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From Random Import Randint Python | Parameter Values
Random Numbers with the Python Standard Library
The Python standard library provides a module called random that offers a suite of functions for generating random numbers.
Python uses a popular and robust pseudorandom number generator called the Mersenne Twister.
In this section, we will look at a number of use cases for generating and using random numbers and randomness with the standard Python API.
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Seed The Random Number Generator
The pseudorandom number generator is a mathematical function that generates a sequence of nearly random numbers.
It takes a parameter to start off the sequence, called the seed. The function is deterministic, meaning given the same seed, it will produce the same sequence of numbers every time. The choice of
seed does not matter.
The seed() function will seed the pseudorandom number generator, taking an integer value as an argument, such as 1 or 7. If the seed() function is not called prior to using randomness, the default is
to use the current system time in milliseconds from epoch (1970).
The example below demonstrates seeding the pseudorandom number generator, generates some random numbers, and shows that reseeding the generator will result in the same sequence of numbers being
# seed the pseudorandom number generator
from random import seed
from random import random
# seed random number generator
# generate some random numbers
print(random(), random(), random())
# reset the seed
# generate some random numbers
print(random(), random(), random())
Running the example seeds the pseudorandom number generator with the value 1, generates 3 random numbers, reseeds the generator, and shows that the same three random numbers are generated.
0.13436424411240122 0.8474337369372327 0.763774618976614
0.13436424411240122 0.8474337369372327 0.763774618976614
It can be useful to control the randomness by setting the seed to ensure that your code produces the same result each time, such as in a production model.
For running experiments where randomization is used to control for confounding variables, a different seed may be used for each experimental run.
Random Floating Point Values
Random floating point values can be generated using the random() function. Values will be generated in the range between 0 and 1, specifically in the interval [0,1).
Values are drawn from a uniform distribution, meaning each value has an equal chance of being drawn.
The example below generates 10 random floating point values.
# generate random floating point values
from random import seed
from random import random
# seed random number generator
# generate random numbers between 0-1
for _ in range(10):
value = random()
Running the example generates and prints each random floating point value.
The floating point values could be rescaled to a desired range by multiplying them by the size of the new range and adding the min value, as follows:
scaled value = min + (value * (max – min))
Where min and max are the minimum and maximum values of the desired range respectively, and value is the randomly generated floating point value in the range between 0 and 1.
Random Integer Values
Random integer values can be generated with the randint() function.
This function takes two arguments: the start and the end of the range for the generated integer values. Random integers are generated within and including the start and end of range values,
specifically in the interval [start, end]. Random values are drawn from a uniform distribution.
The example below generates 10 random integer values between 0 and 10.
# generate random integer values
from random import seed
from random import randint
# seed random number generator
# generate some integers
for _ in range(10):
value = randint(0, 10)
Running the example generates and prints 10 random integer values.
Random Gaussian Values
Random floating point values can be drawn from a Gaussian distribution using the gauss() function.
This function takes two arguments that correspond to the parameters that control the size of the distribution, specifically the mean and the standard deviation.
The example below generates 10 random values drawn from a Gaussian distribution with a mean of 0.0 and a standard deviation of 1.0.
Note that these parameters are not the bounds on the values and that the spread of the values will be controlled by the bell shape of the distribution, in this case proportionately likely above and
below 0.0.
# generate random Gaussian values
from random import seed
from random import gauss
# seed random number generator
# generate some Gaussian values
for _ in range(10):
value = gauss(0, 1)
Running the example generates and prints 10 Gaussian random values.
Note: In the random module, there is a function
that functions the same as
. The former is thread-safe while
is not. However, you rarely run Python in multithread and
is faster.
Randomly Choosing From a List
Random numbers can be used to randomly choose an item from a list.
For example, if a list had 10 items with indexes between 0 and 9, then you could generate a random integer between 0 and 9 and use it to randomly select an item from the list. The choice() function
implements this behavior for you. Selections are made with a uniform likelihood.
The example below generates a list of 20 integers and gives five examples of choosing one random item from the list.
# choose a random element from a list
from random import seed
from random import choice
# seed random number generator
# prepare a sequence
sequence = [i for i in range(20)]
# make choices from the sequence
for _ in range(5):
selection = choice(sequence)
Running the example first prints the list of integer values, followed by five examples of choosing and printing a random value from the list.
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
Random Subsample From a List
We may be interested in repeating the random selection of items from a list to create a randomly chosen subset.
Importantly, once an item is selected from the list and added to the subset, it should not be added again. This is called selection without replacement because once an item from the list is selected
for the subset, it is not added back to the original list (i.e. is not made available for re-selection).
This behavior is provided in the sample() function that selects a random sample from a list without replacement. The function takes both the list and the size of the subset to select as arguments.
Note that items are not actually removed from the original list, only selected into a copy of the list.
The example below demonstrates selecting a subset of five items from a list of 20 integers.
# select a random sample without replacement
from random import seed
from random import sample
# seed random number generator
# prepare a sequence
sequence = [i for i in range(20)]
# select a subset without replacement
subset = sample(sequence, 5)
Running the example first prints the list of integer values, then the random sample is chosen and printed for comparison.
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
[4, 18, 2, 8, 3]
Randomly Shuffle a List
Randomness can be used to shuffle a list of items, like shuffling a deck of cards.
The shuffle() function can be used to shuffle a list. The shuffle is performed in place, meaning that the list provided as an argument to the shuffle() function is shuffled rather than a shuffled
copy of the list being made and returned.
The example below demonstrates randomly shuffling a list of integer values.
# randomly shuffle a sequence
from random import seed
from random import shuffle
# seed random number generator
# prepare a sequence
sequence = [i for i in range(20)]
# randomly shuffle the sequence
Running the example first prints the list of integers, then the same list after it has been randomly shuffled.
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
[11, 5, 17, 19, 9, 0, 16, 1, 15, 6, 10, 13, 14, 12, 7, 3, 8, 2, 18, 4]
Further Reading: Probability Theory and Statistical Analysis
If you’re interested in the theory behind random number generation, consider studying probability theory. And if you want to apply random number generation in data analysis, look into statistical
analysis in Python. Both topics will provide a deeper understanding of the power and potential of Python’s
Further Resources for Python Modules
For a more profound understanding of Python Modules, we have gathered several insightful resources for you:
• Python Modules Fundamentals Covered – Dive deep into Python’s module caching and reload mechanisms.
• Implementing Queues in Python – Dive into various queue types, including FIFO and LIFO, in Python.
• Simplifying Random Data Generation in Python – Learn how to add randomness to your Python programs with “random.”
• Python’s Random Module – Learn about the random module and generating random numbers with this Programiz guide.
• How to Create Random Numbers in Python – A Medium article that delves into generating random numbers in Python.
• Python’s Random Tutorial – A tutorial by Real Python covering topics related to generating random numbers in Python.
Explore these resources, and you’ll be taking another stride towards expertise in Python and taking your coding abilities to the next level.
random numbers in python
Output :
Traceback (most recent call last):
File “/home/fb805b21fea0e29c6a65f62b99998953.py”, line 5, in
r2=random.randint(‘a’, ‘z’)
File “/usr/lib/python3.5/random.py”, line 218, in randint
return self.randrange(a, b+1)
TypeError: Can’t convert ‘int’ object to str implicitly
Applications : The randint() function can be used to simulate a lucky draw situation. Let’s say User has participated in a lucky draw competition. The user gets three chances to guess the number
between 1 and 10. If guess is correct user wins, else loses the competition.
Random Numbers
To generate random numbers, first import the randint command section from Python’s random code library on the first line of the program.
The randint command stands for random integer. In brackets, state the number range to randomly choose from.
The random value should be saved into a variable.
from random import randint
number = randint(1,100)
print(“A random number between 1 and 100 is” , number)
A random number between 1 and 100 is 39
A random number between 1 and 100 is 73
A random number between 1 and 100 is 4
The randint range does not have to be fixed values and could be replaced by variables.
Below is a program where the user selects the upper and lower values of the range:
from random import randint
lower = int(input(“What is the lowest number? “))
upper = int(input(“What is the highest number? “))
number = randint(lower,upper)
print(“A random number between” , lower , “and” , upper , “is” , number)
What is the lowest number? 1What is the highest number? 50A random number between 1 and 50 is 36
What is the lowest number? 500What is the highest number? 1000A random number between 500 and 1000 is 868
Random Numbers Task 1 (Ice Comet)
A special comet made of ice passes the Earth only once every one hundred years, and it hasn’t been seen yet in the 21st century.
Use the randint command to randomly print a year between the current year and 2099.
Example solutions:
Did you know it won’t be until 2032 that the ice comet will next pass Earth!?
Did you know it won’t be until 2075 that the ice comet will next pass Earth!?
Random Numbers Task 2 (Guess the Number)
Use randint to generate a random number between 1 and 5.
Ask the user to enter a guess for the number with int and input.
Print the random number and use an if statement to check if there is a match, printing an appropriate statement if there is and something different if there is not a match.
Example solutions:
Enter a number between 1 and 5: 4Computer’s number: 5No match this time!
Enter a number between 1 and 5: 3Computer’s number: 3Well guessed! It’s a match!
Unit 3 – Built in Functions, Importing Random, and Using randint function – Python
Further Reading: Probability Theory and Statistical Analysis
If you’re interested in the theory behind random number generation, consider studying probability theory. And if you want to apply random number generation in data analysis, look into statistical
analysis in Python. Both topics will provide a deeper understanding of the power and potential of Python’s
Further Resources for Python Modules
For a more profound understanding of Python Modules, we have gathered several insightful resources for you:
• Python Modules Fundamentals Covered – Dive deep into Python’s module caching and reload mechanisms.
• Implementing Queues in Python – Dive into various queue types, including FIFO and LIFO, in Python.
• Simplifying Random Data Generation in Python – Learn how to add randomness to your Python programs with “random.”
• Python’s Random Module – Learn about the random module and generating random numbers with this Programiz guide.
• How to Create Random Numbers in Python – A Medium article that delves into generating random numbers in Python.
• Python’s Random Tutorial – A tutorial by Real Python covering topics related to generating random numbers in Python.
Explore these resources, and you’ll be taking another stride towards expertise in Python and taking your coding abilities to the next level.
Expanding the Range of randint
function isn’t limited to small ranges. In fact, you can generate random integers in any range you like. For example, if you’re simulating a lottery draw, you might need to generate numbers between 1
and 1000:
import random lottery_number = random.randint(1, 1000) print(lottery_number) # Output: # (A random number between 1 and 1000)
In this example, we’ve expanded the range of
to generate a random number between 1 and 1000. This demonstrates the flexibility of the
Python Tutorial: Generate Random Numbers and Data Using the random Module
Understanding Python’s randint Function
is a function that belongs to the
module. It is used to generate a random integer within a defined range. The function takes two parameters: the start and end of the range, inclusive.
Using randint: A Simple Example
Let’s look at a simple code example to understand how
import random number = random.randint(1, 10) print(number) # Output: # (A random number between 1 and 10)
In this example,
import random
is used to import the random module, which contains the randint function. Next,
random.randint(1, 10)
is used to generate a random integer between 1 and 10, inclusive. The result is then stored in the variable
, which is printed out.
Parameters and Return Value
randint(a, b)
function takes two parameters:
• : The lower limit of the range (inclusive).
• : The upper limit of the range (inclusive).
The function returns a random integer such that
a <= N <= b
Understanding the Output
The output of the
function is a random integer within the specified range. In our example, the output is a random number between 1 and 10. Each time you run the code, you might get a different number because the
selection is random.
By understanding the basics of Python’s
function, you can start to harness the power of random number generation in your coding projects.
Random Numbers with NumPy
In machine learning, you are likely using libraries such as scikit-learn and Keras.
These libraries make use of NumPy under the covers, a library that makes working with vectors and matrices of numbers very efficient.
NumPy also has its own implementation of a pseudorandom number generator and convenience wrapper functions.
NumPy also implements the Mersenne Twister pseudorandom number generator.
Let’s look at a few examples of generating random numbers and using randomness with NumPy arrays.
Seed The Random Number Generator
The NumPy pseudorandom number generator is different from the Python standard library pseudorandom number generator.
Importantly, seeding the Python pseudorandom number generator does not impact the NumPy pseudorandom number generator. It must be seeded and used separately.
The seed() function can be used to seed the NumPy pseudorandom number generator, taking an integer as the seed value.
The example below demonstrates how to seed the generator and how reseeding the generator will result in the same sequence of random numbers being generated.
# seed the pseudorandom number generator
from numpy.random import seed
from numpy.random import rand
# seed random number generator
# generate some random numbers
# reset the seed
# generate some random numbers
Running the example seeds the pseudorandom number generator, prints a sequence of random numbers, then reseeds the generator showing that the exact same sequence of random numbers is generated.
[4.17022005e-01 7.20324493e-01 1.14374817e-04]
[4.17022005e-01 7.20324493e-01 1.14374817e-04]
Array of Random Floating Point Values
An array of random floating point values can be generated with the rand() NumPy function.
If no argument is provided, then a single random value is created, otherwise the size of the array can be specified.
The example below creates an array of 10 random floating point values drawn from a uniform distribution.
# generate random floating point values
from numpy.random import seed
from numpy.random import rand
# seed random number generator
# generate random numbers between 0-1
values = rand(10)
Running the example generates and prints the NumPy array of random floating point values.
[4.17022005e-01 7.20324493e-01 1.14374817e-04 3.02332573e-01
1.46755891e-01 9.23385948e-02 1.86260211e-01 3.45560727e-01
3.96767474e-01 5.38816734e-01]
Array of Random Integer Values
An array of random integers can be generated using the randint() NumPy function.
This function takes three arguments, the lower end of the range, the upper end of the range, and the number of integer values to generate or the size of the array. Random integers will be drawn from
a uniform distribution including the lower value and excluding the upper value, e.g. in the interval [lower, upper).
The example below demonstrates generating an array of random integers.
# generate random integer values
from numpy.random import seed
from numpy.random import randint
# seed random number generator
# generate some integers
values = randint(0, 10, 20)
Running the example generates and prints an array of 20 random integer values between 0 and 10.
[5 8 9 5 0 0 1 7 6 9 2 4 5 2 4 2 4 7 7 9]
Array of Random Gaussian Values
An array of random Gaussian values can be generated using the randn() NumPy function.
This function takes a single argument to specify the size of the resulting array. The Gaussian values are drawn from a standard Gaussian distribution; this is a distribution that has a mean of 0.0
and a standard deviation of 1.0.
The example below shows how to generate an array of random Gaussian values.
# generate random Gaussian values
from numpy.random import seed
from numpy.random import randn
# seed random number generator
# generate some Gaussian values
values = randn(10)
Running the example generates and prints an array of 10 random values from a standard Gaussian distribution.
[ 1.62434536 -0.61175641 -0.52817175 -1.07296862 0.86540763 -2.3015387
1.74481176 -0.7612069 0.3190391 -0.24937038]
Values from a standard Gaussian distribution can be scaled by multiplying the value by the standard deviation and adding the mean from the desired scaled distribution. For example:
scaled value = mean + value * stdev
Where mean and stdev are the mean and standard deviation for the desired scaled Gaussian distribution and value is the randomly generated value from a standard Gaussian distribution.
Shuffle NumPy Array
A NumPy array can be randomly shuffled in-place using the shuffle() NumPy function.
The example below demonstrates how to shuffle a NumPy array.
# randomly shuffle a sequence
from numpy.random import seed
from numpy.random import shuffle
# seed random number generator
# prepare a sequence
sequence = [i for i in range(20)]
# randomly shuffle the sequence
Running the example first generates a list of 20 integer values, then shuffles and prints the shuffled array.
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
[3, 16, 6, 10, 2, 14, 4, 17, 7, 1, 13, 0, 19, 18, 9, 15, 8, 12, 11, 5]
Modern Ways of Random Number Generation in NumPy
In newer version of NumPy, you can do random number generation the following way:
import numpy as np
rng = np.random.Generator(np.random.PCG64())
10 rng = np.random.default_rng()
11 # uniform from 0 to 1
12 value = rng.random()
13 # generate 10 Gaussian random number
14 value = rng.standard_normal(10)
15 # generate 20 random integers between 0 and 10
16 value = rng.integers(low=0, high=10, size=20)
# shuffle a sequence in-place
The object
is a random number generator. You can create multiple such generators, or use the default one. The idea is to allow you to have multiple independent random number generator so drawing random numbers
from one generator would not affect another. This would make your code more robust (because you can mitigate the race condition in parallel algorithms) and allows you to fine-tune the pseudo-random
number generation algorithm.
Python Random Module Functions – Coding Examples (random, randint, choice, randrange)
TL;DR: How Do I Use the randint Function in Python?
The randint function is part of Python’s random module, and it’s used to generate a random integer within a specified range. Here’s a simple example:
import random number = random.randint(1, 10) print(number) # Output: # (A random number between 1 and 10)
In this example, we’re using Python’s
function to generate a random number between 1 and 10. The
import random
line at the beginning is necessary because randint is part of the random module in Python. The function
random.randint(1, 10)
then generates a random integer within the range of 1 and 10.
If you’re interested in learning more about the randint function, including its more advanced uses and potential issues you might encounter, keep reading for a comprehensive exploration.
Table of Contents
• Understanding Python’s randint Function
• Expanding the Range of randint
• Using randint in Loops
• Exploring Alternatives to randint
• Common Issues and Solutions with randint
• Best Practices with randint
• Understanding Random Number Generation in Python
• Real-World Applications of Python’s randint
• Further Reading: Probability Theory and Statistical Analysis
• Wrapping Up: Python’s randint Function
Summing It Up
In this article, you learned everything about the randint Python function. You also learned something fun, which was creating a lucky draw game that you can enjoy in your free time. You can also
tweak that code to make it a bit more complex for a bigger lucky draw game.
It is fascinating to see how one can do so much with Python programming. If you want to learn more about Python’s concepts, you can refer to Simplilearn’s Python Tutorial for Beginners. The tutorial
is dedicated to newbies to help them get acquainted with Python’s basics. Once you are done with the basics, you can opt for our Post Graduate Program in Full Stack Web Development course to learn
the advanced concepts and excel in Python development.
Have any questions for us? Leave them in the comments section, and our experts will answer them for you ASAP!
Is “randint” a function? Or is it just imported?
Codeacademy automatically adds this code on the top.
from random import randint
What does this function do? I can pass the sections from 7/9 – 9/9 with or without this function. Here is the complete code.
from random import randint board = [] for x in range(0,5): board.append(["O"] * 5) def print_board(board): for row in board: print " ".join(row) def random_row(board): return randint(0, len
(board) - 1) def random_col(board): return randint(0, len(board[0]) - 1) ship_row = random_row(board) ship_col = random_col(board) print ship_col print ship_row # Add your code below! guess_row =
int(raw_input("Guess Row:")) guess_col = int(raw_input("Guess Col:"))
python from random import randint
Wrapping Up: Python’s randint Function
function is a powerful tool in the
module, providing a straightforward way to generate random integers within a specified range. From simple applications to more complex scenarios,
offers a reliable solution for random number generation.
is generally easy to use, common issues include forgetting to import the
module and using incorrect range values. These can be easily avoided by following best practices such as always importing necessary modules and ensuring correct parameter values.
, Python offers other methods for random number generation. These include the
functions in the
module, and the
function in the NumPy module. Each method has its unique advantages and can be more suitable depending on your specific needs.
Random number generation is a fundamental aspect of programming with diverse applications. By mastering Python’s
function and understanding other random number generation methods, you can harness the power of randomness in your Python projects.
The use of randomness is an important part of the configuration and evaluation of machine learning algorithms.
From the random initialization of weights in an artificial neural network, to the splitting of data into random train and test sets, to the random shuffling of a training dataset in stochastic
gradient descent, generating random numbers and harnessing randomness is a required skill.
In this tutorial, you will discover how to generate and work with random numbers in Python.
After completing this tutorial, you will know:
• That randomness can be applied in programs via the use of pseudorandom number generators.
• How to generate random numbers and use randomness via the Python standard library.
• How to generate arrays of random numbers via the NumPy library.
Kick-start your project with my new book Statistics for Machine Learning, including step-by-step tutorials and the Python source code files for all examples.
Let’s get started.
How to Generate Random Numbers in PythonPhoto by Thomas Lipike. Some rights reserved.
Real-World Applications of Python’s randint
function isn’t just for academic exercises—it has practical applications in real-world scenarios. Let’s explore some of these applications.
In simulations,
can be used to generate random inputs. For example, in a weather simulation,
could generate random temperatures or wind speeds.
import random random_temperature = random.randint(-10, 40) print('Random Temperature:', random_temperature, '°C') # Output: # Random Temperature: (A random number between -10 and 40) °C
In this code, we’re using
to generate a random temperature between -10 and 40 degrees Celsius.
In games,
can be used to create unpredictable elements, making the game more exciting. For instance, in a dice game,
could be used to generate the dice roll.
import random dice_roll = random.randint(1, 6) print('Dice Roll:', dice_roll) # Output: # Dice Roll: (A random number between 1 and 6)
In this code, we’re using
to simulate a dice roll, generating a random number between 1 and 6.
Data Analysis
In data analysis,
can be used to generate random samples from a larger dataset. This can ensure a more representative sample and more accurate analysis.
random.randint error – python
Understanding Random Number Generation in Python
Random number generation is a fundamental concept in programming that has a variety of applications, from game development to data analysis. Python’s
module, which includes the
function, is a powerful tool for generating these random numbers.
The Role of Python’s Random Module
module provides a suite of functions for generating random numbers. These functions include
, and many others. Each function generates a random number in a different way or within a different range.
import random random_integer = random.randint(1, 10) random_float = random.random() random_uniform = random.uniform(1.0, 10.0) print(random_integer, random_float, random_uniform) # Output: # (A
random integer between 1 and 10, a random float between 0.0 and 1.0, a random float between 1.0 and 10.0)
In this code, we’re using three functions from Python’s
module to generate different types of random numbers. Each function provides a unique way to generate random numbers, making the
module a versatile tool for random number generation in Python.
The Importance of Randomness in Programming
Randomness plays a crucial role in many areas of programming. For instance, in game development, randomness can be used to create unpredictable gameplay elements. In data analysis, random sampling
can help ensure a representative sample of data. By understanding how to generate random numbers in Python, you can harness the power of randomness in your own programming projects.
Discrete distributions¶
The following function generates a discrete distribution.
random.binomialvariate(n=1, p=0.5)¶
Binomial distribution. Return the number of successes for n independent trials with the probability of success in each trial being p:
Mathematically equivalent to:
sum(random() < p for i in range(n))
The number of trials n should be a non-negative integer. The probability of success p should be between
0.0 <= p <= 1.0
. The result is an integer in the range
0 <= X <= n
New in version 3.12.
Random number generator in Python
Sample – Random Numbers
You can also use the sample command to choose several integers from a given range.
By implementing the range command you don’t need to individually write out each number.
from random import sample
numbers = sample(range(1,100) , 5)
print(“Five random numbers between 1 and 100 are:” , *numbers)
Five random numbers between 1 and 100 are: 53 42 11 8 20
Five random numbers between 1 and 100 are: 74 52 51 1 6
Random Samples Task 1 (Frost Comets)
The ice comet from a previous task has broken up into four smaller frosty comets that could pass the Earth anytime from next year to the year 2095.
Print four random years in that range.
Example solutions:
I predict the frost comets will be seen in these years: 2093 2036 2027 2091
I predict the frost comets will be seen in these years: 2076 2033 2053 2085
Random Samples Task 2 (Baby Boy)
Aunt Meredith is having a baby boy.
Create a program that randomly selects 3 male names from a list of 10 possible names.
Example solutions:
Hey Aunt Meredith, how about these names: Charlie Eddie Frank
Hey Aunt Meredith, how about these names: George Harold Bill
In this tutorial, you discovered how to generate and work with random numbers in Python.
Specifically, you learned:
• That randomness can be applied in programs via the use of pseudorandom number generators.
• How to generate random numbers and use randomness via the Python standard library.
• How to generate arrays of random numbers via the NumPy library.
Do you have any questions?Ask your questions in the comments below and I will do my best to answer.
Why Is \”random.seed()\” So Important In Python?
Section 5 looks at additional commands that you can import and use from Python’s code libraries.
A library is a collection of different commands that automatically come with Python but are separate from the main file. They can be imported (brought in) to your program by using the import command
at the start of your program.
Imagine Python’s library to be similar to an actual library. There are different sections in a real library (such as History, Geography, Reference) and different sections in Python’s library (such as
random or time). Each real library has many individual books in each section, just like commands in Python.
from random import randint
from time import ctime
You can import a specific command from one of Python’s libraries using the from and import commands at the top of your program.
(random.randint(beg, end))
Program to Demonstrate the ValueError
In this example, we are seeing that if we passes the floating point values as parameters in the randint() function then a ValueError occurs.
Generate random numbers in Python 🎲
Bookkeeping functions¶
random.seed(a=None, version=2)¶
Initialize the random number generator.
If a is omitted or
, the current system time is used. If randomness sources are provided by the operating system, they are used instead of the system time (see the
function for details on availability).
If a is an int, it is used directly.
With version 2 (the default), a
, or
object gets converted to an
and all of its bits are used.
With version 1 (provided for reproducing random sequences from older versions of Python), the algorithm for
generates a narrower range of seeds.
Changed in version 3.2: Moved to the version 2 scheme which uses all of the bits in a string seed.
Return an object capturing the current internal state of the generator. This object can be passed to
to restore the state.
state should have been obtained from a previous call to
, and
restores the internal state of the generator to what it was at the time
was called.
What are the Applications of Randint Python Function?
Since the Python randint() function generates a pseudo-random integer, it is usually helpful in gaming and lottery applications. For instance, let’s say a participant gets three chances to guess the
following random number generated within a range. If the person gets it correct within the three attempts, he wins, or else loses. Let’s simulate this situation using the randint Python function in
the example below.
Example: Creating a Lucky Draw Game with the randint() Function
In the below code, you will generate a random number between 1 and 20 three times. You will then accept input from the user and check if the guess is correct and wins. Let’s begin with our example.
# importing randint function
from random import randint
# Function to generate new random integer
def random_generator():
return randint(1, 20)
# Function to take input from user and show results
def your_guess():
# calling function to generate random numbers
rand_number = random_generator()
# defining the number of
# guesses the user gets
remaining_gus = 3
# Setting win-condition checker flagship variable
flagship = 0
# Using loop to provide only three chances
while remaining_gus > 0:
# Taking user input
guess_num = int(input(“What’s your guess?\n”))
# checking if the guess is correct
if guess_num == rand_number
# setting flagship 1 for correct guess to
# break loop
flagship = 1
# printing the failure message
print(“Oops, you missed!”)
# Decrementing guesses left
remaining_gus -= 1
# your_guess function returns True if win-condition
# is satisfied
if flagship == 1:
return True
# else returns False
return False
# Final output code to decide win or lose
if __name__ == ‘__main__’:
if your_guess() is True:
print(“Wow, you hit the bull’s eye!”)
else :
print(“Sorry, better luck next time!”)
Second attempt output:
Well, as it is evident, you could not get it correct in the first game. But in the second game, you hit the bull’s eye in the very first attempt. Copy-paste the code in any Python compiler, and you
too can enjoy the game. You can compete with your friends and family members, and make a day out of it.
Giáo sư Toán dạy bạn đánh Tài Xỉu thắng 100%, quay Gacha bách phát bách trúng
Using randint in Loops
One of the powerful ways to use
is within loops. This allows you to generate multiple random numbers at once. For instance, if you need to generate a list of 5 random numbers between 1 and 10, you can use a for loop with
import random random_numbers = [random.randint(1, 10) for _ in range(5)] print(random_numbers) # Output: # (A list of 5 random numbers between 1 and 10)
In this code, we’re using a for loop to generate a list of 5 random numbers. The
random.randint(1, 10)
function is called 5 times, once for each iteration of the loop, generating a new random number each time. The result is a list of 5 random integers.
These examples demonstrate how you can use Python’s
function in more complex ways to suit your needs. By adjusting the range and using loops, you can generate a variety of random number sequences.
Choice – Random Word
Rather than just numbers, we can also randomly generate characters or strings from a specified range by using the choice command.
You must first import the choice command from the random library. Choice works well with a list of values, which require square brackets and commas separating each word.
Below is a program that randomly chooses from a list of animals:
from random import choice
animals = [“cat” , “dog” , “horse” , “cow”]
print(“A random animal is” , choice(animals))
A random animal is horse
Máy tính có random thật không?
What Happens in Case of Multiple randint() Method Call?
You have seen that when you call the randint Python function; it generates and returns a random integer from the specified range. But what if you call it multiple times? Can it return the same value
twice? If not, what happens if the number of calls is more than the available range of values? Let’s get all these queries answered.
Example: Calling the Randint Python Function Multiple Times
In this example, you will keep the available range larger. Let’s see the output.
import random
for x in range(5):
print(random.randint(init, end))
As you can see in the output, the randint() function could still give different random values. Now, let’s cut short the available range and see what happens.
import random
for x in range(5):
print(random.randint(init, end))
Since the range was shorter and the number of calls was still 5, the randint Python function gave a repeated value: 5. Thus, the randint() function can provide the same number twice or even more
times. There are also chances that you might get a repeated value, even if the available range is more extensive.
• randint() function is used to generate the random integers between the two given numbers passed as parameters.
• Randint is an in-built function of random library in python. So we have to import the random library at the start of our code to generate random numbers.
• randint function returns:
□ ValueError, when floating point values are passed as parameters.
□ TypeErrorwhen anything other than numeric values is passed as parameters.
The random module in Python allows you to generate pseudo-random variables. The module provides various methods to get the random variables, one of which is the randint method. The randint Python
function is a built-in method that lets you generate random integers using the random module.
Python lists, sets, and tuples explained 🍍
Notes on Reproducibility¶
Sometimes it is useful to be able to reproduce the sequences given by a pseudo-random number generator. By reusing a seed value, the same sequence should be reproducible from run to run as long as
multiple threads are not running.
Most of the random module’s algorithms and seeding functions are subject to change across Python versions, but two aspects are guaranteed not to change:
• If a new seeding method is added, then a backward compatible seeder will be offered.
• The generator’s
method will continue to produce the same sequence when the compatible seeder is given the same seed.
Parameters Used in Randint Python Function
As you can see in the syntax, the Python randint() function accepts two parameters, which are:
• start: It is a required parameter that accepts integer value and determines the starting range from which it will generate the random integer.
• end: It is a required parameter that accepts integer value and defines the ending of the range from which it will generate the random integer.
Absolute and Relative imports – Python Tutorial 28
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The use of randomness is an important part of the configuration and evaluation of machine learning algorithms.
From the random initialization of weights in an artificial neural network, to the splitting of data into random train and test sets, to the random shuffling of a training dataset in stochastic
gradient descent, generating random numbers and harnessing randomness is a required skill.
In this tutorial, you will discover how to generate and work with random numbers in Python.
After completing this tutorial, you will know:
• That randomness can be applied in programs via the use of pseudorandom number generators.
• How to generate random numbers and use randomness via the Python standard library.
• How to generate arrays of random numbers via the NumPy library.
Kick-start your project with my new book Statistics for Machine Learning, including step-by-step tutorials and the Python source code files for all examples.
Let’s get started.
How to Generate Random Numbers in PythonPhoto by Thomas Lipike. Some rights reserved.
Answer 55c1f9dbe0a300455c000072
Hi, @Takeshi Travis Sugiyama,
is a function that is part of the
The official Python documentation on it is at random.randint(a, b). It returns a pseudorandom integer between and , inclusive.
You must
the function in some manner, in order to use it. The statement …
from random import randint
… enables you to use the name
to call the function.
If, instead you had …
import random
… you would need to use dot notation to specify the name of the module, in connection with the name of the function, in order to call it, for example …
return random.randint(0, len(board) - 1)
If you managed to pass without using the
function at all, congratulations :). But, of course the game is less interesting if it does not choose a random location for the battleship.
If you passed with code that used the
function, but after your having removed the
statement, it was because Codecademy remembers names defined by previous submissions of exercises in the same section, during a given session. This is a possible source of confusion, as it can cause
code to be accepted when it is, in fact, incomplete. To make sure your code is actually valid, you can refresh the page and submit the code again.
Avoiding import loops in Python
guess_left >
"Pick your number to "
"enter the lucky draw\n"
"Wrong Guess!!"
"Congrats!! You Win."
"Sorry, You Lost!"
Pick your number to enter the lucky draw
Wrong Guess!!
Pick your number to enter the lucky draw
Wrong Guess!!
Pick your number to enter the lucky draw
Congrats!! You Win.
Don’t miss your chance to ride the wave of the data revolution! Every industry is scaling new heights by tapping into the power of data. Sharpen your skills and become a part of the hottest trend in
the 21st century.
Dive into the future of technology – explore the Complete Machine Learning and Data Science Program by GeeksforGeeks and stay ahead of the curve.
Last Updated :
30 Oct, 2023
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The Quadratic Formula Calculator: A Comprehensive Guide
Introduction to Quadratic Equations
Quadratic equations are fundamental in algebra, representing a cornerstone of mathematical studies. They are often encountered in various branches of mathematics, physics, engineering, and economics.
A quadratic equation typically describes a parabola when graphed, which can model a variety of real-world scenarios, from the trajectory of an object under gravity to the optimization problems in
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation that takes the form:
ax2+bx+c=0ax^2 + bx + c = 0
In this equation, aa, bb, and cc are constants, with aa not equal to zero. The solutions to this equation are known as the roots, and they can be found using various methods, with the Quadratic
Formula being one of the most reliable and widely used.
The Role of the Quadratic Formula
The Quadratic Formula is a mathematical tool that provides a direct way to find the roots of any quadratic equation. It is derived from the process of completing the square and offers a standardized
approach to solving these equations. The formula is invaluable because it works universally for all quadratic equations, regardless of whether the roots are real or complex.
Introduction to Quadratic Formula Calculator
With the advent of technology, solving quadratic equations has become more accessible than ever, thanks to tools like the Quadratic Formula Calculator. This online tool automates the process of
finding the roots of a quadratic equation, making it easier for students, professionals, and enthusiasts to solve these equations quickly and accurately.
What is a Quadratic Formula Calculator?
A Quadratic Formula Calculator is an online computational tool designed to solve quadratic equations using the Quadratic Formula. Users input the coefficients aa, bb, and cc from the quadratic
equation, and the calculator instantly provides the roots. This tool eliminates the need for manual calculations, reducing the potential for errors and saving time.
The Popularity of Online Calculators
Online calculators have gained significant popularity due to their ease of use and accessibility. They allow users to solve complex mathematical problems with just a few clicks, without requiring
in-depth knowledge of the underlying mathematical principles. This accessibility makes them particularly useful for students and professionals who may not have the time or resources to solve
equations manually.
The Mechanics Behind the Quadratic Formula Calculator
Understanding how a Quadratic Formula Calculator works requires a basic knowledge of the mathematical processes it automates. While users don't need to perform these steps manually, knowing the
mechanics can provide a deeper appreciation for the tool’s functionality.
Input Parameters: The Coefficients
The first step in using a Quadratic Formula Calculator is to input the coefficients aa, bb, and cc. These coefficients correspond to the quadratic, linear, and constant terms of the equation,
respectively. The calculator uses these values to apply the Quadratic Formula and compute the roots.
Automated Calculation Process
Once the coefficients are entered, the calculator performs the necessary arithmetic operations to solve the equation. This involves several steps, including calculating the discriminant and then
using it to find the roots. The discriminant determines the nature of the roots, whether they are real and distinct, real and repeated, or complex.
Output: The Roots of the Equation
After the calculations are completed, the Quadratic Formula Calculator provides the roots of the equation. These roots are the solutions to the equation and represent the points where the graph of
the quadratic function intersects the x-axis. The calculator may also display additional information, such as the discriminant value, which offers insight into the nature of the roots.
Advantages of Using a Quadratic Formula Calculator
The Quadratic Formula Calculator offers numerous benefits, making it an essential tool for anyone dealing with quadratic equations. Its advantages extend beyond just simplifying the solving process,
impacting both educational and professional fields.
Accuracy and Precision
One of the primary advantages of using a Quadratic Formula Calculator is the accuracy it provides. Manual calculations can be prone to errors, especially when dealing with complex numbers or when the
coefficients are large or involve decimals. The calculator ensures precision by automating the process, reducing the likelihood of mistakes.
Time Efficiency
Another significant benefit is the time saved by using a Quadratic Formula Calculator. Solving quadratic equations manually can be time-consuming, particularly when multiple equations need to be
solved. The calculator provides instant results, allowing users to focus on other aspects of their work or studies.
Accessibility and Ease of Use
Quadratic Formula Calculators are incredibly user-friendly, making them accessible to a wide range of users, from students to professionals. The interface is typically straightforward, requiring only
the input of the coefficients before delivering results. This ease of use makes the calculator a valuable tool for those who may not have extensive mathematical backgrounds.
Educational Benefits
For students, the Quadratic Formula Calculator can serve as an educational aid. It allows them to check their work and gain confidence in their problem-solving abilities. Additionally, it can help
them understand the relationship between the coefficients and the roots, reinforcing their grasp of quadratic equations.
Limitations and Considerations
While the Quadratic Formula Calculator is a powerful tool, it does have some limitations. Understanding these can help users employ the calculator more effectively and avoid potential pitfalls.
Dependency on Technology
One of the main limitations of relying on a Quadratic Formula Calculator is the dependency it creates on technology. Users may become overly reliant on the tool, which could hinder their ability to
solve quadratic equations manually. This is particularly important in educational settings where understanding the underlying mathematics is crucial.
Lack of Conceptual Understanding
Using a calculator to solve quadratic equations can sometimes lead to a lack of conceptual understanding. While the calculator provides the correct answer, it does not explain the steps involved in
reaching that solution. This can result in users missing out on the opportunity to develop a deeper understanding of the mathematical principles at play.
Limitations in Complex Problem Solving
Quadratic Formula Calculators are designed to solve standard quadratic equations. However, in more complex problems where additional steps or considerations are required, such as when the equation
needs to be simplified first, the calculator may not be sufficient. In these cases, a deeper understanding of the problem is necessary.
History and Evolution of the Quadratic Formula and Calculators
The Quadratic Formula has a rich history that dates back centuries. Understanding its origins and evolution provides context for the development of modern tools like the Quadratic Formula Calculator.
Early Developments in Quadratic Equations
The study of quadratic equations can be traced back to ancient civilizations. The Babylonians, around 2000 BC, were among the first to solve quadratic equations, although their methods were geometric
rather than algebraic. The Greeks, particularly Euclid, also made significant contributions by solving quadratic equations through geometric means.
The Birth of the Quadratic Formula
The algebraic solution to quadratic equations, which would eventually lead to the Quadratic Formula, was first developed by mathematicians in the Islamic Golden Age. The Persian mathematician
Al-Khwarizmi, in the 9th century, wrote a treatise that included methods for solving quadratic equations algebraically. His work laid the foundation for the Quadratic Formula we use today.
Modern-Day Quadratic Formula Calculators
With the advent of computers and the internet, the process of solving quadratic equations was further simplified. Early computer programs could solve quadratic equations, but the development of
online calculators made this functionality accessible to anyone with an internet connection. Today, Quadratic Formula Calculators are available on numerous educational websites, making them a
ubiquitous tool for solving quadratic equations.
Applications of Quadratic Equations in Various Fields
Quadratic equations are not just a theoretical concept; they have numerous practical applications across various fields. Understanding these applications highlights the importance of tools like the
Quadratic Formula Calculator.
Physics and Engineering
In physics and engineering, quadratic equations are used to model a wide range of phenomena. For example, they describe the motion of objects under gravity, where the trajectory of a projectile
follows a parabolic path. Engineers also use quadratic equations in the design of structures and systems, where they help in optimizing performance and ensuring safety.
Economics and Finance
Quadratic equations play a role in economics and finance as well. They are used in modeling profit maximization and cost minimization problems, where the relationship between variables is nonlinear.
Quadratic equations can also describe the behavior of investment portfolios and the pricing of financial derivatives.
Biology and Medicine
In biology and medicine, quadratic equations can model population dynamics, where the growth of a population follows a quadratic relationship. They are also used in pharmacokinetics to describe the
concentration of a drug in the bloodstream over time.
Computer Science and Algorithms
Quadratic equations are also relevant in computer science, particularly in the analysis of algorithms. Certain algorithms have time complexities that can be modeled by quadratic equations, helping
computer scientists understand the efficiency of their solutions.
The Future of Quadratic Formula Calculators
As technology continues to evolve, so too will the tools we use to solve mathematical problems. The future of Quadratic Formula Calculators looks promising, with potential advancements that could
further enhance their capabilities.
Integration with Educational Platforms
One possible development is the integration of Quadratic Formula Calculators with educational platforms. This could provide students with real-time feedback and explanations as they solve quadratic
equations, helping them learn the concepts more effectively.
Enhanced User Interfaces
Future Quadratic Formula Calculators may feature enhanced user interfaces, making them even more accessible and intuitive. This could include voice input, interactive graphs, and step-by-step
solutions that guide users through the solving process.
Artificial Intelligence and Machine Learning
Artificial intelligence (AI) and machine learning could also play a role in the future of Quadratic Formula Calculators. AI could be used to provide personalized learning experiences, where the
calculator adapts to the user's skill level and provides tailored explanations and challenges.
Broader Applications
As quadratic equations continue to be applied in new and emerging fields, Quadratic Formula Calculators may also expand their functionality. This could include solving more complex equations or
integrating with other mathematical tools to provide a comprehensive solution for various types of problems.
The Quadratic Formula Calculator is a powerful tool that has simplified the process of solving quadratic equations. Its impact is felt across education, professional fields, and beyond, making it an
invaluable resource for anyone working with these types of equations. While it has its limitations, the advantages it offers in terms of accuracy, time efficiency, and accessibility are undeniable.
As technology advances, we can expect these calculators to become even more sophisticated, continuing to play a crucial role in the study and application of mathematics.
A Quadratic Formula Calculator is an online tool that quickly solves quadratic equations by calculating their roots.
Enter the coefficients π a, π b, and π c from your quadratic equation, and the calculator will instantly provide the roots.
It saves time and ensures accuracy when solving quadratic equations, especially for complex or multiple problems.
Yes, it can solve any quadratic equation, whether the roots are real, repeated, or complex.
Yes, it can reinforce understanding by allowing you to check your work and see how the roots are calculated. | {"url":"https://hightechtools.online/quadratic-formula-calculator","timestamp":"2024-11-12T10:50:45Z","content_type":"text/html","content_length":"87650","record_id":"<urn:uuid:5e54b310-6127-405f-8468-f993168c6188>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00591.warc.gz"} |
Stan User’s Guide
2.6 Hidden Markov Models
A hidden Markov model (HMM) generates a sequence of \(T\) output variables \(y_t\) conditioned on a parallel sequence of latent categorical state variables \(z_t \in \{1,\ldots, K\}\). These
``hidden" state variables are assumed to form a Markov chain so that \(z_t\) is conditionally independent of other variables given \(z_{t-1}\). This Markov chain is parameterized by a transition
matrix \(\theta\) where \(\theta_k\) is a \(K\)-simplex for \(k \in \{1,\ldots, K\}\). The probability of transitioning to state \(z_t\) from state \(z_{t-1}\) is \[ z_t \sim \mathsf{Categorical}(\
theta_{z[t-1]}). \] The output \(y_t\) at time \(t\) is generated conditionally independently based on the latent state \(z_t\).
This section describes HMMs with a simple categorical model for outputs \(y_t \in \{1,\ldots,V\}\). The categorical distribution for latent state \(k\) is parameterized by a \(V\)-simplex \(\phi_k\).
The observed output \(y_t\) at time \(t\) is generated based on the hidden state indicator \(z_t\) at time \(t\), \[ y_t \sim \mathsf{Categorical}(\phi_{z[t]}). \] In short, HMMs form a discrete
mixture model where the mixture component indicators form a latent Markov chain.
Supervised Parameter Estimation
In the situation where the hidden states are known, the following naive model can be used to fit the parameters \(\theta\) and \(\phi\).
data {
int<lower=1> K; // num categories
int<lower=1> V; // num words
int<lower=0> T; // num instances
int<lower=1,upper=V> w[T]; // words
int<lower=1,upper=K> z[T]; // categories
vector<lower=0>[K] alpha; // transit prior
vector<lower=0>[V] beta; // emit prior
parameters {
simplex[K] theta[K]; // transit probs
simplex[V] phi[K]; // emit probs
model {
for (k in 1:K)
theta[k] ~ dirichlet(alpha);
for (k in 1:K)
phi[k] ~ dirichlet(beta);
for (t in 1:T)
w[t] ~ categorical(phi[z[t]]);
for (t in 2:T)
z[t] ~ categorical(theta[z[t - 1]]);
Explicit Dirichlet priors have been provided for \(\theta_k\) and \(\phi_k\); dropping these two statements would implicitly take the prior to be uniform over all valid simplexes.
Start-State and End-State Probabilities
Although workable, the above description of HMMs is incomplete because the start state \(z_1\) is not modeled (the index runs from 2 to \(T\)). If the data are conceived as a subsequence of a
long-running process, the probability of \(z_1\) should be set to the stationary state probabilities in the Markov chain. In this case, there is no distinct end to the data, so there is no need to
model the probability that the sequence ends at \(z_T\).
An alternative conception of HMMs is as models of finite-length sequences. For example, human language sentences have distinct starting distributions (usually a capital letter) and ending
distributions (usually some kind of punctuation). The simplest way to model the sequence boundaries is to add a new latent state \(K+1\), generate the first state from a categorical distribution with
parameter vector \(\theta_{K+1}\), and restrict the transitions so that a transition to state \(K+1\) is forced to occur at the end of the sentence and is prohibited elsewhere.
Calculating Sufficient Statistics
The naive HMM estimation model presented above can be sped up dramatically by replacing the loops over categorical distributions with a single multinomial distribution.
The data are declared as before. The transformed data block computes the sufficient statistics for estimating the transition and emission matrices.
transformed data {
int<lower=0> trans[K, K];
int<lower=0> emit[K, V];
for (k1 in 1:K)
for (k2 in 1:K)
trans[k1, k2] = 0;
for (t in 2:T)
trans[z[t - 1], z[t]] += 1;
for (k in 1:K)
for (v in 1:V)
emit[k,v] = 0;
for (t in 1:T)
emit[z[t], w[t]] += 1;
The likelihood component of the model based on looping over the input is replaced with multinomials as follows.
model {
for (k in 1:K)
trans[k] ~ multinomial(theta[k]);
for (k in 1:K)
emit[k] ~ multinomial(phi[k]);
In a continuous HMM with normal emission probabilities could be sped up in the same way by computing sufficient statistics.
Analytic Posterior
With the Dirichlet-multinomial HMM, the posterior can be computed analytically because the Dirichlet is the conjugate prior to the multinomial. The following example illustrates how a Stan model can
define the posterior analytically. This is possible in the Stan language because the model only needs to define the conditional probability of the parameters given the data up to a proportion, which
can be done by defining the (unnormalized) joint probability or the (unnormalized) conditional posterior, or anything in between.
The model has the same data and parameters as the previous models, but now computes the posterior Dirichlet parameters in the transformed data block.
transformed data {
vector<lower=0>[K] alpha_post[K];
vector<lower=0>[V] beta_post[K];
for (k in 1:K)
alpha_post[k] = alpha;
for (t in 2:T)
alpha_post[z[t-1], z[t]] += 1;
for (k in 1:K)
beta_post[k] = beta;
for (t in 1:T)
beta_post[z[t], w[t]] += 1;
The posterior can now be written analytically as follows.
model {
for (k in 1:K)
theta[k] ~ dirichlet(alpha_post[k]);
for (k in 1:K)
phi[k] ~ dirichlet(beta_post[k]);
Semisupervised Estimation
HMMs can be estimated in a fully unsupervised fashion without any data for which latent states are known. The resulting posteriors are typically extremely multimodal. An intermediate solution is to
use semisupervised estimation, which is based on a combination of supervised and unsupervised data. Implementing this estimation strategy in Stan requires calculating the probability of an output
sequence with an unknown state sequence. This is a marginalization problem, and for HMMs, it is computed with the so-called forward algorithm.
In Stan, the forward algorithm is coded as follows. First, two additional data variable are declared for the unsupervised data.
data {
int<lower=1> T_unsup; // num unsupervised items
int<lower=1,upper=V> u[T_unsup]; // unsup words
The model for the supervised data does not change; the unsupervised data are handled with the following Stan implementation of the forward algorithm.
model {
real acc[K];
real gamma[T_unsup, K];
for (k in 1:K)
gamma[1, k] = log(phi[k, u[1]]);
for (t in 2:T_unsup) {
for (k in 1:K) {
for (j in 1:K)
acc[j] = gamma[t-1, j] + log(theta[j, k]) + log(phi[k, u[t]]);
gamma[t, k] = log_sum_exp(acc);
target += log_sum_exp(gamma[T_unsup]);
The forward values gamma[t,~k] are defined to be the log marginal probability of the inputs u[1],...,u[t] up to time t and the latent state being equal to k at time t; the previous latent states are
marginalized out. The first row of gamma is initialized by setting gamma[1,~k] equal to the log probability of latent state k generating the first output u[1]; as before, the probability of the first
latent state is not itself modeled. For each subsequent time t and output j, the value acc[j] is set to the probability of the latent state at time t-1 being j, plus the log transition probability
from state j at time t-1 to state k at time t, plus the log probability of the output u[t] being generated by state k. The log_sum_exp operation just multiplies the probabilities for each prior state
j on the log scale in an arithmetically stable way.
The brackets provide the scope for the local variables acc and gamma; these could have been declared earlier, but it is clearer to keep their declaration near their use.
Predictive Inference
Given the transition and emission parameters, \(\theta_{k, k'}\) and \(\phi_{k,v}\) and an observation sequence \(u_1,\ldots,u_T \in \{ 1,\ldots,V \}\), the Viterbi (dynamic programming) algorithm
computes the state sequence which is most likely to have generated the observed output \(u\).
The Viterbi algorithm can be coded in Stan in the generated quantities block as follows. The predictions here is the most likely state sequence y_star[1], ..., y_star[T_unsup] underlying the array of
observations u[1], ..., u[T_unsup]. Because this sequence is determined from the transition probabilities theta and emission probabilities phi, it may be different from sample to sample in the
generated quantities {
int<lower=1,upper=K> y_star[T_unsup];
real log_p_y_star;
int back_ptr[T_unsup, K];
real best_logp[T_unsup, K];
real best_total_logp;
for (k in 1:K)
best_logp[1, k] = log(phi[k, u[1]]);
for (t in 2:T_unsup) {
for (k in 1:K) {
best_logp[t, k] = negative_infinity();
for (j in 1:K) {
real logp;
logp = best_logp[t-1, j]
+ log(theta[j, k]) + log(phi[k, u[t]]);
if (logp > best_logp[t, k]) {
back_ptr[t, k] = j;
best_logp[t, k] = logp;
log_p_y_star = max(best_logp[T_unsup]);
for (k in 1:K)
if (best_logp[T_unsup, k] == log_p_y_star)
y_star[T_unsup] = k;
for (t in 1:(T_unsup - 1))
y_star[T_unsup - t] = back_ptr[T_unsup - t + 1,
y_star[T_unsup - t + 1]];
The bracketed block is used to make the three variables back_ptr, best_logp, and best_total_logp local so they will not be output. The variable y_star will hold the label sequence with the highest
probability given the input sequence u. Unlike the forward algorithm, where the intermediate quantities were total probability, here they consist of the maximum probability best_logp[t,~k] for the
sequence up to time t with final output category k for time t, along with a backpointer to the source of the link. Following the backpointers from the best final log probability for the final time t
yields the optimal state sequence.
This inference can be run for the same unsupervised outputs u as are used to fit the semisupervised model. The above code can be found in the same model file as the unsupervised fit. This is the
Bayesian approach to inference, where the data being reasoned about is used in a semisupervised way to train the model. It is not `cheating" because the underlying states foru` are never observed —
they are just estimated along with all of the other parameters.
If the outputs u are not used for semisupervised estimation but simply as the basis for prediction, the result is equivalent to what is represented in the BUGS modeling language via the cut
operation. That is, the model is fit independently of u, then those parameters used to find the most likely state to have generated u. | {"url":"https://mc-stan.org/docs/2_18/stan-users-guide/hmms-section.html","timestamp":"2024-11-13T21:50:39Z","content_type":"text/html","content_length":"100044","record_id":"<urn:uuid:7f11251a-50c6-4e88-997a-533b3113442c>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00130.warc.gz"} |
WeBWorK Standalone Renderer
You are testing your brand new Ferrari Testarossa. To see how well the brakes work you accelerate to 100 miles per hour, slam on the brakes, and determine that you brought the car to a stop over a
distance of 470 feet. Assuming a constant deceleration you figure that that deceleration is feet per second squared. (Enter a positive number.)
I trust that you don't have the courage to try this, but that night you wonder how long it would take you to stop (with the same constant deceleration) if you were moving at 200 miles per hour. Your
stopping distance would be feet. (Enter a number, not an arithmetic expression.)
You can earn partial credit on this problem. | {"url":"https://wwrenderer.libretexts.org/render-api?sourceFilePath=Library/Utah/AP_Calculus_I/set6_The_Integral/z4.pg&problemSeed=1234567&courseID=anonymous&userID=anonymous&course_password=anonymous&answersSubmitted=0&showSummary=1&displayMode=MathJax&language=en&outputFormat=nosubmit","timestamp":"2024-11-03T20:00:35Z","content_type":"text/html","content_length":"6231","record_id":"<urn:uuid:ef11bf01-20e3-4511-86ed-85892bda51f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00745.warc.gz"} |
Copyright 2010-2012 Johan Tibell
License BSD-style
Maintainer johan.tibell@gmail.com
Stability provisional
Portability portable
Safe Haskell Safe
Language Haskell2010
A map from hashable keys to values. A map cannot contain duplicate keys; each key can map to at most one value. A HashMap makes no guarantees as to the order of its elements.
The implementation is based on hash array mapped tries. A HashMap is often faster than other tree-based set types, especially when key comparison is expensive, as in the case of strings.
Many operations have a average-case complexity of O(log n). The implementation uses a large base (i.e. 16) so in practice these operations are constant time.
Strictness properties
This module satisfies the following strictness properties:
1. Key arguments are evaluated to WHNF;
2. Keys and values are evaluated to WHNF before they are stored in the map.
data HashMap k v Source #
A map from keys to values. A map cannot contain duplicate keys; each key can map to at most one value.
Bifoldable HashMap Source # Since: 0.2.11
Defined in Data.HashMap.Internal
Eq2 HashMap Source #
Defined in Data.HashMap.Internal
Ord2 HashMap Source #
Defined in Data.HashMap.Internal
Show2 HashMap Source #
Defined in Data.HashMap.Internal
NFData2 HashMap Source # Since: 0.2.14.0
Defined in Data.HashMap.Internal
Hashable2 HashMap Source #
Defined in Data.HashMap.Internal
Functor (HashMap k) Source #
Defined in Data.HashMap.Internal
Foldable (HashMap k) Source #
Defined in Data.HashMap.Internal
Traversable (HashMap k) Source #
Defined in Data.HashMap.Internal
Eq k => Eq1 (HashMap k) Source #
Defined in Data.HashMap.Internal
Ord k => Ord1 (HashMap k) Source #
Defined in Data.HashMap.Internal
(Eq k, Hashable k, Read k) => Read1 (HashMap k)
Source #
Defined in Data.HashMap.Internal
Show k => Show1 (HashMap k) Source #
Defined in Data.HashMap.Internal
NFData k => NFData1 (HashMap k) Source # Since: 0.2.14.0
Defined in Data.HashMap.Internal
Hashable k => Hashable1 (HashMap k) Source #
Defined in Data.HashMap.Internal
(Eq k, Hashable k) => IsList (HashMap k v) Source
Defined in Data.HashMap.Internal
Note that, in the presence of hash collisions, equal HashMaps may behave differently, i.e. substitutivity may be violated:
>>> data D = A | B deriving (Eq, Show)
>>> instance Hashable D where hashWithSalt salt _d = salt
>>> x = fromList [(A,1), (B,2)]
>>> y = fromList [(B,2), (A,1)]
(Eq k, Eq v) => Eq (HashMap k v) Source # >>> x == y
>>> toList x
>>> toList y
In general, the lack of substitutivity can be observed with any function that depends on the key ordering, such as folds and traversals.
Defined in Data.HashMap.Internal
(Data k, Data v, Eq k, Hashable k) => Data (
HashMap k v) Source #
Defined in Data.HashMap.Internal
The ordering is total and consistent with the Eq instance. However, nothing else about the ordering is specified, and it may change from version to
(Ord k, Ord v) => Ord (HashMap k v) Source # version of either this package or of hashable.
Defined in Data.HashMap.Internal
(Eq k, Hashable k, Read k, Read e) => Read (
HashMap k e) Source #
Defined in Data.HashMap.Internal
(Show k, Show v) => Show (HashMap k v) Source #
Defined in Data.HashMap.Internal
<> = union
If a key occurs in both maps, the mapping from the first will be the mapping in the result.
(Eq k, Hashable k) => Semigroup (HashMap k v)
Source # Examples
>>> fromList [(1,'a'),(2,'b')] <> fromList [(2,'c'),(3,'d')]
fromList [(1,'a'),(2,'b'),(3,'d')]
Defined in Data.HashMap.Internal
mempty = empty
mappend = union
(Eq k, Hashable k) => Monoid (HashMap k v) Source If a key occurs in both maps, the mapping from the first will be the mapping in the result.
>>> mappend (fromList [(1,'a'),(2,'b')]) (fromList [(2,'c'),(3,'d')])
fromList [(1,'a'),(2,'b'),(3,'d')]
Defined in Data.HashMap.Internal
(NFData k, NFData v) => NFData (HashMap k v)
Source #
Defined in Data.HashMap.Internal
(Hashable k, Hashable v) => Hashable (HashMap k v)
Source #
Defined in Data.HashMap.Internal
type Item (HashMap k v) Source #
Defined in Data.HashMap.Internal
Basic interface
lookup :: (Eq k, Hashable k) => k -> HashMap k v -> Maybe v Source #
O(log n) Return the value to which the specified key is mapped, or Nothing if this map contains no mapping for the key.
(!?) :: (Eq k, Hashable k) => HashMap k v -> k -> Maybe v Source #
O(log n) Return the value to which the specified key is mapped, or Nothing if this map contains no mapping for the key.
This is a flipped version of lookup.
Since: 0.2.11
findWithDefault Source #
:: (Eq k, Hashable k)
=> v Default value to return.
-> k
-> HashMap k v
-> v
O(log n) Return the value to which the specified key is mapped, or the default value if this map contains no mapping for the key.
Since: 0.2.11
lookupDefault Source #
:: (Eq k, Hashable k)
=> v Default value to return.
-> k
-> HashMap k v
-> v
O(log n) Return the value to which the specified key is mapped, or the default value if this map contains no mapping for the key.
DEPRECATED: lookupDefault is deprecated as of version 0.2.11, replaced by findWithDefault.
(!) :: (Eq k, Hashable k, HasCallStack) => HashMap k v -> k -> v infixl 9Source #
O(log n) Return the value to which the specified key is mapped. Calls error if this map contains no mapping for the key.
insert :: (Eq k, Hashable k) => k -> v -> HashMap k v -> HashMap k v Source #
O(log n) Associate the specified value with the specified key in this map. If this map previously contained a mapping for the key, the old value is replaced.
insertWith :: (Eq k, Hashable k) => (v -> v -> v) -> k -> v -> HashMap k v -> HashMap k v Source #
O(log n) Associate the value with the key in this map. If this map previously contained a mapping for the key, the old value is replaced by the result of applying the given function to the new and
old value. Example:
insertWith f k v map
where f new old = new + old
delete :: (Eq k, Hashable k) => k -> HashMap k v -> HashMap k v Source #
O(log n) Remove the mapping for the specified key from this map if present.
adjust :: (Eq k, Hashable k) => (v -> v) -> k -> HashMap k v -> HashMap k v Source #
O(log n) Adjust the value tied to a given key in this map only if it is present. Otherwise, leave the map alone.
update :: (Eq k, Hashable k) => (a -> Maybe a) -> k -> HashMap k a -> HashMap k a Source #
O(log n) The expression (update f k map) updates the value x at k (if it is in the map). If (f x) is Nothing, the element is deleted. If it is (Just y), the key k is bound to the new value y.
alter :: (Eq k, Hashable k) => (Maybe v -> Maybe v) -> k -> HashMap k v -> HashMap k v Source #
O(log n) The expression (alter f k map) alters the value x at k, or absence thereof.
alter can be used to insert, delete, or update a value in a map. In short:
lookup k (alter f k m) = f (lookup k m)
alterF :: (Functor f, Eq k, Hashable k) => (Maybe v -> f (Maybe v)) -> k -> HashMap k v -> f (HashMap k v) Source #
O(log n) The expression (alterF f k map) alters the value x at k, or absence thereof.
alterF can be used to insert, delete, or update a value in a map.
Note: alterF is a flipped version of the at combinator from Control.Lens.At.
Since: 0.2.10
isSubmapOf :: (Eq k, Hashable k, Eq v) => HashMap k v -> HashMap k v -> Bool Source #
O(n*log m) Inclusion of maps. A map is included in another map if the keys are subsets and the corresponding values are equal:
isSubmapOf m1 m2 = keys m1 `isSubsetOf` keys m2 &&
and [ v1 == v2 | (k1,v1) <- toList m1; let v2 = m2 ! k1 ]
>>> fromList [(1,'a')] `isSubmapOf` fromList [(1,'a'),(2,'b')]
>>> fromList [(1,'a'),(2,'b')] `isSubmapOf` fromList [(1,'a')]
Since: 0.2.12
isSubmapOfBy :: (Eq k, Hashable k) => (v1 -> v2 -> Bool) -> HashMap k v1 -> HashMap k v2 -> Bool Source #
O(n*log m) Inclusion of maps with value comparison. A map is included in another map if the keys are subsets and if the comparison function is true for the corresponding values:
isSubmapOfBy cmpV m1 m2 = keys m1 `isSubsetOf` keys m2 &&
and [ v1 `cmpV` v2 | (k1,v1) <- toList m1; let v2 = m2 ! k1 ]
>>> isSubmapOfBy (<=) (fromList [(1,'a')]) (fromList [(1,'b'),(2,'c')])
>>> isSubmapOfBy (<=) (fromList [(1,'b')]) (fromList [(1,'a'),(2,'c')])
Since: 0.2.12
union :: (Eq k, Hashable k) => HashMap k v -> HashMap k v -> HashMap k v Source #
O(n+m) The union of two maps. If a key occurs in both maps, the mapping from the first will be the mapping in the result.
>>> union (fromList [(1,'a'),(2,'b')]) (fromList [(2,'c'),(3,'d')])
fromList [(1,'a'),(2,'b'),(3,'d')]
unionWith :: (Eq k, Hashable k) => (v -> v -> v) -> HashMap k v -> HashMap k v -> HashMap k v Source #
O(n+m) The union of two maps. If a key occurs in both maps, the provided function (first argument) will be used to compute the result.
unionWithKey :: (Eq k, Hashable k) => (k -> v -> v -> v) -> HashMap k v -> HashMap k v -> HashMap k v Source #
O(n+m) The union of two maps. If a key occurs in both maps, the provided function (first argument) will be used to compute the result.
compose :: (Eq b, Hashable b) => HashMap b c -> HashMap a b -> HashMap a c Source #
Relate the keys of one map to the values of the other, by using the values of the former as keys for lookups in the latter.
Complexity: \( O (n * \log(m)) \), where \(m\) is the size of the first argument
>>> compose (fromList [('a', "A"), ('b', "B")]) (fromList [(1,'a'),(2,'b'),(3,'z')])
fromList [(1,"A"),(2,"B")]
(compose bc ab !?) = (bc !?) <=< (ab !?)
Since: 0.2.13.0
map :: (v1 -> v2) -> HashMap k v1 -> HashMap k v2 Source #
O(n) Transform this map by applying a function to every value.
traverseWithKey :: Applicative f => (k -> v1 -> f v2) -> HashMap k v1 -> f (HashMap k v2) Source #
O(n) Perform an Applicative action for each key-value pair in a HashMap and produce a HashMap of all the results. Each HashMap will be strict in all its values.
traverseWithKey f = fmap (map id) . Data.HashMap.Lazy.traverseWithKey f
Note: the order in which the actions occur is unspecified. In particular, when the map contains hash collisions, the order in which the actions associated with the keys involved will depend in an
unspecified way on their insertion order.
mapKeys :: (Eq k2, Hashable k2) => (k1 -> k2) -> HashMap k1 v -> HashMap k2 v Source #
O(n). mapKeys f s is the map obtained by applying f to each key of s.
The size of the result may be smaller if f maps two or more distinct keys to the same new key. In this case there is no guarantee which of the associated values is chosen for the conflicting key.
>>> mapKeys (+ 1) (fromList [(5,"a"), (3,"b")])
fromList [(4,"b"),(6,"a")]
>>> mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")])
fromList [(1,"c")]
>>> mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")])
fromList [(3,"c")]
Since: 0.2.14.0
Difference and intersection
difference :: (Eq k, Hashable k) => HashMap k v -> HashMap k w -> HashMap k v Source #
O(n*log m) Difference of two maps. Return elements of the first map not existing in the second.
differenceWith :: (Eq k, Hashable k) => (v -> w -> Maybe v) -> HashMap k v -> HashMap k w -> HashMap k v Source #
O(n*log m) Difference with a combining function. When two equal keys are encountered, the combining function is applied to the values of these keys. If it returns Nothing, the element is discarded
(proper set difference). If it returns (Just y), the element is updated with a new value y.
intersection :: (Eq k, Hashable k) => HashMap k v -> HashMap k w -> HashMap k v Source #
O(n*log m) Intersection of two maps. Return elements of the first map for keys existing in the second.
intersectionWith :: (Eq k, Hashable k) => (v1 -> v2 -> v3) -> HashMap k v1 -> HashMap k v2 -> HashMap k v3 Source #
O(n+m) Intersection of two maps. If a key occurs in both maps the provided function is used to combine the values from the two maps.
intersectionWithKey :: (Eq k, Hashable k) => (k -> v1 -> v2 -> v3) -> HashMap k v1 -> HashMap k v2 -> HashMap k v3 Source #
O(n+m) Intersection of two maps. If a key occurs in both maps the provided function is used to combine the values from the two maps.
foldMapWithKey :: Monoid m => (k -> v -> m) -> HashMap k v -> m Source #
O(n) Reduce the map by applying a function to each element and combining the results with a monoid operation.
foldr :: (v -> a -> a) -> a -> HashMap k v -> a Source #
O(n) Reduce this map by applying a binary operator to all elements, using the given starting value (typically the right-identity of the operator).
foldl :: (a -> v -> a) -> a -> HashMap k v -> a Source #
O(n) Reduce this map by applying a binary operator to all elements, using the given starting value (typically the left-identity of the operator).
foldr' :: (v -> a -> a) -> a -> HashMap k v -> a Source #
O(n) Reduce this map by applying a binary operator to all elements, using the given starting value (typically the right-identity of the operator). Each application of the operator is evaluated before
using the result in the next application. This function is strict in the starting value.
foldl' :: (a -> v -> a) -> a -> HashMap k v -> a Source #
O(n) Reduce this map by applying a binary operator to all elements, using the given starting value (typically the left-identity of the operator). Each application of the operator is evaluated before
using the result in the next application. This function is strict in the starting value.
foldrWithKey' :: (k -> v -> a -> a) -> a -> HashMap k v -> a Source #
O(n) Reduce this map by applying a binary operator to all elements, using the given starting value (typically the right-identity of the operator). Each application of the operator is evaluated before
using the result in the next application. This function is strict in the starting value.
foldlWithKey' :: (a -> k -> v -> a) -> a -> HashMap k v -> a Source #
O(n) Reduce this map by applying a binary operator to all elements, using the given starting value (typically the left-identity of the operator). Each application of the operator is evaluated before
using the result in the next application. This function is strict in the starting value.
foldrWithKey :: (k -> v -> a -> a) -> a -> HashMap k v -> a Source #
O(n) Reduce this map by applying a binary operator to all elements, using the given starting value (typically the right-identity of the operator).
foldlWithKey :: (a -> k -> v -> a) -> a -> HashMap k v -> a Source #
O(n) Reduce this map by applying a binary operator to all elements, using the given starting value (typically the left-identity of the operator).
filterWithKey :: forall k v. (k -> v -> Bool) -> HashMap k v -> HashMap k v Source #
O(n) Filter this map by retaining only elements satisfying a predicate.
mapMaybe :: (v1 -> Maybe v2) -> HashMap k v1 -> HashMap k v2 Source #
O(n) Transform this map by applying a function to every value and retaining only some of them.
mapMaybeWithKey :: (k -> v1 -> Maybe v2) -> HashMap k v1 -> HashMap k v2 Source #
O(n) Transform this map by applying a function to every value and retaining only some of them.
keys :: HashMap k v -> [k] Source #
O(n) Return a list of this map's keys. The list is produced lazily.
elems :: HashMap k v -> [v] Source #
O(n) Return a list of this map's values. The list is produced lazily.
toList :: HashMap k v -> [(k, v)] Source #
O(n) Return a list of this map's elements. The list is produced lazily. The order of its elements is unspecified.
fromList :: (Eq k, Hashable k) => [(k, v)] -> HashMap k v Source #
O(n*log n) Construct a map with the supplied mappings. If the list contains duplicate mappings, the later mappings take precedence.
fromListWith :: (Eq k, Hashable k) => (v -> v -> v) -> [(k, v)] -> HashMap k v Source #
O(n*log n) Construct a map from a list of elements. Uses the provided function f to merge duplicate entries with (f newVal oldVal).
Given a list xs, create a map with the number of occurrences of each element in xs:
let xs = ['a', 'b', 'a']
in fromListWith (+) [ (x, 1) | x <- xs ]
= fromList [('a', 2), ('b', 1)]
Given a list of key-value pairs xs :: [(k, v)], group all values by their keys and return a HashMap k [v].
let xs = ('a', 1), ('b', 2), ('a', 3)]
in fromListWith (++) [ (k, [v]) | (k, v) <- xs ]
= fromList [('a', [3, 1]), ('b', [2])]
Note that the lists in the resulting map contain elements in reverse order from their occurences in the original list.
More generally, duplicate entries are accumulated as follows; this matters when f is not commutative or not associative.
fromListWith f [(k, a), (k, b), (k, c), (k, d)]
= fromList [(k, f d (f c (f b a)))]
fromListWithKey :: (Eq k, Hashable k) => (k -> v -> v -> v) -> [(k, v)] -> HashMap k v Source #
O(n*log n) Construct a map from a list of elements. Uses the provided function to merge duplicate entries.
Given a list of key-value pairs where the keys are of different flavours, e.g:
data Key = Div | Sub
and the values need to be combined differently when there are duplicates, depending on the key:
combine Div = div
combine Sub = (-)
then fromListWithKey can be used as follows:
fromListWithKey combine [(Div, 2), (Div, 6), (Sub, 2), (Sub, 3)]
= fromList [(Div, 3), (Sub, 1)]
More generally, duplicate entries are accumulated as follows;
fromListWith f [(k, a), (k, b), (k, c), (k, d)]
= fromList [(k, f k d (f k c (f k b a)))]
Since: 0.2.11
keysSet :: HashMap k a -> HashSet k Source #
O(n) Produce a HashSet of all the keys in the given HashMap.
>>> HashSet.keysSet (HashMap.fromList [(1, "a"), (2, "b")]
fromList [1,2]
Since: 0.2.10.0 | {"url":"https://hackage-origin.haskell.org/package/unordered-containers-0.2.14.0/docs/Data-HashMap-Strict.html","timestamp":"2024-11-06T05:21:20Z","content_type":"application/xhtml+xml","content_length":"136644","record_id":"<urn:uuid:9e9e6939-dd9b-413f-9bc0-469682e8b743>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00281.warc.gz"} |
corr | Dataframe
Returns DataFrame with the pairwise correlation between two sets of columns.
It computes the Pearson correlation coefficient.
corr { columns1 } .with { columns2 } | .withItself()
To compute pairwise correlation between all columns in the DataFrame use corr without arguments:
The function is available for numeric- and Boolean columns. Boolean values are converted into 1 for true and 0 for false. All other columns are ignored.
If a ColumnGroup instance is passed as target column for correlation, it will be unpacked into suitable nested columns.
The resulting DataFrame will have n1 rows and n2+1 columns, where n1 and n2 are the number of columns in columns1 and columns2 correspondingly.
The first column will have the name "column" and will contain names of columns in column1. Other columns will have the same names as in columns2 and will contain the computed correlation
If exactly one ColumnGroup is passed in columns1, the first column in the output will have its name.
Last modified: 27 September 2024 | {"url":"https://kotlin.github.io/dataframe/corr.html","timestamp":"2024-11-04T01:43:34Z","content_type":"text/html","content_length":"13331","record_id":"<urn:uuid:0efe3612-959a-4cfd-9290-36bdd7910a60>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00398.warc.gz"} |
Escape velocity to Mars
Not open for further replies.
Could we save fuel to achieve escape velocity, by periodically turn on engine to change orbit to more and more ecliptically and then hyperbolical to Mars. Use Earth gravity like slingshot.
Thats what SMART-1 did so I would have thought so. <br />I wonder how the ESA dual layer plasma motor would change things?<br />
<br /><br />same efficency but several more times power, so a faster journey with less time in the radiation belts??
You cannot use the mass / gravity of the planet / sun you are orbiting to change the orbit. Your orbital energy is 'bound' to it already. You need to use a *third* body to pull off these tricks.<br
/><br />We can use the moons of Jupiter to change our orbit around Jupiter, likewise we can use Earth's moon to change our orbit around Earth.<br /><br />For example, we can get a boost from the Moon
when leaving Earth orbit. Just a boost, though, you still need to power yourself out of Earth's gravity well.<br /><br />My understanding is that the most boost one can hope for from Luna on a Mars
trajectory is ~0.5 km/s, but this invloves something like 3.0 km/s of rocket-provided dV <br /><br />These maneuvers are called "slingshots" but I do not like the term. More descriptive is the term
'fly-by'. Even better is 're-direct', because in all cases what is happening is that you are changing the direction of your velocity vector. The magnitude of your velocity remains unchanged, but you
can redirect it and thus achieve an orbit of a different energy. <div class="Discussion_UserSignature"> </div>
This is getting into an area where I am "knowledge-challenged", but I was under the impression that on a slingshot maneuver, more than a change of direction was involved, and that there was an
increase in energy. There is a coresponding decrease in the energy of the object used (planet or moon's velocity in orbit). The "falling" object (the spacecraft) gets the energy by virtue of it's
being pulled by gravity from the large object. If mass is expelled by thrusting at the "bottom" of the trajectory, there's a boost there (greater mass falling in than climbing out).<br />At least,
that's how I understand it. Someone want to explain it better?<br />
I found something similar article about Hohmann transfer orbit. Could we achieve that orbit by periodically push of engine. For example during 60 days(more and more eliptical). And last push will
move us to the elliptical transfer orbit.
><i>The magnitude of your velocity remains unchanged, but you can redirect it and thus achieve an orbit of a different energy.</i><p>I believe this is incorrect. The momentum of the system
(planet-probe) remains the same, but the planet is robbed of some momentum and the probe gains some, the difference in mass meaning that the change in velocity of the planet is imperceptible.</p>
There is a pretty good top level visualization here:<br /><br />
<br /><br />Wayne <div class="Discussion_UserSignature"> <p>"1) Give no quarter; 2) Take no prisoners; 3) Sink everything." Admiral Jackie Fisher</p> </div>
I do not like that wikipedia explanation and I stand by my post. The planet's orbital velocity does not have anything to do with it, it is a simple matter of the mass of the planet bending the path
of the probe. It's not a slingshot, it's a re-direct. Yes I know that everyone has been told it's like a slingshot, but that's wrong information. Yes the probe 'steals' angular momentum from the
planet but the mechanism shown is wrong wrong wrong. Note that there are no equations given which actually allow you to calculate a trajectory change. When I get home tonight I'll dig up the
equations if y'all want.<br /><br />It's all a matter of frame of reference. The probe has an orbital velocity in the sun's f.o.r., but as it crosses into the planet's sphere of influence, the probe
begins falling toward the planet. If it doesn't hit atmoshere or the surface, it leaves the planet's sphere of influence at the same velocity *magnitude* as it entered - relative to the planet.
Vector arithmetic then lets you find the velocity in the solar f.o.r. and you find that you've boosted your heliocentric orbital energy.<br /><br />Maybe if you google 'impact parameter' you'll find
better math and diagrams. Beware entering an argument with me on this, I wouldn't have said what I said if I wasn't 100% sure. <div class="Discussion_UserSignature"> </div>
><i>The magnitude of your velocity remains unchanged, but you can redirect it and thus achieve an orbit of a different energy.</i><p>If the energy of the orbit changes, then the angular momentum of
the probe has changed.</p>
Absolutely! Energy and momentum must be conserved.<br /><br />It's all about the frames of reference in which you define your velocity. I think I'm writing my sentences too quickly here at work . . .
<br /><br />The velocity magnitude in the planet's f.o.r. is unchanged but the direction has been altered. The velocity magnitude in the Sun's f.o.r. has most definitely been changed along with the
direction, thanks to stealing angular momentum from the planet <div class="Discussion_UserSignature"> </div>
Well that's the 'problem' - if we're talking about inter-planetary missions, the most convenient f.o.r is a sun-centred one.
OK the boss just left so I can slow down a sec, lol<br /><br />These things are calculated by the 'patched conic method'; all trajectories (elliptical, parabolic, hyperbolic) are conic sections. A
swing-by maneuver involves leaving the heliocentric coordinates, entering the planetary coordinates and then returning to heliocentric. This is done because the gravity of the planet becomes more
important than the gravity of the sun when you get close enough to the planet. While in reality this is a gradual process, math lets us make an abrupt transition from one to the other at a certain
distance from the planet known as the 'sphere of influence' which is a function of the masses of the sun and planet.<br /><br />A probe is cruising along in its orbit and you can calculate its
velocity vector using the vis-viva equation for magnitude and the tangent to its elliptical orbit for direction. <br /><br />And then suddenly - wham! - the probe is close to a planet and it crosses
into its sphere of influence. At that instant, we mathematically transpose the heliocentric velocity vector into a planet-centered velocity vector. It's the exact same velocity, just in different
coordinates. We are patching the heliocentric trajectory to the planet-centered trajectory.<br /><br />We now are looking at a hyperbolic trajectory *relative to the planet* - we're going too fast to
actually orbit the planet, BUT we are close enough that the planet influences our flight path. We're in a 'hyperbolic orbit'.<br /><br />Conservation of angular momentum *of the planet-probe system*
dictates that after we make our closest pass to the planet, our flight path *relative to the planet* is a mirror image of our approach. Thus, when we get back to the sphere of influence distance on
our way out, we will have the same velocity magnitude *relative to the planet*. BUT we are going a different direction, because our orbit lasted a finite length of time; the planet has redirected our
velocity vector.<br /><br />So here w <div class="Discussion_UserSignature"> </div>
Good. What about trying to clarify the Wikipedia article <img src="/images/icons/tongue.gif" />.
<font color="yellow"> You can not increase your velocity by orbiting the planet you launched from !</font><br /><br />Well, several spacecraft have made earth flybys (sometimes even more than once)
to increase their energy BUT this only works after you're already in a heliocentric (solar) orbit. Then you can just treat the earth like any other planet. And depending on where you're going, this
may save propellant but most of the time it will end up taking longer than a direct transfer.
Are you sure about that Dave? A flyby is just a hyperbolic orbit and I always understood that the total energy (kinetic + grav potential) remained the same at any point in the orbit. So when the
spacecraft leaves the sphere of influence ("infinite distance") grav potential energy will be zero, i.e. same as when it entered. Therefore kinetic energy must be the same therefore same velocity <b>
relative to the flyby planet</b>.<br /><br />Of course this vector is in a different direction to the initial velocity vector.<br /><br />If we let <b>Vi/p</b> be the inital velocity of the
spacecraft and <b>Vf/p</b> be the final velocity relative to the planet while <b>Vi/s</b> and <b>Vf/s</b> are the velocities relative to the sun, and <b>Vp/s</b> is the velocity of the planet
relative to the sun, then<br /><br /><b>Vi/s</b> = <b>Vi/p</b> + <b>Vp/s</b><br /><br />|<b>Vi/p</b>| = |<b>Vf/p</b>|<br /><br /><b>Vf/s</b> = <b>Vf/p</b> + <b>Vp/s</b><br /><br />Because the
direction of <b>Vf/p</b> may be different (and must be, for a useful flyby) to <b>Vi/p</b>, you can consider the extreme case where they are in opposite directions i.e. <b>Vf/p</b> = -<b>Vi/p</b>.<br
/><br />Then <b>Vf/s</b> = <b>Vi/s</b> + 2<b>Vi/p</b><br /><br />So you can get a boost of up to twice the "encounter velocity". But the magnitude of your velocity <b>with respect to the planet</b>
doesn't change.
Better explanation:<br /><br />
<br /><br />Wayne <div class="Discussion_UserSignature"> <p>"1) Give no quarter; 2) Take no prisoners; 3) Sink everything." Admiral Jackie Fisher</p> </div>
"the probe begins falling toward the planet. If it doesn't hit atmoshere or the surface, it leaves the planet's sphere of influence at the same velocity *magnitude* as it entered - relative to the
planet." <br /><br /><font color="yellow">No, that is not true. The probe will have a velocity vector magnitude change. </font><br /><br />Are you saying that coconuts migrate? <img src="/images/
icons/wink.gif" /> <img src="/images/icons/laugh.gif" /> sorry . . . <br /><br />Um, before we argue, are you sure you're reading my statement carefully? "Relative to the planet." If you don't fire
the engines, and the sphere of influence distance is the same, the flight path is symmetrical. There is nothing to cause your entering velocity - at the sphere of influence distance - to differ from
your departure velocity - at the sphere of influence distance. Conservation of momentum, momentum equals r-vector dot v-vector, r magnitude unchanged, r-v angle identical . . . <br /><br />In actual
practice, I believe engines are fired during the maneuver, which of course changes things.<br /><br />I can find some links if needed, but I'll wait for your response. <div class=
"Discussion_UserSignature"> </div>
<br /><br />I do have to retract a statement I made that "the planet's orbital velocity has nothing to do with it" - you need to use that velocity to transfom the vector from one set of coordinates
to another. <br /><br />I see what you're saying Dave and now I finally understand what is meant by "slingshot". The thing is, mathematically speaking, in the patched conic method, we've already
transformed into a planetary coordinate system, so I cannot accept - in that mathematical context - that the planet is moving. The wikipedia article makes more sense now as well.<br /><br />Looking
at the planet and probe from a heliocentric point of view, yes the magnitude changes. I've just been focused on the only way I know how to get a quantitative result, rather than just a qualitative
description.<br /><br />Thank you for the input, I learned something! <div class="Discussion_UserSignature"> </div>
I remember getting a lot of grief in my orals for working a problem in a non-inertial reference frame.<br /><br />Wayne <div class="Discussion_UserSignature"> <p>"1) Give no quarter; 2) Take no
prisoners; 3) Sink everything." Admiral Jackie Fisher</p> </div>
Adding to the list of things that amaze me about space stuff, is what it takes to plan these orbital flybys years and years in advance of the actual flyby and that just about all of them make it to
where they are supposed to go to (the one to Mars that was lost due to metric/english error is the only one I know of that didn't make it).
It's amazing what you can do with a calculator :/ <img src="/images/icons/smile.gif" />
I couldn't tell you where I will be tomorrow at 6:00pm local time, but I could figure out exactly where the Earth or any other body is, or will be at that time. The lure of astrology, pay no
attention to the man behind the curtain. <br /><br />It's not magic it's numbers and the math is consistant throughout the Universe. <div class="Discussion_UserSignature"> </div>
No, I mean astrology. The seers could point to the repeating paterns of the stars and convince people that it had a meaning to their lives. They knew when Mars and Venus would both show up in the
night sky and they could use this to convince people they knew what they were talking about it proved their point.<br /><br />The fact a star, that is millions of light years away has any influence
on people is because they could show you this same star, or group of stars, in the place and time they said it would be there. Just like magic.<br /><br />So then we have the seers and oracles, and
such, reaping the benefits of their presceince, which lead directly to the Jim Jones and Pat Robertsons we put up with today.<br /><br />I know the difference in Astronomy and astrology. One is real,
the other is mythology. <div class="Discussion_UserSignature"> </div>
excellent explanation on previous page thanks spacester! <div class="Discussion_UserSignature"> </div>
Not open for further replies. | {"url":"https://forums.space.com/threads/escape-velocity-to-mars.2736/","timestamp":"2024-11-02T21:33:59Z","content_type":"text/html","content_length":"217786","record_id":"<urn:uuid:428fb7af-406c-4091-ba2f-facca04bbb4f>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00604.warc.gz"} |
Writing a Linear Function in a Real-World Context
Question Video: Writing a Linear Function in a Real-World Context
Ruby spends $3.88 every day on transportation to and from work. Write a function rule that relates the total amount of money Ruby spends on transportation to the number of Ruby’s working days. Let 𝑥
represent the number of working days and 𝑦 the total amount of money she spends on transportation.
Video Transcript
Ruby spends three dollars and eighty-eight cents every day on transportation to and from work. Write a function rule that relates the total amount of money Ruby spends on transportation to the number
of Ruby’s working days. Let 𝑥 represent the number of working days and 𝑦 the total amount of money she spends on transportation.
We need to write a function rule. Our function rule will have an 𝑥 that represents the number of working days and a 𝑦 that represents the total amount of money Ruby spends on transportation. We can
think of our function rule like a machine; you input some information that you know, the function rule occurs, and then it gives you some kind of output.
In our case, we’ll input the number of working days; these are the days that Ruby uses the transportation. After the function rule occurs in our machine, it will output the total amount spent on
transportation for those working days. The question is what happens here. What happens inside the function rule in our problem? How do we go from the working days to the total money spent? And this
is the key. The key is the three dollars and eighty-eight cents Ruby spends on her working days.
Our function rule is the number of working days multiplied by three dollars and eighty-eight cents and that gives us the total that Ruby spent. Our problem wants us to use 𝑥 to represent the number
of working days and 𝑦 to represent the total amount of money. So we substitute 𝑥 and 𝑦, respectively, for the working days and the total amount. Simplifying it a little bit, three dollars and
eighty-eight cents times 𝑥 equals 𝑦. Using this function rule, we’ll always be able to find out the total amount of money that Ruby spends on transportation if we know 𝑥 — how many days she worked. | {"url":"https://www.nagwa.com/en/videos/752108595861/","timestamp":"2024-11-05T03:26:56Z","content_type":"text/html","content_length":"242672","record_id":"<urn:uuid:f5df77f3-7bd9-4006-ae99-bdb907c376cc>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00270.warc.gz"} |
Comparative visualizations
Let's suppose we'd like to compare the distributions of electorate data between the UK and Russia. We've already seen in this chapter how to make use of CDFs and box plots, so let's investigate an
alternative that's similar to a histogram.
We could try and plot both datasets on a histogram but this would be a bad idea. We wouldn't be able to interpret the results for two reasons:
• The sizes of the voting districts, and therefore the means of the distributions, are very different
• The number of voting districts overall is so different, so the histograms bars will have different heights
An alternative to the histogram that addresses both of these issues is the probability mass function (PMF).
Probability mass functions
The probability mass function, or PMF, has a lot in common with a histogram. Instead of plotting the counts of values falling into bins, though, it instead plots the probability that a number drawn
from a distribution will be exactly equal to a given value. As the function assigns a probability to every value that can possibly be returned by the distribution, and because probabilities are
measured on a scale from zero to one, (with one corresponding to certainty), the area under the probability mass function is equal to one.
Thus, the PMF ensures that the area under our plots will be comparable between datasets. However, we still have the issue that the sizes of the voting districts—and therefore the means of the
distributions—can't be compared. This can be addressed by a separate technique—normalization.
Normalizing the data isn't related to the normal distribution. It's the name given to the general task of bringing one or more sequences of values into alignment. Depending on the context, it could
mean simply adjusting the values so they fall within the same range, or more sophisticated procedures to ensure that the distributions of data are the same. In general, the goal of normalization is
to facilitate the comparison of two or more series of data.
There are innumerable ways to normalize data, but one of the most basic is to ensure that each series is in the range zero to one. None of our values decrease below zero, so we can accomplish this
normalization by simply dividing by the largest value:
(defn as-pmf [bins]
(let [histogram (frequencies bins)
total (reduce + (vals histogram))]
(->> histogram
(map (fn [[k v]]
[k (/ v total)]))
(into {}))))
With the preceding function in place, we can normalize both the UK and Russia data and plot it side by side on the same axes:
(defn ex-1-32 []
(let [n-bins 40
uk (->> (load-data :uk-victors)
(i/$ :turnout)
(bin n-bins)
ru (->> (load-data :ru-victors)
(i/$ :turnout)
(bin n-bins)
(-> (c/xy-plot (keys uk) (vals uk)
:series-label "UK"
:legend true
:x-label "Turnout Bins"
:y-label "Probability")
(c/add-lines (keys ru) (vals ru)
:series-label "Russia")
The preceding example generates the following chart:
After normalization, the two distributions can be compared more readily. It's clearly apparent how—in spite of having a lower mean turnout than the UK—the Russia election had a massive uplift towards
100-percent turnout. Insofar as it represents the combined effect of many independent choices, we would expect election results to conform to the central limit theorem and be approximately normally
distributed. In fact, election results from around the world generally conform to this expectation.
Although not quite as high as the modal peak in the center of the distribution—corresponding to approximately 50 percent turnout—the Russian election data presents a very anomalous result. Researcher
Peter Klimek and his colleagues at the Medical University of Vienna have gone as far as to suggest that this is a clear signature of ballot-rigging.
We've observed the curious results for the turnout at the Russian election and identified that it has a different signature from the UK election. Next, let's see how the proportion of votes for the
winning candidate is related to the turnout. After all, if the unexpectedly high turnout really is a sign of foul play by the incumbent government, we'd anticipate that they'll be voting for
themselves rather than anyone else. Thus we'd expect most, if not all, of these additional votes to be for the ultimate election winners.
Chapter 3, Correlation, will cover the statistics behind correlating two variables in much more detail, but for now it would be interesting simply to visualize the relationship between turnout and
the proportion of votes for the winning party.
The final visualization we'll introduce this chapter is the scatter plot. Scatter plots are very useful for visualizing correlations between two variables: where a linear correlation exists, it will
be evident as a diagonal tendency in the scatter plot. Incanter contains the c/scatter-plot function for this kind of chart with arguments the same as for the c/xy-plot function.
(defn ex-1-33 []
(let [data (load-data :uk-victors)]
(-> (c/scatter-plot (i/$ :turnout data)
(i/$ :victors-share data)
:x-label "Turnout"
:y-label "Victor's Share")
The preceding code generates the following chart:
Although the points are arranged broadly in a fuzzy ellipse, a diagonal tendency towards the top right of the scatter plot is clearly apparent. This indicates an interesting result—turnout is
correlated with the proportion of votes for the ultimate election winners. We might have expected the reverse: voter complacency leading to a lower turnout where there was a clear victor in the
As mentioned earlier, the UK election of 2010 was far from ordinary, resulting in a hung parliament and a coalition government. In fact, the "winners" in this case represent two parties who had, up
until election day, been opponents. A vote for either counts as a vote for the winners.
Next, we'll create the same scatter plot for the Russia election:
(defn ex-1-34 []
(let [data (load-data :ru-victors)]
(-> (c/scatter-plot (i/$ :turnout data)
(i/$ :victors-share data)
:x-label "Turnout"
:y-label "Victor's Share")
This generates the following plot:
Although a diagonal tendency in the Russia data is clearly evident from the outline of the points, the sheer volume of data obscures the internal structure. In the last section of this chapter, we'll
show a simple technique for extracting structure from a chart such as the earlier one using opacity.
In situations such as the preceding one where a scatter plot is overwhelmed by the volume of points, transparency can help to visualize the structure of the data. Since translucent points that
overlap will be more opaque, and areas with fewer points will be more transparent, a scatter plot with semi-transparent points can show the density of the data much better than solid points can.
We can set the alpha transparency of points plotted on an Incanter chart with the c/set-alpha function. It accepts two arguments: the chart and a number between zero and one. One signifies fully
opaque and zero fully transparent.
(defn ex-1-35 []
(let [data (-> (load-data :ru-victors)
(s/sample :size 10000))]
(-> (c/scatter-plot (i/$ :turnout data)
(i/$ :victors-share data)
:x-label "Turnout"
:y-label "Victor Share")
(c/set-alpha 0.05)
The preceding example generates the following chart:
The preceding scatter plot shows the general tendency of the victor's share and the turnout to vary together. We can see a correlation between the two values, and a "hot spot" in the top right corner
of the chart corresponding to close to 100-percent turnout and 100-percent votes for the winning party. This in particular is the sign that the researchers at the Medial University of Vienna have
highlighted as being the signature of electoral fraud. It's evident in the results of other disputed elections around the world, such as those of the 2011 Ugandan presidential election, too.
The district-level results for many other elections around the world are available at http://www.complex-systems.meduniwien.ac.at/elections/election.html. Visit the site for links to the research
paper and to download other datasets on which to practice what you've learned in this chapter about scrubbing and transforming real data.
We'll cover correlation in more detail in Chapter 3, Correlation, when we'll learn how to quantify the strength of the relationship between two values and build a predictive model based on it. We'll
also revisit this data in Chapter 10, Visualization when we implement a custom two-dimensional histogram to visualize the relationship between turnout and the winner's proportion of the vote even
more clearly. | {"url":"https://subscription.packtpub.com/book/data/9781784397180/1/ch01lvl1sec20/comparative-visualizations","timestamp":"2024-11-06T15:25:04Z","content_type":"text/html","content_length":"213060","record_id":"<urn:uuid:c7a5013f-2b44-4383-a127-13dd2dc46a5c>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00665.warc.gz"} |
Golden Rectangle
• geometry
a rectangle whose width divided by height is equal to the golden ratio
The golden rectangle is a rectangle whose width divided by height is equal to the golden ratio (phi), where . The rectangle can be constructed using a compass and straight edge^[1].
The golden ratio, often denoted as φ (phi), is an irrational number that has unique mathematical properties and appears in art, nature and, architecture. It is approximately equal to 1.61803398875.
1. Construct Golden Rectangle Example | {"url":"https://wumbo.net/glossary/golden-rectangle/","timestamp":"2024-11-04T21:09:20Z","content_type":"text/html","content_length":"6356","record_id":"<urn:uuid:0ec82e3d-3d38-4317-88f8-3dde24f9182c>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00619.warc.gz"} |
Course Outline
11/12/2024 10:47:24 AM MATH 156 Course Outline as of Summer 2019 Reinstated Course
Discipline and Nbr: MATH 156 Title: INT ALGEBRA B-STEM
Full Title: Intermediate Algebra for Business and STEM Majors
Last Reviewed:10/22/2018
│ Units │ Course Hours per Week │ │ Nbr of Weeks │ Course Hours Total │
│ Maximum │ 5.00 │ Lecture Scheduled │ 5.00 │ 17.5 max. │ Lecture Scheduled │ 87.50 │
│ Minimum │ 5.00 │ Lab Scheduled │ 0 │ 8 min. │ Lab Scheduled │ 0 │
│ │ Contact DHR │ 0 │ │ Contact DHR │ 0 │
│ │ Contact Total │ 5.00 │ │ Contact Total │ 87.50 │
│ │ Non-contact DHR │ 0 │ │ Non-contact DHR Total │ 0 │
Total Out of Class Hours: 175.00 Total Student Learning Hours: 262.50
Title 5 Category: AA Degree Applicable Grading: Grade or P/NP Repeatability: 00 - Two Repeats if Grade was D, F, NC, or NP Also Listed As: Formerly: MATH 56 Catalog Description:
An intermediate algebra course that incorporates the use of graphing technology. Topics include functions and their graphs, equations and inequalities in one variable, systems of equations in two and
three variables, exponential and logarithmic functions and equations, and conic sections.
Prerequisites/Corequisites: Completion of MATH 150 or MATH 150B or MATH 151 or appropriate placement based on AB 705 mandates Recommended Preparation: Limits on Enrollment: Schedule of Classes
Information Description:
An intermediate algebra course that incorporates the use of graphing technology. Topics include functions and their graphs, equations and inequalities in one variable, systems of equations in two and
three variables, exponential and logarithmic functions and equations, and conic sections.
(Grade or P/NP) Prerequisites:Completion of MATH 150 or MATH 150B or MATH 151 or appropriate placement based on AB 705 mandates Recommended: Limits on Enrollment: Transfer Credit: Repeatability:00 -
Two Repeats if Grade was D, F, NC, or NP ARTICULATION, MAJOR, and CERTIFICATION INFORMATION
Associate Degree: Effective: Summer 2019 Inactive:
Area: B Communication and Analytical Thinking
MC Math Competency
CSU GE: Transfer Area Effective: Inactive:
B4 Math/Quantitative Reasoning Fall 1981 Fall 1988
IGETC: Transfer Area Effective: Inactive:
CSU Transfer: Effective: Inactive:
UC Transfer: Effective: Inactive:
Certificate/Major Applicable: Both Certificate and Major Applicable COURSE CONTENT Student Learning Outcomes: At the conclusion of this course, the student should be able to:
1. Analyze functions and solve equations and inequalities using graphing technology and
algebraic methods.
2. Create mathematical models and solve applications of linear and nonlinear functions.
3. Solve systems of linear equations using matrix methods and graphing technology.
4. Graph conic sections, including parabolas, ellipses, and hyperbolas.
During this course, students will:
1. Define function, domain, and range, and use function notation.
2. Identify basic features of the graphs of polynomial, radical, absolute value, exponential and
logarithmic functions.
3. Use graphing technology to construct graphs, to solve nonlinear equations and inequalities in
one variable, and to locate roots, intersection points, and extrema.
4. Use algebraic methods to solve equations that involve polynomial, radical, absolute value,
rational, exponential and logarithmic expressions.
5. Find algebraic solutions to literal equations.
6. Apply algebraic or graphical methods, as appropriate, to solve application problems
involving polynomial, radical, absolute value, rational, exponential and logarithmic functions.
7. Apply properties of exponents and logarithms.
8. Express an understanding of the number e.
9. Graph conic sections, including parabolas, ellipses, and hyperbolas.
10. Use algebraic and graphical methods to solve linear and nonlinear systems in two variables,
and use Reduced Row Echelon Form (RREF) to solve systems of linear equations in three
11. Solve application and modeling problems that require the use of a system of linear equations.
12. Find graphical solutions to systems of linear inequalities.
Topics and Scope
I. Use of Technology
A. Evaluate and graph functions
B. Solve equations and inequalities graphically
C. Matrices and RREF
II. Functions
A. Definition of relation, function, domain, and range
B. Function notation and evaluation
C. Interval notation, intersection and union
D. Analyze graphs of polynomial, absolute value, radical, exponential, and logarithmic
functions with and without graphing technology
E. Mathematical models and other applications of linear and nonlinear functions
III. Equations and Inequalities
A. Equations
1. Solutions of literal equations
2. Algebraic and graphical solutions of linear, quadratic, radical, rational, absolute value,
exponential, and logarithmic equations
B. Inequalities
1. Algebraic solutions to absolute value inequalities
2. Graphical solutions of linear and nonlinear inequalities using graphing technology
IV. Quadratic Functions
A. Vertex and general forms
B. Discriminant
C. Solutions to quadratic equations using factoring, quadratic formula, and completing the
D. Applications and modeling
V. Rational Expressions and Equations
A. Simplification of rational expressions, including complex fractions
B. Operations on rational expressions
C. Solving rational equations
D. Applications and modeling
VI. Exponential and Logarithmic Functions
A. The number e
B. Common and natural logarithms
C. Laws of logarithms
D. Applications and modeling
VII. Introduction to Conic Sections
A. Midpoint and Distance Formulas, Circles
B. Parabolas
C. Ellipses
D. Hyperbolas
VIII. Systems of Equations/Inequalities
A. Linear and nonlinear systems of equations
B. Matrices and RREF
C. Systems of linear inequalities
D. Applications and modeling
1. Reading outside of class (0-60 pages per week)
2. Problem sets (1-8 per week)
3. Quizzes (0-4 per week)
4. Projects (0-10)
5. Exams (3-8)
6. Final exam
Methods of Evaluation/Basis of Grade.
Writing: Assessment tools that demonstrate writing skill and/or require students to select, organize and explain ideas in writing. Writing
0 - 0%
This is a degree applicable course but assessment tools based on writing are not included because problem solving assessments are more appropriate for this course.
Problem solving: Assessment tools, other than exams, that demonstrate competence in computational or non-computational problem solving skills. Problem Solving
5 - 20%
Problem sets
Skill Demonstrations: All skill-based and physical demonstrations used for assessment purposes including skill performance exams. Skill Demonstrations
0 - 0%
Exams: All forms of formal testing, other than skill performance exams. Exams
70 - 95%
Exams and quizzes
Other: Includes any assessment tools that do not logically fit into the above categories. Other Category
0 - 10%
Representative Textbooks and Materials:
Intermediate Algebra: A STEM Approach. Woodbury, George. Pearson. 2019
Beginning Algebra. 5th ed. Miller, Julie and O'Neill, Molly and Hyde, Nancy. McGraw Hill Publishing. 2017
Print PDF | {"url":"https://portal.santarosa.edu/SRweb/SR_CourseOutlines.aspx?CVID=48766&Semester=20195","timestamp":"2024-11-12T18:47:25Z","content_type":"text/html","content_length":"73689","record_id":"<urn:uuid:6954cf50-16d3-4939-aea9-968c99e3f418>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00740.warc.gz"} |
How to Use a Binary Subtraction Calculator for Simple and Advanced Equations - Age calculator
How to Use a Binary Subtraction calculator for Simple and Advanced Equations
Binary subtraction, as the term suggests, is the subtraction of two binary numbers. This operation is essential in computer science and electronics engineering. Binary subtraction is the foundation
of many complex mathematical models used in computer science. A binary subtraction calculator is a computer program that performs binary subtraction. It is a vital tool in the field of electronics
engineering and computer science.
Using a Binary Subtraction calculator
A binary subtraction calculator is similar to other calculators that you use to perform arithmetic operations. Here are the simple steps to use a binary subtraction calculator effectively:
Step 1: Turn on your calculator.
Step 2: Enter the first binary number.
Step 3: Enter the binary second number.
Step 4: Press the subtract button (-).
Step 5: Read the result on the display screen.
Yes, it’s that simple!
However, if you’re dealing with large binary numbers, the process can become tedious and time-consuming. In such cases, using a binary subtraction calculator with a larger display screen can be
Learning the Basics of Binary Subtraction
To fully understand and use a binary subtraction calculator, you need to understand the basics of binary subtraction. The concept of binary subtraction is fundamental in understanding the concept of
binary operations.
Binary subtraction is the process of subtracting two binary numbers from each other. In binary arithmetic, only two digits, 0 and 1, are used. Subtraction in binary operations works similarly to that
of decimal operations. The only difference is that the base is 2 instead of 10.
To perform binary subtraction, follow this simple rule:
1. If the second digit is 0, keep the first digit as it is.
2. If the second digit is 1, and the first digit is 0, borrow 1 from the next significant bit.
3. Subtract the second digit from the first digit.
4. Repeat this process until all digits are subtracted.
For example, suppose you want to subtract the binary number “101” from “111.”
-1 0 1
First, we start with the rightmost binary digits, subtracting 1 from 1. It gives us 0.
Next, we move left to the next decimal place and subtract 0 from 1. It gives us 1.
Finally, we subtract 1 from 1, and it gives us 0.
Therefore, the result is “010.”
Using a Binary Subtraction calculator for Advanced Equations.
Binary subtraction is an essential arithmetic operation in digital electronics, and it’s used in many complex mathematical computations. In such cases, using a binary subtraction calculator is
essential. Here are some examples of how to use a binary subtraction calculator for advanced equations.
Example 1:
Suppose we want to find the 2’s complement of the binary number 1010.
To do this, we need to subtract the binary number from 11111.
– 1 0 1 0 0
Here, we used the concept of 2’s complement to convert the binary number to its negative representation.
Example 2:
Suppose we want to subtract the binary number 10101 from 11000.
We can use the borrow and subtract method to solve this equation.
– 1 0 1 0 1
Here, we borrow 1 from the 2nd place and subtract 1 from it, therefore converting 0 to 1.
1. What is a binary subtraction calculator?
A binary subtraction calculator is a computer program designed to perform subtraction operations on binary numbers.
2. Can I use a binary subtraction calculator for advanced equations?
Yes, you can use a binary subtraction calculator for complex mathematical computations.
3. How do I subtract two binary numbers from each other?
Following the borrow and subtract method is the easiest way to subtract two binary numbers from each other.
A binary subtraction calculator is an essential tool for any computer science or electronics engineering student or professional. Binary subtraction is the foundation of many complex mathematical
models, and therefore, it’s essential to fully understand how to perform binary subtraction. By using a binary subtraction calculator, you can simplify your calculations and speed up your workflow.
Remember to follow the fundamentals of subtraction, such as the borrow and subtract method, to solve complex equations.
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Convex Hull - (Data Science Numerical Analysis) - Vocab, Definition, Explanations | Fiveable
Convex Hull
from class:
Data Science Numerical Analysis
The convex hull of a set of points in a Euclidean space is the smallest convex set that contains all the points. Imagine stretching a rubber band around the outermost points; when released, it forms
a shape that envelops all the points, which is essentially the convex hull. This concept is fundamental in various fields, such as computational geometry and optimization, as it helps to simplify
problems by focusing on the outer boundary of a dataset.
congrats on reading the definition of Convex Hull. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The convex hull can be computed using algorithms like Graham's scan or Jarvis's march, which are efficient for determining the outer boundary of a set of points.
2. In two dimensions, the convex hull can be visualized as a polygon that wraps around all given points, while in three dimensions, it appears as a polyhedron.
3. The convex hull plays a key role in optimization problems where solutions are often confined to the boundaries of feasible regions defined by linear constraints.
4. Convex hulls have applications in pattern recognition, computer graphics, and geographic information systems (GIS), facilitating tasks like shape analysis and collision detection.
5. The concept of the convex hull extends beyond simple point sets; it can be applied to functions and more complex shapes in higher-dimensional spaces.
Review Questions
• How does understanding the concept of a convex hull assist in solving optimization problems?
□ Understanding the convex hull is crucial for solving optimization problems because it allows us to focus on the extreme points or vertices of feasible regions defined by constraints. In many
optimization scenarios, only these boundary points are relevant for finding optimal solutions. By restricting our attention to the convex hull, we can simplify complex problems and apply
efficient algorithms that leverage this geometric property.
• Discuss how different algorithms for computing the convex hull can impact computational efficiency in data analysis.
□ Different algorithms for computing the convex hull vary significantly in terms of computational efficiency and suitability for various types of data. For example, Graham's scan operates in O
(n log n) time complexity, making it efficient for larger datasets, while Jarvis's march may take O(nh) time complexity where 'h' is the number of vertices in the hull. The choice of
algorithm can affect overall performance in data analysis tasks, particularly when processing large datasets or performing real-time computations.
• Evaluate the significance of convex hulls in fields such as computer graphics and GIS, and provide examples of their application.
□ Convex hulls hold significant importance in fields like computer graphics and GIS because they help simplify complex shapes and optimize calculations related to spatial data. In computer
graphics, convex hulls are used for collision detection, enabling efficient rendering and interaction with objects in virtual environments. In GIS, they assist in area calculations and
spatial queries by outlining geographic features or determining boundaries within datasets. These applications highlight how convex hulls contribute to practical problem-solving across
different domains.
"Convex Hull" also found in:
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Lenz's Law
Trending Questions
How can one increase the strength of induced current?
View Solution
An electron moves on a straight line path 'XY' as shown. The 'abcd' is a coil adjacent to the path of electron. What will be the direction of current, if any, induced in the coil?
A. No current induced
B. abcd
C. adcb
D. The current will reverse its direction as the electron goes past the coil
View Solution
Q. For the given charge distribution on the ring, the net electric field at the centre of non-conducting ring is
(Assume the part of ring in first and third quadrant is neutral, second quadrant is positively charged, fourth quadrant is negatively charged)
View Solution
The current
in an inductance coil varies with time
according to the following graph
Which one of the following plots shows the variations of voltage in the coil magnitude wise? (all vertical lines are not part of graphs)
View Solution
Find the flux through the surface
of the given Gaussian surface containing a point charge
+9 nC
as shown in figure.
is an equilateral triangle and take
ϵo=9×10−12 C2/N-m2
A. 833 N-m2/C
B. 933 N-m2/C
C. 633 N-m2/C
D. 733 N-m2/C
View Solution
The direction of induced current or the polarity of induced emf can be found by
A. Kirchoff's law
B. Amber’s law
C. Lenz’s law
D. Lorentz’s law
View Solution
An electron moves on a straight line path 'XY' as shown. The 'abcd' is a coil adjacent to the path of electron. What will be the direction of current, if any, induced in the coil?
A. No current induced
B. abcd
C. adcb
D. The current will reverse its direction as the electron goes past the coil
View Solution
Consider the situation given in the figure. The wire AB is slid on the fixed rails with a constant velocity. If the wire AB is replaced by a semicircular wire, the magnitude of the induced current
A. decrease
B. increases
C. increase or decrease depending on whether the semicircle bulges towards the resistance or away from it
D. remain same
View Solution
A magnet NS is suspended from a spring and while it oscillates, the magnet moves in and out of the coil C. The coil is connected to a galvanometer G. Then as the magnet oscillates,
A. G shows deflection to the left and right with constant amplitude
B. G shows deflection on one side
C. G shows no deflection.
D. G shows deflection to the left and right but the amplitude steadily decreases.
View Solution
A small magnet
is allowed to fall through a fixed horizontal conducting ring
. Let
be the acceleration due to gravity. The acceleration of
will be
A. <g when it is above R and moving toward R.
B. >g when it is above R and moving toward R.
C. <g when it is below R and moving away from R.
D. >g when it is below R and moving away from R.
View Solution
A short bar magnet is moved with constant velocity along the axis of a short coil. The magnet enters into the coil and then leaves it. The variation of induced emf
with time
is best represented as
View Solution
An electron moves along the line AB, which lies in the same plane as a circular loop of conducting wires as shown in the diagram. What will be the direction of current induced if any, in the loop
A. No current will be induced.
B. The current will be clockwise.
C. The current will be anticlockwise.
D. The current will change direction as the electron passes by.
View Solution
A short bar magnet is moved with constant velocity along the axis of a short coil. The magnet enters into the coil and then leaves it. The variation of induced emf
with time
is best represented as
View Solution
A magnet NS is suspended from a spring and while it oscillates, the magnet moves in and out of the coil C. The coil is connected to a galvanometer G. Then as the magnet oscillates,
A. G shows deflection to the left and right with constant amplitude
B. G shows deflection on one side
C. G shows no deflection.
D. G shows deflection to the left and right but the amplitude steadily decreases.
View Solution
Q. The current (I) in the inductance is varying with time (t) according to the plot shown in the figure.
Which one of the following is the correct variation of voltage with time in the coil?
View Solution
A bar magnet is thrown along the axis of a ring with speed
. After sometime, speed of magnet is
, then
A. v>v0
B. v<v0
C. v=v0
D. Information is incomplete
View Solution
A conducting ring is placed around the core of an electromagnet as shown in fig. When key K is pressed, the ring: (Consider the ring is light weight)
A. Remains stationary.
B. Is attracted towards the electromagnet.
C. Jumps out of the core.
D. Revolves around the core.
View Solution | {"url":"https://byjus.com/question-answer/Grade/Standard-XIII/Physics/None/Lenz's-Law/","timestamp":"2024-11-03T06:44:17Z","content_type":"text/html","content_length":"178948","record_id":"<urn:uuid:8b10a851-c5d9-4139-bb47-047501063f8c>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00285.warc.gz"} |
Estimate OB decomposition
estimate_ob_decompose {ddecompose} R Documentation
Estimate OB decomposition
The function performs the linear Oaxaca-Blinder decomposition.
formula formula object
data_used data.frame with data used for estimation (including weight and group variable)
reference_0 boolean: indicating if group 0 is the reference group and if its coefficients are used to compute the counterfactual mean.
normalize_factors boolean: If 'TRUE', then factor variables are normalized as proposed by Gardeazabal/Ugidos (2004)
compute_analytical_se boolean: If 'TRUE', then analytical standard errors for decomposition terms are calculated (assuming independence between groups).
return_model_fit boolean: If 'TRUE', then model objects are returned.
reweighting boolean: if 'TRUE', then the decomposition is performed with with respect to reweighted reference group.
rifreg boolean: if 'TRUE', then RIF decomposition is performed
rifreg_statistic string containing the distributional statistic for which to compute the RIF.
rifreg_probs a vector of length 1 or more with probabilities of quantiles.
custom_rif_function the RIF function to compute the RIF of the custom distributional statistic.
na.action generic function that defines how NAs in the data should be handled.
vcov unction estimating covariance matrix of regression coefficients if compute_analytical_se == TRUE
... additional parameters passed to custom_rif_function
version 1.0.0 | {"url":"https://search.r-project.org/CRAN/refmans/ddecompose/html/estimate_ob_decompose.html","timestamp":"2024-11-11T03:49:16Z","content_type":"text/html","content_length":"4206","record_id":"<urn:uuid:b0c1f359-30bb-45c1-826c-e9cc60167ea0>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00344.warc.gz"} |
Real Valued Data and the Normal Inverse-Wishart Distribution
One of the most common forms of data is real valued data
Let’s set up our environment and consider an example dataset
import pandas as pd
import seaborn as sns
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
The Iris Flower Dataset is a standard machine learning data set dating back to the 1930s. It contains measurements from 150 flowers, 50 from each of the following species:
• Iris Setosa
• Iris Versicolor
• Iris Virginica
iris = sns.load_dataset('iris')
│ │sepal_length │sepal_width│petal_length │petal_width│species│
│0│5.1 │3.5 │1.4 │0.2 │setosa │
│1│4.9 │3.0 │1.4 │0.2 │setosa │
│2│4.7 │3.2 │1.3 │0.2 │setosa │
│3│4.6 │3.1 │1.5 │0.2 │setosa │
│4│5.0 │3.6 │1.4 │0.2 │setosa │
In the case of the iris dataset, plotting the data shows that indiviudal species exhibit a typical range of measurements
irisplot = sns.pairplot(iris, hue="species", palette='Set2', diag_kind="kde", size=2.5)
irisplot.fig.suptitle('Scatter Plots and Kernel Density Estimate of Iris Data by Species', fontsize = 18)
If we wanted to learn these underlying species’ measurements, we would use these real valued measurements and make assumptions about the structure of the data.
In practice, real valued data is commonly assumed to be distributed normally, or Gaussian
We could assume that conditioned on species, the measurement data follwed a multivariate normal
The normal inverse-Wishart distribution allows us to learn the underlying parameters of each normal distribution, its mean \(\mu_s\) and its covariance \(\Sigma_s\). Since the normal inverse-Wishart
is the conjugate prior of the multivariate normal, the posterior distribution of a multivariate normal with a normal inverse-Wishart prior also follows a normal inverse-Wishart distribution. This
allows us to infer the distirbution over values of \(\mu_s\) and \(\Sigma_{s}\) when we define our model.
Note that if we have only one real valued variable, the normal inverse-Wishart distribution is often referred to as the normal inverse-gamma distribution. In this case, we learn the scalar valued
mean \(\mu\) and variance \(\sigma^2\) for each inferred cluster.
Univariate real data, however, should be modeled with our normal invese-chi-squared distribution, which is optimized for infering univariate parameters.
See Murphy 2007 for derrivations of our normal likelihood models
To specify the joint distribution of a multivariate normal inverse-Wishart distribution, we would import our likelihood model
from microscopes.models import niw as normal_inverse_wishart | {"url":"https://datamicroscopes.github.io/niw.html","timestamp":"2024-11-02T07:44:51Z","content_type":"application/xhtml+xml","content_length":"17622","record_id":"<urn:uuid:70e3ba4e-8df9-4d21-84bf-edf5fb0255b4>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00468.warc.gz"} |
[Term 2] Three cubes each of volume 64cm3 are joined end to end to for
Three cubes each of volume 64 cm ^ 3 are joined end to end to form a cuboid. Find the total surface area of the cuboid so formed?
This question is similar to Ex 13.1, 1 Chapter 13 Class 10 - Quadratic Equations | {"url":"https://www.teachoo.com/16200/3700/Question-2/category/CBSE-Class-10-Sample-Paper-for-2022-Boards---Maths-Basic--Term-2-/","timestamp":"2024-11-11T06:45:49Z","content_type":"text/html","content_length":"131974","record_id":"<urn:uuid:b697f6c5-9850-4f52-bb9d-391bfdbcb4a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00047.warc.gz"} |
POV-Ray: Newsgroups: povray.advanced-users: Making Patterns with functions
<<< Previous 10 Messages Goto Latest 10 Messages Next 10 Messages >>>
"Bald Eagle" <cre### [at] netscape> "Bald Eagle" <cre### [at] netscape
> (Still experimenting and struggling with various aspects of getting patterns
> from other platforms to play well with SDL.)
> Important tips to generate quality patterns from equations
> ----------------------------------------------------------
> Use Mike Williams' "shell" trick for isosurfaces to give an infinitely thin line
> a visible thickness
Say what??!!
That would be really useful.
Do you have a link to this trick, or a short explanation? (--keeping my fingers
Post a reply to this message
"Kenneth" <kdw### [at] gmail> "Bald Eagle" <cre### [at] netscape
> > Use Mike Williams' "shell" trick for isosurfaces to give an infinitely
> > thin line a visible thickness
> Do you have a link to this trick, or a short explanation? (--keeping my fingers
> crossed--)
I just took a look at the Mike Williams 'isosurface tutorial' that you packaged
and sent to me as a .pdf file some years ago. Is the technique you refer to here
the same as Mike's 'thickening' trick using 2 parallel surfaces (page 18 of
149)? Or is it something different?
If we have a function F(x,y,z) we can turn it into two parallel surfaces by
using abs(F(x,y,z))-C where C is some small value. The original function should
be one that works with zero threshold. The two resulting surfaces are what you
would get by rendering the original function with threshold +C and -C, but
combined into a single image. The space between the two surfaces becomes the
"inside" of the object. In this way we can construct things like glasses and
cups that have walls of non-zero thickness.
#declare F = function {y + f_noise3d (x*2, 0, z*2)}
isosurface {
function {abs (F(x, y, z))-0.1}
Post a reply to this message
"Kenneth" <kdw### [at] gmail> Is the technique you refer to here
> the same as Mike's 'thickening' trick using 2 parallel surfaces (page 18 of
> 149)? Or is it something different?
It is the same - yet with a slight difference.
If you're trying to make a black pattern on a white background, you're going to
want a solid black pattern, not a small gradient from -C to 0 to +C.
So you just use select to make a discontinuous function.
#declare Pattern = function {select (abs(F(x,y,z)-C, 0, 0, 1)}
And then when you do plane {z, 0, pigment {function {Pattern (x, y, z)}}}, you
get what you're probably expecting.
- BW
Post a reply to this message
"Bald Eagle" <cre### [at] netscape> "Kenneth" <kdw### [at] gmail
> > Is the technique you refer to here
> > the same as Mike's 'thickening' trick using 2 parallel surfaces (page 18 of
> > 149)? Or is it something different?
> It is the same - yet with a slight difference.
> If you're trying to make a black pattern on a white background, you're going to
> want a solid black pattern, not a small gradient from -C to 0 to +C.
> So you just use select to make a discontinuous function.
> #declare Pattern = function {select (abs(F(x,y,z)-C, 0, 0, 1)}
> And then when you do plane {z, 0, pigment {function {Pattern (x, y, z)}}}, you
> get what you're probably expecting.
Great, thanks. Yes, you were reading my mind, ha. (Actually, my own use would be
the reverse-- a white pattern on a black background.) I'll test it and play...
Post a reply to this message
I played with a few more functions - I think I'm up to about 189 in the
collection, and still have many more to implement.
This one I liked because it was "simple", and would probably find use in any
number of scenes, for different purposes.
Like the ultra-impressive Greek frieze pattern, this one got adapted from
Desmos, and it takes a lot of reading and thinking to grasp Desmos' syntax /
expression structure, unravel what is meant, and discard the superfluous parts.
I got it wrong about a dozen times, but in the process found that sometimes the
mistakes yield some pretty amazing and complex pattern in and of themselves.
So, when experimenting with new patterns, don't give up, and don't be surprised
when the raw pattern seems to bear absolutely no relation to the pattern that
you're trying to code. The raw vs "thickened" pattern can differ - starkly - in
Hopefully we can get a few new patterns posted in this thread, and I'm going to
try to post at least one new pattern a week, to keep some momentum going.
Hope you like.
- BE
Post a reply to this message
Download 'mathpatterns1.png' (59 KB)
Preview of image 'mathpatterns1.png'
"Bald Eagle" <cre### [at] netscape> I'm going to try to post at least one new pattern a week, to keep some momentum
Post a reply to this message
Download 'mathpatterns1.png' (193 KB)
Preview of image 'mathpatterns1.png'
Neat grid pattern.
Post a reply to this message
Download 'mathpatterns1.png' (15 KB)
Preview of image 'mathpatterns1.png'
"Bald Eagle" <cre### [at] netscape> "Bald Eagle" <cre### [at] netscape
> > I'm going to try to post at least one new pattern a week, to keep some momentum
I like this one.
It seems like you're getting pretty good at this.
Tor Olav
Post a reply to this message
"Tor Olav Kristensen" <tor### [at] TOBEREMOVEDgmail> I like this one.
I do too, which is why I posted it.
It brought to mind the vintage Corelle plate rim pattern. (attached)
> It seems like you're getting pretty good at this.
I'm getting a bit better at finding interesting patterns and translating them
from other languages/syntaxes.
I still have a devil of a time with some of them.
Some I can't get to work at all, especially if I have to make lines, use x and y
at the same time, or try to make a parametric function based on atan2(y, x).
I'm also trying to figure out why sometimes I only get a partial pattern.
But they _are_ more than formless grayscale vomit, so there is definitely some
When I understand WHY things work or don't, and know enough to fix them, then I
will be lots better.
When I can invent such functions on my own, then I will be good.
But that was half of the goal of this - to try implementing so many functions
that eventually I found changes that worked, and could be applied to others that
failed, and iteratively get better.
I'm almost at 200 functions, and at some point I'll have (temporarily) run out
of patterns to copy, and can start investigating variations and combinations.
There also seems to be various common methods for making patterns, and perhaps I
can distill them all down into several macros to make rendering each method a
little easier.
Thank you as always for your interest, tutelage, and encouragement. :)
- BW
Post a reply to this message
Download 'p0000016777s000000469617t1.jpg' (81 KB)
Preview of image 'p0000016777s000000469617t1.jpg'
Here's one to torture your eyes. :P
Post a reply to this message
Download 'mathpatterns1.png' (422 KB)
Preview of image 'mathpatterns1.png'
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[Solved] Two lines are respectively perpendicular to two parall... | Filo
Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other.
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According to the figure consider the two parallel lines and
we know that and
so we get
We know that and is a transversal
from the figure we know that and are corresponding angles
so we get
We also know that
We know that and are corresponding angles when the transversal cuts and
so we get
therefore, it is shown that the two lines which are perpendicular to two parallel lines are parallel to each other.
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Question Text Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other.
Updated On Jul 19, 2023
Topic Lines and Angles
Subject Mathematics
Class Class 9
Answer Type Text solution:1 Video solution: 2
Upvotes 257
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Arcsin Scalar
These nodes output the arc cosine, arc sine or arc tangent of the Input scalar. These are the inverse cosine, inverse sine or inverse tangent of the Input scalar, respectively. Using arc sine as an
example, the inverse sine of a value is the number which is the sine of the value. Put another way :
The sine of 1 is 0.017452
The arc sine, or inverse sine, of 0.017452 is 1
The output of this function is in radians. You can use the Radians to degrees scalar node to convert the output to degrees.
It is important to note that for correct results the input to the Arccos scalar node and Arcsin scalar node needs to be between -1 and 1. TG2 doesn't check for this at the time of writing. You can
use the Clamp scalar node with a Min value of -1 and a Max value of 1 to ensure this.
These nodes have no other settings apart from the Input node.
• Name: This setting allows you to apply a descriptive name to the node, which can be helpful when using multiple Arcsin Scalar nodes in a project.
• Enable: When checked, the node is active, and when unchecked the node is ignored.
Error conditions:
• Arccos scalar: It is an error for the input value to be outside the range of -1 to 1. The output value is undefined in this situation.
• Arcsin scalar: It is an error for the input value to be outside the range of -1 to 1. The output value is undefined in this situation.
A scalar is a single number. 1, 200.45, -45, -0.2 are all examples of scalar values.
A single object or device in the node network which generates or modifies data and may accept input data or create output data or both, depending on its function. Nodes usually have their own
settings which control the data they create or how they modify data passing through them. Nodes are connected together in a network to perform work in a network-based user interface. In Terragen 2
nodes are connected together to describe a scene. | {"url":"https://planetside.co.uk/wiki/index.php?title=Arcsin_Scalar","timestamp":"2024-11-15T04:55:50Z","content_type":"text/html","content_length":"20611","record_id":"<urn:uuid:1e94cdc8-2f43-4e1c-89e8-fc2daa911a3b>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00480.warc.gz"} |
The Future of Modeling (Risk Magazine)
What is the purpose of modelling, in any field? Clearly, it is divination whether foretelling the future, or controlling it. So my task here is to foretell the future of a field that itself tries to
foretell the future. To do that, I must first locate the present: what works now, and why. My view is a parochial one; I wasn’t trained as an economist, but as a natural scientist who, for the past
10 years or so, has made a living and had some fun building the models and systems used by people who trade complex, mostly derivative, securities for their living. It is interesting, though limited,
work, but it is what I know about, from the bottom up.
So let me start by giving you one view of the derivatives trading environment today: vast struggles with dispersed data and information and record-keeping, all overlaid with ambitious, sometimes
astonishingly successful, attempts to describe the underlying phenomena with the classical tools of the natural sciences. People worry about model risk, but I think the largest risks are procedural,
administrative and operational.
Given this picture. you can understand why, at Goldman Sachs, despite the models we build, the papers we write and the clients we visit, only four or five of our group of 30 people in quantitative
strategies in equity derivatives are directly involved in modelling: that is, in isolating financial variables, studying their dynamical relationships, formulating them as differential equations or
statistical affinities, solving them and, finally, writing the programs that implement the solution.
How are the models used? In brief, to value listed or over-the-counter options for market making and proprietary trading; to calculate and hedge the exposure of portfolios across many different
countries and currencies; to convert listed prices to the common currency of implied volatilities; to engineer structured derivatives; to run systems that look for mismatches between fair value and
the market; to value and hedge corporate finance instruments for arbitrage purposes; and, finally, to estimate firm-wide value-it-risk. Less frequently, we also use models directly to examine
non-derivative securities.
Models are important, as they lie beneath most of our applications but take few pure resources. Why are there so few modellers compared with programmers and system builders? And, interestingly, why
are there fewer in equities than in fixed income?
Derivatives and non-linearity
According to Professor Stephen Ross in the Palgrave Dictionary of Economics: “… options pricing theory is the most successful theory not only in finance, but in all of economics”. This seems
unquestionable, but why has it worked so well?
I think it is because the fundamental problem of options theory is the valuation of hybrid, nonlinear securities, and options theory is an ingenious but glorified method of interpolation. I don’t
mean that as an insult. Traders use options theory intuitively to understand complex, nonlinear patterns of variation in price in terms of simpler, linear changes in volatility and probability. They
do this by regarding a hybrid as a probability-weighted mixture of simpler securities, with the probability depending on the volatility. They think linearly in terms of perceived changes in
volatility and probability, and use the model to transform their perceptions into non-linear changes in price.
In the real world of traded securities, few of the assumptions of Black, Scholes and Merton are strictly respected. But their view of the hybrid nature of a stock option as a probability-weighted
mixture of stock and bond captures a core of truth that provides the foundation for the model’s robustness.
The same strategy – to think of something complex as a non-linear, probability-weighted mix of simpler things – underpins yield curve models, which let you regard swaptions as bond hybrids subject to
interpolation. Similarly, implied tree models regard exotic options as interpolated mixtures of vanilla options of various strikes and expiries.
Options theory works because it aims at relative, rather than absolute, value. A necessary prerequisite is the notion, sometimes scorned by academics, of value calibration: the effort to ensure that
the derivative value matches the value of each underlyer under conditions where mixtures become pure and certain. Without that, the relativity of value has no foundation.
Underlyers and linearity
Stock options can be likened to molecules made of indivisible atoms, where we understand the basic processes of chemistry and synthesis. The stocks themselves, in contrast, are the atoms: the
supposedly irreducible primitives that comprise the derivatives.
But this analogy is limited. In physics, we have a deep understanding of the fundamental laws of atomic physics that support chemistry, but in finance we understand the laws of options – the
molecular chemistry – much better than we do the laws of stocks. This isn’t unprecedented; advances in nineteenth-century chemistry did precede advances in twentieth-century physics. At present, our
stock model lacks deep structure or firm laws. So most traditional equity modelling resources focus on data.
Not so with bonds. Although they are the underlyers of the fixed-income world, with interest rates extracted from bond prices, people think of interest rates as the underlyer and bonds as the
non-linear derivatives, So, in this case, even the simplest instruments are non-linear and need interpolation and mathematics. That is why there are so many more quantitative modellers and computer
scientists in fixed-income areas than in equities.
Limits of traditional modelling
Where can traditional modelling work? “Theory”, in the natural sciences, has come to mean identifying the primitives and describing the rest of the world in terms of the postulated dynamical
relations between them.
But theories of the natural world involve man playing against God, using ostensibly universal variables, such as position and momentum, and universal laws such as Newton’s, that we pretend to believe
are independent of human existence, holding true forever. (I do not believe that this independence is as obvious as it seems, and furthermore, recent cosmological theories contemplate our universe
consisting of many subuniverses, each pinched off from the others, each with different laws.)
In the financial world, in contrast, it is man playing against man. But mankind’s financial variables are clearly not universal: they are quantities – such as expected return and expected risk – that
do not exist without humans; it is humans doing the expecting. Also, these variables are frequently hidden or unobservable – they are parts of the theory that are observed only as values implied by
some other traded quantity. But human expectations and strategies are transient, unlike those of the God of the physicists. So financial modelling is never going to provide the eight-decimal place
forecasting of some areas of physics.
Advances in engineering have often followed advances in scientific understanding. The industrial revolution exploited mechanics and thermodynamics. The computer revolution needed Boolean algebra and
solid-state physics. The biotech revolution of genetic engineering and immunology, which is just starting up, requires the structure of DNA and the genetic code.
Ultimately, I do not think that physics and genetics are reliable role models for finance and economics. Physics has immutable laws and great predictive power, expressed through mathematics. You
would expect its textbooks to took pure and rigorous. Finance has few dynamical laws and little predictive power and you would expect its textbooks to look discursive.
So why is it that finance books look like pure mathematics, filled with axioms, whereas physics books look like applied mathematics? The degree of axiomatisation seems inversely proportional to
applicability. This unnatural disequilibrium reminds me of an inverted yield curve, or of the put skew in equity markets: how long can it last without the crash it implies?
Black, Scholes and Merton were the Newtons of derivatives. They created and then almost completed the field, the only part of finance ripe for an industrial revolution based on principles. We are now
living in the post-Newtonian world and it will take a long time for Einstein to appear. We will continue to see the extension of derivatives models and the relative-value approach. What more can we
Extensions of ideas that work
Options theory uses the following principles: (1) The law of one price; (2) A dynamic strategy for options replication: (3) Lognormal underlyer evolution; and (4) Calibration of the model to known
market values. What extensions of these principles can we expect?
Rationality rather than voodoo. Options theory is rational and causal, based on logic. It is mathematical but the mathematics is secondary. Mathematics is the language used to express dynamics. There
are still many traders, even options traders, who have a taste for mathematics without reason – for voodoo number-juggling and patterns and curve fitting and forecasting. I think we will continue to
see successful models based on ideas about the real world, expressed in mathematics, as opposed to mathematical-looking formulas alone.
Better adjustments of the theory to the real world. The real world violates most of the option pricing principles. Liquidity issues and transaction costs mitigate the law of one price. Evolution
isn’t lognormal. Volatility is stochastic. Replication is neither continuous nor costless. Consequently, simulation shows that the profit and loss of a “risklessly hedged” option has an astonishingly
large variance when you rehedge intermittently and when you allow for the small, but inevitable, mismatch between realised and hedging volatility. How, you may wonder, do options desks make any
I think the truth is that many desks do not fully understand the source of their profit and loss. I expect to see more realistic analyses of the profit and loss of options books under practical
circumstances. Leland’s 1995 paper on transactions costs was a good start. More recently, a Risk magazine article by Ajay Gupta (July 1997, page 37) started to probe the effects of mismatches between
implied and realised volatility, similar in spirit to some analyses we have been doing at Goldman.
Forwards as a basis. Many of the advances in modelling in the past 20 years have been connected with the efficacy of using forward, rather than spot, values as the appropriate mathematical basis of a
model. This is the essence of the Heath, Jarrow & Morton (1992) approach to yield curve modelling, and similar ideas can also be applied to volatility. Recent work on market models of interest rates
by Brace, Gatarek & Musiela (1977), Jamshidian (1996) and others is also closely connected to this concept.
Calibration. A good trading model must both match the values of known liquid securities and realistically represent the range of future market variables. Very few models manage this. Academics tend
to favour those with a realistic evolution but practitioners who hedge cannot live without well-calibrated models; it is no good valuing an option on a bond with a model that misprices the underlying
bond itself. If I were forced to choose, I would prefer to calibrate determinacy first – that is, to get the values of known securities right – and hope for robustness if I get the stochastics a
little wrong. Obviously, that’s not perfect. I hope to see progress in building models that are both market calibrated and evolutionarily realistic.
The wisdom of implied variables. There is little certain knowledge about future values in finance. Implied values are the rational expectations that make a model fit the market, and provide the best
(and sometimes the only) insight into what people expect. During the recent stock market correction, the pre-crash implied volatilities of options with different strikes gave a good indication of the
level and variation of post-crash, at-the-money implied volatilities. I expect to see modelling based on implied variables – implied forward rates, volatilities, correlations and credit spreads –
continue to grow in applicability and sophistication.
Traded variables as stochastic factors. A few years ago, there was a tendency to build stochastic models based on whatever principal components emerged from the data, no matter how arcane their
groupings. The current fashion, factors that represent traded instruments, seems sensible. Market models of interest rates are an attractive step in this direction. They model directly the evolution
of traded, discrete securities, and intuitively justify simple pricing formulas. I like models whose stochastic factors can be grasped viscerally. Finance is still too immature to rely on esoteric
dynamical variables.
Changes of numeraire. This method, pioneered by Margrabe (1978), seems to keep re-emerging as a tactic for simplifying complex problems by reducing them to simpler, previously solved problems when
viewed in a different currency.
Techniques of limited value
Optimisation. Optimisation sounds vital to people who do not work in the industry, but I don’t find it that useful in practical finance. I am a little embarrassed to admit that we rarely use
optimisation programmes in our equity derivatives options group at Goldman. In engineering, where the laws are exactly understood, or in travelling-salesman-style problems – where one optimises over
many possible paths, each of whose length is exactly known – optimisation is sensible. One is simply trying to isolate the scenario that produces the best well-specified outcome.
In financial theory, in contrast, each scenario is inexact – there is a crude interest rate model, a crude prepayment model and other misspecifications. While averaging may cancel much of the
misspecification, optimisation tends to accentuate it. So I am largely a sceptic about optimisation in finance, although that is not to say that it never makes sense, just that it should be used with
The capital asset pricing model. This provided the original framework for the Black-Scholes equation, and its ideas about risk and return loosely permeate all thoughts about trading. In practice, we
do not use it much.
Large dimension problems. Financial theory seems more solidly successful when applied to problems with a small number of dimensions.
New directions
Underlyer modelling. We need more sophisticated models of underlyers, but we lack any good general laws beyond lognormality. In the real world, there are fat tails, jumps, exchange rate bands and
other so-called anomalies. Classical physics starts with the certainty of single particle dynamics and proceeds to the statistics of ensembles. in finance, even a single stock suffers from
uncertainty. The broadest theoretical advance would be some new theory of underlyers; perhaps there is some way to “derivitify” underlyers by regarding them as dependent on more primitive quantities.
But I know of nothing, from behavioural finance to chaos theory, that is ready to for real applicability.
Computing and electronic markets. Computing will continue to be the driving force behind financial markets. Fast computation will allow electronic market making and automated exchanges for options as
well as stocks. Expect faster trading, fewer intermediaries and more direct access to capital. Trading systems will have to accommodate these changes. Fast access to relevant information is even more
important in electronic markets. Limited artificial intelligence models will find their use in areas where information is vast and logic is limited. Rule-based systems might work well here. It is
easier to see the advantages of computing power than of models. Furthermore, computers will have increasing value in displaying and examining multi-dimensional risk profiles.
Market microstructure. Most financial models assume an economic equilibrium. The approach to equilibrium in models of market microstructure is becoming a fertile area. I recently heard an interesting
talk at the Society for Quantitative Analysts in New York by Charles Plott of the California Institute of Technology on trading experiments that observe the approach to price equilibrium. This type
of work will ultimately help organise market making systems and tie them ever more closely to hardware and software.
Statistical arbitrage. I am unsure what to predict here. I am always struck by the difference between statistics in physics and in finance. First, in theory. In physics, the microscopic laws of
mechanics and the macroscopic laws of thermodynamics were ultimately joined in statistical thermodynamics. In finance, both the macroscopic intuition and the microscopic laws are sometimes missing,
yet modellers still like to apply statistics and optimisation.
Second, experiment. In the natural sciences, theory is compared with experiment via statistics. Kelvin is supposed to have said that if, as an experimenter, you actually need statistics, then you did
the wrong experiment. In finance. researchers sometimes do statistical analysis first and then look for a theory. I am a great believer in thoughtfulness and causality. I would hope to see
researchers think more about the causal dynamics between particular underlyers, propose models of cointegration, and then test them using the data.
Value-at-risk. The VAR problem is mostly operational: how do you get all of a firm’s positions and pricing parameters in one place at one time? With that in place, you can run a Monte Carlo
simulation to calculate expected losses. This is useful, but is no substitute for the much more detailed scenario analysis and common sense and experience necessary to run a derivatives book. There
is no short cut to understanding complexity. For a view of the practicalities of portfolio risk management in a trading environment, see Litterman (1996). From the theoretical point of view, Cornell
University’s David Heath et al have written some interesting notes on the axiomatic requirements for consistent measures of value-at-risk (Risk November 1997, pages 68-71).
Some recent sociological changes in the modelling world. Being a geek is now officially cool. You don’t have to apologise about talking mathematics in the elevator any more.
Financial theory seems to be moving out of business schools in both directions, leftwards to the sciences and rightwards to real businesses. On the one hand, sophisticated financial research now
thrives on Wall Street, perhaps even more than in universities. There has been a mass exodus of skilled financial theorists into the banking arena. Even textbooks refer to theories created by
practitioners. On the other hand, financial theory is also becoming part of an applied mathematics curriculum. Mathematics departments give financial engineering degrees, and mathematicians write
books on options valuation. Applied mathematicians get PhDs in options pricing with transactions costs.
Options valuation models are becoming commoditised and cheaply available. Companies that write risk management systems are going public. Risk consulting is lucrative and commonplace. Big firms still
prefer to do it themselves, but smaller ones can buy or contract most of what they need.
From the viewpoint of someone who works with traders, I like to think of models the way quantum physicists used Gedanken experiments, as a sort of imaginary stress-testing of the physical world done
in your head, or on paper, in order to force your picture of the world into a contradiction. Einstein, in thinking about special relativity, imagined what he would see sitting on the edge of a moving
light beam, while Schr?dinger’s contemplation of quantum mechanics famously led him to imagine a cat in a sealed box with a radioactive atom that would trigger a Geiger counter that would release
I think that is the right way to use mathematical models in finance. In most cases, the world doesn’t really behave in exactly the way you have constructed it. You are trying to make a limited
approximation of reality, with perceptual variables you can think about, so that you can say to yourself “What happens if volatility goes up, or if the slope of the yield curve changes?” Then you can
come up with a value based on what you can understand and describe.
You have to have a sense of wonder, almost a suspension of disbelief, when you observe desks using quantitative models to value and hedge complex securities. Think of a collateralised mortgage
obligation: you use an (at best) quasi-realistic model for interest rate evolution and a crude model for prepayments, and combine both to simulate thousands of future scenarios to value the curvature
of the mortgage. Then you pay for it, based on that value. It’s not quite preposterous, but it is amazing. The strongest reasons for trusting it are that it is rational and thoughtful, and there is
nothing better. It will probably continue to be that way. But I think that’s good news.
Emanuel Derman is a managing director in the quantitative strategies group at Goldman Sachs in New York. This article is based on a speech presented at the Risk Tenth Anniversary Global Summit In
London on November 19 and is © Goldman Sachs, 1997. It reflects the personal views of the author and does not necessarily reflect the views of Goldman Sachs. | {"url":"https://emanuelderman.com/the-future-of-modeling/","timestamp":"2024-11-05T07:19:08Z","content_type":"text/html","content_length":"61872","record_id":"<urn:uuid:74dc2535-2144-4c93-a0ab-a89b9007d86a>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00334.warc.gz"} |
Apply element-wise operation to two arrays with implicit expansion enabled
C = bsxfun(fun,A,B) applies the element-wise binary operation specified by the function handle fun to arrays A and B.
Deviation of Matrix Elements from Column Mean
Subtract the column mean from the corresponding column elements of a matrix A. Then normalize by the standard deviation.
A = [1 2 10; 3 4 20; 9 6 15];
C = bsxfun(@minus, A, mean(A));
D = bsxfun(@rdivide, C, std(A))
D = 3×3
-0.8006 -1.0000 -1.0000
-0.3203 0 1.0000
1.1209 1.0000 0
In MATLAB® R2016b and later, you can directly use operators instead of bsxfun, since the operators independently support implicit expansion of arrays with compatible sizes.
ans = 3×3
-0.8006 -1.0000 -1.0000
-0.3203 0 1.0000
1.1209 1.0000 0
Compare Vector Elements
Compare the elements in a column vector and a row vector. The result is a matrix containing the comparison of each combination of elements from the vectors. An equivalent way to execute this
operation is with A > B.
C = 4x3 logical array
Expansion with Custom Function
Create a function handle that represents the function $f\left(a,b\right)=a-{e}^{b}$.
Use bsxfun to apply the function to vectors a and b. The bsxfun function expands the vectors into matrices of the same size, which is an efficient way to evaluate fun for many combinations of the
a = 1:7;
b = pi*[0 1/4 1/3 1/2 2/3 3/4 1].';
C = bsxfun(fun,a,b)
C = 7×7
0 1.0000 2.0000 3.0000 4.0000 5.0000 6.0000
-1.1933 -0.1933 0.8067 1.8067 2.8067 3.8067 4.8067
-1.8497 -0.8497 0.1503 1.1503 2.1503 3.1503 4.1503
-3.8105 -2.8105 -1.8105 -0.8105 0.1895 1.1895 2.1895
-7.1205 -6.1205 -5.1205 -4.1205 -3.1205 -2.1205 -1.1205
-9.5507 -8.5507 -7.5507 -6.5507 -5.5507 -4.5507 -3.5507
-22.1407 -21.1407 -20.1407 -19.1407 -18.1407 -17.1407 -16.1407
Input Arguments
fun — Binary function to apply
function handle
Binary function to apply, specified as a function handle. fun must be a binary (two-input) element-wise function of the form C = fun(A,B) that accepts arrays A and B with compatible sizes. For more
information, see Compatible Array Sizes for Basic Operations. fun must support scalar expansion, such that if A or B is a scalar, then C is the result of applying the scalar to every element in the
other input array.
In MATLAB^® R2016b and later, the built-in binary functions listed in this table independently support implicit expansion. With these functions, you can call the function or operator directly instead
of using bsxfun. For example, you can replace C = bsxfun(@plus,A,B) with A+B.
Function Symbol Description
plus + Plus
minus - Minus
times .* Array multiply
rdivide ./ Right array divide
ldivide .\ Left array divide
power .^ Array power
eq == Equal
ne ~= Not equal
gt > Greater than
ge >= Greater than or equal to
lt < Less than
le <= Less than or equal to
and & Element-wise logical AND
or | Element-wise logical OR
xor N/A Logical exclusive OR
bitand N/A Bit-wise AND
bitor N/A Bit-wise OR
bitxor N/A Bit-wise XOR
max N/A Binary maximum
min N/A Binary minimum
mod N/A Modulus after division
rem N/A Remainder after division
atan2 N/A Four-quadrant inverse tangent; result in radians
atan2d N/A Four-quadrant inverse tangent; result in degrees
hypot N/A Square root of sum of squares
Example: C = bsxfun(@plus,[1 2],[2; 3])
Data Types: function_handle
A,B — Input arrays
scalars | vectors | matrices | multidimensional arrays
Input arrays, specified as scalars, vectors, matrices, or multidimensional arrays. Inputs A and B must have compatible sizes. For more information, see Compatible Array Sizes for Basic Operations.
Whenever a dimension of A or B is singleton (equal to one), bsxfun virtually replicates the array along that dimension to match the other array. In the case where a dimension of A or B is singleton,
and the corresponding dimension in the other array is zero, bsxfun virtually diminishes the singleton dimension to zero.
Data Types: single | double | uint8 | uint16 | uint32 | uint64 | int8 | int16 | int32 | int64 | char | logical
Complex Number Support: Yes
• It is recommended that you replace most uses of bsxfun with direct calls to the functions and operators that support implicit expansion. Compared to using bsxfun, implicit expansion offers faster
speed of execution, better memory usage, and improved readability of code. For more information, see Compatible Array Sizes for Basic Operations.
Extended Capabilities
Tall Arrays
Calculate with arrays that have more rows than fit in memory.
The bsxfun function supports tall arrays with the following usage notes and limitations:
The specified function must not rely on persistent variables.
For more information, see Tall Arrays.
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
• Code generation does not support sparse matrix inputs for this function.
GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.
Usage notes and limitations:
• Code generation does not support sparse matrix inputs for this function.
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
The bsxfun function supports GPU array input with these usage notes and limitations:
• See bsxfun (Parallel Computing Toolbox).
Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.
This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).
Version History
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A Point About Polygons
An essay on the aesthetics of polygons and algorithms that one might see in a web image map.
Several algorithms exist in the public domain for web servers to determine whether a point is inside a polygon. They are used in the implementation of “image maps”, both of the traditional
server-side variety as well as those of the more modern client-side. So who needs one more? Well, the bone this author wishes to respectfully pick is that most of the point-in-polygon code he could
find is woefully over-complicated. Being a lover of simplicity and simplification, he just could not leave well enough alone.
The resulting C-language routine has just three if statements and no divides. Contrast that with three divides and ten if statements in the corresponding routine that's part of the popular Apache web
server. Get the Apache distribution and search for pointinpoly to see the whole works. The routine from CERN/W3C's httpd is even worse, weighing in at 19 if statements! Search for inside_poly in
their HTImage.c. (The URLs are shown in Table 2.)
Table 1 contrasts five different routines in the public domain for finding out if a point is in a polygon. In all cases, the polygon is specified as an array of X,Y coordinates of the corner points.
Table 1. Comparison of Point-in-Polygon Algorithms
This is a pretty casual analysis of the algorithms. I certainly didn't shy away from showing my inpoly() in a good light. For example, && and || operators in C are often statements in disguise. I
used one of these operators (as did most of the other folks), but that doesn't show up in the table at all. Also, some line counts are inflated slightly by comments and blank lines. But you get a
rough idea.
Table 2. Sources of Public Domain Point-in-Polygon Algorithms
All judgments have a context, and I should explain mine. The primary prerogative in this article is algorithmic simplicity. This, I confess, has very little to do with the practical needs of the Web.
In case I've gotten ahead of myself, a web image map is a way of carving up an image so that clicking in one particular region does one thing, and clicking somewhere else does something else. Web
image maps are such a tiny fraction of the work of web servers and browsers that all of the above routines are just fine as they are. Changing from one to another is not going to make any noticeable
difference in web performance. And once we're sure it works, who's going to look at the code again for 100 years? Thus I don't have any practical considerations of performance or readability to
justify my cause. I'm simply championing the aesthetics of simplicity.
The point I wish to make is that problems are not always what they seem. Sometimes a simple solution exists, but you've got to take a hard look to find it. My buddy Craig had started out by porting
inside_poly() from W3C, I think it was, for use on our web server. When I saw all the floating point math and if statement special cases, I thought there had to be a better way. So Craig and I
started from scratch, wrestled the problem to the ground, and came up with a solution containing no floating point math, which is silly for screen pixels, and no math more tedious than
multiplication. We also got rid of all the pesky special cases, except for one: polygons with fewer than three sides are excluded. What could be inside a two-sided polygon? Apache's pointinpoly()
doesn't even check, and probably makes a big mess with a one-point polygon.
Now, the stated goal is simplicity, not performance, but I did stray from that course on one issue: avoiding divides. Again, performance hardly matters for image map applications, but one day someone
might use this algorithm for some kind of 3D hidden surface algorithm or something. Getting rid of the divide may have, in effect, required me to use an additional if statement. Anyhow, what all this
is leading up to is that Kevin Kenny's algorithm (see Table 1) at 29 lines and two if statements is by far the shortest and simplest. But mine is still better in some sense, because mine doesn't need
a divide and his does.
Now let's discuss the more popular algorithm for determining whether a point is inside a polygon.
Imagine you could detect whether a point was in a polygon or not by placing a friendly trained snail at the point and telling him to head for the North Pole. (We're only concerned with image maps, so
we exclude polygons that extend to the North Pole, and we ignore Coriolis forces.) You'd equip our intrepid friend in Figure 1 with a snail-sized clipboard and instruct him to tick off each time he
crossed an edge of the polygon. He'd call you from the North Pole and report the number of crossings. An even number (including zero) means he started outside the polygon, odd means inside.
This reduces the problem to detecting whether or not line segments intersect. It's even a little better than that, because one of the line segments is simply the positive Y axis. To make that leap,
just declare the snail's starting point to be the origin, (0,0), and translate all of the polygon corners so they're relative to that point.
We'll go into the algorithm a little later, but take a look at the finished code in Listing 1. The very picture of simplicity, right? If you haven't checked out the other versions, you really ought
The test program (Listing 2) draws a random 40-sided polygon and then picks random points to throw at the inpoly() routine. Points the routine says are inside the polygon it draws red, points outside
are blue.
Our first rendition of inpoly() had a subtle flaw which the test program made evident. The full story contains an embarrassing lesson. “It'll work,” we sneered, “We don't need to waste time on a full
graphical test. Besides, it'd be too much fun.” After we found out our image maps had leaks, we wrote the test program. Figure 3 shows a close-up of the flaw.
Along a vertical line, all the colors are wrong. The flaw turned out to be that when our mindless mollusk crosses the bottom corner, the little hummer was counting the crossing of both edges! After
that, he was always exactly wrong—he thought he was in when he was out, and he thought he was out when he was in. The solution must ensure that when our esteemed escargot crosses into the polygon
corner, he counts exactly one crossing. Two is no good, and in fact, zero is just as bad—one is what we need. The reason the flaw in the close-up extends up from the corner is that the positive Y
axis extends downward in screen coordinates.
I suspect this is a problem unique to the fixed-point world. I'm sure my fellow point-in-polygon smiths have either lucked out or dealt with it somehow. At least, I'd like to think so. (A
lie-detector would peg me on that one. This article would be insufferably smug if I had found leaky corners in any of the other algorithms.) In my case, I realized I could not blindly count all
crossings of the end point of each of the edges as a crossing. My first thought was to associate each end point with one—and only one—edge. This sounds fair and equitable, but like many things
fitting that description, it just plain won't work. A problem turns up when Agent Snail just lightly nicks the corner of a polygon he's not inside at all. That's counted as one crossing, hence the
snail report is bunk.
Since I abhor special cases, I sought something that would work in all cases.
The scheme for getting our faithful friend to count corner crossings correctly is to always count a crossing of the right end of each edge, but never the left end (right meaning positive X). In the
figure, the black circles represent points our snail will count if he crosses; the white circles he won't count. When you put the polygon together, everything ends up the way we want. Nicking the
corner means he counts either 0 crossings or two crossings. We don't care which; both are even and our snail knows he's outside. The circles with ones in them represent points counted once if the
snail crosses them. This is fine, just like crossing the nearby sides.
It's time to analyze the guts of the inpoly() routine in Listing 3. This represents a slight modification of the snail's instructions. He plays a bit of a “she loves me, she loves me not” kind of
game rather than counting up the crossings and then reporting whether the total is even or odd. He starts out assuming he's outside, and complements that assumption with each crossing. So much for
the inside=!inside statement.
Listing 3. The “Guts” of the inpoly() Routine
This if test happens inside a for loop that considers all of the edges of the polygon, one at a time. Each edge is a line segment that stretches between the corners (xold,yold) and (xnew,ynew). We've
arranged it so (x1,y1) and (x2,y2) also represent the same edge, but the points are swapped, if necessary, to make it so x1 <= x2.
Now two things must be true for our ever-meticulous snail to count the crossing of this edge. First, the segment must straddle the Y axis (where the right end is counted but the left one is not).
Second, straddling has to happen to the north of the snail's starting point. These are exactly the questions determined by the if statement's two pieces, on either side of the &&.
Now that first expression is a sneaky one, and I confess I might have preferred the less opaque code (x1 < xt && xt <= x2). You can see it does the same thing if you look carefully (very carefully—I
was fooled for a while there). But I hate to fix something unless I've already broken it, if you know what I mean.
That north computation is the one I'm proud of because none of my esteemed fellow polygon smiths made one that doesn't need a divide. It does depend on the knowledge that (x2-x1) is positive. Other
than that, it's just a transmogrification of that famous y=mx+b equation from high school algebra.
By the way, I've left out the case where an edge line segment stands straight up and down above the snail touchdown point. Such an edge would never be counted by Mr. Snail at all! That's because the
== test would always be false, since xnew, xt and xold are all the same value. What's really wild is that's just what we want. In a sense, he's crossing three edges when we only want to count one. It
turns out the adjacent line segment crossings are all we're interested in, and the rules already discussed work perfectly for them.
By the way, who cares whether the points in an image map along the edge of a polygon are technically inside or outside? As you can see in the close-up, some of the originally white pixels
(representing the polygon edge) turned to red, others to blue. If a browsing user clicks on the edge of a region, he may get in, he may not. But being one pixel off is usually not an issue if your
screen resolution is greater than 100 x 100. In the inpoly() routine, some edges are in, some are out. (I don't mind admitting to a crime after convincing everyone it deserves no punishment.)
I haven't discussed the angle-sum method used by Woods Hole Oceanographic Institution for their algorithm written in Matlab. The algorithm needs to compute arc-tangents, so it's mostly just a
laboratory curiosity. The idea is that you add up the angles subtended by lines drawn from the target point to each of the corners of the polygon. If the sum is an even multiple of 360 degrees,
you're out; odd, you're in. Vaguely familiar? Here's the analogy: You're in a pitch-black room with a very, very long snake all over the floor. This is a particularly rare variety of deep sea snake
(Woods Hole knows all about them) with glow-in-the-dark dots every foot or so. Oh, and he reacts to light by instantly constricting in an iron grip of death. Your question is whether you're standing
inside the maze of coils at your feet or outside. You'd like to know before you turn on the light because he gets very annoyed if you step on him.
Face the head of the snake and visually trace his entire body, somehow noting as you do how your feet turn (it's a stretch I know). When you're done, face the head again. Now, if you didn't have to
turn around at all, you're safely outside the snake. If you turned around twice in either direction things are fine too. Four times, and you're still OK. If you turned around an odd number of times
in either direction, you're meat—no wonder folks tend to use the crossing-count algorithm. | {"url":"http://nnc3.com/mags/LM18/LJ/035/2029.html","timestamp":"2024-11-12T03:23:51Z","content_type":"text/html","content_length":"18977","record_id":"<urn:uuid:0c7ddda7-f998-40d0-9165-88e161461647>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00277.warc.gz"} |
Irradiance in GIA and Efficiency | Zemax Community
Something does not addup in the calculation of power and efficiency in my spectrometer setup. I have a following reported Irradiance, using 7pixels and 538 pixels( equivalent to 13um pixel size). I
used the Total Watts of 1 and I used wavelength input of 255 to 345nm with 5nm apart, which all has the weight of 1.
The total efficiency is reported to be 100%. This means 1Watt input should correspond to 1Watt output on the detector to my understanding. However, when I take the numerical data
After adding all the Irradiance together and multiply them by pixel size (13*13*1e-4), I get only 0.35 which is far from 1Watt. How does that work?
Additionaly I have looked at the result for each wavelengths seperately with 1 Watt as total Watts.
I added all the raddiance values over the y-position for each wavelength individualy and multiply that with pixel size (13*13*1e-4). I added at the end all the results together to cover for all the
wavelengths, this gave me 5.5Watts. Therefore, I donot understand the 100% efficiency here and why I have more than 1Watt.
Thank you | {"url":"https://community.zemax.com/got-a-question-7/irradiance-in-gia-and-efficiency-1652?postid=5429","timestamp":"2024-11-12T00:24:37Z","content_type":"text/html","content_length":"170748","record_id":"<urn:uuid:0f234b0f-91d5-41b9-972c-dd3814c94ebd>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00470.warc.gz"} |
Show HN: Probabilistic Tic-Tac-Toe
Nice! And irritating! I would make it a lot faster though. It takes so much time waiting for the animations to finish.
Too much time to load too (ditch the overkill 3D engine, there are lighter frameworks out there).
Cool game though. I am still puzzled by how the probabilities are arrived at. Random?
Agreed that 3D is overkill. I'm fastest at prototyping in Unity though and this was only a couple day project, so I'm unlikely to port it to anything else.
Probabilities are mostly randomized during board generation but skewed in a way to make gameplay feel a bit better. There's a cap on the likelihood of the neutral event, and a bias towards the good
event rather than a bad one.
I got the die to settle on an edge in the corner of the playfield, which triggered a re-roll.
Is this actually simulating a d20 with physics?
Can you please share the specifics? I'm trying to make my own AI for this game, and would like to compare mine against random play to estimate its strength.
Also, in your listing of your ai beating the random, how are you counting drawn games?
The current code for board generation is as follows:
var neutralChances = Random.Range(1, MaxNeutralChances + 1);
square.GoodChances = Random.Range(MinGoodChances, 20 - neutralChances);
square.BadChances = 20 - (square.GoodChances + neutralChances);
MaxNeutralChances and MinGoodChances are both set to 6 in the release build. Note that one chance is equal to one face of the die, so 5%. Also, this overload of Random.Range() has an inclusive min
value but an exclusive max value.
I guess I didn't include ties in that little blurb I wrote up, but the real results of my 10k trials were around 5:1:11.5 (lose:tie:win) for the AI vs random actor.
Would love to see your AI when it's done! Please shoot me an email if you want. My email is in my profile / in the site footer.
Why an AI? Just for the fun of implementing it (totally valid, just curious)? Given the probabilities of the outcomes couldn't you just "solve" for the best way to play it based on expected value?
If there are 65% and 50% to complete a row in one direction, and a 35% and 20% in another direction, you don't really need AI to tell you which one would be more advantageous to go after?
Yes, I've made what is intended to be a perfect solver (Although it in some testing it's clearly making mistakes, so I have some debugging to do yet). I'm making it because I was nerd sniped into
thinking through how to handle some of the trickiness with the solving. It's not an AI in the LLM or machine learning sense, but in the previously common use of the term (eg, deep blue), or in the
video game sense.
Very cool! Best of luck working on it!
That's cool — use what you're comfortable with.
Myself, I am trying to create lightweight 3D code to sit on top of Canvas and HTML5. That may be why I was sensitive to the "overkill", ha ha.
Agreed. Taking the time to roll the dice is important the first few times, to fully cement the idea of the game. After that it gets annoying.
To be specific, you could probably even leave the roll time as-is, to give you that suspense, but the time it takes to move the die to the center, flash it, and flash the result, is too long and gets
That's good feedback, thanks. I've added a fast forward button to the top left.
You overdid it.
1x is right at first, but 2x is really fast.
An extra click that would stop the animation immediately might be helpful.
Or to turn this into a different game: the d20 stops fast by default, but extra click to cheat and keep it rolling if you feel that it's about to stop on an unfavorable face.
I made some updates to speed up the UI, and improved the computer player, as I was interested in finding the optimal strategy: https://keshav.is/coding/pt3
Harder for humans, but easy to make a really strong AI for this. Even overcounting because of illegal board states (multiple winners) and not even bothering to eliminate symmetries, there are at most
2 * 3^9 = 39366 board states.
There are cycles in the board state graph, although they are of a very specific form (the only kind of cycle that exists is for board B with O and X alternating turns). So it is probably possible to
make a completely deterministic and optimal algorithm for this probabilistic game, but it does sound complicated. You can't naively apply expectiminimax.
However after marking the winning board states as 0 or 1 respectively if O or X wins I would expect value iteration to very quickly converge to an optimal strategy here.
>Harder for humans
IDK if it's just me, but I went 6-0. Is something wrong with the computer player logic?
The OP mentioned that the AI they programmed is using a simple heuristic.
What happens when you play against yourself?
i went up 8-1 and 6 games later it was 8-7
1/64 chance.
What happens if you play 6 more?
I think this is doable. Say we assign a win rate W(S) to each board state S, and let W(S, A) denote the win rate after taking action A from state S. Since the transition is probabilistic, we can
W(S, A) = P(good) * (1 - W(S_good)) + P(bad) * (1 - W(S_bad)) + (1 - P(good) - P(bad)) * (1 - W(S))
And obvisouly:
W(S) = max(W(S, A), foreach A in Actions)
max() is troublesome, but we can replace it with a >= sign:
W(S) >= (W(S, A), forall A in Actions)
And then we can expand W(S, A) and move W(S) in each of the inequalities to the left hand side. After all that we will have a bunch of linear inequalities which can be optimized with linear
programming. I think the objective would just be:
<del>maximize</del> minimize W(empty_board)
Yep, this is what I ended up doing as well! With how the game generate boards, the player that goes first always have a ~5% advantage. Since players switch hands each around they should have 50% win
rate if both play optimally.
In practice, playing against author's AI I barely get ~60% win rate (small caveat, I count ties as 0.5 to both players). What about yours?
Edit: nvm I saw you did the same with ties.
I think you have an error in the equation defining V(s).
You have component n_c * V(s) for the 'nothing happened' case, but I don't think that's correct. If you rolled that nothing happens the turn still passes to your opponent, so I think it should be n_c
* V'(s).
Turns out linear programming is not fast... Takes about 90 minutes to find the optimal solution for any board configuration.
I think you will find it extremely difficult to do better than simply checking the probability that each square gives you a spot times the number of victory paths it opens up minus the probability
that it gives your opponent a spot times the number of victory paths it opens up for them. Add another clause for paths closed if you want.
Since chance is involved, you will basically never want to do anything but the greediest highest value next action. Sometimes more than half the board has net value of 0 or less which makes them very
easy to ignore.
Wouldn't you also have to take into account the probability of the following moves also being successful and giving you a win?
No because the odds are symmetric for every slot. If you have two in a row; and the third slot has higher odds that it goes to the non-roller, you should just... not roll in it.
The timing of when it gets rolled won't matter. The need to urgently consider blocking off other routes to victory will be embedded in the scoring described above.
> Since chance is involved, you will basically never want to do anything but the greediest highest value next action. Sometimes more than half the board has net value of 0 or less which makes them
very easy to ignore.
Since passing is not an option, you can't ignore a net value of 0 or less, because all options might have a net value of 0 or less.
Sure. But there's still no conceivable situation where it is advantageous to pursue such an option while a net positive option exists. Ergo, it is easy to ignore.
WRT to computing an exact solution, something something markov chains, transition matrices, eigenvalues. I think it is tractable
Usually those are for additive/linear systems, the problem with game theoretic graphs like these is that you alternate between max and min nodes, so the system is highly nonlinear.
You're right.
I'll work on the simpler problem of :) / :( first. I think that can be done with just minimax
And then maybe win chance for each possible state of a purely random game
If it were just solely :) / :( then it is a freshman's exercise in expectiminimax.
I think this fails to take into account that your opponent can also roll 'meh', making it your turn again.
Brilliant! Makes a simple children's game very interesting. One aspect I really enjoy is that it makes clear the knee-jerk response towards action bias[0]. There are times when your opponent has
two-in-a-row, but the probability of a frown on the third is > 50%, in which case it's in my interest to have my opponent click on the third square instead of me (but even knowing that cognitively,
it's still hard to not action).
[0]: https://en.wikipedia.org/wiki/Action_bias
As someone who often prints boardgames, this strikes me as a game that would be very easy to build a physical version of, just printing some tiles with random distributions printed on, and finding
some tokens and a die to use. It would make a compact travel game. I do not think there would have to be a huge number of tiles. A few more than nine ought to be enough?
you would need different dice for each distribution but you could use a normal d20 and a lookup table for less needed equipment.. I think it could work!
A D20 would be more than enough, you just put the probabilities of the tiles in terms of 20 digits.
I'm saying if you wanted to mimic the happy/meh/sad faces on the die in this game you would need multiple dice. But since all of the percentages are in 5% increments yes a D20 is all you would need.
I suggest a lookup table for the players who are uncomfortable or uninterested in doing the percentage division in their head, and the sum of the two lower partitions. But you've got me thinking you
could probably also have a run of custom-labeled D20s with increments of 5 instead of 1 to eliminate the LUT.
The square would look like:
1-5 sad face 6-9 neutral face 10-20 happy face
more or less like an ability check in DnD
Yes, I think we're saying the same thing. Your second sentence is exactly what I mean by lookup table.
But you don't need a lookup table. The square itself would literally have that printed on it. No percentages.
but that is what a lookup table is -- something to turn the number on the die into the happy/meh/sad outcome. There would be nine on each card.
Neat idea! The computer keeps beating me with its basic strategy.
I also managed to get the die stuck on a side with an edge pointing up, to where the game couldn't choose a face. I thought it was going to brick the game, but it detected this and re-rolled the die.
Great game. I find it interesting how impactful neutral rolls are. Whoever is last to place a mark can be forced to act against their own interest, making a roll that is likely to result in the win
for the opponent. But rolling the neutral face skips the turn, changing who places the last mark.
> So what gives us the right to claim responsibility for our victories? Do we ever truly win? Or do we just get lucky sometimes?
> Well, in any given game of Probabilistic Tic-Tac-Toe you can do everything right and still lose (or do everything wrong and win.) However, the better player always rises to the top over time.
> Bad breaks are inevitable, but good judgment is always rewarded (eventually, and given enough chances.)
This assumes that everyone is on a level playing field with only non-compounding randomness preventing the better player from winning. But as you point out, luck does compound over time:
>The parents we’re born to, societal power structures... so many past events have an invisible impact on each new action we take
This is commonly known as the rich get richer and the poor get poorer, and to economists as the Matthew Effect[1].
You could try to model this in the game by having wins skew the odds of the next game in your favor. It's harder to model in a simple two person game like this... You have to persist state for a
population of players over time.
I've wanted to publish alternate rules for Monopoly, where at the start of the game players don't get the same amount of cash. Cash is instead distributed according to real statistics for "birth
wealth". Alternatively, your cash at the end of a game roles over into the next game.
I'd love to discuss this with you if you are interested. We might even collaborate on a future project.
[1]: https://en.wikipedia.org/wiki/Matthew_effect
As a longtime XCOM veteran, I am constitutionally opposed to making any game move with less than an 85% chance of success.
Took me a minute to catch on - just play the odds! At first I was trying to play tic tac toe, but the winning strategy seems to be to go for the square likeliest to land your own mark.
That's how the AI seems to play it. :)
Yes, the AI mostly just looks for plays that have high certainty and are connected to other potential winning squares (for either team). Then it weights plays positively or negatively based on
whether or not the "bad" chance outweighs "good"
I don’t think this is right though. I watched it pick a 70 it didn’t need over a 60 I used to win.
I love this!
Quick feature request: the die-roll is really cool, but can you make a lower-latency version so I can play more games in less time?
Thanks! I've added a fast forward button to the top left so that you can play faster.
UI suggestion: show the probabilities for a move as a point in a triangle, with your outcome labels on the vertices. (Or maybe as red/green/neutral colors in the triangle's interior.) This
representation is called the "probability simplex". It would look less busy, quicker to scan, I think.
Doesn't have stable positions.
I was able to consistently get about 2:1 lead over the ai by balancing the center and corners as valuable and trying to force it to play bad squares. It's a good amount of randomness tossed in.
The do nothing move is a nice touch
I think the AI should be optimized to not make plays that look obviously bad. It doesn't really need to be any harder, but it kinda ruins it when it makes a play that seems really obviously bad to
Also does it simply never play the center? It seems center is never an outlier probability but also feels like the AI should play it sometime. (edit: After 20 or so games it finally did. Maybe I was
just overvaluing it? Although I'm winning about 80%.)
These suggestions are all about the feel of playing the AI rather than difficulty.
It doesn't require AI to solve the game. It's possible to do with probabilistic theory, dynamic programming and game theory (minimax)
What engine is this using?
Any recommendations for creating simple 3d visualizations of orbiting spheres? Something like the one from the link, but more web-native (instead of python)?: https://trinket.io/glowscript/a90cba5f40
Unity per the logo on the loading screen
Totally changes the game for me. Makes it so that you (almost) never want to play the middle square unless your hand is forced.
Also reminds me of how I was playing Senet last night. I controlled the game until the very end, where by chance, I kept rolling "bad" numbers and my opponent kept rolling "good" numbers.
Middle square is actually pretty good. Solid odds of the opponent giving you a layup with a bad break.
The odds on each square change every game. I didn't realize that at first.
Really cool, lost two games in a row, finally won my third one after the computer got extremely unlucky.
I was wondering if I maybe experienced a bug. Do you shortcut drawn games when neither player can win?)
Yes, those should go straight to a "Tie" result.
I'm not 100% sure, but I think it didn't display the outcome of the dice in this case. And it would be nice to have some hint that shows that this game ends in a Tie.
I just played 100 rounds of this game, winning 47 times, tying 6, and losing 47. Very fun. I think it would be cool if I could look back at my previous games and figure out more optimal strategies so
I could possibly get the slightest edge on the CPU.
This is just awesome, great idea. The computer doesn’t seem to defend against obvious, probabilistic winning moves (doesn’t block a final square). But funnily this works to its advantage sometimes if
you end up rolling frowny
This just reminded me why I hate output randomness in games. Lost 3 75% in a row
Fascinating. I am curious, what other games do you all think this could be extended to and still remains fun? Connect Four seems like a natural extension to me. I'd love to see some of these dynamics
in Battleship.
You can still strategize when the probability of failure and success are equal.
For example, O should choose the lower right because it gives them a greater than 50% chance of winning, whereas choosing another spot gives them a greater than 50% chance of loosing
X X O
X _ O
_ X _
A new rule could be to make a neutral roll prevent the enemy to play that square for one turn (unless it's the last square).
This is a small thing but I really wish it would draw a line through the three in a row when you get one.
This is fantastic!
The dice roll animation is :chefkiss:
Okay. This has a lot more depth than it initially appeared. What a great twist on a simple game!
It's an interesting game, but the AI is making really bad plays.
are you, by chance, giving the computer artificially poor luck? My opponent has been rolling so amusingly poorly that I have to wonder if he's handicapped somehow?
Offline multiplayer over bluetooth would be a great addition!
all I see is a dark grey square? using Firefox on mac
Got the same in chrome but it eventually loaded
Worked for me eventually on Firefox / Linux. Definitely has a slow start on first load.
Thanks for that, I added a loading bar. It should be visible now if you refresh the page.
Me too. Android Chrome and Firefox.
I tried it again and after the loading bar it just went black again. I waited a bit but nothing happened. Android Chrome.
Number of times I selected a square with 5% "meh" chance: 10. Number of times I got "meh" and the computer then selected that square: 8. I know probability is weird but this happens to me when
rolling dice as well (I had a D&D 5E character who nearly always rolled attacks with advantage. I had a streak of 20 attacks in a row (i.e. 40 rolls of a 20 sided die) without getting a double-digit
number, and even got a critical failure (which required two 1s).
Reminds me of Quantum Chess
it's a bit slow with all the animation... but nice idea
I enjoyed playing a few rounds of Probabilistic TTT, and 'Incomplete Information Tic-Tac-Toe' sounded interesting too.
After thinking about it last night, I made a quick version this morning, and I think it's fun to play as well: https://eapl.me/incomplete/
Montecarlo Tree Search (MCTS) would be very ideal for this situation. Since the tree depth is really low, you would not need a neural network estimator. You would just load the entire game tree, and
walk randomly through it, updating visit counts. The walk would be biased by the visit counts, and the biases would then converge to scores for each position.
See the following for a really nice tutorial for a slightly more advanced but more technically correct algorithm, Monte Carlo graph search (MCGS). This exploits the fact that some nodes in the game
tree might are identical positions on the board and can be merged.
For your setup he could easily do either one, but the graph search might give you more mileage in the future:
Once your scores have converged on the entire game tree, you can print out a crib sheet visually showing each position and the correct move. That might be the closest we can get to a human executable
strategy. But the crib sheet might have strategic principles or hard rules that humans can identify
I implemented expectiminimax in the browser which allows for a fairly strong AI player: https://keshav.is/coding/pt3/
I found that once you naively search the game tree beyond a depth of 8, it more or less converges on a particular choice. The presence of a neutral outcome (i.e. neither player claims the selected
square) means the tree depth is technically infinite, but feasible to search thoroughly once the first few moves have been played.
> load the entire game tree, and walk randomly through it
Can't you just multiply all the percentages and just get an expected value for each field? Why the random walk? Can't you just calculate this exhaustively?
I was thinking the same ø, but does it work with the neutral field and changing players in the mix?
I mean there will be non-zero chance that the game could go on forever.
Read the article, my friend. Then you'll see what is so magical about the random walk algorithm - namely, it is easier to implement then other tree evaluation algorithms!
Of course, you can use a number of algorithms to calculate the value, and if you beat me to the punch and it's correct, how about this I'll buy you a burger. But which specific algorithm are you
proposing, and where is its pseudocode and correctness proof?
And is it simpler? If so I'll implement that instead of the random walk!
I picked the random walk algorithm because it is much easier to implement than any other game tree evaluation algorithm I know. | {"url":"https://www.hckrnws.com/stories/40635397","timestamp":"2024-11-06T14:27:35Z","content_type":"text/html","content_length":"221381","record_id":"<urn:uuid:90890b8d-3f1b-4d7c-9086-8ccd0c7cacdc>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00428.warc.gz"} |
Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions which includes one or several terms, each of which has a variable raised to a power. Dividing polynomials is a crucial working in algebra which involves
figuring out the quotient and remainder as soon as one polynomial is divided by another. In this article, we will explore the different approaches of dividing polynomials, involving synthetic
division and long division, and offer scenarios of how to utilize them.
We will also discuss the importance of dividing polynomials and its uses in multiple fields of mathematics.
Significance of Dividing Polynomials
Dividing polynomials is an essential operation in algebra that has many applications in various fields of math, involving calculus, number theory, and abstract algebra. It is applied to work out a
broad range of problems, consisting of working out the roots of polynomial equations, working out limits of functions, and solving differential equations.
In calculus, dividing polynomials is applied to figure out the derivative of a function, that is the rate of change of the function at any moment. The quotient rule of differentiation consists of
dividing two polynomials, that is applied to work out the derivative of a function which is the quotient of two polynomials.
In number theory, dividing polynomials is used to learn the characteristics of prime numbers and to factorize large numbers into their prime factors. It is further utilized to study algebraic
structures such as fields and rings, that are basic theories in abstract algebra.
In abstract algebra, dividing polynomials is used to specify polynomial rings, which are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are applied in various
domains of math, including algebraic geometry and algebraic number theory.
Synthetic Division
Synthetic division is a technique of dividing polynomials that is applied to divide a polynomial with a linear factor of the form (x - c), at point which c is a constant. The approach is on the basis
of the fact that if f(x) is a polynomial of degree n, subsequently the division of f(x) by (x - c) gives a quotient polynomial of degree n-1 and a remainder of f(c).
The synthetic division algorithm involves writing the coefficients of the polynomial in a row, using the constant as the divisor, and carrying out a sequence of workings to figure out the remainder
and quotient. The outcome is a streamlined structure of the polynomial which is easier to work with.
Long Division
Long division is an approach of dividing polynomials that is applied to divide a polynomial by any other polynomial. The method is relying on the reality that if f(x) is a polynomial of degree n, and
g(x) is a polynomial of degree m, at which point m ≤ n, subsequently the division of f(x) by g(x) gives a quotient polynomial of degree n-m and a remainder of degree m-1 or less.
The long division algorithm includes dividing the greatest degree term of the dividend by the highest degree term of the divisor, and then multiplying the outcome by the total divisor. The result is
subtracted from the dividend to obtain the remainder. The procedure is recurring as far as the degree of the remainder is less compared to the degree of the divisor.
Examples of Dividing Polynomials
Here are some examples of dividing polynomial expressions:
Example 1: Synthetic Division
Let's assume we have to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 with the linear factor (x - 1). We can use synthetic division to simplify the expression:
1 | 3 4 -5 2 | 3 7 2 |---------- 3 7 2 4
The outcome of the synthetic division is the quotient polynomial 3x^2 + 7x + 2 and the remainder 4. Thus, we can state f(x) as:
f(x) = (x - 1)(3x^2 + 7x + 2) + 4
Example 2: Long Division
Example 2: Long Division
Let's say we have to divide the polynomial f(x) = 6x^4 - 5x^3 + 2x^2 + 9x + 3 by the polynomial g(x) = x^2 - 2x + 1. We can utilize long division to simplify the expression:
First, we divide the highest degree term of the dividend with the highest degree term of the divisor to obtain:
Then, we multiply the whole divisor by the quotient term, 6x^2, to attain:
6x^4 - 12x^3 + 6x^2
We subtract this from the dividend to obtain the new dividend:
6x^4 - 5x^3 + 2x^2 + 9x + 3 - (6x^4 - 12x^3 + 6x^2)
that streamlines to:
7x^3 - 4x^2 + 9x + 3
We repeat the procedure, dividing the highest degree term of the new dividend, 7x^3, with the highest degree term of the divisor, x^2, to obtain:
Subsequently, we multiply the whole divisor by the quotient term, 7x, to get:
7x^3 - 14x^2 + 7x
We subtract this of the new dividend to achieve the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
which simplifies to:
10x^2 + 2x + 3
We repeat the method again, dividing the highest degree term of the new dividend, 10x^2, with the highest degree term of the divisor, x^2, to obtain:
Next, we multiply the total divisor with the quotient term, 10, to get:
10x^2 - 20x + 10
We subtract this of the new dividend to achieve the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that simplifies to:
13x - 10
Therefore, the result of the long division is the quotient polynomial 6x^2 - 7x + 9 and the remainder 13x - 10. We can state f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
Ultimately, dividing polynomials is a crucial operation in algebra which has several uses in multiple domains of math. Understanding the various methods of dividing polynomials, for example synthetic
division and long division, can help in working out complicated challenges efficiently. Whether you're a learner struggling to understand algebra or a professional operating in a field that includes
polynomial arithmetic, mastering the theories of dividing polynomials is important.
If you require help understanding dividing polynomials or anything related to algebraic concept, think about reaching out to Grade Potential Tutoring. Our adept teachers are accessible online or
in-person to give customized and effective tutoring services to support you succeed. Connect with us right now to plan a tutoring session and take your mathematics skills to the next level. | {"url":"https://www.kansascityinhometutors.com/blog/dividing-polynomials-definition-synthetic-division-long-division-and-examples","timestamp":"2024-11-11T20:51:50Z","content_type":"text/html","content_length":"77850","record_id":"<urn:uuid:7f725965-60c8-423f-98d4-97e4562c8cf4>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00624.warc.gz"} |
Are most lower bounds really upper bounds?
Recently Daniel Apon (grad student of Jon Katz at UMCP, but he also hangs out with me) proved a LOWER BOUND by proving an UPPER BOUND. His paper is
. I have heard it said (I think by Avi W and Lane H) that MOST lower bounds are really upper bounds. Below I use the term
Non-Alg Lower Bound
for a lower bound that is NOT an algorithm. This is not a rigorous notion and the items below are up for debate.
1. Time Hier, Space Hier- Diagonalization. I would call that Non-Alg.
2. Cooks Theorem: this is an ALGORITHM to transform a Non Det TM and a string x to a Boolean Formula.
3. All reductions can be viewed as ALGORITHMS.
4. Parity not in AC0: The Yao-Hastad proof can be viewed as a non-alg lower bound for the depth 1 or 2, and then a randomized ALGORITHM to transform depth d to depth d-1.
5. Parity not in AC0[3]: This is a Non-Alg lower bounds--- you show that parity has some property (not being able to be approx by low degree polys) and then you show that AC0[3] cannot deal with
this property.
6. Comm Complexity: The det lower bound on EQ is a Non-Alg lower bound. I think the randomized lower bound on DISJOINT is a Non-Alg lower bounds. Many others lower bounds are reductions to these
two, and hence are algorithms.
7. Multiparty Comm Comp: I'll just mention one result: Chandra-Furst-Lipton's lower bounds on EXACT-N for k-player Number-on-Forehead. The lower bounds shows that if there is a protocol of t bits
then some structure can be colored in a certain way. Then Ramsey Theory is used. Non-Alg Lower bound I think.
8. Decision Tree Complexity (Comparisons): The lower bounds on SORTING and MAX, are non-Alg lower bounds. The following leave-counting lower bound for
2nd largest is sort-of a reduction to MAX but I still think its non-alg: First note that the lower bound for MAX is very strong- even the best case
requires n-1 comps. Hence any DT for MAX has roughly 2^n-1 leaves.
T is a DT for 2nd largest. For all i=1,...,n let T[i] be the subtree where x[i] WINS. This is a MAX tree for n-1 elements so has 2^n-2 leaves. All these sets of leaves are disjoint so T has n2^
n-2 leaves.Hence T has height n+ log n + \Omega(1).)
9. Decision Tree Complexity (Other queries): Algebraic Queries, k-ary queries have all been studied.
The Algebraic Queries lower bounds use number-of-component arguments and seem non-alg. Some of the k-ary query lower bounds use Ramsey Theory to reduce to the comparison case.
10. Branching programs and other models often reduce to comm complexity. Is that an algorithm.
11. Ryan Williams proof that NEXP is not in ACC is really, at its core, an algorithm that does slightly better than brute force.
My Final Opinion: The above is a random sample, but it seems to be that there are plenty of lower bounds that are non-alg lower bounds. However, as more and more lower bounds are known, more and more
reductions will be used and hence there will be more algorithms.
14 comments:
1. See also the same question on cstheory:
1. ...and also this related question: http://cstheory.stackexchange.com/q/14085
2. What is the difference between a "constructive" proof (possibly a randomized constructive) and an "algorithmic" one in your definition? Proofs have constructive parts (as in Ryan Williams'
algorithm) as well as non-constructive parts (the diagonalization that is at the NEXP not in ACC^0 core). Ditto for the Ajtai-FSS-Yao-Hastad parity not in AC^0 proofs.
3. Petition to ask ACM to join Open Access:
4. There is typo in 5. ACC0 should be AC0[3].
1. Fixed
2. You fixed the wrong instance of ACC0. You are now underselling Ryan Williams' result.
3. Thanks.
NOW I have fixed it. Hopefull.
5. Dear Dr. Gasarch,
I think his definition (2) of Cook's Theorem is incomplete, since it is missing that the reduction must be done in polynomial time, and that the running time of that poly-time NTM must be given
in order to that reduction works (with poly-time construction of a Boolean formula using the description of that NTM and input x).
Notice that this is not preciously, because without such details many computer theorists cannot understand the serious flaw that affects this theorem, as explained at http://
6. this does seem to be a deep principle that might somehow be formalized more generally and rigorously and generally. maybe it is some kind of tradeoff phenomenon between time and space (and other
computational resources). this can be seen in SAT lower bounds that are stated in terms of TISP, time and space. ie the two are interrelated as far as optimal algorithms.
it would seem that the interrelation between lower bounds and upper bounds can be visualized in terms of the inherent tradeoff in compression algorithms. one can compress strings "better"
(shorter) if one has more time. it appears that the concept of compression algorithms is quite fundamental to TCS eg as in kolmogorov complexity and maybe a larger "unifying framework" someday.
this following question is an attempt to formalize some of this via questions on the compression of the (state, symbol) sequence that ensues on TM computatoins.
compression of a TM run sequence
7. The hierarchy theorems are proven by constructing a universal simulating algorithm for the smaller class. For example, by showing that there is an *algorithm* that runs in time O(n^3) that can
simulate all algorithms that run in time O(n^2). Finding a better *algorithm* for simulating k-tapes on 2-tapes is (or at least seems to be) the bottleneck for improving the Time Hierarchy
(Sometimes diagonalization is non-algorithmic - for example, a few oracle constructions are non-computable, yielding non-computable oracles - but I'd say not in the case of the hierarchy
theorems. Many oracle constructions are algorithmic, however, though the underlying algorithms are often very inefficient.)
8. I think also the proof that parity is not in AC0 has an algorithm at its core: given any small size circuit with constant depth, we find (probabilistically) a low degree polynomial which
approximate it.
9. Bill, I think the title of your post has little (if any) to do with its content, with "algorithmic or not". EVERY lower bound proof, which reduces the problem to lower bounding some combinatorial
/algebraic measure, may be viewed as "algorithmic". Most of the proofs in circuit complexity are such. But I don't know here of any result proved via *upper bounding* some measure. I mean
standard "direct" (not Ryan's type) proofs for formulas, branching programs/tree, bounded-depth or monotone circuits. Does somebody know such an "upperbounding" proof there?
1. P.S. Most lower bounds proofs for a complexity measure c(f) first UPPER bound some more tractable combinatorial measure m(f) in terms of c(f), and then LOWER bounds m(f). The first step is
indeed an UPPER bounds problem: after all, m(f) must be not much larger than c(f). This is usually achieved by UPPER bounding the gate-by-gate or level-by-level progress. So, every proof is a
mix of solving upper and lower bounds problems. It depends on which part is harder. In AC^0 stuff, UPPER bounding is harder. In monotone circuits, LOWER bounding is harder (especially, for
the Perfect Matching function). In the paper by your student also LOWER bounding is harder. In some proofs, both steps are equally hard. So, my question actually was: does somebody know a
proof where a lower bound on c(f) is obtained by proving an upper bound on m(f)? | {"url":"https://blog.computationalcomplexity.org/2013/02/are-most-lower-bounds-really-upper.html?m=0","timestamp":"2024-11-05T12:49:13Z","content_type":"application/xhtml+xml","content_length":"200172","record_id":"<urn:uuid:7271bc56-77d3-40a3-a945-190f1ca69a6a>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00832.warc.gz"} |
Lamda (Options Pricing) - Explained
What is LAMDA in Options Pricing?
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What is Lambda?
One of the "Greeks" being lambda refers to the ratio of an option's change in dollar price to a 1% change in the anticipated price volatility (implied volatility), of an underlying asset. Lambda
informs investors of how much the price of an option would change for a specific change in the implied volatility, even if the underlying's actual price remains the same. The value of lambda is
higher the farther away an option's expiry date is and drops as the expiry date approaches. Just as each individual option has a lambda, an options portfolio also has a net lambda that's determined
by summing up each individual position's lambda. In options analysis, terms such as kappa, sigma, and vega are used interchangeably with lambda.
How Does Lambda Work?
Lambda changes when either large price movements occur or there is an increase in the volatility of an underlying asset. For instance, if an option's price moves 10% higher as volatility rises by 5%,
then 2.0 would be its lambda value. Lambda is calculated as the division of the price move by the rise in volatility. Supposing lambda is high, the option value is highly sensitive to little
volatility changes. Supposing lambda is low, volatility changes would have little effect on the option. A positive lambda is linked with a long option which means that as volatility increases, the
option becomes more valuable. On the other hand, a negative lambda is affiliated with a short option which means that as volatility decreases, the options gains more value. One of the core options
Greeks is the lambda. Other major options Greeks include: Gamma - measures the rate of delta's change Delta - measures the effect of a change in the price of the underlying asset Theta - measures the
effect of a change in the time left for expiration, also termed as time decay.
Lambda in Action
If ABC's share of stock trades at exactly $40 in April and a MAY 45 call is selling for $2, then 0.15 is the option's lambda and 20% is its volatility. If there was a 1% to 21% increase in the
underlying volatility, then theoretically, the price of the option should rise to $2 + (1 0.15) = $2.15. On the other hand, if there was a 3% to 17% decline instead, then the option should drop to $2
- (3 0.15) = $1.55. Implied Volatility Implied volatility refers to the gyrations or estimated volatility of the price of a security and is mainly utilized when pricing options. Mostly, but not
always, implied volatility increases in a bear market, or when investors believe the price of the asset would decline eventually. It usually, but not always, declines in the bull market, or when
investors believe that the asset's price would rise over time. This movement is as a result of the general belief that bearish markets are riskier than bullish ones. Implied volatility is a method
used to estimate the future fluctuations of a Security's worth based on specific predictive factors. As stated earlier, lambda the theoretical percentage change in price for each any of every
percentage move in implied volatility. Implied volatility is calculated with an options pricing model and this determines what the present market prices are estimating the future volatility of an
underlying asset to be. However, it's possible for the implied volatility to deviate from the realized future volatility. | {"url":"https://thebusinessprofessor.com/investments-trading-financial-markets/lamda-definition","timestamp":"2024-11-08T18:07:30Z","content_type":"text/html","content_length":"97927","record_id":"<urn:uuid:9f823829-8851-4a30-9f59-221f8daff158>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00122.warc.gz"} |
Quantization Unit Test
When implementing quantization operations, creating unit test is often a headache. Because often the time, there is no ground truth quantization operation reference implementations to compare the
results with.
In this blog post, I would like to quickly discuss how to test quantization with or sometimes without floating point operation reference implementations.
Quantization Unit Test
The Correct Approach
To test quantization using floating point operation reference implementations, the idea is similar to the fake quantization used in the quantization aware training.
1. Create floating point input tensors $x$, usually filled with random values, and compute their scaling factors $s_{x}$.
2. Quantize the floating point input tensors $x$, resulting in the quantized input tensors $x_{q}$.
3. Dequantize the quantized input tensors, resulting in the dequantized input tensors $x^{\prime}$.
4. Feed the dequantized input tensors to the floating point operation reference implementation $f$, and collect the floating point reference output tensors $y^{\prime} = f(x^{\prime})$, and compute
their scaling factors $s_{y}$.
5. Quantize the floating point reference output tensors $y^{\prime}$, resulting in the quantized output tensors $y_{q}$.
To unit test the quantization operation implementation $f_{q}$, we will feed the reference quantized input tensors $x_{q}$, the input tensor scaling factors $s_{x}$, and the output tensor scaling
factors $s_{y}$ to the quantization operation implementation $f_{q}$, and compare the quantized output tensors from the quantization operation implementation $f_{q}$, $y^{\prime}_{q} = f_{q}(x_{q},
s_{x}, s_{y})$, with the reference quantized output tensors $y_{q}$.
If $f_{q}$ is implemented correctly, ideally, the quantized output tensors from the quantization operation implementation $f_{q}$, $y^{\prime}_{q} = f_{q}(x_{q}, s_{x}, s_{y})$, should be
bitwise-identical to the reference quantized output tensors $y_{q}$.
The Incorrect Approach
The above approach is the correct way to unit test quantization operation implementations. However, it is very possible that the developer does not have access to the floating point reference
implementation $f$. All the developer has is the floating point reference input tensors $x$ and the floating point reference output tensors $y$ from the floating point reference implementation $f$.
Can the developer still do anything to test the quantization operation implementation $f_{q}$?
Intuitively, the developer will do the followings.
1. Given the floating point input tensors $x$, compute their scaling factors $s_{x}$.
2. Given the floating point output tensors $x$, compute their scaling factors $s_{y}$.
3. Quantize the floating point input tensors $x$, resulting in the quantized input tensors $x_{q}$.
4. Feed the quantized input tensors $x_{q}$ to the quantization operation implementation $f_{q}$, and collect the quantization output tensors $y_{q} = f_{q}(x_{q}, s_{x}, s_{y})$.
5. Dequantize the quantization output tensors $y_{q} = f_{q}(x_{q}, s_{x}, s_{y})$, resulting in the floating point output tensors $y^{\prime}$.
6. Compare the floating point output tensors $y^{\prime}$ with the floating point reference output tensors $y$.
Unless the developer is extremely lucky, the floating point output tensors $y^{\prime}$ will not be bitwise-identical to the floating point reference output tensors $y$. The difference between the
floating point output tensors $y^{\prime}$ and the floating point reference output tensors $y$ is the test error $\Delta y = | y^{\prime} - y |$. I have seen the developer using the test error $\
Delta y$ to determine whether the quantization operation implementation $f_{q}$ is implemented correctly.
If the quantization operation implementation $f_{q}$ is implemented correctly, the test error $\Delta y$ is the accumulated quantization error and it’s completely normal. However, if the quantization
operation implementation $f_{q}$ is implemented incorrectly, the test error $\Delta y$ will also contain the error due to the incorrect implementation of quantization operation. Even if the
quantization operation implementation $f_{q}$ is implemented correctly, the test error $\Delta y$ can still be very large. Even if the test error $\Delta y$ is very small, it does not guarantee that
the quantization operation implementation $f_{q}$ is implemented correctly. It becomes ambiguous using the value of the test error $\Delta y$ to determine whether the quantization operation
implementation $f_{q}$ is implemented correctly.
So instead of struggling with the test error $\Delta y$, especially when $\Delta y$ is large, the developer should switch to the correct approach mentioned above, either by asking the other
developers to generate the reference quantization tensors and the scaling factors or obtaining the floating point reference implementation $f$. | {"url":"https://leimao.github.io/blog/Quantization-Unit-Test/","timestamp":"2024-11-02T17:37:57Z","content_type":"text/html","content_length":"30386","record_id":"<urn:uuid:2bd840cc-d243-4e6f-9e01-96d6540204de>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00440.warc.gz"} |
High-dimensional Lipschitz functions are typically flat
A homomorphism height function on the d-dimensional torus Z[n]^d is a function on the vertices of the torus taking integer values and constrained to have adjacent vertices take adjacent integer
values. A Lipschitz height function is defined similarly but may also take equal values on adjacent vertices. For each of these models, we consider the uniform distribution over all such functions
with predetermined values at some fixed vertices (boundary conditions). Our main result is that in high dimensions and with zero boundary values, the random function obtained is typically very flat,
having bounded variance at any fixed vertex and taking at most C(log n)^1/d values with high probability. This result matches, up to constants, a lower bound of Benjamini, Yadin and Yehudayoff. Our
results extend to any dimension d ≥ 2; if one replaces the torus Z[n]^d by an enhanced version of it, the torus Z[n]^d ×Z [2] ^d0 for some fixed d0. Consequently, we establish one side of a
conjectured roughening transition in two dimensions. The full transition is established for a class of tori with nonequal side lengths, including, for example, the n × ⌊ 1/10 log n⌋ torus. In another
case of interest, we find that when the dimension d is taken to infinity while n remains fixed, the random function takes at most r values with high probability, where r = 5 for the homomorphism
model and r = 4 for the Lipschitz model. Suitable generalizations are obtained when n grows with d. Our results have consequences also for the related model of uniform 3-coloring and establish that
for certain boundary conditions, a uniformly sampled proper 3-coloring of Z[n]^d will be nearly constant on either the even or odd sublattice. Our proofs are based on the construction of a
combinatorial transformation suitable to the homomorphism model and on a careful analysis of the properties of a class of cutsets which we term odd cutsets. For the Lipschitz model, our results rely
also on a bijection of Yadin. This work generalizes results of Galvin and Kahn, refutes a conjecture of Benjamini, Yadin and Yehudayoff and answers a question of Benjamini, Häggström and Mossel.
Funders Funder number
National Science Foundation OISE 0730136
Office of the Director 0730136
Centre Elile Borel, Institut Henri-Poincaré
• Anti-ferromagnetic Potts model
• Homomorphism height functions
• Kotecký conjecture
• Localization
• Odd cutsets
• Proper 3-colorings
• Random Lipschitz functions
• Random graph homomorphism
• Rigidity
• Roughening transition
Dive into the research topics of 'High-dimensional Lipschitz functions are typically flat'. Together they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/high-dimensional-lipschitz-functions-are-typically-flat","timestamp":"2024-11-10T17:46:09Z","content_type":"text/html","content_length":"55086","record_id":"<urn:uuid:f1a164a4-1685-47f5-ba0a-e69afc5cb632>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00869.warc.gz"} |
Finding the perimeter and physical education
I was tempted to call this lesson plan “Find the perimeter and leave me in peace” because I was asked if I had any lessons students could do on their own. By this point, many parents are getting
tired of interrupting their own work every 5 minutes to help their children with math problems.
Finding the perimeter activities can be fun in school or out
No, really, I’m serious. You need three things for this activity.
1. A piece of paper
2. A pen or pencil
3. A phone
A measuring device like a ruler, measuring tape or yardstick is optional but would be fun to have.
Step 1: Watch the perimeter video
Step 2. Make a table like in the example below
OBJECT Length Width
Seat of chair
For this exercise, every object should be a rectangle. Be prepared for the question,
“Is a square a rectangle?”
– every third-grader , ever
Yes. Yes it is. If you want to get technical about it, a rectangle is a quadrilateral with four 90 degree angles. Or you could just say yes, a rectangle is a shape with four sides that are not
slanted and a square definitely has four sides and is not slanted.
Step 3. Go measure 10 rectangles in the house
This is where the physical education comes in. Tell your child he or she has 10 minutes to complete the table with 10 items. An item can be as small as a box of candy or as big as the floor of a
room. For each rectangle, write down the name of the object, the length and the width. Just put the whole number. If it says 18 1/4 or 18 1/2 just put 18. You may be tempted to tell your child to
just round it but remember, he or she may not have learned fractions yet. That’s a lesson for another day.
If you don’t happen to have a ruler, yardstick or tape measure, your phone probably has a Measure app. This comes by default with an iPhone and if you don’t see it right away look in the Utilities
folder. To use it, point at a surface and click to select a point. Then, move the phone until you are at the end of what you want to measure.
Depending on how much exercise you want your child to get, the size of your house and how much peace you need (I won’t judge you), you may want to add a few rules like:
• None of the objects can come from the room you are currently in.
• They need to find rectangles in at least 3 different rooms
• They need to find at least one rectangle in the backyard/ garage/ basement.
Once you have shown your child how to use the measure app and they have the table and a pencil, set the alarm on your phone and tell them to go. The alarm will go off when the 10 minutes are up.
Check their number of rectangles and if they are a few short give an extra 2- 5 minutes to find the rest.
Step 4 Watch another video on perimeter
Why? Because I believe that kids often don’t remember something if they only heard it once.
Step 5: Compute the perimeter for each object you have measured
OBJECT Length Width Perimeter
Seat of chair 20 17 74
Table 32 17 98
Your child does this, not you. You’ve already completed elementary school.
You can play it on the web or download it from the app store for free. You can download it from Google Play for $1.99 or email info@7generationgames.com and we’ll give you a discount code.
Reason with shapes and their attributes.
Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides) and that the shared attributes can define a larger category
(e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.Apply the area
and perimeter formulas for rectangles in real-world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area
formula as a multiplication equation with an unknown facto | {"url":"https://www.7generationgames.com/finding-the-perimeter-and-physical-education/","timestamp":"2024-11-13T17:57:54Z","content_type":"text/html","content_length":"55715","record_id":"<urn:uuid:5e9d2d9e-66d2-4ed0-88bd-8fdedeeb100c>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00640.warc.gz"} |
Microscopic Electron Dynamics in Metal Nanoparticles for Photovoltaic Systems
Department of Quantum Technologies, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
Madrid Institute for Advanced Studies in Nanoscience (IMDEA Nanoscience), C/Faraday 9, 28049 Madrid, Spain
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Submission received: 15 May 2018 / Revised: 19 June 2018 / Accepted: 20 June 2018 / Published: 25 June 2018
Nanoparticles—regularly patterned or randomly dispersed—are a key ingredient for emerging technologies in photonics. Of particular interest are scattering and field enhancement effects of metal
nanoparticles for energy harvesting and converting systems. An often neglected aspect in the modeling of nanoparticles are light interaction effects at the ultimate nanoscale beyond classical
electrodynamics. Those arise from microscopic electron dynamics in confined systems, the accelerated motion in the plasmon oscillation and the quantum nature of the free electron gas in metals, such
as Coulomb repulsion and electron diffusion. We give a detailed account on free electron phenomena in metal nanoparticles and discuss analytic expressions stemming from microscopic (Random Phase
Approximation—RPA) and semi-classical (hydrodynamic) theories. These can be incorporated into standard computational schemes to produce more reliable results on the optical properties of metal
nanoparticles. We combine these solutions into a single framework and study systematically their joint impact on isolated Au, Ag, and Al nanoparticles as well as dimer structures. The spectral
position of the plasmon resonance and its broadening as well as local field enhancement show an intriguing dependence on the particle size due to the relevance of additional damping channels.
1. Introduction
An accurate description of microscopic properties of metal nanoparticles (metal NPs—MNPs) is important to predict the optical response of e.g., molecules in close proximity to metal surfaces and
resulting field enhancement and quenching effects. Nanoparticles as part of functionalized layers in sensing, spectroscopy [
] and light harvesting applications, photovoltaics [
] and photocatalysis [
], can improve the performance of such devices. They are efficient subwavelength scatterers improving the light trapping effect and MNPs provide, in particular, large local fields enhancing charge
carrier generation, absorption, and light-induced effects from other nanostructures such as spectral conversion [
] or photoluminescence [
For over a hundred years, modeling of the optical properties of MNPs relies on classical electrodynamics. In highly symmetric cases (spherical and cylindrical NPs) analytic solutions are obtained
within Mie scattering theory [
] using corresponding basis functions. The electric part
of the electromagnetic field creates a polarization field
$P = α ( ϵ 0 , ϵ ) E$
in solid matter, expressed in terms of the permittivities
$ϵ 0 ( ω )$
$ϵ ( ω )$
of the environment and the bulk material, respectively. This polarizability
, depending only on the optical response at a frequency
, neglects microscopic electron interaction effects at the ultimate nanoscale arising not only from the quantum nature of the free electron gas in metals, but also from accelerated motion in the
plasmon oscillation.
Light-matter interaction involves processes within the electron subsystem in solids, crystals and molecules. Inhomogeneities on the length scale of the de Broglie wavelength
$λ e = h 2 m E$
produce scattering and interference effects of electrons which mutually interact with incoming light, see
Figure 1
a. Hereby,
is Planck’s constant,
is the (effective) electron mass which depends on the bulk material, and
is the energy of the electron wave. Typically, this wavelength is about 7.5 nm in solids at room temperature
$T = 300$
K, where
$E = k B T$
with the Boltzmann constant
$k B$
. For MNPs, the main source of electron scattering is the particle surface, see
Figure 1
b, where the surface-to-volume ratio indicates the relevance of such scattering events.
Microscopic interaction effects of electrons in metals are accurately described using first-principle methods, e.g., Density Functional Theory (DFT) [
]. These solve Schrödinger’s equation for a large, but finite number of electron wave functions from all atoms in the considered system. Unfortunately, even with strong approximations such as the
Time Dependent Local Density Approximation (TDLDA), time-consuming algorithms limit their applicability to particles of a few nanometers in size [
]. Moreover, advances in fabrication of nanostructures along with experimental access to particle sizes and interparticle spacings below 10 nm led to the possibility of direct or indirect observation
of such effects [
]. The situation described above resulted in increased interest in semi-classical approaches towards the incorporation of damping and interaction effects stemming from the quantum nature of charge
carriers, illustrated in
Figure 1
. In this article, we present two such semi-classical approaches, the Random Phase Approximation (RPA) and Generalized Nonlocal Optical Response (GNOR), and ultimately combine them into a single
framework to study their joint impact on MNPs of different materials, sizes and in different environments.
The original formulation of light scattering by a sphere by Gustav Mie [
] excludes microscopic dynamics of the conduction band electrons in bulk and surface effects. However, efforts to extend have been made since the 1970s [
]. Advanced semi-classical material models can be derived from perturbative theories [
], by separating the free electron dynamics from the core electron polarization via the hydrodynamic equation for an electron plasma [
], and from microscopic theories [
]. It should be noted that a major advantage of ab initio methods lies in their capability to account for the electron spill-out (evanescent tail of the electron wave functions) of the electron
density into the surrounding dielectric medium. It was shown within the hydrodynamic framework that the electron spill-out can be adequately incorporated [
] and a current-dependent potential can be accounted for [
], which is, however, out of scope of the present study.
In this article, we combine two semi-classical approaches towards microscopic electron dynamics into a single feasible framework to address quantum corrections in MNPs allowing the description of
isolated particles, clusters and large-scale (two- or three-dimensional) devices via the integration of analytical expressions into standard procedures. We hereby focus on results on damping in MNPs
derived from the microscopic Random Phase Approximation (RPA), stemming from Lorentz friction, and spatial dispersion (nonlocal) effects obtained with the hydrodynamic approach. We discuss briefly
the separate ingredients of these approaches in the next sections and give more details in the methods section. Moreover, we compare and combine the different processes of mesoscale electron dynamics
stemming from scattering,
Figure 1
a,b, irradiation (Lorentz friction),
Figure 1
c, and nonlocal interaction,
Figure 1
d, and study their impact on the optical response of isolated MNPs and dimers. An emphasis is put on the size regimes where these effects are dominant for the materials silver,
Figure 2
, as well as for aluminum and gold,
Figure 3
2. Results
We briefly discuss classical electrodynamics and mesoscopic electron dynamics obtained from the RPA and GNOR theories. In summary, we compare quantum correction models stemming from microscopic RPA
derivations with the following, semi-classical damping expressions
$( 1 a ) γ = γ p , ( Mie ) ( 1 b ) γ = γ p + C v F a , ( Kreibig ) ( 1 c ) γ = γ p + C v F a + ω 1 3 ω 1 a c 3 , ( perturbative ) ( 1 d ) γ = Im ( Ω 2 ) = − 1 3 l + 1 + 6 l q 2 2 3 3 l A + A 2 1 3 6
l , ( Lorentz ) ,$
and nonlocal interaction effects. Both approaches are described in more detail in the next sections and the methods section. The advantage in the analytic formulation is the straightforward
integration with existing computational tools for nanospheres using modified Mie simulations and multiple scattering techniques [
] for clusters thereof or commercial software such as COMSOL (
2.1. Classical and Phenomenological Approaches
Typically, the optical response of a metal is described with the Drude model via the frequency-dependent permittivity
$ϵ ( ω ) = ϵ b − ω p 2 ω ( ω + i γ p ) ,$
$ϵ b$
is the background permittivity given by bound (valence band) electrons,
$ω p 2 = 4 π n 0 e 2 / m$
is the plasmon frequency, determined by the material dependent electron density
$n 0$
and mass
, and
$γ p$
is the inherent (bulk) damping rate. This widely used Drude model applies only to bulk material and should be modified for nanostructures to include effects due to the finite size of the system. One
of the corrections considered by Kreibig and von Fragstein [
] is the inclusion of an additional damping due to the scattering on the physical particle boundaries, depicted in
Figure 1
b. This is in particular important in particles of sizes equal or smaller than the mean free path
$λ b$
of electrons in bulk metal. In such a case, the electrons will experience (in the classical picture) additional scattering from the boundary of the system. Mathematically, it is described as
$γ K = v F / L e f f$
, where
$v F$
is the Fermi velocity of the electron gas and
$L e f f$
is the effective mean free path of electrons resulting from collisions with the particle surface [
]. The common feature is that
$L e f f$
reflects the volume (proportional to the number of electrons inside the nanoparticle) to surface ratio of the particle. According to this, we get the
$γ K ( a ) = C v F / a$
, where
is the radius of the nanoparticle and
is a constant of the order of unity which depends on the scattering type and particle radius. Similarly, collision effects in the bulk, depicted in
Figure 1
a, can be described via the damping term
$γ p v F / 2 λ b$
2.2. Random Phase Approximation
Nevertheless, this phenomenological approach neglects the microscopic dynamics of electrons inside the MNP. Their accelerated movement (plasmon oscillation) leads to energy loss via irradiation of
the electromagnetic field, see
Figure 1
c. In case of nanoparticles much smaller than the incident wavelength, this effect can be expressed by the Lorentz friction, an effective field stemming from the plasmon induced dipole field
$D ( t )$
$E L = 2 / 3 c 3 ∂ 3 D ( t ) / ∂ t 3$
, with
being the speed of light [
]. The dynamics of the electron density can be described using a driven, damped oscillator, with the incident electromagnetic wave being the driving force and the damping arising form electron
scattering (bulk
$γ p$
and Kreibig damping
$γ K$
) and electromagnetic field irradiation (Lorentz friction).
An analytical form of the exact solution for the damping
and self-frequency
$ω L$
(the exponents
$Ω i$
of solution ∼
$e i Ω i t$
for self-modes
) including Lorentz friction exists [
], which is discussed in more detail in the methods section. They can be summarized as follows
$Ω 1 = − i 3 l − i 2 1 / 3 ( 1 + 6 l q ) 3 l A − i A 2 1 / 3 3 l ∈ Im , Ω 2 = − i 3 l + i ( 1 + i 3 ) ( 1 + 6 l q ) 2 2 / 3 3 l A + i ( 1 − i 3 ) A 2 1 / 3 6 l = ω L + i γ , Ω 3 = − ω L + i γ = − Ω 2
* ,$
$A = B + 4 ( − 1 − 6 l q ) 3 + B 2 1 / 3$
$B = 2 + 27 l 2 + 18 l q$
$q = 1 τ 0 ω 1$
$l = 2 3 ϵ 0 a ω p c 3 3$
$1 / τ 0 = γ p$
. Exact inclusion of the Lorentz friction indicates that the radiative losses and the self-frequencies are a complicated function of particle radius as given by Equation (
), see the methods section for a detailed discussion.
Direct comparison to experimental work for this framework is available within Refs. [
] and good agreement has been found.
2.3. Nonlocal Optical Response
Aside from electron irradiation due to Lorentz friction, we discuss spatial dispersion (nonlocality) which denominates the effects of electron coupling over a short distance, see
Figure 1
d [
]. Such interactions are inherent to the solution for the displacement field
of the Coulomb equation
$∇ D ( ω , r ) = 0 ⇒ D ( ω , r ) = ∫ d r ′ ϵ ( ω , r , r ′ ) E ( ω , r ′ ) .$
In homogeneous media, we can assume a dependence on the distance $r − r ′$ rather than on the specific position of electrons, which allows solving Maxwell’s equations in Fourier space $D ( ω , k ) =
ϵ ( ω , k ) E ( ω , k )$.
The dependence on the wave vector
enables us to describe nonlocal electron-electron interaction (Coulombic force) and electron diffusion effects. It is important to note that the large-
response that originates in the subwavelength oscillations of plasmonic excitations is not only an inherent prerequisite for many intriguing wave phenomena, but also particularly sensitive to
nonlocality. However, the common Mie result has no upper wavelength cut-off and does suppress short-range electron interactions which can strongly dampen the response beyond
$ω / v F$
. We show in the corresponding section below that accounting for nonlocal response leads to longitudinal pressure waves as additional solutions to the combined system of differential equations of the
electromagnetic wave equation and (linearized) Navier-Stokes equation. This is in contrast to the damping expressions derived by Kreibig and for Lorentz friction. Such additional waves offer further
damping channels, however, they can also support resonant enhancement effects [
Experimental work focusing on the blueshift found for nanoparticles decreasing in size, as well as the influence of the electron-spill out has been studied in Refs. [
], including comparisons with the hydrodynamic model.
2.4. Remarks on Retardation, Multipolar Response and Computational Feasibility
Both of the presented semi-classical approaches towards microscopic corrections in the mesoscale electron dynamics in metal nanoparticles have the advantage of analytic expressions fully compatible
with existing computational procedures. For the quantum confinement picture of Kreibig and the mesoscopic RPA result for the Lorentz friction, modified damping terms were derived, see Equations (
)–(1d), which can be used to directly replace the damping in the Drude expression for the permittivity given in Equation (
) and subsequently be used in standard Mie calculations and procedures to calculate optical properties of complex structures, e.g., with a multiple scattering approach [
] or within commercial software such as COMSOL.
It is important to note that although all electrons participate in plasmon oscillations, part of their irradiation is absorbed by other electrons in the system. This is in analogy with the
skin-effect [
] in metals and introduces an effective radiation active electron layer of the depth
$h ∼ 1 / σ ω$
is the conductivity) underneath the particle surface. Therefore, the effective energy transfer outside of the nanoparticle will be reduced by the factor
$4 π 3 a 3 − a − h 3 / 4 π a 3 3$
. According to this, we expect a decrease of radiative damping, especially for larger particles.
The nonlocal theory introduces a novel type of electron motion, longitudinal pressure waves, in addition to the transversal modes stemming from the classical electromagnetic wave equation. This
additional electronic excitation offers further damping channels due to the energy lost in dampened motion. Here, the Mie coefficients are derived from the coupled system of optical and electronic
excitation yielding modified scattering matrices that can again be implemented in existing methods. The properties of the longitudinal wave are given by analytic expressions such as their wave vector
and their importance with respect to the common Mie solution is entirely captured in a single additional term, see the methods section for details.
Retardation is important when either the particle radius or the overall system size becomes large, i.e., for particle dimers, clusters and arrays. Although the presented microscopic effects are
highly localized, they can have a strong impact on a larger particle or system in the interplay with long-range retardation effects. In addition, particle layer modes can couple to nonlocal modes
within particle arrays and thus increase their impact on a larger scale [
]. It is thus noteworthy that the hydrodynamic theory and the damping terms stemming from microscopic analysis within the RPA allow fully retarded calculations; equally for planar geometries
(nonlocal Fresnel coefficients) [
] and regular, two-dimensional particle arrays [
] and even charge carriers in electrolytes (
Nonlocal Soft Plasmonics
) [
3. Discussion
3.1. Single Metal Nanoparticles
We compare the quantum correction models introduced in the previous section, see
Figure 1
, as well as the combined effect of Kreibig damping Equation (1b), Lorentz friction Equation (1d) and spatial dispersion to classical Mie calculations for the materials gold, aluminum and silver in
Figure 2
Figure 3
. Hereby, we show the effect on the Localized Surface Plasmon Resonance (LSPR) for all materials in
Figure 2
a, confirming that the modified damping rates do not alter the resonance position predicted by the classical calculations, whereas nonlocal response—and in combination with any damping model—does
predict an increasing blueshift of the nanoparticle resonance with decreasing particle size. Looking at the extinction cross section as a function of particle radius in
Figure 2
b for silver and
Figure 3
for gold and aluminum, we find that all correction models result in a reduction of the optical response in dependence of both the material and particle size, typically yielding a different optimized
particle size. Hereby, Kreibig damping with a ∼
$1 / a$
dependence drastically attenuates the optical response for the smaller size regime below the maxima (15 nm for Ag, 20 nm for Au, and 10 nm for Al), while the complex size dependence of Lorentz
friction results in a greater effect above this particle size. The diffusion coefficient in the hydrodynamic (GNOR) model (imaginary part of the nonlocal parameter
$β GNOR$
) is chosen thus that its dampening effect captures the Kreibig result [
]. This is best seen in
Figure 3
a for Au. The hydrodynamic pressure (real part of the nonlocal parameter
$β GNOR$
) describes Coulomb interaction between electrons and results in the blueshift observed in
Figure 2
a at very small particle sizes below 5 nm. We can further incorporate the analytical expressions for Lorentz friction. This combined result shows the strongest attenuation since all different damping
channels are included. At a larger particle size (60 nm for Ag, 80 nm for Au, and 40 nm for Al) all material models converge with classical Mie theory where the mesoscale electron dynamics cease to
have an impact.
The damping associated with the Lorentz friction can be approximated to the simpler perturbative expression Equation (1c) in a narrow size window, see the methods section for a detailed discussion.
Since the exact solution can be obtained with analytical expressions which can be incorporated into standard calculation schemes, we discuss exclusively exact Lorentz friction results.
We study the (maximum) field enhancement factor
$EF = | E | 2 / | E 0 | 2$
just outside of the NP (
$r → a +$
) for the different damping models in
Figure 4
for gold nanospheres. Hereby,
Figure 4
a shows the spectral position of the field maximum. The local field enhancement reveals the size dependence of the field resonance with the damping rates. It should be emphasized that Kreibig damping
shows a strong redshift for small particle sizes of the spectral position of local field enhancement maxima in contrast to experimental findings [
] and approaches the Mie result for larger sizes. Nonlocal optical response agrees with the blueshift of the plasmon resonance found experimentally for noble metals, as already seen in the extinction
cross section,
Figure 2
a. However, in order to correctly describe simple metals, the inclusion the electron spill-out region [
] is crucial. Furthermore, advances towards the spatial dispersion found in (doped) semiconductors were made recently [
], which is of further interest when using dielectric nanoparticles to enhance the performance of photovoltaic devices.
Lorentz friction is closest to the classical calculation for smaller sizes and deviates stronger at larger sizes. This is in agreement with the findings of
Figure 2
Figure 3
. The corresponding field enhancement, shown in
Figure 4
b for gold MNPs in water, is strongly suppressed for the considered particle size range when including the damping models while spatial dispersion by itself reduces the predicted field enhancement
mostly for smaller particle sizes and converges with the classical Mie result rapidly with increasing particle size. This behavior is corrected by incorporating Lorentz friction into the GNOR result.
Figure 4
c,d shows the (maximum) field enhancement of gold nanoparticles in dependence of the refractive index (RI) of the surrounding medium (from air
$n = 1$
to Si
$n = 3.4$
) for two particle sizes. This is accompanied with a linear (in case of the nonlocal theory approximately linear) shift in the resonance wavelength towards longer wavelengths (not shown). With
increasing RI of the host medium, the enhancement factor reaches a saturation value which for increasing particle size converges for all material models discussed. The discrepancy between the local
field enhancement values predicted remains similar for small MNPs in different host media spanning several orders of magnitude.
The complexity of the Lorentz friction makes it necessary to restrict ourselves to the dipolar response of the plasmon oscillation. It is therefore important to consider the material, particle size
and wavelength regime in order to assess whether the dipolar response model is adequate for the system under study. We show in
Figure 5
for Au NPs the dipolar and the converged result of local field enhancement obtained from classical Mie calculations at a fixed frequency close to the respective plasmon resonance. Here, the dipolar
approximation is valid up to ca. 100 nm in particle radius which in general covers the discussed microscopic effects well. The inset in
Figure 5
compares this for the combined theories showing small differences already for particles above 25 nm radius.
3.2. Dimers
For particle dimers, in addition to their size, the particle distance becomes important and retardation effects cannot be neglected for larger particles in close proximity. This can transfer the
impact of localized microscopic electron dynamics onto a larger structure.
Figure 6
shows the (maximum) field enhancement at the center of a gold dimer in water as a function of both particle size and distance for the different theories considered. The impact of nonlocal response,
Figure 6
b, on the classical Mie theory,
Figure 6
a, is visible as strong quenching of the local fields. It is worth remembering that one main effect is a blueshift in the position of the maximum enhancement factor, see again
Figure 4
a and Ref. [
]. In addition, the maximum field enhancement within the parametric area of particle and gap size is EF
$≈ 9000$
for the Mie calculations and EF
$≈ 3000$
for the nonlocal theory, showing that indeed there is an impact of the longitudinal waves found. The damping observed within Kreibig theory,
Figure 6
c, is dramatic for the dimer setup and the dominant contribution in the combined theory as seen in
Figure 6
d. This is also evidenced by comparing the Lorentz friction with and without nonlocal damping, see
Figure 6
e,f, respectively. The Lorentz friction has a strong impact on the optical response for larger particle sizes, but also dampens the dimer setup for increasing gap size, which points towards
retardation and the increasing structural size as the main source for this damping effect. This leads to slightly stronger damping when combined with the additional plasmon quenching within GNOR in
Figure 6
The strong field quenching poses limitations to the photovoltaic effect in solar cells. However, considering different materials for MNPs and their environment, the size regimes where local field
quenching is dominant can be avoided with the presented theory of combined damping.
3.3. Summary
In conclusion, we have presented a number of semi-classical corrections to incorporate electron dynamics and non-classical interaction effects into optical response calculations for nanoparticles.
Hereby, pure damping models, such as the Kreibig damping and Lorentz friction, derived from microscopic RPA theory, show an intriguing dependence on the particle size, where the material influences
relevant size regimes. On the other hand, semi-classical nonlocal theories allow evoking additional modes in the system by explicitly considering mesoscopic dynamics of free electrons. This results
in a correction of the spectral position of resonant phenomena and introduces additional, implicit damping channels. The phenomenological Kreibig damping does yield a plasmon broadening that agrees
with experiments [
], however, it also introduces a redshift of the resonance with respect to the classical Mie result contrary to measurements on nanoparticles [
]. This is addressed by using the hydrodynamic GNOR (generalized nonlocal optical response) approach, i.e., by introducing a diffusion parameter, able to reproduce the Kreibig damping while fully
capturing the observed plasmon broadening.
An important aspect is that the resulting analytical expressions can be implemented into existing computational procedures in a straightforward manner, as isolated theories or combined, allowing the
comparison to experiments with little added numerical effort. We have studied the combined effect of these mesoscopic electron interaction effects for single nanospheres and gold dimers and have
evidenced the importance of retardation as a way to communicate localized quantum effects and impact a larger structure.
The straightforward inclusion of electro-optical effects at the nanoscale into (metal) nanoparticle systems is of importance in nanostructures employed for photovoltaics and catalysis as well as in
spectroscopy and sensing applications.
4. Methods
4.1. Electron Dynamics within the RPA
The model of electron dynamics inside MNPs [
] presented here is an extension to the RPA theory developed by Pines and Bohm [
] for bulk metals. In our model, a finite, rigid jellium defines the shape of a nanoparticle. The plasmon oscillations are described as local electron density fluctuations
$ρ ^ ( r , t )$
obtained from the Heisenberg equation
$d 2 ρ ^ ( r , t ) d t 2 = 1 ( i ℏ ) 2 [ [ ρ ^ ( r , t ) , H ^ e ] H ^ e ]$
with a corresponding Hamiltonian
$H ^ e$
for electrons inside the MNP in the jellium model taking the following form
$H ^ e = ∑ j = 1 N e − ℏ 2 ∇ j 2 2 m − e 2 ∫ n e ( r ) d 3 r | r j − r | + 1 2 ∑ j ≠ j ′ e 2 | r j − r j ′ | .$
The operator of the local electron density is defined as
$ρ ( r , t ) = 〈 Ψ e ( t ) | ∑ j δ ( r − r j ) | Ψ e 〉$
$Ψ e$
is the electron wave function,
$N e$
is the number of collective electrons,
$r j$
are their positions and mass. The ion field is approximated as averaged background charge density and described as
$n e ( r ) | e | = n e Θ ( a − r ) | e |$
, where
is the Heaviside step function,
is the radius of the MNP and
$n e = N e / V$
The first term in the Hamiltonian stands for the kinetic energy of electrons, the second for interaction between electrons and positive background charges (approximating the ion lattice potential)
and the last for electron-electron Coulomb interaction.
Taking into account the sharp form of the positive charge density
$n e ( r )$
, one can decompose Equation (
) into two parts corresponding to the inside and outside of the NP, which leads to two separate solutions describing the surface and bulk plasmons. This description is valid for NPs larger than ca. 5
nm for which the surface is well defined and the spill-out effect is negligible.
$δ ρ ˜ ( r , t ) = δ ρ 1 ˜ ( r , t ) , for r < a , δ ρ 2 ˜ ( r , t ) , for r ≥ a , ( r → a + ) .$
The electron density fluctuations are then described with the formulas
$∂ 2 δ ρ 1 ˜ ( r , t ) ∂ t 2 = 2 3 ϵ F m ∇ 2 δ ρ 1 ˜ ( r , t ) − ω p 2 δ ρ 1 ˜ ( r , t )$
$∂ 2 δ ρ 2 ˜ ( r , t ) ∂ t 2 = − [ 2 3 ϵ F m r r ∇ δ ρ 2 ˜ ( r , t ) + ω p 2 4 π r r ∇ ∫ d 3 r 1 1 ∣ r − r 1 ∣ ( δ ρ 1 ˜ ( r 1 , t ) Θ ( a − r 1 ) + δ ρ 2 ˜ ( r 1 , t ) Θ ( r 1 − a ) ) ] δ ( a + ε −
r ) − 2 3 m ∇ 3 5 ϵ F n e + ϵ F δ ρ 2 ˜ ( r , t ) r r δ ( a + ε − r ) .$
$ϵ F$
is the Fermi energy.
The structure of the above equations is of an harmonic oscillator, which allows including a damping term in phenomenological manner by adding to the right hand side $− 2 / τ 0 ∂ ρ ˜ 1 ( 2 ) ( r , t )
/ ∂ t$. The damping $2 / τ 0 = γ p + γ K$ includes collision effects and Kreibig damping due to the particle boundary.
Assuming homogeneity of the external electric field
$E ( t )$
inside the NP (dipole approximation), the solution for surface modes reduces to a single dipole mode
$δ ρ ˜ ( r , t ) = ∑ m = − 1 1 Q 1 m Y 1 m ( Ω ) , for r ≥ a , ( r → a + )$
and for bulk modes
$δ ρ ˜ ( r , t ) = 0$
$r < a$
The function
$Q 1 m ( t ) ( m = − 1 , 0 , 1 )$
represents dipole modes,
$Y l m ( Ω )$
is the spherical function. The former can be related to the vector
$q ( t )$
$Q 11 = 8 π / 3 q x ( t )$
$Q 10 = 4 π / 3 q x ( t )$
$Q 1 − 1 = 8 π / 3 q y ( t )$
satisfying the equation
$∂ 2 ∂ t 2 + 2 τ 0 ∂ ∂ t + ω 1 2 q ( t ) = e n e m E ( t ) .$
Then the plasmon dipole can be defined as
$D ( t ) = e ∫ d 3 r r δ ρ ( r , t ) = 4 π 3 e q ( t ) a 3 .$
Knowing this, the damping caused by electric field irradiation can be simply added to the right hand side of Equation (
) as additional field
$E L = 2 / 3 c 3 ∂ 3 D ( t ) / ∂ t 3$
hampering charge oscillations and can be rewritten in the form
$∂ 2 ∂ t 2 + ω 1 2 D ( t ) = ∂ ∂ t − 2 τ 0 D ( t ) + 2 3 ω 1 ϵ 0 ω p a c 3 3 ∂ 2 ∂ t 2 D ( t ) .$
The above equation is a third order linear differential equation and the exponents ∼
$e i Ω i t$
of its solutions are given in Equation (
). A perturbation approach can be applied to Equation (
) for small particles using
$∂ 2 D ( t ) / ∂ t 2 = − ω 1 2 D ( t )$
. Then the resulting damping term takes the form
$γ = 2 / τ 0 + ω 1 / 3 ϵ 0 ω a / c 3 3$
. The comparison of both damping terms is shown in
Figure 7
justifying the usage of the perturbation formulation for (gold) particles with radii up to ca. 30 nm, where the second term proportional to ∼
$a 3$
still fulfills the perturbation constrain.
For larger radii, the discrepancy between both solutions grows rapidly since the irradiation losses within the perturbation approach scale as
$a 3$
. Therefore, the radiative losses dominate plasmon damping for large nanospheres. On the other hand, scattering is more important for smaller nanospheres scaling as
$1 a$
. One can observe thus the size-dependent crossover in
Figure 3
a of the damping at ca. 12 nm for gold.
4.2. Electron Dynamics with the Hydrodynamic Model
In recent years, a great effort to theoretically [
] describe and subsequently to experimentally [
] verify the effect of spatial dispersion in metals was made. In the hydrodynamic approach, coupling the electromagnetic wave equation
$∇ × ∇ × E − k 2 ϵ b E = 4 π i k 2 ω j ind$
to the (linearized) Navier-Stokes equation
$j ind = i ω + i γ p ω p 2 4 π E − β 2 + D ( γ p − i ω ) ∇ ρ ind$
allows treating the conduction band electrons as a plasma subject to short-ranged interaction such as the Coulomb force included in the pressure term
$p = β 2 ρ ind$
and electron diffusion via the diffusion coefficient
. It is convenient to abbreviate
$β GNOR 2 = β 2 + D ( γ p − i ω )$
(where GNOR refers to the Generalized Nonlocal Optical Response model [
]). With this, we can write the wave equation in a compact form
$∇ 2 E + k 2 ϵ ⊥ E = η ρ ind ,$
$η = 4 π 1 ϵ b − k 2 β GNOR 2 ω ( ω + i γ )$
$ϵ ⊥ = ϵ b − ω p 2 / ω ( ω + i γ p )$
. Together with the continuity equation
$∇ j ind = i ω ρ ind$
, we readily obtain a separate wave equation for the induced charges
$− β GNOR 2 ∇ 2 ρ ind = ϵ ⊥ ϵ b ω ( ω + i γ p ) ρ ind ,$
$∇ E = 4 π / ϵ b ρ ind$
was used. This yields the wave vector of the longitudinal field and motion of electrons
$q = 1 β GNOR ϵ ⊥ ϵ b ω ( ω + i γ p ) .$
Nonlocal theories predict finite distributions of induced charges at an illuminated metal surface—in contrast to classical electrodynamics—with a characteristic penetration depth
$Im ( 1 / q )$
comparable to the electron spill-out [
Thus, this system of coupled equations yields an additional wave solution, longitudinal in character, and can be solved for different geometries leading to nonlocal extensions of Mie [
] and Fresnel coefficients [
], including for charge carriers in electrolytes [
]. Typically, hard-wall boundary conditions are assumed for the additional boundary condition
$n ^ j ind ≡ 0$
prohibiting electrons to trespass through the particle surface into the dielectric surrounding, using a uniform electron density
$n 0 = ω p 2 m / 4 π e 2$
inside the material and neglecting the electron spill-out. However, it was shown that a smooth surface distribution of electrons can be taken into account accurately [
] and that the hydrodynamic model is capable of dealing with the spill-out by solving the above equations with position-dependent material parameters
$ω p ( z ) 2 = 4 π n 0 ( z ) e 2 / m$
The main observations of nonlocal theories are a blueshift of the plasmon resonance with respect to the common local approximation and plasmon broadening, in particular tied to the diffusion
coefficient which can be set to fully capture the broadening found with Kreibig damping [
]. In the present work, we have adopted the diffusion coefficients as deduced in Ref. [
] for the different materials, reflected, for instance, in the correspondence between the Kreibig and GNOR result for gold in
Figure 3
a. Moreover, we add the Lorentz friction result from the RPA technique summarized in Equation (1d) to our GNOR calculations.
Next, we present the derivation of nonlocal Mie scattering coefficients of individual spheres and nanoshells described with the hydrodynamic model [
] starting from Equation (
) which describes the evolution of the electric field, together with Equation (
) which is the wave equation for the induced charge. The resulting scattering matrices can be used to investigate interacting spheres with a multiple scattering method [
]. The hydrodynamic model has no free parameters which makes the resultant nonlocal response for the short distances involved in the interaction (Coulomb force, diffusion) between the charges of MNPs
the sole source of these effects, in contrast to the quantum-confinement picture for plasmon broadening presented by Kreibig.
It is convenient to use an expansion of the electric field into scalar functions [
] as
$E = ( 1 / k ) ∇ ψ L + L ψ M + ∇ × L k i ψ E ,$
$L = − i r × ∇$
is the angular momentum operator, and the superscripts
, and
indicate electric, magnetic, and longitudinal components, respectively. The additional boundary condition, Equation (
), becomes with
$r ^ j = 0$
$β GNOR 2 ∂ ∂ r ρ ind = e 2 n 0 m k ∂ ∂ r ψ L + 1 r l ( l + 1 ) ψ E$
in terms of the scalar functions and the angular momentum number
using the identity
$− r · ( ∇ × L ) = ( − i r × ∇ ) · L = L 2 = l ( l + 1 )$
. The boundary conditions for the electric and magnetic field components result in the continuity of
$ψ M$
$( 1 + r ∂ ∂ r ) ψ M$
$ψ L + ( 1 + r ∂ ∂ r ) ψ E$
, and
$ϵ ψ E$
for the scalar functions.
The magnetic and electric scalar functions
$ψ ν ( ν = { E , M } )$
obey a Helmholtz equation of the form
$( ∇ 2 + k 2 ϵ ⊥ ) ψ ν = 0$
and can therefore be expanded in terms of spherical Bessel functions
$ψ ν = ∑ L ψ L ν j L ( k ⊥ r )$
. Similarly, the electron density is expanded into
$ρ ind ( r , ω ) = ∑ L ρ L j L ( q r )$
, with the longitudinal wave vector
given by Equation (
). The longitudinal scalar function satisfies a different wave equation, namely
$∇ 2 ψ L = 4 π k / ϵ b$
, which we find from the Coulomb law
$∇ ϵ b E = 4 π ρ ind$
Note that the above analysis is needed for the metal region, where the electric ($ν = E$) and magnetic ($ν = M$) field are given by $A l ν j L$, with $j L = j l m ( k ⊥ r )$. Outside the particle,
the longitudinal scalar function vanishes since there are no induced charges in the dielectric surrounding. Therefore, the electric scalar field is given by $j l m ( k 0 r ) + t l ν h l m + ( k 0 r )
$ with unknown parameters $A l ν$ and scattering matrix $t l ν$. Exploiting the boundary conditions stated above, we find a set of linear equations for the magnetic and electric scattering matrices.
Interestingly, the magnetic scattering matrix is unchanged with respect to the local theory, indicating that magnetic modes are not sensitive to the induced longitudinal modes. The scattering matrix
for the electric scalar function is more complicated than in the local approximation due to the appearance of $ψ L$ in the metal region that contains information on the nonlocal response. The
additional boundary condition yields a prescription to calculate $ρ L$.
The local scattering matrix can then be extended by a single parameter describing nonlocal behavior of the electron motion in the conduction band
$g l = l ( l + 1 ) j l ( θ ⊥ ) j l ( q a ) q a j l ′ ( q a ) ϵ ⊥ ϵ b − 1$
and becomes with
$θ 0 = k a ϵ 0$
$θ ⊥ = k a ϵ ⊥$
$t l E = − ϵ ⊥ j l ( θ ⊥ ) [ θ 0 j l ( θ 0 ) ] ′ + ϵ 0 j l ( θ 0 ) ( [ θ ⊥ j l ( θ ⊥ ) ] ′ + g l ) ϵ ⊥ j l ( θ ⊥ ) [ θ 0 h l + ( θ 0 ) ] ′ − ϵ 0 h l + ( θ 0 ) ( [ θ ⊥ j l ( θ ⊥ ) ] ′ + g l ) ,$
where the primes indicate differentiation with respect to the
variables. The scattering coefficients
$t l ν$
fully contain the optical response of the particle for an external observer.
Note that the nonlocal parameter
vanishes under the assumption of local response (
$β GNOR → 0 ⇒ g l → 0$
) fully recovering the original Mie coefficients [
]. This allows us to study the electro-optical properties of NPs with only a small correction in available numerical procedures, see for instance
Figure 5
Likewise, for a nonlocal metal nanoshell, the magnetic response is insensitive to the nonlocal properties of the material. The electric part, however, mixes with the longitudinal components from the
two interfaces of the metal intermediate layer. For the electric scalar functions, we obtain a linear system of six equations and analytical solutions exist for the metal nanoshell [
4.3. Simulations
The modeling presented in this article was obtained by both using the commercial software COMSOL Multiphysics (
) and in-house numerical code to evaluate Mie coefficients, from Equation (
To make predictions that can be compared to experiments, the expressions obtained are used to calculate e.g., the extinction cross section of an individual sphere via
$σ ext = 2 π k 2 ϵ 0 ∑ l ( 2 l + 1 ) Im ( t l E + t l M ) .$
Note that only the electric scattering matrix is sensitive to nonlocal contributions.
The scalar electric field is obtained from
$j l m ( k 0 r ) + t l E h l m + ( k 0 r )$
outside the particle, with the corresponding spherical Bessel and Hankel functions and the related vector field from Equation (
The analytic damping expressions Equations (
)–(1d) are directly introduced as damping terms in the permittivity of the different material permittivities, Equation (
For dimers, we use a multiple elastic scattering approach [
Author Contributions
Investigation, K.K. and C.D.; Methodology, W.J.; Validation, K.K. and C.D.; Visualization, K.K. and C.D.; Writing—original draft, C.D.; Writing—review & editing, L.J. and W.J.
This research was funded by the European Cooperation in Science and Technology (COST) under the COST Action MP1406 MultiscaleSolar.
C.D. thanks the Comunidad de Madrid (Ref. 2017-T2/IND-6092) and acknowledges financial support by the Spanish Ministry of Economy, Industry and Competition (MINECO) via funding of the Centers of
Excellence Severo Ochoa (Ref. SEV-2016-0686).
Conflicts of Interest
The authors declare no conflict of interest.
Figure 1. Illustration of sources of plasmon damping and electron interaction phenomena. (a) Electron-electron collisions in the bulk material; (b) Electron-surface collisions due to confinement; (c)
Electron irradiation due to acceleration during plasmon oscillation; (d) Short-ranged electron-electron interactions, such as Coulomb force and electron diffusion.
Figure 2. Impact of quantum corrections on single nanoparticles. (a) Spectral position of the localized surface plasmon resonance (LSPR) for gold, silver and aluminum; (b) Extinction cross section
normalized to the surface of a hemisphere for silver evaluated at the respective LSPR wavelengths from (a).
Figure 3.
Extinction cross section normalized to the surface of a hemisphere for isolated (
) gold and (
) aluminum nanoparticles evaluated at the respective LSPR wavelengths from
Figure 2
Figure 4. Maximum enhancement factor $EF = | E | 2 / | E 0 | 2$ at the particle surface for gold. Dependence of (a) the maximum EF and (b) its wavelength position for the different quantum
corrections on the particle radius in water. (c), (d) The same as a function of the permittivity $ϵ 0$ of the surrounding medium for nanospheres of (c) $R = 10$ nm and (d) $R = 50$ nm.
Figure 5. Size regime for multipolar response in metal nanoparticles. Enhancement factor $EF = | E | 2 / | E 0 | 2$ at the particle surface, where the EF is maximized, for $λ = 500$ nm close to the
corresponding Mie resonance of gold with classical Mie coefficients and as inset with combined microscopic corrections. The calculations are based on the dipolar response (black), the first three
multipoles (red) and the converged result (blue).
Figure 6. Impact of microscopic electron dynamics on gold dimers in water. We show the maximum field enhancement at the gap center of gold dimers dispersed in water in dependence of their radius ($a
> 0.5$ nm) and separation (>0.1 nm) for (a) classical Mie calculations, (b) spatial dispersion with GNOR, (c) Kreibig damping, (d) all RPA corrections combined, (e) Lorentz friction and (f) GNOR with
Lorentz friction. The incident field is polarized along the dimer axis and the maximum EF is evaluated at the respective resonance frequency calculated for each case.
Figure 7. Comparison of RPA damping rates. The perturbative solution (red) and exact Lorentz friction (blue) for a Au nanoparticle in water.
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Statistical precursor signals for Dansgaard–Oeschger cooling transitions
Articles | Volume 20, issue 3
© Author(s) 2024. This work is distributed under the Creative Commons Attribution 4.0 License.
Statistical precursor signals for Dansgaard–Oeschger cooling transitions
Given growing concerns about climate tipping points and their risks, it is important to investigate the capability of identifying robust precursor signals for the associated transitions. In general,
the variance and short-lag autocorrelations of the fluctuations increase in a stochastically forced system approaching a critical or bifurcation-induced transition, making them theoretically suitable
indicators to warn of such transitions. Paleoclimate records provide useful test beds if such a warning of a forthcoming transition could work in practice. The Dansgaard–Oeschger (DO) events are
characterized by millennial-scale abrupt climate changes during the glacial period, manifesting most clearly as abrupt temperature shifts in the North Atlantic region. Some previous studies have
found such statistical precursor signals for the DO warming transitions. On the other hand, statistical precursor signals for the abrupt DO cooling transitions have not been identified. Analyzing
Greenland ice core records, we find robust and statistically significant precursor signals of DO cooling transitions in most of the interstadials longer than roughly 1500 years but not in the shorter
interstadials. The origin of the statistical precursor signals is mainly related to so-called rebound events, humps in the temperature observed at the end of interstadial, some decades to centuries
prior to the actual transition. We discuss several dynamical mechanisms that give rise to such rebound events and statistical precursor signals.
Received: 10 Jun 2023 – Discussion started: 27 Jun 2023 – Revised: 05 Feb 2024 – Accepted: 08 Feb 2024 – Published: 22 Mar 2024
A tipping point is a critical threshold beyond which a system reorganizes, often abruptly and/or irreversibly (IPCC, 2023). Once a tipping point is passed, a system can abruptly transition to an
alternative stable or oscillatory state (Boers et al., 2022). Empirical and modeling evidence suggests that some components of the Earth system might indeed exhibit tipping behavior, which poses
arguably one of the greatest potential risks in the context of ongoing anthropogenic global warming (Armstrong McKay et al., 2022; Boers et al., 2022). Paleoclimate evidence supports the fact that
abrupt climate changes due to crossing tipping points actually occurred in the past (Dakos et al., 2008; Brovkin et al., 2021; Boers et al., 2022). The Dansgaard–Oeschger events (Dansgaard et al.,
1993) are one of such past abrupt climate changes during the last glacial period and the focus of this study.
Tipping point behavior is mathematically classified into three different types (Ashwin et al., 2012). (1) Bifurcation-induced tipping is an abrupt or qualitative change of a system owing to a
bifurcation of a stable state (more generally a quasi-static attractor). (2) Noise-induced tipping is an escape from a neighborhood of a quasi-static attractor caused by the action of noisy
fluctuations (Ditlevsen and Johnsen, 2010). (3) Rate-induced tipping occurs when a system fails to track a continuously changing quasi-static attractor (Ashwin et al., 2012; Wieczorek et al., 2023;
O'Sullivan et al., 2023). In real-world systems, tipping behaviors often result from a combination of several of the above (Ashwin et al., 2012).
The theory of critical slowing down (CSD) provides a framework to anticipate critical (or bifurcation-induced) transitions (Carpenter and Brock, 2006; Scheffer et al., 2009; Kuehn, 2013; Boers, 2018,
2021; Boers and Rypdal, 2021; Boers et al., 2022; Bury et al., 2020). The framework is based on the fact that the stability of a stable state is gradually lost as the system approaches the
bifurcation point. Theoretically, the variance of the fluctuations around the fixed point diverges and the autocorrelation with a sufficiently small lag increases toward 1 at the critical point of a
codimension-1 bifurcation (Boers et al., 2022; Scheffer et al., 2009; Bury et al., 2020), where the codimension-1 bifurcations are, in simple terms, the bifurcations that can be typically encountered
by the change of a single control parameter (Thompson and Sieber, 2011). Thus, the changes in CSD indicators such as the increase of the variance as well as the autocorrelation can be seen as
statistical precursor signals (SPSs) of critical transitions. See Dakos et al. (2012) as well as Boers et al. (2022) for various methods and CSD indicators for anticipating critical transitions.
Dansgaard–Oeschger (DO) events are millennial-scale abrupt climate transitions during glacial intervals (Dansgaard et al., 1993). They are most clearly imprinted in the δ^18O and calcium ion
concentration [Ca^2+] records from the Greenland ice cores (Fig. 1) (Rasmussen et al., 2014; Seierstad et al., 2014). The δ^18O and [Ca^2+] are interpreted as proxies for site temperature and
atmospheric circulation changes, respectively. DO warmings occur typically within a few decades and are followed by gradual cooling during relatively warm glacial states termed “interstadials”,
before a rapid return to cold states referred to as “stadials”. The amplitude of the abrupt warming transitions ranges from 5 to 16.5°C (Kindler et al., 2014, and references therein). The Greenland
temperatures change concurrently with the North Atlantic temperatures (Bond et al., 1993; Martrat et al., 2004), atmospheric circulation patterns (Yiou et al., 1997), seawater salinity (Dokken et al.
, 2013), sea-ice cover (Sadatzki et al., 2019), and the Atlantic Meridional Overturning Circulation (AMOC), as inferred from indices such as $\mathrm{Pa}/\mathrm{Th}$ and δ^13C (Henry et al., 2016).
The combined proxy evidence suggests that the DO events arise from interactions among these components (Menviel et al., 2020; Boers et al., 2018). The prevailing view is that the mode switching of
the AMOC plays a principal role in generating DO events (Broecker et al., 1985; Rahmstorf, 2002), but it remains debated whether the inferred AMOC changes are a driver of DO events or a response to
the changes in the atmosphere–ocean–sea-ice system in the North Atlantic, Nordic Seas, and the Arctic (Li and Born, 2019; Dokken et al., 2013). Recently, an increasing number of comprehensive climate
models succeeded in simulating DO-like self-sustained oscillations, suggesting that DO events can arise spontaneously from complex interactions between the AMOC, ocean stratification/convection,
atmosphere, and sea ice (Peltier and Vettoretti, 2014; Vettoretti et al., 2022; Brown and Galbraith, 2016; Klockmann et al., 2020; Zhang et al., 2021; Kuniyoshi et al., 2022; Malmierca-Vallet and
Sime, 2023).
The DO events are considered the archetype of climate tipping behavior (Boers et al., 2022; Brovkin et al., 2021). Early works found an SPS based on autocorrelation for one specific DO warming – the
onset of Bølling–Allerød interstadial (Dakos et al., 2008). In subsequent works, the existence of SPS for DO warmings was questioned considering that DO warmings are noise induced rather than
bifurcation induced (Ditlevsen and Johnsen, 2010; Lenton et al., 2012). However, a couple of later studies detected SPS for several DO warmings either by ensemble averaging of CSD indicators for many
events (Cimatoribus et al., 2013) or by using wavelet-transform techniques focusing on a specific frequency band (Rypdal, 2016; Boers, 2018). On the other hand, it has so far not been shown whether
DO coolings are preceded by characteristic CSD-based precursor signals as well.
Recent studies have inferred that the AMOC is currently at its weakest in at least a millennium (Rahmstorf et al., 2015; Caesar et al., 2018) (see also Kilbourne et al. (2022) for possible
uncertainties). The declining AMOC trend is projected to continue in the coming century, although the projections of the AMOC strength in the next 100 years are model dependent (Masson-Delmotte
et al., 2021). Furthermore, the studies applying CSD indicators to observed AMOC fingerprints (Boers, 2021; Ben-Yami et al., 2023; Ditlevsen and Ditlevsen, 2023) as well as a long-term reconstruction
of the Atlantic multidecadal variability (Michel et al., 2022) suggest that the AMOC stability may have declined and the AMOC might be approaching a dangerous tipping point. In this context, it is
important to investigate whether CSD-based precursor signals can be detected for the DO cooling transitions as well, supposing that the DO events reflect past AMOC changes. However, of course,
predictability of past events does not necessarily imply any predictability in the future, especially given that the recent AMOC weakening is presumably driven by global warming and is thus from a
mechanistic point of view different from past AMOC weakenings in the glacial period.
In this study we report SPS for DO cooling transitions recorded in δ^18O and log[10][Ca^2+] (Seierstad et al., 2014; Rasmussen et al., 2014) from three Greenland ice cores: the North Greenland Ice
Core Project (NGRIP), the Greenland Ice Core Project (GRIP), and the Greenland Ice Sheet Project 2 (GISP2) (see Fig. 1 for NGRIP). The important source of observed SPS stems from so-called rebound
events, humps in the temperature proxy occurring at the end of interstadials, some decades to centuries prior to the transition (Capron et al., 2010). When CSD indicators such as variance or lag-1
autocorrelation are used for anticipating a tipping point, we conventionally assume that a system gradually approaches a bifurcation point. However, if DO cycles are spontaneous oscillations as
suggested in the comprehensive climate models (see above), in a strict sense there might not be any bifurcation occurring around the timings of the abrupt transition in the DO cycles. Nevertheless,
with conceptual models of DO cycles, we demonstrate that CSD indicators may show precursor signals for abrupt transitions due to particular dynamics near a critical point or a critical manifold.
The remainder of this paper is organized as follows. In Sect. 2, the data and method are described. In Sect. 3, we identify robust and statistically significant SPSs for several DO cooling
transitions following interstadials with sufficient data length and show that rebound events prior to cooling transition are the source of observed SPSs. In Sect. 4 we discuss the results by using
conceptual models. A summary is given in Sect. 5.
2.1Greenland ice core records
We explore CSD-based precursor signals for DO cooling transitions recorded in δ^18O and log[10][Ca^2+] (Seierstad et al., 2014; Rasmussen et al., 2014) from three Greenland ice cores: NGRIP, GRIP,
and GISP2 (see Fig. 1 for NGRIP). Multiple records are used for a robust assessment because each has regional fluctuations as well as proxy- and ice-core-dependent uncertainties. The six records have
been synchronized and are given at 20-year resolution (Seierstad et al., 2014; Rasmussen et al., 2014). They continuously span the last 104kyrb2k (kiloyears before 2000CE), beyond which only NGRIP
δ^18O is available up to a part of the Eemian Interglacial. In addition, we use a version of the NGRIP δ^18O and dust records at 5cm depth resolution (EPICA community members, 2004; Gkinis et al.,
2014; Ruth et al., 2003) in order to check the dependence of results on temporal resolutions, with the caveat that these high-resolution records span only the last 60kyr.
We follow the classification of interstadials and stadials and associated timings of DO warming and cooling transitions by Rasmussen et al. (2014), where Greenland interstadials (stadials) are
labeled with “GI” (“GS”) with a few exceptions below. A rebound event is an abrupt warming often observed before an interstadial abruptly ends (Capron et al., 2010) (arrows in Figs. 1, 2, and 3).
Generally, a long interstadial accompanies a long rebound event (their durations are correlated with R^2=0.95, Capron et al., 2010). In Capron et al. (2010) and Rasmussen et al. (2014), GI-14 and the
subsequent interstadial, GI-13, are seen as one long interstadial, with GI-13 considered to be a strongly expressed rebound event ending GI-14 because the changes in δ^18O and log[10][Ca^2+] during
the quasi-stadial GS-14 do not reach the baseline levels of adjacent stadials. Similarly GI-23.1 and GI-22 are also seen as one long interstadial, and GI-22 is regarded as a rebound event (and
GS-23.1 as quasi-stadial) (Capron et al., 2010; Rasmussen et al., 2014). GI-20a is also recognized as a rebound event in Rasmussen et al. (2014). Given that the rebound events are warmings following
a colder spell during interstadial conditions that does not reach the stadial levels (Rasmussen et al., 2014), we regard the following nine epochs as rebound-type events: GI-8a, the hump at the end
of GI-11 (42240–∼42500yrb2k), GI-12a, GI-13, the hump at the end of GI-16.1 (56500–∼56900yrb2k), GI-20a, GI-21.1c-b-a (two warming transitions), GI-22, and GI-25a. When we examine the effect
of rebound events on our results, we exclude the entire parts including the cold spells prior to the rebound events.
The start (warming) and end (cooling) of each DO event are identified in 20-year resolution based on both δ^18O and [Ca^2+] in Rasmussen et al. (2014). The estimated uncertainty of event timing
varies from event to event. We remove the 2σ uncertainty range of the event timing (40 to 400 years) estimated in Rasmussen et al. (2014) from our calculation of CSD indicators. It effectively
excludes parts of the transitions themselves from the calculation of the CSD indicators. Since the calculation of CSD indicators requires a minimum length of data points, we mainly focus on
interstadials longer than 1000 years after removing 2σ uncertainty ranges of the transition timings, using 20-year resolution data (Table S1 in the Supplement). This results in 12 DO interstadials to
be investigated (Fig. 1, gray shades). We deal with the interstadials shorter than 1000 years but longer than 300 years using high-resolution records in Sect. 3.2.
2.2Statistical indicators of critical slowing down
Based on the theory of critical slowing down (CSD), we posit that the stability of a dynamical system perturbed by noise is gradually lost as the system approaches a bifurcation point (Boers et al.,
2022; Scheffer et al., 2009; Bury et al., 2020). For the fold bifurcation (also known as the saddle-node bifurcation), the variance of the fluctuations around a local stable state diverges and the
autocorrelation function of the fluctuations increases toward 1 at any lag τ. The same is true for the transcritical as well as the pitchfork bifurcation (Bury et al., 2020). For the Hopf
bifurcation, the variance increases, but the autocorrelation function of the form $C\left(\mathit{\tau }\right)={e}^{\mathit{u }|\mathit{\tau }|}\mathrm{cos}\mathit{\omega }\mathit{\tau }$ may
increase or decrease depending on τ, where ν(≤0) and ±ωi are the real and imaginary parts of the complex eigenvalues of the Jacobian matrix of the local linearized system (Bury et al., 2020).
Nevertheless, the autocorrelation function C(τ) increases for sufficiently small τ. For discrete time series, we follow previous studies and calculate the lag-1 autocorrelation corresponding to a
minimal τ. These characteristics can be used to anticipate abrupt transitions caused by codimension-1 bifurcations. Promisingly previous studies show that these CSD indicators and related measures
can indeed anticipate simulated AMOC collapses (Boulton et al., 2014; Klus et al., 2018; Livina and Lenton, 2007; Held and Kleinen, 2004).
Prior to calculating CSD indicators, we estimate the local stable state by using a local regression method called the locally weighted scatterplot smoothing (LOESS) (Cleveland et al., 1992; Dakos
et al., 2012). In this approach, the time series is seen as a scatter plot and fitted locally by a polynomial function, giving more weight to points near the point whose response is being estimated
and less weight to points further away. Here the polynomial degree is set to 1; i.e., the smoothing is performed with the local linear fit. Nevertheless, the LOESS provides nonlinear smoothed curves.
The smoothing span parameter α that defines the fraction of data points involved in the local regression is set to 50% of each interstadial length in a demonstration case, but we examine the
dependence of results on α over the range 20%–70%. The difference between the record and the smoothed one gives the residual fluctuations. The CSD indicators, i.e., variance and lag-1
autocorrelation, are calculated for the residuals over a rolling window. The size W of this rolling window is set to 50% of each interstadial length in the demonstration case. To test the
robustness, this is changed over the range 20%–60%.
The statistical significance of precursor signals of critical transitions, in terms of positive trends of CSD indicators, is assessed by hypothesis testing (Theiler et al., 1992; Dakos et al., 2012;
Rypdal, 2016; Boers, 2018). We consider as null model a stationary stochastic process with preserved variance and autocorrelation. The n surrogate data are prepared from the original residual series
by the phase-randomization method, thus preserving the variance and autocorrelation function of the original time series via the Wiener–Khinchin theorem. Here we take n=1000. The linear trend (a[o])
of the CSD indicator for the original time series and the linear trends (a[s]) of CSD indicators for the surrogate data are calculated. We consider the precursor signal of the original time series
statistically significant at the 5% level if the probability of a[s]>a[o] (p value) is less than 0.05. The significance level of 0.05 is conservative given that some works analyzing ecological or
paleo-data adopt the significance level of 0.1 (Dakos et al., 2012; Thomas et al., 2015).
3.1Characteristic precursor signals of DO coolings
As CSD indicators we consider the variance and lag-1 autocorrelation, calculated in rolling windows across each interstadial. The 12 interstadials longer than 1000 years are magnified in Figs. 2 and
3 (top rows, blue) for the NGRIP δ^18O record. See Figs. S1–S10 in the Supplement for the other records. For each interstadial, the nonlinear trend is estimated using LOESS smoothing (Figs. 2 and 3,
top row, red). In this case the smoothing span α that defines the fraction of data points involved in the local regression is set to 50% of each interstadial length. Gaussian kernel smoothing gives
similar results. The difference between the record and the nonlinear trend gives the approximately stationary residual fluctuations (second row). The CSD indicators are calculated from the residual
series over a rolling window. In Figs. 2 and 3 the rolling window size W is set to 50% of each interstadial length (a default value in Dakos et al., 2008). The smoothing span α and the rolling
window size W are taken as fractions of individual interstadial length because timescales of local fluctuations (such as the duration of rebound events) change with the entire duration of
interstadial. We examine the dependence of the results on α and W as part of our robustness tests.
The variance is plotted in the third row of Figs. 2 and 3. Positive trends in the variance are observed for 9 out of 12 interstadials; the individual trends are statistically significant in 6 out of
12 cases (p<0.05), based on a null model assuming the same overall variance and autocorrelation, constructed by producing surrogates with randomized Fourier phases. The lag-1 autocorrelation is also
plotted for the same data in the bottom row. Positive trends in lag-1 autocorrelation are observed for 10 out of 12 interstadials but are statistically significant only in two cases (p<0.05). Just a
positive trend without significance cannot be considered a reliable SPS, but if one indicator has a significantly positive trend, the other indicator with a consistently positive trend may at least
support the conclusion (e.g., GI-19.2 and GI-14-13 in Fig. 3). In several cases (GI-24.2, GI-21.1, GI-16.1, GI-14-13, and GI-12), the lag-1 autocorrelation first decreases and then increases. The
initial decreases, harming monotonic increases of CSD indicators, might reflect a memory of the preceding DO warming transition. On the other hand, the drastic increases in both indicators near the
end of the interstadials reflect the rebound events (arrows in Figs. 2 and 3). We obtain similar results for the other ice core records (Figs. S1–S10). While we observe a number of positive trends
for all the records, the number of statistically significant trends detected depends on the record and CSD indicator (Table S2).
We check the robustness of our results against changing smoothing span α and rolling window size W (Dakos et al., 2012). We calculate the p value for the trend of each indicator changing the
smoothing span between 20% and 70% of interstadial length (in steps of 10%) and the rolling window size between 20% and 60% (also in 10% steps). This yields a 6×5 matrix for the p values.
Example results for GI-25 and δ^18O are shown in Fig. 4a (variance) and 4b (lag-1 autocorrelation). The cross mark (x) indicates significant positive trends (p<0.05) and the small open circle (o)
indicates positive trends that are significant at the 10% level but not at the 5% level. Full results for the 12 interstadials, six records, and two CSD indicators are shown in Figs. S11–S22. We
consider positive trends in CSD indicators, i.e., the SPS of the transition, to be overall robust if we obtain significant positive trends (p<0.05) for more than half (>15) of the 30 parameter sets.
The robustness analysis is performed for all the long interstadials of the six records and the two CSD indicators (Fig. 4d). Among the 12 interstadials, we find at least one robust SPS for eight
interstadials (GI-25, GI-23.1, GI-21.1, GI-20, GI-19.2, GI-14, GI-12, and GI-8) and multiple robust SPSs for six (GI-25, GI-23.1, GI-21.1, GI-14, GI-12, and GI-8). If the data series is a stationary
stochastic process, the probability of spuriously observing a robust SPS is estimated to be 5% (Appendix A). In this case, the probability of detecting more than two robust SPSs from 12 independent
stationary samples (i.e., from each row in Fig. 4d) by chance is only ∼2%. Thus, the risk of obtaining the results by chance is quite low. For each interstadial, the detection of robust SPSs depends
on the proxy and core. This is possibly because different proxies from different cores are contaminated by different types and magnitudes of noise (e.g., δ^18O may record local fluctuations of
temperatures and log[10][Ca^2+] turbulent fluctuations of local wind circulations). Robust SPSs are observed for most interstadials longer than roughly 1500 years (GI-25, GI-23.1, GI-21.1, GI-20,
GI-19.2, GI-14, GI-12, and GI-8 except GI-1) but not for the other interstadials, shorter than roughly 1500 years (compare Figs. 4c and 4d).
3.2Further sensitivity analyses
We examine how much the rebound events affect the detection of CSD-based SPS. For this purpose CSD indicators are again calculated excluding the rebound events and their preceding cold spells (see
Sect. 2.1). While eight interstadials (GI-25, GI-23.1, GI-21.1, GI-20, GI-16, GI-14, GI-12, and GI-8) exhibit robust SPS with the rebound events included, only four interstadials (GI-23.1, GI-14,
GI-12, and GI-8) exhibit robust SPS without the rebound events (Fig. S23). The rebound events should hence be considered important, sometimes indispensable, sources for SPS of DO coolings.
We also examine the dependence of the results on the time resolution of the data. Here we use a high-resolution (5cm depth) δ^18O record (EPICA community members, 2004; Gkinis et al., 2014) and a
dust record (Ruth et al., 2003) from the NGRIP over the last 60kyr. Since the data in these records are non-uniform in time, they are linearly resampled every 5 years before calculating CSD
indicators. We focus on 11 interstadials longer than 300 years in order to have enough data points. For the dust record, three interstadials (GI-15, GI-8, and GI-7) are excluded from the analysis
because the original data have long parts of missing values. The CSD indicators, calculated with a smoothing span of α=50% and rolling windows of W=50%, are shown in Figs. S24–S27. Through the
robustness analyses with respect to α and W, we find at least one robust SPS for three out of 11 interstadials (Fig. S28). The robust SPSs for GI-14-13 and GI-12 from the high-resolution records are
consistent with those from the 20-year resolution records. Moreover for GI-1, the high-resolution δ^18O record exhibits a robust SPS in terms of lag-1 autocorrelation, although the 20-year resolution
record does not. Robust SPSs have not been observed again for interstadials shorter than roughly 1500 years (Figs. S28 and 4).
4Possible dynamical mechanisms
We detected robust precursor signals of DO cooling transitions for most interstadials longer than roughly 1500 years but not for shorter interstadials. The results suggest that long interstadials,
the existence of rebound events, and the presence of SPS for the DO cooling transitions are all related (except for GI-19.2, which has no noticeable rebound event). These aspects may be related to
generic properties of nonlinear dynamical systems. On the basis of conceptual mathematical models, we discuss four possible dynamical mechanisms leading to the precursor signals of DO cooling
transitions. In three of four mechanisms, oscillations such as the rebound events can systematically arise prior to the abrupt cooling transitions. These modeling results justify the inclusion of the
rebound events in the search for precursor signals presented above. Unless otherwise mentioned, details on model parameters as well as the hysteresis experiments conducted are given in Appendix B.
1. The fold bifurcation mechanism. Since the pioneering work by Stommel (1961), the AMOC is considered to exhibit bistability depending on the background condition (Rahmstorf, 2002). The bistability
of the AMOC strength x may be conceptually modeled by a double-fold bifurcation model: $\stackrel{\mathrm{˙}}{x}=f\left(x\right)+\mathit{\mu }+\mathit{\xi }\left(t\right)$, where f(x) has two
fold points such as x−x^3 and $|x|\left(\mathrm{1}-x\right)$. Here we take the quadratic from $f\left(x\right)=|x|\left(\mathrm{1}-x\right)$, but the following arguments are qualitatively the
same for x−x^3. The parameter μ represents the surface salinity flux (i.e., negative freshwater flux), and ξ(t) denotes white Gaussian noise. The unperturbed model for ξ(t)=0 has equilibria on an
S-shaped curve: $f\left(x\right)+\mathit{\mu }=\mathrm{0}$ (Fig. 5a, green). The state x(t) initially on the upper stable branch jumps down to the lower stable branch as μ decreases across the
fold bifurcation point at $\mathit{\mu }=-\mathrm{0.25}$. The variance and the autocorrelation of the local fluctuations (i.e., CSD indicators) increase as μ slowly approaches the fold
bifurcation point since the restoring rate toward the stable state decreases, as shown in Fig. 6a (Scheffer et al., 2009; Boers et al., 2022).
2. Stochastic slow–fast oscillation mechanism. The FitzHugh–Nagumo (FHN) system is a prototypical model for slow–fast oscillations and excitability (FitzHugh, 1961; Nagumo et al., 1962). It is often
invoked for conceptual models of DO oscillations (Rial and Yang, 2007; Kwasniok, 2013; Roberts and Saha, 2017; Mitsui and Crucifix, 2017; Lohmann and Ditlevsen, 2019; Riechers et al., 2022;
Vettoretti et al., 2022). An FHN-type model of DO oscillations can be obtained by introducing a slow variable y into the fold bifurcation model: $\stackrel{\mathrm{˙}}{x}=\frac{\mathrm{1}}{{\
mathit{\tau }}_{x}}\left(|x|\left(\mathrm{1}-x\right)+y+\mathit{\mu }\right)+\mathit{\xi }\left(t\right)$, $\stackrel{\mathrm{˙}}{y}=\frac{\mathrm{1}}{{\mathit{\tau }}_{y}}\left(-x-y\right)$,
where τ[x] and τ[y] are timescale parameters (τ[x]≪τ[y]). Invoking the salt-oscillator hypothesis for DO oscillations suggested by the comprehensive climate model simulations that are successful
in reproducing DO cycles (Vettoretti and Peltier, 2018), we may interpret y as the surface mixed-layer salinity in the northern North Atlantic and Labrador Sea, which gradually decreases
(increases) when the AMOC intensity x is strong (weak).
Here we consider the case that the system is excitable. For example, for μ=0.26, the unperturbed system has a stable equilibrium near the upper fold point of the S-shaped critical manifold, $\
mathit{\left\{}\left(x,y\right)\in {\mathbb{R}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}|\phantom{\rule{0.125em}{0ex}}y=-|x|\left(\mathrm{1}-x\right)-\mathit{\mu }\mathit{\right\}}$ (Fig. 5c,
green), but the dynamical noise ξ(t) enables the escape from the barely stable equilibrium and sustains stochastic oscillations (Fig. 5b and c, blue). Due to the timescale separation (τ[x]≪τ[y]),
the oscillations occur along the attracting parts of the critical manifold (Fig. 5c). Because y is much slower than x, the dynamics of x is similar to the dynamics of the fold bifurcation model
with slowly changing y. As a result, SPS can be effectively observed near the upper fold point of the critical manifold (Fig. S29). However, this example is not rigorous bifurcation-induced
tipping. In the example of an excitable system (Fig. 5b and c), the underlying system always has a weakly stable fixed point, and no true bifurcation leading to CSD occurs. In fact, the actual
tipping in this case is noise induced. However, we can effectively observe the SPS in CSD indicators in this case as well, since the system would in each cyclic iteration move from more stable to
less stable conditions until it finally tips to initiate the next cycle, and this partial decrease in stability is imprinted in the CSD indicators (Fig. S29). The increase of the variance prior
to the transitions in the FHN model is also reported in Meisel and Kuehn (2012). Since the unperturbed system has an equilibrium near the upper fold point, the motion is stagnant near the fold
point. This provides favorable conditions for observing SPS. The state jumps from the upper branch of the critical manifold to its lower branch often occur after an upward jump induced by noise.
These upward jumps resemble the rebound events prior to DO cooling transitions. The overall phenomenon is the same in the self-sustained oscillation regime of the FHN model, as long as the
equilibrium is located near the upper fold point of the critical manifold (μ≃0.25).
3. Hopf bifurcation mechanism. In contrast to the fold bifurcation, the Hopf bifurcation manifests oscillatory instability (Strogatz, 2018). In several ocean box models, the thermohaline circulation
is destabilized via a Hopf bifurcation (Alkhayuon et al., 2019; Lucarini and Stone, 2005; Abshagen and Timmermann, 2004; Sakai and Peltier, 1999). It is also considered a potential generating
mechanism of DO oscillations in a low-order coupled climate model (Timmermann et al., 2003) and in a comprehensive climate model (Peltier and Vettoretti, 2014). Assume that the parameter μ
decreases slowly in the FHN-type model (Fig. 5d and e). The underlying dynamics changes from the stable equilibrium to the limit-cycle oscillations at the Hopf bifurcation point $\mathit{\mu }={\
mathit{\mu }}_{\text{Hopf}}\equiv \left(\mathrm{1}-{\mathit{\tau }}_{x}/{\mathit{\tau }}_{y}{\right)}^{\mathrm{2}}/\mathrm{4}$ (Appendix B). If stochastic forcing is added to the system,
noise-induced small oscillations can appear prior to the onset of the limit-cycle oscillations (Fig. 5d and e). The precursor oscillations resemble rebound events, while their shape depends on
the noise as well as the change rate of μ. Again SPS can be observed near the Hopf bifurcation point (Fig. S30) (Bury et al., 2020; Meisel and Kuehn, 2012; Boers et al., 2022). The small
oscillations prior to downward transitions, like the DO rebound events, do not appear if the system goes deeply into the self-sustained oscillation regime away from the Hopf bifurcation point ($\
mathit{\mu }<{\mathit{\mu }}_{\text{Hopf}}\approx \mathrm{0.245}$ in Fig. 5d).
4. Mixed-mode oscillation mechanism. Mixed-mode oscillations (MMOs) are periodic oscillations consisting of small- and large-amplitude oscillations (Koper, 1995; Desroches et al., 2012; Berglund and
Landon, 2012). They often arise in systems with one fast variable and two slow variables (Desroches et al., 2012). In this regard, we can extend the above FHN-type model to exhibit MMOs, for
example, as follows: $\stackrel{\mathrm{˙}}{x}=\frac{\mathrm{1}}{{\mathit{\tau }}_{x}}\left(|x|\left(\mathrm{1}-x\right)+y+\mathit{\mu }\right)$, $\stackrel{\mathrm{˙}}{y}=\frac{\mathrm{1}}{{\
mathit{\tau }}_{y}}\left(-x-y+k\left(z-y\right)\right)$ and $\stackrel{\mathrm{˙}}{z}=\frac{\mathrm{1}}{{\mathit{\tau }}_{z}}\left(-x-z+k\left(y-z\right)\right)$, where z is another slow variable
with timescale τ[z] (≫τ[x]) and k is the diffusive-coupling constant between slow variables. We interpret y as the surface salinity in the northern North Atlantic convection region that directly
affects the AMOC strength x again and z as the surface salinity outside the convection region that affects the surface salinity y in the convection region via mixing. This extended FHN-type model
is introduced here only to demonstrate that MMOs may appear in an FHN-type model with a minimal dimensional extension. For certain parameter settings (Appendix B), the system has an unstable
equilibrium $\left(x,y,z\right)=\left(\sqrt{\mathit{\mu }},-\sqrt{\mathit{\mu }},-\sqrt{\mathit{\mu }}\right)$ of saddle-focus type, with one stable direction with a negative real eigenvalue and
a two-dimensional unstable manifold with complex eigenvalues with a positive real part. The slow–fast oscillations occur along the critical manifold $\mathit{\left\{}\left(x,y,z\right)\in {\
mathbb{R}}^{\mathrm{3}}\phantom{\rule{0.125em}{0ex}}|\phantom{\rule{0.125em}{0ex}}y=-|x|\left(\mathrm{1}-x\right)-\mathit{\mu }\mathit{\right\}}$ (Fig. 5f and g). However, due to the saddle-focus
equilibrium on the critical manifold, the trajectory is attracted toward the saddle from the direction of the stable manifold (black segment) and repelled from it in a spiralling fashion. The
striking point is the systematic occurrence of small-amplitude oscillations prior to the abrupt transition, which also resemble the rebound events prior to the DO cooling transitions. A more
realistic time series is obtained if an observation noise is added on x(t) (Fig. S31). Then SPS can be stably observed near the fold point of the critical manifold.
Based on the four types of simple mathematical models, we have proposed four possible dynamical mechanisms for the DO cooling transitions that can manifest statistical precursor signals (SPS): (1)
strict CSD due to the approaching of a fold bifurcation; (2) CSD in a wider sense, in stochastic slow–fast oscillations; (3) noise-induced oscillations prior to Hopf bifurcations; or (4) the signal
of mixed-mode oscillations. While the details of these mechanisms are different, they are all related to the fold points of the equilibrium curve or the critical manifolds. As a result, the SPS can
be detected by the conventional CSD indicators.
Mechanisms (2), (3), and (4) can generate behavior resembling the rebound events, leading to increases in the classic CSD indicators. In the toy models, rebound event-like behavior is generated when
the trajectory passes by an equilibrium point with marginal stability (i.e., the equilibrium has neither strong stability leading to a permanent state nor strong instability leading to short
interstadials) (Fig. 5b–g). In this case, the duration of the modeled interstadial is relatively long in relation to the marginal stability. By contrast, the absence of equilibrium or the presence of
a strongly unstable equilibrium near the fold point of the critical manifold leads to brief interstadials without a rebound event and consequently a lack of SPS. This provides a possible explanation
of why the rebound events and the robust SPS are simultaneously observed for long interstadials but not for short interstadials.
Another possible explanation for the lack of SPS for short interstadials is the following. The common assumption underlying CSD theory is that the parameter change is much slower than the system's
relaxation time, and the latter is much slower than the correlation time of the noise (Thompson and Sieber, 2011; Ashwin et al., 2012). If this assumption is violated, it is difficult to detect
CSD-based SPS (Clements and Ozgul, 2016; van der Bolt et al., 2021). Consider the fold-bifurcation-induced tipping in the Stommel-type model (1), for example (Fig. 6). If the change in the parameter
μ is faster than the system's relaxation time toward the moving stable equilibrium, it is unlikely to detect significant CSD-based SPS (Fig. 6b) because the trajectory evolves systematically away
from the equilibrium and thus cannot feel the flattening of the potential around the equilibrium even at the true bifurcation point.
In this study we have explored statistical precursor signals (SPSs) and significant increases in critical slowing down (CSD) indicators (variance and lag-1 autocorrelation), for Dansgaard–Oeschger
(DO) cooling transitions following interstadials, using six Greenland ice core records. Among the 12 interstadials longer than 1000 years, we find at least one robust SPS for eight interstadials
longer than roughly 1500 years (GI-25, GI-23.1, GI-21.1, GI-20, GI-19.2, GI-14, GI-12, and GI-8) and multiple robust SPSs for six of them (GI-25, GI-23.1, GI-21.1, GI-14, GI-12, and GI-8) (Fig. 4d).
Robust SPSs are, however, not observed for interstadials shorter than roughly 1500 years. One might link the increase in the proxy variance with the tendency of larger climatic fluctuations in colder
climates (Ditlevsen et al., 1996), but the increases in the lag-1 autocorrelation cannot generally be explained by it. The analysis removing the rebound events from the data shows that the rebound
events prior to the cooling transitions are important in producing the SPS.
We have proposed four different dynamical mechanisms to explain the observed SPSs: (1) strict CSD due to the approaching of a fold bifurcation; (2) CSD in a wider sense, in stochastic slow–fast
oscillations; (3) noise-induced oscillations prior to Hopf bifurcations; or (4) the signal of mixed-mode oscillations. In the last three mechanisms, oscillations such as the rebound events can
systematically arise prior to the abrupt cooling transitions. These precursor oscillations are due to marginally (un)stable equilibria on the critical manifolds that cause a long-lived quasi-stable
state (like long interstadials). This can explain why rebound events and SPSs are simultaneously observed only for long interstadials and are not observed for short ones. While the SPSs for
bifurcation-induced tipping events (mechanisms 1 and 3) are established, detailed properties of SPSs for the stochastic slow–fast oscillations of the excitable system (mechanism 2) and for the
mixed-mode oscillations (mechanism 4) remain to be elucidated.
We should mention the assumptions made in this study as well as alternative scenarios for the DO cooling transitions. First, the four dynamical mechanisms assume slow changes in parameters or slow
variables which cause bifurcations in the fast subsystem. On the other hand, the rate-induced tipping mechanism has also been invoked for a possible AMOC collapse, where the rate of change of the
external forcing (e.g., freshwater flux or atmospheric CO[2] concentration) determines the future AMOC state (Alkhayuon et al., 2019; Lohmann and Ditlevsen, 2021; Ritchie et al., 2023). The lack of
observed SPSs for the interstadials less than roughly 1500 years indicates a rate-dependent aspect of the DO cooling transitions. However, a comprehensive investigation of DO cooling transitions from
the viewpoint of rate-induced tipping is beyond the scope of this work. Second, a recent study using an ocean general circulation model shows that a rebound-event-type behavior of AMOC is caused by a
behavior called “the intermediate tipping”, due to multiple stable ocean circulation states that exist near but prior to the tipping point leading to a significant AMOC weakening (Lohmann et al.,
2023). The intermediate tipping mechanism for rebound events is different from the possible low-dimensional dynamical mechanisms proposed in this study. Further studies are needed to elucidate the
dynamical as well as physical origin of DO coolings and associated rebound events.
We have shown that past abrupt DO cooling transitions in the North Atlantic region can be anticipated based on classic CSD indicators if they are preceded by long interstadials. However, it is found
to be difficult to anticipate DO cooling events, at least from the 20-year-resolution ice core Greenland records, if they occur after a short interstadial. If the DO coolings transitions are actually
associated with AMOC weakening (see the “Introduction”), our results may have an implication for the predicted weakening of the AMOC and its possible collapse in the future: the prediction with CSD
indicators could be more difficult if the forcing changes fast. There is, however, a caveat to this implication because the past DO cooling transitions are different from the presently inferred AMOC
changes. The time resolution (mainly 20 years and additionally 5 years) and the length (mainly >1000 years and additionally >300 years) of the interstadial segment data used in this study are coarser
and mostly longer than the annual data used for analyzing AMOC fingerprints during the industrial period (Boers, 2021; Ben-Yami et al., 2023; Ditlevsen and Ditlevsen, 2023) and the last millennium (
Michel et al., 2022). More crucially, the revealed predictability of past DO cooling events does not necessarily imply predictability of a potential future AMOC collapse since the recent AMOC
weakening, possibly driven by global warming but potentially also part of natural variability, is mechanistically very different from past AMOC weakening in the glacial period.
Appendix A:Probability of observing robust precursor signals
The statistical significance of precursor signals of critical transitions, in terms of positive trends of CSD indicators, is assessed by hypothesis testing (Theiler et al., 1992; Dakos et al., 2012;
Rypdal, 2016; Boers, 2018). We consider a stationary stochastic process with preserved variance and autocorrelation as the null model. The n surrogate data are prepared from the original residual
data series by the phase-randomization method, thus preserving the variance and autocorrelation function of the original time series via the Wiener–Khinchin theorem. Here we take n=1000. The linear
trend (a[0]) of the CSD indicator for the original residual time series and the linear trends (a[s]) of CSD indicators for the surrogate data are calculated. We consider the precursor signal of the
original series statistically significant at the 5% level if the probability of a[s]>a[o] (p value) is less than 0.05. Thus, if the original data are already a stationary stochastic process
(exhibiting no CSD), one should expect spuriously significant results at a probability of 0.05 by definition. In principle this is independent of the smoothing span α as well as the rolling window
size W used for calculating CSD indicators. We consider a precursor signal robust if we find significant cases (p<0.05) for more than half (>15) of 30 combinations of α and W. Then the probability of
observing a robust precursor signal can be shown to be 0.05. In order to check this numerically, we generate 5000 surrogates of the original δ^18O series of interstadial GI-25 and calculate the
probability of finding robust precursor signals. The resulting fractions are 0.041 for the variance and 0.039 for the lag-1 autocorrelation, which are close to 0.05. For the case of GI-12, we obtain
0.038 for the variance and 0.047 for the lag-1 autocorrelation, again close to 0.05. These results support the probability of observing a robust precursor signal being 5% if the data are stationary
stochastic processes.
Appendix B:Details of conceptual models used in Fig. 5
Here we describe specific settings for four conceptual models representing different candidate mechanisms for the DO cooling transitions. Unless otherwise mentioned, the stochastic differential
equations below are solved with the Euler–Maruyama method with a step size of 10^−3.
1. The bistability of the AMOC strength x can be conceptually modeled by a double-fold bifurcation model: $\stackrel{\mathrm{˙}}{x}=f\left(x\right)+\mathit{\mu }+\mathit{\xi }\left(t\right)$, where
f(x) has two fold points; here for f one can use either $f\left(x\right)=x-{x}^{\mathrm{3}}$ or $f\left(x\right)=|x|\left(\mathrm{1}-x\right)$. We take the quadratic function $f\left(x\right)=|x|
\left(\mathrm{1}-x\right)$ that arises in the Stommel (1961) model. μ represents a forcing parameter on the AMOC strength x, e.g., salinity forcing on the North Atlantic (i.e., negative
freshwater forcing). ξ(t) is white Gaussian noise, e.g., freshwater perturbations or weather forcing. In Fig. 5a, we set $\sqrt{〈{\mathit{\xi }}^{\mathrm{2}}〉}=\mathrm{0.03}$, and the initial
condition is taken at x(0)=1.1, near the upper stable fixed point of the unperturbed system. The parameter μ is then slowly decreased from 0.1 to −0.4 over the period from t=0 to t=500, to
trigger the bifurcation-induced transition.
2. The FitzHugh–Nagumo-type (FHN-type) system is a prototypical model of slow–fast oscillators (FitzHugh, 1961; Nagumo et al., 1962) and often invoked for conceptual models of DO oscillations (Rial
and Yang, 2007; Kwasniok, 2013; Roberts and Saha, 2017; Mitsui and Crucifix, 2017; Lohmann and Ditlevsen, 2019; Riechers et al., 2022; Vettoretti et al., 2022). The FHN-type model subjected to
dynamical noise can be obtained by introducing a slow variable y into the fold bifurcation model: $\stackrel{\mathrm{˙}}{x}=\frac{\mathrm{1}}{{\mathit{\tau }}_{x}}\left(|x|\left(\mathrm{1}-x\
right)+y+\mathit{\mu }\right)+\mathit{\xi }\left(t\right)$, $\stackrel{\mathrm{˙}}{y}=\frac{\mathrm{1}}{{\mathit{\tau }}_{y}}\left(-x-y\right)$, where τ[x] and τ[y] are timescale parameters (τ[x]
≪τ[y]). Following the salt-oscillator hypothesis to explain DO cycles (Vettoretti and Peltier, 2018), we may interpret y as the salinity in the polar halocline surface mixed layer, which
decreases (increases) when the AMOC is strong (weak). In the case of Fig. 5b and c, we set μ=0.26, τ[x]=0.01, τ[y]=1, and $\sqrt{〈{\mathit{\xi }}^{\mathrm{2}}〉}=\mathrm{0.3}$, and the initial
condition is taken at $\left(x\left(\mathrm{0}\right),y\left(\mathrm{0}\right)\right)=\left(-\mathrm{0.2},-\mathrm{0.45}\right)$. The x nullcline (critical manifold) of the unperturbed system is
$y=-|x|\left(\mathrm{1}-x\right)-\mathit{\mu }$ (Fig. 5c, green) and the y nullcline is $y=-x$ (Fig. 5c, magenta dashed). The intersection of the x- and y nullclines is the equilibrium point of
the unperturbed system $\left(\sqrt{\mathit{\mu }},-\sqrt{\mathit{\mu }}\right)$, which is near the fold point of the critical manifold in this parameter setting.
3. To demonstrate the Hopf bifurcation mechanism in Fig. 5d and e, the same stochastic FHN-type model is used with τ[x]=0.01, τ[y]=1, and $\sqrt{〈{\mathit{\xi }}^{\mathrm{2}}〉}=\mathrm{0.05}$, but
here μ is gradually decreased from 0.3 to 0.2, over a period of 5 time units. For $\mathrm{0.2}<\mathit{\mu }<\mathrm{0.3}$, the system has a unique equilibrium point at $\left(x,y\right)=\left(\
sqrt{\mathit{\mu }},-\sqrt{\mathit{\mu }}\right)$. The Hopf bifurcation of an equilibrium occurs if the complex eigenvalues of the Jacobian matrix at the equilibrium pass the imaginary axis (
Strogatz, 2018). The eigenvalues of the Jacobian matrix at this equilibrium are ${\mathit{\lambda }}_{±}=\frac{\mathrm{1}}{\mathrm{2}}\mathit{\left\{}\frac{\mathrm{1}-\mathrm{2}\sqrt{\mathit{\mu
}}}{{\mathit{\tau }}_{x}}-\frac{\mathrm{1}}{{\mathit{\tau }}_{y}}±\sqrt{\left(\frac{\mathrm{1}-\mathrm{2}\sqrt{\mathit{\mu }}}{{\mathit{\tau }}_{x}}-\frac{\mathrm{1}}{{\mathit{\tau }}_{y}}{\
right)}^{\mathrm{2}}-\frac{\mathrm{8}\sqrt{\mathit{\mu }}}{{\mathit{\tau }}_{x}{\mathit{\tau }}_{y}}}\mathit{\right\}}$. These eigenvalues λ[±] are complex conjugates for $\frac{\mathrm{1}}{\
mathrm{4}}\left(\mathrm{1}+\frac{{\mathit{\tau }}_{x}}{{\mathit{\tau }}_{y}}-\mathrm{2}\sqrt{\frac{{\mathit{\tau }}_{x}}{{\mathit{\tau }}_{y}}}{\right)}^{\mathrm{2}}<\mathit{\mu }<\frac{\mathrm
{1}}{\mathrm{4}}\left(\mathrm{1}+\frac{{\mathit{\tau }}_{x}}{{\mathit{\tau }}_{y}}+\mathrm{2}\sqrt{\frac{{\mathit{\tau }}_{x}}{{\mathit{\tau }}_{y}}}{\right)}^{\mathrm{2}}$. In this range of μ,
the real part of λ[±] changes from negative to positive at the Hopf bifurcation point: ${\mathit{\mu }}_{\text{Hopf}}=\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{1}-\frac{{\mathit{\tau }}_{x}}{{\
mathit{\tau }}_{y}}{\right)}^{\mathrm{2}}.$ For ${\mathit{\tau }}_{x}/{\mathit{\tau }}_{y}=\mathrm{0.01}$, ${\mathit{\mu }}_{\text{Hopf}}\approx \mathrm{0.245}.$ The initial condition is taken at
the origin.
4. The mixed-mode oscillation model is obtained if the FHN-type model is extended to have multiple interacting slow variables. For example, $\stackrel{\mathrm{˙}}{x}=\frac{\mathrm{1}}{{\mathit{\tau
}}_{x}}\left(|x|\left(\mathrm{1}-x\right)+y+\mathit{\mu }$), $\stackrel{\mathrm{˙}}{y}=\frac{\mathrm{1}}{{\mathit{\tau }}_{y}}\left(-x-y+k\left(z-y\right)\right)$, and $\stackrel{\mathrm{˙}}{z}=\
frac{\mathrm{1}}{{\mathit{\tau }}_{z}}\left(-x-z+k\left(y-z\right)\right)$, where z is another slow variable with timescale τ[z] (≫τ[x]) and k is the diffusive coupling constant between slow
variables. We interpret y as the surface salinity in the northern North Atlantic convection region, which directly affects the AMOC strength x, and z as the surface salinity outside the
convection region that affects the surface salinity y in the convection region via mixing. We set τ[x]=0.02, τ[y]=2, τ[z]=4, μ=0.225, and k=0.8. This system has an unstable equilibrium $\left
(x,y,z\right)=\left(\sqrt{\mathit{\mu }},-\sqrt{\mathit{\mu }},-\sqrt{\mathit{\mu }}\right)$ of saddle-focus type, with one stable direction with a negative real eigenvalue of −0.67 and a
two-dimensional unstable manifold with two complex conjugate eigenvalues with a positive real part of 0.94±4.7i. The initial condition is taken at $\left(x\left(\mathrm{0}\right),y\left(\mathrm
TM conceived the study and conducted the analyses with contributions from NB. Both authors discussed and interpreted the results. TM wrote the manuscript with contributions from NB.
The contact author has declared that neither of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation
in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.
The authors thank Keno Riechers and Maya Ben-Yami for their helpful comments.
The authors acknowledge funding by the Volkswagen Foundation. This is TiPES contribution #243; The TiPES (“Tipping Points in the Earth System”) project has received funding from the European Union's
Horizon 2020 research and innovation programme under grant agreement no. 820970. Niklas Boers acknowledges further funding by the European Union's Horizon 2020 research and innovation programme under
the Marie Sklodowska-Curie grant agreement no. 956170, as well as from the Federal Ministry of Education and Research under grant no. 01LS2001A.
This paper was edited by Bjørg Risebrobakken and reviewed by three anonymous referees.
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Generally, a user of LaplaceApproximation, LaplacesDemon, LaplacesDemon.hpc, PMC, or VariationalBayes does not need to use the LML function, because these methods already include it. However, LML may
be called by the user, should the user desire to estimate the logarithm of the marginal likelihood with a different method, or with non-stationary chains. The LaplacesDemon and LaplacesDemon.hpc
functions only call LML when all parameters are stationary, and only with non-adaptive algorithms.
The GD method, where GD stands for Gelfand-Dey (1994), is a modification of the harmonic mean estimator (HME) that results in a more stable estimator of the logarithm of the marginal likelihood. This
method is unbiased, simulation-consistent, and usually satisfies the Gaussian central limit theorem.
The HME method, where HME stands for harmonic mean estimator, of Newton-Raftery (1994) is the easiest, and therefore fastest, estimation of the logarithm of the marginal likelihood. However, it is an
unreliable estimator and should be avoided, because small likelihood values can overly influence the estimator, variance is often infinite, and the Gaussian central limit theorem is usually not
satisfied. It is included here for completeness. There is not a function in this package that uses this method by default. Given \(N\) samples, the estimator is \(1/[\frac{1}{N} \sum_N \exp(-LL)]\).
The LME method uses the Laplace-Metropolis Estimator (LME), in which the estimation of the Hessian matrix is approximated numerically. It is the slowest method here, though it returns an estimate in
more cases than the other methods. The supplied Model specification must be executed a number of times equal to \(k^2 \times 4\), where \(k\) is the number of parameters. In large dimensions, this is
very slow. The Laplace-Metropolis Estimator is inappropriate with hierarchical models. The IterativeQuadrature, LaplaceApproximation, and VariationalBayes functions use LME when it has converged and
sir=FALSE, in which case it uses the posterior means or modes, and is itself Laplace Approximation.
The Laplace-Metropolis Estimator (LME) is the logarithmic form of equation 4 in Lewis and Raftery (1997). In a non-hierarchical model, the marginal likelihood may easily be approximated with the
Laplace-Metropolis Estimator for model \(m\) as
$$p(\textbf{y}|m) = (2\pi)^{d_m/2}|\Sigma_m|^{1/2}p(\textbf{y}|\Theta_m,m)p(\Theta_m|m)$$
where \(d\) is the number of parameters and \(\Sigma\) is the inverse of the negative of the approximated Hessian matrix of second derivatives.
As a rough estimate of Kass and Raftery (1995), LME is worrisome when the sample size of the data is less than five times the number of parameters, and LME should be adequate in most problems when
the sample size of the data exceeds twenty times the number of parameters (p. 778).
The NSIS method is essentially the MarginalLikelihood function in the MargLikArrogance package. After HME, this is the fastest method available here. The IterativeQuadrature, LaplaceApproximation,
and VariationalBayes functions use NSIS when converged and sir=TRUE. The LaplacesDemon, LaplacesDemon.hpc, and PMC functions use NSIS. At least 301 stationary samples are required, and the number of
parameters cannot exceed half the number of stationary samples. | {"url":"https://www.rdocumentation.org/packages/LaplacesDemon/versions/16.1.6/topics/LML","timestamp":"2024-11-07T00:10:45Z","content_type":"text/html","content_length":"102616","record_id":"<urn:uuid:05c79d4e-3e15-4b80-aedc-900ed2a13a69>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00496.warc.gz"} |
Express Complex Numbers In Polar Form - Precalculus
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Example Questions
Example Question #51 : Polar Coordinates And Complex Numbers
The following equation has complex roots:
Express these roots in polar form.
Every complex number can be written in the form a + bi
The polar form of a complex number takes the form r(cos
Now r can be found by applying the Pythagorean Theorem on a and b, or:
r =
So for this particular problem, the two roots of the quadratic equation
Hence, a = 3/2 and b = 3√3 / 2
Therefore r = 3
And therefore x = r(cos
Example Question #52 : Polar Coordinates And Complex Numbers
Express the roots of the following equation in polar form.
Correct answer:
First, we must use the quadratic formula to calculate the roots in rectangular form.
Remembering that the complex roots of the equation take on the form a+bi,
we can extract the a and b values.
We can now calculate r and theta.
Using these two relations, we get
The angle theta now becomes 150.
You can now plug in r and theta into the standard polar form for a number:
Example Question #1 : Express Complex Numbers In Polar Form
Express the complex number
Correct answer:
The figure below shows a complex number plotted on the complex plane. The horizontal axis is the real axis and the vertical axis is the imaginary axis.
The polar form of a complex number is and is the length of the vector and
We use the Pythagorean Theorem to find
We find
Then we plug
Example Question #4 : Express Complex Numbers In Polar Form
What is the polar form of the complex number
Correct answer:
The correct answer is
The polar form of a complex number
which gives us
Example Question #5 : Express Complex Numbers In Polar Form
Express the complex number in polar form:
Correct answer:
Remember that the standard form of a complex number is:
To find r, we must find the length of the line
To find
Note that this value is in radians, NOT degrees.
Thus, the polar form of this equation can be written as
Example Question #6 : Express Complex Numbers In Polar Form
Express this complex number in polar form.
Possible Answers:
None of these answers are correct.
Correct answer:
Given these identities, first solve for
Example Question #501 : Pre Calculus
Correct answer:
First, find the radius
Then find the angle, thinking of the imaginary part as the height and the radius as the hypotenuse of a right triangle:
We can get the positive coterminal angle by adding
The polar form is
Example Question #3 : Express Complex Numbers In Polar Form
Correct answer:
First find the radius,
Now find the angle, thinking of the imaginary part as the height and the radius as the hypotenuse of a right triangle:
This is an appropriate angle to stay with since this number should be in quadrant I.
The complex number in polar form is
Example Question #1 : Express Complex Numbers In Polar Form
Convert the complex number
Correct answer:
First find
Now find the angle. Consider the imaginary part to be the height of a right triangle with hypotenuse
What the calculator does not know is that this angle is actually located in quadrant II, since the real part is negative and the imaginary part is positive.
To find the angle in quadrant II whose sine is also
The complex number in polar form is
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NCERT Book for Class 11 Maths PDF Download
NCERT Book for Class 11 Maths
CBSE Class 11 NCERT Books for Maths PDF Free Download
The National Council of Educational Research and Training (NCERT) is an independent organisation that was established in 1961 by the Government of India. Its primary mission is to provide the Central
and State Governments of India with assistance and advice on educational policies and programs designed to improve schooling quality.
Teachers recommend students use study materials like the Class 11 Math book PDF because they are highly effective in providing students with a comprehensive understanding of the concepts being
covered in the course. Students who are preparing for at-home examinations such as units, half-yearlies and finals will find the NCERT solutions for Class 11 mathematics to be of tremendous
assistance. These NCERT Solutions for Class 11 Maths are created by licensed math professionals who ensure that the information is delivered gradually with increasing difficulty, enabling students to
build a strong foundation. When working through issues associated with a particular subject area, it may be helpful to consult the below answers, which are arranged in a chapter-based structure.
NCERT Books for CBSE Class 11 Maths
Mathematics is an important subject taught to students from the very beginning of their educational careers. Because success in more advanced levels depends on a solid understanding of foundational
concepts, it is essential to grasp the subject’s fundamentals well.
The math book for class 11th includes a syllabus covering all the CBSE examination topics. This syllabus is fairly comprehensive. If you are looking for study material that is easily accessible, the
NCERT book of Maths Class 11 is available for download from our website. You can also try your hand at solving the questions that are included after each chapter in these NCERT books for Class 11
Chapter 1: Sets
The first chapter of the NCERT textbook for Class 11 Maths covers the topics of Sets and their Representation, Empty sets, Finite and Infinite Sets, Equal Sets, Subsets, Power sets, Universal sets,
Venn Diagrams, Operations on Sets, and Complement of a Set.
A set with no members is referred to as a null set or an empty set, which is covered in the mathematics textbook for Class 11 that NCERT publishes. The sign in mathematical notation represents the
empty set. These phrases are self-explanatory when used for finite and infinite sets. It is possible to determine whether or not a set is finite by determining whether or not it has a beginning and
an endpoint that each contains a finite number of members.
On the other hand, something represents an infinite set when the number of elements can never be exhausted. In addition, you may observe that a subset is essentially a part of another set in the PDF
version of the Maths NCERT textbook for Class 11. For example, set Y will be considered a subset of set X if each element of set Y consists of elements that are also included in set X.
Chapter 2: Relations and Functions
The following topics are covered in this chapter: (1) the Cartesian Product of Sets; (2) Relations; and (3) Functions.
According to the information included in the PDF version of the NCERT Mathematics Class 11 textbook, the Cartesian product of sets is the cross product of two sets, such as X and Y, notated as XxY,
which is the set of all ordered pairs. It is implied that XxY ≠ YxX. If either X or Y is a null set, then the XxY will likewise be a null set, which means that X times Y will be an empty set (XxY=φ).
In their most basic form, relations and functions have an ordered pair, each consisting of a set of inputs and outputs representing a connection between two values. A set of inputs and outputs is
called a relation, and the relationship between any output and one input is what we mean when we talk about function.
Chapter 3: Trigonometric Functions
Angles, trigonometric functions, trigonometric functions of sum and difference of two angles, and trigonometric equations are all covered in this chapter of the NCERT math book for class 11. Angles
are first covered by trigonometric functions and then trigonometric equations.
In the NCERT book of Mathematics for Class 11, Chapter 3 explains angles, degree measure, radian measure, the relation between radian and real numbers, the relation between degree and radian and
other related topics. The real functions that relate the angle of a right-angled triangle to that of the ratio of the side lengths are referred to as trigonometric functions.
Sine, cosine and tangent are the three fundamental operations of trigonometry. In addition to these, further derived trigonometric functions include cosecant, secant, and cotangent. Trigonometric
equations are presented in this chapter as equations of trigonometric functions of a variable.
Chapter 4: Principle of Mathematical Induction
This chapter covers these topics: (1) Motivation and (2) The Principle of Mathematical Induction.
The principle of mathematical induction is defined as the way of proving numerous mathematical propositions noted in terms of n, which is a positive integer and is discussed in Chapter 4 of the NCERT
Mathematics textbook for Class 11 students. Most of it comprises deductive reasoning, which requires one to determine whether or not a particular assertion is true. In this particular setting, the
concept of motivation refers to mathematical induction.
Chapter 5: Complex Numbers and Quadratic Equations
The topics of Complex Numbers, Algebra of Complex Numbers, The Modulus and the Conjugate of a Complex Number, Argand Plane and Polar Representation, Quadratic Equations and The Modulus and the
Conjugate of a Complex Number are covered in the fifth chapter of the mathematics textbook for class 11 that NCERT publishes.
According to the PDF Book for mathematics for Class 11, the solution to the equation x2=-1 is the number that can be written as (a+bi), where a and b are real numbers, and the value of I can be
anything. This equation cannot be satisfied by any real numbers. As a result of this, we refer to it as an imaginary number. On the other hand, if the imaginary part of a complex number is zero, then
the number itself is real. The quadratic equation, written in its standard form, is denoted by the notation NPRNCR+ bx + c = 0. The formula for the quadratic function is represented graphically as a
parabola, and the variables a, b, and c in the equation for the quadratic function are not equal to zero.
Chapter 6: Linear Inequalities
The following topics are covered in this chapter: (1) Inequalities, (2) Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation, (3) Graphical Solutions of
Linear Inequalities in Two Variables, and (4) Solutions of Systems of Linear Inequalities in Two Variables.
In the NCERT Mathematics Class 11 PDF, Chapter 6 demonstrates that a linear inequality fundamentally depicts an equation but has an inequality symbol in place of the equal signs. This signifies that
linear inequality is not equal to anything. The solution to a linear inequality is not a single value but rather a range of values that fall inside its parameters. In the context of algebraic
solutions of linear inequalities with one variable, a solution set is the collection of all possible values for the variable that can transform the inequality into a true statement.
Chapter 7: Permutations and Combinations
The Fundamental Principle of Counting, Permutations and Combinations are covered in the seventh chapter of the NCERT textbook for Class 11 mathematics.
The concepts of permutations and combinations are related to the representation of a collection of items contained within a set and a subset, as well as to various ways of arranging data. A more
precise definition of a permutation would be selecting data from inside a cluster. On the other hand, a combination refers to the sequence in which certain kinds of data are displayed. The notations
of permutations and combinations are described in the chapter that can be found in the PDF textbook for NCERT Class 11 Mathematics. The notation nPr denotes the permutation of r elements from a set
of n elements. The symbol nr denotes the combination of r components extracted from n.
Chapter 8: Binomial Theorem
The following concepts are covered in this chapter: (1) the Binomial Theorem for Positive Integral Indices and (2) General and Middle Terms.
The Binomial Theorem states that for every positive integer (n), the nth power of the sum of two numbers, x, and y, may be written as the sum of n plus 1. This is because the two numbers add up to n.
Pascal’s Triangle is included in elaborating the binomial theorem found in Chapter 8 of the 11th NCERT Maths book in a PDF format. The expansion of the general term for (a + b)n is represented by the
equation Tr + 1 = nCran– r. br. The square of a binomial is the sum of the squares of the first term, the square of the last term and twice the product of the two terms. Binomial factors include
polynomial factors that only have two components.
Chapter 9: Sequence and Series
Sequences, series, arithmetic progression, geometric progression, the relationship between arithmetic and geometric progression and the sum to n terms of special series are all covered in this
According to the explanation provided in Chapter 9 of the PDF version of the mathematics textbook for Class 11, an arithmetic progression can be defined as a sequence of numbers in which each
successive term is created by adding a constant quantity to the terms that came before it. On the other hand, a geometric progression is a sequence of integers in which each number is generated from
the one that came before it by multiplying the same number by a constant. Each number in the progression is formed from the number that came before it. In this situation, the ratio of any two phrases
that follow one another will always be equal.
Chapter 10: Straight Lines
The following topics are covered in the mathematics chapter of the NCERT textbook for class 11: (1) the Slope of a Line, (2) Various Forms of the Equation of a Line, (3) the General Equation of a
Line, and (4) the Distance of a Point from a Line.
In the NCERT textbook for Class 11 Mathematics, Chapter 10 contains not just the definition of the line but also several other related ideas. The degree to which a line is inclined can be determined
by examining its slope. In addition to that, it denotes the path that a line takes.
The slope of a line can be calculated by dividing the difference in y-coordinates between two locations on a C line by the difference in x-coordinates between those same two points. This will give
you the angle at which the line slopes. Equations of straight lines can be written in various formats, including (i) equation of horizontal and vertical lines, (ii) point-slope form equation of a
line, (iii) two-point form equation of a line, (iv) slope-intercept form equation of a line and (v) intercept form.
Chapter 11: Conic Sections
The following topics are covered in this chapter: (1) Sections of Cone; (2) Circle; (3) Parabola; (4) Ellipse; and (5) Hyperbola.
According to the mathematics textbook for Class 11, a circle is a congregation of all points in a plane that remain equidistant from a point placed on the plane. This is one of the definitions of a
circle. The centre of the circle is considered to be the fixed point, and the distance that separates the centre from any other point on the circle is defined as the radius.
A parabola is a set of all points on a plane that are the same distance from both a fixed line and a fixed point on the plane. These points are all located on the same plane. Nevertheless, the line
does not pass through this particular place. Ellipse is the shape that results when all of the points on a plane add up to the same total, and the sum of those totals remains the same between two
fixed points.
Chapter 12: Introduction to the Three Dimensional Geometry
The following topics are covered in this chapter: (1) Coordinate Axes and Coordinate Planes in Three Dimensional Space, (2) the Coordinates of a Point in Space, (3) the Distance between Two Points,
and (4) the Section Formula.
In three-dimensional geometry, the coordinate axes of the rectangular Cartesian coordinate system are defined to be essentially three lines that are mutually perpendicular to one another in Chapter
12 of the NCERT book of Mathematics for Class 11. The axes are labelled x, y and z, in that order. The XY plane, the YZ plane, and the ZX plane are the planes that are determined by a pair of axes.
Octants are the names of the eight individual space regions created by the coordinate planes. A point is denoted by the following notations: (x,0,0), (0,y,0), and (0,0,z) for the x-axis, the y-axis
and the z-axis respectively.
Chapter 13: Limits and Derivatives
The following are some of the topics covered in this chapter: (1) An Intuitive Idea of Derivatives; (2) Limits; (3) Limits of Trigonometric Functions; and (4) Derivatives.
Calculus is introduced in the PDF math book for class 11 through the concepts of limits and derivatives in chapter 13. The value that is approached by a function in a manner that is consistent with
how input is approached towards a value is referred to as the limit of the function. The instantaneous rate of change from one quantity to another is referred to as a derivative and is regarded as a
mathematical concept.
It is responsible for determining the fluctuation of the total amount each and every second. Limits and derivatives can be fundamentally differentiated from one another. A derivative is an example of
a limit. A limit, on the other hand, is a function value close to the input.
Chapter 14: Mathematical Reasoning
This chapter has the following components: (1) Statements, (2) New Statements from Old Statements, (3) Special Words/Phrases, (4) Implications and (5) Validating Statements.
The process of determining whether or not mathematical assertions are correct is an example of mathematical reasoning covered in the Class 11 Maths book PDF. Compound statements are created by
combining one or more statements with the aid of connecting words such as or, and, etc., to produce a new statement that fulfils more than one purpose. Component statements are those that make up a
compound statement and are referred to by that name. Only statements that can be either true or untrue will be included in a mathematically admissible statement, and only those statements will be
Chapter 15: Statistics
Measures of dispersion, range, mean deviation, variance and standard deviation, and an analysis of frequency distributions are the topics covered in this chapter.
The scattered data dispersion measures based on observation and central tendency are discussed in Chapter 15 of the PDF version of the NCERT Mathematics Course for Class 11. The numerous ways in
which dispersion can be measured are as follows: range, standard deviation, quartile deviation and mean deviation. The range is understood as maximum value – minimum value.
Chapter 16: Probability
The following topics are covered in this chapter: (1) Axiomatic Approach to Probability; (2) Event; and (3) Random Experiments.
According to what is covered in the textbook that is used for Maths Class 11 by the NCERT, random experiments involve situations in which the outcome cannot be predicted before the result. The term
“outcome” refers to the whole of the outcomes that could have been obtained from a specific experiment. The sample space is comprised of a collection of such results. A subset of the sample space is
what we refer to as an event.
Why are NCERT books preferred by students and teachers alike?
The NCERT textbooks are highly regarded, not only by educators but also by the students who use them. Even though these books are, for the most part, required by students for reading in schools (for
example, the NCERT books for Class 11 Maths), they are also rather popular among students due to the clear explanations and examples that they provide. There are several reasons why NCERT books are
considered to be the superior choice.
The structure of the textbooks used by NCERT is uncomplicated and exact. The presentation is simple enough for kids to understand. While making it simpler for students to understand the material, it
in no way diminishes the significance or importance of the material in any manner.
The issues are broken down into their component parts and clarified using examples when necessary. Students not only find it easier to read thanks to the inclusion of pictorial representations, but
they also gain a deeper understanding of the material being covered.
At the end of each chapter is a collection of questions for you to consider. Students can review what they have learned in a chapter by practising them, and in the process, they may also obtain a
better grasp of any questions or concerns that they may have had about the material.
The NCERT textbooks are also meticulously crafted to adhere to the curriculum outlined by the CBSE. The material discussed in this manner provides a transparent view of the content that a student is
expected to master to succeed on a curricular exam.
Students need to have an in-depth understanding of the concepts so that they can provide an answer anytime the subject of sustainability is brought up in any part of the chapter. The Central Board of
Secondary Education (CBSE) ensures that the questions are based on the most important concepts discussed in each chapter. If you have a firm grasp of the material, you won’t need to worry about doing
poorly on tests.
The students need to have the principles of each chapter broken down for them in a way that is easy to understand. The NCERT texts should be utilised in as many classrooms as possible, and educators
should make a determined effort to do so. Students should not concentrate on the chapter’s tasks and solutions; they should master the essential concepts and ideas presented in the chapter.
Advice from experts to top in Class 11 maths:
The mathematical concepts taught in CBSE Class 11 differ substantially from those taught in secondary school. Despite the significant drop in mathematical sophistication, there is no need to be
concerned. If you want to pass the mathematics exam for CBSE Class 11, consider using the following tips:
Have the mindset that you will regularly study and work throughout the year, which means accomplishing something, even if it is just a little bit each day. There are a lot of online lectures that
explain all of the principles and answer in-text questions. One example is the lectures provided by Extramarks. Many educators have made their expertise available to students for free on YouTube.
Choosing a teacher or mentor who is a good fit for you is extremely important. Look for someone who can light a fire under you no matter what activity you undertake, as education depends on students
having a positive attitude. Young minds are quite active, and they have tremendous potential if they are allowed to mature appropriately. Extramarks employs a lot of qualified educators and provides
engaging and one-of-a-kind workshop opportunities in addition to this.
It is not as important that you understand the question itself as it is that you can solve any similar questions. Try it once, twice, or even three times, but don’t give up on the concepts; rather,
ask your instructor to clarify them until you fully understand what they mean.
There are a few chapters that you simply cannot skip over due to their significance in mathematics. These chapters include the following: They will follow you up to the Class 12 and are often asked
questions in competitive examinations: 1) Set Theory and Related Topics, 2) Trigonometry, 3) Permutations and Combinations and 4) The Binomial Theorem
There are no shortcuts to becoming proficient in mathematics; if you want to perform well in this study area, you will need to put in a lot of work to get there. Although a plethora of mathematical
literature is available, the NCERT remains the standard reference for educators everywhere and comes highly recommended. As a consequence of this, one is required to finish reading the NCERT of
Mathematics before referring to any other publications.
Achieve Success with Extramarks!
Although there is no replacement for hard work, a little bit of smart work is also important to accomplish your goals. Extramarks is dedicated to providing you with the direction and assistance you
need to succeed in your upcoming exams. We provide assistance from subject experts with years of experience and who offer all assistance you may require for optimal preparation and excellent marks.
E-learning has emerged as a prominent topic of discussion in the world of education in recent years. Simply said, Extramarks has taken the entire social structure that takes place online between a
student and an instructor and incorporated it into its own platform. The effort made by the government to make e-learning and digital education more widespread is an encouraging and forward-thinking
Many students are already using the services offered by Extramarks, demonstrating the company’s continued growth and success. It will not come as a surprise to anyone if Extramarks ends up becoming
one of the most significant assets in the field of digital education. It’s important that you recognise that this could negatively damage traditional teaching techniques like coaching centers and
private tutoring groups. Kids and their parents have increasing confidence in their educational system, reflected in the widespread availability of one-on-one tutoring sessions.
You will also have the opportunity to get any questions you have answered during our live classes. On the web platform, you will also find various other resources, such as NCERT books for class 11
Maths, study notes, question solutions and other materials applied to a wide range of classes and topics.
NCERT Solutions for Class 11 Maths - Chapterwise PDF
NCERT Solutions for Class 11
FAQs (Frequently Asked Questions)
1. What is the NCERT Book Class 11 Maths Syllabus?
Students must understand the format of the exam question paper to perform successfully. The question paper is designed to make the math topic easy to understand and is based on the CBSE Class 11
Maths Syllabus. There are 16 chapters in the Class 11 NCERT Math books. Sets, Relations and Functions, Trigonometric Functions, Complex Numbers and Quadratic Equations, Linear Inequalities,
Permutations and Combinations, Binomial Theorem, Series and Sequences, Straight Lines, Conic Sections, Introduction to Three Dimensional Geometry, Limits and Derivatives, Mathematical Reasoning,
Statistics and Probability are some of the topics covered in this course.
2. I want to attain perfect marks. Thus, is it a good idea to use NCERT books for CBSE Class 11 Maths?
Books for Class 11 from NCERT Maths are a powerful tool necessary for complete learning and getting a flawless math score. The possibility of applying the same technique in daily life enables
students to deal with real-world situations. Maths NCERT books for Class 11 can be very beneficial in many ways, paving the road for success not just in CBSE Class 11 Math but also in Class 12 and
higher education beyond that. This will help students create a mindset that will help them pass challenging tests like the JEE and engineering admissions exams.
3. What is the distance between two straight lines that are parallel?
As explained in the 11th NCERT Maths PDF, the distance between two parallel straight lines is a measurement of the perpendicular line that, when positioned in the Cartesian plane, passes between two
parallel lines. In other words, it is equivalent to the perpendicular distance between a point and a line. Any two straight lines in the Cartesian plane can be in one of several different
relationships to one another, including intersecting, skewed or parallel lines. It’s crucial to remember that there is zero space between any two intersecting lines.
4. What are Sets?
NCERT PDF for Class 11 According to mathematics, sets are defined as collections of items that can be represented in two different ways: (1) in tabular or roster form and (2) in set-builder form. All
set members are enclosed in braces and separated by commas in a Roster form. In contrast, the set builder form sets the properties that each element satisfies. In contemporary mathematics, the idea
of a set is fundamental. This idea is used in almost all branches of mathematics nowadays. Sets are used to define the concepts of relations and functions. Understanding sets is necessary to study
many other topics, including geometry, sequencing, probability, etc. | {"url":"https://www.extramarks.com/studymaterials/ncert-books/ncert-books-class-11-maths/","timestamp":"2024-11-08T14:56:38Z","content_type":"text/html","content_length":"682041","record_id":"<urn:uuid:1f9d3697-e7b6-4746-bf5b-53e3b53ede52>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00041.warc.gz"} |
Theme 3 WP 3.2 and 3.3: Experimental testing methods
Relationship to other projects/themes
The most direct links are to WP 3.1, WP1.1, WP1.2 & WP1.3. This reflects our approach, that for practical nonlinear modal analysis methods to be developed, both the experimental and the theoretical
sides must be addressed together.
To develop the experimental testing methods required to enable nonlinear modal analysis and testing to be viable for full-scale industrial structures.
Progress to date
Within this WP there are three aspects that are being addressed together. Firstly helping develop the theoretical analysis tool of WP3.1 that will allow nonlinear models of structural dynamics to be
linked to the experimental structural response. This response is projected onto the linear modes allowing the modal interactions due to nonlinearity to be identified. Secondly experimental testing
techniques are being developed to identify the nonlinear structural response. Finally, approaches that compare the experimental data and the theoretical form to the response and that allow the
numerical model to be calibrated or corrected are being developed in preparation to start WP3.3.
For the experimental identification of nonlinear structures, a resonant decay approach has been developed which improves upon previous methodologies, in particular tying in with the theoretical
approach of WP3.1, and this has been demonstrated on MDOF systems with coupled modes. The strategy has the potential to be applied at industrial level as it utilises the same hardware and set-up
currently used in standard GVT tests (such as the one performed in MS1) and extend it to the nonlinear regime. Here the system is excited sinusoidally close to resonance and the frequency altered
slowly to ensure a resonant response. The forcing is then switched to zero and the system response becomes an unforced decay of what is approximately a nonlinear normal mode (strictly speaking a
nonlinear normal mode applies to an undamped system) giving resonant decay data.
The data can be analysed to give instantaneous frequency and amplitude response components of each of the linear modes for the nonlinear normal mode. This data can be compared directly to backbone
curves and reveals the nature of the coupling among linear modes due to the nonlinearity. In addition it can be used to identify the type of the nonlinearity acting in the structure and the linear
modes they effect and/or couple.
The experimental backbone curve can be compared directly to theoretical backbone curves developed in WP3.1 to:
1. reveal any coupling among linear modes due to the nonlinearity
2. to identify the type of the nonlinearity acting in the structure
Detailed experimental tests, as part of MS1, have been completed on a structure resembling a scaled aircraft wing with discrete nonlinear pylons [3-5], shown in Figure 1, and also a joined wing half
model structure containing geometric nonlinearities [1,2]. A full set of experimental data enabled an accurate identification of the underlying linear structure, natural frequencies, damping ratios
and mode shapes. For the wing structure, a number of decay responses were recorded for the first four resonance frequencies. Once these responses were projected into the modal space using the matrix
of mode shapes, the decaying modal responses were used to estimate the backbone curves for each resonance frequency.
The backbone curves revealed coupling between two pairs of linear modes. In addition the backbone curves showed that the structure is affected by nonlinearity that is initial softening in stiffness
for small deflections followed by hardening for larger deflections. Experimental data taken from a single pylon in isolation also exhibits this softening then stiffening behaviour.
Using the experimental data, a nonlinear stiffness function that results in the experimentally measured backbone curve has been identified through data fitting [4,5]. An example of this is shown in
Figure 2. The lines in Figure 2 are the experimentally identified backbone curve for an isolated resonance peak in terms of the various linear modes for the engine store structure shown in Figure 1.
The circles show the equivalent curves using the fitted stiffness function. Work is ongoing to identify the behaviour of the full aircraft wing with nonlinear pylons using the experimental backbone
curves, and then to apply the methodologies to the larger joined wing structure.
The third area of research is developing the process to compare the experimental data and the theoretical form to the response allowing the numerical model to be calibrated or corrected. As already
stated some data fitting has been conducted on the experimental data. In addition much work has been conducted on numerical examples where we can assess the accuracy predicted model by direct
comparison with the model used to generate the data set.
The method adopted is to treat the instantaneous frequency during the resonant decay along with the corresponding amplitudes of the various linear modal components at each time point as the data set.
Using these along with the equations from the backbone curves taken from the numerical model, the data set can be used to identify the nonlinear coefficient values within the model, hence calibrating
the nonlinear terms. This identification can be achieved using conventional least-squares fitting, but work is ongoing on investigating the use of the MCMC algorithms being developed in Theme 1 (see
Other ongoing work includes considering the effect of the shaker dynamics on the test structure response. It has been found that unless it is properly accounted for, the shaker’s characteristics
and its location, can cause a shift in the estimated backbone curves resulting in poorer parameter identification. Compensation methods are now being developed. In addition we are now considering the
fitting effectiveness when nonlinear damping and more complex stiffness relationships, including geometric nonlinearities, are present.
Furthermore, the application of the resonance decay approach using multiple shakers will be investigated and extended to nonlinear structures. This can potentially deal with systems where the
underlying linear system contains close interacting modes, where current procedures fail to identify the systems correctly. Finally, with input from Rolls-Royce, under WP 3.3, a handbook has been
developed to give industrial users an approach to tackling localised nonlinearities such as those seen at joints between components, delivered as MS8. These methods will be compared to the resonant
decay/backbone method on the test structures considered in MS1. | {"url":"https://www.engineeringnonlinearity.ac.uk/themes/theme-3-wp-3-2-and-wp-3-3/","timestamp":"2024-11-13T19:37:15Z","content_type":"text/html","content_length":"46826","record_id":"<urn:uuid:69bf2fe4-d31a-4592-8bbc-e6297af8a5b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00679.warc.gz"} |
System of Particles and Collision
A collision between two particles is defined as the mutual interaction between particles for a short interval of time as a result of which energy and momentum of the particle change.
In the collision of two particles, the law of conservation of momentum always holds true but in some collisions, kinetic energy is not always conserved.
Collisions are of two types on the basis of the conservation of energy.
Perfectly elastic collision:
In a perfectly elastic collision, both momentum and kinetic energy of a system are conserved. This type of collision mostly takes place between the atoms, electrons, and protons.
Characteristics of elastic collision:
1. total momentum is conserved.
2. total energy is conserved.
3. total kinetic energy is conserved.
4. total mechanical energy is not converted into any other form of energy.
Consider two particles of masses are $m_{1}$and $m_{2}$ collide with each other with velocities $u_{1}$and $u_{2}$. After collision, their velocities become $v_{1}$and $v_{2}$ respectively.
Considering the collision to be elastic, then from the law of conservation of momentum we have,
and from the law of conservation of energy we have,
Perfectly inelastic collision:
In perfectly inelastic collisions, the momentum of the system is conserved but kinetic energy is not conserved.
Characteristics of inelastic collision:
1. total momentum is conserved.
2. total energy is conserved.
3. total kinetic energy is not conserved.
4. mechanical energy may be converted into other forms of energy.
Consider two particles of masses are $m_{1}$and $m_{2}$ collide with each other with velocities $u_{1}$and $u_{2}$. Considering the collision to be inelastic, then these two particles would stick to
each other and after collision they move with velocity $v$ . Then we have,
Kinetic energy of particles before collisions is,
and kinetic energy after collisions is ,
Using the law of conservation of energy, we get
Where $Q$ is the loss in kinetic energy of particles during the collision.
Head on elastic collision of two particles:
From the law of conservation of momentum we have,
$m_{1}$$u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}$ …………. (1)
and from the law of conservation of energy we have,
Rearranging equation (1) and (2), we get
$m_{1}(u_{1}-v_{1})=$ $m_{2}(v_{2}-u_{2})$ ………………….. (3)
Dividing equation (4) by (3), we get
$u_{2}-u_{1}=-(v_{2}-v_{1})$ …………….. (5)
$u_{2}-u_{1}$ is the relative velocity of the second particle with respect to the first particle before the collision and is the relative velocity of the second particle with respect to the first
particle after the collision.
Equation (5) can be written as, $v_{1}=v_{2}-u_{1}+u_{2}$ ……………… (6)
$v_{2}=v_{1}+u_{1}-u_{2}$ ……………… (6)
Now putting the value of $v_{1}$ in equation (3) we get, $m_{1}(u_{1}-v_{2}+u_{1}-u_{2})=$ $m_{2}(v_{2}-u_{2})$
$-m_{1}v_{2 }-m_{2}v_{2}+2$ $m_{1}u_{1}-m_{1}u_{2}-m_{2}u_{2}=0$
On solving the equation we get the value $v_{2}$ of as
Similarly putting the value of $v_{2}$ in equation (3) we get,
Points to remember:
When $m_{1}=m_{2}$then from equation (7) and (8) , $v_{2}=u_{1}$ and $v_{1}=u_{2}$.
Thus if two particles of equal masses suffer head-on elastic collision then the particles will exchange their velocities. | {"url":"https://www.w3schools.blog/system-of-particles-and-collision","timestamp":"2024-11-13T15:25:57Z","content_type":"text/html","content_length":"158576","record_id":"<urn:uuid:360f265e-8011-48e9-952d-abc470f5889a>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00089.warc.gz"} |
PHYSICS 109N
Return to homework index
PHYSICS 109N
Michael Fowler
Physics 307
Homework Assignment: Due Tuesday 19 September, 2:00p.m.
1. Finding the angular size of the moon: choose some suitable round object, such as a dime, quarter, a tennis ball, ping pong ball or whatever, and while you are looking at the moon, have your
partner hold the ping pong ball (say) at just the right distance that it looks to you exactly the same size as the moon. You could try it directly in front of the moon, to just block it, or side by
side with the moon, whatever works best for you. Now, measure the distance from your eye to the ball, and measure the diameter of the ball. Given that the moon is 230,000 miles away, figure out the
moon's diameter. What is the angular size of the moon? That is, if you take two pencils, point one at the bottom of the moon, and one at the top, as seen from here, what is the angle between them?
You can answer in degrees or radians. (One radian is the angle of a piece of pie having the curved side the same length as the straight sides.)
2. Find the North Star (Polaris). Now, find how high Polaris is above the horizon by pointing a pencil directly at Polaris and having your partner measure the angle between the pencil and a
horizontal line running directly beneath it.
Just after dark, find the Big Dipper, and sketch the Big Dipper together with Polaris. Three hours later, check them again. Has Polaris moved? Has the Big Dipper moved? Draw on your sketch how they
have moved, if at all, in the sky.
3. Explain in a sentence or two, together with a simple diagram of the earth and the sun, why it is warmer in summer than in winter.
4. Some years ago, a very romantic picture of the Rotunda as seen from the middle of the Lawn was published. It depicted the Rotunda at night, and there was a full moon visible in the sky just above
the Rotunda. How can you prove this picture is a fake? | {"url":"https://galileoandeinstein.phys.virginia.edu/more_stuff/homework/95109hw2.html","timestamp":"2024-11-03T20:30:14Z","content_type":"text/html","content_length":"2626","record_id":"<urn:uuid:2ab2c40e-793d-4537-8928-c4bd5b66f38a>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00709.warc.gz"} |
How combinationnal logic works in FPGA ? - Learn FPGA Easily
When I was a child, the back of my cathodic TV amazed me. I wondered what were all those cables and what were their purposes. I learned that it was electricity after. Then that it was only 0 and 1.
And I couldn’t stop asking myself, “how can we do anything with only 0 and 1?”
If you asked yourself this question, you’re in the right place. Because today is the day we speak about combinational logic and LUTs. Of course combinational logic and LUTs aren’t the only way to
process a signal in electronic, but it’s a tiny part that could have satisfied my curiosity back then. And it’s a huge part of how FPGA does it.
In a previous article about DFF, we saw how to store data in our FPGA. Today it’s time to study how to manipulate those data. First we will learn about logic gates, how they work and how it’s
absolutely not what is implemented in an FPGA, and why LUT is used instead.
1. What is a logic gate ?
Logic gates are hardware implementation of the bitwise operations you can do with 1 or 2 bits. They’re the different operation you can do in Boolean algebra.
With one bit you don’t really have the choice, you can :
1. do nothing which is … not a gate, it’s a wire.
2. invert the bit. If it’s a 1, it becomes a 0. If it’s a 0, it becomes a 1.
For a gate with two inputs bit and only one output bit, we can use:
1. The AND gate. The output is a 1, if and only if both input bits are 1.
2. The OR gate. The output is a 1, if at least one of the two input bits is 1.
3. The XOR gate (eXclusive OR). The output is 1 if and only if one of the inputs is 1 (the other one must be 0)
– And that’s it !
– Wait ! I heard there are seven logic gates. You showed only four of them !
Yes, the three missing gates are the NAND, the NOR and the XNOR which are just the three above with and inverters on the output.
1.1 The inverter
The symbol of the inverter. Notice the little circle at the end of the triangle ? This is the symbol of “not” you will find it in NAND, NOR, XNOR too.
1.2 AND gate
1.3 NAND gate
1.4 OR gate
1.5 NOR gate
1.6 XOR gate
1.7 XNOR gate
2. Example : The adder.
Let’s see one of the most simple designs we can do with those gates : the adder. We have two inputs bits and want to add them together.
Given two bits “a” and “b”, then a+b<=2. Since 2 is written “10” in binary, we need two outputs, one for each binary figure.
Knowing the result of the right figure is equivalent to ask : when does the addition’s result is equal to 1 ? If one of the two bits is equal to 1 but not both which is… a XOR gate that’s right.
When does the second figure is equal to 1 = when does the result is equal to 2 ? If both of the bits are 1 which is… a AND gate.
That’s it ! you made an adder:
Combinationnal Adder
Since we had to create another figure to represent the 2 in binary, the second figure is called the “carry” in the same way we have carries in decimal addition. “result” is the right figure. “carry”
is the left figure of the addition.
At this point we could think that our FPGA is full of AND, OR, XOR gates which implement our design. This is actually not how it works and it’s the topic of our next part.
3. What is a LUT ?
LUT stands for Look Up Table. It is the hardware instance of a reconfigurable truth table, which can describe any boolean equation of N variables.
A LUT is instantiated as cascading multiplexers of two inputs and one output. The two inputs are SRAM configured during the power up of your FPGA. Your boolean variables are the “select” input of the
A 3-LUT can be represented like this (is it not how they are design in silicon) :
Any boolean equation with three variables can be instantiated with this 3-LUT. To change the equation we just need to change the value inside the SRAMs. I say “we” but it is actually the synthesizer
and “place and route” tools who are going to configure the LUTs. The picture of the 3-LUT above is made with IceStudio, but you will never have to instantiate a LUT by hand. I made it purely for
educational purpose.
Here is an example of how a LUT could be configured for a XOR and a AND gate inside an FPGA:
Xor as a 2-LUT
And as a 2-LUT
You see ? We only need to change the SRAM value to change the operation. With a 2-LUT you can reproduce all the gates described above. And with a N-LUT, any equation with N variables.
FPGA contains thousands of LUTs that will allow you to implement very complex designs.
The number of inputs and outputs of a LUT can vary depending on the constructor and the version of your FPGA. For example, the Xilinx 7 serie got 6-LUT that can be configured as two 5-LUT as long as
the 5 variables are the same, whereas the Altera ones got 8-LUT that can be configured as a 3-LUT + a 5-LUT (and many other configurations). When you work on FPGA, it is always good to know what
architecture the constructor have chosen. Xilinx also give you coding style advice next to the architecture presentation so you can take advantage of the full potential of their FPGA.
Now, you know how combinational logic works in FPGA and with my previous article on DFF you now know the two most important piece of circuitry in your FPGA.
Again, it’s important that you know what is instantiated with your code !
If you like the blog or post, don’t forget to share it, bookmark it and to comment it 🙂
Hope you enjoyed !
One Response
1. This is elementary circuiting, but the bases are important, because if we don’t get them right, nothing further can be understood.
Thank you four your effort to explain them clearly and simply. | {"url":"https://learn-fpga-easily.com/how-combinationnal-logic-works-in-fpga/","timestamp":"2024-11-07T06:59:25Z","content_type":"text/html","content_length":"162053","record_id":"<urn:uuid:92b032bc-a0d6-4223-bd7d-cde234758fee>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00802.warc.gz"} |
favorite ratios and formulas | Redfield, Blonsky & Starinsky, LLC
The following formulas and such are just some of the methods we use in our investment selection process. Our analysis is always ongoing, which is also known as “dynamic”. We liken the use of
formulas and analysis, as a road map. The map changes continuously. The key to this type of analysis is to understand the big picture. The understanding of the big picture is known as, “seeing the
forest and not the trees”.
1.Return on Equity (ROE) – Some consider this an evaluation of management skill. How does management deploy assets, generate profits, and deal with margins? Please click on
this link to find an interesting article on ROE.
ROE = Net Profit Margin X Asset Turnover X Balance Sheet Leverage
ROE = Net Income X Sales X Assets
Sales Assets Equity
ROE = Net Income
P/E = 8.5 + 2 (Growth Rate) * 4.4/30 year AAA Corporate Bond Rate
I have tailored the above formula and call it the RBCPA Intrinsic Value Formula.
RBCPA Intrinsic Value = eps* ((2*growth rate)+8.5)*4.4/30 AAA corporate bond rate
3. Times Interest Earned (Interest Coverage Ratio) – This evaluates the ability of a company to meet required interest payments.
Times Interest Earned = pretax income + total interest expense / total interest expense.
We like to see interest coverage at > 4 (or 25% on the inverse). We consider 6 (or inverse 16.67%) as a conservative number.
4. Simple formula to determine years it will take to pay debt is discussed on page 445 of Graham’s Security Analysis.
Total debt / net income
5. David Dreman feels that debt should be less than 20% of Equity.
6. Here are just a few of the many items we look at in financial statements
a. Compare earning to consensus estimates
b. Compare earnings, revenues, margins, SG&A, and cash flows to prior periods.
c. Consider the following calculations
1. Allowance for DA / Accounts Receivable
2. Allowance for DA / Sales
3. Change in Net Income / Change in Cash Flow
4. EPS / Debt per Share
d. Look at tax rates. did earnings change because of tax rate changes.
e. Look at shares outstanding. Watch carefully for dilution. Look at Statement of Cash Flows for true operating and free cash flow.
f. Look at “one time charges”. If they are recurring in nature, consider using them as normalized expenses.
7. Earnings Ratio – This is the inverse of the Price Earnings ratio (PE). Intelligent Investor by Benjamin Graham (pg 186 of 4th edition) indicated that this should be as high as the AA 30 year
bond rate.
Earnings Ratio > Current AA 30 Year Corporate Bond Rate
8. Other Graham Criteria from Intelligent Investor . We don’t place as large an emphasis here, yet we certainly look at this.
Current Assets 150% > Current Liabilities
debt < 110% of Current Assets (for industrial companies)
9. Seven Deadly Sins of Corporations
A. Recording revenue to soon
B. Recording bogus revenue
C. Boosting one-time gains
D. Shifting current expenses
E. Improperly recording liabilities
F. Shifting revenue forward
G. Shifting special charges
10. Price / Growth Flow ratio – We will use this ratio when working with companies that have large Research and Development (R&D) expenditures.
Price to Growth Flow Ratio = Price / (EPS + FWD 1Y R&D)
5X Cheap
10 – 12X Normal
> 15 – 20X Expensive
11. Flow Ratio – a measure of working capital efficiency.
Flow Ratio = (Current Assets – Cash and Short Term Investments) s/b < 1.25
Current Liabilities – Short Term Debt
12. Cash King – a measure of cash flow
Cash King = Free Cash Flow s/b > 10%
13. PEG and PEGY ratios – These ratios measure P/E over Growth Rate. The PEGY includes yield in the measurement.
We generally like to invest when PEGS are < 1.
PEG = PE/Growth Rate
PEGY = PE / (Growth Rate + Dividend Yield)
14. Graham Ratio – This is a ratio which I named. Benjamin Graham had a theoretical formula which we refer to. The formula involves book value, hence one needs to consider the differences between
“tangible” and “intangible” book value.
Graham Ratio = (price/book value)* PE s/b < 24
15. Various Liquidity ratios –
Operating Margin = Cash Flow from Operations over Sales
Return on Capital = Net Income over Total Assets at Book Value
Leverage = Total Liabilities over Market Value of Equity
Financing Requirement = Required Debt Financing over Sales
Debt Service Capability = Free Cash Flow over Total Borrowings
Interest Coverage = EBITDA over Interest Expense
ST Liquidity = Net Working Capital over Sales
16. Taxable equivalent yield = tax exempt yield/1 – marginal tax rate
Taxable equivalent yield = interest income /1 – marginal tax rate
17. Interest Rate Change X Duration = Change in bond value
18. Return on Invested Capital = Owners Earnings / Invested Capital
According to “The Intelligent Investor” ROIC is as follows:
Owners Earnings = Operating Profit + Depreciation + amortization +/- Non Recurring Costs – Federal Income Tax – Cost – essential capital expenditures (maintenance) – unsustainable income (such as
rates of returns on pensions) – cost of stock options (if not already deducted from operating profit).
Invested Capital = Total Assets – cash and short term investments + past accounting charges that previously reduced invested capital.
“An ROIC of at least 10% is attractive; even 6% or 7% can be tempting if the company has good brand names, focused management, or is under a temporary cloud.”
According to “Security Analysis”, this is the definition:
Return on Capital = (Net Income + minority interest + tax-adjusted interest) / Tangible Assets – short term accrued payables.
You can read a study of ROIC at this link.
19. Various Cash Flow Ratios
a. Operating Cash Flow (OCF) = CF from Operations/ Current Liabilities
b. Funds Flow Coverage (FFC) = EBITDA / (Interest + Tax adjusted debt repayment + Tax adjusted Preferred Dividends)
c. Cash Interest Coverage = ( CF from Operations + Interest Paid + Taxes Paid) / Interest Paid
d. Cash Current Debt Coverage = ( operating Cash Flow – Cash Dividends) / Current Debt
e. Capital Expenditure = CFO/Capex
f. Total Debt = CFO/ Total Debt
g. Cash Flow Adequacy (CFA) = (EBITDA – taxes paid – interest paid – capex)/ Average annual debt maturities over next 5 years
h. Cash Flow / Enterprise Value
20. Price = Earnings / (Total Return – Earnings Growth) (You can substitute “E” for “D” )
I was thinking about this formula on December 17, 2009, and frankly it makes no sense to me. Feel free to comment. This could be a flawed formula.
P= E/(K-G) or P= D/(K-G)
P= Stock Price
K= Total Return expected (discount rate)
G= Growth rate of earnings
E= Earnings
D= Dividends
I added this formula on the same date 12/17/09 . I have been meaning to study it. I think it is tied into DCF analysis. It too could be flawed.
P = CF (year 0) + (1+G) / R – G
It looks to me, and I could be wrong, that R would be the Discount Rate. I typically will use a discount rate of at least 10% and most often 15% or higher. I got this formula from Hamilton Lin. He
mentioned it only works when R > G.
21. Cap rates:
Value = NOI/Cap Rate
NOI = Revenues less Operating expenses
NOI does not include depreciation, amortization, interest and capex.
21. Al Meyer’s Price to Sales Ratio Rule of Thumb
Net Margin Price/Sales
5% 1X
10% 2X
15% 3X
20% 4X
25% 5X
30% 6X
22. Strong Balance Sheet Rule of Thumb I saw.
CA – CL = WC > LT Liabilities = Strong Balance Sheet
23. Inventory Turnover Calculations:
Inventory Turnover Ratio = Cost of Goods Sold / Average Inventory
Average Days in Inventory = 365 / Inventory Turnover Ratio
24. EBITDA Coverage Ratio:
Apparently Templeton liked using leverage ratios.
EBITDA Coverage Ratio = earnings + interest expense + taxes + depreciation / Interest Expense. He calls 6 a conservative benchmark.
He then used Total Debt to Trailing 12 Months EBITDA and feels a ratio of 3 or less is a conservative benchmark.
25. Goodwill/Tangible Assets > 20% – Watch for potential impairments.
26. I attended this session 5 years ago. I’ve been busy ;-), so I am just getting to transcribing the notes now. Really good stuff. Mulford is good at quality of earnings and cash flows. He has a few
awesome books. One is “The Financial Numbers Game: Detecting Creative Accounting Practices,” and the other is, “Creative Cash Flow Reporting: Uncovering Sustainable Financial Performance.”
Here are notes to this session. There were some areas I may have taken or recalled incorrectly.
A. Look at PPE / Revenue Days.
B. Look for gaps between cash flow and earnings. Always ask why CF is different than EPS. Why are rates of growth different?
C. Look at sustainability of Cash flow.
D. Look at % of costs incurred being capitalized.
E. Cash Flow From Operations – He subtracts Capex. (nothing earth shattering there.)
F. Recast Cash Flow Statement with Balance Sheet changes.
G. Look at Capital Lease disclosure. Perhaps you will find further capex, not identified in Statement of Cash Flows.
H. Look at Securitizations.
I. Don’t add back Tax Benefits of Non-Qualified Stock Options if taxes are not being paid.
J. EQI = Earnings Quality Indicator . Will vary around a general trend. Use normalized CF and NI.
EQI = (CF-Income) / Revenues
or EQI = (OCF/Revenues) – (NI/Revenues)
27. James Montier likes the following Ben Graham Formulas:
A. Earnings Yield should be 2X AAA bond yield. I would use 5 year and 10 year.
B. Dividend Yield should be at least 2/3 AAA bond yield. I would use 5 year and 10 year.
C. Total Debt < 2/3 Tangible Book Value.
28. Selected Formulas from ‘Analysis of Financial Statements’ 5th Edition , Bernstein & Wild.
Total Debt to Total Capital = (Current Liabilities + Long Term Liabilities) / (Equity Capital + Total Liabilities)
Long Term Debt to Equity = Long Term Liabilities / Equity Capital
Return on Total Assets = (NI + Interest Expense (1-Tax Rate) / Average Total Assets
Cash to Current Liabilities Ratio = (Cash + Equivalents + Marketable Securities) / CL | {"url":"https://www.rbcpa.com/favorite-ratios-and-formulas/","timestamp":"2024-11-03T00:38:19Z","content_type":"text/html","content_length":"152827","record_id":"<urn:uuid:2996d9f0-7207-4908-a552-1876f0de228a>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00189.warc.gz"} |
Aaron Schlegel's Notebook of Interesting Things
The Substitution Rule is another technique for integrating complex functions and is the corresponding process of integration as the chain rule is to differentiation. The Substitution Rule is
applicable to a wide variety of integrals, but is most performant when the integral in question is similar to forms where the Chain Rule would be applicable. In this post, the Substitution Rule is
explored with several examples. Python and SymPy are also used to verify our results. | {"url":"https://aaronschlegel.me/tag/python2.html","timestamp":"2024-11-14T15:23:17Z","content_type":"text/html","content_length":"59508","record_id":"<urn:uuid:f78af1f1-9235-4c33-a65c-804cd4fa4e3c>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00643.warc.gz"} |
Multiplication Facts 0 12 Printable
Multiplication facts 0 12 printable - In these multiplication worksheets, the facts are grouped into anchor groups. If you are helping your child learn the multiplication facts, why not make it.
Multiplying by anchor facts 0, 1, 2, 5 and 10 multiplying by facts 3, 4 and 6 multiplying by facts 7, 8 and 9. Best images of printable multiplication tables 0 12 download print. This is the part you
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Table of Contents
Basic Usage
Those users who are not familiar with the predecessor of libRadtran, uvspec, please note the following: The central program of the package is an executable called uvspec which can be found in the bin
directory. If you are interested in a user-friendly program for radiative transfer calculations, uvspec is the software you want to become familiar with. A description of uvspec is provided in the
first part of the manual. Examples of its use, including various input files and corresponding output files for different atmospheric conditions, are provided in the examples directory. For a quick
try of uvspec go to the examples directory and run
../bin/uvspec < UVSPEC_CLEAR.INP > test.out
For the format of the input and output files please refer to the manual.
The bin directory also provides related utilities, like e.g. a mie program (mie), some utilities for the calculation of the position of the sun (zenith, noon, sza2time), a few tools for
interpolation, convolution, and integration (spline, conv, integrate), and some other small tools.
How to setup an input file for your problem (checklist)
There are several steps to consider when setting up an input file for your specific problem. First of all we strongly recommend that you read a radiative transfer textbook to become familiar with
what is required for your problem. After defining your problem you may in principle find all information for setting up the input file and understanding the contents of the output file in the manual
(but who reads manuals anyway?). Below is a short checklist including the steps you need to consider for each problem:
1. Wavelength grid / band parameterization
First you need to think about the spectral range and spectral resolution required for your calculation. Per default the REPTRAN absorption parameterization is used which is available for the full
spectral range from the UV to the far IR. In the ultraviolet or the lower visible spectral range molecular absorption varies smoothly with wavelength in this range and a calculation with 0.5 or 1nm
step width should be sufficient. Above 500nm, however, absorption by water vapour, oxygen, and other trace gases starts; these absorption lines are very narrow, and a spectral calculation which
resolves all lines is not feasible for most applications (such a line-by-line calculation is possible, however, if you provide your own spectral absorption cross sections). For most applications you
have to use a parameterization for molecular absorption, for example the representative wavelengths parameterization, e.g. mol_abs_param reptran which is used by default and which allows
pseudo-spectral calculations (meaning that you still can calculate radiation at any wavelength you want, but the gas absorption is provided only at limited resolution - if you select the wavelengths
too close, you will see the steps in your spectrum). For a spectral or pseudo-spectral calculation, you may define your own wavelength grid with wavelength_grid_file and we recommend to do that
because otherwise you get the default 1nm step which might be too expensive for your application. Finally, in order to calculate integrated shortwave or integrated longwave radiation, please choose
one of the pre-defined correlated-k distributions, e.g. mol_abs_param kato2 or mol_abs_param fu because these are not only much more accurate but also much faster than a pseudo-spectral calculation.
Please read the respective sections in the manual to become familiar with the mol_abs_param options.
2. Quantities
The next point one needs to consider is the desired radiation quantity. Per default, uvspec provides direct, diffuse downward and diffuse upward solar irradiance and actinic flux at the surface.
Thermal quantities can be calculated with source thermal - please note that uvspec currently does either solar or thermal, but not both at the same time. If both components are needed (e.g. for
calculations around 3μm) then uvspec needs to be called twice. To calculate radiances in addition to the irradiances, simply define umu, phi, and phi0 (see next section).
3. Geometry
Geometry includes the location of the sun which is defined with sza (solar zenith angle) and phi (azimuth). The azimuth is only required for radiance calculations. Please note that not only the solar
zenith angle but also the sun-earth-distance change in the course of the year which may be considered with day_of_year (alternatively, latitude, longitude, and time may be used). The altitude of the
location may be defined with altitude which modifies the profiles accordingly. Radiation at locations different from the surface may be calculated with “zout” which gives the sensor altitude above
the ground. For satellites use “zout TOA” (top of atmosphere). For radiance calculations define the cosine of the viewing zenith angle umu and the sensor azimuth phi and don't forget to also specify
the solar azimuth phi0. umu>0 means sensor looking downward (e.g. a satellite), umu<0 means looking upward. phi = phi0 indicates that the sensor looks into the direction of the sun, phi-phi0 = 180°
means that the sun is in the back of the sensor.
4. What do you need to setup the atmosphere?
To define an atmosphere, you need at least an atmosphere_file which usually contains profiles of pressure, temperature, air density, and concentrations of ozone, oxygen, water vapour, carbon dioxide,
and nitrogen dioxide. The set of six standard atmospheres provided with libRadtran is usually a good start: afglms (mid-latitude summer), afglmw (mid-latitude winter), afglss (sub-arctic summer),
afglsw (sub-arctic winter), afglt (tropical), and afglus (US standard). If you don't define anything else, you have an atmosphere with Rayleigh scattering and molecular absorption, but neither
clouds, nor aerosol.
4a. Trace gases?
Trace gases are already there, as stated above. But sometimes you might want to modify the amount. There is a variety of options to do that, e.g. mol_modify O3 which modifies the ozone column, or
mixing_ratio CO2, …
4b. Aerosols?
If you want aerosol, switch it on with aerosol_default and use either the default aerosol or one of the many aerosol_ options to setup whatever you need.
4c. Clouds?
uvspec allows water and ice clouds. Define them with wc_file and ic_file and use one of the many wc_ or ic_ options to define what you need. Please note that for water and ice clouds you also have a
choice of different parameterizations, e.g. ic_properties fu, yang, baum, … - these are used to translate from liquid/ice water content and droplet/particle radius to optical properties. You need
some experience with clouds to define something reasonable. Here are two typical choices for a wc_file 2 0 0
1 0.1 10
and an ic_file
9 0.015 20
The first is a water cloud with effective droplet radius of 10μm between 1 and 2km, and an optical thickness of around 15; the second is an ice cloud with effective particle radius 20μm between 9 and
10km and an optical thickness of about 1.
4d. Surface properties?
Per default, the surface albedo is zero - the surface absorbs all radiation. Define your own monochromatic albedo or spectral albedo_file or a BRDF, e.g. for a water surface which is mainly
determined by the wind speed cox_and_munk_u10.
5. Choice of the radiative transfer equation (rte) solver
The rte-solver is the engine, or heart, in any radiative transfer code. All rte-solvers involve some approximations to the radiative transfer equations, or the solution has some uncertainties due to
the computational demands of the solution method. The choice of rte-solver depends on your problem. For example, if your calculations involves a low sun you should not use a plane-parallel solver,
but one which somehow accounts for the spherical shape of the Earth. You may choose between many rte-solvers in uvspec. The default solution method to the radiative transfer is the discrete ordinate
solver disort which is the method of choice for most applications. There are other solvers like rte_solver twostr (faster but less accurate), rte_solver mystic and mc_polarisation (polarization
included), or rte_solver disort and and pseudospherical to get pseudo-spherical geometry.
6. Postprocessing
The spectral grid of the output is defined by the extraterrestrial spectrum. If you want spectrally integrated results, use either correlated_k kato2/fu and output_process sum or correlated_k lowtran
and output_process integrate. Check also other options like filter_function_file, output_quantity brightness, etc. Instead of calibrated spectral quantities you might also want output_quantity
transmittance or output_quantity reflectivity.
7. Check your input
Last but not least, make always sure that uvspec actually does what you want it to do! A good way to do that is to use verbose which produces a lot of output. To reduce the amount, it is a good idea
to do only a monochromatic calculation. Close to the end of the verbose output you will find profiles of the optical properties (optical thickness, asymmetry parameter, single scattering albedo)
which give you a pretty good idea e.g. if the clouds which you defined are already there, where the aerosol is, etc. As a general rule, never trust your input, but always check, play around, and
improve. For if thou thinkest it cannot happen to me and why bother to use the verbose option, the gods shall surely punish thee for thy arrogance!
Play around with MYSTIC
The Monte Carlo code for the physically correct tracing of photons in cloudy atmospheres (MYSTIC) is fundamentally different from other solvers in the sense that it determines the result by random
tracing of individual photons through the atmosphere. For a simple description of the technique see the publication by Mayer (2009) and the other papers listed here. In the following, we show how to
play around and explore MYSTIC.
First, try a simple uvspec input file:
atmosphere_file ../data/atmmod/afglus.dat
source solar ../data/solar_flux/atlas_plus_modtran
wavelength 450
In this example the default solver (disort) is used and uvspec will provide familar output like
450.000 1.670252e+03 2.048350e+02 -2.314766e-13 1.329144e+02 4.177456e+01 6.935632e-14
If you repeat the simulation, you will get an identical result over and over again. Now let's try MYSTIC by simply adding
rte_solver mystic
to the above input and run uvspec 10 times. You might get
450.000 1.643995e+03 1.997293e+02 0.000000e+00 1.308250e+02 4.676865e+01 0.000000e+00
450.000 1.673167e+03 1.852792e+02 0.000000e+00 1.331464e+02 3.027929e+01 0.000000e+00
450.000 1.704421e+03 1.832073e+02 0.000000e+00 1.356335e+02 4.074436e+01 0.000000e+00
450.000 1.712756e+03 1.977188e+02 0.000000e+00 1.362968e+02 3.850349e+01 0.000000e+00
450.000 1.679417e+03 1.977593e+02 0.000000e+00 1.336438e+02 3.629829e+01 0.000000e+00
450.000 1.652330e+03 1.954993e+02 0.000000e+00 1.314883e+02 3.828460e+01 0.000000e+00
450.000 1.662748e+03 2.040408e+02 0.000000e+00 1.323173e+02 3.629640e+01 0.000000e+00
450.000 1.675250e+03 2.247512e+02 0.000000e+00 1.333122e+02 4.490242e+01 0.000000e+00
450.000 1.681501e+03 2.247862e+02 0.000000e+00 1.338096e+02 4.674322e+01 0.000000e+00
450.000 1.681501e+03 1.811337e+02 0.000000e+00 1.338096e+02 3.694756e+01 0.000000e+00
The result is close to disort, but obviously different each time you run uvspec. The difference is caused by the photon noise. You may compute the noise by calculating the standard deviation of the
10 individual results. For the direct irradiance (column 2) we obtain 1676.7±20.0 and for the diffuse downward 199.4±14.6. In most cases the noise is Gaussian which implies that 68% of the model runs
lie within ±1 standard deviation and 95% within 2 standard deviations. That way you can always determine the statistical noise of your result. The noise is of course determined by the number of
photons run in the simulation. Try increasing the number of photons to 100,000 (the default was 1,000 in the above example) by adding
mc_photons 100000
to the input file. Now we pbtain 1671.1±2.4 for the direct and 203.8±1.9 for the diffuse irradiance. The noise has decreased roughly by a factor of 10. In fact, the noise is proportional to 1 / sqrt
(mc_photons) which means, if you want to reduce the noise by a factor of 10 you need to increase the number of photons and thus the computational time by a factor of 100. Please note that in both
calculations the disort result lies within ±2 standard deviations.
Now let's try something more complicated: Calculate integrated thermal irradiance using the following input file:
atmosphere_file ../data/atmmod/afglus.dat
source thermal
mol_abs_param fu
wavelength_index 7 18
output_process sum
rte_solver mystic
mc_photons 100000
For the diffuse downward irradiance we obtain 267.7±27.6 W/m2 which is unacceptably noisy. When you read the above mentioned publications, you will find that thermal irradiance should rather be
calculated in “backward” mode. Add
to the input file and repeat the calculation. You will obtain something like 283.3±0.5 W/m2. Noise and also computational time have decreased dramatically. The respective disort result is 283.6 W/m2
and the disort computational time is only a factor of 3 faster compared to MYSTIC (the latter was 0.3 s for integrated longwave irradiance). Please note that in backward mode, only one quantity is
calculated at a time. The default is diffuse downward irradiance. If you need diffuse upward instead, please try
mc_backward_output eup
Now let's try radiances with the following input file:
atmosphere_file ../data/atmmod/afglus.dat
source solar ../data/solar_flux/atlas_plus_modtran
wavelength 400
sza 45
rte_solver mystic
mc_photons 100000
umu -1
phi 0
Here we are looking straight upward from the surface in the blue (400 nm). With the default solver disort you get the result directly to stdout while MYSTIC does not provide the radiances there. The
latter are found in mc.rad.spc (see documentation). Here we obtain 56.68±0.18 for the radiance. The respective disort result is 56.53 - again both agree within ±2 standard deviations.
Now something special: Try calculating radiances for several directions by replacing the umu line with
umu -1.0 -0.9 -0.8
You will notice that MYSTIC does the calculation only for the first umu value. In contrast to disort each angle pair (umu, phi) has to be calculated separately for which reason we haven't implemented
the option to calculate several angles at the same time.
So far we have only calculated things which could also have been calculated with disort - usually faster and without noise. Now let's do something which cannot be done with disort. Try the following:
atmosphere_file ../data/atmmod/afglus.dat
source solar ../data/solar_flux/atlas_plus_modtran
wavelength 400
sza 88
As you know, the plane-parallel approximation in disort is not very accurate for low sun (here: SZA 88 degrees). With the default solver we obtain 22.01 for the diffuse downward irradiance. Using the
pseudospherical disort version
we obtain 34.72 instead which is considerably different. Now add the following to the input file:
rte_solver mystic
mc_spherical 1D
mc_photons 100000
in order to obtain 34.47±0.36. MYSTIC includes a fully spherical solver which is invoked with mc_spherical. Here the results of MYSTIC and disort disagree by more than 2 standard deviations. Let's
repeat the experiment and increase the number of photons to 1000000 in order to obtain 34.50±0.09. The result differ in fact. Here you should better trust MYSTIC because MYSTIC is a fully spherical
solver without approximations while the pseudospherical approximation is obviously an approximation. Now let's try a really spherical case: Use
sza 96
instead of 88 degrees. 96 degrees is the onset of nautical twilight (during nautical twilight, sailors can take reliable star sightings of well-known stars). You shouldn't trust the pseudo-spherical
approximation anymore for such low sun, but spherical MYSTIC provides a reliable result of 0.091±0.006 (the pseudospherical disort result was 0.058 in that case which is still the correct order of
magnitude, but we know that the pseudospherical approximation may provide complete nonsense for such SZAs for certain circumstances).
Using spherical MYSTIC you may safely compute radiances and irradiances for any SZA between 0 and 180 degrees. Also, radiances for low viewing angles are correctly computed while those are not
handled correctly with the plane-parallel or pseudo-spherical approximationss. Please note that spherical MYSTIC automatically activates backward mode. If you need quantities other than diffuse
downward irradiance please use mc_backward_output …
MYSTIC also includes a fully vectorized (polarization-dependent) solver. Try
atmosphere_file ../data/atmmod/afglus.dat
source solar ../data/solar_flux/atlas_plus_modtran
wavelength 300
sza 30
rte_solver mystic
mc_photons 1000000
to obtain 1.224±0.002 for the diffuse downward irradiance (disort: 1.224). Now add
and obtain 1.234±0.002 which is 1% higher. The neglect of polarization may cause errors in the radiance of up to 10% according to Mishchenko et al. (1994) while errors in the irradiance are probably
much smaller, as shown in this example. However, the real virtue of the vectorized MYSTIC is the possibility to calculate the full Stokes Vector, required e.g. for a number of modern remote sensing
instruments like POLDER, PARASOL, GOSAT, etc. Simply add
umu -1
phi 0
to the above example and check mc.rad.spc:
300.00000 0 0 0 0.433473
300.00000 0 0 0 -0.0461689
300.00000 0 0 0 -0.000196948
300.00000 0 0 0 0
These are the four components of the Stokes Vector (I,Q,U,V) for the chosen wavelength and geometry.
These examples should be enough to get you started with MYSTIC. It is immediately clear that the required number of photons (and hence the computational time) depends strongly on the problem. Also,
some problems are better solved in forward mode while some (e.g. thermal irradiance) should rather be done in backward mode. Strongly peaked scattering phase functions of aerosol and water clouds and
in particular of ice clouds may cause spikes which can be removed by switching on
It is important to note that all these switches only affect the noise, but not the absolute value since MYSTIC is “physically correct” by definition. The only exception is
which truncates the phase function and may introduce some systematic uncertainty. It's usually not required - use
instead. If you plan to use MYSTIC for your work, make sure you get familiar with the options and check the above mentioned literature!
Mishchenko, M.I., A.A. Lacis, and L.D. Travis, 1994. Errors induced by the neglect of polarization in radiance calculations for Rayleigh-scattering atmospheres. JQSRT 51, 491-510. | {"url":"http://libradtran.org/doku.php?id=basic_usage","timestamp":"2024-11-12T06:55:53Z","content_type":"application/xhtml+xml","content_length":"33289","record_id":"<urn:uuid:a53bcf00-cac7-4938-9c53-0e906fe259f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00853.warc.gz"} |
Neal D. Goldstein, PhD, MBI
May 31, 2018
Assessing the data entry error rate for quality assurance
Many epidemiologists spend a lot of time working with existing data. Not infrequently, these data are derived via an abstraction from other (primary) sources. An example of this is the clinical
epidemiologist working with medical data abstracted from the electronic health record. One question that naturally arises is how well do these observed data capture the true data, assuming the other
data source - in this case the EHR - is the gold standard. There are a whole host of measurement error and misclassification techniques that can be applied to your sampled data; in this simplistic
scenario we just want an idea of the overall error rate (percent that are incorrect). Before we can account for the error (explain, adjust, etc.) we need to understand its presence. To do this we can
create an audit dataset that is then used for comparison against the gold standard to compute the error rate.
Two questions naturally arise:
1. How do I sample from an existing dataset to create an audit dataset?
2. How many data points do I need?
Let's tackle the second question first. This can be thought of as a one-sample test of proportions (the error rate). We want to see if audited data error rate (p) is outside of a threshold of
acceptability (p0). Our null hypothesis is the audited error rate is equivalent to the threshold: p = p0. Our alternative hypothesis is that the audited error rate is less than a threshold for
acceptability: p < p0. Therefore this is a one-sided test. Although we hope not to see it, we can also detected if the audited error rate is greater than a threshold for acceptability: p > p0.
Now for some assumptions. We'll accept a false positive rate of 5% (alpha=0.05) and a false negative rate of 20% (beta=0.20). Our threshold for acceptability is 10% error rate, we hope to see the
calculated error below this value. To specify the effect size we can imagine a window around this threshold, and whether the true error rate will fall below of this window. The more certain we want
to be the actual error is not in this window (by shrinking the window), the larger the sample. For example, if we believe the audited error rate will be 9% and our threshold is 10%, this will require
a much larger sample to detect compared to an audited error rate of 5% and a threshold of 10%. The corollary to this - to reject the null hypothesis and conclude the audited error rate is below a
threshold - will depend on how sure we want to be of the actual error rate. For this exercise, I assume p = 0.07 and p0 = 0.10.
Plugging these numbers into a sample size calculator tells us we need a sample of 557 data points. Users of R can calculate this by plugging in the following code:
ceiling(n) # 557
Now, to return to the first question, this can be a simple random sample from the data. Suppose you have a dataset of 1000 observations with 50 variables. Does the number 557 suggest you check one
variable for 557 people, or do you check all 50 variables for 12 people (rounding up)? This comes down to the independence assumption. The sample size calculation stipulates you need 557 data points,
assuming they are independent from one another. Is there reason to suspect that one observations versus another is more likely to have data entry errors? Or if there were different people abstracting
the data, would that affect the data entry? These are important questions to consider as they may affect the error. If there is some correlation suspected, the net effect is loss of data. A
straightforward solution is to bump up the sample size to account for the correlated data.
In practice, it is probably desirable to sample a range of observations and variables to ensure as complete coverage as possible to fulfill the calculated number of data points. Then the error rate,
p, can be calculated during the audit. With p obtained from the data, one can then calculate a z-statistic and p-value to conclude the hypothesis test. R code as follows:
p_value = 2*pnorm(-abs(z))
Cite: Goldstein ND. Assessing the data entry error rate for quality assurance. May 31, 2018. DOI: 10.17918/goldsteinepi. | {"url":"https://goldsteinepi.com/blog/assessingthedataentryerrorrateforqualityassurance/index.html","timestamp":"2024-11-07T10:34:30Z","content_type":"text/html","content_length":"7000","record_id":"<urn:uuid:40da7ade-c0a4-421b-aef2-e5b020a59a3f>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00328.warc.gz"} |
Departments of Uruguay | Mappr
Uruguay has a total of 19 departments. In this article, we will give some general information about the departments of Uruguay, such as population and area.
Below, you can see Uruguay’s Departments on the map.
Departments of Uruguay
Departments of Uruguay
The population of 2011 is 73,378. It has an area of 11,928 square kilometers.
The population of 2011 is 520,187. It has an area of 4,536 square kilometers.
Cerro Largo
Cerro Largo
The population of 2011 is 84,698. It has an area of 13,648 square kilometers.
The population of 2011 is 123,203. It has an area of 6,106 square kilometers.
The population of 2011 is 57,088. It has an area of 11,643 square kilometers.
2011 yılına ait nüfusu 25,050’dir. 5,144 kilometrekare yüzölçümüne sahiptir.
The population of 2011 is 67,048. It has an area of 10,417 square kilometers.
The population of 2011 is 58,815. It has an area of 10,016 square kilometers.
The population of 2011 is 164,300. It has an area of 4,793 square kilometers.
The population of 2011 is 1,319,108. It has an area of 530 square kilometers.
The population of 2011 is 113,124. It has an area of 13,922 square kilometers.
Río Negro
Río Negro
The population of 2011 is 54,765. It has an area of 9,282 square kilometers.
The population of 2011 is 103,493. It has an area of 9,370 square kilometers.
The population of 2011 is 68,088. It has an area of 10,551 square kilometers.
The population of 2011 is 124,878. It has an area of 14,163 square kilometers.
San José
The population of 2011 is 108,309. It has an area of 4,992 square kilometers.
The population of 2011 is 82,595. It has an area of 9,008 square kilometers.
The population of 2011 is 90,053. It has an area of 15,438 square kilometers.
Treinta y Tres
Treinta y Tres
The population of 2011 is 48,134. It has an area of 9,676 square kilometers. | {"url":"https://www.mappr.co/counties/uruguay-departments/","timestamp":"2024-11-09T15:46:28Z","content_type":"text/html","content_length":"185312","record_id":"<urn:uuid:f88efdd7-3a2f-4617-a219-51f439c8331e>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00708.warc.gz"} |
Basic data preparation for Machine Learning | Mihai AntonMihai Anton
The very core of every learning algorithm is data. The more, the better. Experiments show that for a learning algorithm to reach its full potential, the data that we feed to it must be as qualitative
as it is quantitative. To achieve state of the art results in data science projects, the main material, namely data, has to be ready to be shaped and moulded as our particular situation demands.
Algorithms that accept data as raw and unprocessed as it is are scarce and often fail to leverage the full potential of machine learning and of the dataset itself.
The road between raw data and the actual training of one model is far from straight and often requires various techniques of data processing to reveal insights and to emphasize certain distributions
of the features.
Take for example this dataset from the real estate business. It is by far an easy job to be able to accurately predict the final acquisition price of a house considering only the raw data. There is
no way a generic algorithm could make a difference between ‘1Story’ and ‘2Story’ from the HouseStyle column or differentiate between two different value scales, like the year in the YearBuilt column
and the mark in the OverallQual column.
Data preparation has the duty of building an adequate value distribution for each column so that a generic algorithm could learn features from it. Some examples are rescaling the values, turning text
information into categorical or extracting tokens from continuous string values, like product descriptions. This post will give some insights into how to transform raw data into formats that make the
most out of it.
As the data producing sources are rarely perfect, raw datasets have missing values. Since generic algorithms cannot handle such cases and replacing them with a random value opens the possibility of
obtaining any random output, methods have been implemented to replace them while keeping the data distribution in place, unmodified.
The basic solution is dropping the rows or columns that contain an excessive amount of missing fields, since replacing all the empty fields with the same default value might bias the model rather
than create valuable insights. You might imagine that this approach is not optimal, since we don’t want to delete data that might prove itself valuable. Instead, setting missing values to the median
of the column has, experimentally, provided good results, since it keeps a similar data distribution.
Outliers are data points that lie far away from the majority of data samples in the geometrical space of the distribution. Since these observations are far from the mean, they can influence the
learning algorithm in an unwanted way, biasing it towards the tails of the distribution. A common way to handle outliers is Outlier Capping, which limits the range, casting a value X in the range [ m
- std * delta , m + std * delta ], where m is the median value of the distribution, std the standard deviation and delta an arbitrarily chosen scale factor. This is how we would write this more
There is often the case in machine learning when a feature is not linearly separable. Although more complex algorithms cope with the problem of non linearly separable search spaces, they might
sacrifice accuracy over covering all the nonlinearities. Thus, creating polynomial features can help learning algorithms separate the search space with more ease, yielding better results in the end.
Generating the second degree polynomial of feature 1 and adding it to the dataset yields a better representation of the geometrical data space, thus making it easier to be split. Although this is a
shallow example, it clearly illustrates the importance of polynomial features in machine learning. On a large scale, polynomials of multi-feature combinations are taken into consideration for
generating even more insights from the data.
Those were just a few data processing steps to consider. At aiflow.ltd, we automatically process the data with many more steps, to make sure the prediction quality of our automated algorithms is the
best we can achieve. If you’re curios to find out more, subscribe to our newsletter on aiflow.ltd | {"url":"https://www.antonmihai.com/projects/ml-data-preparation","timestamp":"2024-11-05T22:48:31Z","content_type":"text/html","content_length":"49016","record_id":"<urn:uuid:261f507b-c94f-4c44-9efc-314d5ef82794>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00032.warc.gz"} |
Physics Simulation Python Projects
Physics Simulation with Python is a broad scope of topics from conventional mechanics to electromagnetism and quantum physics. We work on all areas and are updated with best project ideas and topics
that are tailored to your requirements so send us a message to get your work done at right time. Encompassing instance code snippets and descriptions, we offer some extensive project topics for
physics simulations with Python:
Classical Mechanics
1. Projectile Motion Simulation
• Under gravity and resistance of air, we plan to simulate the path of a projectile.
import numpy as np
import matplotlib.pyplot as plt
# Parameters
v0 = 50 # initial velocity (m/s)
theta = 45 # launch angle (degrees)
g = 9.81 # acceleration due to gravity (m/s^2)
dt = 0.01 # time step (s)
# Initial conditions
theta_rad = np.radians(theta)
vx = v0 * np.cos(theta_rad)
vy = v0 * np.sin(theta_rad)
x, y = 0, 0
# Lists to store the results
x_data, y_data = [x], [y]
# Simulation loop
while y >= 0:
x += vx * dt
vy -= g * dt
y += vy * dt
# Plotting the trajectory
plt.plot(x_data, y_data)
plt.xlabel(‘Horizontal Distance (m)’)
plt.ylabel(‘Vertical Distance (m)’)
plt.title(‘Projectile Motion’)
2. Simple Harmonic Motion
• A mass-spring framework ought to be simulated. Our team aims to visualize its oscillatory movement.
import numpy as np
import matplotlib.pyplot as plt
# Parameters
k = 10 # spring constant (N/m)
m = 1 # mass (kg)
x0 = 1 # initial displacement (m)
v0 = 0 # initial velocity (m/s)
dt = 0.01 # time step (s)
t_max = 20 # total simulation time (s)
# Initial conditions
x = x0
v = v0
# Lists to store the results
t_data = np.arange(0, t_max, dt)
x_data = []
# Simulation loop
for t in t_data:
a = -k/m * x # acceleration
v += a * dt
x += v * dt
# Plotting the displacement over time
plt.plot(t_data, x_data)
plt.xlabel(‘Time (s)’)
plt.ylabel(‘Displacement (m)’)
plt.title(‘Simple Harmonic Motion’)
3. Pendulum Simulation
• A fundamental pendulum has to be simulated. Mainly, we focus on examining its movement.
import numpy as np
import matplotlib.pyplot as plt
# Parameters
g = 9.81 # acceleration due to gravity (m/s^2)
L = 1 # length of the pendulum (m)
theta0 = np.pi / 4 # initial angle (radians)
omega0 = 0 # initial angular velocity (rad/s)
dt = 0.01 # time step (s)
t_max = 10 # total simulation time (s)
# Time array
t = np.arange(0, t_max, dt)
# Arrays to store the results
theta = np.zeros_like(t)
omega = np.zeros_like(t)
theta[0] = theta0
omega[0] = omega0
# Function to calculate the derivatives
def derivatives(theta, omega, g, L):
dtheta_dt = omega
domega_dt = -(g / L) * np.sin(theta)
return dtheta_dt, domega_dt
# Runge-Kutta 4th order method
for i in range(1, len(t)):
k1_theta, k1_omega = derivatives(theta[i-1], omega[i-1], g, L)
k2_theta, k2_omega = derivatives(theta[i-1] + 0.5*dt*k1_theta, omega[i-1] + 0.5*dt*k1_omega, g, L)
k3_theta, k3_omega = derivatives(theta[i-1] + 0.5*dt*k2_theta, omega[i-1] + 0.5*dt*k2_omega, g, L)
k4_theta, k4_omega = derivatives(theta[i-1] + dt*k3_theta, omega[i-1] + dt*k3_omega, g, L)
theta[i] = theta[i-1] + (dt/6.0)*(k1_theta + 2*k2_theta + 2*k3_theta + k4_theta)
omega[i] = omega[i-1] + (dt/6.0)*(k1_omega + 2*k2_omega + 2*k3_omega + k4_omega)
# Plotting the results
plt.plot(t, theta, label=’Theta (Angle)’)
plt.plot(t, omega, label=’Omega (Angular Velocity)’)
plt.xlabel(‘Time (s)’)
plt.title(‘Pendulum Simulation’)
1. Electric Field of a Point Charge
• The electric field must be visualized which is produced by a point charge.
import numpy as np
import matplotlib.pyplot as plt
# Parameters
q = 1 # charge (C)
k = 8.99e9 # Coulomb’s constant (N m^2/C^2)
x_range = np.linspace(-10, 10, 400)
y_range = np.linspace(-10, 10, 400)
X, Y = np.meshgrid(x_range, y_range)
# Electric field components
Ex = k * q * X / (X**2 + Y**2)**(3/2)
Ey = k * q * Y / (X**2 + Y**2)**(3/2)
# Plotting the electric field
plt.figure(figsize=(8, 8))
plt.quiver(X, Y, Ex, Ey)
plt.title(‘Electric Field of a Point Charge’)
plt.xlabel(‘x (m)’)
plt.ylabel(‘y (m)’)
2. Magnetic Field of a Current-Carrying Wire
• The magnetic field across a straight current-carrying wire should be simulated.
import numpy as np
import matplotlib.pyplot as plt
# Parameters
I = 1 # current (A)
mu_0 = 4 * np.pi * 1e-7 # permeability of free space (T m/A)
x_range = np.linspace(-10, 10, 400)
y_range = np.linspace(-10, 10, 400)
X, Y = np.meshgrid(x_range, y_range)
# Magnetic field components
Bx = -mu_0 * I * Y / (2 * np.pi * (X**2 + Y**2))
By = mu_0 * I * X / (2 * np.pi * (X**2 + Y**2))
# Plotting the magnetic field
plt.figure(figsize=(8, 8))
plt.quiver(X, Y, Bx, By)
plt.title(‘Magnetic Field of a Current-Carrying Wire’)
plt.xlabel(‘x (m)’)
plt.ylabel(‘y (m)’)
3. Electromagnetic Wave Propagation
• In free space, we intend to simulate the diffusion of an electromagnetic wave.
import numpy as np
import matplotlib.pyplot as plt
# Parameters
c = 3e8 # speed of light (m/s)
wavelength = 1 # wavelength (m)
k = 2 * np.pi / wavelength # wave number (rad/m)
omega = k * c # angular frequency (rad/s)
t = 0 # initial time (s)
dt = 0.01 # time step (s)
x = np.linspace(0, 10, 1000) # spatial domain
# Function to calculate the electric and magnetic fields
def fields(x, t):
E = np.sin(k * x – omega * t)
B = np.sin(k * x – omega * t)
return E, B
# Plotting the fields over time
fig, ax = plt.subplots(figsize=(10, 6))
for i in range(100):
E, B = fields(x, t)
ax.plot(x, E, label=’Electric Field (E)’)
ax.plot(x, B, label=’Magnetic Field (B)’)
ax.set_xlabel(‘Position (m)’)
ax.set_ylabel(‘Field Amplitude’)
ax.set_title(‘Electromagnetic Wave Propagation’)
t += dt
1. Ideal Gas Simulation
• In a container, our team aims to simulate the characteristics of an ideal gas. It is significant to validate the ideal gas law.
import numpy as np
import matplotlib.pyplot as plt
# Parameters
n = 100 # number of particles
L = 10 # size of the container (m)
v_max = 1 # maximum initial speed (m/s)
dt = 0.01 # time step (s)
t_max = 10 # total simulation time (s)
# Initialize particle positions and velocities
positions = np.random.rand(n, 2) * L
velocities = (np.random.rand(n, 2) – 0.5) * v_max * 2
# Simulation loop
for _ in range(int(t_max / dt)):
positions += velocities * dt
# Reflect particles off the walls
velocities[positions[:, 0] < 0] *= [-1, 1]
velocities[positions[:, 0] > L] *= [-1, 1]
velocities[positions[:, 1] < 0] *= [1, -1]
velocities[positions[:, 1] > L] *= [1, -1]
# Plotting the final positions
plt.figure(figsize=(8, 8))
plt.scatter(positions[:, 0], positions[:, 1], s=10)
plt.xlim(0, L)
plt.ylim(0, L)
plt.xlabel(‘x (m)’)
plt.ylabel(‘y (m)’)
plt.title(‘Ideal Gas Simulation’)
2. Heat Diffusion in a Rod
• Along a 1D (One -Dimensional) rod, the dispersion of heat must be simulated.
import numpy as np
import matplotlib.pyplot as plt
# Parameters
L = 10 # length of the rod (m)
T_left = 100 # temperature at the left end (C)
T_right = 0 # temperature at the right end (C)
alpha = 0.01 # thermal diffusivity (m^2/s)
dx = 0.1 # spatial step size (m)
dt = 0.01 # time step (s)
t_max = 2 # total simulation time (s)
# Discretize the rod
x = np.arange(0, L + dx, dx)
T = np.zeros_like(x)
T[0] = T_left
T[-1] = T_right
# Simulation loop
for _ in range(int(t_max / dt)):
T_new = T.copy()
for i in range(1, len(x) – 1):
T_new[i] = T[i] + alpha * dt / dx**2 * (T[i+1] – 2*T[i] + T[i-1])
T = T_new
# Plotting the temperature distribution
plt.plot(x, T)
plt.xlabel(‘Position (m)’)
plt.ylabel(‘Temperature (C)’)
plt.title(‘Heat Diffusion in a Rod’)
Quantum Mechanics
1. Particle in a Box
• In a one-dimensional box, we focus on simulating the wavefunction of a particle.
import numpy as np
import matplotlib.pyplot as plt
# Parameters
L = 10 # length of the box (m)
n = 1 # quantum number
x = np.linspace(0, L, 1000) # spatial domain
# Wavefunction
psi = np.sqrt(2 / L) * np.sin(n * np.pi * x / L)
# Plotting the wavefunction
plt.plot(x, psi)
plt.xlabel(‘Position (m)’)
plt.ylabel(‘Wavefunction (ψ)’)
plt.title(‘Particle in a Box (n=1)’)
2. Quantum Harmonic Oscillator
• Generally, the wavefunctions of a quantum harmonic oscillator ought to be simulated.
import numpy as np
import matplotlib.pyplot as plt
from scipy.special import hermite
# Parameters
m = 1 # mass (kg)
omega = 1 # angular frequency (rad/s)
hbar = 1 # reduced Planck’s constant (J s)
n = 0 # quantum number
x = np.linspace(-5, 5, 1000) # spatial domain
# Wavefunction
Hn = hermite(n)
psi = (1 / np.sqrt(2**n * np.math.factorial(n))) * ((m * omega / (np.pi * hbar))**0.25) * np.exp(-m * omega * x**2 / (2 * hbar)) * Hn(np.sqrt(m * omega / hbar) * x)
# Plotting the wavefunction
plt.plot(x, psi)
plt.xlabel(‘Position (m)’)
plt.ylabel(‘Wavefunction (ψ)’)
plt.title(‘Quantum Harmonic Oscillator (n=0)’)
Advanced Projects
1. Double Pendulum Simulation
• A double pendulum must be simulated. Typically, our team intends to investigate its disruptive features.
import numpy as np
import matplotlib.pyplot as plt
# Parameters
g = 9.81 # acceleration due to gravity (m/s^2)
L1 = 1.0 # length of the first pendulum (m)
L2 = 1.0 # length of the second pendulum (m)
m1 = 1.0 # mass of the first pendulum (kg)
m2 = 1.0 # mass of the second pendulum (kg)
theta1_0 = np.pi / 2 # initial angle of the first pendulum (radians)
theta2_0 = np.pi / 4 # initial angle of the second pendulum (radians)
omega1_0 = 0.0 # initial angular velocity of the first pendulum (radians/s)
omega2_0 = 0.0 # initial angular velocity of the second pendulum (radians/s)
dt = 0.01 # time step (s)
t_max = 20 # total simulation time (s)
# Time array
t = np.arange(0, t_max, dt)
# Arrays to store the results
theta1 = np.zeros_like(t)
omega1 = np.zeros_like(t)
theta2 = np.zeros_like(t)
omega2 = np.zeros_like(t)
theta1[0] = theta1_0
omega1[0] = omega1_0
theta2[0] = theta2_0
omega2[0] = omega2_0
# Function to calculate the derivatives
def derivatives(theta1, omega1, theta2, omega2, g, L1, L2, m1, m2):
delta = theta2 – theta1
denom1 = (m1 + m2) * L1 – m2 * L1 * np.cos(delta) * np.cos(delta)
denom2 = (L2 / L1) * denom1
dtheta1_dt = omega1
dtheta2_dt = omega2
domega1_dt = (m2 * L1 * omega1 * omega1 * np.sin(delta) * np.cos(delta) +
m2 * g * np.sin(theta2) * np.cos(delta) +
m2 * L2 * omega2 * omega2 * np.sin(delta) –
(m1 + m2) * g * np.sin(theta1)) / denom1
domega2_dt = (-m2 * L2 * omega2 * omega2 * np.sin(delta) * np.cos(delta) +
(m1 + m2) * g * np.sin(theta1) * np.cos(delta) –
(m1 + m2) * L1 * omega1 * omega1 * np.sin(delta) –
(m1 + m2) * g * np.sin(theta2)) / denom2
return dtheta1_dt, domega1_dt, dtheta2_dt, domega2_dt
# Runge-Kutta 4th order method
for i in range(1, len(t)):
k1_theta1, k1_omega1, k1_theta2, k1_omega2 = derivatives(theta1[i-1], omega1[i-1], theta2[i-1], omega2[i-1], g, L1, L2, m1, m2)
k2_theta1, k2_omega1, k2_theta2, k2_omega2 = derivatives(theta1[i-1] + 0.5*dt*k1_theta1, omega1[i-1] + 0.5*dt*k1_omega1, theta2[i-1] + 0.5*dt*k1_theta2, omega2[i-1] + 0.5*dt*k1_omega2, g, L1, L2, m1,
k3_theta1, k3_omega1, k3_theta2, k3_omega2 = derivatives(theta1[i-1] + 0.5*dt*k2_theta1, omega1[i-1] + 0.5*dt*k2_omega1, theta2[i-1] + 0.5*dt*k2_theta2, omega2[i-1] + 0.5*dt*k2_omega2, g, L1, L2, m1,
k4_theta1, k4_omega1, k4_theta2, k4_omega2 = derivatives(theta1[i-1] + dt*k3_theta1, omega1[i-1] + dt*k3_omega1, theta2[i-1] + dt*k3_theta2, omega2[i-1] + dt*k3_omega2, g, L1, L2, m1, m2)
theta1[i] = theta1[i-1] + (dt/6.0)*(k1_theta1 + 2*k2_theta1 + 2*k3_theta1 + k4_theta1)
omega1[i] = omega1[i-1] + (dt/6.0)*(k1_omega1 + 2*k2_omega1 + 2*k3_omega1 + k4_omega1)
theta2[i] = theta2[i-1] + (dt/6.0)*(k1_theta2 + 2*k2_theta2 + 2*k3_theta2 + k4_theta2)
omega2[i] = omega2[i-1] + (dt/6.0)*(k1_omega2 + 2*k2_omega2 + 2*k3_omega2 + k4_omega2)
# Plotting the results
plt.figure(figsize=(12, 6))
plt.subplot(2, 1, 1)
plt.plot(t, theta1, label=’Theta1 (Angle of Pendulum 1)’)
plt.plot(t, theta2, label=’Theta2 (Angle of Pendulum 2)’)
plt.xlabel(‘Time (s)’)
plt.ylabel(‘Angle (rad)’)
plt.title(‘Double Pendulum Simulation – Angles’)
plt.subplot(2, 1, 2)
plt.plot(t, omega1, label=’Omega1 (Angular Velocity of Pendulum 1)’)
plt.plot(t, omega2, label=’Omega2 (Angular Velocity of Pendulum 2)’)
plt.xlabel(‘Time (s)’)
plt.ylabel(‘Angular Velocity (rad/s)’)
plt.title(‘Double Pendulum Simulation – Angular Velocities’)
2. Solar System Simulation
• Through the utilization of Newton’s law of motion and gravitation, we plan to simulate the movement of planets in the solar system.
import numpy as np
import matplotlib.pyplot as plt
# Parameters
G = 6.67430e-11 # gravitational constant (m^3 kg^-1 s^-2)
M_sun = 1.989e30 # mass of the Sun (kg)
dt = 1e4 # time step (s)
t_max = 3.154e7 # total simulation time (s)
# Initial conditions for Earth
r_earth = np.array([1.496e11, 0]) # initial position (m)
v_earth = np.array([0, 2.978e4]) # initial velocity (m/s)
# Arrays to store the results
positions = []
# Simulation loop
for _ in range(int(t_max / dt)):
r = np.linalg.norm(r_earth)
a = -G * M_sun / r**3 * r_earth
v_earth += a * dt
r_earth += v_earth * dt
# Convert results to arrays for plotting
positions = np.array(positions)
# Plotting the trajectory
plt.figure(figsize=(8, 8))
plt.plot(positions[:, 0], positions[:, 1])
plt.scatter(0, 0, color=’yellow’, label=’Sun’)
plt.xlabel(‘x (m)’)
plt.ylabel(‘y (m)’)
plt.title(‘Earth\’s Orbit around the Sun’)
physics simulation python Projects
In the current years, numerous project topics on physics simulation are emerging continuously. Encompassing different regions like electromagnetism, quantum mechanics, classical mechanics,
thermodynamics, and more, we suggest 50 extensive physics simulation project topics with Python:
Classical Mechanics
1. Projectile Motion Simulation
• Under the impact of gravity and resistance of air, we plan to simulate the path of a projectile. Generally, various positions, drag coefficients, and preliminary speeds ought to be examined.
2. Simple Harmonic Oscillator
• The movement of a mass-spring model has to be simulated. It is appreciable to investigate the impacts of various spring constants and masses.
3. Damped Harmonic Oscillator
• Our team aims to simulate a damped harmonic oscillator. Typically, the impacts of differing damping coefficients must be explored.
4. Pendulum Simulation
• We focus on designing the fundamental pendulum movement. Crucial impacts of angle of incidence and various lengths ought to be evaluated.
5. Double Pendulum Simulation
• Mainly, a double pendulum has to be simulated. We plan to examine its disruptive features in an appropriate manner.
6. N-Body Gravitational Simulation
• The gravitational communications among numerous bodies like moons and planets should be simulated.
7. Circular Motion
• An object ought to be simulated which is in consistent circular movement. We focus on examining the significant connection among centripetal force, velocity, and mass.
8. Rotational Dynamics
• The rotation of a rigid body must be designed. It is approachable to investigate rotational inertia, torque, and angular momentum.
9. Collision Simulation
• Among particles, our team focuses on simulating flexible and inflexible collisions. The maintenance of energy and momentum has to be examined.
10. Projectile Motion with Drag
• Focusing on air resistance, we intend to simulate movement of the projectile. Mainly, the outcomes have to be contrasted with perfect projectile movement.
1. Electric Field of Point Charges
• The electric field must be visualized which is produced by numerous point charges. It is advisable to investigate magnetic lines and superposition.
2. Electric Potential of Point Charges
• In the case of an arrangement of point charges, we focus on assessing and visualizing the electric potential.
3. Capacitor Simulation
• The characteristics of a parallel plate capacitor ought to be designed. It is appreciable to explore the crucial impacts of various dielectric resources and plate separations.
4. Current-Carrying Wire Magnetic Field
• The magnetic field has to be simulated, which is across a straight current-carrying wire. Our team aims to visualize the magnetic lines in an effective manner.
5. Electromagnetic Wave Propagation
• In free space, we plan to design the diffusion of an electromagnetic wave. Typically, the significant connection among magnetic and electric fields should be explored.
6. RC Circuit Simulation
• An RC circuit has to be simulated. It is advisable to examine the discharging and charging characteristics of the capacitor.
7. RL Circuit Simulation
• Our team aims to simulate an RL circuit. The temporary reaction of the inductor ought to be explored.
8. RLC Circuit Simulation
• Generally, an RLC circuit must be designed. We intend to investigate its damping characteristics and frequency of resonance.
9. Induced EMF in a Moving Conductor
• The induced electromotive force (EMF) in a conductor should be simulated which is travelling across a magnetic field.
10. Faraday’s Law of Induction
• In the case of a varying magnetic field, our team designs the creation of EMF in a coil. Generally, several rates of variation and coil features have to be investigated.
1. Ideal Gas Law Simulation
• In a container, we aim to design the characteristics of an ideal gas. Through differing pressure, temperature, and volume, it is advisable to validate the ideal gas law.
2. Heat Transfer in a Rod
• The heat transmission ought to be simulated, which is across a one-dimensional rod. Periodically, our team examines the temperature dispersion.
3. Heat Transfer in a 2D Plate
• In a two-dimensional plate, we focus on simulating the heat transmission. Typically, balanced and temporary heat transmission has to be investigated.
4. Carnot Cycle Simulation
• The Carnot cycle must be simulated. Our team plans to assess the Carnot engine effectiveness.
5. Maxwell-Boltzmann Distribution
• In an ideal gas, we visualize the Maxwell-Boltzmann dissemination of molecular speeds. Mainly, various temperatures must be examined.
6. Isothermal and Adiabatic Processes
• For an ideal gas, it is significant to simulate procedures of isothermal and adiabatic. In every procedure, our team focuses on contrasting the performed tasks and PV diagrams.
7. Phase Change Simulation
• With the support of latent heat and heat transmission policies, we intend to design the phase transition of a substance such as water to stream.
8. Entropy Change in Thermodynamic Processes
• For different thermodynamic procedures, our team assesses and visualizes the variation in entropy.
9. Heat Engines and Efficiency
• Typically, various heat engine cycles like Diesel, Otto have to be simulated. It is advisable to assess their capabilities.
10. Thermal Expansion
• The thermal expansion of solids should be designed. Among temperature variations and developments, focus on examining the associations or connections.
Quantum Mechanics
1. Particle in a Box
• Consider a particle which is enclosed in a one-dimensional box and simulate their wavefunction. For various energy levels, we visualize the probability density.
2. Quantum Harmonic Oscillator
• Specifically, it is approachable to design the wavefunctions of a quantum harmonic oscillator. Ground states and excited states are required to be examined efficiently.
3. Double-Slit Experiment
• The double-slit experimentation ought to be simulated. Our team plans to visualize the intervention trend of particles.
4. Hydrogen Atom Wavefunctions
• For the hydrogen atom, we visualize the radial probability distribution functions.
5. Tunneling Effect
• Across a possible obstacle, it is approachable to simulate quantum tunnelling of a particle. The reflection and transmission coefficients should be examined.
6. Quantum Entanglement
• The entangled quantum conditions ought to be designed. We plan to examine the destruction of Bell’s discrepancies.
7. Spin-1/2 Particles
• In a magnetic field, our team simulates the characteristics of spin-1/2 particles. The theory of assessment and superposition has to be examined.
8. Time-Dependent Schrödinger Equation
• Consider a particle in a potential well and resolve the time-dependent Schrödinger equation.
9. Quantum Superposition
• The superposition of wavefunctions must be designed. We examine the resultant probability densities in an effective manner.
10. Fermions and Bosons
• In a potential well, we simulate the characteristics of bosons and fermions. Mainly, the Bose-Einstein condensation and Pauli exclusion has to be investigated.
1. Time Dilation
• Time dilation has to be simulated for objects that are traveling at the relativity speeds. We investigate the major connections among time and speed accomplished.
2. Length Contraction
• For objects that are traveling at the relativity speeds, length contraction must be designed. Among speed and examined length, our team investigates the crucial connection.
3. Relativistic Momentum and Energy
• Consider particles that are traveling at extreme speeds and assess their relativistic momentum and energy.
4. Lorentz Transformations
• On time and space scales, our team visualizes the impacts of Lorentz transformations.
5. Relativistic Doppler Effect
• For light and sound waves, we simulate the relativistic Doppler effect. Mainly, for various relative speeds, it is significant to investigate the Doppler effect.
1. Fluid Dynamics Simulation
• By means of employing the Navier-Stokes equations, our team aims to design flow of a fluid. The turbulent and laminar flow should be simulated.
2. Wave Propagation in a Medium
• In a medium, we simulate the diffusion of mechanical waves. Typically, diffraction, reflection, and refraction has to be investigated.
3. Chaos Theory and Lorenz Attractor
• The Lozenz attractor must be simulated. Our team intends to visualize its disruptive features.
4. Solar System Simulation
• Through the utilization of Newton’s laws of gravitation, we design the movement of moons and planets in the solar system.
5. Crystallography and Bragg’s Law
• With the aid of Bragg’s law, it is appreciable to simulate X-ray diffraction trends for crystal architectures.
Encompassing instance code snippets and outlines, we have provided several project topics for physics simulation. Also, 50 widespread physics simulation project topics with Python including numerous
regions like quantum mechanics, electromagnetism, classical mechanics, and thermodynamics, and more are recommended by us in this article. | {"url":"https://matlabsimulation.com/physics-simulation-with-python/","timestamp":"2024-11-10T09:36:13Z","content_type":"text/html","content_length":"91908","record_id":"<urn:uuid:87e64b34-a8ef-4172-b52c-165e8a715304>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00408.warc.gz"} |
Power and domestic appliances
Energy and Thermal Physics
Power and domestic appliances
Physics Narrative for 11-14
Home appliances working
Let's think about making a pot of tea. Boiling enough water for a pot of tea takes 180 second with my kettle. The kettle is marked 3.0 kilowatt, which means that it costs me 3000 joule every second
to run the kettle. To boil the kettle, I must therefore pay the electricity board for 3000 joule / second × 180 second which can be worked out to be 540,000 joule. Other domestic appliances cost me
different numbers of joules, as they work at different rates for different lengths of time. Some are high power, but work only for a short time (cooker, kettle). Others are lower power, but work more
or less continuously (refrigerator) or for long periods of time (lighting).
Here are some typical annual costs:
appliance energy / megajoule
freezer 2380
cooking 2380
dishwasher 1700
lighting 1300
refrigerator 1080
tumble dryer 1010
kettle 900
television 792
washing machine 84
iron 270
vacuum cleaner 90
To find out how much energy each appliance uses, simply keep a log of how long you run it for (time in second – the duration), then multiply this quantity by the power of the appliance (power in
energy = power × duration
energyjoule = powerwatt × durationsecond
energykilojoule = powerkilowatt × durationsecond
Of course the averages given above vary with lifestyle and occupancy. Here are some estimates of how the annual total energy per household might vary:
household energy / megajoule
working couple 14 820
single person 1100
family with two children (parents working, children at school) 19 730 | {"url":"https://spark.iop.org/power-and-domestic-appliances","timestamp":"2024-11-04T10:52:32Z","content_type":"text/html","content_length":"42228","record_id":"<urn:uuid:5c179363-6b7f-4013-820a-fdedbf602d98>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00443.warc.gz"} |
Translation between different logics
+ General Questions (11)
I have 2 questions
1) Is there a precedence when translating between formulas, e.g. what is the precedence when we have !,(),[]
2) In the example above, how do we read the LO1 formulas and how is it translated to LTL, in some of the previous answers you gave you mentioned there are given LTL templates but i am not sure how to
proceed with that.
With templates, I meant the formulas that are generated in the recursive calls of algorithm Tp2Od on slide 48 in Chapter VRS-08-PredLogic.
For instance, consider the given formula where I have done some irrelevant variable renaming (since that form came out of my tool):
∃t1. t0≤t1 ∧ (∃t0. t1≤t0 ∧ (∀t2. (t1≤t2 ∧ t2<t0 → ¬a[t2])) ∧ (∀t1. (t0≤t1 → a[t1] ∧ b[t1])))
Looking at the above formula, we can clearly see that ∀t1. (t0≤t1 → a[t1] ∧ b[t1]) stems from G(a&b) applied at time t0, hence, we may now consider with alpha(t0) := ∀t1. (t0≤t1 → a[t1] ∧ b[t1]) the
following formula:
∃t1. t0≤t1 ∧ (∃t0. t1≤t0 ∧ (∀t2. (t1≤t2 ∧ t2<t0 → ¬a[t2])) ∧ alpha(t0))
Again, (∃t0. t1≤t0 ∧ (∀t2. (t1≤t2 ∧ t2<t0 → ¬a[t2])) ∧ alpha(t0)) is the template of a SU operator with arguments ¬a and alpha, so that the above formula means with beta(t1) = ∃t0. t1≤t0 ∧ (∀t2.
(t1≤t2 ∧ t2<t0 → ¬a[t2])) ∧ alpha(t0)
∃t1. t0≤t1 ∧ beta(t1)
The above means F beta (applied at time t0), and thus
F[¬a SU G(a&b)]
I can't still see how we were able to tell (∃t0. t1≤t0 ∧ (∀t2. (t1≤t2 ∧ t2<t0 → ¬a[t2])) ∧ alpha(t0)) is a template of the SU operator.
The SU operator is given as = ∃t1. t0 ≤ t1 ∧ Tp2Od(t1, ψ) ∧ interval((t0, 1,t1, 0), ϕ);
How do we know that the recursive Tp20d call is !a and also why is the interval in the template for the SU operator is replaced by alpha(t0)
If we evaluate a formula [beta SU alpha] at time t1, then it means that there must be a point of time t0 so that alpha(t0) holds and for all points of time t2 with t1≤t2 ∧ t2<t0, we must have beta
(t2). Having read this, compare it with the formula you are asking for. Matching beta with ¬a makes these statements identical, right?
While I used the semantics of the SU operator to explain this, you get exactly the same when you expand the call to the function Interval where Phi takes the rule of my beta and Psi the role of
i think you skipped my first question,
Is there a precedence when translating between formulas, e.g. what is the precedence when we have !,(),[]
I have doubts about the precedence of operations.
in this question ¬A(GXFb ∧ FXb) the negation was applied at the end after simplifying the inner formula
In this question ¬AXXGF¬E[1 U (Ab)] the negation was applied at the beginning
So is the precedence (),! ?
I guess you are now talking about another problem and you don't clearly say which one which makes it hard to discuss with you. It seems that there is another question for ∃t1. t0≤t1 ∧ (∃t0. t1≤t0 ∧
(∀t2. (t1≤t2 ∧ t2≤t0 → ¬b[t2])) ∧ a[t0]) which has the equivalent LTL formula F[a SB b] and you are wondering whether you could list equivalent LTL formulas also.
If I am right that this is question, then let me say that I would be surprised to read F(a & !b) instead of F[a SB b] (even that is the same) so that you should definitely add a comment about this.
Moreover, the formula you are listing has to be equivalent, and hence, F[!b SU a] is not a correct answer (it is just implied, but the other direction of the implication is not valid).
The solution is F[a SB b] which is not the same as F[!b SU a]. Look, for F[!b SU a], we would get
∃t1. t0≤t1 ∧ (∃t0. t1≤t0 ∧ (∀t2. (t1≤t2 ∧ t2<t0 → ¬b[t2])) ∧ a[t0])
while for F[a SB b], we get
∃t1. t0≤t1 ∧ (∃t0. t1≤t0 ∧ (∀t2. (t1≤t2 ∧ t2≤t0 → ¬b[t2])) ∧ a[t0])
It looks almost the same, but the former has a t2<t0 while the latter has t2≤t0. That is the difference between the two formulas! | {"url":"https://q2a.cs.uni-kl.de/2060/translation-between-different-logics?show=2070","timestamp":"2024-11-14T23:40:26Z","content_type":"text/html","content_length":"77587","record_id":"<urn:uuid:4963ab99-d77e-4d40-9c91-6ee47e7708a7>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00819.warc.gz"} |
Perancangan Generator Induksi 1 Fase dari Motor Induksi 3 Fase
Ambar Pratiwi, Mariska Sari and , Agus Supardi, S.T., M.T and , Aris Budiman, S.T., M.T. (2015) Perancangan Generator Induksi 1 Fase dari Motor Induksi 3 Fase. Skripsi thesis, Universitas
Muhammadiyah Surakarta.
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One of the main components of the power generation system design consideration is the type of generator. The generator is a machine that converts mechanical energy into electrical energy. Electricity
is used to meet the needs of the general public, the electrical equipment used by the general public in the form of electricity 1-phase. So it is more appropriate to use the generator output voltage
1-phase premises. 1 induction generator design phase of 3-phase induction motor by changing the function of 3-phase induction motor becomes a generator induction phase 1 by taking the R phase and the
S phase induction motors 3-phase delta connected and to amplify the output voltage plus the capacitor. Each test 1-phase induction generator using a capacitor with a size 48, 56, and 64 μF. 1-phase
induction generator testing using a resistive load such as incandescent bulb size 5 Watt, 10 Watt and 60 Watt, as well as inductive loads such as fan size of 18 Watt. Data - Data is then analyzed.
The test results 1 phase induction generator without load indicates that a stable voltage value rises according to the rise and fall when the rotational speed is increased capacitor size. Frequency
generated steady rise in accordance with the increase in rotational speed and down when the size of the capacitor is increased because the capacitor becomes a burden for the generator. Resistive load
testing on the rotational speed and the excitation capacitor is fixed by adding a greater burden, the rotational speed to be down, it resulted in the frequency and the voltage drop or voltage drop.
In the inductive load with the size of load power remains, the capacitor has two functions, namely to help the excitation generator and supply an inductive load. The influence of reactive power on
the capacitor resulting rotational speed to be down so that the voltage and frequency down. At 1400 RPM rotational speed - 1550 RPM with a 48 μF capacitor generate excitation frequency 47 Hz - 51 Hz,
the voltage of 145.4 V - 230 V on resistive load with 48 μF capacitor excitation, load 5 Watt produces a frequency 47 Hz - 50 Hz, voltage 142.4 V - 216 V, with load 65 Watt generate the frequency
47.4 Hz - 49 Hz, the voltage of 117.7 V - 185 V. At inductive load rotary speed of 1400 rpm and 1 550 rpm with 48 μF capacitors generate excitation frequency 47.4 Hz - 51 Hz, voltage 128 V - 218 V.
Excitation 64 μF capacitor generates a frequency 41.6 Hz - 49.2 Hz, voltage 104 V - 188 V.
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Students & PostDocs
I have had the privilege of working some very talented students and Post Docs over the years. Below you will find some brief information on my current and past students, as well as something about
the research they are working on.
Clink on their name/image to be taken to their own website or LinkedIn page.
Photo Name Thesis Research Interests
Mathew Cater
Benavides, 2027 TBA TBA
Xuchen Wu, 2026 Partial Information Principal-Agent Problems Xuchen is working on how principal-agent problems can be extended to partial information settings.
Liam is working on developing models characterizing Canadian carbon and GHG financial markets with heterogeneous agent
Liam Welsh Clean Energy Finance, 2025 (expected) participation. Liam’s research interests are in climate finance, renewable energy (market) modelling, mean-field games, and
computational finance.
Generative Machine Learning Models for Vedant is working on generative modelling of financial time series data with an emphasis on synthesizing static arbitrage free
Vedant Choudhary Finance, 2026 (expected) dynamic implied volatility surfaces. He intends to utilize it for downstream applications such as hedging using reinforcement
learning and portfolio allocation via mean-variance optimization.
Emma Kroell Reverse Sensitivity, 2025 (expected) Emma is working on understanding how processes are minimally perturbed when one perturbs their risk at some point in time.
Anthony Coache Risk Aware Reinforcement Learning, 2024 Anthony is working on risk-aware reinforcement learning where an agent accounts not just for total rewards, but the risk
(expected) associated with it.
Yichao Chen Deep learning for Mean Field Type Problems, Yichao is working on developing guarantees on principal-agent problems using neural network function approximation.
Brian Ning Deep Reinforcement Learning, 2022 Brian is working on combining reinforcement learning with deep learning methods and stochastic control and games. One application
area is algorithmic trading.
Arvind Shrivats SREC Markets, 2021 Arvind is developing the theory of how to endogenize prices in Solar Renewable Energy Certificate markets.
Tianyi Jia Algortihmic Trading in Foreign Exchange Tianyi’s work looks at foreign exchange markets. He is developing algorithmic trading schemes for triplets of FX pairs and
Markets, 2021 accounting for order-flow uncertainty.
Zhen Qin Model Uncertainty in Commodity and Energy Zhen’s work focuses on developing commodity models for derivative valuation and trading when the agent accounts for model
Markets, 2021 uncertainty.
Outperformance and Tracking: A Framework for
Ali Al-Aradi Optimal Active and Passive Portfolio Ali focuses on using optimal portfolio allocation problems where the agent has benchmarks that they aim to track.
Management, 2020
Machine Learning in Algorithmic Trading
Tad Ferreira leading to Reinforced Deep Kalman Tad works on machine learning techniques for financial modeling and focuses on deep markov models for reinforcement learning.
Philippe Casgrain Algorithmic trading with latent models and Philippe’s work looks at mean-field games models of optimal trading when there are hidden underlying factors driving the dynamics
mean-field games, 2019 of asset prices.
Mixing Monte Carlo and Partial Differential
Equation Methods For
David Farahany Multi-Dimensional Optimal Stopping Problems David developed a novel approach for valuing path-dependent options using mixed PDE and Monte-Carlo methods.
Under Stochastic
Volatility, 2018
Xuancheng (Bill) Mean-Field Games with Ambiguity Aversion, Bill is developing the theory of incorporating ambiguity aversion (model uncertainty) into mean-field games.
Huang 2017
Dynamic Trading in a Limit Order Book: Luke solved several algorithmic trading problems: trading co-integrated assets, using limit and market orders to hedge options,
Luhui Gan Co-Integration, Option Hedging and Queueing and valuing queue position.
Dynamics, 2017
Ryan Donnelly Ambiguity Aversion in Algorithmic and High Ryan studied how model uncertainty (ambiguity aversion) modifies the trading strategies of algorithmic traders.
Frequency Trading, 2014
Jason Ricci Applied Stochastic Control in High Frequency Jason investigated high-frequency algorithmic trading problems (market making and trading strongly co-dependent assets) using
and Algorithmic Trading, 2014 stochastic control techniques.
Eddie Ng Kernel-based Copula Processes, 2010 Eddie introduced the notation of kernel-based coupla processes, extending the notion of Gaussian process in machine learning to
account for general marginals and co-dependence structure.
Georg Sigloch Utility Indifference Pricing of Credit Georg developed a utility indifference approach to valuing credit risk and accounting for model uncertainty.
Instruments, 2009
pic First Passage Times: Integral Equations, Angel investigated variations of the Skorhod embedding problem, where the goal is to match a given distribution by the
here Angel Valov Randomization and Analytical Approximations, distribution of a stopped Brownian motion.
Vladimir Surkov Option Pricing using Fourier Space Vlad developed an efficient numerical scheme for valuing options with path-dependent features, such as Barrier and American
Time-stepping Framework, 2009 options, using Fourier techniques.
Samuel Hikspoors Multi-Factor Energy Price Models and Exotic Sam worked on projects related to commodity and energy markets. In particular he was intereted in option pricing with stochastic
Derivatives Pricing, 2008 volatility using singular perturbation theory techniques.
Photo Name Research Interests
Ziteng Cheng Ziteng is working on sequential decision making, mean field games, and learning theory.
Dena Firoozi Dena is working on incorporating latent factors and partial information into mean field games.
Omid Namvar Omid is working on using stochastic approximations to solve, in a model free manner, for how to optimally trade assets that exhibit co-integration.
Damir Damir developed a number of algorithmic trading strategies which incorporated both limit and market orders, as well as developed a new class of models based on randomized pinned
Kinzebulatov measures.
Mojtaba Mojtaba’s interest lie in mean-field games (MFGs) when there is one large player and a sea of many small players — called a major-minor MFG. We studied how major-minor MFGs can
Nourian be used to solve the problem of a large institutional investor who is executing an order in an environemnt where there are a number of intelligent HFTs operating. | {"url":"http://sebastian.statistics.utoronto.ca/research/student-team/","timestamp":"2024-11-04T07:20:49Z","content_type":"text/html","content_length":"62040","record_id":"<urn:uuid:a6282f55-d0df-4010-b03c-b60dda8cac6c>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00551.warc.gz"} |
What is the derivative definition of instantaneous velocity? | HIX Tutor
What is the derivative definition of instantaneous velocity?
Answer 1
Instantaneous velocity is the change in position over the change in time. Therefore, the derivative definition of instantaneous velocity is:
instantaneous velocity= #v#= #lim_(Delta t -> 0) ##(Delta x)/(Delta t)#= #dx/dt#
So basically, instantaneous velocity is the derivative of the position function/equation of motion. For example, let's say you had a position function:
Since #v#=#dx/dt#, #v= d/dt 6t^2+t+12= 12t+1#
That is the function of the instantaneous velocity in this case. Note that it is a function because instantaneous velocity is variable- It is dependent on time, or the "instant." For every #t#, there
is a different velocity at that given instant #t#.
Let's say we wanted to know the velocity at #t=10# and the position is measured in meters (m) while the time in measured in seconds (sec).
#v=12(10)+1= 121 (m)/(sec)#
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Answer 2
The derivative definition of instantaneous velocity is the rate of change of position with respect to time at a specific instant. Mathematically, it is expressed as the derivative of the position
function ( s(t) ) with respect to time ( t ) at a particular time ( t_0 ). In symbols, the instantaneous velocity ( v(t_0) ) at time ( t_0 ) is given by:
[ v(t_0) = \lim_{\Delta t \to 0} \frac{s(t_0 + \Delta t) - s(t_0)}{\Delta t} ]
• ( v(t_0) ) is the instantaneous velocity at time ( t_0 ),
• ( s(t) ) is the position function representing the displacement of an object at time ( t ),
• ( \Delta t ) is a small interval of time,
• ( s(t_0 + \Delta t) - s(t_0) ) represents the change in position over the time interval ( \Delta t ),
• The limit as ( \Delta t ) approaches 0 ensures that the velocity is calculated at an infinitesimally small time interval around ( t_0 ), capturing the instantaneous rate of change of position
with respect to time.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Consider the following encoding scheme used in one famous compresion algorithm. Suppose we will code only sequences of lower case letters. Each such sequence of characters can be encoded to a
sequence of pairs (p[i], r[i]), where r[i] is either a character (if p[i] = 0) or an integer greater than zero and less or equal than p[i] (if p[i] > 0).
We describe now the decoding procedure for our encoding scheme. Let (p[1], r[1]), (p[2], r[2]), p[i] = 0 then r[i] is a character and we simply add r[i] to the end of already decoded sequence. If p[i
] > 0 then r[i] is an integer, r[i] letters from this sequence starting at the position p[i] places before the end.
For example, consider the sequence of pairs (0 a), (1, 1), (0, b), (3, 3), (3, 3), (3, 2), (0, c). Decoding (0, a) we get a. Decoding (1, 1) we get aa. (0, b) adds b getting aab. (3, 3) will add aab,
so now we have aabaab. Next pair (3, 3) will again add aab so we have aabaabaab. (3, 2) will add aa, so our sequence is aabaabaabaa and (0, c) adds c. So the decoded sequence is aabaabaabaac. Note
that in general for a given w it can exist more such sequences of pairs.
Let u, v be some sequences. By uv we will understand the sequence created by appending of the sequence v to the end of sequence u. Let C[w] be a sequence of pairs which encodes a sequence of
lowercase letters w. Suppose we have given a sequence of pairs C[w]. The question is how many possibilities does exist for expressing the sequence C[w] in the form C[u] C[v] where u, v are sequences
satisfying the equation w = uv and neither u nor v is empty. Write a program that will answer this question.
The input file consists of blocks of lines. Each block describes one sequence of pairs C[w] to some w in such a way that the i-th line of the block contains either two integers p[i], r[i], (
The output file contains the lines corresponding to the blocks in the input file. Each line contains the number of possibilities of representation of the sequence C[w] in the form C[u] C[v] where u,
v are sequences satisfying the equation w = uv and neither u nor v is empty.
0 a
0 b
0 c | {"url":"https://statement.bacs.cs.istu.ru/statement/get/CitiYWNzL3Byb2JsZW0vMTI2NC9zdGF0ZW1lbnQvdmVyc2lvbnMvQy9odG1sEgYKBEMP47Q/bacs/Q5XnysqlAtS1G62Dle4aCemKnr9ThczJ9Vgt4vtDDa23bzAvX5-8VWJLO24Del3V3ywLBpa3MMPeJnoJ7EHXMDtiWvi5lh3gj73Ypk_QM5sbtHB-P-OAYPwsWNoxNxpGwbLRhcJYT9EoeSRjApVG1qos4NomG3cfEeiHpsLgASArOX8yQdO0nmceRrcD4bpsr3kbdts2rF5V1CPro50Gx1bTk76fVhMAiSV7L-yXeUsggOAI9CjrLSE1v06qtPSo1X6_htj1F519mjBhSheizK_8lQgf7e4ktQON0_x-9WGUaXkJrrVa1oU4sWOPchizFnSXCgg-ficnnnX10eIWKQ/1264.html","timestamp":"2024-11-09T00:46:39Z","content_type":"text/html","content_length":"5589","record_id":"<urn:uuid:5a48b354-db97-4429-90a8-9848b0e35d41>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00309.warc.gz"} |
propagation of singularities theorem
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Solution. Step 1. The function is well defined at (0,0).
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Solution. Step 1. The function is well defined at . Step 2.
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Question Text Solution. Step 1. The function is well defined at . Step 2.
Updated On Oct 27, 2022
Topic Calculus
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 146
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page 3 — eeedg
Remarkably, what you are experiencing maps onto a typical quantum mechanical system, like our universe. In a modern extension of the Copenhagen interpretation of quantum mechanics*, the
characteristic feature of quantum reality is that different observers can have inconsistent histories about what happens in the world. There is fundamentally no single objective reality on which all
observers will agree.**
Glass is based on the premise that while here, the top group of panes looks vertical and the bottom group looks horizontal, ...
... if we see only a small subset of these clusters, we cannot distinguish them.
So the two states exist in superposition, until we reach an edge, where typically one particular orientation is favoured. This represents a measurement in quantum mechanics: the collapse of the wave
* The Consistent Histories Interpretation, proposed by R.B. Griffiths (1984) and elucidated by the late Pierre Hohenberg in his Reviews of Modern Physics article (2009).
** This statement is shared by all interpretations of quantum mechanics. | {"url":"https://eeedg.ca/page-3","timestamp":"2024-11-06T08:10:34Z","content_type":"text/html","content_length":"147308","record_id":"<urn:uuid:5f6a056e-68eb-4247-a96c-055f42bb488d>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00572.warc.gz"} |
Concept RandomAccessIterator
A random access iterator is an iterator that can read through a sequence of values. It can move in either direction through the sequence (by any amount in constant time), and can be either mutable
(data pointed to by it can be changed) or not mutable.
An iterator represents a position in a sequence. Therefore, the iterator can point into the sequence (returning a value when dereferenced and being incrementable), or be off-the-end (and not
dereferenceable or incrementable).
Associated types
• value_type
The value type of the iterator
• category
The category of the iterator
• difference_type
The difference type of the iterator (measure of the number of steps between two iterators)
A type playing the role of iterator-type in the RandomAccessIterator concept.
i, j
Objects of type Iter
Object of type value_type
Object of type difference_type
Object of type int
Valid expressions
Name Expression Type Semantics
Motion i += n Iter & Equivalent to applying i++ n times if n is positive, applying i-- -n times if n is negative, and to a null operation if n
is zero.
Motion (with integer offset) i += Iter & Equivalent to applying i++ n times if n is positive, applying i-- -n times if n is negative, and to a null operation if n
int_off is zero.
Subtractive motion i -= n Iter & Equivalent to i+=(-n)
Subtractive motion (with integer i -= Iter & Equivalent to i+=(-n)
offset) int_off
Addition i + n Iter Equivalent to {Iter j = i; j += n; return j;}
Addition with integer i + int_off Iter Equivalent to {Iter j = i; j += n; return j;}
Addition (count first) n + i Iter Equivalent to i + n
Addition with integer (count first) int_off + i Iter Equivalent to i + n
Subtraction i - n Iter Equivalent to i + (-n)
Subtraction with integer i - int_off Iter Equivalent to i + (-n)
Distance i - j difference_type The number of times i must be incremented (or decremented if the result is negative) to reach j. Not defined if j is not
reachable from i.
Element access i[n] const-if-not-mutable Equivalent to *(i + n)
value_type &
Element access with integer index i[int_off] const-if-not-mutable Equivalent to *(i + n)
value_type &
• T *
• std::vector<T>::iterator
• std::vector<T>::const_iterator
• std::deque<T>::iterator
• std::deque<T>::const_iterator | {"url":"https://live.boost.org/doc/libs/1_78_0/doc/html/RandomAccessIterator.html","timestamp":"2024-11-14T21:47:18Z","content_type":"text/html","content_length":"14708","record_id":"<urn:uuid:96c65df8-b2ad-4451-8b1d-7d7d6038e7f5>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00568.warc.gz"} |
Last change on this file since f1a10a7 was 41634098, checked in by , 7 years ago
Added white paper on user-defined conversions based on resolver design doc
• Property mode set to 100644
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1 ## User-defined Conversions ##
2 C's implicit "usual arithmetic conversions" define a structure among the
3 built-in types consisting of _unsafe_ narrowing conversions and a hierarchy of
4 _safe_ widening conversions.
5 There is also a set of _explicit_ conversions that are only allowed through a
6 cast expression.
7 Based on Glen's notes on conversions [1], I propose that safe and unsafe
8 conversions be expressed as constructor variants, though I make explicit
9 (cast) conversions a constructor variant as well rather than a dedicated
10 operator.
11 Throughout this article, I will use the following operator names for
12 constructors and conversion functions from `From` to `To`:
14 void ?{} ( To*, To ); // copy constructor
15 void ?{} ( To*, From ); // explicit constructor
16 void ?{explicit} ( To*, From ); // explicit cast conversion
17 void ?{safe} ( To*, From ); // implicit safe conversion
18 void ?{unsafe} ( To*, From ); // implicit unsafe conversion
20 [1] http://plg.uwaterloo.ca/~cforall/Conversions/index.html
22 Glen's design made no distinction between constructors and unsafe implicit
23 conversions; this is elegant, but interacts poorly with tuples.
24 Essentially, without making this distinction, a constructor like the following
25 would add an interpretation of any two `int`s as a `Coord`, needlessly
26 multiplying the space of possible interpretations of all functions:
28 void ?{}( Coord *this, int x, int y );
30 That said, it would certainly be possible to make a multiple-argument implicit
31 conversion, as below, though the argument above suggests this option should be
32 used infrequently:
34 void ?{unsafe}( Coord *this, int x, int y );
36 An alternate possibility would be to only count two-arg constructors
37 `void ?{} ( To*, From )` as unsafe conversions; under this semantics, safe and
38 explicit conversions should also have a compiler-enforced restriction to
39 ensure that they are two-arg functions (this restriction may be valuable
40 regardless).
42 Regardless of syntax, there should be a type assertion that expresses `From`
43 is convertable to `To`.
44 If user-defined conversions are not added to the language,
45 `void ?{} ( To*, From )` may be a suitable representation, relying on
46 conversions on the argument types to account for transitivity.
47 On the other hand, `To*` should perhaps match its target type exactly, so
48 another assertion syntax specific to conversions may be required, e.g.
49 `From -> To`.
51 ### Constructor Idiom ###
52 Basing our notion of conversions off otherwise normal Cforall functions means
53 that we can use the full range of Cforall features for conversions, including
54 polymorphism.
55 Glen [1] defines a _constructor idiom_ that can be used to create chains of
56 safe conversions without duplicating code; given a type `Safe` which members
57 of another type `From` can be directly converted to, the constructor idiom
58 allows us to write a conversion for any type `To` which `Safe` converts to:
60 forall(otype To | { void ?{safe}( To*, Safe ) })
61 void ?{safe}( To *this, From that ) {
62 Safe tmp = /* some expression involving that */;
63 *this = tmp; // uses assertion parameter
64 }
66 This idiom can also be used with only minor variations for a parallel set of
67 unsafe conversions.
69 What selective non-use of the constructor idiom gives us is the ability to
70 define a conversion that may only be the *last* conversion in a chain of such.
71 Constructing a conversion graph able to unambiguously represent the full
72 hierarchy of implicit conversions in C is provably impossible using only
73 single-step conversions with no additional information (see Appendix A), but
74 this mechanism is sufficiently powerful (see [1], though the design there has
75 some minor bugs; the general idea is to use the constructor idiom to define
76 two chains of conversions, one among the signed integral types, another among
77 the unsigned, and to use monomorphic conversions to allow conversions between
78 signed and unsigned integer types).
80 ### Appendix A: Partial and Total Orders ###
81 The `<=` relation on integers is a commonly known _total order_, and
82 intuitions based on how it works generally apply well to other total orders.
83 Formally, a total order is some binary relation `<=` over a set `S` such that
84 for any two members `a` and `b` of `S`, `a <= b` or `b <= a` (if both, `a` and
85 `b` must be equal, the _antisymmetry_ property); total orders also have a
86 _transitivity_ property, that if `a <= b` and `b <= c`, then `a <= c`.
87 If `a` and `b` are distinct elements and `a <= b`, we may write `a < b`.
89 A _partial order_ is a generalization of this concept where the `<=` relation
90 is not required to be defined over all pairs of elements in `S` (though there
91 is a _reflexivity_ requirement that for all `a` in `S`, `a <= a`); in other
92 words, it is possible for two elements `a` and `b` of `S` to be
93 _incomparable_, unable to be ordered with respect to one another (any `a` and
94 `b` for which either `a <= b` or `b <= a` are called _comparable_).
95 Antisymmetry and transitivity are also required for a partial order, so all
96 total orders are also partial orders by definition.
97 One fairly natural partial order is the "subset of" relation over sets from
98 the same universe; `{ }` is a subset of both `{ 1 }` and `{ 2 }`, which are
99 both subsets of `{ 1, 2 }`, but neither `{ 1 }` nor `{ 2 }` is a subset of the
100 other - they are incomparable under this relation.
102 We can compose two (or more) partial orders to produce a new partial order on
103 tuples drawn from both (or all the) sets.
104 For example, given `a` and `c` from set `S` and `b` and `d` from set `R`,
105 where both `S` and `R` both have partial orders defined on them, we can define
106 a ordering relation between `(a, b)` and `(c, d)`.
107 One common order is the _lexicographical order_, where `(a, b) <= (c, d)` iff
108 `a < c` or both `a = c` and `b <= d`; this can be thought of as ordering by
109 the first set and "breaking ties" by the second set.
110 Another common order is the _product order_, which can be roughly thought of
111 as "all the components are ordered the same way"; formally `(a, b) <= (c, d)`
112 iff `a <= c` and `b <= d`.
113 One difference between the lexicographical order and the product order is that
114 in the lexicographical order if both `a` and `c` and `b` and `d` are
115 comparable then `(a, b)` and `(c, d)` will be comparable, while in the product
116 order you can have `a <= c` and `d <= b` (both comparable) which will make
117 `(a, b)` and `(c, d)` incomparable.
118 The product order, on the other hand, has the benefit of not prioritizing one
119 order over the other.
121 Any partial order has a natural representation as a directed acyclic graph
122 (DAG).
123 Each element `a` of the set becomes a node of the DAG, with an arc pointing to
124 its _covering_ elements, any element `b` such that `a < b` but where there is
125 no `c` such that `a < c` and `c < b`.
126 Intuitively, the covering elements are the "next ones larger", where you can't
127 fit another element between the two.
128 Under this construction, `a < b` is equivalent to "there is a path from `a` to
129 `b` in the DAG", and the lack of cycles in the directed graph is ensured by
130 the antisymmetry property of the partial order.
132 Partial orders can be generalized to _preorders_ by removing the antisymmetry
133 property.
134 In a preorder the relation is generally called `<~`, and it is possible for
135 two distict elements `a` and `b` to have `a <~ b` and `b <~ a` - in this case
136 we write `a ~ b`; `a <~ b` and not `a ~ b` is written `a < b`.
137 Preorders may also be represented as directed graphs, but in this case the
138 graph may contain cycles.
140 ### Appendix B: Building a Conversion Graph from Un-annotated Single Steps ###
141 The short answer is that it's impossible.
143 The longer answer is that it has to do with what's essentially a diamond
144 inheritance problem.
145 In C, `int` converts to `unsigned int` and also `long` "safely"; both convert
146 to `unsigned long` safely, and it's possible to chain the conversions to
147 convert `int` to `unsigned long`.
148 There are two constraints here; one is that the `int` to `unsigned long`
149 conversion needs to cost more than the other two (because the types aren't as
150 "close" in a very intuitive fashion), and the other is that the system needs a
151 way to choose which path to take to get to the destination type.
152 Now, a fairly natural solution for this would be to just say "C knows how to
153 convert from `int` to `unsigned long`, so we just put in a direct conversion
154 and make the compiler smart enough to figure out the costs" - this is the
155 approach taken by the existing compipler, but given that in a user-defined
156 conversion proposal the users can build an arbitrary graph of conversions,
157 this case still needs to be handled.
159 We can define a preorder over the types by saying that `a <~ b` if there
160 exists a chain of conversions from `a` to `b` (see Appendix A for description
161 of preorders and related constructs).
162 This preorder corresponds roughly to a more usual type-theoretic concept of
163 subtyping ("if I can convert `a` to `b`, `a` is a more specific type than
164 `b`"); however, since this graph is arbitrary, it may contain cycles, so if
165 there is also a path to convert `b` to `a` they are in some sense equivalently
166 specific.
168 Now, to compare the cost of two conversion chains `(s, x1, x2, ... xn)` and
169 `(s, y1, y2, ... ym)`, we have both the length of the chains (`n` versus `m`)
170 and this conversion preorder over the destination types `xn` and `ym`.
171 We could define a preorder by taking chain length and breaking ties by the
172 conversion preorder, but this would lead to unexpected behaviour when closing
173 diamonds with an arm length of longer than 1.
174 Consider a set of types `A`, `B1`, `B2`, `C` with the arcs `A->B1`, `B1->B2`,
175 `B2->C`, and `A->C`.
176 If we are comparing conversions from `A` to both `B2` and `C`, we expect the
177 conversion to `B2` to be chosen because it's the more specific type under the
178 conversion preorder, but since its chain length is longer than the conversion
179 to `C`, it loses and `C` is chosen.
180 However, taking the conversion preorder and breaking ties or ambiguities by
181 chain length also doesn't work, because of cases like the following example
182 where the transitivity property is broken and we can't find a global maximum:
184 `X->Y1->Y2`, `X->Z1->Z2->Z3->W`, `X->W`
186 In this set of arcs, if we're comparing conversions from `X` to each of `Y2`,
187 `Z3` and `W`, converting to `Y2` is cheaper than converting to `Z3`, because
188 there are no conversions between `Y2` and `Z3`, and `Y2` has the shorter chain
189 length.
190 Also, comparing conversions from `X` to `Z3` and to `W`, we find that the
191 conversion to `Z3` is cheaper, because `Z3 < W` by the conversion preorder,
192 and so is considered to be the nearer type.
193 By transitivity, then, the conversion from `X` to `Y2` should be cheaper than
194 the conversion from `X` to `W`, but in this case the `X` and `W` are
195 incomparable by the conversion preorder, so the tie is broken by the shorter
196 path from `X` to `W` in favour of `W`, contradicting the transitivity property
197 for this proposed order.
199 Without transitivity, we would need to compare all pairs of conversions, which
200 would be expensive, and possibly not yield a minimal-cost conversion even if
201 all pairs were comparable.
202 In short, this ordering is infeasible, and by extension I believe any ordering
203 composed solely of single-step conversions between types with no further
204 user-supplied information will be insufficiently powerful to express the
205 built-in conversions between C's types.
for help on using the repository browser. | {"url":"https://cforall.uwaterloo.ca/trac/browser/doc/proposals/user_conversions.md?rev=f1a10a75c241c188c233941aa9f3878408ea2943","timestamp":"2024-11-04T15:39:47Z","content_type":"application/xhtml+xml","content_length":"44483","record_id":"<urn:uuid:0bd5b454-aa86-4b26-9943-82c85d3b7cb0>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00702.warc.gz"} |
FLN 9 Answer Key Nishtha 3.0 Module : Foundational Numeracy
Foundational Numeracy : Nishtha FLN 3.0 Module 9 Answer Key
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FLN 9 Answer Key
This course is designed to help teachers and workers in early childhood education and care centres like anganwadis, standalone nursery schools and nursery schools attached to primary schools in
building their understanding of numeracy. Thus, the course contains the content knowledge and pedagogical processes to form a strong foundation of early mathematical and numeracy skills integrated
with literacy among all children up to the age of 8-9 years. In this post you will be able to know AR, UP, UK, MZ, NL, OD, PB, AP, AS, BH, GJ, HR, HP, JK, JH, KA, MP, CHD, CG, DL, GA, MH, CBSE, KVS,
NVS, MN, ML, RJ, SK, TS, TR, Nishtha FLN 3.0 Module 9 Foundational Numeracy Quiz question and Answer Key PDF in English for Primary School Teachers of all states. यदि आप यह प्रश्नोत्तरी हिन्दी में पढ़ना चाहते हैं, तो
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Nishtha FLN 3.0 Module 9 Answer Key “Foundational Numeracy”
The assessment questionnaire of Nishtha 2.0 and 3.0 training is the same in all states, but the Training Links are different. Out of around 40 questions, you will get only 20 random questions in
an attempt. You will be able to get the certificate by scoring 70% marks in maximum three attempts.
All the Nishtha trainings available on Diksha App are design to improve teacher performance. So, take all the training seriously and solve the assessment quiz at the end. Certificate will issued only
after securing 70% marks in the evaluation quiz. If you face any kind of problem in solving question of Nishtha FLN 3.0 Module 9 Quiz then get complete solution here.
Nishtha FLN 3.0 Module 9 Answer Key
Q. 1: Which of the following scenarios is not involved in the Word problems related to addition and Subtraction?
• Classification of objects
• Combination of two or more objects
• increase or decrease of same quantity
• Comparison of objects
Q. 2: Which of the following is not a correct way of assessment
• A test based on memorisation
• A subjective test according to the learning levels of children
• Use of self-assessment
• Use of audio-visual tool for assessment
Q. 3: Numbers are used to communicate the size of
a group of objects.
• Ordinal numbers
• Cardinal numbers
• Nominal numbers
• All of the above
Q. 4: Which of the following does not involve the ordering a
collection of objects according to the given rule.
• Seriation
• Arrangement
• Classification
• Patterning
Q. 5: How many times should we add 4 to get 16
• Sixty four times
• Twenty times
• Sixteen times
• Four times
Q. 6: Which of the following is not a type and utility of
• Nominal Numbers
• Ordinal Numbers
• Aesthetic Numbers
• Cardinal Numbers
Q. 7: Essential requirement to classify objects is to:
• Read the names of the shapes
• Identify the objects by their characteristics
• Know the name of the objects
• Recite the name of the objects
Q. 8: For building upon the understanding of one-to-one correspondence, children do not need to understand the meaning of
• many and few
• as many as
• numeration
• more than/ less than
Q. 9: Which of the following is not an objective of making a child proficient in numeracy in the foundational years?
• It helps in achieving learning outcomes in later
• stages
• It helps in developing logical thinking and
• reasoning in daily life
• It helps them in dealing with numbers
• It helps them to do fast calculations
Q. 10: Which of the following is not a component of foundational numeracy:
• Data Handling
• Memorizing number names
• Patterns
• Mathematical Communications
Q. 11: The ability to immediately perceive the cardinality of a collection, usually not more than four or five elements without counting is called as
• Classification
• Conservation
• Seriation
• Subitization
Q. 12: What is the right sequence to teach numbers:
1. Opportunities for Counting
2. Writing numerals
3. Reading numerals
4. Developing number sense
• 1,2,3,4
• 1,4,3,2
• 1,4,2,3
• 2,1,3,4
Q. 13: What is subitising?
• Ability to recite number names up to ten
• Ability to count
• Ability to discriminate between objects
• Ability to identify the number of objects by simply looking at them and without actually counting each object.
Q. 14: Which of the following is not a component of Data Handling?
• Representation of Data
• Interpretation of Data
• Construction of Data
• Collection of Data
Q. 15: What are numerals?
• Value of numbers
• Size of numbers
• Number names
• Symbols for numbers
Q. 16: ______Number are used to describe the position of an object when they are arranged in a specific order.
• Ordinal Number
• Cardinal Number
• Aesthetic Number
• Nominal Number
Q. 17: When does a child is said not to acquire understanding of shapes and space?
• When he/she crams the names of shapes like cube, cuboid, sphere, etc. without understanding
• When he/she explores and communicates association between an object and its shape
• When he/she observes the objects in the environment and their geometrical attributes
• When he/she uses own vocabulary to describe space and the shapes
Q. 18: Which of the following is the most appropriate strategy
to teach shapes at a foundational stage?
• Shapes at foundational stage should be limited to the recognition of simple basic shapes
• Development of extensive vocabulary of shapes need to be the primary objective at foundational stage
• Children should be given ample opportunities to develop intuitive understanding of shapes
• Teacher should introduce by giving clear definition of simple shapes
Q. 19: Which of the following pairs are not complementary to each other?
• Multiplication and Division
• Addition and Multiplication
• Addition and Subtraction
• Subtraction and Multiplication
Q. 20: In order to ensure strong FLN the children should be assessed-
• through question paper which have more questions from the textbooks
• through weekly and monthly tests
• continuously through formative/adaptive methods
• Annually by state/district authority
Q. 21: Which of the following is the most crucial aspect of learning multiplication?
• Understanding multiplication as finding “how many times”
• Recall of tables and their recitation
• Memorization of multiplication facts
• learning the multiplication algorithm and solving sums
Q. 22: Child should be able to seriate objects before learning numbers, because seriation is:
• not related to counting
• related with ordination or placing numbers in order
• needed for operations on numbers
• about reciting number names
Q. 23: Which of the following is not a dimension of
assessments of mathematics learning?
• Communication
• Procedural knowledge
• Disposition towards mathematics
• Mathematical reasoning
Q. 24: What should be the appropriate sequence in
earning/understanding multiplication?
i. Applying distributive law of multiplication w.r.t. addition
ii. Understanding the meaning of multiplication
iii. Learning the algorithm of multiplication
iv. Understanding and using the language of multiplication
• ii, iv, i, iii
• iv, ii, iii, i
• iv, iii, i, ii
• i. ii, iii, iv
Q. 25: ‘Seema has 12 roses. Shifa has 15 roses. Who has more and by how much?’ What subtraction context has been used in the above word problem?
• What left
• Complementary addition
• Take away
• Comparison
Q. 26: Which is of the following is not true:
• All squares are rectangles
• All squares are parallelograms
• All rectangles are parallelograms
• All rectangles are squares
Q. 27: Which of the following pre- school teachers should avoid?
• Include items in the classroom and at home that promote mathematical thinking
• Ask children to write numbers before number sense
• Building on everyday activities of children
• Use language focused on mathematical concepts
Q. 28: The concept of ‘zero’ can be introduced best through which of the following operations?
• Subtraction
• Division
• Multiplication
• Addition
Q. 29: During the learning of Mathematics at early stages, a child is not expected to-
• Use the vocabulary for understanding of space and shapes
• Learn Counting before number sense
• Learn Conventions needed for Mathematical techniques
• Think mathematically and taking decisions with reasoning
Q. 30: Which of the following is not a pedagogical process to
enhance foundational Numeracy skill:
• Using poems, rhymes, stories, riddles in mathematics
• Use of manipulative
• Instruction in home language
• Giving lots of practise questions
Q. 31: Which of the following does not involve cne to one correspondence?
• Matching
• Mapping
• Grouping
• Pairing
Q. 32: The process by which information is exchanged
between individuals through mathematical symbols, signs, diagrams, graphs is known as
• first language learning
• language acquisition
• mathematical Language
• mathematical communication
Q. 33: Which of the following is not a mathematical process?
• Visualization
• Estimation
• Spatial understanding
• Rote Memorization
Q. 34: Which of the following is not a key skill to develop under Number sense
• Recitation of number names
• Applications of basic operations in daily life
• Comparison of numbers like bigger than/smaller than
• Fundamental operations like addition/subtraction
Q. 35: Which of the following is not a pre number skill:
• knowing numerals
• seriation
• classification
• one to one correspondence
Q. 36: Activities on matching or pairing of objects will help in che development of which pre-number skill
• Classification
• Counting on
• Seriation
• One to one correspondence
Q. 37: Putting together things that have some characteristics in common enhances the competence of
• mathematical communication
• number sense
• classification
• seriation
Q. 38: Which of the following should not be an approach for teaching measurement?
• Directly introducing standard units of measurements by the teacher and their conversions
• Let children figure out their own units for measurement
• Provide opportunities to use language of comparison
• Children in activities and other experiences that involve measurement
Q. 39: During the process of counting, a child
• classifies into two groups
• recites number names in order
• writes number names
• points object one at a time
Q. 40: Which of the following activities is best suited for the
development of spatial understanding among children?
• Drawing numbers on a number line
• Noting the time of sunset
• Drawing the front view of a bottle
• Memorizing definitions for each basic shape
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Selina ICSE Class 10 Maths Solutions Chapter 19 Constructions Circles
Question 1. Draw a circle of radius 3 cm. Mark a point P at a distance of 5 cm from the centre of the circle drawn. Draw two tangents PA and PB to the given circle and measure the length of each
Construction Procedures:
1. Draw a circle with a radius of 3 cm and a center of O.
2. Take a point P from O such that OP = 5 cm.
3. Draw a bisector of OP that passes through M.
4. Draw a circle with a center M and a radius OM that intersects the provided circle at A and B.
5. Assemble AP and BP.
As a result, the tangents AP and BP are necessary. AP = BP = 4 cm as measured
Question 2. Draw a circle of diameter of 9 cm. Mark a point at a distance of 7.5 cm from the centre of the circle. Draw tangents to the given circle from this exterior point. Measure the length of
each tangent.
Construction Procedures:
1. Make a 9cm radius circle with the letter O in the middle.
2. Outside the circle, draw a line with a point P of 7.5 cm.
3. Draw a circle that intersects the preceding circle at A and B with OP as the diameter.
4. Join PA and PB as a member.
Tangents PA and PB are required as a result. PB = 6 cm.
Question 3. Draw a circle of radius 5 cm. Draw two tangents to this circle so that the angle between the tangents is 45˚.
Construction Procedures
1. Draw a circle with a diameter of 5 cm and a diameter of O.
2. At O, draw arcs with an angle of 180˚-45˚=135˚ so that AOB=135˚
3. Draw two rays at each point A and B that meet at point P, outside the circle, at a 90˚ angle.
4. At P, the requisite tangents AP and BP establish a 45˚ angle with each other.
Question 4. Draw a circle of radius 4.5 cm. Draw two tangents to this circle so that the angle between the tangents is 60˚.
Construction Procedures:
1. Draw a circle with a radius of 4.5 cm and a radius of O.
2. Draw arcs at O with an angle of 180˚–60˚=120˚, so that AOB=120˚.
3. Draw two rays at A and B that intersect at point P, outside the circle, at a 90˚ angle at each point.
4. At P, the requisite tangents AP and BP form a 60˚ angle with each other.
Question 5. Using ruler and compasses only, draw an equilateral triangle of side 4.5 cm and draw its circumscribed circle. Measure the radius of the circle.
Construction Procedures:
1. Draw a BC = 4.5 cm line segment.
2. Draw two arcs of radius 4.5 cm with centers B and C that meet at A.
3. Assemble AC and AB.
4. Make perpendicular bisectors of AC and BC that cross at O.
5. Draw a circle with a center O and a radius OA, OB, or OC that passes through A, B, and C.
This is the triangle ABC’s needed circumcircle. When the radius is measured, it is found to be OA = 2.6 cm.
Question 6. Using ruler and compasses only.
(i) Construct triangle ABC, having given BC = 7 cm, AB – AC = 1 cm and ∠ABC = 45°.
(ii) Inscribe a circle in the ∆ABC constructed in (i) above. Measure its radius.
(i) Construction of triangle:
1. Draw a BC = 7 cm line segment.
2. Cut off BE = AB – AC = 1 cm at B by drawing a ray BX at a 45o angle.
3. Draw the perpendicular bisector of EC intersecting BX at A by joining EC.
4. Become an AC member.
The needed triangle is ABC.
(ii) Construction of in circle:
1. Draw angle bisectors for ABC and ACB that overlap at O.
2. Draw perpendiculars from O to BC.
3. Using O as the center and OL as the radius, draw a circle that touches the ABC’s sides. This is the needed ABC in-circle.
4. When measuring, the OL radius was found to be 1.8 cm.
Question 7. Using ruler and compasses only, draw an equilateral triangle of side 5 cm. Draw its inscribed circle. Measure the radius of the circle.
Construction Procedures:
1. Draw a BC = 5 cm line segment.
2. Draw two arcs of 5 cm radius each, intersecting at A, with centers B and C.
3. Assemble AB and AC.
4. Create angle bisectors for B and C that intersect at O.
5. Draw OL ⊥ BC from O.
6. Now draw a circle with a center O and a radius OL that touches the edges of ABC. OL = 1.4 cm as measured.
Question 8. Using ruler and compasses only,
(i) Construct a triangle ABC with the following data: Base AB = 6 cm, BC = 6.2 cm and ∠CAB – 60°
(ii) In the same diagram, draw a circle which passes through the points A, B and C and mark its centre as O.
(iii) Draw a perpendicular from O to AB which meets AB in D.
(iv) Prove that AD = BD
Construction Procedures:
1. Draw a line segment with the length AB = 6 cm.
2. Draw a ray at A that forms a 60o angle with BC.
3. Draw an arc with B as the center and a radius of 6.2 cm that crosses the AX ray at C.
4. Become a BC member.
The needed triangle is ABC.
1. Draw the crossing perpendicular bisectors of AB and AC at O.
2. Draw a circle that passes through A, B, and C with the center O and the radius OA, OB, or OC.3. From O, draw OD ⊥ AB.
Proof: In right ∆OAD and ∆OBD
OA = OB (radii of same circle)
OD = OD (common)
∆OAD ≅ ∆OBD (RHS)
AD = BD (CPCT)
Question 9. Using ruler and compasses only construct a triangle ABC in which BC = 4 cm, ∠ACB = 45° and perpendicular from A on BC is 2.5 cm. Draw a circle circumscribing the triangle ABC.
Construction Procedures:
1. Draw a BC = 4 cm line segment.
2. Draw a perpendicular line CX at C and cut off CE = 2.5 cm from it.
3. Draw another perpendicular line EY from E.
4. Draw a ray from C that makes a 45˚ angle with CB and intersects EY at A.
5. Become an AB member.
The needed triangle is ABC.
1. Make perpendicular bisectors of the sides AB and BC that intersect at O.
2. With centre O, and radius OB, draw a circle which will pass through A, B and C. Measuring the radius OB = OC = OA = 2 cm
Question 10. Perpendicular bisectors of the sides AB and AC of a triangle ABC meet at O.
(i) What do you call the point O?
(ii) What is the relation between the distances OA, OB and OC?
(iii) Does the perpendicular bisector of BC pass through O?
Construction Procedures:
1. O is the circumcentre of the ABC circumcircle.
2. The circumcircle’s radii are OA, OB, and OC.
3. Yes, BC’s perpendicular bisector will traverse O.
Question 11. The bisectors of angles A and B of a scalene triangle ABC meet at O.
(i) What is the point O called?
(ii) OR and OQ are drawn perpendiculars to AB and CA respectively. What is the relation between OR and OQ?
(iii) What is the relation between angle ACO and angle BCO?
Construction Procedures:
1. The letter O is referred to as the center of the ABC circle.
2. The radii of the in circle are OR and OQ, and OR = OQ.
3. The bisector of angle C is OC.
∠ACO = ∠BCO.
Question 12. (i) Using ruler and compasses only, construct a triangle ABC in which AB = 8 cm, BC = 6 cm and CA = 5 cm.
(ii) Find its in centre and mark it I.
(iii) With I as centre, draw a circle which will cut off 2 cm chords from each side of the triangle.
Construction Procedures:
1. Draw a BC = 6 cm line segment.
2. Draw an arc with a radius of 8 cm and a center B.
3. Draw another arc with a radius of 5 cm that meets the first arc at A.
4. Assemble AB and AC.
The needed triangle is ABC.
1. Draw B and A’s angle bisectors intersecting at I. Then there’s me, who’s in the middle of the ABC triangle.
2. Draw ID AB through I.
3. Now from D, cut off DP=DQ=2/2=1cm
4. With centre I, and radius IP or IQ, draw a circle which will intersect each side of triangle ABC cutting chords of 2 cm each.
Question 13. Construct an equilateral triangle ABC with side 6 cm. Draw a circle circumscribing the triangle ABC.
Construction Procedures:
1. Draw a BC = 6 cm line segment.
2. Draw two circles of radius 6 cm that meet at A, with centers B and C.
3. Assemble AC and AB.
4. Make perpendicular bisectors of AC, AB, and BC that intersect at O.
5. Draw a circle with a center O and a radius OA, OB, or OC that passes through A, B, and C.
This is the triangle ABC’s needed circumcircle.
Question 14. Construct a circle, inscribing an equilateral triangle with side 5.6 cm.
Construction Procedures:
1. Draw a BC = 5.6 cm line segment.
2. Draw two circles of 5.6 cm radius each, intersecting at A, with centers B and C.
3. Assemble AB and AC.
4. Create angle bisectors for B and C that intersect at O.
5. Draw OL BC from O.
6. Now, using the center O and the radius OL, draw a circle that touches the edges of ABC. This is the needed circle.
Question 15. Draw a circle circumscribing a regular hexagon of side 5 cm.
Construction Procedures:
1. Draw a regular hexagon ABCDEF with 5 cm on each side and a 120o interior angle.
2. Join the diagonals AD, BE, and CF that cross at O.
3. Draw a circle with the center at O and the radius at OA that passes through the vertices A, B, C, D, E, and F.
4. The needed circumcircle is this.
Question 16. Draw an inscribing circle of a regular hexagon of side 5.8 cm.
Construction Procedures:
1. Draw a line segment with the length AB = 5.8 cm.
2. Draw rays at an angle of 120° at A and B, and cut off AF = BC = 5.8 cm.
3. Draw rays at an angle of 120° each for F and C, and cut off FE = CD = 5.8 cm.
4. Sign up for DE. The normal hexagon is ABCDEF.
5. Draw the intersection of the bisectors of A and B at O.
6. Draw OL AB from O.
7. Draw a circle with a center O and a radius OL that meets the hexagon’s sides.
This is necessary in the hexagonal circle.
Question 17. Construct a regular hexagon of side 4 cm. Construct a circle circumscribing the hexagon.
Construction Procedures:
1. Draw a circle with a radius of 4 cm and a center of O.
2. Draw radii OA and OB so that AOB = 60°, because the internal angle of a regular hexagon is 60˚.
3. On the provided circle, cut out arcs BC, CD, EF, and each equal to arc AB.
4. Join the letters AB, BC, CD, DE, EF, FA to form the needed regular hexagon ABCDEF in a circle.
The hexagon is circumscribed by the circle, which is the requisite circumcircle.
Question 18. Draw a circle of radius 3.5 cm. Mark a point P outside the circle at a distance of 6 cm from the centre. Construct two tangents from P to the given circle. Measure and write down the
length of one tangent.
Construction Procedures:
1. Draw a line segment with a length of OP = 6 cm.
2. Draw a circle with a radius of 3.5 cm and a center of O.
3. Draw the OP’s middle.
4. Draw a circle with a center M and a diameter OP that intersects the circle at T and S.
5. Become a member of both PT and PS.
The needed tangents are PT and PS. The length of PT = PS = 4.8 cm was measured.
Question 19. Construct a triangle ABC in which base BC = 5.5 cm, AB = 6 cm and m ∠ABC =120˚.
i.) Construct a circle circumscribing the triangle ABC.
ii.) Draw a cyclic quadrilateral ABCD so that D is equidistant from B and C.
Construction Procedures:
1.) Draw a BC = 5.4 cm line.
2.) Make AB = 6 cm such that mABC = 120 degrees.
3.) Construct AB and BC’s perpendicular bisectors such that they overlap at O.
4.) Make a circle with the radius of O.
1.) Extend BC’s perpendicular bisector until it crosses the circle at point D.
2.) Combine the BD and CD.
3.) In this case, BD = DC.
Question 20. Using a ruler and compasses only:
(i) Construct a triangle ABC with the following data: AB = 3.5 cm, BC = 6 cm and ∠ABC = 120°.
(ii) In the same diagram, draw a circle with BC as diameter. Find a point P on the circumference of the circle which is equidistant from AB and BC.
(iii) Measure ∠BCP.
Steps of constructions:
1. Draw a BC = 6 cm line segment.
2. At B, draw a ray BX that forms a 120° angle with BC.
3. Cut-off AB = 3.5 cm with B as the center and a radius of 3.5 cm. Come and join AC.
As a result, the needed triangle is ABC.
4. Draw a perpendicular bisector MN of BC via point O. Draw a circle with O as the center and OB as the radius. Draw an ABC angle bisector that intersects the circle at point P. As a result, point P
is equidistant from points AB and BC.
5. BCP = 30° as measured
Question 21. Construct a ∆ABC with BC = 6.5 cm, AB = 5.5 cm, AC = 5 cm. Construct the in circle of the triangle. Measure and record the radius of the in circle.
Construction Procedures:
1. Make a BC of 6.5 cm.
2. Draw an arc with a radius of 5.5 cm with B as the center.
3. Draw a 5 cm radius arc with C at the center. Allow this arc to intersect with the preceding arc at A.
4. Combine the letters AB and AC to get ABC.
5. Draw the ABC and ACB bi sectors. Allow these bisectors to collide at O.
6. Draw ON ⊥ BC.
7. Draw an in circle that touches all of the edges of ABC with O as the center and radius ON.
8. The radius ON is 1.5 cm when measured.
Question 22. Construct a triangle ABC with AB = 5.5 cm, AC = 6 cm and ∠BAC = 105°. Hence :
(i) Construct the locus of points equidistant from BA and BC.
(ii) Construct the locus of points equidistant from B and C.
(iii) Mark the point which satisfies the above two loci as P. Measure and write the length of PC.
Construction Procedures:
1. Draw AB = 5.5 cm on a piece of paper.
2. Create BAR = 105˚.
3. With a radius of 6 cm and a center of A, aut off arc on AR at C.
4. Become a BC member. The needed triangle is ABC.
5. Draw the BD of ABC’s angle bisector, which is the loss of points equidistant between BA and BC.
6. Draw the perpendicular bisector EF of BC, which is the locus of equidistant points between B and C.
7. At point P, BD and EF cross each other. As a result, P meets the first two lod.
PC = 4.8 cm according to measurement
Question 23. Construct a regular hexagon of side 5 cm. Hence construct all its lines of symmetry and name them.
Construction Procedures:
1. Using a ruler, draw an AF measuring 5 cm.
2. Draw an arc above AF with A as the center and a radius equal to AF.
3. Cut the previous arc at Z with F as the center and the same radius as before.
4. Draw a circle running between A and F with Z as the center and the same radius.
5. Draw an arc to cut the circle above AF at B with A as the center and the same radius.
6. Draw an arc to cut the circle at C using B as the center and the same radius.
7. Repeat step 7 to retrieve the remaining hexagon vertices at D and E.
8. To make the hexagon, connect successive arcs on the circle.
9. Draw the AF, FE, and DE perpendicular bisectors.
10. Extend AF, FE, and DE bisectors to meet CD, BC, and AB at X, L, and O, respectively.
11. Become a member of AD, CF, and EB.
These are the regular hexagon’s six symmetry lines.
Question 24. Draw a line AB = 5 cm. Mark a point C on AB such that AC = 3 cm. Using a ruler and a compass only, construct:
(i) A circle of radius 2.5 cm, passing through A and C.
(ii) Construct two tangents to the circle from the external point B. Measure and record the length of the tangents.
Steps for construction:
1. Using a ruler, draw AB = 5 cm.
2. Cut a 3 cm arc on AB with A as the center to get C.
3. Draw an arc above AB with A as the center and a radius of 2.5 cm.
4. Draw an arc to cut the preceding arc with the same radius and C as the center, and label the junction as O.
5. Draw a circle with O as the center and a radius of 2.5 cm, with points A and C on the circle.
6. Become an OB member.
7. To get the mid-point of OB, M, draw the perpendicular bisector of OB.
8. Draw a circle with the center M and a radius equal to OM to cut the preceding circle at points P and Q.
9. Assemble using PB and QB. The needed tangents to the provided circle from exterior point B are PB and QB = PB = 3 cm.
That is, each tangent is 3 cm long.
Question 25. Using a ruler and a compass construct a triangle ABC in which AB = 7 cm, ∠CAB = 60o and
AC = 5 cm. Construct the locus of
1) points equidistant from AB and AC
2) points equidistant from BA and BC
Hence construct a circle touching the three sides of the triangle internally.
Construction Procedures:
1. Draw a line with the length AB = 7 cm.
2. Draw an arc of a circle that crosses AB at M, using P as the center and the same radius.
3. Draw an arc crossing the previously drawn arc at point N, using M as the center and the same radius as before.
4. Draw the ray AX through N, therefore XAB = 60 degrees.
5. Draw an arc cutting AX at C, with A as the center and a radius of 5 cm.
6. Become a BC member.
7. The triangle ABC that is required is acquired.
8. Create an angle bisector between CAB and ABC.
9. Make an O in the middle of their junction.
10. Draw a circle of radius OD with O at the center.
Question 26. Construct a triangle ABC in which AB = 5 cm, BC = 6.8 cm and median AD = 4.4 cm. Draw in circle of this triangle.
Steps for construction:
1. Make a BC of 6.8 cm.
2. Draw a line from point D to the mid-point of BC, where BD = DC = 3.4 cm.
3. Draw a line from point D to point B to point A, which is the intersection of arcs AD = 4.4 cm and AB = 5 cm.
4. Assemble AB, AD, and AC. The needed triangle is ABC.
5. Draw the bisectors of angles B and C, which are rays BX and CY, where I is the circle’s center.
6. Make an ABC circle in the shape of a triangle.
Question 27. Draw two concentric circles with radii 4 cm and 6 cm. Taking a point on the outer circle, construct a pair of tangents to inner circle. By measuring the lengths of both the tangents,
show that they are equal to each other.
Steps for construction:
1. Draw concentric circles of radius 4 cm and 6 cm with centre of O.
2. Take point P on the outer circle.
3. Join OP.
4. Draw perpendicular bisectors of OP where M is the midpoint of OP.
5. Take a distance of a point O from the point M and mark arcs from M on the inner circle it cuts at point A and B respectively.
6. Join PA and PB.
We observe that PA and PB are tangents from outer circle to inner circle are equal of a length 4.5 cm each.
Question 28. In triangle ABC, ∠ABC = 90°, AB = 6 cm, BC = 7.2 cm and BD is perpendicular to side AC. Draw circumcircle of triangle BDC and then state the length of the radius of this circumcircle
Steps for construction:
1. Make a BC of 7.2 cm.
2. Using a compass, draw an angle ABC = 90°.
3. Using a compass, draw BD perpendicular to AC.
4. Become a BD member.
5. Connect the perpendicular bisectors of AB and BC at I, where I is the circle’s circumcentre.
6. Using circumcentre I, draw a circumcircle with a radius of 4.7 cm. | {"url":"https://www.icseboards.com/selina-icse-class-10-maths-solutions-chapter-19-constructions-circles/","timestamp":"2024-11-11T16:24:23Z","content_type":"text/html","content_length":"125858","record_id":"<urn:uuid:d0582e6f-002b-4cbf-bf09-b5879384f371>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00800.warc.gz"} |
Find a Math tutor in Langley - Teachers' Tutoring Service
Find a Math tutor in Langley
Teachers’ Tutoring Services offers Math tutors in Langley. Our Math tutors offer classes that help your child achieve their academic goals
We help you find the right Math tutor
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Looking for Math tutoring services? We are passionate about finding the best Math tutor. To request a tutor our online forms are a fast and easy way to get started. We also take phone calls!
TTS provides personalized help when you need it. We are happy to discuss your specific needs and provide guidance to make sure you find the right tutor for you. Just give us a call at (604) 730-3410.
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Most of our Math tutors have experience teaching in Langley and are familiar with the curriculum. If not, they have a relevant graduate degree and subject matter expertise. Additionally, all of our
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Our tutors love to teach, and our students benefit from working with them. Our Math tutors provide one-to-one tutoring to help their students understand concepts intuitively and problem-solve at
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Turn struggles into success. Our tutors aim to develop in students skills and habits that translate into success for the years to come.
Private Math tutoring in Langley
Math tutoring in Langley
Find the support you need when looking for private Math classes. We are a non-profit society established over 40 years ago by teachers, for teachers. The tutors we refer to you are experienced
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With TTS, you are not required to commit to any number of lessons, nor do we require you to sign a contract. It is a pay as you go system. You pay for each lesson when it takes place.
Our tutors are available to help with both short term needs (as short as one session) as well ongoing tutoring, for those that are looking for help for a semester or a school year.
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TTS has an extensive list of tutors across the Langley area.
Below are some of the neighborhoods in Langley in which we offer Math in-person tutoring.
Math Tutors in Langley Township
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Cost of tutoring broken down
TTS is a non profit society – as such, our goal is to provide you with affordable, high quality tutoring, while minimizing cost. The bulk of the $55/hr hourly rate is paid to the tutor, while a small
portion covers off TTS administration and staffing charges.
Additionally, a portion of the tutoring rate is donated to TAS Tutoring Aid Society. TAS is our registered BC charity that provides subsidized tutoring services to students that wouldn’t otherwise be
able to afford tutoring. | {"url":"https://tutor.bc.ca/math-tutor-in-langley/","timestamp":"2024-11-05T23:42:12Z","content_type":"text/html","content_length":"230143","record_id":"<urn:uuid:2b342959-c4c0-48c3-ac64-0940f025686d>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00629.warc.gz"} |
We characterize strongly proximinal subspaces of finite codimension in C(K) spaces. We give two applications of our results. First, we show that the metric projection on a strongly proximinal
subspace of finite codimension in C(K) is Hausdorff metric continuous. Second, strong proximinality is a transitive relation for finite-codimensional subspaces of C(K). | {"url":"https://eudml.org/search/page?q=sc.general*op.AND*l_0*c_0author_0eq%253A1.Darapaneni+Narayana&qt=SEARCH","timestamp":"2024-11-13T21:36:53Z","content_type":"application/xhtml+xml","content_length":"53684","record_id":"<urn:uuid:e4739b0b-7c89-450a-bc7a-e751967f98eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00710.warc.gz"} |
Math Problem Statement
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Mathematical Concepts
Series Expansion
Trigonometric Functions
Series expansion for logarithmic and exponential functions
Series expansion for trigonometric functions
Limit properties
Series expansion theorem
Suitable Grade Level
Advanced High School | {"url":"https://math.bot/q/solving-limit-problem-series-expansion-trigonometric-functions-DX8e","timestamp":"2024-11-06T23:25:38Z","content_type":"text/html","content_length":"87244","record_id":"<urn:uuid:2c08e47e-d08d-461b-84bd-d2019efaec46>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00759.warc.gz"} |
20-ohm resistor is connected to a 10 V battery. The battery is then replaced by a battery that provides a larger voltage what happens to the - DocumenTV20-ohm resistor is connected to a 10 V battery. The battery is then replaced by a battery that provides a larger voltage what happens to the
20-ohm resistor is connected to a 10 V battery. The battery is then replaced by a battery that provides a larger voltage what happens to the
20-ohm resistor is connected to a 10 V battery. The battery is then replaced by a battery that provides a larger voltage what happens to the current through the resistor
in progress 0
Physics 3 years 2021-07-25T15:13:44+00:00 2021-07-25T15:13:44+00:00 1 Answers 14 views 0 | {"url":"https://documen.tv/question/20-ohm-resistor-is-connected-to-a-10-v-battery-the-battery-is-then-replaced-by-a-battery-that-pr-17467873-8/","timestamp":"2024-11-03T23:16:48Z","content_type":"text/html","content_length":"80003","record_id":"<urn:uuid:1dbc89fd-24cc-4592-bbaa-af7b1a273a2e>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00759.warc.gz"} |
public class NNLS extends Object
Object used to solve nonnegative least squares problems using a modified projected gradient method.
• Nested Class Summary
Modifier and Type
static class
• Method Summary
Modifier and Type
static double[]
Solve a least squares problem, possibly with nonnegativity constraints, by a modified projected gradient method.
• Method Details
□ solve
public static double[] solve(double[] ata, double[] atb, NNLS.Workspace ws)
Solve a least squares problem, possibly with nonnegativity constraints, by a modified projected gradient method. That is, find x minimising ||Ax - b||_2 given A^T A and A^T b.
We solve the problem
$$ min_x 1/2 x^T ata x^T - x^T atb $$
where x is nonnegative.
The method used is similar to one described by Polyak (B. T. Polyak, The conjugate gradient method in extremal problems, Zh. Vychisl. Mat. Mat. Fiz. 9(4)(1969), pp. 94-112) for bound-
constrained nonlinear programming. Polyak unconditionally uses a conjugate gradient direction, however, while this method only uses a conjugate gradient direction if the last iteration did
not cause a previously-inactive constraint to become active.
ata - (undocumented)
atb - (undocumented)
ws - (undocumented) | {"url":"https://spark.apache.org/docs/3.5.1/api/java/org/apache/spark/mllib/optimization/NNLS.html","timestamp":"2024-11-11T17:48:15Z","content_type":"text/html","content_length":"13704","record_id":"<urn:uuid:399e43e2-e2e4-49e8-b4cb-20ea5840cbc5>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00809.warc.gz"} |
bignum - transparent big number support for Perl
use bignum;
$x = 2 + 4.5; # Math::BigFloat 6.5
print 2 ** 512 * 0.1; # Math::BigFloat 134...09.6
print 2 ** 512; # Math::BigInt 134...096
print inf + 42; # Math::BigInt inf
print NaN * 7; # Math::BigInt NaN
print hex("0x1234567890123490"); # Perl v5.10.0 or later
no bignum;
print 2 ** 256; # a normal Perl scalar now
# for older Perls, import into current package:
use bignum qw/hex oct/;
print hex("0x1234567890123490");
print oct("01234567890123490");
#Literal numeric constants
By default, every literal integer becomes a Math::BigInt object, and literal non-integer becomes a Math::BigFloat object. Whether a numeric literal is considered an integer or non-integers depends
only on the value of the constant, not on how it is represented. For instance, the constants 3.14e2 and 0x1.3ap8 become Math::BigInt objects, because they both represent the integer value decimal
The default use bignum; is equivalent to
use bignum downgrade => "Math::BigInt", upgrade => "Math::BigFloat";
The classes used for integers and non-integers can be set at compile time with the downgrade and upgrade options, for example
# use Math::BigInt for integers and Math::BigRat for non-integers
use bignum upgrade => "Math::BigRat";
Note that disabling downgrading and upgrading does not affect how numeric literals are converted to objects
# disable both downgrading and upgrading
use bignum downgrade => undef, upgrade => undef;
$x = 2.4; # becomes 2.4 as a Math::BigFloat
$y = 2; # becomes 2 as a Math::BigInt
#Upgrading and downgrading
By default, when the result of a computation is an integer, an Inf, or a NaN, the result is downgraded even when all the operands are instances of the upgrade class.
use bignum;
$x = 2.4; # becomes 2.4 as a Math::BigFloat
$y = 1.2; # becomes 1.2 as a Math::BigFloat
$z = $x / $y; # becomes 2 as a Math::BigInt due to downgrading
Equivalently, by default, when the result of a computation is a finite non-integer, the result is upgraded even when all the operands are instances of the downgrade class.
use bignum;
$x = 7; # becomes 7 as a Math::BigInt
$y = 2; # becomes 2 as a Math::BigInt
$z = $x / $y; # becomes 3.5 as a Math::BigFloat due to upgrading
The classes used for downgrading and upgrading can be set at runtime with the "downgrade()" and "upgrade()" methods, but see "CAVEATS" below.
The upgrade and downgrade classes don't have to be Math::BigInt and Math::BigFloat. For example, to use Math::BigRat as the upgrade class, use
use bignum upgrade => "Math::BigRat";
$x = 2; # becomes 2 as a Math::BigInt
$y = 3.6; # becomes 18/5 as a Math::BigRat
The upgrade and downgrade classes can be modified at runtime
use bignum;
$x = 3; # becomes 3 as a Math::BigInt
$y = 2; # becomes 2 as a Math::BigInt
$z = $x / $y; # becomes 1.5 as a Math::BigFlaot
bignum -> upgrade("Math::BigRat");
$w = $x / $y; # becomes 3/2 as a Math::BigRat
Disabling downgrading doesn't change the fact that literal constant integers are converted to the downgrade class, it only prevents downgrading as a result of a computation. E.g.,
use bignum downgrade => undef;
$x = 2; # becomes 2 as a Math::BigInt
$y = 2.4; # becomes 2.4 as a Math::BigFloat
$z = 1.2; # becomes 1.2 as a Math::BigFloat
$w = $x / $y; # becomes 2 as a Math::BigFloat due to no downgrading
If you want all numeric literals, both integers and non-integers, to become Math::BigFloat objects, use the bigfloat pragma.
Equivalently, disabling upgrading doesn't change the fact that literal constant non-integers are converted to the upgrade class, it only prevents upgrading as a result of a computation. E.g.,
use bignum upgrade => undef;
$x = 2.5; # becomes 2.5 as a Math::BigFloat
$y = 7; # becomes 7 as a Math::BigInt
$z = 2; # becomes 2 as a Math::BigInt
$w = $x / $y; # becomes 3 as a Math::BigInt due to no upgrading
If you want all numeric literals, both integers and non-integers, to become Math::BigInt objects, use the bigint pragma.
You can even do
use bignum upgrade => "Math::BigRat", upgrade => undef;
which converts all integer literals to Math::BigInt objects and all non-integer literals to Math::BigRat objects. However, when the result of a computation involving two Math::BigInt objects results
in a non-integer (e.g., 7/2), the result will be truncted to a Math::BigInt rather than being upgraded to a Math::BigRat, since upgrading is disabled.
Since all numeric literals become objects, you can call all the usual methods from Math::BigInt and Math::BigFloat on them. This even works to some extent on expressions:
perl -Mbignum -le '$x = 1234; print $x->bdec()'
perl -Mbignum -le 'print 1234->copy()->binc();'
perl -Mbignum -le 'print 1234->copy()->binc()->badd(6);'
bignum recognizes some options that can be passed while loading it via via use. The following options exist:
#a or accuracy
This sets the accuracy for all math operations. The argument must be greater than or equal to zero. See Math::BigInt's bround() method for details.
perl -Mbignum=a,50 -le 'print sqrt(20)'
Note that setting precision and accuracy at the same time is not possible.
#p or precision
This sets the precision for all math operations. The argument can be any integer. Negative values mean a fixed number of digits after the dot, while a positive value rounds to this digit left
from the dot. 0 means round to integer. See Math::BigInt's bfround() method for details.
perl -Mbignum=p,-50 -le 'print sqrt(20)'
Note that setting precision and accuracy at the same time is not possible.
#l, lib, try, or only
Load a different math lib, see "Math Library".
perl -Mbignum=l,GMP -e 'print 2 ** 512'
perl -Mbignum=lib,GMP -e 'print 2 ** 512'
perl -Mbignum=try,GMP -e 'print 2 ** 512'
perl -Mbignum=only,GMP -e 'print 2 ** 512'
Override the built-in hex() method with a version that can handle big numbers. This overrides it by exporting it to the current package. Under Perl v5.10.0 and higher, this is not so necessary,
as hex() is lexically overridden in the current scope whenever the bignum pragma is active.
Override the built-in oct() method with a version that can handle big numbers. This overrides it by exporting it to the current package. Under Perl v5.10.0 and higher, this is not so necessary,
as oct() is lexically overridden in the current scope whenever the bignum pragma is active.
#v or version
this prints out the name and version of the modules and then exits.
perl -Mbignum=v
#Math Library
Math with the numbers is done (by default) by a backend library module called Math::BigInt::Calc. The default is equivalent to saying:
use bignum lib => 'Calc';
you can change this by using:
use bignum lib => 'GMP';
The following would first try to find Math::BigInt::Foo, then Math::BigInt::Bar, and if this also fails, revert to Math::BigInt::Calc:
use bignum lib => 'Foo,Math::BigInt::Bar';
Using c<lib> warns if none of the specified libraries can be found and Math::BigInt and Math::BigFloat fell back to one of the default libraries. To suppress this warning, use try instead:
use bignum try => 'GMP';
If you want the code to die instead of falling back, use only instead:
use bignum only => 'GMP';
Please see respective module documentation for further details.
#Method calls
Since all numbers are now objects, you can use the methods that are part of the Math::BigInt and Math::BigFloat API.
But a warning is in order. When using the following to make a copy of a number, only a shallow copy will be made.
$x = 9; $y = $x;
$x = $y = 7;
Using the copy or the original with overloaded math is okay, e.g., the following work:
$x = 9; $y = $x;
print $x + 1, " ", $y,"\n"; # prints 10 9
but calling any method that modifies the number directly will result in both the original and the copy being destroyed:
$x = 9; $y = $x;
print $x->badd(1), " ", $y,"\n"; # prints 10 10
$x = 9; $y = $x;
print $x->binc(1), " ", $y,"\n"; # prints 10 10
$x = 9; $y = $x;
print $x->bmul(2), " ", $y,"\n"; # prints 18 18
Using methods that do not modify, but test that the contents works:
$x = 9; $y = $x;
$z = 9 if $x->is_zero(); # works fine
See the documentation about the copy constructor and = in overload, as well as the documentation in Math::BigFloat for further details.
A shortcut to return inf as an object. Useful because Perl does not always handle bareword inf properly.
A shortcut to return NaN as an object. Useful because Perl does not always handle bareword NaN properly.
# perl -Mbignum=e -wle 'print e'
Returns Euler's number e, aka exp(1) (= 2.7182818284...).
# perl -Mbignum=PI -wle 'print PI'
Returns PI (= 3.1415926532..).
bexp($power, $accuracy);
Returns Euler's number e raised to the appropriate power, to the wanted accuracy.
# perl -Mbignum=bexp -wle 'print bexp(1,80)'
Returns PI to the wanted accuracy.
# perl -Mbignum=bpi -wle 'print bpi(80)'
Set or get the accuracy.
Set or get the precision.
Set or get the rounding mode.
Set or get the division scale.
Set or get the class that the downgrade class upgrades to, if any. Set the upgrade class to undef to disable upgrading. See /CAVEATS below.
Set or get the class that the upgrade class downgrades to, if any. Set the downgrade class to undef to disable upgrading. See "CAVEATS" below.
use bignum;
print "in effect\n" if bignum::in_effect; # true
no bignum;
print "in effect\n" if bignum::in_effect; # false
Returns true or false if bignum is in effect in the current scope.
This method only works on Perl v5.9.4 or later.
#The upgrade() and downgrade() methods
Note that setting both the upgrade and downgrade classes at runtime with the "upgrade()" and "downgrade()" methods, might not do what you expect:
# Assuming that downgrading and upgrading hasn't been modified so far, so
# the downgrade and upgrade classes are Math::BigInt and Math::BigFloat,
# respectively, the following sets the upgrade class to Math::BigRat, i.e.,
# makes Math::BigInt upgrade to Math::BigRat:
bignum -> upgrade("Math::BigRat");
# The following sets the downgrade class to Math::BigInt::Lite, i.e., makes
# the new upgrade class Math::BigRat downgrade to Math::BigInt::Lite
bignum -> downgrade("Math::BigInt::Lite");
# Note that at this point, it is still Math::BigInt, not Math::BigInt::Lite,
# that upgrades to Math::BigRat, so to get Math::BigInt::Lite to upgrade to
# Math::BigRat, we need to do the following (again):
bignum -> upgrade("Math::BigRat");
A simpler way to do this at runtime is to use import(),
bignum -> import(upgrade => "Math::BigRat",
downgrade => "Math::BigInt::Lite");
#Hexadecimal, octal, and binary floating point literals
Perl (and this module) accepts hexadecimal, octal, and binary floating point literals, but use them with care with Perl versions before v5.32.0, because some versions of Perl silently give the
wrong result.
#Operator vs literal overloading
bigrat works by overloading handling of integer and floating point literals, converting them to Math::BigRat objects.
This means that arithmetic involving only string values or string literals are performed using Perl's built-in operators.
For example:
use bigrat;
my $x = "900000000000000009";
my $y = "900000000000000007";
print $x - $y;
outputs 0 on default 32-bit builds, since bignum never sees the string literals. To ensure the expression is all treated as Math::BigFloat objects, use a literal number in the expression:
print +(0+$x) - $y;
Perl does not allow overloading of ranges, so you can neither safely use ranges with bignum endpoints, nor is the iterator variable a Math::BigFloat.
use 5.010;
for my $i (12..13) {
for my $j (20..21) {
say $i ** $j; # produces a floating-point number,
# not an object
This method only works on Perl v5.9.4 or later.
bignum overrides these routines with versions that can also handle big integer values. Under Perl prior to version v5.9.4, however, this will not happen unless you specifically ask for it with
the two import tags "hex" and "oct" - and then it will be global and cannot be disabled inside a scope with no bignum:
use bignum qw/hex oct/;
print hex("0x1234567890123456");
no bignum;
print hex("0x1234567890123456");
The second call to hex() will warn about a non-portable constant.
Compare this to:
use bignum;
# will warn only under Perl older than v5.9.4
print hex("0x1234567890123456");
Some cool command line examples to impress the Python crowd ;)
perl -Mbignum -le 'print sqrt(33)'
perl -Mbignum -le 'print 2**255'
perl -Mbignum -le 'print 4.5+2**255'
perl -Mbignum -le 'print 3/7 + 5/7 + 8/3'
perl -Mbignum -le 'print 123->is_odd()'
perl -Mbignum -le 'print log(2)'
perl -Mbignum -le 'print exp(1)'
perl -Mbignum -le 'print 2 ** 0.5'
perl -Mbignum=a,65 -le 'print 2 ** 0.2'
perl -Mbignum=l,GMP -le 'print 7 ** 7777'
Please report any bugs or feature requests to bug-bignum at rt.cpan.org, or through the web interface at https://rt.cpan.org/Ticket/Create.html?Queue=bignum (requires login). We will be notified, and
then you'll automatically be notified of progress on your bug as I make changes.
You can find documentation for this module with the perldoc command.
perldoc bignum
You can also look for information at:
This program is free software; you may redistribute it and/or modify it under the same terms as Perl itself.
#SEE ALSO
bigint and bigrat.
Math::BigInt, Math::BigFloat, Math::BigRat and Math::Big as well as Math::BigInt::FastCalc, Math::BigInt::Pari and Math::BigInt::GMP.
• (C) by Tels http://bloodgate.com/ in early 2002 - 2007.
• Maintained by Peter John Acklam <pjacklam@gmail.com>, 2014-. | {"url":"https://perldoc.perl.org/5.36.2/bignum","timestamp":"2024-11-06T21:38:57Z","content_type":"text/html","content_length":"45292","record_id":"<urn:uuid:d440a4ce-8f00-42f4-91b0-d8e900ab3060>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00340.warc.gz"} |
Do More Consistent Skiers Ski Faster?
{ 2011 04 01 }
Do More Consistent Skiers Ski Faster?
In a word, no. Â But the relationship between consistency and speed is a little subtle. Â To look at this question let’s take the distance results from major international competitions (OWG, WSC or
World Cups) and restrict ourselves to those times when an athlete did at least ten races in a particular season. Â Then for each season we’ll calculate a how variable their results were and also the
average of their best five races.I’m going to use FIS points as my measure, but the adjusted version that I’ve created myself to account for differences between mass start and interval start races
and changes in the F-factors over time. Â The “standard” way to measure variability is the standard deviation (SD). Â This can be a bit sensitive to single outliers, though, and we’re only requiring
a minimum of ten races so we’ll also consider the more robust median absolute deviation (MAD), just for kicks.
Here’s a look at the results using SD:
And here’s the same plot using MAD:
Here’s where we need to start being careful, and where I start doling out statistics lessons. Â Our question is whether consistent skiers (low variability) tend to ski faster (low average of best
five FIS points). Â Technically, then, we’re looking to see whether the slopes of the blue lines are positive.
And they are. Â Just look at ’em! Â Case closed, right?
Not really. Â First of all we can see that there’s clearly a difference between using SD and MAD. Â Namely, this relationship seems less pronounced using the more robust MAD. Â I’m inclined to prefer
MAD in this case. Â The SD might not really capture the “typical” level of variability since it can easily be inflated by a single very bad race.
Even so, the lines are sloping upward even in the MAD version of the plots. Â Now, you might be thinking that I’m going to pull some sort of “this isn’t actually statistically significant” trick out
of my hat. Â But I’m not:
Regressing average of best five races on SD of races.
│ │Estimate│Std. Error │t value│p value│
│ Intercept│ 14.5583│ 1.9744│ 7.37│ 0.0000│
│ SD of FIS Points│ 0.4716│ 0.1051│ 4.49│ 0.0000│
│ Gender (Women)│ 5.8085│ 2.8971│ 2.00│ 0.0452│
│ SD:Gender│ 0.2261│ 0.1493│ 1.51│ 0.1302│
Regressing average of best five races on MAD of races.
│ │Estimate│Std. Error │t value│p value│
│ Intercept│ 18.6618│ 1.7301│ 10.79│ 0.0000│
│ MAD of FIS Points│ 0.2749│ 0.1033│ 2.66│ 0.0079│
│ Gender (Women)│ 11.7077│ 2.5020│ 4.68│ 0.0000│
│ MAD:Gender│ -0.0862│ 0.1424│ -0.61│ 0.5452│
If you aren’t a stat-head, these might be more intimidating than some pretty graphs. Â It’s just a really simple regression model, one each for the SD and MAD versions. Â We only need to focus on the
second and fourth rows of each table for now.
The second row gives us the estimated slope for men, and we get the estimated slope for women by adding this to the estimate in the fourth row. Â So for the SD model, the slope for the men is about
0.4716 and for the women it’s about 0.6977. Â And the p-values suggest that this relationship is quite statistically significant! Â The results for the MAD version suggest a smaller slope, but it
still looks statistically significant. Â (The larger p-values in the fourth row indicates a lack of statistical significance between the slopes for the men and women.)
There’s an enormous amount of variation around this relationship we’ve found, indicated by how spread out the points are around the blue lines in the graphs above. Â The relationship exists, “on
average”, but accounts for very little of the variability in the data (the R-Squared values were 0.12 and 0.08 respectively, if you care for that sort of thing).
This suggests that more consistent skiers are, on average, faster, but only slightly and if we listed the factors influencing how fast skiers go in order of importance, consistency would likely be
way down near the bottom.
This is a great example of how statistics can provide information, but not necessarily an answer. Â Sadly, when people ask statisticians for help, they usually want an answer.
Post a Comment
Posted by Joran on Friday, April 1, 2011, at 6:00 am. Filed under Uncategorized. Tagged Analysis, technical, variability, variation. Follow any responses to this post with its comments RSS feed. You
can post a comment or trackback from your blog. | {"url":"https://www.statisticalskier.com/2011/04/do-more-consistent-skiers-ski-faster/","timestamp":"2024-11-03T06:40:39Z","content_type":"application/xhtml+xml","content_length":"46093","record_id":"<urn:uuid:3aae183d-58e0-4c5e-889a-b38c9e4d56bd>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00659.warc.gz"} |
This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary
that addresses important aspects of the task and its potential use.
Material Type:
Provider Set:
Illustrative Mathematics
Date Added: | {"url":"https://oercommons.org/browse?f.new_mccrs_alignment=MCCRS.Math.Content.7.NS.A.2a","timestamp":"2024-11-13T15:41:44Z","content_type":"text/html","content_length":"176860","record_id":"<urn:uuid:a69422ce-12ca-4706-8b83-e16216186573>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00801.warc.gz"} |
If [m][square_root]108[/square_root]= a[square_root]b[/square_root][/m
Official Answer
Question Stats:
100%0% (00:00)based on 4 sessions
Carcass wrote:
Now, when you practice a question in which you have equality = what you have on the LHS must balance out with the RHS
therefore \(\sqrt{108}\) must be equal to \(a\sqrt{b}\)
Next step is to figure out the factors of \(108=\sqrt{2^2*3^3}\) or \(108=\sqrt{2^2*3^2*3}\)
Therefore a must be a number and \(\sqrt{b}\) must be the numbers or factors of 108 but at the same time we must have the a outside the root and then take the sum
I hope this helps
Hey Carcass,
Could you please explain how did you come up with the highlighted text? | {"url":"https://gre.myprepclub.com/forum/if-108-ab-34527.html","timestamp":"2024-11-13T06:32:11Z","content_type":"application/xhtml+xml","content_length":"214022","record_id":"<urn:uuid:85851539-8463-4e00-ae21-3ebf8b6f2a37>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00471.warc.gz"} |
(4) Let T(x, y, z) be "x has traveled to y in z", where the domain for
(4) Let T(x, y, z) be "x has traveled to y in z", where the domain for x is the collection of all people, the domain for y is the collection
of all countries, and the domain for z is the collection of all years (A.D., let's say so e.g., 1981). Translate the following into formulas. Be careful to use correct syntax. (a) Peter has traveled
to some country in 1999 ('Peter' and '1999' are constants, but 'some country' is NOT use quantifier/variable). (b) Katie traveled to exactly one country in 2004 (so we are saying that in the year
2004, Katie traveled to exactly one country). (c) Mike and John traveled to no common country in 2000 (that is, the collection of countries Mike traveled to in 2000 is disjoint from the collection of
countries that John traveled to in 2000). (d) Beth traveled to no countries in 1954.
Fig: 1 | {"url":"https://tutorbin.com/questions-and-answers/4-let-t-x-y-z-be-x-has-traveled-to-y-in-z-where-the-domain-for-x-is-the-collection-of-all-people-the-domain-for-y-is-the","timestamp":"2024-11-11T22:58:51Z","content_type":"text/html","content_length":"63856","record_id":"<urn:uuid:c37e61bd-4a66-4e25-af0a-45d78bbf41ed>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00089.warc.gz"} |
Showing Two Functions Are Symmetric About A Line
• Thread starter Bashyboy
• Start date
In summary, the two functions y_1 and y_2 are symmetric about the line y = \frac{c}{d} if the distance between y_1 and y, and the distance between y_2 and y, are the same.
Hello everyone,
I have the functions [itex]y_1 = \frac{c}{b} + d e^{-bx}[/itex] and[itex]y_2 = \frac{c}{b} - d e^{-bx} [/itex], where [itex]c \in \mathbb{R}[/itex], and [itex]b,d \in \mathbb{R}^+[/itex].
What I would like to know is how to show that these two functions are symmetric about the line [itex]y = \frac{c}{d}[/itex].
What I thought was that if y_1 and y_2 are symmetric about the line [itex]y = \frac{c}{d}[/itex], then the distance between y_1 and y, and the distance between y_2 and y, will be the same. That is,
[itex]d_1 = \sqrt{(y_1 - y)^2 + (x - x_0)^2}[/itex] and [itex]d_2 = \sqrt{(y_2 - y)^2 + (x - x_0)^2}[/itex], where [itex]d_1 = d_2[/itex].
Is this a correct way of determining symmetry? Is it true in general? Are there any other ways in which I could prove symmetry?
Last edited:
Bashyboy said:
Hello everyone,
I have the functions [itex]y_1 = \frac{c}{b} + d e^{-bx}[/itex] and[itex]y_2 = \frac{c}{b} - d e^{-bx} [/itex], where [itex]c \in \mathbb{R}[/itex], and [itex]b,d \in \mathbb{R}[/itex].
What I would like to know is how to show that these two functions are symmetric about the line [itex]y = \frac{c}{d}[/itex].
What I thought was that if y_1 and y_2 are symmetric about the line [itex]y = \frac{c}{d}[/itex], then the distance between y_1 and y, and the distance between y_2 and y, will be the same. That
is, [itex]d_1 = \sqrt{(y_1 - y)^2 + (x - x_0)^2}[/itex] and [itex]d_2 = \sqrt{(y_2 - y)^2 + (x - x_0)^2}[/itex], where [itex]d_1 = d_2[/itex].
Is this a correct way of determining symmetry? Is it true in general? Are there any other ways in which I could prove symmetry?
Some comments:
(1) Your distance formula should not involve x, because for each x
you want to show that the distance between the points (x,c/b) and (x,y_1(x)) is the same as the distance between the points (x,c/b) and (x,y_2(x)). The x drops out of these distance formulas
(although, of course, they still contain y_1(x) and y_2(x)). After that, what you say would be correct.
(2) There is a much easier way.
Last edited:
And what might this easier method be, Ray?
Bashyboy said:
And what might this easier method be, Ray?
That is for you to think about; I am not allowed to give solutions, nor would I want to. I can make one suggestion, however: think about what you would get if you drew graphs of the two functions on
the same plot.
I have already drawn the plot of these functions, and that was how I made inference I made, that the distances must be the same. I am not sure what else could be concluded from the plots.
Would it perhaps be that the sum of the functions y1 and y2 is identically zero for all x, where is a real number?
Bashyboy said:
Would it perhaps be that the sum of the functions y1 and y2 is identically zero for all x, where is a real number?
Well, how would you write it after correcting your erroneous expressions given before?
FAQ: Showing Two Functions Are Symmetric About A Line
What does it mean for two functions to be symmetric about a line?
When two functions are symmetric about a line, it means that if you were to fold the graph of one function along the line of symmetry, the resulting graph would be identical to the other function.
How can I determine if two functions are symmetric about a line?
To determine if two functions are symmetric about a line, you can follow these steps:
1. Find the line of symmetry by setting the two functions equal to each other and solving for x.
2. Substitute the x-value of the line of symmetry into both functions to get the corresponding y-values.
3. If the y-values are equal, then the functions are symmetric about the line. If they are not equal, then the functions are not symmetric about the line.
What is the significance of two functions being symmetric about a line?
When two functions are symmetric about a line, it shows that they have a special relationship and can be used to understand each other more deeply. It also allows for easier analysis and comparison
of the two functions.
Can two functions be symmetric about more than one line?
Yes, two functions can be symmetric about more than one line. A function can be symmetric about any vertical, horizontal, or diagonal line on a graph.
What are some real-life examples of two functions being symmetric about a line?
One real-life example of two functions being symmetric about a line is the motion of a pendulum. The position of the pendulum can be described by two functions, one for the horizontal displacement
and one for the vertical displacement. These two functions are symmetric about the vertical line that passes through the point of suspension. Another example is the relationship between Celsius and
Fahrenheit temperature scales, where the two functions are symmetric about the line y=x. | {"url":"https://www.physicsforums.com/threads/showing-two-functions-are-symmetric-about-a-line.740956/","timestamp":"2024-11-03T09:36:18Z","content_type":"text/html","content_length":"103324","record_id":"<urn:uuid:f1b9ddc5-53c5-4720-a4ae-a929a04075db>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00248.warc.gz"} |
Performance Analysis of Cryptographic Algorithms in the Information Security
NCISIOT - 2019 (Volume 8 - Issue 02)
Performance Analysis of Cryptographic Algorithms in the Information Security
DOI : 10.17577/IJERTCONV8IS02021
Download Full-Text PDF Cite this Publication
U. Thirupalu, Dr. E. Kesavulu Reddy, 2020, Performance Analysis of Cryptographic Algorithms in the Information Security, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY (IJERT) NCISIOT –
2020 (Volume 8 – Issue 02),
• Open Access
• Authors : U. Thirupalu, Dr. E. Kesavulu Reddy
• Paper ID : IJERTCONV8IS02021
• Volume & Issue : NCISIOT – 2020 (Volume 8 – Issue 02)
• Published (First Online): 21-02-2020
• ISSN (Online) : 2278-0181
• Publisher Name : IJERT
• License: This work is licensed under a Creative Commons Attribution 4.0 International License
Text Only Version
Performance Analysis of Cryptographic Algorithms in the Information Security
1. Thirupalu1
Research Scholar Dept. of Computer Science S V U CM&CS-Tirupati
Dr. E. Kesavulu Reddy2 Ph. D, FCSRC (USA) Assistant Professor
Dept. of Computer Science S V U CM&CS-Tirupati-A.P
Abstract: Information security is the process of protecting information. It protects its availability, privacy and integrity. The two main characteristics that identify and differentiate one
encryption algorithm from another are its ability to secure the protected data against attacks, speed and efficiency. security is the most challenging issue in the world and the various security
threats in the cyber security has to be avoided and to give more confidentiality to the users and to enable high integrity and availability of the data. The encryption of the data by using the
various data encryption algorithms will provide the additional security to the data being transmitted. This paper mainly focuses on comparative analysis of (AES, DES, 3DES, BLOWFISH, RC4),
Asymmetric (RSA, DSA, Diffie-Hellman, EI-Gamal, Pailier), Hashing (MD5, MD6, SHA, SHA256) algorithms.
Keywords: Encryption. Decryption, Data Security, Key size, information security, Symmetric algorithms, Asymmetric Algorithms.
DES is first block symmetric encryption algorithm published by NIST designed by IBM on 1974. The same key is used for both encryption and decryption; DES uses 64-bit key in terms of 8-bits for
error correction and 56-bits as a key but in every byte 1 bit in has been selected as a 'parity' bit, and is not used for encryption mechanism. The 56 bit is permuted into 16 sub- keys each of
48- bit length. It also contains 8 S- boxes and same algorithm is used in reversed for decryption [1]. The implementations of the DES (data encryption standard) algorithm based on hardware are
low cost, flexible and efficient encryption solutions.
DES_ Encrypt (X, Y) where E = (L, R) X IP (M)
1. INTRODUCTION
The cryptosystems are processing with different types of cryptographic algorithms. These cryptographic algorithms are used for encryption of data and decryption of data using of shared keys
or single key.These are fall into two categories i.e. 1. Public key cryptosystem or Asymmetric key cryptosystem.2. Secret key cryptosystem or Symmetric key cryptosystem.
\For round 1 to 16 do Yi SY (Y, round)
L L xor F(R, Ki)
swap (L, R)
2. LITERATURE SURVEY
Encryption algorithm plays a vital role, to provide secure communication over the network. Encryption is the fundamental tool for protecting the data. Encryption algorithm converts the data
into scrambled form by using the key and only user have the key to decrypt the data The techniques in Security algorithms are Symmetric Algorithms and asymmetric Algorithms.
1. Symmetric Algorithms
○ D E S
○ 3 D E S
○ BLOWFISH
○ RC5
○ RC6
○ A E S
○ IDEA
○ Homomorphic Encryption
○ DES
swap (L, R)
return X.
☆ Triple-DES
TDES is an enhanced version of DES is based on Feistel structure. The CPU power consumed by TDES is three times more than DES. The 3DES uses a 64 bit plain text with 48 rounds and a Key
Length of 168-bits permuted into 16 sub- keys each of 48- bit length. It also contains 8 S- boxes and same algorithm is used in reversed for decryption [2]. Triple DES the algorithm is
considered to be practically secure, in spite of having theoretical attacks.
For j = 1 to 3
Cj,0 = IVj
For i = 1 to nj
Cj,i = EK3 (DK2 (EK1 (Pj,i Cj,i-1)))
Output Cj,i
☆ Blowfish
Blowfish algorithm was first introduced in 1993.The Blowfish is highly rated secure variable length key encryption algorithm with different structure and functionality than all other
algorithms. Blowfish is a block cipher that uses a 64 bit plain text with 16 rounds, allowing a variable key length, up to 448 bits, permuted into 18 sub- keys each of 32- bit length and
can be implemented on 32- or 64-bit processors. It also contains 4 S- boxes and same algorithm is used in reversed for decryption [3].
Divide x into two 32-bit lengths : xL, xM For i = 1to 16:
XL= XL XOR Pi
xM = F (XL) XOR xM
Swap XL and xM Next i
Swap XL and xM (Undo the last swap.) xM = xM XOR P17
xL = xL XOR P18 then
Combine XL and xM.
☆ RC5
RC5 was developed in 1994. The key length if RC5 is MAX2040 bit with a block size of 32, 64 or 128. The use of this algorithm shows that it is Secure. The speed of this algorithm is slow.
A = A + S[0];
B = B + S[1];
for i = 1 to r do
A = ((A Xor B) <<< B) + S[ 2 * i ]
B = ((B Xor A) <<< A) + S[ 2 * i + 1]
☆ AES
AES is a block cipher that uses a 128 bit plain text with variable 10, 12, or 14 rounds and a variable Key Length of 128, 192, 256 bit permuted into 10 sub- keys each of 128, 192, 256 bit
length respectively[5].. It only contains a single S- box and same algorithm is used in reversed for decryption. Rijndael's default number of Rounds is dependent on key size i.e. Rounds =
key length/32 + 6. Rijndael AES provides great flexibility for implementing based on parallel structure with effective resistance against attacks [3] [4].
Cipher (byte [] input, byte [] output)
byte[4,4] State;
copy input[] into State[] Add Round Key for (round = 1; round < Nr-1; ++round)
SubBytes ShiftRowsMixColumns AddRoundKey
SubBytes ShiftRows AddRoundKey
copy State[] to output []
B .Asymmetric Key cryptographic Algorithms
☆ RSA
☆ DSA
☆ Diffie-Hellman
☆ ELGAMMAL
☆ RSA algorithm
Rivest-Shamir-Adleman (RSA) is a special type of public key cryptography which over the years has reigned supreme as the most widely accepted and implemented general-purpose approach
public-key encryption techniques [2]. The RSA algorithm follows a block cipher encryption technique, in which the plaintext and the cipher are integers between 0 and n 1 for some n. A
typical size for n is 1024 bits, or 309 decimal digits. That is, n is less than to1024.RSA algorithm has three major steps.
1. Key generation
2. Encryption
3. Decryption Algorithm
1. Key Generation; KeyGen (p,q)
Input : Select two prime integers p ,q.
2. Compute n = p q , (n ) = (p-1)(q-1)
3. Choose e as exponent then gcd (e, p-1) = 1 4. gcd (e, q-1) = 1
5. gcd (e, (p-1) (q-1)) = 1
1. Compute d such that ed =1(mod (n))
2. Compute d = e-1 (mod (n)) Find a unique value d such that
(n) divides 5d-1 value —- pi Key Public Key = (n, e).
Private Key = (n, d). Encryption
C M E (mod N)
3. Encrypt the message M C M E (mod N) Decryption
4. To decrypt the cipher text we have M = Cd (mod n)
M = Plain Text.
MD5 (Message Digest5)
The Message Digest5 (MD5) was developed by Ronald Rivest in 1992 by taking the block sizes as 512 bit and the digest size as 128 bit. The hash function producing the
128 bit hash value. The MD5 can be used as the best solution to impos the brute force attack to act against the extensive vulnerabilities and to provide excessive security. SHA (Secure
Hash Algorithm)
The Secure Hash Algorithm (SHA) is the most prominent hash algorithm used in the cryptographic systems. It uses 160- bit which is also a resemblance of the MD5 algorithm. The SHA-1 was
originally developed by the National Security Agency (NSA) to be part of the Digital Signature Algorithm.
☆ DSA
The Digital Signature Algorithm (DSA) was proposed by the National Institute of Standards and Technology (NIST) in August 1991. DSA, the entropy, secrecy, and uniqueness of the random
signature value k is critical [6]. It is so critical that violating any one of those three requirements can reveal the entire private key to an attacker. Using the same value twice (even
while keeping k secret), using a predictable value, or leaking even a few bits of k in each of several signatures, is enough to break DSA. [5]
☆ Diffie-Hellman Key Exchange (D-H)
DiffieHellman key exchange is a specific method of exchanging cryptographic keys. It is one of the earliest practical examples of key exchange implemented within the field of
cryptography. The DiffieHellman key exchange method allows two parties that have no prior knowledge of each other to jointly establish a shared secret key over an insecure communications
channel. This key can then be used to encrypt subsequent communications using a symmetric key cipher.
☆ EIGamel
In cryptography, the EI Gammel encryption system is an asymmetric key encryption algorithm for public-key cryptography which is based on the DiffieHellman key exchange. It was described
by Taher EI Gammel in 1984. EI Gammel encryption is used in the free GNU Privacy Guard software, recent versions of PGP, and other cryptosystems. The Digital Signature Algorithm is a
variant of the EIGamel signature scheme, which should not be confused with EI Gammel encryption. EIGamel encryption can be defined over any cyclic group .Its security depends upon the
difficulty of a certain problem in related to computing discrete logarithms.
☆ TWOFISH
Bruce Schneier is the person who composed Blowfish and its successor Twofish. The Keys used in this algorithm may be up to 256 bits in length .Twofish is regarded as one of the fastest of
its kind, and ideal for use in both hardware and software environments. Twofish is also freely available to anyone who wants to use it. As a result, well find it bundled in encryption
programs such as Photo Encrypt, GPG, and the popular open source software [6].
☆ IDEA
IDEA stands for International Data Encryption Algorithm which was proposed by James Massey and Xuejia Lai in 1991. IDEA is considered as best symmetric key algorithm. It accepts 64 bits
plain text. The key size is 128 bits. IDEA consists of 8.5 rounds. In IDEA the 64 bits of data is divided into 4 blocks each having size 16 bits. The basic operations are modular,
addition, multiplication, and bitwise exclusive OR (XOR) are applied on sub blocks. There are eight and half rounds in IDEA each round consist of different sub keys. Maximum number of
keys used for performing different rounds is 52 [7].
☆ Homomorphic Encryption
Homomorphic encryption was a one of encryption technique which allows specific types of computations to be carried out on cipher text. It gives an encrypted result which when decrypted
matches the result of operations performed on the plaintext. When the data is transferred to the cloud we use standard encryption methods to secure this data, but when we want to do the
calculations on data located on a remote server, it is necessary that the cloud provider has access to the raw data, and then it will decrypt them [8].
3. RELATIVE WORK
Comparison of symmetric and asymmetric cryptography with existing vulnerabilities and countermeasures gives us theoretical comparison of symmetric and asymmetric cryptography algorithms
[9].compares symmetric and asymmetric cryptography algorithms using parameters key length ,speed , encryption ratio and security attacks[10]. Comparisons DES,3DES and AES algorithms with nine
factors key length , cipher type, block size, developed year
,cryptanalytic resistance , possible keys, possible Ascii keys and time required to check all possible keys[11] . Comparative study of symmetric and asymmetric cryptography techniques using
throughput, key length, tunability, speed, encryption ratio and security attacks [12]. Evaluation of blowfish algorithm based on avalanche effect gives a new performance measuring metric
avalanche effect [13].
4. COMPARISON OF SYMMETRIC AND ASYMMETRIC
S.NO CHARAC TERISTIC S ALGORIT HM BSI ZE BIT S KEY LENG TH SECUR ITY SPEE D
1 DES 64 56 Inadequa te Very slow
2 Blow Fish 64 448 Secure Fast
3 RC2 64 128 High secure Very fast
4 RC5 2040 Secure Slow
4 or 128
5 RC6 128 128 or 256 Secure Fast
6 3DES 64 InSecure Slow
128, 128,19
7 AES 192 2 or High secure Very fast
or 256 256
8 RSA 128 Secure Very slow
9 DSA 256 192 Secure Fast
10 DIffe- Hellman — — In secure Slow
11 Two Fish 128 Secure Very Fast
2or 56
12 IDEA 64 128 Inadequa te Slow
13 Elgammel — — Not secure Fast
14 Homomorp hic Encryption — — Secure Fast
15 SHA 512 160 Secure Slow
16 MD5 512 128 Secure Slow
17 RC4 Secure Very fast
Table. I.. Comparison of Symmetric key and asymmetric key Cryptographic Algorithms
Blowfish is the better than other algorithms in throughput and power consumption [14]. Blowfish encryption algorithm is leading with the security level that they provide and faster encryption
speed. Blowfish was replaced by Two fish.RC6 might be observed as interweaving two parallel RC5 encryption Techniques. The RC6 can use an extra multiplication operation but not present in RC5
in order to make the rotation dependent on each bit, and not the least significant few bits [15]. Triple DES has slow performance in terms of power consumption and
throughput when compared with DES [14] [16]. AES encryption is fast and flexible, it can be implemented on various platforms especially in small devices. AES has been carefully tesed for many
security applications [16][17].
RSA is an asymmetric cryptographic algorithm. Asymmetric means that there are two different keys are used in encryption and decryption process [16]. The RSA algorithm can be used for both
public key encryption and digital signatures. Its security is based on the difficulty of factorization of large integers. The main disadvantage of RSA is that it consumes more time to encrypt
data. Actually this is disadvantage of asymmetric key algorithms because the use of two asymmetric keys. It provides good level of security but it is slow for encrypting files. The strength
of the each encryption algorithm depends upon the key management, type of cryptography, number of keys, number of bits used in a key. Longer the key length and data length more will be the
power consumption that will lead to more heat dissipation. So, it is not advisable to use short data sequence and key lengths.
The DSA is a variant of the EIGamal signature scheme, which should not be confused with EIGamal encryption.
Table. 2.Experimental results using Crypto ++
S.NO Algorithm Megabytes(2^20 bytes) Processed Time Taken MB/Second
1 Blowfish 256 3.976 64.386
2 Rijndael (128-bit key) 256 4.196 61.010
3 Rijndael (192-bit key) 256 4.817 53.145
4 Rijndael (256-bit key) 256 5.308 48.229
5 256 4.436 57.710
(128) CTR
6 256 4.837 52.925
(128) OFB
7 256 5.378 47.601
(128) CFB
8 256 4.617 55.447
(128) CBC
9 DES 128 5.998 21.340
10 (3DES)DES- XEX3 128 6.159 20.783
11 (3DES)DES- EDE3 64 6.499 9.848
The security of EIGamal depends on the difficulty of a particular problem in related to computing discrete logarithms [18]. Homomorphic encryption was a one of encryption technique which
allows specific types of computations to be carried out on cipher text.
AES algorithm is most efficient in terms of speed, time, and throughput. DES algorithm consumes least encryption time and AES algorithm has least memory usage while
encryption time difference is very minor in case of AES and DES algorithm. RSA encryption time and memory usage is also very high but output byte is least in case of RSA algorithm.
The experiment result shows that the memory required for implementation is smallest in blowfish whereas it is largest in RSA. DES and AES require medium size of memory. So the Blowfish is
best option smaller size of memory. AES is the more confident, integrity and highest priority for any application. Blowfish consumes the least time amongst all. Blowfish is efficient in
software, at least on some software platforms. AES is the best suitable for better cryptographic strength. DES is the best suitable for network bandwidth.
5. V.EXPERIMENTAL RESULTS
1. Experimental results using Crypto ++
The experiment are conducted on commonly used cryptographic algorithms based on system parameters Pentium 4 and 2.1 GHZ processor on windows XP compiled C++ code with Microsoft Visual
C++.NET 2003to evaluate the execution time for encryption and speed benchmarks [19].
The result shows that Blowfish and AES have the best performance among others. Compare to both AES the best secure and efficient algorithm with limited key size among the all above
algorithms. Finally AES performs highly secure encryption algorithm and accepted with higher key size. The popular secret key algorithms including DES, 3DES, AES (Rijndael), Blowfish,
were implemented, and their performance was compared by encrypting input files of varying contents and sizes. The algorithms were implemented in a uniform language (Java), using their
standard specifications, and were tested on two different hardware platforms, to compare their performance.
The performance of the Secret key or Symmetric key algorithms comparing by the encrypting input files with various contents and sizes using Java as common language in two different
platforms. The first experiment on P-II 266 MHZ and P-4 2.4 GHZ
File Size Bytes DES 3DES AES BF
20,527 2 7 4 2
36,002 4 13 6 3
45,911 5 17 8 4
59,852 7 23 11 6
69,545 9 26 13 7
137,325 17 51 26 14
158,959 20 60 30 16
166,364 21 62 31 17
191,383 24 72 36 19
232,398 30 87 44 24
Average Time 14 42 21 11
B/Sec 7,988 2,663 5,320 10,167
Table.4.Performance comparison on Symmetric key algorithms
The observations based on the results shows that Blowfish has a very good performance compared to other algorithms. AES has better performance than 3DES and DES. AES is considered among
best secure and efficient algorithm in the above all algorithms [20].
6. CONCLUSION
We discuss the weakness and strength of asymmetric key algorithms and symmetric key algorithms. Based on survey RC5 and RC4 security is questionable but RC4 faster than RC5. These encryption
algorithms AES is more secure, efficient and faster than to all algorithms with allowing 256-bit key sizes and protect against future attacks. Blowfish was replaced by Twofish.
RSA is best Asymmetric key algorithm but it consumes more time for encryption and factorization problem for large Integers in the decryption process.
I am U.Thirupalu joined as a Research Scholar (PT) in the department of computer science ,S V U CM&CS , Tirupati. I am pursuing PhD under the guidance of Dr.E.kesavulu Reddy in the Dept. of Computer
Science, S v u CM&CS, Tirupati.
1. Show the results of their experiments conducted on P-II 266 MHz with Java.
File Size KB DES 3DES AES BF
20,527 24 72 39 19
36,002 48 123 74 35
45,911 57 158 94 46
59,852 74 202 125 58
69,545 83 243 143 67
137,325 160 461 285 136
158,959 190 543 324 158
166,364 198 569 355 162
191,383 227 655 378 176
232,398 276 799 460 219
Average Time 134 383 228 108
B/Sec 835 292 491 1,036
Table 3. The experiment shows the Comparative execution times (in seconds) of encryption algorithms in ECB mode on a P-II 266 MHz machine.
1. Cong Wang, Qian Wang, Kui Ren and Wenjing Lou Ensuring Data Storage Security in Cloud Computing. IEEE 200 9.
2. Yogesh Kumar, Rajiv Munjal andn Harsh Sharma Comparison of Symmetric and Asymmetric Cryptography With Existing Vulnerabilities and Countermeasures IJCSMS International Journal of Computer Science
and anagement Studies, Vol. 11, Issue 03, Oct 2011.
3. D. S. Abdul. Elminaam, H. M. Abdul Kader and M. M. Hadhoud , Performance Evaluation of Symmetric Encryption Algorithms, Communications of the IBIMA Volume 8, 2009.
4. Gurpreet Singh, Supriya KingerIntegrating AES, DES, and 3- DES Encryption Algorithms for Enhanced Data Security International Journal of Scientific & Engineering Research, Volume 4, Issue 7,
5. Uma Somani, Implementing Digital Signature with RSA Encryption Algorithm to Enhance the Data Security of Cloud in Cloud Computing," 2010 1st International Conference on Parallel, Distributed and
Grid Computing (PDGC – 2010).
6. Mr. Mukta Sharma and Mr. Moradabad R. Comparative Analysis of Block Key Encryption Algorithms International Journal of Computer Applications (0975 8887) Volume 145 No.7, July 2016.
7. AshimaPansotra and SimarPreet Singh Cloud Security Algorithms International Journal of Security and Its Applications Vol.9, No.10 (2015), pp.353-360.
8. Iram Ahmad and Archana Khandekar Homomorphic Encryption Method Applied to Cloud Computing International Journal of Information & Computation Technology. ISSN 0974-2239 Volume 4, Number 15 (2014),
pp. 1519-1530.
9. Comparison of symmetric and asymmetric cryptography with existing vulnerabilities and countermeasures by Yogesh Kumar, Rajiv Munjal, and Harsh ,(IJAFRC) Volume 1, Issue 6, June 2014. ISSN 2348
10. Comparative analysis of performance efficiency andsecurity measures of some encryption algorithms y AL.Jeeva, Dr.V.Palanisamy, K.Kanagaram compares symmetric and asymmetric cryptography
algorithms ISSN: 2248-9622.
11. New Comparative Study Between DES, 3DES and AES within Nine Factors Hamdan.O.Alanazi, B.B.Zaidan, . A.Zaidan, Hamid A.Jalab, M.Shabbir and Y. Al- Nabhani JOURNAL OF COMPUTING, VOLUME 2, ISSUE 3,
MARCH 2010,ISSN 2151-9617.
12. Comparative Study of Symmetric and Asymmetric Cryptography Techniques by Ritu Tripathi, SanjayAgrawal compares Symmetric and AsymmetricCryptographyTechniques using throughput, key length,
tunability, speed, encryption ratio and security attacks. IJCSMS International Journal of Computer Science and Management Studies, Vol. 11, Issue 03, Oct 2011 ISSN (Online): 2231-5268
13. Evaluation of Blowfish Algorithm based on Avalanche Effect by Manisha Mahindrakar gives a new performance measuring metricavalanche effect. International Journal of Innovations in Engineering and
Technology (IJIET) 2014.
14. Mr. Gurjeevan Singh, Mr.Ashwani Singla And Mr. K S Sandha Cryptography Algorithm Compassion for Security Enhancement In Wireless Intrusion Detection System International Journal of
Multidisciplinary Research Vol.1 Issue 4, August 2011.
15. Mr.Milind Mathur and Mr. Ayush Kesarwani Comparison between DES, 3DES, RC2, RC6, Blowfish and AES Proceedings of National Conference on New Horizons in IT – NCNHIT 2013.
16. urpreet Singh, SupriyaKinger Integrating AES, DES, and 3-DES Encryption Algorithm for Enhanced Data Security International Journal of Scientific & Engineering Research, Volume 4, Issue 7,
17. Uma Somani, Implementing Digital Signature with RSA Encryption Algorithm to Enhance the Data Security of Cloud in Cloud Computing, 2010 First International Conference On parallel, Distributed and
Grid Computing (PDGC-2010).
18. AnnapoornaShetty , ShravyaShetty K , Krithika K A Review on Asymmetric Cryptography RSA and ElGamal Algorithm International Journal of Innovative Research in Computer and Communication
Engineering Vol.2, Special Issue 5, October 2014
19. RFC2828],"Internet Security http://www.faqs.org/ rfcs/rfc2828.html.
20. Aamer Nadeem et al, "A rformance Comparison of Data Encryption Algorithms", IEEE 2005.
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Printable Figure Drawings
Finding The Discriminant Worksheet
Finding The Discriminant Worksheet - It is in the form ax2 + bx + c = 0, where a = 1, b = 5, and c = 6. The simple and clear as crystal quadratic equations featured in these pdf worksheets are in
their standard form: Web statement about the discriminant. You write down problems, solutions and notes to go back. Solve to find the values of k. 7) 4 p2 + 8 p + 4 = 0.
Discriminant of a quadratic equation. => the equation has two real roots. Web statement about the discriminant. The quadratic formula says that. A x 2 + b x + c = 0.
2 10) 2 + 5. Web this is a worksheet designed in a standard form, perfect for a binder.in this worksheet, students will be guided through: => the equation has two real roots. Web your students will
use these worksheets to learn how to find the discriminant of a quadratic equation. For quadratic equation that is:
Find the discriminant of the quadratic equation. This foldable is designed for interactive math notebooks.here is the link of me teaching this lesson using this resource on youtube! Web understanding
the discriminant date_____ period____ find the value of the discriminant of each quadratic equation. => x 3 + ax 2 + bx + c = 0. Web statement about the.
Web understanding the discriminant date_____ period____ find the value of the discriminant of each quadratic equation. So, the given equation has two real roots. The quadratic formula says that.
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______ identify values for “a”, “b”, and “c”. Practice finding the discriminant and number of solutions for quadratic equations! The discriminant tells us whether there are two solutions, one
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discriminant to.
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discriminant is the part of the quadratic formula underneath the square root symbol: The quadratic formulause the quadratic formula to find the solutions (4 problems).
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discriminant worksheets are a great way to learn algebra basics. Solve to find the values of.
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roots. 1) 6 p2 − 2p − 3 = 0 2) −2x2 − x − 1 = 0 3) −4m2 − 4m + 5 =.
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results for finding the discriminant.
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(total for question 9 is 7 marks) 1 the equation x² + kx + 2 = 0, where k is a constant has no real roots.
Find the set of possible values for k. 7) 4 p2 + 8 p + 4 = 0. X = − b ± b 2 − 4 a c 2 a. Discriminant of a quadratic equation.
When given values for the discriminants. Web results for finding the discriminant. Then we substitute the coefficients in the relevant formula to find the discriminant.
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way to learn algebra basics.
Please Also Find In Sections 2 & 3 Below Videos, Powerpoints, Mind Maps And Worksheets On This Topic To Help Your Understanding.
(total for question 9 is 7 marks) 1 the equation x² + kx + 2 = 0, where k is a constant has no real roots. Find the value of the discriminant of each quadratic equation. Find the set of possible
values for k. Find the discriminant of the quadratic equation.
Web Your Students Will Use These Worksheets To Learn How To Find The Discriminant Of A Quadratic Equation.
Web this is a worksheet designed in a standard form, perfect for a binder.in this worksheet, students will be guided through: Practice finding the discriminant and number of solutions for quadratic
equations! Discriminant is usually found inside of the square root of the quadratic formula in quadratic equations. X = − b ± b 2 − 4 a c 2 a.
For Any Value Of K Less Than 4, The Equation Will Have Two Distinct Real Solutions.
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measurement converter cooking ingredient converter cake pan converter more.
=> Ax 2 + Bx + C = 0.
2 10) 2 + 5. Find all the values of a such that ax 2 + 5x + 3 = 0 has two real roots. Solve to find the values of k. Ax 2 + bx + c = 0, with integer coefficients. | {"url":"https://tunxis.commnet.edu/view/finding-the-discriminant-worksheet.html","timestamp":"2024-11-05T16:24:05Z","content_type":"text/html","content_length":"35345","record_id":"<urn:uuid:c83a7308-0df8-4922-98cd-54316b684ba2>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00516.warc.gz"} |
Representation Learning in Sensory Cortex: a theory.
Title Representation Learning in Sensory Cortex: a theory.
Publication CBMM Memo
Year of 2014
Authors Anselmi F, Poggio T
Number 026
Date 11/2014
We review and apply a computational theory of the feedforward path of the ventral stream in visual cortex based on the hypothesis that its main function is the encoding of invariant
representations of images. A key justification of the theory is provided by a theorem linking invariant representations to small sample complexity for recognition – that is, invariant
representations allows learning from very few labeled examples. The theory characterizes how an algorithm that can be implemented by a set of ”simple” and ”complex” cells – a ”HW module”
– provides invariant and selective representations. The invariance can be learned in an unsupervised way from observed transformations. Theorems show that invariance implies several
Abstract properties of the ventral stream organization, including the eccentricity dependent lattice of units in the retina and in V1, and the tuning of its neurons. The theory requires two stages
of processing: the first, consisting of retinotopic visual areas such as V1, V2 and V4 with generic neuronal tuning, leads to representations that are invariant to translation and
scaling; the second, consisting of modules in IT, with class- and object-specific tuning, provides a representation for recognition with approximate invariance to class specific
transformations, such as pose (of a body, of a face) and expression. In the theory the ventral stream main function is the unsupervised learning of ”good” representations that reduce the
sample complexity of the final supervised learning stage.
URL http://cbmm.mit.edu/sites/default/files/publications/CBMM-Memo-026_neuron_ver45.pdf
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Solution and Colligative properties Formulas
Discover Important Solution and Colligative properties Formulas for Clausius Clapeyron Equation, Depression in Freezing Point, Elevation in Boiling Point and Gibb's Phase Rule. Find step-by-step
solutions for each formula to enhance your Solution and Colligative properties skills. Perfect for students, teachers, and all Solution and Colligative properties enthusiasts. | {"url":"https://www.formuladen.com/en/solution-and-colligative-properties-formulas/FormulaList-873","timestamp":"2024-11-09T10:09:41Z","content_type":"application/xhtml+xml","content_length":"94679","record_id":"<urn:uuid:896ad789-f1cd-4d10-aa0f-1705a87b6d8d>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00487.warc.gz"} |
Functions in polynomials rings
Functions in polynomials rings
I want to define a function in a polynomial ring in several variables.
I am trying to define a function that takes $(i,j)$ to $x_i^j$.
I tried
def f(i,j):
return xi^j
This does not work. I tried replacing xi with x[i], that doesn't work. Can someone please tell me what I am doing wrong and how to fix it? If instead of taking 3 variables I take only 1 variable then
the method works.
2 Answers
Sort by » oldest newest most voted
You want to access the generators of R as a tuple:
sage: R = PolynomialRing(QQ, 3, names='x'); R
Multivariate Polynomial Ring in x0, x1, x2 over Rational Field
sage: x = R.gens(); x
(x0, x1, x2)
sage: x[0]
If you want to use some strange alternative indexing, then you can achieve it with a function.
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One can produce strings and have the polynomial ring eat them.
String formatting is easy thanks to Python.
Define a polynomial ring as in the question:
R.<x1, x3, x5> = PolynomialRing(QQ)
Define a "generator power" function as follows:
def f(i, j):
Return the polynomial variable xi raised to the j-th power.
return R('x{}^{}'.format(i, j))
sage: f(3, 2)
edit flag offensive delete link more | {"url":"https://ask.sagemath.org/question/53502/functions-in-polynomials-rings/","timestamp":"2024-11-09T09:50:47Z","content_type":"application/xhtml+xml","content_length":"57555","record_id":"<urn:uuid:8058609a-1bd1-4dc0-bba7-23bf8beb1761>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00501.warc.gz"} |
VCI2022 - The 16th Vienna Conference on Instrumentation
Philipp Windischhofer (University of Oxford (GB))
The Ramo-Shockley theorem defines an efficient and physically very intuitive method for the computation of the electrical signal induced by moving charged particles on the readout electrodes of a
particle detector.
This theorem, along with its various generalisations and extensions, applies only to situations that are quasi-electrostatic, i.e. where radiation and wave propagation effects do not play an
appreciable role.
In this contribution, I will present a fully general signal theorem that encapsulates all electrodynamic effects without any approximations.
It is similar in spirit to the original theorem by Ramo and Shockley, encoding the geometry of the detector in the form of a (time-dependent) weighting field distribution.
I will show the origin of this result as a direct consequence of Maxwell’s equations and discuss how the original quasi-static theorem emerges as a special case. Due to its significant generality,
this new theorem applies to all devices that detect fields or radiation from charged particles. I will highlight applications ranging from particle physics to cosmic ray physics, where it enables the
computation of the radio signature of cosmic ray induced showers. | {"url":"https://indico.cern.ch/event/1044975/contributions/4663706/","timestamp":"2024-11-11T17:22:15Z","content_type":"text/html","content_length":"110959","record_id":"<urn:uuid:77b4ae3a-c889-4382-82f4-5a3414aed006>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00875.warc.gz"} |
How to run one batch in pytorch?
To run a single batch in PyTorch, you need to follow these steps:
1. Define your batch data: Create a batch of input data and corresponding target labels that you want to pass through your model for inference or training.
2. Convert the batch data into PyTorch tensors: PyTorch uses tensors to represent data. Convert your batch data into PyTorch tensors using torch.tensor() or torch.from_numpy().
3. Pass the batch data through your model: Use your PyTorch model to process the input data and generate the output predictions. You can do this by simply calling your model with the batch data
tensor as input (e.g., output = model(batch_data)).
4. Compute loss (if applicable): If you are training your model, you may need to compute a loss value to measure how well the model's predictions match the target labels. Use a loss function like
torch.nn.CrossEntropyLoss() or torch.nn.MSELoss() to calculate the loss between the model outputs and target labels.
5. Backpropagate gradients (if applicable): If you are training your model, you will need to backpropagate the gradients through the model to update the model parameters. Call loss.backward() to
compute the gradients and then use an optimizer like torch.optim.SGD or torch.optim.Adam to update the model parameters.
6. Run optimization step (if applicable): If you are training your model, use the optimizer to update the model parameters based on the computed gradients. Call optimizer.step() to update the
7. Repeat the above steps for each batch until you have processed all the training or validation data.
By following these steps, you can run a single batch in PyTorch for inference or training purposes. | {"url":"https://devhubby.com/thread/how-to-run-one-batch-in-pytorch","timestamp":"2024-11-02T03:34:46Z","content_type":"text/html","content_length":"114386","record_id":"<urn:uuid:d398bfb6-db62-4ac9-a0ae-4c2a55653a0e>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00360.warc.gz"} |
Loan Amortization Schedule for Excel
Loan Amortization Schedule
Create an Amortization Schedule in Excel or Google Sheets | Updated 9/11/2020
An amortization schedule is a list of payments for a mortgage or loan, which shows how each payment is applied to both the principal amount and the interest. The schedule shows the remaining balance
still owed after each payment is made, so you know how much you have left to pay. To create an amortization schedule using Excel, you can use our free amortization calculator which is able to handle
the type of rounding required of an official payment schedule. You can use the free loan amortization schedule for mortgages, auto loans, consumer loans, and business loans. If you are a small
private lender, you can download the commercial version and use it to create a repayment schedule to give to the borrower.
Loan Amortization Schedule
for Excel and Google Sheets
Over 1.5 million downloads!
⤓ Excel (.xlsx)
For: Excel 2010 or later
This spreadsheet-based calculator creates an amortization schedule for a fixed-rate loan, with optional extra payments.
Start by entering the total loan amount, the annual interest rate, the number of years required to repay the loan, and how frequently the payments must be made. Then you can experiment with other
payment scenarios such as making an extra payment or a balloon payment. Make sure to read the related blog article to learn how to pay off your loan earlier and save on interest.
The payment frequency can be annual, semi-annual, quarterly, bi-monthly, monthly, bi-weekly, or weekly. Values are rounded to the nearest cent. The last payment is adjusted to bring the balance to
Loan Payment Schedules: The workbook also contains 2 other worksheets for basic loan payment tracking. The difference between the two has to do with how unpaid interest is handled. In the first,
unpaid interest is added to the balance (negative amortization). In the second (the one shown in the screenshot), unpaid interest is accrued in a separate interest balance.
Note: In both cases, the Payment Date column is for reference only. This spreadsheet handles loans where calculations are not based on payment date. See the Simple Interest Loan spreadsheet if you
have a loan that accrues interest daily and the payment date matters.
Amortization Calculations
Interest Rate, Compound Period, and Payment Period
Usually, the interest rate that you enter into an amortization calculator is the nominal annual rate. However, when creating an amortization schedule, it is the interest rate per period that you use
in the calculations, labeled rate per period in the above spreadsheet.
Basic amortization calculators usually assume that the payment frequency matches the compounding period. In that case, the rate per period is simply the nominal annual interest rate divided by the
number of periods per year. When the compound period and payment period are different (as in Canadian mortgages), a more general formula is needed (see my amortization calculation article).
Some loans in the UK use an annual interest accrual period (annual compounding) where a monthly payment is calculated by dividing the annual payment by 12. The interest portion of the payment is
recalculated only at the start of each year. The way to simulate this using our Amortization Schedule is by setting both the compound period and the payment frequency to annual.
Negative Amortization
There are two scenarios in which you could end up with negative amortization in this spreadsheet (interest being added to the balance). The first is if your payment isn't enough to cover the
interest. The second is if you choose a compound period that is shorter than the payment period (for example, choosing a weekly compound period but making payments monthly).
A loan payment schedule usually shows all payments and interest rounded to the nearest cent. That is because the schedule is meant to show you the actual payments. Amortization calculations are much
easier if you don't round. Many loan and amortization calculators, especially those used for academic or illustrative purposes, do not do any rounding. This spreadsheet rounds the monthly payment and
the interest payment to the nearest cent, but it also includes an option to turn off the rounding (so that you can quickly compare the calculations to other calculators).
When an amortization schedule includes rounding, the last payment usually has to be changed to make up the difference and bring the balance to zero. This might be done by changing the Payment Amount
or by changing the Interest Amount. Changing the Payment Amount makes more sense to me, and is the approach I use in my spreadsheets. So, depending on how your lender decides to handle the rounding,
you may see slight differences between this spreadsheet, your specific payment schedule, or an online loan amortization calculator.
Extra Payments
With this template, it is really quite simple to handle arbitrary extra payments (prepayments or additional payments on the principal). You simply add the extra payment to the amount of principal
that is paid that period. For fixed-rate loans, this reduces the balance and the overall interest, and can help you pay off your loan early. But, the normal payment remains the same (except for the
last payment required to bring the balance to zero - see below).
This spreadsheet assumes that the extra payment goes into effect on the payment due date. There is no guarantee that this is how your lender handles the extra payment! However, this approach makes
the calculations simpler than prorating the interest.
Zero Balance
One of the challenges of creating a schedule that accounts for rounding and extra payments is adjusting the final payment to bring the balance to zero. In this spreadsheet, the formula in the Payment
Due column checks the last balance to see if a payment adjustment is needed. In words, this is how the payment is calculated:
If you are on your last payment or the normal payment is greater than (1+rate)*balance, then pay (1+rate)*balance, otherwise make the normal payment.
Payment Type
The "payment type" option lets you choose whether payments are made at the beginning of the period or end of the period. Normally, payments are made at the end of the period. If you choose the
"beginning of period" option, no interest is paid in the first payment, and the Payment amount will be slightly different. You may need to change this option if you are trying to match the
spreadsheet up with a schedule that you received from your lender. This spreadsheet doesn't handle prorated or "per diem" periods that are sometimes used in the first and last payments.
Loan Payment Schedule
One way to account for extra payments is to record the additional payment. This spreadsheet includes a second worksheet (the Loan Payment Schedule) that allows you to record the actual payment
instead. (Just in case you find that more convenient.) For example, if the monthly payment is $300, but you pay $425, you can either record this as an additional $125, or use the Loan Payment
Schedule worksheet to record the actual payment of $425.
• 6/9/2014: New Loan Payment Schedule in Beta - Based on frequent requests for a more advanced loan payment tracker, we're experimenting with providing a new spreadsheet - see Bonus #4 above.
• 7/2/2013: Avoid Payday Loans - People in need of fast cash are often tempted by Payday loans but they should be avoided at all costs! Payday loan fees and interest rates are higher than all other
sources of lending, and they can trap you in a vicious cycle of repeat borrowing to pay off previous payday loans. Look for other sources of money if you must borrow.
• 6/6/2013: Student Loan Refinancing - It used to be much easier to consolidate or refinance student loans than it is today. If you meet income requirements, Federal student loans can often be
refinanced with a lower interest rate, but for individuals who are earning higher income or who carry private student loans – the options are much more limited than they used to be.
• 5/22/2013: Understanding Amortization Calculation - The process of paying off a mortgage or loan that includes both a principal balance and interest payments. A free online amortization
calculator will let you see what different payment frequencies mean for paying off your debt.
More Amortization Info | {"url":"https://totalsheets.com/ExcelTemplates/loan-amortization-schedule.html","timestamp":"2024-11-07T17:04:17Z","content_type":"text/html","content_length":"34358","record_id":"<urn:uuid:9c76c125-5f2d-4657-83ff-58c52557064c>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00376.warc.gz"} |
Log Likelihood estimation
Performing likelihood ratio tests and computing information criteria for a given model requires computation of the log-likelihood
${\cal L}{\cal L}_y(\hat{\theta}) = \log({\cal L}_y(\hat{\theta})) \triangleq \log(p(y;\hat{\theta}))$
where $\hat{\theta}$ is the vector of population parameter estimates for the model being considered. The log-likelihood cannot be computed in closed form for nonlinear mixed effects models. It can
however be estimated in a general framework for all kinds of data and models using the importance sampling Monte Carlo method. This method has the advantage of providing an unbiased estimate of the
log-likelihood – even for nonlinear models – whose variance can be controlled by the Monte Carlo size.
Two different algorithms are proposed to estimate the log-likelihood: by linearization and by Importance sampling. The estimated log-likelihoods are computed and stored in the LLInformation folder in
the result folder. In this folder, two files are stored:
• logLikelihood.txt containing the OFV (objective function value), AIC, and BIC.
• individualLL.txt containing the -2LL for each individual.
Log-likelihood by importance sampling
The observed log-likelihood ${\cal LL}(\theta;y)=\log({\cal L}(\theta;y))$ can be estimated without requiring approximation of the model, using a Monte Carlo approach. Since
${\cal LL}(\theta;y) = \log(p(y;\theta)) = \sum_{i=1}^{N} \log(p (y_i;\theta))$
we can estimate $\log(p(y_i;\theta))$ for each individual and derive an estimate of the log-likelihood as the sum of these individual log-likelihoods. We will now explain how to estimate $\log(p(y_i;
\theta))$ for any individual i. Using the $\phi$-representation of the model (the individual parameters are transformed to be Gaussian), notice first that $p(y_i;\theta)$ can be decomposed as
$p(y_i;\theta) = \int p(y_i,\phi_i;\theta)d\phi_i = \int p(y_i|\phi_i;\theta)p(\phi_i;\theta)d\phi_i = \mathbb{E}_{p_{\phi_i}}\left(p(y_i|\phi_i;\theta)\right)$
Thus, $p(y_i;\theta)$ is expressed as a mean. It can therefore be approximated by an empirical mean using a Monte Carlo procedure:
1. Draw M independent values $\phi_i^{(1)}$, $\phi_i^{(2)}$, …, $\phi_i^{(M)}$ from the marginal distribution $p_{\phi_i}(.;\theta)$.
2. Estimate $p(y_i;\theta)$ with $\hat{p}_{i,M}=\frac{1}{M}\sum_{m=1}^{M}p(y_i | \phi_i^{(m)};\theta)$
By construction, this estimator is unbiased, and consistent since its variance decreases as 1/M:
$\mathbb{E}\left(\hat{p}_{i,M}\right)=\mathbb{E}_{p_{\phi_i}}\left(p(y_i|\phi_i^{(m)};\theta)\right) = p(y_i;\theta) ~~~~\mbox{Var}\left(\hat{p}_{i,M}\right) = \frac{1}{M} \mbox{Var}_{p_{\phi_i}}\
We could consider ourselves satisfied with this estimator since we “only” have to select M large enough to get an estimator with a small variance. Nevertheless, it is possible to improve the
statistical properties of this estimator.
The problem is that it is not possible to generate the $\phi_i^{(m)}$ with this conditional distribution, since that would require to compute a normalizing constant, which here is precisely $p(y_i;\
Nevertheless, this conditional distribution can be estimated using the Metropolis-Hastings algorithm described in the Metropolis-Hastings algorithm for simulating the individual parameters and a
practical proposal “close” to the optimal proposal $p_{\phi_i|y_i}$ can be derived. We can then expect to get a very accurate estimate with a relatively small Monte Carlo size M.
The mean and variance of the conditional distribution $p_{\phi_i|y_i}$ are estimated by Metropolis-Hastings for each individual i. Then, the $\phi_i^{(m)}$ are drawn with a noncentral student t-
$\phi_i^{(m)} = \mu_i + \sigma_i \times T_{i,m}$
where $\mu_i$ and $\sigma^2_i$ are estimates of $\mathbb{E}\left(\phi_i|y_i;\theta\right)$ and $\mbox{Var}\left(\phi_i|y_i;\theta\right)$, and $(T_{i,m})$ is a sequence of i.i.d. random variables
distributed with a Student’s t-distribution with $u$ degrees of freedom.
Remark: Even if $\hat{\cal L}_y(\theta)=\prod_{i=1}^{N}\hat{p}_{i,M}$ is an unbiased estimator of ${\cal L}_y(\theta)$, $\hat{\cal LL}_y(\theta)$ is a biased estimator of ${\cal LL}_y(\theta)$.
Indeed, by Jensen’s inequality, we have :
$\mathbb{E}\left(\log(\hat{\cal L}_y(\theta))\right) \leq \log \left(\mathbb{E}\left(\hat{\cal L}_y(\theta)\right)\right)=\log\left({\cal L}_y(\theta)\right)$
However, the bias decreases as M increases and also if $\hat{\cal L}_y(\theta)$ is close to ${\cal L}_y(\theta)$. It is therefore highly recommended to use a proposal as close as possible to the
conditional distribution $p_{\phi_i|y_i}$, which means having to estimate this conditional distribution before estimating the log-likelihood (i.e run task “individual parameter” with “Cond.
distribution” option).
Remark: The standard error of all the draws is proposed. It is a representation of impact of the variability of the draws of the proposed population parameters and not of the uncertainty of the
Advance settings for the log-likelihood
A t-distribution is used as proposal. The number of degrees of freedom of this distribution can be either fixed or optimized. In such a case, the default possible values are 2, 5, 10 and 20 degree of
freedom. A distribution with a small number of degree of freedom (i.e. heavy tails) should be avoided in case of stiff ODE’s defined models. We recommend to set a degree of freedom at 5.
Log-likelihood by linearization
The likelihood of the nonlinear mixed effects model cannot be computed in a closed-form. An alternative is to approximate this likelihood by the likelihood of the Gaussian model deduced from the
nonlinear mixed effects model after linearization of the function f (defining the structural model) around the predictions of the individual parameters $(\phi_i; 1 \leq i \leq N)$.
Notice that the log-likelihood can not be computed by linearization for discrete outputs (categorical, count, etc.) nor for mixture models or when the posterior distribution have been estimated for
some parameters with priors.
Best practices: When should I use the linearization and when should I use the importance sampling?
Firstly, it is only possible to use the linearization algorithm for the continuous data. In that case, this method is generally much faster than importance sampling method and also gives good
estimates of the LL. The LL calculation by model linearization will generally be able to identify the main features of the model. More precise– and time-consuming – estimation procedures such as
stochastic approximation and importance sampling will have very limited impact in terms of decisions for these most obvious features. Selection of the final model should instead use the unbiased
estimator obtained by Monte Carlo. In the warfarin example, the evaluation of the log-likelihood (along with the AIC and BIC) is presented with the CPU time.
│ Method │ -2LL │ AIC │ BIC │ CPU time [s] │
│ Linearization │ 2178.78 │ 2220.78 │ 2251.56 │ 1.5 │
│ Important sampling │ 2119.76 │ 2161.76 │ 2192.54 │ 27.1 │ | {"url":"https://monolix2016.lixoft.com/tasks/log-likelihood-estimation/","timestamp":"2024-11-08T18:59:29Z","content_type":"text/html","content_length":"83106","record_id":"<urn:uuid:e29631fe-6bf0-4fb4-84e4-01a60156a332>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00460.warc.gz"} |
Science:Math Exam Resources/Courses/MATH110/December 2013/Question 01 (a)
MATH110 December 2013
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q2 • Q3 (a) • Q3 (b) • Q4 (a) • Q4 (b) • Q4 (c) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) • Q6 (c) • Q7 • Q8 • Q9 • Q10 (a) • Q10 (b) • QS01 10(a) • QS01 10(b) •
Question 01 (a)
Determine whether the following statement is true or false. If it is true, provide justification. If it is false, provide a counterexample.
a) The graph of ${\displaystyle f(x)={\frac {1}{2}}((x+3)^{2}-2)}$ crosses the x-axis.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is
correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still
stuck, go for the next hint.
Hint 1
There are multiple ways to see this and each hint corresponds to each solution.
For a hint to the first solution, you could always just try to solve this directly using the quadratic formula.
Hint 2
You could also try to solve this be exploiting the fact that the parabola is given in vertex form.
Hint 3
One could also reason by starting with the parabola ${\displaystyle \displaystyle y=x^{2}}$ and from here trying to get the given parabola by a series of transformations. Argue how these
transformations affect the number of roots of the parabola.
Hint 4
One could also use the Intermediate Value Theorem (IVT) to solve this problem.
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
• If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you
are stuck or if you want to check your work.
• If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem
and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
Solution 1
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
Note: The first three parts of this question was identical to that of the October Midterm.
True: There are multiple ways to see this.
Solve the quadratic equation.
The function expands to: ${\displaystyle f(x)={\frac {1}{2}}x^{2}+3x+{\frac {7}{2}}}$. Which then gives:
{\displaystyle {\begin{aligned}x&={\frac {-3\pm {\sqrt {9-4\left({\frac {1}{2}}\right)\left({\frac {7}{2}}\right)}}}{2\left({\frac {1}{2}}\right)}}\\&={\frac {-3\pm {\sqrt {9-7}}}{1}}\\x&=-3\pm {\
sqrt {2}}\\\end{aligned}}}
Solution 2
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
We can also use the vertex formula by seeing that ${\displaystyle f(x)={\frac {1}{2}}(x+3)^{2}-1}$, with vertex ${\displaystyle (-3,-1)}$ which is below the x-axis. Next, observe that a parabola is
continuous (because its a polynomial). Combining this with the fact that the parabola has a positive coefficient of ${\displaystyle {\frac {1}{2}}}$, meaning that the parabola is curving upwards, we
get that it must cross the ${\displaystyle x}$ axis.
Solution 3
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
We can view this as a graph transformation of ${\displaystyle y=x^{2}}$ which touches the ${\displaystyle x}$-axis. Then we shift it ${\displaystyle 3}$ to the left and then ${\displaystyle 2}$ down,
so it still touches the ${\displaystyle x}$-axis. This is then followed by a vertical compression of ${\displaystyle 2}$ (or expansion by ${\displaystyle 0.5}$). This does not change direction of
curvature of the function and so it must still touch the ${\displaystyle x}$-axis.
Solution 4
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
We can also use the IVT. The function is a quadratic and hence is continuous. We then look at ${\displaystyle f(-3)=-1<0}$ and then ${\displaystyle f(0)=3.5>0}$. Thus there must be a point between $
{\displaystyle -3}$ and ${\displaystyle 0}$ that crosses the ${\displaystyle x}$-axis.
Click here for similar questions
MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Function properties, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag | {"url":"https://wiki.ubc.ca/Science:Math_Exam_Resources/Courses/MATH110/December_2013/Question_01_(a)","timestamp":"2024-11-09T13:41:38Z","content_type":"text/html","content_length":"65547","record_id":"<urn:uuid:ee2fe880-3d2f-4c0d-9a4a-d6222a256b21>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00468.warc.gz"} |
Cdml Te X
The element CdmlTeX is used for writing mathematical formulas primarily. This feature uses TeX as a subsystem called from ProWiki to render the formulas.
The text:
x_{1,2} = - { b \over 2 } \pm \sqrt { { b^2 \over 4 } - c }
You can test this on .
• scale: influences the size of the graphic produced (default: 0.5); e.g. [scale=0.8] which produces a picture 60% larger.
Installation of the TeX subsystem
To use this feature you should follow these instructions:
• install TeX (the tetex package, you need tex, gs, dvips)
• install the graphics package netpbm (you need pnmcrop, pbmtopgm,pnmcrop, pnmscale, ppmtogif)
• create seven corresponding symbolic links in the ExecutablesDir, so that the ProWiki script can call these programs.
• install the following Perl script tex2gif in the ExecutablesDir, which is called by the ProWiki script and actually does most of interfacing.
cd $2
tex $1.tex
dvips $1.dvi
time gs -r300 -dNOPAUSE -dBATCH -sDEVICE=pbmraw -sPAPERSIZE=a3 -sOutputFile=$1.pbm $1.ps
pnmcrop $1.pbm >$1.pnm
pbmtopgm 3 3 $1.pnm >$1.pgm
pnmscale $3 $1.pgm >$1-ss.pgm
ppmtogif $1-ss.pgm >$1.gif
rm $1*.p* $1.dvi $1.log
• check all permissions to make sure that the ProWiki script is allowed to call the tex2gif script and the other executables.
Note: If you want to use very special symbols, fonts or font sizes then you must install these according to instructions of the TeX system. ProWiki can only pass commands to TeX, and TeX can only act
with what it has available.
Note: If you have LaTeX installed, then you also have TeX installed. LaTeX is a macro package that makes TeX easier to use. It has no effect on the ProWiki => TeX interface.
Note: TeX produces BlackAndWhite pixel graphics which is not very beautiful on screen. So we use the trick to produce the formula in triple resolution of what we need on screen (300 dpi instead of
100dpi) and then scale this down by this factor 3 going to gray scale, which produces nicer, smooth antialiased pictures.
Implementation notes
The TeX subsystem uses SubSystemPictureCaching.
FolderWikiFeatures FolderCdml | {"url":"http://www.prowiki.org/prowiki/wiki.cgi?CdmlTeX","timestamp":"2024-11-06T09:14:37Z","content_type":"application/xhtml+xml","content_length":"16304","record_id":"<urn:uuid:23867dbf-75f0-4830-b3ae-4a4e5cce0c41>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00549.warc.gz"} |
Installed MathJax-LaTeX WordPress plugin for blog.bigsmoke.us
Soon, I wish to document some statistical issues I’ve been running into lately due to the lack of understanding maintained by my recipe-level statistics training. Also, I’d like to document some of
the things I did learn over the years, and, hopefully, the things I find out while working myself out of the modelling mountain that I currently find so difficult to mount. For this I will need to
use some mathematical language, which is why I just installed the MathJaX-LaTeX WordPress plugin. MathJax-LaTeX uses the MathJax JavaScript library to support LaTeX and MathML math equations in
WordPress without requiring the browser to have MathML support.
As for testing it, my knowledge (\(K()\)) of MathML (\(M\)) is pretty much nonexistant, while I’m quite comfortable with LaTeX (\(L\)) math exations, which is why I’m typing the LaTeX code “K(M) \ll
K(L)” to generate the following simple equation:
\(K(M) \ll K(L)\)
1 Comment
1. To quickly look up the LaTeX command for the much less greater than operator, I used the Detexify LaTeX handwritten symbol recognition, which does exactly what is says it does. | {"url":"https://blog.bigsmoke.us/2015/03/16/mathjax-latex-wordpress-plugin","timestamp":"2024-11-11T11:31:13Z","content_type":"text/html","content_length":"47738","record_id":"<urn:uuid:c08c0f44-0eea-4d94-aea6-368981321a08>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00535.warc.gz"} |
Bounds on the index of an intersection of two subgroups - Solutions to Linear Algebra Done Right
Bounds on the index of an intersection of two subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.10
Let $G$ be a group and let $H,K \leq G$ be subgroups of finite index; say $[G:H] = m$ and $[G:K] = n$. Prove that $$\mathsf{lcm}(m,n) \leq [G: H \cap K] \leq mn.$$ Deduce that if $m$ and $n$ are
relatively prime, then $[G: H \cap K] = [G : H] \cdot [G : K]$.
Lemma 1: Let $A$ and $B$ be sets, $\varphi : A \rightarrow B$ a map, and $\Phi$ an equivalence relation on $A$. Suppose that if $a_1 \,\Phi\, a_2$ then $\varphi(a_1) = \varphi(a_2)$ for all $a_1,a_2
\in A$. Then $\psi : A/\Phi \rightarrow B$ given by $[a]_\Phi \mapsto \varphi(a)$ is a function. Moreover, if $\varphi$ is surjective, then $\psi$ is surjective, and if $\varphi(a_1) = \varphi(a_2)$
implies $a_1 \,\Phi\, a_2$ for all $a_1,a_2 \in A$, then $\psi$ is injective.
Proof: $\psi$ is clearly well defined. If $\varphi$ is surjective, then for every $b \in B$ there exists $a \in A$ such that $\varphi(a) = b$. Then $\psi([a]_\Phi) = b$, so that $\psi$ is surjective.
If $\psi([a_1]) = \psi([a_2])$, then $\varphi(a_1) = \varphi(a_2)$, so that $a_1 \,\Phi\, a_2$, and we have $[a_1] = [a_2]$. $\square$
First we prove the second inequality.
Lemma 2: Let $G$ be a group and let $H,K \leq G$ be subgroups. Then there exists an injective map $\psi : G/(H \cap K) \rightarrow G/H \times G/K$.
Proof: Define $\varphi : G \rightarrow G/H \times G/K$ by $\varphi(g) = (gH,gK)$. Now if $g_2^{-1}g_1 \in H \cap K$, then we have $g_2^{-1}g_1 \in H$, so that $g_1H = g_2H$, and $g_2^{-1}g_1 \in K$,
so that $g_1K = g_2K$. Thus $\varphi(g_1) = \varphi(g_2)$. Moreover, if $(g_1H,g_2K) = (g_1H,g_2K)$ then we have $g_2^{-1}g_1 \in H \cap K$, so that $g_1(H \cap K) = g_2(H \cap K)$. By Lemma 1, there
exists an injective mapping $\psi : G/(H \cap K) \rightarrow G/H \times G/K$ given by $\psi(g(H \cap K)) = (gH,gK)$. $\square$
As a consequence, if $[G : H]$ and $[G : K]$ are finite, $[G : H \cap K] \leq [G : H] \cdot [G : K]$.
Now to the first inequality.
Lemma 3: Let $G$ be a group and $K \leq H \leq G$. Let $S$ be a set of coset representatives of $G/H$. Then the mapping $\psi : S \times H/K \rightarrow G/K$ given by $\psi(g,hK) = ghK$ is bijective.
(Well defined) Suppose $h_2^{-1}h_1 \in K$. Then $h_1K = h_2K$, so that $gh_1K = gh_2K$, and we have $\psi(g,h_1K) = \psi(g,h_2K)$.
(Surjective) Let $gK \in G/K$. Now $g \in \overline{g}H$ for some $\overline{g} \in S$; say $g = \overline{g}h$. Then $\psi(\overline{g},hK) = gK$, so that $\psi$ is surjective.
(Injective) Suppose $\psi(g_1,h_1K) = \psi(g_2,h_2K)$. Then $g_1h_1K = g_2h_2K$; in particular, $g_1h_1 \in g_2h_2K \subseteq g_2H$, so that $g_1 \in g_2H$ and hence $g_2^{-1}g_1 \in H$. So $g_1H =
g_2H$, and in fact $g_2 = g_1$. Thus $h_1K = h_2K$, and $\psi$ is injective. $\square$
As a consequence, we have $[G : H] \cdot [H : K] = [G : K]$.
Now in this case we have $H \cap K \leq H \leq G$. Thus $m$ divides $[G : H \cap K]$ and $n$ divides $[G : H \cap K]$, so that $\mathsf{lcm}(m,n)$ divides $[G : H \cap K]$. In particular, since all
numbers involved are natural, $$\mathsf{lcm}(m,n) \leq [G : H \cap K].$$Finally, if $m$ and $n$ are relatively prime, then $\mathsf{lcm}(m,n) = mn$, and we have $[G : H \cap K] = mn$. | {"url":"https://linearalgebras.com/solution-abstract-algebra-exercise-3-2-10.html","timestamp":"2024-11-13T02:59:03Z","content_type":"text/html","content_length":"56905","record_id":"<urn:uuid:bf391ba0-0bcd-48d7-8704-50553fac5f9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00472.warc.gz"} |
Research | Alex Strang's Homepage | Chicago
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Research: Welcome
I am interested in stochastic processes and modeling in biological systems, the interplay of structure and dynamics in networks, and Bayesian inference and inverse problems. My work relies heavily on
linear algebra, non-equilibrium thermodynamics, optimization, and computational topology.
I currently work on variational inference problems, noise propagation in biological networks, self-organizing edge flows, and functional form game theory (with exciting applications to multi-agent
training and visualization). My published work includes the study of extinction events and large deviations, geometric solutions to moment closure problems, and the characterization of network
structure in tournaments. I also work on data visualization techniques that summarize the interactions of competing agents.
I received the 2022 Suzuki Postdoctoral Fellowship Award in recognition of my research.
Research: About
Research: About
Research: Services
bottom of page | {"url":"https://www.alexanderstrang.com/research","timestamp":"2024-11-13T21:32:10Z","content_type":"text/html","content_length":"494759","record_id":"<urn:uuid:05ec8a28-d24c-4e47-b09c-81b7620e0fb1>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00879.warc.gz"} |
Time fractional Kupershmidt equation: symmetry analysis and explicit series solution with convergence analysis
Time fractional Kupershmidt equation: symmetry analysis and explicit series solution with convergence analysisArticle
In this work, the fractional Lie symmetry method is applied for symmetry analysis of time fractional Kupershmidt equation. Using the Lie symmetry method, the symmetry generators for time fractional
Kupershmidt equation are obtained with Riemann-Liouville fractional derivative. With the help of symmetry generators, the fractional partial differential equation is reduced into the fractional
ordinary differential equation using Erdélyi-Kober fractional differential operator. The conservation laws are determined for the time fractional Kupershmidt equation with the help of new
conservation theorem and fractional Noether operators. The explicit analytic solutions of fractional Kupershmidt equation are obtained using the power series method. Also, the convergence of the
power series solutions is discussed by using the implicit function theorem.
Volume: Volume 27 (2019), Issue 2
Published on: December 31, 2019
Imported on: May 11, 2022
Keywords: General Mathematics,[MATH]Mathematics [math] | {"url":"https://cm.episciences.org/9494","timestamp":"2024-11-09T06:03:16Z","content_type":"application/xhtml+xml","content_length":"50330","record_id":"<urn:uuid:c0d7e601-8ee1-49c1-8aea-76da51de7cc8>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00466.warc.gz"} |
Case Studies
The following case studies are available for download (see the comment in the
root folder of some of the downloaded vf-files for additional information) :
1. Folder Sorting contains the verification of the sorting algorithms
□ Insertionsort
□ Bubblesort
□ Minimumsort
□ Selectionsort
□ Treesort
□ Mergesort
□ Quicksort
□ Heapsort
□ Natural Mergesort
2. Folder Number Theory contains
□ proofs of inequations and series (Textbook Exercises)
□ a proof of the irrationality of the square root of any prime number
□ a proof of the Binomial Theorem and some properties of Binomial Coefficients
□ a verification of Eratosthenes' method for computing all primes in a given interval
□ a proof of Fermat's Little Theorem using the Binomial Theorem (pdf)
□ a proof of Fermat's Little Theorem using reduced residue systems (pdf)
□ a proof of Euler's Theorem (pdf)
□ a proof of Wilson's Theorem (prime modulus) (pdf)
□ a proof of Wilson's Theorem (composite modulus)
□ a verification of the RSA encryption method
□ a proof of the infinitude of primes using Euclid's method
□ a proof of the infinitude of primes using Fermat Numbers
□ two proofs of the infinitude of primes using the factorial function
□ a proof of the infinitude of primes using pronic numbers
□ a proof of soundness, completeness and uniqueness of prime factorization (pdf)
□ a proof of the boundedness of the smallest prime factor by the square root of a composite
□ a proof of Bézout's Lemma (extended Euclidean algorithm)
□ a verification of Montgomery Multiplication (pdf)
□ a verification of Newton-Raphson Iteration for Multiplicative Inverses Modulo Powers of Any Base (pdf)
□ the verification of a test for (non-)divisibility of numbers by considering the residues of their cross sums in a positional numeral system
□ a proof of the Chinese Remainder Theorem (pdf)
3. Folder Propositional Logic contains soundness and completeness proofs for
□ a sequent calculus
□ an implicational calculus
□ unit resolution for Horn clause sets
□ the Boyer-Moore tautology checker
□ the Davis-Putnam procedure
4. Folder Matching & Unification contains proofs of soundness, completeness and most-generality of
□ a first-order matching algorithm
□ a first-order unification algorithm
5. Folder Boyer-Moore contains 3 case studies from the Boyer-Moore corpus, viz.
□ verification of the Boyer-Moore Fast String Search algorithm
□ verification of the Boyer-Moore tautology checker
□ proof of the unsolvability of the Halting Problem
6. Folder VFR 16-01 Proofs contains 3 case studies from the paper Fermat, Euler, Wilson - Three Case Studies in Number Theory.
□ The remaining case studies of the paper are collected in the folder Number Theory above.
7. Folder Miscellaneous contains
□ proofs of properties of the Ackermann function
□ a verification of soundness, completeness and log-boundedness of Binary Search (pdf)
□ the verification of a recursive decent analyzer for a small LL(1)-grammar
□ the verification of a code generator generating machine code from while-programs (pdf)
□ a verification of Dijkstra's Shortest Path Algorithm
□ the verification of two algorithms deciding the word problem for regular languages
Last update 2020-06-12 | {"url":"https://verifun.jimdofree.com/case-studies/","timestamp":"2024-11-07T22:49:34Z","content_type":"text/html","content_length":"99384","record_id":"<urn:uuid:a29f2934-4112-47b8-b23a-31016967ad3d>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00697.warc.gz"} |
ring Archives - Quantum Calculus
The dual multiplication of the ring of networks is topological interesting as Kuenneth holds for this multiplication and Euler characteristic is a ring homomorphism from this dual ring to the ring of
A ring of networks
Assuming the join operation to be the addition, we found a multiplication which produces a ring of oriented networks. We have a commutative ring in which the empty graph is the zero element and the
one point graph is the one element. This ring contains the usual integers as a subring. In the form of positive and negative complete subgraphs. | {"url":"https://www.quantumcalculus.org/tag/ring/","timestamp":"2024-11-02T08:00:40Z","content_type":"text/html","content_length":"53549","record_id":"<urn:uuid:cff746a3-e165-4539-8958-ce67416885fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00423.warc.gz"} |
SOLVED: . Esfandairi Enterprises is considering a new three-year...
| SkillsMatt
Assignment Instructions/ Description
Image transcription textEsfandairi Enterprises is considering a new three-year expansion project that requires an initial fixed asset investment of $2,370,000.
The fixed asset falls into the three-year MACRS class (MACRS schedule). The project is estimated to generate $1,755,000 in annual
sales, with costs of $656,000. The project requires an initial investment in net working capital of $340,000, and the fixed asset will
have a market value of $315,000 at the end of the project.
a. If the tax rate is 24 percent, what is the project's Year 0 net cash flow? Year 1? Year 2? Year 3?
Note: A negative answer should be indicated by a minus sign. Do not round intermediate calculations and round your answers
to two decimal places, e.g., 32.16.
b. If the required return is 9 percent, what is the project's NPV?
Note: Do not round intermediate calculations and round your answer to two decimal places, e.g., 32.16.
a. Year 0 cash flow
Year 1 cash flow
Year 2 cash flow
Year 3 cash flow
b. NPV... Show more� | {"url":"https://www.skillsmatt.com/tutors-problem/39504/esfandairi-enterprises-is-considering-a-new-three-year-3","timestamp":"2024-11-06T14:26:57Z","content_type":"text/html","content_length":"64041","record_id":"<urn:uuid:40652b25-7b65-4736-9351-27d0c63ecf69>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00689.warc.gz"} |
Array manipulation and booleans in numpy, hunting for what went wrong
Recently I was doing some raster processing with
but I was getting some surprising results.
The goal was to convert some continuous value into a binary value based on a treshold and then detect places where only A was True (value: -1), where only B was True (value: 1) and where both A and B
where True (value: 0). With a treshold set to 2 I came with the following formula:
(A > 2) - (B > 2)
But after calling the gdal_calc.py with my formula and inspecting the results I only got values of 0 and 1.
After inspecting gdal_calc.py I noticed that it uses
and more specifically numpy arrays for the raster manipulation.
This how my python shell session went (
>>> import numpy as np
>>> a = np.array([1,2,3,4])
>>> b = np.array([1,5,3,2])
>>> print(a > 2)
[False False True True]
>>> print(b > 2)
[False True True False]
>>> print(True-False)
>>> print(False-True)
>>> print((a>2)-(b>2))
[False True False True]
>>> print((a>2)*1-(b>2)) # we got a winner
[ 0 -1 0 1]
The problem was that boolean substraction in Python does generate the expected numeric results but the results where converted back into a boolean array by numpy after the substraction. And indeed
converting -1, 1 or any other non zero number to a boolean generates a True which when converted back to number for writing the raster to disk gives you the value 1.
The solution was to force at least one of the arrays to be numeric so that we substract numeric values.
>>> bool(1)
>>> bool(2)
>>> bool(-1)
>>> bool(0)
>>> print((a>2)*1)
[0 0 1 1]
If you want to try this yourself on Windows then the easiest way to install gdal is with the
OSGeo4W installer
. A
windows installer for numpy
can be found on the website by Christopher Golke but consider also installing the full SciPy stack with one of the
Scientific Python distributions
What surprising results have you encountered with numpy.array or gdal_calc ? | {"url":"https://www.samuelbosch.com/2014/04/array-manipulation-and-booleans-in.html","timestamp":"2024-11-03T05:39:20Z","content_type":"application/xhtml+xml","content_length":"73285","record_id":"<urn:uuid:9b28ca4a-b4c1-4d82-9f25-70208915fd30>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00614.warc.gz"} |
3.4 Electric Power and Energy
Learning Objectives
Learning Objectives
By the end of this section, you will be able to do the following:
• Calculate the power dissipated by a resistor and the power supplied by a power supply
• Calculate the cost of electricity under various circumstances
The information presented in this section supports the following AP® learning objectives and science practices:
• 5.B.9.8 The student is able to translate between graphical and symbolic representations of experimental data describing relationships among power, current, and potential difference across a
resistor. (S.P. 1.5)
Power in Electric Circuits
Power in Electric Circuits
Power is associated by many people with electricity. Knowing that power is the rate of energy use or energy conversion, what is the expression for electric power? Power transmission lines might come
to mind. We also think of lightbulbs in terms of their power ratings in watts. Let us compare a 25-W bulb with a 60-W bulb. (See Figure 3.17(a)) Since both operate on the same voltage, the 60-W bulb
must draw more current to have a greater power rating. Thus the 60-W bulb's resistance must be lower than that of a 25-W bulb. If we increase voltage, we also increase power. For example, when a 25-W
bulb that is designed to operate on 120 V is connected to 240 V, it briefly glows very brightly and then burns out. Precisely how are voltage, current, and resistance related to electric power?
Electric energy depends on both the voltage involved and the charge moved. This is expressed most simply as $PE=qV,PE=qV, size 12{"PE"= ital "qV"} {}$ where $qq size 12{q} {}$ is the charge moved and
$VV size 12{V} {}$ is the voltage (or more precisely, the potential difference the charge moves through). Power is the rate at which energy is moved, and so electric power is
3.26 $P=PEt=qVt.P=PEt=qVt. size 12{P = { { ital "PE"} over {t} } = { { ital "qV"} over {t} } "."} {}$
Recognizing that current is $I=q/tI=q/t size 12{I = q/t} {}$ (note that $Δt=tΔt=t size 12{Δt=t} {}$ here), the expression for power becomes
3.27 $P=IV.P=IV. size 12{P = ital "IV."} {}$
Electric power ($P ) P)$ is simply the product of current times voltage. Power has familiar units of watts. Since the SI unit for potential energy (PE) is the joule, power has units of joules per
second, or watts. Thus, $1 A⋅V=1 W. 1 A⋅V=1 W.$ For example, cars often have one or more auxiliary power outlets with which you can charge a cell phone or other electronic devices. These outlets may
be rated at 20 A, so that the circuit can deliver a maximum power $P=IV=(20 A)(12 V)=240 W.P=IV=(20 A)(12 V)=240 W.$ In some applications, electric power may be expressed as volt-amperes or even
kilovolt-amperes $(1 kA ⋅V= 1 kW).(1 kA ⋅V= 1 kW). size 12{"1 kA " cdot V=" 1 kW"} {}$
To see the relationship of power to resistance, we combine Ohm's law with $P=IV.P=IV. size 12{P = ital "IV"} {}$ Substituting $I=V/RI=V/R size 12{I = ital "V/R"} {}$ gives $P=(V/R)V=V2/R.P=(V/R)V=V2/
R. size 12{P = \( V/R \) V=V rSup { size 8{2} } R} {}$ Similarly, substituting $V=IRV=IR size 12{V= ital "IR"} {}$ gives $P=I(IR)=I2R.P=I(IR)=I2R. size 12{P =I \( ital "IR" \) = I rSup { size 8{2} }
R} {}$ Three expressions for electric power are listed together here for convenience.
3.28 $P = IV P = IV size 12{P = ital "IV"} {}$
3.29 $P = V 2 R P = V 2 R size 12{P = { {V rSup { size 8{2} } } over {R} } } {}$
3.30 $P= I 2 R P= I 2 R$
Note that the first equation is always valid, whereas the other two can be used only for resistors. In a simple circuit, with one voltage source and a single resistor, the power supplied by the
voltage source and that dissipated by the resistor are identical. In more complicated circuits, $PP size 12{P} {}$ can be the power dissipated by a single device and not the total power in the
Making Connections: Using Graphs to Calculate Resistance
As $p∝ I 2 p∝ I 2$ and $p∝ V 2 p∝ V 2$, the graph for power versus current or voltage is quadratic. An example is shown in the figure below.
Using equations (20.29) and (20.30), we can calculate the resistance in each case. In graph (a), the power is 50 W when current is 5 A; hence, the resistance can be calculated as $R=P/ I 2 =50/ 5 2 =
2 Ω. R=P/ I 2 =50/ 5 2 =2 Ω.$ Similarly, the resistance value can be calculated in graph (b) as $R= V 2 /P= 10 2 /50=2 Ω R= V 2 /P= 10 2 /50=2 Ω$
Different insights can be gained from the three different expressions for electric power. For example, $P=V2/RP=V2/R size 12{P = V rSup { size 8{2} } /R} {}$ implies that the lower the resistance
connected to a given voltage source, the greater the power delivered. Furthermore, since voltage is squared in $P= V 2 /R , P= V 2 /R ,$ the effect of applying a higher voltage is perhaps greater
than expected. Thus, when the voltage is doubled to a 25-W bulb, its power nearly quadruples to about 100 W, burning it out. If the bulb's resistance remained constant, its power would be exactly 100
W, but at the higher temperature its resistance is higher, too.
Example 3.7 Calculating Power Dissipation and Current: Hot and Cold Power
(a) Consider the examples given in Ohm's Law: Resistance and Simple Circuits and Resistance and Resistivity. Then find the power dissipated by the car headlight in these examples, both when it is hot
and when it is cold. (b) What current does it draw when cold?
Strategy for (a)
For the hot headlight, we know voltage and current, so we can use $P=IVP=IV size 12{P = ital "IV"} {}$ to find the power. For the cold headlight, we know the voltage and resistance, so we can use $P=
V2/RP=V2/R size 12{P = V rSup { size 8{2} } /R} {}$ to find the power.
Solution for (a)
Entering the known values of current and voltage for the hot headlight, we obtain
3.31 $P=IV=(2.50 A)(12.0 V)= 30.0 W.P=IV=(2.50 A)(12.0 V)= 30.0 W. size 12{P = ital "IV" = \( 2 "." "50 A" \) \( "12" "." "0 V" \) =" 30" "." "0 W."} {}$
The cold resistance was $0.350 Ω , 0.350 Ω ,$ and so the power it uses when first switched on is
3.32 $P=V2R=(12.0 V)20.350Ω= 411 W.P=V2R=(12.0 V)20.350Ω= 411 W. size 12{P = { {V rSup { size 8{2} } } over {R} } = { { \( "12" "." "0 V" \) rSup { size 8{2} } } over {0 "." "350" %OMEGA } } =" 411
W."} {}$
Discussion for (a)
The 30 W dissipated by the hot headlight is typical. But the 411 W when cold is surprisingly higher. The initial power quickly decreases as the bulb's temperature increases and its resistance
Strategy and Solution for (b)
The current when the bulb is cold can be found several different ways. We rearrange one of the power equations, $P= I 2 R , P= I 2 R ,$ and enter known values, obtaining
3.33 $I=PR=411 W0.350 Ω= 34.3 A.I=PR=411 W0.350 Ω= 34.3 A. size 12{I = sqrt { { {P} over {R} } } = sqrt { { {"411 W"} over {0 "." "350 " %OMEGA } } } =" 34" "." "3 A."} {}$
Discussion for (b)
The cold current is remarkably higher than the steady-state value of 2.50 A, but the current will quickly decline to that value as the bulb's temperature increases. Most fuses and circuit breakers
(used to limit the current in a circuit) are designed to tolerate very high currents briefly as a device comes on. In some cases, such as with electric motors, the current remains high for several
seconds, necessitating special slow blow fuses.
The Cost of Electricity
The Cost of Electricity
The more electric appliances you use and the longer they are left on, the higher your electric bill. This familiar fact is based on the relationship between energy and power. You pay for the energy
used. Since $P=E/t,P=E/t, size 12{P=E/t} {}$ we see that
3.34 $E = Pt E = Pt size 12{E = ital "Pt"} {}$
is the energy used by a device using power $PP size 12{P} {}$ for a time interval $t.t. size 12{t} {}$ For example, the more lightbulbs burning, the greater $PP size 12{P} {}$ used; the longer they
are on, the greater $tt size 12{t} {}$ is. The energy unit on electric bills is the kilowatt-hour ($kW⋅hkW⋅h size 12{"kw" cdot h} {}$), consistent with the relationship $E=Pt.E=Pt. size 12{E = ital
"Pt"} {}$ It is easy to estimate the cost of operating electric appliances if you have some idea of their power consumption rate in watts or kilowatts, the time they are on in hours, and the cost per
kilowatt-hour for your electric utility. Kilowatt-hours, like all other specialized energy units such as food calories, can be converted to joules. You can prove to yourself that $1 kW⋅h = 3.6×106
J.1 kW⋅h = 3.6×106 J. size 12{1"kW" cdot "h = 3" "." 6´"10" rSup { size 8{6} } " J"} {}$
The electrical energy $(E)(E) size 12{E} {}$) used can be reduced either by reducing the time of use or by reducing the power consumption of that appliance or fixture. This will not only reduce the
cost, but it will also result in a reduced impact on the environment. Improvements to lighting are some of the fastest ways to reduce the electrical energy used in a home or business. About 20
percent of a home's use of energy goes to lighting, while the number for commercial establishments is closer to 40 percent. Fluorescent lights are about four times more efficient than incandescent
lights—this is true for both the long tubes and the compact fluorescent lights (CFL). (See Figure 3.17(b)) Thus, a 60-W incandescent bulb can be replaced by a 15-W CFL, which has the same brightness
and color. CFLs have a bent tube inside a globe or a spiral-shaped tube, all connected to a standard screw-in base that fits standard incandescent light sockets. (Original problems with color,
flicker, shape, and high initial investment for CFLs have been addressed in recent years.) The heat transfer from these CFLs is less, and they last up to 10 times longer. The significance of an
investment in such bulbs is addressed in the next example. New white LED lights (which are clusters of small LED bulbs) are even more efficient (twice that of CFLs) and last 5 times longer than CFLs.
However, their cost is still high.
Making Connections: Energy, Power, and Time
The relationship $E=PtE=Pt size 12{E = ital "Pt"} {}$ is one that you will find useful in many different contexts. The energy your body uses in exercise is related to the power level and duration of
your activity, for example. The amount of heating by a power source is related to the power level and time it is applied. Even the radiation dose of an X-ray image is related to the power and time of
Example 3.8 Calculating the Cost Effectiveness of Compact Fluorescent Lights (CFL)
If the cost of electricity in your area is 12 cents per kWh, what is the total cost (capital plus operation) of using a 60-W incandescent bulb for 1,000 hours (the lifetime of that bulb) if the bulb
cost 25 cents? (b) If we replace this bulb with a compact fluorescent light that provides the same light output, but at one-quarter the wattage, and which costs $1.50 but lasts 10 times longer
(10,000 hours), what will that total cost be?
To find the operating cost, we first find the energy used in kilowatt-hours and then multiply by the cost per kilowatt-hour.
Solution for (a)
The energy used in kilowatt-hours is found by entering the power and time into the expression for energy.
3.35 $E=Pt=(60 W)(1,000 h)= 60,000 W ⋅ hE=Pt=(60 W)(1,000 h)= 60,000 W ⋅ h size 12{E = ital "Pt" = \( "60 W" \) \( "1000 h" \) =" 60,000 W " cdot " h."} {}$
In kilowatt-hours, this is
3.36 $E= 60.0 kW ⋅ h.E= 60.0 kW ⋅ h. size 12{E =" 60" "." "0 kW " cdot " h."} {}$
Now the electricity cost is
3.37 $cost = ( 60.0 kW ⋅ h ) ( 0.12 /kW ⋅ h ) = 7.20. cost = ( 60.0 kW ⋅ h ) ( 0.12 /kW ⋅ h ) = 7.20.$
The total cost will be $7.20 for 1,000 hours (about one-half year at 5 hours per day).
Solution for (b)
Since the CFL uses only 15 W and not 60 W, the electricity cost will be $7.20/4 = $1.80. The CFL will last 10 times longer than the incandescent, so that the investment cost will be 1/10 of the bulb
cost for that time period of use, or 0.1($1.50) = $0.15. Therefore, the total cost will be $1.95 for 1,000 hours.
Therefore, it is much cheaper to use the CFLs, even though the initial investment is higher. The increased cost of labor that a business must include for replacing the incandescent bulbs more often
has not been figured in here.
Making Connections: Take-Home Experiment—Electrical Energy Use Inventory
1) Make a list of the power ratings on a range of appliances in your home or room. Explain why something like a toaster has a higher rating than a digital clock. Estimate the energy consumed by these
appliances in an average day (by estimating their time of use). Some appliances might only state the operating current. If the household voltage is 120 V, then use $P=IV.P=IV. size 12{P = ital "IV"}
{}$ 2) Check out the total wattage used in the rest rooms of your school's floor or building. (You might need to assume the long fluorescent lights in use are rated at 32 W.) Suppose that the
building was closed all weekend and that these lights were left on from 6 p.m. Friday until 8 a.m. Monday. What would this oversight cost? How about for an entire year of weekends? | {"url":"https://texasgateway.org/resource/34-electric-power-and-energy?book=79106&binder_id=78811","timestamp":"2024-11-04T11:18:15Z","content_type":"text/html","content_length":"86060","record_id":"<urn:uuid:c3831dcc-336a-4993-bdff-8c6063d85a86>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00575.warc.gz"} |
Levenshtein Distance Algorithm https://www.mzekiosmancik.com/wp-content/uploads/2020/02/Levenshtein_distance_animation-1024x668.gif 1024 668 mezo https://secure.gravatar.com/avatar/
Levenshtein Distance Algorithm
Hello my fellow Padawans
Couple days ago I had to use an algorithm for comparing string and I want to write something about Levenshtein Algorithm. This algorithm is for measure the metric distance between 2 string text.
Informally, the Levenshtein distance between two words is the minimum number of single-character edits (insertions, deletions or substitutions) required to change one word into the other. It is named
after the Soviet mathematician Vladimir Levenshtein, who considered this distance in 1965.^
Mathematically, the Levenshtein distance between two strings a, b (of length |a| and |b| respectively) is given by leva,b(|a|,|b|)
where 1(ai≠bi) is the indicator function equal to 0 when ai≠bi and equal to 1 otherwise, and leva, b(i,j) is the distance between the first i characters of a and the first j characters of b.
Note that the first element in the minimum corresponds to deletion (from a to b), the second to insertion and the third to match or mismatch, depending on whether the respective symbols are the same.
We can use this algorithm for string matching and spell checking
This algorithm calculates the number of edit operation that are necessary to modify one string to another string. Fro using this algorithm for dynamic programming we can use these steps :
1- A matrix is initialized measuring in the (m, n) cells the Levenshtein distance between the m-character prefix of one with the n-prefix of the other word.
2 – The matrix can be filled from the upper left to the lower right corner.
3- Each jump horizontally or vertically corresponds to an insert or a delete, respectively.
4- The cost is normally set to 1 for each of the operations.
5- The diagonal jump can cost either one, if the two characters in the row and column do not match else 0, if they match. Each cell always minimizes the cost locally.
6- This way the number in the lower right corner is the Levenshtein distance between these words.
An example that features the comparison of “HONDA” and “HYUNDAI”,
Following are two representations: Levenshtein distance between “HONDA” and “HYUNDAI” is 3.
The Levenshtein distance can also be computed between two longer strings. But the cost to compute it, which is roughly proportional to the product of the two string lengths, makes this impractical.
Thus, when used to aid in fuzzy string searching in applications such as record linkage, the compared strings are usually short to help improve speed of comparisons.
Here’s the code that you can use the Levenshtein Distance and calculate percentage between 2 string.
See the Pen levenshtein.js by mzekiosmancik (@mzekiosmancik) on CodePen. | {"url":"https://www.mzekiosmancik.com/tag/comparasion/","timestamp":"2024-11-14T01:48:07Z","content_type":"text/html","content_length":"189759","record_id":"<urn:uuid:93ad5a0f-aee5-4d6b-9786-04567db88ed8>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00820.warc.gz"} |
Double Slit Ahead of Single Slit?
This is similar to my earlier query regarding the sequence of topics that are introduced. My earlier post was the order of introducing the concept of energy and the concept of momentum. In this post,
it is the issue of the sequence of introducing the double slit interference ahead of the single-slit diffraction.
This sequence is done in Knight's text "Physics for Scientists and Engineers". I don't follow that sequence because I prefer to introduce the single-slit diffraction first, show the diffraction
pattern, and then introduce the double slit. The fact that the double slit pattern has interference pattern inside a single-slit diffraction envelope is easier to explain after the students already
know about the single-slit diffraction.
What do you think? How did you teach this topic, or how did you learn this topic?
1 comment:
Douglas Natelson said...
I think I learned double-slit first, in a simplified form - each slit acting as a point source of spherical waves in a Huygens-style approach, so that you can look at the phase difference between
the two at the screen and get the overall interference pattern. Then, you build up single slit diffraction as a sum —> integral of such sources over the single slit width. | {"url":"https://physicsandphysicists.blogspot.com/2024/04/double-slit-ahead-of-single-slit.html","timestamp":"2024-11-10T15:47:46Z","content_type":"text/html","content_length":"147136","record_id":"<urn:uuid:c27f4d4d-d55b-443a-a8b2-d74c53cc843b>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00082.warc.gz"} |
Need help Inverse of an operator
Need help Inverse of an operator
• Thread starter einai
• Start date
In summary: Originally posted by Hurkyl I'm not sure if you're off the right track, or if you're on the right track but are missing a detail or two... so I'll give an example of a wrong proof and a
right proof. :smile:Wrong proof:A-1 is given by that formula, and if we plug in a zero for one of the eigenvalues, the sum diverges, so A-1 doesn't exist.
Another quantum question...I feel dumb
A Hermitian operator A has the spectral decomposition
A = Σ a[n]|n><n| (summation in n)
where A|n> = a[n]|n> (the a[n]'s are eigenvalues of A, and |n>'s the eigenstates).
So, how can I find the spectral decomposition of the inverse of A so that
AA^-1 = A^-1A = 1?
My intuition would be A = Σ (1/a[n])|n><n| (summation in n), since 1 = Σ |k><k| (summation in k), but I don't think it's that easy.
Thanks in advance!
Try multiplying the given sum for A and your sum for A^-1 and see if you get 1 as the answer.
Originally posted by Hurkyl
Try multiplying the given sum for A and your sum for A^-1 and see if you get 1 as the answer.
Thanks. I have a question about multiplying the 2 sums though. Should I make n --> n' for one of them, then sum over n and n'? Like this:
A = Σ a
|n><n| (summation in n)
= Σ (1/a
)|n'><n'| (summation in n')
then AA
= ΣΣ a
) |n><n|n'><n'| (summation in n and n') ?
And what is the condition imposed on A so that the inverse exists? I have no idea...
Last edited:
Right, that's how you do the summation. (it will simplify to 1)
As for the condition imposed on A, your formula for A^-1 contains a strong hint as to what that condition might me...
Originally posted by Hurkyl
Right, that's how you do the summation. (it will simplify to 1)
As for the condition imposed on A, your formula for A^-1 contains a strong hint as to what that condition might me...
Thank you!
*scratches head*
Hmm...is it that A and A
must have the same eigenstates, so that <n|n'> gives a delta function? And eigenvalues of A must be none zero otherwise the term in the inverse diverges?
It is true that A and A^-1 have the same eigenstates. For example, if |1> is an eigenstate of A with nonzero eigenvalue λ, then:
|1> = A^-1A|1> = A^-1λ|1>
and thus A^-1|1> = λ^-1|1>
So any eigenstate of A is an eigenstate of A^-1 (and vice-versa, by symmetry).
More importantly, we can always choose an orthonormal eigenbasis, in which <n|n'> would indeed be a delta function. (I would presume the basis used by spectral decomposition would be orthonormal, but
don't quote me on that!)
And yes, the right condition here is that all of the eigenvalues of A be nonzero. However, there is still work to do! At the moment, your formula only proves that if all of the eigenvalues of A are
nonzero then there exists an inverse! You still need to prove that if A has a zero eigenvalue then it cannot have an inverse.
Originally posted by Hurkyl
It is true that A and A^-1 have the same eigenstates. For example, if |1> is an eigenstate of A with nonzero eigenvalue λ, then:
|1> = A^-1A|1> = A^-1λ|1>
and thus A^-1|1> = λ^-1|1>
So any eigenstate of A is an eigenstate of A^-1 (and vice-versa, by symmetry).
That's a very good explanation, thank you.
More importantly, we can always choose an orthonormal eigenbasis, in which <n|n'> would indeed be a delta function. (I would presume the basis used by spectral decomposition would be orthonormal,
but don't quote me on that!)
Yes, they're orthonormal basis since it's the summation over the eigenstates.
And yes, the right condition here is that all of the eigenvalues of A be nonzero. However, there is still work to do! At the moment, your formula only proves that if all of the eigenvalues of A
are nonzero then there exists an inverse! You still need to prove that if A has a zero eigenvalue then it cannot have an inverse.
Yikes, I'm not too sure if I understand this part... I mean, if A has a zero eigenvalue, then its inverse would have an eigenstate with eigenvalue 1/0 which diverges...couldn't that prove A cannot
have an inverse?
I mean, if A has a zero eigenvalue, then its inverse would have an eigenstate with eigenvalue 1/0 which diverges...couldn't that prove A cannot have an inverse?
I'm not sure if you're off the right track, or if you're on the right track but are missing a detail or two... so I'll give an example of a wrong proof and a right proof.
Wrong proof:
is given by that formula, and if we plug in a zero for one of the eigenvalues, the sum diverges, so A
doesn't exist.
This is wrong because you haven't proved that A
must have the form given by your formula; you can only prove that the formula works when the eigenvalues are nonzero.
It might be the case that there's another formula that will give you the inverse when one of the eigenvalues
zero. (It turns out that this is not the case, but from just the information I mentioned above, we can't determine this fact!)
Right proof:
Suppose |1> is an eigenstate of A with eigenvalue 0. Then:
|1> = A
A|1> = A
0|1> = A
0 = 0
But since |1> is not the zero vector, we have a contradiction in assuming A
This is kinda like what you were saying, because we have:
|1> = 0 A
Implying A
has an "infinite" eigenvalue which is impossible... I just wasn't sure if you were aiming at this ponit or not.
Originally posted by Hurkyl
Right proof:
Suppose |1> is an eigenstate of A with eigenvalue 0. Then:
|1> = A^-1A|1> = A^-10|1> = A^-1 0 = 0
But since |1> is not the zero vector, we have a contradiction in assuming A^-1 exists.
This is kinda like what you were saying, because we have:
|1> = 0 A^-1 |1>
Implying A^-1 has an "infinite" eigenvalue which is impossible... I just wasn't sure if you were aiming at this ponit or not.
Oh, I see! Now I understand... I've always been bad at proving things
FAQ: Need help Inverse of an operator
What is an inverse of an operator?
The inverse of an operator is a mathematical concept that refers to the operation that undoes the original operation. It is denoted by the symbol ^-1.
Why is finding the inverse of an operator important?
Finding the inverse of an operator is important in solving equations, simplifying expressions, and understanding the relationship between different operations. It allows us to reverse the effects of
an operation and find the original value.
How do you find the inverse of an operator?
The process of finding the inverse of an operator depends on the type of operator. For basic arithmetic operations, such as addition, subtraction, multiplication, and division, the inverse can be
found by simply performing the opposite operation. For more complex operations, such as logarithms and trigonometric functions, there are specific methods and formulas to find the inverse.
What is the difference between the inverse of an operator and the reciprocal?
The inverse of an operator is the operation that undoes the original operation, while the reciprocal is the multiplicative inverse of a number or expression. In other words, the reciprocal of a
number is the value that, when multiplied by the original number, gives a result of 1.
Can every operator have an inverse?
No, not every operator has an inverse. For an operator to have an inverse, it must be both one-to-one (each input has only one output) and onto (all outputs have a corresponding input). Additionally,
some operations, such as division by zero, do not have an inverse. | {"url":"https://www.physicsforums.com/threads/need-help-inverse-of-an-operator.7777/","timestamp":"2024-11-08T05:54:59Z","content_type":"text/html","content_length":"116114","record_id":"<urn:uuid:cf77cee2-c0de-4865-ae72-10e1c413e51e>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00329.warc.gz"} |
Elementary Geometry for College Students (6th Edition) Chapter 1 - Section 1.3 - Early Definitions and Postulates - Exercises - Page 28 33d
To get eight congruent segments you can use construction to divide the segment in half. Dive each new segment in half two more times to create eight congruent segments.
You can help us out by revising, improving and updating this answer.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | {"url":"https://www.gradesaver.com/textbooks/math/geometry/CLONE-68e52840-b25a-488c-a775-8f1d0bdf0669/chapter-1-section-1-3-early-definitions-and-postulates-exercises-page-28/33d","timestamp":"2024-11-03T10:31:18Z","content_type":"text/html","content_length":"64282","record_id":"<urn:uuid:450af728-5f28-48a9-9634-ffcc0f645f57>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00592.warc.gz"} |
Removal Examinations of Math Service Courses (1st Sem AY 2023-24) | Institute of Mathematics
Removal Examinations of Math Service Courses (1st Sem AY 2023-24)
The removal examinations for Math service courses (Math 21, Math 22, Math 23, Math 30, and Math 40) will be administered on 26 January 2024 (Friday), 1 – 3 PM, at the Institute of Mathematics
Building. Room assignments will be posted at the entrance of the building.
Secure a removal exam permit from your home college. Hard copy of the accomplished permit must be presented and submitted to the proctor on the exam day. NO PERMIT, NO EXAM.
Please bring the following on the day of the exam:
• accomplished removal exam permit
• ID with photo
• bluebooks and non-erasable black or blue pen
For other math courses, kindly contact your instructor. | {"url":"https://math.upd.edu.ph/2024/01/removal-examinations-of-math-service-courses-1st-sem-ay-2023-24","timestamp":"2024-11-11T19:57:55Z","content_type":"text/html","content_length":"51900","record_id":"<urn:uuid:045d85a4-2e52-4841-b712-53648277b97e>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00399.warc.gz"} |
Problem of the Week
Problem B and Solution
Baking Cookies
Felix and Vera are baking cookies. Their recipe bakes \(12\) cookies and uses the following ingredients:
• \(\frac{1}{2}\) cup of butter
• \(\frac{1}{3}\) cup of sugar
• \(1\) cup of flour
• \(\frac{2}{3}\) teaspoon of vanilla
a. Felix and Vera decide to triple the recipe. How much of each ingredient will they need?
b. The cookies are so good that Felix and Vera plan to make \(60\) cookies for a fundraiser.
i. How much butter will they need?
ii. Each batch of cookies takes \(11\) minutes to bake, and their oven can fit only \(24\) cookies at a time. How long will it take to bake all the cookies for the fundraiser?
a. If they triple the recipe, then the amounts of each ingredient will be as follows:
● Butter: \(\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=\frac{3}{2}=1 \frac{1}{2}\) cups
● Sugar: \(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=1\) cup
● Flour: \(1+1+1=3\) cups
● Vanilla: \(\frac{2}{3}+\frac{2}{3}+\frac{2}{3}=\frac{6}{3}=2\) teaspoons
i. Since \(12 \times 5 = 60\), they will need to make \(5\) batches of cookies. So the amount of butter they will need is \(5 \times \frac{1}{2} = \frac{5}{2} = 2\frac{1}{2}\) cups.
ii. Their oven can fit only \(24\) cookies at a time, which is the same as \(2\) batches. Since they need to make \(5\) batches, they will need to bake \(2\) batches, then another \(2\) batches,
then \(1\) batch. So the total baking time would be \(11+11+11=33\) minutes. | {"url":"https://cemc.uwaterloo.ca/sites/default/files/documents/2024/POTWB-24-N-07-S-229.html","timestamp":"2024-11-09T12:11:08Z","content_type":"application/xhtml+xml","content_length":"74985","record_id":"<urn:uuid:40956441-d315-4d6c-9120-7c41fd58c49b>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00499.warc.gz"} |
12+ Sequence Diagram Explanation | Robhosking Diagram12+ Sequence Diagram Explanation
12+ Sequence Diagram Explanation
12+ Sequence Diagram Explanation. Once the user has been retrieved from the database how does the system decide whether to accept or reject a login. A sequence diagram shows interacting individuals
along the top of the diagram and messages passed among them arranged in temporal order down the page.
Object Oriented Design Tools from www.csis.pace.edu
A sequence diagram simply depicts interaction between objects in a sequential order. A sequence diagram shows interacting individuals along the top of the diagram and messages passed among them
arranged in temporal order down the page. Sequence diagrams show object interactions arranged in a time sequence (refer figure 5.10).
The sequence diagram is an interaction diagram of uml.
12+ Sequence Diagram Explanation. The flow of events can be used to determine what objects and interactions are required to accomplish the. Sequence diagram is the most common kind of interaction
diagram , which focuses on the message interchange between a number of lifelines. Learn about sequence diagram notations and messages. You can have several kinds of participants (actors and others),
arrows, notes, groups. | {"url":"https://robhosking.com/12-sequence-diagram-explanation/","timestamp":"2024-11-10T21:01:29Z","content_type":"text/html","content_length":"66118","record_id":"<urn:uuid:9a09afae-a589-4c9e-9c7c-41e1fff9b56b>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00322.warc.gz"} |
Algorithms and Data Structures Books
Advances in Evolutionary Algorithms
by Witold Kosinski, 2008, 284 pages, 40MB, PDF
Advances in Genetic Programming, Vol. 3
edited by L. Spector, W.B. Langdon, U. O'Reilly, P.J. Angeline, 1999, PDF
Algorithmic Algebra
by Bhubaneswar Mishra, 1993, 425 pages, 2.3MB, PDF
Algorithmic Graph Theory
by David Joyner, Minh Van Nguyen, Nathann Cohen, 2010, 105 pages, 760KB, PDF
Algorithmic Information Theory
by Gregory. J. Chaitin, 2003, 236 pages, 0.9MB, PDF
Algorithmic Mathematics
by Leonard Soicher, Franco Vivaldi, 2004, 94 pages, 0.5MB, PDF/PS
Algorithmic Number Theory
by J.P. Buhler, P. Stevenhagen, 2008, 662 pages, PDF
by Robert Sedgewick, Kevin Wayne, 2011
Algorithms and Data Structures
by Niklaus Wirth, 1985, 179 pages, 1.2MB, PDF
Algorithms and Data Structures for External Memory
by Jeffrey Scott Vitter, 2008, 191 pages, 1.1MB, PDF
Algorithms and Data Structures: The Basic Toolbox
by K. Mehlhorn, P. Sanders, 2008, PDF
Algorithms and Data Structures: With Applications to Graphics and Geometry
by Jurg Nievergelt, Klaus Hinrichs, 2011, 299 pp, 3.3MB, PDF
Algorithms for Clustering Data
by Anil K. Jain, Richard C. Dubes, 1988, 334 pages, 39MB, PDF
Algorithms for Modular Elliptic Curves
by J. E. Cremona, 1992, 351 pages, PDF
Algorithms for Reinforcement Learning
by Csaba Szepesvari, 2009, 98 pp, 1.6MB, PDF
Algorithms: Fundamental Techniques
by Macneil Shonle, Matthew Wilson, Martin Krischik, 2006, 68 pages, 1.2MB, PDF
Art Gallery Theorems and Algorithms
by Joseph O'Rourke, 1987, 296 pages, 11MB, PDF
Average Case Analysis of Algorithms on Sequences
by Wojciech Szpankowski, 2000, PS
Behavior of Algorithms
by Daniel Spielman, 2002, PDF
Categories, Types, and Structures
by Andrea Asperti, Giuseppe Longo, 1991, 300 pages, PDF
Clever Algorithms: Nature-Inspired Programming Recipes
by Jason Brownlee, 2011
Combinatorial Algorithms
by Albert Nijenhuis, Herbert S. Wilf, 1978, 316 pages, 5.5MB, PDF
Combinatorial Algorithms
by Jeff Erickson, 2003, 197 pages, 1.9MB, PDF
Communication Complexity (for Algorithm Designers)
by Tim Roughgarden, 2015, 150 pp, 2.8MB, PDF
Computational and Algorithmic Linear Algebra and n-Dimensional Geometry
by Katta G. Murty, 2001, 554 pages, PDF
Computational Geometry: Methods and Applications
by Jianer Chen, 1996, 227 pages, 1.3MB, PDF
Computer Arithmetic of Geometrical Figures: Algorithms and Hardware Design
by Solomon I. Khmelnik, 2013, 150 pp, 890KB, PDF
Data Structures
by Dave Mount, 2001, 123 pages, 730 KB, PDF
Data Structures and Algorithm Analysis
by Clifford A. Shaffer, 2012, 613 pp, 2.6MB, PDF
Data Structures and Algorithms
by Catherine Leung, 2017, 126 pp, multiple formats
Data Structures and Algorithms
by John Morris, 1998
Design and Analysis of Computer Algorithms
by David M. Mount, 2003, 135 pages, 0.8MB, PDF
The Design of Approximation Algorithms
by D. P. Williamson, D. B. Shmoys, 2010, 496 pages, 2.3MB, PDF
Efficient Algorithms for Sorting and Synchronization
by Andrew Tridgell, 1999, 115 pages, 410KB, PDF
Elementary Algorithms
by Larry LIU Xinyu, 2016, 622 pp, 5.8MB, PDF
Essentials of Metaheuristics
by Sean Luke, 2009, 233 pages, 5.3MB, PDF
Evolutionary Algorithms
edited by Eisuke Kita, 2011, 584 pages, 30MB, PDF
Evolved to Win
by Moshe Sipper, 2011, 193 pp, 1.9MB, PDF
From Algorithms to Z-Scores: Probabilistic and Statistical Modeling in Computer Science
by Norm Matloff, 2013, 486 pp, 3.4MB, PDF
Fundamental Data Structures
Wikipedia, 2011, 411 pp, multiple formats
Genetic Algorithms and Evolutionary Computation
by Adam Marczyk, 2004
Genetic Programming: New Approaches and Successful Applications
edited by Sebastian Ventura, 2012, 284 pp, 6.5MB, PDF
Greedy Algorithms
by Witold Bednorz, 2008, 586 pages, 47MB, PDF
Introduction to Algorithms
by Erik Demaine, Srinivas Devadas, Ronald Rivest, 2008, PDF
Introduction to Design Analysis of Algorithms
by K. Raghava Rao, 2013, 142 pp, 4.4MB, PDF
Knapsack Problems: Algorithms and Computer Implementations
by Silvano Martello, Paolo Toth, 1990, 308 pages, 23MB, PDF
Lecture Notes on Bucket Algorithms
by Luc Devroye, 1986, 142 pages, 4MB, PDF
LEDA: A Platform for Combinatorial and Geometric Computing
by K. Mehlhorn, St. Näher, 1999, 1034 pp, multiple PS files
Mathematics for Algorithm and Systems Analysis
by Edward A. Bender, S. Gill Williamson, 2005, 256 pages, PDF
Modern Computer Arithmetic
by Richard P. Brent, Paul Zimmermann, 2009, 239 pages, 1.9MB, PDF
Notes on Data Structures and Programming Techniques
by James Aspnes, 2015, 530 pp, 1.8MB, PDF
Open Data Structures: An Introduction
by Pat Morin, 2013, 336 pp, multiple formats
Optimization Algorithms on Matrix Manifolds
by P.-A. Absil, R. Mahony, R. Sepulchre, 2007, 240 pages, PDF
Planning Algorithms
by Steven M. LaValle, 2006, 842 pages, 13.2MB, PDF
Problem Solving with Algorithms and Data Structures Using Python
by Brad Miller, David Ranum, 2011
Problems on Algorithms, 2nd edition
by Ian Parberry, William Gasarch, 2002, 268 pages, 2.4MB, PDF
Purely Functional Data Structures
by Chris Okasaki, 1996, 162 pp, 620KB, PDF
Quantum Algorithms
by Michele Mosca, 2008, 71 pages, PDF/PS
Quantum algorithms for algebraic problems
by Andrew M. Childs, Wim van Dam, 2008, 52 pages, PDF/PS
Randomized Algorithms
by Wolfgang Merkle, 2001, 46 pp, 370KB, PDF
Search Algorithms and Applications
edited by Nashat Mansour, 2011, 494 pages, 18MB, PDF
Sequential and Parallel Sorting Algorithms
by H. W. Lang, 2000
Sorting and Searching Algorithms: A Cookbook
by Thomas Niemann, 2008, 36 pages, 150KB, PDF
Think Data Structures
by Allen B. Downey, 2016, 187 pp, 780KB, PDF
Topics in Theoretical Computer Science: An Algorithmist's Toolkit
by Jonathan Kelner, 2009, PDF | {"url":"https://www.phdpro.info/2021/05/algorithms-and-data-structures-books.html","timestamp":"2024-11-07T00:50:35Z","content_type":"text/html","content_length":"241020","record_id":"<urn:uuid:a52ed1b9-3fe6-42bf-b944-c74d79f4e3a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00181.warc.gz"} |