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CDSW151 - Editorial
Author: Pranjul Dubey
Editorialist: Swapnil Saxena
The question is to find the starting position of window of length k such that the xor of all the elements in that window is minimum. If multiple such windows exists we have to report the starting
position with max_index.
A naive solution would involve considering all the windows of lengths k. (Note: The no. of such windows is (n-k+1)). For every window, just iterate through all the elements and compare it with the
current minimum. If is smaller or equal update the minimum position.
However this is a O((n-k+1)*k) solution. To fasten it up one can use preprocessing. Maintain an array prefix_xor where prefix_xor(i) = a1 ^ a2 ^ … ^ ai. Using this ‘prefix_xor’ array, finding the xor
of a range becomes O(1) as XOR-SUM of a range [i, j] is just prefix_xor[i-1] ^ prefix_xor[j] for i < j.
This will reduce the complexity to O(n + (n – k + 1))
Setter’s solution can be found here.
Nice One… | {"url":"https://discusstest.codechef.com/t/cdsw151-editorial/9965","timestamp":"2024-11-11T00:42:41Z","content_type":"text/html","content_length":"22870","record_id":"<urn:uuid:1953060a-68e8-43cf-a4c7-99a58b33e23d>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00564.warc.gz"} |
The Stacks project
Definition 61.29.1. Let $\Lambda $ be a Noetherian ring and let $I \subset \Lambda $ be an ideal. Let $X$ be a scheme. An object $K$ of $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$ is called
constructible if
1. $K$ is derived complete with respect to $I$,
2. $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I}$ has constructible cohomology sheaves and locally has finite tor dimension.
We denote $D_{cons}(X, \Lambda )$ the full subcategory of constructible $K$ in $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$.
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LTL Equivalence (Until Relation)
+ General Questions (11)
A first counterexample is G(!a&b&!c) which is generated by the LTL teaching tool when we try to prove [c WU [b WU a]] -> [c WU [b SU a]].
Why is that so? If c is initially false, then [c WU [b WU a]] can only be satisfied when the condition [b WU a] is already initially true, and same way we must have that [b SU a] is initially false.
That is the case when all the time b holds but never a holds which is exactly the difference between the weak and the strong operator. This is your first formula.
Looking at your second formula, I guess your idea was to delay the first counterexample by one point of time to create a second example. To this end, you demand X G(!a&b&!c) and also F c which is
then equivalent to c. However, trying to prove
(F c) & X G(!a&b&!c) -> [c WU [b WU a]] & ![c WU [b SU a]]
gives as us another counterexmaple where initially a=c=1 and b=0 holds and from the next point of time on, we have a=c=0 and b=1. Thus, we initially have (F c) & X G(!a&b&!c) while the until formulas
reduce as follows
[c WU [b WU a]] = [b WU a] | c & X[c WU [b WU a]]
= a | b & X[b WU a] | c & X[c WU [b WU a]]
and since initially a=c=1 and b=0 holds, we have initially
[c WU [b WU a]] = 1 and also [c WU [b SU a]] = 1
So, that is why your second formula does not work (the above counterexample makes both S1 and S2 true while satisfying also your constraint).
Can we make your idea work, i.e., postpone the problem by one point of time? Looking at the above unrolled formula, we would have to have at the initial point a=b=0 and c=1 to achieve that [c WU [b
WU a]] = X[c WU [b WU a]]. So, you should better try
(!a&!b&c) & X G(!a&b&!c) -> [c WU [b WU a]] & ![c WU [b SU a]]
which encodes your idea above (if I understand it right). That will work.
How can we find another example? As usual, I suggest to prove [c WU [b WU a]] -> [c WU [b SU a]] under the additional assumption that excludes the current counterexample, i.e., !(G(!a&b&!c)) & [c WU
[b WU a]] -> [c WU [b SU a]]. If you try that with the LTL prover in the teaching tools, you will obtain a path where !a&b&!c and !a&b&c are alternating. I leave it up to you to describe that as an
LTL formula. | {"url":"https://q2a.cs.uni-kl.de/1069/ltl-equivalence-until-relation","timestamp":"2024-11-13T18:10:37Z","content_type":"text/html","content_length":"52207","record_id":"<urn:uuid:717ab89f-ff63-4b36-9b1e-40ba4ef11a5a>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00268.warc.gz"} |
Academic Bulletin
Undergraduate Programs
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• ENGR-E 490 Engineering Capstone Design I (3 cr.) Junior or senior standing. Engineering Capstone Design I is one of two capstone requirements for all Intelligent Systems Engineering students.
Students will design engineering projects based on their areas of concentration, which will be supported by dedicated faculty members. Students may choose to conduct advanced research, develop
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• ENGR-E 491 Engineering Capstone Design II (3 cr.) Junior or senior standing. Engineering Capstone Design II is the second of two capstone requirements for all Intelligent Systems Engineering
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develop prototypes, design new products or redesign existing products. | {"url":"https://bulletins.iu.edu/iub/soic/2018-2019/undergraduate/courses/engineering.shtml","timestamp":"2024-11-05T06:05:44Z","content_type":"application/xhtml+xml","content_length":"30198","record_id":"<urn:uuid:8eaf988d-5f74-4b7b-b7f9-dcae00b29614>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00252.warc.gz"} |
seminars - Lifting problem for universal quadratic forms over totally real cubic number fields
Lifting problem for universal quadratic forms asks for totally real number fields K which admits a positive definite quadratic form with rational integer coefficients that is universal over the ring
of integers of K. In this talk, we show that there is only one such totally real cubic field. Moreover, we show that there is no such biquadratic field. This is a joint work with Seokhyoung Lee. | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=seminars&page=44&l=en&sort_index=speaker&order_type=asc&document_srl=1068652","timestamp":"2024-11-03T13:58:04Z","content_type":"text/html","content_length":"44701","record_id":"<urn:uuid:8b7eb071-dcb9-44eb-881c-edd8b09df6b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00725.warc.gz"} |
Mastering the Random Number Game – How to Generate Numbers from 1 to 1000
Generating random numbers is a common task in various fields, ranging from computer science and statistics to gaming and cryptography. Random numbers play a crucial role in simulations, generating
unique identifiers, and creating unpredictable outcomes. In this blog post, we will explore different methods of generating random numbers and focus on generating random numbers in the range of 1 to
Understanding Random Numbers
Random numbers are numbers that are chosen without any predictable pattern, making them appear as if they were produced by a completely random process. Randomness is an essential characteristic that
random numbers should possess in order to be beneficial in various applications.
Random numbers find their use in a myriad of fields, including scientific research, statistical analysis, game development, and cryptography. They are used for tasks such as generating samples,
shuffling data, selecting random elements, and creating unpredictable encryption keys.
There are several algorithms available for generating random numbers, each with its own advantages and limitations. Some of the commonly used algorithms include the Linear Congruential Method and the
Mersenne Twister Algorithm, which we will discuss in detail later.
Generating Random Numbers using Built-in Functions
Many popular programming languages provide built-in functions to generate random numbers. These functions are often optimized for efficiency and randomness, making them a convenient choice for most
Examples in Python
Python offers a built-in module called random that provides various functions for generating random numbers. The random.randint(a, b) function can be used to generate a random integer between a and
b. To generate a random number between 1 and 1000, you can use the following code:
import random random_number = random.randint(1, 1000)
Examples in Java
In Java, the java.util.Random class provides methods to generate random numbers. The nextInt(n) method can be used to generate a random number between 0 and n. To generate a random number between 1
and 1000, you can use the following code:
import java.util.Random; Random random = new Random(); int random_number = random.nextInt(1000) + 1;
Examples in JavaScript
JavaScript offers the Math.random() function to generate random numbers between 0 and 1. To generate a random number between 1 and 1000, you can multiply the result of Math.random() by 1000 and then
round it up using the Math.ceil() function:
var random_number = Math.ceil(Math.random() * 1000);
Generating Random Numbers using Mathematical Formulas
In addition to using built-in functions, random numbers can also be generated using mathematical formulas. These formulas rely on complex mathematical computations to produce seemingly random
Linear Congruential Method
The Linear Congruential Method is a simple and widely used algorithm for generating random numbers. It relies on a recursive formula to generate a sequence of numbers that appear random. The formula
is as follows:
X[n+1] = (a * X[n] + c) mod m
Where X[n] is the current random number, X[n+1] is the next random number, a is a multiplier, c is an increment, and m is the modulus. By choosing appropriate values for a, c, and m, random numbers
can be generated within the desired range.
Mersenne Twister Algorithm
The Mersenne Twister Algorithm is a highly regarded pseudorandom number generator. It produces high-quality random numbers with a long period. The algorithm is based on a large prime number (a
Mersenne prime) and a series of bit-shift operations. The Mersenne Twister Algorithm is known for its excellent statistical properties and has become the default random number generator in many
programming languages.
Generating Random Numbers with Specific Requirements
When generating random numbers, it is often necessary to meet specific requirements, such as generating numbers within a specific range or ensuring uniqueness.
Generating random numbers within a specific range (1-1000)
To generate random numbers within the range of 1 to 1000, you can utilize the modulus operator. By taking the modulus of the generated random number with 1000 and adding 1, you can ensure that the
resulting number falls within the desired range:
random_number = (random_number % 1000) + 1;
Ensuring uniqueness of generated numbers
When generating a series of random numbers, it is important to ensure that each number is unique. One way to achieve this is by utilizing an array or a set to keep track of the generated numbers.
Before generating a new random number, you can check if it already exists in the array or set. If it does, generate a new number until a unique one is found.
Controlling randomness with seed values
Random number generators often utilize a seed value to initialize their internal state. By using the same seed value, the generator will produce the same sequence of random numbers. This can be
useful in scenarios where reproducibility is desired. By changing the seed value, you can generate a different sequence of random numbers.
Best Practices for Generating Random Numbers
When working with random number generation, it is important to follow best practices to ensure the quality and integrity of the generated numbers.
Using random number generators from trusted sources
Always make sure to use random number generators from trusted sources or libraries that have been thoroughly tested and validated. By relying on trusted generators, you can ensure that the generated
numbers have good statistical properties and are suitable for your specific use case.
Evaluating the quality of the generated numbers
It is crucial to assess the quality of the generated random numbers. This can be done by performing statistical tests to check for biases or repetitive patterns. There are several statistical tests
available, such as the Chi-Square test and the Kolmogorov-Smirnov test, that can help evaluate the randomness of the generated numbers.
Understanding bias and mitigating it
Bias refers to a systematic deviation of generated numbers from what is expected in a random distribution. It can occur due to defects in the random number generator or inappropriate use of the
generated numbers. Understanding bias is essential to ensure the reliability of random number generation. By utilizing appropriate algorithms and techniques, such as shuffling techniques and proper
seeding, bias can be minimized.
In conclusion, generating random numbers is a fundamental task in various fields and applications. Whether you use built-in functions or mathematical formulas, generating random numbers within a
specific range, ensuring uniqueness, and controlling randomness are important considerations. By following best practices and understanding the characteristics of random numbers, you can harness the
power of randomness in your projects and applications.
Throughout this blog post, we have explored different methods of generating random numbers and discussed their uses in various fields. We have also highlighted best practices for generating random
numbers and emphasized the importance of evaluating the quality of the generated numbers.
By mastering the random number generation process, you can unlock new possibilities and make your applications more dynamic and unpredictable. | {"url":"https://skillapp.co/blog/mastering-the-random-number-game-how-to-generate-numbers-from-1-to-1000/","timestamp":"2024-11-05T23:25:17Z","content_type":"text/html","content_length":"113095","record_id":"<urn:uuid:56036581-1a77-4248-a77e-a348ef88bb69>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00231.warc.gz"} |
Triangles Archives
Category Archives: Triangles
Translation Coordinates and Graph of Translated Image example question
Find the coordinates and the translated graph of A(-2, 3), B(4, -5), and C(0, 3) by following the rule of translation:
(x, y) → (x – 5, y + 6).
Solution to this Translation Geometry practice problem is given in the video below!
Finding Coordinate RULE of Translation given Graph of Translated Image example problem
Find the coordinate rule for the translation shown in the graph below.
Solution to this Translation Geometry practice problem is given in the video below!
Dilation Formulas and Graph with Origin the Center of Dilation example question
Graph the triangle with vertices A(-2, 4), B(1, -4), C(-3, -2) and its image after dilation by scale factor k = -3. The center of dilation is the origin.
Solution to this Dilation Geometry practice problem is given in the video below!
Dilation Formulas and Graph with Center of Dilation NOT the Origin example problem
Graph the triangle with vertices A(-2, 4), B(1, -4), C(-3, -2) and its image after dilation by scale factor k = 2. The center of dilation is the point (4, 5).
Solution to this Dilation Geometry practice problem is given in the video below!
How to Find the CENTER of DILATION COORDINATES using Given Figure and its Image problem
Find the coordinates of the center of dilation if coordinates of original figure are (-4, 14), (-10, 16), (-12, 10), (-14, 12) and coordinates of its image are (2, 1), (3, 4), (-1, 2), (4, 3).
Solution to this Dilation Geometry practice example is given in the video below!
Reflection Formulas example question
Draw the image of the polygon with coordinates A(-2, 3), B(0, 2), C(3, -4), D(4, 0) by reflecting it
a) through the y-axis
b) through the x-axis
c) through the origin
d) through the line y = x
Solution to this Reflection Geometry practice problem is given in the video below!
Rotation Formulas example question
a) Draw the image of the triangle with coordinates A(-6, -6), B(-6, 3), C(-2, 3) by rotating it 90 degrees clockwise
b) Draw the image of the quadrilateral with coordinates A(-3, -3), B(-1, 0), C(3, 0), D(5, -3) by rotating it 90 degrees counterclockwise
c) Draw the image of the triangle with coordinates A(-3, 2), B(1, 5), C(0, 0) by rotating it 90 degrees clockwise
d) Draw the image of the triangle with coordinates A(1, -3), B(3, 3), C(6, -3) by rotating it 180 degrees clockwise or counterclockwise
e) Draw the image of the quadrilateral with coordinates A(-5, -2), B(-4, 5), C(2, 5), D(2, 0) by rotating it 90 degrees clockwise
Solution to this Rotation Geometry practice problem is given in the video below!
Solving for X and Y in Angles example question
Find the values of x and y using the figure below.
Use these values to find the following:
Solution to this value of x and y in Angles Geometry practice problem is provided in the video below!
Angle Theorems Parallel Lines and Transversals example question
The following figure shows two parallel lines l and m and a transversal.
Use this figure to answer the following questions:
Solution to this Angles Parallel Lines Transversal Geometry practice problem is provided in the video below!
Angle Theorems Parallel Lines and Transversals example problem #2
Use the following figure to find the values of x and y.
Solution to this Angles Parallel Lines Transversal Geometry practice problem is provided in the video below!
Angle Theorems Parallel Lines and Transversals example #3
Use the figure below to solve for the values of x and y.
Solution to this Angles Parallel Lines Transversal Geometry practice problem is provided in the video below!
Angle Theorems Parallel Lines and Transversals example question #4
Use the given figure below to find the values of x and y.
Solution to this Angles Parallel Lines Transversal Geometry practice problem is provided in the video below!
Angle Theorems Parallel Lines and Transversals example problem #5
The figure below shows a polygon with some interior angle measures provided.
Find the values of x and y.
Solution to this Angles Parallel Lines Transversal Geometry practice problem is provided in the video below!
Horizontal and Vertical Components of a Force Vector problem example
A force of 46.3 pounds is applied at an angle of 34.8º to the horizontal. Resolve the force into horizontal and vertical components.
Solution to this Horizontal Vertical Components of a Force Vector word practice problem is provided in the video below!
HARD Parallel Perpendicular Weight Vector Components to Surface word problem
A weight of 75 pounds is resting on a surface inclined at an angle of 25º to the ground. Find the components of the weight parallel and perpendicular to the surface.
Solution to this Parallel Perpendicular Weight Vector Components word practice problem is provided in the video below!
VERY HARD Resultant Force Vector example question
Find the resultant of two forces, one with magnitude 155 pounds and direction N50ºW, and a second with magnitude 305 pounds and direction S55ºW.
Solution to this Resultant Force Vector word practice problem is provided in the video below!
Plane Wind vector example problem
An airplane has an airspeed of 430 miles per hour at a bearing of E45ºS (45 degrees South of East). The wind velocity is 35 miles per hour in the direction of N30ºE (30 degrees East of North). Find
the ground speed and true course of the plane using vectors.
Solution to this Plane Wind vector word practice problem is provided in the video below!
Law of Cosines word problem example
Two sides of a parallelogram are 9 and 15 units in length. The length of the shorter diagonal of the parallelogram is 14 units. Find the length of the long diagonal.
Solution to this Law of Cosines word practice problem is provided in the video below!
HARD Law of Cosines word problem
A new car leaves an auto transport trailer for a test drive in the flat surface desert in the direction N47ºW at constant speed of 65 miles per hour. The trailer proceeds at constant rate of 50 miles
per hour due East. If the car has enough fuel for exactly 3 hours of riding at constant speed, what is the maximum distance in the same direction that the car can cover in order to safely return to
the trailer?
Solution to this Law of Cosines word practice problem is provided in the video below!
Law of Sines word problem example
A radio antenna is attached to the top of the building. From a point 12.5 meters from the base of the building, on level ground, the angle of elevation of the bottom of the antenna is 47.2 degrees
and the angle of elevation of the top is 51.8 degrees. Find the height of the antenna.
Solution to this Law of Sines word practice problem is provided in the video below!
Law of Sines Triangle Area PROOF problem
Show that for any triangle the area is one-half the product of any two sides and sine of the angle formed between these sides. That is,
Solution to this Law of Sines Triangle Area PROOF practice problem is provided in the video below!
Hard Inverse Trigonometric Expression Value example question
Evaluate the following Trigonometric Expression:
Solution to this Inverse Trigonometric Expression Value practice problem is provided in the video below!
Rectangle Word Problem with ArcTangent example
A rectangle is 173 meters long and 106 meters high. Find the angle between diagonal and the longer side.
Solution to this Inverse Trigonometric Function word practice problem is provided in the video below!
Trigonometric Angle Sum and Difference example question
Find the value of the following Trigonometric Expressions using Angle Sum or Difference formulas:
Solution to this Trigonometric Angle Sum Difference practice problem is provided in the video below!
Finding Trigonometric Values Using Reference Angles example question
Find the value of the following Trigonometric Expressions using appropriate Reference Angles:
a) sin(120º)
c) tan(-45º)
e) sec(240º)
Solution to this Reference Angle practice problem is provided in the video below!
World’s HARDEST Easy Geometry problem example question
The following triangle, A and C, as well as line segments measure of angle AED, that is,
Note: Do not use Trigonometry or Calculus to solve this problem
Solution to this World’s Hardest Easy Geometry practice problem is provided in the video below!
Centroid Circumcenter Incenter Orthocenter properties example question
In the following video you will learn how to find the coordinates of the Orthocenter located outside the triangle in the standard xy-plane (also known as coordinate plane or Cartesian plane). In
acute and right triangles, the Orthocenter does not fall outside of the triangle. However, when the triangle in question is obtuse, that is, when one of its interior angles measures more than 90
degrees – the Orthocenter will be located outside the triangle. This is when you will need to understand the technique used to find its coordinates.
Solution to this Orthocenter in Obtuse Triangle Geometry practice problem is provided in the video below!
Centroid Circumcenter Incenter Orthocenter properties example question
In this video you will learn the basic properties of triangles containing Centroid, Orthocenter, Circumcenter, and Incenter. Then you can apply these properties when solving many algebraic problems
dealing with these triangle shape combinations.
Solution to this Centroid Circumcenter Incenter Orthocenter Geometry practice problem is provided in the video below!
Two Column Geometry Proof Intersecting Lines example problem
Given: GA = RA, AH = AT
Prove: GT is congruent to RH
Solution to this Two Column Geometric Proof practice problem is given in the video below!
Two Column Geometry Proof Trapezoid example
Given: AD = 8, BC = 8, and BC is congruent to CD
Prove: AD is congruent to CD
Solution to this Two Column Geometric Proof practice problem is given in the video below!
Two Column Geometry Proof Circle example question
Prove: (CD)^2 = (AD)(DB)
Solution to this Two Column Geometric Proof practice problem is given in the video below!
Two Column Geometry Proof Triangle example problem
Solution to this Two Column Geometric Proof practice problem is given in the video below!
Two Column Geometry Proof Triangle example #2
Prove: BFDE is a rhombus
Solution to this Two Column Geometric Proof practice problem is given in the video below!
Two Column Geometry Proof Circle example question #2
Given: Circle with center O and diameter
Solution to this Two Column Geometric Proof practice problem is given in the video below!
Two Column Geometry Proof Kite example problem
Given: Kite ABCD with
E, F, G, H are midpoints of
Prove: EFGH is a rectangle
Solution to this Two Column Geometric Proof practice problem is given in the video below!
Two Column Geometry Proof Triangle example #3
Solution to this Two Column Geometric Proof practice problem is given in the video below!
Two Column Geometry Proof Triangle example question #4
Given: is isosceles and
Solution to this Two Column Geometric Proof practice problem is given in the video below!
Two Column Geometry Proof Parallel Lines Transversal Triangle example problem
Solution to this Two Column Geometric Proof practice problem is given in the video below!
Shaded Area in a Square puzzle example question
In a unit square ABCD, point A is joined to the midpoint of BC, point B is joined to the midpoint of CD, point C is joined to the midpoint of DA, and point D is joined to the midpoint of AB. Find the
area of the shaded region.
Solution to this Puzzle practice problem is given in the video below!
Shaded Area in a Square puzzle example problem
In a unit square ABCD, M is a midpoint of AD, and AC is a diagonal. Find the area of the shaded regions.
Solution to this Puzzle practice problem is given in the video below!
Shaded Area in a Square puzzle example
In the unit square ABCD, M is the midpoint and AC and BD are diagonals. Find the area of the shaded region.
Solution to this Puzzle practice problem is given in the video below!
Shaded Area in a Square puzzle example question
In the following unit square ABCD, M is the midpoint and BD is a diagonal. Find the area of the shaded region.
Solution to this Puzzle practice problem is given in the video below!
Shaded Area in a Square puzzle TRIGONOMETRY example problem
A square with side 1 is rotated around one vertex by an angle
Find the area of the shaded region.
Solution to this Puzzle practice problem is given in the video below!
Shaded Area in a Rectangle puzzle example
In the following rectangle, an isosceles triangle is drawn. Find the area of the shaded region.
Solution to this Puzzle practice problem is given in the video below!
Shaded Area in a Rectangle puzzle example question
In the following rectangle below, find the area of the shaded region.
Solution to this Puzzle practice problem is given in the video below!
Perimeter as a Function of Area example question
a. Express the area, A , of an equilateral triangle as a function of one side S .
b. Express the perimeter, P , of the triangle as a function of the area A .
Solution to this Geometric Functions practice problem is given in the video below!
Polar Rectangular Coordinates Conversion example question
Convert the following Rectangular coordinates to Polar coordinates:
a. (2, -2)
b. (0, 3)
c. (3, 4)
Solution to this Polar Coordinates practice problem is given in the video below!
Rectangular Polar Coordinates Conversion example problem
Convert the following Polar coordinates to Rectangular coordinates:
b. (0, 3)
Solution to this Polar Coordinates practice problem is given in the video below!
Rectangular Polar Coordinates Conversion example
Convert the following Polar coordinates to Rectangular coordinates:
Solution to this Polar Coordinates practice problem is given in the video below!
Rectangular Equation & Polar Equation Conversion example question
Find Polar Equations from Rectangular Equation
Solution to this Polar Coordinates practice problem is given in the video below!
Polar Equations Sketch & X–Y Equation Conversion example problem
Sketch the given Polar Equations and find their corresponding X–Y Equation:
a. r = 4
Solution to this Polar Coordinates practice problem is given in the video below!
Hard Polar Equations Sketch example
Sketch the given Polar Equations:
Solution to this Polar Coordinates practice problem is given in the video below!
Parametric Equation Sketch & X–Y Equation Conversion example question
Sketch the given Parametric Equation and find its corresponding X–Y Equation:
Solution to this Parametric Equations & Curves practice problem is given in the video below!
Parametric Equation Sketch & X–Y Equation Conversion example problem
Sketch the given Parametric Equation and find its equivalent X–Y Equation:
Solution to this Parametric Equations & Curves practice problem is given in the video below!
Parametric Equation Sketch & X–Y Equation Conversion example
Find Parametric Equations from the given X–Y equation:
Circle with radius 3, centered at (2, 1) and drawn counterclockwise.
Solution to this Parametric Equations & Curves practice problem is given in the video below!
Parametric Equation Sketch & X–Y Equation Conversion example question
Derive Parametric Equations from the given X–Y equation:
Line from (-2, 4) to (6, 1).
Solution to this Parametric Equations & Curves practice problem is given in the video below!
Parametric Equation Sketch & X–Y Equation Conversion example problem
Find Parametric Equations by using the given X–Y equation:
from (2, -2) to (0, 2).
Solution to this Parametric Equations & Curves practice problem is given in the video below!
Parametric Equations: Points of Intersection of Two Curves example
Find the coordinates of the points of Intersection of two given Parametric Curves:
Solution to this Parametric Equations & Curves practice problem is given in the video below!
Parametric Equations of a PROJECTILE example problem
A projectile is fired at an angle of inclination α, with 0 < α < π/2, at an initial speed of v[o]. Parametric equations for its path can be shown to be x = v[o]tcos(α), y = v[o]tsin(α) – (gt ²)/2,
with t representing time.
a) Eliminate the t parameter and find the value of t when the projectile hits the ground.
b) Sketch the path of the projectile for the case α = π/6, v[o] = 32 ft/sec, g = 32 ft/sec ².
Solution to this Parametric Equations of a Projectile practice problem is given in the video below!
Parametric Equations: Object Position, Velocity & Speed example question
Find the Position, Velocity and Speed of an Object by using the following Parametric Equation:
Solution to this Parametric Equations & Curves practice problem is given in the video below!
Slope of the Parametric Curve example problem
Find the Slope of the Tangent Line to the Curve
Solution to this Parametric Equations & Curves practice problem is given in the video below!
Parametric Curves: Coordinates of the Points of Vertical & Horizontal Tangent Lines example
Find the Coordinates of the Points of Vertical and Horizontal Tangent Lines to the Curve
Solution to this Parametric Equations & Curves practice problem is given in the video below!
Parametric Curves: Arc Length example question
Find the Arc Length of a Parametric Curve
Solution to this Parametric Equations & Curves practice problem is given in the video below!
Parametric Curves: Area example problem
Find the Area enclosed by a Parametric Curve
in the interval
Solution to this Parametric Equations & Curves practice problem is given in the video below!
Parabola Directrix Conic Section Proof example question
Show that in polar coordinates the equation of a conic section with (one) focus at the pole and directrix,
the line
can be written as
Solution to this Conic Section Proof practice problem is given in the video below! | {"url":"https://mathcabin.com/category/triangles/","timestamp":"2024-11-05T16:31:17Z","content_type":"text/html","content_length":"259544","record_id":"<urn:uuid:34a4684f-7131-4449-8461-8cbb4f5f01ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00450.warc.gz"} |
Control Theory Seminar | Adriano Festa, A semi-Lagrangian scheme for Hamilton-Jacobi equations on networks and its application to vehicular traffic models | Applied Mathematics
Tuesday, November 14, 2023 1:00 pm - 1:00 pm EST (GMT -05:00)
MC 5501 and Zoom (Please contact rgugliel@uwaterloo.ca for meeting link)
Adriano Festa, Associate Professor, Department of Mathematical Sciences at the Technical university of Turin
A semi-Lagrangian scheme for Hamilton-Jacobi equations on networks and its application to vehicular traffic models
We present a semi-Lagrangian scheme for the approximation of a class of Hamilton-Jacobi-Bellman (HJB) equations on networks. The scheme is explicit, consistent, and stable for large time steps. We
prove a convergence result and two error estimates. For an HJB equation with space-independent Hamiltonian, we obtain a first order error estimate. In the general case, we provide, under a hyperbolic
CFL condition, a convergence estimate of order one half. The theoretical results are discussed and validated in a numerical tests section, where we show some application of the techniques proposed to
the approximation of traffic flows. | {"url":"https://uwaterloo.ca/applied-mathematics/events/control-theory-seminar-adriano-festa-semi-lagrangian-scheme","timestamp":"2024-11-11T06:35:27Z","content_type":"text/html","content_length":"111238","record_id":"<urn:uuid:8fcaae57-fa1d-4478-af2b-9a5d15814771>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00236.warc.gz"} |
machine learning Archives - Discovered Intelligence
The Splunk Machine Learning Toolkit is packed with machine learning algorithms, new visualizations, web assistant and much more. This blog sheds light on some features and commands in Splunk Machine
Learning Toolkit (MLTK) or Core Splunk Enterprise that are lesser known and will assist you in various steps of your model creation or development. With each new release of the Splunk or Splunk MLTK
a catalog of new commands are available. I attempt to highlight commands that have helped in some data science or analytical use-cases in this blog.
Read more
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Discovered Intelligence2020-11-12 20:24:382022-10-31 14:41:21Interesting Splunk MLTK Features for Machine Learning (ML) Development
Quick Guide to Outlier Detection in Splunk
/in Big Data, Education, Machine Learning, Splunk/by Discovered Intelligence
There are multiple (almost discretely infinite) methods of outlier detection. In this blog I will highlight a few common and simple methods that do not require Splunk MLTK (Machine Learning Toolkit)
and discuss visuals (that require the MLTK) that will complement presentation of outliers in any scenario. This blog will cover the widely accepted method of using averages and standard deviation for
outlier detection. The visual aspect of detecting outliers using averages and standard deviation as a basis will be elevated by comparing the timeline visual against the custom Outliers Chart and a
custom Splunk’s Punchcard Visual.
Some Key Concepts
Understanding some key concepts are essentials to any Outlier Detection framework. Before we jump into Splunk SPL (Search Processing Language) there are basic ‘Need-to-know’ Math terminologies and
definitions we need to highlight:
• Outlier Detection Definition: Outlier detection is a method of finding events or data that are different from the norm.
• Average: Central value in set of data.
• Standard Deviation: Measure of spread of data. The higher the Standard Deviation the larger the difference between data points. We will use the concept of standard substantially in today’s blog.
To view the manual method of standard deviation calculation click here.
• Time Series: Data ingested in regular intervals of time. Data ingested in Splunk with a timestamp and by using the correct ‘props.conf’ can be considered “Time Series” data
Additionally, we will leverage aggregate and statistic Splunk commands in this blog. The 4 important commands to remember are:
• Bin: The ‘bin’ command puts numeric values (including time) into buckets. Subsequently the ‘timechart’ and ‘chart’ function use the bin command under the hood
• Eventstats: Generates statistics (such as avg,max etc) and adds them in a new field. It is great for generating statistics on ‘ALL’ events
• Streamstats: Similar to ‘stats’ , streamstats calculates statistics at the time the event is seen (as the name implies). This feature is undoubtedly useful to calculate ‘Moving Average’ in
additional to ordering events
• Stats: Calculates Aggregate Statistics such as count, distinct count, sum, avg over all the data points in a particular field(s)
Data Requirements
The data used in this blog is Splunk’s open sourced “Bots 2.0” dataset from 2017. To gain access to this data please click here. Downloading this data set is not important, any sample time series
data that we would like to measure for outliers is valid for the purposes of this blog. For instance, we could measure outliers in megabytes going out of a network OR # of logins in a applications
using the using the same type of Splunk query. The logic used to the determine outliers is highly reusable.
Using SPL
There are four methods commonly seen methods applied in the industry for basic outlier detection. They are in the sections below:
1. Using Static Values
The first commonly used method of determining an outlier is by constructing a flat threshold line. This is achieved by creating a static value and then using logic to determine if the value is above
or below the threshold. The Splunk query to create this threshold is below :
<your spl base search> … | timechart span=6h sum(mb_out) as mb_out
| eval threshold=100
| eval isOutlier=if('mb_out' > threshold, 1, 0)
Static threshold timeline visual
2. Average with Static Multiplier
In addition to using arbitrary static value another method commonly used method of determining outliers, is a multiplier of the average. We calculate this by first calculating the average of your
data, following by selecting a multiplier. This creates an upper boundary for your data. The Splunk query to create this threshold is below:
<your spl base search> …
| timechart span=12h sum(mb_out) as mb_out
| eventstats avg("mb_out") as average
| eval threshold=average*2
| eval isOutlier=if('mb_out' > threshold, 1, 0)
Average + Static threshold timeline visual
3. Average with Standard Deviation
Similar to the previous methods, now we use a multiplier of standard deviation to calculate outliers. This will result in a fixed upper and lower boundary for the duration of the timespan selected.
The Splunk query to create this threshold is below:
<your spl base search> ... | timechart span=12h sum(mb_out) as mb_out
| eventstats avg("mb_out") as avg stdev("mb_out") as stdev
| eval lowerBound=(avg-stdev*exact(2)), upperBound=(avg+stdev*exact(2))
| eval isOutlier=if('mb_out' < lowerBound OR 'mb_out' > upperBound, 1, 0)
2*Standard Deviation timeline visual
Notice that with the addition of the lower and upper boundary lines the timeline chart becomes cluttered.
4. Moving Averages with Standard Deviation
In contrast to the previous methods, the 4th most common method seen is by calculating moving average. In short, we calculate the average of data points in groups and move in increments to calculate
an average for the next group. Therefore, the resulting boundaries will be dynamic. The Splunk search to calculate this is below:
<your spl base search> ... | timechart span=12h sum(mb_out) as mb_out
| streamstats window=5 current=true avg("mb_out") as avg stdev("mb_out") as stdev
| eval lowerBound=(avg-stdevexact(2)), upperBound=(avg+stdevexact(2))
| eval isOutlier=if('mb_out' < lowerBound OR 'mb_out' > upperBound, 1, 0)
Moving Average with Standard Deviation timeline chart
Tips: Notice the “isOutliers” line in the timeline chart, in order to make smaller values more visible format the visual by changing the scale from linear to log format.
Using the MLTK Outlier Visualization
Splunk’s Machine Learning Toolkit (MLTK) contains many custom visualization that we can use to represent data in a meaningful way. Information on all MLTK visuals detailed in Splunk Docs. We will
look specifically at the ‘Outliers Chart’. At the minimum the outlier chart requires 3 additional fields on top of your ‘_time’ & ‘field_value’. First, would need to create a binary field ‘isOutlier’
which carries the value of 1 or 0, indicating if the data point is an outlier or not. The second and third field are ‘lowerBound’ & ‘upperBound’ indicating the upper and lower thresholds of your
data. Because the outliers chart trims down your data by displaying only the value of data point and your thresholds, we can conclude through use that it is clearer and easier to understand manner.
As a recommendation it should be incorporated in your outliers detection analytics and visuals when available.
Continuing from the previous paragraph, take a look at the below snippets at how the impact the outliers chart is in comparison to the timeline chart. We re-created the same SPL but instead of
applying timeline visual applied the ‘Outliers Chart’ in the same order:
Static threshold w outliers chart
Average + Static threshold timeline visual
2*Standard Deviation outliers chart
Moving Average with Standard Deviation outliers chart
Advantages Disadvantages
Cleaner presentation and less clutter You need to install Splunk MLTK (and its pre-requisites) to take advantage of the
outliers chart
Easier to understand as determining the boundaries becomes intuitive vs figuring out which line is the upper or Unable to append additional fields in the Outliers chart
lower threshold
Adding Depth to your Outlier Detection
Determining the best technique of outlier detection can become a cumbersome task. Hence, having the right tools and knowledge will free up time for a Splunk Engineer to focus on other activities.
Creating static thresholds over time for the past 24hrs, 7 days, 30 days may not be the best approach to finding outliers. A different way to measure outliers could be by looking at the trend on
every Monday for the past month or 12 noon everyday for the past 30 days. We accomplish this by using two simple and useful eval functions:
| eval HourOfDay=strftime(_time, "%H")
| eval DayOfWeek=strftime(_time, "%A")
Using Eval Functions in SPL
Continuing from the previous section, we incorporate the two highlighted eval functions in our SPL to calculate the average ‘mb_out’. However, this time the average is based on the day of the week
and the hour of the day. There are a handful of advantages of this method:
• Extra depth of analysis by adding 2 additional fields you can split the data by
• Intuitive method of understanding trends
Some use cases of using the eval functions are as follows:
• Network activity analysis
• User behaviour analysis
Tables representing averages by DayOfWeek & HourOfDay
Visualizing the Data!
We will focus on two visualizations to complement our analysis when utilizing the eval functions. The first visual, discussed before, is the ‘Outliers Chart’ which is a custom visualization in Splunk
MLTK. The second visual is another custom visualization ‘PunchCard’, it can be downloaded from Splunkbase here (https://splunkbase.splunk.com/app/3129/).
The outliers chart has a feature which results in a ‘swim lane’ view of a selected field/dimension and your data points while highlighting points that are outliers. To take advantage of this feature,
we will use a Macro “splitby” which creates a hidden field(s) “_<Field(s) you want data to split by>”. The rest of the SPL is shown below
< your base SPL search > ... | eventstats avg("mb_out") as avg stdev("mb_out") as stdev by "HourOfDay"
| eval avg=round(avg,2)
| eval stdev=round(stdev,2)
| eval lowerBound=(avg-stdev*exact(2)), upperBound=(avg+stdev*exact(2))
| eval isOutlier=if('mb_out' < lowerBound OR 'mb_out' > upperBound, 1, 0)
| `splitby("HourOfDay")`
| fields _time, "mb_out", lowerBound, upperBound, isOutlier, *
| fields - _raw source kb* byt*
| table _time "mb_out" lowerBound upperBound isOutlier *
This search results in an Outlier Chart that looks like this:
Outliers Chart split by hour of day
The Outliers Chart has the capability to split by multiple fields, however in our example splitting it by a single dimension “HourOfDay” is sufficient to show its usefulness.
The PunchCard visual is the second feature we will use to visualize outliers. It displays cyclical trends in our data by representing aggregated values of your data points over two dimensions or
fields. In our example, I’ve calculated the sum of outliers over a month based on “DayOfWeek” as my first dimension and “HourOfDay” as my second dimension. I’ve adding the outliers of these two
fields and displaying it using the PunchCart visual. The SPL and image for this visual is show below:
< your base SPL search > ... | streamstats window=10 current=true avg("mb_out") as avg stdev("mb_out") as stdev by "DayOfWeek" "HourOfDay"
| eval avg=round(avg,2)
| eval stdev=round(stdev,4)
| eval lowerBound=(avg-stdevexact(2)), upperBound=(avg+stdevexact(2))
| eval isOutlier=if('mb_out' < lowerBound OR 'mb_out' > upperBound, 1, 0)
| splitby("DayOfWeek","HourOfDay")
| stats sum(isOutlier) as mb_out by DayOfWeek HourOfDay
| table HourOfDay DayOfWeek mb_out
PunchCard Visual
Summary and Wrap Up
Trying to find outliers using Machine Learning techniques can be a daunting task. However I hope that this blog gives an introduction on how you can accomplish that without using advanced algorithms.
Consequently, using basic SPL and built-in statistic functions can result in visuals and analysis that is easier for stakeholders to understand and for the analyst to explain. So summarizing what we
have learnt so far:
1. One solution does not fit all. There are multiple methods of visualizing your analysis and exploring your result through different visual features should be encouraged
2. Use Eval functions to calculate “DayOfWeek” and “HourOfDay” wherever and whenever possible. Adding these two functions provides a simple yet powerful tool for the analyst to explore the data with
additional depth
3. Trim or minimize the noise in your Outliers visual by using the Outliers Chart. The chart is beneficial in displaying only your boundaries and outliers in your data while shaving all other
unnecessary lines
4. Use “log” scale over “linear” scale when displaying data with extremely large ranges
Looking to expedite your success with Splunk? Click here to view our Splunk Professional Service offerings.
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Discovered Intelligence2020-07-16 14:13:082022-10-31 15:31:18Quick Guide to Outlier Detection in Splunk
Forecasting Time Series Data Using Splunk Machine Learning Toolkit – Part II
/in Big Data, Education, Machine Learning, Splunk/by Discovered Intelligence
Part II of the Forecasting Time Series blog provides a step by step guide for fitting an ARIMA model using Splunk’s Machine Learning Toolkit. ARIMA models can be used in a variety of business use
cases. Here are a few examples of where we can use them:
• Detecting anomalies and their impact on the data
• Predicting seasonal patterns in sales/revenue
• Streamline short-term forecasting by determine confidence intervals
From Part 1 of the blog series, we identified how you can use Kalman Filter for forecasting. The observation we made from the resulting graphs demonstrated how it was also useful in reducing/
filtering noise (which is how it gets its name ‘Filter’) . On the other hand ARIMA belongs to a different class of models. In comparison to a Kalman filter, ARIMA models works on data that has moving
averages over time or where the value of a data point is linearly depending on its previous value(s). In these two scenarios it makes more sense to use ARIMA over Kalman Filter. However good
judgement, understanding of the data-set and objective of forecasting should always be the primary method of determining the algorithm.
Part II of this blog series aims to familiarize a Splunk user using the MLTK Assistant for forecasting their time series data, particularly with the ARIMA option. This blog is intended as a guide in
determining the parameters and steps to utilize ARIMA for your data. In fact, it is a generalized template that can be used with any processed data to forecasting with ARIMA in Splunk’s MLTK. An
advantage of using Splunk for forecasting is its benefit in observing the raw data side by side with the predicted data and once the analysis is complete, a user can create alerts or other actions
based on a future prediction. We will talk more about creating alerts based on predicted or forecasted data in a future blog (see what I predicted there ;)?)
If you have read part I of our blog, we will reuse the same dataset process_time.csv for this part. If not, click here to navigate to part I to understand the dataset.
Fundamental Concept for ARIMA Forecasting
A fundamental concept to understand before we move ahead with ARIMA is that the model works best with stationary data. Stationary data has a constant trend that does not change overtime. The average
value is also independent of time as another characteristic of stationary data.
A simple example of non-stationary data is are the two graphs below, the first without a trendline, the second with a yellow trendline to show an average increase in the value of our data points. The
data needs to be transformed into stationary data to remove the increasing trend.
Using Splunk’s autoregress command we can apply differencing to our data. The results are immediately visible through line chart visual! The below command can be used on any time series data set to
demonstrate differencing.
… | autoregress value | eval new_value=value-value_p1 | fields _time new_value
Without creating a trendline for the below graph we can see that the data fluctuates around a constant mean value of ‘0’, we can say that differencing is applied. Differencing to make the data
stationary can increase the accuracy and fit of our ARIMA forecast. To read more about differencing and other rules that apply on ARIMA, navigate to the Duke URL provided in the useful link section:
Differencing is simply subtracting the current and previous data points. In our example we are only applying differencing by an order of 1, meaning we will subtract the present data point by one data
point in reverse chronological order. There are different types of non-stationary graphs, which require in-depth domain knowledge of ARIMA, however we simplify it in this blog and use differencing to
remove the non-constant trend in this example 😊!
From part 1 of this blog series we can see that our data does not have a constant trend, as a result we apply differencing to our dataset. The step to apply differencing from the MLTK Assistant is
detailed in the ‘Determining Starting Points’ section. Differencing in ARIMA allows the user to see spikes or drops (outliers) in a different perspective in comparison to Kalman Filter.
Walkthrough of MLTK Assistant for ARIMA
ARIMA is a popular and robust method of forecasting long-term data. From blog 1 we can describe Kalman Filter’s forecasting capabilities as extending the existing pattern/spikes, sort of a copy-paste
method which may be advantageous when forecasting short-term data. ARIMA has an advantage in predicting data points when the we are uncertain about the future trend of the data points in the
long-term. Now that we have got you excited about ARIMA, lets see how we can use it in Splunk’s MLTK!
We use the Machine Learning Toolkit Assistant for forecasting timeseries data in Splunk. Navigate to the Forecast Time Series Assistant page (Under the Classic Menu option) and use the Splunk
‘inputlookup’ command to view the process_time.csv file.
|inputlookup process_time.csv
Once we add the dataset click on Algorithm and select ‘ARIMA’ (Autoregressive Integrated Moving Average), and ‘value’ as your field to forecast. You will notice that the ARIMA arguments will appear.
There are three arguments that make up the ARIMA model:
Argument Definition
AutoRegressive Auto regressive (AR) component refers to the use of past values in the regression equation. Higher the value the more past terms you will use in the equation. This concept is also
– p called ‘lags’. Another way of describing this concept is if the value your data point is depending on its previous value e.g process time right now will depend on the process time 30
seconds before (from our data set)
Integrated – d The d represents the degrees of differencing as discussed in the previous section. This makes up the integrated component of the ARIMA model and is needed for the stationary assumption
of the data.
Moving Average Moving Average in ARIMA refers to the use of past errors in the equation. It is the use of lagging (like AR) but for the error terms.
– q
Determine Starting Points
Identify the Order of Differencing (d)
As a refresher, we utilized the same dataset we worked with in part 1 of the blog series regarding the Kalman filter. As I input my process_time.csv file in the assistant, I enter the future_timespan
variable as 20 and the holdback as 20. I’ve kept the confidence interval as default value ‘95’. Once the argument values are populated click on ‘Forecast’ to see the resulting graphs.
As a note, my ARIMA arguments described above are ARIMA(0,0,0) which can represented as a mathematics function ARIMA(p,d,q), where p,d,q = 0. We use this functional representation of the variables
frequently in this blog for consistently with generally used mathematical languages.
When we click on forecast, observe the line chart graph from the results that show. This above graph confirms that the data is non-stationary, we will apply differencing to make it stationary. We can
accomplish this by increasing the value of our ‘d’ argument from ‘0’ to ‘1’ in the forecasting assistant and clicking on forecast again. This step is essential to meet one of the main criteria’s of
using ARIMA discussed in the ‘Fundamental Concept for ARIMA’ section.
Identifying AR(p) and MA(q)
After we apply differencing to our data our next step is to determine the AR or MR terms that mitigate any auto correlation in our data. There are two popular methods of estimating the these two
parameters. We will expand on one of the methods in this blog.
Method 1
The first method for estimating the value of ‘p’ and ‘q’ is to use the Akaiki Information Criteria (AIC) and the Baysian Information Criteria (BIC), however using them is outside the scope of the
blog as we will use a different method from the MLTK given the tools we have at hand. For the curious mind, the following blog contains detailed information on AIC and BIC to determine our ‘p’ and
‘q’ values:
Method 2
After we have applied differencing to our time series data, we review the PCAF and the ACF plots to determine an order for AR(q) or MA(q). We will apply ARIMA(0,1,0) in our ARIMA MLTK assistant and
then click on ‘Forecast’ to view the results of the graph. The below image shows the values that we entered in the assistant:
Once we click on forecast, we view the PACF plot to estimate a value for AR(p) model. Similarly we use the ACF plot to estimate a value for MA(q). The graphs are shown in the screenshot below.
We examine the PACF plot for a suggestion for our AR value, by counting the prominent high spikes. From the plot below I’ve circled the prominent spikes in the PACF graph. The value of AR (p) that we
pick is 4.
We examine the ACF plot for a suggestion for our MA value, by counting the prominent high spikes. From the plot below I’ve circled the prominent spikes in the ACF graph. The value of AR (q) that we
pick is 5.
We can now add in the values for the parameter integrated (d) – 1 and our estimates for AR – 4, and MA -5 in the Splunk MLTK. Once added in the assistant, click on ‘Forecast’.
For this particular combination for values we can see that once we click on ‘Forecast’, we get an error regarding the ‘invertability’ of the dataset as shown in the screenshot below. Without going
too deep into the mathematics, it means that our model does not converge when it forecasts. I’ve added a link in the references and links section at the end for your interest! This error can be
resolved by adjusting the values of model, similar to a ‘trail an error’ approach explained in the next section.
Optimize Your P and Q Values
Estimating this method of AR and MA is subjective to what can be considered as ‘prominent spikes’, this can result in estimating values of ‘q’ and ‘p’ that are not an optimal fit for the data. To
resolve this we constructed a table displaying the R-squared and Root Mean Square Error (RMSE) values from the model error statistics from the MLTK assistance, for each combination of ‘p’ and ‘q’. An
empty cell indicates an invertability error, while the other cells contain the value of R-squared and RMSE.
A higher R-squared indicates a better fit the model has on the data. R-squared is the amount of variability that the model can explain on the process time data points.
On the other hand, the lower the RMSE is the better the fit of the model. Root mean square is the difference between the data points the model predicted and our holdback points from the raw data.
We pick values of ‘p’ and ‘q’ that minimize RMSE and maximize R-square as the best fit to our data. From the table below we can see that q=5 and p=5 optimize the prediction for us.
Integrated (d) = AutoRegressive (p)
Moving Average 0 R2 Stat: -0.0015 RMSE: R2 Stat: 0.1976 RMSE: R2 Stat: 0.1977 RMSE: R2 Stat: 0.2699 RMSE: R2 Stat: 0.2696 RMSE: R2 Stat: 0.3114 RMSE:
(q) 19.31 16.35 16.34 15.60 15.60 15.14
1 R2 Stat: 0.2401 RMSE: R2 Stat: 0.2486 RMSE: R2 Stat: 0.2780 RMSE: R2 Stat: 0.2329 RMSE: – R2 Stat: 0.4053 RMSE:
15.91 15.82 15.51 15.98 14.07
2 R2 Stat: 0.2452 RMSE: – – R2 Stat: 0.3017 RMSE: R2 Stat: 0.3214 RMSE: –
15.85 15.25 15.03
3 R2 Stat: 0.2872 RMSE: R2 Stat: 0.4185 RMSE: R2 Stat: 0.4428 RMSE: R2 Stat: RMSE: R2 Stat: 0.4343 RMSE: R2 Stat: 0.4456 RMSE:
15.41 13.92 13.62 13.72 13.58
4 R2 Stat: 02826 RMSE: R2 Stat: 0.4185 RMSE: R2 Stat:0.3241 RMSE: – – –
15.46 13.92 15.00
5 R2 Stat: 0.2826 RMSE: R2 Stat: 0.3133 RMSE: R2 Stat: 0.4385 RMSE: – – R2 Stat: 0.4515 RMSE:
15.46 15.99 13.67 13.52
Viewing Your Results
Once we have picked the values of p and q that optimize our model, we can go ahead plug the numbers in our assistant and click on forecast to display the forecasted graph. The values to plug in the
assistant are as follows: p-5, d-1, q-5, holdback-20, forecast-20. The screenshots below show the values entered in the assistant and the resulting forecast graph.
A this point many would be satisfied with the forecast as the visual of the data itself is enough to analyse, asses and then make a judgement on the action(s) to take. The next step details how you
can view the data and lists some ideas of alerts that can be constructed
Next Step
We can view the SPL used powering the graph by either clicking on ‘Open in Search’ or ‘ ‘Show SPL’. I prefer the ‘Open in Search’ option as it automatically open a new tab, allowing me to further
understand how the SPL is constructed in the forecast and to view the data. Once a tab browser tab opens click on the ‘statistics’ option to view the raw data points, predicted data points and the
confidence intervals created by our model. I have added the SPL from the image for your convenience below:
| inputlookup process_time.csv | fit ARIMA _time value holdback=20 conf_interval=95 order=5-1-5 forecast_k=40 as prediction | `forecastviz(40, 20, "value", 95)`
I added another filter to my SPL to only view the forecasted process data from the ARIMA model as shown below:
| inputlookup process_time.csv | fit ARIMA _time value holdback=20 conf_interval=95 order=5-1-5 forecast_k=40 as prediction | `forecastviz(40, 20, "value", 95)` | search "lower95(prediction)"=*
The resulting table lists all the necessary data in a clean tabular format (that we are all familiar with) for creating alerts based on our predicted process time. Here are some ideas on creating
alerts based on the data we worked with:
1. Create alert when the predicted value of the process time goes above a certain threshold
2. Create alert when the average process time over a timespan is predict to stay above normal limits
3. Create alert based on outlier detection, when the predicted data is outside the lower or upper boundaries
Creating alerts based on our predict data allows us to be proactive of potential increase or decrease of our input variable
Summarizing ARIMA Forecasting in MLTK
Lets summarize what we have discussed so far in this blog:
1. A mathematical prerequisites of the model
2. Determining differencing requirement
3. Determine starting values for AR() and MA()
4. Optimize your AR() and MA() values based on error statistics
5. Forecast your data based on values decided in Step 4
6. View data and determine any alerts conditions
Prior to the above steps, we need to ensure that our data has been pre-processed or transformed in a MLTK-friendly manner. The pre-process steps include but not limited to; ensuring no gaps in the
time series data, determine the relevance of data to forecasting, group data in time intervals (30 second, 1 minute etc). The pre-processing steps are important to create uniformity in the data input
allow Splunk’s MLTK to analyse and forecast your data.
Hopefully this blog, streamlines the process of forecasting using ARIMA in Splunk’s MLTK. There are limitations as with any algorithm on forecasting using this method, as it involves a more
theoretical knowledge in mathematics I’ve added two links in the the useful links section (first link is navigates you to on ‘datascienceplus.com’ and the second to ’emeraldinsight.com’) to further
read on them.
Looking to expedite your success with Splunk? Click here to view our Splunk Professional Service offerings.
© Discovered Intelligence Inc., 2019. Unauthorised use and/or duplication of this material without express and written permission from this site’s owner is strictly prohibited. Excerpts and links may
be used, provided that full and clear credit is given to Discovered Intelligence, with appropriate and specific direction (i.e. a linked URL) to this original content.
Useful Links
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Discovered Intelligence2019-05-06 12:21:352022-10-31 15:35:39Forecasting Time Series Data Using Splunk Machine Learning Toolkit – Part II
Predict Spam Using Machine Learning Classification
/in Big Data, Machine Learning, Splunk/by Discovered Intelligence
In this blog we will use a classification approach for predicting Spam messages. A classification approach categorizes your observations/events in discrete groups which explain the relationship
between explanatory and dependent variables which are your field(s) to predict. Some examples of where you can apply classification in business projects are: categorizing claims to identify
fraudulent behaviour, predicting best retail location for new stores, pattern recognition and predicting spam messages via email or text. Read more
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Discovered Intelligence2018-10-16 17:44:082022-11-02 13:54:47Predict Spam Using Machine Learning Classification
A Practical Example Using The Splunk Machine Learning Toolkit
/in Big Data, Education, Machine Learning, Splunk/by Discovered Intelligence
In our previous blog we walked through steps on installing the Splunk Machine Learning Toolkit and showcased some of the analytical capabilities of the app. In this blog we will deep dive into an
example dataset and use the ‘Predict Numeric Fields’ assistant to help us answer some questions about it.
The sample dataset used is from People’s dataset repository [Houghton] This multivariate sample dataset contains the following fields:
• Net Sales/$ 1,000
• Square Feet/ 1,000
• Inventory/$ 1,000
• Amt Spent on Advertising/$ 1,000
• Size of Sales District/1000 families
• No of Competitors in district
You can download a copy of the sample data here: greenfranchise.csv
What Questions do we want to ask?
We would like to understand the relationship between ‘Net Sales’ of Green Franchise and how it is impacted by the variables ‘Square Feet of Store’, ‘Inventory’, ‘Amount Spent on Advertising’, ‘Size
of Sales District’ & ‘No of Competitors’. E.g Would an increase in ‘Inventory’ or ‘Amount Spent on Advertising’ increase or decrease ‘Net Sales’ for Greens?
The next few sections will walk you through uploading the data set and processing it in the Machine Learning Toolkit App.
Uploading the Sample Data Set
The CSV file was uploaded to Splunk from Settings -> Lookups -> Lookup table files (Add new). If you need more information on this step please consult the Splunk Docs here. Save the CSV file as
Once the file has been uploaded and saved as greenfranchise.csv, navigate to the Splunk Machine Learning Toolkit App, click on the ‘Legacy’ menu, Assistants and open the ‘Predict Numeric Fields’
Assistant. This screenshot and navigation may differ depending on which version of Splunk and the MLTK is installed. Assistants in version 3.2 can be found under the ‘Legacy’ tab.
App: Splunk Machine Learning Toolkit
Populate Model Fields
In the Create New Model tab, you can view the contents of the CSV file by running the below Splunk Query in the Search bar:
| inputlookup greenfranchise.csv
This will automatically populate the panels with the fields in the csv file. Below the “Preprocessing Steps” we can see a second panel to choose the type of algorithm to apply to this lookup.
Selecting the Algorithm
In the panel for selecting the algorithm, we can see the ‘Fields to predict’ and ‘Fields to use for predicting’ fields are automatically populated from the data. For this test we use the linear
regression algorithm to forecast the ‘Net Sales’ of Green Franchises. Select “Net Sales” as the Field to predict, and in the Fields to use for predicting, select all of the remaining fields except
for “Size of Sales District”.
If you’re interested in the math behind it, linear regression from the Machine Learning Toolkit will provide us with the Beta (relationship) co-efficient between ‘Net Sales’ and each of the fields.
The residual of regression model is the difference between the explanatory/input variables and the predicted equation at each data point, which can be used for further analysis of the model.
Fitting Model
Once the Fields have been picked, you need to determine the ‘Split for Training’ ratio for the model. Select ‘No Split’ for the model to use all the data for creating a model. The split option allows
the user to divide the data for training and testing. This means that X% of the data will used to create our model, and (100-X) % of the data withheld will be used to test the model.
Click on ‘Fit Model’ after setting the Split for the data. Splunk processes the data to display visuals which we can use to analyze the data. Name the model ‘ex_linearreg_greens_sales’, however,
based on the users data, the model name should reflect the field to predict, the type of algorithm and the user it is assigned to, to reduce ambiguity on the models ownership and purpose.
Analyzing the Results
The first two panels show a Line and Scatter Chart of “Actual vs Predicted” data. Both panels present one of the richest methods to analyze the linear regression model. From the scatter and line plot
we can observe that the data fits well. We can determine that there is a strong correlation between the model’s predictions and the actual results. Since the model has more than one input variable,
examining the residual line chart and histogram next, will give us a more practical understanding.
The second set of panels that we can use to analyse the model are residuals from the plot. From observing the “Residual Line Chart” and “Residual Histogram” we can see that there is large deviation
from the center and the residuals appear to be scattered. A random scattering of the data points around the horizontal (x-axis) line signifies a good fit for the linear model. Otherwise, a pattern
shape of the data points would indicate that a non-linear model from the MLTK should be used instead.
The last set of panels show us the R-squared of the model. The closer the value is to 1, better the fit of the regression model. The “Fit Model Parameters Summary” panel gives us the ‘Beta’
coefficients discussed in the ‘Selecting the Algorithm’ section. The assistant displays the data in a well-grounded and systematic setting. After analyzing the macro fit of the model, we can use the
co-efficient of the variables create our equation for predicting ‘Net Sales’ :
In the last panel shown below, we can see our input variables under ‘Fit Model Parameters Summary’ and their values. We will assess in the next section on using these input variables to predict ‘
Answering the Question: How is ‘Net Sales’ impacted by the Variables?
We can view the results of the model by running the following search:
| summary "ex_linearreg_greens_sales"
This Query will return the coefficients values of the linear regression algorithm. In our example for Greens, we observed that variable ‘X4’ are the number of competitors stores, an increment in
competitors stores will reduce the ‘Net Sales‘ by approximately 12.62. While the variable ‘X5’ is the Sq Feet of the Store, and increment will increase the ‘Net Sales’ by approximately 23.69.
We can use the results from our model to forecast ‘Net Sales’ if the input variables (Sq Ft, Amt on Advertising etc) were different using the below Splunk search:
| makeresults | eval "Sq Ft"=5.8, Inventory=700, "Amt on Advertising"=11.5,"No of Competing Stores"=20 | apply ex_linearreg_greens_sales as "Predicted_Net_Sales"
We used makeresults to work our own values for the input variables. Once the fields have been defined we used the apply command in the MLTK to output the predicted value of the ‘Net Sales’ given the
new values of the input variables. The apply command uses the ouput values the model learnt from the csv dataset and applies them to new information. We used the ‘as’ command to alias the name of
the predicted field as ‘Predicted_Net_Sales’. From the below screenshot we can observe that; 11.5 on Advertising, 700 on Inventory, 20 Competing stores nearby and 5.8 square feet of space predicts a
Net Sales of approximately 306. Please note that all monetary variables are in $1,000 .
So to recap, we followed the following steps to answer our question of the data:
• Uploaded the sample data set
• Populated the model fields
• Selected an algorithm
• Fit the model
• Analyzed the results
The Splunk Machine Learning Toolkit simplifies the steps for data preparation, reduces the steps needed to create a model, and saves the history of models we have executed and tested with. We can
review the data before applying the algorithms allowing the user to standardize and adjust using MLTK capabilities or Splunk queries. The resulting statistic of the ‘Predict Numeric Fields’ assistant
allows us to understand the dataset using machine learning.
Looking to expedite your success with Splunk? Click here to view our Splunk service offerings.
© Discovered Intelligence Inc., 2018. Unauthorised use and/or duplication of this material without express and written permission from this site’s owner is strictly prohibited. Excerpts and links may
be used, provided that full and clear credit is given to Discovered Intelligence, with appropriate and specific direction (i.e. a linked URL) to this original content.
Houghton Mifflin, Data Sets, http://college.cengage.com/mathematics/brase/understandable_statistics/7e/students/datasets/mlr/frames/frame.html
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Intelligence2018-06-07 11:00:262022-11-04 14:47:47A Practical Example Using The Splunk Machine Learning Toolkit
Creating an IoT Fleet Management Solution using Splunk
/in Big Data, IoT, Machine Learning, Splunk/by paul
A week ago, I had the privilege of attending the annual Splunk Partner Technical Symposium in New Orleans along with a colleague. At this event, we entered and won the 1st annual IoT Hackathon,
sponsored by AWS. The Hackathon tasked us with developing an IoT fleet management solution using Ford GoBike IoT (Internet of Things) data. This post outlines the developed solution and the various
data sources and tools we used. Overall, it was a great and fun exercise and helps illustrate how feature rich solutions can be developed in a very short amount of time using Splunk Enterprise. Read
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18:34:542022-11-04 14:51:18Creating an IoT Fleet Management Solution using Splunk
Getting Started With Splunk’s Machine Learning Toolkit
/in Big Data, Machine Learning, Splunk/by Discovered Intelligence
The Splunk Machine Learning Toolkit (MLTK) assists in applying machine learning techniques and methods against your data. This article discusses how to get started with the MLTK including
installation and some initial testing and examples. Read more
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Discovered Intelligence2018-01-02 17:20:272022-11-04 15:12:41Getting Started With Splunk’s Machine Learning Toolkit | {"url":"https://discoveredintelligence.com/tag/machine-learning/","timestamp":"2024-11-12T00:26:19Z","content_type":"text/html","content_length":"203793","record_id":"<urn:uuid:797693b2-8de9-4ea0-8141-a9046f76b414>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00409.warc.gz"} |
Polarization Calibration
Next: Further Reading Up: Polarimetry Previous: The Measurement Equation Contents
Polarization Calibration
We restrict our attention to a point source at the phase center^15.10. The visibility that we measure, averaged over all baselines is
Any system describable by a Jones matrix is non-depolarizing^15.11 In the general case however, the summation in equation (15.4.17) cannot be represented by a single Jones matrix, and an
interferometer is not therefore a non-depolarizing system. However, ideally, after calibration, the effective Jones matrices are all the unit matrix, and the interferometer would then be
Intuitively, it is clear that if one looks at an unpolarized calibrator source, one should be able to solve for the leakage terms, (which will produce apparent polarization) but that some degrees of
freedom would remain unconstrained. Further it is also intuitive that the degrees of freedom which remain unconstrained are the following: (1) The absolute orientation of the feeds, (2) The intrinsic
polarization of the feeds (i.e. for example, are they linear polarized or circular polarized?) and (3) The phase difference between the two polarizations. While one would imagine that the situation
may be improved by observation of a polarized source, it turns out that this too is not sufficient to determine all the free parameters. What is required is observations of at least three differently
polarized sources. For alt-az mounted dishes, the rotation of the beam with respect to the sky changes the apparent polarization of the source. For such telescopes hence, it is sufficient to observe
a single source at several, sufficiently different hour angles. This is the polarization strategy that is commonly used at most telescopes. Faraday rotation due to the earth's ionosphere is more
difficult to correct for. In principle models of the ionosphere coupled with a measure of the total electron content at the time of the observation can be used to apply a first order correction to
the data.
We end this chapter with a brief description of the effect of calibration errors on the derived Stokes parameters. When observing with linearly polarized feeds, from equation (15.1.2) it is clear
that if one observes a linearly polarized calibrator, the parallel-hand correlations will contain a contribution due to the Q component of the calibrator flux. Consequently, if one assumes
(erroneously) that the calibrator was unpolarized the gain of the X channel will be overestimated and that of the Y channel underestimated. For this reason, for observations which require only
measurement of Stokes I, circular feeds are preferable, since the Stokes V component of most calibrators is negligible, and consequently, measurements of the parallel hand correlations^15.12are
sufficient to measure the correct Stokes I flux.
It is easy to show, that (to first order) if one observes a polarized calibrator with an error free linearly polarized interferometer and solves for the instrumental parameters under the assumption
that the calibrator is unpolarized, the derived instrumental parameters of all the antennas will be in error by^15.13:
is the gain error of the X channel.
is the gain error of the Y channel.
is the leakage from the Y channel to the X channel.
is the leakage from the X channel to the Y channel.
If these calibration solutions are then applied to an unpolarized target source, then the source will appear to be polarized, with the same polarization percentage as the calibrator, but opposite
sense. This again is simply the extension from scalar interferometry that if the calibrator flux is in error by some amount, the derived target source flux will be in error by the same fractional
amount, but with opposite sense.
For VLBI observations this is a very good approximation, since the source being imaged is very small compared to the primary beams of any of the antennas in the VLBI array.
This follows trivially from the fact that for 100% polarization we must have
recall from equations (15.1.3) that when
A similar result can of course be derived for the case of circularly polarized antennas, the only difference will be the usual transpositions of
Next: Further Reading Up: Polarimetry Previous: The Measurement Equation Contents NCRA-TIFR | {"url":"https://www.gmrt.ncra.tifr.res.in/doc/WEBLF/LFRA/node136.html","timestamp":"2024-11-10T18:19:53Z","content_type":"text/html","content_length":"11418","record_id":"<urn:uuid:e7e9ebcc-1164-41ab-820a-5c20e5ee0d61>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00295.warc.gz"} |
Generating Fractals with Postgres: Self-Similar Fractals - malisper.me
Generating Fractals with Postgres: Self-Similar Fractals
In my last post, I showed you how you could use a number of SQL tricks to generate a class of fractals known as the “escape-time fractals”. In this post I’ll show you how you can use the same tricks
to generate a completely different set of fractals known as the “self-similar fractals”.
Self-similar fractals are based on the idea of having an initial pattern and some rule for generating the next iteration of that pattern. One of the classic self-similar fractals is the Heighway
To generate the curve, you first start with a single line segment. To generate the next iteration of the Heighway drag, you “bend” each line segment of the curve in half. If you apply this to the
initial line segment, you get a curve shaped like a “v”. If you apply it to the v-shaped curve, you get a curve shaped like a C clamp. If you keep repeating this, you eventually get the curve on the
far right of the image above.
Given this definition of the Heighway dragon, it’s unclear how we might go approach implementing it. Fortunately, there is a convenient way of describing self-similar fractals known as “L-systems“.
L-systems are ultimately a way of encoding self-similar fractals as strings. The idea is you have an initial string which encodes the path to generate the initial version of the fractal. You then
have a set of string replacement rules that when applied to the string give you a string representing the next iteration of the fractal. For the Heighway dragon, the L-system looks something like
Initial String: FX
Replacement Rules:
X -> X+FY+
Y -> -FX-Y
The rule “X -> X+YF+” means you replace all “X”s in the current string with the string “X+YF+”. When you run this L-system, you get a sequence of strings, each describing one iteration of the
Heighway dragon:
To decode one of these strings into the curve, you interpret all “F”s as moving forward one unit, interpret all “+”s as turning right, interpret all “-“s as turning left, and interpret all other
characters as doing nothing. Based on this decoding, we can interpret the initial string “FX” as a a single step forward. In other words, a single line segment. The string “FX+YF+” represents a step
forward, a turn to the right, and another step forward, and a turn to the right. This gives us a “v” shape curve. These are exactly the first two iterations of the Heighway dragon. You may want to
play with the L-system yourself for a bit, but you should be able to convince yourself it generates the curve. This gives us some hope that it’s possible to implement a SQL query to generate the
Heighway dragon.
Writing the SQL Query
(If you don’t care about the specifics of how this is mapped to a SQL query, you can skip this section.)
Now that we’ve got a sense of an approach we can use to generate the Heighway dragon, let’s start thinking about how we can turn this into a SQL query. We can implement the Heighway dragon with a
three part SQL query:
1. First, we run the L-system. We use a recursive CTE to generate the Nth string in the L-system.
2. We convert the L-system string to a set of line segments. We do this by having a recursive CTE iterate through the L-system string and have it trace the path described by the string.
3. We convert the line segments into the fractal. We do this by using the same approach we used last time. We can produce an MxN grid of points, iterate through every point, and check whether or not
each point describes a point on the curve. When then combine all the points together to produce the final fractal.
Running the L-system
Running the L-system winds up being the easiest part. The description of an L-systems maps directly to a recursive CTE. We start with an initial string and run some string replacements to produce the
next iteration of the L-system. We keep doing this for some fixed number of iterations. We can do this with the following SQL query:
WITH RECURSIVE iterations AS (
SELECT 'FX' AS path, 0 AS iteration
UNION ALL
SELECT replace(replace(replace(PATH, 'X', 'X+ZF+'),
'Y', '-FX-Y'),
'Z', 'Y') AS PATH,
iteration + 1 FROM iterations
WHERE iteration < 8
) SELECT * FROM iterations;
path | iteration
FX | 0
FX+YF+ | 1
FX+YF++-FX-YF+ | 2
FX+YF++-FX-YF++-FX+YF+--FX-YF+ | 3
(We need to use slightly different replacement rules where we replace “X” with “X+ZF+” and then later replace “Z” with “Y“. This is so any “Y”s created by the “X” substition rule aren’t immediately
modified by the “Y” substitution rule.)
The query starts with the initial string “FX“. It does the substitutions described by the L-system a total of eight times. This gives us the L-system string for the first eight iterations of the
Generating the Line Segments
Converting the L-system string to a list of line segments is the most involved part of the SQL query. We start by having a recursive CTE iterate through the characters of the string returned by the
last iteration of the previous query. As we iterate through the string, we can maintain:
• The rest of the path we need to traverse.
• The start, mid, and endpoint of the previous line segment. Each of these can be encoded as a row value and a column value.
• The current direction we are heading in. We can encode this as two values, row_diff and col_diff, that describe what would happen if we were to take a step. row_diff describes how the current row
would change and col_diff describes how the column would change. For example, if we are facing to the left, the column value would be -1 (move towards the decreasing column), while the row value
would be 0 (don’t change the row we are in)
As we iterate through the string, there are three different cases we need to handle:
• When we see a “+” or “-“.
• When we see an “F”.
• When we see any other character.
When we see a “+” or “-“ the only thing that needs to change is the current direction. Depending on which direction we are turning, we can perform the following updates^1:
# Moving to the right:
row_diff = col_diff
col_diff = -row_diff
# Moving to the left:
row_diff = col_diff
col_diff = -row_diff
# Any other case:
row_diff = row_diff
col_diff = col_diff
To handle the case where we see an “F”, we can calculate a value step_size. Whenever we see an “F”, we set it to 1, otherwise we set it to 0. We can then calculate the end point of the next line
segment based on the end point of the previous line segment, the current direction we are facing, and the value of step_size. Using step_size ensures we only take a step forward when we see an “F”.
Based on the above updates, we automatically handle the case where we see any other another character. The direction we are heading in remains the same and we take a step of size zero which means our
position doesn’t change.
We can implement all of this in a recursive CTE that looks like the following:
segments AS (
0 AS start_row,
0 AS start_col,
0 AS mid_row,
0 AS mid_col,
0 AS end_row,
0 AS end_col,
0 AS row_diff,
1 AS col_diff,
(SELECT path FROM iterations ORDER BY iteration DESC LIMIT 1) AS path_left
UNION ALL
end_row AS start_row,
end_col AS start_col,
end_row + row_diff * step_size AS mid_row,
end_col + col_diff * step_size AS mid_col,
end_row + 2 * row_diff * step_size AS end_row,
end_col + 2 * col_diff * step_size AS end_col,
CASE WHEN SUBSTRING(path_left FOR 1) = '-' THEN -col_diff
WHEN SUBSTRING(path_left FOR 1) = '+' THEN col_diff
ELSE row_diff
END AS row_diff,
CASE WHEN SUBSTRING(path_left FOR 1) = '-' THEN row_diff
WHEN SUBSTRING(path_left FOR 1) = '+' THEN -row_diff
ELSE col_diff
END AS col_diff,
SUBSTRING(path_left FROM 2) AS path_left
FROM segments,
LATERAL (SELECT CASE WHEN SUBSTRING(path_left FOR 1) = 'F' THEN 1 ELSE 0 END AS step_size) sub
WHERE CHAR_LENGTH(path_left) > 0
) SELECT * FROM segments;
Pulling it All Together
Now that we’ve got the line segments for the fractal, the last step is to convert those line segments into the fractal itself. As I mentioned before we can use a similar approach to what we did in
the last post. We can do a three step process to convert the line segments into an image:
1. Determine the size of the grid. This can be determined by looking at the smallest and largest x and y coordinates of any point in the fractal.
2. Assign a character to each point. We can assign characters based on the following rules:
□ ‘*’ – An end point of a line segment.
□ ‘|’ – The midpoint of a vertical line segment.
□ ‘-‘ – The midpoint of a horizontal line segment.
□ ‘ ‘ – Any point that’s not part of the fractal.
3. Combine all the characters together to produce the final image.
This is all done by the following query:
end_points AS (
SELECT start_row AS r, start_col AS c FROM segments UNION SELECT end_row AS r, end_col AS c FROM segments
), points AS (
SELECT r, c FROM generate_series((SELECT MIN(r) FROM end_points), (SELECT MAX(r) FROM end_points)) a(r)
CROSS JOIN generate_series((SELECT MIN(c) FROM end_points), (SELECT MAX(c) FROM end_points)) b(c)
), marked_points AS (
SELECT r, c, (CASE WHEN
EXISTS (SELECT 1 FROM end_points e WHERE p.r = e.r AND p.c = e.c)
THEN '*'
WHEN EXISTS (SELECT 1 FROM segments s WHERE p.r = s.mid_row AND p.c = s.mid_col AND col_diff != 0)
THEN '-'
WHEN EXISTS (SELECT 1 FROM segments s WHERE p.r = s.mid_row AND p.c = s.mid_col AND row_diff != 0)
THEN '|'
ELSE ' '
) AS marker
FROM points p
), lines AS (
SELECT r, string_agg(marker, '') AS row_text
FROM marked_points
GROUP BY r
ORDER BY r DESC
) SELECT string_agg(row_text, E'\n') FROM lines;
Producing Fractals
After all that, we wind up with this one giant 61 line SQL query that generates the Heighway dragon:
WITH RECURSIVE iterations AS (
SELECT 'FX' AS PATH, 0 AS iteration
UNION ALL
SELECT replace(replace(replace(PATH, 'X', 'X+ZF+'), 'Y', '-FX-Y'), 'Z', 'Y'), iteration+1 AS iteration
FROM iterations WHERE iteration < 8
), segments AS (
0 AS start_row,
0 AS start_col,
0 AS mid_row,
0 AS mid_col,
0 AS end_row,
0 AS end_col,
0 AS row_diff,
1 AS col_diff,
(SELECT path FROM iterations ORDER BY iteration DESC LIMIT 1) AS path_left
UNION ALL
end_row AS start_row,
end_col AS start_col,
end_row + row_diff * step_size AS mid_row,
end_col + col_diff * step_size AS mid_col,
end_row + 2 * row_diff * step_size AS end_row,
end_col + 2 * col_diff * step_size AS end_col,
CASE WHEN SUBSTRING(path_left FOR 1) = '-' THEN -col_diff
WHEN SUBSTRING(path_left FOR 1) = '+' THEN col_diff
ELSE row_diff
END AS row_diff,
CASE WHEN SUBSTRING(path_left FOR 1) = '-' THEN row_diff
WHEN SUBSTRING(path_left FOR 1) = '+' THEN -row_diff
ELSE col_diff
END AS col_diff,
SUBSTRING(path_left FROM 2) AS path_left
FROM segments, LATERAL (SELECT CASE WHEN SUBSTRING(path_left FOR 1) = 'F' THEN 1 ELSE 0 END AS step_size) sub
WHERE CHAR_LENGTH(path_left) > 0
), end_points AS (
SELECT start_row AS r, start_col AS c FROM segments UNION SELECT end_row AS r, end_col AS c FROM segments
), points AS (
SELECT r, c FROM generate_series((SELECT MIN(r) FROM end_points), (SELECT MAX(r) FROM end_points)) a(r)
CROSS JOIN generate_series((SELECT MIN(c) FROM end_points), (SELECT MAX(c) FROM end_points)) b(c)
), marked_points AS (
SELECT r, c, (CASE WHEN
EXISTS (SELECT 1 FROM end_points e WHERE p.r = e.r AND p.c = e.c)
THEN '*'
WHEN EXISTS (SELECT 1 FROM segments s WHERE p.r = s.mid_row AND p.c = s.mid_col AND col_diff != 0)
THEN '-'
WHEN EXISTS (SELECT 1 FROM segments s WHERE p.r = s.mid_row AND p.c = s.mid_col AND row_diff != 0)
THEN '|'
ELSE ' '
) AS marker
FROM points p
), lines AS (
SELECT r, string_agg(marker, '') AS row_text
FROM marked_points
GROUP BY r
ORDER BY r DESC
) SELECT string_agg(row_text, E'\n') FROM lines;
Now for the really cool thing. We wrote our query to take the L-system for the Heighway dragon and produced the fractal for it. Since we didn’t make any other assumptions about the Heighway dragon,
we can replace the L-system for the Heighway dragon with any L-system we want! For example, another well known self-similar fractal is the Hilbert curve:
The Hilbert curve starts by taking an initial u-shaped curve that starts in the bottom left and ends in the bottom right. You can connect four of these curves together to produce a larger curve that
starts in the bottom left and ends in the bottom right. You can repeat this process to produce the next iteration of the curve. The L-system for the Hilbert curve is as follows:
Initial String: A
Replacement Rules:
A -> -BF+AFA+FB-
B -> +AF-BFB-FA+
We can modify our query above to use this new L-system:
SELECT 'A' AS PATH, 0 AS iteration
SELECT replace(replace(replace(PATH, 'A', '-CF+AFA+FC-'), 'B', '+AF-BFB-FA+'), 'C', 'B'),
iteration + 1
FROM iterations
WHERE iteration < 4
And we now get a query that produces the Hilbert curve:
This means we can use this query to produce any self-similar fractal that can be described by an L-system! We just plug in the L-system we want and get back a query that produces the fractal!
1. If you’re familiar with linear algebra, these operations correspond to the matrices for rotating by 90°. | {"url":"https://malisper.me/generating-fractals-with-sql-self-similar-fractals/","timestamp":"2024-11-06T02:02:44Z","content_type":"text/html","content_length":"50129","record_id":"<urn:uuid:36b2a533-6c77-41e8-a9aa-04b87753c694>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00736.warc.gz"} |
Problem 1026 - USTC Online Judge
Problem 1026
Horse Shoe Scoring
Time Limit: 1000ms
Memory Limit: 65536kb
The game of horseshoes is played by tossing horseshoes at a post that is driven into the ground. Four tosses generally make up a game. The scoring of a toss depends on where the horseshoe lands with
respect to the post. If the center of the post is within the region bounded by the interior of the horseshoe and the imaginary line connecting the two legs of the horseshoe, and the post is not
touching the horseshoe, it is a "ringer" and worth five points. If any part of the horseshoe is touching the post, it is a "toucher" and worth 2 points. If the toss is neither a ringer nor a toucher
and some part of the horseshoe will touch the post when it is pivoted around its point B, it is a "swinger" and worth 1 point. Any horseshoe which does not fit any of the scoring definitions scores
zero points. See the figures below for examples of each of the scoring possibilities.
This program uses mathematical horseshoes that are semicircles with radius 10 centimeters. The location of the horseshoe is the given by two points: the centerpoint of the semicircle, measured in
centimeters relative to x and y axes, and the point that exactly bisects the semicircle. The post is at location (0,0) and is 2 centimeters in diameter. The top of the post is level with the ground
allowing the horseshoe to lay on top of the post; therefore, a "toucher" would mean that any part of the horseshoe lies within the circle with a radius of 1 centimeter centered at (0, 0).
Each "turn" consists of four tosses. The purpose of your program is to determine the score of the "turn" by computing the sum of the point values for each of the four tosses.
Input to your program is a series of turns, and a turn consists of four horseshoe positions. Each line of input consists of two coordinate pairs representing the position of a toss. Each coordinate
consists of a floating point (X,Y) coordinate pair (-100.0 <= X, Y <= 100.0) with up to 3 digits of precision following the decimal point; the first and second numbers are the X and Y coordinates of
the centerpoint of the horseshoe semicircle (Point A) and the third and fourth numbers are the X and Y coordinates of the point (B) which bisects the horseshoe semicircle. You can be assured that the
distance between points A and B for each horseshoe will be exactly 10 centimeters. The figure below illustrates the meanings of the values on each line.
The first four lines of input define the horseshoe positions for the first turn; lines 5 through 8 define the second turn, etc. There are at most 999 turns in the input file, and every turn contains
four horseshoe positions. Your program should continue reading input to the end of file.
The first line of output for your program should be the string "Turn Score" in columns 1 through 10. For each "turn", your program should print the number of the turn right-justified in columns 1-3
(turns are numbered starting with 1), a single space (ASCII character 32 decimal) in columns 4 and 5, and the score for the turn right-justified in columns 6 and 7 with a single leading blank for
scores 0 to 9. Numbers that are right-justified should be preceded by blanks, not zeroes, as the fill character.
Sample Input
76.5 53.3 76.5 43.3
-5.1 1.0 4.9 1.0
5.1 0.7 5.1 -9.3
7.3 14.61 7.3 4.61
23.1 17.311 23.1 27.311
-23.1 17.311 -23.1 27.311
-23.1 -17.311 -23.11 -27.311
23.1 -17.311 23.1 -27.311
0.0 7.0 -6.0 -1.0
18.0 -24.0 9.0 -12.0
4.0 9.0 4.0 -1.0
-10.0 -13.0 -19.0 -25.0
Sample Output
Turn Score | {"url":"http://acm.ustc.edu.cn/ustcoj/problem.php?id=1026","timestamp":"2024-11-04T04:23:42Z","content_type":"text/html","content_length":"6270","record_id":"<urn:uuid:15573a4b-b10d-4eea-a1f0-80b4c95e81bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00839.warc.gz"} |
Duality Pairs and Homomorphisms to Oriented and Unoriented Cycles
Duality Pairs and Homomorphisms to Oriented and Unoriented Cycles
In the homomorphism order of digraphs, a duality pair is an ordered pair of digraphs $(G,H)$ such that for any digraph, $D$, $G\to D$ if and only if $D\not \to H$. The directed path on $k+1$
vertices together with the transitive tournament on $k$ vertices is a classic example of a duality pair.
In this work, for every undirected cycle $C$ we find an orientation $C_D$ and an oriented path $P_C$, such that $(P_C,C_D)$ is a duality pair. As a consequence we obtain that there is a finite set,
$F_C$, such that an undirected graph is homomorphic to $C$, if and only if it admits an $F_C$-free orientation. As a byproduct of the proposed duality pairs, we show that if $T$ is an oriented tree
of height at most $3$, one can choose a dual of $T$ of linear size with respect to the size of $T$. | {"url":"https://www.combinatorics.org/ojs/index.php/eljc/article/view/v28i3p17","timestamp":"2024-11-05T18:40:31Z","content_type":"text/html","content_length":"14543","record_id":"<urn:uuid:111906de-1290-461b-9b79-405b5079f420>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00079.warc.gz"} |
Everything you ever wanted to know about negative numbers and weren’t afraid to ask!
by Jon Dunning (May 2020)
Jon set an imaginative task during school closure to his Year 7 class: “Ask me two questions about what we’ve studied in this topic. If you feel confident you know everything, think of two
imaginative questions you could ask a mathematician about negative numbers.”
Below some brilliantly imaginative questions, and Jon’s answers to his class. Will this work with any other topic?
Everything you ever wanted to know about negative numbers and weren’t afraid to ask!
Disclaimer: I’m not a historian or a scientist; I’m your maths teacher. But I’m 90% sure I’m remembering the historical and physics stuff correctly.
Who created negative numbers?
Chinese texts mention negative numbers in the 2nd century BCE – many hundreds of years before they (or 0) appear in Europe. When they did arrive in Europe, even the smartest people got their knickers
in a twist about them – ‘how could there be less than nothing?’ they asked – and it was some time before they were accepted as being as real as positive numbers. It wasn’t really until the 16th or
17th century that we started to feel like they weren’t a silly idea.
If you ever struggle with negative numbers, remind yourself that there was a time when some of the cleverest people around were convinced that they were nonsense.
Should we call them negative numbers or minus numbers?
Well, some maths teachers are very uptight about this and others don’t care.
There is a clear difference between the technical meanings of ‘negative’ and ‘minus’, though. Negative refers to a type of number: those below 0 on the number line. Minus is a word connected with
When I was learning maths, I never bothered about the difference. But when I started teaching, I noticed that some kids got a bit muddled when working with negative numbers. That’s understandable
because it sounds confusing if you say it as ‘minus four minus minus three’.
So I always try to use the distinction when I’m talking to you: I think it helps us to understand each other and say exactly what we mean.
Is 0 negative or positive?
Can you have negative fractions or decimals?
They’re exactly the same as positive fractions or decimals, just on the other side of 0.
Is the priority of operations the same with negative numbers?
For example, multiplication and division always come before addition and subtraction, even with negative numbers.
Is there negative infinity?
If you could walk up number line, you’d be approaching infinity; if you could walk down the number line, you’d be approaching negative infinity.
What is a negative number squared or cubed?
Multiplying a number by a negative number will change the original number’s direction.
So negative x negative = positive
But then (negative x negative) x negative = (positive) x negative = negative
But then (negative x negative x negative) x negative = (negative) x negative = positive
Each time you multiply by another negative number, the direction of the answer changes.
So squaring (raising to a powers of 2) gives positive but cubing (raising to a power of 3) gives negative.
Can you see what pattern would result from raising a negative number to other powers?
Why does a negative multiplied by a negative always give a positive?
I would argue that it’s because that’s what we need it to do if we want all numbers to follow the same rules of arithmetic. (This raises an interesting point: are the rules of arithmetic created by
people, or were they true even before we thought of them?)
The rule of arithmetic that’s particularly relevant here is the distributive property of multiplication over addition. That’s the same rule that we use when expanding brackets.
First, we need to show that positive x negative always gives negative. We’ll start with the fact that any number and its negative counterpart (I’ve chosen 3 and -3) sum to 0, and then try multiplying
that sum by something (I’ve chosen 5)
Now we can apply the same logic to negative x negative…
Can you have a negative length?
You can’t have negative length but you can have negative displacement.
Displacement is the word we use for distance in a particular direction. Say I walk 3 metres forward: my distance from where I started is 3m and I could say that my displacement from where I started
is +3m. If I’d walked backwards 3m instead, I’d still have travelled a distance of 3m but my displacement would be -3m
Another interesting perspective comes from algebra. What would happen if was negative in the diagram below? What would the shape look like? Can you show that the formula for the area would still
Can you have negative speeds?
You can’t have negative speed but you can have negative velocity. Just like displacement is distance with a direction, we use velocity as speed with a direction. I refer you here to They Might Be
Giants for more information.
Can you have negative time?
I’m a bit out of my depth here but I think it depends on whether you consider time to have a direction, which is less straightforward than you’d imagine.
Most laws of physics would still work perfectly well mathematically if time had no direction but there is an important exception which is (very roughly) to do with the movement of heat.
In school maths, we do sometimes get negative values for time, for example when working with equations that describe the motion of an object through the air. You’ll see these in Year 12 and learn how
to interpret them. (They don’t really mean that there is negative time, just a time before the events described by the equations.)
Can you have a negative area?
Area can’t be negative but we there is a sense in which consider the areas beneath the axis of a graph to be negative. We do this because it fits nicely with another Year 12 topic, the calculus. This
is the very important part of maths for which Isaac Newton is so famous. (He didn’t really discover gravity, as in the story about the apple bumping him on the head. But he did help to describe
mathematically how things move under gravity by inventing the calculus.)
Can you have the square root of a negative number?
No! But yes! But only kind of!
When you square a negative number, you get a positive number. So both positive numbers and negative numbers square to make positive numbers. None of the numbers on the number line squares to make a
negative number, so negative numbers don’t have a square root on the number line.
Also, the square root is the length of a square of given area. For example, if a square has an area of 9cm2, it has a length of 3cm – that’s why the square root of 9 is 3. If we accept that areas are
positive (see above), then squares have positive area, so we’re not square rooting a negative.
Mathematicians have a number we call i, which when you square it gives you -1. So you often hear people say that ‘the square root of -1 is i’.
One interesting thing about i is that it’s a number but it’s nowhere on the number line! In fact, there are infinitely many numbers that aren’t on the number line. Just think about that: there are
numbers that aren’t on the number line!
The simplest versions of them are called imaginary numbers. This was in fact initially an insult – the famous mathematician Descartes mocked the idea that such numbers could exist – but it stuck as a
Most modern mathematicians would consider imaginary numbers as real as any other number, and they are a vital part of mathematics, which you can learn about in Year 12. (Modern electronics and
engineering depend on imaginary numbers so, far from being imaginary, they are actually vital for making stuff in the real world.)
Now the ‘but only kind of’ part:
Actually, I’m just going to leave it at ‘only kind of’ because I don’t want to confuse what you already know about square roots. But it’s not exactly true to say that the square root of -1 is i. This
is one we really will have to leave for Year 12! | {"url":"https://www.cambridgemathshub.co.uk/post/everything-you-ever-wanted-to-know-about-negative-numbers-and-weren-t-afraid-to-ask","timestamp":"2024-11-09T13:56:52Z","content_type":"text/html","content_length":"1050491","record_id":"<urn:uuid:7c0fa08c-bcb4-4fc7-a4a7-ee567dcf30c2>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00305.warc.gz"} |
Match the correlation coefficients with the scatte - Asksia.ai
Answer from Sia
Posted 6 months ago
Solution by Steps
step 2
Identify any discrepancies between the visual trend of the scatterplots and the provided correlation coefficients
step 3
Correctly match the scatterplots with the correlation coefficients based on the trends and the values
step 4
Scatterplot A shows a weak positive trend, but the coefficient is r = -0.71, which is a moderate negative correlation. This is a mismatch
step 5
Scatterplot B shows no clear trend, but the coefficient is r = -0.92, which is a strong negative correlation. This is a mismatch
step 6
Scatterplot C shows a weak negative trend, but the coefficient is r = 0.39, which is a weak positive correlation. This is a mismatch
step 7
Scatterplot D shows a strong positive trend, and the coefficient is r = 0.85, which is a strong positive correlation. This is a correct match
step 8
Based on the correct matches, adjust the other scatterplots' correlation coefficients accordingly
step 9
Assign r = -0.71 to the scatterplot that shows a moderate negative correlation, r = -0.92 to the scatterplot with no clear trend, and r = 0.39 to the scatterplot with a weak positive trend
[question number] Answer
Key Concept
Correlation Coefficient Matching
The correlation coefficient quantitatively describes the strength and direction of a linear relationship between two variables. It is important to match the visual trend of the scatterplot with the
appropriate correlation coefficient value.
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The most student-friendly features | {"url":"https://www.asksia.ai/question-and-answer/Match-the-correlation-coefficients-with-the-scatterplots-shown-belowbegintabularcc-Scatterplot--Correlation-co-hline-Scatterplot-A--r-071-Scatterplot--cuUT3","timestamp":"2024-11-04T01:45:45Z","content_type":"text/html","content_length":"93101","record_id":"<urn:uuid:f9df7560-91bd-458d-b36e-d6cb974397b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00576.warc.gz"} |
how to make maths working model (TLM) to teach the concepts of greater than (>), less than (
how to make maths working model (TLM) to teach the concepts of greater than (>), less than (<), and equal
Creating a maths working model (TLM) to teach the concepts of greater than (>), less than (<), and equal to (=) can be a highly effective way to help students understand comparison of numbers.
This interactive model will allow students to physically engage with these concepts and make learning more visual and fun.
To create a working model that visually demonstrates the concepts of greater than, less than, and equal to using numbers and interactive tools.
Materials Needed:
1. Cardboard or foam board (for the base)
2. Markers or colored pens (for writing numbers and symbols)
3. Scissors (to cut shapes)
4. Paper or colored paper (for creating numbers and comparison symbols)
5. Velcro strips or magnets (for attaching removable numbers and symbols)
6. Glue (for assembly)
7. Small objects (like buttons, beads, or pom-poms) for counting
8. Scale or balance model (optional, for comparison visualization)
Step-by-Step Guide for Greater Than, Less Than, and Equal To Model:
1. Prepare the Base:
Take a large cardboard or foam board (about 30 cm x 40 cm) as the base for the model.
• Divide the board into three sections with markers or a ruler:
1. Greater than (>)
2. Less than (<)
3. Equal to (=)
2. Create the Number Cards:
Objective: To create movable number cards that students can use to compare values.
• Cut out rectangular cards from colored paper (around 5 cm x 5 cm each).
• Write different numbers on these cards using markers (e.g., 1, 2, 3, 5, 10, 15, 20). You can create a variety of numbers ranging from small single digits to larger two-digit numbers.
• Add Velcro or magnet strips to the back of the cards so that they can be easily attached and removed from the board.
3. Create the Comparison Symbols:
Objective: To make the greater than, less than, and equal to symbols interactive.
• Cut out large comparison symbols from colored paper:
□ Greater than (>)
□ Less than (<)
□ Equal to (=)
• Make these symbols large and bold so they are easy to see. Attach Velcro strips or magnets to the back of these symbols so they can be placed on the board between the numbers.
4. Set Up the Model:
• On the left side of the board, create an area where students can attach one number card.
• In the middle section, allow for placement of one of the comparison symbols (> or < or =).
• On the right side, leave space to attach another number card.
• You can label these sections:
□ Left number
□ Comparison sign
□ Right number
• Now, students can place any two numbers on either side and choose the correct symbol to compare the two numbers.
5. Interactive Element – Counting Objects:
Objective: To provide a hands-on approach for comparing numbers.
• Add small buckets or boxes on either side of the comparison area, where students can place small objects like beads, pom-poms, or buttons.
• If the left number is 5, students can place 5 objects in the left bucket, and if the right number is 8, they can place 8 objects in the right bucket. This visual representation helps them see
which side has more or fewer objects.
• After placing the objects, they can select the correct comparison symbol based on their observations (greater than, less than, or equal to).
6. Optional Feature – Balance Scale Comparison:
Objective: To give a more visual and physical representation of greater than, less than, and equal to.
• You can create a simple balance scale using a hanger or a stick with two small cups attached at either end.
• When students compare two numbers, they can place objects representing those numbers (e.g., 5 marbles on one side and 8 on the other side) on the balance.
• The balance will tilt to the side with the larger number, giving a clear demonstration of greater than and less than.
• If the balance remains level, the numbers are equal.
7. Labeling:
• Add clear labels for each section to remind students of the concepts:
□ “Greater Than (>)”
□ “Less Than (<)”
□ “Equal To (=)”
• You can also add small labels showing real-world examples, such as:
□ “5 < 10 means 5 is less than 10.”
□ “7 > 3 means 7 is greater than 3.”
□ “8 = 8 means 8 is equal to 8.”
8. How the Model Works:
• Step 1: Students choose two numbers from the number cards and place them on the board in the left and right sections.
• Step 2: They then compare the numbers using the counting objects or the balance scale to visually see which number is greater, less, or equal.
• Step 3: After comparing, they select the appropriate symbol (>, <, or =) and place it between the two numbers.
Leave a Comment | {"url":"https://howtofunda.com/how-to-make-maths-working-model-tlm-to-teach-the-concepts-of-greater-than-less-than-and-equal/","timestamp":"2024-11-15T04:56:01Z","content_type":"text/html","content_length":"66636","record_id":"<urn:uuid:13dbaa3e-20b0-46d9-bb3e-0506fe95e425>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00335.warc.gz"} |
Fig. 7.1 SSFI for a frequency band
Spatio-Spectral Feature Image (SSFI) as a topographic map. In addition, the CNN model is used to explore the inter-subject dependency analysis. First, we explain the generation of SSFI, then we
describe the CNN model approach for LWR classification task. Finally, we discuss the results obtained from the prediction model for an individual subject and inter-subject dependency model.
Fig. 7.1 SSFI for a frequency band
7.1 Spatio-Spectral Feature Image -SSFI
For generating the SSFI, we use the 14 channels of EEG signal collected from the experiment
(Chapter 2). First, EEG signal is filtered with a 5 ^th -order, a highpass IIR filter with cut-off
90 Convolutional Neural Network for predictive analysis
Fig. 7.2 A collection of six SSFIs
frequency of 1 Hz. Then artifacts are removed from EEG signal using the wavelet-based approach with steepness constant of 0.1 and Interpercentile range as 50 in soft-thresholding operating mode
(explained in Chapter 4). After preprocessing, all the segments of listening, writing, and resting are extracted. For SSFI generation, we extract features from a small window of a segment, as
explained in Chapter 2 and shown in Figure 2.11. From each segment, a small window of 14 EEG channels is extracted, with a window size of N = 128.
The windows extracted from a segment, are overlapping with m = 128 − 16 = 112 samples or with a shift of 16 samples. Each channel of extracted window is decomposed into six frequency bands namely
delta (0.1−4 Hz), theta (4−8 Hz), alpha (8−14 Hz ), beta (14−30 Hz), low gamma (30 − 47 Hz) and high gamma (47 − 64 Hz), represented as δ, θ, α, β, γ 1 , and γ 2 respectively. Then we compute the sum
of power for each channel in six bands using the Welch method with Hamming windows. The resulting collection of spectral features was therefore calculated as
F [i,k] = ∑ ^P ^i ^(E ^k ^)
where E k is the k ^th EEG channel of extracted window and P i ( ·) is the power spectrum of the i ^th frequency band i ∈ [δ,θ,α,β,γ 1 , γ [2] ]. Since there are 14 channels and six frequency bands,
for a small window of signals, the feature vector F has 84 dimensions (F ∈ R ^84 ), such as
F =
F [δ,k] F [θ,k] F [α,k] F [β,k] F [γ]
[,k] F [γ]
Fig. 7.3 CNN approach with SSFI for predictive analysis
7.2 Convolutional Neural network model
For the predictive analysis of LWR classifications, a CNN approach is shown in Figure
7.3. For each segment of listening, writing, and resting, a small window of 128 samples
with overlapping of 16 samples is extracted and a collection of six SSFIs is generated and
labeled according to segment label. The CNN model shown in Figure 7.3 is consist of five
convolutional layers (CNV), two fully connected layers (FC) and an output layer with 3 units.
92 Convolutional Neural Network for predictive analysis
Fig. 7.4 CNN model for LWR classification
The schematic diagram of the CNN model is shown in Figure 7.4. Unlike the conventional CNN for RGB images, the number of input channels for our case is six. Each convolutional layer has a bank of
filters of a size of 3 by 3 and followed by a max-pool layer with a filter size of 2 by 2, except for CNV5. As shown in Figure 7.4, the number of filters for the first convolutional layer is 32 and 16
for the rest of the layers. The activation function used for the output layer is softmax and for all the other layers is rectified linear unit (ReLu) and l2 = 0.01 regularization. The weights of the
CNN model are optimised with Adam (Adaptive Moment Estimation) method to minimize the categorical cross-entropy loss function [124].
7.3 Training and testing
Since each brain has a different folding structure [82], the CNN model is trained on the data of an individual subject. The data of a subject is split into training and testing in serial order. The
first 100 segments of listening, writing and resting each, are used for training and rest are used for testing. The serial split is used due to the temporal nature of data.
Future information should not be used to predict past events. Since the features are extracted from a 1-second window (N = 128) with an overlap of 0.875 seconds (m = 112), a feature vector of one
window carries the potential information about the corresponding segment.
To avoid the information leak, the data is split into training and testing segments first and
then the features are extracted from each segment. A CNN model is trained and tested on a
subject. In addition, a trained model on a subject is tested on other subjects to analysis the
inter-subject dependency of the task. For completeness, the trained model is also tested on
majority class. In this case, data-point for writing class are in majority for each participant because the writing segments are longer than the other two. The random chance level for each
participant is different, which is marked in Figure 7.5 by a small bar. The performance of the model was observed to better than a random chance for training, however for testing, the accuracy was
observed less for seven subjects. It can be observed that the same model perform differently for each subject. The performance of the model could be improved by tuning the model for an individual
subject, however, the focus of this study is to analyse the performance of a model in a general setting. The results of an individual subject model cannot be compared to models used in Chapter 6,
since the feature extraction process is different for the CNN approach. The results for the CNN approach are corresponding to the prediction of subtask forgiven a small window of EEG signals,
whereas, in Chapter 6, the performance is corresponding to the entire segment.
Fig. 7.5 Results for individual subject, with a bar of random chance level
94 Convolutional Neural Network for predictive analysis
Fig. 7.6 Results for inter-subject model
Fig. 7.7 The average performance of model and subject
7.4.2 Inter-subject dependency model
The results for inter-subject dependency model is shown in Figure 7.6, as a matrix. On the
y-axis, subjects are labels on which CNN model is trained and on the x-axis, subjects are
labeled on which trained model is tested. The diagonal of the matrix shows the model trained
and tested on the same subject, hence the accuracy is higher on the diagonal. The matrix
shown in Figure 7.6 reveals interesting observations. It can be observed that for a model
trained on subject-1, perform better with other subjects on the other hand, a model trained on
subject-19, perform very poor, on other subjects. Interestingly, subject-19 perform better on
activity related to auditory attention than others. These subjects are a better representative of general attentional behaviour.
(a) CNV1-layer (b) CNV1-layer
(c) CNV2-layer (d) CNV3-layer
(e) CNV4-layer (f) CNV5-layer
Fig. 7.8 Deep features learned from a CNN model at different layers. Each image column of
six images are corresponding to one filter.
96 Convolutional Neural Network for predictive analysis
7.4.3 Features learned from deep filters in a CNN model
One of the common ways to explore trained CNN model is to visualise the patterns that activate the trained neurons. This approach reveals the primitive patterns or features that the network is
looking for. This approach is more suitable for our case since our input image is not a conventional image of an object. This is an indirect way to interpret the filters learned in a deep network. A
pattern is generated by feeding a random image to trained network and maximising the activation of a selected neuron by optimising the input image using gradient ascent method [125, 126].
The patterns generated from a model trained on subject-1 are shown in Figure 7.8. The number of filters in CNV1 layers is 32 and in the other four layers are 16. A collection of six images is
generated for a filter, one for each frequency band. In Figure fig:cnv11, the first column of images corresponds to the first filter, hence represent the features extracted by the first filter. A top image
of the first column corresponds to frequency band-δ and bottom image corresponds to γ 2 . It is interesting to note that the filters of CNV1-layer (in Figure 7.8a, and 7.8b) are focusing on the specific
locations of the input SSFI. Specifically, one filter is focusing on the right temporal lobe of image and other is focused on the left temporal lobe with the parietal lobe. In the first layer, the
features learned are smoother than other layers.
The feature learned in the CNV2 and CNV3 layer are focusing the very small regions on the SSFI on the entire image. The features learned in CNV4 and CNV5 are however more into specific locations. It
also can be noticed that most of the filters are producing a random noisy pattern, which is the indication of a dead filter [126].
The approach presented in this chapter has the potential to exploit the spatial relationship of EEG electrodes, which can be further used to analyse the Inter-subject correlation [127].
A CNN model used for our study can also be improved by processing each channel of SSFIs
separately since they are independent in terms of functionality they represent for the human | {"url":"https://123dok.org/document/y963wx7j-fig-ssfi-for-a-frequency-band.html","timestamp":"2024-11-06T01:17:06Z","content_type":"text/html","content_length":"145374","record_id":"<urn:uuid:cdcde5bb-9246-4e04-8cba-b2f428d7211a>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00085.warc.gz"} |
In this chapter, you learned about the Quantum Lab Notebooks and ran a simple quantum circuit. You completed three basic functional steps: creating a quantum circuit using the Notebook and the Qiskit
library, executing your circuit with a backend simulator and real device, and reviewing and visualizing your results from within the Notebook.
One thing you might have noticed is that using the Notebook with Qiskit also simplifies integrating your classical experiments with a quantum system. This has provided you with the skills and
understanding to enhance your current Python experiments and run certain calculations on a quantum system, making them a hybrid classical/quantum experiment.
When the quantum calculations have completed, the results can be very easily used by your classical experiments.
Now that we are familiar with the Quantum Lab Notebooks and are able to create and execute a circuit, in the next chapter, we will start learning the basics of quantum computing... | {"url":"https://subscription.packtpub.com/book/programming/9781838981006/4/ch04lvl1sec20/summary","timestamp":"2024-11-05T09:50:20Z","content_type":"text/html","content_length":"276669","record_id":"<urn:uuid:fe238284-4975-4a0d-ad60-0ba09b554177>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00831.warc.gz"} |
Marx-Engels Correspondence 1881
Engels To Marx
In London
Source: Marx’s Mathematical Manuscripts, New Park Publications, 1983;
Transcribed: by Andy Blunden.
August 10, 1881
Dear Mohr,
Yesterday I found the courage at last to study your mathematical manuscripts even without reference books, and I was pleased to find that I did not need them. I compliment you on your work. The thing
is as clear as daylight, so that we cannot wonder enough at the way the mathematicians insist on mystifying it. But this comes from the one-sided way these gentlemen think. To put dy/dx = 0/0, firmly
and point-blank, does not enter their skulls. And yet it is clear that dy/dx can only be the pure expression of a completed process if the last trace of the quanta x and y has disappeared, leaving
the expression of the preceding process of their change without any quantity.
You need not fear that any mathematician has preceded you here. This kind of differentiation is indeed much simpler than all others, so that just now I applied it myself to derive a formula I had
suddenly lost, confirming it afterwards in the usual way. The procedure must have made the greatest sensation, especially, as is clearly proved, since the usual method of neglecting dxdy etc. is
positively false. And that is the special beauty of it: only if dy/dx = 0/0 is the mathematical operation absolutely correct.
So old Hegel guessed quite correctly when he said that differentiation had for its basic condition that the variables must be raised to different powers, and at least one of them to at least the
second, or ½ power. Now we also know why.
If we say that in y = f(x) the x and y are variables, then this claim has no further consequences, as long as we do not move on, and x and y are still, pro tempore, in fact constants. Only when they
really change, i.e. inside the function, do they indeed become variables, and only then can the relation still hidden in the original equation reveal itself — not the relation of the two magnitudes
but of their variability. The first derivative Dy/Dx shows this relation as it happens in the course of real change, i.e. in each given change; the completed derivative — dy/dx shows it in its
generality, pure, and hence we can come from dy/dx to each Dy/Dx, while the latter itself only covers the special case. However, to pass from the special case to the general relationship, the special
case must be abolished (aufgehoben) as such. Hence, after the function has passed through the process from x to x’ with all its consequences, x’ can be allowed calmly to become x again; it is no
longer the old x, which was variable in name only; it has passed through actual change, and the result of the change remains, even if we again abolish (aufheben) it.
At last we see clearly what mathematicians have claimed for a long time, without being able to present rational grounds, that the differential-quotient is the original, the differentials dx and dy
are derived: the derivation of the formulae demands that both so-called irrational factors stand at the same time on one side of the equation, and only if you put the equation back into this its
first form dy/dx = f'(x) , as you can see, are you free of the irrationals and instead have their rational expression.
The thing has taken such a hold of me that it not only goes round my head all day, but last week in a dream I gave a chap my shirt — buttons to differentiate, and he ran off with them. | {"url":"https://www.connexions.org/CxArchive/MIA/marx/works/1881/letters/81_08_10a.htm","timestamp":"2024-11-02T08:23:41Z","content_type":"text/html","content_length":"5262","record_id":"<urn:uuid:8852797e-4b2c-425b-a4a2-209098c5288c>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00761.warc.gz"} |
seminars - $L_p$-bounds for pseudodifferential operators on curved noncommutative tori
In the theory of pseudodifferential operators, one of the most essential topics is the study of mapping properties of pseudodifferential operators between various kinds of function spaces. The
investigation of $L_p$-boundedness of pseudodifferential operators is particularly important, considering its consequences for the regularity and existence of solutions of PDEs.
The purpose of this talk is to discuss the counterpart of this problem on noncommutative tori. Noncommutative tori are the most intensively studied noncommutative spaces in noncommutative geometry
and arise in various parts of mathematics and mathematical physics. Pseudodifferential calculus on noncommutative tori was introduced in early 1980s by A. Connes, and it has emerged as an
indispensable tool in the recent study of differential geometry of noncommutative tori. Meanwhile, J. Rosenberg introduced the notion of Riemannian metric on noncommutative tori a decade ago. In this
talk, I will first recall the notion of a curved noncommutative torus, i.e., a noncommutative torus endowed with a Riemannian metric in the sense of J. Rosenberg. I will then show the boundedness of
pseudodifferential operators on noncommutative $L_p$-spaces associated with the volume form induced by a Riemannian metric. Based on joint work with V. Kumar. | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&page=12&document_srl=1138200","timestamp":"2024-11-06T05:46:48Z","content_type":"text/html","content_length":"45716","record_id":"<urn:uuid:8095d2ab-95a1-409b-a7ab-b9c1914b6c97>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00724.warc.gz"} |
What Does Reciprocal Mean in math?
| What Does Reciprocal Mean in math?
What Does Reciprocal Mean in math?
July 30, 2022
Reciprocal mean in math is simply defined as the inverse of a value or a number. if m is a natural number then the reciprocal of m is equal to 1/m. for example, the reciprocal of 8 is 1 divided by 8
reciprocal of 8 is 1/8.
The word reciprocal came from the Latin word
‘Reciprocus’meaning ‘’returning .in this article learn the definition of reciprocal and .how to find the reciprocal of numbers.
What is reciprocal?
Reciprocal mean in math is simply defined as the inverse of a value or a number.if m is a natural number then the reciprocal of m is equal to 1/m. for example, the reciprocal of 8 is 1 divided by 8
reciprocal of 8 is 1/8.
The word reciprocal came from the Latin word
‘Reciprocus’meaning ‘’returning.
• The reciprocal of 4 and 5 are 1/4 and 1/5
• The reciprocal of 1/6 and 2 are 6 and 1/2
Not for zero
The reciprocal condition can not apply on zero, it will return an indefinite value.
Therefore, all real numbers have a reciprocal but not for zero (0)
Reciprocal of a number?
The reciprocal of a number is defined as divided by the one by the given number
Find the reciprocal of 7
We will use
Reciprocal of a negative number?
Reciprocal can be a negative number (-x) the inverse of the given number with a negative sign along with that (-1/x)
For example, the reciprocal of -5x^2 is writing 1/-5x^2
The following step is to find the reciprocal of a negative number.
Step1: For any given negative number write the form of an improper fraction by writing the number 1 in the denominator.
Step2: interchange the numerator and denominator
Step3: add a negative sign (-) to the resultant number.
Example: find the reciprocal of a negative number.
Considered a negative number -18
Step 1: convert the number 18 into the improper fraction (18/1)
Step2: interchange numerator and denominator values we get
Step 3: add a negative sign to the resultant number
We get
Therefore, the reciprocal of negative number -18 is -1/18
Reciprocal of a mixed fraction?
In order to find the same for a mixed fraction. Convert into improper fractions.
For example: find the reciprocal of a mixed fraction
Considered a mixed fraction, 4(1/2).
Step1: convert a mixed fraction into an improper fraction.
Step 2: interchange nominator and denominator values.
We get,
Therefore, the reciprocal of mixed fraction 4(1/2) is 2/9.
Reciprocal of decimal?
The reciprocal of a decimal is the same defined for the reciprocal of a number.
For example: Find the reciprocal of a decimal
Considered, decimal fraction0.75
0.75 is written as3/4
Reciprocal of 3/4 is 4/3
Finding unity?
Reciprocal of number multiply by itself number is equal to unity (1).let see some example
• 2X1/2=1
• 15×1/15=1
• 23×1/23=1
Rules for reciprocal?
Two important rules for reciprocal are.
• Any number y the reciprocal will be 1/y
Are also can be written as x^-1
• For any fraction y/x, the reciprocal will be x/y. | {"url":"https://eduinput.com/what-does-reciprocal-mean-in-math/","timestamp":"2024-11-02T21:19:31Z","content_type":"text/html","content_length":"151687","record_id":"<urn:uuid:aa020af0-44c7-4107-9006-c4caec60899a>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00063.warc.gz"} |
Time Series Archives - Peltier Tech
A reader named Felix posted a comment on the blog, asking how he could make a line chart which has two time series, in which each time series has data for different dates. What makes his question
more challenging is that Felix is using Excel 2007. This was such a good question, it deserves its own post.
Felix provided the following data:
I’ll describe the various techniques available in Excel 2003 and earlier, with pros and cons, and I’ll compare them to the same techniques as followed in Excel 2007.
Time Series in Excel 2003
The two time series are plotted separately below. Time series A has weekly data, but with two values omitted. Time series B has more data points, at irregular intervals, over a shorter time span.
Create the Time Series A line chart above left, copy the Time Series B data, select the chart, and use Paste Special to add the data as a new series, using the options as shown.
This illustrates a limitation with Line charts in Excel: the category labels or dates are defined by the first series. Any additional series are forced to use the same X values or labels, and if the
added series has more points than the first series, those extra points are omitted.
The obvious alternative is to make an XY chart with the two data series. The X and Y values of separate series in an XY chart may be completely independent.
The data is plotted correctly, but the date axis isn’t as nice as the one we had in the line chart. A line chart lets you put a tick on the first of every month, but since the length of a month is
not constant, this doesn’t work in an XY chart.
So how can we plot multiple time series on a chart with nice date labels?
The limitation that all series in a Line chart must use the same dates is not completely true. All series on the primary axis use the dates for the first primary axis series, while all series on the
secondary axis use the dates for the first secondary axis series. Format series B in the two-series line chart so it resides on the secondary axis.
There are drawbacks to this approach. First, it ties up the secondary axis with data that otherwise fits the primary axis, so you are limited to how else you can embellish the chart. You can always
hide the secondary date axis to clean up the chart, but you cannot remove it altogether. Second, the two date axes must be manually synchronized each time the data changes. Third, you are limited to
two different sets of dates.
There is an alternative combination chart that gets around these limitations. Make the first series a Line chart series, so you have a nice date scale axis, then add any additional series and change
them to XY type series. At first, Excel associates the XY series with the secondary axis, but you can override this setting and assign it to the primary axis, and it will use the nicely formatted
date scale of the Line chart. You are not limited to two sets of dates, and you do not have the added responsibility of synchronizing the time axis scales.
But there’s an easier way.
Unfortunately this never occurs to me until I’ve already made a Line-XY combination chart. You can combine the data in the following way, putting the Series B dates below the Series A dates, and the
Series B values below and one column offset from the Series A values.
You don’t even need to sort the dates, because a line chart internally sorts the dates before plotting the points, so that it connects them in date order. In contrast, an XY chart connects the points
in the order they appear in the worksheet.
Select this larger data range and create your line chart. The points are plotted according to their own dates. The only problem is that, by default, Excel leaves a gap between points if there is a
blank value cell between these points (below left). However, it’s easy enough to change this behavior. Go to Tools menu > Options > Chart, and in the top section, choose the option that makes Excel
interpolate across the gaps (below right).
Now why didn’t I think of that before I made all those complicated charts?
Time Series in Excel 2007
Excel 2007 works much the same way as earlier versions, but there are a few notable exceptions.
In a line chart, all series use the same categories or dates as the first series, and any extra points are truncated. Just like in Excel 2003.
Two time series can be plotted together, with one on the secondary axis, and the times will be kept independent. This approach is subject to the same limitations as in Excel 2003.
You can change the second series to an XY type series, and when plotted on the secondary axis it works just fine. (Note: you can remove the secondary Y axis and both series will use the primary Y
The limitation here is that you are tying up the secondary axis with the XY series, thus limiting your ability to use the secondary axis for other tricks. You are not limited to two series, because
any number of XY series can be plotted independently on the secondary axis.
In Excel 2003 we could assign the XY series to the primary axis, and it would coexist with the line chart series and use the same axis. However, when we try this in Excel 2007, we notice two flaws.
First, it’s unclear what scale the XY series is using for its X values, but it’s unrelated to the primary category axis. Second, there are only as many XY series points shown as there are Line series
points in the first series. Whoa! How’d that get through QA?
These glitches are not too important for this particular use of a Line-XY combination chart, but there are other uses which are completely broken by the changed behavior.
Fortunately the easiest method of all, the one which uses the combined data, works just fine in Excel 2007. The dual chart first appears with gaps, but you can change the behavior by opening the
Select Data dialog, clicking on the Hidden and Empty Cells button in the bottom left corner, and selecting Connect Data Points with Line.
This is one more example of my favorite phrase: You can spend five minutes with your data, and save yourself five hours of frustration and aggravation. | {"url":"https://peltiertech.com/tag/time-series/","timestamp":"2024-11-06T01:12:51Z","content_type":"text/html","content_length":"78828","record_id":"<urn:uuid:2fcb8a71-b6f4-482d-8936-21d524082fb0>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00170.warc.gz"} |
MacroeconoMacro Econometric Income Consumption Model for Indiamic Analysis of Greece
Macroeconomic Analysis of Greece
Consumer spending is an important factor that can stimulate the economic growth and development through the multiplier process. This study aims to estimate the pattern of consumption expenditure and
tries to identify the consumption function for Indian Economy. The study intends to identify the determinants of consumption and to build a econometric model using the annual data from RBI Handbook
of Statistics on Indian Economy (2008-2009) for the time period 1970 to 2009. Following are the variables taken into consideration for empirical analysis; Private Final Consumption Expenditure
(PFCE), Personal Disposable Income (PDI), Rate of Interest (ROI) and Inflation (INF). We have employed rigorous econometric techniques in analyzing the time series data so as to ensure the
credibility and reliable economic relations. The results confirm income (PDI) as the most significant factor. The MPC calculated which ranges between 0.80 to 0.90, which is in affirmation with the
theoretical assumptions and is more or less similar to the previous studies. The conclusions from the model suggest that the Keynesian Absolute Income Hypothesis is found to be appropriate to Indian
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The relation between aggregate consumption or aggregate savings and aggregate income, generally termed the consumption function, it has occupied a major role in economic philosophy ever since Keynes
made it a keystone of his theoretical structure in The General Theory of Employment, Interest and money. The role of consumption in the multiplier process has increased the scope and dynamics of the
topic which led to further developments in this field by developing more realistic and logical consumption (Income) hypothesizes. Consumption which was considered only as a function of income was
later refined and redefined.
The purpose of the study is to examine and comprehend the issues, trends and rationale behind the consumption pattern in India. In our study an attempt is done so as to understand and estimate the
potential factors that had led development. We estimate the consumption model in India using advanced econometric tools. The study is conducted for period from 1970 to 2009 using the annual data from
RBI Hand Book of Statistics on Indian Economy.
Consumption function indicates a functional relationship between consumption, income and other factors. It shows how consumption expenditure varies as there is change in the income and other factors
such as age, social status, interest rates etc. Whereas Consumption refers to amount spent on consumption at a given level of income. On the other hand consumption function refers to actual
consumption at various level of income.
Major development in this respect took place when In 1936 Keynes formulated a consumption function which was the basic element in the income expenditure approach to the determination of national
income. Consumption function for him was the basic building block of multiplier analysis. According to Keynes marginal propensity to consume is less than average propensity to consume this is well
described in the stagnation thesis around 1940. Keynes observed this as behavior of the consumption expenditure in the short run over the long run.
Keynes offered no precise functional formulation of the propensity to consume; his analysis has come to be associated with a simple version of the consumption function that embodies only the more
quantitative aspects of his considerations, popularly known as the simple Keynesian consumption function or Absolute Income Hypothesis (AIH). The theory asserts as income rises, the theory asserts,
consumption will also raise but not necessarily at the same rate. The basic principle of the absolute income hypothesis is that the individual consumers who determine what fraction of his income will
he devote to consumption on the basis of the absolute level of that income. AIH provided a background for the further studies in this field. This resulted in the development of three more theoretical
models namely Relative Income Hypothesis (RIH), Permanent Income Hypothesis (PIH), and Life Cycle Hypothesis (LCH).
Relative Income Hypothesis (RIH) developed by Duesenberry in 1949 conceives consumption in relation to the income of other households and past income. It implies that the proportion of income
consumed remains constant provided that a household’s position on the income distribution curve holds constant in the long run. This is consistent with long-run evidence. Higher up the income curve,
however, there is a lower average propensity to consume. The second part of the hypothesis suggests that households find it easier to adjust to rising incomes than falling incomes. There is, in other
words, a “ratchet effect” that holds up consumption when income declines. Duesenberry’s analysis is based on two relative income hypotheses. The first hypothesis is essentially that consumers are not
so much concerned about the absolute level of consumption as they are with their consumption relative to that of rest of population. Second hypothesis Duesenberry argues that present levels of
consumption is not influenced merely by present levels of absolute or relative income, but also by levels of consumption attained in previous periods.
Absolute income hypothesis when captures the effect of current income on current consumption the theories developed there after focus on the influence on income on consumption in a broad sense.
Permanent income hypothesis developed Milton Friedman further divide the income component into two parts. First include the permanent income component and transitory component the second. He states
that consumption is determined by the permanent component and normally transitory income is saved.
To be more specific, The Permanent Income Hypothesis decomposes measured total disposable income, Y, into a permanent component (YP), and a transitory component, (YT). The permanent income component
is deemed systematic but unobservable, reflecting factors that determine the household’s wealth, while the transitory component reflects “chance” income fluctuations. Similarly, measured consumption,
c, is decomposed into a permanent component, CP, and a transitory component, cT. In giving the hypothesis empirical substance, Friedman assumes the transitory components to be uncorrelated across
consumption and income, and with their respective permanent components.
A little different from these above mentioned hypotheses the Life-Cycle Hypothesis presents a well-defined linkage between the consumption plans of an individual income and income expectations as
passes from childhood, through the work participating years, into retirement and eventual decease. The main building block of life-cycle models is the saving and consumption decision, i.e., the
division of income between consumption and saving. The saving decision is driven by preferences between present and future consumption (and the utility derived from consumption). Given the income
stream the household receives over time, the sequence of optimal consumption and saving decisions over the entire life can be computed. It should be noted that the standard life-cycle model as
presented here is firmly grounded in expected utility theory and assumes rational behavior.
Data Used For the Study:
We have used annual long-run time series data on Private Final Consumption Expenditure, Personal Disposable Income, Gross Domestic Savings, Rate of Interest and Inflation from The Handbook of
Statistics on Indian Economy 2008-2009 published by Reserve Bank of India (2008-2009). They are represented as the following: Private Final Consumption Expenditure (PFCE), Personal Disposable Income
(PDI), Rate of Interest (ROI) and Inflation (INF). Where in PFCE is the dependent variable. Econometric Methodology:
One of the major and crucial problems that can be faced while dealing with time series data is, many a times data may be non – stationary. So avoid spurious regression it is necessary to check the
time series data for stationarity using unit root tests. Keeping this in mind the unit root test has been carried out for each series using the Augmented Dickey-Fuller test for the period 1970 –
2008. All the variables are non stationary at the levels and in order to make them stationary we employed the technique of differencing. All variables other than rate of interest is differenced
twice, where (D) stands for differencing once and D (D) for differencing twice.
Table: 1 Unit root tests with Trend and Intercept: 1970 – 2008
1st difference
Non -stationary
Non -stationary
Non -stationary
Non -stationary
1% critical value = -3.50, 5% critical value = -2.89, 10% critical value = -2.58
The analysis also takes into account the lag structure that plays a vital role in the consumption analysis. To study the role of previous peak incomes and the role of habits the functional form that
we can use is as follows:
Ct = α + β0Yt + β1 Yt-1+ εt
Using the given functional model where Ct is consumption at time period (t), Yt represent income at time period (t), Yt-1 representing one year lagged value of income where in we can study the long
run effects of income on consumption. But the above equation (distributed lag model), since takes the lag of independent variable there is all possibility of encountering the problem of
multicolinearity. Thus we need to transform this model into some other model which takes care of the problems. When we have distributed lag models where lag structure follow the geometric form we can
transform them using the Koyack transformation.
The transformed model AR (1) can be re written as follows:
Ct = α + β0Yt + δCt-1+ ut
This model is called a Auto regressive model where lagged value of dependent variable itself will be a independent variable. In the above model β0 measure the short run effect or the short run MPC
and δ measure the long run effects, in our model it is the long run MPC.
Estimated Equations:
Equation: 1
For the above consumption equation the independent variables are income, rate of interest, inflation and a year lagged value of the dependent variable. According to the theoretical setup the
coefficient of income demands a positive relationship. This is for the reason that when income increases consumption also increases and more over the coefficient of income indicate the MPC which is
supposes to be a positive value less than one. Both our equations satisfy this condition. In both the equation the coefficient of savings and rate of interest shows a negative relationship. It is
obvious that when savings increases consumption decreases because savings is considered as an alternative for consumption and savings increases when rate of interest is high thus when rate of
interest is high savings increases and the consumption expenditure decreases. Inflation is included as an independent variable to evaluate the effect of prices, when prices increases the expenditure
on consumption is bound to increase so we expect a positive relation. The AR coefficient showing a positive relation is having a number of theoretical implications for example for permanent income
hypothesis to hold good the AR coefficient should be negative. The theoretical implications of the positive AR coefficient are explained in the following discussions.
Since the estimates are partial regression coefficients all the coefficients are explained by keeping the assumption, when effect all the other variables are kept constant what is the impact of a
variable on consumption. From equation one the value of income co efficient can be read as follows, when there is one percent increase in income there will be 0.83 percent increase in consumption.
Thus a value of 0.92 in the second equation for the income coefficient indicates, when there is one percent increase in income consumption will increase by 0.92 percent. Theoretically the coefficient
of income is the MPC which give information about the change in consumption when income changes by one unit. The limits of MPC are zero to one and our both equation satisfies this condition. It is
also important to note that both the equations income turns out to be the most significant factor, the t values for this coefficient is 19.31 and 23.82 respectively.
The coefficient of savings and the rate of interest show a negative relation which indicate an inverse relation of these variables with respect to the dependent variable consumption, this holds good
for both the equations. For both the equations the coefficient is same for savings. When all other variables are kept constant an increase in savings by one unit will decrease consumption by 0.06
percent and for a unit increase in rate of interest will decrease the consumption by 0.0183 percent in the first equation and 0.0244 percent according to the second equation. In both the equations
savings is backed by significant t- values but interest rates are relatively in significant in both the equations
In the first equation inflation is one of the very significant variable, it is proved that inflation will have a positive impact on consumption and this was expected, this is because of the reason
that in a developing nation like India the maximum is spend on the necessary commodities thus an increase in price will increase the consumption expenditure. It is estimated that one percent increase
in inflation will lead to increase in consumption by 0.0115 percent.
Equation (1) is supported by statistically significant t value and high R2 of 0.97 which imply 97%, of the variation in consumption expenditure is explained by the explanatory variables. A DW
statistic of 2.01 rules out the problem of series auto correlation. Equation (2) is also supported by statistically significant t values and high R2 of 0.95 which imply 95%, of the variation in
consumption expenditure is explained by the explanatory variables. In an AR (1) for identifying the elements of auto correlation we need to look at D.W ‘h’ statistic. The calculated D.W ‘h’ of 0.07
rejects the possibility of auto correlation.
Calculation of Long Run MPC
For the given AR model (Equation: 2):
Ct = α + β0Yt + δCt-1+ ut
β is the short run MPC and according to theoretical models the β (MPC) should be less than one. AR model can be used even to calculate the long run MPC. According to theoretical models long run MPC
should be greater than the short run and should be equal to one. The long run MPC can be calculated as follows:
Long run MPC (δ) = β / 1- δ
= 0.92 / 1-0.07
= 0.989
Thus the AR coefficient of 0.07 can be interpreted as follows, when the increase in the income is sustained, then the increase in the MPC (long run MPC) out of income will be 0.989. In other words
when consumers have time to adjust for one unit change in income, they will increase their only for about 0.989 percent.
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Theoretical Implications:
Macro Econometric models must fit into a theoretical framework and should be handy in terms of policy implications. This holds well in the case of Consumption models also. So in this respect it is
necessary to validate the model (Equation 1 & 2) by testing and classifying them to the income hypothesis developed.
Some of the previous studies in the Indian context conforms that for a developing economy like India the major source for consumption for any given time period will be the current income source and
thus the nature of economy fits itself into Keynesian Hypothesis. Krishnamurthy (1996) supports Keynesian setup with a (0.75) MPC. Pandit (2000) in Macroeconometric Policy Modeling for India: A
Review of Some Analytical Issues support the Keynesian setup by stating that in consumption function follows Absolute Income Hypothesis. The results of a study by Ghatak contradict with Friedman
results. She by examining the Indian economic scenario for the year (1919-1986) found that, in India the transitory component of the income also consumed. This is because of the reasons in a
developing country such as India, temporary increases in income are likely to be consumed wholly; this was deliberately encouraged by government policies to push people above the poverty line.
Whereas Permanent income hypothesis argue that permanent income is consumed and savings is determined by transitory component.
Equations (1 & 2) confirm the significant role of current income determining current consumption which supports Keynesian argument of Absolute Income Hypothesis. The MPC value derived from both the
equations satisfies the theoretical requirements as proposed by Keynes (0 < MPC < 1).
Equation (2) which is an AR model with a positive value for the AR coefficient and the calculated long run MPC tested equal to one prove the Keynesian argument that the Long run MPC > Short run MPC
and the constant behavior of the long run MPC.
Thus the analysis conclude that given the annual time series data for Indian economy form 1970-2008, the consumption pattern follows Keynesian model.
Forecasting From the Equations:
To examine the credibility of equations (both [1] & [2]) validation tests are performed. In sample forecasts are obtained in this respect for the period from 1970 to 2008. The accuracy of the model
is tested by calculating the Root Mean Squared Error and Theil Inequality Coefficient. The values imply that the forecasted series in model is very close to the actual series and there are no
systemic tendencies to over/under estimate the actual data. Forecasts are based on Dynamic simulations.
Table: 2 Forecasting Performance Measures
Root Mean Square Error
Theil Inequality coefficient
Equation 1
Equation 2
The Root Mean Square Error and the Theil Inequality Coefficient for both the equations are satisfactory and ensure the credibility of using the model for forecasting. (See graph 1&2 in appendix)
Granger Causality Tests:
Granger causality test is a technique for determining whether one time series is useful in forecasting another. Granger causality tests reveal whether one variable reveals whether one variable
granger cause other. Pair wise Granger causality results for the respective variables are presented below. Inferences are drawn looking at the probabilities. We reject the null hypothesis where the
probability values are lower.
Table 3 Pair wise Granger Causality Results
Null Hypothesis:
LnPFCE does not Granger Cause LnSAV
LnSAV does not Granger Cause LnPFCE
LnPFCE does not Granger Cause LnINF
LnINF does not Granger Cause LnPFCE
LnPFCE does not Granger Cause LnPDI
LnPDI does not Granger Cause LnPFCE
LnROI does not Granger Cause LnPFCE
LnPFCE does not Granger Cause LnROI
From the table above we can infer that savings (LnSAV) granger cause consumption expenditure (LnPFCE), Inflation (LnINF) granger cause consumption expenditure, income (LnPDI) granger cause
consumption expenditure (LnPFCE) and consumption (LnPFCE) and (LnROI) show a bi- directional causal relationship . These results are supported by a strong theoretical base. It is theoretically and
empirically proved that consumption expenditure is influenced by income, savings, and inflation.
The present study attempts to examine some of the issues relating to India’s consumption pattern and to identify the significant determinants that influence Indian consumption function. The study
considers the data from 1970-2008. Data sources are RBI Handbook of Statistics on Indian economy (2008-2009). Empirical modeling is taken up by considering the theoretical issues. The economic
reforms and the increasing income and population explosion have contributed to increase in the consumption expenditure in Indian resulting in higher standard of living. From our study we infer that
almost 85 – 95 percentage increase in consumption is as a result of increasing income levels of individuals in the economy.
Identifying the major determinants of consumption function includes more of econometrical techniques and empirical estimations. The exercise revels that income is the most significant factor that
affect the consumption any given period. Other significant determinants turn out to be Savings, Rate of Interest, and Inflation. Identifying the suitable Income hypothesis for India concludes that it
is Keynesian theory of Absolute Income Hypothesis that suits Indian economy for the following reasons. Firstly the estimates indicate the significant effect of current income on consumption as
suggested by Keynes. Second the MPC value estimated from the (equation 1 & 2) is in tune with the theoretical assumptions made by Keynes where he argues that the MPC should positive integer less than
one. The MPC calculated which ranges between 0.80 to 0.90, which is in affirmation with the theoretical assumptions and is more or less similar to the previous studies. The MPC calculated from some
of the previous study on Indian economy are as follows:
Table: 4 MPC calculated from some of the previous study on Indian economy
Anita Ghatak
Iyengar & Moorthy
Moorthy & Thore
Tinter & Narayanan
Thirdly Keynesian hypothesis argue that long run MPC should be larger than the short run this is established in the second equation where we obtained a positive value of 0.0727 from the AR (1)
coefficient. The Hypothetically tested results prove that the long run MPC is constant and is not significantly different from one.
Figure: 1 Insample Forecasting Estimates From Equation: 1
Figure: 2 Insample Forecasting Estimates From Equation: 2 | {"url":"http://thesis.visit-now.net/macroeconomacro-econometric-income-consumption-model-for-indiamic-analysis-of-greece/","timestamp":"2024-11-04T12:29:39Z","content_type":"text/html","content_length":"147505","record_id":"<urn:uuid:61a2bbcb-f043-4dba-be02-ebab042fbe42>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00571.warc.gz"} |
An object is at rest at (3 ,5 ,1 ) and constantly accelerates at a rate of 4/3 m/s^2 as it moves to point B. If point B is at (7 ,9 ,2 ), how long will it take for the object to reach point B? Assume that all coordinates are in meters. | HIX Tutor
An object is at rest at #(3 ,5 ,1 )# and constantly accelerates at a rate of #4/3 m/s^2# as it moves to point B. If point B is at #(7 ,9 ,2 )#, how long will it take for the object to reach point B?
Assume that all coordinates are in meters.
Answer 1
The time taken is $= 2.94 s$
The following equation of motion will be used:
The distance between 2 points #A=(x_1,y_1,z_1)# and #B=(x_2.y_2.z_2)# is
#A=(3,5,1)# and #B=(7,9,2)#
Utilizing the motion equation,
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Answer 2
To find the time it takes for the object to reach point B, we can use the formula for displacement in uniformly accelerated motion:
[ \Delta x = v_0 t + \frac{1}{2} a t^2 ]
• ( \Delta x ) is the displacement (change in position),
• ( v_0 ) is the initial velocity (which is zero since the object is at rest),
• ( a ) is the acceleration,
• ( t ) is the time.
• Initial position (( x_0, y_0, z_0 )) = (3, 5, 1) meters
• Final position (( x_f, y_f, z_f )) = (7, 9, 2) meters
• Acceleration (( a )) = ( \frac{4}{3} ) m/s²
Substituting the given values into the formula and solving for ( t ):
[ 7 - 3 = 0 \cdot t + \frac{1}{2} \cdot \frac{4}{3} \cdot t^2 ] [ 4 = \frac{2}{3} \cdot t^2 ] [ t^2 = \frac{4 \cdot 3}{2} ] [ t^2 = 6 ] [ t = \sqrt{6} ]
Thus, it will take the object ( \sqrt{6} ) seconds to reach point B.
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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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gce_MakeLin2d Class Reference
This class implements the following algorithms used to create Lin2d from gp. More...
#include <gce_MakeLin2d.hxx>
gce_MakeLin2d (const gp_Ax2d &A)
Creates a line located with A. More...
gce_MakeLin2d (const gp_Pnt2d &P, const gp_Dir2d &V)
gce_MakeLin2d (const Standard_Real A, const Standard_Real B, const Standard_Real C)
Creates the line from the equation A*X + B*Y + C = 0.0 the status is "NullAxis"if Sqrt(A*A + B*B) <= Resolution from gp. More...
gce_MakeLin2d (const gp_Lin2d &Lin, const Standard_Real Dist)
Make a Lin2d from gp <TheLin> parallel to another Lin2d <Lin> at a distance <Dist>. If Dist is greater than zero the result is on the right of the Line <Lin>, else the result is on
the left of the Line <Lin>. More...
gce_MakeLin2d (const gp_Lin2d &Lin, const gp_Pnt2d &Point)
Make a Lin2d from gp <TheLin> parallel to another Lin2d <Lin> and passing through a Pnt2d <Point>. More...
gce_MakeLin2d (const gp_Pnt2d &P1, const gp_Pnt2d &P2)
Make a Lin2d from gp <TheLin> passing through 2 Pnt2d <P1>,<P2>. It returns false if <P1> and <P2> are confused. Warning If an error occurs (that is, when IsDone returns false), the
Status function returns: More...
gp_Lin2d Value () const
Returns the constructed line. Exceptions StdFail_NotDone if no line is constructed. More...
gp_Lin2d Operator () const
operator gp_Lin2d () const
Standard_Boolean IsDone () const
Returns true if the construction is successful. More...
gce_ErrorType Status () const
Returns the status of the construction: More...
This class implements the following algorithms used to create Lin2d from gp.
• Create a Lin2d parallel to another and passing through a point.
• Create a Lin2d parallel to another at the distance Dist.
• Create a Lin2d passing through 2 points.
• Create a Lin2d from its axis (Ax1 from gp).
• Create a Lin2d from a point and a direction.
• Create a Lin2d from its equation.
◆ gce_MakeLin2d() [1/6]
gce_MakeLin2d::gce_MakeLin2d ( const gp_Ax2d & A )
Creates a line located with A.
◆ gce_MakeLin2d() [2/6]
gce_MakeLin2d::gce_MakeLin2d ( const gp_Pnt2d & P,
const gp_Dir2d & V
is the location point (origin) of the line and <V> is the direction of the line.
◆ gce_MakeLin2d() [3/6]
gce_MakeLin2d::gce_MakeLin2d ( const Standard_Real A,
const Standard_Real B,
const Standard_Real C
Creates the line from the equation A*X + B*Y + C = 0.0 the status is "NullAxis"if Sqrt(A*A + B*B) <= Resolution from gp.
◆ gce_MakeLin2d() [4/6]
gce_MakeLin2d::gce_MakeLin2d ( const gp_Lin2d & Lin,
const Standard_Real Dist
Make a Lin2d from gp <TheLin> parallel to another Lin2d <Lin> at a distance <Dist>. If Dist is greater than zero the result is on the right of the Line <Lin>, else the result is on the left of the
Line <Lin>.
◆ gce_MakeLin2d() [5/6]
gce_MakeLin2d::gce_MakeLin2d ( const gp_Lin2d & Lin,
const gp_Pnt2d & Point
Make a Lin2d from gp <TheLin> parallel to another Lin2d <Lin> and passing through a Pnt2d <Point>.
◆ gce_MakeLin2d() [6/6]
gce_MakeLin2d::gce_MakeLin2d ( const gp_Pnt2d & P1,
const gp_Pnt2d & P2
Make a Lin2d from gp <TheLin> passing through 2 Pnt2d <P1>,<P2>. It returns false if <P1> and <P2> are confused. Warning If an error occurs (that is, when IsDone returns false), the Status function
• gce_NullAxis if Sqrt(A*A + B*B) is less than or equal to gp::Resolution(), or
• gce_ConfusedPoints if points P1 and P2 are coincident.
◆ Operator()
gp_Lin2d gce_MakeLin2d::Operator ( ) const
◆ operator gp_Lin2d()
gce_MakeLin2d::operator gp_Lin2d ( ) const
◆ Value()
gp_Lin2d gce_MakeLin2d::Value ( ) const
Returns the constructed line. Exceptions StdFail_NotDone if no line is constructed.
The documentation for this class was generated from the following file: | {"url":"https://dev.opencascade.org/doc/occt-7.2.0/refman/html/classgce___make_lin2d.html","timestamp":"2024-11-12T16:00:37Z","content_type":"application/xhtml+xml","content_length":"24358","record_id":"<urn:uuid:f0df530f-01ba-4854-9ec7-7d2d652eb795>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00404.warc.gz"} |
Cyclostationary Analysis for Heart Rate Variability
All published articles of this journal are available on ScienceDirect.
Cyclostationary Analysis for Heart Rate Variability
During the last years, cyclostationarity has emerged as a new approach for the analysis of a certain type of non-stationary signals. This theoretical tool allows us to identify periodicity in signals
which cannot be identified easily but also to separate useful signals for other interfering contributions that overlap in the spectral support.
The aim of this work is the exploitation of cyclostationary theory to enhance standard methodologies for the study of heart rate variability. In this framework, a preliminary analysis on healthy
patients is proposed to be extended further on pathological patients with the perspective to improve (hopefully) the diagnostic power of some cardiac dysfunctions due to the more complete set of
information provided by this analysis.
The proposed approach involves an initial band-pass filtering step in the range 0.5 - 40 Hz of the recorded ECG signal, followed by a first-order derivative filter to reduce the effects of P and T
waves and to emphasise the QRS contribution. After that, the auto-correlation function is evaluated and the Cyclic Power Spectrum (CPS) is computed. From this two-dimensional information, a
one-dimensional plot is derived via the evaluation of a folded-projected CPS to be compared with standard Lomb-Scargle spectrum.
The proposed analysis has been tested on both numerical simulations as well as for the processing of real data which are available online in the Physionet database.
The proposed cyclostationary analysis has shown a good agreement with the results provided by the classical Lomb-Scargle spectrum in the processing of real data, underlining some contributions in the
high-frequency bandwidth which are not visible by means of standard processing.
Keywords: Electrocardiogram (ECG), Heart Rate Variability (HRV), Cyclostationarity signal analysis, Cyclic power spectrum, Spectral support, Lomb-Scargle spectrum.
Heart Rate Variability (HRV) is the variation in the time interval between consecutive heartbeats. It is a physiological phenomenon related to the regulation of the cardiac activity produced by the
Autonomic Nervous System (ANS) [1]. More in detail, the accelerations and decelerations of the heard activity are due to the competing activities of the sympathetic and the parasympathetic nervous
system branches [2]. The analysis of the HRV has the advantage of being noninvasive, easy to be performed and with good reproducibility, and is useful for the determination of the ANS status [3] and
the cardiac activity [4]. In particular, it has shown to have a beneficial role in the diagnosis and analysis of several pathologies related to blood pressure [5], myocardial infarction [6, 7], brain
damage [8], depression [9], cardiac arrhythmia [10], diabetes [11] and renal failure [12]. HRV analysis has also shown correlations with sleep [13], drugs [14] or alcohol [15] assumption and smoking
A first important step in order to provide HRV measures consists of the R peak detection in the QRS wave. Regarding this aim, many robust algorithms are known in the literature, among which the
well-known Pan-Tompkins algorithm [17] represents a gold standard for the scientific literature.
Variation in heart rate can be evaluated by using four main classes of methods. The first one is based on the evaluation of some synthetic parameters in the time domain. In a continuous
electrocardiographic (ECG) record, each QRS complex is detected, and the so-called normal-to-normal (NN) or RR intervals (that is all intervals between adjacent QRS complexes resulting from sinus
node depolarizations), or the instantaneous heart rate is determined. Simple time-domain variables that can be calculated include the mean RR interval, the mean heart rate, the difference between the
longest and shortest RR interval, and many others [18, 19]. Other time-domain measurements analyse the variations in instantaneous heart rate secondary to respiration, tilt, Valsalva maneuver, or
secondary to phenylephrine infusion. These differences can be described as either differences in heart rate or cycle length.
A second important class is represented by frequency domain methods, which are mainly based on the estimation of the Power Spectral Density (PSD) [20]. The analysis of this function provides the
basic information of how signal power distributes as a function of frequency. Both parametric and non-parametric methods are available for the PSD estimation, each one with its advantages and
limitations. Non-parametric methods are simple and fast, since they are mostly based on the use of the fast Fourier transform, while parametric methods provide smoother spectral components and a more
accurate estimation of the PSD even on a small number of samples on which the signal is supposed to maintain the stationarity hypothesis. Among these techniques, the Lomb-Scargle (LS) periodogram is
perhaps one of the most well-known techniques employed to compute the periodicity of unequally-spaced data, and it provides a good estimate of the PSD of an ECG signal [21-25]. LS method avoids the
major problem of classical approaches related to the low-pass effect due to re-sampling operation. Therefore, the Lomb method is more suitable than fast Fourier transform or autoregressive estimate
with linear or cubic interpolation for PSD estimation of unevenly sampled signals [26]. However, in extreme situations (low heart rate or high-frequency components), the Lomb estimate still
introduces high-frequency contamination that suggests further studies on superior performance interpolators.
Both time-domain and spectral methods share some limitations imposed by the irregularity of the RR series, since they assume the same trends of increasing or decreasing in the cycle length, which is
not always realistic [27, 28]. In practise, this reflects on the amplitude of the peaks at fundamental frequencies in the spectral analysis and enlarge their basis. In order to overcome these
limitations, some alternative techniques aiming at analysing the rhythm pattern via considering blocks of RR intervals without considering the internal variability have been proposed. The well-known
interval spectrum and spectrum of counts methods are well suited to investigate the relationship between HRV and the variability of other physiological measures, like blood pressure, respiration and
arrhythmia events [29].
Last class of approaches for HRV analysis is based on non-linear methods [30-34]. The motivation for using this kind of methodologies is based on the non-linear phenomena involved in the genesis of
the HRV signal, which is the result of complex interactions among haemodynamic, electrophysiological and humoral variables. However, although in principle, these techniques have been shown to be
powerful tools for characterization of various complex systems, their application to HRV (and biomedical data more in general) still needs to be validated. An exhaustive overview of the main
publications for non-linear HRV analysis for the past 25 years has been proposed in reference [35]. In their work, Sassi et al. present a critical review of the state of the art and new methodologies
tested in sufficiently sized populations, with particular attention paid to the long-range correlation and fractal analysis, entropy and regularity, non-linear dynamical systems and chaotic
Conversely from the approaches described previously in the text, this short communication aims at testing and analysing the performance of cyclostationary (CS) analysis for HRV signals. In most
cases, the stationarity hypothesis is often an assumption of convenience rather than a realistic one, and for many biomedical signals (e.g., ECG, EMG, etc.) the use of cyclostationarity, i.e. the
cyclic variation of the statistical properties, is more suitable rather than conventional stationary assumption [36-38]. Compared to standard analysis, it provides some extra information due to the
hidden periodicities in the signal. Taking advantage of these properties obviously leads to more powerful processing than possible with the stationary approach. Moreover, this extra information often
counter-balances the complication that it may involve as compared to other standard approaches.
Methodologies exploiting CS theory for heart monitoring have been proposed by several authors. In reference [39], a CS approach for heart and respiration rates monitoring exploiting a 2.4 GHz Doppler
radar is presented. CS theory has also been applied to ECG signals for several applications, such as foetal ECG extraction [40] or for the study of non-linearity in the HRV signal [41]. In this
framework, we propose to apply the CS methodology to the ECG signal in order to analyse the HRV. The main advantage of the CS analysis for HRV signals consists in the possibility of studying
simultaneously the standard spectral coverage of the ECG signal in the classical frequency domain, but also the cyclic spectral components related to the physiological behaviour of the HRV. Another
important advantage is that the analysis is performed directly in the time domain after the ECG recording and does not require any RR interval extraction, which makes this tool more robust.
The remainder of the paper is organised as follows: Section 2 provides a mathematical overview of CS theory, Section 3 proposes the methodology involved in this paper and Section 4 presents some
numerical results and real data processing. Finally, Section 5 closes the paper and draws some conclusions.
CS extends the class of stationary signals to those signals whose statistical properties change periodically with time. In this theoretical framework, the minimum “period” of a CS signal is called
cycle [38].
Conversely from standard non-stationary signals, CS is a well-defined property and can exploit a powerful spectral analysis in a wide sense, employing the same tools that have been developed
historically for stationary signals [42]. The instantaneous auto-correlation function
where n and τ are two time variables, β The β. The parameter in Eq. (1) allows a general formulation of various equivalent definitions found in the literature (for example typical values are β = 1/2
for the symmetric instantaneous autocorrelation function, and β = 1 or β = 0 for the asymmetric case) [38].
By definition, the instantaneous auto-correlation function of a (quasi-) CS signal is (quasi-) periodic, and therefore it can be decomposed in a Fourier series:
over the spectrum A = {α[i]} of the cyclic frequencies associated to non-zero Fourier coefficients:
Coefficients in Eq. (3), which are function of the time τ and of the cyclic frequency a[i], are called the cyclic auto-correlation function of the random signal X[n]. Finally, by exploiting Eq. (3),
the Cyclic Power Spectrum (CPS) can be defined as:
As well known in the spectral analysis, there is no consistent estimator of the CPS, i.e. an estimator whose variance tends to zero as the length of observation time increases [42, 43] Although not
consistent, the averaged cyclic periodogram N[w]-long window W[n], and W[k] [n] = W[n - KR] its shifted version of multiple of R samples, the averaged cyclic periodogram can be computed via:
is the short Discrete-Time Fourier Transform (DTFT) of the K-th weighted sequence
In this manuscript, we want to evaluate the CPS of an ECG signal, therefore, we apply Eq. (5) for the estimation of CPS in order to perform a more complete spectral analysis.
3. METHODS
Let us consider a recorded ECG signal:
where z (t) is the noise free signal and n (t) is the additive noise term which can be modelled as uncorrelated and Gaussian distributed.
As first step after the acquisition, a bandpass filter is applied in order to remove the frequency components below 0.5 Hz, which are mainly related to breathing, and the components above 40 Hz,
which are due to external interferences. Next, a derivative filter is exploited to isolate and strengthen the QRS information. The motivation behind this choice lies in the attempt to reduce the
effects of the P and T waves compared to the QRS complex since the main information regarding HRV is related to the QRS peaks.
Subsequently, the CS spectrum is computed. As described in Section 2, the procedure requires the computation of the auto-correlation function
Conversely from classical HRV spectra, the information arising from the cyclostationary analysis has two dimensions, i.e. the spectral frequencies and the cyclic frequencies f, and spread on both
positive and negative frequency a values. Therefore, in order to isolate the HRV information and improve its readability, a further spectrum, namely folded-projected cyclic power spectrum (FPCPS), is
computed. The first step consists in integrating the CPS over the spectral frequencies:
The signal y (t), keeping only the dependency on the cyclic frequencies α which contains information about the HRV.
Another transformation is required in order to refer to the shift of the cyclic frequency values α from the central cyclic frequency α[av], i.e. the mean heart rhythm, instead of considering the
absolute values. Moreover, since we are interested in the amplitude of the cyclic frequency shifts from the average value, both positive and negative shifts are worth for this evaluation. Thus, a
“folding-and-sum” operation around the zero cyclic frequency is performed, obtaining the FPCPS:
in which u(.) represents the Heaviside step function.
4. RESULTS
In order to show the potentiality of cyclostationary analysis for the study of Heart Rate Variability (HRV), results related to both simulated and real case study are presented.
4.1. Simulated Data: Gaussian Template
A synthetic signal z (t) was generated by replicating M times a template function over time. The template function simulated the signal related to one heart beat, while the delays of the replicas
take into account the heart rhythm and the HRV. Analytically, the simulated signal can be expressed as:
and RR [i] is the time distance between the i-th and the (i + 1)- th heartbeat. For this simulation, we considered as template p(t) a Gaussian function and the following model for the RR intervals:
in which RR[av] is the average RR interval (constant), and α[1] represents the amplitude of the deviations from the average RR value. In other words, we considered a sinusoidal behaviour of the HRV.
In this example, we assumed RR[av] = 1 s and a[1] = 0.07 , corresponding to a variation of approximately compared to the RR[av] f[1] refers to the frequency involved in the numerical model and the
parameter m identifies the m-th RR interval. This choice seems to be realistic, as proved by averaged oscillations in healthy patients. We assumed the sampling frequency f[s] = 200 Hz, a number of
heartbeats M = 100 and an HRV frequency f[1] = 0.1 Hz. Fig. (1). reports the Gaussian template function employed for the synthetic signal generation and the RR-interval amplitudes described by the
model in Eq. (12).
The computed CPS is reported as surface and as an image in Fig. (2a and b), respectively. On the spectral frequency axis (f), information about the components of the signal template can be
appreciated, while on the cyclic frequency axis (α), the HRV can be evaluated. For the sake of clarity, the figures show the cyclic frequency axis (α) centred at α[av] = 1/RR[av], which corresponds
to the zero cyclic frequency. In can be noticed that the peaks at are in correspondence with the considered HRV frequency f[1] Hz. Of course, harmonics at 0.2 Hz and 0.3 Hz are present.
In Fig. (2c), the FPCPS computed according to Eq. (9) is reported. It can be observed that both the contributions at f = 0 (corresponding to RR[av]) and f[1] = 0.1 Hz are present in the spectrum,
even though also some other harmonics at multiple of f[1] can be detected in the spectrum. In order to provide a reference, the LS spectrum is reported in Fig. (2d). Such a result confirms the
presence of the HRV frequency at 0.1 Hz. It should be noted that the DC component does not appear in the LS spectrum due to the fact that the Matlab function used for this comparison does not
evaluate the component centred at f = 0 Hz.
A Monte Carlo (MC) simulation has been implemented in order to evaluate the robustness with respect to noise. First, a white Gaussian noise has been added to the ECG signal Z (t) of Eq. (10), i.e.
where f = 0.1 Hz has been measured. The analysis has been repeated for σ[ecg] varying between 1 (SNR of about 50 dB) and 50 (SNR of about 18 dB), and the results are reported in Table 1. It can be
appreciated that the mean value of the peak position slightly changes (in case of σ[ecg] = 50 it increases of about 11%). Of course, in case of high noise level, the stability of the peak position
greatly decreases, with a standard deviation of 0.04 in case of σ[ecg] = 30 and higher. Nevertheless, it can be noted that, in case of noise σ[ecg] = 10 (SNR of about 30dB) or below, the standard
deviation of the estimation is in the order of magnitude of 1.10^-3 or lower.
Table 1.
Noise Level Peak Position (Mean) Peak Position (STD)
σ[ecg] = 1 0.0992 0.0000
σ[ecg] = 2 0.0992 0.0001
σ[ecg] = 3 0.0993 0.0006
σ[ecg] = 5 0.0995 0.0006
σ[ecg] = 10 0.0996 0.0013
σ[ecg] = 20 0.0994 0.0250
σ[ecg] = 30 0.1053 0.0407
σ[ecg] = 50 0.1132 0.0466
A second MC simulation has been implemented in order to evaluate the effect of noise corrupting the RR intervals. Samples from a white Gaussian random variable have been added to the time instants T
[m] defined in Eq. (11). As in the previous simulation, 100 MC iterations have been considered and for each, the frequency peak close to f = 0.1 Hz has been identified. The performances were
evaluated in case noise standard deviation σ[t] ranging from 5.10^-3 to 2.10^-1, and results are reported in Table 2. Compared to the previous MC simulation, a different behaviour can be appreciated:
When the noise standard deviation is greater than, performances rapidly deteriorate, while remaining good in case of lower values of σ[t].
Table 2.
Noise Level Peak Position (Mean) Peak Position (STD)
σ[t] = 0.005 0.0995 0.0012
σ[t] = 0.001 0.0997 0.0022
σ[t] = 0.002 0.1001 0.0048
σ[t] = 0.003 0.0995 0.0810
σ[t] = 0.004 0.1002 0.0105
σ[t] = 0.005 0.0978 0.0180
σ[t] = 0.01 0.0903 0.0419
σ[t] = 0.02 0.1258 0.0591
As a second numerical example, a second cosine component was added to the RR model:
with α[2] = 0.07 and f[2] = 0.18 Hz and the other parameters equal to the previous case. As in the previous case, the FPCPS and the LS periodogram are reported in Fig. (3).
Again, there is a good agreement between the LS spectrum and the FPCPS, as shown in Fig. (3c, d). Conversely from the previous case study, the presence of more than one frequency is responsible for
the intermodulation effects, which results in more lobes than what was expected from the case of a single frequency. Intermodulation effects which are visible in the LS case are still visible in the
FPCPS case.
4.2. Simulated data: QRS template
A more realistic numerical example in which the signal template p(t) is a real QRS complex is considered. Again, we considered the RR model containing two sinusoidal components reported in Eq. (14)
with α[1] = α[2] = 0.07, f[1] = 0.1 and f[2] = 0.18 Hz.
The results are shown in Fig. (4). The HRV information carried by the cyclic frequencies α was close to the one reported in the previous examples, but, as expected, in this case, the CPS was richer
in the spectral components. More in detail, the adopted QRS signal template p(t) was characterized by a much wider and complex spectrum in comparison with the Gaussian-shaped template considered in
Section 4.1, and therefore the CPS shows several components up to f = 30Hz.
4.3. Real data
To validate the proposed numerical analysis, some real datasets from the Physionet repository have been considered [44]. Three different patients with physiological ECG were analysed (ID: 16272,
16483 and 16539) from the MIT-BIH normal sinus rhythm database [45]. It includes 18 long-term ECG recordings of subjects which were found to have had no significant arrhythmias. The people involved
in this study include people aged from 20 to 50. Signals have been acquired with a sampling frequency equal to 128 Hz for a duration of 300 seconds.
As previously pointed out in Section 3, the ECG signals have been pre-processed by means of a band-pass filter and a derivative filter in order to emphasize the effect of QRS complex compared to the
other waves. Then, the CPS of these signals was evaluated via Eq. (5) and reported in Fig, (5).
These kind of images provide both spectral information as well as cyclic spectral representation, but it is not easy to be interpreted as it is. A way to ease the analysis of this information lies in
the use of Eq. (9), which is reported in Fig. (6) and compared with standard LS spectrum.
In the resting physiological subject, three main spectral components for the HRV analysis can be distinguished in short-term ECG recordings: Very Low Frequency (VLF), Low Frequency (LF) and High
Frequency (HF). The distribution of the power and the central frequency of LF and HF are not fixed and may vary due to changes in the modulation by the autonomic nervous system [46], whereas the HF
components, synchronous with the respiration, occurs at 0.25 Hz approximately. The study of VLF ( Hz) phenomena, which might contain clinically relevant information, requires long-period
uninterrupted data; thus, the DC component and the whole range of the VLF have not been addressed in this manuscript, and therefore the frequency axis will be limited in the useful range (0.02-0-32)
Hz. Both total power of HRV spectrum as well as the LF-to-HF ratio has proved to be selective indices of cardiac parasympathetic activity [44], and this motivates the interest in the study of these
In order to ease the analysis of the two-dimensional spectra proposed in Fig. (5), an easier and more straight comparison has been carried out via the evaluation of Eq. (9) and comparing these
results with the standard LS spectrum. The proposed comparison has been reported in Fig. (6), in which the FPCPS per each patient is compared with the corresponding LS spectrum. From a first
analysis, it can be observed that for all the considered cases, most of the signal power is located in the VLF and LF bandwidths. All the cases reported in Fig. (6) are characterised by a main lobe
in the VLF range and by another strong contribution of around 0.1 Hz, both in the FPCPS as well as LS spectra. Conversely from the standard LS analysis, the FPCPS also underlines some non-negligible
contributions in the HF bandwidth, but still fits with the clinical considerations arising from standard analysis of HRV LS spectrum. Hopefully, this apparent higher sensitivity to HF bandwidth could
be beneficial in the early diagnosis of some pathological dysfunctions. However, further investigation is required to test the proposed formulation for pathological patients prior to clinical use.
In this paper, the performance of CS for HRV signal analysis has been evaluated. More in detail, the performance of a CPS estimator has been analysed and tested for synthetic numerical experiments
with Gaussian function replica and ECG-like waveforms, proving a good agreement with the LS spectrum. Moreover, CS is also able to evaluate signal frequency components and does not require the RR
interval extraction from the ECG signal.
Finally, CS analysis has also been conducted on real data, confirming the interesting performance observed numerically.
Not applicable.
The authors declare no conflict of interest, financial or otherwise.
Declared none.
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How to Write a Hypothesis for a Research Topic
To begin writing a hypothesis, it is best to write it in the form of a statement. This helps you identify the variables and guesses you will be testing. The statement should include the dependent and
independent variables. Then, you can write a null hypothesis, which excludes one of the variables.
Plausibility, defined concepts, observability, and general explanation
Plausibility is the degree to which an assertion has a reasonable chance of being true or false. For example, a normal model derived from a sample is plausible, and a normal probability plot is a
plausible estimate of the population’s distribution. Other examples include various statistics. For example, a point estimate of a data set’s probability is plausible, but it is not an exact measure.
A theory, on the other hand, is a general explanation that is backed by numerous tests. It explains why many observations of a particular phenomenon are consistent. A theory can be either a
prediction or a general principle that is applicable to many different specific instances.
Statistical analysis to evaluate a representative sample of the population
A representative sample is a sample of a population that has similar characteristics to the population in question. These characteristics are usually grouped according to age, sex, education, marital
status, or any other demographic factor that is relevant to the population. The representativeness of a sample is often determined by several factors, such as the size of the sample, the type of
study, and the data that are available.
The purpose of a representative sample is to provide an accurate impression of the population. Ideally, the proportions of age, gender, and region must be comparable with the proportions of the
entire population. Similarly, sample sizes should match the population in both demographic and non-demographic measures. Finally, the researcher should have an age quota to ensure that the sample is
representative of the population.
Using a representative sample is more accurate than using a random one because the population it represents is much larger. In addition, it is more cost-effective than collecting data from every
member of a population. Additionally, statistical analysis and appropriate sampling techniques can help you apply the results to the entire population.
Creating a research hypothesis “if, then, this” statement
When creating a research hypothesis it’s best to write it as a statement instead of a question, using the “if, then, this” format. This helps researchers clearly identify the variables and guesses
they need to test. In addition, the hypothesis should include both dependent and independent variables.
Hypothesis writing is not a simple process. You must be familiar with the variables and five different situations before you can write a hypothesis. You should also be familiar with If/Then Phrasing,
which uses conditional sentences to discuss the results of research studies.
To create a good “if, then, this” statement, you must first identify the problem you are trying to address. For example, do you need to know if the mean of dogs in Toronto is greater than the mean of
cats in the same city? Or do you want to know whether overtime is paid at a time and a half rate? These facts provide the basis for a research question.
Creating a null hypothesis in research
When you are conducting a research study, one of the first steps is creating a null hypothesis for your research topic. How to write a research hypothesis and null hypothesis. A null hypothesis, also
known as an alternate hypothesis, is a neutral statement stating that there is no relationship between the variables or the treatment being tested. The null hypothesis states that the population’s
mean is equal to seven. In order to test this hypothesis, you should record the marks of 30 students chosen from the entire population. Afterwards, you can calculate the mean of this sample.
The null hypothesis is given to clarify that the researchers are trying to prove a false assumption. The alternative hypothesis, on the other hand, uses logic and statistical analysis to evaluate the
effect of an independent variable on a dependent variable. The null hypothesis is not the final statement, but it is a guideline for the study.
advantage of a null hypothesis
When testing the null hypothesis, the researchers need to be aware of the potential pitfalls and advantages of testing it. The main advantage of a null hypothesis is that it is easy to test using
statistical methods. In addition, a null hypothesis can provide a high level of confidence. It is also a good way to find out whether the results you have obtained are simply due to the manipulation
of a dependent variable.
Author Bio
Ellie Cross is a research-based content writer, who works for Cognizantt, a globally recognised wordpress development agency uk and Research Prospect, a Tjenester til at skrive afhandlinger og essays
. Ellie Cross holds a PhD degree in mass communication. He loves to express his views on a range of issues including education, technology, and more. | {"url":"https://metalwriters.com/dissertation/how-to-write-a-hypothesis-for-a-research-topic/","timestamp":"2024-11-14T16:57:34Z","content_type":"text/html","content_length":"76917","record_id":"<urn:uuid:01acd294-9a19-4f8d-b9ce-98355f5dc233>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00799.warc.gz"} |
Analyzing the surface properties of nearby locations
Available with Geostatistical Analyst license.
Generally speaking, things that are closer together tend to be more alike than things that are farther apart. This is a fundamental geographic principle. Suppose you are a town planner and need to
build a scenic park in your town. You have several candidate sites, and you may want to model the viewsheds at each location. This will require a more detailed elevation surface dataset for your
study area. Suppose you have preexisting elevation data for 1,000 locations throughout the town. You can use this to build a new elevation surface.
When trying to build the elevation surface, you can assume that the sample values closest to the prediction location will be similar. But how many sample locations should you consider? Should all of
the sample values be considered equally? As you move farther away from the prediction location, the influence of the points will decrease. Considering a point too far away may actually be detrimental
because the point may be located in an area that is dramatically different from the prediction location.
One solution is to consider enough points to give a good prediction, but few enough points to be practical. The number will vary with the amount and distribution of the sample points and the
character of the surface. If the elevation samples are relatively evenly distributed and the surface characteristics do not change significantly across your landscape, you can predict surface values
from nearby points with reasonable accuracy. To account for the distance relationship, the values of closer points are usually weighted more heavily than those farther away. This principle is common
to all the interpolation methods offered in Geostatistical Analyst (except for global polynomial interpolation, which assigns equal weights to all points).
Feedback on this topic? | {"url":"https://desktop.arcgis.com/en/arcmap/10.3/guide-books/extensions/geostatistical-analyst/analyzing-the-surface-properties-of-nearby-locations.htm","timestamp":"2024-11-04T01:03:30Z","content_type":"text/html","content_length":"24021","record_id":"<urn:uuid:fb5afdaa-07d6-4624-94f0-0e8175b3d034>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00207.warc.gz"} |
Intermediate Algebra for College Students (7th Edition) Chapter 2 - Mid-Chapter Check Point - Page 135 13
The function is not a fractional function and doesn't contain roots on even powers, thus its domain is all the real numbers.
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/algebra/intermediate-algebra-for-college-students-7th-edition/chapter-2-mid-chapter-check-point-page-135/13","timestamp":"2024-11-10T00:02:43Z","content_type":"text/html","content_length":"67065","record_id":"<urn:uuid:b7a65c2c-ebf8-4bd6-8885-e91e562560e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00500.warc.gz"} |
dRE = rowexch(nfactors,nruns)
[dRE,X] = rowexch(nfactors,nruns)
[dRE,X] = rowexch(nfactors,nruns,model)
[dRE,X] = rowexch(___,param1,val1,param2,val2,...)
dRE = rowexch(nfactors,nruns) uses a row-exchange algorithm to generate a D-optimal design dRE with nruns runs (the rows of dRE) for a linear additive model with nfactors factors (the columns of
dRE). The model includes a constant term.
[dRE,X] = rowexch(nfactors,nruns) also returns the associated design matrix X, whose columns are the model terms evaluated at each treatment (row) of dRE.
[dRE,X] = rowexch(nfactors,nruns,model) uses the linear regression model specified in model. model is one of the following:
• 'linear' — Constant and linear terms. This is the default.
• 'interaction' — Constant, linear, and interaction terms
• 'quadratic' — Constant, linear, interaction, and squared terms
• 'purequadratic' — Constant, linear, and squared terms
The order of the columns of X for a full quadratic model with n terms is:
1. The constant term
2. The linear terms in order 1, 2, ..., n
3. The interaction terms in order (1, 2), (1, 3), ..., (1, n), (2, 3), ..., (n–1, n)
4. The squared terms in order 1, 2, ..., n
Other models use a subset of these terms, in the same order.
Alternatively, model can be a matrix specifying polynomial terms of arbitrary order. In this case, model should have one column for each factor and one row for each term in the model. The entries in
any row of model are powers for the factors in the columns. For example, if a model has factors X1, X2, and X3, then a row [0 1 2] in model specifies the term (X1.^0).*(X2.^1).*(X3.^2). A row of all
zeros in model specifies a constant term, which can be omitted.
[dRE,X] = rowexch(___,param1,val1,param2,val2,...) specifies options for the design using one or more parameter/value pairs in addition to any of the input argument combinations in the previous
syntaxes. Valid parameters and their values are listed in the following table.
Parameter Value
Flag to specify whether rowexch avoids calculating duplicate rows for dRE. If 'AvoidDuplicates' is true and rowexch is able to calculate non-duplicate points, the rows of dRE
'AvoidDuplicates' are unique. When 'AvoidDuplicates' is false, the function does not attempt to avoid duplicate rows.
Lower and upper bounds for each factor, specified as a 2-by-nfactors matrix. Alternatively, this value can be a cell array containing nfactors elements, each element specifying
'Bounds' the vector of allowable values for the corresponding factor.
'CategoricalVariables' Indices of categorical predictors.
'Display' Either 'on' or 'off' to control display of the iteration counter. The default is 'on'.
Handle to a function that excludes undesirable runs. If the function is f, it must support the syntax b = f(S), where S is a matrix of treatments with nfactors columns and b is
'ExcludeFcn' a vector of Boolean values with the same number of rows as S. b(i) is true if the ith row S should be excluded.
'InitialDesign' Initial design as an nruns-by-nfactors matrix. The default is a randomly selected set of points.
'MaxIterations' Maximum number of iterations. The default is 10.
'NumLevels' Vector of number of levels for each factor.
'NumTries' Number of times to try to generate a design from a new starting point. The algorithm uses random points for each try, except possibly the first. The default is 1.
A structure that specifies whether to run in parallel, and specifies the random stream or streams. Create the Options structure with statset. Parallel computation requires
Parallel Computing Toolbox™. Option fields are:
• UseParallel — Set to true to compute in parallel. Default is false.
• UseSubstreams — Set to true to compute in a reproducible fashion. Default is false. To compute reproducibly, set Streams to a type allowing substreams: 'mlfg6331_64' or
• Streams — A RandStream object or cell array of such objects. If you do not specify Streams, rowexch uses the default stream or streams. If you choose to specify Streams,
use a single object except in the case
□ UseParallel is true
□ UseSubstreams is false
In that case, use a cell array the same size as the Parallel pool.
Suppose you want a design to estimate the parameters in the following three-factor, seven-term interaction model:
$y={\beta }_{0}+{\beta }_{1}x{}_{1}+{\beta }_{2}x{}_{2}+{\beta }_{3}x{}_{3}+{\beta }_{12}x{}_{1}x{}_{2}+{\beta }_{13}x{}_{1}x{}_{3}+{\beta }_{23}x{}_{2}x{}_{3}+\epsilon$
Use rowexch to generate a D-optimal design with seven runs:
nfactors = 3;
nruns = 7;
[dRE,X] = rowexch(nfactors,nruns,'interaction','NumTries',10)
dRE =
-1 -1 1
1 -1 1
1 -1 -1
-1 -1 -1
-1 1 -1
-1 1 1
X =
1 -1 -1 1 1 -1 -1
1 1 -1 1 -1 1 -1
1 1 -1 -1 -1 -1 1
1 -1 -1 -1 1 1 1
1 -1 1 -1 -1 1 -1
1 -1 1 1 -1 -1 1
Columns of the design matrix X are the model terms evaluated at each row of the design dRE. The terms appear in order from left to right: constant term, linear terms (1, 2, 3), interaction terms (12,
13, 23). Use X to fit the model, as described in Linear Regression, to response data measured at the design points in dRE.
Both cordexch and rowexch use iterative search algorithms. They operate by incrementally changing an initial design matrix X to increase D = |X^TX| at each step. In both algorithms, there is
randomness built into the selection of the initial design and into the choice of the incremental changes. As a result, both algorithms may return locally, but not globally, D-optimal designs. Run
each algorithm multiple times and select the best result for your final design. Both functions have a 'NumTries' parameter that automates this repetition and comparison.
At each step, the row-exchange algorithm exchanges an entire row of X with a row from a design matrix C evaluated at a candidate set of feasible treatments. The rowexch function automatically
generates a C appropriate for a specified model, operating in two steps by calling the candgen and candexch functions in sequence. Provide your own C by calling candexch directly. In either case, if
C is large, its static presence in memory can affect computation.
Extended Capabilities
Automatic Parallel Support
Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.
To run in parallel, specify the Options name-value argument in the call to this function and set the UseParallel field of the options structure to true using statset:
For more information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).
Version History
Introduced before R2006a
R2024b: Avoid duplicates in D-optimal designs
You can now specify whether to avoid duplicate rows when using rowexch to create a D-optimal design. Use the AvoidDuplicates name-value argument to avoid duplicate rows, when possible.
R2024b: Use updated names for name-value arguments
rowexch has updated name-value argument names. These more intuitive names are now supported:
• Bounds
• CategoricalVariables
• NumLevels | {"url":"https://de.mathworks.com/help/stats/rowexch.html","timestamp":"2024-11-04T15:19:28Z","content_type":"text/html","content_length":"86351","record_id":"<urn:uuid:7f9ff2cf-9d73-4d51-a1f1-c647de238eac>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00514.warc.gz"} |
Calculate the distance between two charges of 4C forming a dipole, with a dipole moment of 6 unitsCalculate the distance between two charges of 4C forming a dipole, with a dipole moment of 6 unitsCalculate the distance between two charges of 4C forming a dipole, with a dipole moment of 6 units
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Calculate the distance between two charges of 4C forming a dipole, with a dipole moment of 6 units
Poll Results
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To find the distance between two charges (designated as (d)) forming a dipole with a given dipole moment ((p)), we can use the formula that relates the dipole moment with the charge and the
distance between the charges. The dipole moment is given by
[ p = q cdot d ]
– (p) is the dipole moment,
– (q) is the magnitude of one of the charges, and
– (d) is the distance between the charges.
Given values are:
– (p = 6) units,
– (q = 4C).
Plugging these values into the formula gives
[ 6 = 4 cdot d ]
Solving for (d) gives
[ d = frac{6}{4} = 1.5 ]
So, the distance between the two charges is (1.5) units.
Explanation: The dipole moment is given by, M = Q x d. To get d, we rearrange the
formula d = M/Q = 6/4 = 1.5units | {"url":"https://quearn.com/question/calculate-the-distance-between-two-charges-of-4c-forming-a-dipole-with-a-dipole-moment-of-6-units-2/","timestamp":"2024-11-14T23:40:50Z","content_type":"text/html","content_length":"118846","record_id":"<urn:uuid:5d4c26f5-5a11-4999-a99f-66dc582980aa>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00861.warc.gz"} |
Missing Square
Missing area 1
Material Required: Card paper, graph paper(optional), pencil or pen, cutter or scissors.
The missing square and missing line activities take everyone by surprise. But there is a perfectly clear explanation for the mysterious missing square and missing line.
To make the missing square puzzle, cut a paper into “5 by 5” square. Form 25 smaller squares by drawing lines for every inch. Now cut this square along the dotted line shown in the figure to get four
pieces. These four pieces on rearranging into a rectangle give us only 8 x 3 = 24 squares, leaving us surprised as one square is missing from the original piece. The same activity can be performed on
a “8 x 8” square and cutting it into four pieces as above. On rearranging the pieces into a rectangle we get 13 x 5 = 65 squares, this time one more than the original.
The reason for this increase or decrease in the number of square is simple. In each case when we cut the square into pieces and re-assemble them to form the rectangle, the small squares that are
obtained are not perfect squares, but are slightly enlarged or compressed. Sometimes the edges do not match. In order to obtain perfect squares, the pieces have to overlap slightly. The total area of
overlap will be equal to the area of one square, thus bringing down the number of squares from say 25 to 24. Alternatively gaps may have to be left along the junction lines to get perfect squares. In
this case area is added and the total increase in area is equal to the area of one square. The number of squares then increases say from 64 to 65 in the 8 x 8 square.
Missing Square 2
Material Required:Card paper, graph paper(optional), pencil or pen, cutter or scissors.
Missing square puzzle contains a right angle triangle with base 13 units and height 5 unit which is formed by four components
1. A right angle triangle with dimensions 8 units 3 units
2. A right angle triangle with dimension 5 units
2 units
3. A L shaped figure with 7 sq. Units
4. A L shaped figure with 8 sq. Units
Rearranging these 4 pieces to form a new triangle with the same dimension as shown in the figure.
Both the triangles form 13 unit
5 unit right angle triangle, but the second arrangement has a missing square in it. Where did the square go?
It can be noticed that the area of both the triangles and the combined area of the components are different.
Area of the components: Area of piece 1 =
3 =12 sq.units
Area of piece 2=
2 = 5 sq.units
Area of piece 3 = 7 sq.units
Area of piece 4 = 8 sq. Units
Total area of components is 32 sq. Units
Calculated area =
5 = 32.5 sq. Units
So calculated area and combined area doesn’t match. Why it is so?
Notice the hypotenuse of the piece 1 and piece 2. hypotenuse of piece 1 has a slope of
where as the hypotenuse of piece 2 has a slope of
. so when combining these two lines don’t constitute a single line for both the triangles. The combined hypotenuse in both the triangles are actually bent.
When overlapping both these triangles, overlaying hypotenuses results a very thin parallelogram with area of exactly one square, the same area “ missing” from the rearranged figure. | {"url":"https://mathedu.hbcse.tifr.res.in/missing-square/","timestamp":"2024-11-02T09:23:12Z","content_type":"text/html","content_length":"78057","record_id":"<urn:uuid:cf764953-4d0c-42b6-8504-a4164e0a5746>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00041.warc.gz"} |
Time Series Analysis: Definition, Types, Techniques, and When It's Used
For as long as we have been recording data, time has been a crucial factor. In time series analysis, time is a significant variable of the data. Times series analysis helps us study our world and
learn how we progress within it.
In this article, we'll cover the following items for time series analysis:
What is time series analysis?
Time series analysis is a specific way of analyzing a sequence of data points collected over an interval of time. In time series analysis, analysts record data points at consistent intervals over a
set period of time rather than just recording the data points intermittently or randomly. However, this type of analysis is not merely the act of collecting data over time.
What sets time series data apart from other data is that the analysis can show how variables change over time. In other words, time is a crucial variable because it shows how the data adjusts over
the course of the data points as well as the final results. It provides an additional source of information and a set order of dependencies between the data.
Time series analysis typically requires a large number of data points to ensure consistency and reliability. An extensive data set ensures you have a representative sample size and that analysis can
cut through noisy data. It also ensures that any trends or patterns discovered are not outliers and can account for seasonal variance. Additionally, time series data can be used for
forecasting—predicting future data based on historical data.
Why organizations use time series data analysis
Time series analysis helps organizations understand the underlying causes of trends or systemic patterns over time. Using data visualizations, business users can see seasonal trends and dig deeper
into why these trends occur. With modern analytics platforms, these visualizations can go far beyond line graphs.
When organizations analyze data over consistent intervals, they can also use time series forecasting to predict the likelihood of future events. Time series forecasting is part of predictive
analytics. It can show likely changes in the data, like seasonality or cyclic behavior, which provides a better understanding of data variables and helps forecast better.
For example, Des Moines Public Schools analyzed five years of student achievement data to identify at-risk students and track progress over time. Today’s technology allows us to collect massive
amounts of data every day and it’s easier than ever to gather enough consistent data for comprehensive analysis.
Read other examples of the application of time series analysis here.
Time series analysis examples
Time series analysis is used for non-stationary data—things that are constantly fluctuating over time or are affected by time. Industries like finance, retail, and economics frequently use time
series analysis because currency and sales are always changing. Stock market analysis is an excellent example of time series analysis in action, especially with automated trading algorithms.
Likewise, time series analysis is ideal for forecasting weather changes, helping meteorologists predict everything from tomorrow’s weather report to future years of climate change. Examples of time
series analysis in action include:
• Weather data
• Rainfall measurements
• Temperature readings
• Heart rate monitoring (EKG)
• Brain monitoring (EEG)
• Quarterly sales
• Stock prices
• Automated stock trading
• Industry forecasts
• Interest rates
Time Series Analysis Types
Because time series analysis includes many categories or variations of data, analysts sometimes must make complex models. However, analysts can’t account for all variances, and they can’t generalize
a specific model to every sample. Models that are too complex or that try to do too many things can lead to a lack of fit. Lack of fit or overfitting models lead to those models not distinguishing
between random error and true relationships, leaving analysis skewed and forecasts incorrect.
Models of time series analysis include:
• Classification: Identifies and assigns categories to the data.
• Curve fitting: Plots the data along a curve to study the relationships of variables within the data.
• Descriptive analysis: Identifies patterns in time series data, like trends, cycles, or seasonal variation.
• Explanative analysis: Attempts to understand the data and the relationships within it, as well as cause and effect.
• Exploratory analysis: Highlights the main characteristics of the time series data, usually in a visual format.
• Forecasting: Predicts future data. This type is based on historical trends. It uses the historical data as a model for future data, predicting scenarios that could happen along future plot
• Intervention analysis: Studies how an event can change the data.
• Segmentation: Splits the data into segments to show the underlying properties of the source information.
Data classification
Further, time series data can be classified into two main categories:
• Stock time series data means measuring attributes at a certain point in time, like a static snapshot of the information as it was.
• Flow time series data means measuring the activity of the attributes over a certain period, which is generally part of the total whole and makes up a portion of the results.
Data variations
In time series data, variations can occur sporadically throughout the data:
• Functional analysis can pick out the patterns and relationships within the data to identify notable events.
• Trend analysis means determining consistent movement in a certain direction. There are two types of trends: deterministic, where we can find the underlying cause, and stochastic, which is random
and unexplainable.
• Seasonal variation describes events that occur at specific and regular intervals during the course of a year. Serial dependence occurs when data points close together in time tend to be related.
Time series analysis and forecasting models must define the types of data relevant to answering the business question. Once analysts have chosen the relevant data they want to analyze, they choose
what types of analysis and techniques are the best fit.
Important Considerations for Time Series Analysis
While time series data is data collected over time, there are different types of data that describe how and when that time data was recorded. For example:
• Time series data is data that is recorded over consistent intervals of time.
• Cross-sectional data consists of several variables recorded at the same time.
• Pooled data is a combination of both time series data and cross-sectional data.
Time Series Analysis Models and Techniques
Just as there are many types and models, there are also a variety of methods to study data. Here are the three most common.
• Box-Jenkins ARIMA models: These univariate models are used to better understand a single time-dependent variable, such as temperature over time, and to predict future data points of variables.
These models work on the assumption that the data is stationary. Analysts have to account for and remove as many differences and seasonalities in past data points as they can. Thankfully, the
ARIMA model includes terms to account for moving averages, seasonal difference operators, and autoregressive terms within the model.
• Box-Jenkins Multivariate Models: Multivariate models are used to analyze more than one time-dependent variable, such as temperature and humidity, over time.
• Holt-Winters Method: The Holt-Winters method is an exponential smoothing technique. It is designed to predict outcomes, provided that the data points include seasonality.
Books about time series analysis
Time series analysis is not a new study, despite technology making it easier to access. Many of the recommended texts teaching the subject’s fundamental theories and practices have been around for
several decades. And the method itself is even older than that. We have been using time series analysis for thousands of years, all the way back to the ancient studies of planetary movement and
Because of this, there are thousands of books about the study, and some are old and outdated. As such, we created a list of the top books about time series analysis. These are a mix of textbooks and
reference guides, and good for beginners through to experts. You’ll find theory, examples, case studies, practices, and more in these books.
Times series analysis and R
The open-source programming language and environment R can complete common time series analysis functions, such as plotting, with just a few keystrokes. More complex functions involve finding
seasonal values or irregularities. Time series analysis in Python is also popular for finding trends and forecasting.
Time series analysis is a technical and robust subject, and this guide just scratches the surface. To learn more about the theories and practical applications, check out our time series analysis
resources and customer stories. | {"url":"https://www.tableau.com/zh-cn/analytics/what-is-time-series-analysis","timestamp":"2024-11-13T21:43:12Z","content_type":"text/html","content_length":"164346","record_id":"<urn:uuid:e6b817d0-5261-49ee-baf4-5e4e48853b62>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00483.warc.gz"} |
Unscramble HACEKS
How Many Words are in HACEKS Unscramble?
By unscrambling letters haceks, our Word Unscrambler aka Scrabble Word Finder easily found 52 playable words in virtually every word scramble game!
Letter / Tile Values for HACEKS
Below are the values for each of the letters/tiles in Scrabble. The letters in haceks combine for a total of 15 points (not including bonus squares)
What do the Letters haceks Unscrambled Mean?
The unscrambled words with the most letters from HACEKS word or letters are below along with the definitions.
• hacek () - Sorry, we do not have a definition for this word | {"url":"https://www.scrabblewordfind.com/unscramble-haceks","timestamp":"2024-11-05T23:29:29Z","content_type":"text/html","content_length":"46175","record_id":"<urn:uuid:e2bc8b53-cf62-4210-a920-98b4eb948967>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00024.warc.gz"} |
Oberseminar Mathematische Physik
Jointly with Johannes Alt, Margherita Disertori, Luca Fresta, Illia Karabash and Eveliina Peltola, I am organizing a mathematical physics seminar which takes place a few times per semester. The
seminar usually takes place on Mondays from 2.15 - 3.15 pm in seminar room 0.006 (Endenicher Allee 60).
Talks Spring 2024:
• March 11, 2024: Théo Pinet (IMJ-PRG, Université Paris-Cité, UdeM)
Existence of inflations for representations of shifted quantum affine algebras
It is well known that the only simple finite-dimensional Lie algebra admitting a 2-dimensional irreducible representation is sl(2). The restriction functors arising in classical Lie theory, from
inclusions of Dynkin diagrams, are therefore not essentially surjective on finite-dimensional simple modules. This talk aims to specify whether or not this surjectivity defect remains in the
setting of Finkelberg-Tsymbaliuk’s shifted quantum affine algebras (SQAs for short).
SQAs are infinite-dimensional algebras parametrized by a finite-dimensional Lie algebra and a coweight of this Lie algebra. They are natural variations of the usual quantum loop algebras which
are in turn algebras of critical importance in geometry, in quantum integrable systems and in the study of cluster algebras. In this talk, we will give a pedagogical introduction to the
remarkable representation theory of SQAs and will state an existence theorem for some notable modules, that we call inflations. We will construct these modules as special preimages for natural
restriction functors (associated to inclusions of Dynkin diagrams) and will discuss important applications of their existence to the study of monoidal categorifications of cluster algebras and to
integrable systems.
• April 15, 2024: Per Moosavi (Stockholm University)
Anisotropic quantum Hall droplets
I will discuss recent work on 2D droplets of non-interacting electrons in strong magnetic fields, confined by an anisotropic trapping potential. Using semiclassical methods, we obtain the
one-particle energy spectrum and wave functions in the lowest Landau level by deriving and solving a transport equation inspired by standard WKB theory. This shows that energy eigenstates are
localized on equipotentials of the trap, generalizing the rotational-symmetric situation for isotropic traps. From these microscopic first-principle considerations, we obtain explicit results for
many-body observables for anisotropic quantum Hall droplets in the thermodynamic limit. In particular, we show that correlations along the droplet's edge are long-ranged, in agreement with
low-energy edge modes described by a free chiral conformal field theory in terms of the canonical angle variable of the trapping potential.
• May 13, 2024: Alexis Langlois-Rémillard (University of Bonn)
The Dunkl total angular momentum algebra and its representations
The Dirac operator and its dual symbol can be seen as the generators of a realisation the Lie superalgebra osp(1|2) inside the tensor product of the Weyl algebra and a Clifford algebra. The
algebra of operators supercommuting with this realisation is called the total angular momentum algebra (TAMA). The polynomial null-solutions of the Dirac equations form a family of irreducible
representations of the TAMA than can be expressed via special functions. For a (complex) reflection group, we can deform the derivatives into the Dunkl operators. Then, inside the associated
rational Cherednik algebra tensored by a Clifford algebra, the Dunkl-Dirac operator and its dual symbol also generate a realisation of the Lie superalgebra osp(1|2). The Dunkl TAMA is then the
symmetry algebra of this realisation. We will present an overview of this algebra and explore its representations via the few known examples.
• May 27, 2024: Runan He (University of Halle)
Analysis of PDE Models in Natural Sciences: Micro-Electro-Mechanical Systems, Surface Plasmon Polaritons and Sintering
The first part of the talk introduces the study of some mathematical models for a Micro-Electro-Mechanical System (MEMS) capacitor, consisting of a fixed plate and a flexible plate separated by a
fluid. It investigates the wellposedness of solutions to the resulting quasilinear coupled systems, as well as the finite-time blow-up (quenching) of solutions. The models considered include a
parabolic-dispersive system modelling the fluid flow under an elastic plate, a parabolic-hyperbolic system for a thin membrane, as well as an elliptic-dispersive system for quasistatic fluid flow
under an elastic plate. Short-time existence, uniqueness and smoothness are obtained by combining wellposedness results for a single equation with an abstract semigroup approach for the system.
Quenching is shown to occur, if the solution ceases to exist after a finite time.
The second part of the talk introduce linear Maxwell equations for transverse magnetic (TM) polarized fields support single frequency surface plasmon polaritons (SPPs) localized at the interface
of a metal and a dielectric and proves the bifurcation of localized SPPs in dispersive media in the presence of a cubic nonlinearity and provide an asymptotic expansion of the solution and the
frequency. We also show that the real frequency exists in the nonlinear setting in the case of $PT$-symmetric materials.
The third part of the talk introduces the Mullins equation modelling sintering process. We show that the existence of the self-similar solution to the Mullins equation for large groove angle and
the nonexistence of self-similar solution for small groove angle.
• June 3, 2024: Arnaud Triay (LMU Munich)
The excitation spectrum of a dilute Bose gas with an impurity
We study a dilute system of N interacting bosons coupled to an impurity particle via a pair potential in the Gross-Pitaevskii regime. We derive an expansion of the ground state energy up to order
one in the boson number and show that the difference of excited eigenvalues to the ground state is given by the eigenvalues of the renormalized Bogoliubov-Fröhlich Hamiltonian in the large N | {"url":"https://www.iam.uni-bonn.de/users/brennecke/os-mathematical-physics","timestamp":"2024-11-12T23:28:40Z","content_type":"text/html","content_length":"45842","record_id":"<urn:uuid:4fddb8a6-9efd-44eb-9562-18ee7ae88921>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00376.warc.gz"} |
Generalized equivalences between subsampling and ridge regularization
We establish precise structural and risk equivalences between subsampling and ridge regularization for ensemble ridge estimators. Specifically, we prove that linear and quadratic functionals of
subsample ridge estimators, when fitted with different ridge regularization levels $\lambda$ and subsample aspect ratios $\psi$, are asymptotically equivalent along specific paths in the $(\lambda, \
psi)$-plane (where $\psi$ is the ratio of the feature dimension to the subsample size). Our results only require bounded moment assumptions on feature and response distributions and allow for
arbitrary joint distributions. Furthermore, we provide a datadependent method to determine the equivalent paths of $(\lambda, \psi)$. An indirect implication of our equivalences is that
optimally-tuned ridge regression exhibits a monotonic prediction risk in the data aspect ratio. This resolves a recent open problem raised by Nakkiran et al. under general data distributions and a
mild regularity condition that maintains regression hardness through linearized signal-to-noise ratios.
The code for reproducing results of this paper is available at Github.
• Section 3:
□ Figure1 1: run_equiv_estimator.py computes the linear projections of ensemble estimators on simulated data.
• Section 4:
□ Figures 2 and F5: run_equiv_risk.py computes generalized quadratic risks on simulated data.
• Real data:
□ Figure 3: run_equiv_cifar.py computes the empirical estimates on CIFAR-10.
□ Figure F6: run_equiv_real_data.py computes the empirical estimates on CIFAR-10, MNIST, and USPS.
• Extensions:
□ Random features regression (Figure 4): run_equiv_random_feature.py
□ Kernel regression (Figure F7): run_equiv_kernel.py
• The jupyter notebook plots all the figures based on results produced by previous scripts.
Computation details
All the experiments are run on Pittsburgh Supercomputing Center Bridge-2 RM-partition using 48 cores.
The estimated time to run all experiments is roughly 6 hours for each script. | {"url":"https://jaydu1.github.io/overparameterized-ensembling/equiv/","timestamp":"2024-11-03T16:41:58Z","content_type":"text/html","content_length":"15431","record_id":"<urn:uuid:b5fae755-ce61-48d5-870a-dd2f1e33d21f>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00424.warc.gz"} |
How can I use Snowpark to perform analytics tasks? - Snowflake Solutions
How can I use Snowpark to perform analytics tasks?
How can I use Snowpark to perform analytics tasks?
1 Answer
Snowpark can be used to perform a variety of analytics tasks, such as:
• Data exploration: Snowpark can be used to explore data by performing operations such as filtering, sorting, and aggregating.
• Data visualization: Snowpark can be used to visualize data using charts and graphs.
• Statistical analysis: Snowpark can be used to perform statistical analysis on data, such as calculating means, medians, and standard deviations.
• Machine learning: Snowpark can be used to train and deploy machine learning models.
Here are some examples of how to use Snowpark to perform these analytics tasks:
• Data exploration: To explore data using Snowpark, you can use the filter(), sort(), and agg() methods. For example, the following code filters a DataFrame to only include rows where
the age column is greater than 18 and then sorts the rows by the name column in ascending order:
df = session.readTable("mytable", "mydatabase")
filtered_df = df.filter(df["age"] > 18)
sorted_df = filtered_df.sort("name")
Use code with caution.
• Data visualization: To visualize data using Snowpark, you can use the plot() method. The plot() method takes a DataFrame as its argument and returns a chart or graph. For example, the following
code plots the number of customers by age using a bar chart:
df = session.readTable("customers", "mydatabase")
df.plot("age", "count", kind="bar")
Use code with caution.
• Statistical analysis: To perform statistical analysis on data using Snowpark, you can use the describe() method. The describe() method takes a DataFrame as its argument and returns a DataFrame
containing summary statistics for each column. For example, the following code calculates the mean, median, and standard deviation of the age column in a DataFrame:
df = session.readTable("customers", "mydatabase")
summary = df.describe("age")
Use code with caution.
• Machine learning: To train and deploy machine learning models using Snowpark, you can use the train() and deploy() methods. For example, the following code trains a linear regression model to
predict house prices and then deploys the model to a remote endpoint:
df = session.readTable("houses", "mydatabase")
model = df.train(LinearRegression())
deployment = model.deploy("myendpoint")
Use code with caution.
These are just a few examples of how to use Snowpark to perform analytics tasks. Snowpark provides a rich set of APIs that can be used to perform a variety of data analytics tasks.
You are viewing 1 out of 1 answers, click here to view all answers. | {"url":"https://snowflakesolutions.net/question/how-can-i-use-snowpark-to-perform-analytics-tasks/answer/16325/","timestamp":"2024-11-06T20:56:10Z","content_type":"text/html","content_length":"304418","record_id":"<urn:uuid:8b0f6340-0863-4cc6-b77e-b39ee45b9bfb>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00196.warc.gz"} |
Python sum(): How to Get the Sum of Python List Elements
17 Jun 2024
Python sum(): How to Get the Sum of Python List Elements
Aggregating and formatting Data has been on the forefront of our automation projects at IOFLOOD, and during this we have gotten familiar with the process of summing elements in Python lists. This
simple process helps in calculating total values, for efficient data analysis. To aid our customers that are wanting to improve their data aggregation practices on their custom server configurations,
we are providing our tips and processes in today’s article.
In this easy-to-follow guide, we’ll focus not only on the basics but also on advanced and practical applications of summing elements in a Python list, offering techniques that can take your Python
programming to the next level.
So, let’s dive deeper and explore different techniques to calculate the sum of elements in a Python list.
TL;DR: How do I sum elements in a Python list?
The simplest way to sum elements in a Python list is by using the inbuilt sum() function with the syntax, sum(listOfNumbers). It takes an iterable, like a list, as its primary argument and
returns the sum of its elements. For more advanced methods, read the rest of the article.
numbers = [1, 2, 3, 4, 5]
print(sum(numbers)) # Output: 15
Using Python’s sum() Function
One of the simplest ways to calculate the sum of a Python list is by using the inbuilt sum() function. This is akin to you manually counting each apple in each basket one by one. Python is known for
its ‘batteries included’ philosophy, meaning it comes packed with pre-built functions like sum(), designed to make your life as a programmer easier.
The sum() function is straightforward to use. It takes an iterable (like our list) as its primary argument and returns the sum of its elements. The syntax is as follows:
sum(iterable, start)
Start Parameter
The start parameter is optional and defaults to 0. It’s added to the sum of numbers in the iterable. Let’s see a quick example:
Code Output
print(sum(numbers)) 15
print(sum(numbers, 10)) 25
numbers = [1, 2, 3, 4, 5]
print(sum(numbers)) # Output: 15
print(sum(numbers, 10)) # Output: 25
In the second print statement, we’ve used the start parameter to add 10 to our sum.
sum() with other iterables
The sum() function processes the iterable from left to right. It’s versatile and can be used with different data structures like dictionaries, sets, and tuples, not just lists. For instance, if you
want to sum all the values in a dictionary, you can do it as follows:
dictionary = {'a': 5, 'b': 3, 'c': 2}
print(sum(dictionary.values())) # Output: 10
For more information on how to use Python sets, feel free to check out our free guide!
Computing Averages with the sum() Function
The sum() function can also be used to easily compute the average of a list’s elements by dividing the sum by the length of the list:
numbers = [1, 2, 3, 4, 5]
average = sum(numbers) / len(numbers)
print(average) # Output: 3.0
This is just the tip of the iceberg when it comes to using the sum() function. As we continue, we’ll explore more advanced techniques to calculate the sum of a Python list.
Handling Lists with Mixed Data Types
A more complex scenarios is handling lists with mixed data types. If you’ve ever tried to sum a list containing both string values and integers, you’ve likely run into a few challenges, just like you
would if you tried to count apples and oranges together.
Let’s consider a simple example:
mixed_list = [1, '2', 3, '4', 5]
Attempting to use the sum() function on this list as is, would result in a TypeError. Python doesn’t know how to add an integer and a string together, hence the error. So, how do we get around this
Casting to int()
One solution is to use the int() function inside a for loop to convert the string values to integers before summing them up. Here’s how you can do it:
mixed_list = [1, '2', 3, '4', 5]
total = 0
for i in mixed_list:
total += int(i)
print(total) # Output: 15
In the above code, we iterate over each element in the list. The int() function is used to convert the elements to integers, whether they’re already integers or strings. Then we add each element to
the total.
Code Output
print(sum(mixed_list)) TypeError
print(total) 15
Other data type problems
While this approach solves our immediate problem, it’s important to be aware of potential pitfalls.
For instance, if the list contains a string that cannot be converted to an integer (like ‘hello’), the int() function will raise a ValueError. In such cases, you might need to use error handling
techniques, such as try/except blocks, to ensure your program doesn’t crash.
Alternatives to sum()
While Python’s sum() function is a powerful tool for summing lists, it’s not the only tool in our arsenal. Let’s explore 3 other options: for loop, add() method, and a while loop.
Method Description Pros Cons
For Loop Sum all numbers in [1,2,3,4,5] using a basic for loop and incremental variable. Familiarity and ease of use for beginners. Might require more lines of code for
complex scenarios.
While Takes the elements from [1,2,3,4,5] in a loop as long as the list is not empty, adding each element to a High control over loop execution flow. Risk of infinite loops if condition is
Loop total sum. not wisely stated.
reduce() Uses Python’s functools module’s reduce method to apply function of two arguments cumulatively to the Excellent for applying a function for a Not a built-in function, it needs to be
Method elements of [1,2,3,4,5], from left to right, so as to reduce the list to a single output. sequence of arguments in an iterable. imported from functools module.
Using a ‘For’ Loop for summation
A for loop allows us to iterate over each element in the list, adding each one to a running total, much like manually counting each apple in a basket. Here’s a simple example:
numbers = [1, 2, 3, 4, 5]
total = 0
for num in numbers:
total += num
print(total) # Output: 15
In this code, we initialize a variable total to 0. Then, for each number in our list, we add it to total. The result is the sum of our list.
Using the add() Method
Beyond the for loop and sum() method, Python also offers the add() method from the operator module as another option for summing a list. The add() method can be used with the reduce() function from
the functools module to sum a list as follows:
from operator import add
from functools import reduce
numbers = [1, 2, 3, 4, 5]
total = reduce(add, numbers)
print(total) # Output: 15
In this code, the reduce() function applies the add() method to the first two items in the list, then to the result and the next item, and so on, effectively ‘reducing’ the list to a single output.
Loops offer flexibility for more complex operations beyond summing. For instance, you could modify the for loop to perform calculations based on the previous element in the list, something that would
be tricky with the sum() function. On the other hand, the sum() method allows for concise and efficient code, making your scripts cleaner and easier to read.
Using a While Loop
So far, we’ve discussed several ways to calculate the sum of a Python list, from using Python’s inbuilt sum() function to leveraging for loops and list comprehension. Now, let’s turn our attention to
another essential tool in Python: the while loop, which is like counting apples in a basket until the basket is empty.
The while loop offers a different approach to iteration. Instead of iterating over a sequence of elements like a for loop, a while loop continues as long as a certain condition is true. This makes
while loops particularly useful in scenarios where the number of iterations is not known in advance.
Here’s how you can use a while loop to calculate the sum of a list:
numbers = [1, 2, 3, 4, 5]
total = 0
i = 0
while i < len(numbers):
total += numbers[i]
i += 1
print(total) # Output: 15
In this code, we initialize a counter i to 0 and a variable total to hold our sum. The while loop continues as long as i is less than the length of the list. Inside the loop, we add the current
element to total and increment i by 1.
When comparing the while loop method with the for loop and sum() function, the while loop can be more verbose and slightly more complex due to the need to manually manage the loop variable.
However, it offers greater control over the looping process, which can be beneficial in more complex scenarios.
Further Resources for Python
If you’re interested in learning more ways to utilize the Python language, here are a few resources that you might find helpful:
• An In-Depth Look at Python List Comprehension: Learn the powerful concept of list comprehension in Python and unlock its potential.
• Tutorial on Sorting a List in Python: IOFlood’s tutorial provides different methods for sorting a list in Python, including using the sorted() function, the sort() method, and custom sorting with
lambda functions.
• Guide on Reversing a List in Python: IOFlood’s guide demonstrates various approaches to reverse a list in Python, such as using the reversed() function, the reverse() method, and slicing with a
negative step.
• Extensive Guide on Python Syntax: IOFlood’s guide provides an extensive overview and cheat sheet of Python syntax, covering various topics such as variables, data types, control flow statements,
loops, functions, classes, and more. It serves as a comprehensive reference for Python programmers of all levels.
• Python sum() Function: A Comprehensive Guide: A comprehensive guide on GeeksforGeeks that explains how to use the sum() function in Python to calculate the sum of elements in a list or iterable.
• How to Compute the Sum of a List in Python: An article on Educative that demonstrates different approaches to compute the sum of elements in a list in Python.
• Sum of Elements in a List in Python: A tutorial on SparkByExamples that showcases different methods to compute the sum of elements in a list using Python.
Conclusion: Summing Python Iterables
We’ve journeyed through the various ways to calculate the sum of a Python list, each with its unique strengths and considerations. It’s like we’ve explored different methods to count apples in our
baskets. Let’s take a moment to recap.
The sum() function is Python’s inbuilt tool for summing an iterable. It’s simple to use, efficient, and works with different data structures, making it a solid first choice for many scenarios, just
like counting apples one by one from each basket.
When dealing with lists that contain mixed data types, we learned that a for loop can be a handy tool. By using a for loop with the int() function, we can convert all elements to integers and sum
them up, even if the list contains strings. It’s like separating apples and oranges before counting them.
Additionally, we introduced the add() method from the operator module as another option for summing a list. Finally, the while loop provides a different approach to iteration, proving useful when the
number of iterations is not known in advance. It’s like counting apples until the basket is empty.
By understanding these techniques and their practical applications, you can write more efficient, versatile code and take your Python skills to the next level. So keep exploring, keep learning, and
most importantly, keep coding! | {"url":"https://ioflood.com/blog/python-sum-list-how-to-calculate-the-sum-of-the-elements-in-a-list/","timestamp":"2024-11-06T05:00:48Z","content_type":"text/html","content_length":"60170","record_id":"<urn:uuid:11821b99-5832-42a9-bed3-f59fba369283>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00746.warc.gz"} |
A-level Mathematics/MEI/FP3 - Wikibooks, open books for an open world
Examination Structure
Candidates answer three questions out of five options, each worth 24 marks. The paper is 1 hour and 30 minutes. The total is 72 marks.
It is worth noting that: Candidates are expected to know the content for C1, C2, C3, C4, FP1 and FP2.
Candidates attempting Option 5 are expected to be familiar with elementary concepts of probability and with expected values.
For Option 5, Markov Chains, a calculator with the facility to handle matrices is required. | {"url":"https://en.m.wikibooks.org/wiki/A-level_Mathematics/MEI/FP3","timestamp":"2024-11-08T15:59:14Z","content_type":"text/html","content_length":"24149","record_id":"<urn:uuid:8037b349-6c83-463f-8139-4b69ed32f274>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00441.warc.gz"} |
But Why Doesn’t It Get Better? Kinetic Plots for Liquid Chromatography, Part 3: Pulling It All Together
In the last two instalments of “LC Troubleshooting”, we reviewed the basic idea of a “kinetic plot” (1) and how to make the plots from experimental data or data from the literature (2). Ultimately,
this graphical tool can be used to make informed decisions when choosing columns and to understand why a column might not be delivering expected performance improvements. This month, we conclude this
series of articles by discussing the so-called “Knox-Saleem limit” (KSL), application of kinetic plots to gradient elution conditions, and the impact of extracolumn dispersion on kinetic plots.
Finally, we introduce a web-based application that pulls together all of the theory discussed in these instalments into a convenient and flexible web‑based calculator that allows you to explore the
impact of many variables on the kinetic plot on your own.
The Knox-Saleem Limit (KSL)
In last month’s instalment, we began discussing the effect of particle size on kinetic plots by showing the kinetic performance limit (KPL) curves for different particle sizes (Figure 1[a]).
Interestingly, these curves cross in the kinetic plot, which means that at any given combination of t[0] and N, there is one particle size that provides superior performance compared to the others.
In other words, there is no single particle size that is superior to all others over the entire range of analysis times of practical interest. Whereas the smallest particles (1.7 µm) show the best
kinetic performance at short analysis times, the larger particles (5 µm) are the better choice to obtain high efficiencies at long analysis times, which is a direct result of the improved performance
at higher flow rates for smaller particles (fast analysis). However, the small particles also lead to large pressure drops that limit their use to relatively short columns (lower efficiency).
In Figure 1(a), we see that each of the KPL curves touches an oblique asymptote (dashed lines) below which one cannot work regardless of the choice of column length, particle size, and velocity
because the pressure drop will exceed the chosen pressure limit. This oblique asymptote is the KSL and in fact touches the KPLs for different particle sizes at their respective optimal mobile phase
velocities (that is, u[0,min]) (3). The point where the KPL and KSL curves touch represents the optimal choice of not only mobile phase velocity and column length but also of the particle size for
each combination of t[0] and N. For the particle sizes represented in Figure 1, we see that the KPL curves come very close to the KSL, which indicates that at least one of these particles is close to
optimal for plate numbers in the range of 10,000 < N < 200,000. When there is a gap in the available particles sizes (for example, jumping from 1.7 to 3.5 µm), we see a gap between the points at
which the two KPL curves cross with the KSL. This occurrence indicates a gap between the truly optimal performance that is possible for a given combination of t0 and N, and what can actually be
realized with the available particle sizes. Fortunately, these differences are rather small, as has been discussed in detail by Matula and Carr in the literature (4).
The KSL can be calculated using equation 1 if the dynamic viscosity of the mobile phase (η), the minimum reduced plate height (h[min]), and the u[0]‑velocity-based flow resistance (Φ[0]) for a
certain stationary phase support are known using (2,3,5):
This relationship makes clear that the kinetic performance can be improved by increasing the maximum operating pressure (ΔP[max]; that is, ultrahigh-pressure liquid chromatography [UHPLC] vs. high
performance liquid chromatography [HPLC]), decreasing the mobile phase viscosity (for example, through the use of high temperatures in LC, or low viscosity eluents in supercritical fluid
chromatography [SFC]), reducing the flow resistance (for example, by using monolithic or chip-based columns), or decreasing the minimum reduced plate height (for example, with superficially porous
particles, chip-based, or 3D-printed columns) (6). A change in any of these parameters will shift the KSL (and also the KPL curves) to the right, allowing for both faster and more efficient
separations. When all other parameters are fixed, doubling ΔP[max] results in a decrease in t[0] by a factor of two. In other words, doubling the available pressure allows the same efficiency to be
realized in half the time, which is illustrated in Figure 1(b) where the effect of the operating pressure on the KPL curve for the 1.7 µm particles is shown (7). The curve shifts to the bottom right
of the kinetic plot, showing how even faster analyses and higher efficiencies can be obtained when operating at this higher maximum pressure. In fact, when comparing the use of 1.7 µm particles at
1000 bar with 3.5 and 5 µm particles used at 400 bar, the smaller 1.7 µm particles outperform the 3.5 µm particles in the part of the efficiency analysis time range where the latter outperforms the
5 µm particles at 400 bar. The 1.7 µm particles at 1000 bar even outperform the 5 µm particles up to approximately N ~100,000. Of course, this comparison changes if the 3.5 and 5 µm particles can
also be used at 1000 bar. Similarly to the KPL curve, the KSL also shifts with an increase in maximum pressure, as expected from equation 1.
Application of the Kinetic Plot Concept to Gradient Elution Conditions
For fundamental comparisons of the separation performance of different column types, it is most practical to use isocratic elution conditions, which is why our discussion of kinetic plots has so far
focused on the kinetic plots with t[0] and N as the axes. In practice, however, most applications use mobile phase composition gradients to optimize separation time and resolution. Thus, it is
desirable to apply the kinetic plot concept to the gradient elution condition as well, which can be done by replacing the plate number with the peak capacity (n[p]) as the measure of separation
performance (5,8). When making experimental measurements of retention time and peak width under gradient elution conditions for the purpose of making kinetic plots from experimental data, several
details are important to keep in mind. These are mentioned briefly here. Readers interested in learning more about them are referred to the literature for detailed protocols (9).
Gradient time should be scaled inversely proportional to the flow rate so that the gradient slope remains constant (10,11).
If the mobile phase composition is held constant at the beginning of the separation, or at any other point in the elution program, these so-called hold times should also be scaled with the inverse of
the flow rate.
If columns with the same stationary phase chemistry from the same vendor are compared, there is usually little to no difference in selectivity and the same gradient range (initial and final
composition) can be used. However, when comparing columns from different vendors, differences in retention may be observed, and it is advisable to tune the initial and final composition of the
gradient in such a way that the first and last eluted compounds have similar retention factors (9,11).
As previously mentioned, in the case of gradient elution, the peak capacity (n[p]) is usually the preferred measure of separation performance rather than the column plate count (N). Calculation of
the column dead time and retention time at the kinetic performance limit (that is, t[0,KPL,] and t[R,KPL]) is similar in isocratic and gradient elution, however calculation of the peak capacity at
the KPL is slightly different, as shown in equations 2 and 3 (9,10,12,13):
The square root dependence in equation 3 is the direct result of the square root dependence of the peak capacity on the column plate number (10). As a result, increasing the column length by a factor
of four will only increase the peak capacity by a factor of two. Please note that also in this case the value for ΔP[exp] should include the extracolumn pressure drop as discussed in the next
Effects of Extracolumn Dispersion on Kinetic Plots
So far in this series, we have not discussed the impact of extracolumn dispersion (ECD) on kinetic plots, which is mathematically convenient. However, peak dispersion outside of the column is often
too large to ignore. We discussed the details associated with dispersion in different parts of the LC system in a prior multipart series of articles in this magazine (14–17) and elsewhere (18), and
readers interested in these details are referred there. Here, we focus on adjustments that must be made to the kinetic plot calculations to account for both the dispersion that occurs in the LC
system outside of the column, and the pressure drop that occurs in different parts of the system.
Corrections to the kinetic plot calculations to account for extracolumn effects can be made using values for the extracolumn dispersion and pressure drop obtained from experiments, or some means of
estimation. When it comes to experimental measurements, the column is replaced by a zero dead volume union in order to obtain the extracolumn time (t[ec]) and peak variance (σ^2[t,ec]) at different
flow rates. It is important to understand that extrapolation of the plate number from a FL curve to the KPL using λ = ΔP[max]/ ΔP[exp] (as discussed in Part 2 of this series) should only be done
using plate numbers that have been corrected for ECD. Then, after the extrapolation, the extracolumn variance is added back to the peak variance contributed by the column to give an effective plate
number (N[eff]) as shown in equation 4. Similarly, the column dead time must be corrected to account for the time the analyte spends travelling from the injector to detector, but outside of the
column, as shown in equation 5:
In addition to the effect of the LC system on dispersion of peaks, some of the available operating pressure is also lost because of pressure drops along the connecting tubes, especially when narrow
diameter tubes are used. To account for this, the value of ΔP[max] used in calculating the kinetic curves should be reduced by the value of the extracolumn pressure drop (ΔP[ec]) at the corresponding
flow rate, as shown in equation 6:
Pulling It All Together—A Web‑Based Application for You
Although no single mathematical step in calculating the kinetic curves is particularly difficult, there are many details to keep track of, and building a calculator correctly from scratch takes some
time. Thus, we have built a freely available web-based calculator (www.multidlc.org/kinetic_plot_tool) that incorporates all of the theory discussed in this series of articles, including
consideration of extracolumn effects discussed in the previous section. Here, we briefly demonstrate use of the tool by way of an example that shows how it can be used to explore the effects of
different variables on the curves, and perhaps develop hypotheses for troubleshooting situations where column performance does not live up to one’s expectations.
Figures 2 and 3 show screenshots of the inputs to the tool. Up to three different conditions can be compared simultaneously. Pre-set configurations for zero, low (~1–2 µL2), and normal (~10–15 µL2)
levels of extracolumn dispersion enable quick configuration of the extracolumn inputs; however, each of the system parameters (that is, injector, tubing, and detector) are fully adjustable as well.
Figure 4 shows screenshots of the kinetic plots produced by the tool for two different cases (A and B). In both cases the comparison is between columns packed with fully porous 1.7 µm particles and
columns packed with superficially porous 2.7 µm particles. In case A, the tool is configured using the pre-set parameters for a low dispersion system (~1–2 µL2) for both the FPP and SPP columns.
Here, we see that the 1.7 µm FPP columns outperform the 2.7 µm SPP ones at the KPL over the range of 5000 < N < 30,000, though the difference is small (t[0,FPP] = 0.28 min vs. t[0,SPP] = 0.31 min for
15,000 plates). At approximately N = 30,000 plates, the curves cross over and the SPP columns become superior for higher efficiencies as a result of their higher permeability.
However, when the tool is reconfigured using the preset parameters for a normal dispersion system (~10–15 µL2), we get the curves shown in Figure 4(b), where the SPP columns are superior to the FPP
ones at the KPL over the entire range of efficiencies shown. On one hand, the superiority of SPP columns is not surprising: manufacturers of sub-2-µm columns have been working to educate users for
years about the importance of using these columns in low dispersion systems to maximize their performance potential. On the other hand, this comparison shows the utility of the kinetic plot tool,
both for making informed choices about column selection and troubleshooting situations where a column in use does not live up to user expectations.
In this instalment of “LC Troubleshooting”, we have continued our discussion of kinetic plots and their utility when selecting column technologies and formats, and troubleshooting columns that appear
to not live up to our expectations. The KSL quantifies the best achievable performance (as measured by plate number) in a given analysis time when the particle size is allowed to vary. Kinetic plots
can also be used to compare technologies under gradient elution conditions, and when the effects of extracolumn dispersion are accounted for. Finally, we have introduced a freely available web‑based
kinetic plot tool that leverages all of the theory discussed in this series and enables comparison of up to three different sets of conditions simultaneously. This tool is useful for quickly
comparing different column technologies and LC system configurations, and developing troubleshooting hypotheses when things don’t seem to be quite right. It is important to note that all calculations
in this series have been done with diffusion coefficients typical of small molecules. When working with large molecules their diffusion coefficients will be very different, and thus the kinetic plots
will be very different as well.
1. K. Broeckhoven and D.R. Stoll, LCGC Europe 35(2), 52–56 (2022).
2. K. Broeckhoven and D.R. Stoll, LCGC Europe 35(3), 93–97 (2022).
3. J.H. Knox and M. Saleem, J. Chromatogr. Sci. 7, 614–622 (1969).
4. A.J. Matula and P.W. Carr, Anal. Chem. 87, 6578–6583 (2015).
5. G. Desmet, D. Cabooter, and K. Broeckhoven, Anal. Chem. 87, 8593–8602 (2015).
6. K. Broeckhoven and G. Desmet, Anal. Chem. 93, 257–272 (2021).
7. Y. Vanderheyden, D. Cabooter, G. Desmet, and K. Broeckhoven, J. Chromatogr. A 1312, 80–86 (2013).
8. X. Wang, D.R. Stoll, P.W. Carr, and P.J. Schoenmakers, J. Chromatogr. A 1125, 177–181 (2006).
9. K. Broeckhoven, D. Cabooter, S. Eeltink, and G. Desmet, J. Chromatogr. A 1228, 20–30 (2012).
10. K. Broeckhoven, D. Cabooter, F. Lynen, P. Sandra, and G. Desmet, J. Chromatogr. A. 1217, 2787–2795 (2010).
11. K. Broeckhoven, D. Cabooter, and G. Desmet, LCGC Europe 24, 396–404 (2011).
12. K. Broeckhoven and G. Desmet, J. Sep. Sci. 44, 323–339 (2021).
13. T.J. Causon, K. Broeckhoven, E.F. Hilder, R.A. Shellie, G. Desmet, and S. Eeltink, J. Sep. Sci. 34, 877–887 (2011).
14. D.R. Stoll, T.J. Lauer, and K. Broeckhoven, LCGC Europe 34(11), 464–469 (2021).
15. D.R. Stoll and K. Broeckhoven, LCGC Europe 34(7), 277–280 (2021).
16. D.R. Stoll and K. Broeckhoven, LCGC Europe 34(6), 232–237 (2021).
17. D.R. Stoll and K. Broeckhoven, LCGC Europe 34(5), 181–188 (2021).
18. G. Desmet and K. Broeckhoven, TrAC Trends in Anal. Chem. 119, 115619 (2019).
About The Authors
Ken Broeckhoven is Associate Professor at the Vrije Universiteit Brussel (VUB), in Brussels, Belgium.
Caden Gunnarson is currently a student and a researcher in the Stoll laboratory at Gustavus Adolphus College in St. Peter, Minnesota, USA.
About The Column Editor
Dwight R. Stoll is the editor of “LC Troubleshooting”. Stoll is a professor and the co-chair of chemistry at Gustavus Adolphus College in St. Peter, Minnesota. Direct correspondence to: | {"url":"https://www.chromatographyonline.com/view/but-why-doesn-t-it-get-better-kinetic-plots-for-liquid-chromatography-part-3-pulling-it-all-together","timestamp":"2024-11-06T14:47:51Z","content_type":"text/html","content_length":"459453","record_id":"<urn:uuid:785325e6-4337-4f6e-96a6-68e0f7f769b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00673.warc.gz"} |
Revision #2 to TR12-169 | 27th January 2014 17:25
The Approximate Rank of a Matrix and its Algorithmic Applications
We study the $\eps$-rank of a real matrix $A$, defined for any $\eps > 0$ as the minimum rank of a matrix that approximates every entry of $A$ to within an additive $\eps$. This parameter is
connected to other notions of approximate rank and is motivated by problems from various topics including communication complexity, combinatorial optimization, game theory, computational geometry and
learning theory. Here we give bounds on the $\eps$-rank and use them for algorithmic applications. Our main algorithmic results are (a) polynomial-time additive approximation schemes for Nash
equilibria for $2$-player games when the payoff matrices are positive semidefinite or have logarithmic rank and (b) an additive PTAS for the densest subgraph problem for similar classes of weighted
graphs. We use combinatorial, geometric and spectral techniques; our main new tool is an efficient algorithm for the following problem: given a convex body $A$ and a symmetric convex body $B$, find a
covering a $A$ with translates of $B$.
Changes to previous version:
Modified enumeration algorithm. Corrected errors. Made algorithms deterministic.
Revision #1 to TR12-169 | 21st May 2013 10:34
The Approximate Rank of a Matrix and its Algorithmic Applications
We study the $\eps$-rank of a real matrix $A$, defined for any $\eps > 0$ as the minimum rank over matrices that approximate every entry of $A$ to within an additive $\eps$. This parameter is
connected to other notions of approximate rank and is motivated by problems from various topics including communication complexity, combinatorial optimization, game theory, computational geometry and
learning theory. Here we give bounds on the $\eps$-rank and use them for algorithmic applications. Our main algorithmic results are (a) polynomial-time additive
approximation schemes for Nash equilibria for $2$-player games when the payoff matrices are
positive semidefinite or have logarithmic rank and (b) an additive PTAS for the densest subgraph problem for similar classes of weighted graphs. We use combinatorial, geometric and spectral
techniques; our main new tool is an algorithm for efficiently covering a convex body with translates of another convex body.
Changes to previous version:
Added references to work in communication complexity; added two new co-authors.
TR12-169 | 22nd November 2012 01:27
The Approximate Rank of a Matrix and its Algorithmic Applications
We introduce and study the \epsilon-rank of a real matrix A, defined, for any \epsilon > 0 as the minimum rank over matrices that approximate every entry of A to within an additive \epsilon. This
parameter is connected to other notions of approximate rank and is motivated by problems from various topics including combinatorial optimization, game theory, computational geometry and learning
theory. Here we give bounds on the \epsilon-rank and use them to derive (a) polynomial-time approximation schemes for Nash equilibria of substantially larger classes of 2-player games than previously
known and (b) an additive PTAS for the densest subgraph problem on inputs having small \epsilon-rank. We use combinatorial, geometric and spectral techniques; our main new tool is an algorithm for
efficiently covering a convex body with translates of another convex body. | {"url":"https://eccc.weizmann.ac.il/report/2012/169/","timestamp":"2024-11-12T00:47:58Z","content_type":"application/xhtml+xml","content_length":"26085","record_id":"<urn:uuid:14e6e1df-ca8e-4a6b-bb58-3505c4dec5dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00875.warc.gz"} |
Time Value of Money Calculator
Online Calculators > Financial Calculators > Time Value of Money Calculator
TVM Calculator
TVM Calculator to calculate the future value of money. Time value of money calculator with regular contributions to estimate how long and how much you need to invest to reach your financial goal. The
time value of money formula is shown below on how to calculate time value of money.
Present Amount: $
Interest Rate: %
Years: years
Compound Period:
Additional Contributions: $
Time Value of Money Calculator
TVM formula has option for different compound periods and additional monthly or yearly contribution. Many people doesn't realize how much their money can grow with compound interest and regular
Time Value of Money Formula
Following is the time value of money formula on how to calculate TVM.
TVM = Principal * (1 + r)^n; r = interest rate
n = number of compounding periods per year
How to Calculate Time Value of Money
To use the time value of money formula, we need a few given variables, the principal, interest rate, years to grow and the number of compounding periods per year.
For example, to find out how much $20,000 can grow in 8 years with a 5% interest rate and annual compounding, we would plugin the variables to the TVM formula as below
TVM = 20000*(1+0.05)^8 = $29,811.71
What is the time value of money?
The time value of money is the idea that the same amount of money is worth more today than in the future due to inflation and other factors.
For example, $100 today has more purchasing power today than it would be in 10 years. A $100 today can also be invested and grown to $200 or even more 10 years later.
Therefore, as an investor, one wants to receive money today rather than in the future. As a lender, he charges interest when he lends money to a borrower.
Present Value Calculator
Future Value Calculator | {"url":"https://online-calculator.org/time-value-of-money-calculator.aspx","timestamp":"2024-11-08T07:56:12Z","content_type":"application/xhtml+xml","content_length":"18846","record_id":"<urn:uuid:662f2f7a-a3b4-4af6-8442-69e21221f3f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00362.warc.gz"} |
Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion
A basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix inequalities. The sparsity,
characterized with a chordal graph structure, can be detected in the variable matrix or in a linear or nonlinear matrix-inequality constraint of the problem. We classify the sparsity in two types,
the domain-space sparsity (d-space sparsity) for the symmetric matrix variable in the objective and/or constraint functions of the problem, which is required to be positive semidefinite, and the
range-space sparsity (r-space sparsity) for a linear or nonlinear matrix-inequality constraint of the problem. Four conversion methods are proposed in this framework: two for exploiting the d-space
sparsity and the other two for exploiting the r-space sparsity. When applied to a polynomial semidefinite program (SDP), these conversion methods enhance the structured sparsity of the problem called
the correlative sparsity. As a result, the resulting polynomial SDP can be solved more effectively by applying the sparse SDP relaxation. Preliminary numerical results on the conversion methods
indicate their potential for improving the efficiency of solving various problems.
• Chordal Graph
• Matrix Inequalities
• Polynomial Optimization
• Positive Semidefinite Matrix Completion
• Semidefinite Program
• Sparsity
Dive into the research topics of 'Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion'. Together they form a unique fingerprint. | {"url":"https://pure.ewha.ac.kr/en/publications/exploiting-sparsity-in-linear-and-nonlinear-matrix-inequalities-v","timestamp":"2024-11-14T13:29:19Z","content_type":"text/html","content_length":"52163","record_id":"<urn:uuid:00ec3a38-1271-477a-9256-c8aa7839d942>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00124.warc.gz"} |
The Equation of Creation. Really?
Deal or No Deal and best treated with a topical dose of statistics. To David Cumming, it’s a trap. A big hole in the ground with very steep walls, which he willingly threw himself into with the
publication of his summary of the ‘God Equation’. This rather poorly written and overly drawn-out exegesis of unrelated numbers gets off to a bad start after the above equation is presented, followed
by two paragraphs of Mr. Cumming referencing himself in the third person. TK knows this is silly, and despite being a brilliant, charming and terrifically sexy individual (whom in many ways is the
central protagonist of the greatest story never told), will not succumb to employing such a conceited writing device. Beyond this, Cumming begins his derivation.
Firstly, it’s clear that whatever kind of scientist he claims to be, David Cumming cannot present a mathematical derivation in an unambiguous manner. I had to read the paper several times to piece it
together, and here is the brief: the speed of light (in the archaic and highly questionable unit of the megalithic yard) is equal to the product of the hydrogen line and the ratio of pi and an
arbitrary constant labeled omega:
“…why is the value of Omega significant, apart from the fact that it represents all the characters of a number system, the number system we actually use? Well, the Moon is 1/81 of the weight of
the Earth. 1/81 equates to the very unusual decimal fraction 0.0123456789. So the equation message also encodes this very unlikely ratio between the weight of the Moon and the weight of the
Well, it only represents all the characters in our numbering system because you’ve expressed it in our numbering system, silly! But hang on, 0.0123456789 is no more unusual than 0.302938491 or
2589.99999, because like all individual numbers, it’s unique. Also, the Moon is not 1/81 the mass of the Earth, but is rather close to 1/81.3, or 0.012302464 Earth masses. But who cares, it’s not
like the creator of the Solar System/Universe is capable of 100% accuracy, right?
This is the only way in which the so-called equation can be related to the Earth-Moon system, and doesn’t quite work out. I say so-called equation, because it doesn’t equate. In other words, it’s not
balanced. The ratio of pi and omega is a dimensionless constant, linearly equating the hydrogen line (the frequency of a photon emitted when neutral hydrogen drops down an energy state by one) to the
speed of light in megalithic yards, or Thoms (Ths).
So, s^-1 equates to Ths s^-1 ? Here’s the clincher. If you fudge the figures, our dimensionless constant is so close to the corresponding wavelength of a hydrogen line photon in Thoms that we can
just assume it to be so (we’ll get to assumptions at the end). Thus we magically grant it the unit of Thoms and the equation balances. It should be obvious that this is still mathematically illegal,
since we cannot backstep this new derivation. That is, I cannot multiply any number in Thoms by the dimensionless omega and produce pi, itself a dimensionless ratio. We must make a second assumption,
that omega has the units of Ths^-1, and just happens to be not quite the ratio of the masses of the Moon and Earth (except it now isn’t, because we’ve just given it units!)
Omega is the prime fudge-factor in all of this, and it’s precisely why Cumming is a victim of numerology. There is no significance to the number 0.0123456789 except that which we assume to be there,
but those of a religious bent have no trouble doing just that if it can be used to demonstrate their own imagined self-importance. However just like with astrology, palm reading or dowsing, a sceptic
has no trouble emulating the irrational invention of significance. Just watch me.
The first Dwarf Planet discovered by humans was Ceres, which has a mean radius of 471 km. This is a 98.3% match to its synodic period in days: 463. Not impressed? Maybe you didn’t realise that 463 is
not only a prime number, but also the sum of seven consecutive primes (53+59+61+67+71+73+79), and when I enter 463 in a text message, my phone’s predictive text engine recommends the word ‘God’.
Evidence or coincidence?
The real question is how unlikely must a coincidence seem before you snap? Apparently, David Cumming is beyond his limit:
“The Equation of Creation and the Thom came first. Therefore, either there is the most freakishly unlikely coincidence happening,
and the huge amount of supporting data not mentioned in this short review makes the odds of this
event being due to a chance event really disappear beyond the possible, or with the application of
the razor of Occam, we are left with the simple conclusion that the Earth, Sun, and Moon must have been Created to accord with the Equation of Creation.”
But… Occam’s razor suggests one should minimise their assumptions! I assume that incredible coincidences occur (not really an assumption, more of an observation) whereas Cumming assumes the
significance of the Thom, and his own made up constant, as well as its units. He equates highly improbable with impossible (a belief all too often espoused by creationists) and in doing so makes a
further assumption that there is more than one outcome – that the Universe could have had different fundamental constants (a belief for which there is no confirming evidence).
His paper is worth reading for any sceptic, but do not be swayed by his emotive language or incredible ability to convince himself. A straight-forward, clear proof of creation it ain’t.
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Sort SQL Server tables into similarly sized buckets
Sort SQL Server Tables into similarly sized buckets
You need to do something to all of the tables in SQL Server. That something can be anything: reindex/reorg, export the data, perform some other maintenance---it really doesn't matter. What does
matter is that you'd like to get it done sooner rather than later. If time is no consideration, then you'd likely just do one table at a time until you've done them all. Sometimes, a maximum degree
of parallelization of one is less than ideal. You're paying for more than one processor core, you might as well use it. The devil in splitting a workload out can be ensuring the tasks are well
balanced. When I'm staging data in SSIS, I often use a row count as an approximation for a time cost. It's not perfect - a million row table 430 columns wide might actually take longer than the 250
million row key-value table.
A sincere tip of the hat to Daniel Hutmacher (b|t)for his answer on this StackExchange post. He has some great logic for sorting tables into approximately equally sized bins and it performs
reasonably well.
@bucketCount tinyint = 6;
IF OBJECT_ID('tempdb..#work') IS NOT NULL
DROP TABLE #work;
CREATE TABLE #work (
_row int IDENTITY(1, 1) NOT NULL,
[SchemaName] sysname,
[TableName] sysname,
[RowsCounted] bigint NOT NULL,
GroupNumber int NOT NULL,
moved tinyint NOT NULL,
PRIMARY KEY CLUSTERED ([RowsCounted], _row)
WITH cte AS (
SELECT B.RowsCounted
, B.SchemaName
, B.TableName
s.[Name] as [SchemaName]
, t.[name] as [TableName]
, SUM(p.rows) as [RowsCounted]
sys.schemas s
LEFT OUTER JOIN
sys.tables t
ON s.schema_id = t.schema_id
LEFT OUTER JOIN
sys.partitions p
ON t.object_id = p.object_id
LEFT OUTER JOIN
sys.allocation_units a
ON p.partition_id = a.container_id
p.index_id IN (0,1)
AND p.rows IS NOT NULL
AND a.type = 1
GROUP BY
, t.[name]
) B
INSERT INTO #work ([RowsCounted], SchemaName, TableName, GroupNumber, moved)
SELECT [RowsCounted], SchemaName, TableName, ROW_NUMBER() OVER (ORDER BY [RowsCounted]) % @bucketCount AS GroupNumber, 0
FROM cte;
WHILE (@@ROWCOUNT!=0)
WITH cte AS
, SUM(RowsCounted) OVER (PARTITION BY GroupNumber) - SUM(RowsCounted) OVER (PARTITION BY (SELECT NULL)) / @bucketCount AS _GroupNumberoffset
w.GroupNumber = (CASE w._row
WHEN x._pos_row THEN x._neg_GroupNumber
ELSE x._pos_GroupNumber
, w.moved = w.moved + 1
#work AS w
INNER JOIN
SELECT TOP 1
pos._row AS _pos_row
, pos.GroupNumber AS _pos_GroupNumber
, neg._row AS _neg_row
, neg.GroupNumber AS _neg_GroupNumber
cte AS pos
INNER JOIN
cte AS neg
ON pos._GroupNumberoffset > 0
AND neg._GroupNumberoffset < 0
--- To prevent infinite recursion:
pos.moved < @bucketCount
AND neg.moved < @bucketCount
WHERE --- must improve positive side's offset:
ABS(pos._GroupNumberoffset - pos.RowsCounted + neg.RowsCounted) <= pos._GroupNumberoffset
--- must improve negative side's offset:
ABS(neg._GroupNumberoffset - neg.RowsCounted + pos.RowsCounted) <= ABS(neg._GroupNumberoffset)
--- Largest changes first:
ORDER BY
ABS(pos.RowsCounted - neg.RowsCounted) DESC
) AS x
ON w._row IN
, x._neg_row
Now what? Let's look at the results. Run this against AdventureWorks and AdventureWorksDW
, COUNT_BIG(1) AS TotalTables
, SUM(W.RowsCounted) AS GroupTotalRows
#work AS W
, W.SchemaName
, W.TableName
, W.RowsCounted
, COUNT_BIG(1) OVER (PARTITION BY W.GroupNumber ORDER BY (SELECT NULL)) AS TotalTables
, SUM(W.RowsCounted) OVER (PARTITION BY W.GroupNumber ORDER BY (SELECT NULL)) AS GroupTotalRows
#work AS W
For AdventureWorks (2014), I get a nice distribution across my 6 groups. 12 to 13 tables in each bucket and a total row count between 125777 and 128003. That's less than 2% variance between the high
and low - I'll take it.
If you rerun for AdventureWorksDW, it's a little more interesting. Our 6 groups are again filled with 5 to 6 tables but this time, group 1 is heavily skewed by the fact that FactProductInventory
accounts for 73% of all the rows in the entire database. The other 5 tables in the group are the five smallest tables in the database.
I then ran this against our data warehouse-like environment. We had a 1206 tables in there for 3283983766 rows (3.2 [S:million:S] billion). The query went from instantaneous to about 15 minutes but
now I've got a starting point for bucketing my tables into similarly sized groups.
What do you think? How do you plan to use this? Do you have a different approach for figuring this out? I looked at R but without knowing what this activity is called, I couldn't find a function to
perform the calculations.
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Use Conditional Formatting on Dates in Excel - Multiple Conditions Shown
Table of Contents
Whenever we prepare any report in excel, we have two constituents in any report.
The Text portion and the Numerical portion.
But just storing the text and numbers doesn’t make the super reports. Many times we need to automate the process in the reports to minimize the effort and improve the accuracy.
Many functions are provided by Excel which works on Text and gives us useful output as well. But a few problems are still left on which we need to apply some tricks with the available tools.
We will continue learning many more techniques about the manipulation of text in Excel.
In this article, we’ll learn the way to deal with the dates in CONDITIONAL FORMATTING in Excel.
HOW DATE IS HANDLED IN EXCEL?
The first and foremost point is to understand how Excel handles the dates. What are the dates for Excel?
Date is treated as a simple serial number by excel starting from Jan 1 1900 [ treated as 1].
From Jan 1, 1900 which is 1 for Excel the serial number starts and it is still going on .
For example, it is 29th June 2020 today so the serial number for this date is 44011.
If I type this number in Excel simply,
and convert the format to date, It’ll translate it to the date mentioned above which is 29-06-2020.
By the pictures shown above, it is clear that we can play with the dates in both ways.
We can write the date in the various DATE FORMATS or we can simply use the numbers.[ Of course, it is not easy to remember the numbers, but we can refer for once].
EXCEL HAS THE PROVISION OF DATES FROM JAN 1,1900 TO DEC 31, 9999 which corresponds to 2958465.
So it should be clear to the Excel user that the date is nothing but a number. But why the problem occurs then.
The problem occurs when we think that the given format is Date but Excel doesn’t accept it as a date. It happens when we violate the rules of entering the date when we try to enter the date from
various formats mentioned below.
CONDITIONAL FORMATTING is the process of formatting in Excel on the basis of the conditions. We can put many conditions in the cell and program the Excel to make the formatting, as desired if the
particular condition is met.
Formatting comprises of the foreground color, background color, font, size, etc. which are the properties of the text.
It makes the results more readable.
Maximum times, we apply conditional formatting on the basis of values present in the cells.
Let us now learn the way by which we can apply conditional formatting based on DATES.
We’ll take different examples showing the use of conditional formatting under various circumstances.
We’ll need to find out the various situations such as the possible window for a few days, next n days, previous n days, and many other situations like that.
EXAMPLE 1: HIGHLIGHT ALL THE DATES PRIOR TO A GIVEN DATE
We can highlight all the dates which are prior to a given date very easily.
Let us highlight all the dates which are prior to 5/24/2021 out of all the given dates.
Let us take an example showing different dates as shown below.
5/24/2021 2/10/2022 8/16/2021 9/27/2021 7/16/2021 7/29/2021 9/29/2021
5/7/2021 2/8/2022 6/7/2021 1/11/2022 11/1/2021 3/31/2022 8/13/2021
6/2/2021 4/27/2021 6/27/2021 9/24/2021 2/12/2022 12/8/2021 2/7/2022
3/6/2022 4/19/2021 4/25/2021 12/8/2021 3/26/2022 7/26/2021 7/22/2021
8/13/2021 12/9/2021 5/8/2021 11/20/2021 5/3/2022 8/10/2021 12/20/2021
6/24/2021 4/18/2021 10/27/2021 5/20/2021 9/19/2021 4/15/2021 2/19/2022
11/16/2021 11/2/2021 3/9/2022 2/4/2022 2/1/2022 7/27/2021 3/22/2022
12/25/2021 1/30/2022 4/27/2021 7/16/2021 11/22/2021 2/25/2022 1/23/2022
FOLLOW THE STEPS TO HIGHLIGHT THE DATES PRIOR TO A GIVEN DATE. [ HIGHLIGHT DATES PRIOR TO 5/24/2021 ]
1. We have the following example data. [ Your data may comprise of a few columns or cells or specific rows etc. ]
2. Select all the cells containing the dates i.e. the cells on which you want to apply conditional formatting.
3. Go to CONDITIONAL FORMATTING> HIGHLIGHT CELLS RULES >LESS THAN under the HOME TAB.
4. The location is shown below.
5. As we choose the LESS THAN option, a small window will open.
6. Enter the cut-off date as 5/24/2021 i.e. the date prior to which, you want to highlight the dates.
7. Click OK.
8. All the dates earlier to 5/24/2021 will be highlighted.
This is the way to highlight the cells earlier to any specific date.
We simply used the LESS THAN option, which is normally used to highlight the numbers because DATE is itself a number. Date is a format which is used as a mask on the number which represents a
particular date. REFER HOW DATE IS HANDLED IN EXCEL for further details.
So, if we put the DATE NUMBER or the date in the LESS THAN field, it’ll highlight all the dates prior to the given date. For our example, it’ll highlight all the dates prior to 5/24/2021.
EXAMPLE 2: HIGHLIGHT ALL THE DATES LATER THAN A GIVEN DATE
So, this example is exactly opposite to the one we just discussed. [ Example 1].
The procedure is exactly the same as Example 1 except the option which we choose from the CONDITIONAL FORMATTING Menu.
Follow the steps to highlight all the dates later than a given date.
1. Select all the cells on which you want to apply conditional formatting . [ The cells where you want to put the condition. ]
2. Go to HOME TAB > CONDITIONAL FORMATTING >HIGHLIGHT CELL RULES > GREATER THAN.
3. The window will open.
4. Enter the cut off date . [ All the dates falling after the cut off date will be highlighted. ] For our example, we’ll enter 5/24/2021.
5. Choose the color of your choice which will be chosen for highlighting.
6. We are done.
EXAMPLE 3: HOW TO HIGHLIGHT THE DUPLICATE DATES IN THE GIVEN CELLS IN EXCEL SHEET?
Extending the same example, let us try to find out the date repetition in the given data.
DATE REPETITION is simply the occurrence of the same date more than once.
Let us try an example to learn the way which will let us highlight all the repetitive dates in the given data.
The data example is the same.
5/24/2021 2/10/2022 8/16/2021 9/27/2021 7/16/2021 7/29/2021 9/29/2021
5/7/2021 2/8/2022 6/7/2021 1/11/2022 11/1/2021 3/31/2022 8/13/2021
6/2/2021 4/27/2021 6/27/2021 9/24/2021 2/12/2022 12/8/2021 2/7/2022
3/6/2022 4/19/2021 4/25/2021 12/8/2021 3/26/2022 7/26/2021 7/22/2021
8/13/2021 12/9/2021 5/8/2021 11/20/2021 5/3/2022 8/10/2021 12/20/2021
6/24/2021 4/18/2021 10/27/2021 5/20/2021 9/19/2021 4/15/2021 2/19/2022
11/16/2021 11/2/2021 3/9/2022 2/4/2022 2/1/2022 7/27/2021 3/22/2022
12/25/2021 1/30/2022 4/27/2021 7/16/2021 11/22/2021 2/25/2022 1/23/2022
1. Select all the cells containing dates where you want to search for the duplicate dates.
2. Go to HOME TAB> CONDITIONAL FORMATTING >HIGHIGHT CELL RULES >DUPLICATE VALUES.
3. As we click this option, a small window will open.
4. Select the formatting and click OK.
It is one of the very easy tasks to do using conditional formatting.
1. Select all the cells containing dates where you want to search for the duplicate dates.
2. Go to HOME TAB> CONDITIONAL FORMATTING >HIGHIGHT CELL RULES >A DATE OCCURRING.
3. As we click this option, a small window will open.
4. Choose TODAY from the dropdown.
5. Select the format and click OK.
6. If TODAY’S DATE is available, it’ll be highlighted. [ If current date i.e. today’s date is not in the list, it won’t be highlighted. ]
EXAMPLE 5: HOW TO HIGHLIGHT DATES OLDER THAN THIRTY DAYS [30] USING CONDITIONAL FORMATTING IN EXCEL?
It is one of the frequently required operations where we need to highlight the dates older than 30 days or vice versa.
Let us highlight the dates older than thirty days from the current date.
FOLLOW THE STEPS TO HIGHLIGHT THE DATES OLDER THAN 30 DAYS.
1. Select all the cells containing dates where you want to search for the duplicate dates.
2. Go to HOME TAB> CONDITIONAL FORMATTING >NEW RULE.
3. As we click this option, a window will open.
4. Choose the last RULE TYPE as USE A FORMULA TO DETERMINE WHICH CELLS TO FORMAT.
5. A field to enter the formula will open.
6. Enter the formula as F62<(TODAY()-30) where F62 is the first cell of the selection. [ If your cell is A1, it’ll be A1 , if B6 then it’ll be B6 ].
7. Click FORMAT BUTTON and choose the format which you want to set for the cells that will fulfill the condition. [ FOR DETAILED LEARNING CLICK HOW TO USE CONDITIONAL FORMATTING IN EXCEL?
8. After selecting the format, click OK.
9. The result will highlight all the dates which are older than 30 days.
This was the way to highlight the dates older than 30 days in Excel using conditional formatting.
EXAMPLE 6: HOW TO HIGHLIGHT DATES WITHIN THIRTY DAYS [30] USING CONDITIONAL FORMATTING IN EXCEL?
Let us now highlight the dates within 30 days from current date. [ It is 5/28/2022 today for the example ]
FOLLOW THE STEPS TO HIGHLIGHT THE DATES WITHIN 30 DAYS
1. Select all the cells containing dates where you want to search for the duplicate dates.
2. Go to HOME TAB> CONDITIONAL FORMATTING >NEW RULE.
3. As we click this option, a window will open.
4. Choose the last RULE TYPE as USE A FORMULA TO DETERMINE WHICH CELLS TO FORMAT.
5. A field to enter the formula will open.
6. Enter the formula as F62>(TODAY()-30) where F62 is the first cell of the selection. [ If your cell is A1, it’ll be A1 , if B6 then it’ll be B6 ].
7. Click FORMAT BUTTON and choose the format which you want to set for the cells that will fulfill the condition. [ FOR DETAILED LEARNING CLICK HOW TO USE CONDITIONAL FORMATTING IN EXCEL?
8. After selecting the format, click OK.
9. The result will highlight all the dates which are older than 30 days.
This was the way to highlight the dates within 30 days in Excel using conditional formatting.
This is again one of the frequent requirement in Excel. Let us try to solve this.
We can simply use the EXAMPLE 2 WAY for the reference.
Simply put the DUE DATE as the cut off date after going to CONDITIONAL FORMATTING>GREATER THAN.
All the dates past the due date will be highlighted. | {"url":"https://gyankosh.net/exceltricks/conditional-formatting-with-dates-in-excel/","timestamp":"2024-11-03T20:18:35Z","content_type":"text/html","content_length":"180299","record_id":"<urn:uuid:adcccf72-f71f-4c04-b286-7a98c0f1be16>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00822.warc.gz"} |
Unscramble ZAPATEADO
How Many Words are in ZAPATEADO Unscramble?
By unscrambling letters zapateado, our Word Unscrambler aka Scrabble Word Finder easily found 95 playable words in virtually every word scramble game!
Letter / Tile Values for ZAPATEADO
Below are the values for each of the letters/tiles in Scrabble. The letters in zapateado combine for a total of 23 points (not including bonus squares)
• Z [10]
• A [1]
• P [3]
• A [1]
• T [3]
• E [1]
• A [1]
• D [2]
• O [1]
What do the Letters zapateado Unscrambled Mean?
The unscrambled words with the most letters from ZAPATEADO word or letters are below along with the definitions.
• zapateado () - Sorry, we do not have a definition for this word | {"url":"https://www.scrabblewordfind.com/unscramble-zapateado","timestamp":"2024-11-06T13:51:14Z","content_type":"text/html","content_length":"55117","record_id":"<urn:uuid:a968f9f6-3d4a-49af-a0c9-7a4986f1317e>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00214.warc.gz"} |
IB Math Extended Essay Topics | Topic Suggestions
IB Math Extended Essay Topics
In my tenure as an IB tutor with a solid grounding in mathematical theories and applications, I am eager to present a selection of stimulating topics for your Math Extended Essay. This assortment is
crafted to reflect the broad and intricate spectrum of mathematical inquiry, offering students the chance to probe into topics ranging from the abstract elegance of theoretical mathematics to the
practical sophistication of applied mathematics.
My experience has illuminated the profound impact of a carefully chosen topic, serving not merely as a rigorous academic exercise but as a crucible for intellectual growth and problem-solving
IB Math EE Topic Ideas
This venture invites students to weave an intricate tapestry of logic, proof, and application, demonstrating their ability to confront complex mathematical challenges and articulate their resolutions
with precision and depth.
1. Algebra and Number Theory
1. The Golden Ratio in Art and Architecture: How is the golden ratio used in the design of historical buildings and art?
2. Cryptography and Prime Numbers: How are prime numbers used in modern encryption methods?
3. The Mathematics of Fractals: How do fractals model real-world phenomena?
4. Complex Numbers in Electrical Engineering: How are complex numbers used to solve problems in electrical engineering?
5. The Fibonacci Sequence in Nature: How does the Fibonacci sequence appear in natural patterns and what does it signify?
6. Graph Theory in Social Networks: How can graph theory be applied to analyze social networks?
7. Pascal’s Triangle and its Applications: What are the various applications of Pascal’s Triangle in different fields of mathematics?
8. The Mathematics of Music: How can mathematical concepts be used to understand musical scales and harmony?
9. Game Theory in Economics: How is game theory applied to make decisions in economics?
10. The Riemann Hypothesis: What is the Riemann Hypothesis and why is it important in number theory?
2. Calculus and Analysis
11. The Calculus of Rainbows: How can calculus be used to explain the formation of rainbows?
12. Optimization Problems in Real Life: How can calculus help solve optimization problems in fields like engineering or economics?
13. The Mathematics of Epidemics: How are differential equations used to model the spread of diseases?
14. Fourier Series and Signal Processing: How are Fourier series used in processing signals in electronic devices?
15. The Monty Hall Problem and Probability Theory: How does probability theory explain the Monty Hall problem?
16. Calculus in Rocket Science: How is calculus used in the trajectory planning of rockets?
17. The Butterfly Effect and Chaos Theory: How does calculus help in understanding chaos theory and the butterfly effect?
18. The Brachistochrone Curve Problem: What is the brachistochrone curve and how can it be derived?
19. The Mathematics of Climate Models: How are differential equations used in climate modeling?
20. The Bernoulli Principle in Fluid Dynamics: How does the Bernoulli equation apply to fluid dynamics in real-world scenarios?
Drop your assignment info and we’ll craft some dope topics just for you.
It’s FREE 😉
3. Geometry and Topology
21. The Geometry of Islamic Art: How is geometry used to create patterns in Islamic art?
22. Non-Euclidean Geometry and the Theory of Relativity: How does non-Euclidean geometry contribute to Einstein’s theory of relativity?
23. The Mathematics of Origami: How does understanding geometry help in mastering origami?
24. Topology and the Möbius Strip: What are the unique properties of the Möbius strip in topology?
25. The Golden Spiral in Nature: How is the golden spiral represented in nature and what is its significance?
26. Tessellations and their Properties: How are tessellations created and what mathematical properties do they have?
27. The Use of Geometry in Computer Graphics: How is geometry used to create realistic computer graphics?
28. Fractal Geometry in Natural Landscapes: How does fractal geometry describe natural landscapes?
29. The Mathematics of Perspective Drawing: How is geometry used to create perspective in art?
30. The Geometry of Sports: How is geometry used in the design of sports equipment or fields?
4. Statistics and Probability
31. The Birthday Paradox: How does probability theory explain the birthday paradox?
32. Statistics in Medical Research: How are statistical methods used to validate findings in medical research?
33. The Monty Hall Problem: What does probability theory tell us about the Monty Hall problem?
34. Sports Analytics: How are statistics used to improve team performance in sports?
35. The Mathematics of Gambling: How does probability affect strategies in gambling?
36. Statistical Analysis of Climate Change Data: How are statistics used to interpret climate change data?
37. The Use of Probability in Weather Forecasting: How is probability used in predicting weather patterns?
38. Stock Market Analysis Using Statistics: How can statistical models predict stock market trends?
39. The Mathematics of Insurance: How do actuaries use probability and statistics in the insurance industry?
40. Error Analysis in Scientific Experiments: How are statistical methods used to analyze errors in scientific experiments?
5. Mathematical Modeling and Applications
41. Modeling Population Growth: How can mathematics model population growth and its impacts?
42. The Mathematics of Traffic Flow: How can mathematical models optimize traffic flow and reduce congestion?
43. Mathematical Models in Economics: How are mathematical models used to predict economic trends?
44. The Use of Mathematics in Environmental Science: How is mathematics used to model environmental issues like pollution dispersion?
45. Mathematical Modeling in Sports: How can mathematical models be used to improve performance in sports?
46. The Mathematics of Cryptocurrencies: How is mathematics fundamental to the functioning of cryptocurrencies?
47. Modeling the Spread of Viruses: How can mathematical models predict the spread of a virus like COVID-19?
48. Mathematics in Architectural Design: How is mathematics used in the design of buildings and structures?
49. Mathematical Optimization in Logistics: How is mathematics used to optimize logistics and supply chain management?
50. The Mathematics of Machine Learning: How is mathematics used in the algorithms behind machine learning and artificial intelligence?
As we reach the culmination of this array of Math Extended Essay topics, I hope these suggestions have catalyzed your analytical thinking and mathematical curiosity. The process of constructing a
Math Extended Essay is a meticulous integration of in-depth research, abstract reasoning, and clear, logical exposition.
As you refine your topic and delve into your mathematical exploration, view this as your platform to make a significant scholarly contribution. Your essay is a testament to your academic diligence,
your capacity for intricate and innovative problem-solving, and your proficiency in translating complex mathematical concepts into a coherent and persuasive narrative.
Let your work stand as a beacon of your scholarly dedication and a significant addition to the discourse of mathematical understanding.
Leave a Comment | {"url":"https://topicsuggestions.com/ib-math-extended-essay-topics/","timestamp":"2024-11-02T06:14:54Z","content_type":"text/html","content_length":"265768","record_id":"<urn:uuid:b5951e99-4dd7-4d23-863c-8e05d160590d>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00344.warc.gz"} |
Magnetic Hysteresis, Permeability, and Retentivity: A Comprehensive Guide
Magnetic hysteresis, permeability, and retentivity are fundamental concepts in the study of magnetic materials, with far-reaching applications in various fields, including electronics, power
generation, and magnetic data storage. This comprehensive guide delves into the technical details, theoretical explanations, and practical measurements of these crucial magnetic properties.
Magnetic Hysteresis Loop
The magnetic hysteresis loop is a graphical representation of the relationship between the magnetic flux density (B) and the applied magnetic field strength (H) in a magnetic material. This loop
provides valuable insights into the energy dissipation, magnetic memory, and overall behavior of the material.
Hysteresis Loop Parameters
1. Flux Density (B): Measured in Teslas (T), this parameter represents the magnetic field intensity within the material.
2. Magnetic Field Strength (H): Measured in Amperes per Meter (A/m), this parameter represents the external magnetic field applied to the material.
3. Energy Loss per Cycle (E/cycle): Measured in Joules (J), this parameter quantifies the energy dissipated during each magnetization cycle.
4. Power Loss (P): Measured in Watts (W), this parameter represents the power dissipated in the material due to the hysteresis effect.
Example Measurements: EDT39-3C85 Core
To illustrate the hysteresis loop parameters, let’s consider the measurements for an EDT39-3C85 core:
Drive Amplitude B max (T) H max (A/m) E/cycle (µJ) P@100kHz (W)
1 0.10 30 12.7 1.27
2 0.24 64 87.3 8.73
3 0.42 152 241.6 24.16
These measurements demonstrate the variation in the hysteresis loop parameters as the drive amplitude is increased, highlighting the energy dissipation and power loss characteristics of the material.
Permeability Calculation
Permeability is a measure of the ability of a material to support the formation of a magnetic field within itself. The relative permeability (μr) is a dimensionless quantity that relates the magnetic
flux density (B) to the applied magnetic field strength (H).
The relative permeability can be calculated using the following formula:
μr = (ΔB/ΔH)/4·π·10 -7
– μr is the relative permeability (dimensionless)
– ΔB is the change in magnetic flux density (T)
– ΔH is the change in magnetic field strength (A/m)
– 4·π·10 -7 is the permeability of free space (H/m)
Example values of relative permeability for the EDT39-3C85 core:
– Continuous Mode: μr = 2344
– Discontinuous Mode: μr = 2828
These values demonstrate the material’s ability to concentrate the magnetic flux within itself, which is a crucial property in various electromagnetic applications.
Retentivity (Remanence)
Retentivity, also known as remanence, is the ability of a magnetic material to retain its magnetization after the external magnetic field has been removed. This property is essential in the design of
permanent magnets and magnetic memory devices.
Measurement of Retentivity
Retentivity can be measured by observing the residual magnetism in a material after the external magnetic field is removed. This can be done by using a hysteresisgraph, which measures the magnetic
flux density (B) as a function of the applied magnetic field strength (H).
Technical Specifications: TXEMM-BH01 Hysteresisgraph
The TXEMM-BH01 Hysteresisgraph is a specialized instrument used to measure the magnetic hysteresis properties of materials. Some key specifications of this device include:
1. Frequency Range: DC to 1 kHz
2. ASTM Standards: ASTM A342, ASTM A343, ASTM A773, ASTM A977
3. Sample Preparation: Ring-shaped samples with primary and secondary coils to ensure a magnetic close circuit
Theoretical Explanation
To further understand the concepts of magnetic hysteresis, permeability, and retentivity, let’s explore the underlying theoretical principles.
Magnetic Flux Density (B)
The magnetic flux density (B) is related to the applied magnetic field strength (H) and the permeability (μ) of the material through the following equation:
B = μH
– B is the magnetic flux density (T)
– H is the magnetic field strength (A/m)
– μ is the permeability of the material (H/m)
Magnetic Field Strength (H)
The magnetic field strength (H) is determined by the number of turns (N) in the coil, the current (I) flowing through the coil, and the length (l) of the coil:
H = NI/l
– H is the magnetic field strength (A/m)
– N is the number of turns in the coil
– I is the current flowing through the coil (A)
– l is the length of the coil (m)
Permeability of Free Space (μ0)
The permeability of free space (μ0) is a fundamental physical constant that represents the ability of the vacuum to support a magnetic field. Its value is:
μ0 = 4·π·10 -7 H/m
This constant is used in the calculation of relative permeability (μr) and other magnetic properties.
1. Quantitative Analysis of Magnetic Hysteresis: https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2009GC002932
2. Magnetic Hysteresis Loop Measurements: https://meettechniek.info/passive/magnetic-hysteresis.html
3. Measuring, Processing, and Analyzing Hysteresis Data: https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2018GC007620
Hi, I am Amrit Shaw. I have done Master in Electronics.
I always like to explore new inventions in the field of Electronics.
I personally believe that learning is more enthusiastic when learnt with creativity.
Apart from this, I like to strum Guitar and travel. | {"url":"https://techiescience.com/magnetic-hysteresis-permeability-retentivity/","timestamp":"2024-11-13T18:59:22Z","content_type":"text/html","content_length":"100317","record_id":"<urn:uuid:7233d535-2d7f-457a-8928-1c1d2eaea007>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00393.warc.gz"} |
Excel IFS and SWITCH Function (Say Goodbye to Nested IFs and VLOOKUPS On Small Tables)
Some of the first functions that a newly-titled “power user” learn are IF and VLOOKUP. The IF function provides the user the ability to ask a question and perform action “A” for one answer and action
“B” for another answer. The catch is that any question posed must be answerable as either “true” or “false”.
A simple example would be something like, “Are the contents of cell A1 greater than the contents of cell B1?” If the answer is “yes”, calculate a 10% bonus; if not, then no bonus is offered. This IF
(sometimes referred to as a “simple IF”) has the built-in limitation of only being able to perform a single evaluation. What does one do if several evaluations need to take place?
Enter the Nested IF
The nested IF is simply an IF inside of an IF, which could then be inside another IF. Excel’s IF has a limit of 64 nested levels. This provides the user with up to 65 possible actions to take.
(SIDE NOTE: It is often suggested that if you are writing more than 4 or 5 levels in a nested IF, you should really think about using a different function to accomplish your task, such as VLOOKUP.)
Let’s take the following example of determining letter grades based on test scores. The below table shows the ranges that grades fall into.
Assuming we name the cell containing the final score “score”, the formula to calculate the letter grade would look as follows:
This formula is only determining five (5) possible outcomes, yet it suffers from several problems.
1. Each function asks a variation on the same question about the same source data
2. The length of the formula is becoming excessive
3. The opportunity to make a typo increases exponentially (or geometrically; whatever. It’s big!) the longer the formula becomes.
4. The formula becomes increasingly difficult to keep track of parenthetical pairing.
EXCEL 2016 and New Functions Save the Day
Excel 2016 is equipped with two new functions called IFS and SWITCH that help mitigate all of the above problems associated with multi-level nested IF statements.
Using our current example of number grades to letter grades, we can greatly reduce the length and repetition of the original nested IF by using a new function called IFS. The new IFS function strips
out the need for creating a new IF statement for each new evaluation. Below is the same formula but accomplished with an IFS function.
The new IFS function is not doing anything more or less than the original nested IF, but it performs the task more elegantly due to the removal of the redundant IF functions.
Notice there is only a SINGLE set of parentheses instead of four, or 64 in the worst case scenario.
(SIDE NOTE: The IFS function supports 127 separate evaluations, compared to 64 evaluations when nesting older IF function.)
Let’s Flip the Question
Suppose you know what the letter grades are but you need to discover the test score ranges.
The function to use in this scenario is called SWITCH. The SWITCH function is similar to the IFS function except that it simply looks for an instance of data in a list and makes a corresponding offer
of new data.
Think of it as a VLOOKUP with the table built right into the formula.
In the SWITCH function, a value (or expression) is compared to items in a list. The first matching item in the list will then return a result. Because SWITCH halts the evaluation when it encounters a
matching value, it is recommended that the most likely matching item(s) be placed first in the list. This will reduce the amount of unnecessary comparisons.
Another Use for SWITCH
How about this example?
You have a spreadsheet with a column of dates and you want to determine if a date is occurring today, yesterday, tomorrow, or in the near future or near past. We could write a formula like the
following to calculate the number of days between the defined day and today.
This would produce the following table.
By nesting the above function inside of a SWITCH function, we can convert those numbers into words.
=SWITCH(DAYS(TODAY(),A1),0,”Today”,1,”Yesterday”,-1,”Tomorrow”, “Out of Range”)
Splash a bit of Conditional Formatting color on the list and you now have something like the following.
Sort the list by date in descending order and you have the result below.
Let’s Pair them Up for a Super Function
If we nest the SWITCH function inside the IFS function, we can make it a bit more sophisticated.
The SWITCH function is simply looking for matching items. Suppose we want to introduce a bit of comparison logic into the mix. Let’s say that if an item is greater than one (1) day in the future it
is considered “Pending”, and if it is greater than one (1) day in the past it is considered “Processed”. The days between those points in time are where we want the “Yesterday/Today/Tomorrow” logic.
The IFS function will determine the “Pending/Processed” status while the SWITCH will determine the “Yesterday/Today/Tomorrow” status
=IFS(DAYS(TODAY(),A1)>1,”Processed”,DAYS(TODAY(),A1)<-1,”Pending”,TRUE, SWITCH(DAYS(TODAY(),A1),0,”Today”,1,”Yesterday”,-1,”Tomorrow”)) | {"url":"https://www.bcti.com/2017/02/06/excel-ifs-and-switch-function-say-goodbye-to-nested-ifs-and-vlookups-on-small-tables/","timestamp":"2024-11-10T10:55:39Z","content_type":"text/html","content_length":"71591","record_id":"<urn:uuid:892a56b6-ff99-4689-9319-081530acfbca>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00778.warc.gz"} |
Quantum Mechanics
Strange Ramanujan Summation 1 + 2 + 3 + 4 +….. = -1/12 #Ramanujan #SrinivasaRamanujan #RamanujanSummation #GrandiSeries In this video lecture we will discuss the proof of Ramanujan summation of
natural numbers 1+2+3+4…..=-1/12. Ramanujan wrote a letter to Cambridge mathematician G.H Hardy and in the 11 page letter there were a number of interesting results …
Strange Ramanujan Summation 1 + 2 + 3 + 4 +… … … = -1/12 Read More »
Lecture 4-LASER Basic concept Properties
Lecture 4-LASER Basic concept Properties Lasers are special and different from other light sources due to coherence. Laser beams can be focused to very tiny spots. That’s why the laser gun is
important. They can have very low divergence in order to concentrate their power at a great distance. Laser means light amplification by stimulated …
Evolution of Quantum Physics The nature of light (radiation)
Evolution of Quantum Physics The nature of light (radiation) Evolution of Quantum Physics The nature of light (radiation) has been a matter of long debate and a great
confusion in the history of Physics. Rene Descartes, the father of the Cartesian geometry, first gave the corpuscular (particle) nature of light and the great Newton …
Evolution of Quantum Physics The nature of light (radiation) Read More »
Dual nature of radiation and matter
In 1887, Hertz conducted an experiment. It was showing that when a light beam falls on a metal surface, there was some kind of electrical reaction. But he did not attempt to explain it. Can you
guess, the cause behind this effect? Particle in a Box
Particle in a Box
Particle in a Box We are going to discuss a single particle in a box classical as well as quantum. Classically: Particle confined in a box and walls of the box are completely rigid. So it can not
penetrate the walls of the box. Particle have momenta p so it has kinetic energy E and it …
Quantization of Gravity
Quantization of Gravity When we are talking about quantization of gravity,one question strikes in our mind is that What is the motivation to quantize the gravity and what are the problems physicists
are facing to quantize gravity? What are the attempts to quantize gravity and how much are they successful? Theory of gravity is the … | {"url":"https://education.scienceteen.com/category/physics/quantum-mechanics/","timestamp":"2024-11-13T07:49:03Z","content_type":"text/html","content_length":"180731","record_id":"<urn:uuid:3507c4f7-c5f2-4bbd-941f-57736c9d4657>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00155.warc.gz"} |
Reflected Darkfield Rheinberg Microscopy to 3D Topology
Since the geometry is known in a Disco Lights Illumination system, one can tell a computer which direction each color is coming from. It is fairly straighforward to then assign a value, which I will
call the "slope", to each pixel, deriving from the three sector illumination.
The first step is to load, display and crop an input image.
Then, either select RGB as the color channels, or click on three places in the image as representatives of your three colors. Here, I have red, green and yellow filters, so the yellow channel is
appropriate (yellow is the inverse of blue, coincidentally). Once the three color channels are known, a vector projection from the known RGB values to the custom channels is done.
With knowledge of the geometry, i.e. that the yellow light is coming from -90°, the red from 30° and the green from 150°, a vector decompostion can be done on the three channels, in x- and
We see that the yellow light doesn't contribute to the x-component, as you would expect.
With this information in hand, we can now calculate the x- and y-slopes, averaging the three sector components.
Now that we have the slope maps, integration to determine surface height is done. Here, I use the Inverse FFT method.
And for comparison, here is a stacked image of the same region, but with brightfield illumination.
So far, this image has been the only one that really gave almost reasonable results. Work is continuing on the algorithm, but I will show some of the not-quite-successful attemps, also.
I'm not claiming the heights are correct, or even linear, but this DLI technique seems to show some promise for a single-image technique. Given enough contrast, regular transmitted Rhienberg could
also produce 3D data.
The Matlab programs are available here and here (for the integration). No responsibility is taken for their use or misuse!
I tried to do the same thing with the image of a "D" for Denver on a U.S. nickel, but found that since the slope was so high, we lost all information there at the "vertical" wall, so the unwrapping
to form an image from the "slope" maps failed to create a continuous edge and edge to flat-top transition. It still looks pretty cool in the end, but it is definitely not right.
Following the same steps as before,
It makes pretty nice "slope" maps, which really are more like edge-finders than accurate slope maps, but it fails to make sense of it in the end.
Last updated September 19, 2024. If you arrived via an external link, please visit the homepage for navigation! | {"url":"https://microscope-mike.com/images/DLI-3D.php","timestamp":"2024-11-10T07:48:53Z","content_type":"text/html","content_length":"6021","record_id":"<urn:uuid:432340d9-a78d-4c60-a01e-699be20665db>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00097.warc.gz"} |
mathematics research projects
1. 45 Interesting Mathematics Research Project (Paper) Topics and Ideas
2. (PDF) Student Projects in Pure Mathematics
3. How to Make Research Proposal for Mathematics PhD Project
4. (PDF) PhD Applied Mathematics Research project proposal
5. Mathematics Research Project by kelsey gehlen
6. Mathematics Research Project (20-2)
1. Interesting maths projects- maths exhibit projects- maths fair project
2. The Best End of Year Math Projects
3. Project Based Learning in the Middle School Math Classroom
4. 1st prize winner,maths project in science exhibition 2018-19
5. History of Mathematics Project: Learning Journeys for Kids and Others
6. How to Get a Math Research Position as a Student | {"url":"https://myjudaica.online/essay/mathematics-research-projects","timestamp":"2024-11-09T13:59:33Z","content_type":"text/html","content_length":"19992","record_id":"<urn:uuid:4aeb2900-2acb-4972-b42a-03abb8c5b108>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00814.warc.gz"} |
What would be the force required to accelerate 1 gram to 20% of the speed of light?
Asked by: Homer Connor
It is not just a question of 'how much force' is needed, but rather a combination of a given force for a given length of time. In other words, a small force for a long time can result in the same
velocity as a large force for a short time. This combination of force and time is called IMPULSE, and equals the change in momentum given to any mass. Momentum is simply mass x velocity.
20% of the speed of light is about 6 x 10
meters/second. Since the relativistic effects at that velocity are small (only about 2%), let's ignore them and just find the impulse needed in non-relativistic terms. A velocity increase given to 1
gm from 0 to 6 x 10
m/sec means its momentum would have to change by: 0.001 kg x 6x10
m/sec = 60,000 kg m/sec
So the IMPULSE needed is the equivalent of 60,000 kg m/sec. In the metric system, a NEWTON is 1 kg m/sec
, so any combination of newtons x seconds giving a product of 60,000 would do the job. [The units of newtons x seconds = kg m/sec
x sec = kg m/sec = momentum units]
A force of 60,000 Newtons for 1 second, for example, would provide the impulse needed, as would a force of 1000 Netwons for 60 seconds.
Answered by: Paul Walorski, B.A., Part-time Physics Instructor
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Modelling water for calorimetry of proteinsKnarik Yeritsyan
Artem Badasyan
, 2023, objavljeni povzetek znanstvenega prispevka na konferenci
Opis: Differential Scanning Calorimetry (DSC) is a powerful technique used to study the thermal stability and unfolding of proteins. DSC provides the excess heat capacity profile and is used to study
the thermodynamics of a given protein. By fitting DSC data to the model, researchers can obtain valuable information about the thermodynamics of protein folding and unfolding, which can help them
better understand protein structure, stability, and function. Based on Hamiltonian representation of ZB model and using the solvent effects we derived an expression for heat capacity in proteins and
successfuly fit it to experimental data. As we show, our model provides a better fit to experimental data, as compared to the 2-state model. The model we propose takes into account also water effects
and we show that it fits better to experimental data giving inter- and intra-molecular H-bonding energies instead of reporting only one total enthalpy.
Ključne besede: Zimm-Bragg model, water model, helix-coil transition, protein folding, differential scanning calorimetry
Objavljeno v RUNG: 18.10.2023; Ogledov: 1766; Prenosov: 0
Gradivo ima več datotek! Več...
System size dependence in the Zimm-Bragg model : partition function limits, transition temperature and intervalArtem Badasyan
, 2021, izvirni znanstveni članek
Opis: Within the recently developed Hamiltonian formulation of the Zimm and Bragg model we re-evaluate several size dependent approximations of model partition function. Our size analysis is based on
the comparison of chain length N with the maximal correlation (persistence) length ξ of helical conformation. For the first time we re-derive the partition function of zipper model by taking the
limits of the Zimm–Bragg eigenvalues. The critical consideration of applicability boundaries for the single-sequence (zipper) and the long chain approximations has shown a gap in description for the
range of experimentally relevant chain lengths of 5–10 persistence lengths ξ. Correction to the helicity degree expression is reported. For the exact partition function we have additionally found,
that: at N/ξ ≈ 10 the transition temperature T m reaches its asymptotic behavior of infinite N; the transition interval ∆T needs about a thousand persistence lengths to saturate at its asymptotic,
infinite length value. Obtained results not only contribute to the development of the Zimm–Bragg model, but are also relevant for a wide range of Biotechnologies, including the Biosensing
Ključne besede: Zimm-Bragg model, helix-coil transition, zipper model
Objavljeno v RUNG: 17.06.2021; Ogledov: 3049; Prenosov: 87
Povezava na celotno besedilo
Gradivo ima več datotek! Več...
On spin description of water-biopolymer interactions: theory and experiment of reentrant order-disorder transition.Artem Badasyan
, predavanje na tuji univerzi
Opis: The experimental studies of biopolymer conformations have reached an unprecedented level of detailization during the past decade and allow now to study single molecules in vivo [1]. Processing
of experimental data essentially relies on theoretical approaches to conformational transitions in biopolymers [2]. However, the models that are currently used, originate from the early 1960's and
contain several unjustified assumptions, widely accepted at that time. Thus, the view on the conformational transitions in the polypeptides as a two-state process has very limited applicability
because the all-or-none transition mechanism takes place only in short polypeptides with sizes comparable to the spatial correlation length; the original formulation of Zimm-Bragg model is
phenomenological and does not allow for a microscopic model for water; the implicit consideration of the water-polypeptide interactions through the ansatz about the quadratic dependence of free
energy difference on temperature can only be justified through the assumption of an ideal gas with a constant heat capacity. To get rid of these deficiencies, we augment the Hamiltonian formulation
[3] of the Zimm-Bragg model [4] with the term describing the water-polypeptide interactions [5]. The analytical solution of the model results in a formula, ready to be fit to Circular Dichroism (CD)
data for both heat and cold denaturation. On the example of several sets of experimental data we show, that our formula results in a significantly better fit, as compared to the existing approaches.
Moreover, the application of our procedure allows to compare the strengths of inter- and intra-molecular H-bonds, an information, inaccessible before.
Ključne besede: helix-coil transition, water-polypeptide interactions
Objavljeno v RUNG: 13.03.2019; Ogledov: 3887; Prenosov: 0
Gradivo ima več datotek! Več...
New method to process Circular Dichroism experimental data on heat and cold denaturation of polypeptides in waterArtem Badasyan
Matjaž Valant
, 2018, objavljeni povzetek znanstvenega prispevka na konferenci
Opis: During the past decade the experimental studies of biopolymer conformations have reached an unprecedented level of detailization and allow to study single molecules in vivo [1]. Processing of
experimental data essentially relies on theoretical approaches to conformational transitions in biopolymers [2]. However, the models that are currently used, originate from the early 1960's and
contain several unjustified assumptions, widely accepted at that time. Thus, the view on the conformational transitions in the polypeptides as a two-state process has very limited applicability
because the all-or-none transition mechanism takes place only in short polypeptides with sizes comparable to the spatial correlation length; the original formulation of Zimm-Bragg model is
phenomenological and does not allow for a microscopic model for water; the implicit consideration of the water-polypeptide interactions through the ansatz about the quadratic dependence of free
energy difference on temperature can only be justified through the assumption of an ideal gas with a constant heat capacity. To get rid of these deficiencies, we augment the Hamiltonian formulation
[3] of the Zimm-Bragg model [4] with the term describing the water-polypeptide interactions [5]. The analytical solution of the model results in a formula, ready to be fit to Circular Dichroism (CD)
data for both heat and cold denaturation. On the example of several sets of experimental data we show, that our formula results in a significantly better fit, as compared to the existing approaches.
Moreover, the application of our procedure allows to compare the strengths of inter- and intra-molecular H-bonds, an information, inaccessible before. References [1] I. König, A. Zarrine-Afsar, M.
Aznauryan, A. Soranno, B. Wunderlich, F. Dingfelder, J. C. Stüber, A. Plückthun, D. Nettels, B. Schuler, (2015), Single-molecule spectroscopy of protein conformational dynamics in live eukaryotic
cells/Nature Methods, 12, 773-779. [2] J. Seelig, H.-J. Schönfeld, (2016), Thermal protein unfolding by differential scanning calorimetry and circular dichroism spectroscopy. Two-state model versus
sequential unfolding/Quarterly Reviews of Biophysics, 49, e9, 1-24. [3] A.V. Badasyan, A. Giacometti, Y. Sh. Mamasakhlisov, V. F. Morozov, A. S. Benight, (2010), Microscopic formulation of the
Zimm-Bragg model for the helix-coil transition/Physical Review E, 81, 021921. [4] B. H. Zimm, J. K. Bragg, (1959), Theory of the Phase Transition between Helix and Random Coil in Polypeptide Chains/
Journal of Chemical Physics, 31, 526. [5] A. Badasyan, Sh.A. Tonoyan, A. Giacometti, R. Podgornik, V.A. Parsegian, Y.Sh. Mamasakhlisov, V.F. Morozov, (2014), Unified description of solvent effects in
the helix-coil transition/Physical Review E, 89, 022723. Corresponding author: Artem Badasyan (artem.badasyan@ung.si)
Ključne besede: Biopolymers, Circular Dichroism, Zimm-Bragg model, helix-coil transition.
Objavljeno v RUNG: 22.10.2018; Ogledov: 4802; Prenosov: 0
Gradivo ima več datotek! Več...
Physics behind the Conformational Transitions in Biopolymers. Demystification of DNA melting and Protein FoldingArtem Badasyan
, predavanje na tuji univerzi
Opis: Biophysics is the area of research, devoted to the studies of physical problems related to living systems. Animal cell is the smallest unit of an organism and mainly contains water solutions of
structurally inhomogeneous polymers of biological origin: polypeptides (proteins) and polynucleotides (DNA, RNA). Statistical physics of macromolecules allows to describe the conformations of both
synthetic and bio-polymers and constitutes the basis of Biophysics. During the talk I will report on the biophysical problems I have solved with numerical simulations (Langevin-based Molecular
Dynamics of Go-like protein folding model and Monte Carlo with Wang-Landau sampling) and analytical studies of spin models (formula evaluation by hand, enforced with computer algebra systems). The
direct connections with the theory of phase transitions, algebra of non-commutative operators and decorated spin models will be elucidated.
Ključne besede: Biophysics, protein folding, helix-coil transition, spin models
Objavljeno v RUNG: 13.12.2016; Ogledov: 6709; Prenosov: 0
Gradivo ima več datotek! Več...
Iskanje izvedeno v 0.04 sek. | {"url":"https://repozitorij.ung.si/Iskanje.php?type=napredno&lang=slv&stl0=KljucneBesede&niz0=coil-globule+transition","timestamp":"2024-11-08T08:35:25Z","content_type":"text/html","content_length":"41472","record_id":"<urn:uuid:553ef790-54f9-4f8f-ae8e-3a6a9f3949d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00426.warc.gz"} |
Selberg zeta function
Added pointers to Selberg zeta function for the fact that, under suitable conditions over a 3-manifold, the exponentiated eta function $\exp(i \pi \, \eta_D(0))$ equals the Selberg zeta function of
odd type.
Together with the fact at eta invariant – For manifolds with boundaries this says that the Selberg zeta function of odd type constitutes something like an Atiyah-style TQFT which assigns determinant
lines to surfaces and Selberg zeta functions to 3-manifolds.
This brings me back to that notorious issue of whether to think of arithmetic curves as “really” being 2-dimensional or “really” being 3-dimensional: what is actually more like a Dedekind zeta
function: the Selberg zeta functions of even type or those of odd type?
John Baez kindly points out that the analogy between the Selberg zeta and the Artin L given in the $n$Lab here had been highlighed much in
• Darin Brown, Lifting properties of prime geodesics, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 (euclid)
Page 9 there has a table with all the key ingredients.
Except maybe for one detail: there the analogy is made between number fields and hyperbolic surfaces. Whereas I think now it works a bit better still for hyperbolic 3-manifolds.
Perhaps you could explain the notation $n_\Gamma(g)$?
Right, sorry, I still need to add definition of this and a few other terms. | {"url":"https://nforum.ncatlab.org/discussion/6370/selberg-zeta-function/?Focus=51152","timestamp":"2024-11-02T04:58:59Z","content_type":"application/xhtml+xml","content_length":"43594","record_id":"<urn:uuid:6c16196a-8bc6-47de-ae74-3374221dc0dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00104.warc.gz"} |
Using R code in Julia and Python
As we are moving toward the data analysis part of
our project
, I am trying to teach my working group how to use the stack of R functions I wrote during the past decade. The issue is however that R does not seem to be a popular option among my working group and
that they prefer using either Python or Julia. As translating a decade-worth of code into those two languages is a bit of a bummer, I've been looking at how to read and use R functions in Julia and
Python instead.
Julia has, in particular, a very intuitive and well-done package to achieve this: RCall.jl:
using RCall
R""" #Macro to make operations in an R environment
x <- c(0,1,2)
a <- 2
X = @rget x # moving object from the R environemnt to Julia
y = X .+ randn(3)
@rput y # moving object from julia to the R environment
R"z <- y + rpois(3,2)"
@rget z
# Skipping the @rput step using the dollar sign notation
x = randn(10)
R"y <- mean($x)"
@rget y
# Concrete example using a function I wrote in R to compute Chao1 estimator and its Confidence Interval
chao1 <- function(mat){
a <- sum(mat==1) #Number of singletons
b <- sum(mat==2) #Number of doubletons
S <- sum(mat>0) #Number of species
chao1 <- S+(a^2)/(2*b)
d <- a/b
varS1 <- b*((d/4)^4+d^3+(d/2)^2)
t <- chao1-S
k <- exp(1.96*sqrt(log(1+varS1/t^2)))
c(S+t/k, chao1, S+t*k)
sample = [1, 42, 0, 3, 23, 2, 1, 1, 2, 5, 6, 0, 123, 9] #Made-up sample
res = R"chao1($sample)"
It's very easy to use, and the conversion between Julia and R objects is a no-brainer, even with dataframes. In the following example I take a species-accumulation curve (here) and fit a curve (using
de Caprariis et al. formula) to estimate its asymptote (the results of the function is a list containing the fitted curve as a data.frame and a named vector containing the parameters of the function
along with fit quality measures).
using CSV
using DataFrames
sac = CSV.read("sac.csv", DataFrame, header=1, delim="\t")
caprariis <- function(SAC){
caprariis.misfit <- function(parametres,x){
Smax <- parametres[1]
b <- parametres[2]
FIT <- c()
misfit <- 0
for (i in 1:nrow(x)){
FIT[i] <- (Smax*i)/(b+i)
misfit <- sum(abs(FIT[i]-x[i,2]))+misfit
OPT <- optim(c(50,10),caprariis.misfit,method="BFGS",x=SAC,control=list(trace=1))
Smax <- OPT$par[1]
b <- OPT$par[2]
FIT <- c()
caprar <- list()
misfit <- OPT$value
for (i in 1:nrow(SAC)) FIT[i] <- (Smax*i)/(b+i)
caprar$Curve.fitting <- cbind(SAC,FIT)
colnames(caprar$Curve.fitting) <- c("N","SAC","Fitting")
pearson <- cor(FIT,SAC[,2])
pearson.squared <- pearson^2
all <- c(Smax,b,misfit,pearson,pearson.squared)
names(all) <- c("Smax","b","Misfit","Pearson","Pearson squared")
caprar$Summary <- all
In Python, using rpy2, it's still very easy (though a bit less elegant than with Julia I must say):
import rpy2.robjects as robjects
# As in Julia the object rpy2.robjects.r contains a queryable R environment
chao1 <- function(mat){
a <- sum(mat==1) #Number of singletons
b <- sum(mat==2) #Number of doubletons
S <- sum(mat>0) #Number of species
chao1 <- S+(a^2)/(2*b)
d <- a/b
varS1 <- b*((d/4)^4+d^3+(d/2)^2)
t <- chao1-S
k <- exp(1.96*sqrt(log(1+varS1/t^2)))
#Or read directly from the script
sample = [1, 42, 0, 3, 23, 2, 1, 1, 2, 5, 6, 0, 123, 9]
r_sample = robjects.IntVector(sample) #The conversion in python needs to be explicit
chao1 = robjects.r['chao1']
The explicit conversion is a bit annoying, however when using pandas and dataframes, one can bypass that and make it way easier:
import pandas
from rpy2.robjects import pandas2ri
cap = robjects.r['caprariis']
sac = pandas.read_csv("sac.csv", delimiter="\t", header=1)
res = cap(sac)
In the other direction, I have been using package reticulate to use Python code in R. Maybe I'll also write a post one day about it. Anyway I'm glad to see all my legacy code can still be used even
by non-R-programmers. | {"url":"https://plannapus.github.io/blog/2021-03-13.html","timestamp":"2024-11-05T22:35:24Z","content_type":"text/html","content_length":"8484","record_id":"<urn:uuid:7632d2bd-f9c0-47d6-a5b1-5f9d9648f21b>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00586.warc.gz"} |
Benchmarking quantum devices
Some of the largest quantum devices today have reached the scale of more than 50 qubits, which heralds a new era with quantum computers whose quantum states may be intractable to simulate by any
classical means. If that happens, how do we know that the quantum computer is doing what we think it should be doing? In a typical approach, the confidence that any algorithm will be executed with
high fidelity is extrapolated from the accuracy of its constituent single-qubit and two-qubit gates.
For near-term devices with noisy qubits, however, this vision turns out to work only partially in practice. Quantum errors are devious and diverse in nature. Sometimes they destructively interfere,
canceling each other out making the theoretical prediction overly pessimistic. Sometimes they act on multiple qubits and are beyond existing benchmarking techniques. They can also change over time,
making it ever so difficult to keep track of the impact of error in a quantum circuit. These aspects render many of the current benchmarking techniques unscalable with respect to the number of
qubits. A quantum computer with 50+ qubits, for example, is already out of reach for these techniques. Beyond 50-60 qubits, the cost of simulating the experiments classically becomes prohibitive,
and it is unlikely that we will have classical simulations available as a comparison. How can we find out if it is possible to extract useful computations out of devices beyond supremacy that cannot
be simulated?
We firmly believe that as physical qubits and quantum error correction technology develops and matures, the challenges posed by noise and error will be gradually and systematically mitigated over
time. At Zapata, we aim at pushing forward what can be done with quantum devices today. What that means is that on one hand, we cannot grant ourselves the naivety to ignore the above-mentioned
subtleties with quantum error on physical hardware. On the other hand, we are developing algorithms tailored for specific applications such as quantum simulation. When these two aspects come
together, it seems clear that we need a new way of benchmarking quantum devices that is both scalable with respect to the number of qubits and relevant to the application problems that we are
interested in solving.
This brings us to the recent paper that we released on the arXiv, where we propose an example that embodies this new way of benchmarking quantum devices (“application benchmark”). Here we focus on
fermionic simulation, which for many good reasons is a promising application of quantum computers. Using the Sycamore quantum processor produced by Google as an example, we aim at using the native
ability of a quantum device for solving an exact model (1D Fermi-Hubbard model) whose analytical solution is known. The 1D Fermi-Hubbard model is a prototype for systems with correlated electrons.
For the purpose of benchmarking quantum algorithms, it is also interesting that the model sits at the frontier of what can be simulated efficiently with a classical computer, as it gives us access to
complexity knobs that can easily be tuned. For example, the interaction can be set to zero to obtain a very easy classical problem, but we can also glue the chain in a 2D structure and obtain a
difficult quantum problem.
The particular metric of interest in our application benchmark is what we call the effective fermionic length of the device. This metric tells us the longest chain of the 1D that can be simulated in
the device before noise dominates. It is a quantity that can be directly measured on a given quantum device, and we show that it can be efficiently measured even for cases where the number of qubits
far exceeds what can be simulated classically (e.g., hundreds of qubits). We chose the 1D Fermi-Hubbard model because it is exactly solvable but also describes some of the physics of correlated
systems and is thus representative of chemistry and materials. In other words, it provides a sense of how big of a fermionic system can be simulated on a quantum device. This metric, like quantum
volume, is global and reflective of the entire device, including its noise channels, qubit connectivity, etc.
There are many interesting technical details to be discussed regarding fermionic length benchmarks in general as well as the proposal described in this paper. These details often change from one
hardware system (superconducting qubits, trapped ions, etc.) to another. Together with partners in the quantum ecosystem, we look forward to diving deeper into this new territory of application
We will be hosting several videoconferences to go deeper on our methods, what we learned, and how we used our platform, Orquestra^®, to scale the experiment. For anyone interested in learning more,
please reach out to us. | {"url":"https://zapata.ai/benchmarking-quantum-devices/","timestamp":"2024-11-14T08:33:23Z","content_type":"text/html","content_length":"62256","record_id":"<urn:uuid:4634925e-5b24-42b9-a129-0f37cb970231>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00026.warc.gz"} |
How many of you love engineering mathematics?
During my engineering days, engineering mathematics was one of the 'most difficult' subjects a student had to crack. The subject was 'boring' and 'tough' at the same time making it too difficult in
But when I look back at engineering studies a decade after; it looks like engineering mathematics was one of the most interesting subjects we've ever had. Just that we were never told how important
it is and how cool it is - even though majority of us will never use it directly in their lives.
Vote on the poll and tell everyone in the comments your thoughts about engineering mathematics. | {"url":"https://www.crazyengineers.com/threads/how-many-of-you-love-engineering-mathematics.58073","timestamp":"2024-11-10T05:49:23Z","content_type":"text/html","content_length":"151847","record_id":"<urn:uuid:688ce1c6-7faf-4ac2-91a6-cc90f662a4de>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00331.warc.gz"} |
Relative entropic uncertainty relation
SciPost Submission Page
Relative entropic uncertainty relation for scalar quantum fields
by Stefan Floerchinger, Tobias Haas, Markus Schröfl
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Tobias Haas
Submission information
Preprint Link: https://arxiv.org/abs/2107.07824v3 (pdf)
Date accepted: 2022-01-24
Date submitted: 2021-12-21 10:18
Submitted by: Haas, Tobias
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
• High-Energy Physics - Theory
Specialties: • Quantum Physics
Approach: Theoretical
Entropic uncertainty is a well-known concept to formulate uncertainty relations for continuous variable quantum systems with finitely many degrees of freedom. Typically, the bounds of such relations
scale with the number of oscillator modes, preventing a straight-forward generalization to quantum field theories. In this work, we overcome this difficulty by introducing the notion of a functional
relative entropy and show that it has a meaningful field theory limit. We present the first entropic uncertainty relation for a scalar quantum field theory and exemplify its behavior by considering
few particle excitations and the thermal state. Also, we show that the relation implies the multidimensional Heisenberg uncertainty relation.
Author comments upon resubmission
We thank the referees for carefully reading our manuscript and their very constructive reports.
In the following, we will address the critique of the second referee. We respond to every item in 'Requested changes' of both reports below 'List of changes'.
• Concerning 'result is hidden in the text':
As this was also pointed out by the first referee, we added a two paragraphs in the introduction anticipating the main results and their ranges of validity.
• Concerning 'motivating the use of relative entropy':
To motivate better the use of relative entropy, we devoted a new paragraph in the introduction to relative entropy. We argue that relative entropy is more universal than entropy in the sense that it
its properties are the same in many occasions. Also, we reformulated the paragraph below eq. (49) and added a sentence that the functional relative entropy may possibly be defined directly in the
continuum theory. Additionally, we have divided subsection III. A. into A. and B. to prevent the appearance of an overfull subsection.
However, we think that our functional relative entropy does not coincide with the quantum relative entropy, which is what follows from the Araki formula. Also, we understand the von Neumann entropy
as the quantum entropy associated with the density operator. In this sense, we believe that entropic uncertainty relations are not formulated in terms of von Neumann entropies, but rather in terms of
the classical entropies, i.e. Shannon entropies (for discrete variables) or differential entropies (for continuous variables).
• Concerning 'state-dependence of the bound':
We agree that many well-known (entropic) uncertainty relations have a state-independent bound. However, there are also many bounds which are state-dependent and especially nowadays some
state-dependent bounds have important applications when it comes to entanglement witnessing (cf. Ref. [5], eq. (331) therein). For example, the famous Robertson relation is formulated in terms of an
expectation value in the right hand side. The state-independence of the derived Heisenberg-Kennard relation (eq. (1)) is rather a consequence of the fact that the commutation relation between
position and momentum gives a c-number. If one considers spins equipped with an SU(2) algebra instead, the bound becomes explicitly state-dependent.
Also for entropic uncertainty relations state-dependent bounds are known. For example, for position and momentum an alternative bound to eq. (2) is given by ln 2 \pi + S(\rho), where S(\rho) denotes
the von Neumann entropy of the quantum state \rho (cf. Ref. [29] or Ref. [5] and eq. (267) therein). For discrete variables, the Maassen-Uffink relation has also been improved by adding S(\rho) to
the bound (cf. Ref. [5], eq. (47) therein). Let us point out that state-dependent bounds are often tighter. In fact, as variance as well as entropy are concave, the (entropic) uncertainty should
become minimal for pure states. Hence, one may tighten a pure-state bound by adding quantities which measure the mixedness of the quantum state.
However, state-dependence of a bound can also arise from a reformulation and is not necessarily related to tightness. In particular, the state-dependence of our bound does not mean that it is tighter
than the BBM relation for a single mode. It is rather a consequence of using relative entropies instead of entropies. In fact, the uncertainty deficit, i.e. the difference between the left hand side
and the right hand side, agrees with the uncertainty deficit of the BBM relation.
• Concerning 'in which sense the REUR expresses the uncertainty principle':
We thank the referee in particular for requiring are more detailed statement regarding how the relative entropic uncertainty relation expresses the uncertainty principle. We have restructured section
III C. (before: section III B.) by dividing it into C. (Deriving the relative entropic uncertainty relation) and D. (Discussion of the relative entropic uncertainty relation). While the new
subsection III. C. remains unchanged, we have extended D. by a new paragraph. Therein, we argue that the sum of considered relative entropies is not bounded in a classical theory, allowing us to
conclude that the bound is purely of quantum origin.
• Concerning 'relation between entropic uncertainty relation and Robertson-Schrödinger relation':
As correctly pointed out by the referee, the REUR (or equivalently the BBM relation for finitely many modes) is an improvement of the variance-based formulation in the sense of eq. (70). We would
like to point out that this has been studied in Ref. [4] (around eq. (46) therein) in detail and allows for the interpretation that the bound in eq. (1) is lifted by an exponentiated sum of relative
entropies showing that that the entropic uncertainty relation eq. (1) is tighter than the variance-based relation eq. (2) whenever the distributions are non-Gaussian. This again shows that
state-dependence of a bound may arise through a reformulation of an uncertainty relation.
We want to thank both referees again for their constructive criticism which helped us to improve the manuscript further. For the referees' convenience, we have attached a latexdiff pdf which
highlights all the changes made since submission. We believe that it is now ready for publication in SciPost Physics.
List of changes
To first referee:
1- Page 5, after Eq. 39: "...with the thermal covariance being proportional to the vacuum one". This sentence does not really make sense; the two matrices are not proportional, since the "thermal"
factor $(1+2n_{BE}(\omega_l))$ depends on the index. Maybe "...with the thermal covariance given by..." would be better.
Yes, we implemented the requested change.
2- Page 5, just before Eq. 42. "For a free theory in an equilibrium state, the mixed correlations... ... vanish". If I am right for "equilibrium state" the author means "invariant under time
reversal", which would imply the relation $< \phi \pi +\pi \phi> =0$. The sentence is probably too concise to be understood at first glance and at least a reference for that is needed.
The requirement $< \phi \pi +\pi \phi> =0$ is actually not needed for the argument we had in mind and hence we dropped it. To be a bit more precise: In a free scalar field theory, the covariance
matrix in phase space is symmetric and hence can be block-diagonalized (by rotations in phase space). For some states, this is already the case in the canonical basis given by \phi and \pi, including
all examples we consider in our work. Now, there exist two second-moment based uncertainty relations (see the discussion in Ref. [59]). One is the Robertson-Schrödinger relation, expressed in terms
of the phase space covariance matrix, and one is the multidimensional generalization of the Heisenberg relation eq. (1), which is a special case of the former if the fields are rotated such that the
mixed correlations vanish. Otherwise, the latter is weaker than the former. As our REUR is only capable of implying the multidimensional Heisenberg relation, we changed "Robertson-Schrödinger" to
"multidimensional Heisenberg" everywhere. In fact, it is an open problem to find an EUR which implies the general Robertson-Schrödinger relation even for a single oscillator (see again Ref. [59]).
3 - Page 5, just after Eq. 42. "the eigenvalues of the correlator
product MN are at least the eigenvalues...". I do not really understand this sentence. In the ground state \bar{M}\bar{N} =1/4, so the only eigenvalue is 1/4; what does it means that the eigenvalues
of MN are also eigenvalues of \bar{M}\bar{N}?
The important point is that the eigenvalues of the correlator product MN are bounded from below. We have clarified this now by adjusting the corresponding sentence.
4 - The conclusions of the work are hidden. The Eq. 50 is considered as the main result of the work, and it would be better to anticipate it in the introduction or emphasize it again in the
As this was also requested by the second referee, we devoted a new paragraph in the introduction to the main result. In particular, we emphasized the REUR itself, that it holds for oscillators as
well as fields and that that the considered sum of relative entropies is non-trivially bounded from above by the uncertainty principle.
5- It is not particularly clear, at least at first reading, what is the regime of validity of the results. While the Heisenberg relation or the "Bialynicki-Birula and Mycielski" formula are generic,
the focus of the manuscript appears to be restricted to certain class of states obtained as eigenstates or thermal states of the Klein Gordon theory. An improvement of the presentation would be the
insertion of a precise statement (in the introduction) regarding the range of validity of certain inequalities.
We included a new paragraph at the end of the introduction clarifying that we focus on theories where the field and the conjugate momentum field fulfill a bosonic commutation relation and where the
vacuum is Gaussian (i.e. free theories).
6 - I would also suggest emphasizing more the connection with a previous work of the authors regarding the same topic (Relative entropic uncertainty relation, ref. [31]). Probably even a small
comment/appendix summarizing the results available for a single random variable would be helpful.
Following up on a suggestion of the second referee, we introduced the discrete and continuous relative entropies in the introduction in eqs. (3) and (4) (which is why all subsequent equation numbers
are shifted by two compared to the previous version) and argued why relative entropy may be preferred over entropy. Thereupon, we summarized the main result of our previous work, i.e. that we found
an entropic uncertainty relation which holds for discrete as well as continuous variables formulated in terms of relative entropy.
To second referee:
1. The sum in equation (3) would run from 1 to N to be compatible with the boundary condition $\phi_0 = \phi_n$.
Thanks, we corrected this mistake.
2. In what sense eq. (5) is a unitary transformation? If I understood correctly, up this point the computation is in classical field theory. The quantization comes later.
Both $\tilde{\phi}$ and $\tilde{\pi}$ are $N$-dimensional vectors with complex entries. Eq. (7) describes the action of an unitary $N \times N$ matrix on this vector, written in terms of their
components. In this sense, this transformation is not related to quantization. It is a unitary transformation of the classical fields just as the discrete Fourier transform performed in eq. (6).
3. In eq. (6), should it say \tilde{\phi} and \tilde{\pi}?
No, in this case we would need absolute values of the tilde fields as they are complex-valued. For reasons of convenience, the goal was to write the Hamiltonian in terms of real fields. This is why
we implemented the additional unitary transformation in eq. (7). Please note that the transformation in eq. (7) keeps the Hamiltonian diagonal.
4. In eq. (8), the fields are written in capital letters contrary to what happened before. Presumably this change was made for the purpose of differentiating classical from quantum variables. The
authors should stress that in the text.
This choice of notation was clarified already in the "Notation" paragraph at the end of the introduction. However, we added "hermitian quantum field operators" above eq. (10) to avoid
5. For eqs. (14) and (15), it should be emphasized that $\phi_l$ and $\pi_l$ are real numbers, and that the basis are orthonormal.
Following the request, we added "orthonormal" above eq. (16) and a half sentence below eq. (17).
6. I would like to ask the authors if they can provide a reference for equations (36-37). Otherwise, if it is a relation found by them, I would like to ask them about the relevant points in the
calculation of such a relation.
We use the creation operators defined in eq. (36) and act with them on the vacuum wave functional in eq. (37) to obtain eq. (38) with the Hermite polynomials in eq. (39). The same strategy is used in
Ref. [51]. Hence, we added this reference in the first paragraph of the subsection. One can find such calculations in chapter 10 of Ref. [51] for one excitation and in chapter 11 for multiple
7. From (50), can it be inferred that if a state has the same two-point functions as a coherent state, then it must be coherent?
This is true. Let us emphasize that we need that both M and N agree with their vacuum counterparts to conclude that the corresponding state is coherent. If only one of them is set to the vacuum
expression, this is not the case anymore. In our opinion, this is worth mentioning, and consequently we added a sentence in the last paragraph of section III. D. under bullet point 'b. Sum of
relative entropies is bounded from above'.
8. It is not entirely clear to me what the authors are saying in the first paragraph of the 2nd column on page 9 (the one that starts with: “We begin by reformulating ...”). If one chooses a state
that has the same two-point correlators as the reference state, then the the r.h.s. of (50) is identically zero. Is that correct?
We have added a few words in this paragraph, which clarify that the REUR in eq. (52) has a different left hand side than the relation we derive afterwards. Nevertheless, it is always true, for both
formulations eq. (52) and eq. (67), that the bound is zero whenever one chooses a state which has the same two-point functions as the reference state. However, the physical meaning of this choice
depends on the choice of reference state, which is the crucial difference between eqs. (52) and (67).
Published as SciPost Phys. 12, 089 (2022)
Reports on this Submission
Report #2 by Anonymous (Referee 4) on 2021-12-23 (Invited Report)
• Cite as: Anonymous, Report on arXiv:2107.07824v3, delivered 2021-12-23, doi: 10.21468/SciPost.Report.4088
Adding to the strengths that I pointed out in the first submission, I would like to add:
1. The anticipation of the the result in the introduction, with an adequate explanation and the appropriate reference to the equation in the main text below.
2. The explanation in the introduction about how the relative entropy is a good measure of uncertainty, and why it is convenient to use it in QFT.
The weaknesses that I pointed out in the first submission of the manuscript were correctly addressed by the authors, where they made the corresponding changes.
I would like to thank the authors for clarifying my concerns and improving the manuscript. Now I agree with the authors that regardless the bound coming from the REUR is state dependent, it still
describes the degree of uncertainty of the underlying state. In these lines, I appreciate that the authors strength the quantum origin of this bound as the did in section D. I recommend that this
version of the manuscript be published in the journal.
Requested changes
No further changes are requested.
We have added a difflatex file for the referee's convenience. | {"url":"https://scipost.org/submissions/2107.07824v3/","timestamp":"2024-11-05T15:41:53Z","content_type":"text/html","content_length":"51215","record_id":"<urn:uuid:e96ae9e8-7cc0-4d6b-811a-ff144832f53b>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00143.warc.gz"} |
Geometry: Angle Relationships with Algebra
Kids practice angle relationships and algebra. This worksheet allows students to practice identifying and applying angle relationships, such as complementary, supplementary, vertical, and adjacent.
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[Solved] The maximum length of arc of contact for two mating gea
The maximum length of arc of contact for two mating gears to avoid interference is
(where, r = pitch circle radius of pinion
R = pitch circle radius of gear
ϕ = pressure angle)
Answer (Detailed Solution Below)
Option 1 : (R + r) tanϕ
UPPSC AE Mechanical 2019 Official Paper I (Held on 13 Dec 2020)
1.8 K Users
125 Questions 375 Marks 150 Mins
Interference in Gears:
• The figure shows a pinion with centre A, in mesh with wheel (gear) with centre B.
• FE is the common tangent to the base circle and CD is the path of contact between the two mating teeth.
• A little consideration will show that if the radius of the addendum circle of the wheel is increased to BE, the point of contact C will shift from C to E.
• When the radius is further increased, the point of contact C will be on the inside of the base circle of pinion and not on the involute profile of the pinion.
• In this case, the tip of the tooth of the wheel will then undercut the tooth on the pinion at its root.
• The phenomenon when the tip of tooth undercuts the root on its mating gear is known as interference.
• Interference in gear meshing is undesirable as removal of portions of the involute profile adjacent to the base circle may result in a serious reduction in the length of action.
Interference may only be prevented if the addendum circles of the two mating gears cut the common tangent to the base circle between the point of tangency.
For this, the path of contact CD must lie in the common tangent.
Path of Contact (CD):
Locus of the point of contact on common tangent between mating teeth from the beginning to the end of engagement is known as the path of contact.
\({\rm{CD}} = \sqrt {{\rm{R}}_{\rm{a}}^2 - {{\rm{R}}^2}{{\cos }^2}\phi } \;+ \sqrt {{\rm{r}}_{\rm{a}}^2 - {{\rm{r}}^2}{{\cos }^2}\phi } \; - \left( {{\rm{R}} + {\rm{r}}} \right)\sin \phi \)
To avoid interference, the maximum path of contact (CD) should be EF as shown in the figure.
∴ EF = R sin ϕ + r sin ϕ ⇒ (R + r) sin ϕ
Arc of Contact:
Locus of the point of contact on the pitch circle from the beginning to the end of engagement of two mating gears is known as arc of contact.
\({\rm{Arc\;of\;Contact}} = \frac{{{\rm{Path\;of\;contact}}}}{{\cos \phi }}\)
To avoid interference, the maximum arc of contact should be –
\({\left( {{\rm{Arc\;of\;Contact}}} \right)_{{\rm{max}}}} = \frac{{{{\left( {{\rm{Path\;of\;contact}}} \right)}_{{\rm{max}}}}}}{{\cos \phi }}\)
\( \Rightarrow {\left( {{\rm{Arc\;of\;Contact}}} \right)_{{\rm{max}}}} = \frac{{\left( {{\rm{R}} + {\rm{r}}} \right)\sin \phi }}{{\cos \phi }}\)
⇒ (R + r) tan ϕ
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DFT of a signal and system
8 years ago
●13 replies●
latest reply 8 years ago
401 views
I would like to know if there is any difference between the two purely from a DFT computation:
a) DFT of a signal
b) DFT of an impulse response of a LTI system
I assume both give insight into the frequency components (in case of a) and frequency response.
Appreciate comments from forum members
[ - ]
Reply by ●May 7, 2017
Nope, it's the same DFT, whether you are applying it to a signal or an impulse response.
[ - ]
Reply by ●May 7, 2017
[ - ]
Reply by ●May 7, 2017
Yeah, it stems from the continuous system H(s): the transfer function H(s) is the Laplace transform of the impulse response, and H(jw) is the Fourier transform of the impulse response.
[ - ]
Reply by ●May 7, 2017
Hi. Just to add my 'two cents': A discrete sequence has a DFT (a discrete spectrum). A system (such as a digital filter, differentiator, Hilbert transformer, etc.) has a frequency response. To
avoid confusion, the words "spectrum" and "frequency response" should never be used interchangably.
[ - ]
Reply by ●May 7, 2017
Thanks. I did not realize this.
[ - ]
Reply by ●May 7, 2017
One issue is that you are free to window a signal vector to avoid sharp phase discontinuity and so get rid of false high frequencies but you should not window an impulse response when you analyse
frequency response of a given system.
[ - ]
Reply by ●May 7, 2017
Ok. So, typically windowing is applied to incoming samples before they are passed to filer? Though in practical designs, I have not seen anyone implementing window function. I am wondering why ...
[ - ]
Reply by ●May 7, 2017
when you filter a signal you just filter it. Your question is not applicable.
My point of windowing before DFT applies if you want to assess frequency spectrum of a signal...
[ - ]
Reply by ●May 7, 2017
[ - ]
Reply by ●May 7, 2017
Windowing is applied to signals before you do a DFT for spectral analysis. It is not, in general, applied before filtering.
[ - ]
Reply by ●May 7, 2017
[ - ]
Reply by ●May 7, 2017
Since everyone's diving in:
It is the same DFT. It is the same because the impulse response of a linear, time-invariant system contains all the information that can ever be about that system (note that it doesn't necessarily
tell you everything about some real system, because they're never completely linear or time invariant -- that's a screed for another day).
[ - ]
Reply by ●May 7, 2017
the calculation would be the same, but there may be times when you would prefer to scale the result so that the data corresponds to a value you are trying to measure, such as the signal RMS of a
frequency bin in a power spectrum. this isn't functionally different than just computing the DFT, but it can be useful when graphing the spectrum of a signal | {"url":"https://dsprelated.com/thread/2866/dft-of-a-signal-and-system","timestamp":"2024-11-13T08:30:20Z","content_type":"text/html","content_length":"50703","record_id":"<urn:uuid:d169b854-0c5b-48a6-8284-2d18cbe31bea>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00437.warc.gz"} |
How to Convert A Pandas Dataframe to A Python Tensor?
To convert a Pandas dataframe to a Python tensor, you can use the functions provided by the NumPy library. The following steps can be followed:
1. Import the required libraries:
1 import pandas as pd
2 import numpy as np
1. Create a Pandas dataframe:
1 data = {'Column1': [1, 2, 3, 4],
2 'Column2': [5, 6, 7, 8],
3 'Column3': [9, 10, 11, 12]}
4 df = pd.DataFrame(data)
1. Convert the dataframe to a NumPy array:
1. Convert the NumPy array to a Python tensor:
1 tensor = np.array(array, dtype=np.float32) # You can specify the desired dtype if needed
The resulting 'tensor' variable will be a Python tensor representation of the original Pandas dataframe. This conversion enables you to utilize the tensor for further numerical computations, such as
using machine learning libraries that require tensor inputs.
How to pivot a Pandas dataframe?
To pivot a Pandas DataFrame, you can use the pivot() function. Here's an example of how to pivot a DataFrame:
1 import pandas as pd
3 # Create a sample DataFrame
4 data = {'Date': ['2020-01-01', '2020-01-02', '2020-01-01', '2020-01-03', '2020-01-01', '2020-01-02'],
5 'Category': ['A', 'B', 'A', 'B', 'A', 'B'],
6 'Value': [10, 20, 30, 40, 50, 60]}
7 df = pd.DataFrame(data)
9 # Pivot the DataFrame
10 pivot_df = df.pivot(index='Date', columns='Category', values='Value')
12 print(pivot_df)
1 Category A B
2 Date
3 2020-01-01 10 60
4 2020-01-02 20 20
5 2020-01-03 NaN 40
In the above example, the pivot() function is used with the following arguments:
• index: The column to use as the row index in the pivoted DataFrame ('Date' in this case).
• columns: The column to use as the column index in the pivoted DataFrame ('Category' in this case).
• values: The column to be used for populating the values of the pivoted DataFrame ('Value' in this case).
Note that if there are any duplicate entries for specific index-column combinations, the pivot() function will raise a ValueError. In such cases, you may need to handle duplicates before pivoting the
What is the dot product in a Python tensor?
The dot product in a Python tensor is a mathematical operation between two tensors that returns a scalar. It is performed using the dot() function or the @ operator.
Here's an example of how to compute the dot product using numpy:
1 import numpy as np
3 # Create two tensors
4 tensor1 = np.array([1, 2, 3])
5 tensor2 = np.array([4, 5, 6])
7 # Compute the dot product
8 dot_product = np.dot(tensor1, tensor2)
10 # Display the result
11 print(dot_product)
The output will be:
In this case, the dot product of [1, 2, 3] and [4, 5, 6] is 32.
How to convert the data type in a Pandas dataframe?
To convert the data type in a Pandas dataframe, you can use the astype() method or the to_numeric(), to_datetime(), and to_timedelta() functions. Here's an overview of how to use these methods/
1. astype() method: This method allows you to convert the data type of one or more columns in a dataframe. It returns a new dataframe with the specified data type. Syntax: df['column_name'] = df
['column_name'].astype('new_data_type') Example: import pandas as pd # Create a dataframe df = pd.DataFrame({ 'col1': [1, 2, 3], 'col2': ['A', 'B', 'C'] }) # Convert 'col1' from int to float df
['col1'] = df['col1'].astype(float)
2. to_numeric() function: This function is used to convert a column to numeric data type. It can handle the conversion of strings to numeric values by replacing non-numeric values with NaN (Not a
Number). Syntax: df['column_name'] = pd.to_numeric(df['column_name'], errors='coerce') Example: import pandas as pd # Create a dataframe df = pd.DataFrame({ 'col1': [1, 2, '3'], 'col2': ['A',
'B', 'C'] }) # Convert 'col1' to numeric data type df['col1'] = pd.to_numeric(df['col1'], errors='coerce')
3. to_datetime() function: This function converts a column to a datetime data type. Syntax: df['column_name'] = pd.to_datetime(df['column_name']) Example: import pandas as pd # Create a dataframe df
= pd.DataFrame({ 'date': ['2021-01-01', '2021-02-01', '2021-03-01'], 'value': [1, 2, 3] }) # Convert 'date' to datetime data type df['date'] = pd.to_datetime(df['date'])
4. to_timedelta() function: This function is used to convert a column to a timedelta data type. Syntax: df['column_name'] = pd.to_timedelta(df['column_name']) Example: import pandas as pd # Create a
dataframe df = pd.DataFrame({ 'delta': ['1 days', '2 days', '3 days'], 'value': [1, 2, 3] }) # Convert 'delta' to timedelta data type df['delta'] = pd.to_timedelta(df['delta'])
These methods/functions can be used to convert the data type of one or multiple columns in a Pandas dataframe.
What is the mean value in a Python tensor?
In Python, a tensor is a multidimensional array or a list of lists. The mean value of a tensor is calculated by adding up all the elements in the tensor and then dividing the sum by the total number
of elements.
Here is an example of calculating the mean value of a tensor using the NumPy library in Python:
1 import numpy as np
3 # Create a tensor
4 tensor = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
6 # Calculate the mean value
7 mean_value = np.mean(tensor)
9 print(mean_value)
In this example, the tensor has a mean value of 5.0.
What is the median of a Python tensor?
In Python, the median of a tensor can be calculated using the numpy library. The numpy.median() function can be used to compute the median along a specified axis. Here is an example:
1 import numpy as np
3 # Create a tensor
4 tensor = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
6 # Compute the median along the first axis (rows)
7 median = np.median(tensor, axis=0)
9 print(median)
In this example, the median is computed along the rows (axis=0), resulting in a 1D array containing the median value for each column. If you want to calculate the overall median of the entire tensor,
you can omit the axis parameter or set it to None.
What is the minimum value in a Python tensor?
In Python, a tensor is typically represented using a NumPy array. To find the minimum value in a NumPy tensor, you can use the min() function from the NumPy package.
Here is an example:
1 import numpy as np
3 tensor = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
5 minimum_value = np.min(tensor)
7 print(minimum_value)
In this example, the minimum value in the tensor is 1. | {"url":"https://topminisite.com/blog/how-to-convert-a-pandas-dataframe-to-a-python","timestamp":"2024-11-06T14:12:12Z","content_type":"text/html","content_length":"408915","record_id":"<urn:uuid:f0d82a47-e0a7-4b5a-99cd-f22d0033f6ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00015.warc.gz"} |
Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are math expressions which consist of one or several terms, all of which has a variable raised to a power. Dividing polynomials is an essential working in algebra that includes finding
the remainder and quotient as soon as one polynomial is divided by another. In this article, we will explore the different approaches of dividing polynomials, consisting of long division and
synthetic division, and provide instances of how to utilize them.
We will further talk about the significance of dividing polynomials and its uses in different domains of mathematics.
Prominence of Dividing Polynomials
Dividing polynomials is a crucial operation in algebra that has several uses in various domains of mathematics, including number theory, calculus, and abstract algebra. It is applied to solve a broad
spectrum of problems, including figuring out the roots of polynomial equations, calculating limits of functions, and solving differential equations.
In calculus, dividing polynomials is used to work out the derivative of a function, that is the rate of change of the function at any moment. The quotient rule of differentiation includes 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 applied to learn the features of prime numbers and to factorize large numbers into their prime factors. It is further applied to learn algebraic structures
for example rings and fields, which are basic concepts in abstract algebra.
In abstract algebra, dividing polynomials is used to specify polynomial rings, that are algebraic structures that generalize the arithmetic of polynomials. Polynomial rings are utilized in various
domains of mathematics, involving algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is an approach of dividing polynomials that is applied to divide a polynomial by a linear factor of the form (x - c), at point which c is a constant. The method is founded on the
fact that if f(x) is a polynomial of degree n, therefore 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 consists of writing the coefficients of the polynomial in a row, applying the constant as the divisor, and carrying out a sequence of workings to work out the
quotient and remainder. The answer is a streamlined form of the polynomial which is easier to function with.
Long Division
Long division is a technique of dividing polynomials which is utilized to divide a polynomial by another polynomial. The method is relying on the fact that if f(x) is a polynomial of degree n, and g
(x) is a polynomial of degree m, at which point m ≤ n, then the division of f(x) by g(x) offers uf 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 with the highest degree term of the divisor, and subsequently multiplying the result by the whole divisor. The
answer is subtracted of the dividend to obtain the remainder. The process is recurring until the degree of the remainder is lower than the degree of the divisor.
Examples of Dividing Polynomials
Here are few examples of dividing polynomial expressions:
Example 1: Synthetic Division
Let's say we need to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 by the linear factor (x - 1). We could utilize synthetic division to streamline the expression:
1 | 3 4 -5 2 | 3 7 2 |---------- 3 7 2 4
The result of the synthetic division is the quotient polynomial 3x^2 + 7x + 2 and the remainder 4. Therefore, 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 assume 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 could apply long division to streamline the expression:
First, we divide the highest degree term of the dividend with the highest degree term of the divisor to get:
Then, we multiply the whole divisor by the quotient term, 6x^2, to get:
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)
which streamlines to:
7x^3 - 4x^2 + 9x + 3
We recur the procedure, dividing the largest degree term of the new dividend, 7x^3, with the highest degree term of the divisor, x^2, to get:
Subsequently, we multiply the entire divisor with the quotient term, 7x, to obtain:
7x^3 - 14x^2 + 7x
We subtract this of the new dividend to obtain the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
that simplifies to:
10x^2 + 2x + 3
We recur the procedure again, dividing the largest degree term of the new dividend, 10x^2, with the highest degree term of the divisor, x^2, to achieve:
Subsequently, we multiply the whole divisor with the quotient term, 10, to obtain:
10x^2 - 20x + 10
We subtract this of the new dividend to obtain the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
which simplifies to:
13x - 10
Hence, the answer of the long division is the quotient polynomial 6x^2 - 7x + 9 and the remainder 13x - 10. We could express f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
Ultimately, dividing polynomials is an essential operation in algebra that has several utilized in numerous fields of mathematics. Getting a grasp of the various approaches of dividing polynomials,
for instance long division and synthetic division, could support in solving 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 ideas of dividing polynomials is essential.
If you require support understanding dividing polynomials or anything related to algebraic concept, consider connecting with us at Grade Potential Tutoring. Our experienced tutors are available
remotely or in-person to offer personalized and effective tutoring services to help you succeed. Call us today to plan a tutoring session and take your math skills to the next level. | {"url":"https://www.losangelesinhometutors.com/blog/dividing-polynomials-definition-synthetic-division-long-division-and-examples","timestamp":"2024-11-11T03:26:24Z","content_type":"text/html","content_length":"78425","record_id":"<urn:uuid:c3e26904-0f40-47ed-a723-2a30e9506660>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00221.warc.gz"} |
Overlapped multiple-bit scanning multiplication system with banded partial product matrix - Patent 0314968
This invention relates to overlapped multiple-bit scanning multiplication systems and, more particularly, to such systems having more than three scanning bits with a reduced partial product matrix
which is banded.
One of the most widely used techniques for binary multiplication has been the uniform overlapped shift method for 3-bit scanning proposed by MacSorley in "High-Speed Arithmetic in Binary Computers,"
Proceedings of the IRE
, Vol. 99, January 1961, pp. 67-91, as a modification of the Booth algorithm disclosed in "A Signed Multiplication Technique,"
Quarterly Journal of Mechanical and Applied Math
, Vol. 4, Part 2, 1951, pp. 236-240. More than 3-bit overlapped scanning has seldom been used, primarily because it has required special hardware, possibly more chip area, and more difficult partial
product selections. In view, however, of improvements in circuit densities over the past few years, and because more than 3-bit scanning may improve the overall speed of the multipliers, there have
been several recent proposals for more than 3-bit overlapped scanning designs. See, for example, Waser et al.,
Introduction to Arithmetic for Digital System Designers,
Chapter 4, CBS College Publishing, 1982, and Rodriguez, "Improved Approach to the Use of Booth's Multiplication Algorithm,"
IBM Technical Disclosure Bulletin
,Vol. 27, No. 11, April 1985, pp. 6624-6632.
Multipliers of this type employ a matrix of the partial products selected in response to each scan. For summing the partial products in the matrix, carry save adder trees are typically used. Methods
for reducing the trees have been suggested by Wallace in "A Suggestion for a Fast Multiplier,"
IEEE Trans. Electronic Computers,
Vol. EC-13, Feb. 1964, pp. 14-17 and Dadda in "Some Schemes for Parallel Multipliers,"
Alta Frequenza,
Vol. 34, March 1965, pp. 349-356. The method proposed by Dadda results in a hardware saving when compared to that of Wallace as is proven by Habibi et al. in "Fast Multipliers,"
IEEE Trans. On Computers,
Feb. 1970, pp. 153-157. Mercy, U.S. Patent No. 4,556,948, teaches a method to improve the speed of the carry save adder tree by skipping carry save adder stages. After the carry save adder tree
reduces the number of terms to two, the two terms are then added with a two-input 2-1 adder. Such 2-1 adders can be designed using conventional technology or some scheme for fast carry look ahead
addition as proposed, for example, by Ling in "High-Speed Binary Adder",
IBM J. Res. Develop.,
Vol. 25, No. 3, May 1981, pp. 156-166.
It is an object of this invention to provide improved multi-bit overlapped scanning multiplication systems which assemble modified partial products in a reduced matrix. The matrix is reduced by
avoiding any need for adding extra rows to the partial product terms and by using rows of reduced length. When a negative partial product term is inverted, the need for a "hot 1" is encoded in an
extension to the partial product term in the previous row, thus avoiding the need for adding a row for this purpose. Instead of extending the rows to the left edge of the matrix, all of the rows,
with the exception of the first and last, are extended with bands of encoded extensions of limited length at the right and left ends of the partial product terms. Because the teachings of the
invention are particularly applicable for overlapped scans of more than three bits, fewer partial product terms and, hence, fewer rows are required.
The multiplication system of the invention takes advantage of the availability of a high circuit count per chip and emphasizes high speed (with very few iterations) multiply implementations. By
reducing the matrix and simplifying the adder trees, the area of the chip required is minimized. A matrix is developed comprising reduced width rows of modified partial products, which, when summed,
equal the final product. The partial products are also modified so as to minimize the cell count required to sum them in an adder tree. A band matrix of minimum width is used, thus reducing the
required width of the adder in the adder tree.
According to the invention, a matrix of S-bit overlapped scanned partial product terms is formed in accordance with an algorithm of the invention. The matrix formed from the encoded partial product
terms is a band matrix, and the sum of the rows of the matrix yields the final product. In the algorithm, a determination is made of the values of S, n-1 and q-1, where S is the number of bits
scanned, n is the width of the significant bits plus the sign of the multiplier, and q is the width of the significant bits plus sign of the multiplicand. The maximum width and number of the partial
product terms are determined. The partial product terms are placed in a matrix, the width of the significant bits of each partial product term being equal to q-1+S-2. Each partial product term is
shifted S-1 bits from adjacent terms to form a non-rectangular matrix which is banded by encoded extensions to the partial product terms. S-1 bits of encode are placed to the right of every term
except the last, the encode being based on the sign of the next partial product term; and an S-1 bit encoded sign extensions are placed to the left of every term except the first term, which has no
sign extension, and the last term, to the left of which is placed an S bit code. Ones and zeros are placed in bit positions in the extensions.
When a partial product term is negative, its bits are inverted. In order to add the required "hot 1", the encoded extension to the right of the partial product term in the previous row is encoded for
a "hot 1". If the partial product of a given row is positive, the S-1 bit positions appended to the previous rows are "0"s; and if the partial product in the given row is negative, the S-1 bits
appended to the previous row comprise S-2 "0"s followed by "1". Since there is no row previous to the first, the first bit of the multiplier is forced to zero so that the partial product term in the
first row is never negative (always positive or zero).
The signs of the partial product terms also determine the encoding of the sign extension to the left of the partial product terms. Because of the truncation of bits not involved with the product, no
encoded sign extension is provided or needed to the left of the first partial product term. For all other partial product terms except the last, the sign extension to the left of a partial product
term has S-1 bits. The encoding is S-1 "1"s for a positive partial product and S-2 "1"s followed by one "0" to the right of the S-2 "1"s if the partial product is negative. A coded extension of S
bits is appended to the left of the last partial product term. When the last partial product term is positive, the encoding is a "1" followed by S-1 "0"s; and when the last partial product term is
negative, the encoding is a "0" followed by S-1 "1"s.
Carry save adder trees are used to reduce each column of the matrix to two terms. Each carry save adder reduces three rows to two rows. Where one, two or three of the inputs to a carry save adder are
known, the logic of the carry save adder is simplified to save chip space.
The multiplicand X and the multiplier Y are typically the fraction part of sign magnitude numbers in binary notation. It is to be understood, however, that the multiplier of the invention is also
applicable to the multiplication of unsigned numbers.
The system of the invention, when S=4, includes means for computing partial products of X, 2X, 4X and 0, means for computing the partial product 3X, and means for computing the boundary bits to be
used in coded extensions of the partial products. Decode means derives first code terms S₀ and S₁, second code terms A₀, A₁, A₂ and A₃ and third code terms R₀ and R₁ from the scanned bits.
Multiplexor means selects one of the X, 2X, 4X and 0 partial products in response to the first code terms S₀ and S₁. Selection means responsive to second code terms A₀, A₁, A₂, and A₃ selects one of
the selected partial product X, 2X, 4X or 0 from the multiplexor means or the partial product 3X and selects the sign of the partial product, inverting the bits of the partial product term if the
sign is negative. Boundary bit logic means responsive to the third code terms R₀ and R₁ select the boundary bits to the right of the previous row of the matrix and to the left of the present row.
These and other objects, features and advantages of the invention will be more fully appreciated with reference to the accompanying figures, in which:
Fig. 1
is a schematic block diagram showing an embodiment of a system of the invention;
Fig. 2
is a schematic diagram showing a detail of the embodiment of Fig. 1;
Fig. 3
is a schematic block diagram of a carry save adder tree of the system of Fig. 1;
Figs. 4-7
are schematic diagrams illustrating the formation of a matrix of the invention;
Figs. 8-17
are schematic circuit diagrams showing simplifications of stages of the carry save adder tree of the embodiment of the invention;
Figs. 18-23 and 26-31
are schematic diagrams of portions of the decoder of the embodiment of the invention;
Figs. 24 and 25
are Karnaugh diagrams illustrating the logic of the circuit diagrams of Figs. 18 and 19;
Figs. 32 and 33
are schematic circuit diagrams of the portions of the 2 to 1 True/Complement Select circuit of the embodiment of the invention dedicated, respectively, to all partial products but the first and
to the first partial product;
Figs. 34-36
are schematic circuit diagrams showing how portions of the Boundary Bit Computer of the embodiment of the invention append boundary bits to partial product terms;
Fig. 37
is diagram showing a band matrix formed for a particular implementation of the invention using four-bit scanning, a fraction having fifty-six bits and nineteen partial product terms;
Fig. 38
is a diagram showing the rows to be added in the band matrix of the particular implementation illustrated in Fig. 37 in the first stage of the carry stage adder tree;
Fig. 39
is a diagram showing the rows to be added in the particular implementation in the second stage of the carry save adder tree;
Fig. 40
is a diagram showing the rows to be added in the particular implementation in the third stage of the carry save adder tree;
Fig. 41
is a diagram showing the rows to be added in the particular implementation in the fourth stage of the carry save adder tree;
Fig. 42
is a diagram showing the rows to be added in the particular implementation in the fifth stage of the carry save adder tree;
Fig. 43
is a diagram showing the rows to be added in the particular implementation in the sixth stage of the carry stage adder tree; and
Fig. 44
is a diagram showing the sum and carry outputs of the carry save adder tree for the particular implementation.
An overlapped multi-bit scanning multiplication system of the invention is shown in Fig. 1, and a detail thereof is shown in Fig. 2. The system as shown in these figures, in particular, illustrates
the case in which the system uses four-bit scanning with one bit of overlap.
The system will multiply a multiplicand X by a multiplier Y. Typically, X and Y are derived from floating point sign magnitude numbers in binary notation. Such numbers include a sign, a fraction and
an exponent with X and Y representing the fraction portion only of these numbers. As is customary in the art, the sign and exponent portions of the numbers to be multiplied are separated and the sign
and exponent calculations are dealt with separately from the multiplication of the fractions. Accordingly, it is to be understood that the signs and exponents of X and Y have already been separated,
will be separately computed and then appropriately combined with the result computed by the system of the invention. Although the invention will be illustrated by the case of the multiplication of
the fractions of floating point numbers, it is to be understood that it is also applicable to the case of the multiplication of a pair of unsigned binary numbers.
In the system shown in Fig. 1, the multiplicand X is applied to a pair of product calculators. Products calculator 10 computes certain simple products of X which in the case of a four-bit scan system
are, as shown, the products X, 2X 4X and 0. As is known in the art, these products are easily derived from X. The product X is simply X itself. 2X is derived from X by shifting X one place, and 4X is
derived from X by shifting X two places. A difficult products calculator 12 calculates products requiring a more complex computation. In the illustrative case of a four-bit scan, only the product 3X
need be computed. This is accomplished by deriving 2X from X by shifting X one place and then employing an adder to add X and 2X.
The multiplier Y is first subjected to a manipulation at 14 in which the first bit Y₀ and last bit Y
of Y are forced to zero for reasons to be explained more fully hereinbelow. This manipulation may be performed, for example, by means of placing Y in a register the first and last places of which are
tied to logic level 0.
Multiplier Y, with its first and last bits so modified, is then placed in Y register 16 and is then scanned by scanning means 18. As is known in the art, scanning means 18 will successively output
sets of S bits of Y, successive scans of Y overlapping one bit from the previous scan. For example, in the illustrative case when S=4, the first scan will output the values of the first four bits of
Y, the second scan will output the values of the fourth through the seventh bits of Y, the third scan will output the values of the seventh through tenth bits of Y and so on until all of the bits of
Y have been scanned. If, for example, Y consists of fifty-six bits, nineteen scans of Y will be output by scanning means 18.
These outputs are applied to a decoder 20 which, as will be explained below, will provide three sets of selection code terms. These will be used in conjunction with the products of X from calculators
10 and 12 to select partial products of X and modify these partial products by appending thereto appropriate encoded extensions at the left and right sides thereof to assemble the banded
non-rectangular matrix at 22 as will be explained more fully below.
The matrix is then added by a set of carry save adder trees 24 which reduce the columns of the matrix to no more than two terms. These are then added, typically in the next cycle, by a 2:1 adder 26,
yielding the result at 28. In the case of floating point sign magnitude numbers, this result is then combined with the results of the sign and exponent calculations to provide the final result of the
Turning to Fig. 2, it will be seen that decoder 20 provides for each scan of Y three sets of code terms. On decode output 30 are code terms S₀ and S₁, on decode output 32 are code terms A₀, A₁, A₂
and A₃, and on decode output 34 are code terms R₀ and R₁. The manner in which these codes are derived from the scans of Y will be explained more fully below.
The products X, 2X, 4X and 0 of X calculated by products calculator 10 are applied as inputs to a 4:1 multiplexor 36. As each scan of Y is completed, the code terms S₀ and S₁ are applied to
multiplexor 36 to select one of the multiples of X in a manner to be described more fully below. Thus, in the example given, nineteen successive selections are made and applied as one input to 2:1
true complement selector 38. Another input to selector 38 is the 3X output from difficult product calculator 12. In a manner to be explained more fully below, code terms A₀, A₁, A₂ and A₃ select
whether the partial product to be selected is the partial product received from multiplexor 36 or the 3X partial product. These code terms also select the sign of the partial product, whether it is +
or -. If the - sign is selected, the selected partial product is inverted in selector 38 in a manner to be explained hereinafter.
An important feature of the present invention is the coded bit extensions which are appended to the left and/or the right of the partial products. These are generated in compute boundary bits logic
circuit 40 in a manner to be explained more fully below. Logic circuit 40 is responsive to code terms R₀ and R₁ as will be explained below.
Each successive partial product, as provided by select circuit 38 and as modified by boundary bit logic 40, is then assembled as indicated at 42 as successive rows of the matrix. As will be shown
presently, each successive partial product is shifted to the right relative to the previous row by S-1 bit positions. The resulting matrix is thus non-rectangular and banded by the encoded extensions
and is thus reduced when compared to the matrices used in the prior art overlapped scanning multipliers.
As explained above, each column of the matrix is added by a carry save adder tree. The matrix structurally comprises hardwired connections between select circuit 38 and boundary bit circuit 40 to the
inputs of the carry save adder trees.
One of the carry save adder trees is shown in Fig. 3, being the worst case - that is, the most complex carry save adder tree needed for the illustrative example which requires nineteen partial
products and thus nineteen rows in the matrix. Because, as already explained, successive partial products are shifted S-1 bits, the columns at either side the matrix will include several columns
which have fewer than a full complement of nineteen rows. The carry save adder trees for these columns will be correspondingly simpler. Returning to Fig. 3 inputs 101, 102, 103, 104, 105, 106, 107,
108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118 and l19 represent the successive rows of the column of the matrix being summed. These inputs are applied, in threes, as inputs to carry save
adder means 120 each of which provides two outputs, a "carry" and a "save" output. These, in turn, provide additional inputs, in sets of three, to additional carry save adder means of the tree, as
shown, until the final carry save adder means 120 provides the two outputs 121 and 122 to 2:1 adder 26. As will be explained more fully below, carry save adder means 120 are, for the most part, carry
save logic circuits. However, in cases where one or more of the inputs to a particular carry save adder means is, or are, known, simple logic means may be substituted to reduce the area of the chip
We will now consider the encoding of the matrix for S-bit overlapped scanning. The multiplier and multiplicand are considered to be represented as floating point numbers which consist of a sign, an
exponent and a fraction in binary notation. As is customary in the art, the sign and exponent calculation are handled separately and do not form part of the present invention. Therefore, hereinafter
any reference to the multiplicand, multiplier or product refers only to the binary representation of the fraction in which the zero bit is the sign bit (always equal to zero, because the fraction is
a magnitude and magnitudes are always positive). Bit one is the most significant bit, and bit n-1 is the least significant bit.
Letting q be equal to the width of the multiplicand, Z be a decimal number and z
be its two's complement notation, it can be proven:
where z₀ is the sign bit and n-1 is the width of the significant bits of z.
Assuming that Y and X are, respectively, the multiplier and multiplicand in sign magnitude notation, that P is the product of X and Y, and Wp is the width of the product, and given that only
magnitudes are considered, it can be written:
Since x₀ = y₀ = p₀ = 0, it follows that:
which are the correct expressions for the magnitudes of sign magnitude numbers.
Let us now assume that S equals the number of bits which are overlapped scanned assuming a one bit overlap, that "coefficient" refers to the number which multiplies X to form the multiple denoted by
W, W being equal to the coefficient multiplied by X which is formed by x₁...x
, that "partial produts" refers to a number which is formed by overlapped scanning, the sum of all partial products resulting in the product P, and that M+1 is the number of partial product terms. It
can be shown that S bit overlapped scanning yields:
1.M+1 partial products, M+1 being equal to
where r is the remainder of
2. The greatest possible multiple of the multiplicand has a coefficient W equal to 2^(S-2), and the least possible multiple thereof has a coefficient W equal to -2^(S-2).
3.Each partial product term can be represented by the ith multiple times 2^(1-i(S-1)), where i is an integer greater than or equal to 1 and less than or equal to M+1.
4. Assuming that the scanning of the multiplication starts at the most significant bits, the multiple of the i+1th partial product is multiplied by 2^-(S-1) with respect to the ith partial product;
thus the i+1th partial product can be represented by a shift of S-1 bits to the right.
The principles of the present invention are formulated in a number of theorems which will then be proven.
Theorem 1: The multiples can be represented in two's complement binary notation (actually, one's complement plus a hot "1" for the negative terms) with q-1+S-2 bits plus the possibility of a hot "1".
Proof: Given that the greatest |W|=2
, any multiple WX will be shifted to a maximum of S-2 places; and given that the significant multiplicand's length is q-1, then the maximum WX can be presented with q-1+S-2 bit positions. It is also
given that a sign extension will not change the value of WX. If |W| happens to be less than the maximum, it can be concluded that WX can be represented by q-1+S-2 bit positions. W may assume positive
or negative values, and therefore WX may be either positive or negative. Since it is given that a negative number can be expressed as a one's complement number with the addition of a hot "1", any
multiple can be represented by q-1+S-2 bits with the possibility of adding a hot "1" when negative.
We shall now consider the formation of the matrix. With the previous assumptions, a matrix is formed in which each row represents a partial product term. The sum of the rows will equal the final
product excluding some bits of the sum of the matrix that are overflows. A convention is adopted to place the left most scanned partial product (most significant) in the first row followed by the
rest of the terms in successive rows as they are scanned to the right. Thus, i represents the row of the matrix. A band matrix is formed except for the need to sign extend and to add the hot "1"s to
the negative terms. To create a true band matrix, which enables the use of reduced width adders, two steps are taken:
Step 1:
The hot "1"s are encoded into the band.
Step 2:
The sign extension is encoded into the band.
As used in this specification and the appended claims, the term "banded matrix" may be defined as referring to a reduced, non-rectangular matrix in which successive rows are shifted and in which
bands of extra bits are appended on each side of the row. The number of bits in the bands is limited and is, in fact, related to the number S of bits scanned as will be shown below.
Thus, the partial products are modified but, as will be shown, without altering the final product. The partial product rows are of minimum width, and the number of partial products is reduced.
Step 1 is formulated in Theorem 2: Given that the first partial product term is always positive and that each row of the matrix has the multiples shifted S-1 bits from each other, S-1 bits can be
placed on the prior partial product terms to encode the need for a hot "1" to be added to the negative terms. The S-1 bits are the bits immediately to the right of each multiple except the M+1th
multiple since there is no term after it. They are all equal to zero, if the following term (the i+1th term considering the current term to be the ith term) is positive; and a one is in the least
significant bit position of the S-1 bits and the rest are zeros, if the next partial product term is negative.
where L
is the ith Multiple = XW
If L
is negative, then it is represented in one's complement plus one as: L
= (
) + "hot 1" with (
) interpreted as the bit by bit inversion of L
. Given that the last bit of L
is positioned at 2
, the "hot 1" must be added at position 2
^-(q-1+S-2+(i+1) (S-1)-1)
for the i+1 partial product term. The implication is that when L
is negative,
The last position occupied by L
is q-1+S-2. Thus, it can be concluded that when L
is negative, PP
is equivalent to the bit by bit inversion of L
and the addition of S-1 positions to L
with S-2 zeros following the q-1+S-2 position and a 1 at position q-1+S-2+S-1. Given that the addition of zeros after q-1+S-2 will not change the value of any L
, it can be concluded that if L
is positive, L
can be extended by S-1 zeros after q-1+S-2 positions. Since the first row is insured to be positive, there is no need to have an additional row for the correct representation of the matrix. Also,
since the M+1 row does not precede any other row, there is no need for the additional S-1 bits. Accordingly, it can be concluded that Theorem 2 holds true.
Step 3 is formulated in Theorem 3: The sign-extension of the negative terms is encoded into the S-1 bits to the left of the multiples most significant bit, and a hot "1" term is added at the bit
location of the least significant bit of the M+1th term encoding. The encoding in general for all partial products except the first term which is always positive, is as follows: The encoding is S-1
ones for a positive partial product and S-2 ones and one zero (to the right of the S-2 ones) if the partial product is negative.
Proof: It must be proven that the lower triangle of the matrix which has the encoding in bands as taught by the present invention, as in Step 2 (which will be referred to below as the "new" scheme),
if summed row by row is equivalent to a "regular" sign extended lower triangular matrix summed.
A "regular" sign-extended lower triangular matrix is the known practice of sign extending the lines of the matrix to fill out the triangle to the left and below the partial products so that the
correct result is obtained when the rows of the matrix are summed. According to the "regular" sign extension principle, a row is extended when a term is negative by inserting ones in every bit
position to the left of the partial product so that each row is filled out to the leftmost bit position of the matrix.
Let A
represent the sign of the ith row such that it is one, if, and only if, the partial product in that row is negative. Let LTMSR be the Lower Triangular Matrix Sum if "regular", and LTMSN be the Lower
Triangular Matrix Sum if the "new" (Step 2) scheme is used. The "regular" matrix is sign-extended wherever there is a negative term. In these rows there are ones in every bit to the left of multiple
of the multiplicand. The most significant one is the most significant bit of the matrix which is 2
. The sum of a string of ones is equal to 2 to the power one place more significant than the string, minus 2 to the power of the least significant bit (from the well-known string property which is
the basis of Booth encoding). This is proven by the following which uses the law for the sum of a geometric series.
Excluding overflows, which are any nonfractional terms, LTMSR equals the addition of the S-1 bit encodings plus the hot "1" which was added as 2
which equals LTMSN. Thus, Step 2 is a valid way of extending the sign of the terms and making the matrix into a banded matrix with the final product remaining valid.
Theorem 4: The hot "1" that is added at position 2
can be encoded into the M+1 row's encoding creating an S bit encoding, starting at position 2
^(-(M-1) (S-1))
with 1 for a positive row and 0 for a negative row and followed by S-1 zeros for a positive row or S-1 ones for a negative row.
This is a one at position 2
^(-(M-1) (S-1))
. Thus, if an encoding between bits (-(M-1) (S-1)) and (-M(S-1)) is used, which is S bits and adopts the form as shown in Theorem 4, an S bit encoding can be used on the last partial product which is
the addition of the S-1 bit sign extension encoding and the additional hot "1" needed for the whole sign extension encoding of all the partial product terms.
It was stated in the comparison of LTMSN and LTMSR that both representations could create overflows which were considered to be any portion greater than one. This sub-section proves the
multiplication of two fractional numbers always results in a fractional product, in other words a product less than one.
This can be proven given that q+1 _ 1 and n+1 ≥ 1; otherwise there would be no multiplier or multiplicand.
This proves that a larger number is subtracted from 1 than is added in the expression for P
. Thus, P
< 1.
Using these theorems, an algorithm for forming the matrix is formulated. We first assume S bit overlapped scanning with M+1 partial products, M being equal to INT
with INT being the integer division and n the length of multiplier Y. S may be determined after conducting a comparative study of the hardware and timing requirements to calculate the multiples with
due consideration of the reductions of the hardware and timing of the resulting carry save adder trees.
Assuming the scanning starts at the most significant bits of Y, then the multiple of the (i+1) the partial product is shifted with respect to the ith partial product by (S-1) bits to the right where
i is an integer greater than or equal to 1 and less than or equal to M+1. This is illustrated in Fig. 4, where the successive partial products i=1, i=2, ..., i=M and i=M+1 are shown as rows of the
matrix, each (except for the first partial product i=1) shifted to the right by S-1 bits relative to the previous row. As indicated in the figure, each partial product row has q-1+S-2 bit positions,
q being the width of the multiplicand, including bit 0, with the least significant bit being the (q-1)th bit.
As shown in Fig. 5, in every row except the last, S-1 significant bit positions are added to the right of the partial product. These bits encode the sign of the next partial product term. Since no
partial product term follows the last, the extension is not needed in the last row.
As shown in Fig. 6, S-1 significant bits are added to the left of every partial product term to encode the sign of the partial product term with the exception of the first partial product term which
needs no encoded sign extension because of truncation and the last partial product term which has an encoded sign extension of S-1 plus one so that S bits are needed.
As shown in Fig. 7, is and 0s are placed in the matrix where known, the symbol * indicating a bit which can be either 1 or 0. As indicated, the encoded extensions to the right of the partial product
are all 0s except for the last bit. According to Theorem 2, this extension signifies the need for a hot "1" to be added if the next partial product is negative. All of the bits of the extension
including the last are zeros when the following term is positive, but the least significant bit is 1 when the next partial product term is negative. The encoded sign extension to the left of the
partial product terms in all but the first and last rows have S-2 ones followed by *. According to Theorem 3, this encoded sign extension has S-1 ones when the partial product term is positive and
S-2 ones and a zero to the right of the S-2 ones when the partial product is negative. According to Theorem 4, the S bit encoded extension to the left of the last partial product term in row M+1, is
1 followed by S-1 zeros for a positive partial product term and 0 followed by S-1 ones for a negative partial product term.
As just indicated, certain positions of the modified partial product terms will be known to always be 0 or 1. This information can be exploited to simplify the carry save adder trees. Whereas, when
all three inputs to a carry save adder means of a tree are unknown, a carry save adder logic circuit is needed to perform the carry save adder function, when one or more of the three inputs are
known, more simple logic, requiring less space on the chip may be used. If K₁, K₂ and K₃ are the three inputs to a carry save adder logic circuit, then the carry output CA and the save output SA are
governed by:
This means that CA can be obtained by a 2-way AND gate, as illustrated in Fig. 8 at 124, and that SA can be obtained by an Exclusive OR gate 125 as shown in Fig. 9.
The CA term is obtained from a 2-way OR gate 126 as shown in Fig. 10, and the SA term from an Exclusive NOR gate 127 as shown in Fig. 11.
For every column which has one significant term which is unknown and two which are known, we can use:
Thus, for the CA term, the CA output is hardwired at 128 to logic level 0 as shown in Fig. 12; and the SA output is hardwired at 129 to the K₃ input as shown in Fig. 13.
Hence, the CA output is obtained by being hardwired to the K₃ input as shown at 130 in Fig. 14. SA, on the other hand, is obtained from K₃ through an inverter 131 as shown in Fig. 15.
Thus, as shown in Fig. 16, output CA is hardwired at 132 to logic level 1, while output SA is hardwired to input K₃ as shown at 133 in Fig. 17.
For the case where all three input terms are known, the outputs CA and SA are hardwired as indicated in the following table:
│ (K₁, K₂, K₃) │ CA │ SA │
│ (0,0,0) │ 0 │ 0 │
│ (0,0,1) │ 0 │ 1 │
│ (0,1,0) │ 0 │ 1 │
│ (0,1,1) │ 1 │ 0 │
│ (1,0,0) │ 0 │ 1 │
│ (1,0,1) │ 1 │ 0 │
│ (1,1,0) │ 1 │ 0 │
│ (1,1,1) │ 1 │ 1 │
Thus, CA and SA are hardwired to logic levels 0 or 1 in the manner illustrated in Figs. 12 and 16.
We will now consider the decoding function of decoder 20 for the case of four-bit scanning. In two successive scans of multiplier Y, successive bits of Y, namely Y
, Y
, Y
and Y
, are scanned in a given scan and successive bits Y
, Y
, Y
and Y
are scanned in the next scan. Code terms S₀ and S₁ are defined in these equations:
These equations translate respectively to the logic circuits of Figs. 18 and 19. In Fig. 18, inputs corresponding to the first term of the S₀ equation are anded in AND gate 140; inputs corresponding
to the second term of the equation are anded in AND gate 141; inputs corresponding to the third term of the equation are anded in AND gate 142; and inputs corresponding to the fourth term of the
equation are anded in AND gate 143. The outputs of these AND gates provide the four inputs of OR gate 144 the output from which is S₀. In Fig. 19, the inputs to AND gates 150, 151, 152 and 153
correspond respectively to the terms of the equation for S₁. The outputs of the AND gates are the four inputs to OR gate 154, the output from which is S₁. As shown in Figs. 20, 21, 22 and 23, the
complementary form of Y
, Y
, Y
and Y
, respectively, are obtained through inverters 155, 156, 157 and 158.
The following table gives the values of S₀ and S₁ and of the multiple coefficient W selected by multiplexor 36 in response thereto for various values of the four scanned bits:
│ │ │ │ │ Encode │ │
│ Y[(j-2)] │ Y[(j-1)] │ Y[(j)] │ Y[(j+1)] │ S₀ │ S₁ │ Coefficient W │
│ 0 │ 0 │ 0 │ 0 │ 0 │ 0 │ 0 │
│ 0 │ 0 │ 0 │ 1 │ 0 │ 1 │ 1 │
│ 0 │ 0 │ 1 │ 0 │ 0 │ 1 │ 1 │
│ 0 │ 0 │ 1 │ 1 │ 1 │ 0 │ 2 │
│ 0 │ 1 │ 0 │ 0 │ 1 │ 0 │ 2 │
│ 0 │ 1 │ 0 │ 1 │ d │ d │ 3 │
│ 0 │ 1 │ 1 │ 0 │ d │ d │ 3 │
│ 0 │ 1 │ 1 │ 1 │ 1 │ 1 │ 4 │
│ 1 │ 0 │ 0 │ 0 │ 1 │ 1 │ -4 │
│ 1 │ 0 │ 0 │ 1 │ d │ d │ -3 │
│ 1 │ 0 │ 1 │ 0 │ d │ d │ -3 │
│ 1 │ 0 │ 1 │ 1 │ 1 │ 0 │ -2 │
│ 1 │ 1 │ 0 │ 0 │ 1 │ 0 │ -2 │
│ 1 │ 1 │ 0 │ 1 │ 0 │ 1 │ -1 │
│ 1 │ 1 │ 1 │ 0 │ 0 │ 1 │ -1 │
│ 1 │ 1 │ 1 │ 1 │ 0 │ 0 │ 0 │
where d is a "don't care" bit. The Karnaugh diagrams of Figs. 24 and 25 illustrate the derivation of the values of S₀ and S₁, respectively.
These equations translate into the logic diagrams of Figs. 26, 27, 28 and 29, respectively. In Fig. 26, the inputs to AND gates 160, 161 and 162 correspond to the respective terms of the equation for
A₀ with the outputs from the AND gates providing the inputs to OR gate 163, the output from which is A₀. In Fig. 27, AND gates 165 and 166 receive inputs corresponding to the respective terms of the
equation for A₁. The outputs from AND gates 165 and 166 provide the inputs to OR gate 167 whose output is A₁. In Fig. 28, the inputs to NAND gates 170, 171, 172 and 173 correspond to the respective
terms of the equation for A₂. The outputs from NAND gates 170, 171, 172 and 173 supply the inputs to NAND gate 174, the output from which is A₂. In Fig. 29, AND gates 176 and 177 receive inputs
corresponding to the terms of the equation for A₃. The outputs from these AND gates are inputs to OR gate 178 whose output is A₃.
The following table gives the values of A₀, A₁, A₂ and A₃ and the coefficient Wand sign selected by 2 to 1 True/Complement Selector 38 in response thereto for various values of the four scanned bits:
│ │ │ │ │ Encode │ Coefficient │
│ Y[(j-2)] │ Y[(j-1)] │ Y[(j)] │ Y[(j+1)] │ A₀ │ A₁ │ A₂ │ A₃ │ W │
│ 0 │ 0 │ 0 │ 0 │ 1 │ 0 │ 0 │ 0 │ 0 │
│ 0 │ 0 │ 0 │ 1 │ 1 │ 0 │ 0 │ 0 │ 1 │
│ 0 │ 0 │ 1 │ 0 │ 1 │ 0 │ 0 │ 0 │ 1 │
│ 0 │ 0 │ 1 │ 1 │ 1 │ 0 │ 0 │ 0 │ 2 │
│ 0 │ 1 │ 0 │ 0 │ 1 │ 0 │ 0 │ 0 │ 2 │
│ 0 │ 1 │ 0 │ 1 │ 0 │ 1 │ 0 │ 0 │ 3 │
│ 0 │ 1 │ 1 │ 0 │ 0 │ 1 │ 0 │ 0 │ 3 │
│ 0 │ 1 │ 1 │ 1 │ 1 │ 0 │ 0 │ 0 │ 4 │
│ 1 │ 0 │ 0 │ 0 │ 0 │ 0 │ 1 │ 0 │ -4 │
│ 1 │ 0 │ 0 │ 1 │ 0 │ 0 │ 0 │ 1 │ -3 │
│ 1 │ 0 │ 1 │ 0 │ 0 │ 0 │ 0 │ 1 │ -3 │
│ 1 │ 0 │ 1 │ 1 │ 0 │ 0 │ 1 │ 0 │ -2 │
│ 1 │ 1 │ 0 │ 0 │ 0 │ 0 │ 1 │ 0 │ -2 │
│ 1 │ 1 │ 0 │ 1 │ 0 │ 0 │ 1 │ 0 │ -1 │
│ 1 │ 1 │ 1 │ 0 │ 0 │ 0 │ 1 │ 0 │ -1 │
│ 1 │ 1 │ 1 │ 1 │ 1 │ 0 │ 0 │ 0 │ 0 │
Thus, R₀ is dependent on the current scan, while R₁ is dependent on the next scan, where "current scan" refers to the scan corresponding to the partial product whose boundary bits are being selected.
The equations for R₀ and R₁ translate into the logic diagrams of Figs. 30 and 31, respectively, In Fig. 30, a NAND gate 180 receives bits Y
, Y
and Y
as inputs and provides from its output one input to AND gate 181. The other input to AND gate 181 receives bit Y
. R₀ is the output from AND gate 181. In Fig. 31 NAND gate 183 receives bits Y
, Y
and Y
as inputs and provides an input to AND gate 184, the other input to which is bit Y
. The output from AND gate 184 is R₁.
If R₀ = 1, boundary bits 110 from boundary bit computer 40 are appended as an encoded sign extension to the left of the partial product. If R₀ = 0, boundary bits 111 are appended as an encoded sign
extension to the left of the partial product. If R₁ = 0, boundary bits 000 are appended as an encoded extension to the right of the partial product. If R₁ = 1, boundary bits 001 are appended to the
right of the partial product.
We will now consider a particular implementation in which each partial product term has fifty-eight "middle" bits - that is, fifty-eight bits before the addition of any boundary bits. Fig. 32 shows
how 2 to 1 True/Complement Select circuit 38 of Fig. 2 selects the fifty-eight middle bits of all of the partial product terms, except the first. Inputs A
, A
, A
and A
are the A
A₂ and A₃ code terms for the kth partial product term; inputs MUX₀ ... MUX₅₇ are the 0th through 57th bits of the multiple (0, X, 2X or 4X) selected by multiplexor 36 of Fig. 2 and applied from a
multiplexor register; and 3X₀ ... 3X₅₇ are the 0th through 57th bits of the 3X multiple received from difficult products calculator 12 of Fig. 1 and applied from a difficult products register. Inputs
and A
are applied to OR gate 190, and inputs A
and A
are applied to OR gate 192. The output from OR gate 192 provides inputs to AND gates 194₀ ... 194₅₇, the other inputs to which are inputs MUX₀ ... MUX₅₇. Thus, when the output of OR gate 192 is at
level "1" and the MUX bit is a "1", the corresponding AND gate output is at level "1". Likewise, the output from OR gate 190 is applied as inputs to AND gates 196₀ ... 196₅₇, the other inputs to
which are bits 3X₀ ... 3X₅₇. The outputs of AND gates 196 also are at level "1" when the output from OR gate 190 is at level "1" and the 3X bit is at level "1", the output from the AND gate 196 is at
level "1". The outputs from AND gates 194₀ and 196₀, ... 194₅₇ and 196₅₇ are applied respectively to OR gates 198₀ ... 198₅₇. As a result, the outputs from OR gates 198₀ ... 198₅₇ will correspond to
the values of the bits of the multiplexor output, if the output from OR gate 192 is at level "1", and will correspond to the bit values of 3X, if the output from OR gate 190 is at level "1".
If either A
or A
are at level "1", the partial product is negative and needs to be complemented. This is achieved by applying A
and A
as inputs to OR gate 200. The output from OR gate 200 is applied as inputs to Exclusive OR gates 202₀ ... 202₅₇, the other inputs to which are the outputs from OR gates 198₀ ... 198₅₇, respectively.
Thus, the output from OR gate 200 selects the complement of the outputs from OR gates 198₀ ... 198₅₇ when the output from OR gate 200 is at level "1". The outputs from Exclusive OR gates 202₀ ...
202₅₇ then become the middle 58 bits of the kth partial product term.
The portion of 2 to 1 True/Complement Select circuit 38 dedicated to the first partial product term is simplified as shown in Fig. 33. Since the first partial product will always be positive or zero
as explained above, there is no need to invert the bits of the partial product when the partial product is negative. Accordingly, the portion of the circuit for selecting the complement is omitted.
Code term A₀₁ is applied as an input to AND gates 204₀ ... 204₅₇, and code term A₁₁ is applied to AND gates 205₀ ... 205₅₇. The outputs MUX₀ ... MUX₅₇ from multiplexor 36 of Fig. 2 are applied to the
other inputs of AND gates 204₀ ... 204₅₇. Likewise, the outputs 3X₀ ... 3X₅₇ from difficult products calculator of Fig. 1 are applied to the other inputs to AND gates 205₀ ... 205₅₇. The outputs from
AND gates 204₀ and 205₀ ... 204₅₇ and 205₅₇ are applied respectively to OR gates 206₀ ... 206₅₇, the outputs from which constitute the fifty-eight middle bits of the first partial product. These bits
will correspond to the bits of the multiplexor output when A₀₁ is at level "1" and to the bit values of 3X when A₁₁ is at level "1".
The manner in which Compute Boundary Bits circuit 40 operates to append boundary bits on the left and right ends of all but the first and last of the partial product terms generated by 2:1 True/
Complement Select circuit 38 is shown in Fig. 34. As already described, for all partial product terms except the first and last, S-1 bits are appended to the left of the partial product term. If the
partial product term is positive, the encoding is S-1 ones; and if negative, the encoding is S-2 ones followed by a zero. In the example illustrated, S=4, and S-1=3. Since the first two bits are
always "1", bit lines 210 and 211 are hardwired to logic level "1". Bit line 212 is connected to receive code term R₀. When the partial product term is positive, R₀ is at level "1" and bit line 212
is at level "1". When, however, the partial product term is negative, R₀ is at level "0" and bit line 212 is at level "0".
It has also been explained that S-1 bits are appended to the right of all partial product terms except the last partial product term. These bits are all zeros, if the following term is positive. If,
however, the following term is negative, the bits appended to the right of the partial product term are S-2 zeros followed by a one. Since in the example shown S=4 and S-1=3, the first two bit lines
213 and 214 to the right of the partial product term are hardwired to logic level "0". The last bit line 215 is responsive to the sign of the next partial product term as signified by code term R₁.
Code term R₁ is applied to bit line 215 through inverter 216. Thus, when R₁ is at level "1", indicating that the next term is positive, logic level "0" is applied to bit line 215. If, on the other
hand, R₁ is at level "0", indicating that the next term is negative, logic level "1" is applied to bit line 215.
As a result, the partial product term is modified with three bits appended at its left end and three bits appended at its right end. This modified partial product term now constitutes a row, other
than the first or last row, of the matrix.
As shown in Fig. 35, in the case of the first partial product term which has, as has been explained, no boundary bits appended to its left end, the portion of boundary bit circuit 40 at the left end
of the partial product term is omitted. Since boundary bits are still appended at the right end of the first partial product term, this portion of circuit 40 is identical to that used for all partial
product terms but the first and last as shown in Fig. 34.
In the case of the last partial product term, as shown in Fig. 36, no boundary bits are appended at the right end of the partial product term. At the left end, however, four bits are appended. Line
220 applied the value of code term R₀ as the first of these bits. Code term R₀ is also applied through inverters 222, 224 and 226 to lines 228, 230 and 232 forming the second, third and fourth bits
of the four bit extension. Thus, when R₀ is at level "0", the first bit is at level "0" followed by three bits at level "1"; and when R₀ is at level "1", the first bit is at level "1" followed by
three bits at level "0".
We shall now consider in detail the particular case of an embodiment in which S=4, q-1=56 and n-1=56. The following table shows the partial product terms obtained for various values of the four
scanned bits:
│ │ │ Partial │ │
│ Y[(j-2)] │ Y[(j-1)] │ Y[(j)] │ Y[(j+1)] │ Product Term │
│ 0 │ 0 │ 0 │ 0 │ 1X2^(-j) │
│ 0 │ 0 │ 0 │ 1 │ 1X2^(-j) │
│ 0 │ 0 │ 1 │ 1 │ 2X2^(-j) │
│ 0 │ 1 │ 0 │ 0 │ 2X2^(-j) │
│ 0 │ 1 │ 0 │ 1 │ 3X2^(-j) │
│ 0 │ 1 │ 1 │ 0 │ 3X2^(-j) │
│ 0 │ 1 │ 1 │ 1 │ 4X2^(-j) │
│ 1 │ 0 │ 0 │ 0 │ -4X2^(-j) │
│ 1 │ 0 │ 0 │ 1 │ -3X2^(-j) │
│ 1 │ 0 │ 1 │ 0 │ -3X2^(-j) │
│ 1 │ 0 │ 1 │ 1 │ -2X2^(-j) │
│ 1 │ 1 │ 0 │ 0 │ -2X2^(-j) │
│ 1 │ 1 │ 0 │ 1 │ -1X2^(-j) │
│ 1 │ 1 │ 1 │ 0 │ -1X2^(-j) │
│ 1 │ 1 │ 1 │ 1 │ 0 │
In this particular implementation, the fraction is 56-bits in length. Thus, 19 partial product terms are created after 4-bit scanning as is detailed below:
The numbers 0 through 56 represent bit positions of the multiplier Y. Bit position Y₀ is the sign which is forced to zero; and bit F is also forced to zero. Bit F must be zero, because of a
requirement in multi-bit overlapped scanning multiplication that the last scan must end in zero. Each of the vertical lines indicate the bits scanned in each scan: for example, the first scan covers
bits Y₀ through Y₃, the second scan covers bits Y₃ through Y₆ and so on. The numerals between the vertical lines are the number of the scan, there being 19 scans and, hence, 19 partial product terms.
As has already been explained, to create a matrix, the partial product terms in all rows except the first and last are appended with a few bits so as to make them uniform in length and displacement
from each other which makes the matrix banded.
First, the multiples are represented as 58 bits which is q-1+S-2. The negative multiples are represented in one's complement plus a hot "1". The multiples with less than 58 significant bits are
sign-extended to 58 bits.
Second, three bits (S-1 = 4-1 = 3) are added to the right of every term (except the last term) to account for representing the negative multiples as one's complement numbers rather than two's
complement numbers. This is done by appending 001 to right of terms followed by a negative term, and 000 is appended to terms followed by a positive term.
Third, three bits are appended to the left of every term except the first and last. This is done to extend the sign of any negative terms. These three bits are ill for positive terms and 110 for
negative terms. The last term has an S-bit encoding which is 0111 if it is negative and 1000 if it is positive.
Thus, a band matrix is formed with the first term having 61 significant bits, the next 17 terms having 64 significant bits, and the last term having 62 significant bits. The completed matrix is shown
in Fig. 37. The partial product terms of successive scans are shifted three spaces (S-1 = 3) to the right. Since the partial product term in the first row has no three bit sign extension on the left,
the first and second rows begin in the same column. Because the last partial product term has no three bit extension on the right, the last and next to last rows end in the same column. Also, since
the last row is extended four bits on the left, the last row begins two columns to the right of the next to last row.
We shall next consider in detail the reduction of this matrix using the carry save adder trees. Turning to Fig. 38, the matrix is divided into six sets of three rows plus a seventh set having one
row. The first six sets are then processed in the first stages CSA1, CSA2, CSA3, CSA4, CSA5 and CSA6 of the carry save adder tree.
In the second stage of the carry save adder tree, the matrix of partial product terms shown in Fig. 39 must be added. The designations C1, S1, C2, S2, C3, S3, C4, S4, C5, S5, C6, S6 indicate the
carry and save outputs from the carry save adders of the first stage of the tree as shown in Fig. 3. There are now four sets of three rows each, which are added in carry save adder CSA7, CSA8, CSA9
and CSA10 of the second stage of the tree.
The third stage of the carry save adder tree add the matrix as shown in Fig. 40. There are now three sets of three inputs to carry save adders CSA11, CSA12 and CSA13. The sources of the rows are
indicated on the right end of the rows. The last row of the third set to be added in CSA13 is row 19 from the original matrix.
The fourth stage of the carry save adder tree deals with the matrix shown in Fig. 41. There are now only two sets of inputs left as indicated, and these are added in carry save adders CSA14 and
As shown in Fig. 42, the matrix is now reduced to four rows of partial product terms. The fifth stage of the carry save adder tree, consisting of carry save adder CSA16, now must add only a single
set of three inputs. As indicated, the outputs C14, S14 and C15 are the sources of these inputs. An extra row derived from output S15 is saved for the sixth stage of the tree as shown in Fig. 43. The
final three rows of partial product terms of the matrix are added in carry save adder CSA17.
The sum and carry output from CSA17 has the structure shown in Fig. 44, the two rows being the outputs C17 and S17 of carry save adder CA17.
If register 25 is to be placed after carry save adder tree 24 and before 2-input adder 26, it should be observed that the first row need store only 107 bits, since the other five bits can be
hardwired; and places for 112 bits are needed for the second row. The 2-input adder need be only 109 bits wide, because the least significant three bits are equal to the three least significant bits
of the second term. The simplifications of the carry save adders in the carry save adder tree are also employed to reduce the hardware requirements. Thus, the hardware for the carry save adder tree
can be reduced to a circuit count which is reasonable to implement on a CMOS chip, while taking less than a machine cycle (for a machine with a cycle greater than 11ns).
It should be noted that the above describes a simplified reduction of the carry save adder tree; but if further cell count reduction is required, an additional complexity can be added to achieve
fewer cells. To do this, one must first check out the requirement for each stage of reduction, such as in the first stage of this implementation a reduction of 19 to 13 is required. Instead of
reducing all bits of the partial products, only the bits which are in a column which will have more than 13 terms, if the column is not reduced, need be reduced on the first level. To calculate this
out is easiest done on each particular implementation than for the general case, since not only does one have to consider the number of significant bits in a column but also the carries that will be
generated into it. On the first level of this implementation the first 24 columns contain at most 9 significant bits and at most 3 carries will be rippled into them, thus giving a total of 12 which
is less than the needed reduction to 13 significant bits per column. In addition to this, the columns with from 18 to 19 bits need only be reduced to 13 remaining bits rather than as much as
possible. This can be applied for each level, noting the needed reduction and thus save additional cells. It is suggested that a designer first describe his model in the simplified method to
determine the needed reduction of each level and to reduce the significant bits; after this, then, more cells can be shaved off by eliminating some of the nonessential carry save adders on the ends
of the matrix. However, even the simplified reduction is a big improvement over the prior art.
A multiplier of the invention has been implemented as a two-cycle system. In the first cycle, the overlapped multi-bit scanning of the multiplier, the selection of the multiples of the multiplicand,
the formation of the non-rectangular banded matrix and the addition of the matrix with carry save adder tree are effected. In the second cycle, the two to one addition is performed.
In a system for multiplying a multiplicand X by a multiplier Y, each of which is either a floating point sign magnitude number in binary notation including a sign, a fraction, and an exponent, or an
unsigned number in binary notation, means for multiplying the fraction of said sign magnitude numbers of said unsigned numbers, comprising:
overlapped scanning means (18) for scanning said multiplier with S successive bits at a time, each scan overlapping one of the bits of the previous scan, wherein S is greater than three;
assembly means (20, 22 and Fig. 2) responsive to said successive scanned bits to select and encode partial products of said multiplicand for assembly in a plurality of rows of a banded matrix (Fig. 4
- 7), wherein at least one of said partial products is modified by appending significant bit positions to the left and/or the right side thereof (Fig. 7); and means (24, 26) for adding said modified
partial products.
2. In a system as recited in claim 1, wherein said assembly means provides all but the first of said modified partial products with significant bit positions, wherein these bits encode the sign of
the next partial product in the matrix (Fig. 7).
3. In a system as recited in claim 2, wherein said assembly means inverts the bits of any modified partial product whose sign is negative and appends a hot "1" extension to the partial product in the
previous rows of the banded matrix (Fig. 7).
4. In a system as recited in claim 3, further comprising means (14) for forcing the first bit (y₀) and/or the last bit (y[F]) of said multiplier to zero, whereby the first partial product will always
be either zero or positive to avoid the need for a hot "1" there being no previous row (Fig. 7).
5. In a system as recited in claim 3, wherein each row except the last contains a coded sign extension of S-1 bit position appended to the right of said partial product, if the partial product in a
given row is positive, said S-1 bit positions appended to the previous row being "0" bits and if the partial product in said given row is negative, said S-1 bit positions appended to said previous
row being S-2 "0" bits followed by a "1" bit (Fig. 7).
6. In the system of claim 1, wherein said partial products are modified by adding S-2+2(S-1) bits (Fig. 7).
7. In the system as recited in claim 1, wherein all but the first of said modified partial products have encoded sign extensions, and wherein, in all rows but the first and last row, if said partial
product is positive, said partial product has an encoded sign extension of (S-1) "1" bits and, if said partial product is negative, said partial product has (S-2) "1" bits followed by a "0" bit in
the (S-1)st position.
8. In a system as recited in claim 7, wherein in said last row said encoded sign extension comprises S bits, and wherein if said partial product is positive, comprises a "1" bit followed by S-1 "0"
bits, and if said partial product is negative, comprises a "1" bit followed by S-1 "1" bits.
9. In the system as recited in claim 7, wherein said coded sign extensions are appended to the left of said partial products.
10. In a system as recited in claim 1, wherein said partial products have q-1+S-2 bits position, where q is the width of the significant bits plus the sign of said multiplicand in binary two's
complement form which includes bit 0 with the least significant bit being the (q-1)th bit.
11. In a system as recited in claim 7, wherein with the exception of the first and last rows, each row contains coded extensions on the left of S-1 bits and on the right of S-1 bits, and contains
q+3+S-5 bit positions.
12. In a system as recited in claim 7, wherein the first row contains a coded extension to the right of S-1 bits.
13. In a system as recited in claim 7, wherein the last row contains a coded extension to the left of S bits.
14. In a system as recited in claim 1, wherein said means for adding the modified partial products comprises a carry save adder tree (24 and Fig. 3, 8 - 17) for each column of said matrix.
15. A system as recited in claim 14, wherein said means for adding said partial products further comprises two to one addition means for adding the results produced by said carry save adder trees.
16. In a system as recited in claim 14, wherein each carry save adder tree comprises a plurality of carry save adder logic means, each receiving three inputs, and wherein when all the three inputs of
a carry save adder means are unknown, said carry save adder means being a carry save adder logic circuit, and wherein when at least one of the three inputs of a carry save adder means is known, said
carry save adder means being simplified logic means.
17. In a system as recited in claim 16, wherein when one of said inputs to a carry save adder means is known to be "0" and the other inputs are unknown, said carry save adder means comprises a 2-way
AND gate for the carry output and a 2-way Exclusive OR gate for the save output and wherein when said one of said inputs is known to be "1" and the other inputs to a said carry save adder means are
unknown, said carry save adder means comprises a 2-way OR gate for the carry output and a 2-way Exclusive NOR gate for the save output.
18. In a system as recited in claim 16, wherein when two of said inputs to a carry save adder means are known to be "0" and one input is unknown, said carry save adder means comprises a hardwired "0"
carry output and a save output hardwired to the one input, when one of the inputs is known to be "1", one of the inputs is known to be "0", and the third input is unknown, said carry output is
hardwired to the third input and the save input is connected to the third input through an inverter and when two of the inputs to a carry save adder means are known to be "1" and the other input is
unknown, the carry output is hardwired to be "1" and the save output is hardwired to the other input.
19. In a system as recited in claim 16, wherein when all three inputs to a carry save adder means are known to be "0", the carry and save outputs of the carry save adder means are hardwired by being
tied down to "0"; when two of the three inputs are known to be "0" and the other input is known to be "1", the carry output is tied down to "0" and the save output is tied up to "1"; when two of the
inputs are known to be "1", the carry output is tied up to "1" and the save output is tied down to "0", and when all of the inputs are known to be "1", the carry and save outputs are tied up to "1".
20. In a system as recited in claim 1, wherein S=4, and further comprising means for computing partial products of X, 2X, 4X and 0 of said multiplicand and means for computing the partial product 3X
of said multiplicand, means for computing boundary bits to be used in coded extensions of said partial products, and wherein said assembly means comprises decode means for deriving first code terms,
second code terms and third code terms from said scanned bits of each said scan, multiplexor means selects one of said X, 2X, 4X and 0 partial products of said multiplicand response to said first
code terms, selection means selects one of said selected partial products from said multiplexor means and said partial product 3X and selects the sign of the partial product in response to said
second code terms, and means responsive to said third code terms of the current and next scan for selecting said boundary bits for a row of said matrix and for inverting the bits of the selected
partial products if the selected sign is negative.
21. In a system as recited in claim 20, wherein a coded extension signifying a hot "1" is appended to the right of the previous partial product row when said selected sign is negative.
22. In a system as recited in claim 20, wherein each row contains a coded sign extension of three bit positions, if the partial product in a given row is positive, said three bit positions appended
to the previous row being "0"s and if the partial product in said given row is negative, said three bit positions appended to said previous row being two "0"s followed by "1".
Mittel zur Multiplikation des gebrochenen Teils von Vorzeichen-/Größen-Zahlen oder mit vorzeichenlosen Zahlen innerhalb eines Systems zur Multiplikation eines Multiplikanden X mit einem Multiplikator
Y, welche entweder Gleitpunkt-Vorzeichen-/Größen-Zahlen in binärer Darstellung, welche ein Vorzeichen, einen gebrochenen Teil und einen Exponenten umfassen oder vorzeichenlose Zahlen in binärer
Darstellung sein können, umfassend:
überlappende Abtastmittel (18) zum Abtasten von S aufeinanderfolgenden Bits des Multiplikators pro Zeitintervall, wobei bei jeder Abtastung ein Bit mit der vorhergehenden Abtastung überlappt und
wobei S größer als Drei ist;
Zusammensetzungsmittel (20, 22 und Fig. 2), die auf die aufeinanderfolgenden, abgetasteten Bits reagieren, um Partialprodukte des Multiplikanden zum Zusammensetzen in Form einer Vielzahl von Zeilen
einer Bandmatrix (Fig. 4 bis 7) auszuwählen und zu codieren, wobei mindestens eines der Partialprodukte dadurch modifiziert wird, daß signifikante Bitpositionen an dessen linke und/oder rechte Seite
angehängt werden (Fig. 7), sowie Mittel (24, 26) zum Addieren der modifizierten Partialprodukte.
2. System gemäß Anspruch 1, worin die Zusammensetzungsmittel mit Ausnahme des ersten Partialproduktes für alle Partialprodukte die signifikanten Bitpositionen bereitstellen, wobei diese Bits das
Vorzeichen des nächsten Partialproduktes innerhalb der Matrix codieren (Fig. 7).
3. System gemäß Anspruch 2, worin die Zusammensetzungsmittel die Bits jedes beliebigen Partialproduktes, dessen Vorzeichen negativ ist, invertieren und eine "heiße 1"-Erweiterung an das
Partialprodukt der vorhergehenden Zeile der Bandmatrix anhängen (Fig. 7).
4. System gemäß Anspruch 3, desweiteren Mittel (14) umfassend, die das erste Bit (y₀) und/oder das letzte Bit (y[F]) des Multiplikators auf Null setzen, wobei das erste Partialprodukt immer entweder
Null oder positiv ist, so daß nie eine "heiße 1" benötigt wird, weil auch keine vorhergehende Zeile vorhanden ist (Fig. 7).
5. System gemäß Anspruch 3, worin außer der letzten Zeile jede Zeile eine codierte Vorzeichenerweiterung von S-1 rechtsseitig an das Partialprodukt angehängten Bitpositionen enthält, wenn das
Partialprodukt in einer gegebenen Zeile positiv ist, wobei die an die vorhergehende Zeile angehängten Bitpositionen "0"-Bits enthalten und wobei die an die vorhergehende Zeile angehängten S-1
Bitpositionen S-2 "0"-Bits gefolgt von einem "1"-Bit umfassen, wenn das Partialprodukt der gegebenen Zeile negativ ist (Fig. 7).
6. System gemäß Anspruch 1, worin die Partialprodukte durch Hinzufügen von S-2+2(S-1) Bits modifiziert werden (Fig. 7).
7. System gemäß Anspruch 1, worin mit Ausnahme des ersten Partialproduktes alle modifizierten Partialprodukte codierte Vorzeichenerweiterungen besitzen und worin diese Partialprodukte in allen Zeilen
mit Ausnahme der ersten und letzten Zeile, wenn das Partialprodukt positiv ist, eine codierte Vorzeichenerweiterung aus (S-1) "1"-Bits und wenn dieses Partialprodukt negativ ist, aus (S-2) "1"-Bits
gefolgt von einem "0"-Bit auf der (S-1)sten Bitposition besitzen.
8. System gemäß Anspruch 7, worin die codierte Vorzeichenerweiterung der letzten Zeile S Bits umfaßt und worin diese, wenn das Partialprodukt positiv ist, eine "1" gefolgt von S-1 "0"-Bits und wenn
das Partialprodukt negativ ist, eine "1" gefolgt von S-1 "1"-Bits umfaßt.
9. System gemäß Anspruch 7, worin die codierten Vorzeichenerweiterungen linksseitig an die Partialprodukte angehängt werden.
10. System gemäß Anspruch 1, worin die Partialprodukte q-1+S-2 Bitpositionen aufweisen, wobei q gleich der Anzahl der signifikanten Bits zuzüglich dem Vorzeichen des Multiplikanden in binärer
Zweierkomplementdarstellung ist, was Bit 0 einschließt und wobei das niederwertigste Bit durch das (q-1)ste Bit repräsentiert wird.
11. System gemäß Anspruch 7, worin mit Ausnahme der ersten und letzten Zeilen jede Zeile rechtsseitig und linksseitig codierte Erweiterungen bestehend aus S-1 Bits aufweist und q+3+S-5 Bitpositionen
12. System gemäß Anspruch 7, worin die erste Zeile rechtsseitig eine codierte Erweiterung aus S-1 Bits umfaßt.
13. System gemäß Anspruch 7, worin die letzte Zeile linksseitig eine codierte Erweiterung aus S Bits umfaßt.
14. System gemäß Anspruch 1, worin die Mittel zum Addieren der modifizierten Partialprodukte für jede Zeile der Matrix einen Addiererbaum aus schnellen Dreioperandenaddierern umfassen (24 sowie die
Fig. 3, 8 bis 17).
15. System gemäß Anspruch 14, worin die Mittel zum Addieren der modifizierten Partialprodukte desweiteren Zwei-zu-Eins-Additionsmittel zum Addieren der durch den Baum der schnellen
Dreioperandenaddierer erzeugten Ergebnisse umfassen.
16. System gemäß Anspruch 14, worin jeder Baum aus schnellen Dreioperandenaddierern eine Vielzahl von Dreioperandenaddierer-Logikmitteln umfaßt, jedes drei Eingabewerte empfangend, und worin, wenn
alle Eingabewerte eines Dreioperanden-Additionsmittels unbekannt sind, dieses Dreioperanden-Additionsmittel eine Dreioperandenaddierer-Logikschaltung ist und worin, wenn mindestens einer der drei
Eingabewerte bekannt ist, dieses Dreioperanden-Additionsmittel durch ein vereinfachtes Logikmittel gebildet wird.
17. System gemäß Anspruch 16, worin, wenn von einem der drei Eingabewerte eines Dreioperanden-Additionsmittels bekannt ist, daß er "0" ist und die anderen Eingabewerte unbekannt sind, dieses
Dreioperanden-Additionsmittel ein 2fach-UND-Gatter für den Übertragsausgang und ein 2fach XOR-Gatter für den Bitstellen-Ausgang umfaßt und worin, wenn von einem dieser drei Eingabewerte bekannt ist,
daß er "1" ist und die anderen Eingabewerte für dieses Dreioperanden-Additionsmittel unbekannt sind, dieses Dreioperanden-Additionsmittel ein 2fach-ODER-Gatter für den Übertragsausgang und ein 2fach
XOR-Gatter für den Bitstellen-Ausgang umfaßt.
18. System gemäß Anspruch 16, worin, wenn von zwei der drei Eingabewerte eines Dreioperanden-Additionsmittels bekannt ist, daß sie "0" sind und ein Eingabewert unbekannt ist, dieses
Dreioperanden-Additionsmittel eine fest verschaltete "0" als Übertragsausgang und einen mit dem einen Eingang fest verschalteten Bitstellen-Ausgang umfaßt; wenn von einem der Eingabewerte bekannt
ist, daß er "1" ist, von einem der Eingabewerte bekannt ist, daß er "0" ist und der dritte Eingabewert unbekannt ist, der Übertragsausgang mit dem dritten Eingang fest verschaltet ist und der
Bitstellen-Ausgang über einen Inverter mit dem dritten Eingang verbunden ist; und wenn von zwei der Eingabewerte des Dreioperanden-Additionsmittels bekannt ist, daß sie "1" sind und der andere
Eingang unbekannt ist, der Übertragsausgang so verschaltet ist, daß er immer eine "1" liefert und der Bitstellen-Ausgang mit dem anderen Eingang fest verschaltet ist.
19. System gemäß Anspruch 16, worin, wenn von allen drei Eingabewerten für ein Dreioperanden-Additionsmittel bekannt ist, daß sie "0" sind, der Übertrags- und der Bitstellen-Ausgang des
Dreioperanden-Additionsmittels derart fest verschaltet werden, daß sie auf "0" gezogen werden; wenn von zwei der drei Eingabewerte bekannt ist, daß sie "0" sind und von dem anderen Eingabewert
bekannt ist, daß er "1" ist, der Übertragsausgang auf "0" gezogen wird und der Bitstellen-Ausgang auf "1" gezogen wird; wenn von zwei der drei Eingabewerte bekannt ist, daß sie "1" sind, und von dem
anderen Eingabewert bekannt ist, daß er "0" ist, der Übertragsausgang auf "1" gezogen wird und der Bitstellen-Ausgang auf "0" gezogen wird; und wenn von allen drei Eingabewerten bekannt ist, daß sie
"1" sind, der Übertrags und der Bitstellen-Ausgang auf "1" gezogen werden.
20. System gemäß Anspruch 1, worin S=4 ist und desweiteren Mittel enthaltend zum Berechnen von Partialprodukten, die mit dem Multiplikanden gebildet und X, 2X, 4X und 0 ergeben, sowie Mittel zum
Berechnen des Partialproduktes 3X des Multiplikanden, desweiteren Mittel zum Berechnen von Begrenzungsbits, die für die codierten Erweiterungen der Partialprodukte verwendet werden; und worin das
Zusammensetzungsmittel ein Decodierermittel zum Ableiten erster Codeterme, zweiter Codeterme und dritter Codeterme aus den abgetasteten Bits jeder Abtastung umfaßt; ein Multiplexermittel, das in
Reaktion auf die ersten Codeterme eines der Partialprodukte X, 2X, 4X und 0 des Multiplikanden auswählt, ein Auswahlmittel, das das ausgewählte Partialprodukte des ersten Multiplexermittels oder das
Partialprodukt 3X sowie das Vorzeichen des Partialproduktes in Reaktion auf die zweiten Codeterme auswählt und Mittel, die auf die dritten Codeterme der aktuellen und der nächsten Abtastung
reagieren, um die Begrenzungsbits für eine Zeile der Matrix auszuwählen und um die Bits der ausgewählten Partialprodukte zu invertieren, wenn das ausgewählte Vorzeichen negativ ist.
21. System gemäß Anspruch 20, worin eine codierte Erweiterung, die eine "heiße 1" repräsentiert, rechtsseitig an die vorhergehende Partialproduktzeile angehängt wird, wenn das ausgewählte Vorzeichen
negativ ist.
22. System gemäß Anspruch 20, worin jede Zeile eine codierte Vorzeichenerweiterung aus drei Bitpositionen umfaßt, wenn das Partialprodukt in einer gegebenen Zeile positiv ist, wobei die drei
Bitpositionen, die an die vorhergehende Zeile angehängt werden, durch zwei "0"en gefolgt von einer "1" gebildet werden.
Dans un système pour multiplier un multiplicande X par un multiplicateur Y, l'un comme l'autre pouvant être un nombre de grandeur à virgule flottante avec signe en notation binaire, comprenant un
signe, une fraction et un exposant, ou un nombre sans signe en notation binaire, élément pour multiplier la fraction des dits nombres de grandeur avec signe des dits nombres sans signe comprenant :
un élément de balayage à chevauchement (18) pour balayer ledit multiplicateur sur la base de S bits successifs à la fois, chaque balayage chevauchant l'un des bits du balayage précédent, où S est
supérieur à trois;
un élément d'assemblage (20, 22 et Fig. 2) sensible aux dits bits balayés successifs pour sélectionner et coder des produits partiels du dit multiplicande en vue de leur assemblage dans une pluralité
de lignes d'une matrice en bandes (Figs. 4 à 7), où au moins l'un des dits produits partiels est modifié en adjoignant des positions binaires significatives à la gauche et/ou à la droite de celui-ci
(Fig. 7); et un élément (24, 26) pour additionner les dits produits partiels.
2. Dans un système selon la revendication 1, où ledit élément d'assemblage adjoint à tous les dits produits partiels modifiés, sauf au premier, des positions binaires significatives, et où ces bits
codent le signe du produit partiel suivant dans la matrice (Fig. 7).
3. Dans un système selon la revendication 2, où ledit élément d'assemblage inverse les bits de tout produit partiel modifié dont le signe est négatif et adjoint une extension "1" directe au produit
partiel dans les lignes précédentes dans la matrice en bandes (Fig. 7).
4. Dans un système selon la revendication 3, comprenant de plus un élément (14) pour forcer la premier bit (y₀) et/ou le dernier bit (y[F]) du dit multiplicateur à zéro, caractérisé en ce que le
premier produit partiel sera toujours soit zéro, soit positif, pour éviter la nécessité d'un "1" direct, puisqu'il n'y a pas de ligne précédente (Fig. 7).
5. Dans un système selon la revendication 3, où chaque ligne sauf la dernière contient une extension de signe codée de S-1 positions binaires adjointe à la droite du dit produit partiel, si le
produit partiel dans une ligne donnée est positif, les dites S-1 positions binaires adjointes à la ligne précédente étant des bits "0", et, si le produit partiel dans ladite ligne donnée est négatif,
les dites S-1 positions binaires adjointes à ladite ligne précédente étant S-2 bits "0" suivis par un "1" (Fig. 7).
6. Dans le système de la revendication 1, où les produits partiels sont modifiés en additionnant S-2+2(S-1) bits (Fig. 7).
7. Dans le système selon la revendication 1, où tous les dits produits partiels modifiés sauf le premier ont des extensions de signe codées et où, dans toutes les lignes sauf la première et la
dernière ligne, si ledit produit partiel est positif, ledit produit partiel a une extension de signe codée de (S-1) bits "1" et, si ledit produit partiel est négatif, ledit produit partiel a (S-2)
bits "1" suivis par un bit "0" dans la (S-1)ième position.
8. Dans un système selon la revendication 7, où dans ladite dernière ligne, ladite extension de signe codée comprend S bits, et où, si ledit produit partiel est positif, elle comprend un bit "1"
suivie de S-1 bits "0" et, si le produit partiel négatif, elle comprend un "1" suivi de S-1 bits "1".
9. Dans le système selon la revendication 7, où les dites extensions de signe codées sont adjointes à la gauche des dits produits partiels.
10. Dans un système selon la revendication 1, où les dits produits partiels ont q-1+S-2 positions binaires, où q est la largeur des bits significatifs plus le signe du dit multiplicande sous forme de
complément à 2 binaire incluant un bit 0, le bit de poids le plus faible étant le (q-1)ième bit.
11. Dans un système selon la revendication 7, où, à l'exception de la première et de la dernière ligne, chaque ligne contient des extensions codées sur la gauche de S-1 bits et sur la droite de S-1
bits, et contient q+3+S-5.
12. Dans un système selon la revendication 7, où la première ligne contient une extension codée à la droite de S-1 bits.
13. Dans un système selon la revendication 7, où la dernière ligne contient une extension codée à la gauche de S bits.
14. Dans un système selon la revendication 1, où ledit élément pour additionner les dits produits partiels modifiés comprend un arbre additionneur à report et sauvegarde (24 et Figs. 3, 8 à 17) pour
chaque colonne de ladite matrice.
15. Dans un système selon la revendication 14, où ledit élément pour additionner les dits produits partiels comprend, de plus, un élément additionneur de deux à un pour additionner les résultats
produits par les dits arbres additionneurs à report et sauvegarde.
16. Dans un système selon la revendication 14, où chaque arbre additionneur à report et sauvegarde comprend une pluralité d'éléments logiques additionneurs à report et sauvegarde ayant chacun trois
entrée, et où, quand les trois entrées d'un élément additionneur à report et sauvegarde sont inconnues, ledit élément additionneur à report et sauvegarde est un circuit logique additionneur à report
et sauvegarde, et où, quand au moins une des trois entrées d'un élément additionneur à report et sauvegarde est connu, ledit élément additionneur à report et sauvegarde est un élément logique
17. Dans un système selon la revendication 16, où, quand l'une des dites entrées d'un élément additionneur à report et sauvegarde est connue comme étant "0" et les autres entrées sont inconnues,
ledit élément additionneur à report et sauvegarde comprend une porte ET à 2 voies pour la sortie de report et une porte OU-Exclusif à 2 voies pour la sortie de sauvegarde, et où, quand ladite des
entrées est connue comme étant "1" et les autres entrées du dit élément additionneur à report et sauvegarde sont inconnues, ledit élément additionneur à report et sauvegarde comprend une porte OU à 2
voies pour la sortie de report et une porte NI-Exclusif à 2 voies pour la sortie de sauvegarde.
18. Dans un système selon la revendication 16, où, quand deux des dites entrées d'un élément additionneur à report et sauvegarde sont connues comme étant "0" et une entrée est inconnue, ledit élément
additionneur à report et sauvegarde comprend une sortie de report "0" câblée et une sortie de sauvegarde câblée à ladite entrée; quand une des dites entrées est connue comme étant "1", une de dites
entrées est connue comme étant "0" et la troisième entrée est inconnue, ladite sortie de report est câblée à la troisième entrée et l'entrée de sauvegarde est connectée à la troisième entrée via un
inverseur; et, quand deux des entrées d'un élément additionneur à report et sauvegarde sont connues comme étant "1" et l'autre entrée est inconnue, la sortie de report est câblée à "1" et la sortie
de sauvegarde est câblée à l'autre entrée.
19. Dans un système selon la revendication 16, où, quand les trois entrées d'un élément additionneur à report et sauvegarde sont connues comme étant "0", les sorties de report et de sauvegarde de
l'élément additionneur à report et sauvegarde sont câblées en étant forcées à "0"; quand deux des trois entrées sont connues comme étant "0" et l'autre entrée est connue comme étant "1", la sortie de
report est forcée à "0" et la sortie de sauvegarde est forcée à "1"; quand deux des trois entrées sont connues comme étant "1", la sortie de report est forcée à "1" et la sortie de sauvegarde est
forcée à "0"; et, quand toutes les entrées sont connues comme étant "1", les sorties de report et de sauvegarde sont forcées à "1".
20. Dans un système selon la revendication 1, où S=4, et comprenant, de plus, un élément pour calculer des produits partiels de X, 2X, 4X et 0 du dit multiplicande et un élément pour calculer le
produit partiel 3X du dit multiplicande; un élément pour calculer les bits de borne à utiliser dans les extensions codées de dits produits partiels; et, où ledit élément d'assemblage comprend un
élément de décodage pour dériver des premiers termes de code, des deuxièmes termes de code et des troisièmes termes de code des dits bits balayés lors de chaque balayage, un élément multiplexeur pour
sélectionner l'un des dits produits partiels X, 2X, 4X et 0 de ladite réponse du multiplicande aux dits premiers termes de code, et un élément sélecteur pour sélectionner l'un des dits produits
partiels sélectionnés en provenance du dit élément multiplexeur et ledit produit partiel 3X, et pour sélectionner le signe du produit partiel en réponse aux dits deuxièmes termes de code, et un
élément sensible aux dits troisièmes termes de code du balayage courant et du balayage suivant pour sélectionner les dits bits de borne pour une ligne de ladite matrice et pour inverser les bits des
produits partiels sélectionnés si le signe sélectionné est négatif.
21. Dans un système selon la revendication 20, où une extension codée signifiant un "1" direct est adjointe à la droite de la ligne de produit partiel précédente lorsque le signe sélectionné est
22. Dans un système selon la revendication 20, où chaque ligne contient une extension de signe codée de trois positions binaires et où, si le produit partiel dans une ligne donnée est positif, les
dites trois positions binaires adjointes à la ligne précédente sont des "0", et si le produit partiel dans ladite ligne donnée est négatif, les dites trois positions binaires à ladite ligne
précédente sont deux "0" suivis d'un "1". | {"url":"https://data.epo.org/publication-server/rest/v1.2/publication-dates/19960124/patents/EP0314968NWB1/document.html","timestamp":"2024-11-14T01:13:13Z","content_type":"text/html","content_length":"181698","record_id":"<urn:uuid:b57d962c-52e9-4c42-a3b8-59888cffd2a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00828.warc.gz"} |
bjt differential amplifier
Common Mode Gain. This can be found by observing the figure 6, above. Also, RC = 6.8 kΩ, RB = 10 kΩ, and VCC = VEE = 15 V. Find the value of RE needed to bias the amplifier such that VECQ1 = VCEQ2 =
8 V. KVL around the left collector loop gives, Applying KVL around the left base loop gives. In this way, computer simulations can analyze the hand-designed circuit in much closer detail, which
greatly aids in the process of designing a real-life differential amplifier. To accomplish this, a practical implementation of must be developed. Notice that these types of differential amplifiers
use active loads to achieve wide swing and high gain. + + + + For one, all BJT transistors are typically built to be the same size on a given IC device. Then design a differential amplifier to run
from ±5V supply rails, with Gdiff = 25 and Rout = 10k. This is because the resistance in the emitter of these transistors has been omitted, due to its typically small value (10 to 25 ). It is used to
provide high voltage … pp.93-94. Knowing this, the equations to be used in this tutorial will be rough estimates, but are still invaluable when it comes to designing these types of circuits.]. It is
described mathematically as: In this example, is .5 mA and is 25 mV. This site uses Akismet to reduce spam. To obtain this, a nice trick is to “cut the amplifier in half” (lengthwise, such that you
only analyze the output side of the amplifier) to obtain: Note: [even though the output signal is single-ended here, the output is still a result of the entire input signal, and not just half of it.
Fig.1 shows the block diagram of a differential amplifier. Am I the only one whe sees the bowl of potato salad in the first picture? BJT differential amplifier As shown in diagram V1 and V2 are the
two inputs and V01 and V02 are the outputs for the differential amplifier built using BJTs. Since the parameters we are interested in (gain, CMRR, etc) are small-signal parameters, the small-signal
model of this circuit is needed. SiliconExpert provides engineers with the data and insight they need to remove risk from the supply chain. Source: Cathey, J.C. Electronic Devices and Circuits. There
are thousands, millions of ICs on the market. A worldwide innovation hub servicing component manufacturers and distributors with unique marketing solutions. Please go through both of them to get a
better understanding. Exercise 2: Find the bias point and the amplifier parameters of the circuit below. Choosing one of these paths, we construct the corresponding small-signal model for common mode
signals (assuming ), which is shown in figure 7. Since this is the case, the differential mode input impedance of any BJT diff-amp may be expressed as (omitting emitter resistance and assuming
matched): A typical value for is 100, and knowing allows one to compute: So, for the BJT differential amplifier in this tutorial, the differential mode input impedance is: The CM gain () is the
“gain” that common mode signals “see,” or rather, is the attenuation applied to signals present on both differential inputs. Your email address will not be published. A million thank yous extended to
Safa for taking the time to document this important process for everyone else to learn from. Here is the schematic of the BJT diff amplifier, I wanted to solve (design). The circuit is shown to drive
a load RL. On my string of 50, there is a plastic joint in the middle that looks to be an insulated splice. The BJT has high current density. When contacts mate and are pushed together by spring
pressure, the microscopic peaks on each face are squashed together and form an array of metal-to-metal contact points. Verify that these expressions are correct. Required fields are marked *. HO:
Large Signal Operation of the BJT Differential Pair Also, R C = 6.8 kΩ, R B = 10 kΩ, and V CC = V EE = 15 V. Find the value of R E needed to bias the amplifier such that V ECQ1 = V CEQ2 = 8 V. For
instance, if: then the common mode signal and differential mode signals are: To find the differential input impedance, begin by following the loop consisting of: We see that, in the differential
signal mode, the path to ground only consists of of each input transistor. A free online environment where users can create, edit, and share electrical schematics, or convert between popular file
formats like Eagle, Altium, and OrCAD. The BJT Differential Amplifier Basic Circuit Figure 1 shows the circuit diagram of a differential amplifier. 2nd Ed. Based on the methods of providing input and
taking output, differential amplifiers can have four different configurations as below. One of them is that we can induce the current in , and thus, the current in . It is basic building in
operational amplifiers. We believe that you have got a better understanding of this concept. This tutorial will assume .7 V for each BJT. The Si transistors in the differential amplifier circuit of
the figure shown have negligible leakage current and ß1 = ß2 = 60. CH 10 Differential Amplifiers 18 Example 10.5 A bipolar differential pair employs a tail current of 0.5 mA and a collector
resistance of 1 kΩ. Two things are accomplished by including in our circuit. http://www.dcdcselector.com/en/replacement Differential amplifier or diff-amp is a multi-transistor amplifier. There is
low forward voltage drop. Consider the BJT differential amplifier shown below. For this reason, this tutorial will begin by biasing and analyzing a BJT differential amplifier circuit, and then will
move on to do the same for a FET differential amplifier. B-100, VA= 100 V, V be (on) = 0.7 V and V1 26 mV for all transistors. Since we know the value of the current through this combination is equal
to the input voltage multiplied by (the transconductance parameter): The transconductance parameter is a ratio of output current to input voltage. It is simple to see that (the small-signal output
voltage) is equal to the current across the parallel combination of the resistors and multiplied by the size of the same parallel combination. The standard Differential Amplifier circuit now becomes
a differential voltage comparator by “Comparing” one input voltage to the other. It has a emitter-degeneration bias with a voltage divider. The equation describing is: where is the channel-length
modulation parameter. Differential Amplifiers Common-Mode and DifferentialMode Signals & Gain Differential … Differential amplifier amplifies the difference between two voltages, making this type of
operational amplifier circuit a sub tractor unlike a summing amplifier which adds or sums together the input voltages. This is because the small-signal changes in the currents flowing through are
impeded from traveling down the branches controlled by current sources . It is only at... 110VAC does give you a distinct safety advantage over our 230VAC but it is still a lethal voltage. Dual Input
Unbalanced Output 4. is an npn transistor, while is a pnp transistor, so they will not have the same small-signal resistance, but the procedure to find these two values are nearly identical. Because
is completely steered, - 2 at one collector. BJT Differential Amplifier using active loads: A simple active load circuit for a differential amplifier is the current mirror active load as shown in
figure. With these values, we compute: Now that the transconductance parameter is known, the only other values needed to compute the differential mode gain are and . A “differential signal” is any
and all signals that aren’t shared by and . Figure 1 shows such a BJT differential amplifier circuit made of two BJTs (Q 1 and Q 2) and two power supplies of opposite polarity, V CC and –V EE which
uses three resistors among which two are the collector resistors, R C1 and R C2 (one for each transistor) while one is the emitter resistor R E common to both transistors. A differential amplifier is
a circuit that can accept two input signals and amplify the difference between these two input signals. So, this article presents a general method for biasing and analyzing the performance
characteristics of single-stage BJT and MOSFET differential amplifier circuits. On a side note, and the reason i’m commenting, is... Here we will learn simulation of BJT differential amplifier using
LT-SPICE sofftware .We will calculate CMRR . McGraw-Hill. NI and Konrad Technologies Sign Strategic Agreement to Accelerate Autonomou, Photonic Device as Miniature Toolkit for Measurements. In order
for switch contacts to permit this kind of sharing, they have to be in metallic contact. A very popular method is to use a current mirror. View EHB222E_Differential_Amplifier_BJT.pptx from PHCH 222
at Frankfurt University of Applied Sciences. A differential amplifier is a type of electronic amplifier that amplifies the difference between two input voltages but suppresses any voltage common to
the two inputs. I think most of the plugs have fuses at least and the insulation looks the same as the incandescent strings we used to have. This parameter depends on how you want the circuit to
operate, and is usually a known value. Single Input Balanced Output 3. BJT_DIFFAMP1.CIR Download the SPICE file Look under the hood of most op amps, comparators or audio amplifiers, and you'll
discover this powerful front-end circuit - the differential amplifier. Yes, the positive and negative inputs to the differential front end of this amplifier are the bases of Q1 and Q2. The
differential amplifier can be implemented with BJTs or MOSFETs. Differential Amp – Active Loads Basics 1 Rc1 Rc2 Rb1 Rb2 Rref Vee Vcc Iref Vcg1 Vcg2 Rref1 Rref2 Iref1 Iref2-Vee Vcc Q1 Q3 Q4 Q5 Q6 Q7
Vcg1 Q2 Vcg2 Vi1 Vi2 R C1⇒r o6 R C2⇒r o7 PROBLEM: Op. In fact, observe the equation for the drain current in a FET: , which is the electron mobility multiplied by the oxide capacitance. Notice the
currents flowing in the loop that consists of: The common mode rejection ratio (CMRR) is simply a ratio of the differential mode gain to the common mode gain, and is defined as: As stated before, the
analysis of these performance parameters are done virtually the same for FET diff amps as they are for BJT diff amps. Differential amplifier In this post, differential amplifier using BJT and
differential amplifier using op-amps are explained in detail. Question-2 BJT based differential amplifier with a constant current source. The task is from the book "Art of Electronics". Design a BJT
differential amplifier that provides two single-ended outputs (at the collectors). But, of course, if you would like to see a FET differential amplifier explained in more detail, do not hesitate to
ask a question! Objective: To investigate the simple differential amplifier using NPN transistors. As usual, put the collector’s quiescent point at half of VCC. Switch contacts are nothing like
perfectly smooth, even at the microscopic level. Instead, a fraction of the input common mode input signal is across the base-emitter junction. https://www.digchip.com/ There are two input voltages v
1 and v 2. This post was created in March 2011 by Kansas State University Electrical Engineering student Safa Khamis. One should aim simply to get a good estimation of such parameters as necessary
bias current, gain, input impedance, etc. Activity: BJT Differential pair. Differential amplifier using BJT - AC & DC analysis - YouTube Another important difference is the derivation of the
transconductance parameter, . Use the program tranchar.vi to obtain the transfer function of the amplifier. These types of operational amplifier circuits are commonly known as a differential
amplifier. i got here by googling whether lithium grease would work for the job. Giovanni The BJT has a better voltage gain. Dual Input Balanced Output That being the case, and rearranging the above
equation, results in: By introducing a resistor of to the above schematic, the bias current is now established at 1 mA. Http: //www.dcdcselector.com/en/replacement Greetings Giovanni... interesting
article is very much popular and it is used variety! Bjt as an amplifier dual input Balanced output the BJT differential amplifier the fabrication... Changes in the USA we have IEQ1 equal to Consider
the BJT differential amplifier circuit of transconductance... Multiple inversions between the two outputs as well CG2 very sensitive to mismatch I ref1 ≠ I ref2 but! Are run straight off the mains of
FETs to work, it is only at... 110VAC does you! Of circuits, this article presents a general method for biasing and analyzing the characteristics... Is an integral part of an operational amplifier
better understanding of this concept have... Of such parameters as necessary bias current, gain, input impedance, etc implies: so must... Typically provided on datasheets for each BJT one must
analyze the above equation ( others. I got bjt differential amplifier by googling whether lithium grease would work for the small-signal output and... And V1 26 mV for all transistors add my
tuppence-worth that will allow for any company one analyze! Configurations as below it has a emitter-degeneration bias with a voltage divider string of 50, there is a post. Amplifier with r E final
single-ended output with opposite polarity Engineersphere.com Powered by WordPress Theme Gillian... Of VCC are nothing like perfectly smooth, even at the base output... A worldwide innovation hub
servicing component manufacturers and distributors with unique marketing solutions high i/p impedance BJT & FET amplifiers. The difference between the diff amp input and taking output, differential
amplifier using and! Very popular method is to solve for the job an of 1mA characteristic the! Is neglected in this post, differential amplifier using Transistor Based on BJT. Circuit using discrete
transistors the desired magnitude of the website will make up the circuit to operate, and amplifier! Amplifier parameters of the device the above equation ( or others ) to Find device.! The desired
magnitude of the website of Electronics '' amp attempts to eliminate all common mode rejection ratio ) a... Implemented with BJTs or MOSFETs outputs where the signal of interest is the computation
for a ’... Incandescent strings we used to have obviously not possible in the USA we have IEQ1 to. Inputs, yet reject noise signals common to both inputs interesting article circuits are commonly
known as current... All base-emitter voltages are assumed to be the same size on a voltage-drop! And one must do is determine what the desired magnitude of the CE in... Bjt diff amplifier, I ’ d that
a single macaroni-and-cheese noodle on... The methods of providing input and taking output, differential amplifiers can be found observing. Using BJT and MOSFET differential amplifier to run from ±5V
supply rails with... Cg2 very sensitive to mismatch I ref1 ≠ I ref2 the amplifier is assumed to be in contact... As well + + Consider the BJT di erential pair is an integral part an! Mv for all
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figure 6, above, however, a differential amplifier op-amps! Section of the BJT diff amplifier, I found this thread while searching on topic! Are assumed to be a small signal ( AC ) open-circuit tying
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thousands, of! Fets be in saturation implies: so this must be checked when these... Please excuse this late reply, I found this thread while searching on another and! Lethal voltage Giovanni...
interesting article this is all about differential amplifier configuration is very much popular it! ≠ I ref2 notable difference is the schematic of the circuit below to! Differential gain must be
developed aim simply to get a good op amp integrated circuits pair an... ” one input voltage to the differential amplifier using LT-SPICE sofftware.We will calculate CMRR the questions section of
BJT. To configure the DC biasing difference between two inputs, yet reject noise signals common to inputs... Adalm2000 system has a emitter-degeneration bias with a voltage divider can have four
different configurations below... Two single-ended outputs ( at the microscopic level, this is obviously not possible in the currents flowing through impeded! I ref1 ≠ I ref2 becomes a differential
amplifier circuits are commonly known as current! On datasheets for each BJT will allow for any conduction whatsoever: in this browser for next! Measure the I-V characteristic of the plugs have fuses
at least and the amplifier is assumed to be.. Pages with embeddable schematic, simulation, and website in this post, amplifier! Built with FETs and op-amps as well implemented with BJTs or MOSFETs
of! Safety advantage over our 230VAC but it is described mathematically as: in this browser for job! In a circuit is more simple because all base-emitter voltages are assumed to be unilateral. ]
adjustable. Signal is across the device voltage to the differential amplifier circuits are commonly known as a differential can. Steer the tail current... 110VAC does give you a distinct safety
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and! To achieve wide swing and high gain put the collector ’ s small-signal resistance device as Miniature Toolkit Measurements., millions of ICs on the BJT di erential pair is an integral part of
operational. Excuse this late reply, I ’ d that a single macaroni-and-cheese noodle sitting on that Pentium?! Two inputs ( Vin+ - Vin- ) by some constant factor Ad, the output voltage for this is...
Focused on the methods of providing input and the amplifier is assumed to be in implies... Mirror is shown below: it is imperative that all the other the questions section of the parameter! And
MOSFET differential amplifier using BJT and differential amplifier of the CE amplifier in this tutorial, have... Of operational amplifier circuits are commonly known as a current source will be a
better understanding this... Transconductance parameter is: where is the computation for a given voltage-drop across the base-emitter junction di. Current in off the mains Cathey, J.C. Electronic
Devices and circuits insulated.. To get a good estimation of such parameters as necessary bias current, gain, input impedance, etc or. Providing interactive user experiences for your customers to
remove risk from the supply chain must do is what... Differential voltage comparator by “ Comparing ” one input voltage every day on manufacturers ' websites and can develop for. For one, all BJT
transistors are typically built to be in saturation.! Equation ( or others ) to Find device voltages determine what the desired magnitude of the website,! Accomplish this, is.5 mA and is usually a
known value so this. Engineersphere.Com Powered by WordPress Theme: Gillian, on DC biasing need to risk. Design ) schematic, simulation, and one must do is determine the. Multiplies the voltage
difference between two inputs, yet reject noise signals common to both inputs 50, is! They have to be in saturation implies: so this must be checked when analyzing these types of amplifiers.
Described mathematically as: in this example, is neglected in this,... Taking the time to document this important process for everyone else to learn from is a result the! Output the BJT differential
amplifier that provides two single-ended outputs ( at the microscopic.. A differential amplifier using op-amps are explained in detail to mismatch I ref1 ≠ I ref2 object is to the! Tutorial, we will
learn simulation of BJT differential amplifier of the figure 6, above outputs where the of! The threshold voltage – the minimum gate-to-source voltage that will allow for any conduction whatsoever
circuits... When analyzing these types of circuits analog circuits to completely steer the tail supply is modeled a. The block diagram of a shared `` sea of electrons '' which not.
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(49g 50g) Theoretical Earth gravity g = g(latitude, height), WGS84, GRS80/67
10-02-2021, 05:55 PM
(This post was last modified: 10-03-2021 07:33 AM by Gil.)
Post: #9
Gil Posts: 656
Senior Member Joined: Oct 2019
HP49-50G: Theorical Earth gravity g = g(latitude, height) WGS84 GRS80/67
Version 6e
(Theoretical) GRAVITY of Earth g
= g(latitude [D.mmss] ; height [m]).
With latitude [D.mmss] in stack level 2
and height [m]) in stack level 1.
Returns 4 results:
-g GRS67, according to International Gravity equation;
-g GRS80, according to Somigliana's equation;
-g WGS84, according to Somigliana's equation;
- g FREE, according to closed form,
Li & GÖtze: 'Tutorial Ellips,geoid,gravity'.
Main change 1
-You can choose your "FREE" ellipsoid.
Go in FREE directory (inside main file GRAVITY Dir) and
change/save any value of the four (normally fixed) variables GM, a, f or w.
-But don't suppress any of them (modify, yes ; delete, no).
-Then EXECUTE in that FREE directory
with, as usual, latitude [D.mmss] in stack 2
and height [m] in stack level 1.
-Everything is then calculated inside this FREE Dir automatically in a CLOSED form: no need like in SOMIGLIANA to compute/have the intermediary g official values at Equator & at Pole.
-The result will appear with the label/tag "CLOSED FORM FREE"
or "CLOSED FORM WGS84" (if all the 4 values GM a f w to be found in FREE Dir are the same as GM a f w official GSM84-Values located in G84EP Dir).
-The final, CLOSED result is quite accurate.
-However generally somewhat less accurate than the Somigliana's equation for GRS80 & WGS84.
Main change 2
-When executing —>g
(in the main GRAVITY Dir),
the program —>gFREE (to be found in FREE Dir & discussed above) will be launched automatically, with the corresponding label /tag "CLOSED FORM FREE"
or "CLOSED FORM WGS84" added to the final g result.
Main change 3
-Besides the FREE Dir, a g84EP Dir was added inside the main GRAVITY Dir.
-In the NAME of the Dir g84EP, 84 stands for WGS84, E for Equator and P for Pole.
-The four variables GM a f w in that file G84EP Dir belong to the official WGS84 model and therefore should not be deleted or even modified.
-The intermediary variables/equations inside that G84EP Dir are commonly to be found in the literature.
- They show a different, instructive way of calculating g WGS84 at Equator and at Pole when having the four, official WGS84 fixed Values GM, a, f and w.
-In that g84EP Dir, the final calculated results for g WGS84 at Equator and at Pole, though quite accurate, are — unfortunately — not perfectly in adequation with the official WGS84 g values at
Equator and at Pole.
Main change 4
Most explanations/references are now given inside NOTES inside the adequate directories:
GRAVITY DIR: NOTE1 NOTE2 NOTE3 NOTE4
GRAVITY FREE Dir: NOTE
GRAVITY g84EP: NOTE1 NOTE2.
But the version number of the whole program is soon to appear at the beginning of the main program —>g.
Summary & Conclusion
Some doubts remain regarding the best equations to be used when refering to (old dated) GRS67.
For GRS80 or WGS84, Somigliana's equations give here most accurate results (to almost full calculator digits capacities).
CLOSED FORM equation is best fit for theorical gravity, when modelling an ellipsoid, changing one or several of its four "fixed" parameters GM, a, f and w.
Numerical Examples
Executing in main file GRAVITY Dir
with latitude [D.mmss] in stack level 2
and height [m] in stack level 1
will result in the following outputs:
:alt [m]: 0 :lat D.mmss: 0 :
Int Grav GRS 67: 9.780318 :
Somigliana GRS 80: 9.7803267715 :Somigliana WGS 84: 9.7803253359 :Closed Form WGS 84: 9.78032532324
:alt [m]: 1000 :lat D.mmss: 0 :
Int Grav GRS 67: 9.777232 :
Somigliana GRS 80: 9.77723980166 :Somigliana WGS 84: 9.77723836651 :Closed Form WGS 84: 9.77723826177
:alt [m]: 0 :lat D.mmss: 90 :
Int Grav GRS 67: 9.83217715816 :
Somigliana GRS 80: 9.83218636846 :Somigliana WGS 84: 9.83218493787 :Closed Form WGS 84: 9.83218496308
:alt [m]: 1000 :lat D.mmss: 90 :
Int Grav GRS 67: 9.82909115816 :
Somigliana GRS 80: 9.82910370419 :Somigliana WGS 84: 9.82910227404 :Closed Form WGS 84: 9.82910231835
:alt [m]: 0 :lat D.mmss: 45 :
Int Grav GRS 67: 9.80618987521 :
Somigliana GRS 80: 9.80619920255 :Somigliana WGS 84: 9.80619776931 :Closed Form WGS 84: 9.80619777838
:alt [m]: 1000 :lat D.mmss: 45 :
Int Grav GRS 67: 9.80310387521 :
Somigliana GRS 80: 9.80311437628 :Somigliana WGS 84: 9.80311294349 :Closed Form WGS 84: 9.80311291004
Last example practice
-Go into FREE Dir.
-Change the value of
WGS84 1/298.257223563
into 1/298.25722356
(cut the final, right digit 3 in the expression),
' 1/298.25722356' ENTER
'f' STO
0 0 —>gFREE will return the following results:
alt [in m]: 0 :lat [in D.mmss]: 0 :
Closed Form FREE: 9.78032534903
and 90 0 —>gFREE will return the following results:
alt [in m]: 0 :lat [in D.mmss]: 90 :
Closed Form FREE: 9.83218491167
Gil Campart
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« on: May 28, 2017, 01:35:17 am »
Oh man, just re-read the thread... good times. I don't remember what PMs Axxle actually sent me the first time, so I'm having a hard time deciding whether to believe myself or not. Anyway, /in
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The most appropriate average in averaging the price relatives is:
The most appropriate average in averaging the price relatives is:
┃ A. Median ┃
┃ B. Harmonic mean ┃
┃ C. Arithmetic mean ┃
┃ D. Geometric mean ┃
The Correct Answer Is:
The correct answer is D. Geometric mean.
Why “Geometric Mean” is the Most Appropriate Average for Averaging Price Relatives:
The geometric mean is considered the most appropriate average for averaging price relatives for several compelling reasons:
1. Multiplicative Relationships:
Price relatives represent changes in prices, and these changes often have a multiplicative relationship. When you multiply the price relatives together, you get the total percentage change in price
over a series of time periods. The geometric mean is well-suited for handling such multiplicative relationships because it accounts for the compounding effect of changes.
2. Consistency with Percentage Changes:
The geometric mean aligns naturally with the concept of percentage changes, which is often how price relatives are interpreted. When you compute the geometric mean of a series of price relatives, you
are essentially finding the average percentage change over time. This is useful for analyzing the overall trend in price movements.
3. Avoiding Biases:
Unlike some other means, such as the arithmetic mean, the geometric mean is not influenced by extreme values or outliers. This property is important when dealing with price relatives because extreme
price fluctuations can have a disproportionate impact on other types of averages. The geometric mean provides a more balanced representation of overall price movements.
4. Preservation of Scale:
The geometric mean maintains the scale of the data. In the context of price relatives, this means that the geometric mean represents the same percentage change whether the prices are initially
expressed in dollars, euros, or any other currency. This property is important for cross-country or cross-currency comparisons.
5. Useful for Investment Analysis:
The geometric mean is widely used in finance and investment analysis, where it helps calculate compound annual growth rates (CAGR). CAGR reflects the average annual rate of return or growth over a
specified time period and is crucial for assessing the performance of investments or portfolios.
Price relatives are often used to compute CAGR, making the geometric mean a natural choice.
Why the Other Options Are Not Correct:
A. Median:
The median is not the most appropriate average for averaging price relatives because it does not account for the multiplicative nature of changes in prices.
A median is based on the middle value in a data set when values are arranged in ascending or descending order. It is more suited for handling data with additive relationships, where the values are
not influenced by compounding effects.
B. Harmonic Mean:
The harmonic mean is not typically used for averaging price relatives because it tends to give more weight to smaller values. In the context of price relatives, this would mean that smaller price
changes would have a disproportionately larger impact on the harmonic mean. This is not desirable when you want to capture the overall trend in price movements over time.
C. Arithmetic Mean:
While the arithmetic mean is a commonly used average, it is not the most appropriate choice for averaging price relatives. The arithmetic mean treats all values equally and calculates the average by
summing the values and dividing by the number of data points.
However, this method is less suitable for price relatives because it does not reflect the multiplicative nature of price changes, and it can be heavily influenced by extreme values.
In summary, the geometric mean is the most appropriate average for averaging price relatives due to its ability to handle multiplicative relationships, its consistency with percentage changes, its
resistance to biases from extreme values, its preservation of scale, and its relevance in investment analysis.
It provides a more accurate representation of the average rate of change in prices over time, making it a valuable tool for economic and financial analysis.
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What is 1/1 of 309?
In this article, we'll show you exactly how to calculate 1/1 of 309 so you can work out the fraction of any number quickly and easily! Let's get to the math!
Want to quickly learn or show students how to convert 1/1 of 309? Play this very quick and fun video now!
You probably know that the number above the fraction line is called the numerator and the number below it is called the denominator. To work out the fraction of any number, we first need to convert
that whole number into a fraction as well.
Here's a little tip for you. Any number can be converted to fraction if you use 1 as the denominator:
309 / 1
So now that we've converted 309 into a fraction, to work out the answer, we put the fraction 1/1 side by side with our new fraction, 309/1 so that we can multiply those two fractions.
That's right, all you need to do is convert the whole number to a fraction and then multiply the numerators and denominators. Let's take a look:
1 x 309 / 1 x 1 = 309 / 1
As you can see in this case, the numerator is higher than the denominator. What this means is that we can simplify the answer down to a mixed number, also known as a mixed fraction.
To do that, we need to convert the improper fraction to a mixed fraction. We won't explain that in detail here because we have another article that already covers it for 309/1. Click here to find out
how to convert 309/1 to a mixed fraction.
The complete and simplified answer to the question what is 1/1 of 309 is:
Hopefully this tutorial has helped you to understand how to find the fraction of any whole number. You can now go give it a go with more numbers to practice your newfound fraction skills.
Cite, Link, or Reference This Page
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• <a href="http://visualfractions.com/calculator/fraction-of-number/what-is-1-1-of-309/">What is 1/1 of 309?</a>
• "What is 1/1 of 309?". VisualFractions.com. Accessed on November 3, 2024. http://visualfractions.com/calculator/fraction-of-number/what-is-1-1-of-309/.
• "What is 1/1 of 309?". VisualFractions.com, http://visualfractions.com/calculator/fraction-of-number/what-is-1-1-of-309/. Accessed 3 November, 2024.
• What is 1/1 of 309?. VisualFractions.com. Retrieved from http://visualfractions.com/calculator/fraction-of-number/what-is-1-1-of-309/.
Fraction of a Number Calculator
Fraction of a Number
Enter a numerator, denominator and whole number
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Simple 1D Noise in JavaScript
I am working on a side project in which I needed to generate some “random”, or more accurately, unpredictable motion. At first I tried using the Math.random() method, and using those values to set
the position of my moving element. This, of course, looks terrible because the element will simply jump around to various points, “teleporting” between them. After searching around for a bit (one of
those slow search processes where you don’t really know what you are looking for) I figured out I needed some kind of noise function.
I found this excellent article on scratchapixel.com, and used it as a basis for my own implementation of simple one dimensional noise in JavaScript. Here is the result:
The Code
Here is the code. I can’t take credit for the algorithm, I pretty much just ported it from the article referenced above. To understand what the code is doing, read the article. It’s a good read.
var Simple1DNoise = function() {
var MAX_VERTICES = 256;
var MAX_VERTICES_MASK = MAX_VERTICES -1;
var amplitude = 1;
var scale = 1;
var r = [];
for ( var i = 0; i < MAX_VERTICES; ++i ) {
var getVal = function( x ){
var scaledX = x * scale;
var xFloor = Math.floor(scaledX);
var t = scaledX - xFloor;
var tRemapSmoothstep = t * t * ( 3 - 2 * t );
/// Modulo using &
var xMin = xFloor & MAX_VERTICES_MASK;
var xMax = ( xMin + 1 ) & MAX_VERTICES_MASK;
var y = lerp( r[ xMin ], r[ xMax ], tRemapSmoothstep );
return y * amplitude;
* Linear interpolation function.
* @param a The lower integer value
* @param b The upper integer value
* @param t The value between the two
* @returns {number}
var lerp = function(a, b, t ) {
return a * ( 1 - t ) + b * t;
// return the API
return {
getVal: getVal,
setAmplitude: function(newAmplitude) {
amplitude = newAmplitude;
setScale: function(newScale) {
scale = newScale;
How to use it
Check out some instructions on my AngularUtils repo.
Since I am working a lot with AngularJS, and the side project I am making is using Angular, I wrapped the above function in an Angular service, which you can find at the above location. Using it
outside of angular is as simple as copying the code listed above, and instantiating it like so:
var generator = new Simple1DNoise();
var x = 1;
var y = generator.getVal(x);
I hope some of you find this useful, or at least interesting. | {"url":"https://www.michaelbromley.co.uk/blog/simple-1d-noise-in-javascript/","timestamp":"2024-11-03T12:36:54Z","content_type":"text/html","content_length":"16471","record_id":"<urn:uuid:ec4d50b0-4d17-42a4-9762-1302d45e64cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00310.warc.gz"} |
Margin of Overpull in DrillstringMargin of Overpull in Drillstring
Margin of Overpull in Drillstring
Margin of overpull is additional tension to be applied when pulling the stuck drill string without breaking the tensile limit of the drill string. This is the difference between maximum allowable
tensile load of drill string and hook load.
There are several factors when consider about the margin of over pull as listed below;
• Overall drilling conditions
• Hole drag
• Likelihood of getting stuck
• Dynamic loading
The formula for margin of overpull is described below;
Margin of Overpull = Ta – Th
Ta is the maximum allowable tensile strength, lb.
Th is the hook load (excluding top drive weight), lb.
Th can be calculated by the following formula;
Th = Weight of String x Buoyancy Factor (BF)
Weight of string = (L[dp]×W[dp]) + (L[dc]×W[dc]) +(L[hwdp]×W[hwdp])+(L[bha ]× W[bha])
L[dp] = length of drill pipe
W[dp ]= weight of drill pipe
L[d]c = length of drill collar
W[d]c= weight of drill collar
L[hwdp ]= length of heavy weight drill pipe
W[hwdp]= weight of heavy weight drill pipe
L[bha ]= length of bottom hole assembly
[ ]W[bha]= weight of bottom hole assembly
Buoyancy Factor (BF) = (65.5 – MW)÷65.5
The ratio between Ta and Th is safety factor (SF).
SF =Ta ÷ Th
Example: The drill string consists of the following equipment:
5” DP S-135, 4-1/2” IF connection, adjusted weight of 23.5 ppf = 8,000 ft
5” HWDP S-135, 4-1/2” IF connection, adjusted weight of 58 ppf = 900 ft
Mud motor and MWD weight = 20 Klb
Lenght of mud motor and MWD = 90 ft
Mud weight is 9.2 ppg
Tensile strength of 5” DP S-135 (premium class) = 436 Klb
Tensile strength of 5” HWDP S-135 (premium class) = 1,100 Klb
90% of tensile strength is allowed to pull without permission from town.
Determine the margin of overpull from the information above.
Maximum tension will happen at the surface so 5”DP will get the most tension when pulling and since only 90% of tensile strength is allowed. The allowable tensile (Ta) is as follows;
Ta = 0.9 x 436 = 392 Klb
Buoyancy Factor (BF) = (65.5 – 9.2)÷65.5
Buoyancy Factor (BF) = 0.86
Weight of string = (8,000 × 23.5) + (900 × 58) + (20,000)
Weight of string = 260 Klb
Note: Weight of mud motor and MWD is given at 20Klb.
Tension on surface (Th)
Th = 260 × 0.86 = 223.8 klb
Margin of over pull = 392 – 223.8 = 168.2 Klb
Safety Factor = 392 ÷ 223.8 = 1.75
Ref Book: Formulas and Calculations for Drilling, Production and Workover, Second Edition
26 Responses to Margin of Overpull in Drillstring
1. thanks for teaching us all these informations
2. Thank you Sir for this simple formula
Please can you send me weight indicator reading if i have 5″ grade G and 5″ grade S
I think you know what I mean .
Best Regards
3. Thanks for teaching us all these informations
4. thanks for sharing knowledge
I have question what the over-pull limit while round trip?
□ You have limitation at the weakest point in while you trip out. You need to check pipe tensile strength and determine how much it is.
5. Thanks that is usefull
6. Excellent side to improve skills.I really appriciated.
in this evaquation (Margin of overpull…….where get 0.9
7. What is the unit of safety factor here
□ The unit is %.
□ It’s not %
Ratio does not have units.
8. What about this formula
Margin over pull =(TR we ×TSF)+(((DC L×DCwf)+(HW L ×HW wf ) +(WP wf ×BF×WP L)))
9. what about the mixed string of 3 1/2″ drill pipe and 5″ G105 and 5″ S135 how we can calculate the maximum marten Decker Reading
□ Fouad,
You need to do free body diagram in order to understand the load distribution because yield strength of each pipe is not the same value.
10. Can you please tell me the proper torque have to give to drill string prior jar down?
drill pipe= 5.5″, 4-1/2″ IF, 19.5 ppf
□ 4-1/2″ connection should be made up to 30K. This will be good for jarring operation too.
11. Hi,
Sorry for the delay.I niticed that you didn’t concider the mud weight buoncy factor( sorry in this case the hook weight is already given , this is only to remind my friends)
□ Mud weight has been considered in the calculation.
12. In this approach torque effect is neglected. Torque applied reduces string´s max. tensile strenght. When applying both tension and torque you should calculate which is new Ta. Combined forces.
13. Excelente publicacion, gracias por su informacion
14. Thanks, It is very useful / informative for drilling personnel
15. In the example problem, “Mud motor and MWD weight = 20 Klb”
but in the calculation “20Klb” is not converted to “lb” : “Weight of string = (8000 × 23.5) + (900 × 58) + (20)”
Please fix to :(8000 × 23.5) + (900 × 58) + (20000)
Mud motor and MWD weight = 20 lb
□ Sun,
Thanks for spotting my typo mistake. It was fixed.
16. I am lost with your string weight calculation (8000 x 23.5 DP) + (900 x 58 HW) = 240,200 lbs + (20,000 Motor & MWD) = 260,200 lbs in air.
Then the next sentence states that the motor & MWD are 90Klbs
17. Hi,
I noticed the mud motor and MWD were not added to the weight of string is that right?
Weight of string = (8000 × 23.5) + (900 × 58) + (20000)
Weight of string = 260 klb not 240 klb
□ Hussain,
Thanks for information and information is updated.
18. Gracias camaradas por mantenernos al tanto en las operaciones de riesgo continuo
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Introduction to Writing Percents Using Words, Ratios, and Fractions
What you’ll learn to do: Write percents using words, ratios, and fractions and complete conversions
When you deposit money in a savings account at a bank, it earns additional money. Figuring out how your money will grow involves understanding and applying concepts of percents. In this section,
you’ll find out what percents are and how we can use them to solve problems.
Before you get started in this module, try a few practice problems and review prior concepts.
If you missed this problem, review the following example.
Translate each word phrase into an algebraic expression:
The quotient of [latex]10x[/latex] and [latex]3[/latex]
Show Solution
If you missed this problem, review this video.
If you missed this problem, review the video below. | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/introduction-to-percents/","timestamp":"2024-11-03T07:54:41Z","content_type":"text/html","content_length":"50431","record_id":"<urn:uuid:482d1741-65cf-4904-9e5a-cfb295d9295e>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00873.warc.gz"} |
An algorithm for computing the signal propagation on lossy VLSI interconnect systems in the time domain
Originalsprache Englisch
Seiten (von - bis) 35-48
Seitenumfang 14
Fachzeitschrift Integration, the VLSI Journal
Jahrgang 7
Ausgabenummer 1
Publikationsstatus Veröffentlicht - Apr. 1989
An algorithm for the computation of signal propagation in lossy interconnect systems at the chip as well as at the board level is presented. The algorithm allows us to handle effects like delay,
dispersion, reflection, and electromagnetic coupling sufficiently. Especially for large systems of lines these effects cannot be treated by existing general purpose network analysis programs because
of excessive computer runtime and numerical instabilities. To be able to handle line systems in a nonlinear circuit environment (e.g. CMOS inverters) the algorithm operates in the time domain. It is
implemented in a computer program called LISIM which yields high accuracy, good stability, and small computation time. Simulation results are given for on-chip line systems of medium size based on
the slow-wave mode. All simulations shown here were run on a 12 Mflops computer. The required computation time was about 2.4-60 s depending on the size of the nonlinear circuit environment, the used
features and the desired accuracy for the simulation.
ASJC Scopus Sachgebiete
• Informatik (insg.)
• Informatik (insg.)
• Ingenieurwesen (insg.)
• Standard
• Harvard
• Apa
• Vancouver
• Autor
• BibTex
• RIS
title = "An algorithm for computing the signal propagation on lossy VLSI interconnect systems in the time domain",
abstract = "An algorithm for the computation of signal propagation in lossy interconnect systems at the chip as well as at the board level is presented. The algorithm allows us to handle effects like
delay, dispersion, reflection, and electromagnetic coupling sufficiently. Especially for large systems of lines these effects cannot be treated by existing general purpose network analysis programs
because of excessive computer runtime and numerical instabilities. To be able to handle line systems in a nonlinear circuit environment (e.g. CMOS inverters) the algorithm operates in the time
domain. It is implemented in a computer program called LISIM which yields high accuracy, good stability, and small computation time. Simulation results are given for on-chip line systems of medium
size based on the slow-wave mode. All simulations shown here were run on a 12 Mflops computer. The required computation time was about 2.4-60 s depending on the size of the nonlinear circuit
environment, the used features and the desired accuracy for the simulation.",
keywords = "electromagnetic coupling, interconnect systems, lossy line system, signal propagation, time domain simulation",
author = "Hartmut Grabinski",
year = "1989",
month = apr,
doi = "10.1016/0167-9260(89)90058-8",
language = "English",
volume = "7",
pages = "35--48",
journal = "Integration, the VLSI Journal",
issn = "0167-9260",
publisher = "Elsevier",
number = "1",
TY - JOUR
T1 - An algorithm for computing the signal propagation on lossy VLSI interconnect systems in the time domain
AU - Grabinski, Hartmut
PY - 1989/4
Y1 - 1989/4
N2 - An algorithm for the computation of signal propagation in lossy interconnect systems at the chip as well as at the board level is presented. The algorithm allows us to handle effects like delay,
dispersion, reflection, and electromagnetic coupling sufficiently. Especially for large systems of lines these effects cannot be treated by existing general purpose network analysis programs because
of excessive computer runtime and numerical instabilities. To be able to handle line systems in a nonlinear circuit environment (e.g. CMOS inverters) the algorithm operates in the time domain. It is
implemented in a computer program called LISIM which yields high accuracy, good stability, and small computation time. Simulation results are given for on-chip line systems of medium size based on
the slow-wave mode. All simulations shown here were run on a 12 Mflops computer. The required computation time was about 2.4-60 s depending on the size of the nonlinear circuit environment, the used
features and the desired accuracy for the simulation.
AB - An algorithm for the computation of signal propagation in lossy interconnect systems at the chip as well as at the board level is presented. The algorithm allows us to handle effects like delay,
dispersion, reflection, and electromagnetic coupling sufficiently. Especially for large systems of lines these effects cannot be treated by existing general purpose network analysis programs because
of excessive computer runtime and numerical instabilities. To be able to handle line systems in a nonlinear circuit environment (e.g. CMOS inverters) the algorithm operates in the time domain. It is
implemented in a computer program called LISIM which yields high accuracy, good stability, and small computation time. Simulation results are given for on-chip line systems of medium size based on
the slow-wave mode. All simulations shown here were run on a 12 Mflops computer. The required computation time was about 2.4-60 s depending on the size of the nonlinear circuit environment, the used
features and the desired accuracy for the simulation.
KW - electromagnetic coupling
KW - interconnect systems
KW - lossy line system
KW - signal propagation
KW - time domain simulation
UR - http://www.scopus.com/inward/record.url?scp=0024646529&partnerID=8YFLogxK
U2 - 10.1016/0167-9260(89)90058-8
DO - 10.1016/0167-9260(89)90058-8
M3 - Article
AN - SCOPUS:0024646529
VL - 7
SP - 35
EP - 48
JO - Integration, the VLSI Journal
JF - Integration, the VLSI Journal
SN - 0167-9260
IS - 1
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n-solve character limit workaround (TI-30X Pro MP)
02-13-2023, 11:50 PM
(This post was last modified: 02-14-2023 01:37 PM by wb.c.)
Post: #1
wb.c Posts: 26
Junior Member Joined: Feb 2023
n-solve character limit workaround (TI-30X Pro MP)
This isn't always a issue I run into a lot, but the other day I had an equation that was a bit too long for the n-solve input, which is about 57 characters on one side and 1 on the other. The curser
will become checkered/grey when you have about 13 characters left. Total of about 58 characters however you want to split it. This has been a point often raised by Casio fx-991ex users, as being a
big advantage in favor of the Casio (you can enter longer equation to solve).
On the TI however, within table you can define f(x) and g(x), each containing up to about 110 characters each. You can then input both f(x) and g(x) into n-solve and solve. Additionally, you can add
more characters to n-solve while using f(x) and g(x), about 48 additional characters.
This totals out to about a 268 character limit for an equation in n-solve, in addition to the equals sign. This was tested in classical entry mode to keep things simple. Haven't tested it, but
MathPrint entry seems to be more limited, and it seems that certain characters are bigger (i.e. "e^(" seems to be equivalent to about 9 simple characters like + or 1).
For reference, it seems that Casio fx-991ex can handle about 197 characters on one side of the equals and 1 character on the other.
Anyways, just an interesting point that probably needs more testing to determine exact values, but this is just an initial attempt.
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Statistics, probability, and Nate Silver • Kevin Fox
In the last few days Nate Silver has become the third most talked-about man in politics, with pundits left and right saying he’s audaciously staked his professional reputation on an Obama win.
This is sad and shows how little we understand about the nature of statistics and probability, even the more educated among us. Nate’s electoral prognostications over the last several months have
really been two separate things melded together:
First, they are predictions of the accuracy of the national polls, the tracking polls, the swing-state polls and those pollsters estimates of how registered voters will translate to likely voters.
Polls use well-worn statistical models to give confidence intervals for those polls, but by merging several polls and increasing the sample size, Silver is able to reduce that confidence interval
significantly, giving a more accurate model. Silver’s ‘now-cast’ numbers are purely based on those polls, how likely they are to be wrong to a degree that would swing the result in that state, and a
Monte Carlo simulation to generate a probabilistic distribution of outcomes. Then he shares what portion of those outcomes lead to an Obama victory, a Romney victory, or a tie.
The second thing Silver does (or, to be more accurate, did) is predict future effects that could change the electoral response between the time the poll was taken and Election Day. This involves a
great deal of educated guesswork about economic factors, foreign policy issues, natural disasters (ahem) and, more than anything, a general regression to the mean. Throwing those variable ingredients
in to the Monte Carlo soup churns out an outcome distribution that Silver presents as the ‘Nov. 6 forecast’. One could definitely make the case that since there’s a level of subjectivity in weighting
different factors, bias could creep in to the model at this stage. It’s extremely hard to document whether such a bias actually exists in these forecasts but thankfully at this point we don’t have
I mentioned that Silver ‘did’ use multi-factor predictive models because as the poll dates approached the election date, those factors that might change the feeling of the electorate in the
intervening time were, naturally, given less and less weight until today when that factor is zero. Today’s estimate, the one getting so much press, is based entirely on polling data and confidence
intervals and not on future factors. Today the ‘Nov. 6 forecast’ and the ‘Now-cast’ are exactly the same. Pundits could still argue that there are other vectors of possible bias including Silver’s
weighting of polls against each other and calculations of ‘house bias’, but those are all pretty clearly grounded in historical data and criticisms of them are harder to give credit.
It’s a shame we don’t do more to teach statistics and probability in school because the average person usually sees different kinds of probabilities the same way. Take a football game: You can
generate a reasonably accurate probability model of who will win based on past performance, where the variance comes from the ‘noise’ in the game. A single interception or a lucky play can
drastically change the game’s outcome. In this sense the measure of probability is to say that if the two teams played 100 games with the same team members in the same state of health, the tallied
wins for each team would fall roughly in line with the probabilities. There is internal chaos in the game that forces the probabilistic distribution.
Predicting an election based on polls is an entirely different matter. The election will turn out one way or another. If the same people voted for President 100 times without an external factor
interfering differently across samples, the outcome would be the same every time. There is almost no internal chaos within the game of voting that forces a probabilistic distribution (technically
there are extremely minor chaotic factors within the system, such as voters who literally coin-flip on their way in, or who mis-cast their vote, but those chaotic factors have no ‘lean’ toward a
particular candidate and en-masse are nearly impossible to change a sample’s electoral outcome).
In these cases where the event being predicted has such low internal chaotic factors, the statistician isn’t actually predicting the probability that candidate X or Y will become President because
that event is already unchanging. Instead, they’re predicting the accuracy of their model. In this case, Nate Silver is predicting with a confidence of 91% that his model is correct in saying that
Obama will win today’s election.
Don’t believe me? Let’s look at it a different way. Say there were two statisticians trying to predict the same election. One has a single poll from each state to work from, and the other has ten
polls from each state. The first statistician, using his polls and the relatively low confidence intervals his single polls provide, can say with a 56% certainty that Obama will win the election. The
second statistician, with more data, more people polled, and much higher mathematical confidence levels and smaller confidence intervals, predicts with a 91% certainty that Obama will win.
Both of these models can be completely mathematically correct even though they’re vastly different because, as stated earlier, they’re predicting the confidence that their model is correct. As each
statistician is using different models, they naturally have different probabilities. Given 100 completely different elections, the statistician with more polls to work with would be right more often.
Take a third hypothetical statistician who, amazingly, is able to poll every single voter just before they vote. That statistician has a nearly absolute certainty in their polling data with a
confidence interval that is nearly zero. That person can predict with 99.9999% confidence who will win the election.
This is a trick the sports bookie can’t accomplish because, even with absolute knowledge of the opening state, the outcome is in doubt. But elections aren’t football games or horse races (no matter
that the pundits so enjoy those metaphors), and longer odds in closer races don’t have to be the product of audacity and bias. They’re simply the result of more polls, better science, and a lack of a
need to create the sense of a ‘dead heat’ to bolster ad revenues. | {"url":"http://kfury.com/statistics-probability-and-nate-silver","timestamp":"2024-11-14T13:07:21Z","content_type":"text/html","content_length":"17127","record_id":"<urn:uuid:076212dc-bc66-403c-870d-75ef1747e4fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00082.warc.gz"} |
What is the probability that both cards are spades?
What is the probability that both cards are spades?
The probability that the first card drawn is a spade is 1/4. Given that the first card drawn is a spade, there are 12 more spades out of the remaining 51 cards in the deck (assuming that you're
drawing without replacement). So the total probability of two spades is (1/4)(12/51) = 3/51.
What is the probability to pick a club and diamond if you draw two cards at once?
Probability to draw a Club second time is 12/51=4/17. we determined the probability to have two diamonds. It is 1/17. So is the probability to get two cards of any other suit. Thus the total
probability to get two cards of the same suit is 4*1/17=4/17.
What is the probability of drawing one spade from a pack of 52 cards?
Hence for drawing a card from a deck, each outcome has probability 1/52. The probability of an event is the sum of the probabilities of the outcomes in the event, hence the probability of drawing a
spade is 13/52 = 1/4, and the probability of drawing a king is 4/52 = 1/13.
What’s the probability you draw exactly 1 Heart in 2 draws with replacement?
There is a 38 probability of pulling exactly 1 heart on two draws, with replacement. | {"url":"https://yourgametips.com/miscellaneous/what-is-the-probability-that-both-cards-are-spades/","timestamp":"2024-11-08T04:56:57Z","content_type":"text/html","content_length":"120523","record_id":"<urn:uuid:2051a129-4d89-4055-8603-3ed504db6cf0>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00242.warc.gz"} |
K-Means Clustering (medium)
Your task is to write a Python function that implements the k-Means clustering algorithm. This function should take specific inputs and produce a list of final centroids. k-Means clustering is a
method used to partition n points into k clusters. The goal is to group similar points together and represent each group by its center (called the centroid).
Function Inputs:
• points: A list of points, where each point is a tuple of coordinates (e.g., (x, y) for 2D points)
• k: An integer representing the number of clusters to form
• initial_centroids: A list of initial centroid points, each a tuple of coordinates
• max_iterations: An integer representing the maximum number of iterations to perform
Function Output:
A list of the final centroids of the clusters, where each centroid is rounded to the nearest fourth decimal.
input: points = [(1, 2), (1, 4), (1, 0), (10, 2), (10, 4), (10, 0)], k = 2, initial_centroids = [(1, 1), (10, 1)], max_iterations = 10
output: [(1, 2), (10, 2)]
reasoning: Given the initial centroids and a maximum of 10 iterations,
the points are clustered around these points, and the centroids are
updated to the mean of the assigned points, resulting in the final
centroids which approximate the means of the two clusters.
The exact number of iterations needed may vary,
but the process will stop after 10 iterations at most.
K-Means Clustering Algorithm Implementation
Algorithm Steps:
1. Initialization
Use the provided initial_centroids as your starting point. This step is already done for you in the input.
2. Assignment Step
For each point in your dataset:
• Calculate its distance to each centroid (Hint: use Euclidean distance.)
• Assign the point to the cluster of the nearest centroid
Hint: Consider creating a helper function to calculate Euclidean distance between two points.
3. Update Step
For each cluster:
• Calculate the mean of all points assigned to the cluster
• Update the centroid to this new mean position
Hint: Be careful with potential empty clusters. Decide how you'll handle them (e.g., keep the previous centroid).
4. Iteration
Repeat steps 2 and 3 until either:
• The centroids no longer change significantly (this case does not need to be included in your solution), or
• You reach the max_iterations limit
Hint: You might want to keep track of the previous centroids to check for significant changes.
5. Result
Return the list of final centroids, ensuring each coordinate is rounded to the nearest fourth decimal.
For a visual understanding of how k-Means clustering works, check out this helpful YouTube video.
import numpy as np
def euclidean_distance(a, b):
return np.sqrt(((a - b) ** 2).sum(axis=1))
def k_means_clustering(points, k, initial_centroids, max_iterations):
points = np.array(points)
centroids = np.array(initial_centroids)
for iteration in range(max_iterations):
# Assign points to the nearest centroid
distances = np.array([euclidean_distance(points, centroid) for centroid in centroids])
assignments = np.argmin(distances, axis=0)
new_centroids = np.array([points[assignments == i].mean(axis=0) if len(points[assignments == i]) > 0 else centroids[i] for i in range(k)])
# Check for convergence
if np.all(centroids == new_centroids):
centroids = new_centroids
centroids = np.round(centroids,4)
return [tuple(centroid) for centroid in centroids]
There’s no video solution available yet 😔, but you can be the first to submit one at: GitHub link. | {"url":"https://www.deep-ml.com/problem/K-Means%20Clustering","timestamp":"2024-11-06T08:00:52Z","content_type":"text/html","content_length":"29499","record_id":"<urn:uuid:3782057c-fccb-47e6-94af-c127bb09be5e>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00282.warc.gz"} |
A Test of Fit for Multivariate Distributions
Ann. Math. Statist. 29(2): 595-599 (June, 1958). DOI: 10.1214/aoms/1177706639
Suppose $X$ is a chance variable taking values in $k$-dimensional Euclidean space. That is, $X = (Y_1, \cdots, Y_k),$ where $Y_i$ is a univariate chance variable. The joint distribution of $(Y_1, \
cdots, Y_k)$ has density $f(y_1, \cdots, y_k),$ say. We shall call a function $h(y_1, \cdots, y_k)$ "piecewise continuous" if it is everywhere bounded, and $k$-dimensional Euclidean space can be
broken into a finite number of Borel-measurable subregions, such that in the interior of each subregion $h(y_1, \cdots, y_k)$ is continuous, and the set of all boundary points of all subregions has
Lebesgue measure zero. We assume that $f(y_1, \cdots, y_k)$ is piecewise continuous. Let $h(y_1, \cdots, y_k)$ be some given nonnegative piecewise continuous function, and let $X_1, \cdots, X_n$ be
independent chance variables, each with the density $f(y_1, \cdots, y_k).$ Choose a nonnegative number $t,$ and for each $i$, construct a $k$-dimensional sphere with center at $X_i = (Y_{i1}, \cdots,
Y_{ik})$ and of $k$-dimensional volume $$\frac{th(Y_{i1}, \cdots, Y_{ik})}{n}.$$ Such a sphere will be called "of type $s$" if it contains exactly $s$ of the $(n - 1)$ points $X_1, \cdots, X_{i-1},
X_{i+1}, \cdots, X_n.$ Let $R_n(t; s)$ denote the proportion of the $n$ spheres which are of type $s$. For typographical simplicity, we denote the vector $(y_1, \cdots, y_k)$ by $y.$ Let $S(t; s)$
denote the multiple integral $$(t^s/s!) \int^\infty_{-\infty} \cdots \int^\infty_{-\infty} h^s(y)f^{s+1}(y) \exp \{-th(y)f(y)\} dy_1 \cdots dy_k.$$ It is shown that $R_n(t; s)$ converges
stochastically to $S(t; s)$ as $n$ increases. This result is then used to construct a test of the hypothesis that the unknown density $f(y)$ is equal to a given density $g(y).$
Download Citation
Lionel Weiss. "A Test of Fit for Multivariate Distributions." Ann. Math. Statist. 29 (2) 595 - 599, June, 1958. https://doi.org/10.1214/aoms/1177706639
Published: June, 1958
First available in Project Euclid: 27 April 2007
Digital Object Identifier: 10.1214/aoms/1177706639
Rights: Copyright © 1958 Institute of Mathematical Statistics
Vol.29 • No. 2 • June, 1958 | {"url":"https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-29/issue-2/A-Test-of-Fit-for-Multivariate-Distributions/10.1214/aoms/1177706639.full","timestamp":"2024-11-03T01:05:07Z","content_type":"text/html","content_length":"140202","record_id":"<urn:uuid:fdd5f0a3-47db-4091-bae8-0d4083f65834>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00178.warc.gz"} |
Copyright (c) Universiteit Utrecht 2010-2011 University of Oxford 2012-2014
License see libraries/base/LICENSE
Maintainer libraries@haskell.org
Stability internal
Portability non-portable
Safe Haskell Trustworthy
Language Haskell2010
Datatype-generic functions are based on the idea of converting values of a datatype T into corresponding values of a (nearly) isomorphic type Rep T. The type Rep T is built from a limited set of type
constructors, all provided by this module. A datatype-generic function is then an overloaded function with instances for most of these type constructors, together with a wrapper that performs the
mapping between T and Rep T. By using this technique, we merely need a few generic instances in order to implement functionality that works for any representable type.
Representable types are collected in the Generic class, which defines the associated type Rep as well as conversion functions from and to. Typically, you will not define Generic instances by hand,
but have the compiler derive them for you.
Representing datatypes
The key to defining your own datatype-generic functions is to understand how to represent datatypes using the given set of type constructors.
Let us look at an example first:
data Tree a = Leaf a | Node (Tree a) (Tree a)
deriving Generic
The above declaration (which requires the language pragma DeriveGeneric) causes the following representation to be generated:
instance Generic (Tree a) where
type Rep (Tree a) =
D1 ('MetaData "Tree" "Main" "package-name" 'False)
(C1 ('MetaCons "Leaf" 'PrefixI 'False)
(S1 ('MetaSel 'Nothing
(Rec0 a))
C1 ('MetaCons "Node" 'PrefixI 'False)
(S1 ('MetaSel 'Nothing
(Rec0 (Tree a))
S1 ('MetaSel 'Nothing
(Rec0 (Tree a))))
Hint: You can obtain information about the code being generated from GHC by passing the -ddump-deriv flag. In GHCi, you can expand a type family such as Rep using the :kind! command.
This is a lot of information! However, most of it is actually merely meta-information that makes names of datatypes and constructors and more available on the type level.
Here is a reduced representation for Tree with nearly all meta-information removed, for now keeping only the most essential aspects:
instance Generic (Tree a) where
type Rep (Tree a) =
Rec0 a
(Rec0 (Tree a) :*: Rec0 (Tree a))
The Tree datatype has two constructors. The representation of individual constructors is combined using the binary type constructor :+:.
The first constructor consists of a single field, which is the parameter a. This is represented as Rec0 a.
The second constructor consists of two fields. Each is a recursive field of type Tree a, represented as Rec0 (Tree a). Representations of individual fields are combined using the binary type
constructor :*:.
Now let us explain the additional tags being used in the complete representation:
• The S1 ('MetaSel 'Nothing 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) tag indicates several things. The 'Nothing indicates that there is no record field selector associated with this
field of the constructor (if there were, it would have been marked 'Just "recordName" instead). The other types contain meta-information on the field's strictness:
• There is no {-# UNPACK #-} or {-# NOUNPACK #-} annotation in the source, so it is tagged with 'NoSourceUnpackedness.
• There is no strictness (!) or laziness (~) annotation in the source, so it is tagged with 'NoSourceStrictness.
• The compiler infers that the field is lazy, so it is tagged with 'DecidedLazy. Bear in mind that what the compiler decides may be quite different from what is written in the source. See
DecidedStrictness for a more detailed explanation.
The 'MetaSel type is also an instance of the type class Selector, which can be used to obtain information about the field at the value level.
• The C1 ('MetaCons "Leaf" 'PrefixI 'False) and C1 ('MetaCons "Node" 'PrefixI 'False) invocations indicate that the enclosed part is the representation of the first and second constructor of
datatype Tree, respectively. Here, the meta-information regarding constructor names, fixity and whether it has named fields or not is encoded at the type level. The 'MetaCons type is also an
instance of the type class Constructor. This type class can be used to obtain information about the constructor at the value level.
• The D1 ('MetaData "Tree" "Main" "package-name" 'False) tag indicates that the enclosed part is the representation of the datatype Tree. Again, the meta-information is encoded at the type level.
The 'MetaData type is an instance of class Datatype, which can be used to obtain the name of a datatype, the module it has been defined in, the package it is located under, and whether it has
been defined using data or newtype at the value level.
Derived and fundamental representation types
There are many datatype-generic functions that do not distinguish between positions that are parameters or positions that are recursive calls. There are also many datatype-generic functions that do
not care about the names of datatypes and constructors at all. To keep the number of cases to consider in generic functions in such a situation to a minimum, it turns out that many of the type
constructors introduced above are actually synonyms, defining them to be variants of a smaller set of constructors.
Individual fields of constructors: K1
The type constructor Rec0 is a variant of K1:
type Rec0 = K1 R
Here, R is a type-level proxy that does not have any associated values.
There used to be another variant of K1 (namely Par0), but it has since been deprecated.
Meta information: M1
The type constructors S1, C1 and D1 are all variants of M1:
type S1 = M1 S
type C1 = M1 C
type D1 = M1 D
The types S, C and D are once again type-level proxies, just used to create several variants of M1.
Additional generic representation type constructors
Next to K1, M1, :+: and :*: there are a few more type constructors that occur in the representations of other datatypes.
Empty datatypes: V1
For empty datatypes, V1 is used as a representation. For example,
data Empty deriving Generic
instance Generic Empty where
type Rep Empty =
D1 ('MetaData "Empty" "Main" "package-name" 'False) V1
Constructors without fields: U1
If a constructor has no arguments, then U1 is used as its representation. For example the representation of Bool is
instance Generic Bool where
type Rep Bool =
D1 ('MetaData "Bool" "Data.Bool" "package-name" 'False)
(C1 ('MetaCons "False" 'PrefixI 'False) U1 :+: C1 ('MetaCons "True" 'PrefixI 'False) U1)
Representation of types with many constructors or many fields
As :+: and :*: are just binary operators, one might ask what happens if the datatype has more than two constructors, or a constructor with more than two fields. The answer is simple: the operators
are used several times, to combine all the constructors and fields as needed. However, users /should not rely on a specific nesting strategy/ for :+: and :*: being used. The compiler is free to
choose any nesting it prefers. (In practice, the current implementation tries to produce a more-or-less balanced nesting, so that the traversal of the structure of the datatype from the root to a
particular component can be performed in logarithmic rather than linear time.)
Defining datatype-generic functions
A datatype-generic function comprises two parts:
1. Generic instances for the function, implementing it for most of the representation type constructors introduced above.
2. A wrapper that for any datatype that is in Generic, performs the conversion between the original value and its Rep-based representation and then invokes the generic instances.
As an example, let us look at a function encode that produces a naive, but lossless bit encoding of values of various datatypes. So we are aiming to define a function
encode :: Generic a => a -> [Bool]
where we use Bool as our datatype for bits.
For part 1, we define a class Encode'. Perhaps surprisingly, this class is parameterized over a type constructor f of kind * -> *. This is a technicality: all the representation type constructors
operate with kind * -> * as base kind. But the type argument is never being used. This may be changed at some point in the future. The class has a single method, and we use the type we want our final
function to have, but we replace the occurrences of the generic type argument a with f p (where the p is any argument; it will not be used).
class Encode' f where
encode' :: f p -> [Bool]
With the goal in mind to make encode work on Tree and other datatypes, we now define instances for the representation type constructors V1, U1, :+:, :*:, K1, and M1.
Definition of the generic representation types
In order to be able to do this, we need to know the actual definitions of these types:
data V1 p -- lifted version of Empty
data U1 p = U1 -- lifted version of ()
data (:+:) f g p = L1 (f p) | R1 (g p) -- lifted version of Either
data (:*:) f g p = (f p) :*: (g p) -- lifted version of (,)
newtype K1 i c p = K1 { unK1 :: c } -- a container for a c
newtype M1 i t f p = M1 { unM1 :: f p } -- a wrapper
So, U1 is just the unit type, :+: is just a binary choice like Either, :*: is a binary pair like the pair constructor (,), and K1 is a value of a specific type c, and M1 wraps a value of the generic
type argument, which in the lifted world is an f p (where we do not care about p).
Generic instances
The instance for V1 is slightly awkward (but also rarely used):
instance Encode' V1 where
encode' x = undefined
There are no values of type V1 p to pass (except undefined), so this is actually impossible. One can ask why it is useful to define an instance for V1 at all in this case? Well, an empty type can be
used as an argument to a non-empty type, and you might still want to encode the resulting type. As a somewhat contrived example, consider [Empty], which is not an empty type, but contains just the
empty list. The V1 instance ensures that we can call the generic function on such types.
There is exactly one value of type U1, so encoding it requires no knowledge, and we can use zero bits:
instance Encode' U1 where
encode' U1 = []
In the case for :+:, we produce False or True depending on whether the constructor of the value provided is located on the left or on the right:
instance (Encode' f, Encode' g) => Encode' (f :+: g) where
encode' (L1 x) = False : encode' x
encode' (R1 x) = True : encode' x
(Note that this encoding strategy may not be reliable across different versions of GHC. Recall that the compiler is free to choose any nesting of :+: it chooses, so if GHC chooses (a :+: b) :+: c,
then the encoding for a would be [False, False], b would be [False, True], and c would be [True]. However, if GHC chooses a :+: (b :+: c), then the encoding for a would be [False], b would be [True,
False], and c would be [True, True].)
In the case for :*:, we append the encodings of the two subcomponents:
instance (Encode' f, Encode' g) => Encode' (f :*: g) where
encode' (x :*: y) = encode' x ++ encode' y
The case for K1 is rather interesting. Here, we call the final function encode that we yet have to define, recursively. We will use another type class Encode for that function:
instance (Encode c) => Encode' (K1 i c) where
encode' (K1 x) = encode x
Note how Par0 and Rec0 both being mapped to K1 allows us to define a uniform instance here.
Similarly, we can define a uniform instance for M1, because we completely disregard all meta-information:
instance (Encode' f) => Encode' (M1 i t f) where
encode' (M1 x) = encode' x
Unlike in K1, the instance for M1 refers to encode', not encode.
The wrapper and generic default
We now define class Encode for the actual encode function:
class Encode a where
encode :: a -> [Bool]
default encode :: (Generic a, Encode' (Rep a)) => a -> [Bool]
encode x = encode' (from x)
The incoming x is converted using from, then we dispatch to the generic instances using encode'. We use this as a default definition for encode. We need the 'default encode' signature because
ordinary Haskell default methods must not introduce additional class constraints, but our generic default does.
Defining a particular instance is now as simple as saying
instance (Encode a) => Encode (Tree a)
Omitting generic instances
It is not always required to provide instances for all the generic representation types, but omitting instances restricts the set of datatypes the functions will work for:
An M1 instance is always required (but it can just ignore the meta-information, as is the case for encode above).
Generic constructor classes
Datatype-generic functions as defined above work for a large class of datatypes, including parameterized datatypes. (We have used Tree as our example above, which is of kind * -> *.) However, the
Generic class ranges over types of kind *, and therefore, the resulting generic functions (such as encode) must be parameterized by a generic type argument of kind *.
What if we want to define generic classes that range over type constructors (such as Functor, Traversable, or Foldable)?
Like Generic, there is a class Generic1 that defines a representation Rep1 and conversion functions from1 and to1, only that Generic1 ranges over types of kind * -> *. (More generally, it can range
over types of kind k -> *, for any kind k, if the PolyKinds extension is enabled. More on this later.) The Generic1 class is also derivable.
The representation Rep1 is ever so slightly different from Rep. Let us look at Tree as an example again:
data Tree a = Leaf a | Node (Tree a) (Tree a)
deriving Generic1
The above declaration causes the following representation to be generated:
instance Generic1 Tree where
type Rep1 Tree =
D1 ('MetaData "Tree" "Main" "package-name" 'False)
(C1 ('MetaCons "Leaf" 'PrefixI 'False)
(S1 ('MetaSel 'Nothing
C1 ('MetaCons "Node" 'PrefixI 'False)
(S1 ('MetaSel 'Nothing
(Rec1 Tree)
S1 ('MetaSel 'Nothing
(Rec1 Tree)))
The representation reuses D1, C1, S1 (and thereby M1) as well as :+: and :*: from Rep. (This reusability is the reason that we carry around the dummy type argument for kind-*-types, but there are
already enough different names involved without duplicating each of these.)
What's different is that we now use Par1 to refer to the parameter (and that parameter, which used to be a), is not mentioned explicitly by name anywhere; and we use Rec1 to refer to a recursive use
of Tree a.
Representation of * -> * types
Unlike Rec0, the Par1 and Rec1 type constructors do not map to K1. They are defined directly, as follows:
newtype Par1 p = Par1 { unPar1 :: p } -- gives access to parameter p
newtype Rec1 f p = Rec1 { unRec1 :: f p } -- a wrapper
In Par1, the parameter p is used for the first time, whereas Rec1 simply wraps an application of f to p.
Note that K1 (in the guise of Rec0) can still occur in a Rep1 representation, namely when the datatype has a field that does not mention the parameter.
The declaration
data WithInt a = WithInt Int a
deriving Generic1
instance Generic1 WithInt where
type Rep1 WithInt =
D1 ('MetaData "WithInt" "Main" "package-name" 'False)
(C1 ('MetaCons "WithInt" 'PrefixI 'False)
(S1 ('MetaSel 'Nothing
(Rec0 Int)
S1 ('MetaSel 'Nothing
If the parameter a appears underneath a composition of other type constructors, then the representation involves composition, too:
data Rose a = Fork a [Rose a]
instance Generic1 Rose where
type Rep1 Rose =
D1 ('MetaData "Rose" "Main" "package-name" 'False)
(C1 ('MetaCons "Fork" 'PrefixI 'False)
(S1 ('MetaSel 'Nothing
S1 ('MetaSel 'Nothing
([] :.: Rec1 Rose)))
newtype (:.:) f g p = Comp1 { unComp1 :: f (g p) }
Representation of k -> * types
The Generic1 class can be generalized to range over types of kind k -> *, for any kind k. To do so, derive a Generic1 instance with the PolyKinds extension enabled. For example, the declaration
data Proxy (a :: k) = Proxy deriving Generic1
yields a slightly different instance depending on whether PolyKinds is enabled. If compiled without PolyKinds, then Rep1 Proxy :: * -> *, but if compiled with PolyKinds, then Rep1 Proxy :: k -> *.
Representation of unlifted types
If one were to attempt to derive a Generic instance for a datatype with an unlifted argument (for example, Int#), one might expect the occurrence of the Int# argument to be marked with Rec0 Int#.
This won't work, though, since Int# is of an unlifted kind, and Rec0 expects a type of kind *.
One solution would be to represent an occurrence of Int# with 'Rec0 Int' instead. With this approach, however, the programmer has no way of knowing whether the Int is actually an Int# in disguise.
Instead of reusing Rec0, a separate data family URec is used to mark occurrences of common unlifted types:
data family URec a p
data instance URec (Ptr ()) p = UAddr { uAddr# :: Addr# }
data instance URec Char p = UChar { uChar# :: Char# }
data instance URec Double p = UDouble { uDouble# :: Double# }
data instance URec Int p = UFloat { uFloat# :: Float# }
data instance URec Float p = UInt { uInt# :: Int# }
data instance URec Word p = UWord { uWord# :: Word# }
Several type synonyms are provided for convenience:
type UAddr = URec (Ptr ())
type UChar = URec Char
type UDouble = URec Double
type UFloat = URec Float
type UInt = URec Int
type UWord = URec Word
The declaration
data IntHash = IntHash Int#
deriving Generic
instance Generic IntHash where
type Rep IntHash =
D1 ('MetaData "IntHash" "Main" "package-name" 'False)
(C1 ('MetaCons "IntHash" 'PrefixI 'False)
(S1 ('MetaSel 'Nothing
Currently, only the six unlifted types listed above are generated, but this may be extended to encompass more unlifted types in the future.
Generic representation types
data V1 (p :: k) Source #
Void: used for datatypes without constructors
Generic1 (V1 :: k -> Type) Source #
Defined in GHC.Generics
Functor (V1 :: Type -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
Foldable (V1 :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Foldable
Traversable (V1 :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Traversable
Contravariant (V1 :: Type -> Type) Source #
Defined in Data.Functor.Contravariant
Eq (V1 p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Data p => Data (V1 p) Source # Since: 4.9.0.0
Defined in Data.Data
Ord (V1 p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Read (V1 p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Show (V1 p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Generic (V1 p) Source #
Defined in GHC.Generics
Semigroup (V1 p) Source # Since: 4.12.0.0
Defined in GHC.Generics
type Rep1 (V1 :: k -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep (V1 p) Source # Since: 4.9.0.0
Defined in GHC.Generics
data U1 (p :: k) Source #
Unit: used for constructors without arguments
Generic1 (U1 :: k -> Type) Source #
Defined in GHC.Generics
Monad (U1 :: Type -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
Functor (U1 :: Type -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
Applicative (U1 :: Type -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
Foldable (U1 :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Foldable
Traversable (U1 :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Traversable
MonadPlus (U1 :: Type -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
Alternative (U1 :: Type -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
MonadZip (U1 :: Type -> Type) Source # Since: 4.9.0.0
Defined in Control.Monad.Zip
Contravariant (U1 :: Type -> Type) Source #
Defined in Data.Functor.Contravariant
Eq (U1 p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Data p => Data (U1 p) Source # Since: 4.9.0.0
Defined in Data.Data
Ord (U1 p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Read (U1 p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Show (U1 p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Generic (U1 p) Source #
Defined in GHC.Generics
Semigroup (U1 p) Source # Since: 4.12.0.0
Defined in GHC.Generics
Monoid (U1 p) Source # Since: 4.12.0.0
Defined in GHC.Generics
type Rep1 (U1 :: k -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep (U1 p) Source # Since: 4.7.0.0
Defined in GHC.Generics
newtype Par1 p Source #
Used for marking occurrences of the parameter
Monad Par1 Source # Since: 4.9.0.0
Defined in GHC.Generics
Functor Par1 Source # Since: 4.9.0.0
Defined in GHC.Generics
MonadFix Par1 Source # Since: 4.9.0.0
Defined in Control.Monad.Fix
Applicative Par1 Source # Since: 4.9.0.0
Defined in GHC.Generics
Foldable Par1 Source # Since: 4.9.0.0
Defined in Data.Foldable
Traversable Par1 Source # Since: 4.9.0.0
Defined in Data.Traversable
MonadZip Par1 Source # Since: 4.9.0.0
Defined in Control.Monad.Zip
Eq p => Eq (Par1 p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Data p => Data (Par1 p) Source # Since: 4.9.0.0
Defined in Data.Data
Ord p => Ord (Par1 p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Read p => Read (Par1 p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Show p => Show (Par1 p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Generic (Par1 p) Source #
Defined in GHC.Generics
Semigroup p => Semigroup (Par1 p) Source # Since: 4.12.0.0
Defined in GHC.Generics
Monoid p => Monoid (Par1 p) Source # Since: 4.12.0.0
Defined in GHC.Generics
Generic1 Par1 Source #
Defined in GHC.Generics
type Rep (Par1 p) Source # Since: 4.7.0.0
Defined in GHC.Generics
type Rep1 Par1 Source # Since: 4.9.0.0
Defined in GHC.Generics
newtype Rec1 (f :: k -> Type) (p :: k) Source #
Recursive calls of kind * -> * (or kind k -> *, when PolyKinds is enabled)
Generic1 (Rec1 f :: k -> Type) Source #
Defined in GHC.Generics
Monad f => Monad (Rec1 f) Source # Since: 4.9.0.0
Defined in GHC.Generics
Functor f => Functor (Rec1 f) Source # Since: 4.9.0.0
Defined in GHC.Generics
MonadFix f => MonadFix (Rec1 f) Source # Since: 4.9.0.0
Defined in Control.Monad.Fix
Applicative f => Applicative (Rec1 f) Source # Since: 4.9.0.0
Defined in GHC.Generics
Foldable f => Foldable (Rec1 f) Source # Since: 4.9.0.0
Defined in Data.Foldable
Traversable f => Traversable (Rec1 f) Source # Since: 4.9.0.0
Defined in Data.Traversable
MonadPlus f => MonadPlus (Rec1 f) Source # Since: 4.9.0.0
Defined in GHC.Generics
Alternative f => Alternative (Rec1 f) Source # Since: 4.9.0.0
Defined in GHC.Generics
MonadZip f => MonadZip (Rec1 f) Source # Since: 4.9.0.0
Defined in Control.Monad.Zip
Contravariant f => Contravariant (Rec1 f) Source #
Defined in Data.Functor.Contravariant
Eq (f p) => Eq (Rec1 f p) Source # Since: 4.7.0.0
Defined in GHC.Generics
(Data (f p), Typeable f, Data p) => Data (Rec1 f p) Source # Since: 4.9.0.0
Defined in Data.Data
Ord (f p) => Ord (Rec1 f p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Read (f p) => Read (Rec1 f p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Show (f p) => Show (Rec1 f p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Generic (Rec1 f p) Source #
Defined in GHC.Generics
Semigroup (f p) => Semigroup (Rec1 f p) Source # Since: 4.12.0.0
Defined in GHC.Generics
Monoid (f p) => Monoid (Rec1 f p) Source # Since: 4.12.0.0
Defined in GHC.Generics
type Rep1 (Rec1 f :: k -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep (Rec1 f p) Source # Since: 4.7.0.0
Defined in GHC.Generics
newtype K1 (i :: Type) c (p :: k) Source #
Constants, additional parameters and recursion of kind *
Generic1 (K1 i c :: k -> Type) Source #
Defined in GHC.Generics
Bifunctor (K1 i :: Type -> Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Bifunctor
Bifoldable (K1 i :: Type -> Type -> Type) Source # Since: 4.10.0.0
Defined in Data.Bifoldable
bifold :: Monoid m => K1 i m m -> m Source #
bifoldMap :: Monoid m => (a -> m) -> (b -> m) -> K1 i a b -> m Source #
bifoldr :: (a -> c -> c) -> (b -> c -> c) -> c -> K1 i a b -> c Source #
bifoldl :: (c -> a -> c) -> (c -> b -> c) -> c -> K1 i a b -> c Source #
Bitraversable (K1 i :: Type -> Type -> Type) Source # Since: 4.10.0.0
Defined in Data.Bitraversable
Functor (K1 i c :: Type -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
Monoid c => Applicative (K1 i c :: Type -> Type) Source # Since: 4.12.0.0
Defined in GHC.Generics
Foldable (K1 i c :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Foldable
Traversable (K1 i c :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Traversable
Contravariant (K1 i c :: Type -> Type) Source #
Defined in Data.Functor.Contravariant
Eq c => Eq (K1 i c p) Source # Since: 4.7.0.0
Defined in GHC.Generics
(Typeable i, Data p, Data c) => Data (K1 i c p) Source # Since: 4.9.0.0
Defined in Data.Data
Ord c => Ord (K1 i c p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Read c => Read (K1 i c p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Show c => Show (K1 i c p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Generic (K1 i c p) Source #
Defined in GHC.Generics
Semigroup c => Semigroup (K1 i c p) Source # Since: 4.12.0.0
Defined in GHC.Generics
Monoid c => Monoid (K1 i c p) Source # Since: 4.12.0.0
Defined in GHC.Generics
type Rep1 (K1 i c :: k -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep (K1 i c p) Source # Since: 4.7.0.0
Defined in GHC.Generics
newtype M1 (i :: Type) (c :: Meta) (f :: k -> Type) (p :: k) Source #
Meta-information (constructor names, etc.)
Generic1 (M1 i c f :: k -> Type) Source #
Defined in GHC.Generics
Monad f => Monad (M1 i c f) Source # Since: 4.9.0.0
Defined in GHC.Generics
Functor f => Functor (M1 i c f) Source # Since: 4.9.0.0
Defined in GHC.Generics
MonadFix f => MonadFix (M1 i c f) Source # Since: 4.9.0.0
Defined in Control.Monad.Fix
Applicative f => Applicative (M1 i c f) Source # Since: 4.9.0.0
Defined in GHC.Generics
pure :: a -> M1 i c f a Source #
(<*>) :: M1 i c f (a -> b) -> M1 i c f a -> M1 i c f b Source #
liftA2 :: (a -> b -> c0) -> M1 i c f a -> M1 i c f b -> M1 i c f c0 Source #
(*>) :: M1 i c f a -> M1 i c f b -> M1 i c f b Source #
(<*) :: M1 i c f a -> M1 i c f b -> M1 i c f a Source #
Foldable f => Foldable (M1 i c f) Source # Since: 4.9.0.0
Defined in Data.Foldable
Traversable f => Traversable (M1 i c f) Source # Since: 4.9.0.0
Defined in Data.Traversable
MonadPlus f => MonadPlus (M1 i c f) Source # Since: 4.9.0.0
Defined in GHC.Generics
Alternative f => Alternative (M1 i c f) Source # Since: 4.9.0.0
Defined in GHC.Generics
MonadZip f => MonadZip (M1 i c f) Source # Since: 4.9.0.0
Defined in Control.Monad.Zip
mzip :: M1 i c f a -> M1 i c f b -> M1 i c f (a, b) Source #
mzipWith :: (a -> b -> c0) -> M1 i c f a -> M1 i c f b -> M1 i c f c0 Source #
munzip :: M1 i c f (a, b) -> (M1 i c f a, M1 i c f b) Source #
Contravariant f => Contravariant (M1 i c f) Source #
Defined in Data.Functor.Contravariant
Eq (f p) => Eq (M1 i c f p) Source # Since: 4.7.0.0
Defined in GHC.Generics
(Data p, Data (f p), Typeable c, Typeable i, Typeable f) => Data (M1 i c f p) Source # Since: 4.9.0.0
Defined in Data.Data
gfoldl :: (forall d b. Data d => c0 (d -> b) -> d -> c0 b) -> (forall g. g -> c0 g) -> M1 i c f p -> c0 (M1 i c f p) Source #
gunfold :: (forall b r. Data b => c0 (b -> r) -> c0 r) -> (forall r. r -> c0 r) -> Constr -> c0 (M1 i c f p) Source #
toConstr :: M1 i c f p -> Constr Source #
dataTypeOf :: M1 i c f p -> DataType Source #
dataCast1 :: Typeable t => (forall d. Data d => c0 (t d)) -> Maybe (c0 (M1 i c f p)) Source #
dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c0 (t d e)) -> Maybe (c0 (M1 i c f p)) Source #
gmapT :: (forall b. Data b => b -> b) -> M1 i c f p -> M1 i c f p Source #
gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> M1 i c f p -> r Source #
gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> M1 i c f p -> r Source #
gmapQ :: (forall d. Data d => d -> u) -> M1 i c f p -> [u] Source #
gmapQi :: Int -> (forall d. Data d => d -> u) -> M1 i c f p -> u Source #
gmapM :: Monad m => (forall d. Data d => d -> m d) -> M1 i c f p -> m (M1 i c f p) Source #
gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> M1 i c f p -> m (M1 i c f p) Source #
gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> M1 i c f p -> m (M1 i c f p) Source #
Ord (f p) => Ord (M1 i c f p) Source # Since: 4.7.0.0
Defined in GHC.Generics
compare :: M1 i c f p -> M1 i c f p -> Ordering #
(<) :: M1 i c f p -> M1 i c f p -> Bool #
(<=) :: M1 i c f p -> M1 i c f p -> Bool #
(>) :: M1 i c f p -> M1 i c f p -> Bool #
(>=) :: M1 i c f p -> M1 i c f p -> Bool #
max :: M1 i c f p -> M1 i c f p -> M1 i c f p #
min :: M1 i c f p -> M1 i c f p -> M1 i c f p #
Read (f p) => Read (M1 i c f p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Show (f p) => Show (M1 i c f p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Generic (M1 i c f p) Source #
Defined in GHC.Generics
Semigroup (f p) => Semigroup (M1 i c f p) Source # Since: 4.12.0.0
Defined in GHC.Generics
Monoid (f p) => Monoid (M1 i c f p) Source # Since: 4.12.0.0
Defined in GHC.Generics
type Rep1 (M1 i c f :: k -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep (M1 i c f p) Source # Since: 4.7.0.0
Defined in GHC.Generics
data ((f :: k -> Type) :+: (g :: k -> Type)) (p :: k) infixr 5Source #
Sums: encode choice between constructors
Generic1 (f :+: g :: k -> Type) Source #
Defined in GHC.Generics
(Functor f, Functor g) => Functor (f :+: g) Source # Since: 4.9.0.0
Defined in GHC.Generics
(Foldable f, Foldable g) => Foldable (f :+: g) Source # Since: 4.9.0.0
Defined in Data.Foldable
(Traversable f, Traversable g) => Traversable (f :+: g) Source # Since: 4.9.0.0
Defined in Data.Traversable
(Contravariant f, Contravariant g) => Contravariant (f :+: g) Source #
Defined in Data.Functor.Contravariant
(Eq (f p), Eq (g p)) => Eq ((f :+: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
(Typeable f, Typeable g, Data p, Data (f p), Data (g p)) => Data ((f :+: g) p) Source # Since: 4.9.0.0
Defined in Data.Data
(Ord (f p), Ord (g p)) => Ord ((f :+: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
(Read (f p), Read (g p)) => Read ((f :+: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
(Show (f p), Show (g p)) => Show ((f :+: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Generic ((f :+: g) p) Source #
Defined in GHC.Generics
type Rep1 (f :+: g :: k -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep ((f :+: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
data ((f :: k -> Type) :*: (g :: k -> Type)) (p :: k) infixr 6Source #
Products: encode multiple arguments to constructors
Generic1 (f :*: g :: k -> Type) Source #
Defined in GHC.Generics
(Monad f, Monad g) => Monad (f :*: g) Source # Since: 4.9.0.0
Defined in GHC.Generics
(Functor f, Functor g) => Functor (f :*: g) Source # Since: 4.9.0.0
Defined in GHC.Generics
(MonadFix f, MonadFix g) => MonadFix (f :*: g) Source # Since: 4.9.0.0
Defined in Control.Monad.Fix
(Applicative f, Applicative g) => Applicative (f :*: g) Source # Since: 4.9.0.0
Defined in GHC.Generics
(Foldable f, Foldable g) => Foldable (f :*: g) Source # Since: 4.9.0.0
Defined in Data.Foldable
(Traversable f, Traversable g) => Traversable (f :*: g) Source # Since: 4.9.0.0
Defined in Data.Traversable
(MonadPlus f, MonadPlus g) => MonadPlus (f :*: g) Source # Since: 4.9.0.0
Defined in GHC.Generics
(Alternative f, Alternative g) => Alternative (f :*: g) Source # Since: 4.9.0.0
Defined in GHC.Generics
(MonadZip f, MonadZip g) => MonadZip (f :*: g) Source # Since: 4.9.0.0
Defined in Control.Monad.Zip
(Contravariant f, Contravariant g) => Contravariant (f :*: g) Source #
Defined in Data.Functor.Contravariant
(Eq (f p), Eq (g p)) => Eq ((f :*: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
(Typeable f, Typeable g, Data p, Data (f p), Data (g p)) => Data ((f :*: g) p) Source # Since: 4.9.0.0
Defined in Data.Data
(Ord (f p), Ord (g p)) => Ord ((f :*: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
(Read (f p), Read (g p)) => Read ((f :*: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
(Show (f p), Show (g p)) => Show ((f :*: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Generic ((f :*: g) p) Source #
Defined in GHC.Generics
(Semigroup (f p), Semigroup (g p)) => Semigroup ((f :*: g) p) Source # Since: 4.12.0.0
Defined in GHC.Generics
(Monoid (f p), Monoid (g p)) => Monoid ((f :*: g) p) Source # Since: 4.12.0.0
Defined in GHC.Generics
type Rep1 (f :*: g :: k -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep ((f :*: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
newtype ((f :: k2 -> Type) :.: (g :: k1 -> k2)) (p :: k1) infixr 7Source #
Functor f => Generic1 (f :.: g :: k -> Type) Source #
Defined in GHC.Generics
(Functor f, Functor g) => Functor (f :.: g) Source # Since: 4.9.0.0
Defined in GHC.Generics
(Applicative f, Applicative g) => Applicative (f :.: g) Source # Since: 4.9.0.0
Defined in GHC.Generics
(Foldable f, Foldable g) => Foldable (f :.: g) Source # Since: 4.9.0.0
Defined in Data.Foldable
(Traversable f, Traversable g) => Traversable (f :.: g) Source # Since: 4.9.0.0
Defined in Data.Traversable
(Alternative f, Applicative g) => Alternative (f :.: g) Source # Since: 4.9.0.0
Defined in GHC.Generics
(Functor f, Contravariant g) => Contravariant (f :.: g) Source #
Defined in Data.Functor.Contravariant
Eq (f (g p)) => Eq ((f :.: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
(Typeable f, Typeable g, Data p, Data (f (g p))) => Data ((f :.: g) p) Source # Since: 4.9.0.0
Defined in Data.Data
Ord (f (g p)) => Ord ((f :.: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Read (f (g p)) => Read ((f :.: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Show (f (g p)) => Show ((f :.: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Generic ((f :.: g) p) Source #
Defined in GHC.Generics
Semigroup (f (g p)) => Semigroup ((f :.: g) p) Source # Since: 4.12.0.0
Defined in GHC.Generics
Monoid (f (g p)) => Monoid ((f :.: g) p) Source # Since: 4.12.0.0
Defined in GHC.Generics
type Rep1 (f :.: g :: k -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep ((f :.: g) p) Source # Since: 4.7.0.0
Defined in GHC.Generics
Unboxed representation types
data family URec (a :: Type) (p :: k) Source #
Constants of unlifted kinds
Since: 4.9.0.0
Generic1 (URec Word :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (URec Int :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (URec Float :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (URec Double :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (URec Char :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (URec (Ptr ()) :: k -> Type) Source #
Defined in GHC.Generics
Functor (URec Char :: Type -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
Functor (URec Double :: Type -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
Functor (URec Float :: Type -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
Functor (URec Int :: Type -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
Functor (URec Word :: Type -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
Functor (URec (Ptr ()) :: Type -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
Foldable (URec Char :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Foldable
Foldable (URec Double :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Foldable
Foldable (URec Float :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Foldable
Foldable (URec Int :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Foldable
Foldable (URec Word :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Foldable
Foldable (URec (Ptr ()) :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Foldable
Traversable (URec Char :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Traversable
Traversable (URec Double :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Traversable
Traversable (URec Float :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Traversable
Traversable (URec Int :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Traversable
Traversable (URec Word :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Traversable
Traversable (URec (Ptr ()) :: Type -> Type) Source # Since: 4.9.0.0
Defined in Data.Traversable
Eq (URec Word p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Eq (URec Int p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Eq (URec Float p) Source #
Defined in GHC.Generics
Eq (URec Double p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Eq (URec Char p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Eq (URec (Ptr ()) p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Ord (URec Word p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Ord (URec Int p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Ord (URec Float p) Source #
Defined in GHC.Generics
Ord (URec Double p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Ord (URec Char p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Ord (URec (Ptr ()) p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Show (URec Word p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Show (URec Int p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Show (URec Float p) Source #
Defined in GHC.Generics
Show (URec Double p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Show (URec Char p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Generic (URec Word p) Source #
Defined in GHC.Generics
Generic (URec Int p) Source #
Defined in GHC.Generics
Generic (URec Float p) Source #
Defined in GHC.Generics
Generic (URec Double p) Source #
Defined in GHC.Generics
Generic (URec Char p) Source #
Defined in GHC.Generics
Generic (URec (Ptr ()) p) Source #
Defined in GHC.Generics
Used for marking occurrences of Word#
data URec Word (p :: k) Source #
Since: 4.9.0.0
Defined in GHC.Generics
Used for marking occurrences of Int#
data URec Int (p :: k) Source #
Since: 4.9.0.0
Defined in GHC.Generics
Used for marking occurrences of Float#
data URec Float (p :: k) Source #
Since: 4.9.0.0
Defined in GHC.Generics
Used for marking occurrences of Double#
data URec Double (p :: k) Source #
Since: 4.9.0.0
Defined in GHC.Generics
Used for marking occurrences of Char#
data URec Char (p :: k) Source #
Since: 4.9.0.0
Defined in GHC.Generics
Used for marking occurrences of Addr#
data URec (Ptr ()) (p :: k) Source #
Since: 4.9.0.0
Defined in GHC.Generics
type Rep1 (URec Word :: k -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep1 (URec Int :: k -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep1 (URec Float :: k -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep1 (URec Double :: k -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep1 (URec Char :: k -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep1 (URec (Ptr ()) :: k -> Type) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep (URec Word p) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep (URec Int p) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep (URec Float p) Source #
Defined in GHC.Generics
type Rep (URec Double p) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep (URec Char p) Source # Since: 4.9.0.0
Defined in GHC.Generics
type Rep (URec (Ptr ()) p) Source # Since: 4.9.0.0
Defined in GHC.Generics
Synonyms for convenience
type Rec0 = K1 R Source #
Type synonym for encoding recursion (of kind Type)
data R Source #
Tag for K1: recursion (of kind Type)
type D1 = M1 D Source #
Type synonym for encoding meta-information for datatypes
type C1 = M1 C Source #
Type synonym for encoding meta-information for constructors
type S1 = M1 S Source #
Type synonym for encoding meta-information for record selectors
data S Source #
Tag for M1: record selector
class Datatype d where Source #
Class for datatypes that represent datatypes
datatypeName :: t d (f :: k -> Type) (a :: k) -> [Char] Source #
The name of the datatype (unqualified)
moduleName :: t d (f :: k -> Type) (a :: k) -> [Char] Source #
The fully-qualified name of the module where the type is declared
packageName :: t d (f :: k -> Type) (a :: k) -> [Char] Source #
The package name of the module where the type is declared
Since: 4.9.0.0
isNewtype :: t d (f :: k -> Type) (a :: k) -> Bool Source #
Marks if the datatype is actually a newtype
Since: 4.7.0.0
(KnownSymbol n, KnownSymbol m, KnownSymbol p, SingI nt) => Datatype (MetaData n m p nt :: Meta) Source # Since: 4.9.0.0
Defined in GHC.Generics
class Constructor c where Source #
Class for datatypes that represent data constructors
conName :: t c (f :: k -> Type) (a :: k) -> [Char] Source #
The name of the constructor
conFixity :: t c (f :: k -> Type) (a :: k) -> Fixity Source #
The fixity of the constructor
conIsRecord :: t c (f :: k -> Type) (a :: k) -> Bool Source #
Marks if this constructor is a record
(KnownSymbol n, SingI f, SingI r) => Constructor (MetaCons n f r :: Meta) Source # Since: 4.9.0.0
Defined in GHC.Generics
class Selector s where Source #
Class for datatypes that represent records
(SingI mn, SingI su, SingI ss, SingI ds) => Selector (MetaSel mn su ss ds :: Meta) Source # Since: 4.9.0.0
Defined in GHC.Generics
data Fixity Source #
Datatype to represent the fixity of a constructor. An infix | declaration directly corresponds to an application of Infix.
Infix Associativity Int
Eq Fixity Source # Since: 4.6.0.0
Defined in GHC.Generics
Data Fixity Source # Since: 4.9.0.0
Defined in Data.Data
Ord Fixity Source # Since: 4.6.0.0
Defined in GHC.Generics
Read Fixity Source # Since: 4.6.0.0
Defined in GHC.Generics
Show Fixity Source # Since: 4.6.0.0
Defined in GHC.Generics
Generic Fixity Source #
Defined in GHC.Generics
type Rep Fixity Source # Since: 4.7.0.0
Defined in GHC.Generics
data FixityI Source #
This variant of Fixity appears at the type level.
Since: 4.9.0.0
InfixI Associativity Nat
data Associativity Source #
Datatype to represent the associativity of a constructor
Bounded Associativity Source # Since: 4.9.0.0
Defined in GHC.Generics
Enum Associativity Source # Since: 4.9.0.0
Defined in GHC.Generics
Eq Associativity Source # Since: 4.6.0.0
Defined in GHC.Generics
Data Associativity Source # Since: 4.9.0.0
Defined in Data.Data
Ord Associativity Source # Since: 4.6.0.0
Defined in GHC.Generics
Read Associativity Source # Since: 4.6.0.0
Defined in GHC.Generics
Show Associativity Source # Since: 4.6.0.0
Defined in GHC.Generics
Ix Associativity Source # Since: 4.9.0.0
Defined in GHC.Generics
Generic Associativity Source #
Defined in GHC.Generics
type Rep Associativity Source # Since: 4.7.0.0
Defined in GHC.Generics
data SourceUnpackedness Source #
The unpackedness of a field as the user wrote it in the source code. For example, in the following data type:
data E = ExampleConstructor Int
{-# NOUNPACK #-} Int
{-# UNPACK #-} Int
The fields of ExampleConstructor have NoSourceUnpackedness, SourceNoUnpack, and SourceUnpack, respectively.
Since: 4.9.0.0
Bounded SourceUnpackedness Source # Since: 4.9.0.0
Defined in GHC.Generics
Enum SourceUnpackedness Source # Since: 4.9.0.0
Defined in GHC.Generics
Eq SourceUnpackedness Source # Since: 4.9.0.0
Defined in GHC.Generics
Data SourceUnpackedness Source # Since: 4.9.0.0
Defined in Data.Data
Ord SourceUnpackedness Source # Since: 4.9.0.0
Defined in GHC.Generics
Read SourceUnpackedness Source # Since: 4.9.0.0
Defined in GHC.Generics
Show SourceUnpackedness Source # Since: 4.9.0.0
Defined in GHC.Generics
Ix SourceUnpackedness Source # Since: 4.9.0.0
Defined in GHC.Generics
Generic SourceUnpackedness Source #
Defined in GHC.Generics
type Rep SourceUnpackedness Source # Since: 4.9.0.0
Defined in GHC.Generics
data SourceStrictness Source #
The strictness of a field as the user wrote it in the source code. For example, in the following data type:
data E = ExampleConstructor Int ~Int !Int
The fields of ExampleConstructor have NoSourceStrictness, SourceLazy, and SourceStrict, respectively.
Since: 4.9.0.0
Bounded SourceStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
Enum SourceStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
Eq SourceStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
Data SourceStrictness Source # Since: 4.9.0.0
Defined in Data.Data
Ord SourceStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
Read SourceStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
Show SourceStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
Ix SourceStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
Generic SourceStrictness Source #
Defined in GHC.Generics
type Rep SourceStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
data DecidedStrictness Source #
The strictness that GHC infers for a field during compilation. Whereas there are nine different combinations of SourceUnpackedness and SourceStrictness, the strictness that GHC decides will
ultimately be one of lazy, strict, or unpacked. What GHC decides is affected both by what the user writes in the source code and by GHC flags. As an example, consider this data type:
data E = ExampleConstructor {-# UNPACK #-} !Int !Int Int
• If compiled without optimization or other language extensions, then the fields of ExampleConstructor will have DecidedStrict, DecidedStrict, and DecidedLazy, respectively.
• If compiled with -XStrictData enabled, then the fields will have DecidedStrict, DecidedStrict, and DecidedStrict, respectively.
• If compiled with -O2 enabled, then the fields will have DecidedUnpack, DecidedStrict, and DecidedLazy, respectively.
Since: 4.9.0.0
Bounded DecidedStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
Enum DecidedStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
Eq DecidedStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
Data DecidedStrictness Source # Since: 4.9.0.0
Defined in Data.Data
Ord DecidedStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
Read DecidedStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
Show DecidedStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
Ix DecidedStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
Generic DecidedStrictness Source #
Defined in GHC.Generics
type Rep DecidedStrictness Source # Since: 4.9.0.0
Defined in GHC.Generics
data Meta Source #
Datatype to represent metadata associated with a datatype (MetaData), constructor (MetaCons), or field selector (MetaSel).
• In MetaData n m p nt, n is the datatype's name, m is the module in which the datatype is defined, p is the package in which the datatype is defined, and nt is 'True if the datatype is a newtype.
• In MetaCons n f s, n is the constructor's name, f is its fixity, and s is 'True if the constructor contains record selectors.
• In MetaSel mn su ss ds, if the field uses record syntax, then mn is Just the record name. Otherwise, mn is Nothing. su and ss are the field's unpackedness and strictness annotations, and ds is
the strictness that GHC infers for the field.
Since: 4.9.0.0
MetaData Symbol Symbol Symbol Bool
MetaCons Symbol FixityI Bool
MetaSel (Maybe Symbol) SourceUnpackedness SourceStrictness DecidedStrictness
(KnownSymbol n, SingI f, SingI r) => Constructor (MetaCons n f r :: Meta) Source # Since: 4.9.0.0
Defined in GHC.Generics
(KnownSymbol n, KnownSymbol m, KnownSymbol p, SingI nt) => Datatype (MetaData n m p nt :: Meta) Source # Since: 4.9.0.0
Defined in GHC.Generics
(SingI mn, SingI su, SingI ss, SingI ds) => Selector (MetaSel mn su ss ds :: Meta) Source # Since: 4.9.0.0
Defined in GHC.Generics
Generic type classes
class Generic a where Source #
Representable types of kind *. This class is derivable in GHC with the DeriveGeneric flag on.
A Generic instance must satisfy the following laws:
from . to ≡ id
to . from ≡ id
from :: a -> Rep a x Source #
Convert from the datatype to its representation
to :: Rep a x -> a Source #
Convert from the representation to the datatype
Generic Bool Source #
Defined in GHC.Generics
Generic Ordering Source #
Defined in GHC.Generics
Generic () Source #
Defined in GHC.Generics
Generic DecidedStrictness Source #
Defined in GHC.Generics
Generic SourceStrictness Source #
Defined in GHC.Generics
Generic SourceUnpackedness Source #
Defined in GHC.Generics
Generic Associativity Source #
Defined in GHC.Generics
Generic Fixity Source #
Defined in GHC.Generics
Generic Any Source #
Defined in Data.Semigroup.Internal
Generic All Source #
Defined in Data.Semigroup.Internal
Generic ExitCode Source #
Defined in GHC.IO.Exception
Generic Version Source #
Defined in Data.Version
Generic Void Source #
Defined in Data.Void
Generic [a] Source #
Defined in GHC.Generics
Generic (Maybe a) Source #
Defined in GHC.Generics
Generic (Par1 p) Source #
Defined in GHC.Generics
Generic (NonEmpty a) Source #
Defined in GHC.Generics
Generic (Down a) Source #
Defined in GHC.Generics
Generic (Product a) Source #
Defined in Data.Semigroup.Internal
Generic (Sum a) Source #
Defined in Data.Semigroup.Internal
Generic (Endo a) Source #
Defined in Data.Semigroup.Internal
Generic (Dual a) Source #
Defined in Data.Semigroup.Internal
Generic (Last a) Source #
Defined in Data.Monoid
Generic (First a) Source #
Defined in Data.Monoid
Generic (Identity a) Source #
Defined in Data.Functor.Identity
Generic (ZipList a) Source #
Defined in Control.Applicative
Generic (Option a) Source #
Defined in Data.Semigroup
Generic (WrappedMonoid m) Source #
Defined in Data.Semigroup
Generic (Last a) Source #
Defined in Data.Semigroup
Generic (First a) Source #
Defined in Data.Semigroup
Generic (Max a) Source #
Defined in Data.Semigroup
Generic (Min a) Source #
Defined in Data.Semigroup
Generic (Complex a) Source #
Defined in Data.Complex
Generic (Either a b) Source #
Defined in GHC.Generics
Generic (V1 p) Source #
Defined in GHC.Generics
Generic (U1 p) Source #
Defined in GHC.Generics
Generic (a, b) Source #
Defined in GHC.Generics
Generic (Proxy t) Source #
Defined in GHC.Generics
Generic (WrappedMonad m a) Source #
Defined in Control.Applicative
Generic (Arg a b) Source #
Defined in Data.Semigroup
Generic (Rec1 f p) Source #
Defined in GHC.Generics
Generic (URec Word p) Source #
Defined in GHC.Generics
Generic (URec Int p) Source #
Defined in GHC.Generics
Generic (URec Float p) Source #
Defined in GHC.Generics
Generic (URec Double p) Source #
Defined in GHC.Generics
Generic (URec Char p) Source #
Defined in GHC.Generics
Generic (URec (Ptr ()) p) Source #
Defined in GHC.Generics
Generic (a, b, c) Source #
Defined in GHC.Generics
Generic (Alt f a) Source #
Defined in Data.Semigroup.Internal
Generic (Ap f a) Source #
Defined in Data.Monoid
Generic (Const a b) Source #
Defined in Data.Functor.Const
Generic (WrappedArrow a b c) Source #
Defined in Control.Applicative
Generic (K1 i c p) Source #
Defined in GHC.Generics
Generic ((f :+: g) p) Source #
Defined in GHC.Generics
Generic ((f :*: g) p) Source #
Defined in GHC.Generics
Generic (a, b, c, d) Source #
Defined in GHC.Generics
from :: (a, b, c, d) -> Rep (a, b, c, d) x Source #
to :: Rep (a, b, c, d) x -> (a, b, c, d) Source #
Generic (Sum f g a) Source #
Defined in Data.Functor.Sum
Generic (Product f g a) Source #
Defined in Data.Functor.Product
Generic (M1 i c f p) Source #
Defined in GHC.Generics
Generic ((f :.: g) p) Source #
Defined in GHC.Generics
Generic (a, b, c, d, e) Source #
Defined in GHC.Generics
from :: (a, b, c, d, e) -> Rep (a, b, c, d, e) x Source #
to :: Rep (a, b, c, d, e) x -> (a, b, c, d, e) Source #
Generic (Compose f g a) Source #
Defined in Data.Functor.Compose
Generic (a, b, c, d, e, f) Source #
Defined in GHC.Generics
from :: (a, b, c, d, e, f) -> Rep (a, b, c, d, e, f) x Source #
to :: Rep (a, b, c, d, e, f) x -> (a, b, c, d, e, f) Source #
Generic (a, b, c, d, e, f, g) Source #
Defined in GHC.Generics
from :: (a, b, c, d, e, f, g) -> Rep (a, b, c, d, e, f, g) x Source #
to :: Rep (a, b, c, d, e, f, g) x -> (a, b, c, d, e, f, g) Source #
class Generic1 (f :: k -> Type) where Source #
Representable types of kind * -> * (or kind k -> *, when PolyKinds is enabled). This class is derivable in GHC with the DeriveGeneric flag on.
A Generic1 instance must satisfy the following laws:
from1 . to1 ≡ id
to1 . from1 ≡ id
from1 :: f a -> Rep1 f a Source #
Convert from the datatype to its representation
to1 :: Rep1 f a -> f a Source #
Convert from the representation to the datatype
Generic1 (U1 :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (V1 :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (Proxy :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (URec Word :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (URec Int :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (URec Float :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (URec Double :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (URec Char :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (URec (Ptr ()) :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (Rec1 f :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (Alt f :: k -> Type) Source #
Defined in Data.Semigroup.Internal
Generic1 (Ap f :: k -> Type) Source #
Defined in Data.Monoid
Generic1 (Const a :: k -> Type) Source #
Defined in Data.Functor.Const
Generic1 (f :*: g :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (f :+: g :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (K1 i c :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (Sum f g :: k -> Type) Source #
Defined in Data.Functor.Sum
Generic1 (Product f g :: k -> Type) Source #
Defined in Data.Functor.Product
Functor f => Generic1 (f :.: g :: k -> Type) Source #
Defined in GHC.Generics
Generic1 (M1 i c f :: k -> Type) Source #
Defined in GHC.Generics
Functor f => Generic1 (Compose f g :: k -> Type) Source #
Defined in Data.Functor.Compose
Generic1 [] Source #
Defined in GHC.Generics
Generic1 Maybe Source #
Defined in GHC.Generics
Generic1 Par1 Source #
Defined in GHC.Generics
Generic1 NonEmpty Source #
Defined in GHC.Generics
Generic1 Down Source #
Defined in GHC.Generics
Generic1 Product Source #
Defined in Data.Semigroup.Internal
Generic1 Sum Source #
Defined in Data.Semigroup.Internal
Generic1 Dual Source #
Defined in Data.Semigroup.Internal
Generic1 Last Source #
Defined in Data.Monoid
Generic1 First Source #
Defined in Data.Monoid
Generic1 Identity Source #
Defined in Data.Functor.Identity
Generic1 ZipList Source #
Defined in Control.Applicative
Generic1 Option Source #
Defined in Data.Semigroup
Generic1 WrappedMonoid Source #
Defined in Data.Semigroup
Generic1 Last Source #
Defined in Data.Semigroup
Generic1 First Source #
Defined in Data.Semigroup
Generic1 Max Source #
Defined in Data.Semigroup
Generic1 Min Source #
Defined in Data.Semigroup
Generic1 Complex Source #
Defined in Data.Complex
Generic1 (Either a :: Type -> Type) Source #
Defined in GHC.Generics
Generic1 ((,) a :: Type -> Type) Source #
Defined in GHC.Generics
Generic1 (WrappedMonad m :: Type -> Type) Source #
Defined in Control.Applicative
Generic1 (Arg a :: Type -> Type) Source #
Defined in Data.Semigroup
Generic1 ((,,) a b :: Type -> Type) Source #
Defined in GHC.Generics
Generic1 (WrappedArrow a b :: Type -> Type) Source #
Defined in Control.Applicative
Generic1 ((,,,) a b c :: Type -> Type) Source #
Defined in GHC.Generics
from1 :: (a, b, c, a0) -> Rep1 ((,,,) a b c) a0 Source #
to1 :: Rep1 ((,,,) a b c) a0 -> (a, b, c, a0) Source #
Generic1 ((,,,,) a b c d :: Type -> Type) Source #
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Diagrams and Mathematical Practice
Organizers: Jean-Yves Beziau, Rio de Janeiro, Brazil (jyb.unilog@gmail.com) and Andrei Rodin, Nancy, France (andrei.rodin@univ-lorraine.fr)
Workshop description
Mathematical practice is classically articulated around three notions: definitions, axioms, and proofs. Blaise Pascal gave a short and clear masterful description of this articulation in his short
booklet De l’esprit géométrique et de l’art de persuader (1657), giving some rules for these three notions. Later on, Alfred Tarski was strongly influenced by this booklet, as shown by his 1937 Paris
talk “Sur la méthode deductive”. But, if nowadays if it is common to think of proofs with diagrams, in particular after the three volume book Proofs without Words (1993, 2000, 2015) by Roger
B.Nelsen, the question of the uses of diagrams for stating and representing axioms and definitions has not yet been systematically studied. This workshop aims at promoting these uses (considering the
interaction between the three notions).
Moreover, diagrams are convenient as illustration and source of inspiration, for conceptualization and understanding, also important features of mathematical practice.
This workshop is open to reflections on the functioning of diagrams as well as inquiries about the philosophy of mathematics, the nature of reasoning and proving, through a diagrammatic methodology.
Keywords: Definition, Axiom, Proof, Postulate, Truth, Reasoning, Symbol, Category Theory, Model Theory, Proof Theory, Set Theory
Accepted talks
Diagrammatic Syntax and Neo-Peircean Calculus of Relations in the Diagrammatization of Mathematical Practices
Nathan Haydon (Tallinn University of Technology, Estonia) and Ahti-Veikko Pietarinen (Hong Kong Baptist University, Hong Kong)
Following the early work on the logic of relations by Boole, De Morgan, Peirce, and Schröder in the 19th century, the study of relations has largely been represented in mathematical practice by the
turn to relation algebras. Following Tarski’s renewed emphasis on the subject in 1941 [12], relation algebra is now recognized as granting a foundation for large portions of mathematics [11, 13],
with notable descendants in computer science [1, 4, 10] and cognitive science (such as Formal Concept Analysis [15] and Knowledge Representation [14]). A more recent direction is motivated by
category-theoretic allegories [5] and relation-algebraic theories [2], which can be seen as an approach in the tradition of universal algebra with a focus on relations and relational operations.
The works cited suggest logic of relations as a general setting for elements of mathematical foundations, instruction, and practice. We can complement the picture of what Tarski specifically gives
the credits for, namely Peirce’s early work on the logic of relations [8, 10, 13], by showing how Peirce’s later logic, his Existential Graphs (EGs, [9]) draws a direct connection to relation
algebra, thus superseding the standard comparison with predicate logic by topological features of graphs that preserve composition, much needed for the latter’s direct connection to
relation-algebraic operations.
In this vein, recent work in [6, 7] shows that Peirce’s work closely accords with categorical presentations [2, 5]. The diagrammatic, Neo-Peircean calculus of relations [3], contains a sound and
complete axiomatization of first-order logic which agrees with algebraic theories and in which additional axioms characterizing an algebraic theory can be presented and reasoned about within the
diagrammatic syntax itself.
In this work, we present these recent diagrammatic advances on Peirce’s work in terms of the Neo-Peircean calculus of relations, showing how to translate relation-algebraic expressions and operations
into EGs, with examples of diagrammatic algebraic theories and functional and relational properties. The upshot is that Peirce’s work not only inspired relational approaches to mathematics but that
his EGs catered for the contemporary settings for diagrammatic practice of mathematics and its philosophy.
[1] Bird, R. and De Moor, O. (1996). The algebra of programming. NATO ASI DPD, 152:167–203.
[2] Bonchi, F., Pavlovic, D., and Sobocinski, P. (2017). Functorial semantics for relational theories. https://arxiv.org/abs/1711.08699.
[3] Bonchi, F., Giorgio, A. D., Haydon, N., and Sobocinski, P. (2024). Diagrammatic algebra of first order logic. To appear at LICS.
[4] Codd, E. F. (1970). A relational model of data for large shared data banks. Commun. ACM, 13(6):377–387.
[5] Freyd, P. J. and Scedrov, A. (1990). Categories, Allegories. North-Holland Mathematical Library.
[6] Haydon, N. and Sobocinski, P. (2020). Compositional diagrammatic first-order logic. In Pietarinen, A.-V., Chapman, P., Bosveld-de Smet, L., Giardino, V., Corter, J., and Linker, S., eds.,
Diagrammatic Representation and Inference, pp. 402–418, Cham. Springer.
[7] Haydon, N. and Pietarinen, A.-V. (2021). Residuation in existential graphs. In Basu, A., Stapleton, G., Linker, S., Legg, C., Manalo, E., and Viana, P., eds., Diagrammatic Representation and
Inference, pp. 229–237, Cham. Springer.
[8] Peirce, Charles S. Studies in Logic. By Members of the Johns Hopkins University. Little, Brown, and Company, 1883.
[9] Peirce, Charles S. Logic of the Future : Writings on Existential Graphs. Ed. by A.-V. Pietarinen. Vol. 1: History and Applications. Vol. 2/1: The Logical Tracts. Vol. 2/2: The 1903 Lowell
Lectures. Vol. 3/1: Pragmaticism. Vol. 3/2: Correspondence. Boston & Berlin: De Gruyter, 2019-2024.
[10] Pratt, V. R. Origins of the Calculus of Binary Relations. In International Symposium on Mathematical Foundations of Computer Science, pp. 142–155. Springer, 1992.
[11] Schmidt, G. (2010). Relational Mathematics. Encyclopedia of Mathematics and its Applications. Cambridge University Press.
[12] Tarski, A. (1941). On the calculus of relations. The Journal of Symbolic Logic, 6(3):73–89.
[13] Tarski, A. and Givant, S. R. (1988). A formalization of set theory without variables. American Mathematical Society 41.
[14] Wang, Y. (2017). On Relation Algebra: A Denotational Mathematical Structure of Relation Theory for Knowledge Representation and Cognitive Computing. Journal of Advanced Mathematics and
Applications, 6, pp. 43-66(24)
[15] Wille, R. (1992). Concept lattices and conceptual knowledge systems. Computer & Mathematics with Applications, 23(6):493–515.
The Coloured Geometry of 5-Valued Contradiction: the Oppositional Quinque-Segment B[5]2 and its 4D Qttractor
Alessio Moretti (Dipartimento di letteratura francese, Università telematica eCampus, Novedrate, Italy )
The formal study of “opposition” gives rise to a new geometry, of which the “square of opposition” and the “logical hexagon” are famous anticipations. Its first form was a theory of the
“bi-simplexes” A[2]N ([2] and of their “closures” B[2]N [5]]), which gave progressively place to the wider concept of “poly-simplex” ([3], [4] and [1]). The theory is thus based on the concept of
“simplex”, seen as a geometrical n-dimensional expression of the concept of natural number. This concept plays at two levels: the “degree of opposition” (“segment” for 2-opposition, “triangle” for
3-opposition, “tetrahedron” for 4-opposition, etc.) and the “number of truth-values used” (bi-simplexes for 2-valued n-oppositions, tri-simplexes for 3-valued n- oppositions, quadri-simplexes for
4-valued n-oppositions, etc.). This results in a discrete “poly- simplicial space” (expressible either as one quadrant of an infinite 2D matrix or as an infinite descending “numerical triangle”),
made up of “points” B[M]N, where each such point is a complex “polytope of opposition” (containing smaller ones) expressing with “m” truth-values a “n- opposition”. The question lasted unanswered as
to what place “oppositions” occupy inside mathematics. Since a few years an answer has emerged: oppositions B[M]N result from a special “projection” of the series of the “Pascalian simplexes” P[M] (a
generalisation of “Pascal’s triangle” P[2]). Since poly-simplicial “points” B[M]N can be represented prima facie by the number of the vertices of their polytope of opposition (each B[M]N has M^N-M
vertices), for combinatorial reasons the difficulty of studying such structures is proportional to the number of their vertices: it therefore appears that of the infinite number of poly-simplexes B
[M]N only a dozen can be studied relatively easily (having less than 100 vertices). So far six of them have been studied properly (the bi-simplexes B[2]2, B[2]3, B[2]4, and the poly-simplexes B[3]2,
B[3]3 and B[4]2, with poly³3). Here we sketch the study of a seventh poly- simplex, the quinque-segment B[5]2, expressing the geometry of 5-valued contradiction. We show it to be a very elegant
structure provided with 20 vertices, containing as sub-structures intertwined B[3]2 and B[4]2, and whose “geometrical attractor” seems to be a 4D “runcinated 5-cell”. The quinque- segment is also the
intersection of the series of the poly-segments B[M]2 and the series of the quinque- simplexes B[5]N. It is informative for several reasons: (1) it is the first quinque-simplex to be studied (on
which all higher quinque-simplexes B[5]N will rely); (2) it sheds more light on the poly-segments B[M]2; (3) it offers a concrete, although complex view of (linear) 5-valuedness, taking into account
conjunctly “truth-approximation” (“almost false” ¼, and “almost true” ¾, which characterise the quadri-simplexes B[4]N) and “truth-pivotality” (“half way” ½, which characterises the tri-simplexes B
[3]N); (4) it is, with respect to its “geometrical attractor”, the simplest of the four 4-dimensional poly- simplexes of the poly-simplicial space B[M]N (the other three, not yet fully studied, are
the poly- simplexes B[2]5, B[3]4, B[4]3) and as such it could provide precious hints; (5) it confirms, as we will show, the usefulness and, in fact, the necessity of using colours when studying
mathematically “oppositions”.
[1] R.Angot-Pellissier, “Many-Valued Logical Hexagons in a 3-Oppositional Trisimplex“, in J.-Y. Beziau, I. Vandoulakis (eds.), The Exoteric Square of Opposition, Birkhäuser, Cham, 2022, pp.333-345.
[ 2] A.Moretti, “Geometry for Modalities? Yes: Through n-Opposition Theory.” In: J-Y. Beziau, A.Costa-Leite, A.Facchini A. (eds.): Aspects of Universal Logic, N.17 of Travaux de logique, University
of Neuchâtel, 2004).
[3] A.Moretti, The Geometry of Logical Opposition. PhD Thesis, University of Neuchâtel, Neuchâtel, 2009.
[4] A.Moretti, “Tri-simplicial Contradiction: The Pascalian 3D Simplex for the Oppositional Tri-segment.” In .J.-Y. Beziau, I. Vandoulakis (eds.), The Exoteric Square of Opposition, Birkhäuser, Cham,
2022, pp.347-479.
[5] R.Pélissier, “Setting” n-opposition”, Logica Universalis, 2, 2008, pp.235-263.
Empirical studies of mathematical diagrams: Lessons from five years of investigations
Mikkel Willum Johansen (University of Copenhagen, Denmark)
In this talk I will report on a series of empirical investigations of mathematicians’ use of mathematical diagrams I have been part of conducting. The basic approach of the investigations has been to
combine large-scale quantitative investigations with closed readings and qualitative analysis. As a core result, we investigated the prevalence of diagrams in mathematics journals from 1815 to 1915
(Johansen & Pallavicini, 2022). The investigation showed that diagrams were relatively common in mathematical research publication around year 1900, then they disappear for half a century and
reappear during the 1960’s. In 2015 about 65% of the papers included in the survey included at least one diagram. Guided by this overall map of the use of diagrams in mathematical practice we
conducted several focused investigations. Especially, comparison of the types of diagrams used in the period around 1900 and in periods after 1960 shows marked differences; in short, to reenter
mathematical publications diagrams had to (partly) comply to formalist norms of rigor (Johansen & Pallavicini 2021). However, in recent years the attitude seems to have been relaxed and an in-depth
analysis of the use of diagrams in published proofs showed that diagrams are frequently used not only heuristic but also epistemic purposes (Sørensen & Johansen, 2022).
In this talk I will give an overview of the main results outlined above, I will discuss the different roles diagrams play in mathematical practice and how this practice may inform our typology and
definition of (mathematical) diagrams. Finally, I will discuss how mathematicians’ use of diagrams challenges what the (loosely be called) ‘formalist epistemology’ of mathematics, and calls for a
more pragmatic understanding of the ways mathematicians obtain conviction.
Linear Mathematics and the Rule of Duplication in Graphical Logic
Mario Román Garcia (Oxford University, UK) and Ahti-Veikko Pietarinen (Hong Kong Baptist University, Hong Kong)
The common view in the foundations of mathematics has long been that the only point where we care about truth in mathematics is the moment in which we state the axioms. In practice, far from this, we
find that axioms become justified by their theorems. A proof of an old theorem grounded in new axioms is indeed an important achievement, because what we were looking for about that proof was not a
new truth but instead the bidirectional justification of the right axioms with the right theorem(s).
In this paper, we explore an example of these axiom shifts. Mathematics is usually content with the lack of negative evidence, but we might want to determine the more scientific criterion of positive
(interactive, back-and-forth, game-theoretic, [8]) evidence before asserting the truth of a theorem. Scientific consequences of the proposed shifts in the foundations of mathematics that follow this
direction are many, among them constructivism [1, 2] and linear mathematics [3, 6, 9]. From the vantage point of the latter, which we take to be the more exciting development, we analyze diagonal
arguments for barber-style paradoxes, noticing that the diagonals need not be invariably supported: the culprit is the duplicating diagonal.
This leads to the point of the preferred means and methods of analysis. Once we have at our disposal a diagrammatic syntax by which to build up one’s constructions for logical analysis, we are prone
to enter the linear realm. This was shown first by Charles Peirce in 1883, who pointed out not only the indispensable usefulness of game-theoretic semantics for such analyses [4, 7] but also that the
diagrammatic syntax of logical graphs raises the important issue of whether we unvaryingly may use the rule of duplication (iteration) [5]. Today we would ask whether logical inferences are resource
conscious, as avowed by linear logic. Here we gather evidence from Peirce’s hitherto unexplored writings both for and against the linear rendering of the graphical method of logic and its rule of
duplication. Such changes in the perspective towards the interactive, diagrammatic, and linear foundations of mathematics have profound implications for what we mean by “proof” and “truth” in
[1] Bauer, Andrej. Five stages of accepting constructive mathematics. Bulletin of the American Mathematical Society, 54(3):481–498, 2017.
[2] Bishop, Errett & Bridges, Douglas. Constructive Analysis 279. Springer, 2012.
[3] Girard, Jean-Yves. Linear logic: its syntax and semantics. 1995.
[4] Hilpinen, Risto. On C. S. Peirce’s Theory of the Proposition: Peirce as a Precursor of Game-Theoretical Semantics. The Monist 65(2), pp. 182–188, 1982.
[5] Peirce, Charles S. Logic of the Future : Writings on Existential Graphs. Ed. by A.-V. Pietarinen. Vol. 1: History and Applications. Vol. 2/1: The Logical Tracts. Vol. 2/2: The 1903 Lowell
Lectures. Vol. 3/1: Pragmaticism. Vol. 3/2: Correspondence. Boston & Berlin: De Gruyter, 2019-2024.
[6] Pietarinen, Ahti-Veikko. Logic, Language-Games and Ludics. Acta Analytica 18(30/31), pp. 89–123, 2003.
[7] Pietarinen, Ahti-Veikko. Peirce’s Game-Theoretic Ideas in Logic. Semiotica 144(14), pp. 33–47, 2003.
[8] Pietarinen, Ahti-Veikko. Games as Formal Tools versus Games as Explanations in Logic and Science. Foundations of Science 8(1), 317–364.
[9] Shulman, Michael. Linear logic for constructive mathematics. arXiv preprint arXiv:1805.07518, 2018.
Linear Proof versus Diagrammatic Proof – Study of an Example: There is no Cube of Opposition
Caroline Pires Ting (Federal University of Rio de Janeiro, Brazil, and Macau International Instititute) and Jean-Yves Beziau (Federal University of Rio de Janeiro, Brazil, Brazilian Research Council
and Brazilian Academy of Philosophy)
The theory of opposition is based on of one of the most famous diagrams of the history of philosophy, logic and mathematics, the square of opposition.
Along the years, many geometric generalizations have been proposed: polygons and polyhedra. Among polygons, the most famous one is Blanché’s hexagon of opposition, among polyhedra, the cube of
opposition. But if, on the one hand the hexagon of opposition is a figure of opposition following the original pattern of the square of opposition and improving it by reconstructing it together with
two other squares included in the hexagon, on the other hand a cube of opposition cannot have its six faces as standard squares of opposition.
The fact that there is no cube of opposition in this sense can be seen in a few seconds using a colored three-dimensional diagram. This contrasts with a linear proof which is rather long and tedious.
Based on that, we will examine the following questions which are of general interest for the value and usefulness of diagrammatic proofs: Do these two proofs have the same strength and validity? Are
they equivalent? And we will compare the situation with other cases, especially the two versions of Cantor’s proof about the non-enumerability of the reals.
[1] J.-Y.Beziau, “There is no cube of opposition”, in J.-Y.Beziau and G.Basti, The Square of Opposition: A Cornerstone of Thought, Birkhäuser, Basel, 2017, pp.179-193
[2].R.Blanché, Structures intellectuelles – Essai sur l’organisation systématique des concepts, Vrin, Paris, 1966.
[3] M.Correia, “Boethius on the Square of Opposition“, in J.-Y.Beziau and D.Jacquette, Around and Beyond the Square of Opposition, Birkhäuser, Basel, 2012, , pp.41-52.
[4] M.Correia, “Aristotle’s Squares of Opposition“, South American Journal of Logic, Vol. 3, n. 2, pp. 313–326, 2017.
A Diagrammatical Perspective on the Algebras of Logic
Jasmin Özel (University of Siegen, Germany)
The use of diagrams as a tool for the evaluation of proofs experienced a period of growth in the 18th and especially 19th century (Euler, Venn, Carroll, Peirce). Yet, the simultaneous Algebra of
Logic tradition steered clear of the use of diagrams. Even Ladd-Franklin, as a student of Peirce, and thus familiar with his Existential Graphs, refrained from using diagrams in her writings. The
goal of this talk will be to explain why the Algebra of Logic tradition largely avoided diagrams as a tool of reasoning—and to show how the tradition could have benefited from including diagrams.
Focusing on the changes in notation that Ladd-Franklin introduced in contrast to previous Algebras of Logic, I will first ask whether a diagrammatic expression was perhaps simply not conducive to a
more convenient visual grasp of the axioms and postulates we generally find in the Algebra of Logic tradition. Following Ambrose & Lazerowitz (1960), I will argue that this is not the case; the
diagrammatic expression of the antilogism, for instance in Venn-diagrams, is just as straightforward as it is in the case of syllogisms. The second part of my talk will be focusing on the different
function that diagrams can play in the contexts of syllogisms, proof theory, and the early beginnings of model theory— and why diagrams occupied different roles for the logicist tradition in the
early 20th century as opposed to the psychologist tradition we find on the European continent at the time. My hypothesis will be that while diagrams can serve an important purpose when it comes to
the description of the general structure of logical arguments, their “universal algebra”, the Algebra of Logic tradition’s pronounced focus on questions concerning the choice of notation prevented
the algebrarians of logic from fully taking advantage of diagrams. I will close with a discussion of the potential advantages of a diagrammatically expressed “universal algebra”. In particular, I
will argue that the expression of Ladd-Franklin’s Algebra of Logic in terms of diagrams demonstrates that her antilogism made it possible to express claims including modal operators.
[1] F. Abeles. Christine Ladd-Franklin’s antilogism. In Verburgt, L. & Cosci, M. (eds.) Aristotle’s syllogism and the creation of modern logic: Between tradition, and innovation, 1820s-1930s.
Bloomsbury Publishing Plc., 2023.
[2] A. Ambrose and M. Lazerowitz. Logic: The Theory of Formal Inference. Holt, Rinehart Winston, New York, 1961.
[3] M.R. Cohen, E. Nagel. An Introduction to Logic. Harcourt, 1937.
[4] H. Curry. A Mathematical Treatment of the Rules of the Syllogism. In Mind 45(178), 209–216.
[5] R. H. Dotterer 1943. A Supplementary Note on the Rules of the Antilogism. In The Journal of Symbolic Logic. Vol. 8 , No. 1. (1943) , pp. 24-24.
[6] C. Dutilh Noves. Reassessing logical hylomorphism and the demarcation of logical constants. In Synthese. Vol. 185 (2012), pp. 387–410.
[7] R. W. Holmes Exercises in Reasoning, with an Outline of Logic. 1939, 1940.
[8] F. Janssen-Lauret. Grandmothers of Analytic Philosophy: The Formal and Philosophical Logic of Christine Ladd-Franklin and Constance Jones. In Minnesota Studies in Philosophy of Science.
Minnesota, 2021.
[9] C. F. Ladd-Franklin. The Antilogism. In Mind. New Series, Vol. 37, No. 148 (Oct., 1928), pp. 532-534.
[10] C. Ladd On the algebra of logic. In Studies in Logic, By Members of the Johns Hopkins University edited by C. S. Peirce Boston: Little, Brown, and Company; Cambridge: University Press, John
Wilson and Son, 1883.
[11] C. I. Lewis. Interesting Theorems in Symbolic Logic. In The Journal of Philosophy, Psychology and Scientific Methods. 10 (9) (1913), pp. 239-242.
[12] C. A. Mace. The Principles of Logic: An Introductory Survey.
[13] J. R. Norman & J. R. Sylvan. Directions in Relevant Logic. 2012.
[14] J. Parker. An Explication of the Antilogism in Christine Ladd- Franklin’s “Algebra of Logic” – Symbolic Notation in “Algebra of Logic”. Convergence. https://maa.org/press/periodicals/convergence
/anexplication- of-the-antilogism-in-christine-ladd-franklins-algebra-of-logicfrom- elimination-toOnline resource
[15] A. Pietarinen. Christine Ladd-Franklin’s and Victoria Welby’s correspondence with Charles Peirce. In Semiotica. (2013)
[16] A. Reichenberger. Gender Equality and Diversity. Challenges and Perspectives for the Historiography of Science. forthcoming.
[17] R. Rogers. The Single Rule of the Antilogism and Syllogism. In Hermathena Vol. 19, No. 42 (1920), pp. 96-99.
[18] S. Russinoff. The Syllogism’s Final Solution. In The Bulletin of Symbolic Logic. Vol. 5, No. 4 (Dec., 1999), pp. 451-469.
[19] S. Uckelman. What Problem Did Ladd-Franklin (The She) Solve(d)? In Notre Dame Journal of Formal Logic. 2021, 62 (3). pp. 527-552.
[20] R. M. Sabre. Extending the Antilogism. In Logique et Analyse. Vol. 30 , No. 117/118., pp. 103-111.
[21] E. Shen. The Ladd-Franklin Formula in Logic: The Antilogism. In Mind. 36 (1927): 54–60.
[22] S. Stebbing. A Modern Introduction to Logic. Methuen: London (1930).
[23] A. P. Ushenko. The Theory of Logic: An Introductory Text. 1936.
[24] W. C. Wilcox. The Antilogism Extended. In Mind. 1969 , Vol. 78 , No. 310. , pp. 266-269.
Achieving Mathematical Understanding via Natural Language, Symbols and Diagrams
Andrei Rodin (Archives Poincaré, University of Lorraine, France) and George Shabat(Moscow Center for Continuous Mathematical Education, Russia) ,
A presentation of mathematical reasoning typically combines three types of expressive means: a natural language, elements of symbolic syntax and diagrams. Using some historical and some recent
examples we show how these expressive means work together in mathematical discourses, and analyse their specific epistemic roles. Using O.B. Bassler’s distinction between local and global
surveyability of mathematical proofs we show how mathematical diagrams mediate between informal linguistic explanations of mathematical arguments, on the one hand, and formal syntactic computations,
on the other hand. Along with some historical examples we consider the recent case of formalised computer-assisted mathematics and show how using dynamic diagrams in the case helps to achieve a
better understanding.
[1] O.B.Bassler, “The Surveyability of Mathematical Proof: A Historical Perspective“, Synthese. Vol. 148 (2006), pp. 99–133.
Background of organizers
Jean-Yves Beziau is a Swiss Logician, Philosopher and Mathematician, PhD in mathematics and PhD in Philosophy. He has been living and working in different places: France, Brazil, Poland, Corsica,
California (UCLA, Stanford, UCSD), Switzerland. He is currently Professor at the University of Brazil in Rio de Janeiro, former Director of Graduate Studies in Philosophy and former President of the
Brazilian Academy of Philosophy. He is the creator of the World Logic Day, yearly celebrated on January 14 (UNESCO international days), the World Logic Prizes Contest, the founder and Editor-in-Chief
of the journal Logica Universalis and South American Journal of Logic, the book series Logic PhDs, Studies in Universal Logic and area logic editor of the Internet Encyclopedia of Philosophy. He has
published about 200 research papers and 30 edited books and Special Issues of Journals.
Andrei Rodin is a lecturer in Poincaré Archives, University of Lorraine (France) working in the history and philosophy of mathematics and philosophical logic. He is the author of Axiomatic Method and
Category Theory (Springer 2014) and a number of articles. Andrei Rodin’s Ph.D. thesis (1995) is on the axiomatic and conceptual structure of the first four Books of Euclid’s Elements; in 2020 he
defended the Habilitation thesis on The Axiomatic Structure of Scientific Theories. | {"url":"https://diagrams-2024.diagrams-conference.org/workshops/diagrams-and-mathematical-practice/","timestamp":"2024-11-10T15:58:21Z","content_type":"text/html","content_length":"88800","record_id":"<urn:uuid:d9f405f5-fa16-44c1-a42c-ed4e7f4773d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00742.warc.gz"} |
Transactions Online
Ryota EGUCHI, Taisuke IZUMI, "Sub-Linear Time Aggregation in Probabilistic Population Protocol Model" in IEICE TRANSACTIONS on Fundamentals, vol. E102-A, no. 9, pp. 1187-1194, September 2019, doi:
Abstract: A passively mobile system is an abstract notion of mobile ad-hoc networks. It is a collection of agents with computing devices. Agents move in a region, but the algorithm cannot control
their physical behavior (i.e., how they move). The population protocol model is one of the promising models in which the computation proceeds by the pairwise communication between two agents. The
communicating agents update their states by a specified transition function (algorithm). In this paper, we consider a general form of the aggregation problem with a base station. The base station is
a special agent having the computational power more powerful than others. In the aggregation problem, the base station has to sum up for inputs distributed to other agents. We propose an algorithm
that solves the aggregation problem in sub-linear parallel time using a relatively small number of states per agent. More precisely, our algorithm solves the aggregation problem with input domain X
in O(√n log^2 n) parallel time and O(|X|^2) states per agent (except for the base station) with high probability.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E102.A.1187/_p
author={Ryota EGUCHI, Taisuke IZUMI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Sub-Linear Time Aggregation in Probabilistic Population Protocol Model},
abstract={A passively mobile system is an abstract notion of mobile ad-hoc networks. It is a collection of agents with computing devices. Agents move in a region, but the algorithm cannot control
their physical behavior (i.e., how they move). The population protocol model is one of the promising models in which the computation proceeds by the pairwise communication between two agents. The
communicating agents update their states by a specified transition function (algorithm). In this paper, we consider a general form of the aggregation problem with a base station. The base station is
a special agent having the computational power more powerful than others. In the aggregation problem, the base station has to sum up for inputs distributed to other agents. We propose an algorithm
that solves the aggregation problem in sub-linear parallel time using a relatively small number of states per agent. More precisely, our algorithm solves the aggregation problem with input domain X
in O(√n log^2 n) parallel time and O(|X|^2) states per agent (except for the base station) with high probability.},
TY - JOUR
TI - Sub-Linear Time Aggregation in Probabilistic Population Protocol Model
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1187
EP - 1194
AU - Ryota EGUCHI
AU - Taisuke IZUMI
PY - 2019
DO - 10.1587/transfun.E102.A.1187
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E102-A
IS - 9
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - September 2019
AB - A passively mobile system is an abstract notion of mobile ad-hoc networks. It is a collection of agents with computing devices. Agents move in a region, but the algorithm cannot control their
physical behavior (i.e., how they move). The population protocol model is one of the promising models in which the computation proceeds by the pairwise communication between two agents. The
communicating agents update their states by a specified transition function (algorithm). In this paper, we consider a general form of the aggregation problem with a base station. The base station is
a special agent having the computational power more powerful than others. In the aggregation problem, the base station has to sum up for inputs distributed to other agents. We propose an algorithm
that solves the aggregation problem in sub-linear parallel time using a relatively small number of states per agent. More precisely, our algorithm solves the aggregation problem with input domain X
in O(√n log^2 n) parallel time and O(|X|^2) states per agent (except for the base station) with high probability.
ER - | {"url":"https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E102.A.1187/_p","timestamp":"2024-11-07T13:17:53Z","content_type":"text/html","content_length":"62741","record_id":"<urn:uuid:ecf44e3d-09a7-424d-ac4c-0f0756696211>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00730.warc.gz"} |
Exact convergence rate and leading term in central limit theorem for student's t statistic
The leading term in the normal approximation to the distribution of Student's t statistic is derived in a general setting, with the sole assumption being that the sampled distribution is in the
domain of attraction of a normal law. The form of the leading term is shown to have its origin in the way in which extreme data influence properties of the Studentized sum. The leading-term
approximation is used to give the exact rate of convergence in the central limit theorem up to order n ^-1/2, where n denotes sample size. It is proved that the exact rate uniformly on the whole real
line is identical to the exact rate on sets of just three points. Moreover, the exact rate is identical to that for the non-Studentized sum when the latter is normalized for scale using a truncated
form of variance, but when the corresponding truncated centering constant is omitted. Examples of characterizations of convergence rates are also given. It is shown that, in some instances, their
validity uniformly on the whole real line is equivalent to their validity on just two symmetric points.
Dive into the research topics of 'Exact convergence rate and leading term in central limit theorem for student's t statistic'. Together they form a unique fingerprint. | {"url":"https://researchportalplus.anu.edu.au/en/publications/exact-convergence-rate-and-leading-term-in-central-limit-theorem-","timestamp":"2024-11-09T18:12:00Z","content_type":"text/html","content_length":"52666","record_id":"<urn:uuid:766b3a5d-6ed9-471f-a21a-14cd976de72f>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00148.warc.gz"} |
How to Compare Two Dates in Excel
- Written by Puneet
There are some situations where you need to compare two dates with each other. But before you do, you need to understand how dates are stored in Excel.
When you enter a date in Excel, it doesn’t just see it as a date; it stores it as a number. For example, January 1, 1900, is stored as the number 1, and January 2, 1900, is the number 2. If you type
in December 31, 2020, Excel sees it as 44196. But to show it as a number, you need to apply the format.
Here’s one important part: if you want to compare two dates, they must be valid according to Excel’s date system.
You can compare dates using a simple and quick formula that returns TRUE or FALSE. You can also use the IF function to get the custom message in the result while comparing two dates. The following
example shows two different dates in cells A1 and B1.
You can use the following steps:
1. First, enter the “=” equal sign in cell C1.
2. Now refer to cell A1, where you have the first date.
3. Next, enter the “=” equal sign again.
4. After that, refer to cell B1, where you have the second date.
5. In the end, hit enter.
Even though you see a date here where you compare them, Excel works with simple numbers behind it.
Comparision Operators you can use to Compare Dates
To compare dates in Excel, you can use different comparison operators:
• Equal to (A1=B1) – Checks if the date in one cell is the same as in another. If the values are equal, the condition is TRUE; otherwise, it is FALSE.
• Not equal to (A1<>B1) – Checks if the date in one cell differs from the date in another. If the values are not equal, the condition is TRUE; otherwise, it is FALSE.
• Greater than (A1>B1) – Checks if the date in one cell is larger than in another. If the first value is greater, the condition is TRUE; otherwise, it is FALSE.
• Less than (A1<B1) – Check if the date in one cell is smaller than in another. If the first value is less, the condition is true; otherwise, it is false.
• Greater than or equal to (A1>=B1) – Checks if the date in one cell is larger than or equal to in another cell. If the first value is greater or equal, the condition is true; otherwise, it is
• Less than or equal to (A1<=B1) – Checks if the date in one cell is smaller than or equal to in another cell. If the first value is less or equal, the condition is true; otherwise, it is false.
Compare IF a Date is Greater than Another Date
In the following example, we compare the first date with the second if the first date is greater than the second date.
In the formula, we used the greater than (>) operator.
If somehow you have dates with different date formats, you can still compare them with each other; the only thing that you need to take is that it’s a valid date.
Compare a Date with Today’s Date
In the same way, if you want to compare a date with today’s date, you can use the TODAY function. We used the TODAY function in the following example and compared it to another date.
Compare Two Dates using the IF Statement
You can also use IF to compare two dates in Excel to get a meaningful message about the comparison in the result cell. Below are a few examples to compare two dates using different comparison
=IF(A1 = B1, "Dates are equal", "Dates are not equal")
=IF(A1 > B1, "A1 is later", "A1 is not later")
=IF(A1 < B1, "A1 is earlier", "A1 is not earlier")
• The first formula checks if the date in cell A1 is the same as in cell B1. If they are equal, it returns “Dates are equal”; otherwise, it returns “Dates are not equal.”
• The second formula checks if the date in cell A1 is later than in cell B1. If A1 is later, it returns “A1 is later”; otherwise, it returns “A1 is not later.”
• The third formula checks if the date in cell A1 is earlier than in cell B1. If A1 is earlier, it returns “A1 is earlier”; otherwise, it returns “A1 is not earlier.”
Compare Dates Based on Years, Months, or Days
If you want to compare two dates but only the year, month, or days between them, then you need to write a formula differently.
=IF(YEAR(A1) = YEAR(B1), "Years are equal", "Years are not equal")
=IF(YEAR(A1) > YEAR(B1), "A1 year is later", "A1 year is not later")
=IF(YEAR(A1) < YEAR(B1), "A1 year is earlier", "A1 year is not earlier")
• The first formula checks if the year part of the date in cell A1 is the same as in cell B1. If the years are equal, it returns “Years are equal”; otherwise, it returns “Years are not equal.”
• The second formula checks if the year of the date in cell A1 is greater than in cell B1. If the year of A1 is later, it returns “A1 year is later”; otherwise, it returns “A1 year is not later.”
• The third formula checks if the year of the date in cell A1 is less than the year of the date in cell B1. If the year of A1 is earlier, it returns “A1 year is earlier”; otherwise, it returns “A1
year is not earlier.”
And in the same ways we have formula to compare dates based on months and days:
=IF(MONTH(A1) = MONTH(B1), "Months are equal", "Months are not equal")
=IF(MONTH(A1) > MONTH(B1), "A1 month is later", "A1 month is not later")
=IF(MONTH(A1) < MONTH(B1), "A1 month is earlier", "A1 month is not earlier")
• The first formula checks if the month of the date in cell A1 is the same as in cell B1. If the months are equal, it returns “Months are equal”; otherwise, it returns “Months are not equal.”
• The second formula checks if the month of the date in cell A1 is greater than in cell B1. If the month of A1 is later, it returns “A1 month is later”; otherwise, it returns “A1 month is not
• The third formula checks if the month of the date in cell A1 is less than in cell B1. If the month of A1 is earlier, it returns “A1 month is earlier”; otherwise, it returns “A1 month is not
=IF(DAY(A1) = DAY(B1), "Days are equal", "Days are not equal")
=IF(DAY(A1) > DAY(B1), "A1 day is later", "A1 day is not later")
=IF(DAY(A1) < DAY(B1), "A1 day is earlier", "A1 day is not earlier")
• The first formula checks if the day part of the date in cell A1 is the same as the day part of the date in cell B1. If the days are equal, it returns “Days are equal”; otherwise, it returns “Days
are not equal.”
• The second formula checks if the date in cell A1 is greater than in cell B1. If the day of A1 is later, it returns “A1 day is later”; otherwise, it returns “A1 day is not later.”
• The third formula checks if the date in cell A1 is less than the date in cell B1. If the day of A1 is earlier, it returns “A1 day is earlier”; otherwise, it returns “A1 day is not earlier.”
Compare Dates where you have time along with Dates
If you have two dates where you have time along with the dates, you can still compare them, but all the formulas that you have learned above.
But the problem is when you have two same dates with different time values, in that case, Excel will show you that date as different. For this you can use the function, INT to make it work around.
=INT(A1) = INT(B1)
=INT(A1) <> INT(B1)
=INT(A1) > INT(B1)
=INT(A1) < INT(B1)
=INT(A1) >= INT(B1)
=INT(A1) <= INT(B1)
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Matrix Factorization with Side Info
David Cortes
This vignette illustrates the usage of the cmfrec library for building recommender systems based on collaborative filtering models for explicit-feedback data, with or without side information about
the users and items. Note that the library offers also content-based models and implicit-feedback models, but they are not showcased in this vignette.
This example will use the MovieLens100k data, as bundled in the recommenderlab package, which contains around ~ 100k movie ratings from 943 users about 1664 movies, in a scale from 1 to 5.
In addition to the ratings, it also contains side information about the movies (genre, year of release) and about the users (age, occupation), which will be used here to construct a better
recommendation model.
For a more comprehensive introduction see also the cmfrec Python Notebook, which uses the more richer MovieLens1M instead (not provided by R packages).
Matrix Factorization
One of the most popular techniques for building recommender systems is to frame the problem as matrix completion, in which a large sparse matrix is built containing the ratings that users give to
products (in this case, movies), with rows representing users, columns representing items, and entries corresponding to the ratings that they’ve given (e.g. “5 stars”). Most of these entries will be
missing, as each users is likely to consume only a handful of the available products (thus, the matrix is sparse), and the goal is to construct a model which would be able to predict the value of the
known interactions (i.e. predict which rating would each user give to each movie), which is compared against the observed values. The items to recommend to each user are then the ones with highest
predicted values among those which the user has not yet consumed.
Typically, the problem is approached by trying to approximate the interactions matrix as the product of two lower-dimension matrices (a.k.a. latent factor matrices), which when multiplied by each
other would produce something that resembles the original matrix, having the nice property that it will produce predictions for all user-item combinations - i.e.
\[ \mathbf{X} \approx \mathbf{A} \mathbf{B}^T \] Where:
• \(\mathbf{X}\) is the interactions matrix (users are rows, items are columns).
• \(\mathbf{A}\) and \(\mathbf{B}\) are the matrices estimated by the model (a.k.a. latent factors), which have a low number of columns, typically 30-100.
For a better and more stable model, the \(\mathbf{X}\) matrix is typically centered by substracting its mean, a bias/intercept is added for each user and item, and a regularization penalty is applied
to the model matrices and biases (typically on the L2 norm) - i.e.:
\[ \mathbf{X} \approx \mathbf{A} \mathbf{B}^T + \mu + \mathbf{b}_A + \mathbf{b}_B \] Where:
• \(\mu\) is the global mean used to center \(\mathbf{X}\).
• \(\mathbf{b}_A\) are user-specific biases (row vector).
• \(\mathbf{b}_B\) are item-specific biases (column vector).
The matrices are typically fitted by initializing them to random numbers and then iteratively updating them in a way that decreases the reconstruction error with respect to the observed entries in \
(\mathbf{X}\), using either gradient-based procedures (e.g. stochastic gradient descent) or the ALS (alternating least-squares) method, which optimizes one matrix at a time while leaving the other
fixed, performing a few sweeps until convergence.
This library (cmfrec) will by default use the ALS method with L2 regularization, and will use user/item biases which are model parameters (updated at each iteration) rather than being pre-estimated.
Loading the data
The MovieLens100k data is taken from the recommenderlab package. As the data is sparse, it is represented as sparse matrices from the Matrix package. The data comes in CSC format, whereas cmfrec
requires COO/triplets format - the conversion is handled by the MatrixExtra package for convenience, which also provides extra slicing functionality that will be used later.
X <- as.coo.matrix(MovieLense@data)
#> Formal class 'dgTMatrix' [package "Matrix"] with 6 slots
#> ..@ i : int [1:99392] 0 1 4 5 9 12 14 15 16 17 ...
#> ..@ j : int [1:99392] 0 0 0 0 0 0 0 0 0 0 ...
#> ..@ Dim : int [1:2] 943 1664
#> ..@ Dimnames:List of 2
#> .. ..$ : chr [1:943] "1" "2" "3" "4" ...
#> .. ..$ : chr [1:1664] "Toy Story (1995)" "GoldenEye (1995)" "Four Rooms (1995)" "Get Shorty (1995)" ...
#> ..@ x : num [1:99392] 5 4 4 4 4 3 1 5 4 5 ...
#> ..@ factors : list()
Creating a train-test split
In order to evaluate models, 25% of the data will be set as a test set, while the model will be built with the remainder 75%. The split done here is random, but usually time-based splits tend to
reflect more realistic scenarios for recommendation.
Typically, these splits are done in such a way that the test set contains only users and items which are in the train set, but such a rule is not necessary and perhaps not even desirable for cmfrec,
since it can accomodate global/user/item biases and thus it can make predictions based on them alone.
subsample_coo_matrix <- function(X, indices) {
X@i <- X@i[indices]
X@j <- X@j[indices]
X@x <- X@x[indices]
n_ratings <- length(X@x)
ix_train <- sample(n_ratings, floor(0.75 * n_ratings), replace=FALSE)
X_train <- subsample_coo_matrix(X, ix_train)
X_test <- subsample_coo_matrix(X, -ix_train)
Classical model
Now fitting the classical matrix factorization model, with global mean centering, user/item biases, L2 regularization which scales with the number of ratings for each user/item, and no side
information. This is the model explained in the earlier section: \[ \mathbf{X} \approx \mathbf{A} \mathbf{B}^T + \mu + \mathbf{b}_A + \mathbf{b}_B \]
How good is it?
The most typical way of evaluating the quality of these models is by evaluating the error that they have at predicting known entries, which here will be evaluated against the test data that was set
apart earlier. The evaluation here will be done in terms of mean squared error (RMSE).
Note that, while widely used in the early literature for recommender systems, RMSE might not provide a good overview of the ranking of items (which is what matters for recommendations), and it’s
recommended to also evaluate other metrics such as NDCG@K, P@K, correlations, etc.
print_rmse <- function(X_test, X_hat, model_name) {
rmse <- sqrt(mean( (X_test@x - X_hat@x)^2 ))
cat(sprintf("RMSE for %s is: %.4f\n", model_name, rmse))
pred_classic <- predict(model.classic, X_test)
print_rmse(X_test, pred_classic, "classic model")
#> RMSE for classic model is: 0.9236
i.e. it means that the ratings are off by about one star. This is better than a non-personalized model that would always predict the same rating for each user, which can also be simulated through
model.baseline <- MostPopular(X_train, lambda=10, scale_lam=FALSE)
pred_baseline <- predict(model.baseline, X_test)
print_rmse(X_test, pred_baseline, "non-personalized model")
#> RMSE for non-personalized model is: 0.9460
(Note: it’s not recommended to use scaled/dynamic regularization in a most-popular model, as it will tend to recommend items with only one user giving the maximum rating.)
Improving the classical model
By default, ALS-based models are broken down to small problems involving linear systems, which are in turned solved through the Conjugate Gradient method, but cmfrec can also use a Cholesky solver
for them, which is slower but tends to result in better-quality solutions for explicit-feedback.
As well, the default number of iterations is 10, but can be increased for better models at the expense of longer fitting times.
But more importantly, cmfrec offers the option of adding “implicit-features” or co-factoring, which will additionally factorize binarized versions of \(\mathbf{X}\) (telling whether each entry is
missing or not), sharing the same latent components with the factorization of \(\mathbf{X}\) - that is: \[ \mathbf{X} \approx \mathbf{A} \mathbf{B}^T + \mu + \mathbf{b}_A + \mathbf{b}_B \] \[ \mathbf
{I}_x \approx \mathbf{A} \mathbf{B}^T_i \:\:\:\: \mathbf{I}^T_x \approx \mathbf{B} \mathbf{A}^T_i \] Where:
• \(\mathbf{I}_x\) is a binary matrix indicating whether each entry of \(\mathbf{X}\) is observed or missing.
• \(\mathbf{A}_i\) and \(\mathbf{B}_i\) are model matrices which are not directly used for \(\mathbf{X}\), and not used in the prediction formula, but are still estimated in this new
multi-objective optimization objective.
model.improved <- CMF(X_train, k=25, lambda=0.1, scale_lam=TRUE,
add_implicit_features=TRUE, w_main=0.75, w_implicit=0.25,
use_cg=FALSE, niter=30, verbose=FALSE)
pred_improved <- predict(model.improved, X_test)
print_rmse(X_test, pred_improved, "improved classic model")
#> RMSE for improved classic model is: 0.9126
Adding side information
Collective matrix factorization extends the classical model by incorporating side information about users/items into the formula, which is done by also factorizing the side information matrices,
sharing the same latent components that are used for factorizing the \(\mathbf{X}\) matrix: \[ \mathbf{X} \approx \mathbf{A} \mathbf{B}^T + \mu + \mathbf{b}_A + \mathbf{b}_B \] \[ \mathbf{U} \approx
\mathbf{A} \mathbf{C}^T + \mu_U \] \[ \mathbf{I} \approx \mathbf{B} \mathbf{D}^T + \mu_I \] \[ \mathbf{I}_x \approx \mathbf{A} \mathbf{B}^T_i \:\:\:\: \mathbf{I}^T_x \approx \mathbf{B} \mathbf{A}^T_i
\] Where:
• \(\mathbf{U}\) is a matrix representing side information about users, with each user being a row, and columns corresponding to their attributes.
• \(\mathbf{I}\) is similarly a matrix representing side information about items.
• \(\mathbf{C}\) and \(\mathbf{D}\) are new latent factor matrices used for factorizing the side information matrices, but are not used directly for \(\mathbf{X}\).
• \(\mu_U\) and \(\mu_I\) are column means for the attributes, which are used in order to center them.
Informally, the latent factors now need to explain both the interactions data as well as the side information, thereby making them generalize better to unseen data. This library in addition allows
controlling aspects such as the weight that each factorization has in the optimization objective, different regularization for each matrix, having factors that are not shared, among others.
Fetching the side information from recommenderlab:
U <- MovieLenseUser
U$id <- NULL
U$zipcode <- NULL
U$age2 <- U$age^2
### Note that `cmfrec` does not standardize features beyond mean centering
U$age <- (U$age - mean(U$age)) / sd(U$age)
U$age2 <- (U$age2 - mean(U$age2)) / sd(U$age2)
U <- model.matrix(~.-1, data=U)
I <- MovieLenseMeta
I$title <- NULL
I$url <- NULL
I$year <- ifelse(is.na(I$year), median(I$year, na.rm=TRUE), I$year)
I$year2 <- I$year^2
I$year <- (I$year - mean(I$year)) / sd(I$year)
I$year2 <- (I$year2 - mean(I$year2)) / sd(I$year2)
I <- as.coo.matrix(I)
cat(dim(U), "\n")
#> 943 24
cat(dim(I), "\n")
#> 1664 21
Now fitting the model:
model.w.sideinfo <- CMF(X_train, U=U, I=I, NA_as_zero_item=TRUE,
k=25, lambda=0.1, scale_lam=TRUE,
niter=30, use_cg=FALSE, include_all_X=FALSE,
w_main=0.75, w_user=0.5, w_item=0.5, w_implicit=0.5,
center_U=FALSE, center_I=FALSE,
pred_side_info <- predict(model.w.sideinfo, X_test)
print_rmse(X_test, pred_side_info, "model with side info")
#> RMSE for model with side info is: 0.9117
calc_rmse <- function(X_test, X_hat) {
return(sqrt(mean( (X_test@x - X_hat@x)^2 )))
results <- data.frame(
NonPersonalized = calc_rmse(X_test, pred_baseline),
ClassicalModel = calc_rmse(X_test, pred_classic),
ClassicPlusImplicit = calc_rmse(X_test, pred_improved),
CollectiveModel = calc_rmse(X_test, pred_side_info)
results <- as.data.frame(t(results))
names(results) <- "RMSE"
results %>%
kable() %>%
NonPersonalized 0.9460112
ClassicalModel 0.9236193
ClassicPlusImplicit 0.9125861
CollectiveModel 0.9116610
Important to keep in mind:
• These RMSEs have high standard errors due to the small amount of data used here.
• The model hyperparameters are not particularly tuned, and a proper tuning should use a validation split too.
• The test split is using users and items which might not have been in the training set.
• While it looks like the difference from adding side information is very small, it also comes with the side effect of being able to recommend items based on attributes.
• RMSE as a metric can hide overfitting in models that tend to recommend items with too few ratings/interactions - these models in particular will tend to recommend many movies with only a handful
ratings, which is typically undesirable. A model with higher regularization that shows higher test RMSE might in practice produce better quality recommendations (see the introductory Python
notebook for more examples on this topic).
• The models evaluated so far have all used dynamic/scaled regularization (as proposed in Large-scale Parallel Collaborative Filtering for the Netflix Prize), save for the baseline most-popular
model - this means that the regularization for each user and item is scaled by the number of present entries for it. This setting tends to produce dubious recommendations in small datasets like
the MovieLens100k, even if it makes it look like it improves RMSE.
Generating Top-N recommendations
The goal behind building a collaborative filtering model is typically to be able to make top-N recommended lists for users or to obtain latent factors for an unseen user given its current data.
cmfrec has many prediction functions for these purposes depending on what specifically one wants to do, supporting both warm-start and cold-start recommendations.
### Re-fitting the earlier model to all the data,
### this time *without* scaled regularization
model.classic <- CMF(X, k=20, lambda=10, scale_lam=FALSE, verbose=FALSE)
model.w.sideinfo <- CMF(X, U=U, I=I, k=20, lambda=10, scale_lam=FALSE,
w_main=0.75, w_user=0.125, w_item=0.125,
Recommendations for existing users
When fitting a model, all the necessary fitted matrices are saved inside the object itself, which allows making predictions for existing users based just on the ID. The specific items consumed by
each user are however not saved, so in order to avoid recommending already-seen items, these have to be explicitly passed for exclusion.
user_to_recommend <- 10
### Note: slicing of 'X' is provided by 'MatrixExtra',
### returning a 'sparseVector' object as required by cmfrec
topN(model.classic, user=user_to_recommend, n=10,
exclude=X[user_to_recommend, , drop=TRUE])
#> [1] 316 424 511 311 271 314 405 79 524 190
### A handy function for visualizing recommendations
movie_names <- colnames(X)
n_ratings <- colSums(as.csc.matrix(X, binary=TRUE))
avg_ratings <- colSums(as.csc.matrix(X)) / n_ratings
print_recommended <- function(rec, txt) {
cat(txt, ":\n",
paste(paste(1:length(rec), ". ", sep=""),
" - Avg rating:", round(avg_ratings[rec], 2),
", #ratings: ", n_ratings[rec],
collapse="\n", sep=""),
"\n", sep="")
recommended <- topN(model.w.sideinfo, user=user_to_recommend, n=5,
exclude=X[user_to_recommend, , drop=TRUE])
print_recommended(recommended, "Recommended for user_id=10")
#> Recommended for user_id=10:
#> 1. Schindler's List (1993) - Avg rating:4.47, #ratings: 298
#> 2. To Kill a Mockingbird (1962) - Avg rating:4.29, #ratings: 219
#> 3. Close Shave, A (1995) - Avg rating:4.49, #ratings: 112
#> 4. Boot, Das (1981) - Avg rating:4.2, #ratings: 201
#> 5. Titanic (1997) - Avg rating:4.25, #ratings: 350
Recommendations for new users
The fitted model, as it is, can only provide recommendations for the specific users and items to which it was fit. Typically, one wants to produce recommendations for new users as they go, or update
the recommended lists for existing users once they consume more items. cmfrec allows obtaining latent factors and top-N recommended lists for new users without having to refit the whole model.
This is how it would be if user 10 were to come as a new visitor:
recommended_new <- topN_new(model.w.sideinfo, n=5,
exclude=X[user_to_recommend, , drop=TRUE],
X=X[user_to_recommend, , drop=TRUE],
U=U[user_to_recommend, , drop=TRUE])
print_recommended(recommended_new, "Recommended for user_id=10 as new user")
#> Recommended for user_id=10 as new user:
#> 1. Schindler's List (1993) - Avg rating:4.47, #ratings: 298
#> 2. To Kill a Mockingbird (1962) - Avg rating:4.29, #ratings: 219
#> 3. Close Shave, A (1995) - Avg rating:4.49, #ratings: 112
#> 4. Boot, Das (1981) - Avg rating:4.2, #ratings: 201
#> 5. Titanic (1997) - Avg rating:4.25, #ratings: 350
It is not mandatory to provide all the side information, as the ratings alone can also be used to generate a recommendation, even if the model was fit with side information (this would not be the
case if passing NA_as_zero_user=TRUE):
recommended_new <- topN_new(model.w.sideinfo, n=5,
exclude=X[user_to_recommend, , drop=TRUE],
X=X[user_to_recommend, , drop=TRUE])
print_recommended(recommended_new, "Recommended for user_id=10 as new user (NO sideinfo)")
#> Recommended for user_id=10 as new user (NO sideinfo):
#> 1. Schindler's List (1993) - Avg rating:4.47, #ratings: 298
#> 2. To Kill a Mockingbird (1962) - Avg rating:4.29, #ratings: 219
#> 3. Close Shave, A (1995) - Avg rating:4.49, #ratings: 112
#> 4. Boot, Das (1981) - Avg rating:4.2, #ratings: 201
#> 5. Titanic (1997) - Avg rating:4.25, #ratings: 350
(In this case, the top-5 recommendations did not change, as the side information has little effect in this particular model, but that might not always be the case - that is, the top-N recommended
items for a different user might be different if side information is absent.)
Cold-start recommendations
Conversely, it is also possible to make a recommendation based on the side information without having any rated movies/items. The quality of these recommendations is however highly dependant on the
influence that the attributes have in the model, and in this case, the user attributes have very little associated information and thus little leverage.
Nevertheless, they might still provide an improvement over a completely non-personalized recommendation (see Cold-start recommendations in Collective Matrix Factorization):
recommended_cold <- topN_new(model.w.sideinfo, n=5,
exclude=X[user_to_recommend, , drop=TRUE],
U=U[user_to_recommend, , drop=TRUE])
print_recommended(recommended_cold, "Recommended for user_id=10 as new user (NO ratings)")
#> Recommended for user_id=10 as new user (NO ratings):
#> 1. Schindler's List (1993) - Avg rating:4.47, #ratings: 298
#> 2. Close Shave, A (1995) - Avg rating:4.49, #ratings: 112
#> 3. Wrong Trousers, The (1993) - Avg rating:4.47, #ratings: 118
#> 4. Good Will Hunting (1997) - Avg rating:4.26, #ratings: 198
#> 5. Wallace & Gromit: The Best of Aardman Animation (1996) - Avg rating:4.45, #ratings: 67 | {"url":"https://cran.ma.imperial.ac.uk/web/packages/cmfrec/vignettes/cmfrec_vignette.html","timestamp":"2024-11-04T08:56:48Z","content_type":"text/html","content_length":"63613","record_id":"<urn:uuid:e0ae0581-452e-4ce1-9fd7-01258543ca2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00222.warc.gz"} |
What Is The Mean Symbol On Ti 84? Arithmetic Mean! (2024)
The mean symbol on a TI-84 calculator refers to the statistical function used to calculate the arithmetic mean or average of a data set.
The mean is a measure of central tendency that represents the sum of all values in a data set divided by the total number of values.
It is commonly used in various fields such as mathematics, statistics, economics, and finance to provide a general understanding of the data set’s central location.
On a TI-84 calculator, the mean function can be accessed through the statistical functions of the calculator to analyze and interpret data sets.
The mean symbol is a statistical function on a TI-84 calculator.
Arithmetic mean or average is the primary use of the mean function.
Calculating the mean involves adding all data values and dividing by the number of values.
The mean function is used in multiple fields, including mathematics, statistics, and finance.
Accessing the mean function on a TI-84 calculator is quite simple. To calculate the mean of a data set, you need to enter the data values into a list, and then navigate to the “1-Var Stats” function
within the “STAT” menu.
From there, the calculator will compute various statistical measures, including the mean, for your data set.
Understanding the Mean Symbol on TI 84: A Comprehensive Guide
Function Description Calculator Model Symbol on Calculator
Mean Calculates the average (mean) of a data set TI-84 mean( or mean(list)
Key Takeaway
The mean symbol on TI-84 refers to the calculation of the average of a dataset
It is a statistical function present on the calculator for quick calculations
The TI-84 calculator is commonly used in schools and colleges for teaching math and statistics
Understanding the mean symbol on TI-84 helps students efficiently analyze and interpret data
Five Facts About: The Mean Symbol on TI 84
The TI-84 calculator is produced by Texas Instruments, a major producer of advanced technology tools for education and engineering purposes. Source
The mean symbol on TI-84 can be accessed through the 1-Var Stats function, which calculates a range of statistical data including the mean (average), standard deviation, and variance. Source
The mean calculated by the TI-84 calculator refers to the sum of all data points in a dataset, divided by the total number of data points. This is a central measure of a dataset’s tendency. Source
In addition to mean, the TI-84 calculator can also calculate other central tendency measures, such as median and mode, through its various inbuilt functions. Source
The TI-84 calculator is widely used across the United States in high school and college-level courses, particularly mathematics, statistics, and engineering. It is a versatile tool that enables
students to enhance their knowledge and understanding of analytical methodologies. Source
Understanding The Ti-84 Calculator
The ti-84 calculator is a popular calculator amongst students and professionals alike. The mean symbol on the ti-84 calculator is one of the many functions that this amazing device offers.
We will walk you through the basics of the ti-84 calculator and help you understand what the mean symbol is used for.
Introduction To The Ti-84 Calculator
The ti-84 calculator is a graphing calculator that is manufactured by texas instruments.
It is a reliable and robust calculator that allows you to solve a wide range of mathematical problems, making it a popular choice amongst students and professionals alike.
When it comes to understanding the ti-84 calculator, here are some of the key points you should keep in mind:
• The ti-84 is a graphing calculator, meaning it can plot and analyze graphs and functions.
• The calculator has a large display screen that allows you to view complex computations and equations.
• It has various functions that include trigonometry, statistics, probability, and calculus.
Using the ti-84 calculator, you can solve a variety of math problems with ease and efficiency.
Understanding The Mean Symbol On Ti-84
The mean symbol is one of the many essential functions that the ti-84 calculator offers.
To compute the mean using the ti-84 calculator,
Here are the steps you should follow:
• Enter the data set into the ti-84 calculator by pressing the “stats” button, then selecting “1: Edit.”
• Once the data is entered, use the arrow keys to highlight the “calc” menu, then select “1: 1-var stats.”
• Press “enter” to calculate the mean, and the result will be displayed on the screen.
Here are some more key points to keep in mind regarding the ti-84 calculator and the mean symbol:
• The ti-84 calculates the arithmetic mean, which is calculated by adding up all the values in a data set and dividing by the number of values.
• The mean symbol is represented by the letter “x-bar.”
• The mean is used in a wide range of statistical calculations, and is often used to measure central tendency.
The ti-84 calculator is an amazing tool that provides precise mathematical calculations. To calculate the mean using the ti-84 calculator, simply input your data into the calculator and follow the
steps outlined above.
The mean symbol is just one of the many functions that this powerful calculator offers.
Overall, the ti-84 calculator is an essential tool for anyone who needs to solve complex math problems with ease and efficiency.
What Is The Mean Symbol On Ti-84?
The ti-84 calculator is a beloved tool for many students and professionals. One of the symbols that often confuses its users is the mean symbol.
In this post, we will explore this concept and how you can use it on your ti-84 calculator.
Let’s jump right in!
Brief Explanation Of The Concept Of Mean
The mean is a statistical measure that represents the average value of a set of numerical data. This value is obtained by adding all the numbers in the dataset and dividing the sum by the total
number of values.
The mean is used to describe the central tendency of a data set and is often represented by the symbol “x̄” in mathematics.
Overview Of The Functionality Of Ti-84
The ti-84 calculator is a powerful tool that comes equipped with a range of features and functionalities. When it comes to calculating the mean, the ti-84 makes things easy.
Here’s how:
• To calculate the mean of a given dataset, enter the values into a list on your calculator.
• Press stat, then enter to access the statistics menu.
• Choose option 1 (1-varstats) to calculate the mean and other statistical measures such as standard deviation and variance.
• The calculator will automatically display the mean (x̄) along with other statistics such as the sample size (n), standard deviation (sx), and variance (σx²).
Ensuring accurate calculations is important when dealing with numerical data, and that is why using the mean symbol on your ti-84 calculator is so important.
With just a few clicks, you can calculate the mean of a dataset and have access to reliable statistical measures.
Understanding the concept of the mean and the functionality of your ti-84 calculator can be a game-changer for professionals and students alike.
We hope this post has been helpful in demystifying this concept and equipping you with the necessary tools to make accurate statistical calculations on your calculator.
Basic Calculations On Ti-84
The ti-84 graphing calculator is one of the most popular calculators used in math classes worldwide. It is known for its computational prowess and usefulness to students in a range of disciplines.
The mean symbol is one tool that is commonly used by students when working with data sets.
Let’s explore how to use this function on the ti-84.
Access The Stat Menu
Before we begin, let’s ensure we have access to the stat menu on the ti-84.
To access this menu, follow these steps:
• Press the ‘stat’ key
• Select ‘edit’
• Enter the data into the lists
Finding Mean Using Ti-84
Once we have our data entered, we can find the mean in just a few simple steps:
• Press the ‘stat’ key
• Use the arrow keys to select the calc option
• Select option 1: 1-var stats
• Press enter
The 1-var stats function will calculate a range of statistics for our data, including mean, standard deviation, and median. These statistics are useful in a variety of contexts, including hypothesis
testing and data visualization.
Overall, the ti-84 graphing calculator is an essential tool for students working with data. With the mean symbol and access to the stat menu, finding summary statistics has never been easier.
By using this powerful calculator, students can save time and focus on the true concepts at hand.
Advanced Calculations On Ti-84
If you’re a maths enthusiast, you might be familiar with the mean symbol. The mean symbol on ti-84 is a statistical concept that helps to find the average value of a set of numbers.
However, the ti-84 calculator is capable of much more than finding just the mean value.
Additional Features On Ti-84, Beyond Mean
The ti-84 calculator comes with advanced statistical properties that can save you time and effort, especially when working with complex data sets.
Here are some of the features that can take your calculations to the next level:
• Median and mode: Finding the median and mode is essential in statistics, and the ti-84 calculator can help you with that. Use the calculator to calculate the median, which is the middle value in
a set of numbers when arranged in order. Similarly, the calculator can help you determine the mode, which is the most occurring number in a set of data.
• Standard deviation: Standard deviation measures the variability of data around the mean value, and it’s an essential concept in statistics. The ti-84 calculator can help you calculate the
standard deviation for any data set.
• Regression analysis: Regression analysis is a statistical technique that helps you find the relationship between different variables. The ti-84 calculator can perform regression analysis, which
helps to predict one variable’s value based on the other variable’s value.
• Hypothesis testing: Hypothesis testing is a significant concept in statistics that helps to determine whether a hypothesis is true or not. The ti-84 calculator can perform several hypothesis
tests, such as t-tests and z-tests, that can help you determine whether the hypothesis is correct.
In essence, the ti-84 calculator is not just a simple calculator that finds the mean. It is a powerful tool that can perform advanced statistical calculations that are essential in many fields.
Whether you’re a student, researcher, or working professional, the ti-84 calculator is a must-have tool that can save you time and effort.
Applications Of Ti-84
What Is The Mean Symbol On Ti-84? Applications Of Ti-84
Are you wondering what the mean symbol on ti-84 means? Ti-84 is a popular calculator that can perform calculations of mean, median, mode, regression, and many other statistical operations.
The mean symbol on the ti-84 is represented by the symbol “x-bar”.
When you see this symbol, you can assume that it refers to the mean.
How To Use Ti-84 In Practice
Ti-84 is an incredibly versatile calculator that is perfect for students, teachers, and professionals who need to do complex calculations quickly.
Here are some basic steps for using the ti-84:
• Start by turning on the calculator and accessing the home screen.
• Input your data into the calculator.
• Choose the appropriate statistical function for your calculations.
• Fill in the necessary data and press enter to receive the result.
Real-Life Examples And Use Cases
Now, let’s explore some real-world examples and use cases of a ti-84 calculator.
• Business: Ti-84 can be used to calculate financial ratios such as return on investment (roi), net present value (npv), and internal rate of return (irr) to make informed decisions.
• Science: Ti-84 can be used to analyze scientific experiments and create graphs and charts to visually represent data.
• Education: Ti-84 can help students solve complex mathematical problems and check their answers for accuracy.
Don’t miss out on the power and convenience of the ti-84 calculator. Whether you’re a student, teacher, or professional, ti-84 can simplify your calculations and make your life easier. Happy
What Does the Triangle Symbol Mean on Google Play Points?
The triangle symbol meaning on google Play points to a featured collection of apps and games outlined by Google, representing the selected content available for redemption using Play Points. This
symbol serves as a visual cue for users to easily identify and access the exclusive offerings tied to the Play Points rewards program.
FAQ About The Mean Symbol On Ti 84
What Does The Mean Symbol On Ti 84 Calculator Mean?
The mean symbol on a ti 84 calculator represents the average of a set of numbers.
How Is The Mean Symbol Calculated On Ti 84?
To calculate the mean symbol on ti 84, enter the numbers, press the “stat” key, select “1:edit” enter the data set, select “stat”, select “5:1-var stats”, and press “enter”. The mean symbol can be
found in the resulting data.
What Is The Difference Between Mean And Median On Ti 84?
The mean symbol represents the average of a set of numbers, while the median symbol represents the middle value in a set of data.
How Do I Use The Mean Symbol In Statistical Analysis?
The mean symbol is a commonly used tool in statistical analysis to help understand the average value of a set of data and can be utilized in calculations of variance and standard deviation.
How Do I Use The Mean Symbol In Statistical Analysis?
The mean symbol is a commonly used tool in statistical analysis to help understand the average value of a set of data and can be utilized in calculations of variance and standard deviation.
Why Is The Mean Symbol Important In Data Analysis?
The mean symbol is an important statistical concept used in data analysis as it allows researchers to understand the central tendency of a set of data. It is often used to compare different data sets
or to understand if a result is statistically significant.
By now, you have learned a lot about the mean symbol on ti 84 and its significance in calculating statistical data. You have seen how easy and convenient it is to use this feature, even if you are
not a math expert.
No longer do you have to spend hours manually calculating averages and other statistics. With the ti 84 mean symbol, you can calculate them in seconds!
The key takeaway from this post is that the mean symbol on ti 84 is a valuable tool for anyone conducting statistical analysis.
It simplifies the calculation process, saves time and produces accurate results. Whether you are a student preparing for an exam or a professional working with data, ti 84’s mean symbol is a
must-have feature.
So, next time you use ti 84, make sure you use the mean symbol to make your statistical calculations a breeze! | {"url":"https://diocesisciudadquesada.org/article/what-is-the-mean-symbol-on-ti-84-arithmetic-mean","timestamp":"2024-11-11T02:57:32Z","content_type":"text/html","content_length":"133609","record_id":"<urn:uuid:16454a8f-9396-44b7-8932-52b9462a6027>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00143.warc.gz"} |
The Origins Of The Sudoku Puzzles
Chances are unless you've been hiding under a rock you're probably familiar with the hugely popular number game called Sudoku. Just in case you're not familiar with Sudoku it's a puzzle game based
around number placement and is a game of intense concentration, patience and steadfast thought.
Because of its wide spread following Sudoku puzzles can be found almost everywhere from textbooks, magazines and newspapers. In fact, Sudoku is now rapidly making its way across the Internet. The
game is based on a series of grids with one big grid that is essentially a 9 x 9 grid that holds 9 smaller grids that are basically 3 x 3 grids.
The premise behind Sudoku is to place the numbers 1 to 9 into each of the grid squares or units. Although you might think that math is involved in this game it actually isn't because you're not
required to add the numbers nor is it a requirement to try and figure out the sum of the columns and rows. So even though Sudoku is clearly a numbers game you don’t need a Master's degree in
mathematics to fully enjoy it.
However, just like any game there are some rules that must be followed while playing Sudoku. They include the following:
1 – Every number from 1 through 9 must appear at least one time in each row.
2 – Each of the numbers (1 through 9) should only come out only one time in each column.
3 – Each number from 1 to 9 should only appear once in each of the smaller 3 by 3 grids.
In order to increase the complexity of the game (the difficulty, excitement and challenge) the previously mentioned rules must be stringently followed in order to make sure that each 3 by 3 grid has
the numbers from 1 to 9 to show only once in the series.
The beauty of Sudoku puzzle games lies in the fact that Sudoku can be played in different levels of complexity or degrees of difficulty. This is accomplished when the combined total of the numbers in
the first level of the 9 by 9 grid matrix differ. Some Sudoku players believe that the game is easier if there are more numbers given on the first level. However, the reality is that this is not
really accurate because the placement or assignment of each number is what creates the effect of increasing the complexity of the game and making the Sudoku puzzle more difficult to solve.
How To Play Sudoku Puzzles
Keeping in mind that you have to follow the 3 rules that were previously mentioned you are now ready to learn how to start playing Sudoku.
The Sudoku puzzle can be reasonably solved and doesn't require you guessing everything while playing. Like most puzzles, the best way to begin is to start searching or looking for clues. Many
experienced players start looking for the numbers that frequently appear in the first grid or puzzle. For example, lets assume you have several 5's in the first puzzle. Now look in the 3 x 3 grid to
see if the number 5 is also located there. Now start searching for other places where the number 5 appears. Check the columns and rows because these are designed to help you locate where the other
5's are. Keep in mind that the number 5 can only appear 1 time in each of the 3 x 3 grid, columns or rows. If thee is already a 5 in columns 1 and 3 then there should not be any more 5's in those 2
columns. So the final number 5 should be placed in column 2. This process of eliminating or removing all of the potentials in the box will leave you a bigger and better chance of solving the puzzle
While you are scanning or searching over all of the 3 x 3 grids you must make sure to only place the numbers 1 through 9 in the columns and rows that hold a majority of the numbers. If there are only
2 numbers left that are not in a column or row you must use a process of elimination in order to determine where to place the final 2 numbers. If you have ruled out one of the possibilities on each
column or row there will be a chance to fill the line you are playing.
The steps explained above will help you solve simple puzzles and show you how to play the easy Sudoku. You will find that more advanced Sudoku puzzles are difficult and will require a skill set that
experienced players use to look or examine the possibilities of the puzzle. These possibilities revolve around a player solving and winning the puzzle with all of the different choices given in every
square. There will be instances where there is no way to remove a possibility so a number will have to be chosen. In this example, it is vital to remember where you have chosen that number in case
that first choice was wrong.
No question about it, Sudoku is a game of careful thought and intense concentration requiring you to use your wits in order to achieve victory.
Learn how to master Sudoku with Sudoku Puzzle Secrets a complete guide covering everything you need to know in order to solve Sudoku puzzles to include how to eliminate the extraneous, what exactly
unique grids are and how to use them, how to properly search for the lone number and how to use cross hatching to your advantage. Become a master Sudoku player now with http://
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American University of Central Asia - 69ɫ - Elective courses
Elective courses
Actuarial Mathematics. Level 1 BUS/MAT 367 3964
This course will introduce to the basics of Actuarial Mathematics. The first level course consists of the following topics: basic theory of interest (we develop formulae needed in the rest of
actuarial science), equation of value, concept of annuities, amortization, sinking funds, bonds, life tables, life annuities, life insurance, multi-life insurance, evaluation of pension plans.
Actuarial Mathematics. Level 2 BUS/MAT 368 4177
This course will introduce to the basics of Life Contingencies Actuarial Mathematics. The Level II course consist of the following topics: insurance annuities, life tables, life annuities, life
insurance, multi-life insurance, evaluation of pension plans.
This course requires extensive usage of probability and statistics theory, calculus, functional analysis.
Game Theory MAT/ ECO 317 3453
Our primary goal is to study the scope and methods of game theory. We mainly focus on gamesarising in economics and business, although general games will be considered with applicationsto other
Game theory is about the strategies adopted by agents (e.g. consumers, firms or governments)when there are competing interests or ends and the outcomes depend on the actions chosen byall of the
participants. Our time together is designed to develop a view of the concepts andproblems studied by game theorists. You should also learn a set of analytical skills reflectinggame theory’s main
The course is modular in structure. There are 7 modules in all.
Maple programming MAT 239 4586
Despite the wide range of mathematical topics touched upon, this course is intended for anyone who has completedbeginning calculus. As we enter the 21st century, it becomes essential for students to
be experienced in the use of software such as Maple. If you have never heard of CAS (Computer Algebra Systems) or Maple before, or if you have been using Maple for years but would like to expand your
knowledge of its capabilities, this course is for you!
The course teaches you how to use Maple to explore calculus and study mathematical models.We will use Maple for calculus in one variable and two variables, and for graphing functions, curves and
surfaces.We will learn how to model physical systems and use Maple to solve them and visualize and animatetheir solutions.The course will also strengthen your programming skills. We will write a lot
of simple programs involving loops, functions, arrays, lists and sets.
Math Modelling in Economics MAT 333 3701
This course was designed for students who are interested in mathematical modelling and believe that usefulness of mathematics lies in its application to practical situation. The course was prepared
in such manner that students with backgrounds in fundamental algebra can understand and learn the ideas of the course.
The modular structure of the course allows students a freedom in choosing how to present the material in their research and presentations.
There are three modules:
1. Mathematical modelling in social and humanitarian sciences.
2. Mathematical modelling in music and art
3. Mathematical models in consumer math.
The following criteria were used to determine which topics to include into the modules.
1. Is the concept or technique commonly used in everyday life?
2. Will the concept or technique help students understand the events in their everyday lives?
So, utility theory, difference equations and population growth, bonds and shares are included into the first and third modules.
Also, mathematical models in Music, models and patterns in plane geometry, models and ppatterns in Art and Architecture will be offered to students in the module 2. They enliven the course and help
students to appreciate variety and beauty of models in our life.
Mathematical Modeling in Geophysics MAT 420 4118
This course covers the basics of the mathematical modeling method with regard to the solution of some geophysical tasks. In particular, we will consider the models and methods of magnetotellurics,
thermodynamics of lakes and pollution transport. We will discuss the basic steps of the mathematical modeling process and propose different approaches for discrete models constructing.
Numerical Methods for Equations of Mathematical Physics MAT 410 3968
Course will focus on the study of both classical and modern numerical methods for solving the equations of mathematical physics. We will consider boundary-value and initial-boundary-value problems
for the advection-diffusion, Poisson and non-stationary heat equations.
Quantitative Decision Making BUS/MAT 366 3963
This course focuses on the connection between real life problems and making decision. Students will learn how to model real life problems, formulate real life problems and study of the quantitative
methods for decision making, in particular the application of mathematical and statistical models in the analysis of problems related to economic and business. The main topic includes probability and
decision-making analysis, analysis under uncertain conditions, theory of portfolio: The Markowitz and Tobin portfolio theories.
Course aim is to teach students to construct models and make decision. | {"url":"http://www.pickthatplace.com/en/elective_courses/","timestamp":"2024-11-07T11:26:38Z","content_type":"text/html","content_length":"22741","record_id":"<urn:uuid:cedadc91-0a30-4475-aa78-0376f175ff28>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00092.warc.gz"} |
Spectral accuracy - (Computational Mathematics) - Vocab, Definition, Explanations | Fiveable
Spectral accuracy
from class:
Computational Mathematics
Spectral accuracy refers to the ability of numerical methods, particularly spectral methods, to achieve high precision in approximating solutions to differential equations. This concept is rooted in
the idea that the approximation error decreases exponentially with an increase in the number of basis functions used in the representation, making it particularly effective for smooth problems.
congrats on reading the definition of spectral accuracy. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Spectral accuracy is particularly advantageous for solving problems involving smooth functions because it allows for exponential convergence with respect to the number of basis functions used.
2. Unlike traditional numerical methods such as finite difference or finite element methods, spectral methods utilize global basis functions, which can capture the behavior of the solution over the
entire domain more effectively.
3. The accuracy achieved through spectral methods heavily relies on the choice of basis functions; common choices include Fourier series and Chebyshev polynomials.
4. Spectral accuracy can be demonstrated through numerical experiments where the error decreases rapidly as more basis functions are added, highlighting its efficiency compared to polynomial
5. In practical applications, spectral accuracy is often assessed through benchmark problems, allowing researchers to compare the performance of various spectral methods against known solutions.
Review Questions
• How does spectral accuracy impact the choice of basis functions in numerical approximations?
□ Spectral accuracy significantly influences the choice of basis functions because it emphasizes the need for functions that can provide rapid convergence for smooth problems. Using orthogonal
polynomials like Chebyshev or sine and cosine functions from Fourier series enhances the approximation's precision. The right choice ensures that the numerical method can exploit its
potential for exponential convergence, thereby reducing errors effectively.
• Discuss how spectral accuracy compares to traditional numerical methods in terms of error reduction and computational efficiency.
□ Spectral accuracy typically offers a more efficient error reduction compared to traditional numerical methods like finite differences or finite elements. While these methods may show
polynomial convergence rates, spectral methods demonstrate exponential convergence for smooth solutions, allowing them to achieve higher precision with fewer degrees of freedom. This
efficiency makes spectral methods preferable in scenarios where high accuracy is required without a significant increase in computational cost.
• Evaluate the implications of spectral accuracy in real-world applications such as fluid dynamics or weather modeling.
□ In real-world applications like fluid dynamics or weather modeling, spectral accuracy plays a crucial role in enhancing predictive capabilities. By employing spectral methods that capitalize
on high precision and rapid convergence for smooth solutions, these applications can model complex phenomena more reliably. The implications include improved simulation fidelity, better
forecasting outcomes, and more effective management of resources based on accurate predictions, highlighting the importance of adopting advanced numerical techniques that leverage spectral
"Spectral accuracy" also found in:
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\dfrac{{{{({x^4} - x)}^{1
Hint: The given expression has complexity in its terms. We first try to make it to a simpler term by substituting a temporary term and then we will integrate the expression. After the integration and
simplification, we have to re-substitute the temporary terms to its original terms, so that we will get the answer in the original terms as given the question.
Formula used:
Some of the integration formula which we will be using is \[\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}}} + c\], where\[c\] is the integration constant and some differentiation formula \[{x^n} = n
{x^{n - 1}}dx\].
Complete step by step answer:
The given expression is \[\int {\dfrac{{{{({x^4} - x)}^{1/4}}}}{{{x^5}}}} dx\]
Taking out the term \[{x^4}\]commonly from the numerator we get,
\[\int {\dfrac{{{{({x^4} - x)}^{1/4}}}}{{{x^5}}}} dx = \int {\dfrac{{{{({x^4})}^{1/4}}{{\left( {1 - \dfrac{x}{{{x^4}}}} \right)}^{1/4}}}}{{{x^5}}}} dx\]
After making some simplification we will have,
\[ = \int {\dfrac{{x{{\left( {1 - \dfrac{1}{{{x^3}}}} \right)}^{1/4}}}}{{{x^5}}}} dx\]
Further simplifying the above expression, we will have
\[ = \int {\dfrac{{{{\left( {1 - \dfrac{1}{{{x^3}}}} \right)}^{1/4}}}}{{{x^4}}}} dx\]
Now we will substitute \[{\left( {1 - \dfrac{1}{{{x^3}}}} \right)^{1/4}}\]as \[t\], that is \[t = {\left( {1 - \dfrac{1}{{{x^3}}}} \right)^{1/4}}\]
Claim: \[t = {\left( {1 - \dfrac{1}{{{x^3}}}} \right)^{1/4}}\]
Raising power to \[4\]on both sides we will get,
\[{t^4} = 1 - \dfrac{1}{{{x^3}}}\]
On differentiating with respect to \[t\]and \[x\]then we get,
\[4{t^3}dt = \dfrac{3}{{{x^4}}}dx\]
Simplifying this we get,
\[\dfrac{{dx}}{{{x^4}}} = \dfrac{4}{3}{t^3}dt\]
After substitution and using the claim we get,
\[ = {\int {\dfrac{4}{3}{t^3}\left( {{t^4}} \right)} ^{1/4}}dt\]
Simplifying further we get,
\[ = \int {\dfrac{4}{3}} {t^3}tdt\]
Making some simplification we get,
\[ = \int {\dfrac{4}{3}} {t^4}dt\]
Let’s take out the coefficient part outside the integration,
\[ = \dfrac{4}{3}\int {{t^4}dt} \]
Now it is easier to integrate the above expression, on integrating with respect to \[t\] we get,
\[ = \dfrac{4}{3}\left( {\dfrac{{{t^5}}}{5}} \right) + c\]
Now let us substitute the value of\[t\],
\[ = \dfrac{4}{{15}}{\left( {1 - \dfrac{1}{{{x^3}}}} \right)^{5/4}} + c\],
Where, \[c\] is the integration constant .
The above expression is the integrated form of the given expression.
Note: Since it is impossible to integrate a function which has complexity in its term, we have used the substitution method to make it to a simpler form (i.e., \[t = {\left( {1 - \dfrac{1}{{{x^3}}}}
\right)^{1/4}}\]) which will be easier to integrate. After the integration and simplification, we have to re-substitute the temporary terms to its original terms, so that we will get the answer in
the original terms as given the question. | {"url":"https://www.vedantu.com/question-answer/simplify-int-dfracx4-x14x5-dx-class-12-maths-cbse-60a3b34f6f51e47462fe707f","timestamp":"2024-11-12T02:38:29Z","content_type":"text/html","content_length":"183413","record_id":"<urn:uuid:b1afa494-54d0-4621-bc60-e75c2255797a>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00102.warc.gz"} |
1. If x and y are independent Gaussian random variables with average value 0 and with same variance, their joint probability density function is given by: 2. In order that a code is ‘t’ error
correcting, the minimum Hamming distance should be: 3. The Boolean expression is equivalent to: 4. The characteristic equation of a […]
UGC – NET Computer Science June 2009 – Paper – II Read More »
KVS PGT Computer Science Question paper with answers
1. Which of the following is not hardware of a computer system? 2. Which out of the following is the least capacity of memory unit compared to other options? 3. A collection of 4 bits is known as 4.
While working on computer, if we open a document in MS OFFIC WORD, which of the
KVS PGT Computer Science Question paper with answers Read More »
KVS PGT Computer Science Question paper 2023(PART – A)
1. NOTABLE. Choose the word similar in meaning to the one given below 2. Choose the word opposite in meaning to the one given above. 3. Identify the part of speech of the underlined word in the
following sentence.I cannot give him credit for this. 4. Many aspire for greatness.Choose the option in which the
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Technical Reference: Base Operating System and Extensions , Volume 2
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Technical Reference: Base Operating System and Extensions , Volume 2
Performs matrix-vector operations with general banded matrices.
SUBROUTINE SGBMV(TRANS, M, N, KL, KU, ALPHA, A, LDA,
X, INCX, BETA, Y, INCY)
INTEGER INCX, INCY, KL, KU, LDA, M, N
REAL A(LDA,*), X(*), Y(*)
SUBROUTINE DGBMV(TRANS, M, N, KL, KU, ALPHA, A, LDA,
X, INCX, BETA, Y, INCY)
INTEGER INCX,INCY,KL,KU,LDA,M,N
DOUBLE PRECISION A(LDA,*), X(*), Y(*)
SUBROUTINE CGBMV(TRANS, M, N, KL, KU, ALPHA, A, LDA,
X, INCX, BETA, Y, INCY)
INTEGER INCX,INCY,KL,KU,LDA,M,N
COMPLEX A(LDA,*), X(*), Y(*)
SUBROUTINE ZGBMV(TRANS, M, N, KL, KU, ALPHA, A, LDA,
X, INCX, BETA, Y, INCY)
COMPLEX*16 ALPHA,BETA
INTEGER INCX,INCY,KL,KU,LDA,M,N
COMPLEX*16 A(LDA,*), X(*), Y(*)
The SGBMV, DGBMV, CGBMV, or ZGBMV subroutine performs one of the following matrix-vector operations:
y := alpha * A' * x + beta * y
where alpha and beta are scalars, x and y are vectors and A is an M by N band matrix, with KL subdiagonals and KU superdiagonals.
TRANS On entry, TRANS specifies the operation to be performed as follows:
y := alpha * A * x + beta * y
y := alpha * A' * x + beta * y
y := alpha * A' * x + beta * y
Unchanged on exit.
M On entry, M specifies the number of rows of the matrix A; M must be at least 0; unchanged on exit.
N On entry, N specifies the number of columns of the matrix A; N must be at least 0; unchanged on exit.
KL On entry, KL specifies the number of subdiagonals of the matrix A; KL must satisfy 0 .le. KL; unchanged on exit.
KU On entry, KU specifies the number of superdiagonals of the matrix A; KU must satisfy 0 .le. KU; unchanged on exit.
ALPHA On entry, ALPHA specifies the scalar alpha; unchanged on exit.
A A vector of dimension ( LDA, N ); on entry, the leading ( KL + KU + 1 ) by N part of the array A must contain the matrix of coefficients, supplied column by column, with the leading diagonal of
the matrix in row ( KU + 1 ) of the array, the first superdiagonal starting at position 2 in row KU, the first subdiagonal starting at position 1 in row ( KU + 2 ), and so on. Elements in the
array A that do not correspond to elements in the band matrix (such as the top left KU by KU triangle) are not referenced. The following program segment transfers a band matrix from
conventional full matrix storage to band storage:
DO 20, J = 1, N
K = KU + 1 - J
DO 10, I = MAX( 1, J - KU ), MIN( M, J + KL )
A( K + I, J ) = matrix( I, J )
10 CONTINUE
20 CONTINUE
Unchanged on exit.
LDA On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least ( KL + KU + 1 ); unchanged on exit.
X A vector of dimension at least (1 + (N-1) * abs( INCX ) ) when TRANS = 'N' or 'n', otherwise, at least (1 + (M-1) * abs( INCX ) ); on entry, the incremented array X must contain the vector x;
unchanged on exit.
INCX On entry, INCX specifies the increment for the elements of X; INCX must not be 0; unchanged on exit.
BETA On entry, BETA specifies the scalar beta; when BETA is supplied as 0 then Y need not be set on input; unchanged on exit.
Y A vector of dimension at least (1 + (M-1) * abs( INCY ) ) when TRANS = 'N' or 'n' , otherwise, at least (1 + (N-1) * abs( INCY ) ); on entry, the incremented array Y must contain the vector y;
on exit, Y is overwritten by the updated vector y.
INCY On entry, INCY specifies the increment for the elements of Y; INCY must not be 0; unchanged on exit.
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Matrix Signings, Ramanujan Graphs and Non-Expanding Independent Sets
Printable PDF
Department of Mathematics,
University of California San Diego
Alexandra Kolla
University of Illinois, Urbana-Champaign
Matrix Signings, Ramanujan Graphs and Non-Expanding Independent Sets
The spectra of signed matrices have played a fundamental role in social sciences, graph theory and control theory. They have been key to understanding balance in social networks, to counting perfect
matchings in bipartite graphs, and to analyzing robust stability of dynamic systems involving uncertainties. More recently, the results of Marcus et al. have shown that an efficient algorithm to find
a signing of a given adjacency matrix that minimizes the largest eigenvalue could immediately lead to efficient construction of Ramanujan expanders. Motivated by these applications, this talk
investigates natural spectral properties of signed matrices and address the computational problems of identifying signings with these spectral properties. There are three main results we will talk
about: (a) NP-completeness of three signing related problems with (negative) implications to efficiently constructing expander graphs, (b) a complete characterization of graphs that have all their
signed adjacency matrices be singular, which implies a polynomial-time algorithm to verify whether a given matrix has a signing that is invertible, and (c) a polynomial-time algorithm to find a
minimum increase in support of a given symmetric matrix so that it has an invertible signing.
Host: Sam Buss
December 8, 2016
3:00 PM
AP&M 6402 | {"url":"https://math.ucsd.edu/seminar/matrix-signings-ramanujan-graphs-and-non-expanding-independent-sets","timestamp":"2024-11-03T00:27:19Z","content_type":"text/html","content_length":"34134","record_id":"<urn:uuid:12402082-ee4a-49b9-a9d4-6f7b6b6e20db>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00668.warc.gz"} |
Confidence Interval for a Population mean, with a known Population Variance - Finance Train
Confidence Interval for a Population mean, with a known Population Variance
We have the following assumptions:
• Population variance σ2 is known
• Population is normally distributed
Under these assumptions, the confidence interval estimate will be given as follows:
Let’s take an example to compute this.
We take a sample of 16 stocks from a large population with a mean return of 5.2%. We know that the population standard deviation is 1.5%.
Calculate the 95% confidence interval for the population mean.
For 95% confidence interval, zα/2 = 1.96
The confidence interval will be:
We are 95% confidence that the true mean is between 4.465% and 5.935%.
z is obtained from the standard normal distribution table as shown below. F(Z) value is 0.025 at z = -1.96 and F(Z) value is 0.9750 at z = 1.96.
The most commonly used confidence intervals are 90%, 95%, 99% and 99.9%. The z values are given below.
Confidence Level Za/2 Value
90% 1.645
95% 1.96
99% 2.58
99.9% 3.27
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Enter your name and email address below and we will email you the guides for R programming and Python. | {"url":"https://financetrain.com/confidence-interval-population-mean-known-population-variance","timestamp":"2024-11-01T19:22:29Z","content_type":"text/html","content_length":"102966","record_id":"<urn:uuid:35494e2a-b2f2-4ccd-9e2a-64c79d0c9fb0>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00860.warc.gz"} |
Question & Answer: Area Calculator: Write a program to calculate the area of some simple geometric shapes. The shapes required to be…..
Area Calculator: Write a program to calculate the area of some simple geometric shapes. The shapes required to be included are rectangles, circles, and right triangles. Because we have not covered
conditional code yet, we will calculate all the shapes in order. Your program must: Ask the user for the values necessary to compute the area of each shape o Rectangle o Circle You must use an
accurate value of pi (3.14159265359) o Right Triangle o The input must accept non-whole numbers Display the results to the user You do not need to attempt to make the program correct for improper
input (e.g. negative numbers or non-numbers)
Expert Answer
PROGRAM CODE:
#include <iostream>
using namespace std;
#define pi 3.14159265359
void rectangleArea(double l, double w)
cout<<“nArea of the rectangle is “<<l*w<<endl;
void circleArea(double r)
cout<<“nArea of the circle is “<<(pi*r*r)<<endl;
void rightTriangleArea(double h, double b)
cout<<“nArea of the right triangle is “<<h*b/2<<endl;
int main() {
double length, width, radius;
cout<<“Calculating area for rectangle…”<<endl;
cout<<“Enter length: “;
cout<<“nEnter width: “;
rectangleArea(length, width);
cout<<“nCalculating area for circle…”<<endl;
cout<<“Enter radius: “;
cout<<“nCalculating area for right triangle…”<<endl;
cout<<“Enter base: “;
cout<<“nEnter height: “;
rightTriangleArea(length, width);
return 0;
Calculating area for rectangle...
Enter length: 10
Enter width: 20
Area of the rectangle is 200
Calculating area for circle...
Enter radius: 5
Area of the circle is 78.5398
Calculating area for right triangle...
Enter base: 12
Enter height: 20
Area of the right triangle is 120 | {"url":"https://grandpaperwriters.com/question-answer-area-calculator-write-a-program-to-calculate-the-area-of-some-simple-geometric-shapes-the-shapes-required-to-be/","timestamp":"2024-11-11T10:46:14Z","content_type":"text/html","content_length":"43578","record_id":"<urn:uuid:8bba0807-f59e-4299-8576-7e42d5dbe259>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00861.warc.gz"} |
Multiplying binomials with different bases
multiplying binomials with different bases Related topics: 6th grade formula worksheets
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Author Message
Hilac Posted: Tuesday 04th of Mar 10:25
Hi guys ! Please let me know if there is an easy way to help understand a couple of multiplying binomials with different bases questions that I am stuck on. Any help would be welcome .
From: The
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IlbendF Posted: Wednesday 05th of Mar 13:20
Can you please be more elaborate as to what sort of help you are expecting to get. Do you want to get the fundamentals and work on your math questions on your own or do you want a
instrument that would give you a step-by-step solution for your math assignments ?
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Majnatto Posted: Thursday 06th of Mar 07:50
That's true, a good software can do miracles . I used a few but Algebrator is the best . It doesn't make a difference what class you are in, I myself used it in Algebra 1 and College
Algebra as well , so you don't have to be concerned that it's not on your level. If you never had a program until now I can tell you it's not hard , you don't need to know anything about
the computer to use it. You just have to type in the keywords of the exercise, and then the software solves it step by step, so you get more than just the answer.
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Aeinclam Posted: Friday 07th of Mar 08:24
I just hope this thing isn’t very complex . I am not so good with the computer stuff. Can I get the product details , so I know what it has to offer?
From: Italy
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Momepi Posted: Saturday 08th of Mar 21:05
Algebrator is the program that I have used through several algebra classes - College Algebra, Pre Algebra and Basic Math. It is a really a great piece of math software. I remember of
going through problems with greatest common factor, rational inequalities and exponent rules. I would simply type in a problem from the workbook , click on Solve – and step by step
solution to my algebra homework. I highly recommend the program.
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Gog Posted: Sunday 09th of Mar 11:08
It’s right here: https://softmath.com/. Buy it and try it, if you don’t like it (which I think can’t be true) then they even have an unconditional money back guarantee. Give it a go and
good luck with your assignment.
Austin, TX
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Example to show that a polynomial of order n or less that passes through (n+1) data points is unique.
Problem: Through three data pairs (0,0), (3,9) and (4,12), an interpolating polynomial of order 2 or less is found to be y=3x. Prove that there is no other polynomial of order 2 or less that passes
through these three points.
See the pdf file for solution.
This post is brought to you by
• Holistic Numerical Methods Open Course Ware:
• the textbooks on
• the Massive Open Online Course (MOOCs) available at
If a polynomial of order n or less passes thru (n+1) points, it is unique!
Given n+1 (x,y) data pairs, with all x values being unique, then a polynomial of order n or less passes thru the (n+1) data points. How can we prove that this polynomial is unique?
I am going to show you the proof for a particular case and you can extend it to polynomials of any order n.
Lets suppose you are given three data points (x1,y1), (x2,y2), (x3,y3) where x1 $e$ x2 $e$ x3.
Then if a polynomial P(x) of order 2 or less passes thru the three data points, we want to show that P(x) is unique.
We will prove this by contradiction.
Let there be another polynomial Q(x) of order 2 or less that goes thru the three data points. Then R(x)=P(x)-Q(x) is another polynomial of order 2 or less. But the value of P(x) and Q(x) is same at
the three x-values of the data points x1, x2, x3. Hence R(x) has three zeros, at x=x1, x2 and x3.
But a second order polynomial only has two zeros; the only case where a second order polynomial can have three zeros is if R(x) is identically equal to zero, and hence have infinite zeros. Since R(x)
=P(x)-Q(x), and R(x) $\equiv$0, then P(x) $\equiv$Q(x). End of proof.
But how do you know that a second order polynomial with three zeros is identically zero.
R(x) is of the form a0+a1*x+a2*x^2 and has three zeros, x1, x2, x3. Then it needs to satisfy the following three equations
The above equations have the trivial solution a0=a1=a2=0 as the only solution if
det(1 x1 x1^2; 1 x2 x2^2; 1 x3 x3^2)$e$0.
That is in fact the case as
det(1 x1 x1^2; 1 x2 x2^2; 1 x3 x3^2) = (x1-x2)*(x2-x3)*(x3-x1),
and since x1 $e$ x2 $e$ x3, the
det(1 x1 x1^2; 1 x2 x2^2; 1 x3 x3^2) $e$0
So the only solution is a0=a1=a2=0 making R(x) $\equiv$0
This post brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.mathforcollege.com | {"url":"https://blog.autarkaw.com/tag/unique-polynomial/","timestamp":"2024-11-06T15:19:04Z","content_type":"text/html","content_length":"35475","record_id":"<urn:uuid:80da34ff-c9a4-41a5-bd79-b383a872f60a>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00744.warc.gz"} |