠A + ∠B + ∠C = 180°.
What is the Angle Sum Property of a Hexagon?
According to the angle sum property of a hexagon, the sum of all the interior angles of a hexagon is 720°. In order to find the sum of the interior angles of a hexagon, we multiply the number of
triangles in it by 180°. This is expressed by the formula: S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. In this case, 'n' = 6. Therefore, the sum of the interior
angles of a hexagon = S = (n − 2) × 180° = (6 − 2) × 180° = 4 × 180° = 720°.
What is the Angle Sum Property of a Quadrilateral?
According to the angle sum property of a quadrilateral, the sum of all its four interior angles is 360°. This can be calculated by the formula, S = (n − 2) × 180°, where 'n' represents the number of
sides in the polygon. In this case, 'n' = 4. Therefore, the sum of the interior angles of a quadrilateral = S = (4 − 2) × 180° = (4 − 2) × 180° = 2 × 180° = 360°.
What is the Exterior Angle Sum Property of a Triangle?
The exterior angle theorem says that the measure of each exterior angle of a triangle is equal to the sum of the opposite and non-adjacent interior angles.
What is the Formula of Angle Sum Property?
The formula for the angle sum property is, S = ( n − 2) × 180°, where 'n' represents the number of sides in the polygon. For example, if we want to find the sum of the interior angles of an octagon,
in this case, 'n' = 8. Therefore, we will substitute the value of 'n' in the formula, and the sum of the interior angles of an octagon = S = (n − 2) × 180° = (8 − 2) × 180° = 6 × 180° = 1080°.
What is the Angle Sum Property of a Pentagon?
As per the angle sum property of a pentagon, the sum of all the interior angles of a pentagon is 540°. In order to find the sum of the interior angles of a pentagon, we multiply the number of
triangles in it by 180°. This is expressed by the formula: S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. In this case, 'n' = 5. Therefore, the sum of the interior
angles of a pentagon = S = (n − 2) × 180° = (5 − 2) × 180° = 3 × 180° = 540°.
How to Find the Third Angle in a Triangle?
We know that the sum of the angles of a triangle is always 180°. Therefore, if we know the two angles of a triangle, and we need to find its third angle, we use the angle sum property. We add the two
known angles and subtract their sum from 180° to get the measure of the third angle. For example, if two angles of a triangle are 70° and 60°, we will add these, 70 + 60 = 130°, and we will subtract
it from 180°, which is the sum of the angles of a triangle. So, the third angle = 180° - 130° = 50°.
How to Find the Exterior Angle of a Polygon?
The exterior angle of a polygon is the angle formed between any side of a polygon and a line that is extended from the adjacent side. In order to find the measure of an exterior angle of a regular
polygon, we divide 360 by the number of sides 'n' of the given polygon. For example, in a regular hexagon, where 'n' = 6, each exterior angle will be 60° because 360 ÷ n = 360 ÷ 6 = 60°. It should be
noted that the corresponding interior and exterior angles are supplementary and the exterior angles of a regular polygon are equal in measure. | {"url":"https://infraszaunaepites.com/article/angle-sum-property-theorem-proof-examples-cuemath","timestamp":"2024-11-06T14:59:55Z","content_type":"text/html","content_length":"70335","record_id":"<urn:uuid:d2bdcc20-8238-438b-a610-8bfc14c2de13>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00806.warc.gz"} |
Some of you have asked me previously whether or not we can share any test documents to demonstrate Calc’s new OpenCL-based formula engine. Thanks to AMD, we can now make available 3 test documents
that showcase the performance of the new engine, and how it compares to Calc’s existing engine as well as Excel’s.
These files are intentionally in Excel format so that they can be used both in Calc and Excel. They also contain VBA script to automate the execution of formula cell recalculation and measure the
recalculation time with a single button click.
All you have to do is to open one of these files, click “Recalculate” and wait for it to finish. It should give you the number that represents the duration of the recalculation in milliseconds.
Note that the 64-bit version of Excel requires different VBA syntax for calling native function in DLL, which is why we have a separate set of documents just for that version. You should not use
these documents unless you want to test them specifically in the 64-bit version of Excel. Use the other one for all the rest.
On Linux, you need to use a reasonably recent build from the master branch in order for the VBA macro to be able to call the native DLL function. If you decide to run them on Linux, make sure your
build is recent enough to contain this commit.
Once again, huge thanks to AMD for allowing us to share these documents with everyone!
Shared formula to reduce memory usage
This week I have finally finished implementing a true shared formula framework in Calc core which allows Calc to share token array instances between adjacent formula cells if they contain identical
set of formula tokens. Since one of the major benefits of sharing formula token arrays is reduced memory footprint, I decided to measure the trend in Calc’s memory usages since 4.0 all the way up to
the latest master, to see how much impact this shared formula work has made in Calc’s overall memory footprint.
Test document
Here is the test document I used to measure Calc’s memory usage
This ODF spreadsheet document contains 100000 rows of cells in 4 columns of which 399999 are formula cells. Column A contains a series of integers that grow linearly down the column. Here, only the
first cell (A1) is a numeric cell while the rest are all formula cells that reference their respective immediate upper cell. Cells in Column B all reference their immediate left in Column A, cells in
Column C all reference their immediate left in Column B, and so on. References used in this document are all relative references; no absolute references are used.
Tested builds
I’ve tested a total of 4 builds. One is the 4.0.1 build packaged for openSUSE 11.4 (x64) from the openSUSE repository, one is the 4.0.6 build built from the 4.0 branch, one is the 4.1.1 build built
from the 4.1 branch, and the last one is the latest from the master branch. With the exception of the packaged 4.0.1 build, all builds are built locally on my machine running openSUSE 11.4 (x64).
Also on the master build, I’ve tested memory usage both with and without shared formulas.
In each tested build, the memory usage was measured by directly opening the test document from the command line and recording the virtual memory usage in GNOME system monitor. After the document was
loaded, I allowed for the virtual memory reading to stabilize by waiting several seconds before recording the number. The results are presented graphically in the following chart.
The following table shows the actual numbers recorded.
Build Virtual memory
4.0.1 (packaged by openSUSE) 4.0 GiB
4.0.6 892.1 MiB
4.1.1 858.4 MiB
master (no shared formula) 842.2 MiB
master (shared formula) 763.9 MiB
Additionally, I’ve also measured the number of token array instances between the two master builds (one with shared formula and one without), and the build without shared formula created 399999 token
array instances (exactly 4 x 100000 – 1) upon file load, whereas the build with shared formula created only 4 token array instances. This likely accounts for the difference of 78.3 MiB in virtual
memory usage between the two builds.
Effect of cell storage rework
One thing worth noting here is that, even without shared formulas, the numbers clearly show a steady decline of Calc’s memory usage from 4.0 to 4.1, and to the current master. While we can’t clearly
infer from these numbers alone what caused the memory usage to shrink, I can say with reasonable confidence that the cell storage rework we did during the same period is a significant factor in such
memory footprint shrinkage. I won’t go into the details of the cell storage rework here; I’ll reserve that topic for another blog post.
Oh by the way, I have absolutely no idea why the 4.0.1 build packaged from the openSUSE repository shows such high memory usage. To me this looks more like an anomaly, indicative of earlier memory
leaks we had later fixed, different custom allocator that only the distro packaged version uses that favors large up-front memory allocation, or anything else I haven’t thought of. Either way, I’m
not counting this as something that resulted from any of our improvements we did in Calc core.
Ixion – threaded formula calculation library
I spent my entire last week on my personal project, by taking advantage of Novell’s HackWeek. Officially, HackWeek took place two weeks ago, but because I had to travel that week I postponed mine
till the following week.
Ixion is the project I worked on as part of my HackWeek. This project is an experimental effort to develop a stand-alone library that supports parallel computation of formula expressions using
threads. I’d been working on this on and off in my spare time, but when the opportunity came along to spend one week of my paid time on any project of my choice (personal or otherwise), I didn’t
hesitate to pick Ixion.
So, what’s Ixion? Ixion aims to provide a library for calculating the results of formula expressions stored in multiple named targets, or “cells”. The cells can be referenced from each other, and the
library takes care of resolving their dependencies automatically upon calculation. The caller can run the calculation routine either in a single-threaded mode, or a multi-threaded mode. The library
also supports re-calculation where the contents of one or more cells have been modified since the last calculation, and a partial calculation of only the affected cells gets performed. It is written
entirely in C++, and makes extensive use of the boost library to achieve portability across different platforms. It has currently been tested to build on Linux and Windows.
The goal is to eventually bring this library up to the level where it can serve as a full-featured calculation engine for spreadsheet applications. But right now, this project remains as an
experimental, proof-of-concept project to help me understand what is required to build a threaded calculation engine capable of performing all sorts of tasks required in a typical spreadsheet app.
I consider this project a library project; however, building this project only creates a single stand-alone console application at the moment. I plan to separate it into a shared library and a
front-end executable in the future, to allow external apps to dynamically link to it.
How it works
Building this project creates an executable called ixion-parser. Running it with a -h option displays the following help content:
Usage: ixion-parser [options] FILE1 FILE2 ...
The FILE must contain the definitions of cells according to the cell definition rule.
Allowed options:
-h [ --help ] print this help.
-t [ --thread ] arg specify the number of threads to use for calculation.
Note that the number specified by this option
corresponds with the number of calculation threads i.e.
those child threads that perform cell interpretations.
The main thread does not perform any calculations;
instead, it creates a new child thread to manage the
calculation threads, the number of which is specified
by the arg. Therefore, the total number of threads
used by this program will be arg + 2.
The parser expects one or more cell definition files as arguments. A cell definition file may look like this:
%mode init
%mode result
%mode edit
%mode result
%mode edit
I hope the format of the cell definition rule is straightforward. The definitions are read from top down. I used the so-called A1 notation to name target cells, but it doesn’t have to be that way.
You can use any naming scheme to name target cells as long as the lexer recognizes them as names. Also, the format supports a command construct; a line beginning with a ‘%’ is considered a command.
Several commands are currently available. For instance the mode command lets you switch input modes. The parser currently supports three input modes:
• init – initialize cells with specified contents.
• result – pick up expected results for cells, for verification.
• edit – modify cell contents.
In addition to the mode command, the following commands are also supported:
• calc – perform full calculation, by resetting the cached results of all involved cells.
• recalc – perform partial re-calculation of modified cells and cells that reference modified cells, either directly or indirectly.
• check – verify the calculation results.
Given all this, let’s see what happens when you run the parser with the above cell definition file.
./ixion-parser -t 4 test/01-simple-arithmetic.txt
Using 4 threads
Number of CPUS: 4
parsing test/01-simple-arithmetic.txt
A1: 1
A1: result = 1
A2: A1+10
A2: result = 11
A3: A2+A1*30
A3: result = 41
A4: (10+20)*A2
A4: result = 330
A5: A1-A2+A3*A4
A5: result = 13520
A8: 10/0
result = #DIV/0!
A6: A1+A3
A6: result = 42
result = #DIV/0!
A7: result = #REF!
checking results
A2 : 11
A8 : #DIV/0!
A3 : 41
A9 : #DIV/0!
A4 : 330
A5 : 13520
A6 : 42
A7 : #REF!
A1 : 1
A6: A1+A2
A6: result = 12
checking results
A2 : 11
A8 : #DIV/0!
A3 : 41
A9 : #DIV/0!
A4 : 330
A5 : 13520
A6 : 12
A7 : #REF!
A1 : 1
A1: 10
A1: result = 10
A2: A1+10
A2: result = 20
A3: A2+A1*30
A3: result = 320
A4: (10+20)*A2
A4: result = 600
A5: A1-A2+A3*A4
A5: result = 191990
A6: A1+A2
A6: result = 30
(duration: 0.00113601 sec)
Notice that at the beginning of the output, it displays the number of threads being used, and the number of “CPU”s it detected. Here, the “CPU” may refer to the number of physical CPUs, the number of
cores, or the number of hyper-threading units. I’m well aware that I need to use a different term for this other than “CPU”, but anyways… The number of child threads to use to perform calculation can
be specified at run-time via -t option. When running without the -t option, the parser will run in a single-threaded mode. Now, let me go over what the above output means.
The first calculation performed is a full calculation. Since no cells have been calculated yet, we need to calculate results for all defined cells. This is followed by a verification of the initial
calculation. After this, we modify cell A6, and perform partial re-calculation. Since no other cells depend on the result of cell A6, the re-calc only calculates A6.
Now, the third calculation is also a partial re-calculation following the modification of cell A1. This time, because several other cells do depend on the result of A1, those cells also need to be
re-calculated. The end result is that cells A1, A2, A3, A4, A5 and A6 all get re-calculated.
Under the hood
Cell dependency resolution
There are several technical aspects of the implementation of this library I’d like to cover. The first is cell dependency resolution. I use a well-known algorithm called topological sort to sort
cells in order of dependency so that cells can be calculated one by one without being blocked by the calculation of precedent cells. Topological sort is typically used to manage scheduling of tasks
that are inter-dependent with each other, and it was a perfect one to use to resolve cell dependencies. This algorithm is a by-product of depth first search of directed acyclic graph (DAG), and is
well-documented. A quick google search should give you tons of pseudo code examples of this algorithm. This algorithm work well both for full calculation and partial re-calculation routines.
Managing threaded calculation
The heart of this project is to implement parallel evaluation of formula expressions, which has been my number 1 goal from the get-go. This is also the reason why I put my focus on designing the
threaded calculation engine as my initial goal before I start putting my focus into other areas. Programming with threads was also very new to me, so I took extra care to ensure that I understand
what I’m doing, and I’m designing it correctly. Also, designing a framework that uses multiple threads can easily get out-of-hand and out-of-control when it’s overdone. So, I made an extra effort to
limit the area where multiple threads are used while keeping the rest of the code single-threaded, in order to keep the code simple and maintainable.
As I soon realized, even knowing the basics of programming with threads, you are not immune to falling into many pitfalls that may arise during the actual designing and debugging of concurrent code.
You have to go extra miles ensuring that access to thread-global data are synchronized, and that one thread waits for another thread in case threads must be executed in certain order. These things
may sound like common sense and probably are in every thread programming text book, but in reality they are very easy to overlook, especially to those who have not had substantial exposure to
concurrency before. Parallelism seemed that un-orthodox to conventional minds like myself. Having said all that, once you go through enough pain dealing with concurrency, it does become less painful
after a while. Your mind can simply adjust to “thinking in parallel”.
Back to the topic. I’ve picked the following pattern to manage threaded calculation.
First, the main thread creates a new thread whose job is to manage cell queues, that is, receiving queues from the main thread and assigning them to idle threads to perform calculation. It is also
responsible for keeping track of which threads are idle and ready to take on a cell assignment. Let’s call this thread a queue manager thread. When the queue manager thread is created, it spawns a
specified number of child threads, and waits until they are all ready. These child threads are the ones that perform cell calculation, and we call them calc threads.
Each calc thread registers itself as an idle thread upon creation, then sleeps until the queue manager thread assigns it a cell to calculate and signals it to wake up. Once awake, it calculates the
cell, registers itself as an idle thread once again and goes back to sleep. This cycle continues until the queue manager thread sends a termination request to it, after which it breaks out of the
cycle and reaches the end of its execution path to terminate.
The role of the queue manager thread is to receive cell calculation requests from the main thread and pass them on to idle calc threads. It keeps doing it until it receives a termination request from
the main thread. Once receiving the termination request from the main thread, it sends all the remaining cells in queue to the calc threads to finish up, then sends termination requests to the calc
threads and wait until all of them terminate.
Thanks to the cells being sorted in topological order, the process of putting a cell in queue and having a calc thread perform calculation is entirely asynchronous. The only exception is that when
referencing another cell during calculation, the result of that referenced cell may not be available at the time of the value query due to concurrency. In such cases, the calculating thread needs to
block its execution until the result of the referenced cell becomes available. When running in a single-threaded mode, on the other hand, the result of a referenced cell is guaranteed to be available
as long as cells are calculated in topological order and contain no circular references.
What I accomplished during HackWeek
During HackWeek, I was able to accomplish quite a few things. Before the HackWeek, the threaded calculation framework was not even there; the parser was only able to reliably perform calculation in a
single-threaded mode. I had some test code to design the threaded queue management framework, but that code had yet to be integrated into the main formula interpreter code. A lot of work was still
needed, but thanks to having an entire week devoted for this, I was able to
• port the test threaded queue manager framework code into the formula interpreter code,
• adopt the circular dependency detection code for the new threaded calculation framework,
• test the new framework to squeeze lots of kinks out,
• put some performance optimization in the cell definition parser and the formula lexer code,
• implement result verification framework, and
• implement partial re-calculation.
Had I had to do all this in my spare time alone, it would have easily taken months. So, I’m very thankful for the event, and I look forward to having another opportunity like this in hopefully
not-so-distant future.
What lies ahead
So, what lies ahead for Ixion? You may ask. There are quite a few things to get done. Let me start first by saying that, this library is far from providing all the features that a typical spreadsheet
application needs. So, there are still lots of work needed to make it even usable. Moreover, I’m not even sure whether this library will become usable enough for real-world spreadsheet use, or it
will simply end up being just another interesting proof-of-concept. My hope is of course to see this library evolve into maturity, but at the same time I’m also aware that it would be hard to advance
this project with only my scarce spare time to spend in.
With that said, here are some outstanding issues that I plan on addressing as time permits.
• Add support for literal strings, and support textural formula results in addition to numerical results.
• Add support for empty cells. Empty cells are those cells that are not defined in the model definition file but can still be referenced. Currently, referencing a cell that is not defined causes a
reference error.
• Add support for cell ranges. This implies that I need to make cell instances addressable by 3-dimensional coordinates rather than by pointer values.
• Split the code into two parts: a shared library and an executable.
• Use autoconf to make the build process configurable.
• Make the expression parser feature-complete.
• Implement more functions. Currently only MAX and MIN are implemented.
• Support for localized numbers.
• Lots and lots more.
This concludes my HackWeek report. Thank you very much, ladies and gentlemen.
Two more enhancements are in
Today, I’d like to talk about two minor enhancements I just checked in to ooo-build master. They are not really earth-shuttering per se, but still worth mentioning & may be interesting to some users.
Insert new sheet tab
Here is the first enhancement. In Calc, you’ll see a new tab at the right end of the sheet tabs, to allow quick insertion of new sheets. Each time you click this tab, a new sheet gets inserted to the
right end. The sheet names are automatically assigned.
Previously, inserting a new sheet has to be done by opening the Insert sheet dialog, selecting the position of the new sheet and how many new sheets are to be inserted etc. But if you always append a
single sheet at the right end and don’t care to name the new sheet (or name it after the sheet is inserted), this enhancement will save you a few clicks. Implementing this was actually not that hard
since I was able to re-use the existing code for most of its functionality. I personally wanted to give it a little more visual appeal, but that will be a future project.
Anyway, I hope some of you will find this useful.
English function names in non-English locale
The second enhancement is related to cell functions. If you use a localized version of OOo, you probably know that the function names are localized. But there has been quite a few requests to support
English function names even if the UI is localized. This is where this enhancement comes in.
First, there is now an additional check box in the Formula options page:
By default, the check box is off, which means the localized function names are used. Checking this check box will swap localized function names with the English ones across the board. You can of
course uncheck it to go back to the localized function names.
For example, in French locale, the name of the function that calculates a summation of a cell range is called SOMME, but when the English function name option is enabled, this becomes SUM as you can
see in the following screenshot:
This change takes effect in all of the following areas:
• formula input and display,
• function wizard, and
• formula tips.
As always, please test this thoroughly, and report any bugs. Thanks! | {"url":"http://kohei.us/tag/formula/","timestamp":"2024-11-10T02:16:11Z","content_type":"text/html","content_length":"46068","record_id":"<urn:uuid:d9b02f0a-1fbb-47b1-b046-6f52469304d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00516.warc.gz"} |
micro-optimizing loops (was Help with casts)
Chris Torek torek at elf.ee.lbl.gov
Fri Mar 1 15:40:48 AEST 1991
>In article <10191 at dog.ee.lbl.gov> I wrote:
>>Of course, the best optimization for:
>> for (i = 1; i < 100; i++)
>> x += i;
>> x += 4950;
In article <14522 at ganymede.inmos.co.uk> conor at inmos.co.uk (Conor O'Neill)
> { x += 4950; i = 100; }
I was assuming (perhaps unwisely) that everyone understood that the
variable `i' was `dead' after the loop. The alternative `optimizations'
under discussion were
for (i = 100; --i > 0;)
x += i;
for (i = 99; i; i--)
x += i;
both of which leave i==0. For these to be suitable `optimizations' for
the original (count-from-1-to-99) loop, the final value of `i' would
have to be irrelevant. (Actually, it is conceivable that the final
value of `i' might have to be exactly 0 or exactly 100; in real code,
as opposed to pedagogic examples, one should check.)
(You might also note that article <10191 at dog.ee.lbl.gov> touched lightly
on compiler transformations that changed the final value of `i' in a
count-down loop from 0 to -1. I *did* consider it....)
(Richard O'Keefe also pointed out some of the dangers in transforming
code sequences. One must always be careful of semantics. The final
value of `i' is such a semantic, and Conor O'Neill is right to be wary
of anything that alters it. In this particular example, however, the
final value should be considered irrelevant.)
In-Real-Life: Chris Torek, Lawrence Berkeley Lab EE div (+1 415 486 5427)
Berkeley, CA Domain: torek at ee.lbl.gov
More information about the Comp.lang.c mailing list | {"url":"http://tuhs.vtda.org/Unix_Usenet/comp.lang.c/1991-March/041680.html","timestamp":"2024-11-12T12:18:03Z","content_type":"text/html","content_length":"4726","record_id":"<urn:uuid:7084dfe1-6269-4c08-a485-d1bae619d385>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00507.warc.gz"} |
Chapter 1 – Algebraic Notation
This chapter recaps the prerequisite material necessary for the study of algebraic concepts in preparation for taking a course in Calculus. The assumption is that students are already familiar with
this material, therefore the presentation is intentionally brief. The major focus is working with polynomials and rational expressions, specifically expanding and factoring polynomials, and the
arithmetic of fractions/rational expressions. Laws of Exponents are also introduced. These ideas are expanded upon in later chapters: Polynomials and Rational Expressions are connected to their
graphs in the chapter on Quadratics, Polynomials and Rational Expressions and the Laws of Exponents are crucial for the chapter on Exponential and Logarithmic Functions. | {"url":"https://open.lib.umn.edu/algebra/part/chapter-1-algebraic-notation/","timestamp":"2024-11-14T18:30:46Z","content_type":"text/html","content_length":"75734","record_id":"<urn:uuid:f8ae4d1f-367b-401e-940d-2acf2559d444>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00055.warc.gz"} |
The brand new assignment expression of Python 3.8
Python 3.8 is coming and not surprisingly it comes with a bag of new features. In this post, I’d like to present only one that I’ve been really waiting for: assignment expressions!
The problem
Whenever we see a new solution, we have to understand the problem or if there is a problem at all in the first place.
Let’s take this piece of code
1 def f(s):
2 result = s
3 # ... do some stuff
4 return result
6 def g():
7 return True # for the sake of the example
9 t=input()
10 if f(t) and g():
11 p = f(t)
12 # do something with p
Here we can identify at least two problems:
• We have code duplication as we wrote f(s) twice, while it’s part of the same logical branch
• What is evident, but it is even worse, we don’t just write, but we call f(s) twice. Even if we assume that with the same input we’ll always have the same outputs, so the function is deterministic
and might even be pure, we might face some issues. What if it triggers expensive calculations? Well, then we do it twice, which might have bad consequences.
There is a solution, we can call f(s) once before the if block, and save the result in a variable!
1 def f(s):
2 result = s
3 # ... do some stuff
4 return result
6 def g():
7 return True # for the sake of the example
9 t=input()
10 p = f(t)
11 if p and g():
12 # do something with p
Is that better? Well, it depends.
On the one hand, you type a bit less and if the calculations in f(s) are costly, you eliminated that expensive function call, that’s great!
On the other hand, now you have a variable that is accessible outside the if block where you orientally wanted to use it. This might be unsafe. Imagine that you create your variable as a reference to
something that takes a long-chained command to retrieve.
But before you use it, you want to make a validity check.
If you create the variable outside the if block, so before making the validity check, you have to remember that if you want to use that variable later on in that function (which would probably be a
bad practice anyway), you must do the validity check again.
Python 3.8 and PEP 572 provides us with the ultimate solution you probably always wanted for such issues.
You can create a variable in the if expression whose scope is only the whole if block. Isn’t that awesome?! You can do things like this, just to continue with the previous example:
1 def f(s):
2 result = s
3 # ...
4 return result
6 def g():
7 return True # for the sake of the example
9 t=input()
10 if p:=f(t) and g():
11 # do something with p
12 print(p)
By the “whole if block”, I meant that else is also included. To generalize, we can say that the scope of the variable assigned in the assignment expression is just the current scope. If it’s an if,
then an if, if it’s the whole function, it’s the whole function.
What I also like a lot is that we now can simplify list comprehensions as well. Look at this example:
1 inputs = [1, 2, 3, 56, 78, 42, 36, 54, 35, 99]
2 numbers_and_int_square_roots = {k: int(math.sqrt(k)) for k in inputs if int(math.sqrt(k)) == math.sqrt(k)}
3 print(*numbers_and_int_square_roots.items())
So, we take a list of numbers and we want to keep the ones that are squares of an integers and we also want to keep their squared and non-squared values. We have to calculate the square roots twice!
Only if I could save the square root in that generator expression! Lo and behold, now I can:
1 inputs = [1, 2, 3, 56, 78, 42, 36, 54, 35, 99]
2 numbers_and_int_square_roots = {k: v for k in inputs if (v:=int(math.sqrt(k))) == math.sqrt(k)}
3 print(*numbers_and_int_square_roots.items())
That is just super cool to me! Less typing, less calculations, faster runtime!
What do you think?
Python 3.8 introduces assignment expressions which lets us create new variables in places where we always wanted but never could in a usable way. Probably the best way to use this new feature is to
create new variables in an if and use it in the scope of that block.
For more information, you can read the specs here.
As an online interpreter, for the moment you can use this.
And if you want to install python3.8 locally, you can refer to this article
Happy coding! | {"url":"https://www.sandordargo.com/blog/2019/10/16/python-assignment-expression","timestamp":"2024-11-09T22:58:32Z","content_type":"text/html","content_length":"35623","record_id":"<urn:uuid:f3a74bf5-fc83-4887-aae9-57d5802018b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00226.warc.gz"} |
Only the lowest mass bin is outside the range, so we won't use that. One thing the concentration module does not check is the cosmology: if a model was calibrated only for a particular cosmology, it
can still be evaluated even if a different cosmology is set. For example: | {"url":"https://bdiemer.bitbucket.io/colossus/_static/tutorial_halo_concentration.html","timestamp":"2024-11-12T02:35:14Z","content_type":"text/html","content_length":"329595","record_id":"<urn:uuid:7a88517f-210b-4d3d-990d-636bd67d6b5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00450.warc.gz"} |
CMPT 280– Assignment 8 solved
2 Background
In this section we present material required for Question 1.
2.1 Union-find ADT
A union-find ADT (also called a disjoint-set ADT) keeps track of a set of elements which are partitioned into
disjoint subsets. It is useful for establishing equivalencies of groups of items in a set about which nothing
is known initially. For example, suppose we have an initial set of cities:
Vancouver, Edmonton, Regina, Saskatoon, Winnipeg, Toronto, Montreal, Calgary
Let’s then suppose that we decide that Vancouver and Edmonton are “equivalent” (this can be defined
in any number of ways), that Regina, Saskatoon, and Winnipeg are equivalent, and that Montreal and
Calgary are equivalent. Now we would have four subsets of equivalent elements of our overall set:
{Vancouver, Edmonton}, {Regina, Saskatoon,Winnipeg}, {Toronto}, {Montreal, Calgary}
Note that since Toronto was not deemed equivalent to anything, it is in its own subset by itself. Now, let’s
suppose we want to find out which set a particular city is in. This is done by choosing from each subset a
representative (also called an equivalence-class label) which acts as the identifier for that set. Suppose for the
sake of simplicity, that we choose the first item in each set as its representative (shown in bold):
{Vancouver, Edmonton}, {Regina, Saskatoon,Winnipeg}, {Toronto}, {Montreal, Calgary}
If we were to now ask which subset Winnipeg belongs to, the answer would be Regina. Asking which
subset an element belongs to is called the find operation. The find operation applied to an element returns
the representative of the set to which it belongs, for example, find(Winnipeg) = Regina, or find(Calgary) =
Montreal, or find(Vancouver) = Vancouver. The find operation is one of the two main operations supported
by the Union-Find ADT.
The Union-Find ADT unsurprisingly supports a second operation called union. The union operation
takes two elements as arguments, and establishes them as being “equivalent”, meaning, they should be in
the same set. So union(Edmonton, Calgary) would place Calgary and Edmonton in the same subset. But if
Edmonton and Calgary are equivalent, then by transitivity, everything in the subsets to which Edmonton
and Calgary belong must also be equivalent, so the union operation actually merges two subsets into one
— so this is just familiar set union operation!. Thus, union(Edmonton, Calgary) would alter our group of
subsets so they look like this:
{Vancouver, Edmonton, Montreal, Calgary}, {Regina, Saskatoon,Winnipeg}, {Toronto}
So now Find(Calgary) would result in an answer of Vancouver. You may be wondering why we chose Vancouver as the representative element of the merged subset instead of Montreal. This is an
implementationlevel decision. In principle, either one could be chosen.
In summary, the Union-Find data structure keeps track of a set of disjoint subsets of a set of elements.
It supports the operations find(X) (look up the name of the subset to which element X belongs) and
union(X,Y) (merge the subsets containing X and Y). In this assignment we will implement the union-find
ADT using a directed, unweighted graph.
2.2 Minimum Spanning Tree
Given a connected, weighted, undirected graph, its minimum spanning tree consists of the subset of the
graph’s edges of smallest total weight such that the graph remains connected. Such a set of edges always
forms a tree because if it weren’t a tree there would be a cycle, which implies that it wouldn’t be the
minimum cost set of edges that keeps the graph connected because you could remove one edge from the
cycle and the graph would still be connected. Here is a weighed, undirected graph, and its minimum
spanning tree (denoted by thicker, red edges):
No other set of edges that keeps the above graph connected has a smaller sun of weights.
The minimum spanning tree has many applications since many optimization problems can be reduced
to a minimum spanning tree algorithm. Suppose you have identified several sites at which to build
network routers and you know what it would cost to connect each pair of network routers by a physical
wire. You would like to know what is the cheapest possible way to connect all your routers. This is an
instance of the minimum spanning tree problem.
Finding the minimum spanning tree isn’t as straightforward as it might seem. There are various algorithms for finding the minimum spanning tree. We will be using Kruskal’s algorithm which,
can be implemented efficiently with a union-find ADT.
3 Your Tasks
Question 1 (16 points):
For this problem you will implement Kruskal’s algorithm for finding the minimum spanning tree of
an undirected weighted graph. Kruskal’s algorithm uses a union-find data structure to keep track
of subsets of vertices of the input graph G. Initially, every vertex of G is in a subset by itself. The
intuition for Kruskal’s algorithm is that the edges of the input graph G are sorted in ascending order
of weight (smallest weights first), then each such edge (a, b) is examined in order, and if a and b are
currently in different subsets we merge the two sets containing a and b and add (a, b) to the graph
of the minimum spanning tree. This works because vertices in the same subset in the union-find
structure are all connected. Once all of the vertices are in the same subset, we know that they are
all connected. Since we always add the next smallest edge possible to the minimum spanning tree,
the result is the smallest-cost set of edges that cause the graph to be completely connected, i.e. the
minimum spanning tree! Here’s Kruskal’s algorithm, in pseudocode:
Algoirthm minimumSpanningTreeKruskal ( G )
G – A weighted , undirected graph .
minST = an undirected , weighted graph with the same node set as G ,
but no edges .
UF = a union – find data structure containing the node set of G in which
each node is initially in its own subset .
Sort the edges of G in order from smallest to largest weight .
for each edge e =( a , b ) in sorted order
if UF . find ( a ) != UF . find ( b )
minST . addEdge (a , b )
set the weight of (a , b ) in minST to the weight of (a , b ) in G
UF . union (a , b )
return minST
In order to implement Kruskal’s algorithm you will first need to implement a union-find ADT. We can
implement union-find with a directed (unweighted) graph F. Initially the graph has a node for each
item in the set, and no edges. This makes the union operation very easy. The operation union(a,b) can
be completed simply by adding the edge (find(a), find(b)) to F, that is, we add an edge that connects
the representative elements of the subsets containing a and b. The find(a) operation then works by
checking node a to see if it has an outgoing edge, if it does, we follow it and check the node we get to
to see if it has an outgoing edge. We continue going in this fashion until we find a node that does not
have an outgoing edge. That node is the representative element of the subset that contains a, and we
would return that node. Here’s an example of a directed graph that represents a set of subsets of the
elements 1 through 8:
If we were to call find(7) on this graph, we would see that 7 has an edge to 3, which has an edge to 2,
but 2 has no outgoing edge, so find(7) = 2. Similarly if we called find(4), we would follow the edge to
node 6, then its outgoing edge to node 5, and find that 5 has no outgoing edge, so find(4) = 5. Overall,
this graph represents that 1, 2, 3, and 7 are in the same subset, which has 2 as its representative
element; that 4, 5, and 6 are in the same subset with representative element 5, and 8 is in a subset by
itself. Now, suppose we do union(6, 1). This causes an edge to be added from find(6)=5 to find(1)=2,
that is an edge from 5 to 2:
This causes the subsets containing 6 and 1 to be merged, and the new merged subset has representative
element 2. Convince yourself that if you call find() on any element except 8, you will get a result of 2
– follow the arrows from the starting node and you’ll always end up at 2.
Here are the algorithms for the union and find operations using a graph as the underlying data
Algorithm union (a , b )
a , b – elements whose subsets are to be merged
// If a and b are already in the same set , do nothing .
if find ( a ) == find ( b )
// Otherwise , merge the sets
add the edge ( find ( a ) , find ( b )) to the union – find graph .
Algorithm find ( a )
a – element for which we want to determine set membership
// Follow the chain of directed edges starting from a
x = a
while x has an outgoing edge (x , y ) in the union – find graph
x = y
// Since at this point x has no outgoing edge , it must be the
// representative element of the set to which a belongs , so …
return x
These are the simplest possible algorithms for union() and find(), and they don’t result in the most
efficient implementations. There are improvements that we could make, but to keep things simple,
we won’t bother with them. Eventually, I’ll provide solutions that use these algorithms, as well as an
improved, more efficient solution for those who are interested.
Well, that was a lot of stuff. Now we can finally get to what you actually have to do:
1. Import the project Kruskal-Template (provided) module into IntelliJ workspace. You may need
to add the lib280-asn8 project (also provided) as a module depdnency of the Kruskal-Template
module (this process is covered in the self-guided tutorials on Moodle).
2. In the UnionFind280 class in the Kruskal-Template project, complete the implementation of the
methods union() and find(). Do not modify anything else. You may add a main method to the
UnionFind class for testing purposes.
3. In Kruskal.java complete the implementation of the minSpanningTree method. Do not modify
anything else.
4. Run the main program in Kruskal.java. The pre-programmed input graph is the same as the
one shown in Section 2.2. The input graph and the minimum spanning tree as computed by the
minSpanningTree() method are displayed as output. Check the output to see if the minimum
spanning tree that is output matches the one in Section 2.2.
Implementation Hints
When implementing Kruskal’s algorithm, you should be able to avoid having to write your own
sorting algorithm, or putting the edges into an array to sort the edges by their weights. You can take
advantage of ADTs already in lib280-asn8a. All you need is to put the edges in a dispenser which,
when you remove an item, will always give you the edge with the smallest weight (hint: look in the
lib280.tree package for ArrayedMinHeap280). Conveniently, WeightedEdge280 objects are Comparable
based on their weight.
Question 2 (20 points):
For this question you will implement Dijkstra’s algorithm. The implementation will be done within
the NonNegativeWeightedGraphAdjListRep280 class which you can find in the lib280-asn8.graph
package. This class is an extension of WeightedGraphAdjListRep280 which restricts the graph edges
to have nonnegative weights. This works well for us since Dijkstra’s algorithm can only be used on
graphs with nonnegative weights.
1. Implement the shortestPathDijkstra method in NonNegativeWeightedGraphAdjListRep280.
The method’s javadoc comment explains the inputs and outputs of the method.
2. Implement the extractPath method in NonNegativeWeightedGraphAdjListRep280. The method’s
javadoc comment explains the inputs and outputs of the method.
The pseudocode for Dijkstra’s algorithm is reproduced below.
Algoirthm dijkstra (G , s )
G is a weighted graph with non – negative weights .
s is the start vertex .
Postcondition : v . tentativeDistance is the length of the
shortest path from s to v .
v . predecessorNode is the node that appears before v
on the shortest path from s to v .
Let V be the set of vertices in G.
For each v in V
v . tentativeDistance = infinity
v . visited = false
v . predecessorNode = null
s . tentativeDistance = 0
while there is an unvisited vertex
cur = the unvisited vertex with the smallest tentative distance .
cur . visited = true
// update tentative distances for adjacent vertices if needed
// note that w(i,j) is the cost of the edge from i to j.
For each z adjacent to cur
if ( z is unvisited and z . tentativeDistance >
cur . tentativeDistance + w ( cur , z ) )
z . tentativeDistance = cur . tentativeDistance + w ( cur , z )
z . predecessorNode = cur
Implementation Hints
Even though the pseudocode implies that tentativeDistance, visited and predecessorNode are
properties of vertices and perhaps should be stored in vertex objects, it is easiest to just use a set of
parallel arrays in the implementation of Dijstra’s algorithm, much like the way we represented these
as arrays during the in-class examples. E.g. an array boolean visited[] such that if visisted[i]
is true, it means that vertex i has been visited. This is quite easy to use since vertices are always
numbered 1 through n.
Sample Output
If you done things right, then you should get the following outputs for start vertices 1 and 9 respectively.
Enter the number of the start vertex :
The length of the shortest path from vertex 1 to vertex 1 is : 0.0
Not reachable .
The length of the shortest path from vertex 1 to vertex 2 is : 1.0
The path to 2 is : 1 , 2
The length of the shortest path from vertex 1 to vertex 3 is : 3.0
The path to 3 is : 1 , 3
The length of the shortest path from vertex 1 to vertex 4 is : 23.0
The path to 4 is : 1 , 3 , 5 , 6 , 4
The length of the shortest path from vertex 1 to vertex 5 is : 7.0
The path to 5 is : 1 , 3 , 5
The length of the shortest path from vertex 1 to vertex 6 is : 16.0
The path to 6 is : 1 , 3 , 5 , 6
The length of the shortest path from vertex 1 to vertex 7 is : 42.0
The path to 7 is : 1 , 3 , 5 , 6 , 4 , 8 , 9 , 7
The length of the shortest path from vertex 1 to vertex 8 is : 31.0
The path to 8 is : 1 , 3 , 5 , 6 , 4 , 8
The length of the shortest path from vertex 1 to vertex 9 is : 36.0
The path to 9 is : 1 , 3 , 5 , 6 , 4 , 8 , 9
Enter the number of the start vertex :
The length of the shortest path from vertex 9 to vertex 1 is : 36.0
The path to 1 is : 9 , 8 , 4 , 6 , 5 , 3 , 1
The length of the shortest path from vertex 9 to vertex 2 is : 35.0
The path to 2 is : 9 , 8 , 4 , 6 , 5 , 3 , 2
The length of the shortest path from vertex 9 to vertex 3 is : 33.0
The path to 3 is : 9 , 8 , 4 , 6 , 5 , 3
The length of the shortest path from vertex 9 to vertex 4 is : 13.0
The path to 4 is : 9 , 8 , 4
The length of the shortest path from vertex 9 to vertex 5 is : 29.0
The path to 5 is : 9 , 8 , 4 , 6 , 5
The length of the shortest path from vertex 9 to vertex 6 is : 20.0
The path to 6 is : 9 , 8 , 4 , 6
The length of the shortest path from vertex 9 to vertex 7 is : 6.0
The path to 7 is : 9 , 7
The length of the shortest path from vertex 9 to vertex 8 is : 5.0
The path to 8 is : 9 , 8
The length of the shortest path from vertex 9 to vertex 9 is : 0.0
Not reachable .
Question 3 (25 points):
For this problem you will write a method (or methods) to sort an array of strings using the MSD Radix
Sort. For purposes of this assignment, you may assume that strings contain only the uppercase letters
A through Z.
You have been provided with an IntelliJ module RadixSortMSD-Template which includes a short main
program that will load a data file containing strings to be sorted. There are several files provided
named words-XXXXXX.txt where “XXXXXX” denotes the number of words in the file. The file format
starts with the number of words in the file, followed by one word per line. There is also a file
words-basictest.txt which is a good set of words to use to determine whether your sort is running
The pseudocode for MSD Radix Sort from the notes is duplicated on the next page for your convenience. Note that we are removing the optimization of sorting short lists with insertion sort, as
indicated by the strikethrough text. You may just always recursively radix sort any list with more
than one element on it.1
Complete the following tasks:
1. Write your sort method(s) within the RadixSortMSD class. It should accept an array of strings as
input, and return nothing. When the method returns, the array that was passed in should be in
lexicographic order (i.e. dictionary order).
2. Call your sort at the spot indicated in the main() function.
3. Record in a file called a8q1.txt/doc/pdf the time in milliseconds it takes to sort 50, 100, 500,
1000, 10000, 50000, and 235884 items (there are input files with each of these numbers of words
provided). Include this file in your assignment submission.
4. When you hand in RadixSortMSD.java, leave the input file set at words-basictest.txt so that
it is easy for the markers to run your program on this input file to see that it works.
There will be marks allotted to the following aspects of your solution:
Correctness. As always, the solution has to work!
Design and speed of your implementation. Design and speed will be considered together because
they influence each other — a poor design choice may result in a slower runtime. Design-wise,
any reasonable design will be accepted; marks will be deducted for especially poor choices.
Speed-wise, the bar to get full marks here will be fairly low, but if your sort is egregiously slow
we will deduct some points.
Javadoc and inline commenting. As usual, include a javadoc comment with each header, and document meaningful blocks of code with inline comments to enhance understanding of the code.
For full details, consult the grading rubric on Moodle.
1You can, however, use the insertion-sort optimization if you want to, but you will probably have to implement your own insertion
sort. If you choose to attempt this optional optimization, you are permitted to use resources from the internet to implement insertion
sort, but only for the insertion sort, and only if you properly attribute any code you use to its original author or website.
Implementation Hints
• One of the most important decisions you have to make in your implementation is the choice of
data structure to represent the array of lists used in the sortByDigit() helper method. Choose
carefully! Your choice will have an impact on the speed of your sort, and the ease of implementing it. When considering which data structure to use, you may select from containers in either
lib280 or the standard Java API. Case in point: on my first attempt, I made a bad decision that
caused the sort of 10000 words to take several minutes. Now it takes 24ms (on my computer).
• Although runtimes will vary from machine to machine, on a decent machine you should be able
to sort even the 235884-word input file in less than one second (1000ms). Even on slower machines
if it is taking more than a few seconds, then you’ve done something particularly inefficient.
• Don’t take the pseudocode below too literally. This is very high-level pseudocode which is intended to describe the operation of the algorithm, but intentionally glosses over a lot of details
that become important at implementation-time (that’s what pseudocode is for!). You need to fill
in those details as you go. Don’t be afraid to do what you need to do to get the job done, but that
said, you should not need to write hundreds of lines of code (if you are, you should seek help
and advice from Mark or a TA).
Algoirthm MsdRadixSort ( keys , R )
keys – keys to be sorted
R – the radix
sortByDigit ( keys , R , 0)
Algorithm sortByDigit ( keys , R , i )
keys – keys to be sorted
R – the radix
i – digit on which to partition — i = 0 is the left – most digit
for k = 0 to R -1
list [ k ] = new list // Make a new list for each digit
for each key
if the i – th digit of the key has value k add the key to list k
for k = 0 to R -1
if there is another digit to consider
if list[k] is small
use an insertion sort to sort the items in list[k]
sortByDigit ( list [ k] , i +1)
keys = new list // empty the input list
For k = 0 to R -1
keys = keys append list [ k ]
4 Files Provided
lib280-asn8: A copy of lib280 which includes:
• solutions to assignment 7;
• graph classes necessary for questions 1 and 2.
GraphAdjListRep280 and WeightedGraphAdjListRep280 which you’ll use in Question 1
Kruskal-template An IntelliJ module with templates template for question 1.
NonNegativeWeightedGraphAdjListRep280 class for Question 2.
RadixSortMSD-Template: The project template for question 3.
5 What to Hand In
UnionFind280.java Your completed union-find class from Question 1
Kruskal.java Your completed implementation of Kruskal’s algorithm from Question 1.
NonNegativeWeightedGraphAdjListRep280.java Your completed implementation of Dijkstra’s algorithm
from Question 2.
RadixSortMSD.java: Your completed radix sort from question 3.
a8q1.txt/doc/pdf: Your timing observations from your sort in question 3. | {"url":"https://codeshive.com/questions-and-answers/cmpt-280-assignment-8-solved/","timestamp":"2024-11-04T01:13:19Z","content_type":"text/html","content_length":"139718","record_id":"<urn:uuid:f4d1b87f-828a-405f-a6eb-402381759460>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00309.warc.gz"} |
Using Machine Learning to Predict the Weather: Part 3
This is the final article on using machine learning in Python to make predictions of the mean temperature based off of meteorological weather data retrieved from Weather Underground as described in
part one of this series.
The topic of this final article will be to build a neural network regressor using Google's Open Source TensorFlow library. For a general introduction into TensorFlow, as well a discussion of
installation methods, please see Mihajlo Pavloski's excellent post TensorFlow Neural Network Tutorial.
Topics I will be covering in this article include:
• Understanding Artificial Neural Networks Theory
• TensorFlow's High Level Estimator API
• Building a DNNRegressor to Predict the Weather
Understanding Artificial Neural Networks Theory
In the last article (part 2) I described the process of building a linear regression model, a venerable machine learning technique that underlies many others, to predict the mean daily temperature in
Lincoln, Nebraska. Linear regression models are extremely powerful and have been used to make numerical, as well as categorical, predictions since well before the term "machine learning" was ever
coined. However, the technique has some criticisms, mostly around its ridged assumption of a linear relationship between the dependent variable and the independent variable(s).
An uncountable number of other algorithms exist in the data science and machine learning industry which overcome this assumption of linearity. One of the more popular areas of focus in recent years
has been to apply neural networks to a vast array of machine learning problems. Neural networks have a powerful way of utilizing learning techniques based on both linear and non-linear operations.
Neural networks are inspired by biological neurons in the brain which work in a complex network of interactions to transmit, collect, and learn information based off a history of the information that
has already been collected. The computational neural networks we are interested in are similar to the neurons of the brain in that they are a collection of neurons (nodes) that receive input signals
(numerical quantities), process the input, and transmits the processed signals to other downstream agents in the network. The processing of signals as numerical quantities that pass through the
neural network is a very powerful feature that is not limited to linear relationships.
In this series I have been focusing on a specific type of machine learning called supervised learning, which simply means that the models being trained are built using data that has known target
outcomes that the model is trying to learn to predict. Furthermore, the type of predictions being made are numerical real values, which means we are dealing with regressor prediction algorithms.
Graphically, a neural network similar to the one being described in this article is shown in the image below.
The neural network depicted above contains an input layer on the far left representing two features, x1 and x2, that are feeding the neural network. Those two features are fed into the neural
network, which are processed and transmitted through two layers of neurons, which are referred to as hidden layers. This depiction shows two hidden layers with each layer containing three neurons
(nodes). The signal then exits the neural network and is aggregated at the output layer as a single numerical predicted value.
Let me take a moment to explain the meaning behind the arrows signifying data being processed from node to node across the layers. Each arrow represents a mathematical transformation of a value,
beginning at the arrow's base, which is then multiplied by a weight specific to that path. Each node within a layer will be fed a value in this way. Then all the values converging at the node are
summed. It is this aggregate of multiplying by weights and summing the products that define the linear operations of a neural network I mentioned earlier.
After summation is carried out at each node a special, non-linear, function is applied to the sum, which is depicted in the image above as Fn(...). This special function that introduces non-linear
characteristics into a neural network is called an activation function. It is this non-linear characteristic brought about by activation functions that give multi-layer neural networks their power.
If it was not for the non-linearity added to the process then all layers would effectively just algebraically combine into one constant operation consisting of multiplying the inputs by some flat
coefficient value (ie, a linear model).
Alright, so that is all fine and dandy, but I hope you are wondering in the back of your mind... ok, Adam, but how does this translate into a learning algorithm? Well the most straight forward answer
to that is to evaluate the predictions being made, the output of the model "y", to the actual expected values (the targets) and make a series of adjustments to the weights in a manner that improves
the overall prediction accuracy.
In the world of regressor machine learning algorithms one evaluates the accuracy by using a cost (aka "loss", or "objective") function, namely the sum of squared errors (SSE). Notice that I
generalized that statement to the whole continuum of machine learning, not just neural networks. In the prior article the Ordinary Least Squares algorithm accomplished just that, it found the
combinations of coefficients that minimized the sum of the squared errors (ie, least squares).
Our neural network regressor will do the exact same thing. It will iterate over the training data feeding in feature values, calculate the cost function (using SSE) and make adjustments to the
weights in a way that minimizes the cost function. This process of iteratively pushing features through the algorithm and evaluating how to adjust the weights based off the cost function is, in
essence, what is known as model optimization.
Model optimization algorithms are very important in building robust neural networks. As examples are fed through the networks architecture (ie, the width and depth) then evaluated against the cost
function, the weights are adjusted. The models is said to be "learning" when the optimizer function identifies that a weight adjustment was made in a way that does not improve (lower) the cost
function, which is registered with the optimizer so that it does not adjust the weights in that direction again.
TensorFlow's High Level Estimator API
Google's TensorFlow library consists a few API's, with the most popular being the Core API, which gives the user a low level set of tools to define and train essentially any machine learning
algorithm using symbolic operations. This is referred to as TensorFlow Core. While TensorFlow Core is an amazing API with vast application capability, I will be focusing on a newer, higher level, API
the TensorFlow team developed that is collectively referred to as the Estimator API.
The TensorFlow team developed the Estimator API to make the library more accessible to the everyday developer. This high level API provides a common interface to train(...) models, evaluate(...)
models, and predict(...) outcomes of unknown cases similar to (and influenced by) the popular Sci-Kit Learn library, which is accomplished by implementing a common interface for various algorithms.
Also, built into the high level API are a load of machine learning best practices, abstractions, and ability for scalability.
All of this machine learning goodness brings about a set of tools implemented in the base Estimator class as well as multiple pre-canned model types that lowers the barrier to entry for using
TensorFlow so it can be applied to a host of everyday problems (or opportunities). By abstracting away much of the mundane and manual aspects of things like writing training loops or dealing with
sessions, the developer is able to focus on more important things like rapidly trying multiple models and model architectures to find the one that best fits their need.
In this article I will be describing how to use one of the very powerful deep neural network estimators, the DNNRegressor.
Building a DNNRegressor to Predict the Weather
Let me start by importing a number of different libraries that I will use to build the model:
import pandas as pd
import numpy as np
import tensorflow as tf
from sklearn.metrics import explained_variance_score, \
mean_absolute_error, \
from sklearn.model_selection import train_test_split
Now let us get our hands on the data and take a couple of peaks at it again to familiarize ourselves with it. I have placed all the code and data in my GitHub repo here so that readers can follow
# read in the csv data into a pandas data frame and set the date as the index
df = pd.read_csv('end-part2_df.csv').set_index('date')
# execute the describe() function and transpose the output so that it doesn't overflow the width of the screen
count mean std min 25% 50% 75% max
meantempm 997.0 13.129388 10.971591 -17.0 5.0 15.0 22.00 32.00
maxtempm 997.0 19.509529 11.577275 -12.0 11.0 22.0 29.00 38.00
mintempm 997.0 6.438315 10.957267 -27.0 -2.0 7.0 16.00 26.00
meantempm_1 997.0 13.109328 10.984613 -17.0 5.0 15.0 22.00 32.00
meantempm_2 997.0 13.088265 11.001106 -17.0 5.0 14.0 22.00 32.00
meantempm_3 997.0 13.066199 11.017312 -17.0 5.0 14.0 22.00 32.00
meandewptm_1 997.0 6.440321 10.596265 -22.0 -2.0 7.0 16.00 24.00
meandewptm_2 997.0 6.420261 10.606550 -22.0 -2.0 7.0 16.00 24.00
meandewptm_3 997.0 6.393180 10.619083 -22.0 -2.0 7.0 16.00 24.00
meanpressurem_1 997.0 1016.139418 7.582453 989.0 1011.0 1016.0 1021.00 1040.00
meanpressurem_2 997.0 1016.142427 7.584185 989.0 1011.0 1016.0 1021.00 1040.00
meanpressurem_3 997.0 1016.151454 7.586988 989.0 1011.0 1016.0 1021.00 1040.00
maxhumidity_1 997.0 88.107322 9.280627 47.0 83.0 90.0 93.00 100.00
maxhumidity_2 997.0 88.106319 9.280152 47.0 83.0 90.0 93.00 100.00
maxhumidity_3 997.0 88.093280 9.276775 47.0 83.0 90.0 93.00 100.00
minhumidity_1 997.0 46.025075 16.108517 9.0 35.0 45.0 56.00 92.00
minhumidity_2 997.0 46.021063 16.105530 9.0 35.0 45.0 56.00 92.00
minhumidity_3 997.0 45.984955 16.047081 9.0 35.0 45.0 56.00 92.00
maxtempm_1 997.0 19.489468 11.588542 -12.0 11.0 22.0 29.00 38.00
maxtempm_2 997.0 19.471414 11.603318 -12.0 11.0 22.0 29.00 38.00
maxtempm_3 997.0 19.455366 11.616412 -12.0 11.0 22.0 29.00 38.00
mintempm_1 997.0 6.417252 10.974433 -27.0 -2.0 7.0 16.00 26.00
mintempm_2 997.0 6.394183 10.988954 -27.0 -2.0 7.0 16.00 26.00
mintempm_3 997.0 6.367101 11.003451 -27.0 -2.0 7.0 16.00 26.00
maxdewptm_1 997.0 9.378134 10.160778 -18.0 1.0 11.0 18.00 26.00
maxdewptm_2 997.0 9.359077 10.171790 -18.0 1.0 11.0 18.00 26.00
maxdewptm_3 997.0 9.336008 10.180521 -18.0 1.0 11.0 18.00 26.00
mindewptm_1 997.0 3.251755 11.225411 -28.0 -6.0 4.0 13.00 22.00
mindewptm_2 997.0 3.229689 11.235718 -28.0 -6.0 4.0 13.00 22.00
mindewptm_3 997.0 3.198596 11.251536 -28.0 -6.0 4.0 13.00 22.00
maxpressurem_1 997.0 1019.913741 7.755590 993.0 1015.0 1019.0 1024.00 1055.00
maxpressurem_2 997.0 1019.917753 7.757705 993.0 1015.0 1019.0 1024.00 1055.00
maxpressurem_3 997.0 1019.927783 7.757805 993.0 1015.0 1019.0 1024.00 1055.00
minpressurem_1 997.0 1012.317954 7.885743 956.0 1008.0 1012.0 1017.00 1035.00
minpressurem_2 997.0 1012.319960 7.886681 956.0 1008.0 1012.0 1017.00 1035.00
minpressurem_3 997.0 1012.326981 7.889511 956.0 1008.0 1012.0 1017.00 1035.00
precipm_1 997.0 2.593180 8.428058 0.0 0.0 0.0 0.25 95.76
precipm_2 997.0 2.593180 8.428058 0.0 0.0 0.0 0.25 95.76
precipm_3 997.0 2.573049 8.410223 0.0 0.0 0.0 0.25 95.76
# execute the info() function
<class 'pandas.core.frame.DataFrame'>
Index: 997 entries, 2015-01-04 to 2017-09-27
Data columns (total 39 columns):
meantempm 997 non-null int64
maxtempm 997 non-null int64
mintempm 997 non-null int64
meantempm_1 997 non-null float64
meantempm_2 997 non-null float64
meantempm_3 997 non-null float64
meandewptm_1 997 non-null float64
meandewptm_2 997 non-null float64
meandewptm_3 997 non-null float64
meanpressurem_1 997 non-null float64
meanpressurem_2 997 non-null float64
meanpressurem_3 997 non-null float64
maxhumidity_1 997 non-null float64
maxhumidity_2 997 non-null float64
maxhumidity_3 997 non-null float64
minhumidity_1 997 non-null float64
minhumidity_2 997 non-null float64
minhumidity_3 997 non-null float64
maxtempm_1 997 non-null float64
maxtempm_2 997 non-null float64
maxtempm_3 997 non-null float64
mintempm_1 997 non-null float64
mintempm_2 997 non-null float64
mintempm_3 997 non-null float64
maxdewptm_1 997 non-null float64
maxdewptm_2 997 non-null float64
maxdewptm_3 997 non-null float64
mindewptm_1 997 non-null float64
mindewptm_2 997 non-null float64
mindewptm_3 997 non-null float64
maxpressurem_1 997 non-null float64
maxpressurem_2 997 non-null float64
maxpressurem_3 997 non-null float64
minpressurem_1 997 non-null float64
minpressurem_2 997 non-null float64
minpressurem_3 997 non-null float64
precipm_1 997 non-null float64
precipm_2 997 non-null float64
precipm_3 997 non-null float64
dtypes: float64(36), int64(3)
memory usage: 311.6+ KB
Note that we have just under a 1000 records of meteorological data and that all the features are numerical in nature. Also, because of our hard work in the first article, all of the records are
complete in that they are not missing (no non-nulls) any values.
Now I will remove the "mintempm" and "maxtempm" columns as they have no meaning in helping us predict the average mean temperatures. We are trying to predict the future so we obviously can not have
data about the future. I will also separate out the features (X) from the targets (y).
# First drop the maxtempm and mintempm from the dataframe
df = df.drop(['mintempm', 'maxtempm'], axis=1)
# X will be a pandas dataframe of all columns except meantempm
X = df[[col for col in df.columns if col != 'meantempm']]
# y will be a pandas series of the meantempm
y = df['meantempm']
As with all supervised machine learning applications, I will be dividing my dataset into training and testing sets. However, to better explain the iterative process of training this neural network I
will be using an additional dataset I will refer to as a "validation set". For the training set I will be utilizing 80 percent of the data and for the testing and validation set they will each be 10%
of the remaining data.
To split out this data I will again be using Sci-Kit Learn's train_test_split(...).
# split data into training set and a temporary set using sklearn.model_selection.traing_test_split
X_train, X_tmp, y_train, y_tmp = train_test_split(X, y, test_size=0.2, random_state=23)
# take the remaining 20% of data in X_tmp, y_tmp and split them evenly
X_test, X_val, y_test, y_val = train_test_split(X_tmp, y_tmp, test_size=0.5, random_state=23)
X_train.shape, X_test.shape, X_val.shape
print("Training instances {}, Training features {}".format(X_train.shape[0], X_train.shape[1]))
print("Validation instances {}, Validation features {}".format(X_val.shape[0], X_val.shape[1]))
print("Testing instances {}, Testing features {}".format(X_test.shape[0], X_test.shape[1]))
Training instances 797, Training features 36
Validation instances 100, Validation features 36
Testing instances 100, Testing features 36
The first step to take when building a neural network model is to instantiate the tf.estimator.DNNRegressor(...) class. The class constructor has multiple parameters, but I will be focusing on the
• feature_columns: A list-like structure containing a definition of the name and data types for the features being fed into the model
• hidden_units: A list-like structure containing a definition of the number width and depth of the neural network
• optimizer: An instance of tf.Optimizer subclass, which optimizes the model's weights during training; its default is the AdaGrad optimizer.
• activation_fn: An activation function used to introduce non-linearity into the network at each layer; the default is ReLU
• model_dir: A directory to be created that will contain metadata and other checkpoint saves for the model
I will begin by defining a list of numeric feature columns. To do this I use the tf.feature_column.numeric_column() function which returns a FeatureColumn instance for numeric, continuous-valued
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feature_cols = [tf.feature_column.numeric_column(col) for col in X.columns]
With the feature columns defined I can now instantiate the DNNRegressor class and store it in the regressor variable. I specify that I want a neural network that has two layers deep where both layers
have a width of 50 nodes. I also indicate that I want my model data stored in a directory called tf_wx_model.
regressor = tf.estimator.DNNRegressor(feature_columns=feature_cols,
hidden_units=[50, 50],
INFO:tensorflow:Using default config.
INFO:tensorflow:Using config: {'_tf_random_seed': 1, '_save_checkpoints_steps': None, '_save_checkpoints_secs': 600, '_model_dir': 'tf_wx_model', '_log_step_count_steps': 100, '_keep_checkpoint_every_n_hours': 10000, '_save_summary_steps': 100, '_keep_checkpoint_max': 5, '_session_config': None}
The next thing that I want to do is to define a reusable function that is generically referred to as an "input function", which I will call wx_input_fn(...). This function will be used to feed data
into my neural network during the training and testing phases. There are many different ways to build input functions, but I will be describing how to define and use one based off the
tf.estimator.inputs.pandas_input_fn(...) since my data is in a pandas data structures.
def wx_input_fn(X, y=None, num_epochs=None, shuffle=True, batch_size=400):
return tf.estimator.inputs.pandas_input_fn(x=X,
Notice that this wx_input_fn(...) function takes in one mandatory and four optional parameters which are then handed off to a TensorFlow input function specifically for pandas data, which is
returned. This is a very powerful feature of the TensorFlow API (and Python and other languages that treat functions as first class citizens).
The parameters to the function are defined as follows:
• X: The input features to be fed into one of the three DNNRegressor interface methods (train, evaluate, and predict)
• y: The target values of X, which are optional and will not be supplied to the predict call
• num_epochs: An optional parameter. An epoch occurs when the algorithm executes over the entire dataset one time.
• shuffle: An optional parameter, specifies whether to randomly select a batch (subset) of the dataset each time the algorithm executes
• batch_size: The number of samples to include each time the algorithm executes
With our input function defined we can now train our neural network on our training dataset. For readers who are familiar with the TensorFlow high-level API you will probably notice that I am being a
little unconventional about how I am training my model. That is, at least from the perspective of the current tutorials on the TensorFlow website and other tutorials on the web.
Normally you will see something like the following when one trains one of these high level API pre-canned models.
regressor.train(input_fn=input_fn(training_data, num_epochs=None, shuffle=True), steps=some_large_number)
lots of log info
Then the author will jump right into demonstrating the evaluate(...) function and barely hint at describing what it does or why this line of code exists.
regressor.evaluate(input_fn=input_fn(eval_data, num_epochs=1, shuffle=False), steps=1)
less log info
And after this they'll jump straight into executing the predict(...) function assuming all is perfect with the trained model.
predictions = regressor.predict(input_fn=input_fn(pred_data, num_epochs=1, shuffle=False), steps=1)
For the ML newcomer reading this type of a tutorial I cringe. There is so much more thought that goes into those three lines of code that warrants more attention. This, I feel, is the only downside
to having a high level API - it becomes very easy to throw together a model without understanding the key points. I hope to provide a reasonable explanation of how to train and evaluate this neural
network in a way that will minimize the risk of dramatically under fitting or overfitting this model to the training data.
So, without further delay let me define a simple training loop to train the model on the training data and evaluate it periodically on the evaluation data.
evaluations = []
STEPS = 400
for i in range(100):
regressor.train(input_fn=wx_input_fn(X_train, y=y_train), steps=STEPS)
INFO:tensorflow:Create CheckpointSaverHook.
INFO:tensorflow:Saving checkpoints for 1 into tf_wx_model/model.ckpt.
INFO:tensorflow:step = 1, loss = 1.11335e+07
INFO:tensorflow:global_step/sec: 75.7886
INFO:tensorflow:step = 101, loss = 36981.3 (1.321 sec)
INFO:tensorflow:global_step/sec: 85.0322
... A WHOLE LOT OF LOG OUTPUT ...
INFO:tensorflow:step = 39901, loss = 5205.02 (1.233 sec)
INFO:tensorflow:Saving checkpoints for 40000 into tf_wx_model/model.ckpt.
INFO:tensorflow:Loss for final step: 4557.79.
INFO:tensorflow:Starting evaluation at 2017-12-05-13:48:43
INFO:tensorflow:Restoring parameters from tf_wx_model/model.ckpt-40000
INFO:tensorflow:Evaluation [1/1]
INFO:tensorflow:Finished evaluation at 2017-12-05-13:48:43
INFO:tensorflow:Saving dict for global step 40000: average_loss = 10.2416, global_step = 40000, loss = 1024.16
INFO:tensorflow:Starting evaluation at 2017-12-05-13:48:43
INFO:tensorflow:Restoring parameters from tf_wx_model/model.ckpt-40000
INFO:tensorflow:Finished evaluation at 2017-12-05-13:48:43
INFO:tensorflow:Saving dict for global step 40000: average_loss = 10.2416, global_step = 40000, loss = 1024.16
The above loop iterates 100 times. In the body of the loop I call the train(...) method of the regressor object, passing it my reusable wx_input_fn(...) which is in turn passed my training feature
set and targets. I purposefully left the default parameters num_epochs equal to None, which basically says "I don't care how many times you pass over the training set just keep going training the
algorithm against each default batch_size of 400" (roughly half the size of the training set). I also left the shuffle parameter equal to its default value of True so that while training, the data is
selected randomly to avoid any sequential relationships in the data. The final parameter to the train(...) method is steps which I set to 400, which means the training set will be batched 400 times
per loop.
This gives me a good time to explain in a more concrete numerical way what the meaning of an epoch is. Recall from the bullets above that an epoch occurs when all the records of a training set is
passed through the neural network to train exactly one time. So, if we have about 800 (797 to be exact) records in our training set and each batch selects 400, then for every two batches we have
accomplished one epoch. Thus, if we iterate over the over the training set for 100 iterations of 400 steps each with a batch size of 400 (one half an epoch per batch) we get:
(100 x 400 / 2) = 20,000 epochs
Now you might be wondering why I executed and evaluate(...) method for each iteration of the loop and captured its output in a list. First let me explain what happens each time time the train(...)
method is fired. It selects a random batch of training records and pushes them through the network until a prediction is made, and for each record the loss function is calculated. Then based off the
loss calculated the weights are adjusted according to the optimizer's logic, which does a pretty good job at making adjustments towards the direction that reduces the overall loss for the next
iteration. These loss values, in general as long as the learning rate is small enough, decline over time with each iteration or step.
However, after a certain amount of these learning iterations the weights start to be influenced not just by the overall trends in the data but, also by the uninformative noise inherit in virtually
all real data. At this point the network is over-influenced by the idiosyncrasies of the training data and becomes unable to generalize predictions about the overall population of data (ie, data it
has not yet seen).
This relates to the issue I mentioned earlier where many other tutorials on the high-level TensorFlow API have fallen short. It is quite important to break periodically during training and evaluate
how the model is generalizing to an evaluation, or validation, dataset. Lets take a moment to look at what the evaluate(...) function returns by looking at the first loop iteration's evaluation
{'average_loss': 31.116383, 'global_step': 400, 'loss': 3111.6382}
As you can see it outputs the average loss (Mean Squared Error) and the total loss (Sum of Squared Errors) for the step in training which for this one is the 400th step. What you will normally see in
a highly trained network is a trend where both the training and evaluation losses more or less constantly decline in parallel. However, in an overfitted model at some point in time, actually at the
point where over fitting starts to occur, the validation training set will cease to see reductions in the output of its evaluate(...) method. This is where you want to stop further training the
model, preferably right before that change occurs.
Now that we have a collection of evaluations for each of the iterations let us plot them as a function of training steps to ensure we have not over-trained our model. To do so I will use a simple
scatter plot from matplotlib's pyplot module.
import matplotlib.pyplot as plt
%matplotlib inline
# manually set the parameters of the figure to and appropriate size
plt.rcParams['figure.figsize'] = [14, 10]
loss_values = [ev['loss'] for ev in evaluations]
training_steps = [ev['global_step'] for ev in evaluations]
plt.scatter(x=training_steps, y=loss_values)
plt.xlabel('Training steps (Epochs = steps / 2)')
plt.ylabel('Loss (SSE)')
Cool! From the chart above it looks like after all those iterations I have not overfitted the model because the evaluation losses never exhibit a significant change in direction toward an increasing
value. Now I can safely move on to making predictions based off my remaining test dataset and assess how the model does as predicting mean weather temperatures.
Similar to the other two regressor method I have demonstrated, the predict(...) method requires an input_fn which I will pass in using the reusable wx_input_fn(...), handing it the test dataset,
specifying the num_epochs to be one and shuffle to be false so that it is sequentially feeding all the data to test against.
Next, I do some formatting of the iterable of dicts that are returned from the predict(...) method so that I have a numpy array of predictions. I then use the array of predictions with the sklearn
methods explained_variance_score(...), mean_absolute_error(...), and median_absolute_error(...) to measure how well the predictions fared in relation to the known targets y_test. This tells the
developer what the predictive capabilities of the model are.
pred = regressor.predict(input_fn=wx_input_fn(X_test,
predictions = np.array([p['predictions'][0] for p in pred])
print("The Explained Variance: %.2f" % explained_variance_score(
y_test, predictions))
print("The Mean Absolute Error: %.2f degrees Celcius" % mean_absolute_error(
y_test, predictions))
print("The Median Absolute Error: %.2f degrees Celcius" % median_absolute_error(
y_test, predictions))
INFO:tensorflow:Restoring parameters from tf_wx_model/model.ckpt-40000
The Explained Variance: 0.88
The Mean Absolute Error: 3.11 degrees Celcius
The Median Absolute Error: 2.51 degrees Celcius
I have used the same metrics as the previous article covering the Linear Regression technique so that we can not only evaluate this model, but we can also compare them. As you can see the two models
performed quite similarly with the more simple Linear Regression model being slightly better. However, an astute practitioner would certainly run several experiments varying the hyper-parameters
(learning rate, width, and depth) of this neural network to fine tune it a bit, but in general this is probably pretty close to the optimal model.
This brings up a point worth mentioning, it is rarely the case, and definitely not advisable, to simply rely on one model or the most recent hot topic in the machine learning community. No two
datasets are identical and no one model is king. The only way to determine the best model is to actually try them out. Then once you have identified the best model there are other trade-offs to
account for, such as interpretability.
This article has demonstrated how to use the TensorFlow high-level API for the pre-canned Estimator subclass DNNRegressor. Along the way I have described, in a general sense, the theory of neural
networks, how they are trained, and the importance of being cognizant of the dangers of overfitting a model in the process.
To demonstrate this process of building neural networks I have built a model that is capable of predicting the mean temperature for the next day based off numerical features collected in the first
article of this series. That being said, I would like to take a moment to clarify my intentions for this series. My primary objective has been not to actually build state of the art forecasting
models in either the Linear Regression article or the current one on neural networks, but my goals have been to accomplish the following:
1. Demonstrate the general process for undertaking an analytics (machine learning, data science, whatever...) project from data collection, data processing, exploratory data analysis, model
selection, model building, and model evaluation.
2. Demonstrate how to select meaningful features that do not violate key assumptions of the Linear Regression technique using two popular Python libraries, StatsModels and Scikit Learn.
3. Demonstrate how to use the high level TensorFlow API and give some intuition into what is happening under all those layers of abstraction.
4. Discuss the issues associated with over fitting a model.
5. Explain the importance of experimenting with more than one model type to best solve a problem.
Thank you for reading. I hope you enjoyed this series as much as I did and, as always I welcome comments and criticism. | {"url":"https://stackabuse.com/using-machine-learning-to-predict-the-weather-part-3/","timestamp":"2024-11-06T08:12:44Z","content_type":"text/html","content_length":"175866","record_id":"<urn:uuid:4e08551b-d0cc-4989-9acf-7b6a6f8333c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00105.warc.gz"} |
Binary Save System
Hi everyone! Here's a binary save system so you can turn the 64 cartdata values into 2048 boolean values, or any other data you want by converting it to binary. I had a lot of trouble understanding
other binary save system information I found for Pico-8 so I think I've done a good job here of making the code understandable.
I've also written a full article to accompany this cartridge on my website: https://ultiman3rd.wordpress.com/2018/02/01/pico-8-binary-save-system/
I hope you all find this useful!
I figured I might as well post the article text here too:
Pico-8 has a simple save system which allows you to set a cartridge ID:
Then store/read 64 number values, each 32 bits:
dset(index, value)
var = dget(index)
This works well enough if all you want to store is the level a player has reached and a high score. But what if you want to store more?
The simplest way to expand this storage is by peeking and poking the memory directly. After running cartdata as above you can access your saved data in memory from 0x5e00 to 0x5eff
value = peek(0x5e00)
poke(0x5e00, value)
Peek and poke read and write a single byte (8 bits), so by using these you can quadruple the number of values you can store! However these values are limited to 8 bit integers. You can mix and match
32-bit and 8-bit numbers by also utilizing peek4 and poke4 which are the same as peek and poke but use 4 bytes each instead of 1.
However, what if 256 8-bit numbers aren’t enough for you? What if, hypothetically, you were working on a Pokemon-inspired monster-catching RPG, having to store the level, selected moves and
experience for each mon in your party, and keeping track of the players items, having a customizable character and multiple save slots? Well then you’d probably need a binary save system.
The binary save system I’ve come up with works like this:
• Start with a binary table of 2048 boolean (true/false) values.
• Convert various kinds of data to binary and store it in the table.
• Convert the table to 8-bit chunks.
• Save/load these 8-bit chunks with peek/poke.
With the above system you can store any kind of data you want, as long as you can convert it to and from binary. Booleans are simple; you set a value in the binary table:
Now the first thing to do is to figure out how we can take our binary table and save it into the cartdata, and load that cartdata back into the binary table. Let’s start with saving.
function commit_bintable()
for i=0,127 do
poke(0x5e00+i, get_poker(i))
This function is pretty simple. It iterates 128 times and pokes 128 “pokers” into the cartdata. The tougher part is creating those pokers. Here’s the code:
function get_poker(_i)
local _poker=0
for n=0,7 do
if(bintable[_i+n]) _poker += 2^n
return _poker
First we take our _i value (0-127) which is the poker index and convert it to a starting position in the binary table. Then we loop over 8 bits in the table to calculate _poker. Starting from bit 1
and going to bit 8 in the table, each bit represents 2^n. So to switch the nth bit from 0 to 1 in our integer poker we add 2^n to it. This results in some meaningless number whose bits each
correspond to the 8 bits in this section of the binary table.
Binary: 0 0 0 0 0 0 0 0
Represent: 1 2 4 8 16 32 64 128
By adding any of those 2^n numbers we can switch a 0 bit to a 1.
And that’s saving the binary table done! On to loading it:
function load_bintable()
for i=0,127 do
local _poker=peek(0x5e00+i)
for j=0,7 do
bintable[i*8+1+j] = get_bit(_poker,j)
We iterate over our 128 “pokers” – though we’re peeking them this time – and extract each of their 8 bits into the binary table. The key to all this is the get_bit function. So here it is:
function get_bit(_value,_n)
return flr(shr(_value,_n))%2 == 1
Let’s break that down. We take our _value which in this case will be an 8-bit integer. We shift its bits _n spaces to the right, remove everything after the decimal point and check whether it’s even
or odd. Sound a bit complicated? Let’s take a look at what’s happening in binary. Say we’ve retrieved a poker and it’s 203. In binary:
Then we shift the bits to the right 1 space:
We no longer have an integer so we floor it:
Then we use the % (modulus) operator to check if 2 divides into it evenly or if there is a remainder.
If the rightmost bit is 1 then the number is odd and our value%2 will return 1. If the rightmost bit is 0 then the number is even and value%2 will return 0. To convert this to a boolean value (true/
false) we compare it with ==1 and return the result. Just like that we can take our 8-bit pokers and extract each bit into our binary table!
Now that we’ve got a binary table being saved and loaded let’s fill it with useful information! The most important thing to start with is integers. I want each integer to take up the minimum number
of bits necessary for that particular piece of data so I’ve created these functions:
function numtobintable(_value,_dest,_nbits)
for i=0,_nbits-1 do
function bitstonum(_addr,_nbits)
local _p=0
for i=0,_nbits-1 do
return _p
In numtobintable we take an integer value, destination in the binary table and a number of bits to take up. All we have to do is iterate over each bit in the number with get_bit and put those bits
into the binary table. Easy!
Loading the integers is a little more difficult but it should be familiar. We start with the number 0 then iterate over each bit, adding 2^i whenever the bit is true and not adding anything for
false. Pretty simple really, aye?
But hold on, how do you know how many bits to use for each integer? You’ll have to figure out what the maximum value is you might be saving for each variable, then find the lowest power of 2 number
it fits into, minus 1. With a single bit you can store 2 values, with 2 bits you can store 4, then 8, 16, etc. Say you want to save a Notemon’s level for example. Notemon can be any level from 1 to
50. That’s 50 values, so the lowest power of 2 greater than or equal to that is 64. 64 is 2^7. Minus 1 equals 6, so we need 6 bits to store values from 1-50. To verify, our 6 bits will represent
multiples of 1, 2, 4, 8, 16, 32. If all of those bits were true they’d add up to 63. Including 0 as a possible value that means 64 total values so we’ve found the minimum number of bits necessary to
save this piece of information!
So with that we can store useful information in our binary table. That really is all you need to make a pretty comprehensive save/load system! One last thing before I sign off from my first article
though: strings! I’m not currently using this feature in Notemon, but at one point I wanted players to be able to name their mon so I made the following code:
function save_string(_str,_addr,_len)
for i=0,_len-1 do
for i=1,#_str do
function load_string(_addr,_l)
local _s=""
for i=0,_l-1 do
local _c=inttochar[bitstonum(_addr+i*5,5)]
if(_c=="_"or _c==nil)break
return _s
chartoint and inttochar aren’t even functions – they’re tables! Since Pico-8 doesn’t have any ord() function to convert characters to and from integers I just made a couple of tables to go back and
forth. Simple stuff.
savestring has two for loops which might seem odd at first. The first loop starts at the address and basically clears all the characters to the “” character which I chose as a code meaning “end
string here”. It could be anything, or you could have fixed length strings and not need this first loop but Notemon names needed to be variable in length.
Next we loop over each character in the string and add it to the binary table. See all that *5 stuff going on? That’s because I allowed for 27 characters. As above I calculated that would require 5
bits to store.
In loadstring we start with an empty string then copy each character from the binary table into the string until we hit the “” character.
That’s all, folks! I really hope this is helpful to the Pico-8 community. I have previously searched for a Pico-8 binary save system and found something on the BBS but it was esoteric. Instead I
muddled my way until I finally understood binary and how to use it in Pico-8. The cart above contains all the code, along with some bonus code for the naming screen I made for Notemon and the code to
generate that matrix background you see in the cart image 🙂
Thanks for reading my post. Expect more Pico-8 tech posts as I continue creating Notemon!
Amazing effort!
I've had this bookmarked since it was first posted - I'm sure it'll come in handy one day.
How would I save my entire map? or, at least everything the camera is showing?
@Squidkidd Maybe you could just save the position of the character at the map. Each tile of the map is a coordinate, and the tile your character is on represents being at that coordinate.
@aKidCalledAris Good idea, but in my game, I am changing the map so that would not work. also, I don't have a player. maybe it will help if I post the link. What i want to do is save the chickens
(which i put on the map using mset()) here is the game https://squidkidd.itch.io/merge-chickens
My mind is blowing right now. soo... what function do I call to save and load?
Why in commit_bintable(), the i goes from 0 to 127?
Shouldn't it go from 0 to 255?
[Please log in to post a comment] | {"url":"https://www.lexaloffle.com/bbs/?tid=30711","timestamp":"2024-11-09T04:36:39Z","content_type":"text/html","content_length":"172762","record_id":"<urn:uuid:93d5987b-0d35-4d35-a31a-7ef24ca2244e>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00244.warc.gz"} |
Scaffolding Proof to Cultivate Intellectual Need in Geometry
This year I'm teaching proof much more the way I have taught writing in previous years in English programs, and I have to say that the scaffolding and assessment techniques I learned as a part of a
very high-performing ELA/Writing program are helping me (and benefiting my Geo students) a lot this year.
I should qualify that my school places an extremely high value on proof skills in our math sequence. Geometry is only the first place where our students are required to use the techniques of formal
proof in our math courses. So I feel a strong duty to help my regular mortal Geometry students to leverage their strengths wherever possible in my classes. Since a huge number of my students are
outstanding writers, it has made a world of difference to use techniques that they "get" about learning and growing as writers and apply them to learning and growing as mathematicians.
We are still in the very early stages of doing proofs, but the very first thing I have upped is the frequency of proof. We now do at least one proof a day in my Geometry classes; however, because of
the increased frequency, we are doing them in ways that are scaffolded to promote fluency.
Here's the thing: the hardest thing about teaching proof in Geometry, in my opinion, is to constantly make sure that it is the STUDENTS who are doing the proving of essential theorems.
Most of the textbooks I have seen tend to scaffold proof by giving students the sequence of "Statements" and asking them to provide the "Reasons."
While this seems
to me at times and for many students (especially during the early stages), it also seems dramatically
because it removes the burden of sequencing and identifying logical dependencies and interdependencies between and among "Statements."
So my new daily scaffolding technique for October takes a page from Malcolm Swan and Guershon Harel (by way of Dan Meyer).
I give them the diagram, the Givens, the Prove statement, and a batch of unsorted, tiny Statement cards to cut out.
Every day they have to discuss and sequence the statements, and then justify each statement as a step in their proof.
This has led to some amazing discussions of argumentation and logical dependencies.
An example of what I give them (copied 2-UP to be chopped into two handouts, one per student) can be found here:
Sample proof to be sequenced & justified
Now students are starting to understand why congruent triangles are so useful and how they enable us to make use of their corresponding parts! The conversations about intellectual need have been
I am grateful to Dan Meyer for being so darned persistent and for pounding away on the notion of developing intellectual need in his work!
4 comments:
1. Michael, I confess that I haven't read it, and so I am curious to hear what you value about it and why. I have read a bit about it and done some investigation of this kind of approach to
geometric inquiry, but I've run into a few of problems so far. The first one is that my school is so far from 1-1 with the needed technologies it has been difficult to explore. The second is that
I have two blind students (plus some visually impaired) who would be completely unable to access GSP given the state of available tactile display technology (though we are cheering on the U of
Michigan and their effort to come up with a full-page refreshable tactile display!). Finally, there seems to be a difference between understanding the geometry interactively (as through GSP or
Geogebra) and affirmatively constructing the written argument/proof that takes advantage of that interactive understanding.
Can you say a little more about your thoughts and/or experience with the de Villiers book/method?
- Elizabeth (@cheesemonkeysf)
2. Heya Cheese, this seems like really useful scaffolding for proof but I don't see the need angle. Based on this treatment, how do you hope students would respond to the question, "What's the point
of proof?"
1. Hi Dan, That's a reasonable question, but it's a bit more meta than we are working with right now. We're operating in a state of 'suspended disbelief' around the need for proof itself. My
students here accept the a priori need for argumentation or proof. But they have trouble sometimes seeing why you would need a particular new theorem (for example). So the essential question
might be, what would enable you to make a connection between congruent triangles, say, and parallel lines?
We are working at a more modest, micro-level than justifying proof itself!
- Elizabeth (@cheesemonkeysf)
2. Gotcha gotcha gotcha. This makes sense to me now:
"Now students are starting to understand why congruent triangles are so useful and how they enable us to make use of their corresponding parts!" | {"url":"http://cheesemonkeysf.blogspot.com/2016/10/scaffolding-proof-to-cultivate.html","timestamp":"2024-11-07T06:35:48Z","content_type":"text/html","content_length":"141270","record_id":"<urn:uuid:497eb639-708a-4c70-aa17-4ac47d28493f>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00862.warc.gz"} |
Captum · Model Interpretability for PyTorch
To compute the integrated gradients, we use the attribute method of the IntegratedGradients object. The method takes tensor(s) of input examples (matching the forward function of the model), and
returns the input attributions for the given examples. For a network with multiple outputs, a target index must also be provided, defining the index of the output for which gradients are computed.
For this example, we provide target = 1, corresponding to survival.
The input tensor provided should require grad, so we call requires_grad_ on the tensor. The attribute method also takes a baseline, which is the starting point from which gradients are integrated.
The default value is just the 0 tensor, which is a reasonable baseline / default for this task.
The returned values of the attribute method are the attributions, which match the size of the given inputs, and delta, which approximates the error between the approximated integral and true | {"url":"https://captum.ai/tutorials/Titanic_Basic_Interpret","timestamp":"2024-11-06T01:38:33Z","content_type":"text/html","content_length":"180212","record_id":"<urn:uuid:544857e4-1e8e-455d-939b-f864334e4a00>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00210.warc.gz"} |
Fractions on a Number Line | sofatutor.com
Fractions on a Number Line
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Basics on the topic Fractions on a Number Line
How to Do Fractions on a Number Line
Imagine you are on a treasure hunt, but to open the secret door to the hidden treasure you must complete two fraction number line puzzles. In order to complete the puzzles, you must first learn about
plotting benchmark fractions on a number line 0 to 1. Not to worry – the following text will teach you how to become the treasure master!
How to Put Fractions on a Number Line
To represent fractions on a number line, you can follow some simple steps. Fraction bars can help when showing unit fractions on a number line. The infographic below shows the different steps to
represent fractions on a number line:
Step # What to do
1 Identify the denominator
2 Divide the number line into equal parts
3 Count forward from zero the numerator
4 Plot the fraction on the number line
Plotting Fractions on a Number Line – Examples
Below are two examples of labelling fractions on a number line. The steps above can be followed to help plot fractions on a number line.
Show Fractions on a Number Line – First Example
The fraction is $\frac{1}{2}$. We see the denominator, the bottom number, is two. Using a fraction bar to help us, we can divide the number line into two equal parts.
Now we look at the numerator, the top number. This is a one, so we can count forward one part from the zero.
We have located where $\frac{1}{2}$ is on the number line, so we can plot it at the correct location.
Show Fractions on a Number Line – Second Example
The fraction is $\frac{2}{3}$. We see the denominator, the bottom number, is three. Using a fraction bar to help us, we can divide the number line into three equal parts.
Now we look at the numerator, the top number. This is a two, so we can count forward two parts from the zero.
We have located where $\frac{2}{3}$ is on the number line, so we can plot it at the correct location.
It is treasure time! Do you understand fractions on a number line now? If you would like to practise some more, there are fractions on a number line practice problems and fractions on a number line
worksheets after the video.
Transcript Fractions on a Number Line
Axel and Tank find a treasure map that shows there's a secret door nearby. Let's join them to find out more! "Oh, look Tank!" "I think we found it!" "Hmm, but how does the door open?" "It looks like
we need to solve two puzzles by moving the sliders to represent Fractions on a Number Line." Remember, a fraction represents a PART of a WHOLE. A fraction bar like THIS can help when writing
fractions on a number line. A fraction bar represents the WHOLE, and each section is a PART of the whole. We can set up a number line from zero to one where the ONE represents our whole! Now we are
ready to represent one-third on a number line. First, "identify the denominator". Next, "divide the number line into equal parts" matching the denominator. Then "count the number of parts from zero"
shown in the numerator. Finally, "write the fraction on the number line". Now we know how to write fractions on a number line, let's try an example together before we help Axel and Tank! Here we have
the fraction one half. First we draw our number line. Now we look at the denominator and see we have two parts. We can use a half fraction bar to help us divide the number line into two equal parts
right HERE. Now we look at the numerator and see we have one part. We count one part from zero, and write one half HERE. Okay, let's help Axel and Tank crack the code!
What is your first step? Draw your number line, then identify the denominator. Can you identify the denominator? FOUR is the denominator. Do you remember what the next step is? Use a quarters, or
four parts, fraction bar to help you divide the number line into equal parts. What is your final step? You count from zero the number of parts you have as shown by the numerator. What is the
numerator? The numerator is one, so you count one part from zero. Finally, you write the fraction on the number line like THIS. Last, we need to show the fraction TWO-THIRDS on a number line. What is
your first step? Draw your number line, then identify the denominator. Can you identify the denominator? The denominator is three. What is your next step? Divide the number line into equal parts,
using a thirds fraction bar to help. Can you remember the final step? You count from zero the number of parts you have as shown by the numerator. What is the numerator? The numerator is two, so you
count two parts from zero. Finally, you write the fraction on the number line like THIS. Remember, to write fractions on a number line, first, "identify the denominator". Next, "divide the number
line into equal parts" matching the denominator. Then "count the number of parts from zero" shown in the numerator. Finally, "write the fraction on the number line". Yay, we helped Axel and Tank
solve the puzzles! "I wonder where this will take us." "Let's go this way!"
"Wait, that was it?" "But Axel, the treasure IS our home!"
Fractions on a Number Line exercise
Would you like to apply the knowledge you’ve learnt? You can review and practice it with the tasks for the video Fractions on a Number Line.
• Describe how to place a fraction on a number line.
To know how many equal parts to divide the number line into, what do we need to identify first?
After we divide the number line into equal parts, we can count the number of parts from zero.
1. Identify the denominator.
2. Divide the number line into equal parts.
3. Count the number of parts from zero.
4. Write the fraction on the number line.
• Fractions on a number line.
If we are placing $\frac{1}{3}$ on the number line, how many parts should the number line be divided into?
What is the numerator in $\frac{1}{3}$? Count this number of parts from 0.
Step 1: Identify the denominator, 3.
Step 2: Divide the number line into 3 equal parts.
Step 3: Count 1 part (the numerator) from 0.
Step 4: Place $\frac{1}{3}$ on the number line.
• Match the fraction to the number line.
How many parts is the number line divided into? This will match the denominator.
Count the number of parts from 0 shown in the numerator (top number), and place the fraction on the number line.
Step 1: Identify the denominator, 4.
Step 2: Divide the number line into 4 equal parts.
Step 3: Count 3 parts (the numerator) from 0.
Step 4: Place $\frac{3}{4}$ on the number line. ___________________________________________________________________________________________________ For the other fractions:
Step 1: Identify the denominator, 2.
Step 2: Divide the number line into 2 equal parts.
Step 3: Count 1 part (the numerator) from 0.
Step 4: Place $\frac{1}{2}$ on the number line.
Step 1: Identify the denominator, 3.
Step 2: Divide the number line into 3 equal parts.
Step 3: Count 3 parts (the numerator) from 0.
Step 4: Place $\frac{3}{3}$ on the number line.
Step 1: Identify the denominator, 4.
Step 2: Divide the number line into 4 equal parts.
Step 3: Count 1 part (the numerator) from 0.
Step 4: Place $\frac{1}{4}$ on the number line.
• Match the number line with the fraction.
How many parts is the number line divided into? This will match the denominator.
Count the number of jumps from zero. This will match the numerator.
Remember the denominator is the bottom number and the numerator is the top number.
Step 1: Identify the denominator, 5.
Step 2: Divide the number line into 5 equal parts.
Step 3: Count 2 parts (the numerator) from 0.
Step 4: Place $\frac{2}{5}$ on the number line. ___________________________________________________________________________________________________ For the other fractions:
Step 1: Identify the denominator, 4.
Step 2: Divide the number line into 4 equal parts.
Step 3: Count 1 part (the numerator) from 0.
Step 4: Place $\frac{1}{4}$ on the number line.
Step 1: Identify the denominator, 3.
Step 2: Divide the number line into 3 equal parts.
Step 3: Count 2 parts (the numerator) from 0.
Step 4: Place $\frac{2}{3}$ on the number line.
Step 1: Identify the denominator, 4.
Step 2: Divide the number line into 4 equal parts.
Step 3: Count 2 parts (the numerator) from 0.
Step 4: Place $\frac{2}{4}$ on the number line.
• Which number line shows $\frac{1}{2}$ placed correctly?
If we are placing $\frac{1}{2}$ on a number line, how many parts should the number line be divided into?
What is the numerator in $\frac{1}{2}$? Count this number of parts in from zero.
Remember, the numerator is the top number and the denominator is the bottom number.
Step 1: Identify the denominator, 2.
Step 2: Divide the number line into 2 equal parts.
Step 3: Count 1 part (the numerator) from 0.
Step 4: Place $\frac{1}{2}$ on the number line.
• Place the fractions on the number line.
Identify the denominator and divide the number line into equal parts matching the denominator. Then, count the number of parts from 0 shown in the numerator, and place the fraction on the number
Look at the fractions with the same denominator, but different numerators, and place them on the number line.
Look at the fractions with the same numerator, but different denominators. Which piece would be bigger, a piece divided into 4 parts or 3 parts? Whichever one you think is bigger will be closer
to 1.
Step 1: Identify the denominator, 2.
Step 2: Divide the number line into 2 equal parts.
Step 3: Count 1 part (the numerator) from 0.
Step 4: Place $\frac{1}{2}$ on the number line.
Step 1: Identify the denominator, 3.
Step 2: Divide the number line into 3 equal parts.
Step 3: Count 1 part (the numerator) from 0.
Step 4: Place $\frac{1}{3}$ on the number line.
Step 1: Identify the denominator, 3.
Step 2: Divide the number line into 3 equal parts.
Step 3: Count 3 parts (the numerator) from 0.
Step 4: Place $\frac{3}{3}$ on the number line.
Step 1: Identify the denominator, 4.
Step 2: Divide the number line into 4 equal parts.
Step 3: Count 1 part (the numerator) from 0.
Step 4: Place $\frac{1}{4}$ on the number line.
Step 1: Identify the denominator, 4.
Step 2: Divide the number line into 4 equal parts.
Step 3: Count 3 parts (the numerator) from 0.
Step 4: Place $\frac{3}{4}$ on the number line.
More videos in this topic Writing fractions | {"url":"https://www.sofatutor.co.uk/maths/videos/fractions-on-a-number-line","timestamp":"2024-11-06T13:40:54Z","content_type":"text/html","content_length":"151840","record_id":"<urn:uuid:4da8fee0-1715-4741-bb6a-bb2b4c4afd04>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00400.warc.gz"} |
Use exact comparision for doubles (!8) · Merge requests · Informatik 8 / CoPaR · GitLab
Use exact comparision for doubles
requested to merge
Comparision with (a-b < ɛ) is not transitive and thus doesn't induce an equality relation. Since the whole job of the algorithm is to partition according to an equality relation, this may be
Instead, to compare doubles a and b, we now round both values to floats and compare those exactly. This means that we get a proper equality relation, even if it fails to distinguish some states and
wrongly distinguishes others when compared to operations on true real numbers.
Merge request reports | {"url":"https://gitlab.cs.fau.de/i8/copar/-/merge_requests/8","timestamp":"2024-11-05T07:11:07Z","content_type":"text/html","content_length":"63060","record_id":"<urn:uuid:bf1ec518-a051-465f-8bea-2c28b9d417f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00129.warc.gz"} |
Function Description
Advanced | SIFREQ Computes the exact fundamental frequency of a signal.
Advanced | SIMAX1 Computes the absolute or algebraic maximum on a specific part of a signal.
Advanced | SIMAX2 Computes the absolute or algebraic maximum of a signal resulting from a mathematical operation between two signals.
Advanced | SIMAX3 Computes the absolute and algebraic maximum on a specific part of a signal of a phase to phase or a phase to neutral voltage or a current.
Advanced | SIHARMO Computes the harmonics of a signal.
Advanced | SIMXHART Computes the maximum of the amplitudes of a given harmonic.
/wiki/spaces/RD/pages/45893731Advanced | SIMXSOM1 Computes the absolute maximum of an integral, a mean or a RMS value on a specific part of a signal and in a window moving.
Advanced | SIMXSOM2 Computes the absolute maximum of an integral, a mean or a RMS value in a window moving over a signal resulting from a mathematical operation.
Advanced | SIDYN2 Computes the dynamic response of a control signal following a disturbance.
Advanced | SIMXSYMT Compute the absolute maximum amplitudes of the harmonic components in a window moving over time.
Advanced | SIPZ Compute the power, impedance, addition or subtraction of two signals for a given harmonic.
Advanced | SISEUIL Compute overshoot statistics of a signal using a threshold condition.
Advanced | SISYM3 Compute the symmetric components of three signals for a given harmonic.
Advanced | SITENV Compute the time in milliseconds during which the envelope of a signal overshoots a specific threshold value.
Advanced | SITOP Compute the operating time of a relay.
Advanced | SITOPREF Compute the operating time of a relay relative to the rise or fall of a reference signal.
Advanced | SITSEUIL Compute the time at which a threshold value is reached.
Advanced | SIGCMP Compare the signal with a reference. | {"url":"https://opal-rt.atlassian.net/wiki/pages/diffpagesbyversion.action?pageId=150242211&selectedPageVersions=3&selectedPageVersions=4","timestamp":"2024-11-03T04:20:47Z","content_type":"text/html","content_length":"64548","record_id":"<urn:uuid:965eb09b-e6e6-4f8d-9a76-5a1f2efd8afa>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00790.warc.gz"} |
A probability distribution on the real line is a mixture of ... - Ask Spacebar
A probability distribution on the real line is a mixture of two classes +1 and−1 with density N(1,2) (normal distribution with mean1 and variance 2) andN(3,1), with prior probabilities (probability
of eachclass) 0.4 and 0.6 respectively. What is the Bayes decision rule? Give anestimate for the Bayes risk.
Views: 0 Asked: 12-02 16:45:12
On this page you can find the answer to the question of the mathematics category, and also ask your own question
Other questions in category | {"url":"https://ask.spacebarclicker.org/question/24","timestamp":"2024-11-09T06:47:56Z","content_type":"text/html","content_length":"27254","record_id":"<urn:uuid:dee5b6bc-d96b-463d-a1ad-ba545029924a>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00877.warc.gz"} |
NCERT Solutions for Miscellaneous Exercise Chapter 13 Class 12 - Probability
It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and
they can be opportunities for growth.
Possible steps:
1. Re-evaluate Your Study Strategies:
□ Identify Weak Areas: Pinpoint the specific topics or concepts that caused difficulties.
□ Seek Clarification: Reach out to teachers, tutors, or online resources for additional explanations.
□ Practice Regularly: Consistent practice is key to mastering chemistry.
2. Consider Professional Help:
□ Tutoring: A tutor can provide personalized guidance and support.
□ Counseling: If you're feeling overwhelmed or unsure about your path, counseling can help.
3. Explore Alternative Options:
□ Retake the Exam: If you're confident in your ability to improve, consider retaking the chemistry compartment exam.
□ Change Course: If you're not interested in pursuing chemistry further, explore other academic options that align with your interests.
4. Focus on NEET 2025 Preparation:
□ Stay Dedicated: Continue your NEET preparation with renewed determination.
□ Utilize Resources: Make use of study materials, online courses, and mock tests.
5. Seek Support:
□ Talk to Friends and Family: Sharing your feelings can provide comfort and encouragement.
□ Join Study Groups: Collaborating with peers can create a supportive learning environment.
Remember: This is a temporary setback. With the right approach and perseverance, you can overcome this challenge and achieve your goals.
I hope this information helps you.
Kanishka kaushikii 17 September,2024
Hello there! Thanks for reaching out to us at Careers360.
Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.
Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help
you practice!
Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're
great for getting a feel for the types of questions that might come up.
If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.
Let me know if you need any other tips for your math prep. Good luck with your studies!
Jefferson 25 September,2024
8. Where can I get NCERT solutions ?
Here you can get NCERT Solutions.
7. Give an example of impossible event ?
The event of getting number 0 we roll a die is an example of an impossible event.
6. Give an example of certain event .
The event of getting a number less than 7 when we roll a die is an example of a certain event.
5. What is the probability of getting number less than 7 when we roll a die ?
The probability of a number less than 7 when we roll a die is one.
4. What is the probability of getting 0 when we roll a die ?
The probability of getting 0 when we roll a die is zero.
3. What is the probability of an impossible event ?
The probability of an impossible event is 0.
2. What are the topics in miscellaneous exercise probability Class 12 ?
Miscellaneous exercise is consists of different questions from all the topics of probability Class 12 Maths.
1. Do questions from the miscellaneous exercises comes in 12th board exams?
More than 95% of questions in the board's exam are not asked from miscellaneous exercises but it is very important for competitive exams.
Age: As of the last registration date, you must be between the ages of 16 and 40.
Qualification: You must have graduated from an accredited board or at least passed the tenth grade. Higher qualifications are also accepted, such as a diploma, postgraduate degree, graduation, or
11th or 12th grade.
How to Apply:
Get the Medhavi app by visiting the Google Play Store.
Register: In the app, create an account.
Examine Notification: Examine the comprehensive notification on the scholarship examination.
Sign up to Take the Test: Finish the app's registration process.
Examine: The Medhavi app allows you to take the exam from the comfort of your home.
Get Results: In just two days, the results are made public.
Verification of Documents: Provide the required paperwork and bank account information for validation.
Get Scholarship: Following a successful verification process, the scholarship will be given. You need to have at least passed the 10th grade/matriculation scholarship amount will be transferred
directly to your bank account.
Scholarship Details:
Type A: For candidates scoring 60% or above in the exam.
Type B: For candidates scoring between 50% and 60%.
Type C: For candidates scoring between 40% and 50%.
Cash Scholarship:
Scholarships can range from Rs. 2,000 to Rs. 18,000 per month, depending on the marks obtained and the type of scholarship exam (SAKSHAM, SWABHIMAN, SAMADHAN, etc.).
Since you already have a 12th grade qualification with 84%, you meet the qualification criteria and are eligible to apply for the Medhavi Scholarship exam. Make sure to prepare well for the exam to
maximize your chances of receiving a higher scholarship.
Hope you find this useful!
Yuvan 30 August,2024
hello mahima,
If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should
display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.
hope this helps.
Ashvadha 12 July,2024
Hello Akash,
If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and
after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.
You can get the Previous Year Questions (PYQs) on the official website of the respective board.
I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.
Thank you and wishing you all the best for your bright future.
kumardurgesh1802 10 July,2024 | {"url":"https://school.careers360.com/ncert/ncert-solutions-for-miscellaneous-exercise-chapter-13-class-12-probability","timestamp":"2024-11-12T01:45:43Z","content_type":"text/html","content_length":"841161","record_id":"<urn:uuid:06f2e10e-dcc0-4fd4-a8bb-e0fba87fa0a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00044.warc.gz"} |
Smooth cohomology of $ C^* $-algebras
Smooth cohomology of C^* -algebras
We define a notion of smooth cohomology for C^* -algebras which admit a faithful trace. We show that if \A\subseteq B(\h) is a C^* -algebra with a faithful normal trace \tau on the ultra-weak closure
\bar{\A} of \mathcal{A} , and $... Show more | {"url":"https://synthical.com/article/Smooth-cohomology-of-%24-C%5E*-%24-algebras-043e873e-ffb8-11ed-9b54-72eb57fa10b3?","timestamp":"2024-11-03T06:46:17Z","content_type":"text/html","content_length":"55147","record_id":"<urn:uuid:d5001d8c-4d29-4eaa-a3a0-7739ce1ac728>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00563.warc.gz"} |
Release notes for SCIP 2.1
SCIP 2.1.2
Performance improvements
• fixed performance issue in debug mode, where SCIPvarGetLPSol_rec() calculated a value to often, which in the end lead to exponential growth in running time
• force cuts from linearizations of convex constraint in NLP relax solution into LP, thus allowing faster proving of optimality for convex NLPs
Fixed bugs
• fixed bug in varAddTransitiveImplic() in var.c, when adding implications on special aggregated, namely negated, variables
• fixed issue if a primal solution leads to a cutoff of the current focus node
• fix compilation issues with zlib 1.2.6
• fixed bug in SCIPsolveKnapsackExactly(), trying to allocate too much memory which led to an overflow and later to a segmentation fault
• fixed bug in sepa_rapidlearning, carrying on the optimization process, when already solved
• Heuristics:
□ fixed bug in heur_undercover.c, where a variable with fixed bounds but not of status SCIP_VARSTATUS_FIXED was wrongly handled
□ fixed bug in heur_oneopt.c which forgot to check LP rows if local rows are present
• Constraints:
□ fixed bug in SCIPsolveKnapsackExactly()
□ fixed bug in cons_quadratic where bounds on activity of quadratic term were not always invalidated when quadratic variables were removed
□ fixed bug in cons.c, where after a restart the arrays for all initial constraints were corrected in the initsol process instead of the initpre process, this was to late because you might
change the status in presolving which lead to an assert()
□ fixed bug in NLP representation of abspower constraints handling (x+a)^2 with nonzero a
□ fixed bug parsing an and-constraint in cip format
□ fixed bug in cons_setppc, did not handle new constraints with inactive variables
□ fixed bug in cons_xor.c which did not copy the artificial integer variable (used for the lp relaxation)
SCIP 2.1.1
• the pseudo objective propagator can be forced to propagate if active pricers are present; this can be done if for all (known or unknown) variables follows that: they have positive (negative)
objective coefficient and the global lower (upper) bound is zero.
Performance improvements
• improvements in undercover heuristic
• improve SCIPintervalSolveBivariateQuadExpressionAllScalar() for ax=0 case if x has 0 in the interval for the linear coef.
• better domain propagation for quadratic constraints that consist of non-overlapping bilinear terms only
• ensure that a fixing of a variable in an abspower constraint is propagated to a fixing of the other variable
• improvements in undercover heuristic, e.g., bound disjunction constraints are considered when setting up the covering problem
Interface changes
Changed parameters
• changed parameter propagating/pseudoobj/maxcands to propagating/pseudoobj/minuseless (see prop_pseudoobj.c) due to revision of the pseudo objective propagator
New parameters
• added parameters heuristics/undercover/coverbd and heuristics/undercover/fixingorder
Fixed bugs
• fixed numeric issue in aggregations
• fixed pseudo cost computation
• fixed bug with setting type of slack variables to be implicitly integral
• fixed bug when copying problem data in c++ case returned with the result SCIP_DIDNOTRUN
• fixed computation of counter which state the changes since the last call of a presolver
• fixed handling of unbounded solutions, including double-checking their feasibility and that the primal ray is a valid unboundedness proof and reoptimizing the LP with modified settings if the
solution is not feasible
• fixed compilation issues with negate() function in intervalarith.c on exotic platforms
• fixed bug in SCIPsortedvecDelPos...() templates
• pseudo objective propagator does not propagate it active pricers are present
• fixed bug in heur_shiftandpropagate.c concerning the treatment of unbounded variables
• workaround for trying to add variable bounds with to small coefficients
• Reading and Writing:
□ gams writer now also substitutes $-sign from variable/equation names
□ fixed bug in reader_mps.c: INTEND marker is now also written, if COLUMNS section ends with non-continous variables
□ fixed bug in flatzinc reader w.r.t. boolean expressions
• Constraints:
□ fixed constraint flags evaluation within the ZIMPL reader (reader_zpl.c)
□ fixed bug in SCIPmakeIndicatorFeasible() in cons_indicator.c
□ fixed bug with conflict clause modification in cons_indicator
□ fixed bug in cons_bounddisjunction with uninitialized return values
□ fixed bug in cons_orbitope with calling conflict analysis
□ fixed bug in nlpi_oracle w.r.t. changing linear coefs in a NLP constraint
SCIP 2.1.0
• New original solution storage capability, which allows transfering solutions between SCIP runs
• SCIP-CPX is now threadsafe
• comparison of solutions now also works for original solutions
• can now compute the relative interior point of the current LP
• interval arithmetics for power, log, exp, bivariate quadratic expressions should be rounding safe now
• LP iterations in resolving calls can now be limited w.r.t. the average number of LP iterations in previous calls (after the root node); this is currently only done for the initial LP solve at a
node and the corresponding parameter resolveiterfac is set to -1 (no limit) per default
• it is now possible in SCIP_STAGE_TRANSFORMED to call SCIPaddVarLocks() (i.e. to lock variables in initialization methods)
• changed computation of optimality gap which is now done in the same way as described in the MIPLIB 2010 paper: the gap is 0, if primalbound (pb) and dualbound (db) are equal (within tolerances),
it is infinity if pb and db have opposite signs and (this changed), if both have the same sign, the difference between pb and db is devided by the minimum of the absolute values of pb and db
(instead of always the dual bound)
• functionality to use the bound flipping ratio test of SoPlex is available (requires at least version 1.5.0.7)
• there exists now a solution candidate store for the original problem; during transformation these solutions are tried; during free the transformed problem the best feasible solution of the
transformed problem are copied to the solution candidate store of the original problem; this useful if you solve several problems iteratively, solutions get now carried over automatically.
• reworked concept of lazy bounds: they can now also be used for problems where constraints and objective together ensure the bounds; to allow this also for diving heuristics that might change the
objective and thus destroy this property, lazy bounds are explicitly put into the LP during diving and removed afterwards
• SCIP_HASHMAP now works also without block memory
• The variable deletion event is now a variable specific event and not global, anymore.
• All timing flags are now defined type_timing.h.
• all C template files are now called <plugintype>_xyz.{c,h} instead of <plugintype>_xxx.{c,h}
• Separators and Cuts:
□ reorganized computation of scores in cut filtering: instead of the computation at the time of addition, scores are now only computed w.r.t. the current LP solution and when cut filtering is
performed; one can now fill the cut storage with cuts that were separated for different solutions
□ New separator for close cuts and a new function to compute relative interior points of the LP
□ added first version of sepa_closecuts.{c,h} to separate cuts w.r.t. a point that is closer to the integral polyhedron
• Constraints:
□ implement possibility to force a restart in cons_indicator if enough indicator variables have been fixed
□ the xor constraint handler can now parse its constraints
□ the bounddisjunction constraint handler can now parse its constraints
□ the knapsack, setppc and soc constraint handler can now parse their constraints
□ the varbound constraint handler can now parse its constraints
□ added beta version of variable deletion: for branch-and-price application, variables can now be completely deleted from the problem; variables that are deletable have to be marked with
SCIPvarMarkDeletable(), constraint handlers can implement the new SCIP_DECL_DELVARS callback that should remove variables from the constraints; at the moment, only the linear, the setppc and
the knapsack constraint handler support this callback; furthermore, when using this feature, all used plugins have to capture and release variables they store in their data, this is currently
only done for the aforementioned constraint handlers as well as the and, the varbound and the logicor constraint handler; for more details about this feature, see the FAQ
□ added pseudoboolean constraint handler (cons_pseudoboolean.{c,h})
□ added first version of cons_disjunction.{c,h} which allows a disjunction of constraints
□ added constraint handler for (absolute) power constraints (cons_abspower.{c,h}) to handle equations like z = sign(x)abs(x)^n, n > 1
• Heuristics:
□ new heuristic vbounds which use the variables lower and upper bounds to fix variable and performs a neighborhood search
□ added vbound heuristic (heur_vbounds.{c,h})
□ added clique heuristic (heur_clique.{c,h})
• Reading and Writing:
□ added writing for wbo files
□ added writing for pip files (linear, quadratic, polynomial nonlinear, polynomial abspower, polynomial bivariate, and and constraints)
□ CIP format variable characters defined, e.g. SCIP_VARTYPE_INTEGER_CHAR
□ Improved support for wbo format for weighted PBO problems, IBM's xml-solution format and pip and zimpl format for polynomial mixed-integer programs
□ New reader for (standard) bounds on variables
□ Extended reader for CIP models to handle various new constraints, including all types of linear constraints
□ flatzinc reader is now capable to read cumulative constraints
□ changed opb(/wbo) reader which now creates pseudoboolean constraints instead of linear- and and-constraints, only a non-linear objective will create and-constraints inside the reader and
while reading a wbo file the topcost constraint is created as well
□ added clock for determine the time for reading
□ added reader for variable bounds (reader_bnd.{c,h})
□ Removed method SCIPreadSol(); call solution reading via SCIPreadProb() which calls the solution reader for .sol files.
• Nonlinear:
□ Major extensions for nonlinear CIP, new option for n-ary branching on nonlinear variables (within pseudocost branching rule)
□ added BETA version of constraint handler for nonlinear constraints (cons_nonlinear.{c,h}) to handle nonlinear equations given by algebraic expressions using operands like addition,
multiplication, power, exp, log, bivariate nonlinear constraints; currently no trigonometric functions
□ added BETA version of constraint handler for bivariate nonlinear constraints (cons_bivariate.{c,h}) to compute tight estimators for 1-convex and convex-concave bivariate nonlinear functions
(given as expression tree)
□ the gams writer can now write nonlinear, abspower and bivariate constraints
□ Extended writer for GAMS and pip format to write more types of nonlinear constraints
□ the pip and zimpl reader now create nonlinear constraints for polynomials of degree > 2
• Presolving:
□ new dual presolving methods in cons_setppc and cons_logicor
□ new presolving step removeConstraintsDueToNegCliques in locigor constraint handler which updates logicor constraints to setppc constraints if a negated clique inside this constraint exist, by
default is off
□ new presolving step in cons_knapsack (detectRedundantVars, deleteRedundantVars) which determines redundant variables in knapsack constraint with or without using clique information
□ cons_logicor is now able to replace all aggregated variables in presolving by there active or negation of an active variable counterpart
□ prop_pseudoobj is now working in presolving as well
□ implement presolving in exitpre() in cons_orbitope and cons_indicator
• Propagators:
□ added counter for number calls and timing for resolve propagation calls for constraint handler and propagators
□ Propagators are now also called in node presolving
□ the probing presolver presol_probing.{c.h} is now a propagator prop_probing.{c,h}, all corresponding parameters moved as well
□ the redcost separator sepa_redcost.{c.h} is now a propagator prop_redcost.{c,h}, all corresponding parameters moved as well
□ outsourced propAndSolve() method in solve.c which calls domain propagation and solving of the lp and relaxation
• Statistic:
□ solutions which are given by the user from the outside are now marked by # in the output
□ the Solving Time is now spitted into presolving, solving and reading time
□ Presolvers section has new column AddCons which states the number of added constraint
□ Constraints section has new column named #ResProp which show the number of resolve propagation calls of certain constraint handler
□ Constraint Timing section has a new column #ResProp which states the time spend in resolve propagation method of the constraint handler
□ improved output of propagators in display statistics
□ new section Propagator Timing which shows the time spend in different callbacks of the propagator
□ rearranged first two columns of Propagators section; #Propagate and #ResProp stating the number of call for propagation and resolve propagation; the Time column is moved into the new section
Propagator Timings
□ Constraints section has new column named MaxNumber which the maximum number of active constraint of a certain constraint handler
□ added two columns Time-0-It and Calls-0-It in the LP section which states the number of LP call and time spend for solving LPs with zero iterations (only refactorization)
□ The display of statistics for presolvers, propagators, constraints and LP solving has changed.
Performance improvements
• Reorganized filtering process of separation storage (allows adding cuts for different solutions)
• Improved presolving for various constraint handlers
• Improved propagation methods for variable bound constraints
• Improved performance for quadratic constraints
• performance improvements in prop_vbounds
• child selection rules now get also applied when the relaxation value is equal to the bound changed in branching
• added dual reduction to cons_cumulative.c
• for continuous variables, the pseudo costs update and the pscost branching rule now use the same strategies for updating the pseudo costs and estimating the improvement in the LP bound
• only perform probing if the variables are locked
• performance and memory consumption improvements in xmlparse.c
• Improved knapsack cover cuts
• avoid very long separation times of LEWIs in cons_knapsack for very large minimal covers
• used SCIPallocMemoryArray() instead of SCIPallocBlockMemoryArray() which leads to fewer memory consumption in getLiftingSequence() in cons_knapsack, also improved cache use bei using an extra
array instead blockmemory chunks
• switched FASTMIP from 1 to 2 for CPLEX and changed default pricing rule back to steepest edge pricing instead of quickstart steepest edge pricing
• made sorting method more robust
• LNS heuristics now use SCIPcopy() by default
• considering inactive variables in undercover heuristic; limiting effort for solving covering problem
• if during probing mode the LP relaxation is solved from scratch, e.g., when calling the shiftandpropagate heuristic before root node solving, then we clear the resulting LP state, since it might
be a bad starting basis for the next solve of the LP relaxation (controlled by new parameter lp/clearinitialprobinglp)
• included LP simplifier into SoPlex LP interface, applied when solving from scratch (lpi_spx.cpp)
• new presolving steps in varbound constraint handler, tightening bounds, coefficients, sides and pairwise presolving
Interface changes
• Miscellaneous:
□ The emphasis setting types now distinguish between plugin-type specific parameter settings (default, aggressive, fast, off), which are changed by SCIPsetHeuristics/Presolving/Separating(),
and global emphasis settings (default, cpsolver, easycip, feasibility, hardlp, optimality, counter), which can be set using SCIPsetEmphasis().
New and changed callbacks
• added propagator timings SCIP_PROPTIMING_BEFORELP, SCIP_PROPTIMING_DURINGLPLOOP and SCIP_PROPTIMING_AFTERLPLOOP for all propagation callbacks (see propagators and constraint handlers) which lead
to calling the propagation methods of a propagator before the lp is solved, during the lp loop and after the lp solving loop
• Conflict Analysis:
□ Added parameter separate to conflict handler callback method SCIP_DECL_CONFLICTEXEC() that defines whether the conflict constraint should be separated or not.
• Constraint Handler:
□ The new constraint handler callback SCIP_DECL_CONSDELVARS() is called after variables were marked for deletion. This method is optional and only of interest if you are using SCIP as a
branch-and-price framework. That means, you are generating new variables during the search. If you are not doing that just define the function pointer to be NULL. If this method gets
implemented you should iterate over all constraints of the constraint handler and delete all variables that were marked for deletion by SCIPdelVar().
• NLP Solver Interface:
• Presolving:
□ New parameters isunbounded and isinfeasible for presolving initialization (SCIP_DECL_CONSINITPRE(), SCIP_DECL_PRESOLINITPRE(), SCIP_DECL_PROPINITPRE()) and presolving deinitialization (
SCIP_DECL_CONSEXITPRE(), SCIP_DECL_PRESOLEXITPRE(), SCIP_DECL_PROPEXITPRE()) callbacks of presolvers, constraint handlers and propagators, telling the callback whether the problem was already
declared to be unbounded or infeasible. This allows to avoid expensive steps in these methods in case the problem is already solved, anyway.
Note, that the C++ methods
□ Propagators are now also called in during presolving, this is supported by the new callback methods SCIP_DECL_PROPINITPRE(), SCIP_DECL_PROPEXITPRE(), and SCIP_DECL_PROPPRESOL().
□ The new parameters nnewaddconss and naddconss were added to the constraint handler callback method SCIP_DECL_CONSPRESOL() and the presolver callback method SCIP_DECL_PRESOLEXEC(). These
parameters were also added to corresponding C++ wrapper class methods (scip_presol() in objconshdlr.h and scip_exec() in objpresol.h)
• Problem Data:
□ The callback SCIP_DECL_PROBCOPY() got a new parameter global to indicate whether the global problem or a local version is copied.
Deleted and changed API methods
• implemented SCIPlpiGetPrimalRay() in SoPlex interface that has become available with SoPlex version 1.5.0.2
• allowed calling SCIPgetRowSolActivity() in SCIP_STAGE_SOLVED, since LP is still available
• various extensions and modifications for expressions and expression trees (too much to state here)
• The result value SCIP_NEWROUND has been added, it allows a separator/constraint handler to start a new separation round (without previous calls to other separators/conshdlrs).
• SCIPcalcNodeselPriority() got a new parameter branchdir, which defines the type of branching that was performed: upwards, downwards, or fixed.
• Constraint Handlers:
□ Method SCIPincludeQuadconsUpgrade() of quadratic constraint handler got new parameter active to indicate whether the upgrading method is active by default.
□ Method SCIPseparateRelaxedKnapsack() in knapsack constraint handler got new parameter cutoff, which is a pointer to store whether a cutoff was found.
• Nonlinear expressions, relaxation, and solver interface:
□ SCIPcreateNLPSol() now creates a SCIP_SOL that is linked to the solution of the current NLP relaxation
□ Various types and functions dealing with polynomial expressions have been renamed to use the proper terms monomial and polynomial in nonlinear expressions (nlpi/∗expr*); results in many
renamings of types, structs and methods.
□ The methods SCIPnlpGetObjective(), SCIPnlpGetSolVals(), and SCIPnlpGetVarSolVal() have been removed, use SCIPgetNLPObjval(), SCIPvarGetNLPSol() and SCIPcreateNLPSol() to retrieve NLP solution
values instead. SCIPcreateNLPSol() now returns an error if NLP or NLP solution is not available
□ Removed methods SCIPmarkRequireNLP() and SCIPisNLPRequired(), because the NLP is now always constructed if nonlinearities are present.
□ SCIPgetNLP() has been removed and NLP-methods from pub_nlp.h have been moved to scip.h, which resulted in some renamings, too.
□ renamed SCIPexprtreeEvalSol() to SCIPevalExprtreeSol() and now located in scip.h.
□ renamed SCIPexprtreeEvalIntLocalBounds() to SCIPevalExprtreeLocalBounds() and now located in scip.h.
□ renamed SCIPexprtreeEvalIntGlobalBounds() to SCIPevalExprtreeGlobalBounds() and now located in scip.h.
□ The functions SCIPnlpiGetSolution() and SCIPnlpiSetInitialGuess() got additional arguments to get/set dual values.
□ The method SCIPgetNLPI() got a new parameter nlpiproblem, which is a pointer to store the NLP solver interface problem.
• Timing:
• Problem Data:
□ The method SCIPcopyProb() got a new parameter global to indicate whether the global problem or a local version is copied.
• Writing and Parsing Constraints:
□ The methods SCIPwriteVarName(), SCIPwriteVarsList(), and SCIPwriteVarsLinearsum() got a new boolean parameter type that indicates whether the variable type should be written or not.
□ The methods SCIPparseVarName() and SCIPparseVarsList() got a new output parameter endptr that is filled with the position where the parsing stopped.
□ The method SCIPwriteVarsList() got additionally a new parameter delimiter that defines the character which is used for delimitation.
• Variables:
New API functions
• information about the quality of the solution of an LP (currently the condition number of the basis matrix) can now be:
□ requested from the LPI (currently only available for CPLEX): methods SCIPlpiGetRealSolQuality() and
□ SCIPprintLPSolutionQuality() command display lpsolquality in interactive shell display column lpcond to show
□ estimate on condition number, if available
• SCIPround() and SCIPfeasRound() to round to nearest integer
• SCIPsortRealRealIntInt() and corresponding sorting/inserting/deleting methods in pub_misc.h and necessary defines in misc.c
• SCIPsortRealIntLong(), SCIPsortPtrPtrRealInt() and corresponding sorting/inserting/deleting methods in pub_misc.h and necessary defines in misc.c
• SCIPcomputeLPRelIntPoint() to compute relative interior point of the current LP
• SCIPstartSolvingTime() and SCIPstopSolvingTime() which can be used to start or stop the solving time clock
• SCIPstrToRealValue() and SCIPstrCopySection() in pub_misc.h; these methods can be used to convert a string into a SCIP_Real value and to copy a substring.
• SCIPgetBinvarRepresentatives() which gets binary variables that are equal to some given binary variables, and which are either active, fixed, or multi-aggregated, or the negated variables of
active, fixed, or multi-aggregated variables
• SCIPhasPrimalRay() and SCIPgetPrimalRayVal() that return whether a primal ray is stored and which value a given variable has in the primal ray, respectively
• SCIPsetParam() which is a generic parameter setter method, independent of the parameter type
• SCIPpropInitpre(), SCIPpropExitpre(), SCIPpropPresol() which initializes, exists and executes the presolving phase
• SCIProwGetAge() to access the age of a row (pub_lp.h/lp.c)
• SCIPsolGetOrigObj() in pub_sol.h which returns for a solution in the original problem space the objective value
• SCIPretransformSol() in scip.h that allows to retransform a solution to the original space
• SCIPlpiClearState() to LP interfaces for clearing basis information in the LP solver
• SCIPgetSubscipDepth() to access the depth of the current SCIP as a copied subproblem
• SCIPdebugAddSolVal() and SCIPdebugGetSolVal() to add/get values to/from a debug solution
• SCIPsepastoreRemoveInefficaciousCuts() to remove non-efficious cuts from the separation storage
• Nodes:
□ SCIPnodeGetParent() to get parent node of a node
□ SCIPnodesSharePath() in pub_tree.h that determines whether two nodes are on the same leaf-root path
□ SCIPnodesGetCommonAncestor() in pub_tree.h that finds the common ancestor node for two given nodes
• Read and Write:
• Memory:
□ SCIPcreateMesshdlrPThreads() and SCIPfreeMesshdlrPThreads() for allocating and deleting necessary memory for message handlers for parallel pthread version
□ SCIPallocClearMemoryArray() and BMSallocClearMemoryArray() for allocating cleared memory arrays in scip.h and memory.h
• Intervals:
• Variables:
□ SCIPcomputeVarCurrent{L,U}b{Local,Global}() to compute local or global lower or upper bounds of a multiaggregated variable from the bounds of the aggregation variables
□ SCIPbranchVarValNary() for n-ary variable branching
□ SCIPgetNegatedVars() which returns all negated variables for a given array of variables, if the negated variables are not existing yet, they will be created
□ SCIPgetNTotalVars() that returns the total number of created vars, icluding variables that were deleted in the meantime
□ SCIPvarGetHashkey(), SCIPvarIsHashkeyEq(), SCIPvarGetHashkeyVal() in pub_var.h which can be used for SCIP_HASHTABLE of variables
□ SCIPvarGetNBdchgInfosLb() and SCIPvarGetNBdchgInfosUb() in pub_var.h returning the number of lower or upper bound changes on the active path
□ SCIPvarGetBdchgInfoLb() and SCIPvarGetBdchgInfoUb() returning the bound change information at the given position
□ SCIPvarMarkDeletable() to mark a variable to be deletable completely from the problem (for branch-and-price); can only be called before the variable is added to the problem
□ SCIPvarMarkNotDeletable() that marks a variable to be non-deleteable (used within SCIP for forbidding deletion of variables contained in solution, LP bases, (multi)aggregation, ...)
□ SCIPvarIsDeletable() that returns whether a variable is marked to be deletable (each variable is per default non-deletable)
• NLP:
• Propagator:
• Constraints:
□ added to linear constraint handler SCIPsetUpgradeConsLinear(), which (de-)activates the possibility to upgrade a linear constraint to a specialized linear constraint (e.g. knapsack)
□ SCIPconvertCutsToConss() and SCIPcopyCuts() to scip.{c,h} for copying cuts to linear constraints
□ SCIPaddCoefLogicor() to add a variable to a logic or constraint
□ SCIPfindOrigCons() which return a original constraint with the given name or NULL
□ SCIPconshdlrGetNAddConss() which returns the number of added constraints during presolving by a given constraint handler
□ SCIPpresolGetNAddConss() which returns the number of added constraints during presolving by a given presolver
Command line interface
• New funtionalities in the interactive shell (modify current CIP instance, write NLP relaxation)
• added dialog write nlp to write current NLP relaxation to a file
• new dialog change freetransproblem to free transformed problem in the interactive shell before changing the problem
• it is possible to change bounds of a variable in the interactive shell
• it is possible to add a constraint to a problem in the interactive shell
Interfaces to external software
• Improved SOPLEX interface (LP simplifier)
• Improved CPLEX interface, including measures for numerical stability
Changed parameters
• change default value of parameter nodeselection/restartdfs/selectbestfreq 100
• moved parameters for pseudoboolean constraints from opb-reader to pseudoboolean constraint handler
• changed possible parameter values of branching/pscost/strategy from bri to cdsu: default is now u, i.e., to estimate the LP gain by a branching for external branching candidates (esp. continuous
variables) the same way as their pseudo costs are updated
• added possible value d for constraints/soc/nlpform to choose a convex division form for SOC constraint representation in NLP
• renamed parameter constraints/quadratic/linearizenlpsol to constraints/quadratic/linearizeheursol and do linearizations in every solution found by some heuristic
• renamed parameter constraints/quadratic/mincutefficacyenfo to constraints/quadratic/mincutefficacyenfofac and interpret it as a factor of the feasibility tolerance
• removed fastmip setting 2, which means the dualsolution would not be calculated but because SCIP always asks for the dual solution, the lp would be reoptimized to calculate them; so it had no
real effect
• all parameters in cons_indicator and cons_sos1 have been converted to lower case!
• changed default value of parameter separating/gomory/maxroundsroot to 10
• changed default value of parameter separating/gomory/maxsepacutsroot to 50
• removed parameter heuristics/subnlp/nlpsolver, use nlp/solver instead
New parameters
• branching/delaypscostupdate to delay the update of pseudo costs for continuous variables behind the separation round: default is TRUE
• branching/lpgainnormalize to set the strategy how the LP gain for a continuous variable is normalized when updating the variables pseudocosts: default is to divide LP gain by reduction of
variable's domain in sibling node
• branching/pscost/nchildren and branching/pscost/nary* to enable and customize n-ary branching on external branching candidates (e.g., in spatial branching for MINLP)
• conflict/bounddisjunction/continuousfrac which defines the maximum percantage of continuous variables within a conflict create by the bounddisjunction conflict handler
• conflict/separate which enables or disables the separation of conflict constraints
• constraints/{nonlinear,quadratic,soc,abspower}/sepanlpmincont to specify minimal required fraction of continuous variables in problem to enable linearization of convex constraints in NLP
relaxation solution in root
• constraints/indicator/forcerestart and constraints/indicator/restartfrac to control forced restart in cons_indicator
• constraints/indicator/generatebilinear to generate bilinear (quadratic) constraints instead of indicator constraints
• constraints/indicator/maxconditionaltlp to enable a quality check for the solution of the alternative LP
• constraints/indicator/removeindicators to remove indicator constraints if corresponding vub has been added
• constraints/linear/nmincomparisons and constraints/linear/mingainpernmincomparisons to influence stopping criterium for pairwise comparison of linear constraints
• constraints/pseudoboolean/decompose, for pseudoboolean constraints to transform pseudoboolean constraints into linear- and and-constraints
• constraints/quadratic/binreforminitial to indicate whether linear (non-varbound) constraints added due to reformulation of products with binary variables in a quadratic constraints should be
initial (if the quadratic constraint is initial), default is FALSE
• constraints/quadratic/checkfactorable to disable check for factorable quadratic functions (xAx = (ax+b)*(cx+d)) in quadratic constraints and not to use of this information in separation
(generates lifted tangent inequalities according to Belotti/Miller/Namazifar if also linear vars are present)
• constraints/quadratic/disaggregate to split a block-separable quadratic constraint into several quadratic constraint
• constraints/quadratic/maxproprounds and constraints/quadratic/maxproproundspresolve to limit the number of propagations rounds for quadratic constraints within one propagation round of SCIP solve
or during SCIP presolve
• constraints/varbound/presolpairwise that allows pairwise presolving of varbound constraints, default is TRUE
• heuristics/shiftandpropagate/onlywithoutsol to switch whether the heuristic should be called in case a primal solution is already present
• limit/maxorigsol which defines the size of the solution candidate store (default value is 10)
• lp/resolverestore controlling how LP solution is restored after diving: if TRUE by resolving them, if FALSE by buffering them; if lp/freesolvalbuffers is TRUE, we free the buffer memory each time
(FALSE by default)
• lp/clearinitialprobinglp to clear LP state at end of probing mode, if LP was initially unsolved
• lp/resolveitermin and lp/resolveiterfac to limit the number of LP iterations in resolving calls: resolveiterfac is a factor by which the average number of iterations per call is multiplied to get
the limit, but the limit is at least resolveitermin; default is -1 (no limit) for resolveiterfac and 1000 for resolveitermin
• lp/resolverestore and lp/freesolvalbuffers possibility to buffer and restore LP solution after diving without having to resolve the LP; currently turned off, because performance impact is
• misc/improvingsols which states whether only solutions which have a better (or equal) primal bound as the best known are checked; this is of interest if the check of a solution is expensive;
default value is FALSE
• misc/resetstat which state if the statistics should be reseted if the transformed problem is freed (in case of a benders decomposition this parameter should be set to FALSE) default value is TRUE
• nodeselection/restartdfs/countonlyleafs in node selector restart dfs which can be used to select the counting process of processed nodes
• presolving/donotaggr to deactivate aggregation of variables globally
• pricing/delvars and pricing/delvarsroot that define, whether variables created at a node / the root node should be deleted when the node is solved in case they are not present in the LP anymore
• propagating/s/maxprerounds for all propagators which allows to change to maximal number of rounds of presolving where this propagator participates in
• propagating/s/presoldelay for all propagators which allows to change if the presolving call of the given propagator should be delayed
• propagating/s/presolpriority for all propagators which allows to change the priority of calling the given propagator
• propagating/pseudoobj/propfullinroot for allowing to propagate all variables in the root node, instead of stopping after maxcands which is set by a parameter as well
• reading/gmsreader/bigmdefault and reading/gmsreader/indicatorreform reader_gms is now able to write indicator constraints (reformulated either via big-M or sos1)
• reading/gmsreader/signpower to enable writing sign(x)abs(x)^n as the rarely used gams function signpower(x,n)
• separating/closecuts/maxunsucessful to turn off separation if we can not find cuts
• timing/reading to add reading time to solving time
Data structures
• split off PARAMEMPHASIS from PARAMSETTING (in pub_paramset.c/paramset.c)
• new data structure SCIP_STAIRMAP
• add expression graph data structures and methods for reformulation, domain propagation, simple convexity check on nonlinear expressions and simplification for expression trees and graphs
• New scripts for running tests with GAMS
• added scripts check_gams.sh, evalcheck_gams.sh and check_gams.awk and target testgams in Makefile
• adjusted all test scripts to use the same new optimality gap computation as in SCIP
• added Makefile option VALGRIND=true to enable running the SCIP checks (make test) through valgrind; valgrind errors and memory leaks are reported as fails
• moved *.test and *.solu files to subdirectory testset in check directory and adjusted test scripts
Build system
• Variables:
□ via PARASCIP=true as a Makefile option it is possible to compile SCIP threadsafe in DEBUG-mode, (in OPT-mode it's only necessary if non-default messagehandler or CppAD is used)
□ the make parameter PARASCIP=true leads to threadsafe message handlers where you need to call SCIPcreateMesshdlrPThreads() and SCIPmessageSetHandler()/SCIPmessageSetDefaultHandler() and
SCIPfreeMesshdlrPThreads(); therefore we need to link with pthread library
□ new variable in Makefile which define installation directory for the libraries, /lib/, binary, /bin and include headers, /include, the default value is the empty string
• Linking:
□ Linking against Clp and Ipopt has been simplified. Only the directory where the package has been installed need to be provided now. For details see the INSTALL file.
□ to link against IPOPT, only the base directory of an Ipopt installation need to be specified now; additionally, if building with gnu compilers, the Ipopt libraries directory is stored in the
SCIP binary, which should make it easier to run with Ipopt shared libraries
□ to link against Clp, only the base directory of an Clp installation needs to be specified now
• Targets:
□ New targets (un)install in Makefile, support for valgrind in testing environment
□ new target make libs which compiles only the libraries
□ new target in Makefile install performs make and copies using the install command the include headers, binary, and libraries
□ new target in Makefile uninstall removes libraries, binary and include headers form INSTALLDIR
□ removed target lintfiles, this target is now imitated by the lint target and a none empty variable FILES
Fixed bugs
• fixed bug in copying if the target SCIP already is in solving stage: it might be that the copy of a variable cannot be found/created
• fixed bug trying to print big messages bigger than SCIP_MAXSTRLEN
• fixed bug w.r.t. counting feasible solutions and turned of sparse solution test
• LP solution status is now checked when checking root LP solution. Otherwise, due to different time measurements, it might happen that the LP solving was stopped due to the time limit, but SCIP
did not reach the limit, yet.
• fixed bug trying to tighten multiaggregated variables, which have only one active representation and this variable is already tightened
• fixed possible buffer overrun in tclique_graph.c
• fixed issue with interactive shell in case (user) plugins are included after the default plugins
• fixed bug where mutiaggregating leads to an aggregation and both variables were of implicit or integral type
• fixed bug in conflict.c, where LPi was manipulated, but not marked as not solved
• Tree:
□ fixed assertion in tree.c w.r.t. node estimation
□ fixed bug in debug.c: removed tree nodes had not been checked if they were pruned due to an incumbent solution found by a diving heuristic
• Bounds:
□ fixed bug which occured when changing a bound in the solving stage when this variables got upgraded from continuous to a integer type, where the bounds of this variable were still not
integral; due to that SCIPchgVarType() has changed (see above)
□ fixed bug in handling of lazy bounds that resulted in putting the bounds explicitly into the LP
• Separation:
□ fixed assert in sepa_clique.c which is currently not valid because implicit binary variables in cliques are ignored
□ fixed bug in sepa_zerohalf.c concerning inconsistent construction of solution array of variables and fixed wrong assert about variable bounds
• Constraints:
□ fixed not correct merging of variable in logicor constraint handler and changed the name of the method to a common name used by other constraint handlers too(findPairsAndSets->mergeMultiples)
□ fixed bugs in changing the initial and checked flags for constraints in original problem
□ fixed bug in cons_linear.c, when scaling a constraint maxabscoef was not set correctly, furthermore the correction of maxabscoef was not handled correctly
□ fixed bug in cons_indicator.c trying to copy a constraint where the pointer to the linear constraint did not point to the already transformed linear constraint (, happend when SCIPcopy() is
used after transforming before presolving)
□ fixed numerical bug in linear constraint handler: polishing of coefficients after fixing variables led to wrong results for continuous variables fixed to a close-to-zero value.
□ fixed bug in cons_bounddisjunction where branching on multiaggregated variables was tried while all aggregation variables are fixed
□ fixed bug in presolving of cons_logicor.c: adding variable implications can lead to further reductions; added call to applyFixings()
□ fixed bug in cons_countsols.c w.r.t. none active variables
□ fixed bug in cons_linear.c, scaling could have led to wrong values
• Reader:
□ fixed bug in reader_fzn.c w.r.t. cumulative constraints
□ fixed bug in reader_mps.c: if a variables first occurence is in the bounds section, then the corresponding variable bound was lost
□ fixed several issues in flatzinc reader
□ deactived checking of zero solution in Zimpl reader when no starting values are provided
□ reader_lp is now able to read lines longer than 65534 characters
• Memory:
□ fixed bug in copying NLPI interfaces that use block-memory (NLPI copy used block memory from source SCIP)
□ fixed memory leak in reader_pip.c
□ fixed memory leak in coloring part of maximum clique algorithm (tclique_coloring.c)
□ fixed memory leak in coloring part of maximum clique algorithm (tclique_coloring.c) in a better way
• Numerics:
□ fixed bug which occured when the dual farkas multipliers were not available in the lpi because the LP could only be solved with the primal simplex due to numerical problems
□ fixed bug in ZI round heuristic that led to infeasible shiftings for numerically slightly infeasible rows with close-to-zero coefficients
□ fixed numerical issue in octane heuristic: close-to-zero values for ray direction could have led to bad computations
• Propagation:
□ fixed bug in propagation of indicator constraints: cannot fix slack variable to 0 if linear constraint is disabled/not active
□ fixed bug in cons_linear.c while sorting the eventdatas during the binary variable sorting for propagation
□ fixed bug and wrong assert in heur_shiftandpropagate.c when relaxing continuous variables from the problem
□ fixed bug in cons_orbitope:resprop() for the packing case
□ fixed wrong changing of wasdelayed flag for propagators
□ fixed bug using wrong sign in infinity check in prop_pseudoobj
□ fixed bug in redcost propagator: can only be called if the current node has an LP
□ fixed bug w.r.t. infinity loop during propagation
• The interface contains several additional callback functions and parameters for plugins. Some effort may be required to compile your old projects with SCIP 2.1. For details see section Changes
between version 2.0 and 2.1 in the doxygen documentation. | {"url":"https://www.scipopt.org/doc/html/RN2_1.php","timestamp":"2024-11-05T03:19:57Z","content_type":"text/html","content_length":"77731","record_id":"<urn:uuid:cfe00640-70d2-402b-9894-05854d631f37>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00761.warc.gz"} |
Volume Oscillator
VOLUME OSCILLATOR
The Volume Oscillator displays the difference between two moving averages of a security's volume. The difference between the moving averages can be expressed in either points or percentages.
You can use the difference between two moving averages of volume to determine if the overall volume trend is increasing or decreasing. When the Volume Oscillator rises above zero, it signifies that
the shorter-term volume moving average has risen above the longer-term volume moving average, and thus, that the short-term volume trend is higher (i.e., more volume) than the longer-term volume
There are many ways to interpret changes in volume trends. One common belief is that rising prices coupled with increased volume, and falling prices coupled with decreased volume, is bullish.
Conversely, if volume increases when prices fall, and volume decreases when prices rise, the market is showing signs of underlying weakness.
The theory behind this is straight forward. Rising prices coupled with increased volume signifies increased upside participation (more buyers) that should lead to a continued move. Conversely,
falling prices coupled with increased volume (more sellers) signifies decreased upside participation.
The following chart shows Xerox and 5/10-week Volume Oscillator.
I drew linear regression trendlines on both the prices and the Volume Oscillator.
This chart shows a healthy pattern. When prices were moving higher, as shown by rising linear regression trendlines, the Volume Oscillator was also rising. When prices were falling, the Volume
Oscillator was also falling.
The Volume Oscillator can display the difference between the two moving averages as either points or percentages. To see the difference in points, subtract the longer-term moving average of volume
from the shorter-term moving average of volume:
To display the difference between the moving averages in percentages, divide the difference between the two moving averages by the shorter-term moving average: | {"url":"https://www.marketinout.com/technical_analysis.php?t=Volume_Oscillator&id=115","timestamp":"2024-11-06T18:48:58Z","content_type":"text/html","content_length":"24142","record_id":"<urn:uuid:ed3420e2-e018-4066-8dc4-a2a5bc3f7f00>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00374.warc.gz"} |
andhill, Inc. leased equipment from Tower Company under a 4-year lease requiring equal annual payments of $304152, with the first payment due at...
Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.
andhill, Inc. leased equipment from Tower Company under a 4-year lease requiring equal annual payments of $304152, with the first payment due at...
andhill, Inc. leased equipment from Tower Company under a 4-year lease requiring equal annual payments of $304152, with the first payment due at lease inception. The lease does not transfer
ownership, nor is there a bargain purchase option. The equipment has a 4 year useful life and no salvage value. Sandhill, Inc.'s incremental borrowing rate is 9% and the rate implicit in the lease
(which is known by Sandhill, Inc.) is 7%. Assuming that this lease is properly classified as a capital lease, what is the amount of principal reduction recorded when the second lease payment is made
in Year 2?
PV Annuity Due
PV Ordinary Annuity
7%, 4 periods3.62432
9%, 4 periods3.53129
Show more
Homework Categories
Ask a Question | {"url":"https://studydaddy.com/question/andhill-inc-leased-equipment-from-tower-company-under-a-4-year-lease-requiring-e","timestamp":"2024-11-06T05:54:51Z","content_type":"text/html","content_length":"26665","record_id":"<urn:uuid:82c0e180-ef64-49ee-b687-c86bdb734d39>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00118.warc.gz"} |
NVIDIA DRIVE OS Linux SDK API Reference
Defines the float-precision location of a point on a two-dimensional object.
Definition at line 80 of file NvSIPLCommon.hpp.
float_t x
Holds the horizontal location of the point. More...
float_t y
Holds the vertical location of the point. More...
float_t nvsipl::NvSiplPointFloat::x
Holds the horizontal location of the point.
Definition at line 82 of file NvSIPLCommon.hpp.
float_t nvsipl::NvSiplPointFloat::y
The documentation for this struct was generated from the following file: | {"url":"https://developer.nvidia.com/docs/drive/drive-os/6.0.6/public/drive-os-linux-sdk/api_reference/structnvsipl_1_1NvSiplPointFloat.html","timestamp":"2024-11-15T00:14:25Z","content_type":"application/xhtml+xml","content_length":"10623","record_id":"<urn:uuid:473af1dc-92ac-4d1c-a1c6-10f6bb6a0fb6>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00198.warc.gz"} |