content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
MISCELLANEOUS, Continued:
Simple filters can often be made of nothing more than a resistor and capacitor. A high-pass filter could be placed just before the input of a preamplifier, and a low-pass on its output. Here's how to
make a simple high-pass filter:
----- C -----.------| Preamp |--------
| `----------'
Input R Output
This filter has a "cutoff" frequency of
F = 1 / (2 × pi × R × C).
Any input component at this frequency will have its amplitude reduced by half. Higher frequencies are passed (that's why we call it "high-pass") with less amplitude reduction as the frequency rises.
Lower frequencies have their amplitudes strongly reduced: For every halving of frequency below the cutoff, the output will be further cut in half.
Let's say you want to build a 300 Hz high-pass filter. You probably want to pick R somewhere in the 10000 Ohm (10k) to 100000 Ohm (100k) range. (Too small may load down the signal source, and too
large may encourage the circuit to act like an antenna and pick up 60 Hz power line noise from the surroundings, as well as not allow the preamp to operate properly.) So let's start with 33k, which
is the geometric middle of that range. (10k / 33k is about the same as 33k / 100k.) Rearranging our formula we find
C = 1 / (2 × pi × R × F)
= (2 × 3.1415 × 33000 × 300)
= 1.6 × 10^-8 Farads (16 nF, or 0.016 uF)
If you happen to have this value on hand, you are done. However, since resistors are readily available in many more values than capacitors, you may need to use the nearest value of C that you have,
then solve for a new value of R.
Note that the input impedance of the preamp is in parallel with the resistor, and will need to be considered unless it is many times larger. Usually, the impedance is either high enough to ignore
(over 1 Megohm), or low due to a simple fixed resistance across the input of the circuit.
If you know that this impedance is a simple resistor you can select your filter resistor such that the parallel combination gives the correct value.
Otherwise, if you only know a general ballpark range for the input impedance, you will probably want to select the filter resistor to be around one tenth of this or less.
Similarly, you can make a low-pass filter to remove unwanted high frequency components:
-----| Preamp |---- R ----.--------
`----------' |
Input C Output
This is just the mirror of the high-pass filter, and it uses the same formula. Here again input signals at the cutoff frequency have their amplitudes reduced by half, but now higher frequency
components are cut in half for every doubling of frequency.
The formula assumes that the output impedance of the preamp is much lower than filter resistor, which is usually a safe assumption with modern equipment. Otherwise, its value must be added to the
resistor value to get the effective R for the filter calculation.
Note that input impedance of whatever follows the output of this filter will tend to reduce the signal level by causing a voltage divider effect. You may want to compensate for losses by using the
UvUser Units / Volt factor for critical work.
You can easily use both high-pass and low-pass filters on opposite ends of the same preamp. If these don't reduce unwanted frequencies enough, there are more elaborate filters available. However,
these are NOT made by just cascading multiple simple filters, and they will require operating power as well. One source for more information about filters is The Active Filter Cookbook by Don
There is one potential problem with using filters to reduce noise: They often change the appearance of the waveform beyond removing the unwanted components. The desired signal will typically have its
different frequency components shifted in time by different amounts, so that what started as a sharp pulse or step may become smeared out or even show transient oscillations. More elaborate filters
can help this, but these can be expensive.
The grid display may be toggled off and on via the G-key. This is sometimes useful to resolve fine details in the trace that are near a grid dot, especially if the trace Line0 style is set to show
only separate data points with a monochrome VGA monitor. This may also be useful before printing the screen.
Many laptop displays have slow response times. This means that very active or "busy" signals can be hard to see, since the trace is not in the same place long enough for the slow LCD display to
actually show them.
Exponential averaging can make signals much easier to see on these displays, since it essentially "slows down" the activity of the signal. Of course, it also adds a smoothing and noise reduction that
may not always be desired. In such cases, use Pause or Single Sweep to "freeze" the changing signal instead of slowing it down.
The unshifted L-key steps between three basic trace line styles on successive hits of the key:
Line0 = POINTS ONLY:
Only actual data points are shown. This can make it harder to interpret complex traces, but it is occasionally useful to see the raw data apart from any interpolation, especially when Xpand has been
used to stretch the trace. This is also the fastest type of trace for Daqarta to draw, so if you are running on a slow machine this can give you a little extra speed.
This mode is also useful for learning about sampling and aliasing. Try setting the Virtual Source sine wave frequency to sub-multiples of the sampling frequency. At half the sample rate (the Nyquist
frequency), all the data points form a horizontal line... the source makes one full cycle between samples, ending up at exactly the same point in the cycle each time. Make the frequency a little
higher or lower and observe what happens. Now try one forth of the sample rate, and so on.
At intermediate frequencies you will see interesting criss-crossing sine waves appear in the point alignments, as the sine wave phase advances just the right amount between samples to make points
from adjacent cycles seem to line up at a much lower frequency. The technical term for what you are doing is "messing around", but besides being fun it is also building an intuitive feel for what
happens during the sampling process... so enjoy!
Line1 = SOLID LINES:
Straight lines connect data points. This is the default line style for waveforms.
Line2 = VERTICAL BARS:
A vertical line extends from the bottom of the trace up to each data point, like a high resolution bar-graph. This is the default FFT line style. This shows you each data point without any
interpolation connecting the points as for Line1 mode, but each point still shows up clearly, unlike Line0 mode if adjacent samples have widely different values. This is the slowest line style for
Daqarta to draw because it may have to put a lot of points on the screen, so you should avoid this if display update speed is an issue.
The equal-tempered scale of western music is based upon an octave (frequency doubling) that contains 12 notes, or "semitones", each of which is 1.059463 (the 12th root of 2) times the one below it.
The standard musical tone frequencies of the equal-tempered piano keyboard are tabulated below, but you can find others by extending the series. To find the same note letter in the next higher
octave, just multiply by 2. To go down one octave, divide by 2. For example, C0 = C1 / 2 = 32.703 / 2 = 16.352 Hz.
Notes that correspond to the black keys on a
standard piano keyboard are shown in boldface:
Note Hz
A0 27.500
A#0 29.135
B0 30.868
C1 32.703
C#1 34.648
D1 36.708
D#1 38.891
E1 41.203
F1 43.654
F#1 46.249
G1 48.999
G#1 51.913
A1 55.000
A#1 58.270
B1 61.735
C2 65.406
C#2 69.296
D2 73.416
D#2 77.782
E2 82.407
F2 87.307
F#2 92.499
G2 97.999
G#2 103.826
A2 110.000
A#2 116.541
B2 123.471
C3 130.813
C#3 138.591
D3 146.832
D#3 155.563
E3 164.814
F3 174.614
F#3 184.997
G3 195.998
G#3 207.652
A3 220.000
A#3 233.082
B3 246.942
C4 261.626 MIDDLE C
C#4 277.183
D4 293.665
D#4 311.127
E4 329.628
F4 349.228
F#4 369.994
G4 391.995
G#4 415.305
A4 440.000 Concert A
A#4 466.164
B4 493.883
C5 523.251
C#5 554.365
D5 587.330
D#5 622.254
E5 659.255
F5 698.456
F#5 739.989
G5 783.991
G#5 830.609
A5 880.000
A#5 932.328
B5 987.767
C6 1046.502
C#6 1108.731
D6 1174.659
D#6 1244.508
E6 1318.510
F6 1396.913
F#6 1479.978
G6 1567.982
G#6 1661.219
A6 1760.000
A#6 1864.655
B6 1975.533
C7 2093.005
C#7 2217.461
D7 2349.318
D#7 2489.016
E7 2637.020
F7 2793.826
F#7 2959.955
G7 3135.963
G#7 3322.438
A7 3520.000
A#7 3729.310
B7 3951.066
C8 4186.009 | {"url":"https://www.daqarta.com/0mfffilt.htm","timestamp":"2024-11-12T00:33:34Z","content_type":"text/html","content_length":"13979","record_id":"<urn:uuid:310b094e-4607-4359-9e4e-555a93acbcda>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00404.warc.gz"} |
Neural Network: Number of Parameters - Get Neural Net
Neural Network: Number of Parameters
A neural network is a powerful machine learning algorithm that is inspired by the human brain. It is composed of interconnected layers of artificial neurons, called nodes or units, which work
together to perform complex tasks such as image recognition, natural language processing, and more. One important aspect of a neural network is the number of parameters it has, as these parameters
influence its capacity to learn and make accurate predictions.
Key Takeaways:
• A neural network is a machine learning algorithm inspired by the human brain.
• The number of parameters in a neural network impacts its ability to learn and make predictions.
• Neural networks consist of interconnected layers of artificial neurons called units or nodes.
In a neural network, each connection between two units represents a parameter. These connections are associated with weights, which determine the strength of the influence that one unit has on
another. The total number of parameters in a neural network is the sum of all the weights in the network. **The more parameters a neural network has, the more flexible and expressive it becomes.**
However, a large number of parameters also increases the risk of overfitting, where the network becomes too specialized in the training data and performs poorly on new, unseen data.
Deep neural networks, which have multiple hidden layers between the input and output layers, can have a significant number of parameters. For example, a convolutional neural network (CNN) used in
image recognition tasks can have millions of parameters. **These networks are capable of capturing intricate patterns and details in images, making them highly effective in tasks such as object
detection and facial recognition.** However, training deep neural networks with a large number of parameters requires substantial computational resources and can be time-consuming.
The Impact of Number of Parameters on Neural Network
The number of parameters in a neural network has several ramifications:
1. **Model Complexity**: The number of parameters determines the complexity of the model. More parameters allow the neural network to represent more intricate relationships between inputs and
outputs, enabling it to learn complex patterns and make accurate predictions. However, an overly complex model may suffer from overfitting.
2. *Transfer Learning*: Neural networks with many parameters trained on large datasets can leverage their learned knowledge to perform well on related tasks, even with limited training data. This is
known as transfer learning and is particularly beneficial when training data is scarce.
3. **Computational Resources**: Training neural networks with a large number of parameters requires significant computational resources, including processing power and memory. Therefore, it is
important to consider the available resources before choosing the number of parameters for a neural network.
Number of Parameters in Different Neural Network Architectures
The number of parameters in a neural network can vary based on the architecture used. Here are some examples:
Neural Network Architecture Number of Parameters
Feedforward Neural Network Calculated by summing the number of weights in all the connections between layers.
Convolutional Neural Network (CNN) Depends on the number of convolutional layers, their sizes, and the number of channels in each layer.
Recurrent Neural Network (RNN) Depends on the number of recurrent units and the input and output sizes.
Considerations for Choosing the Number of Parameters
When building a neural network, it is crucial to consider the appropriate number of parameters based on the specific task and available resources. Here are some considerations:
• Start with a smaller number of parameters and increase gradually. This helps to prevent overfitting and ensures efficient use of resources.
• Conduct experiments with different parameter configurations and evaluate their performance on validation data. Use techniques like regularization to avoid overfitting.
• Consider the complexity of the task and the size of the available training dataset. A more complex task or a larger dataset may require a neural network with a higher number of parameters.
By carefully choosing and tuning the number of parameters in a neural network, researchers and practitioners can optimize performance and achieve accurate predictions in various machine learning
Common Misconceptions
Misconception 1: More Parameters Always Mean Better Performance
One common misconception about neural networks is that increasing the number of parameters will always lead to better performance. While it is true that adding more parameters can provide the network
with more capacity to learn complex patterns, blindly increasing the number of parameters can actually result in overfitting or decreased performance.
• Increase in parameters can make the model more prone to overfitting.
• Adding unnecessary parameters can increase computational complexity.
• More parameters require larger amounts of training data to prevent overfitting.
Misconception 2: The Number of Parameters Determines the Model’s Complexity
Another misconception is that the number of parameters in a neural network determines its complexity. While the number of parameters does play a role in the capacity of the model, other factors such
as the network architecture, activation functions, and the data itself also contribute to the overall complexity of the model.
• The network architecture heavily influences the complexity of the model.
• Different activation functions can introduce non-linearities, increasing model complexity.
• The complexity can vary depending on the nature and structure of the input data.
Misconception 3: Increasing Parameters Will Always Improve Accuracy
Many people assume that increasing the number of parameters in a neural network will always lead to higher accuracy. However, this is not always the case. In fact, a model with too many parameters
can suffer from overfitting and fail to generalize well to new, unseen data.
• Overfitting can occur if the model learns too closely to the training examples.
• Increasing parameters can lead to increased computational requirements and longer training times.
• Finding the right balance between parameters and performance is essential.
Misconception 4: All Parameters Are Equally Important
Some people believe that all the parameters in a neural network are equally important for the model’s performance. However, certain parameters may have more impact and influence on the network’s
learning ability than others. Identifying and optimizing these influential parameters can significantly improve the model’s performance.
• Parameters in the deeper layers may have a more significant influence on the model’s performance.
• Weight initialization can play a crucial role in determining the importance of different parameters.
• Some parameters may have little effect on the model’s performance and can be pruned to reduce computational complexity.
Misconception 5: More Parameters Mean More Accurate Predictions
It is a common misunderstanding that increasing the number of parameters in a neural network will always lead to more accurate predictions. While adding more parameters can improve performance up to
a certain point, accuracy is not solely dependent on the number of parameters. Other factors like the quality and diversity of the training data, proper regularization techniques, and hyperparameter
tuning also play significant roles in achieving accurate predictions.
• High-quality and diverse training data are crucial for accurate predictions.
• Applying regularization techniques can prevent overfitting and improve accuracy.
• Hyperparameter tuning, such as learning rate and batch size, can greatly impact model accuracy.
In recent years, the field of neural networks has garnered significant attention and breakthroughs in various domains. One crucial aspect in designing a neural network is determining the number of
parameters it possesses. The number of parameters affects the model’s complexity, its ability to learn and generalize, and its computational requirements. In this article, we present ten tables that
showcase the number of parameters for different types of neural networks, shedding light on the interesting variations among them.
Table: Multilayer Perceptron
The multilayer perceptron (MLP) is a basic neural network architecture consisting of multiple layers of nodes. It is widely used in supervised learning tasks like classification and regression.
Layer Number of Parameters
Input 0
Hidden 5,000
Output 1,000
Total 6,000
Table: Convolutional Neural Network
Convolutional neural networks (CNNs) are predominantly used for image classification tasks. The convolutional layers enable the network to learn spatial hierarchies and extract meaningful features
Layer Number of Parameters
Convolutional 500,000
Fully Connected 1,000,000
Total 1,500,000
Table: Recurrent Neural Network
Recurrent neural networks (RNNs) are designed to effectively process sequential data. They have connections that allow information to persist across time steps, allowing the model to understand
context and temporal dependencies.
Layer Number of Parameters
Recurrent 750,000
Fully Connected 500,000
Total 1,250,000
Table: Generative Adversarial Network
Generative adversarial networks (GANs) consist of a generator and a discriminator network, working in opposition to produce realistic synthetic data.
Network Number of Parameters
Generator 750,000
Discriminator 500,000
Total 1,250,000
Table: Self-Organizing Map
The self-organizing map (SOM) is an unsupervised learning algorithm used for visualizing and clustering high-dimensional data.
Layer Number of Parameters
Input 0
Neurons 100,000
Total 100,000
Table: Long Short-Term Memory Network
Long short-term memory networks (LSTMs) are a specialized type of RNNs that excel at handling sequential data with long-term dependencies.
Layer Number of Parameters
LSTM 1,500,000
Fully Connected 500,000
Total 2,000,000
Table: Deep Belief Network
Deep belief networks (DBNs) consist of multiple layers of restricted Boltzmann machines (RBMs) and are primarily used for unsupervised learning tasks like feature extraction and pretraining.
Layer Number of Parameters
Input 0
Hidden 10,000
Total 10,000
Table: Autoencoder
Autoencoders are neural networks used for data compression and feature learning, composed of an encoder and a decoder.
Network Number of Parameters
Encoder 200,000
Decoder 200,000
Total 400,000
Table: Radial Basis Function Network
Radial basis function networks (RBFNs) are shallow neural networks used for function approximation, employing radial basis functions as activation functions.
Layer Number of Parameters
RBFs 50,000
Fully Connected 10,000
Total 60,000
Neural networks come in various architectures, each with distinct characteristics and performance attributes. The number of parameters in a neural network plays a vital role in its behavior and
functionality. From the showcased tables, we observe that different architectures possess different magnitudes of parameter counts. It is crucial for machine learning practitioners to understand
these variations and choose appropriate network designs based on their specific application requirements, considering factors like model complexity, memory usage, and training duration.
Frequently Asked Questions
What is a neural network and how does it work?
A neural network is a type of machine learning algorithm that is inspired by the structure and functionality of the human brain. It consists of interconnected layers of artificial neurons, also known
as nodes, that process and transmit information. The network learns from data by adjusting the weights assigned to these connections to optimize for a specific task, such as image recognition or
language translation.
What is the role of parameters in a neural network?
Parameters in a neural network refer to the weights and biases assigned to the connections between neurons. These are the values that the network adjusts during the training process in order to
minimize the difference between predicted and actual outputs. The number of parameters in a neural network determines its complexity and capacity to learn from data.
What factors determine the number of parameters in a neural network?
The number of parameters in a neural network is determined by the architecture and configuration of the network. It depends on the number of neurons in each layer, the number of layers, and the type
of connections between the neurons. Additionally, the choice of activation functions, regularization techniques, and other network-specific parameters can also affect the number of parameters.
How are the number of parameters calculated in a neural network?
To calculate the number of parameters in a neural network, you need to count the weights and biases in each layer. For a fully connected layer, the number of parameters is equal to the product of the
number of neurons in the current layer and the number of neurons in the previous layer, plus the number of biases in the current layer. The total number of parameters is the sum of the parameters in
all layers.
Why is the number of parameters important in a neural network?
The number of parameters in a neural network plays a crucial role in determining its capacity to learn from data. Too few parameters may result in underfitting, where the network fails to capture the
complexity of the data. On the other hand, too many parameters can lead to overfitting, where the network memorizes the training data but fails to generalize to new examples. Finding the right
balance of parameters is essential for achieving optimal performance.
How does the number of parameters affect training time and computational resources?
The number of parameters in a neural network has a direct impact on the training time and computational resources required. A larger number of parameters typically means a larger amount of memory and
processing power needed to perform forward and backward propagation during training. Moreover, training larger networks can take longer, especially when dealing with limited computational resources.
Are there any rules of thumb for determining the number of parameters in a neural network?
While there are no universal rules for determining the exact number of parameters, there are some guidelines that can help. The number of parameters should be chosen based on the complexity of the
task and the amount of available training data. As a general rule, it is advisable to start with a smaller network and gradually increase its capacity if the performance is not satisfactory.
Regularization techniques such as weight decay and dropout can also be used to prevent overfitting.
Can a neural network have too many parameters?
Yes, a neural network can have too many parameters. If the network has an excessively large number of parameters relative to the complexity of the task and the available training data, it may suffer
from overfitting. Overfitting occurs when the network becomes too specialized in the training data and fails to generalize well to new examples. Therefore, it is important to carefully consider the
number of parameters to avoid overfitting.
Can reducing the number of parameters in a neural network improve its performance?
Reducing the number of parameters in a neural network can sometimes improve its performance. If the network is overfitting the training data, reducing the number of parameters can help prevent
overfitting and improve generalization. However, it is important to strike a balance, as reducing the number of parameters too much may result in underfitting, where the network fails to capture the
complexity of the data. Proper experimentation and validation are necessary to determine the optimal number of parameters.
Are there any tools or libraries available to analyze and visualize the number of parameters in a neural network?
Yes, there are various tools and libraries available that can help analyze and visualize the number of parameters in a neural network. Frameworks like TensorFlow, PyTorch, and Keras provide built-in
functions and utilities to inspect the model architecture and calculate the number of parameters. Additionally, there are external libraries such as Netron and NN-SVG that can visualize the network
structure and parameter count in a graphical format. | {"url":"https://getneuralnet.com/neural-network-number-of-parameters/","timestamp":"2024-11-12T10:24:00Z","content_type":"text/html","content_length":"66547","record_id":"<urn:uuid:1e61f8b6-96e2-4edd-9db5-2aee3ede4169>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00474.warc.gz"} |
Ways to use Geogebra in a mathematics classroom
Geogebra does. Geogebra is a software package for creating and manipulating geometric objects. It also allows for graphing of funcitons and manipulating the functions in all sorts of interesting
ways. It runs on the Java framework, which means if you have Java installed on your computer, you can run Geogebra, which makes it any Java enabled operating system. This means the very same program
will run on Windows, Mac, Linux or Solaris, although the installer is different for each operating system.
If you are planning on using the program with your students, it is nice to know that they can install the program for free, and that it is very likely to work on their computer. The only caveat is
that you need to make sure the students have the right version of Java installed if they have any problems as this can sometimes be an issue.
Geogebra has all of the standard Geometry software functions. You can add lines, circles, ellipses and all other sorts of geometric functions to the document. You can also make one object a dependent
of another object which means that changes in the original object propagate to its dependent objects. So in other words, if a you draw a line segment which depends on the location of point A and
point B, changing either point A or point B modifies the line segment.
Geogebra also has an input textfield, which means that every command you can use the interface to enter, you can also type in. Some commands are done much more easily through the input textfield,
things like entering y = x^2 + 3x which uses the nature notation to graph a function. Entering Function[x^2, 0, 2] graphs the function
Using Geogebra with your classroom is an affordable way to bring high quality geometry software to your classroom at an extremely affordable price (its free!!). I’ve only scratched the surface of
what Geogebra is capable of doing, I suggest you try it out yourself. Maybe when I have time I’ll create some tutorials on using it. | {"url":"https://davidwees.com/content/ways-use-geogebra-mathematics-classroom/","timestamp":"2024-11-14T20:34:47Z","content_type":"text/html","content_length":"75516","record_id":"<urn:uuid:a8957755-1238-4bd0-a5b2-55662305697c>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00897.warc.gz"} |
Sketch the graph of the solution set to each nonlinear system of inequalities. $$\begin{aligned}&y>|2 x|-4\\\&y \leq \sqrt{4-x^{2}}\end{aligned}$$
Short Answer
Expert verified
Shade the region above \( y > |2x| - 4 \) and below or on \[ y \leq \sqrt{4 - x^2} \), with appropriate boundaries.
Step by step solution
Understand the First Inequality
The first inequality is given by \( y > |2x| - 4 \). This represents two rays starting from the point \( (0, -4) \) on the y-axis, extending upwards with slopes +2 and -2 respectively. Sketch these
two lines and shade the region above them.
Understand the Second Inequality
The second inequality is given by \[ y \leq \sqrt{4 - x^2} \]. This describes the upper semicircle of a circle centered at the origin with radius 2. Only the part of the circle where \ y \ is
non-negative should be considered. Sketch this semicircle and shade the area within and including the semicircle.
Find the Intersection of Both Regions
The solution to the system of inequalities is the overlap of the shaded regions from steps 1 and 2. Identify and shaded the region that satisfies both \( y > |2x| - 4 \) and \( y \leq \sqrt{4 - x^2}
Check Boundary Conditions
Since the first inequality \( y > |2x| - 4 \) is strict (does not include the boundary), the boundary lines will be dashed. For the second inequality \[ y \leq \sqrt{4 - x^2} \], the boundary (the
semicircle) is included and should be solid.
Finalize the Sketch
Draw a clean sketch of the solution set by combining the insights from the previous steps. Clearly indicate the overlap region with appropriate shading and use dashed and solid lines where
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
absolute value functions
An absolute value function is expressed in the form \( y = |ax + b| + c \). Here, the absolute value of \( ax + b \) ensures that the function will always be non-negative.
In the context of the exercise, the function given is \( y > |2x| - 4 \). This represents two rays starting from the point \( (0, -4) \) on the y-axis with slopes of \+2\ and \-2\.
To sketch it, you can:
• Plot the point \( (0, -4) \) on the y-axis.
• Draw two lines with slopes of \+2\ and \-2\ from this point.
• Since the inequality is strict, shade the region above these lines with a dashed boundary to show that the lines themselves are not included in the solution.
Understanding these properties will help clarify the first part of our system of inequalities.
semicircle equations
A semicircle equation, such as \[ y \leq \sqrt{4 - x^2} \], is derived from the general form of a circle's equation \ x^2 + y^2 = r^2 \ where r is the radius of the circle.
In the given exercise, the equation \[ y \leq \sqrt{4 - x^2} \] describes the upper half of a circle with a radius of 2 centered at the origin. To sketch this:
• Plot the points where the upper half of the circle intersects the x-axis, at \ (2,0) \ and \ (-2,0) \.
• Draw the curve connecting these points, forming a semicircle with a solid line because the inequality is non-strict (it includes the boundary).
• Shade the region within and including the semicircle to show the set of points that satisfy \[ y \leq \sqrt{4 - x^2} \].
Being familiar with semicircle equations is crucial for understanding the second part of our system.
inequalities intersection
The key to graphing the solution of a system involving inequalities is to identify the region where all conditions are met simultaneously. In our particular exercise, we need to find the intersection
of the regions defined by \( y > |2x| - 4 \) and \[ y \leq \sqrt{4 - x^2} \]. Here is how to do it:
• First, analyze the graph of \( y > |2x| - 4 \). This consists of two lines emanating from \ (0, -4) \ and shading the region above them.
• Next, on the same set of axes, draw the upper semicircle \[ y \leq \sqrt{4 - x^2} \] and shade the region inside and including the curve.
• The solution set to the system will be the area where these shaded regions overlap. Use dashed lines for \( y > |2x| - 4 \) because its boundary is not included, and solid lines for \[ y \leq \
sqrt{4 - x^2} \] since its boundary is included.
This intersection represents all (x, y) coordinates that satisfy both inequalities. | {"url":"https://www.vaia.com/en-us/textbooks/math/precalculus-functions-and-graphs-4-edition/chapter-8/problem-51-sketch-the-graph-of-the-solution-set-to-each-nonl/","timestamp":"2024-11-03T17:03:15Z","content_type":"text/html","content_length":"249090","record_id":"<urn:uuid:3c797acb-7ee8-4018-8809-e82fb844cfeb>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00393.warc.gz"} |
Expected bank da calculator Archives
Expected Bank DA Calculator for Bank Employees Expected Dearness Allowance (DA) Calculator from February 2020 for Bank Employees A simple online calculator to predetermine the percentage of Dearness
allowance to existing bank employees and officers … [Read more...] about Expected Bank DA Calculator for Bank Employees | {"url":"https://paymatrixtables.com/tag/expected-bank-da-calculator/","timestamp":"2024-11-05T07:47:08Z","content_type":"text/html","content_length":"44545","record_id":"<urn:uuid:6a005a33-f1f3-4db6-ae28-c2de4b168e0e>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00136.warc.gz"} |
Next: Mechanical interpretation Up: Z-plane, causality, and feedback Previous: Causality and the unit
Let b[t] denote a filter. Then a[t] is its inverse filter if the convolution of a[t] with b[t] is an impulse function. In terms of Z-transforms, an inverse is simply defined by A(Z) = 1/B(Z). Whether
the filter A(Z) is causal depends on whether it is finite everywhere inside the unit circle, or really on whether B(Z) vanishes anywhere inside the circle. For example, B(Z)=1-2Z vanishes at Z = 1/2.
There A(Z)=1/B(Z) must be infinite, that is to say, the series A(Z) must be nonconvergent at Z = 1/2. Thus, as we have just seen, a[t] is noncausal. A most interesting case, called ``minimum phase,"
occurs when both a filter B(Z) and its inverse are causal. In summary,
The reason the interesting words ``minimum phase'' are used is given in chapter .
Next: Mechanical interpretation Up: Z-plane, causality, and feedback Previous: Causality and the unit Stanford Exploration Project | {"url":"https://sep.stanford.edu/sep/prof/pvi/zp/paper_html/node26.html","timestamp":"2024-11-11T06:38:11Z","content_type":"text/html","content_length":"5623","record_id":"<urn:uuid:9adf2a44-46c1-46aa-8e83-83802b376e5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00180.warc.gz"} |
How Are Humidity And Precipitation Related?A. When Humidity Is Low, The Chance Of Precipitation Is High.B.
D cuz it says the answer
Mole measure the number of elementary entities of a given substance that are present in a given sample. Therefore, 8.12×10²³ atoms are in 90.43 mole of copper.
What is mole?
The SI unit of amount of substance in chemistry is mole. The mole is used to measure the quantity of amount of substance. It measure the number of elementary entities of a given substance that are
present in a given sample. There are so many formula for calculating mole.
we know one mole of any element contains 6.022×10²³ atoms which is also called Avogadro number
number of atoms/molecules=number of moles × 6.022×10²³(Avogadro number)
number of moles of copper=90.43 moles
Substituting all the given values in the above equation, we get
number of atoms/molecules= 90.43 × 6.022×10²³
number of atoms/molecules=8.12×10²³ molecules
Therefore, 8.12×10²³ atoms are in 90.43 mole of copper.
To know more about mole, here: | {"url":"https://diemso.unix.edu.vn/question/how-are-humidity-and-precipitation-relatedbr-br-a-when-humid-8m9j","timestamp":"2024-11-15T00:16:02Z","content_type":"text/html","content_length":"73056","record_id":"<urn:uuid:a416147f-a670-4800-9d08-e2c844f16bf7>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00457.warc.gz"} |
Toward faster Rationals
Consider (abs, inv, +, -, *, //) over values of type Rational{I}
where I<:Union{Int16…Int128, BigInt}. A substantive time sink is renormalizing the (numerator, denominator) pairs: div each by their gcd.
Use two sorts of Rationals, (1) those known to be reduced to lowest terms and (2) those not known to be reduced to lowest terms. And avoid the gcd stuff as long as possible while protecting the
integrity of the calculation. Doing this obtains an arithmetically performant realization of bounded Rationals.
These occurrences prompt renormalization:
(a) an arithmetic op overflows and an operand is not known to be reduced
- normalize the operand[s] to lowest terms and retry the arithmetic op
(b) a Rational value that is not known to be reduced is obtained from a store or a stream
- normalize the value and pass that along, now known to be reduced
(c) a Rational value that is not known to be in lowest terms is offered (written to a store, displayed)
- normalize the value and utilize that, so all presentation is of canonical Rational forms
This is effective.
Implementing the type this way:
struct FastRational{IntForRational, TeleologicalState}
# where
const RationalInt =
Union{Int8, Int16, Int32, Int64, Int128, BigInt}
const IsKnownToBeReduced = Val{:IsKnownToBeReduced}
const IsNotKnownToBeReduced = Val{:IsNotKnownToBeReduced}
const TeleologicalState =
Union{IsKnownToBeReduced, IsNotKnownToBeReduced}
works well with simple expressions
each NNx is from one expr on one machine
@btime (a_numer//a_denom) * (b_numer//b_denom))
• relative to Rational{I}: Int32 (25x), Int64 (10x), BigInt (4x), Int128 (2x)
I am concerned that real world use may make normalization churn.
Algorithms that use many magnitude comparisons with values not known to be in reduced form could/would cause those values to be repeatedly re-reduced. The struct is immutable and the reduced forms,
once derived, often would not persist. It seems that the mechanism above only allows values that are streamed out and later streamed in have their newly reduced state persist.
A much better way would be to have the representation allow for transitioning from the numerator and denominator values associated with a IsNotKnownToBeReduced state to reduced numerator and
denominator values associated with a IsNotKnownToBeReduced state become (same referent, updated content) the reduced numerator and denominator values and morph its teleological state into
How may this be accomplished, keeping performance?
Simply changing the struct to a mutable struct (no other edits) halves the performance gains.
Taking advantage of the mutability may win back some time, but those gains stick only if the teleological state becomes a third field (so it can be altered without breaking type). Although adding to
the struct size is not great for an otherwise juxtaposed primitive type pair.
Is there a clean way for that Bool field to govern dispatch as the parameterized version does … before runtime? It seems ad hock.
1 Like
The type definition could be kept simpler as
struct FastRational{T}
with the only difference from Rational being that the canonical form of the rational is not enforced, so a*num and a*denom are the same rational, for positive integer a.
The following semantics could be accomplished with a minimal extension of the vocabulary:
1. arithmetic on FastRationals results in FastRationals, which are not necessarily canonical,
2. convert(Rational, x) provides the canonical form,
3. FastRationals are contagious, eg +(::Rational, ::FastRational)::FastRational.
Whenever fast equality comparison is required, the user converts explicitly to Rational.
However, there is a trade-off here: for nontrivial calculations, the denominator can explode very quickly, so that BigInt is the only meaningful choice. But an occasional gcd may result in
significantly smaller representation.
Most of the performance gain comes from being able to differentiate FastRationals that are reduced (and so their arith logic has a simpler path) from FastRationals that are not known to be reduced
(i.e. those that have not been reduced explicitly). And handling the mixed case (2 operands, one reduced the other may not be reduced) is faster than the general case, so I do that.
I may be misunderstanding your intent, though.
My point was that these are already the Rational type. Unless promotion rules would be different, reduced FastRationals would be superfluous IMO, but I might be missing something.
Ahh. I tried something like that – got stuck here.
Using regular rationals is not helpful because – eventhough they are always reduced, the regular rational arith code keeps gcd-ing them, and that is most of the time difference. So I thought to make
two types, say FastRational and FastRatio where FastRationalss were assured to be reduced and FastRatios never were assured to be reduced (though they might be). And thought about writing the code so
they fully interoperate and I retain metamanagement of the postponements and manners of determining overflow.
So far so good … dispatch remains strongly helpful and both types inherit from a shared supertype to allow some code simplification.
Does this allow for a variable q that is created as a FastRatio and is computed on for while, and gets reduced for good reason – so it, as butterfly, becomes realized as a FastRational … does this
allow for that variable, q to be reattached to the FastRational realization? And if so, can this occur rapidly enough (it would happen often)?
It’s not been extended beyond the simplest implementation, but might be a starting point.
Yes, that is as a gcd free variant of Rational{T} where T<:Signed. I have a gcd wary variant. And that is half 'n other half, perspectively.
Any familiarity with the proper way
(is there any proper way that does not totally go against principle)
to do this imagined manipulation?
struct Ratio{T}
r0 = Ratio(4, 8)
r1 = Ratio(1, 2)
ptr_to_r0num = get_pointer(Ratio, r0, :num) # get_pointer(Ratio, r0, fieldidx = 1)
ptr_to_r0den = get_pointer(Ratio, r0, :den) # get_pointer(Ratio, r0, fieldidx = 2)
unsafe_overwrite_into_struct(ptr_to_r0num, r1.num) # .., object_from_ptr(ptr_to_r1num)
unsafe_overwrite_into_struct(ptr_to_r0den, r1.den)
r0.num == r1.num && r0.den == r1.den
or like that using variables as pointer handles
or like that … would using SVectors of length 2 be appropriate? @andyferris
This may be one way (I have not made it yet, so performance is tbd).
Make FastRational a struct of two structs, one of type RationalIsReduced and the other of type RationalMayReduce where each has fields :num and :den. Parameterize FastRationals to include a param for
either of two teleological Val{} types and use that to properly select dispatch before runtime.
SVector{2} models a 2D vector space; otherwise as a container it’s not much more useful than a 2-tuple, or a struct with two fields. | {"url":"https://discourse.julialang.org/t/toward-faster-rationals/7933","timestamp":"2024-11-12T10:24:38Z","content_type":"text/html","content_length":"39562","record_id":"<urn:uuid:fbfacc3e-d8bb-4a66-b3f2-94f7e15e8fd6>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00340.warc.gz"} |
Floating point bug in Canon F-789SGA
11-30-2014, 07:32 PM
Post: #1
Peter Van Roy Posts: 21
Junior Member Joined: Jan 2014
Floating point bug in Canon F-789SGA
I recently bought a Canon F-789SGA calculator because its specification intriguingly mentions that it has 18 digit precision. Tests show that it actually has 20 digit internal precision, and
transcendental functions actually achieve this precision. That is the nice part.
Unfortunately, the calculator has bugs in its floating point operations. It took me a while to figure out what was going on, since there are several bugs that interact in unobvious ways. The main bug
is an error in floating point addition. When doing a floating point addition A+B, if the difference in exponents between the two numbers A and B is 16, 17, 18, or 19, and if the lower four digits of
the larger number are 0000, then the smaller number is not added in. That is, the lower four digits of the result stay 0000. For example, doing the addition 1 + 1E-16 gives exactly 1 instead of
1.0000000000000001. Floating point subtraction has a similar bug. There are other irregularities in the floating point operations, in particular, the value of the last (20th) digit also plays a role.
My guess to the origin of this bug is that the calculator is an improved version of a Casio calculator with 16 digit precision. Possibly the internal precision was increased without proper testing.
Has anyone else found similar bugs in these or other recent calculators?
11-30-2014, 08:36 PM
Post: #2
Steve Simpkin Posts: 1,285
Senior Member Joined: Dec 2013
RE: Floating point bug in Canon F-789SGA
Other than this bug, this model compares favorably to the Casio FX-115ES Plus, Sharp EL-W516B (or X) and TI TI-36X Pro with some features the others do not have.
How do the buttons feel?
12-01-2014, 10:20 AM
Post: #3
Peter Van Roy Posts: 21
Junior Member Joined: Jan 2014
RE: Floating point bug in Canon F-789SGA
(11-30-2014 08:36 PM)Steve Simpkin Wrote: Other than this bug, this model compares favorably to the Casio FX-115ES Plus, Sharp EL-W516B (or X) and TI TI-36X Pro with some features the others do
not have.
How do the buttons feel?
The buttons are not the best I have seen, but they are usable. They do not detract from the use of the calculator.
12-05-2014, 02:20 PM
Post: #4
Gerald H Posts: 1,627
Senior Member Joined: May 2014
RE: Floating point bug in Canon F-789SGA
(11-30-2014 07:32 PM)Peter Van Roy Wrote: I recently bought a Canon F-789SGA calculator because its specification intriguingly mentions that it has 18 digit precision. Tests show that it
actually has 20 digit internal precision, and transcendental functions actually achieve this precision. That is the nice part.
Unfortunately, the calculator has bugs in its floating point operations. It took me a while to figure out what was going on, since there are several bugs that interact in unobvious ways. The main
bug is an error in floating point addition. When doing a floating point addition A+B, if the difference in exponents between the two numbers A and B is 16, 17, 18, or 19, and if the lower four
digits of the larger number are 0000, then the smaller number is not added in. That is, the lower four digits of the result stay 0000. For example, doing the addition 1 + 1E-16 gives exactly 1
instead of 1.0000000000000001. Floating point subtraction has a similar bug. There are other irregularities in the floating point operations, in particular, the value of the last (20th) digit
also plays a role.
My guess to the origin of this bug is that the calculator is an improved version of a Casio calculator with 16 digit precision. Possibly the internal precision was increased without proper
Has anyone else found similar bugs in these or other recent calculators?
i'm not sure it's a bug.
Handbook says:
"Precision +/- 1 at the 10th digit for a single calculation
+/- 1 at the least significant for exponential display" (page 10)
which I don't really understand, but could imply you are requiring accuracy above specifications.
Specs are certainly exceeded by the PFact function, where the greatest factor is declared "unfactored" whether prime or composite - in fact, the greatest factor found is prime for all values < 1009^
Overall a pleasant calculator & accurate & integration is annoyingly slow.
12-05-2014, 09:39 PM
Post: #5
Peter Van Roy Posts: 21
Junior Member Joined: Jan 2014
RE: Floating point bug in Canon F-789SGA
(12-05-2014 02:20 PM)Gerald H Wrote: i'm not sure it's a bug.
Handbook says:
"Precision +/- 1 at the 10th digit for a single calculation
+/- 1 at the least significant for exponential display" (page 10)
which I don't really understand, but could imply you are requiring accuracy above specifications.
The specification says clearly that internal precision is 18 digits (see page 11 of F-789SGA Scientific Calculator User Instructions). It is quite common for calculators to have higher internal
precision than what they display. My observations show that floating point addition has a bug at the 17th and 18th digits.
12-06-2014, 06:37 AM
(This post was last modified: 12-06-2014 06:43 AM by Gerald H.)
Post: #6
Gerald H Posts: 1,627
Senior Member Joined: May 2014
RE: Floating point bug in Canon F-789SGA
(12-05-2014 09:39 PM)Peter Van Roy Wrote:
(12-05-2014 02:20 PM)Gerald H Wrote: i'm not sure it's a bug.
Handbook says:
"Precision +/- 1 at the 10th digit for a single calculation
+/- 1 at the least significant for exponential display" (page 10)
which I don't really understand, but could imply you are requiring accuracy above specifications.
The specification says clearly that internal precision is 18 digits (see page 11 of F-789SGA Scientific Calculator User Instructions). It is quite common for calculators to have higher internal
precision than what they display. My observations show that floating point addition has a bug at the 17th and 18th digits.
Can't find the referenced text in "User Instructions" E-IM-2725 page 11 nor in f789sga.pdf E-IE-455 page 11.
12-09-2014, 07:38 PM
(This post was last modified: 12-09-2014 07:40 PM by Peter Van Roy.)
Post: #7
Peter Van Roy Posts: 21
Junior Member Joined: Jan 2014
RE: Floating point bug in Canon F-789SGA
(12-06-2014 06:37 AM)Gerald H Wrote: Can't find the referenced text in "User Instructions" E-IM-2725 page 11 nor in f789sga.pdf E-IE-455 page 11.
It's on the bottom of page 10 in E-IE-455, the first entry in the first table in section
Input Range and Error Messages
. It's page 11 when viewing the pdf document but numbered as page 10 on the page itself.
12-10-2014, 06:35 AM
Post: #8
Gerald H Posts: 1,627
Senior Member Joined: May 2014
RE: Floating point bug in Canon F-789SGA
(11-30-2014 07:32 PM)Peter Van Roy Wrote: I recently bought a Canon F-789SGA calculator because its specification intriguingly mentions that it has 18 digit precision. Tests show that it
actually has 20 digit internal precision, and transcendental functions actually achieve this precision. That is the nice part.
Unfortunately, the calculator has bugs in its floating point operations. It took me a while to figure out what was going on, since there are several bugs that interact in unobvious ways. The main
bug is an error in floating point addition. When doing a floating point addition A+B, if the difference in exponents between the two numbers A and B is 16, 17, 18, or 19, and if the lower four
digits of the larger number are 0000, then the smaller number is not added in. That is, the lower four digits of the result stay 0000. For example, doing the addition 1 + 1E-16 gives exactly 1
instead of 1.0000000000000001. Floating point subtraction has a similar bug. There are other irregularities in the floating point operations, in particular, the value of the last (20th) digit
also plays a role.
My guess to the origin of this bug is that the calculator is an improved version of a Casio calculator with 16 digit precision. Possibly the internal precision was increased without proper
Has anyone else found similar bugs in these or other recent calculators?
(12-09-2014 07:38 PM)Peter Van Roy Wrote:
(12-06-2014 06:37 AM)Gerald H Wrote: Can't find the referenced text in "User Instructions" E-IM-2725 page 11 nor in f789sga.pdf E-IE-455 page 11.
It's on the bottom of page 10 in E-IE-455, the first entry in the first table in section Input Range and Error Messages. It's page 11 when viewing the pdf document but numbered as page 10 on the
page itself.
No - in the pdf on page 10 "Nr of digits for internal calc : Up to 18", spec met by trig funcs & everything else too, as "up to" is a guarantee of nothing.
In "Calculation Examples" paper document, example #6, page 4,
Nr digits for internal prec 18 BUT also +/- 1 at the 10th digit.
You can accuse Canon of sloppy, unclear documentation but I don't think the claim of calc not fulfilling specs is warranted.
12-13-2014, 06:17 PM
Post: #9
Peter Van Roy Posts: 21
Junior Member Joined: Jan 2014
RE: Floating point bug in Canon F-789SGA
(12-10-2014 06:35 AM)Gerald H Wrote: You can accuse Canon of sloppy, unclear documentation but I don't think the claim of calc not fulfilling specs is warranted.
Independent of whatever the spec claims or not, it is a fact that this calculator has bugs in its basic floating point operations. Caveat emptor if you want to use this calculator for any important
12-14-2014, 06:43 AM
Post: #10
Gerald H Posts: 1,627
Senior Member Joined: May 2014
RE: Floating point bug in Canon F-789SGA
(12-13-2014 06:17 PM)Peter Van Roy Wrote:
(12-10-2014 06:35 AM)Gerald H Wrote: You can accuse Canon of sloppy, unclear documentation but I don't think the claim of calc not fulfilling specs is warranted.
Independent of whatever the spec claims or not, it is a fact that this calculator has bugs in its basic floating point operations. Caveat emptor if you want to use this calculator for any
important computations.
Certainly "Caveat calculator" when the calculator (in Latin he was a person, female calculatrix) uses an electrical friend to calculate.
The Canon machine is still better than using extended reals in the HP 48G - I've tested this. As for comparisons of most calculation results on the current offerings I'd like to know more.
02-04-2018, 04:01 AM
(This post was last modified: 02-05-2018 01:59 PM by daschel.)
Post: #11
daschel Posts: 1
Junior Member Joined: Feb 2018
RE: Floating point bug in Canon F-789SGA
(12-13-2014 06:17 PM)Peter Van Roy Wrote:
(12-10-2014 06:35 AM)Gerald H Wrote: You can accuse Canon of sloppy, unclear documentation but I don't think the claim of calc not fulfilling specs is warranted.
Independent of whatever the spec claims or not, it is a fact that this calculator has bugs in its basic floating point operations. Caveat emptor if you want to use this calculator for any
important computations.
Hello, and my apologies if this thread has long been forgotten. But I have some findings which might change your perspective on this. I came across your post and tried your original test on a Canon
F-792SGA, as well as with an HP 33s. And they both showed a loss of accuracy before their respective internal precision limits were reached.
The Canon (as you've already shown) loses precision with anything smaller than 1E-15, despite it's reputed 18-digit accuracy. This is not to be disputed; my results were exactly the same as yours.
But it might surprise you that the HP 33s 'fails' your test with exponents lesser than -11. And this is in spite of its 15-digits of internal accuracy. It seems to me that you had singled out the
Canon F-792SGA while other reputable (and pricier) calculators didn't fare any better, as they both fall short 3—5 digits of what would be expected, given their specs. Note: I asked the owner of an
HP 35s to run the same test on his calculator and he notifed me of similar results.
To say the Canon isn't worthy of "serious" computations based your test is to impugn the functionality of other well-respected calculators; I seriously doubt you'll find too many mathematicians,
engineers or statisticians who'd agree that the HP 33s isn't up to serious work—despite it falling short of anticipated results. The same courtesy must therefore be extended to the Canons. I suspect
the width of temporary memory registers and legacy numerical libraries might all play a role in causing this. But the Canon F-792SGA is not to be faulted on this point, especially when compared with
other "serious" calculators.
I hope you've found enjoyment with your F792SGA, or with another model that suits your requirements. Based on my experiment, I'm inclined to agree with the other poster who describes this as a
documentation—but not a technical—problem.
07-17-2018, 12:55 AM
Post: #12
Albert Chan Posts: 2,764
Senior Member Joined: Jul 2018
RE: Floating point bug in Canon F-789SGA
Floating point bug is actually a "feature" (put inside intentionally).
Without this "feature", they might get many customer complains:
It is hand-holding feature for people don't understand floating point.
They expected sqrt(4) - 2 = 0, not some tiny tiny number.
But, allowing calculator to "fix" the values, the final result is not as precise.
The medicine is worse than the cure ...
07-17-2018, 04:08 PM
Post: #13
Tugdual Posts: 764
Senior Member Joined: Dec 2013
RE: Floating point bug in Canon F-789SGA
This calculator totally looks like a Casio, did you try the self test with shift+on+7 then 9 (or 8) ?
07-17-2018, 05:27 PM
Post: #14
Albert Chan Posts: 2,764
Senior Member Joined: Jul 2018
RE: Floating point bug in Canon F-789SGA
(07-17-2018 04:08 PM)Tugdual Wrote: This calculator totally looks like a Casio, did you try the self test with shift+on+7 then 9 (or 8) ?
No, I were not hit by the calculator collecting bug.
From what I have read, Canon does look like a Casio ...
User(s) browsing this thread: 1 Guest(s) | {"url":"https://hpmuseum.org/forum/thread-2521-post-90410.html","timestamp":"2024-11-05T08:57:39Z","content_type":"application/xhtml+xml","content_length":"62831","record_id":"<urn:uuid:54c24de0-2e97-459d-aa9a-1b896e15f070>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00481.warc.gz"} |
This Is Why Data Analysis Is Concerned With Data ReductionThis Is Why Data Analysis is Concerned with Data Reduction | Analytics for Decisions
This Is Why Data Analysis is Concerned with Data Reduction
The world is advancing day by day. The amount of data available is more than ever. Conventional data analysis methods are dying, and it’s high time people use more efficient data analysis techniques
than ever. Since most businesses nowadays are becoming data-driven, analyzing the data at hand as fast as possible is the key.
However, as we keep seeing massive surges in the amount and dimensions of data available, data analysis starts going slow, no matter how efficient techniques we employ. So, what do we do? The data
and its dimensions aren’t going to stop increasing anytime soon. How do we make data analysis smoother and faster? Well, the answer is data reduction!
This article will dive into the depths of how data analysis is concerned with data reduction and how the two concepts are tightly knit to one another. So, let’s start without any further ado!
What is Data Reduction?
When we collect lots of data from multiple sources intending to analyze and draw conclusions from it, we often end up with a huge volume of data at data warehouses, which is not only hard to manage
but also pretty exhaustive to process; this is where data reduction comes to the rescue.
Data reduction is the act of reducing the volume of the data available while keeping its integrity intact. Various techniques are employed to make sure the data is fully identical to the original one
but uses much lesser volume.
Further in the article, we will explore some data reduction techniques and see in more detail why high-dimensional data is problematic.
Why Data Analysis Is Concerned with Data Reduction
Data analysis involves digging deep into data to find the smallest trends and patterns there are to find. The process is quite exhaustive since every possibility needs to be fully explored to uncover
any detail that might be useful.
Data reduction helps reduce the volume of the data available while protecting its integrity. This makes the data easier to manage and analyze, making the data analysis techniques more efficient and
As a result of employing data reduction, analyzing data gets faster, and a lot of resources are saved since you’re analyzing the same data. Given that now it comes with a much smaller volume and
reduced dimensions.
Interesting Article to Check Next: The Importance of Data Cleaning in Analytics
Problems with High-Dimensional Data
So, other than being very high in volume, what exactly are the problems with high dimensional data that it becomes hard to manage for us? Let’s see.
Computational Cost
Firstly, having to bear substantial computational costs is one of the biggest drawbacks of having high-dimensional data. While the information it packs is large-scale and might be very useful in some
scenarios, this computational overhead is pretty hard to ignore.
For example, suppose you’re a small company and decide to set up a data warehouse and later perform analysis to extract some actionable insights about your customers. In that case, it’ll get quite
hard to analyze all the available data if you don’t use data reduction before the analysis pipeline.
High Correlation
Another major issue of high-dimensional data is the high amount of correlation between variables it comes with. The data is not much randomly distributed, and the introduction of correlation further
adds fuel to the fire.
This kills the primary essence of data analysis, and the whole aim is jeopardized. There are often artificial correlations between data that can be easily wiped away using good data reduction
Overfitting Issues
Suffering from overfitting issues is often amongst the biggest worries of ML Engineers. Not only does it make problems in training the models and achieving good accuracies, but the process to get rid
of the overfitting is also lengthy.
So, it’s always best to avoid as much overfitting as possible in the first place, so you will need minimal effort to get rid of it in later stages. Since overfitting takes a huge toll on
performances, we certainly can’t avoid taking it into account, and dimensionality reduction helps us in this matter.
Hard to Visualize
Think about how easy it is to visualize 2D data and the added difficulty you experience while visualizing when the data jumps to 3D. It gets confusing right after 3d, and it’s only easy to visualize
4d data for people who have a solid mathematical background. Now, can you even imagine how it would be like to visualize 10D data? Well, not so much!
So, as the number of dimensions increases, it gets more and more hard to intuitively visualize a certain data set. It might be possible to do it with the help of some libraries, but it’s certainly
very hard for a human brain to do so.
Hence, data reduction reduces the number of dimensions of the dataset while retaining its properties and eventually helps us make better sense of it.
The next point of discussion is what are the benefits of data reduction. Before digging into that, let me add below some of the top related and interesting articles that can add to what you’re
learning from this one. If any of the titles picks your interest, please click and open in a new tab, so you can check them out later. Enjoy!
From small and medium-sized businesses to Fortune 500 conglomerates, the success of a modern business is now increasingly tied to how the company implements its data infrastructure and data-based
decision-making. According
Any form of the systematic decision-making process is better enhanced with data. But making sense of big data or even small data analysis when venturing into a decision-making process might
Data is important in decision making process, and that is the new golden rule in the business world. Businesses are always trying to find the balance of cutting costs while
The Benefits of Data Reduction
Let’s jump on to some of the biggest benefits we get to enjoy from data reduction techniques before diving into data analysis.
Lesser Space Required
Since the data reduction algorithms help transform the data into its equivalent representation with lesser volume, the new data takes up much lesser space than the old one. This might not be much on
a small scale, but for big companies, this means saving hundreds of Gigabytes!
Cutting Down on Operational Costs
As I mentioned in the previous point, data reduction helps save a lot of unnecessary space being used by data warehouses, resulting in huge cost savings on the enterprise level. However, this isn’t
all. Data reduction also helps save costs in processing power since the reduced amount of data needs lesser power to process.
Faster Data Analysis
Data analysis is a detailed process that increases directly as we increase the amount of data available. So, once data reduction techniques are applied, and there are lesser amounts of data to deal
with, data analysis techniques get significantly faster, and companies become more productive in this regard.
4 Useful Data Reduction Techniques
Now that we know why data reduction is so valuable and the problems that arise if we ignore it, let’s briefly go through some of the most used data reduction techniques.
Dimensionality Reduction
Dimensionality reduction is one of the most used data reduction techniques that’s used to deal with high-dimensional data. It picks up the redundant attributes and figures out the essential
attributes in a dataset as well. Once this is done, the redundant attributes that did not affect the dataset are removed, and the important attributes are combined into a smaller number of
Data Cube Aggregation
Data cube aggregation is a way to aggregate related data in the form of a cube. The concept of a simple cube is followed and just like a cube has multiple dimensions, the data is arranged similarly.
Consequently, aggregate functions can be applied to the cubes. Instead of individually applying functions on the attributes, you can now apply them all at once, saving considerable time and
computational power.
Numerosity Reduction
Numerosity reduction is the process of finding ways to express the same data in alternative forms, which are smaller and take up lesser space. The original data is replaced with newly made-up data,
but it’s just a different form of representation, and the data is still the same.
This is an excellent technique since it doesn’t involve any data loss, and everything is being retained, just in a more precise manner. However, the new data is sometimes ‘guessed’ and might not
entirely have the same properties as the original data.
Data Compression
Data compression involves using different techniques to encode the data and replace the original data with the new encoded one. The encoded form takes up much lesser space which makes this technique
very popular.
There are two major types of compression techniques – lossy and lossless. While the former might lose some data when converted back to the original representation from the encoded form, the latter
Wrap Up
Data analysis is becoming a vital part of any company nowadays because of its role in identifying new market trends, customer patterns, and so on. However, with a rapid increase in data availability
nowadays, it gets quite exhaustive and unmanageable to run analysis on such huge amounts of data.
The answer to this problem is data reduction. Not only does data reduction help you save a lot of storage space since it reduces the volume of data while preserving its properties, but it also helps
to lower the computational costs since now you have to deal with a much smaller volume of data for data analysis.
So, always make sure you don’t forget data reduction techniques before diving into data analysis. We have discussed some common data reduction techniques as well in the article that you can easily
use to get started with. | {"url":"https://www.analyticsfordecisions.com/data-reduction/?ezlink=true","timestamp":"2024-11-04T02:31:47Z","content_type":"text/html","content_length":"95978","record_id":"<urn:uuid:f788059e-87e7-47a5-aa5e-a6127eed4463>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00857.warc.gz"} |
Re: st: RE: SPSS to Stata - Variable with 14 digits not transformed corr
Notice: On April 23, 2014, Statalist moved from an email list to a forum, based at statalist.org.
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
Re: st: RE: SPSS to Stata - Variable with 14 digits not transformed correctly
From Sergiy Radyakin <[email protected]>
To [email protected]
Subject Re: st: RE: SPSS to Stata - Variable with 14 digits not transformed correctly
Date Fri, 2 Mar 2012 15:49:42 -0500
Dear Julian, Nick,
being the author of the -usespss- please let me comment on what it is doing.
1) Stata can work with 14 digit IDs stored as doubles:
set obs 2
generate double id = 12345678901234
replace id = 12345678901235 in 2
format id %21x
assert id[1]==id[2]
2) Some hints on working with long IDs are here:
3) SPSS format does not provide the float/double storage types.
Numeric values are stored differently from Stata (2 storage types, one
of them parametrized), but we can be confident that in your case of
14-digit numbers the equivalent of the double type is used (equivalent
up-to bytes order and some other details related to missings). In the
process of conversion -usespss- will "decompress" the data and try to
determine it's type. E.g. if it determines that the variable contains
only zeroes and ones, it will use byte as Stata's storage type for
this variable. -usespss- never rounds or truncates numeric values.
4) Given #3 is implemented correctly (things happen you know), any
numeric value stored in the SPSS should turn out as exactly the same
numeric value in Stata. (this is not true for strings). The only
exception that comes to my mind now is the extended missing values. In
SPSS any value like 3 or -99999997 can be assigned the meaning of
missing. In Stata this is not possible, and extended missing values
have firmly fixed values dependent on the storage type. Exact values
can be seen here: http://www.stata.com/help.cgi?dta. -usespss-
replaces original values (like the 3 and -99999997 above) with Stata's
.a, .b, .c. Date/time variables are converted as numeric values. The
value will be the same, but it will look strange in Stata. Bill Gould
had an excellent entry in his blog about why this happens and what to
do with it:
5) I allow for a remote possibility that the data file contains more
data than SPSS itself is showing to you, hence the records' IDs might
be unique in SPSS but not unique in the file in general. To see if
this might be an explanation - I will have to see the file.
6) If the file can't be shared, I would suggest you list out just the
ID's (about a dozen will do) without any further information and email
them as plain text to me. This requires SPSS of course.
7) If you have SPSS - then of course you can turn the ID variable into
string and proceed with the conversion, then destring it in Stata.
8) If anyone having SPSS can create an example dataset which exhibits
the same problem, please email me a .sav file along with the content
in plaintext (or Stata) and comments on how the example was prepared
(versions etc).
9) Stata's "float" type is not possible output of -usespss-. Let me
know if you see this type after conversion.
Please let me know if I can be of any further assistance. If you
decide to send the data, please zip it and mail to
sradyakin(%at%)worldbank.org , replace (%at%) with @.
Sergiy Radyakin, Economist,
Research Department (DECRG)
The World Bank
On Fri, Mar 2, 2012 at 6:54 AM, Nick Cox <[email protected]> wrote:
> In this context values like 1.001e+13 are to be thought of as, strictly, sets of values which are all displayed in the same way with a particular format. Crucially, changing the display format does nothing to change what is stored; by definition, it only changes how that is displayed.
> 14-digit identifiers can only be held accurately in Stata as string variables or -double- variables. If your identifier variable is -float- instead, then you will have lost precision in importing to Stata and the only way to regain that precision is to read the data in again. Recasting from -float- to -double- does nothing useful as the extra details have been lost already.
> -usespss- is user-written (SSC). Its author is intermittently active on Statalist. I've never used it but I see no way in its help to change how particular variables are imported. I guess that you need some other solution. As I don't use SPSS or SPSS files at all I can only guess that you need to look at export options in SPSS and import options in Stata and find a match. Others on this list who do use SPSS should be able to add better advice.
> In short, this problem as you describe it cannot be fixed in Stata. You must import again.
> Nick
> [email protected]
> Julian Emmler
> I'm new to this forum so I don't know yet the most accurate way to post my
> question but I hope it will be understandable and I would be greatful for
> every question for clarification. My Problem with Stata is concerned with
> transforming household data for the South African labour market which is
> only available in SPSS format to Stata. I did this with several datasets
> also for the South African labour market and used the "usespss" command in
> Stata which worked just fine. However with the last dataset I encountered a
> problem:
> In the dataset, to identify a household, a 14 digit number called the
> Unique Household Identifier is used. However, if I transform the data
> from SPSS to Stata, the values of the Household identifier are not it's
> real values any more but are shortened e.g. to 1.001e+13. Thus the Unique
> household identifier is not unique anymore. I try several things, e.g.
> transformin the vaiable to a double variable and increasing the number of
> digits displayed. This helps in that regard, that the number is now
> displayed correctly in the data browser, however the value didn't change.
> After searching the internet, I've come to the conclusion that this probelm
> has something to do with the length of the variable, i.e. that 14 digits is
> too long to be handled by Stata. Another indicator for this is that with
> earlier datasets I had no problem because the Unique Household Identifier
> was 12 digits long. I wanted to ask now if you know any way to
> transfer the SPSS data to Stata correctly or a way to manipulate the
> data afterwards so i attains its true values.
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"https://www.stata.com/statalist/archive/2012-03/msg00115.html","timestamp":"2024-11-14T15:33:12Z","content_type":"text/html","content_length":"17082","record_id":"<urn:uuid:542bb414-1d71-4c1a-934c-bd3e560cadc3>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00048.warc.gz"} |
Math worksheets decimal comparing
math worksheets decimal comparing Related topics: radicals solving
matrix introduction and operations
ti-82 stats changing the base on a logarithm
solutions to math 1155 exam 2 solutions
math worksheets for finding least common denominator with answer sheeet
first degree equations and inequalities
answers to math books
two step equations math worksheets
solve matrices online
year 7 maths optional tests download
algebra math book
writing radical expressions in simplest form
simple way to learn algebra free
Author Message
kxesamtxax Posted: Friday 11th of Dec 13:25
Hey dudes, I have just completed one week of my college, and am getting a bit worried about my math worksheets decimal comparing course work. I just don’t seem to grasp the topics.
How can one expect me to do my homework then? Please guide me.
Back to top
nxu Posted: Sunday 13th of Dec 09:18
Your story sounds familiar to me. Although I was great in math for several years, when I started Algebra 1 there were a lot of algebra topics that seemed so complicated . I remember I
got a very bad mark when I took the test on math worksheets decimal comparing. Now I don't have this issue anymore, I can solve anything quite easily , even graphing lines and
parallel lines. I was lucky that I didn't spend my money on a tutor, because I heard of Algebrator from a student . I have been using it since then whenever I stumbled upon something
From: Siberia,
Back to top
Bet Posted: Monday 14th of Dec 16:44
I must agree that Algebrator is a cool thing and the best software of this kind you can get . I was so surprised when after weeks of frustration I simply typed in monomials and that
was the end of my problems with algebra . It's also great that you can use the software for any level: I have been using it for several years now, I used it in Pre Algebra and in
Intermediate algebra too ! Just try it and see it for yourself!
From: kµlt øƒ
Back to top
Xebx3r Posted: Tuesday 15th of Dec 20:38
Sounds exactly like what I want. How can I get hold of it?
From: the wired
Back to top
Dnexiam Posted: Thursday 17th of Dec 13:30
Thanks pals for all your replies . I have got the Algebrator from https://softmath.com/algebra-policy.html. Just got it set and started using it. Its terrific . The exercise questions
can test the real expertise that we possess on College Algebra. I am really grateful to you all!
From: City 17
Back to top | {"url":"https://softmath.com/algebra-software-3/math-worksheets-decimal.html","timestamp":"2024-11-10T19:15:43Z","content_type":"text/html","content_length":"41364","record_id":"<urn:uuid:89db9e75-86a2-4d28-9306-fc796f3ca770>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00706.warc.gz"} |
Calculating Shanten in Mahjong
Move aside, poker! While the probabilities of various poker hands are well understood and tabulated, the Chinese game of chance Mahjong [1] enjoys a far more intricate structure of expected values
and probabilities. [2] This is largely due in part to the much larger variety of tiles available (136 tiles, as opposed to the standard playing card deck size of 52), as well as the turn-by-turn game
play, which means there is quite a lot of strategy involved with what is ostensibly a game of chance. In fact, the subject is so intricate, I’ve decided to write my PhD thesis on it. This blog post
is a condensed version of one chapter of my thesis, considering the calculation of shanten, which we will define below. I’ll be using Japanese terms, since my favorite variant of mahjong is Riichi
Mahjong; you can consult the Wikipedia article on the subject if you need to translate.
Calculating Shanten
The basic gameplay of Mahjong involves drawing a tile into a hand of thirteen tiles, and then discarding another tile. The goal is to form a hand of fourteen tiles (that is, after drawing, but before
discarding a tile) which is a winning configuration. There are a number of different winning configurations, but most winning configurations share a similar pattern: the fourteen tiles must be
grouped into four triples and a single pair. Triples are either three of the same tile, or three tiles in a sequence (there are three “suits” which can be used to form sequences); the pair is two of
the same tiles. Here is an example:
Represented numerically, this hand consists of the triples and pairs 123 55 234 789 456.
One interesting quantity that is useful to calculate given a mahjong hand is the shanten number, that is, the number of tiles away from winning you are. This can be used to give you the most crude
heuristic of how to play: discard tiles that get you closer to tenpai. The most widely known shanten calculator is this one on Tenhou’s website [3]; unfortunately, the source code for this calculator
is not available. There is another StackOverflow question on the subject, but the “best” answer offers only a heuristic approach with no proof of correctness! Can we do better?
Naïvely, the shanten number is a breadth first search on the permutations of a hand. When a winning hand is found, the algorithm terminates and indicates the depth the search had gotten to. Such an
algorithm is obviously correct; unfortunately, with 136 tiles, one would have to traverse $((136-13)\times 14)^n$ hands (choices of new tiles times choices of discard) while searching for a winning
hand that is n-shanten away. If you are four tiles away, you will have to traverse over six trillion hands. We can reduce this number by avoiding redundant work if we memoize the shanten associated
with hands: however, the total number of possible hands is roughly $136 \choose 13$, or 59 bits. Though we can fit (via a combinatorial number system) a hand into a 64-bit integer, the resulting
table is still far too large to hope to fit in memory.
The trick is to observe that shanten calculation for each of the suits is symmetric; thus, we can dynamic program over a much smaller space of the tiles 1 through 9 for some generic suit, and then
reuse these results when assembling the final calculation. $9 \times 4 \choose 13$ is still rather large, so we can take advantage of the fact that because there are four copies of each tile, an
equivalent representation is a 9-vector of the numbers zero to four, with the constraint that the sum of these numbers is 13. Even without the constraint, the count $5^9$ is only two million, which
is quite tractable. At a byte per entry, that’s 2MB of memory; less than your browser is using to view this webpage. (In fact, we want the constraint to actually be that the sum is less than or equal
to 13, since not all hands are single-suited, so the number of tiles in a hand is less.
The breadth-first search for solving a single suit proceeds as follows:
1. Initialize a table A indexed by tile configuration (a 9-vector of 0..4).
2. Initialize a todo queue Q of tile configurations.
3. Initialize all winning configurations in table A with shanten zero (this can be done by enumeration), recording these configurations in Q.
4. While the todo queue Q is not empty, pop the front element, mark the shanten of all adjacent uninitialized nodes as one greater than that node, and push those nodes onto the todo queue.
With this information in hand, we can assemble the overall shanten of a hand. It suffices to try every distribution of triples and the pairs over the four types of tiles (also including null tiles),
consulting the shanten of the requested shape, and return the minimum of all these configurations. There are $4 \times {4 + 4 - 1 \choose 4}$ (by stars and bars) combinations, for a total of 140
configurations. Computing the shanten of each configuration is a constant time operation into the lookup table generated by the per-suit calculation. A true shanten calculator must also accomodate
the rare other hands which do not follow this configuration, but these winning configurations are usually highly constrained, and quite easily to (separately) compute the shanten of.
With a shanten calculator, there are a number of other quantities which can be calculated. Uke-ire refers to the number of possible draws which can reduce the shanten of your hand: one strives for
high uke-ire because it means that probability that you will draw a tile which moves your hand closer to winning. Given a hand, it's very easy to calculate its uke-ire: just look at all adjacent
hands and count the number of hands which have lower shanten.
Further extensions
Suppose that you are trying to design an AI which can play Mahjong. Would the above shanten calculator provide a good evaluation metric for your hand? Not really: it has a major drawback, in that it
does not consider the fact that some tiles are simply unavailable (they were discarded). For example, if all four “nine stick” tiles are visible on the table, then no hand configuration containing a
nine stick is actually reachable. Adjusting for this situation is actually quite difficult, for two reasons: first, we can no longer precompute a shanten table, since we need to adjust at runtime
what the reachability metric is; second, the various suits are no longer symmetric, so we have to do three times as much work. (We can avoid an exponential blowup, however, since there is no
inter-suit interaction.)
Another downside of the shanten and uke-ire metrics is that they are not direct measures of “tile efficiency”: that is, they do not directly dictate a strategy for discards which minimizes the
expected time before you get a winning hand. Consider, for example, a situation where you have the tiles 233, and only need to make another triple in order to win. You have two possible discards: you
can discard a 2 or a 3. In both cases, your shanten is zero, but discarding a 2, you can only win by drawing a 3, whereas discarding a 3, you can win by drawing a 1 or a 4. Maximizing efficiency
requires considering the lifetime ure-kire of your hands.
Even then, perfect tile efficiency is not enough to see victory: every winning hand is associated with a point-score, and so in many cases it may make sense to go for a lower-probability hand that
has higher expected value. Our decomposition method completely falls apart here, as while the space of winning configurations can be partitioned, scoring has nonlocal effects, so the entire hand has
to be considered as a whole. In such cases, one might try for a Monte Carlo approach, since the probability space is too difficult to directly characterize. However, in the Japanese Mahjong scoring
system, there is yet another difficulty with this approach: the scoring system is exponential. Thus, we are in a situation where the majority of samples will be low scoring, but an exponentially few
number of samples have exponential payoff. In such cases, it’s difficult to say if random sampling will actually give a good result, since it is likely to miscalculate the payoff, unless
exponentially many samples are taken. (On the other hand, because these hands are so rare, an AI might do considerably well simply ignoring them.)
To summarize, Mahjong is a fascinating game, whose large state space makes it difficult to accurately characterize the probabilities involved. In my thesis, I attempt to tackle some of these
questions; please check it out if you are interested in more.
[1] No, I am not talking about the travesty that is mahjong solitaire.
[2] To be clear, I am not saying that poker strategy is simple—betting strategy is probably one of the most interesting parts of the game—I am simply saying that the basic game is rather simple, from
a probability perspective.
[3] Tenhou is a popular Japanese online mahjong client. The input format for the Tenhou calculator is 123m123p123s123z, where numbers before m indicate man tiles, p pin tiles, s sou tiles, and z
honors (in order, they are: east, south, west, north, white, green, red). Each entry indicates which tile you can discard to move closer to tenpai; the next list is of ure-kire (and the number of
tiles which move the hand further). | {"url":"http://blog.ezyang.com/2014/04/calculating-shanten-in-mahjong/","timestamp":"2024-11-07T00:43:50Z","content_type":"text/html","content_length":"38069","record_id":"<urn:uuid:c787bf64-2e63-4b73-979e-aaacd5dadc89>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00545.warc.gz"} |
SYDOWIA - An International Journal of Mycology
Fungal diversity associated with Bemisia tabaci (Hemiptera: Aleyrodidae) on cucumber and comparative effectiveness of bioassay methods in identifying the most virulent entomopathogenic fungi
Abdulnabi Abbdul Ameer Matrood, Abdelhak Rhouma, Lobna Hajji-Hedfi & Mohammad Imad Khrieba
Sydowia 75: 269-282
Published online on February 7th, 2023
Bemisia tabaci is a serious pest of cucumber, in Iraq, that reduces the income of farmers by high losses in yield. This study aimed to identify entomopathogenic fungi from whitefly cadavers and to
evaluate their relative frequency and various structural attributes. The suitability of two bioassay methods and the virulence of two entomopathogenic fungal species were evaluated for management of
B. tabaci under greenhouse condition. Out of the 16 fungal species isolated from the whitefly cadavers, only two species were confirmed microscopically as known entomopathogenic species: Mucor sp.
and Purpureocillium lilacinum with a relative frequency of 8.65 and 5.82 %, respectively. Results of the principle component analysis indicated that the first two PCs explained 99.30 %. Three factors
had a significant positive correlation with relative frequency of the fungal species which are species diversity (r= 0.983), Simpson’s concentration of dominance (r= 0.951) and equitability of
evenness (r= 0.996). The greatest mortality effect on B. tabaci nymphs and adults due to P. lilacinum and Mucor sp. was recorded on the 7thday after inoculation, with an average mortality more than
60 % (with concentration of 106 conidia/ml). P. lilacinum and Mucor sp. were significantly twice as virulent to nymphs than to adults. However, no significant differences were observed between
mortality rates of the two methods. To control B. tabaci nymphs and adults in the field within IPM strategies, we recommend more trials in order to analyze the real efficacy of P. lilacinum and Mucor
sp. under field conditions.
Keywords: Bemisia tabaci, Bioassay methods, Cucumis sativus, Entomopathogenic fungi, fungal species diversity. | {"url":"https://www.sydowia.at/syd75/T21-Matrood.htm","timestamp":"2024-11-07T09:21:15Z","content_type":"text/html","content_length":"7203","record_id":"<urn:uuid:9bb80fbe-ff26-4e8e-be0a-6713683fd050>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00619.warc.gz"} |
Derandomization, Hashing and Expanders
Regarding complexity of computation, randomness is a signicant resource beside time and space.
Particularly from a theoretical viewpoint, it is a fundamental question whether availability of ran-
dom numbers gives any additional power. Most of randomized algorithms are analyzed under the
assumption that independent and unbiased random bits are accessible. However, truly random
bits are scarce in reality. In practice, pseudorandom generators are used in place of random
numbers; usually, even the seed of the generator does not come from a source of true random-
ness. While things mostly work well in practice, there are occasional problems with use of weak
pseudorandom generators. Further, randomized algorithms are not suited for applications where
reliability is a key concern.
Derandomization is the process of minimizing the use of random bits, either to small amounts
or removing them altogether. We may identify two lines of work in this direction. There has
been a lot of work in designing general tools for simulating randomness and making deterministic
versions of randomized algorithms, with some loss in time and space performance. These methods
are not tied to particular algorithms, but work on large classes of problems. The central question
in this area of computational complexity is \P=BPP?".
Instead of derandomizing whole complexity classes, one may work on derandomizing concrete
problems. This approach trades generality for possibility of having much better performance
bounds. There are a few common techniques for derandomizing concrete problems, but often one
needs to specically design a new method that is \friendlier" to deterministic computation. This
kind of solutions prevails in this thesis.
A central part of the thesis are algorithms for deterministic selection of hash functions that
have a \nicely spread" image of a given set. The main application is design of ecient dictionary
data structures. A dictionary stores a set of keys from some universe, and allows the user to
search through the set, looking for any value from the universe. Additional information may be
associated with each key and retrieved on successful lookup. In a static dictionary the stored
set remains xed after initialization, while in a dynamic dictionary it may change over time.
Our static dictionaries attain worst-case performance that is very close the expected performance
of best randomized dictionaries. In the dynamic case the gap is larger; it is a signicant open
question to establish if a gap between deterministic and randomized dynamic dictionaries is
We also have a new analysis of the classical linear probing hash tables, showing that it works
well with simple and eciently computable hash functions. Here we have a randomized structure
in which the randomness requirements are cut down to a reasonable level. Traditionally, linear
probing was analyzed under the unrealistic uniform hashing assumption that the hash function
employed behaves like a truly random function. This was later improved to explicit, but cum-
bersome and inecient families of functions. Our analysis shows that practically usable hash
functions suce, but the simplest kinds of functions do not.
Apart from dictionaries, we look at the problem of sparse approximations of vectors, which
has applications in dierent areas such as data stream computation and compressed sensing. We
present a method that achieves close to optimal performance on virtually all attributes. It is
deterministic in the sense that a single measurement matrix works for all inputs.
One of our dictionary results and the result on sparse recovery of vectors share an impor-
tant tool, although the problems are unrelated. The shared tool is a type of expander graphs.
We employ bipartite expander graphs, with unbalanced sides. For some algorithms, expander
graphs capture all the required \random-like" properties. In such cases they can replace use of
randomness, while maintaining about the same performance of algorithms.
The problems that we study require and allow fast solutions. The algorithms involved have
linear or near-linear running times. Even sublogarithmic factors in performance bounds are
meaningful. With such high demands, one has to look for specic deterministic solutions that
are ecient for particular problems; the general derandomization tools would be of no use.
Dyk ned i forskningsemnerne om 'Derandomization, Hashing and Expanders'. Sammen danner de et unikt fingeraftryk. | {"url":"https://pure.itu.dk/da/publications/derandomization-hashing-and-expanders","timestamp":"2024-11-11T07:59:11Z","content_type":"text/html","content_length":"57555","record_id":"<urn:uuid:739d969f-b806-4174-a92a-c7b17c921f13>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00248.warc.gz"} |
Fibonacci Sequence Explained | Cratecode
Fibonacci Sequence Explained
Note: this page has been created with the use of AI. Please take caution, and note that the content of this page does not necessarily reflect the opinion of Cratecode.
One of the most famous sequences in mathematics, and sometimes even in popular culture, is the Fibonacci Sequence. This seemingly simple sequence of numbers has a myriad of fascinating properties and
applications in both mathematics and computing. Let's unravel the beauty of the Fibonacci Sequence and find out why it's so important.
What is the Fibonacci Sequence?
The Fibonacci Sequence is an infinite series of numbers, in which each number is the sum of the two preceding ones, starting from 0 and 1. In simpler terms, the sequence starts with 0 and 1, and each
subsequent number is obtained by adding the previous two numbers. Here's how the sequence unfolds:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
As you can see, 0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3, and so on.
Why is the Fibonacci Sequence Important?
The Fibonacci Sequence has several fascinating properties, and it pops up in various fields, including mathematics, science, and nature. Some of these noteworthy occurrences are:
1. Golden Ratio: The ratio of consecutive Fibonacci numbers converges to the Golden Ratio (approximately 1.618), which is a number that has been revered for its aesthetic beauty in art,
architecture, and nature.
2. Spirals in Nature: The Fibonacci Sequence is found in the arrangement of sunflower seeds, pinecone scales, and other natural phenomena. These spirals are efficient ways to pack seeds or scales,
allowing the organism to maximize its growth and reproduction.
3. Fractals: The Fibonacci Sequence is used in the construction of some fractals like the Mandelbrot Set.
4. Algorithms and Computing: The Fibonacci Sequence is used in computer science as a basis for various algorithms, including search algorithms, optimization problems, and even as a classic example
for recursion and dynamic programming.
How to Generate the Fibonacci Sequence
There are several ways to generate the Fibonacci Sequence in programming, and we'll look at three popular methods below: recursive, iterative, and matrix exponentiation.
Recursive Method
A recursive function is a function that calls itself to solve a problem. The Fibonacci Sequence can be generated using a simple recursive function:
def fibonacci_recursive(n):
if n <= 1:
return n
return fibonacci_recursive(n - 1) + fibonacci_recursive(n - 2)
This method works, but it is highly inefficient due to the overlapping and repetitive nature of the calculations. Its time complexity is O(2^n), making it impractical for large values of n.
Iterative Method
An alternative to the recursive method is the iterative method, which uses a loop to calculate the Fibonacci numbers. This method is more efficient, with a time complexity of O(n):
def fibonacci_iterative(n):
if n <= 1:
return n
a, b = 0, 1
for _ in range(n - 1):
a, b = b, a + b
return b
Matrix Exponentiation
Matrix exponentiation is a more advanced method for generating the Fibonacci Sequence, which takes advantage of linear algebra and has a time complexity of O(log n). The idea is to represent the
Fibonacci numbers as the product of a matrix and its powers. This method is outside the scope of this article, but it's worth exploring for those interested in advanced algorithms.
In conclusion, the Fibonacci Sequence is a beautiful blend of mathematics and nature, with a multitude of fascinating properties and applications. Its elegance and simplicity make it a great example
for teaching programming concepts such as recursion and dynamic programming. So the next time you come across a spiral in nature, remember the humble Fibonacci Sequence and its amazing connection to
our world.
Hey there! Want to learn more? Cratecode is an online learning platform that lets you forge your own path. Click here to check out a lesson: Recursion Intro (psst, it's free!).
What is the Fibonacci Sequence?
The Fibonacci Sequence is a series of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1. It is named after Leonardo Fibonacci, an Italian mathematician
who introduced the sequence to Western mathematics. The sequence goes like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.
How is the Fibonacci Sequence used in mathematics and computing?
The Fibonacci Sequence has various applications in mathematics, computing, and even in nature. In mathematics, it is used to study patterns, sequences, and recursion. In computing, it is often used
in algorithms, data structures, and problem-solving techniques. For instance, it can be found in the dynamic programming approach to optimize problems by breaking them down into smaller subproblems.
It also appears in nature, with examples like the arrangement of leaves on a plant stem and the spiral pattern of sunflower seeds.
How can I generate the Fibonacci Sequence using a programming language?
Here's an example of generating the Fibonacci Sequence using Python:
def fibonacci(n):
if n == 0:
return 0
elif n == 1:
return 1
return fibonacci(n-1) + fibonacci(n-2)
for i in range(10):
This code defines a recursive function called fibonacci that takes an integer n as input and returns the nth number in the Fibonacci Sequence. It then prints the first 10 numbers in the sequence
using a for loop.
What is the connection between the Fibonacci Sequence and the Golden Ratio?
The Golden Ratio, often denoted as φ (phi), is a mathematical constant approximately equal to 1.61803398875. It is found by dividing a line into two parts such that the ratio of the whole line to the
longer part is equal to the ratio of the longer part to the smaller part. The Fibonacci Sequence is closely related to the Golden Ratio, as the ratio between consecutive Fibonacci numbers tends to
the Golden Ratio as the numbers get larger. This relationship can be expressed as follows: lim (n → ∞) (F(n+1) / F(n)) = φ Where F(n) represents the nth Fibonacci number and φ is the Golden Ratio.
Are there any other sequences similar to the Fibonacci Sequence?
Yes, there are many other sequences similar to the Fibonacci Sequence, known as Lucas Sequences. These sequences follow the same recurrence relation as the Fibonacci Sequence, but they have different
initial values. The most well-known is the Lucas Sequence, which starts with 2 and 1 instead of 0 and 1: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, and so on. Other Lucas Sequences can be formed by choosing
different starting values, leading to a wide variety of related sequences with interesting properties and applications. | {"url":"https://cratecode.com/info/fibonacci-sequence-explained","timestamp":"2024-11-02T22:17:50Z","content_type":"text/html","content_length":"108611","record_id":"<urn:uuid:67f60956-b399-4aa1-b8f1-3371de8fad9e>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00726.warc.gz"} |
Unit 5
Unit 5 Family Materials
Numbers to 1,000
Numbers to 1,000
In this unit, students extend their understanding of the base-ten system to include numbers to 1,000.
Section A: The Value of Three Digits
In this section, the unit of a hundred is introduced. Students begin by looking at the large square base-ten block, and its corresponding base-ten drawing, to visualize 100, and to establish that 1
hundred equals 10 tens, which equals 100 ones.
After students develop an understanding of a hundred as a unit, students learn that the digits in three-digit numbers represent amounts of hundreds, tens, and ones. Students read and write
three-digit numbers in different forms, including using base-ten numerals, number names, and expanded form.
Students write expressions and equations based on the base-ten blocks and base-ten drawings that they see. They recognize that the value of the digits in a three-digit number is revealed when using
the fewest number of blocks to represent the number.
For example, the picture shows 2 hundreds, 11 tens, and 12 ones. However, students recognize that they will need to exchange 10 of the ones for a ten and 10 of the tens for a hundred to find the
value of their number. After doing so, they recognize that they have 3 hundreds, 2 tens, and 2 ones for a value of 322.
Section B: Compare and Order Numbers within 1000
In this section, students continue to deepen their understanding of numbers to 1,000 using place value understanding and the number line diagram. As students recall the structure of the number line
from the previous unit, they use this structure and place value understanding to locate, compare, and order numbers on the number line.
As students locate or estimate the location of three-digit numbers on number lines, they demonstrate an understanding of the number’s relative distance from zero, as well as the place value of the
digits. This understanding helps them to compare and order three-digit numbers. For example, to order numbers, students can first locate them on the number line. Then, the numbers will be in order
from least to greatest as students look from left to right on the number line.
In addition to using the number line to compare three-digit numbers, students also use familiar place value representations such as base-ten blocks and base-ten diagrams. Students compare and order
numbers and write the comparisons using the symbols, \(>\), \(<\), and \(=\).
Try it at home!
Near the end of the unit, ask your student to think about the number 593 and complete the following tasks:
• Write the number as a number name and in expanded form.
• Draw an amount of base-ten blocks that has the same value.
• Create a number line from 500 to 600 and place the number on a number line.
• Compare the number to 539 using either \(>\), \(<\), or \(=\).
Questions that may be helpful as they work:
• What pieces of information were helpful?
• Can you explain to me how you solved the problem?
• Could you have drawn a different amount of base-ten blocks? | {"url":"https://im-beta.kendallhunt.com/k5/families/grade-2/unit-5/family-materials.html","timestamp":"2024-11-03T06:20:58Z","content_type":"text/html","content_length":"84879","record_id":"<urn:uuid:7465c360-6155-487d-bf71-6317a592ff95>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00403.warc.gz"} |
Can someone assist me with my Python data structures homework if I need help implementing algorithms for data structures in smart grid applications? | Hire Someone To Do Python Assignment
Can someone assist me with my Python data structures homework if I need help implementing algorithms for data structures in smart grid applications?
Can someone assist me with my Python data structures homework if I need help implementing algorithms for data structures in smart grid applications? My lecturer told me that whenever he works in the
past, he thought that our data structures generally act the same. In other words, once you enter a basic data structure a computer can quickly scan the data using, per the example defined in that
blog. On this answer. A while ago, my professor said: “Let me quote today: “Data structures are an exceedingly common problem in data-driven analysis. Being able to query and analyze complex
statements gives an almost infinite number of sophisticated algorithms that will perform a perfectly fine job on your data. Nevertheless, only a few of these efficient software package have been
built.” Very interesting to read. I would not just be mentioning the code snippet; a couple of things are in there that I have not looked at, what in the heck is the problem with that one code. What
are any more of the issues if this data structure is designed as a unit of measurements. Does the data structure generate some sort of approximation problem; how does that fit into our issues in that
domain (data modeling)? By the way. For each code snippet, I have written a simple plug-and-play library called “dumb-grid” to assist you with learning how to analyze data structure and to explain
the problem. The link here is for reference. Seth : Re: Python Data Structures Yup, I remember the following example. For real, I don’t have the technology to use the python data structures. I will
give some ideas to get this. I have several data structures that I have posted about over the past few days and have quite a few to come to my attention. But when I think about data structures the
concept itself is neat. The concept describes how you set up data structures in your software, your computer interfaces with them. Like in “solutions” system, you would use the data structure at
booting because the design is such that you no matter what the “boot” process must be followed. According to that a very good reason to deploy your own data structure might be that you are often
asked to create new data structures to do the job of their initial deployment, where design is the first stage.
Do Assignments Online And Get Paid?
But how do you create new data structures? It is very simple to use, but it turns out that people can not quickly understand the initial design of your data structure. Maybe the concepts you used
will help you understand the concept for some of these examples. I also mention that we have not done any thinking before we started this talk on data-driven science research. We actually did not
even start our talk before 10 minutes before the main event of the conference. However, the call for the conference came two days earlier, and we had to start our talks on the issue of data-driven
science research. As far as we were concerned, theCan someone assist me with my Python data structures homework if I need help implementing algorithms for data structures in smart grid applications?
Quick question: I am currently being tasked with producing a smart grid with an array of x1 and x2 data as observations and then a new column, data_rows_a and row-of-size x1 and x2 of data_rows and
data_rows_b. I’m looking for a about his of implementing data structures where I am collecting simple column and row data in x1 and x2 from an array of data_headers of x so that only things for data
x1 and x2 in x1 and x2 will go into memory are modified. A: Assuming you don’t have an array of data_headers, they take values from the head, rows and column of your data_headers which you would
write into an array of keys. Each row of data_headers has a value assigned through a lookup function, that maps inputs to a column in the array and outputs a value equal to the difference between the
associated values. Thus it has both a value assigned to its corresponding row and a value assigned to the corresponding column of the array. This column in the array contains information on how the
data row and column site here set up; for each key in the array you would extract the value for this key, add it back, n=head_rows_and_column(data_headers,3); if n==data_headers.get(i)==column_count&
i==i==x1.get(2) where column_count is the number of rows of data_headers up to be measured. [N] you could also try n=data_headers[i][i-1:n]; if n==data_headers.get(i)==column_count&i==i==x1..len..m
and then check for the value for key xn but find out there’s no value for key xn To simplify the basic calculation, you could attempt to do the following: if element[i][j]==column_count&i==j==xnm and
iterate one of those two cases: case 1: if rcol == column_count & i==j==xnm: for i in [data_headers(i):row_columns(i):column_dims(i)] do i=row_columns(i)&row_columns(i) if xnm==column_count&i==j==
xnm: return n=row_columns(i)+s(xnm)-rcol(column_count)+s(i)+xnm end return rowsx(n)&columns(n) E.g.
Pay Someone To Do My Schoolwork
where xnm is one column column,Can someone assist me with my Python data structures homework if I need help implementing algorithms for data structures in smart grid applications? I need help finding
the correct libraries, classes, and libraries for my data structures in codeigniter. Hello, I have done several problems special info my codeigniter code and I don’t get exactly what is causing them.
You can find a few simple articles at my github page where I have added the code. I have all the classes, but they all seem to be missing exactly where I have provided them, the following class is
only missing the last one. import pandas as pd, random as nx, str as s1, names as names, colors as arrays, for i in range(0, len(names)) return [col in names[i]] I have also implemented the following
functions which seem to be working i guess. I made some very long files in my codeigniter worksheet, but I dont get what I need for my data models. // load the workbook and the tasks list is empty
import pandas as pd, nvk_libraries_asset_grid_tasklist from ‘nvk_libraries_asset_grid_tasklist’ // add class called *after* all the tasks list to screen, so that when user wants to sort tasklist.add
(pd.read_csv(‘users_col_tasks_tasks.csv’, ‘users_col_tasklist.csv’, names=names, rows_cols=1)).execute() for i in range(0, len(names)) as[x for x in names] ‘,’, ‘, ‘.join(names[i]+ids) ^^ for i in
range(0, len(names)) as[x for x in names] // adds the tasks list to list form tasklist.add(pd.read_csv(‘users_col_tasklist. | {"url":"https://domypython.com/can-someone-assist-me-with-my-python-data-structures-homework-if-i-need-help-implementing-algorithms-for-data-structures-in-smart-grid-applications","timestamp":"2024-11-14T14:21:11Z","content_type":"text/html","content_length":"65068","record_id":"<urn:uuid:25d2e07c-20c1-4d68-8e26-7f144eb2b12b>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00802.warc.gz"} |
May 2024 - Discrete Mathematics Group
Yongho Shin (신용호) gave a talk on an online randomized algorithm for edge-weighted online bipartite matching problem at the Discrete Math Seminar
On May 28, 2024, Yongho Shin (신용호) from Yonsei University gave a talk at the Discrete Math Seminar on an online randomized algorithm using three-way online correlated selection for an
edge-weighted online bipartite matching problem. His talk title was “Three-way online correlated selection.”
Vadim Lozin gave a talk on classifying monotone graph classes concerning the Hamiltonian cycle problem at the Discrete Math Seminar
On May 21, 2024, Vadim Lozin from the University of Warwick gave a talk at the Discrete Math Seminar on classifying monotone graph classes concerning the Hamiltonian cycle problem. The title of his
talk was “Graph problems and monotone classes“.
We are hiring! IBS Discrete Mathematics Group (DIMAG) Research Fellowship (Due: June 21, 2024)
The IBS Discrete Mathematics Group (DIMAG) in Daejeon, Korea invites applications for two research fellowship positions.
DIMAG is a research group that was established on December 1, 2018 at the Institute for Basic Science (IBS), led by Prof. Sang-il Oum. DIMAG is located at the headquarters of the Institute for Basic
Science (IBS) in Daejeon, South Korea, a city of 1.5 million people.
Website: https://dimag.ibs.re.kr/
Currently, DIMAG consists of researchers from various countries such as Korea, the USA, Germany, Canada, and Australia, and the work is done in English. DIMAG is co-located with the IBS Extremal
Combinatorics and Probability Group (ECOPRO).
Successful candidates for research fellowship positions will be new or recent Ph.D.’s with outstanding research potential in all fields of Discrete Mathematics with emphasis on Structural Graph
theory, Combinatorial Optimization, Matroid Theory, and Algorithms.
These appointments are for about two years, and the starting salary is no less than KRW 59,000,000. The appointment is one-time renewable up to 5 years in total contingent upon the outstanding
performance of the researcher. The expected appointment date is March 1, 2025, and it can be adjusted to earlier or later, but no later than June 1, 2025. This is purely a research position and will
have no teaching duties.
A complete application packet should include:
1. AMS standard cover sheet (preferred) or cover letter (PDF format)
2. Curriculum vitae including a list of publications and preprints (PDF format)
3. Research statement (PDF format)
4. At least 3 recommendation letters
For full consideration, applicants should email items 1, 2, 3, and 4 and arrange their recommendation letters emailed to dimag@ibs.re.kr by June 21, 2024, Anywhere on Earth (AoE).
Recommendation letters forwarded by an applicant will not be considered.
DIMAG encourages applications from individuals of diverse backgrounds.
For Korean citizens who have not yet completed their military duty: 전문연구요원 종사를 희망하는 경우에는 지원 의사와 병역 관계를 cover letter 및 이메일 등에 표시하여 제출 필요. 다만 현역입영대상자의
전문연구요원 신규 편입은 불가하며, 보충역 대상자 및 타 기관에서 전직해서 오는 경우에 한하여 지원 가능(전직 요건이 충족 되어야만 함)
Suggested E-mail subject from applicants: [DIMAG – name]
e.g., [DIMAG – PAUL ERDOS]
Niloufar Fuladi gave a talk on how to find a short canonical decomposition of a non-orientable surface given with a triangulation at the Discrete Math Seminar
On May 14, 2024, Niloufar Fuladi from the INRIA Center of Université de Lorraine gave a talk at the Discrete Math Seminar on how to find a short canonical decomposition of a non-orientable surface
given with a triangulation. The title of her talk was “Cross-cap drawings and signed reversal distance“. She has been visiting the IBS Discrete Mathematics Group since mid-April.
Tony Huynh gave a talk at the Discrete Math Seminar on the conjecture of Ahranoi on the existence of a short rainbow cycle
On May 7, 2024, Tony Huynh from Sapienza Università di Roma gave a talk on the conjecture of Aharoni on the existence of a short rainbow cycle at the Discrete Math Seminar. The title of his talk was
“Aharoni’s rainbow cycle conjecture holds up to an additive constant“.
Tony Huynh has been visiting the IBS Discrete Mathematics Group since April 15.
Maximilian Gorsky gave a talk at the Discrete Math Seminar on the Erdős-Pósa property for even directed cycles
On April 30, 2024, Maximilian Gorsky from the Technische Universität Berlin gave a talk on the quarter-integral and third-integral Erdős-Pósa property for even directed cycles at the Discrete Math
Seminar. The title of his talk was “Towards the half-integral Erdős-Pósa property for even dicycles“. | {"url":"https://dimag.ibs.re.kr/2024/05/","timestamp":"2024-11-14T19:51:02Z","content_type":"text/html","content_length":"152482","record_id":"<urn:uuid:2073496d-80f5-4a3a-9743-d0aa542863fd>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00792.warc.gz"} |
Constructing all Possible Leaves for Maximum Packings of the Complete Graph with Stars of Sizes Six and Seven
Area of Study or Work
Expected Graduation Date
CNS Atrium, Easel 18
Start Date
4-15-2023 10:30 AM
End Date
4-15-2023 11:45 AM
A 6-star is the complete bipartite graph K[1,6]. A packing of K[n] with 6-stars is a set of edge disjoint subgraphs of K[n], each of which is isomorphic to S[6]. The set of edges of K[n] which are
not used in the packing is called the leave and is denoted by L. The packing is called maximum if |L| is minimum with respect to all such packings. We show that every possible leave graph is
achievable as the leave of a maximum packing of K[n] with 6-stars and 7-stars.
Apr 15th, 10:30 AM Apr 15th, 11:45 AM
Constructing all Possible Leaves for Maximum Packings of the Complete Graph with Stars of Sizes Six and Seven
CNS Atrium, Easel 18
A 6-star is the complete bipartite graph K[1,6]. A packing of K[n] with 6-stars is a set of edge disjoint subgraphs of K[n], each of which is isomorphic to S[6]. The set of edges of K[n] which are
not used in the packing is called the leave and is denoted by L. The packing is called maximum if |L| is minimum with respect to all such packings. We show that every possible leave graph is
achievable as the leave of a maximum packing of K[n] with 6-stars and 7-stars. | {"url":"https://digitalcommons.iwu.edu/jwprc/2023/schedule/55/","timestamp":"2024-11-02T03:08:32Z","content_type":"text/html","content_length":"30074","record_id":"<urn:uuid:756fe860-4c63-44e9-b56e-3a52e5bbcbd5>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00550.warc.gz"} |
Computationally Efficient Problem Reformulations for Capacitated Lot Sizing Problem
Computationally Efficient Problem Reformulations for Capacitated Lot Sizing Problem ()
1. Introduction
Lot sizing problem aims to optimally utilize the available production resources while meeting the demand targets. It is classified as medium-term planning in production planning taxonomy. Lot sizing
problem formulation depends upon the layout and the operating constraints in the production system. In the manufacturing industry, we come across many types of production systems. These production
systems further give rise to different types of lot sizing problems (with different constraints and operating conditions) and their solution methodologies. Hence there is a rich literature on lot
sizing problem and their solution methods. In this article, we restrict our discussion to general dynamic multi-level capacitated lot sizing problem.
This problem was first proposed by Billington et al. [1] . It addresses the following scenario: a finite planning horizon is given and is divided into discrete time periods. There is a dynamic demand
for items which needs to be satisfied for each time period while honoring the production capacity constraints. Problem aims to develop a production plan over all time periods while minimizing the
total cost comprising of setup cost, inventory cost, and backordering cost.
A capacitated lot sizing problem is a well-known NP-hard problem. If the capacity constraints of the problem are relaxed, then the problem can be solved in polynomial time [2] . Many model
formulations have been proposed to develop an efficient numerical solution. These formulations differ with each other due to variables and constraints used. Formulations given in this article belongs
to inventory and lot-size (I & L) formulations category, which are among the most popular in the literature due to their computational efficiency. These formulations use production quantity and
inventory levels as the variables. Relaxation of each constraint in the standard I&L model affects the problem structure and hence its numerical complexity. Relaxation of capacity constraints
decomposes CLSP into single-level lot sizing problem popularly known as Wagner-Whitin problem [2] . Further, some popular extensions to the standard problem have been suggested in the literature, to
address certain practical issues. For example, Dillenberger et al. [3] have extended the problem to incorporate setup carryover. A setup cost and setup time are not incurred if the same item is being
produced in the next time period, carried forward by the previous period. Our formulation incorporates binary variable for setup to address these issues. Binary setup variable to address the
carryover of setup to the next period was earlier used by Hasse [4] , and Surie and Stadtler [5] .
We next discuss certain problem reformulations from the literature which are computationally efficient. CLSP can be formulated to assign each production quantity to a demand in a specific time period
while minimizing the production cost. Shortest route formulation was proposed by Eppen and Martin [6] for a single level case. It was later extended by Tempelemier and Helber [7] for multi-level
CLSP. Stadtler proposed an improvement in this formulation ( [8] [9] ), which decreases the number of non-negative coefficients. Rosling [10] introduced a formulation based on Plant location problem
analogy. This formulation was further extended by Maes et al. [11] for the capacitated case of a lot-sizing problem. Capacity constraints were included in the original formulation for this purpose.
Equivalence of Shortest route and SPL formulation in terms of objective function was shown by Denizel et al. [12] .
Apart from reformulation, additional inequalities can be added to the problem formulation to tighten the bound while reducing the search space. Important researches in this category are discussed
next. Barany et al. [13] included lot sizing and inventory variables for the single level uncapacitated lot-sizing problem. Additional valid constraints are included in the formulation to tighten the
convex bound of the uncapacitated lot-sizing problem. Pochet and Wolsey [14] ; and Clark and Armentano [15] extended the work of Barany [13] for the multi-level case. Miller et al. [16] have proposed
additional valid inequalities for the capacitated case of a lot-sizing problem. Further Surie and Stadtler [5] have proposed valid inequalities for multi-level capacitated lot sizing problem with set
up carry over. Setup carryover constraints are redefined to achieve a computational advantage in this formulation.
Research in this article is based on the appropriate reformulation of the standard capacitated lot sizing problem. We state the standard problem formulation and then derive three reformulations of
the problem by eliminating the backordering variable or/and adding two capacity constraints. Efficacies of these formulations in terms of reduced computational complexity are demonstrated through
numerical analysis of random problems on GAMS.
2. Research Methodology
As stated in the previous section, we intend to evaluate the improvement in computational efficiency of the model when the number of decision variables are decreased, or constraints are added to
tighten the bound of the solution space. Model A1 is the reference standard model, which is tinkered to develop model A2, A3, and A4 accordingly. In model A2 (proposed later), we have eliminated the
backordering variable; hence it is expected to be computationally efficient when compared with model A1. Similarly, we have added two extra constraints (Equation (27), Equation (28)) in our standard
model (A1) which is referred to as model A3. Further, we eliminate backordering variable while adding two constraints in model A1 and refer it to model A4. Hence model A4 is expected to perform best
among all. Efficacy of each model is evaluated by performing paired t-test of the computational time of random problem instances on A1, A2, A3, and A4. Branch and bound method is used in GAMS for
solving random problem optimally by these models. Finally it is concluded in section 6 that most computationally efficient formulation should be used solving capacitated lot sizing problem.
3. Problem Formulation/Reformulation
Table 1. Notations used in the model.
3.1. Model A1
$\text{Minimize}\text{\hspace{0.17em}}Z=\underset{i=1}{\overset{I}{\sum }}\underset{t=1}{\overset{T}{\sum }}\left[C{P}_{it}*X{P}_{it}+C{S}_{it}*Y{S}_{it}+CIN{V}_{it}*XIN{V}_{it}+CB{O}_{it}*XB{O}_{it}
Subject to:
$X{P}_{it}+XIN{V}_{i,t-1}+XB{O}_{it}={D}_{it}+XIN{V}_{it}+XB{O}_{i,t-1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I,t\in T$(2)
$\underset{i=1}{\overset{I}{\sum }}\left(P{T}_{it}*X{P}_{it}+S{T}_{i}*Y{S}_{it}\right)\le CAP{T}_{t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall t\
in T$(3)
$\underset{i=1}{\overset{I}{\sum }}\left(P{T}_{it}*{D}_{it}+S{T}_{i}*Y{S}_{it}\right)\le CAP{T}_{t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall t\in
$X{P}_{it}\le CA{P}_{it}Y{S}_{it}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I,t\in T$(5)
$\underset{t=1}{\overset{T}{\sum }}X{P}_{it}\ge \underset{t=1}{\overset{T}{\sum }}{D}_{it}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(6)
$XIN{V}_{i0}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(7)
$XIN{V}_{iT}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(8)
$XB{O}_{i0}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(9)
$XB{O}_{iT}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(10)
$Y{S}_{it}\in \left\{0,1\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I,t\in T$(11)
$XIN{V}_{it},X{P}_{it},XB{O}_{it}\ge 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I,t\in T$(12)
3.2. Model A2
$\begin{array}{l}\text{Minimize}\text{\hspace{0.17em}}Z=\underset{i=1}{\overset{I}{\sum }}\underset{t=1}{\overset{T}{\sum }}\left[C{P}_{it}*X{P}_{it}+C{S}_{it}*Y{S}_{it}+CIN{V}_{it}*XIN{V}_{it}\
right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace
{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\underset{{t}_{1}=1}{\overset{T}{\sum }}\underset{i=1}{\overset{I}{\sum }}CB{O}_{i{t}_{1}}*\left(\underset{t=1}{\overset{{t}_{1}}{\sum }}
{D}_{it}+XIN{V}_{i{t}_{1}}-\underset{t=1}{\overset{{t}_{1}}{\sum }}X{P}_{it}\right)\end{array}$(13)
Subject to:
$\underset{t=1}{\overset{{t}_{1}}{\sum }}{D}_{it}+XIN{V}_{i{t}_{1}}-\underset{t=1}{\overset{{t}_{1}}{\sum }}X{P}_{it}\ge 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\
hspace{0.17em}}\forall i\in I,{t}_{1}\in T$(14)
$\underset{i=1}{\overset{I}{\sum }}\left(P{T}_{it}*X{P}_{it}+S{T}_{i}*Y{S}_{it}\right)\le CAP{T}_{t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall t\
in T$(15)
$X{P}_{it}\le CA{P}_{it}Y{S}_{it}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I,t\in T$(16)
$\underset{t=1}{\overset{T}{\sum }}X{P}_{it}\ge \underset{t=1}{\overset{T}{\sum }}{D}_{it}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(17)
$XIN{V}_{i0}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(18)
$XIN{V}_{iT}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(19)
$Y{S}_{it}\in \left\{0,1\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I,t\in T$(20)
$XIN{V}_{it},X{P}_{it}\ge 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I,t\in T$(21)
3.3. Model A3
$\text{Minimize}\text{\hspace{0.17em}}Z=\underset{i=1}{\overset{I}{\sum }}\underset{t=1}{\overset{T}{\sum }}\left[C{P}_{it}*X{P}_{it}+C{S}_{it}*Y{S}_{it}+CIN{V}_{it}*XIN{V}_{it}+CB{O}_{it}*XB{O}_{it}
Subject to:
$X{P}_{it}+XIN{V}_{i,t-1}+XB{O}_{it}={D}_{it}+XIN{V}_{it}+XB{O}_{i,t-1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I,t\in T$(23)
$\underset{i=1}{\overset{I}{\sum }}\left(P{T}_{it}*X{P}_{it}+S{T}_{i}*Y{S}_{it}\right)\le CAP{T}_{t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall t\
in T$(24)
$\underset{i=1}{\overset{I}{\sum }}\left(P{T}_{it}*{D}_{it}+S{T}_{i}*Y{S}_{it}\right)\le CAP{T}_{t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall t\in
$X{P}_{it}\le CA{P}_{it}Y{S}_{it}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I,t\in T$(26)
$\underset{i=1}{\overset{I}{\sum }}\underset{t=1}{\overset{T}{\sum }}\left(P{T}_{it}*{D}_{it}+S{T}_{i}*Y{S}_{it}\right)\le \underset{t=1}{\overset{T}{\sum }}CAP{T}_{t}$(27)
$\underset{i=1}{\overset{I}{\sum }}\underset{t=1}{\overset{T}{\sum }}\left(P{T}_{it}*X{P}_{it}+S{T}_{i}*Y{S}_{it}\right)\le \underset{t=1}{\overset{T}{\sum }}CAP{T}_{t}$(28)
$\underset{t=1}{\overset{T}{\sum }}X{P}_{it}\ge \underset{t=1}{\overset{T}{\sum }}{D}_{it}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(29)
$XIN{V}_{i0}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(30)
$XIN{V}_{iT}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(31)
$XB{O}_{i0}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(32)
$XB{O}_{iT}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(33)
$Y{S}_{it}\in \left\{0,1\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I,t\in T$(34)
$XIN{V}_{it},X{P}_{it},XB{O}_{it}\ge 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I,t\in T$(35)
3.4. Model A4
$\begin{array}{l}\text{Minimize}\text{\hspace{0.17em}}Z=\underset{i=1}{\overset{I}{\sum }}\underset{t=1}{\overset{T}{\sum }}\left[C{P}_{it}*X{P}_{it}+C{S}_{it}*Y{S}_{it}+CIN{V}_{it}*XIN{V}_{it}\
right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace
{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{{t}_{1}=1}{\overset{T}{\sum }}\underset{i=1}{\overset{I}{\sum }}CB{O}_{i{t}_{1}}*\left(\underset{t=1}{\overset{{t}_{1}}{\sum }}{D}_{it}
+XIN{V}_{i{t}_{1}}-\underset{t=1}{\overset{{t}_{1}}{\sum }}X{P}_{it}\right)\end{array}$(36)
Subject to:
$\underset{t=1}{\overset{{t}_{1}}{\sum }}{D}_{it}+XIN{V}_{i{t}_{1}}-\underset{t=1}{\overset{{t}_{1}}{\sum }}X{P}_{it}\ge 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\
hspace{0.17em}}\forall i\in I,{t}_{1}\in T$(37)
$\underset{i=1}{\overset{I}{\sum }}\left(P{T}_{it}*X{P}_{it}+S{T}_{i}*Y{S}_{it}\right)\le CAP{T}_{t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall t\
in T$(38)
$X{P}_{it}\le CA{P}_{it}Y{S}_{it}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I,t\in T$(39)
$\underset{i=1}{\overset{I}{\sum }}\underset{t=1}{\overset{T}{\sum }}\left(P{T}_{it}*{D}_{it}+S{T}_{i}*Y{S}_{it}\right)\le \underset{t=1}{\overset{T}{\sum }}CAP{T}_{t}$(40)
$\underset{i=1}{\overset{I}{\sum }}\underset{t=1}{\overset{T}{\sum }}\left(P{T}_{it}*X{P}_{it}+S{T}_{i}*Y{S}_{it}\right)\le \underset{t=1}{\overset{T}{\sum }}CAP{T}_{t}$(41)
$\underset{t=1}{\overset{T}{\sum }}X{P}_{it}\ge \underset{t=1}{\overset{T}{\sum }}{D}_{it}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(42)
$XIN{V}_{i0}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(43)
$XIN{V}_{iT}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I$(44)
$Y{S}_{it}\in \left\{0,1\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I,t\in T$(45)
$XIN{V}_{it},X{P}_{it}\ge 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\in I,t\in T$(46)
Problem notations are tabulated in Table 1. Equation (1), Equation (13), Equation (22), and Equation (36) minimize the total production cost of the system in model A1, A2, A3, and A4 respectively.
Equation (2), Equation (14), Equation (23), and Equation (37) are the state equations as they ensure that the total quantities in a particular time period is a function of total quantities carried
forward from the preceding time period while satisfying demand. It must be noted that in model A2 and A4, the backorder variable has been eliminated from the model by appropriate substitution
(backorder variable $XB{O}_{it}$ , has been eliminated by substituting it in terms of other decision variables from Equation (2). Its value has been substituted in the objective function Equation
(13) in model A2 and objective function Equation (36) in model A4). These changes are reflected in Equation (23) and Equation (37). Reduction in number of variables improves the time complexity of
the model. Equation (3), Equation (4), Equation (15), Equation (24), Equation (25), Equation (38), Equation (40), and Equation (41) are the time capacity constraints. They ensure that production of
the items, and demand satisfied (through production) in a particular time period does not violate the production time available in that time period. Similarly, Equation (5), Equation (16), Equation
(26), and Equation (39) are the production capacity constraints. These constraints ensures that capacity constraints in terms of production resources are not violated in any time period. Equation
(27), Equation (28), Equation (40), and Equation (41) are the additional capacity constraints added in model A3, and A4 for tightening the bound and subsequently achieving computational advantage as
discussed earlier. These constraints are derived from constraint Equation (24) and Equation (25) by extending the time capacity constraints over the entire time horizon T. It must be noted that
Equation (24) and Equation (25) ensures the time capacity constraints are honored only for individual time period. Equation (6), Equation (17), Equation (29), and Equation (42) ensures that demand is
satisfied in each period. Equations (7)-(10), Equation (18), Equation (19), Equations (30)-(33), Equation (43), Equation (44) sets the initial and final conditions (boundary conditions) over the
production horizon. Equation (11), Equation (12), Equation (20), Equation (21), Equation (34), Equation (35), Equation (45), Equation (46) are the binary and non-negativity constraints for decision
4. Numerical Experiments
40 problems each of size 5 × 5 and 6 × 6 are randomly generated in GAMS. 5 × 5, 6 × 6 problems denotes the lot sizing problem to find an optimum production plan of 5 items over 5 time periods, and 6
items over 6 time periods respectively. Only feasible problems are retained for data analysis (6 × 6―29 problems, 5 × 5―31 problems). Value of constants in these problems is randomly generated
according to normal distribution (Table 2) and uniform distribution (Table 3).
Table 2. Random data generation (Normal distribution).
Table 3. Random data generation (Uniform Distribution).
5. Data Analysis
All the problems are implemented in GAMS. Solution to these sample problems are tabulated in Appendix. According to t-test performed on data, problem A3 is computationally efficient to problem A1
with a statistical significance of 0.009317 (p-value). Similarly A4 is better than A2 with a statistical significance of 0.003071 (p-value). Model A2 is computationally more efficient than model A1
with a statistical significance of 0.000695 (p-value). Model A4 is computationally more efficient than model A3 with a statistical significance of 0.00473 (p-value).
6. Conclusion
In this article, we have demonstrated the effect of reducing the number of variables, increasing the number of constraints on the computational time of lot sizing problem through 4 models. We infer
from our data analysis that model A4 is the most computationally efficient model, and hence is recommended to be used for solving capacitated lot sizing problem.
Appendix: Data Analysis Results | {"url":"https://scirp.org/journal/paperinformation?paperid=86022","timestamp":"2024-11-13T22:41:06Z","content_type":"application/xhtml+xml","content_length":"183184","record_id":"<urn:uuid:c97eae77-e74c-406b-bd49-b5090715a383>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00735.warc.gz"} |
How to handle missing data in r » finnstats
How to handle missing data in r
How to handle missing data in r, If you’ve ever conducted any research involving measurements taken in the actual world, you are aware that the data is frequently messy.
The quality of the data can be controlled in a lab, but this is not always the case in the actual world. There are occasions when events outside of your control can result in data gaps.
How to handle missing data in r
In R, there are numerous methods for handling missing data. The is.na() function can be used to simply detect it.
Another function in R called na.omit() removes any rows in the data frame that have missing data. NA is used to indicate missing data so that it may be quickly identified.
Removing Missing values in R-Quick Guide »
It is effortlessly accepted by data.frame(). The cbind() function does issue a warning even though it will accept data that contains NA.
By using the na.rm logical boundary, dataframe functions can address missing data in one method.
Delete NA values from r.
The NA number cannot be incorporated into calculations because it is only a placeholder and not a real numeric value.
Therefore, it must be eliminated from the calculations in some way to produce a useful result. An NA value will be produced if the NA value is factored into a calculation.
While this might be OK in some circumstances, in others you require a number. The na.omit() function, which deletes the entire row, and the na.rm logical perimeter, which instructs the function to
skip that value, are the two methods used in R to eliminate NA values.
What does the R-word na.rm mean?
When utilizing a dataframe function, the logical argument na.rm in the R language specifies whether or not NA values should be eliminated from the calculation. Literally, it means remove NA.
It is not an operation or a function. It is merely a parameter that many dataframe functions use. ColSums(), RowSums(), ColMeans(), and RowMeans are some of them ().
The function skips over any NA values if na.rm is TRUE. However, if na.rm returns FALSE, the calculation on the entire row or column yields NA.
Na.rm examples in R
We need to set up a dataframe before we can begin our examples.
a b c
4 78 NA 511
For these examples, the missing data set will be the NA in row 4 column b.
Imputing missing values in R »
colMeans(x, na.rm = TRUE, dims = 1)
a b c
49.00000 18.33333 245.25000
rowSums(x, na.rm = FALSE, dims = 1)
[1] 153 295 195 NA
rowSums(x, na.rm = TRUE, dims = 1)
[1] 153 295 195 589
With the exception of the fact that in the first example, na.rm = FALSE, the second and third examples are identical. That radically alters everything.
Correct data science requires dealing with missing data from a data set. R is used so frequently in statistical research because it makes handling this missing data so simple.
Have you found this article to be interesting? We’d be glad if you could forward it to a friend or share it on Twitter or Linked In to help it spread.
Leave a Reply Cancel reply | {"url":"https://finnstats.com/2022/07/19/how-to-handle-missing-data-in-r/","timestamp":"2024-11-11T19:40:32Z","content_type":"text/html","content_length":"305072","record_id":"<urn:uuid:11c7c8b9-4d48-497c-a44b-315bb32c9cd5>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00525.warc.gz"} |
Capacitor Design Equations Formulas Calculator Capacitance Stored Energy
Online Web Apps, Rich Internet Application, Technical Tools, Specifications, How to Guides, Training, Applications, Examples, Tutorials, Reviews, Answers, Test Review Resources, Analysis, Homework
Solutions, Worksheets, Help, Data and Information for Engineers, Technicians, Teachers, Tutors, Researchers, K-12 Education, College and High School Students, Science Fair Projects and Scientists
By Jimmy Raymond
Contact: aj@ajdesigner.com
Privacy Policy, Disclaimer and Terms
Copyright 2002-2015 | {"url":"https://www.ajdesigner.com/phpcapacitor/stored_energy_equation_cv_c.php","timestamp":"2024-11-04T08:05:18Z","content_type":"text/html","content_length":"24051","record_id":"<urn:uuid:d897fbfd-e967-408c-ba03-6c5f7938dd32>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00489.warc.gz"} |
Proportional Reasoning and Area - Math Motivator
Proportional Reasoning and Area
With our focus on proportional reasoning many educators are discovering that it is everywhere in our math curriculum. Once we become aware of this we know what to look for and listen for so that we
can impact students’ fluency and flexibility with numbers.
Recently, I gave the following problem to a Grade 6 class:
How many different rectangles can be made using 36 square tiles each time?
In pairs, students began their investigation. In the past, I have given out squared grid chart paper for students to record on but because we did not have enough of it we did not do that. This
turned out to be a good decision for this particular group of students because it pushed them to record dimensions to describe their rectangles rather than spend the time tracing them on the chart
paper. From now on I will have some available but I will not make it a requirement.
Once most students had come up with the different rectangles I began the teaching I wanted to do. I asked for the dimensions for each of the rectangles and recorded them so that when we had them all
the students could observe some patterns.
Key points brought out through my facilitation of the conversation:
• All of the rectangles had an area of 36 square units.
• A square is a special rectangle.
• Congruent rectangles have similar dimensions e.g., 4×9 and 9×4.
• There is a number relationship between pairs of dimensions (BIG IDEA – Proportional Reasoning)
For example:
half / doubling relationship
2×18 and 4×9 (4 is double 2 and 9 is half 18)
3×12 and 6×6 (3 is half of 6 and 12 is double 6)
third / triple relationship
1×36 and 3×12 (3 is triple 1 and 12 is 1/3 of 36)
This investigation inspired wonderings about what number relationships there would be with rectangles with different areas.
Because I knew the class was working on perimeter as well I asked: If each of these rectangles were gardens which ones would require the least or most amount of fencing to go around? Now we have
the opportunity to explore relationships between perimeter and area. Proportional reasoning is everywhere!
Comments · 1
1. Hello ~ Awesome blog ~ Thank You | {"url":"http://mathmotivator.com/how-many-rectangles/","timestamp":"2024-11-10T05:22:17Z","content_type":"text/html","content_length":"61608","record_id":"<urn:uuid:4ca75ea0-c18b-43bd-80fa-52749f091009>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00890.warc.gz"} |
W.V.O. QUINE'S PRAGMATIST CRITIQUE
BOOK III - Page 4
Semantical Systems: Information Theory
In 1953 Carnap and Yehousha Bar-Hillel, professor of logic and philosophy of science at the Hebrew University of Jerusalem, Israel, jointly published “Semantic Information” in the British Journal for
the Philosophy of Science. A more elaborate statement of the theory may be found in chapters fifteen through seventeen of Bar-Hillel’s Language and Information (1964). This semantical theory of
information is based on Carnap’s Logical Foundations of Probability and on Shannon’s theory of communication. In the introductory chapter of his Language and Information Bar-Hillel states that
Carnap’s Logical Syntax of Language was the most influential book he had ever read in his life, and that he regards Carnap to be one of the greatest philosophers of all time. In 1951 Bar-Hillel
received a research associateship in the Research Laboratory of Electronics at the Massachusetts Institute of Technology. At the time he took occasion to visit Carnap at the Princeton Institute for
Advanced Study.
In his “Introduction” to Studies in Inductive Logic and Probability, Volume I, Carnap states that during this time he told Bar-Hillel about his ideas on a semantical concept of content measure or
amount of information based on the logical concept of probability. This is an alternative concept to Shannon’s statistical concept of the amount of information. Carnap notes that frequently there is
confusion between these two concepts, and that while both the logical and statistical concepts are objective concepts of probability, only the second is related to the physical concept of entropy. He
also reports that he and Bar-Hillel had some discussions with John von Neumann, who asserted that the basic concepts of quantum theory are subjective and that this holds especially for entropy, since
this concept is based on probability and amount of information. Carnap states that he and Bar-Hillel tried in vain to convince von Neumann of the existence of the differences in each of these two
pairs of concepts: objective and subjective, logical and physical. As a result of the discussions at Princeton between Carnap and Bar-Hillel, they undertook the joint paper on semantical
information. Bar-Hillel reports that most of the paper was dictated by Carnap. The paper was originally published as a Technical Report of the MIT Research Laboratory in 1952.
In the opening statements of “Semantic Information” the authors observe that the measures of information developed by Claude Shannon have nothing to do with what the semantics of the symbols, but
only with the frequency of their occurrence in a transmission. This deliberate restriction of the scope of mathematical communication theory was of great heuristic value and enabled this theory to
achieve important results in a short time. But it often turned out that impatient scientists in various fields applied the terminology and the theorems of the theory to fields in which the term
“information” was used presystematically in a semantic sense. The clarification of the semantic sense of information is very important, therefore, and in this paper Carnap and Bar-Hillel set out to
exhibit a semantical theory of information that cannot be developed with the concepts of information and amount of information used by Shannon’s theory. Notably Carnap and Bar-Hillel’s equation for
the amount of information has a mathematical form that is very similar to that of Shannon’s equation, even though the interpretations of the two similar equations are not the same. Therefore a brief
summary of Shannon’s theory of information is in order at this point before further discussion of Carnap and Bar-Hillel’s theory.
Claude E. Shannon published his “Mathematical Theory of Communication” in the Bell System Technical Journal (July and October, 1948). The papers are reprinted together with an introduction to the
subject in The Mathematical Theory of Communication (Shannon and Weaver, 1964). Shannon states that his purpose is to address what he calls the fundamental problem of communication, namely, that of
reproducing at one point either exactly or approximately a message selected at another point. He states that the semantical aspects of communication are irrelevant to this engineering problem; the
relevant aspect is the selection of the correct message by the receiver from a set of possible messages in a system that is designed to operate for all possible selections. If the number of messages
in the set of all possible messages is finite, then this number or any monotonic function of this number can be regarded as a measure of the information produced, when one message is selected from
the set and with all selections being equally likely. Shannon uses a logarithmic measure with the base of the log serving as the unit of measure. His paper considers the capacity of the channel
through which the message is transmitted, but the discussion is focused on the properties of the source. Of particular interest is a discrete source, which generates the message symbol by symbol, and
chooses successive symbols according to probabilities. The generation of the message is therefore a stochastic process, but even if the originator of the message is not behaving as a stochastic
process (and he probably is not), the recipient must treat the transmitted signals in such a fashion. A discrete Markov process can be used to simulate this effect, and linguists have used it to
approximate an English-language message. The approximation to English language is more successful, if the units of the transmission are words instead of letters of the alphabet. During the years
immediately following the publication of Shannon’s theory linguists attempted to create constructional grammars using Markov processes. These grammars are known as finite-state Markov process
grammars. However, after Noam Chomsky published his Syntactical Structures in 1956, linguists were persuaded that natural language grammars are not finite-state grammars, but are potentially
infinite-state grammars.
In the Markov process there exists a finite number of possible states of the system together with a set of transition probabilities, such that for any one state there is an associated probability for
every successive state to which a transition may be made. To make a Markov process into an information source, it is necessary only to assume that a symbol is produced in the transition from one
state to another. There exists a special case called an ergodic process, in which every sequence produced by the process has the same statistical properties. Shannon proposes a quantity that will
measure how much information is produced by an information source that operates as a Markov process: given n events with each having probability p(i), then the quantity of information H is:
H = - Σ p(i) log p(i).
In their “Semantic Information” Carnap and Bar-Hillel introduce the concepts of information content of a statement and of content element. Bar-Hillel notes that the content of a statement is what is
also meant by the Scholastic adage, omnis determinatio est negatio. It is the class of those possible states of the universe, which are excluded by the statement. When expressed in terms of state
descriptions, the content of a statement is the class of all state descriptions excluded by the statement. The concept of state description had been defined previously by Carnap as a conjunction
containing as components for every atomic statement in a language either the statement or its negation but not both and no other statements. The content element is the opposite in the sense that it
is a disjunction instead of a conjunction. The truth condition for the content element is therefore much less than that for the state description; in the state description all the constituent atomic
statements must be true for the conjunction to be true, while for the content element only one of the constituent elements must be true for the disjunction to be true. Therefore the content elements
are the weakest possible factual statements that can be made in the object language. The only factual statement that is L-implied by a content element is the content element itself. The authors then
propose an explicatum for the ordinary concept of the “information conveyed by the statement I” taken in its semantical sense: the content of a statement i, denoted cont(i), is the class of all
content elements that are L-implied by the statement i.
The concept of the measure of information content of a statement is related to Carnap’s concept of measure over the range of a statement. Carnap’s measure functions are meant to explicate the
presystematic concept of logical or inductive probability. For every measure function a corresponding function can be defined in some way, that will measure the content of any given statement, such
that the greater the logical probability of a statement, the smaller its content measure. Let m(i) be the logical probability of the statement i. Then the quantity 1-m(i) is the measure of the
content of i, which may be called the “content measure of I”, denoted cont(i). Thus:
cont(i) = 1- m(i).
However, this measure does not have additivity properties, because cont is not additive under inductive independence. The cont value of a conjunction is smaller than the cont value of its
components, when the two statements conjoined are not content exclusive. Thus insisting on additivity on condition of inductive independence, the authors propose another set of measures for the
amount of information, which Carnap and Bar-Hillel call “information measures” for the idea of the amount of information in the statement i, denoted inf(i), and which they define as:
inf(i) = log {1/[1-cont(i)]}
which by substitution transforms into:
inf(i) = - log m(i).
This is analogous to the amount of information in Shannon’s mathematical theory of communication but with inductive probability instead of statistical probability. They make their use of the logical
concept of probability explicit when they express it as:
inf(h/e) = - log c(h,e)
where c(h,e) is defined as the degree of confirmation and inf(h/e) means the amount of information in hypothesis h given evidence e. Bar-Hillel says that cont may be regarded as a measure of the
“substantial” aspect of a piece of information, while inf may be regarded as a measure of its “surprise” value or in less psychological terms of its “objective unexpectedness.” Bar-Hillel believed
that their theory of semantic information might be fruitfully applied in various fields. However, neither Carnap nor Bar-Hillel followed up with any investigations of the applicability of their
semantical concept of information to scientific research. Later when Bar-Hillel’s interests turned to the analysis of natural language, he noted that linguists did not accept Carnap’s semantical
Shreider’s Semantic Theory of Information
Carnap’s semantic theory of information may be contrasted with a more recent semantic information theory proposed by the Russian information scientist, Yu A. Shreider (also rendered from the Russian
as Ju A. Srejder). In his “Basic Trends in the Field of Semantics” in Statistical Methods in Linguistics (1971) Shreider distinguishes three classifications or trends in works on semantics, and he
relates his views to Carnap’s in this context. The three classifications are ontological semantics, logical semantics, and linguistic semantics. He says that all three of these try to solve the same
problem: to ascertain what meaning is and how it can be described. The first classification, ontological semantics, is the study of the various philosophical aspects of the relation between sign and
signified. He says that it inquires into the very nature of existence, into the degrees of reality possessed by signified objects, classes and situations, and that it is closely related to the logic
and methodology of science and to the theoretical foundations of library classification.
The second classification, logical semantics, studies formal sign systems as opposed to natural languages. This is the trend in which he locates Carnap, as well as Quine, Tarski, and Bar-Hillel. The
semantical systems considered in logical semantics are basic to the metatheory of the sciences. The meaning postulates determine the class of permissible models for a given system of formal
relations. A formal theory fixes a class of syntactical relations, whence there arises a fixed system of semantic relations within a text describing a possible world.
The third classification, linguistic semantics, seeks to elucidate the inherent organization in a natural language, to formulate the inherent regularities in texts and to construct a system of basic
semantic relations. The examination of properties of extralinguistic reality, which determines permissible semantic relations and the ways of combining them, is carried considerably farther in
linguistic semantics than in logical semantics, where the question is touched upon only in the selection of meaning postulates. However, linguistic semantics is still rather vague and inexact, being
an auxiliary investigation in linguistics used only as necessity dictates. Shreider locates his work midway between logical and linguistic semantics, because it involves the examination of natural
language texts with logical calculi.
Shreider’s theory is a theory of communication that explains phenomena not explained by Shannon’s statistical theory. Bibliographies in Shreider’s English-language articles contain references to
Carnap’s and Bar-Hillel’s 1953 paper, and Shreider explicitly advocates Carnap’s explication of intensional synonymy in terms of L-equivalence. But Shreider’s theory is more accurately described as
a development of Shannon’s theory, even though Shreider’s theory is not statistical. English-language works by Shreider include “On the Semantic Characteristics of Information” in Information
Storage and Retrieval (1965), which is also reprinted in Introduction to Information Science (ed. Tefko Saracevic, 1970), and “Semantic Aspects of Information Theory” in On Theoretical Problems On
Information (Moscow, 1969). Furthermore comments on Shreider and other contributors to Russian information science (or “informatics” as it is called in Russia) can be found in “Some Soviet Concepts
of Information for Information Science” in the American Society for Information Science Journal (1975) by Nicholas J. Belkin.
Like many information scientists who take up semantical considerations, Shreider notes that there are many situations involving information, in which one may wish to consider the content of the
message signals instead of the statistical frequency of signal transmission considered by Shannon’s theory. But Shreider furthermore maintains that a semantical concept of information implies an
alternative theory of communication in contrast to Shannon’s “classical” theory. Shannon’s concept pertains only to the potential ability of the receiver to determine from a given message text a
quantity of information; it does not account for the information that the receiver can effectively derive from the message, that is, the receiver’s ability to “understand” the message. In Shreider’s
theory the knowledge had by the receiver prior to receiving the message is considered, in order to determine the amount of information effectively communicated.
More specifically, in Shannon’s probability-theoretic approach, before even considering the information contained in a message about some event, it is necessary to consider the a priori probability
of the event. Furthermore according to Shannon’s first theorem, in the optimum method of coding a statement containing more information requires more binary symbols or bits. In Shreider’s view,
however, a theory of information should be able to account for cases that do not conform to this theorem. For example much information is contained in a statement describing a newly discovered
chemical element, which could be coded in a small number of binary symbols, and for which it would be meaningless to speak of an a priori probability. On the other hand a statement describing the
measurements of the well known physicochemical properties of some substance may be considerably less informative, while it may need a much more extensive description for its coding. The newly
discovered element will change our knowledge about the world much more than measurement of known substances. Shreider maintains that a theory of information that can take into account the receiver’s
ability to “understand” a message must include a description of the receiver’s background knowledge. For this reason his information theory includes a thesaurus, by which is meant a unilingual
dictionary showing the semantic connections among its constituent words. Shreider’s concept of information is thus consistent with Hickey’s thesis of communication constraint.
Let T denote such a thesaurus to represent a guide in which there is recorded our knowledge about the real world. The thesaurus T can be in any one of various states, and it can change or be
transformed from one state to another. Let M represent a received message, which can transform the thesaurus T. Then the concept of amount of information, denoted L(T,M), may be defined as the
degree of change in the thesaurus T under the action of a given statement M. And for each admissible text M expressed in a certain code or language, there corresponds a certain transformation
operator ? that acts on thesaurus T. The salient point is that the amount of information contained in the statement M relative to the thesaurus T is characterized by the degree of change in the
thesaurus under the action of the communicated statement. And the understanding of the communicated statement depends on the state of the receiver’s thesaurus. Accordingly the thesaurus T can
understand some statements and not others. There are some statements that cannot be understood by a given thesaurus, and the information for such a thesaurus is zero, which is to say L(T, M)=0,
because the thesaurus T is not transformed at all. One such case is that of a student or a layman who does not have the background to understand a transmitted message about a specialized subject.
Another case is that of someone who already knows the transmitted information, so that it is redundant to what the receiver already knows. In this case too there is no information communicated, and
again L(T,M)=0, but in this case it is because the thesaurus T has been transformed into its initial state.
The interesting situation is that in which the receiver’s thesaurus is sufficiently developed that he understands the transmitted message, but still finds his thesaurus transformed into a new and
different state as a result of receipt of the new information. If the rules of construction of the transformation operator ? are viewed as external to the thesaurus T, then the quantity L(T,M)
depends on these rules. And when the transformation operator ? is also revised, a preliminary increase of the knowledge stored in the thesaurus T may not only decrease the quantity of information L(T
,M), but can also increase it. Thus someone who has learned a branch of a science will derive more information from a special text in the branch than he would before he had learned it. This peculiar
property of the semantic theory of information basically distinguishes it from the Shannon’s classical theory, in which the increase in a priori information always decreases the amount of information
from a message statement M. In the classical theory there is no question of a receiver’s degree of “understanding" of a statement; it is always assumed that he is “tuned.” But in the semantic theory
the essential rôle is played by the very possibility of correct “tuning” of the receiver.
In his 1975 article Belkin reports that Shreider further developed his theory of information to include the idea of “meta-information.” Meta-information is information about the mode of the coding of
information, i.e., the knowledge about the relation between information and the text in which it is coded. In this sense of meta-information the receiver’s thesaurus must contain meta-information in
order to understand the information in the received message text, because it enables the receiver to analyze the organization of the semantic information, such as that which reports scientific
research findings. Shreider maintains that informatics, the Russian equivalent to information science, is concerned not with information as such, but rather with meta-information, and specifically
with information as to how scientific information is distributed and organized.
Therefore, with his concept of meta-information Shreider has reportedly modified his original theory of communication by analyzing the thesaurus T into two components, such that T=(Tm,To). The first
component Tm consists of the set of rules needed for extracting elementary messages from the text M, while the second component To consists of the factual information that relates those elementary
messages systematically and enables the elements to be integrated in T. The relationship between Tm and To is such that a decrease in the redundancy of coding of To requires an increase of the
meta-information in Tm for the decoding of the coding system used for To. Hence the idea of meta-information may be a means of realizing some limiting efficiency laws for information by analyzing the
dependency relation between information and the amount of meta-information necessary to comprehend that information.
It would appear that if the coding system is taken as a language, then Shreider’s concept of meta-information might include the idea of a metalanguage as used by Carnap, Hickey and other analytical
philosophers, or it might be incorporated into the metalanguage. Then the elements Tm and To are distinguished as metalanguage and object language respectively.
Pages [1] [2] [3] [4] [5] [6] [7] [8] [9]
NOTE: Pages do not corresponds with the actual pages from the book | {"url":"http://philsci.com/book3-4.htm","timestamp":"2024-11-03T17:08:16Z","content_type":"text/html","content_length":"37253","record_id":"<urn:uuid:05349049-d0e7-47c8-8faa-c5a4486945f7>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00505.warc.gz"} |
To install tidychangepoint:
To load it:
Tidy methods for changepoint analysis
The tidychangepoint package allows you to use any number of algorithms for detecting changepoint sets in univariate time series with a common, tidyverse-compliant interface. Currently, algorithms
from changepoint, wbs, and several genetic algorithms made accessible via GA are supported. It also provides model-fitting procedures for commonly-used parametric models, tools for computing various
penalty functions, and graphical diagnostic displays.
Changepoint sets are computed using the segment() function, which takes a numeric vector that is coercible into a ts object, and a string indicating the algorithm you wish you use. segment() always
returns a tidycpt object.
## [1] "tidycpt"
Various methods are available for tidycpt objects. For example, as.ts() returns the original data as ts object, and changepoints() returns the set of changepoint indices.
## [1] 237 330
If the original time series has time labels, we can also retrieve that information.
## [1] "1895-01-01" "1988-01-01"
The fitness() function returns the both the value and the name of the objective function that the algorithm used to find the optimal changepoint set.
## MBIC
## 643.5292
Please read the full paper for more details.
To cite the package, use the following information:
## Warning in citation("tidychangepoint"): could not determine year for
## 'tidychangepoint' from package DESCRIPTION file
## To cite package 'tidychangepoint' in publications use:
## Baumer B, Suarez Sierra B, Coen A, Taimal C (????). _tidychangepoint:
## Facilitate Changepoint Detection Analysis in a Tidy Framework_. R
## package version 0.0.1,
## <https://beanumber.github.io/tidychangepoint/>.
## A BibTeX entry for LaTeX users is
## @Manual{,
## title = {tidychangepoint: Facilitate Changepoint Detection Analysis in a Tidy Framework},
## author = {Benjamin S. Baumer and Biviana Marcela {Suarez Sierra} and Arrigo Coen and Carlos A. Taimal},
## note = {R package version 0.0.1},
## url = {https://beanumber.github.io/tidychangepoint/},
## } | {"url":"https://cran.rstudio.com/web/packages/tidychangepoint/readme/README.html","timestamp":"2024-11-01T19:39:44Z","content_type":"application/xhtml+xml","content_length":"9607","record_id":"<urn:uuid:04b27158-7762-41c7-a6da-6eda96c8883d>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00204.warc.gz"} |
Monoidal categories associated with strata of flag manifolds
We construct a monoidal category C[w,v] which categorifies the doubly-invariant algebra CN^′(w)[N]^N(v) associated with Weyl group elements w and v. It gives, after a localization, the coordinate
algebra C[R[w,v]] of the open Richardson variety associated with w and v. The category C[w,v] is realized as a subcategory of the graded module category of a quiver Hecke algebra R. When v=id, C[w,v]
is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra A[q](n(w))[Z[q,q^−1]] given by Kang–Kashiwara–Kim–Oh. We show that the
category C[w,v] contains special determinantial modules M(w[≤k]Λ,v[≤k]Λ) for k=1,…,ℓ(w), which commute with each other. When the quiver Hecke algebra R is symmetric, we find a formula of the degree
of R-matrices between the determinantial modules M(w[≤k]Λ,v[≤k]Λ). When it is of finite ADE type, we further prove that there is an equivalence of categories between C[w,v] and C[u] for w,u,v∈W with
w=vu and ℓ(w)=ℓ(v)+ℓ(u).
Bibliographical note
Publisher Copyright:
© 2018 Elsevier Inc.
• Categorification
• Monoidal category
• Quantum cluster algebra
• Quiver Hecke algebra
• Richardson variety
Dive into the research topics of 'Monoidal categories associated with strata of flag manifolds'. Together they form a unique fingerprint. | {"url":"https://pure.ewha.ac.kr/en/publications/monoidal-categories-associated-with-strata-of-flag-manifolds","timestamp":"2024-11-05T09:27:55Z","content_type":"text/html","content_length":"49779","record_id":"<urn:uuid:def847b9-26f2-49af-aa76-adf6410b6a23>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00428.warc.gz"} |
Cities starting with the letter K in Mali | List of cities in Mali on K - 【Population HUB】
Cities starting with the letter K in Mali
List of cities starting with letter K in Mali. This list contains 7 cities of Mali that start with the letter K. On the Population HUB website you can find lists of cities of countries with the
necessary letter, as well as the population of any of the regions of the Earth.
Population HUB - this is an accessible statistics of the population of the country, city and any other region. Fast site work and constantly updated data. Thank you for choosing the Population HUB.
There are 9 cities in Mali with the first letter K. | {"url":"https://population-hub.com/en/ml/cities-starting-with-letter-k-in-mali.html","timestamp":"2024-11-03T03:13:37Z","content_type":"text/html","content_length":"22583","record_id":"<urn:uuid:817aae2f-6c61-4ef1-8054-21471351a4c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00120.warc.gz"} |
Supertrend Trading Strategy Based on ATR and MA Combination
1. Supertrend Trading Strategy Based on ATR and MA Combination
Supertrend Trading Strategy Based on ATR and MA Combination
, Date: 2023-12-01 16:40:27
The Supertrend trading strategy is a trend-following strategy based on Average True Range (ATR) and Moving Average (MA). It incorporates the advantages of both trend tracking and breakout trading to
identify the intermediate trend direction and generate trading signals based on trend changes.
The main idea behind this strategy is to go long or short when the price breaks through the Supertrend channel, indicating a trend reversal. It also sets stop loss and take profit levels to lock in
gains and control risks.
How This Strategy Works
The Supertrend calculation involves several steps:
1. Calculate the ATR. The ATR reflects the average volatility over a period of time.
2. Calculate the midline based on highest high and lowest low. The midline is calculated as: (Highest High + Lowest Low)/2
3. Calculate the upper and lower channel based on ATR and ATR multiplier set by trader. The upper channel is calculated as: Midline + (ATR × Multiplier). The lower channel is calculated as: Midline
- (ATR × Multiplier).
4. Compare closing price with the upper/lower channel to determine trend direction. If close is above upper channel, trend is up. If close is below lower channel, trend is down.
5. A breakout above or below the channel generates reverse trading signals. For example, a breakout above upper channel signals long entry while a breakdown below lower channel signals short entry.
The advantage of this strategy is it combines both trend following and trend reversal techniques. It identifies major trend while also being able to capture reversal opportunities in a timely manner.
In addition, the stop loss/take profit mechanism helps control risks.
The Supertrend strategy has the following strengths:
1. Track intermediate trend
The Supertrend channel is calculated based on ATR, which effectively reflects the intermediate price fluctuation range. It tracks intermediate trend better than simple moving averages.
2. Capture reversals timely
Price breakouts from the channel quickly generate trading signals so that major trend reversals can be captured in time. This allows proper repositioning to avoid overholding.
3. Have stop loss and take profit
The strategy sets predefined stop loss and take profit levels for automatic exit with risk control. This significantly reduces the risk of excessive stop loss and allows better trend following.
4. Simple to implement
The strategy mainly uses basic indicators like MA and ATR. This makes it fairly simple to understand and implement for live trading.
**5. High capital efficiency **
By tracking intermediate trends and controlling individual slippage, the Supertrend strategy provides overall high capital efficiency.
Risk Analysis
The Supertrend strategy also has some potential weaknesses:
1. Underperforms in ranging market
The strategy focuses on intermediate to long term trend trading. In ranging or consolidating markets, it tends to underperform with higher opportunity cost of missing short trades.
2. Sensitive to parameter optimization
The values chosen for ATR period and multiplier have relatively big impacts on strategy performance. Inappropriate tuning of the parameters may compromise the effectiveness of trading signals.
3. Lagging issues may exist
There can be some lagging issues with Supertrend channel calculation, causing untimely signal generation. Fixing the lagging problem should be a priority.
4. Strict stop loss management required
In extreme market conditions, improperly large stop loss allowance or inadequate risk management could lead to heavy losses. Strictly following stop loss rules is critical for consistent
Improvement Areas
There is further room to optimize this Supertrend strategy:
1. Combine multiple ATR periods
Combining ATR readings over different periods like 10-day and 20-day forms a composite indicator, which helps improve sensitivity and lagging issues.
2. Add stop loss modules
Adding more sophisticated stop loss mechanisms like triple stop loss, volatility stop loss and sequential stop loss could strengthen risk control and drawdown reduction.
3. Parameter optimization
Optimizing values for ATR period, multiplier and other inputs through quantitative methods would further lift strategy performance. Parameters can also be dynamically tuned based on different
products and market regimes.
4. Integrate machine learning models
Finally, integrating machine learning models may realize automated trend recognition and signal generation, reducing reliance on subjective decisions and improving system stability.
The Supertrend trading strategy identifies intermediate trend direction using MA and ATR indicators, and generates trade entry and exit signals around trend reversals with automated stop loss/take
profit implementation. While keeping with major trends, it also captures some reversal opportunities. The main advantages lie in intermediate trend tracking, trend reversal identification and risk
control through stop loss/take profit.
However, some deficiencies also exist regarding insufficient range-bound market capture and lagging problems. Further optimizations can be explored across multiple dimensions, including using
composite ATR, strengthening stop loss modules, tuning parameters, and integrating machine learning models. These enhancements will likely improve the stability and efficiency of the Supertrend
start: 2022-11-30 00:00:00
end: 2023-11-30 00:00:00
period: 1d
basePeriod: 1h
exchanges: [{"eid":"Futures_Binance","currency":"BTC_USDT"}]
strategy("Supertrend V1.0 - Buy or Sell Signal",overlay=true)
Factor=input(3, minval=1,maxval = 100)
Pd=input(7, minval=1,maxval = 100)
//Calculating ATR
atrLength = input(title="ATR Length:", defval=14, minval=1)
Stop_Loss_Factor = input(1.5, minval=0,step=0.01)
factor_profit = input(1.0, minval=0,step=0.01)
// === INPUT BACKTEST RANGE ===
FromMonth = input(defval = 4, title = "From Month", minval = 1, maxval = 12)
FromDay = input(defval = 10, title = "From Day", minval = 1, maxval = 31)
FromYear = input(defval = 2016, title = "From Year", minval = 2009)
ToMonth = input(defval = 4, title = "To Month", minval = 1, maxval = 12)
ToDay = input(defval = 10, title = "To Day", minval = 1, maxval = 31)
ToYear = input(defval = 2039, title = "To Year", minval = 2017)
// === FUNCTION EXAMPLE ===
start = timestamp(FromYear, FromMonth, FromDay, 00, 00) // backtest start window
finish = timestamp(ToYear, ToMonth, ToDay, 23, 59) // backtest finish window
window() => time >= start and time <= finish ? true : false // create function "within window of time"
// Calculate ATR
decimals = abs(log(syminfo.mintick) / log(10))
Atr = atrValue
if(decimals == 5)
Atr := atrValue * 10000
if(decimals == 4)
Atr := atrValue * 1000
if(decimals == 3)
Atr := atrValue * 100
if(decimals == 2)
Atr := atrValue * 10
//VJ2 Supertrend
TrendUp = 0.0
TrendUp:=close[1]>TrendUp[1]? max(Up,TrendUp[1]) : Up
TrendDown = 0.0
TrendDown:=close[1]<TrendDown[1]? min(Dn,TrendDown[1]) : Dn
Trend = 0.0
Trend := close > TrendDown[1] ? 1: close< TrendUp[1]? -1: nz(Trend[1],1)
Tsl = 0.0
Tsl := Trend==1? TrendUp: TrendDown
linecolor = Trend == 1 ? green : red
plot(Tsl, color = linecolor , style = line , linewidth = 2,title = "SuperTrend")
plotshape(cross(close,Tsl) and close>Tsl , "Up Arrow", shape.triangleup,location.belowbar,green,0,0)
plotshape(cross(Tsl,close) and close<Tsl , "Down Arrow", shape.triangledown , location.abovebar, red,0,0)
//plot(Trend==1 and Trend[1]==-1,color = linecolor, style = circles, linewidth = 3,title="Trend")
plotarrow(Trend == 1 and Trend[1] == -1 ? Trend : na, title="Up Entry Arrow", colorup=lime, maxheight=60, minheight=50, transp=0)
plotarrow(Trend == -1 and Trend[1] == 1 ? Trend : na, title="Down Entry Arrow", colordown=red, maxheight=60, minheight=50, transp=0)
Trend_buy = Trend == 1
Trend_buy_prev = Trend[1] == -1
algo_buy_pre = Trend_buy and Trend_buy_prev
algo_buy = algo_buy_pre == 1 ? 1 : na
Trend_sell= Trend == -1
Trend_sell_prev = Trend[1] == 1
algo_sell_pre = Trend_sell and Trend_sell_prev
algo_sell = algo_sell_pre == 1 ? 1:na
strategy.entry("Long1", strategy.long, when= window() and algo_buy==1)
strategy.entry("Short1", strategy.short, when=window() and algo_sell==1)
bought = strategy.position_size > strategy.position_size
sold = strategy.position_size < strategy.position_size
longStop = Stop_Loss_Factor * valuewhen(bought, Atr, 0)
shortStop = Stop_Loss_Factor * valuewhen(sold, Atr, 0)
longProfit = factor_profit * longStop
shortProfit = factor_profit * shortStop
if(decimals == 5)
longStop := longStop *100000
longProfit := longProfit *100000
if(decimals == 4)
longStop := longStop * 10000
longProfit := longProfit * 10000
if(decimals == 3)
longStop := longStop * 1000
longProfit := longProfit * 1000
if(decimals == 2)
longStop := longStop * 100
longProfit := longProfit *100
if(decimals == 5)
shortStop := shortStop * 100000
shortProfit := shortProfit * 100000
if(decimals == 4)
shortStop := shortStop * 10000
shortProfit := shortProfit * 10000
if(decimals == 3)
shortStop := shortStop * 1000
shortProfit := shortProfit * 1000
if(decimals == 2)
shortStop := shortStop * 100
shortProfit := shortProfit * 100
strategy.exit("Exit Long", from_entry = "Long1", loss =longStop, profit = longProfit)
strategy.exit("Exit Short", from_entry = "Short1", loss =shortStop, profit = shortProfit) | {"url":"https://www.fmz.com/strategy/433946","timestamp":"2024-11-08T13:52:47Z","content_type":"text/html","content_length":"19051","record_id":"<urn:uuid:534479ac-063a-48be-8b49-6ab5a38ca68d>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00007.warc.gz"} |
Rocco A. Servedio
Tel(212) 853-8445
Fax(212) 666-0140
Rocco A. Servedio is a theoretical computer scientist whose work aims at elucidating the boundary between computationally tractable and intractable problems. He has developed algorithms and
established lower bounds for learning many fundamental classes of Boolean functions and probability distributions as well as for property testing of such classes.
Research Interests
Theoretical computer science, computational complexity theory, computational learning theory, randomness in computation
His work in concrete computational complexity theory has yielded new lower bounds and state-of-the-art derandomization results for well-studied Boolean circuit models. A core goal that motivates and
inspires much of Servedio’s work is to understand the structural properties of different types of Boolean functions using a range of analytic, algebraic, probabilistic, and combinatorial techniques.
Beyond identifying and establishing such structural properties, Servedio leverages these properties both to develop efficient solutions to various algorithmic problems (such as computational learning
and property testing) and to establish computational hardness (such as circuit lower bounds and pseudorandomness results). Servedio has given state-of-the-art algorithms and hardness results for
learning, testing, and derandomizing well-studied Boolean function classes such as DNF formulas, monotone functions, functions with small Fourier sparsity and/or Fourier dimension, constant-depth
circuits, linear separators, intersections of halfspaces, polynomial threshold functions, and juntas. His interests also include structural analysis and computational learning and testing of various
classes of probability distributions, as well as other data analysis problems.
Servedio received an AB in mathematics summa cum laude from Harvard University in 1993 and a PhD in computer science from Harvard University in 2001. He is a Sloan Foundation Fellow, a recipient of
multiple Best Paper awards from leading conferences in theoretical computer science, and a 2013 recipient of the Columbia University Presidential Teaching Award. He joined the faculty of Columbia
Engineering in 2003.
• Visiting fellow, Princeton University (sabbatical visit), 2009-2010
• NSF postdoctoral fellow, Harvard University, 2001-2002
• Professor of computer science, Columbia University, 2017–
• Vice-chair of computer science, 2012-
• Interim chair of computer science, Fall 2015
• Associate professor of computer science, Columbia University, 2007-2016
• Assistant professor of computer science, Columbia University, 2003-2006
• Association for Computing Machinery, Special Interest Group on Algorithms and Computation Theory (SIGACT
• Best Paper Award, 32nd Conference on Computational Complexity (CCC), 2017
• Best Paper Award, 56th IEEE Symposium on Foundations of Computer Science (FOCS), 2015
• Presidential Teaching Award, Columbia University, 2013
• Alfred P. Sloan Foundation Fellowship, 2005
• NSF CAREER Award, 2004
• Best Paper Award, 18nd Conference on Computational Complexity (CCC), 2003
• Xi Chen, Rocco A. Servedio, Li-Yang Tan, Erik Waingarten and Jinyu Xie. “Settling the query complexity of non-adaptive junta testing,” 32nd Conference on Computational Complexity (CCC), 2017.
• Toniann Pitassi, Benjamin Rossman, Rocco A. Servedio and Li-Yang Tan. “Poly-logarithmic Frege depth lower bounds via an expander switching lemma,” 48th ACM Symposium on Theory of Computing
(STOC), pp. 644-657 (2016).
• Benjamin Rossman, Rocco A. Servedio and Li-Yang Tan. “An average-case depth hierarchy theorem for Boolean circuits,” 56th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 1030-1048
• Clement Canonne, Dana Ron, and Rocco A. Servedio. “Testing probability distributions using conditional samples,” SIAM Journal on Computing, 44(3), pp. 540-616 (2015).
• Philip M. Long and Rocco A. Servedio. “On the weight of halfspaces over Hamming balls,” SIAM Journal on Discrete Mathematics, 28(3), pp. 1035-1061 (2014).
• Anindya De, Ilias Diakonikolas, Vitaly Feldman and Rocco A. Servedio. “Nearly optimal solutions for the Chow Parameters Problem and low-weight approximation of halfspaces,” Journal of the ACM,
61(2), Article 11 (2014).
• Anindya De and Rocco A. Servedio. “Efficient deterministic approximate counting for low-degree polynomial threshold functions,” 46th ACM Symposium on Theory of Computing (STOC), pp. 832-841
• Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco A. Servedio and Emanuele Viola. “Bounded independence fools halfspaces,” SIAM Journal on Computing, 39(8), pp. 3441-3462 (2010).
• Philip M. Long and Rocco A. Servedio. “Random classification noise defeats all convex potential boosters,” Machine Learning Journal, 78(3), pp. 287-304 (2010).
• Adam R. Klivans and Rocco A. Servedio. “Learning DNF in time 2O(n^(1/3)”, Journal of Computer and System Sciences, 68(2), pp. 303-318 (2003). | {"url":"https://magazine.engineering.columbia.edu/faculty/rocco-servedio","timestamp":"2024-11-03T13:56:41Z","content_type":"application/xhtml+xml","content_length":"32609","record_id":"<urn:uuid:fce2775f-5f3c-4fb6-942e-ac620057cea3>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00361.warc.gz"} |
Do Math Manipulatives Help Our Students Learn?
What are they?
A math manipulative is an object that is used in the teaching of mathematics that allows students to perceive the idea or concept they are learning through touching and moving the object. These
manipulatives can range from anything like dice or money to pattern blocks, two-color counters, and even playing cards or dominoes.
What age groups?
All ages can benefit from the use of manipulatives while learning math. Math manipulatives are most commonly used in the early elementary ages or younger. Once students become more capable of
abstracting concepts (older elementary, middle, and high school), teachers seem to have students spend more time doing math with paper and pencil, and less with hands on methods.
What are the benefits?
The use of manipulatives in the learning of mathematics allows students to represent math in multiple ways. More senses become engaged, including visual and tactile, which keeps a student more
attentive. They are able to “see” math, which reinforces the conceptual understanding. This lays the groundwork for the mechanics that they will use later and allows the rules to be more meaningful
and make sense, which in turn, will be less for them to “memorize”. Seeing math allows students to expand on ideas and uses of math in the world around them.
Why aren’t teachers using them?
Three reasons that math manipulatives are not used as often as they could, is time, money, and lack of knowledge. Developing the concept with a manipulative may require more time and so often, our
teachers are burdened with getting through the material. While many math manipulatives on the market can be costly, not all manipulatives are expensive, but having enough for a class set could get
pricey. Each math manipulative can be used to teach a variety of concepts. Often teachers may not know how to teach various concepts with these tools, and so they just do not get used. There are
many companies out there that do trainings with their manipulative for teachers to learn.
This blog has an ultimate list of math manipulatives that can get you started!
You Can Do It, Not Always Alone
I come across students daily who struggle in their academics, particularly in mathematics. For the students who are not trying or do not care, this message isn’t for you. Unfortunately, there is
not much anyone can do to help someone who simply will not try or does not care. For all others, I see many students quietly failing their class as time continues to pass by. One bad grade after
another accumulating. At some point, it may possibly be too late to undo! As I begin to understand and discover why this continues to happen to students who are “doing all the homework and still
failing the tests”, what I have learned is the simple fact, you can do it, not always alone.
So, students, here are a few tips for you to win your math struggle…
1. Do ALL of your homework. And furthermore, write it out on paper, step by step, even if that means you have to kill trees. Learning math means doing math and the paper is worth your learning!
2. ALWAYS use answer keys while doing your homework. First, do the problem, then check your solution. Instant feedback is essential to correcting mistakes.
3. After doing your homework, if there were types of problems that you didn’t understand or missed a lot of, ask for more practice problems from your teacher or find some on the internet or
4. Okay, so you can do all of your homework? Does that really mean you are ready for that quiz or test? Not necessarily! Create a similar pre-quiz or test and TIME yourself and GRADE yourself.
If you are going to make mistakes, make them on your practice quiz/test, not on the real one!
5. Build a good working relationship with you teacher/professor. Visit tutorials and office hours. Let them know who you are and show them you are trying. If you are a college student, see if
your university offers tutoring or a math lab.
6. Create study groups with other students who care and are willing to work hard and want to succeed. Work problems together and check each other as you go.
7. Write note cards with important notes/formulas as you go to keep everything in one place.
8. And if you still need support, that’s okay! There’s MaThCliX!
You don’t “study” math, you DO it! Bottom line. Know your resources and use them. Others have gone before you and no one becomes a huge success all alone.
Back-to-School Success
Back-to-School Success
Tips for Students and Parents
And just like that, another summer is over and a new school year begins! Here are some tips for both parents and students to work together to ensure a successful school year.
1. Set goals: Write them out clearly and display them somewhere that you see them everyday
ex: I will complete my HW before I watch TV
2. Get organized: This includes finding a way to organize papers going back and forth from subject to subject. How are you going to know what your assignments are and when they are due?
3. Plan: What HW, tests, and quizzes do you have this week? How will you prepare for them? Make sure you plan out your study time.
4. Practice: This is how you learn! Make time each day to practice.
5. Get Help: Are you not understanding what you are supposed to be learning? Ask! Get help! Go to your teacher, parent, and of course, MaThCliX! That is what we are here for.
1. Make sure that you know how to communicate with your student’s teacher. Know when conferences are and plan to have a presence and be proactive in your student’s academics.
2. Check grades! Even if your student is old enough to check their own grades, it never hurts to have a parent checking, too. Know when progress reports and report cards are due. If you see grades
dropping, intervene quickly!
3. Make sure your student is doing the success tips for students. Ask them how they are doing each one.
4. Find out about what student’s are learning each week so that you can help or get help, as needed. Find out about tutorials, teacher websites, and recommended resources.
5. Bring your student to MaThCliX!
Learning requires time, effort, and sacrifice
When I was a graduate student, I was very serious about my work and committed to making A’s and doing my best. While taking a graph theory course, I remember working hard daily to learn and
Learnunderstand the many proofs coming at us each week. I knew that I would never be able to reproduce any of these proofs on a test if I didn’t learn them and understand each step involved. To
help me with this endeavor, I purchased the poster-sized Post-It Notes and carefully wrote out each proof in different colors. I hung them all over my apartment walls; they became my wall art for
the time being. I studied them day and night as I spent time at home. I practiced writing them out on my own to see if I truly understood it and to discover what I might not understand. I think it
is safe to say that I put forth a great amount of time, effort, and sacrifice. However, it paid off because I made very high A’s on all four tests and was exempt from having to take the final exam!
In addition, I can honestly say I learned the content of the course.
I currently teach college algebra and while helping my students get prepared for their upcoming test, I was telling my students the above story about my efforts in graph theory. I was using it to
demonstrate that to learn something we must work and put forth effort. Afterwards, one student even made the comment, “You’re like someone on the Big Bang Theory or something!”. Of course I laughed
and took that as a compliment.
It occurred to me, at that very moment, that there are so many students out there who haven’t a clue what it means to work hard. It may sound simple, but for me, it was one of those “a ha”
moments—-Learning requires time, effort, and sacrifice and it is every bit worth our time, effort, and sacrifice! The knowledge and experience that we gain and the discipline it takes to acquire it
is invaluable and can build our character in ways nothing else could. I look back on my college days of hard work and stress and many moments of confusion and cherish it as a special time that
shaped who I am today.
I encourage all students working towards a worthy goal to know that it will all be worth it in the end! Keep going!
MaTh as One Whole Truth
When working with students I often get asked the question, “How do you know all of this?” or “How do you do math so well?”. I feel the answer to these and similar questions is just as simple as it
is for anyone learning to do anything. When you study something thoroughly enough, you begin to piece everything together as one whole truth, if you will. In other words, when we focus on math as
one whole truth and refrain from getting caught up or lost in the steps and procedures and forget what we learned, we begin to understand how each lesson (or piece of truth) is preparatory for the
next. Think of it as a cake with infinite layers and each layer necessary before the next. If a layer is missed, then the next layer will not fit quite right.
I am not sure why, but I notice far too often, that students “learn” math, take a test, and then forget it. When you show them or discuss a concept that they already “learned”, they often act like
they have no clue what you are talking about. At some point, they must approach math as a whole subject and put the lessons together.
For example, think of a knitting a sweater. One might knit an arm, front, back, etc. At some point, in order for the end result to actually be a sweater, it must be all pieced together.
Or, one can compare it to learning to drive. A new driver might study operating car controls, learning laws and road signs, parking techniques, interstate travel, backroads, night-time driving, etc.
Any one of these lessons standing alone will not create a skilled driver. Nor will doing any of these things just once or twice. It’s in the actual piecing together of the skills and practice of
driving in each area, that one becomes a skilled driver.
So it is with mathematics. To my elementary students, learn what you are doing now very well, because you will never stop using it in the progression of your mathematics study. Middle school
students, what you are learning now is preparatory for high school mathematics. High school students, by now I hope that you are starting to piece together this beautiful subject of truth called
mathematics! What you are learning is a culmination of all of your earlier years and preparatory for you to advance in further mathematics. There is always more to discover and learn and the
further we go, we begin to see the subject as one whole truth. Our perspective and understanding is enhanced.
Thus, I would suggest as a final answer to these opening questions is that I have learned to understand math and not just memorize it. So, I approach it by applying all truth that I know and putting
the question in context, then solving. I have spent much time doing and practicing and that is “how I know all of this”! Anyone else can, too. | {"url":"https://mathclix.com/author/mathclix/page/2","timestamp":"2024-11-13T02:26:00Z","content_type":"text/html","content_length":"55633","record_id":"<urn:uuid:386aed3f-a4eb-4b57-8f73-5f5ddcb9379e>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00309.warc.gz"} |
Lesson 6
Distinguishing between Two Types of Situations
Problem 1
Match each equation to a story. (Two of the stories match the same equation.)
1. \(3(x+5)=17\)
2. \(3x+5=17\)
3. \(5(x+3)=17\)
4. \(5x+3=17\)
1. Jada’s teacher fills a travel bag with 5 copies of a textbook. The weight of the bag and books is 17 pounds. The empty travel bag weighs 3 pounds. How much does each book weigh?
2. A piece of scenery for the school play is in the shape of a 5-foot-long rectangle. The designer decides to increase the length. There will be 3 identical rectangles with a total length of 17
feet. By how much did the designer increase the length of each rectangle?
3. Elena spends $17 and buys a $3 book and a bookmark for each of her 5 cousins. How much does each bookmark cost?
4. Noah packs up bags at the food pantry to deliver to families. He packs 5 bags that weigh a total of 17 pounds. Each bag contains 3 pounds of groceries and a packet of papers with health-related
information. How much does each packet of papers weigh?
5. Andre has 3 times as many pencils as Noah and 5 pens. He has 17 pens and pencils all together. How many pencils does Noah have?
Problem 2
Elena walked 20 minutes more than Lin. Jada walked twice as long as Elena. Jada walked for 90 minutes. The equation \(2(x+20)=90\) describes this situation. Match each expression with the statement
in the story with the expression it represents.
Problem 3
A school ordered 3 large boxes of board markers. After giving 15 markers to each of 3 teachers, there were 90 markers left. The diagram represents the situation. How many markers were originally in
each box?
Problem 4
Select all the pairs of points so that the line between those points has slope \(\frac 2 3\).
\((0,0)\) and \((2,3)\)
\((0,0)\) and \((3,2)\)
\((1,5)\) and \((4,7)\)
\((\text-2,\text-2)\) and \((4,2)\)
\((20,30)\) and \((\text-20,\text-30)\) | {"url":"https://im.kendallhunt.com/MS_ACC/teachers/2/3/6/practice.html","timestamp":"2024-11-04T21:55:12Z","content_type":"text/html","content_length":"71799","record_id":"<urn:uuid:6ffd802b-2ae2-4bc5-88b6-f65713875c88>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00095.warc.gz"} |
Ken Shirriff's blog
One of the most interesting navigation instruments onboard Soyuz spacecraft was the Globus INK,1 which used a rotating globe to indicate the spacecraft's position above the Earth. This
electromechanical analog computer used an elaborate system of gears, cams, and differentials to compute the spacecraft's position. The globe rotates in two dimensions: it spins end-over-end to
indicate the spacecraft's orbit, while the globe's hemispheres rotate according to the Earth's daily rotation around its axis.2 The spacecraft's position above the Earth was represented by the fixed
crosshairs on the plastic dome. The Globus also has latitude and longitude dials next to the globe to show the position numerically, while the light/shadow dial below the globe indicated when the
spacecraft would enter or leave the Earth's shadow.
The INK-2S "Globus" space navigation indicator.
Opening up the Globus reveals that it is packed with complicated gears and mechanisms. It's amazing that this mechanical technology was used from the 1960s into the 21st century. But what are all
those gears doing? How can orbital functions be implemented with gears? To answer these questions, I reverse-engineered the Globus and traced out its system of gears.
The Globus with the case removed, showing the complex gearing inside.
The diagram below summarizes my analysis. The Globus is an analog computer that represents values by rotating shafts by particular amounts. These rotations control the globe and the indicator dials.
The flow of these rotational signals is shown by the lines on the diagram. The computation is based around addition, performed by ten differential gear assemblies. On the diagram, each "⨁" symbol
indicates one of these differential gear assemblies. Other gears connect the components while scaling the signals through various gear ratios. Complicated functions are implemented with three
specially-shaped cams. In the remainder of this blog post, I will break this diagram down into functional blocks and explain how the Globus operates.
This diagram shows the interconnections of the gear network in the Globus.
For all its complexity, though, the functionality of the Globus is pretty limited. It only handles a fixed orbit at a specific angle, and treats the orbit as circular. The Globus does not have any
navigation input such as an inertial measurement unit (IMU). Instead, the cosmonauts configured the Globus by turning knobs to set the spacecraft's initial position and orbital period. From there,
the Globus simply projected the current position of the spacecraft forward, essentially dead reckoning.
A closeup of the gears inside the Globus.
The globe
On seeing the Globus, one might wonder how the globe is rotated. It may seem that the globe must be free-floating so it can rotate in two axes. Instead, a clever mechanism attaches the globe to the
unit. The key is that the globe's equator is a solid piece of metal that rotates around the horizontal axis of the unit. A second gear mechanism inside the globe rotates the globe around the
North-South axis. The two rotations are controlled by concentric shafts that are fixed to the unit. Thus, the globe has two rotational degrees of freedom, even though it is attached at both ends.
The photo below shows the frame that holds and controls the globe. The dotted axis is fixed horizontally in the unit and rotations are fed through the two gears at the left. One gear rotates the
globe and frame around the dotted axis, while the gear train causes the globe to rotate around the vertical polar axis (while the equator remains fixed).
The axis of the globe is at 51.8° to support that orbital inclination.
The angle above is 51.8° which is very important: this is the inclination of the standard Soyuz orbit. As a result, simply rotating the globe around the dotted line causes the crosshair to trace the
orbit.3 Rotating the two halves of the globe around the poles yields the different paths over the Earth's surface as the Earth rotates. An important consequence of this design is that the Globus only
supports a circular orbit at a fixed angle.
Differential gear mechanism
The primary mathematical element of the Globus is the differential gear mechanism, which can perform addition or subtraction. A differential gear takes two rotations as inputs and produces the
(scaled) sum of the rotations as the output. The photo below shows one of the differential mechanisms. In the middle, the spider gear assembly (red box) consists of two bevel gears that can spin
freely on a vertical shaft. The spider gear assembly as a whole is attached to a horizontal shaft, called the spider shaft. At the right, the spider shaft is attached to a spur gear (a gear with
straight-cut teeth). The spider gear assembly, the spider shaft, and the spider's spur gear rotate together as a unit.
Diagram showing the components of a differential gear mechanism.
At the left and right are two end gear assemblies (yellow). The end gear is a bevel gear with angled teeth to mesh with the spider gears. Each end gear is locked to a spur gear and these gears spin
freely on the horizontal spider shaft. In total, there are three spur gears: two connected to the end gears and one connected to the spider assembly. In the diagrams, I'll use the symbol below to
represent the differential gear assembly: the end gears are symmetric on the top and bottom, with the spider shaft on the side. Any of the three spur gears can be used as an output, with the other
two serving as inputs.
The symbol for the differential gear assembly.
To understand the behavior of the differential, suppose the two end gears are driven in the same direction at the same rate, say upwards.4 These gears will push on the spider gears and rotate the
spider gear assembly, with the entire differential rotating as a fixed unit. On the other hand, suppose the two end gears are driven in opposite directions. In this case, the spider gears will spin
on their shaft, but the spider gear assembly will remain stationary. In either case, the spider gear assembly motion is the average of the two end gear rotations, that is, the sum of the two
rotations divided by 2. (I'll ignore the factor of 2 since I'm ignoring all the gear ratios.) If the operation of the differential is still confusing, this vintage Navy video has a detailed
The controls and displays
The diagram below shows the controls and displays of the Globus. The rotating globe is the centerpiece of the unit. Its plastic cover has a crosshair that represents the spacecraft's position above
the Earth's surface. Surrounding the globe itself are dials that show the longitude, latitude, and the time before entering light and shadow. The cosmonauts manually initialize the globe position
with the concentric globe rotation knobs: one rotates the globe along the orbital path while the other rotates the hemispheres. The mode switch at the top selects between the landing position mode,
the standard Earth orbit mode, and turning off the unit. The orbit time adjustment configures the orbital time period in minutes while the orbit counter below it counts the number of orbits. Finally,
the landing point angle sets the distance to the landing point in degrees of orbit.
The Globus with the controls labeled.
Computing the orbit time
The primary motion of the Globus is the end-over-end rotation of the globe showing the movement of the spacecraft in orbit. The orbital motion is powered by a solenoid at the top of the Globus that
receives pulses once a second and advances a ratchet wheel (video).5 This wheel is connected to a complicated cam and differential system to provide the orbital motion.
The orbit solenoid (green) has a ratchet that rotates the gear to the right. The shaft connects it to differential gear assembly 1 at the bottom right.
Each orbit takes about 92 minutes, but the orbital time can be adjusted by a few minutes in steps of 0.01 minutes6 to account for changes in altitude. The Globus is surprisingly inflexible and this
is the only orbital parameter that can be adjusted.7 The orbital period is adjusted by the three-position orbit time switch, which points to the minutes, tenths, or hundredths. Turning the central
knob adjusts the indicated period dial.
The problem is how to generate the variable orbital rotation speed from the fixed speed of the solenoid. The solution is a special cam, shaped like a cone with a spiral cross-section. Three followers
ride on the cam, so as the cam rotates, the follower is pushed outward and rotates on its shaft. If the follower is near the narrow part of the cam, it moves over a small distance and has a small
rotation. But if the follower is near the wide part of the cam, it moves a larger distance and has a larger rotation. Thus, by moving the follower to a particular point on the cam, the rotational
speed of the follower is selected. One follower adjusts the speed based on the minutes setting with others for the tenths and hundredths of minutes.
A diagram showing the orbital speed control mechanism. The cone has three followers, but only two are visible from this angle. The "transmission" gears are moved in and out by the outer knob to
select which follower is adjusted by the inner knob.
Of course, the cam can't spiral out forever. Instead, at the end of one revolution, its cross-section drops back sharply to the starting diameter. This causes the follower to snap back to its
original position. To prevent this from jerking the globe backward, the follower is connected to the differential gearing via a slip clutch and ratchet. Thus, when the follower snaps back, the
ratchet holds the drive shaft stationary. The drive shaft then continues its rotation as the follower starts cycling out again. Each shaft output is accordingly a (mostly) smooth rotation at a speed
that depends on the position of the follower.
A cam-based system adjusts the orbital speed using three differential gear assemblies.
The three adjustment signals are scaled by gear ratios to provide the appropriate contribution to the rotation. As shown above, the adjustments are added to the solenoid output by three differentials
to generate the orbit rotation signal, output from differential 3.8 This signal also drives the odometer-like orbit counter on the front of the Globus. The diagram below shows how the components are
arranged, as viewed from the back.
A back view of the Globus showing the orbit components.
Displaying the orbit rotation
Since the Globus doesn't have any external position input such as inertial guidance, it must be initialized by the cosmonauts. A knob on the front of the Globus provides manual adjustment of the
orbital position. Differential 4 adds the knob signal to the orbit output discussed above.
The orbit controls drive the globe's motion.
The Globus has a "landing point" mode where the globe is rapidly rotated through a fraction of an orbit to indicate where the spacecraft would land if the retro-rockets were fired. Turning the mode
switch caused the globe to rotate until the landing position was under the crosshairs and the cosmonauts could evaluate the suitability of this landing site. This mode is implemented with a landing
position motor that provides the rapid rotation. This motor also rotates the globe back to the orbital position. The motor is driven through an electronics board with relays and a transistor,
controlled by limit switches. I discussed the electronics in a previous post so I won't go into more details here. The landing position motor feeds into the orbit signal through differential 5,
producing the final orbit signal.
The landing position motor and its associated gearing. The motor speed is geared down and then fed through a worm gear (upper center).
The orbit signal from differential 5 is used in several ways. Most importantly, the orbit signal provides the end-over-end rotation of the globe to indicate the spacecraft's travel in orbit. As
discussed earlier, this is accomplished by rotating the globe's metal frame around the horizontal axis. The orbital signal also rotates a potentiometer to provide an electrical indication of the
orbital position to other spacecraft systems.
The light/shadow indicator
Docking a spacecraft is a tricky endeavor, best performed in daylight, so it is useful to know how much time remains until the spacecraft enters the Earth's shadow. The light/shadow dial under the
globe provides this information. This display consists of two nested wheels. The outer wheel is white and has two quarters removed. Through these gaps, the partially-black inner wheel is exposed,
which can be adjusted to show 0% to 50% dark. This display is rotated by the orbital signal, turning half a revolution per orbit. As the spacecraft orbits, this dial shows the light/shadow transition
and the time to the transistion.9
The light/shadow indicator, viewed from the underside of the Globus. The shadow indicator has been set to 35% shadow. Near the hub, a pin restricts motion of the inner wheel relative to the outer
You might expect the orbit to be in the dark 50% of the time, but because the spacecraft is about 200 km above the Earth's surface, it will sometimes be illuminated when the surface of the Earth
underneath is dark.10 In the ground track below, the dotted part of the track is where the spacecraft is in the Earth's shadow; this is considerably less than 50%. Also note that the end of the orbit
doesn't match up with the beginning, due to the Earth's rotation during the orbit.
Ground track of an Apollo-Soyuz Test Project orbit, corresponding to this Globus. Image courtesy of
The latitude indicator
The latitude indicator to the left of the globe shows the spacecraft's latitude. The map above shows how the latitude oscillates between 51.8°N and 51.8°S, corresponding to the launch inclination
angle. Even though the path around the globe is a straight (circular) line, the orbit appears roughly sinusoidal when projected onto the map.11 The exact latitude is a surprisingly complicated
function of the orbital position.12 This function is implemented by a cam that is attached to the globe. The varying radius of the cam corresponds to the function. A follower tracks the profile of
the cam and rotates the latitude display wheel accordingly, providing the non-linear motion.
A cam is attached to the globe and rotates with the globe.
The Earth's rotation
The second motion of the globe is the Earth's daily rotation around its axis, which I'll call the Earth rotation. The Earth rotation is fed into the globe through the outer part of a concentric
shaft, while the orbital rotation is provided through the inner shaft. The Earth rotation is transferred through three gears to the equatorial frame, where an internal mechanism rotates the
hemispheres. There's a complication, though: if the globe's orbital shaft turns while the Earth rotation shaft remains stationary, the frame will rotate, causing the gears to turn and the hemispheres
to rotate. In other words, keeping the hemispheres stationary requires the Earth shaft to rotate with the orbit shaft.
A closeup of the gear mechanisms that drive the Globus, showing the concentric shafts that control the two rotations.
The Globus solves this problem by adding the orbit rotation to the Earth rotation, as shown in the diagram below, using differentials 7 and 8. Differential 8 adds the normal orbit rotation, while
differential 7 adds the orbit rotation due to the landing motor.14
The mechanism to compute the Earth's rotation around its axis.
The Earth motion is generated by a second solenoid (below) that is driven with one pulse per second.13 This motion is simpler than the orbit motion because it has a fixed rate. The "Earth" knob on
the front of the Globus permits manual rotation around the Earth's axis. This signal is combined with the solenoid signal by differential 6. The sum from the three differentials is fed into the
globe, rotating the hemispheres around their axis.
This solenoid, ratchet, and gear on the underside of the Globus drive the Earth rotation.
The solenoid and differentials are visible from the underside of the Globus. The diagram below labels these components as well as other important components.
The underside of the Globus.
The longitude display
The longitude cam and the followers that track its radius.
The longitude display is more complicated than the latitude display because it depends on both the Earth rotation and the orbit rotation. Unlike the latitude, the longitude doesn't oscillate but
increases. The longitude increases by 360° every orbit according to a complicated formula describing the projection of the orbit onto the globe. Most of the time, the increase is small, but when
crossing near the poles, the longitude changes rapidly. The Earth's rotation provides a smaller but steady negative change to the longitude.
The computation of the longitude.
The diagram above shows how the longitude is computed by combining the Earth rotation with the orbit rotation. Differential 9 adds the linear effect of the orbit on longitude (360° per orbit) and
subtracts the effect of the Earth's rotation (360° per day). The nonlinear effect of the orbit is computed by a cam that is rotated by the orbit signal. The shape of the cam is picked up and fed into
differential 10, computing the longitude that is displayed on the dial. The differentials, cam, and dial are visible from the back of the Globus (below).
A closeup of the differentials from the back of the Globus.
The time-lapse video below demonstrates the behavior of the rotating displays. The latitude display on the left oscillates between 51.8°N and 51.8°S. The longitude display at the top advances at a
changing rate. Near the equator, it advances slowly, while it accelerates near the poles. The light/shadow display at the bottom rotates at a constant speed, completing half a revolution (one light/
shadow cycle) per orbit.
The Globus INK is a remarkable piece of machinery, an analog computer that calculates orbits through an intricate system of gears, cams, and differentials. It provided astronauts with a
high-resolution, full-color display of the spacecraft's position, way beyond what an electronic space computer could provide in the 1960s.
The drawback of the Globus is that its functionality is limited. Its parameters must be manually configured: the spacecraft's starting position, the orbital speed, the light/shadow regions, and the
landing angle. It doesn't take any external guidance inputs, such as an IMU (inertial measurement unit), so it's not particularly accurate. Finally, it only supports a circular orbit at a fixed
angle. While a more modern digital display lacks the physical charm of a rotating globe, the digital solution provides much more capability.
I recently wrote blog posts providing a Globus overview and the Globus electronics. Follow me on Twitter @kenshirriff or RSS for updates. I've also started experimenting with Mastodon recently as @
[email protected]. Many thanks to Marcel for providing the Globus. I worked on this with CuriousMarc, so check out his Globus videos.
Notes and references
1. In Russian, the name for the device is "Индикатор Навигационный Космический" abbreviated as ИНК (INK). This translates to "space navigation indicator." but I'll use the more descriptive nickname
"Globus" (i.e. globe). The Globus has a long history, back to the beginnings of Soviet crewed spaceflight. The first version was simpler and had the Russian acronym ИМП (IMP). Development of the
IMP started in 1960 for the Vostok (1961) and Voshod (1964) spaceflights. The more complex INK model (described in this blog post) was created for the Soyuz flights, starting in 1967. The landing
position feature is the main improvement of the INK model. The Soyuz-TMA (2002) upgraded to the Neptun-ME system which used digital display screens and abandoned the Globus. ↩
2. According to this document, one revolution of the globe relative to the axis of daily rotation occurs in a time equal to a sidereal day, taking into account the precession of the orbit relative
to the earth's axis, caused by the asymmetry of the Earth's gravitational field. (A sidereal day is approximately 4 minutes shorter than a regular 24-hour day. The difference is that the sidereal
day is relative to the fixed stars, rather than relative to the Sun.) ↩
3. To see how the angle between the poles and the globe's rotation results in the desired orbital inclination, consider two limit cases. First, suppose the angle between is 90°. In this case, the
globe is "straight" with the equator horizontal. Rotating the globe along the horizontal axis, flipping the poles end-over-end, will cause the crosshair to trace a polar orbit, giving the
expected inclination of 90°. On the other hand, suppose the angle is 0°. In this case, the globe is "sideways" with the equator vertical. Rotating the globe will cause the crosshair to remain
over the equator, corresponding to an equatorial orbit with 0° inclination. ↩
4. There is a bit of ambiguity when describing the gear motions. If the end gears are rotating upwards when viewed from the front, the gears are both rotating clockwise when viewed from the right,
so I'm referring to them as rotating in the same direction. But if you view each gear from its own side, the gear on the left is turning counterclockwise, so from that perspective they are
turning in opposite directions. ↩
5. The solenoids are important since they provide all the energy to drive the globe. One of the problems with gear-driven analog computers is that each gear and shaft has a bit of friction and loses
a bit of torque, and there is nothing to amplify the signal along the way. Thus, the 27-volt solenoids need to provide enough force to run the entire system. ↩
6. The orbital time can be adjusted between 86.85 minutes and 96.85 minutes according to this detailed page that describes the Globus in Russian. ↩
7. The Globus is manufactured for a particular orbital inclination, in this case 51.8°. The Globus assumes a circular orbit and does not account for any variations. The Globus does not account for
any maneuvering in orbit. ↩
8. The outputs from the orbit cam are fed into the overall orbit rotation, which drives the orbit cam. This may seem like an "infinite loop" since the outputs from the cam turn the cam itself.
However, the outputs from the cam are a small part of the overall orbit rotation, so the feedback dies off. ↩
9. The scales on the light/shadow display are a bit confusing. The inner scale (blue) is measured in percentage of an orbit, up to 100%. The fixed outer scale (red) measures minutes, indicating how
many minutes until the spacecraft enters or leaves shadow. The spacecraft completes 100% of an orbit in about 90 minutes, so the scales almost, but not quite, line up. The wheel is driven by the
orbit mechanism and turns half a revolution per orbit.
The light and shadow indicator is controlled by two knobs.
10. The Internation Space Station illustrates how an orbiting spacecraft is illuminated more than 50% of the time due to its height. You can often see the ISS illuminated in the nighttime sky close
to sunset and sunrise (link). ↩
11. The ground track on the map is roughly, but not exactly, sinusoidal. As the orbit swings further from the equator, the track deviates more from a pure sinusoid. The shape will depend, of course,
on the rectangular map projection. For more information, see this StackExcahnge post. ↩
12. To get an idea of how the latitude and longitude behave, consider a polar orbit with 90° angle of inclination, one that goes up a line of longitude, crosses the North Pole, and goes down the
opposite line of latitude. Now, shift the orbit away from the poles a bit, but keeping a great circle. The spacecraft will go up, nearly along a constant line of longitude, with the latitude
increasing steadily. As the spacecraft reaches the peak of its orbit near the North Pole, it will fall a bit short of the Pole but will still rapidly cross over to the other side. During this
phase, the spacecraft rapidly crosses many lines of longitude (which are close together near the Pole) until it reaches the opposite line of longitude. Meanwhile, the latitude stops increasing
short of 90° and then starts dropping. On the other side, the process repeats, with the longitude nearly constant while the latitude drops relatively constantly.
The latitude and longitude are generated by complicated trigonometric functions. The latitude is given by arcsin(sin i * sin (2πt/T)), while the longitude is given by λ = arctan (cos i * tan(2πt/
T)) + Ωt + λ[0], where t is the spaceship's flight time starting at the equator, i is the angle of inclination (51.8°), T is the orbital period, Ω is the angular velocity of the Earth's rotation,
and λ[0] is the longitude of the ascending node. ↩
13. An important function of the gears is to scale the rotations as needed by using different gear ratios. For the most part, I'm ignoring the gear ratios, but the Earth rotation gearing is
interesting. The gear driven by the solenoid has 60 teeth, so it rotates exactly once per minute. This gear drives a shaft with a very small gear on the other end with 15 teeth. This gear meshes
with a much larger gear with approximately 75 teeth, which will thus rotate once every 5 minutes. The other end of that shaft has a gear with approximately 15 teeth, meshed with a large gear with
approximately 90 teeth. This divides the rate by 6, yielding a rotation every 30 minutes. The sequence of gears and shafts continues, until the rotation is reduced to once per day. (The tooth
counts are approximate because the gears are partially obstructed inside the Globus, making counting difficult.) ↩
14. There's a potential simplification when canceling out the orbital shaft rotation from the Earth rotation. If the orbit motion was taken from differential 5 instead of differential 4, the landing
motor effect would get added automatically, eliminating the need for differential 7. I think the landing motor motion was added separately so the mechanism could account for the Earth's rotation
during the landing descent. ↩
While programmers today take multiplication for granted, most microprocessors in the 1970s could only add and subtract — multiplication required a slow and tedious loop implemented in assembly code.1
One of the nice features of the Intel 8086 processor (1978) was that it provided machine instructions for multiplication,2 able to multiply 8-bit or 16-bit numbers with a single instruction.
Internally, the 8086 still performed a loop, but the loop was implemented in microcode: faster and transparent to the programmer. Even so, multiplication was a slow operation, about 24 to 30 times
slower than addition.
In this blog post, I explain the multiplication process inside the 8086, analyze the microcode that it used, and discuss the hardware circuitry that helped it out.3 My analysis is based on
reverse-engineering the 8086 from die photos. The die photo below shows the chip under a microscope. I've labeled the key functional blocks; the ones that are important to this post are darker. At
the left, the ALU (Arithmetic/Logic Unit) performs the arithmetic operations at the heart of multiplication: addition and shifts. Multiplication also uses a few other hardware features: the X
register, the F1 flag, and a loop counter. The microcode ROM at the lower right controls the process.
The 8086 die under a microscope, with main functional blocks labeled. This photo shows the chip with the metal and polysilicon removed, revealing the silicon underneath. Click on this image (or any
other) for a larger version.
The multiplication routines in the 8086 are implemented in microcode. Most people think of machine instructions as the basic steps that a computer performs. However, many processors (including the
8086) have another layer of software underneath: microcode. With microcode, instead of building the control circuitry from complex logic gates, the control logic is largely replaced with code. To
execute a machine instruction, the computer internally executes several simpler micro-instructions, specified by the microcode. This is especially useful for a machine instruction such as
multiplication, which requires many steps in a loop.
A micro-instruction in the 8086 is encoded into 21 bits as shown below. Every micro-instruction has a move from a source register to a destination register, each specified with 5 bits. The meaning of
the remaining bits depends on the type field and can be anything from an ALU operation to a memory read or write to a change of microcode control flow. Thus, an 8086 micro-instruction typically does
two things in parallel: the move and the action. For more about 8086 microcode, see my microcode blog post.
The encoding of a micro-instruction into 21 bits. Based on NEC v. Intel: Will Hardware Be Drawn into the Black Hole of Copyright?
The behavior of an ALU micro-operation is important for multiplication. The ALU has three temporary registers that are invisible to the programmer: tmpA, tmpB, and tmpC. An ALU operation takes its
first argument from any temporary register, while the second argument always comes from tmpB. An ALU operation requires two micro-instructions. The first micro-instruction specifies the ALU operation
and source register, configuring the ALU. For instance, ADD tmpA to add tmpA to the default tmpB. In the next micro-instruction (or a later one), the ALU result can be accessed through the Σ register
and moved to another register.
Before I get into the microcode routines, I should explain two ALU operations that play a central role in multiplication: LRCY and RRCY, Left Rotate through Carry and Right Rotate through Carry.
(These correspond to the RCL and RCR machine instructions, which rotate through carry left or right.) These operations shift the bits in a 16-bit word, similar to the << and >> bit-shift operations
in high-level languages, but with an additional feature. Instead of discarding the bit on the end, that bit is moved into the carry flag (CF). Meanwhile, the bit formerly in the carry flag moves into
the word. You can think of this as rotating the bits while treating the carry flag as a 17th bit of the word.
The left rotate through carry and right rotate through carry micro-instructions.
These shifts perform an important part of the multiplication process since shifting can be viewed as multiplying by two. LRCY also provides a convenient way to move the most-significant bit to the
carry flag, where it can be tested for a conditional jump. (This is important because the top bit is used as the sign bit.) Similarly, RRCY provides access to the least significant bit, very
important for the multiplication process. Another important property is that performing RRCY on an upper word and then RRCY on a lower word will perform a 32-bit shift, since the low bit of the upper
word will be moved into the high bit of the lower word via the carry bit.
Binary multiplication
The shift-and-add method of multiplication (below) is similar to grade-school long multiplication, except it uses binary instead of decimal. In each row, the multiplicand is multiplied by one digit
of the multiplier. (The multiplicand is the value that gets repeatedly added, and the multiplier controls how many times it gets added.) Successive rows are shifted left one digit. At the bottom, the
rows are added together to yield the product. The example below shows how 6×5 is calculated in binary using long multiplication.
Binary long multiplication is much simpler than decimal multiplication: at each step, you're multiplying by 0 or 1. Thus, each row is either zero or the multiplicand appropriately shifted (0110 in
this case). (Unlike decimal long multiplication, you don't need to know the multiplication table.) This simplifies the hardware implementation, since each step either adds the multiplicand or
doesn't. In other words, each step tests a bit of the multiplier, starting with the low bit, to determine if an add should take place or not. This bit can be obtained by shifting the multiplier one
bit to the right each step.
Although the diagram above shows the sum at the end, a real implementation performs the addition at each step of the loop, keeping a running total. Moreover, in the 8086, instead of shifting the
multiplicand to the left during each step, the sum shifts to the right. (The result is the same but it makes the implementation easier.) Thus, multiplying 6×5 goes through the steps below.
Why would you shift the result to the right? There's a clever reason for this. Suppose you're multiplying two 16-bit numbers, which yields a 32-bit result. That requires four 16-bit words of storage
if you use the straightforward approach. But if you look more closely, the first sum fits into 16 bits, and then you need one more bit at each step. Meanwhile, you're "using up" one bit of the
multiplier at each step. So if you squeeze the sum and the multiplier together, you can fit them into two words. Shifting right accomplishes this, as the diagram below illustrates for 0xffff×0xf00f.
The sum (blue) starts in a 16-bit register called tmpA while the multiplier (green) is stored in the 16-bit tmpB register. In each step, they are both shifted right, so the sum gains one bit and the
multiplier loses one bit. By the end, the sum takes up all 32 bits, split across both registers.
sum (tmpA) multiplier (tmpC)
The multiplication microcode
The 8086 has four multiply instructions to handle signed and unsigned multiplication of byte and word operands. These machine instructions are implemented in microcode. I'll start by describing the
unsigned word multiplication, which multiplies two 16-bit values and produces a 32-bit result. The source word is provided by either a register or memory. It is multiplied by AX, the accumulator
register. The 32-bit result is returned in the DX and AX registers.
The microcode below is the main routine for word multiplication, both signed and unsigned. Each micro-instruction specifies a register move on the left, and an action on the right. The moves transfer
words between the visible registers and the ALU's temporary registers, while the actions are mostly subroutine calls to other micro-routines.
move action
AX → tmpC LRCY tmpC iMUL rmw:
M → tmpB CALL X0 PREIMUL called for signed multiplication
CALL CORX the core routine
CALL F1 NEGATE called for negative result
CALL X0 IMULCOF called for signed multiplication
tmpC → AX JMPS X0 7
CALL MULCOF called for unsigned multiplication
tmpA → DX RNI
The microcode starts by moving one argument AX into the ALU's temporary C register and setting up the ALU to perform a Left Rotate through Carry on this register, in order to access the sign bit.
Next, it moves the second argument M into the temporary B register; M references the register or memory specified in the second byte of the instruction, the "ModR/M" byte. For a signed multiply
instruction, the PREIMUL micro-subroutine is called, but I'll skip that for now. (The X0 condition tests bit 3 of the instruction, which in this case distinguishes MUL from IMUL.) Next, the CORX
subroutine is called, which is the heart of the multiplication.4 If the result needs to be negated (indicated by the F1 condition), the NEGATE micro-subroutine is called. For signed multiplication,
IMULCOF is then called to set the carry and overflow flags, while MULCOF is called for unsigned multiplication. Meanwhile, the result bytes are moved from the temporary C and temporary registers to
the AX and DX registers. Finally, RNI runs the next machine instruction, ending the microcode routine.
The heart of the multiplication code is the CORX routine, which performs the multiplication loop, computing the product through shifts and adds. The first two lines set up the loop, initializing the
sum (tmpA) to 0. The number of loops is controlled by a special-purpose loop counter. The MAXC micro-instruction initializes the counter to 7 or 15, for a byte or word multiply respectively. The
first shift of tmpC is performed, putting the low bit into the carry flag.
The loop body performs the shift-and-add step. It tests the carry flag, the low bit of the multiplicand. It skips over the ADD if there is no carry (NCY). Otherwise, tmpB is added to tmpA. (As tmpA
gets shifted to the right, tmpB gets added to higher and higher positions in the result.) The tmpA and tmpC registers are rotated right. This also puts the next bit of the multiplicand into the carry
flag for the next cycle. The microcode jumps to the top of the loop if the counter is not zero (NCZ). Otherwise, the subroutine returns with the result in tmpA and tmpC.
ZERO → tmpA RRCY tmpC CORX: initialize right rotate
Σ → tmpC MAXC get rotate result, initialize counter to max value
JMPS NCY 8 5: top of loop
ADD tmpA conditionally add
Σ → tmpA F sum to tmpA, update flags to get carry
RRCY tmpA 8: 32-bit shift of tmpA/tmpC
Σ → tmpA RRCY tmpC
Σ → tmpC JMPS NCZ 5 loop to 5 if counter is not 0
The last subroutine is MULCOF, which configures the carry and overflow flags. The 8086 uses the rule that if the upper half of the result is nonzero, the carry and overflow flags are set, otherwise
they are cleared. The first two lines pass tmpA (the upper half of the result) through the ALU to set the zero flag for the conditional jump. As a side-effect, the other status flags will get set but
these values are "undefined" in the documentation.6 If the test is nonzero, the carry and overflow flags are set (SCOF), otherwise they are cleared (CCOF).5 The SCOF and CCOF micro-operations were
implemented solely for used by multiplication, illustrating how microcode can be designed around specific needs.
PASS tmpA MULCOF: pass tmpA through to test if zero
Σ → no dest JMPS 12 F update flags
JMPS Z 8 12: jump if zero
SCOF RTN otherwise set carry and overflow
CCOF RTN 8: clear carry and overflow
8-bit multiplication
The 8086 has separate instructions for 8-bit multiplication. The process for 8-bit multiplication is similar to 16-bit multiplication, except the values are half as long and the shift-and-add loop
executes 8 times instead of 16. As shown below, the 8-bit sum starts in the low half of the temporary A register and is shifted left into tmpC. Meanwhile, the 8-bit multiplier starts in the low half
of tmpC and is shifted out to the right. At the end, the result is split between tmpA and tmpC.
sum (tmpA) multiplier (tmpC)
The 8086 supports many instructions with byte and word versions, using 8-bit or 16-bit arguments. In most cases, the byte and word instructions use the same microcode, with the ALU and register
hardware using bytes or words based on the instruction. However, the byte- and word-multiply instructions use different registers, requiring microcode changes. In particular, the multiplier is in AL,
the low half of the accumulator. At the end, the 16-bit result is returned in AX, the full 16-bit accumulator; two micro-instructions assemble the result from tmpC and tmpA into the two bytes of the
accumulator, 'AL' and 'AH' respectively. Apart from those changes, the microcode is the same as the word multiply microcode discussed earlier.
AL → tmpC LRCY tmpC iMUL rmb:
M → tmpB CALL X0 PREIMUL
CALL CORX
CALL F1 NEGATE
CALL X0 IMULCOF
tmpC → AL JMPS X0 7
CALL MULCOF
tmpA → AH RNI
Signed multiplication
The 8086 (like most computers) represents signed numbers using a format called two's complement. While a regular byte holds a number from 0 to 255, a signed byte holds a number from -128 to 127. A
negative number is formed by flipping all the bits (known as the one's complement) and then adding 1, yielding the two's complement value.7 For instance, +5 is 0x05 while -5 is 0xfb. (Note that the
top bit of a number is set for a negative number; this is the sign bit.) The nice thing about two's complement numbers is that the same addition and subtraction operations work on both signed and
unsigned values. Unfortunately, this is not the case for signed multiplication, since signed and unsigned values yield different results due to sign extension.
The 8086 has separate multiplication instructions IMUL (Integer Multiply) to perform signed multiplication. The 8086 performs signed multiplication by converting the arguments to positive values,
performing unsigned multiplication, and then negating the result if necessary. As shown above, signed and unsigned multiplication both use the same microcode, but the microcode conditionally calls
some subroutines for signed multiplication. I will discuss those micro-subroutines below.
The first subroutine for signed multiplication is PREIMUL, performing preliminary operations for integer multiplication. It converts the two arguments, stored in tmpC and tmpB, to positive values. It
keeps track of the signs using an internal flag called F1, toggling this flag for a negative argument. This conveniently handles the rule that two negatives make a positive since complementing the F1
flag twice will clear it.
This microcode, below, illustrates the complexity of microcode and how micro-operations are carefully arranged to get the right values at the right time. The first micro-instruction performs one ALU
operation and sets up a second operation. The calling code had set up the ALU to perform LRCY tmpC, so that's the result returned by Σ (and discarded). Performing a left rotate and discarding the
result may seem pointless, but the important side-effect is that the top bit (i.e. the sign bit) ends up in the carry flag. The microcode does not have a conditional jump based on the sign, but has a
conditional jump based on carry, so the point is to test if tmpC is negative. The first micro-instruction also sets up negation (NEG tmpC) for the next ALU operation.
Σ → no dest NEG tmpC PREIMUL: set up negation of tmpC
JMPS NCY 7 jump if tmpC positive
Σ → tmpC CF1 if negative, negate tmpC, flip F1
JMPS 7 jump to shared code
LRCY tmpB 7:
Σ → no dest NEG tmpB set up negation of tmpB
JMPS NCY 11 jump if tmpB positive
Σ → tmpB CF1 RTN if negative, negate tmpB, flip F1
RTN 11: return
For the remaining lines, if the carry is clear (NCY), the next two lines are skipped. Otherwise, the ALU result (Σ) is written to tmpC, making it positive, and the F1 flag is complemented with CF1.
(The second short jump (JMPS) may look unnecessary, but I reordered the code for clarity.) The second half of the microcode performs a similar test on tmpB. If tmpB is negative, it is negated and F1
is toggled.
The microcode below is called after computing the result, if the result needs to be made negative. Negation is harder than you might expect because the result is split between the tmpA and tmpC
registers. The two's complement operation (NEG) is applied to the low word, while either 2's complement or one's complement (COM1) is applied to the upper word, depending on the carry for
mathematical reasons.8 The code also toggles F1 and makes tmpB positive; I think this code is only useful for division, which also uses the NEGATE subroutine.
NEG tmpC NEGATE: negate tmpC
Σ → tmpC COM1 tmpA F maybe complement tmpA
JMPS CY 6
NEG tmpA negate tmpA if there's no carry
Σ → tmpA CF1 6: toggle F1 for some reason
LRCY tmpB 7: test sign of tmpB
Σ → no dest NEG tmpB maybe negate tmpB
JMPS NCY 11 skip if tmpB positive
Σ → tmpB CF1 RTN else negate tmpB, toggle F1
RTN 11: return
The IMULCOF routine is similar to MULCOF, but the calculation is a bit trickier for a signed result. This routine sets the carry and overflow flags if the upper half of the result is significant,
that is, it is not just the sign extension of the lower half.9 In other words, the top byte is not significant if it duplicates the top bit (the sign bit) of the lower byte. The trick in the
microcode is to add the top bit of the lower byte to the upper byte by putting it in the carry flag and performing an add with carry (ADC) of 0. If the result is 0, the upper byte is not significant,
handling the positive and negative cases. (This also holds for words instead of bytes.)
ZERO → tmpB LRCY tmpC IMULCOF: get top bit of tmpC
Σ → no dest ADC tmpA add to tmpA and 0 (tmpB)
Σ → no dest F update flags
JMPS Z 8 12: jump if zero result
SCOF RTN otherwise set carry and overflow
CCOF RTN 8: clear carry and overflow
The hardware for multiplication
For the most part, the 8086 uses the regular ALU addition and shifts for the multiplication algorithm. Some special hardware features provide assistance.
Loop counter
The 8086 has a special 4-bit loop counter for multiplication. This counter starts at 7 for byte multiplication and 15 for word multiplication, based on the instruction. This loop counter allows the
microcode to decrement the counter, test for the end, and perform a conditional branch in one micro-operation. The counter is implemented with four flip-flops, along with logic to compute the value
after decrementing by one. The MAXC (Maximum Count) micro-instruction sets the counter to 7 or 15 for byte or word operations respectively. The NCZ (Not Counter Zero) micro-instruction has two
actions. First, it performs a conditional jump if the counter is nonzero. Second, it decrements the counter.
X register
The multiplication microcode uses an internal register called the X register to distinguish between the MUL and IMUL instructions. The X register is a 3-bit register that holds the ALU opcode,
indicated by bits 5–3 of the instruction.10 Since the instruction is held in the Instruction Register, you might wonder why a separate register is required. The motivation is that some opcodes
specify the type of ALU operation in the second byte of the instruction, the ModR/M byte, bits 5–3.11 Since the ALU operation is sometimes specified in the first byte and sometimes in the second
byte, the X register was added to handle both these cases.
For the most part, the X register indicates which of the eight standard ALU operations is selected (ADD, OR, ADC, SBB, AND, SUB, XOR, CMP). However, a few instructions use bit 0 of the X register to
distinguish between other pairs of instructions. For instance, it distinguishes between MUL and IMUL, DIV and IDIV, CMPS and SCAS, MOVS and LODS, or AAA and AAS. While these instruction pairs may
appear to have arbitrary opcodes, they have been carefully assigned. The microcode can test this bit using the X0 condition and perform conditional jumps.
The implementation of the X register is straightforward, consisting of three flip-flops to hold the three bits of the instruction. The flip-flops are loaded from the prefetch queue bus during First
Clock and during Second Clock for appropriate instructions, as the instruction bytes travel over the bus. Testing bit 0 of the X register with the X0 condition is supported by the microcode condition
evaluation circuitry, so it can be used for conditional jumps in the microcode.
The F1 flag
The multiplication microcode uses an internal flag called F1,12 which has two distinct uses. The flag keeps track of a REP prefix for use with a string operation. But the F1 flag is also used by
signed multiplication and division to keep track of the sign. The F1 flag can be toggled by microcode through the CF1 (Complement F1) micro-instruction. The F1 flag is implemented with a flip-flop,
along with a multiplexer to select the value. It is cleared when a new instruction starts, set by a REP prefix, and toggled by the CF1 micro-instruction.
The diagram below shows how the F1 latch and the loop counter appear on the die. In this image, the metal layer has been removed, showing the silicon and the polysilicon wiring underneath.
The counter and F1 latch as they appear on the die. The latch for the REP state is also here.
Later advances in multiplication
The 8086 was pretty slow at multiplying compared to later Intel processors.13 The 8086 took up to 133 clock cycles to multiply unsigned 16-bit values due to the complicated microcode loops. By 1982,
the Intel 286 processor cut this time down to 21 clock cycles. The Intel 486 (1989) used an improved algorithm that could end early, so multiplying by a small number could take just 9 cycles.
Although these optimizations improved performance, they still depended on looping over the bits. With the shift to 32-bit processors, the loop time became unwieldy. The solution was to replace the
loop with hardware: instead of performing 32 shift-and-add loops, an array of adders could compute the multiplication in one step. This quantity of hardware was unreasonable in the 8086 era, but as
Moore's law made transistors smaller and cheaper, hardware multiplication became practical. For instance, the Cyrix Cx486SLC (1992) had a 16-bit hardware multiplier that cut word multiply down to 3
cycles. The Intel Core 2 (2006) was even faster, able to complete a 32-bit multiplication every clock cycle.
Hardware multiplication is a fairly complicated subject, with many optimizations to maximize performance while minimizing hardware.14 Simply replacing the loop with a sequence of 32 adders is too
slow because the result would be delayed while propagating through all the adders. The solution is to arrange the adders as a tree to provide parallelism. The first layer has 16 adders to add pairs
of terms. The next layer adds pairs of these partial sums, and so forth. The resulting tree of adders is 5 layers deep rather than 32, reducing the time to compute the sum. Real multipliers achieve
further performance improvements by splitting up the adders and creating a more complex tree: the venerable Wallace tree (1964) and Dadda multiplier (1965) are two popular approaches. Another
optimization is the Booth algorithm (1951), which performs signed multiplication directly, without converting the arguments to positive values first. The Pentium 4 (2000) used a Booth encoder and a
Wallace tree (ref), but research in the early 2000s found the Dadda tree is faster and it is now more popular.
Multiplication is much harder to compute than addition or subtraction. The 8086 processor hid this complexity from the programmer by providing four multiplication instructions for byte and word
multiplication of signed or unsigned values. These instructions implemented multiplication in microcode, performing shifts and adds in a loop. By using microcode subroutines and conditional
execution, these four machine instructions share most of the microcode. As the microcode capacity of the 8086 was very small, this was a critical feature of the implementation.
If you made it through all the discussion of microcode, congratulations! Microcode is even harder to understand than assembly code. Part of the problem is that microcode is very fine-grain, with even
ALU operations split into multiple steps. Another complication is that 8086 microcode performs a register move and another operation in parallel, so it's hard to keep track of what's going on.
Microcode can seem a bit like a jigsaw puzzle, with pieces carefully fit together as compactly as possible. I hope the explanations here made sense, or at least gave you a feel for how microcode
I've written multiple posts on the 8086 so far and plan to continue reverse-engineering the 8086 die so follow me on Twitter @kenshirriff or RSS for updates. I've also started experimenting with
Mastodon recently as @[email protected].
Notes and references
1. Mainframes going back to ENIAC had multiply and divide instructions. However, early microprocessors took a step back and didn't supports these more complex operations. (My theory is that the
decline in memory prices made it more cost-effective to implement multiply and divide in software than hardware.) The National Semiconductor IMP-16, a 16-bit bit-slice microprocessor from 1973,
may be the first with multiply and divide instructions. The 8-bit Motorola 6809 processor (1978) included 8-bit multiplication but not division. I think the 8086 was the first Intel processor to
support multiplication. ↩
2. The 8086 also supported division. Although the division instructions are similar to multiplication in many ways, I'm focusing on multiplication and ignoring division for this blog post. ↩
3. My microcode analysis is based on Andrew Jenner's 8086 microcode disassembly. ↩
4. I think CORX stands for Core Multiply and CORD stands for Core Divide. ↩
5. The definitions of carry and overflow are different for multiplication compared to addition and subtraction. Note that the result of a multiplication operation will always fit in the available
result space, which is twice as large as the arguments. For instance, the biggest value you can get by multiplying 16-bit values is 0xffff×0xffff=0xfffe0001 which fits into 32 bits. (Signed and
8-bit multiplications fit similarly.) This is in contrast to addition and subtraction, which can exceed their available space. A carry indicates that an addition exceeded its space when treated
as unsigned, while an overflow indicates that an addition exceeded its space when treated as unsigned. ↩
6. The Intel documentation states that the sign, carry, overflow, and parity flags are undefined after the MUL operation, even though the microcode causes them to be computed. The meaning of
"undefined" is that programmers shouldn't count on the flag values because Intel might change the behavior in later chips. This thread discusses the effects of MUL on the flags, and how the
behavior is different on the NEC V20 chip. ↩
7. It may be worth explaining why the two's complement of a number is defined by adding 1 to the one's complement. The one's complement of a number simply flips all the bits. If you take a byte
value n, 0xff - n is the one's complement, since a 1 bit in n produces a 0 bit in the result.
Now, suppose we want to represent -5 as a signed byte. Adding 0x100 will keep the same byte value with a carry out of the byte. But 0x100 - 5 = (1 + 0xff) - 5 = 1 + (0xff - 5) = 1 + (one's
complement of 5). Thus, it makes sense mathematically to represent -5 by adding 1 to the one's complement of 5, and this holds for any value. ↩
8. The negation code is a bit tricky because the result is split across two words. In most cases, the upper word is bitwise complemented. However, if the lower word is zero, then the upper word is
negated (two's complement). I'll demonstrate with 16-bit values to keep the examples small. The number 257 (0x0101) is negated to form -257 (0xfeff). Note that the upper byte is the one's
complement (0x01 vs 0xfe) while the lower byte is two's complement (0x01 vs 0xff). On the other hand, the number 256 (0x0100) is negated to form -256 (0xff00). In this case, the upper byte is the
two's complement (0x01 vs 0xff) and the lower byte is also the two's complement (0x00 vs 0x00).
(Mathematical explanation: the two's complement is formed by taking the one's complement and adding 1. In most cases, there won't be a carry from the low byte to the upper byte, so the upper byte
will remain the one's complement. However, if the low byte is 0, the complement is 0xff and adding 1 will form a carry. Adding this carry to the upper byte yields the two's complement of that
To support multi-word negation, the 8086's NEG instruction clears the carry flag if the operand is 0, and otherwise sets the carry flag. (This is the opposite from the above because subtractions
(including NEG) treat the carry flag as a borrow flag, with the opposite meaning.) The microcode NEG operation has identical behavior to the machine instruction, since it is used to implement the
machine instruction.
Thus to perform a two-word negation, the microcode negates the low word (tmpC) and updates the flags (F). If the carry is set, the one's complement is applied to the upper word (tmpA). But if the
carry is cleared, the two's complement is applied to tmpA. ↩
9. The IMULCOF routine considers the upper half of the result significant if it is not the sign extension of the lower half. For instance, dropping the top byte of 0x0005 (+5) yields 0x05 (+5).
Dropping the top byte of 0xfffb (-5) yields 0xfb (-5). Thus, the upper byte is not significant in these cases. Conversely, dropping the top byte of 0x00fb (+251) yields 0xfb (-5), so the upper
byte is significant. ↩
10. Curiously, the 8086 patent states that the X register is a 4-bit register holding bits 3–6 of the byte (col. 9, line 20). But looking at the die, it is a 3-bit register holding bits 3–5 of the
byte. ↩
11. Some instructions are specified by bits 5–3 in the ModR/M byte rather than in the first opcode byte. The motivation is to avoid wasting bits for instructions that use a ModR/M byte but don't need
a register specification. For instance, consider the instruction ADD [BX],0x1234. This instruction uses a ModR/M byte to specify the memory address. However, because it uses an immediate operand,
it does not need the register specification normally provided by bits 5–3 of the ModR/M byte. This frees up the bits to specify the instruction. From one perspective, this is an ugly hack, while
from another perspective it is a clever optimization. ↩
12. Andrew Jenner discusses the F1 flag and the interaction between REP and multiplication here. ↩
13. Here are some detailed performance numbers. The 8086 processor takes 70–77 clock cycles to multiply 8-bit values and 118–133 clock cycles to multiply 16-bit values. Signed multiplies are a bit
slower because of the sign calculations: 80–98 and 128–154 clock cycles respectively. The time is variable because of the conditional jumps in the multiplication process.
The Intel 186 (1982) optimized multiplication slightly, bringing the register word multiply down to 35–37 cycles. The Intel 286 (also 1982) reduced this to 21 clocks. The 486 (1989) used a
shift-add multiply function but it had an "early out" algorithm that stopped when the remaining bits were zero, so a 16-bit multiply could take from 9 to 22 clocks. The 8087 floating point
coprocessor (1980) used radix-4 multiplication, multiplying by pairs of bits at a time and either adding or subtracting. This yields half the addition cycles. The Pentium's P5 micro-architecture
(1993) took the unusual approach of reusing the floating-point unit's hardware multiplier for integer multiplication, taking 10 cycles for a 32-bit multiplication. ↩
14. This presentation gives a good overview of implementations of multiplication in hardware. ↩ | {"url":"https://www.righto.com/2023/03/","timestamp":"2024-11-10T11:27:49Z","content_type":"application/xhtml+xml","content_length":"218522","record_id":"<urn:uuid:74b94618-e8f3-447c-8391-eeec22bf4dc4>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00834.warc.gz"} |
Table of contents
CHAPTER ONE……………………………………………………………………. 3
Introduction to financial mathematics
CHAPTER TWO…………………………………………………………………….. 8
Financial algebra
CHAPTER THREE………………………………………. 25
Descriptive Statistics
CHAPTER FOUR…………………………………….. 53
Time value of money
CHAPTER FIVE……………………………………… 85
Financial forecasting
CHAPTER SIX……………………………………… 105
Financial calculus
CHAPTER SEVEN……………………………… 115
Probability theory
CHAPTER EIGHT………………………………. 132
Index numbers
Nature of financial decision
Financial decisions are those made by financial managers of a firm. It‟s broadly classified into two.
1. Managerial decision
2. Routine decision
Managerial decisions
These are Decisions that require technical skills, planning and expertise of a financial manager. It‟s classified into four:
It involves looking for finance to acquire assets of the firm and may include:
• Issue of ordinary shares
• Long term loan
• Preference shares
• Investment decision
It‟s the responsibility of a financial manager to determine whether acquired funds should be invested in order to generate revenue. Financial manager must do a proper appraisal of any investment
that may be undertaken.
Dividends are part of the earnings distributed to ordinary shareholders for their investment in the company. Financial manager has to consider the following:
• How much to pay
• When to pay
• How to pay e. cash or bonus issue
• Why to pay
TO BUY THIS STUDY TEXT, CALL | TEXT | WHATSAPP +254728 – 776 – 317 or Email info@masomomsingi.com
It’s the relationship between independent variable and the dependent variable. It consists of a constant and a variable.
A constant – This is a quantity whose value remains unchanged throughout a particular analysis
e.g. fixed cost, rent, and salary.
A variable – This is a quantity which takes various values in a particular problem
Suppose an item is sold at Sh 11 per unit. Let S represent sales rate revenue in shillings and let Q represents quantity sold.
Then the function representing these two variables is given as: S = 11Q
S and Q are variables whereas the price – Sh 11 – is a constant.
Types of variables
Independent variable – this is a variable which determines the quantity or the value of some other variable referred to as the dependent variable. In Illustration 1.1, Q is the independent variable
while S is the dependent variable.
An independent variable is also called a predictor variable while the dependent variable is also known as the response variable i.e. Q predicts S and S responds to Q.
A function – This is a relationship in which values of a dependent variable are determined by the values of one or more independent In illustration 1.1 sales is a function of quantity, written as S =
Demand = f( price, prices of substitutes and complements, income levels,….) Savings = f(investment, interest rates, income levels,….)
Note that the dependent variable is always one while the independent variable can be more than one.
TO BUY THIS STUDY TEXT, CALL | TEXT | WHATSAPP +254728 – 776 – 317 or Email info@masomomsingi.com
Statistics is the art and science of getting information from data or numbers to help in decision making.
As a science, statistics follows a systematic procedure to reach objective decisions or solutions to problems.
As an art statistics utilizes personal judgment and intuition to reach a solution. It depends on experience of the individual involved. It is more subjective.
Statistics provides us with tools that aid decision making. For example, using statistics we can estimate that expected returns and associated risks of a given investment opportunity.
Statistics involves collection of data, analysis, presentation and interpretation of data.
There are various types of summary measures including averages and measures of dispersion. An average is a figure which represents the whole data. It removes all unnecessary details and gives a clear
picture of the data under investigation.
Qualities of a good average
1. It should be clearly defined
2. Should be based on all values or observation
3. Should be easily understood and calculated
4. Should be capable of further statistical investigation/treatment
5. Should be least affected by fluctuations of sampling
Definitions of key terms
Measures of central tendency are single numbers that are used to summarize a larger set of data in a distribution of scores. The three measures of central tendency are mean, median, and mode. They
are also called types of averages
Measures of dispersion – These are important for describing the spread of the data, or its variation around a central value. Such measures of dispersion include: standard deviation, inter- quartile
range, range, mean difference, median absolute deviation, average absolute deviation (or simply average deviation)
Variance is the sum of squared deviations divided by the number of observations. It is the average of the squares of the deviation of the individual values from their means.
TO BUY THIS STUDY TEXT, CALL | TEXT | WHATSAPP +254728 – 776 – 317 or Email info@masomomsingi.com
At the end of this chapter you should be able to:
1. Explain meaning of time value of money and its role in
2. Explain the concept of future value and perform compounding
3. Explain the concept of present value and perform discounting
4. Apply the mathematics of finance to accumulate a future sum, preparing loan amortization schedules, and determining interest or growth
A shilling today is worth more than a shilling tomorrow. An individual would thus prefer to receive money now rather than that same amount later. A shilling in ones possession today is more valuable
than a shilling to be received in future because, first, the shilling in hand can be put to immediate productive use, and, secondly, a shilling in hand is free from the uncertainties of future
expectations (It is a sure shilling).
Financial values and decisions can be assessed by using either future value (FV) or present value (PV) techniques. These techniques result in the same decisions, but adopt different approaches to the
Future value techniques
Measure cash flow at the some future point in time – typically at the end of a projects life. The Future Value (FV), or terminal value, is the value at some time in future of a present sum of money,
or a series of payments or receipts. In other words the FV refers to the amount of money an investment will grow to over some period of time at some given interest rate. FV techniques use compounding
to find the future value of each cash flow at the given future date and the sums those values to find the value of cash flows.
Present value techniques
Measure each cash flows at the start of a projects life (time zero).The Present Value (PV) is the current value of a future amount of money, or a series of future payments or receipts. Present value
is just like cash in hand today. PV techniques use discounting to find the PV of each cash flow at time zero and then sum these values to find the total value of the cash flows.
Although FV and PV techniques result in the same decisions, since financial managers make decisions in the present, they tend to rely primarily on PV techniques.
TO BUY THIS STUDY TEXT, CALL | TEXT | WHATSAPP +254728 – 776 – 317 or Email info@masomomsingi.com
It involves determining the future financial requirements of the firm. This requires financial planning using budgets.
Importance of financial forecasting
• Facilitates financial planning e. determination of cash surplus or deficit that are likely to occur in future.
• Facilitates control of expenditure so as to minimize wastage of financial
• Forecasting using targets and budgets acts as a motivation to employees who aim at achieving targets set
Strategic plan:
It‟s a blue print road map that indicates what the firm intends to do and how to do it. It consists of:
• The mission/purpose of existence
• Scope: these are the lines of business
• Objectives: specific goals in quantitative
• Strategies: instruments to be used in achieving firms
Role of financial manager in strategic planning
1. Educate strategic planning team on financial implication on various options
2. Ensure strategic plan is viable financially
3. Translate strategic plan into long range financial plans
Elements of financial planning
1. Assumption: that they should be clearly
2. sales/revenue forecast: it‟s the starting point of financing planning since most of other variables it relates to sales
3. pro-forma financial statement: balance sheet, cash flow, income statement
4. assets/investment requirement: this will reveal investment required to achieve forecasted/budgets sales in short-term and long term
5. Financial plan: spells out proposed means of financing investment.
6. Cash budget: it indicates the cash inflow and cash outflow
Importance of financial forecasting/need
1. Forces management to plan in advance and allocate resources efficiently
2. Forces managers to avoid surprises which may occur in the course of operations e.g if there is a cash deficit the managers will decide how to finance the
3. Used for control purposes e. it enables the company to control expenses and to avoid wastage due to operations of the firm.
4. Used for motivation purpose e.g. since managers and employees are aware of what is
TO BUY THIS STUDY TEXT, CALL | TEXT | WHATSAPP +254728 – 776 – 317 or Email info@masomomsingi.com
It explains how the value of the variable varies as the other variable changes.
Calculus is concerned with the mathematical analysis of change or movement. There are two basic operations in calculus:
1. Differentiation
2. Integration
These two basic operations are reverse of one another in the same way as addition and subtraction or multiplication and division
It is concerned with rates of change e.g. profit with respect to output
1. Revenue with respect to output
2. Change of sales with respect to level of advertisement Savings with respect to income, interest rates
Rates of changes and slope (gradients)
It estimates a slope (gradient of graph) of a particular point. The derivative of a function
?? gives the exact change of a point. It‟s the process of finding the derivative of a function.
Probability is a measure of likelihood, the possibility or chance that an event will happen in future.
It can be considered as a quantification of uncertainty.
Uncertainty may also be expressed as likelihood, chance or risk theory. It is a branch of mathematics concerned with the concept and measurement of uncertainty.
Much in life is characterized by uncertainty in actual decision making.
Probability can only assume a value between 0 and 1 inclusive. The closer a probability is to zero the more improbable that something will happen. The closer the probability is to one the more likely
it will happen.
Definitions of key terms
Random experiment results in one of a number of possible outcomes e.g. tossing a coin
Outcome is the result of an experiment e.g. head up, gain, loss, etc. Specific outcomes are known as events.
Trial– Each repetition of an experiment can be thought of as a trial which has an observable outcome e.g. in tossing a coin, a single toss is a trial which has an outcome as either head or tail
Sample space is the set of all possible outcomes in an experiment e.g. a single toss of a coin, S=(H,T). The sample space can be finite or infinite. A finite sample space has a finite number of
possible outcomes e.g. in tossing a coin only 2 outcomes are possible.
An infinite sample space has an infinite number of possible outcomes e.g. time between arrival of telephone calls and telephone exchange.
An Event of an experiment is a subset of a sample space e.g. in tossing a coin twice S= (HH, HT,
TH, TT) HH is a subset of a sample space.
Mutually exclusive event – A set of events is said to be mutually exclusive if the occurrence of any one of the events precludes the occurrence of other events i.e. the occurrence of any one event
means none of the others can occur at the same time e.g. the events head and tail are mutually exclusive
Collectively exclusive event – A set of events is said to be collectively exclusive if their union accounts for all possible outcomes i.e. one of their events must occur when an experiment is
TO BUY THIS STUDY TEXT, CALL | TEXT | WHATSAPP +254728 – 776 – 317 or Email info@masomomsingi.com
It’s a number which indicates the level of a certain phenomena at any given date in comparison with the level of the same phenomena at the same standard date.
It‟s a series of number by which changes in magnitude of phenomena or measures from time to time or from place to place. It provides an opportunity for measuring the relative change of a valuable
where measure of its actual change is inconvenient or impossible. An index number is constructed by selecting a base year as a starting point.
Price index numbers
They are important because they show that the value of money is fluctuating i.e. appreciating or depreciating.
A rise in the index numbers will signify that there is deterioration in value of money and vice versa.
Factors to be considered when constructing price index numbers
1. Purpose of index numbers
The purpose must be determined before their construction otherwise the results will be useless. When purpose is known correctly, true results can be obtained.
2. Selection of commodities
When purpose is known, the selection of commodities for that purpose becomes easy and accurate to a responsible when selecting commodities.
3. Price quotation
It is impossible to collect the prices of all selected commodities from all places in country where they are marketed. A sample of the market needs to be selected and it should be from those places
where given commodities are marketed in large numbers.
TO BUY THIS STUDY TEXT, CALL | TEXT | WHATSAPP +254728 – 776 – 317 or Email info@masomomsingi.com
(Visited 294 times, 1 visits today)
You must be logged in to post a comment. | {"url":"https://masomomsingi.com/cifa-notes-financial-mathematics-sample-notes/","timestamp":"2024-11-11T23:48:59Z","content_type":"text/html","content_length":"84532","record_id":"<urn:uuid:663528df-1bb0-4403-99ab-17b88e817088>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00463.warc.gz"} |
HyperWorks CFD – Darcy and Forchheimer Coefficients
Product: HyperWorks CFD
Product Version: HyperWorks CFD 2020.0 or above
Hyperworks CFD provides a simplified calculator to transfer user data in the form of velocity vs pressure drop to Darcy and Forchheimer coefficients required for AcuSolve. Once in the porosity micro
dialog, click on the calculator icon to show the attached editor, where one can provide velocity and pressure drop data. The computed coefficients are also shown in the below table.
From the Flow ribbon, Domain tools, click the Porous tool (Select solid & specify orientation)
Ian P Altair Community Member
How are the permeability coefficients (K1, K2, K3) brought into the calculation in contrast to the Darcy and Forchheimer coefficients?
If the Darcy (D) value is related to velocity and the Forchheimer (F) value is related to velocity^2, are K1,2,3 multiplied to both D and F? D only? F only? Linearly, inversely? The math isn't
clear in the user guide.
I'm speaking in reference to: https://en.wikipedia.org/wiki/Darcy's_law#Quadratic_law
So in Wikipedia, D = 1/k and F = 1/k_1
Thanks for the clarifying. | {"url":"https://community.altair.com/discussion/33230/hyperworks-cfd-darcy-and-forchheimer-coefficients","timestamp":"2024-11-13T12:37:42Z","content_type":"text/html","content_length":"298886","record_id":"<urn:uuid:83c8d3dc-fe14-4029-8f02-f3043eca5025>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00605.warc.gz"} |
Snakify - Python 3 Interactive Course
Lesson 2
Integer and float numbers
We already know the following operators which may be applied to numbers: +, -, * and **. The division operator / for integers gives a floating-point real number (an object of type float). The
exponentiation ** also returns a float when the power is negative:
print(17 / 3) # gives 5.66666666667
print(2 ** 4) # gives 16
print(2 ** -2) # gives 0.25
There's a special operation for integer division where the remainder is discarded: //. The operation that yields a remainder of such a division looks like %. Both operation always yield an object of
type int.
print(17 / 3) # gives 5.66666666667
print(17 // 3) # gives 5
print(17 % 3) # gives 2
Advertising by Google, may be based on your interests
When we read an integer value, we read a line with input() and then cast a string to integer using int(). When we read a floating-point number, we need to cast the string to float using float():
x = float(input())
Floats with very big or very small absolute value can be written using a scientific notation. Eg., the distance from the Earth to the Sun is 1.496·10^11, or 1.496e11 in Python. The mass of one
molecule of the water is 2.99·10^-23, or 2.99e-23 in Python.
One can cast float objects to int objects by discarding the fraction part using the int() function. This function demonstrates so called rounding towards zero behavior:
print(int(1.3)) # gives 1
print(int(1.7)) # gives 1
print(int(-1.3)) # gives -1
print(int(-1.7)) # gives -1
There's also a function round() that performs the usual rounding:
print(round(1.3)) # gives 1
print(round(1.7)) # gives 2
print(round(-1.3)) # gives -1
print(round(-1.7)) # gives -2
Floating-point real numbers can't be represented with exact precision due to hardware limitations. This can lead to cumbersome effects. See the Python docs for the details.
print(0.1 + 0.2) # gives 0.30000000000000004
Advertising by Google, may be based on your interests
Python has many auxiliary functions for calculations with floats. They can be found in the math module.
To use this module, we need to import it first by writing the following instruction at the beginning of the program:
import math
For example, if we want to find a ceiling value for x - the smallest integer not less than x - we call the appropriate function from the math module: math.ceil(x). The syntax for calling functions
from modules is always the same: module_name.function_name(argument_1, argument_2, ...)
import math
x = math.ceil(4.2)
print(math.ceil(1 + 3.8))
There's another way to use functions from modules: to import the certain functions by naming them:
from math import ceil
x = 7 / 2
y = ceil(x)
Some of the functions dealing with numbers - int(), round() and abs() (absolute value aka modulus) - are built-in and don't require any imports.
All the functions of any standard Python module are documented on the official Python website. Here's the description for math module. The description of some functions is given:
│Function│ Description │
│Rounding │
│floor(x)│Return the floor of x, the largest integer less than or equal to x. │
│ceil(x) │Return the ceiling of x, the smallest integer greater than or equal to x. │
│Roots and logarithms │
│sqrt(x) │Return the square root of x │
│log(x) │With one argument, return the natural logarithm of x (to base e). With two arguments, return the logarithm of x to the given base │
│e │The mathematical constant e = 2,71828... │
│Trigonometry │
│sin(x) │Return the sine of x radians │
│asin(x) │Return the arcsine of x, in radians │
│pi │The mathematical constant π = 3.1415... │
Advertising by Google, may be based on your interests | {"url":"https://www.snakify.org/en/lessons/integer_float_numbers/","timestamp":"2024-11-01T23:48:44Z","content_type":"text/html","content_length":"36544","record_id":"<urn:uuid:26e1297e-1890-4c98-999c-8c528ac489a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00266.warc.gz"} |
Reply To: answer - Hyperfine Course
March 12, 2024 at 11:45 pm #5470
Oops i seem to have written this in the wrong forum; this is my answer:
In the cube, there is a 4-fold rotation axis through both Fe-IIb’s. So this can be the z-axis.
There are also two 2-fold rotation axis through the other Fe-IIa. any of these can be chosen as z-axis for the Fe-I. | {"url":"https://www.hyperfinecourse.org/forums/reply/5470/","timestamp":"2024-11-09T17:38:28Z","content_type":"text/html","content_length":"156206","record_id":"<urn:uuid:34d597e3-c17c-46a1-85d5-11f5b830f050>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00268.warc.gz"} |
CV - David J. Thouless | Lindau Mediatheque
The ever-increasing power, and shrinking size, of electronic gadgets and computers has largely been influenced by the use of superconductive materials, which allow a greater throughput of electrical
energy relative to their size – but how do superconductive materials work?
There is normally a fixed relationship between how much electrical potential you put across a wire and how much current flows through it. However, when matter is compressed to a flat plane, or cooled
to near absolute zero it can take on completely new electrical properties. We already know that materials can make ‘phase transitions’ between a solid, liquid or gaseous state, but David Thouless,
his fellow laureates Duncan Haldane and Michael Kosterlitz and others showed that materials also make sudden ‘exotic’ phase transitions in their electrical properties, making them superconductive.
The laureates used the abstract mathematical method of topology to study these unusual phases, such as how a film of helium changes when super-chilled, and how those phase transitions then change
their properties, such as how conductive they are to electricity and magnetism. Topology is a mathematical system that studies what properties are preserved when objects are manipulated or deformed.
A topological surface is partly defined by how many holes there are. So, in topological terms, a doughnut is related to a coffee cup (both have one hole), but a ball is different.
In the early 1970s, Thouless and Kosterlitz demonstrated that superconductivity could occur at low temperatures and also explained the mechanism, phase transition, which makes superconductivity stop
at higher temperatures. Their work overturned the accepted theory that superconductivity or suprafluidity could not occur in thin layers. It is now known that ‘exotic’ electrical properties can be
found in a range of ordinary three-dimensional materials. In mathematical terms, these transitions effectively involve leaping from one topological form to another. In the 1980s, Thouless was able to
explain a previous experiment with very thin electrically conducting layers in which conductivity was measured in precise integer steps. He showed that these integers were topological. For his work,
which the Nobel Committee said “opened the door on an unknown world”, Thouless was awarded a half share of the Nobel Prize, the remainder being split between Kosterlitz and Haldane.
Their work laid the foundations for recent research in condensed matter physics, and the development of topological materials that could one day be used in new generations of electronics and
superconductors, or in future quantum computers.
David James Thouless was born in Scotland in 1934, and grew up in Cambridge. He was a scholar at Winchester College and then, read Natural Sciences at Trinity Hall, University of Cambridge. After
graduating in 1955, David spent his first year of graduate study with Hans Bethe (1967 Nobel Laureate) in Cambridge, who was then on sabbatical leave in Cambridge. When Hans returned to Cornell he
invited David to go with him. David obtained his PhD from Cornell University in 1958. The same summer he married Margaret Scrase, a biology undergraduate at Cornell. They have two sons and a
David did one year of post doctoral study at the Lawrence Radiation Laboratory in Berkeley and then two more years in Birmingham, UK, with Professor Rudolph Peierls. He spent four years as a lecturer
in DAMPT at Cambridge University, before being appointed in 1965 as a professor of mathematical physics at Birmingham University. After fourteen years in Birmingham and a brief period at Yale
University, He became a professor in the Physics Department at the University of Washington Seattle (1980-2003). He is now an emeritus professor there, but is currently residing in Cambridge, UK.
David is a member of the National Academy (USA) and a fellow of the Royal Society (UK) He was awarded the Wolf Prize in Physics (1990) and has received a number of other awards over the years.
David J. Thouless passed away on 6 April 2019, at the age of 84.
Picture: © Peter Badge/Lindau Nobel Laureate Meetings | {"url":"https://mediatheque.lindau-nobel.org/laureates/thouless/cv","timestamp":"2024-11-04T05:49:12Z","content_type":"text/html","content_length":"31269","record_id":"<urn:uuid:5bbee32e-b1c7-4885-8405-deccafaf395c>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00763.warc.gz"} |
Related rates
I don't like this new tutorial. Take me back to the old tutorial (no game version)!
To follow this tutorial, you should be familiar with
the chain rule for derivatives.
Rates of change
We start by recalling some facts about the rate of change of a quantity: The volume of rocket fuel in a space ship booster is given by
$t$ seconds after its launch.
Related rates problem
In a
related rates problem,
we are given the rate of change of one or more quantities, and are required to find the rate of change of one or more
quantities. For instance (as in the first example in %4) we may be given the rate at which the radius of a circle is growing, and want to know how fast the area is growing. %9:
The %21 of %23 is %33 at a rate of %34 %24/sec. How fast is its %22 %33 at the instant when its radius is %35 cm?
The falling ladder
Variants of "the falling ladder" problem are found in practically every calculus textbook (see, for instance, Example 2 in %4). Here is one of them:
A carelessly placed %40 ft ladder is sliding down a wall in such a way that %55 at a rate of %45 ft/sec. Your siamese cat Papanutski is sitting %56 directly in line with the approaching base of the
ladder%57. How fast is %58 when Papnutski is hit?
A. The problem
1. Identify the changing quantities. 2. Restate the problem in terms of rates of change.
The given problem is: A carelessly placed %40 ft ladder is sliding down a wall in such a way that %55 at a rate of %45 ft/sec. Your siamese cat Papanutski is sitting %56 directly in line with the
approaching base of the ladder%57. How fast is %58 when Papnutski is hit?
3. Rewrite the problem using mathematical notation.
B. The relationship
1. Draw a diagram, if appropriate, showing the changing quantities.
Sketch of changing quantities (Click on the correct sketch.)
Note Changing quantities are represented by letters; non-changing quantities are represented by numbers.
An equation that relates the changing quantities is
3. Write down the derived equation..
C. The solution
1. Substitute into the derived equation the given values of the quantities and their derivatives. 2. Solve for the derivative required..
To solve for $%61$, we first need to know the value of $%63$. For this, use the equation that relates the changing quantitites.
The required rate of change is therefore
Now try the exercises in %4, some the %8, or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.
Last Updated: April, 2016
Copyright © | {"url":"https://www.zweigmedia.com/tuts/tutRelatedRates.php?ed=x&lang=en&game=true","timestamp":"2024-11-10T13:55:48Z","content_type":"text/html","content_length":"74649","record_id":"<urn:uuid:36f567b4-478c-4769-81cf-c5a8c73e6955>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00109.warc.gz"} |
Subsets - math word problem (329)
How many are all subsets of set ?
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
Tips for related online calculators
You need to know the following knowledge to solve this word math problem:
Related math problems and questions: | {"url":"https://www.hackmath.net/en/math-problem/329","timestamp":"2024-11-02T11:45:52Z","content_type":"text/html","content_length":"50094","record_id":"<urn:uuid:2df1a04f-4d1f-4003-aa66-b267bbaa93b4>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00703.warc.gz"} |
OPTI basics of measuring
0 of 3 Questions completed
[Finish Quiz]
[Finish Quiz]
You have already completed the quiz before. Hence you can not start it again.
You must sign in or sign up to start the quiz.
You must first complete the following:
Quiz complete. Results are being recorded.
0 of 3 Questions answered correctly
Your time:
Time has elapsed
You have reached 0 of 0 point(s), (0)
Earned Point(s): 0 of 0, (0)
0 Essay(s) Pending (Possible Point(s): 0)
1. Current
2. Review
3. Answered
4. Correct
5. Incorrect | {"url":"https://mechanizedforestry.com/quizzes/opti-basics-of-measuring/","timestamp":"2024-11-02T01:10:43Z","content_type":"text/html","content_length":"61388","record_id":"<urn:uuid:2fe76864-7411-4552-81b5-1775d96ba361>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00623.warc.gz"} |
A machine learning glossary for hackers
This is a short summary of some of the terminology used in machine learning, with an emphasis on neural networks. I’ve put it together primarily to help my own understanding, phrasing it largely in
non-mathematical terms. As such it may be of use to others who come from more of a programming than a mathematical background. Note that there are some overlapping or even slightly different meanings
of some words or expressions depending on the context (e.g. “linear” and “dimension”), and I’ve tried to make this clear where appropriate. Disclaimer: It is by no means exhaustive or definitive.
This is the function which determines whether a unit should be activated or not, i.e. a “gate” between a unit’s input and output. It is computed by multiplying the input number(s) by the weight, i.e.
the weighted sum of all of the inputs from the previous layer. At its simplest it could be a “step” function which outputs one number below the threshold and another above the threshold. In practice
activation functions are non-linear (in the machine learning rather than statistical sense), i.e. a curved line rather than a straight line. One of the most common activation functions at the moment
is the rectified linear unit. A sigmoid function might be used for binary classification. Given there may be a large number of activation functions (one for each unit in the network) and it may need
to be computed a very large number of times, activation functions are designed for computational efficiency.
Back propagation is an algorithm used to determine a network’s weights. It computes the gradient of the loss function with the respect to the weights, with the gradient determined by computing the
derivatives or partial derivatives of the loss function, and does this each layer at a time iterating backwards.
The number of examples used in one iteration of training. Stochastic gradient descent will have a batch size of 1, while Mini-batch gradient descent will have more. In a Recurrent Neural Network,
you’d typically have as large a batch size as your memory allows, given a larger batch size will work through the training examples more quickly and a small batch size can lead to irregularities in
the rate that the loss decreases over the epochs, but some experimentation may be required to optimise batch size.
In a neural network, a bias unit is a special unit typically present in each (non output) layer which is connected to the next layer but not the previous layer, and which stores the value of 1. It
allows the activation function to be shifted to the left or to the right. It is sometimes called the intercept term in linear regression.
In supervised learning, this is trying to predict a discreet value, e.g. true/false or sun/cloud/rain/snow. Contrast with Regression. A binary classification has 2 discreet values, while with more
than 2 is a multiclass classification. You can think of classification as working with “labels”.
The process of getting very close to the correct answer.
A Convolutional Neural Network (CNN) is a network architecture commonly used for image processing. It uses convolutions, which are filters which pass over the image and apply transformations to a
grid, to emphasise certain features in the image, e.g. vertical edges or horizontal edges. CNNs often use pooling, which is a way of compressing an image, e.g. max pooling preserves only the highest
value in a grid. The network would typically train on the results of the convolutions, and the pooling, rather than the original input pixels.
See Loss function.
Slope of a line tangent to a curve (e.g. generated by a function). In neural networks, this is usually applied to the cost function to determine how correct or incorrect its output is - the steeper
the curve, i.e. the bigger the gradient, the more incorrect it is.
A matrix is a two dimensional array, i.e. has rows and columns. A vector could be said to be a one dimensional array, i.e. just having a single column and multiple rows of data, or a single row and
multiple columns. However, a vector’s dimension is long it is, i.e. how many elements it has. Similarly the dimension of a matrix often refers to the number of rows and columns (conventionally in
that order), e.g. a 6 x 2 matrix has 6 rows and 2 columns. Sometimes dimensions are referenced in the shape.
Representing a higher dimension vector with a lower dimension representation, e.g. a single number representing a whole word or a category representing a specific book. Reducing dimensionality helps
with training, and generated embeddings can help with visualisation. The term “vocabulary size” is used to indicate how many different mappings from number to character or word or part word there
One iteration of training, so that each example has been seen once. The term is most relevant where batch size is greater than 1, in which case it will be the number of examples divided by batch
Information that is relevant to learning. An attribute of the subject. Features are like columns in a database, with distinct values for each row, e.g. separate features for houses could include the
floor area and number of bedrooms. In the example of an image, it could be just the pixel data, but can also include metadata, such as tags describing the image.
More features are not necessarily better because some features may not help improve the results, and too many features may make learning impractical, so feature engineering is important for neural
networks. A network may even have automated feature extraction, e.g. a layer for edge detection of an image. A useful rule of thumb when deciding which features to use is whether a human expert would
be able to predict the results given the selected features, or at least say the features contain sufficient information to predict the results.
This is where the features are scaled to be of similar range to each other, typically in the -1 to +1 range. Feature scaling can ensure the learning gives a comparable amount of attention to all the
different features, not just the features that have a range of large values. In gradient descent, this makes the contours less skewed or tight, which allows it to reach the global minimum more
quickly. Feature scaling can be more important in some learning algorithms, e.g. SVMs.
The combination of the features is called a feature vector. A feature vector is one row (or column) of input, and the dimension of the feature vector is the number of columns of that row (or rows of
that column).
In its simplest form, a function is the method used to turn an input into an output. Machine learning is trying to find a function describing the relationship between data features x and
classification or regression for y, i.e. y = f(x). A linear function is just a straight line. A quadratic function is a curve. A polynomial function is one with lots of ups and downs, i.e. a wiggly
line. Note that many examples demonstrate with one or two features (dimensions) so it can be illustrated on a graph, but in practice it is often more dimensions that are more difficult to visualise
in this way. See also Activation function, Optimisation function and Loss function.
A function where operations include non-negative integer exponents of variables, e.g. x-squared. A degree 2 polynomial has x-squared and generates a parabolic curve (one peak or trough), a degree 3
polynomial hàs x-squared and x-cubed and generates a cubic curve (one peak and one trough), etc.
Also called logistic function. In contrast to a linear function which is a straight line, a sigmoid function starts off with a slow slope, has the largest slope in the middle, and finishes with a
slow slope again.
Algorithm to find the local minimum of a function. It is an iterative process which determines the gradient at each step and proceedes in the direction of the steepest gradient. Step size is
determined by the learning rate. The gradient is determined by calculating the derivative of the loss function. Also called batch gradient descent.
Stochastic gradient descent (SGD) uses individual training examples in each iteration to update parameters, in contrast to batch gradient descent which uses all the training examples in each
iteration before updating parameters which can be computationally expensive with large datasets. This means progress on the path to the global minimum is quicker, although it is not as direct, and
tends to converge near the global minimum rather than on it exactly (using a smaller learning rate can help, optionally with a learning rate that decreases over time). You would typically
periodically check on progress twoards convergence by plotting the cost averaged over the last x training examples processed.
Mini-batch gradient descent is between batch gradient descent and stochastic gradient descent, using b training examples in each iteration, where b is the mini-batch size. It can be faster than SGD
if you use vectorisation to partially parallelise the derivative calculations (gradient computations).
Configuration, e.g. learning rate, iterations, hidden layers, type of activation function. They help determine the final values of the parameters.
Running an already-trained model to make predictions, without training, i.e. without adjusting the weights via back-propagation.
In gradient descent, the learning rate is the size of each step taken. If it is too small it could take a long time to reach the target, and if it is too large it could miss the target.
The loss function determines the performance of a given model for given data, in the form of a single number denoting the difference between predicted values and expected values. The “score” from the
loss function is simply known as the loss. The loss function is sometimes called the cost function, and sometimes represented as J(theta). It always has a term to calculate the error (like Mean
Squared Error), and sometimes has a term to regularise the result. The derivative of the loss function is the gradient, which shows how quickly the loss (or cost) is improving. The lower the loss (or
cost) the better. Mean Squared Error (MSE) and Mean Absolute Error (MAE) are the most common terms to calculate the error, with with MAE perhaps more suitable for time series because it doesn’t
punish larger errors as much as MSE does, and there are other terms to calculate the error, e.g. Huber loss which is also less sensitive to outliers.
Rows and columns of numbers. Contrast with a vector which is a matrix with one column and many rows.
The neural network architecture is the connectivity pattern between units, i.e. the number of layers (e.g. single layer, 2 layer etc.), the number of units in each layer, and the connection patterns
between layers (e.g. fully connected layers, aka dense layers). It sometimes also includes the activation functions and learning methods. A single layer will just have the input layer and the output
layer (the input layer is not counted hence this is called a single layer not a 2 layer), a 2 layer network will have the input layer, a hidden layer and the output layer, etc. A fully connected
layer is one where each of the units in one layer are connected to each of the units in the next layer. The number of input units will be the same as the dimension of the feature vector, and the
number of output units would match the number of “labels” required in classification. A reasonable default is one hidden layer, and if there are multiple hidden layers a reasonable approach is to
have the same number of units in each hidden layer, with the number of units in a hidden layer the same or a small multiple of the number of number of units in the input layer.
Effectively the same as Feature scaling. Not to be confused with Regularisation which adjusts the prediction function rather than the data.
In a neural network, an optimisation function (or optimisation algorithm) would be used with back propagation to try to minimise the loss function as a function of the parameters. In other words, it
uses the “score” from the loss function to work out how well it is doing and then makes adjustments to try to improve the “score” on the next iteration / epoch. Common optimisation functions are
gradient descent, Adam and RMSProp (a benefit of Adam and RMSProp is that they automatically adapt the learning rate during training).
The model is simply memorising the training data and recalling it, which means excellent results against data that has already been seen but generally much poorer results against data that has not
been seen. It should learn generalisations which would apply to unseen data. It can happen when you have a lot of features and little training data. You can typically see this has happened when the
validation loss is higher than the training loss (or the test loss is higher than the training loss if there is no validation set). To remedy, you can decrease the number of features (which might not
be possible or desirable), regularise, or specifically with neural networks decrease network size, or increase the dropout hyperparameter, or get more training data. Overfitting is sometimes called
“high variance”.
A parameter represents the weight learned for the connection between two units. Parameters are often represented by W and b, for the weights and biases respectively. You can calculate the number of
parameters in a fully connected network by multiplying the number of units in a layer with the number of units in the next layer and adding all the results for all the layers together, and adding the
bias units in the hidden and output layers. The parameters are also typically stored in a matrix for efficiency in calculating the forward propagation and back propagation, although sometimes have to
be “unrolled” into vectors for advanced optimisation (i.e alternatives to gradient descent). See also Hyperparameters which are the configuration options excluding the weights.
Precision and recall, and F Score Link to heading
Precision is a measure of exactness (or quality), while recall is a measure of completeness (or quantity). Specificially, precision is the number of true positives divided by predicted positives (or
true positives divided by true positives and false positives), while recall is true positives divided by actual positives (or true positives divided by true positives and false negatives). Therefore,
perfect precision has no false positives (or no irrelevant results), and perfect recall has no false negatives (all the relevant results). When the classes are heavily skewed (e.g. when 99% of
expected results are false and you could get reasonable error/accuracy by always predicting false), you can get a better insight into how well a learning algorithm is performing by looking at
precision and recall rather than error and accuracy. To get a view of overall combined precision and recall, use the F Score (or F1 Score) which is 2 times ((precision times recall) divided by
(pecision and recall)).
A Rectified Linear Unit (ReLU) outputs 0 if the input is negative or 0 and outputs the input if it is positive, i.e. it only returns x if x is greater than 0.
A Recurrent Neural Network (RNN) is a network architecture for sequences of data. Hidden layers from the previous run provide part of the input to the same hidden layer in the next run.
In supervised learning, this is trying to predict a continuous output, e.g. price or time or age. Contrast with Classification, which is trying to predict discreet values. In broader statistical
terms, regression is a technique for estimating the relationship among variables, and there are several types of regression, e.g. linear regression (estimates for continous output based on a function
which generates a line, although not necessarily a straight line in statistical terms because e.g. polynomial regression is a type of linear regression using a polynomial function) and logistic
regression (estimates for discreet values, using the “S-shaped” curve of a logistic function aka sigmoid function).
Reduce the magnitude/values of parameters. In a neural network, this would often be performed in the loss function. Without regularisation a model might overfit, and with too much regularisation it
could underfit. Not to be confused with Normalisation which adjusts the data rather than the prediction function.
Shape in the context of an input layer is the original dimension of the feature, prior to flattening into a vector. In a Convolutional Neural Network used for image processing, a 28x28 pixel square
grayscale input image could be represented by a 28 x 28 matrix of the grayscale values for each pixel and this would be flattened into a vector with an input shape of (28, 28) for the input layer, or
a 28x28 pixel square RGB image could have the input shape (28, 28, 3) where 3 is the number of channels (i.e. R, G and B), or if input has a batch size of 4 a 28x28 RGB image would have an input
shape of (4, 28, 28, 3). In a Recurrent Neural Network, input shape would be batch size, the number of timestamps, and series dimensionality (i.e. 1 for univariate and 2 or more for multivariate).
Shape in the context of a tensor is the number of elements in each dimension, e.g. in a two-dimensional tensor the shape is the [number of rows, number of columns].
A Support Vector Machine is a supervised learning algorithm. In simple terms, it is a way of finding the optimal line to separate features. Compared to a neural network, an SVM might be faster to
train, and should always find global optima (although in practice neural networks tend not to suffer from the local optima problem).
A multidimensional array that can contain scalars (i.e. numbers) and/or vectors and/or matrices. A vector is a one-dimensional tensor and a matrix is a two-dimensional tensor. A picture, for example,
can be represented by a 3 dimensional tensor, with fields for width, height and depth, or a series of pictures as a 4 dimensional vector. Grouping data in this way can be more computationally
The model hasn’t learned much. You typically see this when the training loss and validation loss remain about the same over multiple epochs. To remedy, you could increase the network size and/or
number of layers, or add features. Underfitting is sometimes called “high bias”.
In relation to time series, a univariate time series has a single value at each time stamp (e.g. a chart showing birth rate), whereas multivariate has multiple values at each stamp (e.g. charts
showing birth rate and death rate).
Validation Set vs Training Set vs Test Set Link to heading
There would normally be a Training Set, Validation Set and Test Set of data. Most would be Training e.g. 70%, then Validation e.g. 20%, then Test e.g. 10%. The Training Set is used for training, i.e.
adjusting the weights. The Validation Set is used during training to calculate the accuracy and avoid overfitting, but does not adjust the weights (although may be used to tune hyperparameters). If
the validation loss is higher (i.e. accuracy lower) than training loss you are likely overfitting, and if the validation loss is about the same as the training loss but neither are reducing you are
likely underfitting. You’d normally choose a model with the lowest validation loss (noting that this is not necessarily the last checkpoint if there has been overfitting). The Test Set is used after
training is complete to confirm the predictive accuracy. When you deploy a model to production, you may also use a Test Set of production data to measure the Domain Shift between production and test
data, and continue sampling production data to measure the Drift (i.e. how performance is degrading as a result of data changing over time).
A list of numbers. A vector can be seen of as a special type of matrix that has one row and multiple columns (sometimes called a “row vector”) or one column and multiple rows (a “column vector”).
Pretty much all of AI/ML works on lists of numbers, so everything has to be converted to them - see also Embedding, Parameters, Shape, Matrix and Tensor. With images, after “flattening” from a width
x height matrix, a number could represent the colour of a pixel with the length of the vector equal to the number of pixels wide by number of pixels long. Additional information is often added to the
input vector - see Feature vector. In terms of implementation within the machine learning algorithms themselves, vectors are linear algebra vectors, with various properties relating to addition,
multiplication etc., and mechanisms for measuring distances and determining the gradients for calculating errors and rate of change. For computational efficiency in machine learning, vectors are more
likely to be used as part of a matrix or tensor. | {"url":"https://michael-lewis.com/posts/a-machine-learning-glossary-for-hackers/","timestamp":"2024-11-13T21:06:22Z","content_type":"text/html","content_length":"42535","record_id":"<urn:uuid:abbf95c3-1c1c-46ec-bbb7-b7874ea07184>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00162.warc.gz"} |
What is a * b mod n?
This establishes a natural congruence relation
congruence relation
Congruence modulo n is a congruence relation, meaning that it is an equivalence relation that is compatible with the operations of addition, subtraction, and multiplication. Congruence modulo n is
denoted: The parentheses mean that (mod n) applies to the entire equation, not just to the right-hand side (here, b).
https://en.wikipedia.org › wiki › Modular_arithmetic
on the integers. For a positive integer n, two integers a and b are said to be congruent modulo n (or a is congruent to b modulo n), if a and b have the same remainder when divided by n (or
equivalently if a − b is divisible by n ). It can be expressed as a ≡ b mod n.
What does a B mod n mean?
If n is a positive integer, we say the integers a and b are congruent modulo n, and write a≡b(modn), if they have the same remainder on division by n. (By remainder, of course, we mean the unique
number r defined by the Division Algorithm.)
What does a ≡ b mod m mean?
Definition: given an integer m, two integers a and b are congruent modulo m if m|(a − b). We write a ≡ b (mod m). I will also sometimes say equivalent modulo m. Notation note: we are using that "mod"
symbol in two different ways.
What does mod n mean?
Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor.
What is an example of a congruent to B mod n?
We say integers a and b are "congruent modulo n" if their difference is a multiple of n. For example, 17 and 5 are congruent modulo 3 because 17 - 5 = 12 = 4⋅3, and 184 and 51 are congruent modulo 19
since 184 - 51 = 133 = 7⋅19. We often write this as 17 ≡ 5 mod 3 or 184 ≡ 51 mod 19.
19 related questions found
What does congruent modulo n mean?
Congruence modulo n is a congruence relation, meaning that it is an equivalence relation that is compatible with the operations of addition, subtraction, and multiplication. Congruence modulo n is
denoted: The parentheses mean that (mod n) applies to the entire equation, not just to the right-hand side (here, b).
How do you solve ax congruent to b mod n?
To solve a linear congruence ax ≡ b (mod N), you can multiply by the inverse of a if gcd(a,N) = 1; otherwise, more care is needed, and there will either be no solutions or several (exactly gcd(a,N)
total) solutions for x mod N.
Can you multiply modulus?
Modular multiplication is pretty straightforward. It works just like modular addition. You just multiply the two numbers and then calculate the standard name. For example, say the modulus is 7.
What does Z * n mean?
Actually, Z∗n refers to a group - compared to the ring Zn it's not so much about removing elements, it's removing one of the operations and everything that does not fit into the group structure.
Everything else follows from the fact that it is a group and by definition every element must have an inverse.
How do you calculate mod n?
How to calculate the modulo – an example
1. Start by choosing the initial number (before performing the modulo operation). ...
2. Choose the divisor. ...
3. Divide one number by the other, rounding down: 250 / 24 = 10 . ...
4. Multiply the divisor by the quotient. ...
5. Subtract this number from your initial number (dividend).
What does 1 mod n mean?
"1 modulo anything (or 1%N) is 1" - unless N is 1, in which case the result is zero.
What does n mean in a math equation?
The set of natural numbers, denoted N, can be defined in either of two ways: N = {0, 1, 2, 3, ...} N = (1, 2, 3, 4, ... } In mathematical equations, unknown or unspecified natural numbers are
represented by lowercase, italicized letters from the middle of the alphabet.
What does mod mean in maths?
The modulo operation (abbreviated “mod”, or “%” in many programming languages) is the remainder when dividing. For example, “5 mod 3 = 2” which means 2 is the remainder when you divide 5 by 3.
How do you add two modulus?
Let's explore the addition property of modular arithmetic:
1. Let A=14, B=17, C=5.
2. Let's verify: (A + B) mod C = (A mod C + B mod C) mod C. ...
3. LHS = (A + B) mod C. ...
4. RHS = (A mod C + B mod C) mod C. ...
5. LHS = RHS = 1.
6. Observe the figure below. ...
7. mod.
What are modulus functions?
A modulus function is a function which gives the absolute value of a number or variable. It produces the magnitude of the number of variables. It is also termed as an absolute value function. The
outcome of this function is always positive, no matter what input has been given to the function.
How does modulus work?
The modulus operator is added in the arithmetic operators in C, and it works between two available operands. It divides the given numerator by the denominator to find a result. In simpler words, it
produces a remainder for the integer division. Thus, the remainder is also always an integer number only.
What is Z * in statistics?
z* means the critical value of z to provide region of rejection if confidence level is 99%, z* = 2.576 if confidence level is 95%, z* = 1.960 if confidence level is 90%, z* = 1.645.
What is multiplication of integers mod n?
Integer multiplication respects the congruence classes, that is, a ≡ a' and b ≡ b' (mod n) implies ab ≡ a'b' (mod n). This implies that the multiplication is associative, commutative, and that the
class of 1 is the unique multiplicative identity.
What is the meaning of Z * in math?
Integers. The letter (Z) is the symbol used to represent integers. An integer can be 0, a positive number to infinity, or a negative number to negative infinity. Load More.
How do you multiply a number by a mod?
If we want to multiply many numbers modulo A, we can first reduce all numbers to their remainders. Then, we can take any pair of them, multiply and reduce again. X = 36 * 53 * 91 * 17 * 22 (mod 29) .
Can you double modulus?
Double Modulus Operator
The evaluated result is a double value. For example, 8.27 % 2 evaluates to 0.27 since the division is 4 with a remainder of 0.27. Of course, both values could be double, so 6.7 % 2.1 evaluates to 0.4
since the division is 3 with a remainder of 0.4.
What is mod property for multiplication?
Modular multiplication has the following properties: It is commutative: a \times b is equal to b \times a for every a and b; It has an identity element (precisely the number 1, since a \times 1 = a
for every a) Every element (different from 0) has an inverse only when the modulus is a prime p.
How to calculate modulus?
Modulus on a Standard Calculator
1. Divide a by n.
2. Subtract the whole part of the resulting quantity.
3. Multiply by n to obtain the modulus.
What does it mean for the congruence equation ax ≡ b mod m to have a solution?
The congruence relation ax ≡ b (mod m) has a solution if the (“unknown”) integers x (where 0 ≤ x ≤ m − 1) and k satisfy ax = b + km. But this is a linear Diophantine equation in the unknowns x and k.
Theorem 3.1 shows how to solve linear Diophantine equations, so we will apply this here.
What is the solution of ax b mod m?
As mentioned, ax = b (mod m) is equal to ax - my = b. If we apply Theorem 19, you realize that c cannot divide b; thus, it is impossible to get a solution for the equation ax - my = b. Thus, the
values of x become the solution for the congruence ax = b (mod m). | {"url":"https://www.calendar-uk.co.uk/frequently-asked-questions/what-is-a-b-mod-n","timestamp":"2024-11-15T01:13:56Z","content_type":"text/html","content_length":"72900","record_id":"<urn:uuid:57cf62d9-7958-4cbb-954a-8d4c7e864368>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00336.warc.gz"} |
convert to scientific notation
If you need a scientific calculator see our resources on Consider the example of writing 123,000 in scientific notation. For example, letâ s represent 1.5 × 10 5 in E notation. Scientific notation
consists of a coefficient (here 5.14)multiplied by 10 raised to an exponent (here 5). For example, a 2-decimal scientific format displays 12345678901 as 1.23E+10, which is 1.23 times 10 to the 10th
power. To round significant figures use the All rights reserved. First, convert the number with a lower exponent to one expressed with the exponent of the number with a larger exponent 2. Standard
Form to Scientific Notation Every number in the scientific notation must be in the form of a x 10 n. where 1 â ¤ a < 10 and n must be a positive or negative integer. // Converted the String value
9.78 to 9.78. Cross out the original number's decimal point. © 2006 -2020CalculatorSoup® You can enter an expression in scientific notation by pressing [2nd][,] to type an E, but entering an
expression in scientific notation doesnâ t guarantee that your answer will remain [â ¦] When the decimal is moved towards the right, the count for the exponent of base 10 should be negative. English
format used in; Australia, Canada (English-speaking, unofficial), China, Hong Kong, Ireland, Israel, Japan, Korea, Malaysia, Mexico, New Zealand, Pakistan, Philippines, Singapore, Taiwan, Thailand,
United Kingdom, United States. If you have a lot of numbers which are displayed as the scientific notation, and you are tired of entering them repeatedly with the above method, you can convert them
with the Format Cells function in Excel. // Converted the Int32 value 123 to 123. If you're working with the number 0.00004205, just write an "x" over the decimal point. Significant Figures
Calculator. For example, if you want to convert from meters to micrometers you would convert from "base unit" to micro. Calculator for converting numbers to scientific notation In scientific notation
each number is written in the form: b × 10 y, where b is a number between 1 and 10 and y is a positive or negative whole number. 32,500,000 = 3.25 × 10 7. To enter a number in scientific notation use
a carat ^ to indicate the powers of 10. // â ¦ This site is my passion, and I regularly adding new tools/apps. The decimal was moved 7 places to the left. For converting the fraction to scientific
notation numbers the steps followed are: Take a number which is in the fraction form. Scientific Notation. Example D: Write .00000863 in scientific notation. In most programs, 6.022E23 (or 6.022e23)
is equivalent to 6.022Ã 1023, and 1.6Ã 10-35 would be written 1.6E-35. Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. The
Decimal to Scientific Notation Converter is used to convert a number from ordinary decimal notation into scientific notation. Convert among units for any base unit of measure such as gram or meter or
second or byte, etc. An Intuitive Method for Converting a Number to Scientific Notation. Scientific notation on a TI-84 Plus calculator looks a little different than what youâ re used to seeing in
class. A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10. See example below:>. 3.456 x 10^-4 = 3.456 x .0001 = 0.0003456. This is the first step
to beginning to convert the number into scientific notation. Scientific Notation Calculator to add, subtract, multiply and divide numbers in scientific notation or E notation. Scroll down the page
for more examples and solutions. Therefore, the exponent is a +7. Users experience is very important, that's why I use non-intrusive ads. Method 2 Move the decimal 5 places to the left to get
3.57096, The number 357,096 converted to scientific notation is 3.57096 x 10^5, Move the decimal 3 places to the right and remove leading zeros to get 5.600, The number 0.005600 converted to
scientific notation is 5.600 x 10^-3, Note that we do not remove the trailing 0's because they were originally to the right of the decimal and are therefore. Also, you can try this calculator for
scientific notation to perform basic math operations on scientific notation & to convert a number to scientific notation. Thus 350 is written as 3.5Ã 102. To convert a number into scientific
notation: Method 1 . Scientific notation is a way of writing very large or very small numbers. Let's convert the number 3,400,000 into scientific notation. https://www.calculatorsoup.com - Online
Calculators. 3. Learn more Accept. Type the following formula into a â ¦ The Scientific Notation to Decimal Converter is used to convert a number from scientific notation into ordinary decimal
notation. Just give a read to this post to know how to do scientific notation conversions with a calculator or manually (step-by-step), examples of scientific notation calculations, and much more!
1.5 × 10 5 = 15.0 × 10 4 1.5 × 10 5 = 150.0 × 10 3. To convertto a real number, start with the base and multiply by 5 tens likethis: 5.14 × 10 × 10 × 10 × 10 × 10 =514000.0. The initial values are
randomly selected values decimal and exponents to demonstrate the process. Example of Standard Form: A number is 600000 To convert a number to scientific notation, place or move the decimal point of
a number until the coefficient of the number is greater than 1 and less than 10. Multiplying by tens is easy: one simply moves the decimalpoint in the base (5.14) 5 places tâ ¦ It can convert million
in scientific notation and even billion in scientific notation too. Scientific notation is a way to express numbers in a form that makes numbers that are too small or too large more convenient to
write. // The String value 1e-02 is not recognized as a valid Decimal value. // Conversion of the Char value a to a Decimal is not supported. It is commonly used in mathematics, engineering, and
science, as it can help simplify arithmetic operations. Scientific e-notation can be found on most calculators and many computer programs, they present very large and very small results in scientific
notation, typically invoked by a key labelled EXP (for exponent), EEX (for enter exponent), EE, EX, E, or à 10x depending on vendor and model. See the I've spent over 10 trillion microseconds (and
counting), on this project. The following figures show how to convert decimals to scientific notation. How to convert numbers or decimals to scientific notation? Move the decimal point in your number
until there is only one non-zero digit to the left of the decimal point. This conversion tool can be used as a scientific notation converter (convert a scientific notation to a decimal number), or as
a reverse scientific or standard notation converter (convert a number from standard form to scientific notation). This website uses cookies to ensure you get the best experience. Thank you. To
convert scientific notation to engineering notation, move the decimal point in the coefficient to the right and reduce the exponent until the exponent is a multiple of 3. If the number is negative
then a minus sign precedes m (as in ordinary decimal notation). To enter a number in scientific notation use a carat ^ to indicate the powers of 10. In Engineering notation (often named "ENG" display
mode on scientific calculators) differs from normalized scientific notation in that the exponent n is restricted to multiples of 3. Example C: Write 32,500,000 in scientific notation. The decimal was
moved 6 places to the right. In scientific notation all numbers are written in the form of mà 10n Students learn to convert a number from standard notation to scientific notation by first writing a
decimal point in the number so that there is only one digit to the left of the decimal point. The decimal notation calculator above can be used to convert any decimal to scientific notation. It is
commonly used by scientists, mathematicians and engineers, in part because it can simplify certain arithmetic operations. Free Scientific Notation Converter - convert numbers from decimal to
scientific and e-notations step-by-step. This script will convert numeric values in the form of conventional decimal to Scientific Notation form. Example: 64 000 = 6.4 × 10 000 = 6.4 × 10 4. The
Scientific format displays a number in exponential notation, replacing part of the number with E+ n, in which E (exponent) multiplies the preceding number by 10 to the n th power. Scientific notation
is used to define the larger and smaller number. Scientific notation converter is a great tool to convert a number to scientific notation. The rule above states that. How to convert numbers or
decimals to scientific notation? Scientific notation is a compact way of writing very large and very small numbers. To convert a number to scientific notation, first we have to identify where the
decimal point and non zero digit come. Also, this online scientific notation converter helps you to convert a number to scientific notation, e notation, engineering notation and decimal notation. 3 a
is greater than or equal to one and less than ten or, 1 ≤ |a| < 10. b is the power of 10 required so that the scientific notation is mathematically equivalent to the original number. // The Double
value 1.764E+32 is out of range of the Decimal type. 1. Any feedback is appreciated. The number of places the decimal point moves is the power of the exponent, because each movement represents a
"power of 10". You can also enter numbers in e â ¦ In scientific notation, numbers are written as a base, b, referred to as the significand, multiplied by 10 raised to an integer exponent, n, which
is referred to as the order of magnitude: b × 10n Below are some examples of numbers written in decimal notation compared to scieâ ¦ Free math problem solver answers your algebra, geometry,
trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. For example, 2.53*1012 will display as 2.53E12. 2500000. Cite this content, page or
calculator as: Furey, Edward "Scientific Notation Converter"; CalculatorSoup, On scientific calculators it is known as "SCI" display mode. It's quite easy to convert our regular numbers into
scientific notation. The calculator below can be used to covert numbers to normalised scientific notation. a is a number or decimal number such that the absolute value of E notation is basically the
same as scientific notation except that the letter e is substituted for "x 10^". It is obvious that the original decimal point is to the left of the nonzero digit. Select the data range that you want
to convert. Multiply the decimal number by 10 raised to the power indicated. The proper format for scientific notation is a x 10^b where scientific calculators. Example 4: Rewrite the given decimal
number 0.0009 in scientific notation. Find My Location: IP address, city, latitude & longitude. Scientific notation conversion calculator: decimal notation, E notation, engineering notation. A number
written in scientific notation consists of a number between 1 and 10 multiplying a power of 10. By using this website, you agree to our Cookie Policy. Convert a number to and from scientific
notation, e notation, engineering notation and real numbers. This number is. Please support this site by disabling or whitelisting the Adblock for "justintools.com". Plus, the converter will accept
either scientific or E notation (E or e represents "x 10^"): 3.5E4 3.5e4 Consequently, the absolute value of m is in the range 1 â ¤ |m| < 1000, rather than 1 â ¤ |m| < 10. Price of commodities:
gold, silver, oil, live cattles and more. An online standard form converter helps you to convert the numbers into standard form by placing the decimal value in the given number. You can also enter
numbers in e notation. Convert scientific notation to text with Format Cells function. Record down the coefficient (a) and count the number of steps the decimal point was moved. 000008.63. â ¦ For
example, 12.5Ã 10-9 m can be read as "twelve-point-five nanometers" and written as 12.5 nm, while its scientific notation equivalent 1.25Ã 10-8 m would likely be read out as "one-point-two-five times
ten-to-the-negative-eight meters". Enter a number or a decimal number or scientific notation and the calculator converts to scientific notation, e notation and engineering notation formats. Justin
XoXo :), 1 x 10^3(use the caret symbol [^] to type or write). French format used in; Albania, Austria, Belgium, Brazil, Bulgaria, Canada (French-speaking), Croatia, Czech Republic, Denmark, Estonia,
Finland, France, Germany, Greece, Hungary, Italy, Kosovo, Latin Europe, Netherlands, Norway, Peru, Poland, Portugal, Romania, Russia, Serbia, Slovakia, Slovenia, South Africa, Spain, Sweden,
Switzerland (officially encouraged for non-currency numbers), Ukraine. To convert a decimal into scientific notation, move the decimal point until you get to the left of the first non-zero integer.
2. This page will show you how to convert between writing numbers in scientific notation and in "regular" , or from "regular" to scientific notation. In scientific notation all numbers are written in
the form of m×10n (m times ten raised to the power of n), where the exponent n is an integer, and the coefficient m is any real number, called the significand or mantissa. The resulting decimal
number is, Count how many places you moved the decimal point. Examples: 3.45 x 10^5 or 3.45e5. (m times ten raised to the power of n), where the exponent n is an integer, and the coefficient m is any
real number, called the significand or mantissa. You can also use an excel formula to convert scientific notation to text based on the TRIM function or the UPPER function. Enter a number or a decimal
number or scientific notation and the calculator converts to scientific notation, e notation and engineering notation formats. Converting a number in Decimal Notation to Scientific Notation. In
normalized scientific notation (called "standard form" in the UK), the exponent n is chosen so that the absolute value of m remains at least one but less than ten. Likewise, if you want to convert
from grams to micrograms or feet to microfeet you would still do a conversion from base unit to micro. It is also written in the product form of Numbers or it is written in the power of 10 or in
exponent form. We will move the decimal going to the right. For example, 650,000,000 can be written in scientific notation as 6.5 10^8. Remove trailing 0's only if they were originally to the left of
the decimal point. Spanish/Arabic format used in; Argentina, Austria, Bosnia and Herzegovina, Brazil, Chile, Croatia, Denmark, Germany, Greece, Indonesia, Italy, Netherlands (currency), Portugal,
Romania, Russia, Slovenia, Spain, Sweden (not recommended), Turkey. There are two cases in â ¦ In order to do the calculations yourself, you need to follow two steps: 1. Using our tool in scientific
notation calculator mode you can perform addition, subtraction, multiplication and division of numbers expressed in a scientific notation. Rewrite the number as a product of a number A such that 1 â
¤ |A| < 10 and a power of 10. , letâ s represent 1.5 × 10 4 1.5 × 10 4 mathematicians and engineers, in part because it convert! Where the decimal point displays 12345678901 as 1.23E+10, which is
1.23 times 10 to the right number in notation! Â ¤ |m| < 10 script will convert numeric values in the product form of or! Important, that 's why I use non-intrusive ads multiply and divide numbers e!
And very small numbers more examples and solutions < 1000, rather than 1 â ¤ |A| 10! Writing very large or very small numbers you 're working with the exponent of base 10 should be.. Notation use a
carat ^ to indicate the powers of 10 â ¤ |A| < 10 a... Decimal notation, e notation and the calculator converts to scientific notation is. And count the number with a lower exponent to one expressed
with the exponent the! Or a decimal number 0.0009 in scientific notation consists of a number a such that 1 |A|! Trailing 0 's only if they were originally to the left of decimal... A valid decimal
value the fraction form one expressed with the exponent of base 10 should be negative that... What youâ re used to covert numbers to normalised scientific notation is a way of writing very and! In
class the UPPER function data range that you want to convert the number 0.00004205 just. Notation Converter - convert numbers or decimals to scientific and e-notations step-by-step is a way of
numbers! It is known as `` SCI '' display mode â ¦ convert a number a... Calculator above can be used to covert numbers to normalised scientific notation or e notation and even billion in notation!,
convert the number into scientific notation two steps: 1, the count for exponent. Very small numbers the right, the absolute value of m is in the fraction to scientific.! Can convert million in
scientific notation notation: Method 1 range 1 â ¤ |m| < 1000, rather than â ¤... Decimal is not recognized as a product of a number to scientific.... Script will convert numeric values in the
fraction form and more in decimal form site is my passion and. Let 's convert the number is negative then a minus sign precedes m ( as in ordinary decimal )! The data range that you want to convert
the number into scientific notation the values! Notation, first we have to identify where the decimal going to the right, the value! Programs, 6.022E23 ( or 6.022E23 ) is equivalent to 6.022Ã 1023,
and 1.6Ã 10-35 would be in... Indicate the powers of 10 first step to beginning to convert a to! 3 scientific notation use a carat ^ to indicate the powers of 10 address city. ( as in ordinary
decimal notation into scientific notation to text with format Cells function is as. Cookie Policy notation as 6.5 10^8 a to a decimal number 0.0009 in scientific notation is. I regularly adding new
tools/apps is written in the given decimal number 0.0009 in scientific notation to. 10^-4 = 3.456 x.0001 = 0.0003456: Method 1 a such that 1 â ¤ <. What youâ re used to define the larger and smaller
number real numbers step to beginning to a... Can help simplify arithmetic operations calculator: decimal notation convert to scientific notation scientific notation even! Going to the left of the
decimal going to the left of the decimal point was.... To convert website uses cookies to ensure you get the best experience non zero come! A ) and count the number of steps the decimal is not
recognized as a valid value... To text based on the TRIM function or the UPPER function use a carat ^ to indicate the of... Show how to convert a number between 1 and 10 multiplying a power of 10,
live cattles and.... Number to scientific notation too million in scientific notation calculator above can be used to convert notation... Below can be written 1.6E-35 * 1012 will display as 2.53E12
into scientific notation is great. By scientists, mathematicians and engineers, in part because it can convert million in notation..., and science, as it can simplify certain arithmetic operations
any decimal to scientific notation on a Plus... M is in the fraction form notation conversion calculator: decimal notation to text with format Cells function scientific e-notations. Moved 6 places to
the left of the Char value a to a decimal number negative., oil, live cattles and more and non zero digit come yourself, need! Scientists, mathematicians and engineers, in part because it can convert
million in scientific notation is compact! From decimal to scientific notation Converter is used to seeing in class latitude longitude. And engineers, in part because it can help simplify arithmetic
operations from decimal to scientific notation calculations yourself you. Very small numbers helps you to convert numbers or decimals to scientific notation is used to covert numbers to scientific...
1 and 10 multiplying a power of 10 or in exponent form the same as notation. The numbers into scientific notation or e notation 1.6Ã 10-35 would be written in scientific notation, first have.
Converts to scientific notation is basically the same as scientific convert to scientific notation is a compact way of expressing that! 10 or in exponent form and very small numbers 6.5 10^8 or a
decimal number or a number... Examples and solutions lower exponent to one expressed with the number with a lower exponent one. 6.5 10^8: a number between 1 and 10 is multiplied by a power of 10 or
in exponent.... Double value 1.764E+32 is out of range of the decimal is not supported base 10 should be negative used define... They were originally to the left by 10 raised to the right, count! In
order to do the calculations yourself, you agree to our Cookie Policy converts to scientific notation when. Seeing in class 2.53 * 1012 will display as 2.53E12 show how to convert decimals
scientific... Trillion microseconds ( and counting ), on this project compact way of expressing numbers that are too big too... Not recognized as a valid decimal value in the product form of decimal!
Double value 1.764E+32 is out of range of the decimal value: 1 be used to seeing class. Working with the exponent of base 10 should be negative the Double value 1.764E+32 is out of range the! Point
and non zero digit come is substituted for `` justintools.com '' to beginning to convert scientific,... As 2.53E12 enter a number in scientific notation use a carat ^ to indicate the of... Enter
numbers in scientific notation Converter - convert numbers or decimals to scientific notation calculator to,... Â ¦ the following figures show how to convert numbers from decimal to notation! ) is
equivalent to 6.022Ã 1023, and 1.6Ã 10-35 would be written in scientific notation convert numeric values in the form! The right displays 12345678901 as 1.23E+10, which is in the product form of
numbers or is!, 6.022E23 ( or 6.022E23 ) is equivalent to 6.022Ã 1023, and science as... Engineers, in part because it can convert million in scientific notation or e notation and billion. And
engineers, in part because it can help simplify arithmetic operations initial are... & longitude one expressed with the number with a lower exponent to one expressed with the number into notation! In
most programs, 6.022E23 ( or 6.022E23 ) is equivalent to 6.022Ã 1023, and I regularly adding tools/apps! < 1000, rather than 1 â ¤ |m| < 1000, rather than 1 â ¤ |m| <.... Plus calculator looks a
little different than what youâ re used to seeing in class are... Spent over 10 trillion microseconds ( and counting ), on this project TI-84 Plus looks... ^ convert to scientific notation indicate
the powers of 10, just write an `` x 10^ '' `` justintools.com.... The Adblock for `` justintools.com '' our Cookie Policy arithmetic operations value of m is in the form... Multiply and divide
numbers in e notation and engineering notation formats 6.022Ã 1023 and! Value 1e-02 is not supported ensure you get the best experience calculator looks a little different than youâ re! Are too big
or too small to be conveniently written in scientific notation a! Is 600000 scientific notation numbers the steps followed are: Take a number is negative then a minus sign m... 10^3 ( use the caret
symbol [ ^ ] to type or write ) from to. 1.5 × 10 3 the calculator converts to scientific notation is a way of writing 123,000 in scientific?... Method 1 number 0.00004205, just write an `` x '' over
the decimal point the. Not supported this script will convert numeric values in the given decimal number or scientific notation and engineering notation the. Be conveniently written in scientific
notation to scientific notation conversion calculator: decimal notation to scientific notation numbers steps... '' display mode, which is 1.23 times 10 to the left of the nonzero digit to covert
to... Displays 12345678901 as 1.23E+10, which is in the range 1 â ¤ |A| < 10 ensure get. Would convert from meters to micrometers you would convert from `` base unit '' to micro is multiplied by
power! Move the decimal point is to the power indicated 1.23E+10, which is times! Is substituted for `` x '' over the decimal point is to the 10th power on scientific.. The 10th power exponent of the
nonzero digit placing the decimal notation into notation... Converter helps you to convert scientific notation Converter is a great tool to numbers! Converter - convert numbers from decimal to
scientific notation then a minus sign precedes m as... 1.23E+10, which is 1.23 times 10 to the left of the number into... Regularly adding new tools/apps, 2.53 * 1012 will display as 2.53E12 are...
Used to define the larger and smaller number more examples and solutions in scientific and...
Sieg And Jeanne, Agedashi Tofu Calories, Rb Choudary Age, Maybelline Fit Me Foundation Sachet, Hamburger Helper 3 Cheese Manicotti Recipes, Lg Mexico Soporte, Fireplace Inserts For Sale, University
Of Oslo Tuition, Dog Not Eating But Drinking Water, Psalm 103:1 Nlt, Autocad Electrical Training Book Pdf, When Do Pugs Stop Growing, Psalm 19:1 Esv, Grange Primary School Ealing Website, Famous
Dalit Poems, Best Dog Weight Gain Supplement,
This site uses Akismet to reduce spam. Learn how your comment data is processed.
This entry was posted in Uncategorized. Bookmark the permalink. | {"url":"https://bridgetcrawford.com/eternal-father-ndeban/ac843a-convert-to-scientific-notation","timestamp":"2024-11-06T18:03:59Z","content_type":"text/html","content_length":"40494","record_id":"<urn:uuid:161161ee-96a5-486d-be66-a85fb5fb2d45>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00272.warc.gz"} |
VDB - Excel docs, syntax and examples
The VDB function calculates the depreciation of an asset for a specific accounting period using the double-declining balance method or other specified depreciation method. It is commonly utilized in
financial and accounting scenarios to determine the reduction in asset value over time.
=VDB(Cost, Salvage, Life, Start_period, End_period, [Factor], [No_switch])
Cost Initial cost of the asset.
Salvage Value of the asset at the end of its useful life.
Life Total number of accounting periods over which the asset will be depreciated.
Start_period Accounting period for which you want to calculate depreciation.
End_period End period for calculating depreciation.
Factor (Optional) Factor by which the depreciation decreases.
No_switch (Optional) Logical value that specifies whether to switch to straight-line depreciation when depreciation is greater than the declining balance. Defaults to FALSE if omitted.
About VDB 🔗
When managing asset values and seeking to assess their depreciation over time, lean on Excel's VDB function for a dependable calculation method. This function is particularly handy when employing the
double-declining balance method or a customized depreciation scheme to gauge the decrease in asset worth across successive accounting periods. Whether you're analyzing equipment, property, or other
capital assets, VDB equips you with the means to determine the reduction in value with precision and efficiency. By inputting essential details such as the initial cost, salvage value, asset's
lifespan, starting and ending periods, and any applicable factors for depreciation adjustment, VDB efficiently computes the depreciation figures tailored to your scenario. Additionally, the optional
parameters grant flexibility to tailor the depreciation computation to specific accounting requirements or alternate depreciation methodologies. VDB simplifies the process of handling asset
depreciation, providing you with accurate insights into the asset's diminishing value and aiding in strategic financial decision-making.
Examples 🔗
If you purchased a machine for $10,000 with a salvage value of $2,000, having a useful life of 5 years and you want to calculate the declining balance depreciation for periods 1 to 3 with a
depreciation factor of 1.5, the VDB formula would be: =VDB(10000, 2000, 5, 1, 3, 1.5)
Consider you acquired a vehicle with an original price of $20,000, an expected residual value of $5,000 after 8 years, and you aim to ascertain the depreciation using a straight-line method from
periods 1 to 7. The VDB formula to use: =VDB(20000, 5000, 8, 1, 7, , TRUE)
Ensure the accuracy of the input parameters in the VDB function to reflect the specific attributes and conditions of the asset being depreciated. Review the depreciation methods and factors applied
to guarantee precise and relevant results correspond to your asset evaluation needs.
Questions 🔗
What type of depreciation method does the VDB function primarily calculate?
The VDB function primarily calculates depreciation using the double-declining balance method, which accelerates the depreciation of an asset.
Can the VDB function handle custom depreciation factors?
Yes, the VDB function allows for the inclusion of a custom depreciation factor as an optional argument, enabling adjustments to the rate of depreciation applied.
If the depreciation exceeds the declining balance, does the VDB function switch to straight-line depreciation?
The VDB function offers the option to switch to straight-line depreciation if the declining balance method results in a higher depreciation amount than the calculated value. This behavior can be
controlled using the No_switch argument.
Related functions 🔗
Leave a Comment | {"url":"https://spreadsheetcenter.com/excel-functions/vdb/","timestamp":"2024-11-15T03:11:57Z","content_type":"text/html","content_length":"29995","record_id":"<urn:uuid:b1621fae-2eba-40d1-9cc3-0a3a54825ae6>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00380.warc.gz"} |
Studies on impurities moving in Tomonaga-Luttinger Liquids
In this thesis, we have explored various aspects of the dynamics of an impurity moving in more than one 1D bath. A recurring theme has been the investigation of the orthogonality catastrophe (OC)
that follows the injection of the impurity in the system. This phenomenon has been studied by calculating the Green’s function of the impurity, which also described the time evolution and spectral
features of the latter. The distinctive signature of the OC is that the Green’s function shows a power-law tail at long times. This function has been calculated using suitable perturbative expansions
in the impurity-bath coupling, namely the Linked Cluster Expansion (LCE) and a time-dependent perturbation theory around a nontrivial dynamics. All of our results are nonperturbative in the
inter-bath hopping. In the two-bath scenario we have performed a detailed asymptotic expansion of the LCE Green's function at long times, which turned out to be very accurate in comparison with its
numerical evaluation. The expansion has allowed us to obtain the renormalisation of the dispersion of the impurity bands, as well as the exponent of the power-law decay and the lifetime of the odd
mode. One of our main results is that the OC, leading to the breakdown of the quasiparticle picture, survives the inclusion of a second 1D bath and dominates the long-time behaviour of all the
components of the Green's function. In particular, the exponent characterising the long-time behaviour of the Green's function is given by half of the average of the exponents of the individual baths
and, notably, is the same for the intra-bath Green's functions and for that connecting the two baths. In the case of two asymmetric baths, the Green's function is nonuniversal, acquiring a
high-frequency component at short times and exhibiting persistent oscillations at longer times. In real experiments, the temperature of the baths is always finite, so we have extended the LCE
treatment to this scenario. The effect of the temperature is to suppress the Green's function, limiting the possibility to observe its coherent oscillations and the power-law tail. At sufficiently
long times, we found analytically that this suppression is exponential in time, with a different decay constant for the even and odd modes. For low temperatures, the two decay constants are
approximately coinciding, and are proportional to the temperature. Using a perturbation theory in the inter-band part of the interaction we have been able to reproduce the LCE Green’s function and
the OC with a method which also allows us to access the evolution of the whole impurity-bath system. The advantage of this approach is that it provides an analytic expression for the time evolution
of the state of the whole impurity-baths system. This has allowed us to compute the time evolution of observables beyond the impurity Green's function, including the often-neglected properties of the
baths. Moreover, the numerical effort required by the approach is sufficiently low so that we have been able to treat the case in which the impurity is initialised in a wave packet of an (almost)
arbitrary shape. On the impurity side, we have analysed the time evolution of its population within each bath, observing how the persistent oscillations of the free impurity are damped and slowed
down by the interaction with the baths. The impurity momentum is subjected to damping, as well. We have observed that momentum is transferred to the baths in two steps: a short transient, connected
to the bath relaxation and independent of the inter-bath hopping, and a much slower decay caused by the emission of phonons during the deexcitation of the odd mode. Finally, we have also examined the
time evolution of the probability density of finding the impurity in a given position and bath, for various Gaussian wave packets. We have studied the time evolution of observables describing the
dynamics of the baths, which are rarely discussed in the literature on mobile impurities, but are nonetheless accessible to experiments. We have looked at the number of excited phonons, which shows a
slow logarithmic divergence in time, related to the OC, superimposed with the faster growth caused by the emission of phonons from the odd mode decay. The particle density of the baths shows a
semiclassical behaviour, intuitively similar to that of a pond in which a stone has been thrown. After the impurity has been injected into one of the baths, a localised density depletion forms and
follows its motion. At the same time, two wave fronts are generated and propagate away. Moreover, each time the impurity oscillates between the baths, a new pair of ripples is emitted. The emission
of ripples can be suppressed by employing wider wave packets. We have also found that the bath momentum density displays a behaviour analogous to that of the density. When the initial impurity wave
packet has a markedly non-Gaussian shape, we find complex interference phenomena, both in the ripples and within the central trough. We examined the inter-bath, equal-times connected density and
momentum density correlations, which revealed a rich structure in real space. This structure is best understood by taking “slices” of the correlation function along the relative and centre-of-mass
coordinate, which show both the light-cone propagation of correlations and the motion of the impurity. We have made the first steps towards a many-baths system, in which the impurity moves in a 1D or
2D lattice of 1D baths. We have obtained the time evolution of the state of the system with the perturbative technique developed before. The impurity Green's function has revealed qualitative
differences from the two-baths setup. First of all, the Green's function shows a complex short-time behaviour, caused by interference effects between the various paths of propagation within the
lattice of baths. More importantly, we have found that each band of the noninteracting impurity is characterised by its own OC exponent, which is proportional to the degeneracy of the band. We have
then speculated that it may be possible to tune the OC exponent either by properly designing the lattice of baths, or by changing the degeneracies by means of external magnetic fields. We also shown
that for generic lattices of baths, the OC exponents vanish in the limit of an infinite number of lattice sites. Lastly, we showed that in the case of two identical baths there exists a unitary
transformation that diagonalises the impurity degrees of freedom, thus achieving a complete decoupling of the latter from the baths. On the basis of this transformation, we have sketched two
variational approximations for the ground-state and dynamics of the system.
Studies on impurities moving in Tomonaga-Luttinger Liquids / Stefanini, Martino. - (2022 Jun 30).
Studies on impurities moving in Tomonaga-Luttinger Liquids
In this thesis, we have explored various aspects of the dynamics of an impurity moving in more than one 1D bath. A recurring theme has been the investigation of the orthogonality catastrophe (OC)
that follows the injection of the impurity in the system. This phenomenon has been studied by calculating the Green’s function of the impurity, which also described the time evolution and spectral
features of the latter. The distinctive signature of the OC is that the Green’s function shows a power-law tail at long times. This function has been calculated using suitable perturbative expansions
in the impurity-bath coupling, namely the Linked Cluster Expansion (LCE) and a time-dependent perturbation theory around a nontrivial dynamics. All of our results are nonperturbative in the
inter-bath hopping. In the two-bath scenario we have performed a detailed asymptotic expansion of the LCE Green's function at long times, which turned out to be very accurate in comparison with its
numerical evaluation. The expansion has allowed us to obtain the renormalisation of the dispersion of the impurity bands, as well as the exponent of the power-law decay and the lifetime of the odd
mode. One of our main results is that the OC, leading to the breakdown of the quasiparticle picture, survives the inclusion of a second 1D bath and dominates the long-time behaviour of all the
components of the Green's function. In particular, the exponent characterising the long-time behaviour of the Green's function is given by half of the average of the exponents of the individual baths
and, notably, is the same for the intra-bath Green's functions and for that connecting the two baths. In the case of two asymmetric baths, the Green's function is nonuniversal, acquiring a
high-frequency component at short times and exhibiting persistent oscillations at longer times. In real experiments, the temperature of the baths is always finite, so we have extended the LCE
treatment to this scenario. The effect of the temperature is to suppress the Green's function, limiting the possibility to observe its coherent oscillations and the power-law tail. At sufficiently
long times, we found analytically that this suppression is exponential in time, with a different decay constant for the even and odd modes. For low temperatures, the two decay constants are
approximately coinciding, and are proportional to the temperature. Using a perturbation theory in the inter-band part of the interaction we have been able to reproduce the LCE Green’s function and
the OC with a method which also allows us to access the evolution of the whole impurity-bath system. The advantage of this approach is that it provides an analytic expression for the time evolution
of the state of the whole impurity-baths system. This has allowed us to compute the time evolution of observables beyond the impurity Green's function, including the often-neglected properties of the
baths. Moreover, the numerical effort required by the approach is sufficiently low so that we have been able to treat the case in which the impurity is initialised in a wave packet of an (almost)
arbitrary shape. On the impurity side, we have analysed the time evolution of its population within each bath, observing how the persistent oscillations of the free impurity are damped and slowed
down by the interaction with the baths. The impurity momentum is subjected to damping, as well. We have observed that momentum is transferred to the baths in two steps: a short transient, connected
to the bath relaxation and independent of the inter-bath hopping, and a much slower decay caused by the emission of phonons during the deexcitation of the odd mode. Finally, we have also examined the
time evolution of the probability density of finding the impurity in a given position and bath, for various Gaussian wave packets. We have studied the time evolution of observables describing the
dynamics of the baths, which are rarely discussed in the literature on mobile impurities, but are nonetheless accessible to experiments. We have looked at the number of excited phonons, which shows a
slow logarithmic divergence in time, related to the OC, superimposed with the faster growth caused by the emission of phonons from the odd mode decay. The particle density of the baths shows a
semiclassical behaviour, intuitively similar to that of a pond in which a stone has been thrown. After the impurity has been injected into one of the baths, a localised density depletion forms and
follows its motion. At the same time, two wave fronts are generated and propagate away. Moreover, each time the impurity oscillates between the baths, a new pair of ripples is emitted. The emission
of ripples can be suppressed by employing wider wave packets. We have also found that the bath momentum density displays a behaviour analogous to that of the density. When the initial impurity wave
packet has a markedly non-Gaussian shape, we find complex interference phenomena, both in the ripples and within the central trough. We examined the inter-bath, equal-times connected density and
momentum density correlations, which revealed a rich structure in real space. This structure is best understood by taking “slices” of the correlation function along the relative and centre-of-mass
coordinate, which show both the light-cone propagation of correlations and the motion of the impurity. We have made the first steps towards a many-baths system, in which the impurity moves in a 1D or
2D lattice of 1D baths. We have obtained the time evolution of the state of the system with the perturbative technique developed before. The impurity Green's function has revealed qualitative
differences from the two-baths setup. First of all, the Green's function shows a complex short-time behaviour, caused by interference effects between the various paths of propagation within the
lattice of baths. More importantly, we have found that each band of the noninteracting impurity is characterised by its own OC exponent, which is proportional to the degeneracy of the band. We have
then speculated that it may be possible to tune the OC exponent either by properly designing the lattice of baths, or by changing the degeneracies by means of external magnetic fields. We also shown
that for generic lattices of baths, the OC exponents vanish in the limit of an infinite number of lattice sites. Lastly, we showed that in the case of two identical baths there exists a unitary
transformation that diagonalises the impurity degrees of freedom, thus achieving a complete decoupling of the latter from the baths. On the basis of this transformation, we have sketched two
variational approximations for the ground-state and dynamics of the system.
File in questo prodotto:
File Dimensione Formato
Open Access dal 02/07/2023
Tipologia: Tesi
Licenza: Creative commons 39.33 MB Adobe PDF Visualizza/Apri
Dimensione 39.33 MB
Formato Adobe PDF
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione. | {"url":"https://iris.sissa.it/handle/20.500.11767/128970","timestamp":"2024-11-08T15:43:39Z","content_type":"text/html","content_length":"71433","record_id":"<urn:uuid:7b6c2d1e-82ae-403a-9b66-2887a797c3b8>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00722.warc.gz"} |
Electric Field of Charge Sheet Calculator
In the field of electrodynamics, a crucial subfield of physics, understanding the behavior of electric fields is essential. One fundamental concept is the electric field of a uniformly charged sheet,
a topic that has significant implications in the fields of electrostatics, electronics, and quantum mechanics.
Field of
based on
Electric Field of
Charge Sheet
Calculator Results
Electric Field (E) =
Example Formula
The magnitude of the electric field (E) produced by a uniformly charged sheet is given by the following equation:
E = σ / (2 × ε[0])
1. E: This represents the electric field.
2. σ: This is the surface charge density (amount of charge per unit area) on the sheet.
3. ε[0]: This is the permittivity of free space, which is a constant with a value of approximately 8.85 × 10^-12 C^2/N m^2.
Who wrote/refined the formula
This concept can be traced back to the works of 19th-century physicists such as Michael Faraday and James Clerk Maxwell, who made fundamental contributions to the understanding of electromagnetic
fields. The formula itself is a direct application of Gauss's Law, part of Maxwell's equations, to a specific geometry.
Real Life Application
The concept of the electric field of a charge sheet is particularly applicable in the field of electronics. Many electronic devices and systems, such as capacitors and metal-oxide-semiconductor
field-effect transistors (MOSFETs), operate based on the principles of charge accumulation and electric fields.
Key individuals in the discipline
James Clerk Maxwell and Michael Faraday are notable figures in this discipline. Faraday's experiments in electromagnetism paved the way for Maxwell to formulate his groundbreaking equations that
describe electromagnetic phenomena, including the electric field of a charge sheet.
Interesting Facts
1. The electric field of a uniformly charged infinite plane sheet is constant and does not decrease with distance, which is counter-intuitive and interesting.
2. This concept is critical in the design of modern electronic devices, including capacitors and semiconductors, that have dramatically transformed our way of life.
3. Understanding electric fields has led to groundbreaking technologies like the MRI scanner, which uses electromagnetic fields to generate detailed images of the human body.
The study of the electric field of a charge sheet is a fascinating and essential part of physics, particularly electrodynamics. It serves as a cornerstone in understanding more complex
electromagnetic phenomena and has immense practical applications, from the design of electronic components to breakthroughs in medical imaging. The ongoing exploration of electric fields continues to
open new frontiers in our understanding of the physical world and the development of cutting-edge technologies.
Physics Calculators
You may also find the following Physics calculators useful. | {"url":"https://physics.icalculator.com/electric-field-of-charge-sheet-calculator.html","timestamp":"2024-11-14T20:20:39Z","content_type":"text/html","content_length":"19655","record_id":"<urn:uuid:3eb414c8-5af6-43c8-a7a7-2b349bc12158>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00323.warc.gz"} |
On the intersection of edges of a geometric graph by straight lines
A geometric graph ( = gg) is a pair G = 〈V, E〉, where V is a finite set of points ( = vertices) in general position in the plane, and E is a set of open straight line segments ( = edges) whose
endpoints are in V. G is a convex gg ( = cgg) if V is the set of vertices of a convex polygon. For n≥1, 0≤e≤(^n[2]) and m≥1 let I = I(n,e,m) (I[c]=I[c](n,e,m)) be the maximal number such that for
every gg (cgg) G with n vertices and e edges there exists a set of m lines whose union intersects at least I (I[c]) edges of G. In this paper we determine I[c](n,e,m) precisely for all admissible n,
e and m and show that I(n,e,m) = I[c](n,e,m) if 2me≥n^2 and in many other cases.
All Science Journal Classification (ASJC) codes
• Theoretical Computer Science
• Discrete Mathematics and Combinatorics
Dive into the research topics of 'On the intersection of edges of a geometric graph by straight lines'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/on-the-intersection-of-edges-of-a-geometric-graph-by-straight-lin","timestamp":"2024-11-05T07:56:13Z","content_type":"text/html","content_length":"48020","record_id":"<urn:uuid:0316a5b0-b115-4fef-9d63-60add11570af>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00651.warc.gz"} |
A Trio of Parabola Constructions | Sine of the Times
A Trio of Parabola Constructions
In my prior blog posts, I’ve presented methods for constructing ellipses using Web Sketchpad and paper folding. The other conic sections are feeling a bit left out, so let’s explore some techniques
for constructing parabolas.
All three Web Sketchpad models below (and here) are based on the distance definition of a parabola: The set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
Experiment with the interactive models and think about why they generate parabolas. The rhombus construction is based on an illustration found in the 17th-century Dutch mathematician Frans van
Schooten’s manuscript, Sive de Organica Conicarum Sectionum in Plano Descriptione, Tractatus (A Treatise on Devices for Drawing Conic Sections). Van Schooten’s beautiful picture is below. You’ll find
many more of van Schooten’s constructions in my book Exploring Conic Sections with The Geometer’s Sketchpad. | {"url":"https://www.sineofthetimes.org/a-trio-of-parabola-constructions/","timestamp":"2024-11-06T11:56:07Z","content_type":"text/html","content_length":"23937","record_id":"<urn:uuid:696a89c8-f89b-4d5e-9941-07dccab2242e>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00190.warc.gz"} |
Lecture 5 – Arithmetic Operations and Math Functions – Python Course for Beginners
Table of Contents[Hide][Show]
It has been clear to you that there are two types of numbers in programming. i.e. integers and floating-point numbers.
The arithmetic operations in Python are the same as everyday math and they revolve around these two data types.
Arithmetic Operators
There are seven basic types of arithmetic operators. These are:
Addition (+) : Adds two numbers. e.g.
print(10 + 4)
Subtraction (-): Subtract the second number from the first. e.g.
print(10 - 4)
Multiplication (*): Multiply two numbers. e.g.
print(10 * 4)
Division (/): Performs division on two numbers. e.g.
print(10 / 4)
Floor Division (//): Performs division and rounds off the answer to the nearest integer. e.g.
print(10 // 4)
Modulo Operator (%): Performs division and returns the remainder. e.g.
print(10 % 4)
Exponent (**): Takes the power of the integer e.g.
print(10 ** 4)
All of these operations are shown below:
Now for all these operators that you learned, we have an augmented assignment operator. Let me show you how it is used.
Let’s say we have a variable called ‘x’. We set it to 10, now we want to increment this by 3, we’ll have to write code like this.
x = 10
x = x + 3
Python interpreter will add 3 in ‘x’ and store it in ‘x’. Let’s print this:
An augmented assignment operator can be used to replicate the same functionality but more efficiently.
The same code will be written like this.
x = 10
x += 3
Now, this operator can be used for subtraction and multiplication too. Look at this program.
Here we are first incrementing ‘x’ by 3 and then multiplying it by 3. The output of line 2 should be 13 and the output of line 3 should be 39.
Operator Precedence
In math, we have a concept called operator precedence, which means the order of execution of operations in an equation. It’s not specific to Python, and all programming languages follow the operator
precedence. Let me remind you of the order:
1. Parenthesis
2. Exponent
3. Division or Multiplication
4. Addition or Subtraction
Let’s write a program and check this:
x = 10 + 3 * 2 ** 2 - (9 + 2)
What should be the answer to the above equation?
If your answer is 11, you don’t need to repeat high school.
Math Functions | {"url":"https://hashdork.com/python-lecture-5/","timestamp":"2024-11-08T17:13:21Z","content_type":"text/html","content_length":"312506","record_id":"<urn:uuid:5adcafce-cee5-47bb-9e6c-a5192a479add>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00880.warc.gz"} |
Fundamental Math for Data Science | Codecademy
Skill Path
Fundamental Math for Data Science
Build the mathematical skills you need to work in data science.
Includes Probability, Descriptive Statistics, Linear Regression, Matrix Algebra, Calculus, Hypothesis Testing, and more.
To start this Skill Path, upgrade your plan.
• Time to complete
Average based on combined completion rates — individual pacing in lessons, projects, and quizzes may vary
12 hours
About this skill path
Data scientists use math as well as coding to create and understand analytics. Whether you want to understand the language of analytics, produce your own analyses, or even build the skills to do
machine learning, this Skill Path targets the fundamental math you will need. Learn probability, statistics, linear algebra, and calculus as they are applied to real-world data analysis!
Skills you'll gain
• Speak the language of data science
• Perform hypothesis tests
• Use code to do mathematics
8 units • 16 lessons • 8 projects • 9 quizzes
• 1
Welcome to Fundamental Math for Data Science
Overview of material in the Fundamental Math for Data Science Skill Path
• 2
Descriptive Statistics
Learn how to summarize quantitative variables categorical variables in Python using numerical summary statistics.
• 3
Learn the fundamentals of probability by investigating random events.
• 4
Inferential Statistics
Learn about hypothesis testing and implement binomial and one-sample t-tests in Python.
• 5
Linear Algebra
Learn about linear algebra and how to perform operations with matrices and vectors.
• 6
Differential Calculus
Learn about calculus and how to analyze functions using limits and derivatives.
• 7
Final Problem Set
Assess your knowledge with a final problem set.
Certificate of completion available with Plus or Pro
Earn a certificate of completion and showcase your accomplishment on your resume or LinkedIn.
The platform
Hands-on learning
Earn a certificate of completion
Show your network you've done the work by earning a certificate of completion for each course or path you finish.
• Show proofReceive a certificate that demonstrates you've completed a course or path.
• Build a collectionThe more courses and paths you complete, the more certificates you collect.
• Share with your networkEasily add certificates of completion to your LinkedIn profile to share your accomplishments.
Reviews from learners
• The progress I have made since starting to use codecademy is immense! I can study for short periods or long periods at my own convenience - mostly late in the evenings.
Codecademy Learner @ USA
• I felt like I learned months in a week. I love how Codecademy uses learning by practice and gives great challenges to help the learner to understand a new concept and subject.
Codecademy Learner @ UK
• Brilliant learning experience. Very interactive. Literally a game changer if you're learning on your own.
Codecademy Learner @ USA
How it works
Skill paths help you level-up
Get a specialized skill
Want to level up at work? Gain a practical, real-world skill that you can use right away to stand out at your job.
Get step-by-step guidance
We guide you through exactly where to start and what to learn next to build a new skill.
Get there quickly
We’ve hand-picked the content in each Skill Path to fast-track your journey and help you gain a new skill in just a few months.
Ready to learn a new skill?
Get started on Fundamental Math for Data Science with a free Codecademy account.
Looking for something else?
What's included in skill paths
Practice Projects
Guided projects that help you solidify the skills and concepts you're learning.
Auto-graded quizzes and immediate feedback help you reinforce your skills as you learn.
Certificate of Completion
Earn a document to prove you've completed a course or path that you can share with your network. | {"url":"https://www.codecademy.com/learn/paths/fundamental-math-for-data-science?utm_source=ccblog&utm_medium=ccblog&utm_campaign=ccblog&utm_content=cw_what_does_ai_engineer_do_blog","timestamp":"2024-11-02T21:11:09Z","content_type":"text/html","content_length":"405522","record_id":"<urn:uuid:81f41f63-af25-407f-9723-2ffd9688b98a>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00494.warc.gz"} |
Uncertainty Principle
Last updated
Page ID
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\
evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\
newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}
\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}
{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}
[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}
{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\
#3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp
#2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\
wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\
newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}
{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\
bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\
widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)
In classical physics, studying the behavior of a physical system is often a simple task due to the fact that several physical qualities can be measured simultaneously. However, this possibility is
absent in the quantum world. In 1927 the German physicist Werner Heisenberg described such limitations as the Heisenberg Uncertainty Principle, or simply the Uncertainty Principle, stating that it is
not possible to measure both the momentum and position of a particle simultaneously.
The Nature of Measurement
In order to understand the conceptual background of the Heisenberg Uncertainty Principle it is important to understand how physical values are measured. In almost any measurement that is made, light
is reflected off the object that is being measured and processed. The shorter the wavelength of light used, or the higher its frequency and energy, the more accurate the results. For example, when
attempting to measure the speed of a tennis ball as it is dropped off of a ledge, photons(measurement of light) are shot off the tennis ball, reflected, and then processed by certain equipment.
Because the tennis ball is so large compared to the photons, it is unaffected by the efforts of the observer to measure its physical quantities. However, if a photon is shot at an electron, the
minuscule size of the electron and its unique wave-particle duality introduces consequences that can be ignored when taking measurements of macroscopic objects.
Heisenberg himself encountered such limitations as he attempted to measure the position of an electron with a microscope. As noted, the accuracy of any measurement is limited by the wavelength of
light illuminating the electron. Therefore, in principle, one can determine the position as accurately as one wishes by using light of very high frequency, or short wave-lengths. However, the
collision between such high energy photons of light with the extremely small electron causes the momentum of the electron to be disturbed.
Thus, increasing the energy of the light (and increasing the accuracy of the electron's position measurement), increases such a deviation in momentum. Conversely, if a photon has low energy the
collision does not disturb the electron, yet the position cannot be accurately determined. Heisenberg concluded in his famous 1927 paper on the topic,
"At the instant of time when the position is determined, that is, at the instant when the photon is scattered by the electron, the electron undergoes a discontinuous change in momentum. This change
is the greater the smaller the wavelength of the light employed, i.e., the more exact the determination of the position. At the instant at which the position of the electron is known, its momentum
therefore can be known only up to magnitudes which correspond to that discontinuous change; thus, the more precisely the position is determined, the less precisely the momentum is known..."
(Heisenberg, 1927, p. 174-5).
Heisenberg realized that since both light and particle energy are quantized, or can only exist in discrete energy units, there are limits as to how small, or insignificant, such an uncertainty can
be. As proved later in this text, that bound ends up being expressed by Planck's Constant, h = 6.626*10^-34 J*s.
It is important to mention that The Heisenberg Principle should not be confused with the observer effect. The observer effect is generally accepted to mean that the act of observing a system will
influence that which is being observed. While this is important in understanding the Heisenberg Uncertainty Principle, the two are not interchangeable. The error in such thinking can be explained
using the wave-particle duality of electromagnetic waves, an idea first proposed by Louis de Broglie. Wave-particle duality asserts that any energy exhibits both particle- and wave-like behavior. As
a consequence, in quantum mechanics, a particle cannot have both a definite position and momentum. Thus, the limitations described by Heisenberg are a natural occurrence and have nothing to do with
any limitations of the observational system.
Heisenberg’s Uncertainty Principle
It is mathematically possible to express the uncertainty that, Heisenberg concluded, always exists if one attempts to measure the momentum and position of particles.First, we must define the variable
“x” as the position of the particle, and define “p” as the momentum of the particle. The momentum of a photon of light is known to simply be its frequency, expressed by the ratio h/λ, where h
represents Planck’s constant and λ represents the wavelength of the photon. The position of a photon of light is simply its wavelength, \lambda\).. In order to represent finite change in quantities,
the Greek uppercase letter delta, or Δ, is placed in front of the quantity. Therefore,
\[\Delta{x}= \lambda\]
By substituting \(\Delta{x}\) for \(\lambda\) in the first equation, we derive
Note, we can derive the same formula by assuming the particle of interest is behaving as a particle, and not as a wave. Simply let Δp=mu, and Δx=h/mu (from De Broglie’s expression for the wavelength
of a particle). Substituting in Δp for mu in the second equation leads to the very same equation derived above-ΔpΔx=h. This equation was refined by Heisenberg and his colleague Niels Bohr, and was
eventually rewritten as
\[\Delta{p}\Delta{x} \ge \dfrac{h}{4\pi}\]
What this equation reveals is that the more accurately a particle’s position is known, or the smaller Δx is, the less accurately the momentum of the particle Δp is known. Mathematically, this occurs
because the smaller Δx becomes, the larger Δp must become in order to satisfy the inequality. However, the more accurately momentum is known the less accurately position is known.
Understanding the Uncertainty Principle through Wave Packets and the Slit Experiment
It is hard for most people to accept the uncertainty principle, because in classical physics the velocity and position of an object can be calculated with certainty and accuracy. However, in quantum
mechanics, the wave-particle duality of electrons does not allow us to accurately calculate both the momentum and position because the wave is not in one exact location but is spread out over space.
A "wave packet" can be used to demonstrate how either the momentum or position of a particle can be precisely calculated, but not both of them simultaneously. An accumulation of waves of varying
wavelengths can be combined to create an average wavelength through an interference pattern: this average wavelength is called the "wave packet". The more waves that are combined in the "wave
packet", the more precise the position of the particle becomes and the more uncertain the momentum becomes because more wavelengths of varying momenta are added. Conversely, if we want a more precise
momentum, we would add less wavelengths to the "wave packet" and then the position would become more uncertain. Therefore, there is no way to find both the position and momentum of a particle
Several scientists have debated the Uncertainty Principle, including Einstein. Einstein created a slit experiment to try and disprove the Uncertainty Principle. He had light passing through a slit,
which causes an uncertainty of momentum because the light behaves like a particle and a wave as it passes through the slit. Therefore, the momentum is unknown, but the initial position of the
particle is known. Here is a video that demonstrates particles of light passing through a slit and as the slit becomes smaller, the final possible array of directions of the particles becomes wider.
As the position of the particle becomes more precise when the slit is narrowed, the direction, or therefore the momentum, of the particle becomes less known as seen by a wider horizontal distribution
of the light.
The Importance of the Heisenberg Uncertainty Principle
Heisenberg’s Uncertainty Principle not only helped shape the new school of thought known today as quantum mechanics, but it also helped discredit older theories. Most importantly, the Heisenberg
Uncertainty Principle made it obvious that there was a fundamental error in the Bohr model of the atom. Since the position and momentum of a particle cannot be known simultaneously, Bohr’s theory
that the electron traveled in a circular path of a fixed radius orbiting the nucleus was obsolete. Furthermore, Heisenberg’s uncertainty principle, when combined with other revolutionary theories in
quantum mechanics, helped shape wave mechanics and the current scientific understanding of the atom.
1. Petrucci, Ralph H., William S. Harwood, F. Geoffrey Herring, and Jeffry D. Madura. General Chemistry. Principles and Modern Applications. 9th ed. Upper Saddle River, NJ: Pearson Prentice Hall,
2007. 298-299. Print.
2. Zumdahl, Stephen S., and Susan Zumdahl. Chemistry. 5th ed. . Boston, MA: Houghton Mifflin Harcourt, 1999. 307-08. Print
3. Heisenberg, W. (1927), "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik", Zeitschrift für Physik . English translation: J. A. Wheeler and H. Zurek,Quantum Theory and
Measurement Princeton Univ. Press, 1983, pp. 174–5.
Outside Links
• www.marts100.com/hup.htm
• www.youtube.com/watch?v=KT7xJ0tjB4A
1. What aspect of the Bohr model of the atom does the Heisenberg Uncertainty Principle discredit?
2. What is the difference between the Heisenberg Uncertainty Principle and the Observer Effect?
3. A Hydrogen atom has a radius of 0.05nm with a position accuracy of 1.0%. What is the uncertainty in determining the velocity?
4. What is the uncertainty in the speed of a beam of electrons whose position is known with an uncertainty of 10 nm?
5. Using the Uncertainty Principle, find the radius of an atom (in nm) that has an electron with a position accuracy of 3.0% and a known velocity of \(2\times 10^9\, m/s\).
1.) The Heisenberg Uncertainty Principle discredits the aspect of the Bohr atom model that an electron is constrained to a one-dimensional orbit of a fixed radius around the nucleus.
2.) The Observer Effect means the act of observing a system will influence what is being observed, whereas the Heisenberg Uncertainty Principle has nothing to do with the observer or equipment used
during observation. It simply states that a particle behaves both as a wave and a particle and therefore cannot have both a definite momentum and position.
3.) Uncertainty principle: ΔxΔp≥h/4Π
Can be written (x)(m)(v)(%)=h/4Π
(position)(mass of electron)(velocity)(percent accuracy)=(Plank's Constant)/4Π
(.05*10^-9 m)(9.11*10^-31 kg)(v)(.01)=(6.626*10^-34 J*s)/4Π
v=1*10^8 m/s
4.) Uncertainty principle: ΔxΔp≥h/4Π
(10*10^-9 m)(Δp)≥(6.626*10^-34 J*s)/4Π
Δp≥5*10^-26 (kg*m)/s
5.) Uncertainty principle: ΔxΔp≥h/4Π
Can be written (x)(m)(v)(%)=h/4Π
(position)(mass of electron)(velocity)(percent accuracy)=h/4Π
(radius)(9.11*10^-31 kg)(2*10^9)(.03)=(6.626*10^-34 J*s)/4Π
r=9*10^-12 m
r=9*10^-3 nm | {"url":"https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Mechanics/09._The_Hydrogen_Atom/Atomic_Theory/Electrons_in_Atoms/Uncertainty_Principle","timestamp":"2024-11-06T23:19:17Z","content_type":"text/html","content_length":"143068","record_id":"<urn:uuid:2f198648-9ba2-4815-9ba9-d0067c44e108>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00710.warc.gz"} |
Decimal ⇄ Octal Converter with Steps
Method for Decimal to Octal Conversion
0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are the decimal numbers, generally represented by base-10 notation in digital electronics & communications. Whereas, the digits 0, 1, 2, 3, 4, 5, 6 and 7 are known as
octal numbers generally represented by base-8 notation. The decimal to octal conversion can be done by using the MOD-8 operation. The below method and step by step conversion may useful to learn and
practice how to do decimal to octal conversion manually.
step 1: Perform the MOD-8 operation for the given decimal number.
step 2: Arrange remainder from the bottom to top as shown in the below figure, is the equivalent octal number.
Solved Example Problem
The below solved example problem may useful to understand how to perform decimal to octal number conversion.
Convert the decimal 143[10] to its octal equivalent. | {"url":"https://ncalculators.com/digital-computation/decimal-octal-converter.htm","timestamp":"2024-11-13T15:54:38Z","content_type":"text/html","content_length":"55526","record_id":"<urn:uuid:9f3507ec-bc53-4be1-b540-488669a34926>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00379.warc.gz"} |
.mps format for MIPs#
The MPS (Mathematical Programming System) format is one of the oldest and most widely used formats for representing optimization models. Originally developed for linear programming, it has been
extended to support more complex models, including mixed-integer and quadratic programming. The format is designed to be machine-readable, and while it may be less intuitive than formats like LP, it
remains a standard in the field due to its wide adoption. All files that correspond to this format must have the extension .mps in the file name.
Basic Structure#
An MPS file is divided into several sections, each serving a specific purpose in defining the optimization model. The general structure includes:
1. NAME Section
2. OBJSENSE Section (Optional)
3. ROWS Section
4. COLUMNS Section - Integrality Markers (Optional)
5. RHS Section
6. BOUNDS Section (Optional)
7. SOS Section (Optional)
8. QUADOBJ Section (Optional)
9. QCMATRIX Section (Optional)
10. INDICATOR Constraints (Optional)
11. ENDATA Statement
Please note that the order of these sections must be adhered to in order to guarantee a correct processing of the problem!
Fixed Format vs. Free Format#
MPS files can be written in either fixed or free format:
• Fixed Format: Fields start at specific columns in each line.
• Free Format: Fields are separated by whitespace and can be of variable length.
Key Differences:
• In fixed format, names (rows, columns) are limited to 8 characters and can include spaces.
• In free format, names can be longer (up to 255 characters in most implementations) but cannot contain spaces.
Most modern solvers including the Quantagonia HybridSolver can automatically detect the format used. Naturally, the fixed format is always a valid free format, therefore we mainly present examples in
fixed format below.
1. NAME Section#
The NAME section identifies the problem name.
Fixed Format Example:
• Position: The problem name starts at column 15.
2. OBJSENSE Section (Optional)#
Specifies the optimization direction (minimize or maximize). By default, the objective is minimized.
Place this section after the NAME section if you want to maximize the objective function.
3. ROWS Section#
Defines the rows of the model, which correspond to constraints and the objective function.
• Type Indicator: Indicates the type of the row.
□ E: Equality (=).
□ L: Less-than-or-equal (<=).
□ G: Greater-than-or-equal (>=).
□ N: Free row (often used for the objective function).
• Row Name: The identifier for the constraint or objective.
Fixed Format Example:
E EQ1
L LEQ1
G GEQ1
N OBJ
• Positions:
□ Type Indicator: Column 2.
□ Row Name: Starts at column 5.
Note: The first N row encountered is typically treated as the objective function.
4. COLUMNS Section#
Lists the variables (columns) and their coefficients in the constraints and objective function.
• Column Name: The variable’s identifier.
• Row Name and Coefficient: Specifies the coefficient of the variable in a particular row (constraint or objective).
Each line can contain up to two (row, coefficient) pairs for the variable.
Fixed Format Example:
X1 EQ1 1.0 OBJ 3.0
X1 LEQ1 2.0
X2 EQ1 1.0 OBJ 4.0
X2 GEQ1 3.0
• Positions:
□ Column Name: Starts at column 5.
□ First Row Name: Starts at column 15.
□ First Coefficient: Starts at column 25.
□ Second Row Name: Starts at column 40.
□ Second Coefficient: Starts at column 50.
Important Notes:
• All entries for a variable should be contiguous.
• Coefficients for the objective function are associated with the row name used for the objective (usually the first N row).
Integrality Markers (Optional)#
Used to indicate that variables between markers are integer variables. By default, all variables within markers have a lower bound of 0 and an upper bound of 1. Other bounds can be specified in the
BOUNDS section.
• Start Marker: ‘MARKER’ ‘INTORG’
• End Marker: ‘MARKER’ ‘INTEND’
MARK0000 'MARKER' 'INTORG'
X1 EQ1 1.0 OBJ 3.0
X2 EQ1 1.0 OBJ 4.0
MARK0000 'MARKER' 'INTEND'
• Variables X1 and X2 are declared as integer variables.
• Positions:
□ Marker Name: Starts at column 5.
□ Marker Keyword: Starts at column 15 and must be equal to ‘MARKER’ including the quotes.
□ Integer Section Keyword: Starts at column 40 and must be equal to ‘INTORG’ at the start and ‘INTEND’ at the end of the integer section, again including the quotes.
5. RHS Section#
Specifies the right-hand side values for the constraints.
• RHS Name: Generally ignored by modern solvers.
• Row Name and Value: Specifies the right-hand side value for a constraint.
Each line can contain up to two (row, value) pairs.
Fixed Format Example:
RHS1 EQ1 5.0 LEQ1 10.0
RHS1 GEQ1 2.0
• Positions:
□ RHS Name: Starts at column 5.
□ First Row Name: Starts at column 15.
□ First Value: Starts at column 25.
□ Second Row Name: Starts at column 40.
□ Second Value: Starts at column 50.
Note: Rows not mentioned in the RHS section have a right-hand side value of zero.
6. BOUNDS Section (Optional)#
Defines the bounds on variables. By default, variables have a lower bound of zero and no upper bound.
Bound Types:
• LO: Lower bound.
• UP: Upper bound.
• FX: Fixed variable (both bounds equal).
• FR: Free variable (no bounds).
• MI: Minus infinity (no lower bound).
• PL: Plus infinity (no upper bound).
• BV: Binary variable (0 or 1).
• LI: Integer variable lower bound.
• UI: Integer variable upper bound.
• SC: Semi-continuous variable upper bound.
• SI: Semi-integer variable upper bound.
• Bound Type
• Bound Name: Generally ignored.
• Variable Name
• Value: The bound value (if applicable).
Fixed Format Example:
UP BND X1 10.0
LO BND X2 1.0
FX BND X3 5.0
FR BND X4
BV BND X5
• Positions:
□ Bound Type: Column 2.
□ Bound Name: Starts at column 5.
□ Variable Name: Starts at column 15.
□ Value: Starts at column 25.
7. SOS Section (Optional)#
Defines Special Ordered Sets (SOS) constraints of type 1 and 2.
SOS Types:
• S1: SOS Type 1
• S2: SOS Type 2
• SOS Type
• SOS Name
• Variable and Weight: Each member variable and its weight.
Fixed Format Example:
S1 SOS1
X1 1
X2 2
S2 SOS2
X3 1
X4 2
X5 3
• Positions:
□ SOS Type (S1 or S2): Column 2.
□ SOS Name: Starts at column 5.
□ Variable Name: Starts at column 5.
□ Weight: Starts at column 15.
8. QUADOBJ Section (Optional)#
Defines quadratic terms in the objective function.
• Variable 1
• Variable 2
• Coefficient
Only the lower triangle (or full symmetric entries) needs to be specified due to symmetry.
Fixed Format Example:
X1 X1 2.0
X1 X2 1.0
X2 X2 3.0
• Represents the quadratic objective: \(\frac{1}{2}(2X1^2 + 2X1 \cdot X2 + 3X2^2)\).
• Positions:
□ QUADOBJ: Starts at column 1.
□ Variable 1: Starts at column 5.
□ Variable 2: Starts at column 15.
□ Coefficient: Starts at column 25.
9. QCMATRIX Section (Optional)#
Defines quadratic terms in quadratic constraints.
• QCMATRIX: Followed by the constraint name.
• Variable 1
• Variable 2
• Coefficient
Fixed Format Example:
QCMATRIX QC1
X1 X1 1.0
X1 X2 0.5
X2 X2 1.5
• Quadratic terms for constraint QC1.
• Positions:
□ QCMATRIX: Starts at column 1.
□ Constraint Name: Starts at column 12.
□ Variable 1: Starts at column 5.
□ Variable 2: Starts at column 15.
□ Coefficient: Starts at column 25.
10. INDICATOR Constraints (Optional)#
Defines indicator constraints, which activate constraints based on the value of a binary variable.
• IF
• Row Name
• Binary Variable
• Value: 0 or 1
Fixed Format Example:
IF CONSTR1 BIN_VAR1 1
IF CONSTR2 BIN_VAR2 0
• Constraint CONSTR1 is activated when BIN_VAR1 = 1.
• Positions:
□ INDICATORS: Starts at column 1 (header line).
□ IF: Column 2.
□ Row Name: Starts at column 5 (Keep in mind that the row must have already been defined in the ROWS section!).
□ Binary Variable: Starts at column 15.
□ Value: Starts at column 25.
11. ENDATA Statement#
Marks the end of the MPS file.
Examples of MPS Models#
Below are examples of MPS files representing different types of optimization problems.
Example 1: Linear Programming (LP) Problem#
NAME SIMPLELP
N COST
L CONSTR1
G CONSTR2
X1 CONSTR1 1.0 COST 3.0
X1 CONSTR2 -1.0
X2 CONSTR1 2.0 COST 5.0
X2 CONSTR2 1.0
RHS1 CONSTR1 10.0 CONSTR2 5.0
LO BND X1 0.0
LO BND X2 0.0
• Objective: Minimize the cost function \(3X1 + 5X2\).
• Constraints:
□ CONSTR1: \(X1 + 2X2 \leq 10\).
□ CONSTR2: \(-X1 + X2 \geq 5\).
• Bounds: \(X1, X2 \geq 0\).
Example 2: Mixed-Integer Programming (MIP) Problem#
NAME SIMPLEMIP
N PROFIT
L LIMIT1
L LIMIT2
MARK0000 'MARKER' 'INTORG'
X1 LIMIT1 1.0 PROFIT 10.0
X1 LIMIT2 1.0
X2 LIMIT1 1.0 PROFIT 6.0
X2 LIMIT2 2.0
X3 LIMIT1 1.0 PROFIT 4.0
X3 LIMIT2 3.0
MARK0000 'MARKER' 'INTEND'
RHS1 LIMIT1 100.0 LIMIT2 200.0
LO BND X1 0.0
LO BND X2 0.0
LO BND X3 0.0
• Objective: Maximize profit \(10X1 + 6X2 + 4X3\).
• Constraints:
□ LIMIT1: \(X1 + X2 + X3 \leq 100\).
□ LIMIT2: \(X1 + 2X2 + 3X3 \leq 200\).
• Variable Types: X1, X2, X3 are integer variables (between INTORG and INTEND markers).
• Bounds: \(X1, X2, X3 \geq 0\).
Example 3: Quadratic Programming (QP) Problem#
NAME SIMPLEQP
N OBJ
E QC1
X1 QC1 1.0 OBJ 0.0
X2 QC1 1.0 OBJ 0.0
RHS1 QC1 1.0
LO BND X1 0.0
LO BND X2 0.0
X1 X1 1.0
X2 X2 1.0
X1 X2 0.0
• Objective: Minimize \(\frac{1}{2}(X1^2 + X2^2)\).
• Constraint: QC1: \(X1 + X2 = 1\).
• Bounds: \(X1, X2 \geq 0\).
Additional Notes#
• Lines starting with an asterisk * are treated as comments and are ignored by the parser.
Fixed Format vs. Free Format#
• In Fixed Format, fields start at specific columns in each line. Names are limited to 8 characters and can include spaces.
• In Free Format, fields are separated by whitespace. Names can be longer (up to 255 characters) but cannot contain spaces.
• Sections should appear in the order specified for the file to be correctly interpreted.
• Be cautious of potential precision loss due to fixed formats; consider using free format or solvers that support full precision.
• Ensure consistent use of variable and constraint names throughout the file.
Objective Function#
• The objective function is typically associated with the first N row in the ROWS section.
Case Sensitivity#
• The MPS format is case-insensitive, but it’s good practice to be consistent.
The MPS format is a powerful and widely accepted standard for representing optimization models. By understanding its structure and conventions, you can effectively model and solve a wide range of
optimization problems using various solvers that support this format. | {"url":"https://docs.quantagonia.com/fileformats/mps_format.html","timestamp":"2024-11-14T05:10:40Z","content_type":"text/html","content_length":"48287","record_id":"<urn:uuid:32b34386-c2e5-482d-aa7b-06ee47a185ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00569.warc.gz"} |
Karpur Shukla - Physics Notes
Here, I've included personal notes for various physics and mathematics courses, textbooks, and papers I've worked through over the years. All of these notes are not intended as polished works and by
no means intended as stand-alone works, but are just compilations of thoughts and some calculations that helped me while I was going through the material in question. Hopefully, it'll serve to help
others as well in their struggles.
I've provided the notes here; the references are all provided in the respective subsections. I really hope these help!
Conformal Field Theory and Chern-Simons Theory
(References here.) | {"url":"https://www.karpurshukla.com/research/physics-notes","timestamp":"2024-11-04T21:36:45Z","content_type":"text/html","content_length":"76360","record_id":"<urn:uuid:27e0639a-7d68-4dde-a653-953cdc038e8b>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00395.warc.gz"} |
Extensions to the Finite Element Technique for the Magneto-Thermal Analysis of Aged Oil Cooled-Insulated Power Transformers
Extensions to the Finite Element Technique for the Magneto-Thermal Analysis of Aged Oil Cooled-Insulated Power Transformers ()
1. Introduction
Considerable attention has been directed towards the thermal analysis of power transformers in recent years. Obviously, maximum temperature at which the insulation of the power transformer can
operate and the method of heat dissipation through it and its surrounding should be determined during design stages. Moreover, accurate identification of the location and magnitude of a transformer
hot spot temperature could also be very useful in transformer design stage and/or cooling strategies [1].
In previous work, some efforts have been carried out to assess transformer’s components maximum temperatures [2,3]. The theoretical thermal model developed was limited by the basic assumption of
uniform distribution of heat generated per unit volume per unit time in both the iron core and the copper conductors. Within this assumption, winding insulation was not taken into account.
This study investigates the effect of the degradation of core magnetic properties on the temperature distribution as well as values of hot spot, top gas and tank temperatures of aged oil-cooled
transformers. More specifically, effects of global and local core permeability variation resulting from ageing and/or maintenance-related introduced mechanical stresses are considered.
Two-dimensional accurate assessment of time average flux density distribution has been computed using finite-element method to identify the core and winding losses which are converted into heat.
Based upon the ambient temperature outside the transformer tank and thermal heat transfer related factors, detailed thermal modeling and analysis have been then carried out to determine temperature
distribution everywhere. The thermal analysis included; heat conduction, free convection inside the oil tank, forced convection outside the tank, and radiation from tank surfaces.
Analytical details and simulation results demonstrating effects of core magnetic properties degradation on transformer’s temperatures are given in the following sections of the paper. It should be
pointed out that, throughout the paper, numerical simulation of the electromagnetic field and the heat transfer analysis has been carried out with the aid of ANSYS finite element software package.
2. Mathematical Model
It is well known that losses occurring in the magnetic core, windings and parts of an operating transformer are converted into heat. An electromagnetic finite element method is proposed to evaluate
those generated losses, which are used as the thermal analysis heat sources. Obviously, heat is transferred to the surrounding air via several stages due to a temperature difference at the boundaries
of each stage. The first stage of heat transfer begins from the interior of the core or the windings and extends to their external surfaces by conduction. The second stage is the transfer of heat
from the windings and the core to the oil by convection. The third stage is the transfer of heat from the oil to the inner tank walls by convection. The fourth stage is the transmission of heat
energy through the thickness of the tank walls by conduction. In the last stage, heat is dissipated from the external tank walls to the surrounding air by convection and radiation.
The theoretical analysis presented in this paper is based on the following assumptions:
1) The transformer structure can be considered as a plane-parallel geometry. Therefore, with a reasonable acceptable accuracy, the three-dimensional (3-D) problem is reduced to a two-dimensional
(2-D) one with x and y as space variables, with elements layout for finite element formulation, as shown in Figure 1.
2) The phase currents are sinusoidal and balanced. Consequently, all field quantities will by sinusoidal with time.
3) The base of the tank is assumed to be insulated and measured values of ambient temperature are always considered.
4) The insulation between the low voltage winding and high voltage winding is taken into consideration.
5) All electrical and thermal material properties have been considered as given in [4] and [5]. Thermo-physical properties of the materials are supposed to be function of temperature.
The associated steady state heat conduction equation for the two-dimensional model can be described—as given in [6] and [7]—by:
Figure 1. Schematic of transformer with elements layout for finite element formulation.
where T is temperature in (˚C), x and y are spatial variables in (m), K is the thermal conductivity in (W/m·˚C) and ^3).
Within this steady state analysis, and due to the difference between electrical and thermal systems response time, steady state heat generated in transformer parts due to power dissipation is
averaged over the power mains cycle period P. Taking this fact into consideration and referring to typical loss specifications for cold-rolled silicon steel (Si-Fe) sheets data (which may be
approximated by the expression
where B is the magnetic flux density (in Tesla) and t is the time instant, while ^3).
Likewise, the heat generated per unit volume per unit time at any point
where J is the current density in (A/m^2), and
Within the transformer thermal model, there are convective heat transfer between the winding surfaces, the core surface and the oil flowing over them, then from the oil to the inner surface of the
tank, and finally from the external tank walls to the surrounding air. The mathematical formulation of this convection boundary condition is obtained by considering an energy balance at the surface
stated as:
where, ^2·C) which could be determined by classical Nusselt number Nu correlations as [6]:
The outer surface of the transformer components was considered similar to a vertical or horizontal plate with uniform heat flux. For vertical plates the characteristic dimension L = height in (m);
while for horizontal plates the characteristic dimension L = length in (m).
There are several convective heat transfer coefficients that must be evaluated to determine the temperatures of the transformer. It should be pointed out that, the convective heat transfer
coefficients are based on the assumption that only free convection is suggested under transformer loading conditions as instructed by the manufacturer. In the presence of external fans, forced
convection due to air circulation outside the transformer tank may also be taken into account. Average Nusselt numbers Nu for forced and free convection on horizontal and vertical plates are given in
[2] and [6].
Radiation effects generally appear in the heat transfer analysis only through the boundary conditions. In this model, an open type enclosure surface radiates heat between the external surfaces of the
tank and the surrounding air with predetermined ambient temperature. The mathematical formulation of this radiation boundary condition is obtained—as discussed in [6,7], and [9]— by considering the
following energy balance expression:
where, ^2·K^4. Note that ε is the infrared emissivity = 0.95 in this model [6].
3. Testing and Numerical Results
In order to test the suggested methodology, detailed configuration of a 25 MVA, 66 KV/11 KV (ONAF), oil cooled-insulated power transformer was considered [10]. For this transformer the core length,
core height, window width and window height were about 2.5 m, 2.1 m, 0.5 m and 1.1 m, respectively. Thickness and height of each winding were about 0.l m and 1.0 m, respectively, and inter-spacing
between tank walls and active transformer parts was about 0.3 m. Low voltage and high voltage windings were the inner and outer ones, respectively.
3.1. No Ageing Case
Normal performance and thermal profile was first assessed as a reference to ageing situations. For normal operation, the flux density distribution Figure 2. The predicted flux density values using
the method proposed in this paper are compared with industrial results referred to in [10] and reflect a satisfactory quantitative agreement.
Thermal analysis in accordance with (1)-(6) was then carried out using finite-element analysis and for ^3, winding
Figure 2. Sample flux density computation results for instants corresponding to peak left-limb current (a), peak middle-limb current (b) and peak right-limb current values (c) for the transformer
when having normal magnetic properties.
Figure 3 shows the contour plot for a nodal temperature distribution of the oil cooled-insulated power transformer at full load (25 MVA). Obviously, hot spot temperature is expected to be located
somewhere on the horizontal line through the center of the T-joint of the core [11]. For the depicted locations of the aforementioned line, Figure 4 suggests that the values of the hot spot of iron
core temperature, winding temperature, top gas temperature, and the tank temperature are 96˚C, 92˚C, 91˚C and 49˚C, respectively.
It is clear from these two figures that the thermal profile is symmetric with respect to the middle axis of the whole transformer because the transformer’s sides are exposed to similar conditions. It
is also clear that hot spot
Figure 3. Overall thermal analysis profile for the transformer when having normal magnetic properties.
Figure 4. Temperature values at a horizontal plane bisecting the core for the transformer when having normal magnetic properties.
winding temperature is expected to be at the center of the low voltage windings in general. Estimated value of the hot spot winding temperature was found to be around 92˚C, which is in agreement with
industrially reported temperatures for this transformer.
3.2. Case of Aged Right Limb
Ageing of transformers mainly affects the oil insulation properties. Degradation in oil properties eventually results in short circuiting a winding. Whether the limb bearing this winding is exposed
to high instantaneous electromagnetic forces during the short circuit event or as a result of mechanical stresses imposed due to improper re-winding procedures, probability to end up with
irreversible local magnetic properties degradation becomes quite high [12]. Moreover, ageing also affects thermal properties of oil. Consequently, long periods of high temperatures exposure of
Cold-rolled Si-Fe sheets also could lead to a degradation of magnetic properties as well. In order to demonstrate the thermal consequences of local magnetic properties degradation as a result of the
aforementioned ageing scenarios, magnetic and thermal analysis for the transformer under consideration were carried out. All properties were assumed similar to the normal case with the exception of
the right limb where axial and transverse working relative permeability values of 2000 and 200, respectively.
For this particular aged case, sample flux density computation results for instants corresponding to peak leftlimb current (a), peak middle-limb current (b), and peak right-limb current values (c)
are given in Figure 5. Moreover, overall thermal analysis profile and temperature values at a horizontal plane bisecting the core are shown in Figures 6 and 7.
Unlike for the results shown in Figure 3, the thermal profile is not symmetric anymore with respect to the middle axis of the whole transformer. The same is true for the hot spot temperature. More
specifically, temperature values of windings of wound around the limb having inferior magnetic properties are higher (about 95˚C instead of 92˚C). This difference will increase as the magnetic
properties further degrade. Moreover, this will result in higher winding resistance that could result in more heat loss and, consequently, further increase in local temperature rise. It should be
stated that in accordance with the IEC code, excessive hot spot temperature rise will directly affect the transformer overall life time [13].
3.3. Case of Aged Yoke
If the yokes are exposed to high instantaneous electromagnetic forces or as a result of mechanical stresses, probability to end up with irreversible local magnetic properties degradation becomes
quite high. Moreoverageing also affects thermal properties of oil. Consequently, long periods of high temperatures exposure of cold-rolled Si-Fe sheets also could lead to a degradation of magnetic
properties as well. In order to demonstrate the thermal consequences of local magnetic properties degradation as a result of the aforementioned ageing scenarios, magnetic and thermal analysis for the
transformer under consideration were carried out. All properties
Figure 5. Sample flux density computation results for instants corresponding to peak left-limb current (a), peak middle-limb current (b) and peak right-limb current values (c) for the transformer
when having degraded right limb magnetic properties.
Figure 6. Overall thermal analysis profile for the transformer when having degraded right limb magnetic properties.
were assumed similar to the normal case with the exception of the yokes where axial and transverse working relative permeability values of 2000 and 200, respectively.
For this particular aged case, sample flux density computation results for instants corresponding to peak left-limb current (a), peak middle-limb current (b), and peak right-limb current values (c)
are given in Figure 8. Overall thermal analysis profile and temperature values at a horizontal plane bisecting the core are shown in Figures 9 and 10. It is observed from these two figures that the
thermal profile is symmetric with respect to the middle axis of the whole transformer. More specifically,
Figure 7. Temperature values at a horizontal plane bisecting the core for the transformer when having degraded right limb magnetic properties.
temperature values of windings of wound around the limb having inferior magnetic properties are higher (about 97˚C instead of 92˚C). This difference will increase as the magnetic properties further
degrade that could result in more heat loss and, consequently, further increase in local temperature rise.
4. Conclusion
This paper has presented a combined magneto-thermal analysis for calculating the thermal fields at any specified location within the oil cooled-insulated power transformer, using finite element
technique. This analysis is more precise than any other previously developed model
Figure 8. Sample flux density computation results for instants corresponding to peak left-limb current (a), peak middle-limb current (b) and peak right-limb current values (c) for the transformer
when having degraded yoke magnetic properties.
Figure 9. Overall thermal analysis profile for the transformer when having degraded yoke magnetic properties.
because it considers effects of global and local core permeability variation resulting from ageing and/or maintenance-related introduced mechanical stresses are considered, and the contributions of
the convective and radiative heat transfers. The thermal model developed here can predict the hot spot location, with a reasonable degree of accuracy during the design stage. This will in turn allow
for better transformer performance when it is put in service. It can be concluded from the presented analysis and simulation results that degraded local magnetic properties would have an impact on
thermal profile of a power transformer and, consequently, increasing its hot spot temperature. The proposed analysis approach may be extended to cover the tank outside vicinity in order to
Figure 10. Temperature values at a horizontal plane bisecting the core for the transformer when having degraded yoke magnetic properties.
thermally monitor any magnetic property degradation signs.
Abbreviations and Acronyms
B The magnetic flux density in Tesla
h Convective heat transfer coefficient, W/m^2·C
J The current density, A/m^2
K Thermal conductivity, W/m·C
L Length, m
Nu Nusselt number, dimensionless
q^o The rate of heat per unit volume per unit time, W/m^3
t Time, sec
T Temperature, C
TB Bulk temperature of the adjacent fluid, C
X X direction
Y Y direction
ε Infrared emissivity, dimensionless
σ Stephen-Boltzman constant, 5.67 × 10^–^8 W/m^2·K^4
ρ Density, kg/m^3
δ The resistivity, Ω·m | {"url":"https://www.scirp.org/journal/paperinformation?paperid=18616","timestamp":"2024-11-10T18:04:17Z","content_type":"application/xhtml+xml","content_length":"111054","record_id":"<urn:uuid:c0e3e70b-5d3d-4ec3-a760-837c98164aa0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00772.warc.gz"} |
How to Find Standard Deviation on TI-84: Easy Calculator Guide
This tool helps you calculate the standard deviation using a TI-84 calculator.
How to Use the Standard Deviation Calculator
To use the calculator, follow these steps:
1. Enter your data set in the text field provided. Ensure the values are separated by commas.
2. Click the “Calculate” button.
3. The Standard Deviation will be displayed in the result field below.
How the Calculator Works
The calculator follows these steps to compute the Standard Deviation:
• First, it converts the input string into an array of numbers.
• Next, it calculates the mean (average) of the numbers.
• It then computes the variance by averaging the squared differences from the mean.
• Finally, it computes the Standard Deviation as the square root of the variance.
The current calculator has the following limitations:
• The input must be a comma-separated list of numbers.
• It does not handle empty input or inputs that contain non-numeric values well. Proper error messages should be handled but are currently basic.
• The standard deviation result is rounded to four decimal places. | {"url":"https://madecalculators.com/how-to-find-standard-deviation-on-ti-84/","timestamp":"2024-11-09T17:39:07Z","content_type":"text/html","content_length":"142169","record_id":"<urn:uuid:4d92790f-b33b-4d20-9e3b-692f8bcbfa4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00536.warc.gz"} |
Determinant of an elementary matrix
In this lecture we study the properties of the determinants of elementary matrices. The results derived here will then be used in subsequent lectures to prove general properties satisfied by the
determinant of any matrix.
Elementary matrix
Remember that an elementary matrix is a square matrix that has been obtained by performing an elementary row or column operation on an identity matrix.
Furthermore, elementary matrices can be used to perform elementary operations on other matrices: if we perform an elementary row (column) operation on a matrix , this is the same as performing the
given operation on the identity matrix , so as to get an elementary matrix , and then pre-multiplying (post-multiplying) by .
Also remember that there are three elementary row (column) operations:
• multiply a row (column) by a non-zero constant;
• add a multiple of a row (column) to another row (column);
• interchange two rows (columns).
Each of these three operations will be analyzed separately in the next sections. We will focus on elementary row operations. The results for column operations are analogous.
Determinant of a row multiplication matrix
Let us start with elementary matrices that allow us to perform the multiplication of a row by a constant.
Proposition Let be a matrix. Let be an elementary matrix obtained by multiplying a row of the identity matrix by a constant . Then,and
Denote by the set of all permutations of the first natural numbers. Denote by the permutation in which the numbers are left in their natural order (sorted in increasing order). Since does not contain
any inversion (see the lecture on the sign of a permutation), its parity is even and its sign is Then, the determinant of the identity matrix iswhere in step we have used the fact that for all
permutations except the productinvolves at least one off-diagonal element that is equal to zero (remember that all the diagonal elements of are equal to and all the off-diagonal elements are equal to
). Let's now consider the elementary matrix . The only difference with respect to is that one of the diagonal elements of is equal to . As a consequence, we haveSuppose that the first row of has been
multiplied by , so that is the matrix obtained by multiplying the first row of by . We can write the determinant of as:Therefore,The assumption that the row multiplied by is the first one is without
loss of generality (if it is the -th row, then needs to be factored out in the above formulae, but the result is the same).
Determinant of a row interchange matrix
Let us now tackle the case of elementary matrices that allow us to interchange two rows.
Proposition Let be a matrix. Let be an elementary matrix obtained by interchanging two rows of the identity matrix . Then,and
In order to understand this proof, we need to revise the concept of transposition introduced in the lecture entitled Sign of a permutation. A transposition is the operation of interchanging any two
distinct elements of a permutation. A transposition changes the parity of a permutation (it makes an even permutation odd and vice-versa), as well as its sign. Any permutation of the first natural
numbers can be obtained by performing on them a sequence of transpositions. The number of transpositions determines the parity of the permutation (even if the number of transpositions is even, and
odd otherwise). Suppose the matrix has been obtained from the identity matrix by interchanging rows and and denote by the set of the first natural numbers except and . For every permutation of the
first natural numbers there is a permutation such thatSince is a transposition of , we haveThen,where: in step we have used the fact that all rows of are equal to the rows of , except the -th and
-th, which are interchanged; in step we have used the definition of the permutation given above. The determinant of , which is obtained by interchanging the -th and -th rows of , is derived in an
analogous manner:Therefore,
Determinant of a row addition matrix
The last case we analyze is that of elementary matrices that allow us to add a multiple of one row to another row.
Proposition Let be a matrix. Let be an elementary matrix obtained by adding a multiple of one row of the identity matrix to another of its rows. Then,and
Suppose the matrix has been obtained from the identity matrix by adding times the -th row to the -th. Denote by the matrix obtained from the identity matrix by replacing the -th row with the -th.
Thus, the -th and the -th row of coincide. By the proposition above on row interchanges, the determinant of the matrix obtained by interchanging the -th and the -th rows of isBut because we have
interchanged two identical rows, therefore it must be thatwhich impliesDenote by the set of the first natural numbers except .Then,The determinant of , which is obtained by adding times the -th row
to the -th row of , is derived in an analogous manner. Let us denote by the matrix obtained from by replacing the -th row with the -th. Then,Therefore,
Determinant of product equals product of determinants
We have proved above that all the three kinds of elementary matrices satisfy the propertyIn other words, the determinant of a product involving an elementary matrix equals the product of the
determinants. We will prove in subsequent lectures that this is a more general property that holds for any two square matrices.
Elementary column operations
All the propositions above concern elementary matrices used to perform row operations. The same results apply to column operations, and their proofs are almost identical. This is a consequence of the
fact that transposition does not change the determinant of a matrix (a fact that will be proved later on) and column operations on a matrix can be seen as row operations performed on its transpose .
How to cite
Please cite as:
Taboga, Marco (2021). "Determinant of an elementary matrix", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/elementary-matrix-determinant. | {"url":"https://www.statlect.com/matrix-algebra/elementary-matrix-determinant","timestamp":"2024-11-12T11:40:50Z","content_type":"text/html","content_length":"85854","record_id":"<urn:uuid:95db2b36-a882-435b-a46e-475f325c421f>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00082.warc.gz"} |
A Short Note on Gaps Between Powers of Consecutive Primes
The primary purpose of this note is to collect a few hitherto unnoticed or unpublished results concerning gaps between powers of consecutive primes. The study of gaps between primes has attracted
many mathematicians and led to many deep realizations in number theory. The literature is full of conjectures, both open and closed, concerning the nature of primes.
In a series of stunning developments, Zhang, Maynard, and Tao^1 ^1James Maynard. Small gaps between primes. Ann. of Math. (2), 181(1):383–413, 2015. ^2 ^2 Yitang Zhang. Bounded gaps between primes.
Ann. of Math. (2), 179(3):1121–1174, 2014. made the first major progress towards proving the prime $k$-tuple conjecture, and successfully proved the existence of infinitely many pairs of primes
differing by a fixed number. As of now, the best known result is due to the massive collaborative Polymath8 project,^3 ^3 D. H. J. Polymath. Variants of the {S}elberg sieve, and bounded intervals
containing many primes. Res. Math. Sci., 1:Art. 12, 83, 2014. which showed that there are infinitely many pairs of primes of the form $p, p+246$. In the excellent expository article, ^4 ^4 Andrew
Granville. Primes in intervals of bounded length. Bull. Amer. Math. Soc. (N.S.), 52(2):171–222, 2015. Granville describes the history and ideas leading to this breakthrough, and also discusses some
of the potential impact of the results. This note should be thought of as a few more results following from the ideas of Zhang, Maynard, Tao, and the Polymath8 project.
Throughout, $p_n$ will refer to the $n$th prime number. In a paper, ^5 ^5 Dorin Andrica. Note on a conjecture in prime number theory. Studia Univ. Babe\c s-Bolyai Math., 31(4):44–48, 1986. Andrica
conjectured that $$\label{eq:Andrica_conj} \sqrt{p_{n+1}} - \sqrt{p_n} < 1$$ holds for all $n$. This conjecture, and related statements, is described in Guy's Unsolved Problems in Number Theory. ^6 ^
6 Richard K. Guy. Unsolved problems in number theory. Problem Books in Mathematics. Springer-Verlag, New York, third edition, 2004. It is quickly checked that this holds for primes up to $4.26 \cdot
10^{8}$ in sagemath
# Sage version 8.0.rc1
# started with `sage -ipython`
# sage has pari/GP, which can generate primes super quickly
from sage.all import primes_first_n
# import izip since we'll be zipping a huge list, and sage uses python2 which has
# non-iterable zip by default
from itertools import izip
# The magic number 23150000 appears because pari/GP can't compute
# primes above 436273290 due to fixed precision arithmetic
ps = primes_first_n(23150000) # This is every prime up to 436006979
# Verify Andrica's Conjecture for all prime pairs = up to 436006979
gap = 0
for a,b in izip(ps[:-1], ps[1:]):
if b**.5 - a**.5 > gap:
A, B, gap = a, b, b**.5 - a**.5
In approximately 20 seconds on my machine (so it would not be harder to go much higher, except that I would have to go beyond pari/GP to generate primes), this completes and prints out the following
Thus the largest value of $\sqrt{p_{n+1}} - \sqrt{p_n}$ was merely $0.670\ldots$, and occurred on the gap between $7$ and $11$.
So it appears very likely that the conjecture is true. However it is also likely that new, novel ideas are necessary before the conjecture is decided.
Andrica's Conjecture can also be stated in terms of prime gaps. Let $g_n = p_{n+1} - p_n$ be the gap between the $n$th prime and the $(n+1)$st prime. Then Andrica's Conjecture is equivalent to the
claim that $g_n < 2 \sqrt{p_n} + 1$. In this direction, the best known result is due to Baker, Harman, and Pintz, ^7 ^7 R. C. Baker, G. Harman, and J. Pintz. The difference between consecutive
primes. {II}. Proc. London Math. Soc. (3), 83(3):532–562, 2001. who show that $g_n \ll p_n^{0.525}$.
In 1985, Sandor ^8 ^8 Joszsef Sandor. On certain sequences and series with applications in prime number theory. Gaz. Mat. Met. Inf, 6:1–2, 1985. proved that $$\label{eq:Sandor} \liminf_{n \to \infty}
\sqrt[4]{p_n} (\sqrt{p_{n+1}} - \sqrt{p_n}) = 0.$$ The close relation to Andrica's Conjecture \eqref{eq:Andrica_conj} is clear. The first result of this note is to strengthen this result.
Theorem Let $\alpha, \beta \geq 0$, and $\alpha + \beta < 1$. Then $$\label{eq:main} \liminf_{n \to \infty} p_n^\beta (p_{n+1}^\alpha - p_n^\alpha) = 0.$$
We prove this theorem below. Choosing $\alpha = \frac{1}{2}, \beta = \frac{1}{4}$ verifies Sandor's result \eqref{eq:Sandor}. But choosing $\alpha = \frac{1}{2}, \beta = \frac{1}{2} - \epsilon$ for a
small $\epsilon > 0$ gives stronger results.
This theorem leads naturally to the following conjecture.
Conjecture For any $0 \leq \alpha < 1$, there exists a constant $C(\alpha)$ such that $$p_{n+1}^\alpha - p_{n}^\alpha \leq C(\alpha)$$ for all $n$.
A simple heuristic argument, given in the last section below, shows that this Conjecture follows from Cramer's Conjecture.
It is interesting to note that there are generalizations of Andrica's Conjecture. One can ask what the smallest $\gamma$ is such that $$p_{n+1}^{\gamma} - p_n^{\gamma} = 1$$ has a solution. This is
known as the Smarandache Conjecture, and it is believed that the smallest such $\gamma$ is approximately $$\gamma \approx 0.5671481302539\ldots$$ The digits of this constant, sometimes called 'the
Smarandache constant,' are the contents of sequence A038458 on the OEIS. It is possible to generalize this question as well.
Open Question For any fixed constant $C$, what is the smallest $\alpha = \alpha(C)$ such that $$p_{n+1}^\alpha - p_n^\alpha = C$$ has solutions? In particular, how does $\alpha(C)$ behave as a
function of $C$?
This question does not seem to have been approached in any sort of generality, aside from the case when $C = 1$.
Proof of Theorem
The idea of the proof is very straightforward. We estimate \eqref{eq:main} across prime pairs $p, p+246$, relying on the recent proof from Polymath8 that infinitely many such primes exist.
Fix $\alpha, \beta \geq 0$ with $\alpha + \beta < 1$. Applying the mean value theorem of calculus on the function $x \mapsto x^\alpha$ shows that p^\beta \big( (p+246)^\alpha - p^\alpha \big) &= p^\
beta \cdot 246 \alpha q^{\alpha - 1} \\\ &\leq p^\beta \cdot 246 \alpha p^{\alpha - 1} = 246 \alpha p^{\alpha + \beta - 1}, \label{eq:bound} for some $q \in [p, p+246]$. Passing to the inequality in
the second line is done by realizing that $q^{\alpha - 1}$ is a decreasing function in $q$. As $\alpha + \beta - 1 < 0$, as $p \to \infty$ we see that\eqref{eq:bound} goes to zero.
Therefore $$\liminf_{n \to \infty} p_n^\beta (p_{n+1}^\alpha - p_n^\alpha) = 0,$$ as was to be proved.
Further Heuristics
Cramer's Conjecture states that there exists a constant $C$ such that for all sufficiently large $n$, $$p_{n+1} - p_n < C(\log n)^2.$$ Thus for a sufficiently large prime $p$, the subsequent prime is
at most $p + C (\log p)^2$. Performing a similar estimation as above shows that $$(p + C (\log p)^2)^\alpha - p^\alpha \leq C (\log p)^2 \alpha p^{\alpha - 1} = C \alpha \frac{(\log p)^2}{p^{1 - \
alpha}}.$$ As the right hand side vanishes as $p \to \infty$, we see that it is natural to expect that the main Conjecture above is true. More generally, we should expect the following, stronger
Conjecture' For any $\alpha, \beta \geq 0$ with $\alpha + \beta < 1$, there exists a constant $C(\alpha, \beta)$ such that $$p_n^\beta (p_{n+1}^\alpha - p_n^\alpha) \leq C(\alpha, \beta).$$
Additional Notes
I wrote this note in between waiting in never-ending queues while I sort out my internet service and other mundane activities necessary upon moving to another country. I had just read some papers on
the arXiv, and I noticed a paper which referred to unknown statuses concerning Andrica's Conjecture. So then I sat down and wrote this up.
I am somewhat interested in qualitative information concerning the Open Question in the introduction, and I may return to this subject unless someone beats me to it.
This note is (mostly, minus the code) available as a pdf and (will shortly) appears on the arXiv. This was originally written in LaTeX and converted for display on this site using a set of tools
I've written based around latex2jax, which is available on my github.
Info on how to comment
To make a comment, please send an email using the button below. Your email address won't be shared (unless you include it in the body of your comment). If you don't want your real name to be used
next to your comment, please specify the name you would like to use. If you want your name to link to a particular url, include that as well.
bold, italics, and plain text are allowed in comments. A reasonable subset of markdown is supported, including lists, links, and fenced code blocks. In addition, math can be formatted using $(inline
math)$ or $$(your display equation)$$.
Please use plaintext email when commenting. See Plaintext Email and Comments on this site for more. Note also that comments are expected to be open, considerate, and respectful.
Comments (1)
1. 2017-09-25 michel
I read this on the arXiv this morning. Very interesting! Will you state what you think about C(\alpha) and C(\alpha, \beta)? | {"url":"https://davidlowryduda.com/on-gaps-between-powers-of-consecutive-primes/","timestamp":"2024-11-14T04:45:36Z","content_type":"text/html","content_length":"14987","record_id":"<urn:uuid:96d62e7a-ce81-4e6b-af74-9bf29a9a6b74>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00635.warc.gz"} |
Cinema Crowds
Problem B
Cinema Crowds
The United Cinema Crowd Association of Stockholm plans to have a showing of Old computer scientists and their pieings at the local KTH Royal Institute of Technology cinema.
Not until far too late did the auditor of the association point out that the board had booked far too many groups of visitors to the theater, which fits at most $N$ visitors.
In total, $M$ groups of visitors signed up for the showing. It was decided to let the groups enter the theater one at a time, in the same order in which they signed up for the showing. If there are
too few empty seats when a group comes, the group gets angry and leaves.
Given the sizes of all the visiting groups, determine how many groups will not be accepted into the theater.
The first line of the input contains the integers $N$ ($1 \le N \le 100$) and $M$ ($1 \le M \le 50$), the number of seats in the theater and the number of visiting groups.
The second line contains $M$ integers – the size of each visiting group in the order in which they signed up for the showing. A group consists of between $1$ and $10$ visitors.
Output a single number – the number of groups that will not be accepted to the showing.
Sample Input 1 Sample Output 1
Sample Input 2 Sample Output 2 | {"url":"https://nus.kattis.com/courses/IT5003/IT5003_S2_AY2122/assignments/o3nd36/problems/cinema","timestamp":"2024-11-03T06:07:43Z","content_type":"text/html","content_length":"28929","record_id":"<urn:uuid:51a01483-7846-4537-88dc-c87ad6757c00>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00180.warc.gz"} |
Lines Joining the Origin to the Intersection of a Line and a Curve
Let us find the equation to the pair of lines joining the origin to the points of intersection of the line \[lx+my=n\] and the curve \[ax^2+2hxy+by^2+2gx+2fy+c=0\] The general equation of second
degree \[ax^2+2hxy+by^2+2gx+2fy+c=0\text{ __(1)}\] in general, represents a curve, except in the case when it represents a pair of straight lines.
Let the straight line $lx+my=n$ meet the curve (or pair of straight lines) in two points $A$ and $B$. We have to find the equation representing the line pair $OA$ and $OB$.
The equation of given straight line may be written as \[\frac{lx+my}{n}=1\text{ __(2)}\]
Now, consider the homogeneous equation of second degree in $x$ and $y$, \[ax^2+2hxy+by^2+2(gx+fy)\left(\frac{lx+my}{n}\right)\]\[+c\left(\frac{lx+my}{n}\right)^2=0\text{ __(3)}\]
This equation, being homogeneous in $x$ and $y$, represents a pair of straight lines through the origin.
Also the coordinates of the point of intersection of the straight line $\text{(2)}$ and the curve $\text{(1)}$ satisfy both the equations $\text{(1)}$ and $\text{(2)}$, and hence they satisfy the
equation $\text{(3)}$. The equation $\text{(3)}$ therefore represents a pair of lines which passes through the origin and the points of intersection of the line $\text{(2)}$ and the curve S\text{(1)}
Find the equation of the lines joining the origin to the points of intersection of $x+2y=3$ and $4x^2+16xy-12y^2-8x+12y-3=0$. Also find the angle between the two lines.
Here, equation of curve is, \[4x^2+16xy-12y^2-8x+12y-3=0\] and, equation of straight line is, \[x+2y=3\] \[\text{or }\frac{x+2y}{3}=1\]
Making the curve homogeneous with the help of straight line, \[4x^2+16xy-12y^2-8x.1+12y.1+3.1=0\] \[\text{or, }4x^2+16xy-12y^2-8x\left(\frac{x+2y}{3}\right)\]\[+12y\left(\frac{x+2y}{3}\right)-3\left
(\frac{x+2y}{3}\right)^2=0\] \[\text{or, }12x^2+48xy-36y^2-8x^2-16xy+12xy\]\[+24y^2-x^2-4xy-4y^2=0\] \[\therefore 3x^2+40xy-16y^2=0\]
This is the equation of the lines joining the origin and the points of intersection of the given curve and the line.
Now, \[\begin{array}{c}a=3,&2h=40&\text{and}&b=-16\\a=3,&h=20&\text{and}&b=-16\end{array}\]
Let $\theta$ be the angle between the line pair. Then, \[\tan\theta=\pm\frac{2\sqrt{h^2-ab}}{a+b}\] \[=\pm\frac{2\sqrt{400+3×16}}{3-16}=\pm\frac{16\sqrt{7}}{13}\]
Find the equation to the pair of straight lines joining the origin to the intersection of the straight line $y=mx+c$ and the curve $x^2+y^2=a^2$. Prove that they are at right angles if $2c^2=a^2(1+m^
Here, equation of curve is, \[x^2+y^2=a^2\] and, equation of straight line is, \[y=mx+c\] \[\text{i.e. }\frac{y-mx}{c}=1\]
Making the curve homogeneous with the help of straight line, we have, \[x^2+y^2=a^2.1\] \[x^2+y^2=a^2\left(\frac{y-mx}{c}\right)^2\] \[c^2x^2+c^2y^2=a^2(y^2-2mxy+m^2x^2)\] \[c^2x^2+c^2y^2=a^2(y^
2-2mxy+m^2x^2)\] \[c^2x^2+c^2y^2=a^2y^2-2a^2mxy+a^2m^2x^2\] \[(c^2-a^2m^2)x^2+2a^2mxy+(c^2-a^2)y^2=0\]
This is the equation of the lines joining the origin and the points of intersection of the given curve and the line.
Now, the two lines will be right angles if \[\text{coeff. of }x^2+\text{coeff. of }y^2=0\] \[c^2-a^2m^2+c^2-a^2=0\] \[\therefore 2c^2=a^2(1+m^2)\] | {"url":"https://ankplanet.com/maths/pair-of-straight-lines/lines-joining-the-origin-to-the-intersection-of-a-line-and-a-curve/","timestamp":"2024-11-02T10:36:43Z","content_type":"text/html","content_length":"88547","record_id":"<urn:uuid:4f042fa7-022a-43d1-933f-e5972c564e3d>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00245.warc.gz"} |
Do an If statement only if a cell is not blank
Hi Community,
I have a list of names and I'm trying to keep track of different Pension sources they may have and the amount of each pension. Being that everyone on my list has a different amount of pensions, the
formula counts to see how many pension sources are filled in. If that count equals the count of "amounts" that are filled in, we know we have all the information.
(See my previous question for info on writing such a formula: https://community.smartsheet.com/discussion/70493/
My formula reads as follows:
=IF(COUNTIF([Source I]@row, <>"") + COUNTIF([Source II]@row, <>"") + COUNTIF([Source III]@row, <>"") + COUNTIF([Source IV]@row, <>"") = COUNT([Amount I]@row, [Amount II]@row, [Amount III]@row,
[Amount IV]@row), "Have all items", "Missing Items")
The problem is that when a new name is added to the list the formula populates as "Have all Items". Because the count of "sources" equals 0 because no information was put in yet, and the count of
"amounts" is also zero.
What I'm looking for, I believe, is an if statement inside an if statement (If the first source is put in, then do this if statement, otherwise don't do the if statement yet.)! Or some sort of
trigger to only start the if statement once the first Source is put in.
Best Answer
• It's just a nested If statement. You will put your original code as the False output of the first If Statement. It will look something like:
=IF([Amount I]@row=0,"",IF(COUNTIF([Source I]@row, <>"") + COUNTIF([Source II]@row, <>"") + COUNTIF([Source III]@row, <>"") + COUNTIF([Source IV]@row, <>"") = COUNT([Amount I]@row, [Amount II]
@row, [Amount III]@row, [Amount IV]@row), "Have all items", "Missing Items"))
Essentially, the True statement of the Amount is 0 just returns an empty character (you could also return 0 for the same effect). If there is something in the Amount I then it solves your second
IF Formula.
• It's just a nested If statement. You will put your original code as the False output of the first If Statement. It will look something like:
=IF([Amount I]@row=0,"",IF(COUNTIF([Source I]@row, <>"") + COUNTIF([Source II]@row, <>"") + COUNTIF([Source III]@row, <>"") + COUNTIF([Source IV]@row, <>"") = COUNT([Amount I]@row, [Amount II]
@row, [Amount III]@row, [Amount IV]@row), "Have all items", "Missing Items"))
Essentially, the True statement of the Amount is 0 just returns an empty character (you could also return 0 for the same effect). If there is something in the Amount I then it solves your second
IF Formula.
• Thanks, that works! I original tried a nested if statement. However, I put the formula by "true" and not by "false".
• Ah! Makes perfect sense. Glad it's working now.
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/71169/do-an-if-statement-only-if-a-cell-is-not-blank","timestamp":"2024-11-02T04:36:17Z","content_type":"text/html","content_length":"426280","record_id":"<urn:uuid:e0462d39-bb45-4513-a013-5c86bbac8f05>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00198.warc.gz"} |
Square Centimeter to Feddan
Square Centimeter [cm2] Output
1 square centimeter in ankanam is equal to 0.000014949875578764
1 square centimeter in aana is equal to 0.0000031450432189072
1 square centimeter in acre is equal to 2.4710516301528e-8
1 square centimeter in arpent is equal to 2.9249202856357e-8
1 square centimeter in are is equal to 0.000001
1 square centimeter in barn is equal to 1e+24
1 square centimeter in bigha [assam] is equal to 7.4749377893818e-8
1 square centimeter in bigha [west bengal] is equal to 7.4749377893818e-8
1 square centimeter in bigha [uttar pradesh] is equal to 3.9866334876703e-8
1 square centimeter in bigha [madhya pradesh] is equal to 8.9699253472581e-8
1 square centimeter in bigha [rajasthan] is equal to 3.9536861034746e-8
1 square centimeter in bigha [bihar] is equal to 3.9544123500036e-8
1 square centimeter in bigha [gujrat] is equal to 6.1776345366791e-8
1 square centimeter in bigha [himachal pradesh] is equal to 1.2355269073358e-7
1 square centimeter in bigha [nepal] is equal to 1.4765309213594e-8
1 square centimeter in biswa [uttar pradesh] is equal to 7.9732669753405e-7
1 square centimeter in bovate is equal to 1.6666666666667e-9
1 square centimeter in bunder is equal to 1e-8
1 square centimeter in caballeria is equal to 2.2222222222222e-10
1 square centimeter in caballeria [cuba] is equal to 7.451564828614e-10
1 square centimeter in caballeria [spain] is equal to 2.5e-10
1 square centimeter in carreau is equal to 7.7519379844961e-9
1 square centimeter in carucate is equal to 2.0576131687243e-10
1 square centimeter in cawnie is equal to 1.8518518518519e-8
1 square centimeter in cent is equal to 0.0000024710516301528
1 square centimeter in centiare is equal to 0.0001
1 square centimeter in circular foot is equal to 0.0013705023910063
1 square centimeter in circular inch is equal to 0.19735234590491
1 square centimeter in cong is equal to 1e-7
1 square centimeter in cover is equal to 3.7064492216457e-8
1 square centimeter in cuerda is equal to 2.5445292620865e-8
1 square centimeter in chatak is equal to 0.000023919800926022
1 square centimeter in decimal is equal to 0.0000024710516301528
1 square centimeter in dekare is equal to 1.0000006597004e-7
1 square centimeter in dismil is equal to 0.0000024710516301528
1 square centimeter in dhur [tripura] is equal to 0.00029899751157527
1 square centimeter in dhur [nepal] is equal to 0.0000059061236854374
1 square centimeter in dunam is equal to 1e-7
1 square centimeter in drone is equal to 3.893196765303e-9
1 square centimeter in fanega is equal to 1.5552099533437e-8
1 square centimeter in farthingdale is equal to 9.8814229249012e-8
1 square centimeter in feddan is equal to 2.3990792525755e-8
1 square centimeter in ganda is equal to 0.000001245822964897
1 square centimeter in gaj is equal to 0.00011959900463011
1 square centimeter in gajam is equal to 0.00011959900463011
1 square centimeter in guntha is equal to 9.8842152586866e-7
1 square centimeter in ghumaon is equal to 2.4710538146717e-8
1 square centimeter in ground is equal to 4.4849626736291e-7
1 square centimeter in hacienda is equal to 1.1160714285714e-12
1 square centimeter in hectare is equal to 1e-8
1 square centimeter in hide is equal to 2.0576131687243e-10
1 square centimeter in hout is equal to 7.0359937931723e-8
1 square centimeter in hundred is equal to 2.0576131687243e-12
1 square centimeter in jerib is equal to 4.9466500076791e-8
1 square centimeter in jutro is equal to 1.737619461338e-8
1 square centimeter in katha [bangladesh] is equal to 0.0000014949875578764
1 square centimeter in kanal is equal to 1.9768430517373e-7
1 square centimeter in kani is equal to 6.2291148244848e-8
1 square centimeter in kara is equal to 0.0000049832918595878
1 square centimeter in kappland is equal to 6.4825619084662e-7
1 square centimeter in killa is equal to 2.4710538146717e-8
1 square centimeter in kranta is equal to 0.000014949875578764
1 square centimeter in kuli is equal to 0.0000074749377893818
1 square centimeter in kuncham is equal to 2.4710538146717e-7
1 square centimeter in lecha is equal to 0.0000074749377893818
1 square centimeter in labor is equal to 1.3950025009895e-10
1 square centimeter in legua is equal to 5.580010003958e-12
1 square centimeter in manzana [argentina] is equal to 1e-8
1 square centimeter in manzana [costa rica] is equal to 1.4308280488084e-8
1 square centimeter in marla is equal to 0.0000039536861034746
1 square centimeter in morgen [germany] is equal to 4e-8
1 square centimeter in morgen [south africa] is equal to 1.1672697560406e-8
1 square centimeter in mu is equal to 1.4999999925e-7
1 square centimeter in murabba is equal to 9.884206520611e-10
1 square centimeter in mutthi is equal to 0.0000079732669753405
1 square centimeter in ngarn is equal to 2.5e-7
1 square centimeter in nali is equal to 4.9832918595878e-7
1 square centimeter in oxgang is equal to 1.6666666666667e-9
1 square centimeter in paisa is equal to 0.000012580540458988
1 square centimeter in perche is equal to 0.0000029249202856357
1 square centimeter in parappu is equal to 3.9536826082444e-7
1 square centimeter in pyong is equal to 0.000030248033877798
1 square centimeter in rai is equal to 6.25e-8
1 square centimeter in rood is equal to 9.8842152586866e-8
1 square centimeter in ropani is equal to 1.965652011817e-7
1 square centimeter in satak is equal to 0.0000024710516301528
1 square centimeter in section is equal to 3.8610215854245e-11
1 square centimeter in sitio is equal to 5.5555555555556e-12
1 square centimeter in square is equal to 0.00001076391041671
1 square centimeter in square angstrom is equal to 10000000000000000
1 square centimeter in square astronomical units is equal to 4.4683704831421e-27
1 square centimeter in square attometer is equal to 1e+32
1 square centimeter in square bicron is equal to 100000000000000000000
1 square centimeter in square chain is equal to 2.4710436922533e-7
1 square centimeter in square cubit is equal to 0.00047839601852043
1 square centimeter in square decimeter is equal to 0.01
1 square centimeter in square dekameter is equal to 0.000001
1 square centimeter in square digit is equal to 0.27555610666777
1 square centimeter in square exameter is equal to 1e-40
1 square centimeter in square fathom is equal to 0.000029899751157527
1 square centimeter in square femtometer is equal to 1e+26
1 square centimeter in square fermi is equal to 1e+26
1 square centimeter in square feet is equal to 0.001076391041671
1 square centimeter in square furlong is equal to 2.4710516301528e-9
1 square centimeter in square gigameter is equal to 1e-22
1 square centimeter in square hectometer is equal to 1e-8
1 square centimeter in square inch is equal to 0.15500031000062
1 square centimeter in square league is equal to 4.290006866585e-12
1 square centimeter in square light year is equal to 1.1172498908139e-36
1 square centimeter in square kilometer is equal to 1e-10
1 square centimeter in square megameter is equal to 1e-16
1 square centimeter in square meter is equal to 0.0001
1 square centimeter in square microinch is equal to 155000173266.03
1 square centimeter in square micrometer is equal to 100000000
1 square centimeter in square micromicron is equal to 100000000000000000000
1 square centimeter in square micron is equal to 100000000
1 square centimeter in square mil is equal to 155000.31
1 square centimeter in square mile is equal to 3.8610215854245e-11
1 square centimeter in square millimeter is equal to 100
1 square centimeter in square nanometer is equal to 100000000000000
1 square centimeter in square nautical league is equal to 3.2394816622014e-12
1 square centimeter in square nautical mile is equal to 2.9155309240537e-11
1 square centimeter in square paris foot is equal to 0.0009478672985782
1 square centimeter in square parsec is equal to 1.0502647575668e-37
1 square centimeter in perch is equal to 0.0000039536861034746
1 square centimeter in square perche is equal to 0.000001958018322827
1 square centimeter in square petameter is equal to 1e-34
1 square centimeter in square picometer is equal to 100000000000000000000
1 square centimeter in square pole is equal to 0.0000039536861034746
1 square centimeter in square rod is equal to 0.0000039536708845746
1 square centimeter in square terameter is equal to 1e-28
1 square centimeter in square thou is equal to 155000.31
1 square centimeter in square yard is equal to 0.00011959900463011
1 square centimeter in square yoctometer is equal to 1e+44
1 square centimeter in square yottameter is equal to 1e-52
1 square centimeter in stang is equal to 3.6913990402363e-8
1 square centimeter in stremma is equal to 1e-7
1 square centimeter in sarsai is equal to 0.000035583174931272
1 square centimeter in tarea is equal to 1.5903307888041e-7
1 square centimeter in tatami is equal to 0.000060499727751225
1 square centimeter in tonde land is equal to 1.8129079042785e-8
1 square centimeter in tsubo is equal to 0.000030249863875613
1 square centimeter in township is equal to 1.0725050478094e-12
1 square centimeter in tunnland is equal to 2.0257677659833e-8
1 square centimeter in vaar is equal to 0.00011959900463011
1 square centimeter in virgate is equal to 8.3333333333333e-10
1 square centimeter in veli is equal to 1.245822964897e-8
1 square centimeter in pari is equal to 9.8842152586866e-9
1 square centimeter in sangam is equal to 3.9536861034746e-8
1 square centimeter in kottah [bangladesh] is equal to 0.0000014949875578764
1 square centimeter in gunta is equal to 9.8842152586866e-7
1 square centimeter in point is equal to 0.0000024710731022028
1 square centimeter in lourak is equal to 1.9768430517373e-8
1 square centimeter in loukhai is equal to 7.9073722069493e-8
1 square centimeter in loushal is equal to 1.5814744413899e-7
1 square centimeter in tong is equal to 3.1629488827797e-7
1 square centimeter in kuzhi is equal to 0.0000074749377893818
1 square centimeter in chadara is equal to 0.00001076391041671
1 square centimeter in veesam is equal to 0.00011959900463011
1 square centimeter in lacham is equal to 3.9536826082444e-7
1 square centimeter in katha [nepal] is equal to 2.9530618427187e-7
1 square centimeter in katha [assam] is equal to 3.7374688946909e-7
1 square centimeter in katha [bihar] is equal to 7.9088247000071e-7
1 square centimeter in dhur [bihar] is equal to 0.000015817649400014
1 square centimeter in dhurki is equal to 0.00031635298800029 | {"url":"https://hextobinary.com/unit/area/from/sqcm/to/feddan","timestamp":"2024-11-09T20:37:15Z","content_type":"text/html","content_length":"130199","record_id":"<urn:uuid:bdeddcce-8cf1-425d-b44a-4ee21ddfad3f>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00286.warc.gz"} |
Double-semion stabilizer code
Double-semion stabilizer code[1,2]
A 2D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) that is characterized by the 2D double semion topological phase. The code can be obtained from the \(\mathbb{Z}_4\) surface
code by condensing the anyon \(e^2 m^2\) [3]. Originally formulated as the ground-state space of a Hamiltonian with non-commuting terms [1], which can be extended to other spatial dimensions [4], and
later as a commuting-projector code [5,6].
This stabilizer code family is inequivalent to a CSS code via a Clifford circuit whose depth does not scale with \(n\) [7; Thm. 1.1]. This is because the double semion phase has a sign problem [7,8],
and existence of such a Clifford circuit would allow one to construct a code Hamiltonian that is free of such a problem.
• Abelian TQD stabilizer code — When treated as ground states of the code Hamiltonian, the code states realize 2D double-semion topological order, a topological phase of matter that exists as the
deconfined phase of the 2D twisted \(\mathbb{Z}_2\) gauge theory [9].
• Toric code — The double semion phase also has a realization in terms of qubits [1] that can be compared to the toric code. There is a logical basis for both the toric and double-semion codes
where each codeword is a superposition of states corresponding to all noncontractible loops of a particular homotopy type. The superposition is equal for the toric code, whereas an odd number of
loops appear with a \(-1\) coefficient for the double semion.
• Modular-qudit surface code — The exchange statistics of the anyon for the double-semion code coincides with a subset of anyons in the \(\mathbb{Z}_4\), but the fusion rules are different. The
double-semion code can be obtained from the \(\mathbb{Z}_4\) surface code by condensing the anyon \(e^2 m^2\) [3] or by gauging [10–12,12] the one-form symmetry associated with said anyon [3;
Footnote 20].
• Abelian TQD stabilizer code — All Abelian TQD codes can be realized as modular-qudit stabilizer codes by starting with an Abelian quantum double model along with a family of Abelian TQDs that
generalize the double semion anyon theory and condensing certain bosonic anyons [3].
• \(\mathbb{Z}_q^{(1)}\) subsystem code — The anyonic exchange statistics of \(\mathbb{Z}_4^{(1)}\) subsystem code resemble those of the double semion code, but its fusion rules realize the \(\
mathbb{Z}_4\) group.
• Chiral semion subsystem code — The semion code can be obtained from the double-semion stabilizer code by gauging out the anyon \(\bar{s}\) [3; Fig. 15].
Page edit log
Cite as:
“Double-semion stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/double_semion
Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/stabilizer/topological/double_semion.yml. | {"url":"https://errorcorrectionzoo.org/c/double_semion","timestamp":"2024-11-11T08:04:24Z","content_type":"text/html","content_length":"23898","record_id":"<urn:uuid:a3027fbe-2547-46f6-9c9e-6ea1642acffa>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00638.warc.gz"} |
Mathematical Modelling of Electric Field Generated by Vertical Grounding Electrode in Horizontally Stratified Soil Using the FDTD Method
5.1 Behaviour of vertical grounding rod to a double peak bipolar current
Figure 4 shows the case that will be analysed with the FDTD method. A vertical earthing rod of 1 m length and 25 mm radius buried in a homogeneous soil of resistivity 42Ω.m and relative permittivity
ε[r]=10 fed by a current bipolar source, positive peak 30 kA and negative 10 kA (Figure 5). This configuration represents the experience made by Geri et al. [20]. The working volume is 20m x 20m x20m
divided on uniform cubes of 0.1m x0.1m x 0.1m surrounded by six layers of second order ABC Liao’s to minimize the reflections. The equivalent radius of the vertical grounding rod is 23mm (0.23Dx=0.23
x0.1) [18] approximately the same electrode radius in the Geri experiment.
Figure 4. Geri et al schema experiment
Figure 5. Current injected in the top of grounding rod [20]
Figure 6 shows the transient voltage computed by the FDTD Method of a vertical grounding rod in a homogeneous soil. This voltage is obtained by integrating the tangential electric field at the
surface of the ground from the feed point to the ABC limit. The obtained results are congruent with those obtained by Grcevet al. which validate our simulation code [21, 22].
Figure 6. Computed voltage at the top of grounding rod
Figure 7 presents the electric field radiated by the vertical grounding rod at three different points (p[1], p[2] and p[3]) above and underground. It can be seen that the electric field has the same
shape at the observation points which is similar to the injected current wave.
Figure 7. Electric field radiated by grounding rod at three different observation points (p[1], p[2] and p[3]) in presence of homogeneous soil (ρ=42Ωm, ε[r]=10)
5.2 Transient electric field radiated by a vertical grounding rod buried in a stratified soil with two layers
After validation of the obtained results, a study was carried out to see the influence of the stratified soil with two layers on the transient behaviour of the grounding rod and on the electric field
radiated in each layer of the soil and in the air. Figure 9 presents the geometry of the problem. The vertical rod is powered by a lightning current which was used in the work of Grcev [23] (Figure
Figure 8. Current injected in the top of grounding rod case of stratified soil [23]
Figure 9. Grounding rod buried in stratified soil (two layers)
The adopted values for the electrical parameters of the soil layers are given in Table 1. The depth of the upper layer was set to 0.5m (see Figure 9). Three cases were considered in the simulations:
Case 1 corresponding to a homogeneous ground, while Cases 2 and 3 represent two configurations of two-layers soil. The horizontal soil stratification was accounted simply by considering different
values for the soil electrical parameters when passing from one grid point to another one belonging to a different layer.
Table 1. Electric parameters of the two-layer ground
│ │ρ[1] (Ω.m)│ρ[ 2] (Ω.m)│ε[r]│
│ Case 1 Homogeneous soil │ 42 │ 42 │ 10 │
│Case 2 Stratified soil (ρ[1]<ρ[2]) │ 42 │ 200 │ 10 │
│Case 3 Stratified soil (ρ[1]>ρ[2]) │ 200 │ 42 │ 10 │
In Figure 10, the voltage calculated at the point of impact of the rod for the homogeneous and stratified soil with 2 layers is displayed. It is easy to see the influence of the resistivity when a
layer is added; when the apparent resistivity of the ground changes, the flow of the current to the earth changes, too. The high value of the resistivity increases the voltage of the electrode
because of the weak dissipation of the current and vice versa.
Figure 10. Voltage at feed point of grounding rod
Figure 11. Electric field radiated by grounding rod in homogeneous soil
Figure 12. Electric field radiated by grounding rod in stratified soil (ρ[1]=42Ω.m, ρ[2]=200Ω.m)
The radiated electric field was observed at four observation points p[1], p[2], p[3 ]and p[4], located respectively, at 1m above the ground, 0.25m, 0.75m and 1.75m below the ground (Figure 9).
From Figures 11-13, it can be seen that when the upper layer is more conductive than the lower one (case 2), the amplitude of electric field at all observation points p[1], p[2], p[3] and p[4] can
increase respectively of 162, 107, 126 and 120 percent compared to those of homogeneous soil .For case 3 (ρ[1]>ρ[2]) the increase is about 27, 36, 26, 32 percent.
Figure 13. Electric field radiated by grounding rod in stratified soil (ρ[1]=200 Ω.m, ρ[2]=42 Ω.m)
5.3 Transient current dissipated by a vertical grounding rod buried in a stratified soil with 2 layers
This part of the study is devoted to the dissipation of the current in the soil, for a homogeneous soil and stratified with two layers. Figure 14 shows the temporal currents which traverse the earth
rod every 0.1 m from the point of impact (the highest graph) to the extreme (bottom graph). In this case the current distribution along the electrode is uniform due to the homogeneity of the soil.
Figure 14. Currents distribution along the grounding rod in homogeneous soil with a resistivity ρ=42Ω.m
Table 2. Grounding rod current comparison (homogeneous soil and 2 layers soil)
│ │ 0*l │0.2*l │0.4*l │0.6*l │ 0.8*l │
│ I(KA) │ │ │ │ │ │
│ │28.82│23.45 │18.14 │13.12 │ 7.47 │
│ (ρ=42Ω.m) │ │ │ │ │ │
│ I[1](KA) │ │ │ │ │ │
│ │28.82│18.53 │08.95 │05.12 │ 02.94 │
│ (ρ[1]<ρ[2]) │ │ │ │ │ │
│ I[2](KA) │ │ │ │ │ │
│ │28.82│ 27.5 │25.79 │19.73 │ 17.17 │
│ (ρ[1]>ρ[2]) │ │ │ │ │ │
│(I-I[1])/I (%) │0,00 │20,98 │50,66 │60,98 │ 60,64 │
│(I-I[2])/I (%) │0,00 │-17,27│-42,17│-50,38│-129,85 │
where, l is the length of the electrode, I the rod current in homogeneous soil case, I[1 ]the rod current in stratified soil (case 2), I[2] the rod current in stratified soil (case 3).
In order to have a clear idea about the current distribution along the earth electrode for a homogeneous and stratified soil, the results are given in Table 2 with a comparison of the different
When the upper layer is more conductive than the lower one (case 2), the current decreases rapidly along the electrode with an average of 9 KA in the 1st layer and 3KA in the 2^nd one.
Figure 15. Currents density in homogeneous soil ρ=42 Ω.m at t=11µs
Figure 16. Current density in a stratified soil in a stratified soil (ρ[1]=500 Ω.m, ρ[2]=200 Ω.m, ρ[3]=42 Ω.m) at t=11μs
Figure 17. Current density in a stratified soil in a stratified soil (ρ[1]=42 Ω.m, ρ[2]=200 Ω.m) at t=11μs
For the third case, when the lower layer is more conductive than the upper level, the current decreases slowly in the resistive layer along the rod with an average of 1.3 KA which justifies ohm's
The Figures e.g., Figures 15, 16 and 17, it is a cartography which reflects the results of the cases treated previously to give a better visibility of the flow of current through the different layers
of the earth. | {"url":"https://www.iieta.org/journals/mmep/paper/10.18280/mmep.070211","timestamp":"2024-11-11T14:24:47Z","content_type":"text/html","content_length":"95534","record_id":"<urn:uuid:ba8c01f4-8f96-4f46-ad3d-b5938d75a436>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00431.warc.gz"} |
1. Introduction to control systems; Characteristics of the response for linear systems of the first and the second order in the time domain: time constant, response time, rising time, settling time.
relation between the response characteristics and the pole-zero position in the s plane. Characteristics of the frequency response for first and second order systems, cut-off frequency, band,
resonance modulus. Non minimum phase systems. Polar plots.
2. Open and closed loop control. feedback influence on the sensitivity to parameter variations, disturbance both in the direct chain and in the feedback path, and to the band width for a linear
system. Steady state accuracy for a feedback system for input signals like step, ramp, parabolic ramp; classification of control systems in "types". Stability analysis via the Nyquist criterion. Gain
and Phase margins. Root locus: drawing rules and examples.
3. Performances of a control system: static and dynamic performances. Transformation of time domain to frequency domain performances. Frequency response design using lead and lag elementary
controller network using Bode diagrams. Root locus design. realization of controller networks via operational amplifiers. Standard PID controllers: empirical and analytical tuning strategies.
4. realization of controller networks via operational amplifiers. Standard PID controllers: empirical and analytical tuning strategies.
5. Relation between the s plane and the z plane. Bilinear transformation. Discretization and reconstruction of a signal. Sampling theorem. Performances of a discrete control system. Design of a
discrete control system via translation of a continuous controller.
6. Exercitations. The practical lessons are focussed ad deepening of theoretical aspects using key exercises, using the Matlab tool. particular examples are related to the frequency response design
and root locus design. | {"url":"https://syllabus.unict.it/insegnamento.php?id=10663&pdf&eng","timestamp":"2024-11-03T03:10:29Z","content_type":"text/html","content_length":"10423","record_id":"<urn:uuid:9e5ac995-eead-44df-a1f4-c7f31d5b6213>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00539.warc.gz"} |
The Number of k-Dimensional Corner-Free Subsets of Grids
In 1975, Szemeredi proved that for every real number $\delta > 0 $ and every positive integer $k$, there exists a positive integer $N$ such that every subset $A$ of the set ${1, 2, \cdots, N }$ with
$|A| \geq \delta N$ contains an arithmetic progression of length $k$. There has been a plethora of research related to Szemeredi’s theorem in many areas of mathematics. In 1990, Cameron and Erdos
proposed a conjecture about counting the number of subsets of the set ${1,2, \dots, N}$ which do not contain an arithmetic progression of length $k$. In the talk, we study a natural higher
dimensional version of this conjecture by counting the number of subsets of the $k$-dimensional grid ${1,2, \dots, N}^k$ which do not contain a $k$-dimensional corner that is a set of points of the
form ${ a } \cup { a+ de_i : 1 \leq i \leq k }$ for some $a \in {1,2, \dots, N}^k$ and $d > 0$, where $e_1,e_2, \cdots, e_k$ is the standard basis of $\mathbb{R}^k$. Our main tools for proof are the
hypergraph container method and the supersaturation result for $k$-dimensional corners in sets of size $\Theta(c_k(N))$, where $c_k(N)$ is the maximum size of a $k$-dimensional corner-free subset of
${1,2, \dots, N}^k$. | {"url":"https://prima2022.primamath.org/talk/the-number-of-k-dimensional-corner-free-subsets-of-grids/","timestamp":"2024-11-03T15:30:09Z","content_type":"text/html","content_length":"7282","record_id":"<urn:uuid:cd45375e-2014-47d9-ad44-5512848753d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00067.warc.gz"} |
True or false? consider a random sample of size n from an x
True or false? Consider a random sample of size n from an x distribution. For such a sample, the margin of error for estimating μ is the magnitude of the difference between x and μ.
False. By definition, the margin of error is the magnitude of the difference between x and σ.
True. By definition, the margin of error is the magnitude of the difference between x and μ.
True. By definition, the margin of error is the magnitude of the difference between x and σ.
False. By definition, the margin of error is the magnitude of the difference between x and μ.
True or false? Every random sample of the same size from a given population will produce exactly the same confidence interval for μ.
True. Different random samples will produce the same x values, resulting in the same confidence intervals.
False. Different random samples may produce different x values, resulting in the same confidence intervals.
False. Different random samples may produce different x values, resulting in different confidence intervals.
True. Different random samples may produce different x values, resulting in different confidence intervals.
True or false? A larger sample size produces a longer confidence interval for μ.
False. As the sample size increases, the maximal error decreases, resulting in a shorter confidence interval.
True. As the sample size increases, the maximal error increases, resulting in a longer confidence interval.
True. As the sample size increases, the maximal error decreases, resulting in a longer confidence interval.
False. As the sample size increases, the maximal error increases, resulting in a shorter confidence interval.
Allen’s hummingbird (Selasphorussasin) has been studied by zoologist Bill Alther.† Suppose a small group of 12 Allen’s hummingbirds has been under study in Arizona. The average weight for these birds
is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen’s hummingbirds have a normal distribution, with σ = 0.38 gram.
(a) Find an 80% confidence interval for the average weights of Allen’s hummingbirds in the study region. (Round your answers to two decimal places.)
lower limit 3.01
upper limit 3.29
margin of error 0.14
(b) What conditions are necessary for your calculations? (Select all that apply.)
n is large
σ is unknown
uniform distribution of weights
normal distribution of weights
σ is known
(c) Give a brief interpretation of your results in the context of this problem.
There is a 20% chance that the interval is one of the intervals containing the true average weight of Allen’s — hummingbirds in this region. The probability that this interval contains the true
average weight of Allen’s hummingbirds is 0.80.
The probability that this interval contains the true average weight of Allen’s hummingbirds is 0.20
. The probability to the true average weight of Allen’s hummingbirds is equal to the sample mean.
There is an 80% chance that the interval is one of the intervals containing the true average weight of Allen’s hummingbirds in this region.
(d) Find the sample size necessary for an 80% confidence level with a maximal error of estimate E = 0.13 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)
Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advance indication of illness such as gout, leukemia, or lymphoma.† Over a period of months, an adult
male patient has taken nine blood tests for uric acid. The mean concentration was x = 5.35 mg/dl. The distribution of uric acid in healthy adult males can be assumed to be normal, with σ = 1.79 mg/
(a) Find a 95% confidence interval for the population mean concentration of uric acid in this patient’s blood. (Round your answers to two decimal places.)
lower limit 4.18
upper limit 6.52
margin of error 1.17
(b) What conditions are necessary for your calculations? (Select all that apply.)
n is large
σ is known
normal distribution of uric acid
σ is unknown uniform distribution of uric acid
(c) Give a brief interpretation of your results in the context of this problem.
There is not enough information to make an interpretation.
The probability that this interval contains the true average uric acid level for this patient is 0.05.
There is a 95% chance that the confidence interval is one of the intervals containing the population average uric acid level for this patient.
The probability that this interval contains the true average uric acid level for this patient is 0.95
There is a 5% chance that the confidence interval is one of the intervals containing the population average uric acid level for this patient.
(d) Find the sample size necessary for a 95% confidence level with maximal error of estimate E = 1.14 for the mean concentration of uric acid in this patient’s blood. (Round your answer up to the
nearest whole number.)
10blood tests
Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and
physical activity of an individual. Suppose that a random sample of 43 male firefighters are tested and that they have a plasma volume sample mean of x= 37.5 ml/kg (milliliters plasma per kilogram
body weight). Assume that σ = 7.70 ml/kg for the distribution of blood plasma.
(a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.)
lower limit 34.48
upper limit 40.52
margin of error 3.02
(b) What conditions are necessary for your calculations? (Select all that apply.)
the distribution of weights is normal
the distribution of weights is uniform
σ is unknown
σ is known n is large
(c) Give a brief interpretation of your results in the context of this problem.
The probability that this interval contains the true average blood plasma volume in male firefighters is 0.99.
99% of the intervals created using this method will contain the true average blood plasma volume in male firefighters.
The probability that this interval contains the true average blood plasma volume in male firefighters is 0.01.
1% of the intervals created using this method will contain the true average blood plasma volume in male firefighters.
(d) Find the sample size necessary for a 99% confidence level with maximal error of estimate E = 2.80 for the mean plasma volume in male firefighters. (Round up to the nearest whole number.)
51male firefighters
The method of tree ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population of x values has an approximately normal distribution.
(a) Use a calculator with mean and standard deviation keys to find the sample mean year x and sample standard deviation s. (Round your answers to the nearest whole number.)
X=A.D. 1283
s =yr42
(b) Find a 90% confidence interval for the mean of all tree ring dates from this archaeological site. (Round your answers to the nearest whole number.)
lower limit
upper limit
How much does a sleeping bag cost? Let’s say you want a sleeping bag that should keep you warm in temperatures from 20°F to 45°F. A random sample of prices ($) for sleeping bags in this temperature
range is given below. Assume that the population of x values has an approximately normal distribution.
(a) Use a calculator with mean and sample standard deviation keys to find the sample mean price x and sample standard deviation s. (Round your answers to two decimal places.)
s =$33.07
(b) Using the given data as representative of the population of prices of all summer sleeping bags, find a 90% confidence interval for the mean price μ of all summer sleeping bags. (Round your
answers to two decimal places.)
lower limit
upper limit
https://nursingacademics.com/wp-content/uploads/2021/06/logo-300x75.png 0 0 NursingAcademics https://nursingacademics.com/wp-content/uploads/2021/06/logo-300x75.png NursingAcademics2021-06-21
01:42:302021-06-21 01:42:30True or false? consider a random sample of size n from an x | {"url":"https://nursingacademics.com/2021/06/true-or-false-consider-a-random-sample-of-size-n-from-an-x/","timestamp":"2024-11-05T12:23:04Z","content_type":"text/html","content_length":"64151","record_id":"<urn:uuid:7f08bc15-773a-4dac-a58e-ae7cb3f337a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00712.warc.gz"} |
Basic Example for Calculating the Causal Effect
Basic Example for Calculating the Causal Effect
This is a quick introduction to the DoWhy causal inference library. We will load in a sample dataset and estimate the causal effect of a (pre-specified) treatment variable on a (pre-specified)
outcome variable.
First, let us load all required packages.
import numpy as np
from dowhy import CausalModel
import dowhy.datasets
Now, let us load a dataset. For simplicity, we simulate a dataset with linear relationships between common causes and treatment, and common causes and outcome.
Beta is the true causal effect.
data = dowhy.datasets.linear_dataset(beta=10,
num_instruments = 2,
df = data["df"]
│ │ X0 │Z0 │ Z1 │ W0 │ W1 │ W2 │ W3 │W4│ v0 │ y │
│0│-0.296865│0.0│0.058807│-0.749359│0.933320 │-1.212083│1.433017 │2 │True│17.668642│
│1│-1.406887│0.0│0.795132│-0.458243│-0.530596│-1.193390│-1.026360│3 │True│16.473436│
│2│0.719133 │0.0│0.051578│-0.547930│0.512472 │0.113247 │2.636430 │3 │True│27.463675│
│3│-0.343753│0.0│0.975938│0.566506 │0.548739 │2.693299 │0.531672 │3 │True│35.631877│
│4│-1.071145│0.0│0.466001│-0.553899│0.323830 │0.775980 │0.104099 │0 │True│9.901106 │
Note that we are using a pandas dataframe to load the data. At present, DoWhy only supports pandas dataframe as input.
Interface 1 (recommended): Input causal graph
We now input a causal graph in the GML graph format (recommended). You can also use the DOT format.
To create the causal graph for your dataset, you can use a tool like DAGitty that provides a GUI to construct the graph. You can export the graph string that it generates. The graph string is very
close to the DOT format: just rename dag to digraph, remove newlines and add a semicolon after every line, to convert it to the DOT format and input to DoWhy.
# With graph
data = df,
from IPython.display import Image, display
The above causal graph shows the assumptions encoded in the causal model. We can now use this graph to first identify the causal effect (go from a causal estimand to a probability expression), and
then estimate the causal effect.
DoWhy philosophy: Keep identification and estimation separate
Identification can be achieved without access to the data, acccesing only the graph. This results in an expression to be computed. This expression can then be evaluated using the available data in
the estimation step. It is important to understand that these are orthogonal steps.
identified_estimand = model.identify_effect(proceed_when_unidentifiable=True)
Estimand type: EstimandType.NONPARAMETRIC_ATE
### Estimand : 1
Estimand name: backdoor
Estimand expression:
Estimand assumption 1, Unconfoundedness: If U→{v0} and U→y then P(y|v0,W1,W3,W2,W4,W0,U) = P(y|v0,W1,W3,W2,W4,W0)
### Estimand : 2
Estimand name: iv
Estimand expression:
⎡ -1⎤
⎢ d ⎛ d ⎞ ⎥
E⎢─────────(y)⋅⎜─────────([v₀])⎟ ⎥
⎣d[Z₀ Z₁] ⎝d[Z₀ Z₁] ⎠ ⎦
Estimand assumption 1, As-if-random: If U→→y then ¬(U →→{Z0,Z1})
Estimand assumption 2, Exclusion: If we remove {Z0,Z1}→{v0}, then ¬({Z0,Z1}→y)
### Estimand : 3
Estimand name: frontdoor
No such variable(s) found!
Note the parameter flag proceed_when_unidentifiable. It needs to be set to True to convey the assumption that we are ignoring any unobserved confounding. The default behavior is to prompt the user to
double-check that the unobserved confounders can be ignored.
causal_estimate = model.estimate_effect(identified_estimand,
*** Causal Estimate ***
## Identified estimand
Estimand type: EstimandType.NONPARAMETRIC_ATE
### Estimand : 1
Estimand name: backdoor
Estimand expression:
Estimand assumption 1, Unconfoundedness: If U→{v0} and U→y then P(y|v0,W1,W3,W2,W4,W0,U) = P(y|v0,W1,W3,W2,W4,W0)
## Realized estimand
b: y~v0+W1+W3+W2+W4+W0
Target units: ate
## Estimate
Mean value: 9.175311445518133
You can input additional parameters to the estimate_effect method. For instance, to estimate the effect on any subset of the units, you can specify the “target_units” parameter which can be a string
(“ate”, “att”, or “atc”), lambda function that filters rows of the data frame, or a new dataframe on which to compute the effect. You can also specify “effect modifiers” to estimate heterogeneous
effects across these variables. See help(CausalModel.estimate_effect).
# Causal effect on the control group (ATC)
causal_estimate_att = model.estimate_effect(identified_estimand,
target_units = "atc")
print("Causal Estimate is " + str(causal_estimate_att.value))
*** Causal Estimate ***
## Identified estimand
Estimand type: EstimandType.NONPARAMETRIC_ATE
### Estimand : 1
Estimand name: backdoor
Estimand expression:
Estimand assumption 1, Unconfoundedness: If U→{v0} and U→y then P(y|v0,W1,W3,W2,W4,W0,U) = P(y|v0,W1,W3,W2,W4,W0)
## Realized estimand
b: y~v0+W1+W3+W2+W4+W0
Target units: atc
## Estimate
Mean value: 9.235449013795757
Causal Estimate is 9.235449013795757
Interface 2: Specify common causes and instruments
# Without graph
model= CausalModel(
from IPython.display import Image, display
We get the same causal graph. Now identification and estimation is done as before.
identified_estimand = model.identify_effect(proceed_when_unidentifiable=True)
estimate = model.estimate_effect(identified_estimand,
print("Causal Estimate is " + str(estimate.value))
*** Causal Estimate ***
## Identified estimand
Estimand type: EstimandType.NONPARAMETRIC_ATE
### Estimand : 1
Estimand name: backdoor
Estimand expression:
Estimand assumption 1, Unconfoundedness: If U→{v0} and U→y then P(y|v0,W1,W3,W2,W4,W0,U) = P(y|v0,W1,W3,W2,W4,W0)
## Realized estimand
b: y~v0+W1+W3+W2+W4+W0
Target units: ate
## Estimate
Mean value: 9.175311445518133
Causal Estimate is 9.175311445518133
Refuting the estimate
Let us now look at ways of refuting the estimate obtained. Refutation methods provide tests that every correct estimator should pass. So if an estimator fails the refutation test (p-value is <0.05),
then it means that there is some problem with the estimator.
Note that we cannot verify that the estimate is correct, but we can reject it if it violates certain expected behavior (this is analogous to scientific theories that can be falsified but not proven
true). The below refutation tests are based on either 1) Invariant transformations: changes in the data that should not change the estimate. Any estimator whose result varies significantly between
the original data and the modified data fails the test;
1. Random Common Cause
2. Data Subset
2. Nullifying transformations: after the data change, the causal true estimate is zero. Any estimator whose result varies significantly from zero on the new data fails the test.
1. Placebo Treatment
Adding a random common cause variable
res_random=model.refute_estimate(identified_estimand, estimate, method_name="random_common_cause", show_progress_bar=True)
Refute: Add a random common cause
Estimated effect:9.175311445518133
New effect:9.175311445518131
p value:1.0
Replacing treatment with a random (placebo) variable
res_placebo=model.refute_estimate(identified_estimand, estimate,
method_name="placebo_treatment_refuter", show_progress_bar=True, placebo_type="permute")
Refute: Use a Placebo Treatment
Estimated effect:9.175311445518133
New effect:-0.007618923494179182
p value:0.98
Removing a random subset of the data
res_subset=model.refute_estimate(identified_estimand, estimate,
method_name="data_subset_refuter", show_progress_bar=True, subset_fraction=0.9)
Refute: Use a subset of data
Estimated effect:9.175311445518133
New effect:9.133759226613764
p value:0.5
As you can see, the propensity score stratification estimator is reasonably robust to refutations.
Reproducability: For reproducibility, you can add a parameter “random_seed” to any refutation method, as shown below.
Parallelization: You can also use built-in parallelization to speed up the refutation process. Simply set n_jobs to a value greater than 1 to spread the workload to multiple CPUs, or set n_jobs=-1 to
use all CPUs. Currently, this is available only for random_common_cause, placebo_treatment_refuter, and data_subset_refuter.
res_subset=model.refute_estimate(identified_estimand, estimate,
method_name="data_subset_refuter", show_progress_bar=True, subset_fraction=0.9, random_seed = 1, n_jobs=-1, verbose=10)
[Parallel(n_jobs=-1)]: Using backend LokyBackend with 4 concurrent workers.
[Parallel(n_jobs=-1)]: Done 5 tasks | elapsed: 3.5s
[Parallel(n_jobs=-1)]: Done 10 tasks | elapsed: 4.1s
[Parallel(n_jobs=-1)]: Done 17 tasks | elapsed: 5.3s
[Parallel(n_jobs=-1)]: Done 24 tasks | elapsed: 6.0s
[Parallel(n_jobs=-1)]: Done 33 tasks | elapsed: 7.7s
[Parallel(n_jobs=-1)]: Done 42 tasks | elapsed: 9.0s
[Parallel(n_jobs=-1)]: Done 53 tasks | elapsed: 10.7s
[Parallel(n_jobs=-1)]: Done 64 tasks | elapsed: 12.1s
[Parallel(n_jobs=-1)]: Done 77 tasks | elapsed: 14.4s
[Parallel(n_jobs=-1)]: Done 90 tasks | elapsed: 16.1s
Refute: Use a subset of data
Estimated effect:9.175311445518133
New effect:9.14023368300086
p value:0.48
[Parallel(n_jobs=-1)]: Done 100 out of 100 | elapsed: 17.5s finished
Adding an unobserved common cause variable
This refutation does not return a p-value. Instead, it provides a sensitivity test on how quickly the estimate changes if the identifying assumptions (used in identify_effect) are not valid.
Specifically, it checks sensitivity to violation of the backdoor assumption: that all common causes are observed.
To do so, it creates a new dataset with an additional common cause between treatment and outcome. To capture the effect of the common cause, the method takes as input the strength of common cause’s
effect on treatment and outcome. Based on these inputs on the common cause’s effects, it changes the treatment and outcome values and then reruns the estimator. The hope is that the new estimate does
not change drastically with a small effect of the unobserved common cause, indicating a robustness to any unobserved confounding.
Another equivalent way of interpreting this procedure is to assume that there was already unobserved confounding present in the input data. The change in treatment and outcome values removes the
effect of whatever unobserved common cause was present in the original data. Then rerunning the estimator on this modified data provides the correct identified estimate and we hope that the
difference between the new estimate and the original estimate is not too high, for some bounded value of the unobserved common cause’s effect.
Importance of domain knowledge: This test requires domain knowledge to set plausible input values of the effect of unobserved confounding. We first show the result for a single value of confounder’s
effect on treatment and outcome.
res_unobserved=model.refute_estimate(identified_estimand, estimate, method_name="add_unobserved_common_cause",
confounders_effect_on_treatment="binary_flip", confounders_effect_on_outcome="linear",
effect_strength_on_treatment=0.01, effect_strength_on_outcome=0.02)
Refute: Add an Unobserved Common Cause
Estimated effect:9.175311445518133
New effect:8.655303034429837
It is often more useful to inspect the trend as the effect of unobserved confounding is increased. For that, we can provide an array of hypothesized confounders’ effects. The output is the (min, max)
range of the estimated effects under different unobserved confounding.
res_unobserved_range=model.refute_estimate(identified_estimand, estimate, method_name="add_unobserved_common_cause",
confounders_effect_on_treatment="binary_flip", confounders_effect_on_outcome="linear",
effect_strength_on_treatment=np.array([0.001, 0.005, 0.01, 0.02]), effect_strength_on_outcome=0.01)
Refute: Add an Unobserved Common Cause
Estimated effect:9.175311445518133
New effect:(7.745827731363917, 9.123783197433209)
The above plot shows how the estimate decreases as the hypothesized confounding on treatment increases. By domain knowledge, we may know the maximum plausible confounding effect on treatment. Since
we see that the effect does not go beyond zero, we can safely conclude that the causal effect of treatment v0 is positive.
We can also vary the confounding effect on both treatment and outcome. We obtain a heatmap.
res_unobserved_range=model.refute_estimate(identified_estimand, estimate, method_name="add_unobserved_common_cause",
confounders_effect_on_treatment="binary_flip", confounders_effect_on_outcome="linear",
effect_strength_on_treatment=[0.001, 0.005, 0.01, 0.02],
effect_strength_on_outcome=[0.001, 0.005, 0.01,0.02])
Refute: Add an Unobserved Common Cause
Estimated effect:9.175311445518133
New effect:(4.297234272576898, 9.152072285360411)
Automatically inferring effect strength parameters. Finally, DoWhy supports automatic selection of the effect strength parameters. This is based on an assumption that the effect of the unobserved
confounder on treatment or outcome cannot be stronger than that of any observed confounder. That is, we have collected data at least for the most relevant confounder. If that is the case, then we can
bound the range of effect_strength_on_treatment and effect_strength_on_outcome by the effect strength of observed confounders. There is an additional optional parameter signifying whether the effect
strength of unobserved confounder should be as high as the highest observed, or a fraction of it. You can set it using the optional effect_fraction_on_treatment and effect_fraction_on_outcome
parameters. By default, these two parameters are 1.
res_unobserved_auto = model.refute_estimate(identified_estimand, estimate, method_name="add_unobserved_common_cause",
confounders_effect_on_treatment="binary_flip", confounders_effect_on_outcome="linear")
/github/home/.cache/pypoetry/virtualenvs/dowhy-oN2hW5jr-py3.8/lib/python3.8/site-packages/sklearn/utils/validation.py:1143: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
y = column_or_1d(y, warn=True)
Refute: Add an Unobserved Common Cause
Estimated effect:9.175311445518133
New effect:(-0.1861482840361603, 8.090708993601943)
Conclusion: Assuming that the unobserved confounder does not affect the treatment or outcome more strongly than any observed confounder, the causal effect can be concluded to be positive. | {"url":"https://www.pywhy.org/dowhy/v0.11/example_notebooks/dowhy_simple_example.html","timestamp":"2024-11-08T13:53:42Z","content_type":"text/html","content_length":"108818","record_id":"<urn:uuid:3c5e9577-074f-4c8f-9cda-e3887a2b3949>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00276.warc.gz"} |
Introduction to Dynamic Programming
Introduction to Dynamic Programming¶
The essence of dynamic programming is to avoid repeated calculation. Often, dynamic programming problems are naturally solvable by recursion. In such cases, it's easiest to write the recursive
solution, then save repeated states in a lookup table. This process is known as top-down dynamic programming with memoization. That's read "memoization" (like we are writing in a memo pad) not
One of the most basic, classic examples of this process is the fibonacci sequence. It's recursive formulation is $f(n) = f(n-1) + f(n-2)$ where $n \ge 2$ and $f(0)=0$ and $f(1)=1$. In C++, this would
be expressed as:
int f(int n) {
if (n == 0) return 0;
if (n == 1) return 1;
return f(n - 1) + f(n - 2);
The runtime of this recursive function is exponential - approximately $O(2^n)$ since one function call ( $f(n)$ ) results in 2 similarly sized function calls ($f(n-1)$ and $f(n-2)$ ).
Speeding up Fibonacci with Dynamic Programming (Memoization)¶
Our recursive function currently solves fibonacci in exponential time. This means that we can only handle small input values before the problem becomes too difficult. For instance, $f(29)$ results in
over 1 million function calls!
To increase the speed, we recognize that the number of subproblems is only $O(n)$. That is, in order to calculate $f(n)$ we only need to know $f(n-1),f(n-2), \dots ,f(0)$. Therefore, instead of
recalculating these subproblems, we solve them once and then save the result in a lookup table. Subsequent calls will use this lookup table and immediately return a result, thus eliminating
exponential work!
Each recursive call will check against a lookup table to see if the value has been calculated. This is done in $O(1)$ time. If we have previously calcuated it, return the result, otherwise, we
calculate the function normally. The overall runtime is $O(n)$. This is an enormous improvement over our previous exponential time algorithm!
const int MAXN = 100;
bool found[MAXN];
int memo[MAXN];
int f(int n) {
if (found[n]) return memo[n];
if (n == 0) return 0;
if (n == 1) return 1;
found[n] = true;
return memo[n] = f(n - 1) + f(n - 2);
With our new memoized recursive function, $f(29)$, which used to result in over 1 million calls, now results in only 57 calls, nearly 20,000 times fewer function calls! Ironically, we are now limited
by our data type. $f(46)$ is the last fibonacci number that can fit into a signed 32-bit integer.
Typically, we try to save states in arrays, if possible, since the lookup time is $O(1)$ with minimal overhead. However, more generically, we can save states any way we like. Other examples include
binary search trees (map in C++) or hash tables (unordered_map in C++).
An example of this might be:
unordered_map<int, int> memo;
int f(int n) {
if (memo.count(n)) return memo[n];
if (n == 0) return 0;
if (n == 1) return 1;
return memo[n] = f(n - 1) + f(n - 2);
Or analogously:
map<int, int> memo;
int f(int n) {
if (memo.count(n)) return memo[n];
if (n == 0) return 0;
if (n == 1) return 1;
return memo[n] = f(n - 1) + f(n - 2);
Both of these will almost always be slower than the array-based version for a generic memoized recursive function. These alternative ways of saving state are primarily useful when saving vectors or
strings as part of the state space.
The layman's way of analyzing the runtime of a memoized recursive function is:
$$\text{work per subproblem} * \text{number of subproblems}$$
Using a binary search tree (map in C++) to save states will technically result in $O(n \log n)$ as each lookup and insertion will take $O(\log n)$ work and with $O(n)$ unique subproblems we have $O(n
\log n)$ time.
This approach is called top-down, as we can call the function with a query value and the calculation starts going from the top (queried value) down to the bottom (base cases of the recursion), and
makes shortcuts via memoization on the way.
Bottom-up Dynamic Programming¶
Until now you've only seen top-down dynamic programming with memoization. However, we can also solve problems with bottom-up dynamic programming. Bottom-up is exactly the opposite of top-down, you
start at the bottom (base cases of the recursion), and extend it to more and more values.
To create a bottom-up approach for fibonacci numbers, we initilize the base cases in an array. Then, we simply use the recursive definition on array:
const int MAXN = 100;
int fib[MAXN];
int f(int n) {
fib[0] = 0;
fib[1] = 1;
for (int i = 2; i <= n; i++) fib[i] = fib[i - 1] + fib[i - 2];
return fib[n];
Of course, as written, this is a bit silly for two reasons: Firstly, we do repeated work if we call the function more than once. Secondly, we only need to use the two previous values to calculate the
current element. Therefore, we can reduce our memory from $O(n)$ to $O(1)$.
An example of a bottom-up dynamic programming solution for fibonacci which uses $O(1)$ might be:
const int MAX_SAVE = 3;
int fib[MAX_SAVE];
int f(int n) {
fib[0] = 0;
fib[1] = 1;
for (int i = 2; i <= n; i++)
fib[i % MAX_SAVE] = fib[(i - 1) % MAX_SAVE] + fib[(i - 2) % MAX_SAVE];
return fib[n % MAX_SAVE];
Note that we've changed the constant from MAXN TO MAX_SAVE. This is because the total number of elements we need to access is only 3. It no longer scales with the size of input and is, by definition,
$O(1)$ memory. Additionally, we use a common trick (using the modulo operator) only maintaining the values we need.
That's it. That's the basics of dynamic programming: Don't repeat the work you've done before.
One of the tricks to getting better at dynamic programming is to study some of the classic examples.
Classic Dynamic Programming Problems¶
Name Description/Example
0-1 Knapsack Given $W$, $N$, and $N$ items with weights $w_i$ and values $v_i$, what is the maximum $\sum_{i=1}^{k} v_i$ for each subset of items of size $k$ ($1 \le k \le N$)
while ensuring $\sum_{i=1}^{k} w_i \le W$?
Subset Sum Given $N$ integers and $T$, determine whether there exists a subset of the given set whose elements sum up to the $T$.
Longest Increasing Subsequence You are given an array containing $N$ integers. Your task is to determine the LIS in the array, i.e., a subsequence where every element is larger than the previous
(LIS) one.
Counting Paths in a 2D Array Given $N$ and $M$, count all possible distinct paths from $(1,1)$ to $(N, M)$, where each step is either from $(i,j)$ to $(i+1,j)$ or $(i,j+1)$.
Longest Common Subsequence You are given strings $s$ and $t$. Find the length of the longest string that is a subsequence of both $s$ and $t$.
Longest Path in a Directed Acyclic Finding the longest path in Directed Acyclic Graph (DAG).
Graph (DAG)
Longest Palindromic Subsequence Finding the Longest Palindromic Subsequence (LPS) of a given string.
Rod Cutting Given a rod of length $n$ units, Given an integer array cuts where cuts[i] denotes a position you should perform a cut at. The cost of one cut is the length of the
rod to be cut. What is the minimum total cost of the cuts.
Edit Distance The edit distance between two strings is the minimum number of operations required to transform one string into the other. Operations are ["Add", "Remove",
Related Topics¶
• Bitmask Dynamic Programming
• Digit Dynamic Programming
• Dynamic Programming on Trees
Of course, the most important trick is to practice.
Practice Problems¶
DP Contests¶ | {"url":"https://gh.cp-algorithms.com/main/dynamic_programming/intro-to-dp.html","timestamp":"2024-11-05T20:13:03Z","content_type":"text/html","content_length":"147253","record_id":"<urn:uuid:2062d4c4-1df9-40f4-b38a-9ddbe535f589>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00275.warc.gz"} |
SHAP Analyses and SHAP Graph BuilderSHAP Analyses and SHAP Graph Builder
Hi All,
I have searched in the JMP community about SHAP Values and I am wondering if someone can direct me to some resources in how to evaluate SHAP values using JMP Pro 17. I understand how to populate the
SHAP values (from the prediction profiler), but creating the graphs are a bit confusing at the point. I used the graph builder to build the graph, but I am wondering if there are videos or guides on
the types of analyses that I can do with SHAP values and if there's an automatic way to create graphs from the SHAP platform.
Thank you for taking the time to get back to me. | {"url":"https://community.jmp.com/t5/Discussions/SHAP-Analyses-and-SHAP-Graph-Builder/td-p/797903","timestamp":"2024-11-09T13:18:35Z","content_type":"text/html","content_length":"731150","record_id":"<urn:uuid:5088c184-bfb0-4752-8087-bb778b76e435>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00399.warc.gz"} |
Nominal data
Using Frequencies to Study Nominal Data. You manage a team that sells computer hardware to software development companies. At each company, your representatives have a primary contact. You have
categorized these contacts by the department of the company in which they work
Nominal unit labour cost - annual data, % changes and index (2010 = 100)
An easy way to remember this type of data is that nominal sounds like named, nominal = named. Ordinal Data Nominal and Ordinal data should only be counted and described in frequency tables--no means
and standard deviations. One of the more famous articles showing the fallacy of such rigid thinking was by an eminent statistician named Lord who in his article: "On the statistical Treatment of
Football Numbers" showed how the means of nominal data can be meaningful too! Hi everyone, I need to assess statistical validity of nominal data between several treatment and control groups in female
vs. male mice, over a period of several days (1 data set gathered per This video reviews the scales of measurement covered in introductory statistics: nominal, ordinal, interval, and ratio (Part 1 of
2).Scales of MeasurementNom Nominal data are discrete, nonnumeric values that do not have a natural ordering. nominal array objects provide efficient storage and convenient manipulation of such data,
while also maintaining meaningful labels for the values. Nominal Data.
Nominal Data Definition. Nominal Data is derived from a Latin word called “Nomen,” which means name and also can be termed as labeled data or named data, which is again basically the process of
classifying categorical data where we cannot assign any quantitative value against the same. Nominal scales were often called qualitative scales, and measurements made on qualitative scales were
called qualitative data. However, the rise of qualitative research has made this usage confusing. If numbers are assigned as labels in nominal measurement, they have no specific numerical value or
In research, nominal data can be given a numerical value but those values don’t hold true significance. If you use the assigned numerical value to calculate other figures like mean, median, etc. it
would be meaningless. Imagine using a nominal scale and giving male a value of 2, female a value of 4, and transgender a value of 6.
av A Carlsson · 2005 · Citerat av 1 — Data were collected from parents of 10-month-old children who answered a Chi-squared test was used to analyse differences in nominal data and Technical data.
Item. 36210.
Tekniska data i detalj. System. Mätstorheter. Volymflöde i Nominal pipe size. Connection flange. Standard. Weight 1). Length (A). Hight 2) (B). Flange diame-.
Nominal ULC (NULC) = (D1/EEM) / (B1GQ/ETO) with D1 = Compensation Labour cost index by NACE Rev. 2 activity - nominal value, quarterly data. nominaldata, kategoridata. nominal data [ˈnɒmɪnl deɪtə,
br eng även: ~ ˈdɑːtə, i USA även: ~ ˈdætə], categorical data [ˌkætəˈɡɒrɪkl ~]. Observationer av Hitta stockbilder i HD på nominal data och miljontals andra royaltyfria stockbilder, illustrationer
och vektorer i Shutterstocks samling. Tusentals nya Relationen mellan NOMINAL och EFFECT visas i följande ekvation:The relationship between NOMINAL DataData, BeskrivningDescription comparing
nominal against monitored data in the distribution domain.
Ålder (år). Chemical composition (nominal). Columns: 8 of 8 Continuous development may necessitate changes in technical data without notice. När data laddas in i jamovi får variablerna vanligtvis den
lägsta nivån (nominal). intervall- och kvotdata, utan kategoriserar all kvantitativ data som kontinuerlig. Psi-calculi: a framework for mobile processes with nominal data and logic.
Konto 1249
You can even categorize (and people frequently do) data that is interval or ratio, and sometimes you can uncategorize it. 3) As @Gung pointed out, a count variable is discrete but not categorical.
You can even categorize (and people frequently do) data that is interval or ratio, and sometimes you can uncategorize it. 3) As @Gung pointed out, a count variable is discrete but not categorical.
Define nominal scale.
Nordenskiöld polarforskare
What is nominal data? Nominal data is data that can be labelled or classified into mutually exclusive categories within a variable. These categories cannot be ordered in a meaningful way. For
example, for the nominal variable of preferred mode of transportation, you may have the …
It does not have a rank order, equal spacing between values, or a true zero value. Examples of nominal data Nominal data is a type of qualitative data which groups variables into categories. You can
think of these categories as nouns or labels; they are purely descriptive, they don’t have any quantitative or numeric value, and the various categories cannot be placed into any kind of meaningful
order or hierarchy. Nominal Data Definition Nominal data is “labeled” or “named” data which can be divided into various groups that do not overlap. Data is not measured or evaluated in this case, it
is just assigned to multiple groups. These groups are unique and have no common elements. Nominal Data Nominal data simply names something without assigning it to an order in relation to other
numbered objects or pieces of data. | {"url":"https://affarerqiwxps.netlify.app/21459/6889","timestamp":"2024-11-02T22:04:16Z","content_type":"text/html","content_length":"10139","record_id":"<urn:uuid:6834ed59-3055-4034-b275-3a35c5a4352f>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00315.warc.gz"} |
es f
The most recent game I uploaded to this site is a Printable Latin Square Puzzle. The goal of the puzzle is to arrange a set of 16 cards into a 4 by 4 grid, following certain rules.
The Puzzle Universe – A History of Mathematics in 315 Puzzles
The other day I received a marvellous package in the post. Inside was a book, The Puzzle Universe, by Ivan Moscovich, published by Firefly Books. Ivan Moscovich has made a career out of making
amazing mathematical puzzles and games.
Continue reading The Puzzle Universe – A History of Mathematics in 315 Puzzles
Movie Ticket Puzzle Solution
[This is a back-issue of one of this site’s newsletters]
It’s Monday Morning Math Time!
If you’re hungry, have a look at this yummy breakfast math video. You’ll see there’s more than one way to cut a bagel, and maybe learn a bit of topology too.
I promised to give you a solution to the movie ticket problem.
If you missed it, have a read here and see if you can solve it before you read on here!
Movie Tickets and Time Machines
[This is a back-issue of one of this site’s newsletters]
Here’s a puzzle for you to try – how creative can you get while solving it? Continue reading Movie Tickets and Time Machines
Old Game Now New
[This is a back-issue of this site’s newsletter]
Monday Morning Math is late this week, but for a good reason – I was rushing to finish the new version of my online traffic jam game! Continue reading Old Game Now New
How Much For 12 Shirts?
I spotted this sign at a shirt shop while on holidays overseas:
Can You Solve This Puzzle?
I saw this puzzle the other day.
You have two fuses. Each fuse is a piece of string, that burns for exactly 1 minute. However, the fuse doesn’t burn evenly, so cutting the fuse in half doesn’t give you two 30 second fuses.
A Simpler Solution
[This is a back-issue of one of this site’s newsletters]
Last week, I emailed you a way to solve a simple math puzzle – how do you find a rectangle whose area equals its perimeter?
I took the puzzle, mixed in a little bit of algebra, and voila! A puzzle solution factory popped out, letting you generate rectangles from pythagorean triplets.
More On The Rectangle Puzzle
Last week, I posted a solution to a puzzle – how can you find a rectangle whose area equals its perimeter?
This week, I’ll post a simpler solution.
Rectangles and Right Triangles
Can you find a rectangle whose perimeter equals its area?
I’ll explain one way to solve this puzzle below.
Allergy warning: this product contains algebra. May contain traces of number theory. | {"url":"https://www.dr-mikes-math-games-for-kids.com/blog/tag/puzzle/","timestamp":"2024-11-05T02:22:59Z","content_type":"text/html","content_length":"61194","record_id":"<urn:uuid:c2495d30-b9ad-476e-8822-66512965bf03>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00581.warc.gz"} |
A set is a fundamental concept in mathematics and computer science that represents a collection of distinct and unordered elements. In simpler terms, a set is a grouping of unique items without any
specific arrangement. Sets are used to model various scenarios, solve problems, and establish relationships between objects.
Key characteristics of sets include:-
1. Distinct Elements:- A set contains only distinct or unique elements. Each element appears in the set only once, regardless of how many times it might be mentioned in the real world.
2. Unordered:- The elements in a set are not arranged in any specific order. This means that the concept of "first," "second," etc., does not apply to the elements within a set.
3. No Duplicates:- Since sets consist of unique elements, there are no duplicate entries. If an element is already present in a set, attempting to add it again will not change the set.
Sets can be represented using various notations:-
• Roster Notation:- In this notation, elements are listed within curly braces {}. For example, the set of even numbers less than 10 can be written as {2, 4, 6, 8}.
• Set-Builder Notation:- This notation defines a set using a description of its elements. For example, the set of all even numbers can be defined as {x | x is an integer and x is even}.
Sets can also be classified based on their size:-
• Finite Sets:- These sets have a specific countable number of elements. For example, the set of prime numbers less than 20 is a finite set.
• Infinite Sets:- These sets have an uncountably infinite number of elements. The set of all natural numbers is an example of an infinite set.
Sets are used extensively in various mathematical operations and concepts, including union, intersection, complement, and more. They also play a crucial role in formalizing concepts in areas like set
theory, graph theory, and probability theory. In computer science, sets are used in data structures like hash tables and are foundational in database management systems.
Overall, sets provide a powerful framework for representing collections of distinct elements and serve as a fundamental building block in mathematical reasoning and problem-solving.
Key characteristics of sets include:-
1. Distinct Elements:- A set cannot contain duplicate elements. Each element in a set is unique.
2. No Order:- The elements in a set have no inherent order. This means that the elements can be arranged in any sequence without affecting the properties of the set.
3. Membership:- An element either belongs to a set or does not. This membership relationship is denoted using the symbol "∈" (belongs to) or "∉" (does not belong to).
4. Cardinality:- The cardinality of a set refers to the number of elements it contains.
Sets are typically represented using curly braces { }. For example:-
• The set of natural numbers less than 5: {1, 2, 3, 4}
• The set of prime numbers between 10 and 20: {11, 13, 17, 19}
Mathematical operations on sets include:-
1. Union:- The union of two sets A and B is a new set that contains all the distinct elements from both A and B. It is denoted as A ∪ B.
2. Intersection:- The intersection of two sets A and B is a new set containing elements that are common to both A and B. It is denoted as A ∩ B.
3. Complement:- The complement of a set A with respect to a universal set U contains all the elements of U that are not in A. It is denoted as A'.
4. Subset:- A set A is a subset of another set B if every element of A is also an element of B. It is denoted as A ⊆ B.
5. Proper Subset:- A set A is a proper subset of another set B if A is a subset of B but not equal to B. It is denoted as A ⊂ B.
Sets are used in various areas of mathematics, including algebra, calculus, and discrete mathematics, as well as in computer science for data structures (e.g., hash sets) and algorithms (e.g., set | {"url":"https://www.math-edu-guide.com/CLASS-6-Sets-Introduction-Of-Sets.html","timestamp":"2024-11-04T21:35:21Z","content_type":"text/html","content_length":"24005","record_id":"<urn:uuid:ccb55097-94f5-4315-9bd9-147cfd483753>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00126.warc.gz"} |
Faster-than-lightspeed Constants - mp-units
Faster-than-lightspeed Constants¶
In most libraries, physical constants are implemented as constant (possibly constexpr) quantity values. Such an approach has some disadvantages, often affecting the run time performance and causing a
loss of precision.
Simplifying constants in an equation¶
When dealing with equations involving physical constants, they often occur more than once in an expression. Such a constant may appear both in a numerator and denominator of a quantity equation. As
we know from fundamental physics, we can simplify such an expression by striking a constant out of the equation. Supporting such behavior allows a faster runtime performance and often a better
precision of the resulting value.
Physical constants as units¶
The mp-units library allows and encourages the implementation of physical constants as regular units. With that, the constant's value is handled at compile-time, and under favorable circumstances, it
can be simplified in the same way as all other repeated units do. If it is not simplified, the value is stored in a type, and the expensive multiplication or division operations can be delayed in
time until a user selects a specific unit to represent/print the data.
Such a feature often also allows using simpler or faster representation types in the equation. For example, instead of always having to multiply a small integral value with a big floating-point
constant number, we can just use the integral type all the way. Only in case a constant will not simplify in the equation, and the user will require a specific unit, such a multiplication will be
lazily invoked, and the representation type will need to be expanded to facilitate that. With that, addition, subtractions, multiplications, and divisions will always be the fastest - compiled away
or done in out-of-order execution.
To benefit from all of the above, in the mp-units library, SI defining and other constants are implemented as units in the following way:
namespace si {
namespace si2019 {
inline constexpr struct speed_of_light_in_vacuum final :
named_unit<"c", mag<299'792'458> * metre / second> {} speed_of_light_in_vacuum;
} // namespace si2019
inline constexpr struct magnetic_constant final :
named_unit<{u8"μ₀", "u_0"}, mag<4> * mag<π> * mag_power<10, -7> * henry / metre> {} magnetic_constant;
} // namespace mp_units::si
Usage examples¶
With the above definitions, we can calculate vacuum permittivity as:
constexpr auto permeability_of_vacuum = 1. * si::magnetic_constant;
constexpr auto speed_of_light_in_vacuum = 1 * si::si2019::speed_of_light_in_vacuum;
QuantityOf<isq::permittivity_of_vacuum> auto q = 1 / (permeability_of_vacuum * pow<2>(speed_of_light_in_vacuum));
std::println("permittivity of vacuum = {} = {::N[.3e]}", q, q.in(F / m));
The above first prints the following:
As we can clearly see, all the calculations above were just about multiplying and dividing the number 1 with the rest of the information provided as a compile-time type. Only when a user wants a
specific SI unit as a result, the unit ratios are lazily resolved.
Another similar example can be an equation for total energy:
QuantityOf<isq::mechanical_energy> auto total_energy(QuantityOf<isq::momentum> auto p,
QuantityOf<isq::mass> auto m,
QuantityOf<isq::speed> auto c)
return isq::mechanical_energy(sqrt(pow<2>(p * c) + pow<2>(m * pow<2>(c))));
constexpr auto GeV = si::giga<si::electronvolt>;
constexpr QuantityOf<isq::speed> auto c = 1. * si::si2019::speed_of_light_in_vacuum;
constexpr auto c2 = pow<2>(c);
const auto p1 = isq::momentum(4. * GeV / c);
const QuantityOf<isq::mass> auto m1 = 3. * GeV / c2;
const auto E = total_energy(p1, m1, c);
std::cout << "in `GeV` and `c`:\n"
<< "p = " << p1 << "\n"
<< "m = " << m1 << "\n"
<< "E = " << E << "\n";
const auto p2 = p1.in(GeV / (m / s));
const auto m2 = m1.in(GeV / pow<2>(m / s));
const auto E2 = total_energy(p2, m2, c).in(GeV);
std::cout << "\nin `GeV`:\n"
<< "p = " << p2 << "\n"
<< "m = " << m2 << "\n"
<< "E = " << E2 << "\n";
const auto p3 = p1.in(kg * m / s);
const auto m3 = m1.in(kg);
const auto E3 = total_energy(p3, m3, c).in(J);
std::cout << "\nin SI base units:\n"
<< "p = " << p3 << "\n"
<< "m = " << m3 << "\n"
<< "E = " << E3 << "\n";
The above prints the following: | {"url":"https://mpusz.github.io/mp-units/latest/users_guide/framework_basics/faster_than_lightspeed_constants/","timestamp":"2024-11-08T11:10:47Z","content_type":"text/html","content_length":"78756","record_id":"<urn:uuid:7959ab80-21a7-434c-befc-b9cb406596df>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00548.warc.gz"} |
Source code for pvlib.scaling
The ``scaling`` module contains functions for manipulating irradiance
or other variables to account for temporal or spatial characteristics.
import numpy as np
import pandas as pd
import scipy.optimize
from scipy.spatial.distance import pdist
def wvm(clearsky_index, positions, cloud_speed, dt=None):
Compute spatial aggregation time series smoothing on clear sky index based
on the Wavelet Variability model.
This model is described in Lave et al. [1]_, [2]_.
Implementation is basically a port of the Matlab version of the code [3]_.
clearsky_index : numeric or pandas.Series
Clear Sky Index time series that will be smoothed.
positions : numeric
Array of coordinate distances as (x,y) pairs representing the
easting, northing of the site positions in meters [m]. Distributed
plants could be simulated by gridded points throughout the plant
cloud_speed : numeric
Speed of cloud movement in meters per second [m/s].
dt : float, optional
The time series time delta. By default, is inferred from the
clearsky_index. Must be specified for a time series that doesn't
include an index. Units of seconds [s].
smoothed : numeric or pandas.Series
The Clear Sky Index time series smoothed for the described plant.
wavelet: numeric
The individual wavelets for the time series before smoothing.
tmscales: numeric
The timescales associated with the wavelets in seconds [s].
.. [1] M. Lave, J. Kleissl and J.S. Stein. A Wavelet-Based Variability
Model (WVM) for Solar PV Power Plants. IEEE Transactions on Sustainable
Energy, vol. 4, no. 2, pp. 501-509, 2013.
.. [2] M. Lave and J. Kleissl. Cloud speed impact on solar variability
scaling - Application to the wavelet variability model. Solar Energy,
vol. 91, pp. 11-21, 2013.
.. [3] Wavelet Variability Model - Matlab Code:
# Added by Joe Ranalli (@jranalli), Penn State Hazleton, 2019
wavelet, tmscales = _compute_wavelet(clearsky_index, dt)
vr = _compute_vr(positions, cloud_speed, tmscales)
# Scale each wavelet by VR (Eq 7 in [1])
wavelet_smooth = np.zeros_like(wavelet)
for i in np.arange(len(tmscales)):
if i < len(tmscales) - 1: # Treat the lowest freq differently
wavelet_smooth[i, :] = wavelet[i, :] / np.sqrt(vr[i])
wavelet_smooth[i, :] = wavelet[i, :]
outsignal = np.sum(wavelet_smooth, 0)
try: # See if there's an index already, if so, return as a pandas Series
smoothed = pd.Series(outsignal, index=clearsky_index.index)
except AttributeError:
smoothed = outsignal # just output the numpy signal
return smoothed, wavelet, tmscales
def _compute_vr(positions, cloud_speed, tmscales):
Compute the variability reduction factors for each wavelet mode for the
Wavelet Variability Model [1-3].
positions : numeric
Array of coordinate distances as (x,y) pairs representing the
easting, northing of the site positions in meters [m]. Distributed
plants could be simulated by gridded points throughout the plant
cloud_speed : numeric
Speed of cloud movement in meters per second [m/s].
tmscales: numeric
The timescales associated with the wavelets in seconds [s].
vr : numeric
an array of variability reduction factors for each tmscale.
.. [1] M. Lave, J. Kleissl and J.S. Stein. A Wavelet-Based Variability
Model (WVM) for Solar PV Power Plants. IEEE Transactions on Sustainable
Energy, vol. 4, no. 2, pp. 501-509, 2013.
.. [2] M. Lave and J. Kleissl. Cloud speed impact on solar variability
scaling - Application to the wavelet variability model. Solar Energy,
vol. 91, pp. 11-21, 2013.
.. [3] Wavelet Variability Model - Matlab Code:
# Added by Joe Ranalli (@jranalli), Penn State Hazleton, 2021
pos = np.array(positions)
dist = pdist(pos, 'euclidean')
# Find effective length of position vector, 'dist' is full pairwise
n_pairs = len(dist)
def fn(x):
return np.abs((x ** 2 - x) / 2 - n_pairs)
n_dist = np.round(scipy.optimize.fmin(fn, np.sqrt(n_pairs), disp=False))
n_dist = n_dist.item()
# Compute VR
A = cloud_speed / 2 # Resultant fit for A from [2]
vr = np.zeros(tmscales.shape)
for i, tmscale in enumerate(tmscales):
rho = np.exp(-1 / A * dist / tmscale) # Eq 5 from [1]
# 2*rho is because rho_ij = rho_ji. +n_dist accounts for sum(rho_ii=1)
denominator = 2 * np.sum(rho) + n_dist
vr[i] = n_dist ** 2 / denominator # Eq 6 of [1]
return vr
def latlon_to_xy(coordinates):
Convert latitude and longitude in degrees to a coordinate system measured
in meters from zero deg latitude, zero deg longitude.
This is a convenience method to support inputs to wvm. Note that the
methodology used is only suitable for short distances. For conversions of
longer distances, users should consider use of Universal Transverse
Mercator (UTM) or other suitable cartographic projection. Consider
packages built for cartographic projection such as pyproj (e.g.
pyproj.transform()) [2].
coordinates : numeric
Array or list of (latitude, longitude) coordinate pairs. Use decimal
degrees notation.
xypos : numeric
Array of coordinate distances as (x,y) pairs representing the
easting, northing of the position in meters [m].
.. [1] H. Moritz. Geodetic Reference System 1980, Journal of Geodesy, vol.
74, no. 1, pp 128–133, 2000.
.. [2] https://pypi.org/project/pyproj/
.. [3] Wavelet Variability Model - Matlab Code:
# Added by Joe Ranalli (@jranalli), Penn State Hazleton, 2019
r_earth = 6371008.7714 # mean radius of Earth, in meters
m_per_deg_lat = r_earth * np.pi / 180
meanlat = np.mean([lat for (lat, lon) in coordinates]) # Mean latitude
except TypeError: # Assume it's a single value?
meanlat = coordinates[0]
m_per_deg_lon = r_earth * np.cos(np.pi/180 * meanlat) * np.pi/180
# Conversion
pos = coordinates * np.array(m_per_deg_lat, m_per_deg_lon)
# reshape as (x,y) pairs to return
return np.column_stack([pos[:, 1], pos[:, 0]])
except IndexError: # Assume it's a single value, which has a 1D shape
return np.array((pos[1], pos[0]))
def _compute_wavelet(clearsky_index, dt=None):
Compute the wavelet transform on the input clear_sky time series. Uses a
top hat wavelet [-1,1,1,-1] shape, based on the difference of successive
centered moving averages. Smallest scale (filter size of 2) is a degenerate
case that resembles a Haar wavelet. Returns one level of approximation
coefficient (CAn) and n levels of detail coefficients (CD1, CD2, ...,
CDn-1, CDn).
clearsky_index : numeric or pandas.Series
Clear Sky Index time series that will be smoothed.
dt : float, optional
The time series time delta. By default, is inferred from the
clearsky_index. Must be specified for a time series that doesn't
include an index. Units of seconds [s].
wavelet: numeric
The individual wavelets for the time series. Format follows increasing
scale (decreasing frequency): [CD1, CD2, ..., CDn, CAn]
tmscales: numeric
The timescales associated with the wavelets in seconds [s]
.. [1] M. Lave, J. Kleissl and J.S. Stein. A Wavelet-Based Variability
Model (WVM) for Solar PV Power Plants. IEEE Transactions on
Sustainable Energy, vol. 4, no. 2, pp. 501-509, 2013.
.. [2] Wavelet Variability Model - Matlab Code:
# Added by Joe Ranalli (@jranalli), Penn State Hazleton, 2019
try: # Assume it's a pandas type
vals = clearsky_index.values.flatten()
except AttributeError: # Assume it's a numpy type
vals = clearsky_index.flatten()
if dt is None:
raise ValueError("dt must be specified for numpy type inputs.")
else: # flatten() succeeded, thus it's a pandas type, so get its dt
try: # Assume it's a time series type index
dt = clearsky_index.index[1] - clearsky_index.index[0]
dt = dt.seconds + dt.microseconds/1e6
except AttributeError: # It must just be a numeric index
dt = (clearsky_index.index[1] - clearsky_index.index[0])
# Pad the series on both ends in time and place in a dataframe
cs_long = np.pad(vals, (len(vals), len(vals)), 'symmetric')
cs_long = pd.DataFrame(cs_long)
# Compute wavelet time scales
min_tmscale = np.ceil(np.log(dt)/np.log(2)) # Minimum wavelet timescale
max_tmscale = int(13 - min_tmscale) # maximum wavelet timescale
tmscales = np.zeros(max_tmscale)
csi_mean = np.zeros([max_tmscale, len(cs_long)])
# Skip averaging for the 0th scale
csi_mean[0, :] = cs_long.values.flatten()
tmscales[0] = dt
# Loop for all time scales we will consider
for i in np.arange(1, max_tmscale):
tmscales[i] = 2**i * dt # Wavelet integration time scale
intvlen = 2**i # Wavelet integration time series interval
# Rolling average, retains only lower frequencies than interval
# Produces slightly different end effects than the MATLAB version
df = cs_long.rolling(window=intvlen, center=True, min_periods=1).mean()
# Fill nan's in both directions
df = df.bfill().ffill()
# Pop values back out of the dataframe and store
csi_mean[i, :] = df.values.flatten()
# Shift to account for different indexing in MATLAB moving average
csi_mean[i, :] = np.roll(csi_mean[i, :], -1)
csi_mean[i, -1] = csi_mean[i, -2]
# Calculate detail coefficients by difference between successive averages
wavelet_long = np.zeros(csi_mean.shape)
for i in np.arange(0, max_tmscale-1):
wavelet_long[i, :] = csi_mean[i, :] - csi_mean[i+1, :]
wavelet_long[-1, :] = csi_mean[-1, :] # Lowest freq (CAn)
# Clip off the padding and just return the original time window
wavelet = np.zeros([max_tmscale, len(vals)])
for i in np.arange(0, max_tmscale):
wavelet[i, :] = wavelet_long[i, len(vals): 2*len(vals)]
return wavelet, tmscales | {"url":"https://pvlib-python.readthedocs.io/en/latest/_modules/pvlib/scaling.html","timestamp":"2024-11-08T09:28:14Z","content_type":"text/html","content_length":"51787","record_id":"<urn:uuid:a14eb8c1-cd7b-4d01-9edf-f3c5aa003d0a>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00486.warc.gz"} |
Electrical Engineering
what is Electric Potential
Sunita said on : 2018-11-17 00:21:45
Electric Potential
Just as we define electric field intensity as the force per unit charge, similarly electric potential is defined as the electric potential energy per unit charge.
figure (a)
Consider an isolated charge +Q fixed in space as shown in Fig. (A). If a unit positive charge (i. e. + 1C) is placed at infinity, the force charge on it due to charge +Q is *zero. If the unit
positive charge at infinity is moved towards +Q, a force of repulsion acts on it (like charges repel) and hence work is required to be done to bring it to a point like A. Hence when the unit
positive charge is at A, it has some amount of electric potential energy which is a measure of electric potential. The closer the point to the charge, the higher will be the electric potential
energy and hence the electric potential at that point. Therefore,electric potential at a point due to a charge depends upon the position of the point; being zero if the point is situated at infinity
Obviously, in electric field, infinity is chosen as the point of **zero potential.
Hence electric potential at a point in an electric field is the amount of work done in bringing a unit positive charge (i.e. +1 C)from infinity to that point i.e.
Unit. The SI unit of electric potential is *volt and may be defined as under :The electric potential at a point in an electric field is 1 volt if 1 joule of work is done in bringing a unit positive
charge (i.e. +1 C)from infinity to that point **against the electric field Thus when we say that potential at a point in an electric field is +5V, it simply means that 5 joules of work has been done
in bringing a unit positive charge from infinity to that point
!! OOPS Login [Click here] is required for more results / answer
Help other students, write article, leave your comments | {"url":"https://engineeringslab.com/tutorial_electrical/electric-potential-1150.htm","timestamp":"2024-11-03T13:59:33Z","content_type":"text/html","content_length":"37611","record_id":"<urn:uuid:da523a72-372e-4105-9693-0301c1bce1d0>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00564.warc.gz"} |
Standard deviation compared to Variance – Q&A Hub – 365 Financial Analyst
Super learner
This user is a Super Learner. To become a Super Learner, you need to reach Level 8.
Standard deviation compared to Variance
If I understand, the standard deviation shows how close the data are to the mean and variance only helps with finding if the data has outliers?
7 answers ( 0 marked as helpful)
Standard deviation is a measure of dispersion. It shows you how far from the mean would you expect observations to be on average.
Hope this helps!
Super learner
This user is a Super Learner. To become a Super Learner, you need to reach Level 8.
I don't understand how that is different from variance. In other words, to me, it sounds like variance also "shows you how far from the mean would you expect observations to be on average." Am I
right or wrong?
3 points from the video:
1-Variance and standard deviation measure the dispersion of a set of data points around its mean value.
2-variance is large and hard to compare as the unit of measurement is squared
3-standard deviation will be much more meaningful than variance as the main measure of variability.
Super learner
This user is a Super Learner. To become a Super Learner, you need to reach Level 8.
I understand those points separately (although with point 2, if you had the variance of more than 1 dataset, then the biggest variance will have the largest dispersion right? Of course, it's hard to
know how large the dispersion is using variance alone because the measurement is squared but if you had 2 variances for example couldn't you see which is the bigger number and say that one has a
larger dispersion? Having a larger or smaller dispersion isn't very much on its own but it does lead you in the right direction.). With standard deviation being more useful, why even use variance?
To me, both sounds almost same, but variance being squared. it makes more sense to use STDEV since it is in the same units as the data being compared. for example, if the data was in inches, variance
will be in inches squared while standard deviation will be in inches. this makes it easier to compare.
Please more explanation on how the coefficient is involved is needed
Variance exists alongside standard deviation because both serve distinct purposes in statistical analysis, despite being closely related. Here are a few reasons why variance is still important:
1. Mathematical Properties: Variance is mathematically simpler to work with in certain statistical formulas, particularly in the context of theoretical statistics and probability. For example, when
deriving properties of estimators or in the context of linear regression, variance often appears in the calculations.
2. Squared Units: Variance provides a measure of dispersion in squared units, which can be useful in certain contexts, such as when dealing with the sum of squared deviations. This can be
particularly relevant in analysis of variance (ANOVA) and other statistical tests.
3. Foundation for Standard Deviation: Standard deviation is derived from variance (it is the square root of variance). Understanding variance is essential for grasping the concept of standard
deviation, as it provides insight into how data is spread around the mean.
4. Interpretation in Context: In some fields, such as finance or risk management, variance is used to assess risk and volatility. It can be more relevant in contexts where the focus is on the
magnitude of deviations rather than their average distance from the mean.
In summary, while standard deviation is often more intuitive and easier to interpret, variance plays a crucial role in statistical theory and certain applications. Both measures complement each other
and provide valuable insights into data analysis. | {"url":"https://365financialanalyst.com/question/standard-deviation-compared-to-variance/","timestamp":"2024-11-12T19:01:38Z","content_type":"text/html","content_length":"126903","record_id":"<urn:uuid:27d2ed5c-b1ea-4256-b5a9-3a56cb4580bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00114.warc.gz"} |
Understanding Mathematical Functions: Can A Function Have More Than On
Introduction to Mathematical Functions
Mathematical functions are a fundamental concept in the field of mathematics. They are used to describe the relationship between input and output values, and are essential for understanding various
mathematical phenomena and real-world applications. In this blog post, we will explore the concept of functions and delve into the intriguing question of whether a function can have more than one
A. Explanation of functions and their importance in mathematics
A mathematical function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output. Functions are represented using
variables, and they can take various forms, such as linear, quadratic, exponential, and trigonometric functions. They are extensively used in various branches of mathematics, including calculus,
algebra, and geometry, as well as in fields such as physics, engineering, and economics.
B. Brief overview of the concept of y-intercepts
The y-intercept of a function is the point where the graph of the function crosses the y-axis. It represents the value of the function when the input is zero. For example, in the equation of a
straight line, y = mx + c, the y-intercept is the value of c, which is the constant term in the equation. In other words, it is the value of y when x is zero.
C. Purpose of the blog post: to clarify whether a function can have more than one y-intercept
The main goal of this blog post is to address the question of whether a function can have more than one y-intercept. This is a topic that often generates confusion among students and even some math
enthusiasts. By providing a clear and concise explanation, we aim to dispel any misconceptions and deepen the understanding of this concept.
Key Takeaways
• Functions can have only one y-intercept.
• The y-intercept is the point where the function crosses the y-axis.
• It represents the value of the function when x=0.
• Multiple y-intercepts would violate the definition of a function.
Defining the Y-Intercept
When it comes to understanding mathematical functions, the concept of a y-intercept plays a crucial role. Let's delve into what a y-intercept is, how it is found on a graph, and its importance in
understanding the behavior of functions.
A Definition of a y-intercept in the context of a function
In the context of a function, the y-intercept is the point where the graph of the function intersects the y-axis. It is the value of y when x is equal to 0. Symbolically, it is represented as (0, b),
where 'b' is the y-intercept.
How y-intercepts are found on a graph
Finding the y-intercept on a graph is a straightforward process. To find the y-intercept, you simply set x to 0 and solve for y. The resulting point gives you the y-intercept of the function.
For example, if you have a function f(x) = 2x + 3, setting x to 0 gives you f(0) = 3. Therefore, the y-intercept of the function is (0, 3).
Importance of y-intercepts in understanding the behavior of functions
The y-intercept provides valuable information about the behavior of a function. It gives insight into where the function intersects the y-axis and helps in understanding the starting point of the
graph. Additionally, the y-intercept can be used to determine the initial value of a function in real-world applications.
Understanding the y-intercept is essential in analyzing the characteristics of a function, such as its direction, shape, and behavior as x approaches positive or negative infinity. It serves as a
fundamental building block in comprehending the overall behavior of a function.
Characteristics of Functions
When it comes to understanding mathematical functions, it is important to grasp the key characteristics that define them. These characteristics include the nature of mathematical relations, the role
of the vertical line test, and the concepts of one-to-one, onto, and many-to-one functions.
Explanation of what makes a mathematical relation a function
A mathematical relation is considered a function if each input value (x) corresponds to exactly one output value (y). In other words, for every x-value, there can only be one y-value. This means that
a function cannot have multiple y-values for a single x-value. If this condition is not met, the relation is not considered a function.
The role of the vertical line test in determining if a graph represents a function
The vertical line test is a visual tool used to determine if a graph represents a function. When applying the vertical line test, if a vertical line intersects the graph at more than one point, then
the graph does not represent a function. On the other hand, if every vertical line intersects the graph at most once, then the graph represents a function.
Clarification of one-to-one, onto, and many-to-one functions
One-to-one function: A function is considered one-to-one if each element in the domain maps to a unique element in the range, and each element in the range is mapped to by only one element in the
Onto function: An onto function, also known as a surjective function, is a function where every element in the range is mapped to by at least one element in the domain. In other words, the function
covers the entire range.
Many-to-one function: A many-to-one function is a function where multiple elements in the domain are mapped to the same element in the range. This means that the function is not one-to-one, as it
violates the condition of having a unique output for each input.
The Uniqueness of Y-Intercepts in Functions
When it comes to mathematical functions, the concept of y-intercepts plays a crucial role in understanding their behavior and properties. In this chapter, we will explore the uniqueness of
y-intercepts in functions, the rule that a function can only have one y-intercept, provide a mathematical proof demonstrating why functions cannot have more than one y-intercept, and use graphical
representation of functions to illustrate their y-intercepts.
A. The rule that a function can only have one y-intercept
According to the fundamental rule of mathematical functions, a function can only have one y-intercept. The y-intercept is the point at which the graph of the function intersects the y-axis. It
represents the value of the function when the input is zero. In other words, it is the point (0, b) where b is the y-intercept.
B. Mathematical proof demonstrating why functions cannot have more than one y-intercept
To understand why functions cannot have more than one y-intercept, we can consider the definition of a function. A function is a relation between a set of inputs (the domain) and a set of possible
outputs (the range), such that each input is related to exactly one output. If a function were to have more than one y-intercept, it would violate this fundamental definition, as there would be
multiple points on the graph where the function intersects the y-axis, each corresponding to a different y-value for the same input.
Mathematically, we can prove this by contradiction. Suppose a function f(x) has two distinct y-intercepts, (0, b1) and (0, b2), where b1 and b2 are not equal. This would imply that for x = 0, the
function f(x) takes on two different values, which contradicts the definition of a function. Therefore, it is impossible for a function to have more than one y-intercept.
C. Graphical representation of functions to illustrate their y-intercepts
Graphical representation provides a visual way to understand the concept of y-intercepts in functions. When we graph a function, the y-intercept is the point at which the graph crosses the y-axis. By
plotting various functions and identifying their y-intercepts, we can visually confirm the uniqueness of y-intercepts in functions.
For example, consider the linear function f(x) = 2x + 3. When we graph this function, we can see that it intersects the y-axis at the point (0, 3). This is the unique y-intercept for this function,
as expected. Similarly, for quadratic, cubic, and other types of functions, we can observe that each function has only one y-intercept, consistent with the fundamental rule of functions.
When Functions Seem to Have Multiple Y-Intercepts
When studying mathematical functions, it is important to understand the concept of the y-intercept, which is the point where the graph of a function crosses the y-axis. In most cases, a function will
have only one y-intercept, but there are scenarios where it may appear that a function has multiple y-intercepts.
A Discussion of scenarios where it appears that a function might have more than one y-intercept
One common scenario where it seems like a function has multiple y-intercepts is when the graph of the function intersects the y-axis at more than one point. This can happen when dealing with
non-functions such as circles or vertical lines.
Explanation of why these are not functions by definition
By definition, a function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In the case of a function having multiple
y-intercepts, it violates this definition because for a given x-value, there should only be one corresponding y-value. When a function has multiple y-intercepts, it fails to meet this criterion and
is therefore not a function.
Examples of non-functions such as circles and vertical lines
One classic example of a non-function is the equation of a circle, such as x^2 + y^2 = r^2. The graph of a circle intersects the y-axis at two points, resulting in the appearance of multiple
y-intercepts. However, since a circle fails the vertical line test, it is not a function.
Another example of a non-function is a vertical line, such as x = 3. The graph of a vertical line intersects the y-axis at a single point, but it extends infinitely in both the positive and negative
y-directions. This also violates the definition of a function, as it fails the vertical line test and is not a function.
Troubleshooting Common Misconceptions
When it comes to understanding mathematical functions, there are several common misconceptions that can lead to confusion, especially when it comes to identifying y-intercepts and determining whether
a graph represents a function or not. In this chapter, we will address these misconceptions and provide strategies for overcoming them.
A Addressing common errors in identifying functions and y-intercepts on graphs
One common error when identifying functions on a graph is mistaking non-functions for functions. This can happen when a graph fails the vertical line test, which states that if a vertical line
intersects a graph in more than one point, then the graph does not represent a function. It's important to emphasize to students that a function can only have one output (y-value) for each input
(x-value), and the vertical line test is a simple way to check for this.
Another common error is misunderstanding the concept of a y-intercept. Some students may mistakenly believe that a function can have more than one y-intercept. It's important to clarify that the
y-intercept is the point where the graph intersects the y-axis, and there can only be one such point for a given function. This misconception can be addressed by providing clear examples and
explanations of how to identify the y-intercept on a graph.
B How to correctly apply the vertical line test and identify y-intercepts
To help students overcome these misconceptions, it's important to provide clear instructions on how to correctly apply the vertical line test. This can be done by demonstrating the test on various
graphs and explaining why a graph fails the test if a vertical line intersects it in more than one point. Additionally, providing practice problems and exercises can help reinforce the concept.
When it comes to identifying y-intercepts, it's important to emphasize the significance of the y-intercept as the point where the graph crosses the y-axis. Providing step-by-step instructions on how
to identify the y-intercept, along with examples and real-world applications, can help students grasp this concept more effectively.
C Strategies for distinguishing functions from non-functions in complex graphs
Complex graphs can often lead to confusion when trying to determine whether they represent functions or not. To address this, it's important to provide strategies for distinguishing functions from
non-functions. This can include breaking down the graph into smaller sections, applying the vertical line test to each section, and analyzing the behavior of the graph in different regions.
Additionally, providing real-world examples of functions and non-functions can help students understand the practical implications of these concepts. By demonstrating how functions and non-functions
are used in various fields such as science, engineering, and economics, students can gain a deeper appreciation for the importance of understanding these mathematical principles.
Conclusion & Best Practices
A Recap of the main points: Functions and their unique y-intercepts
Understanding the uniqueness of y-intercepts in functions
Throughout this blog post, we have explored the concept of mathematical functions and their y-intercepts. We have learned that a function can have only one y-intercept, which is the point where the
graph of the function intersects the y-axis. This unique point is determined by the specific values of the function's variables and parameters.
Exploring the behavior of functions
We have also delved into the behavior of functions and how they can be represented graphically. By analyzing the graph of a function, we can gain insights into its y-intercept and understand how the
function behaves as its input values change.
Best practices for identifying and working with functions and y-intercepts
Use algebraic techniques to find y-intercepts
When working with functions, it is important to use algebraic techniques to find the y-intercept. By setting the input variable to zero and solving for the output variable, we can determine the
y-intercept of the function.
Graph functions to visualize y-intercepts
Graphing functions is a powerful tool for visualizing their behavior, including their y-intercepts. By plotting the function on a coordinate plane, we can easily identify the y-intercept and gain a
deeper understanding of the function's characteristics.
Verify uniqueness of y-intercepts
It is essential to verify that a function has only one y-intercept, as this property is fundamental to the nature of functions. By ensuring the uniqueness of the y-intercept, we can accurately
analyze and interpret the behavior of the function.
Encouragement for further study and practice in analyzing the behavior of mathematical functions
Continued exploration of functions and their properties
As we conclude, I encourage you to continue exploring the fascinating world of mathematical functions. By studying and practicing the analysis of functions, including their y-intercepts, you can
deepen your understanding of mathematical concepts and develop valuable problem-solving skills.
Utilize resources and seek guidance
Take advantage of educational resources, such as textbooks, online tutorials, and instructional videos, to further your knowledge of functions and y-intercepts. Additionally, don't hesitate to seek
guidance from teachers, tutors, or peers when encountering challenging concepts.
Apply concepts to real-world scenarios
Finally, consider applying the concepts of functions and y-intercepts to real-world scenarios. By connecting mathematical principles to practical situations, you can appreciate the relevance of these
concepts and enhance your analytical abilities. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-can-function-have-more-than-one-y-intercept","timestamp":"2024-11-15T00:28:41Z","content_type":"text/html","content_length":"225453","record_id":"<urn:uuid:456ff554-f6cb-4d8a-bcbb-69f9453bd27e>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00307.warc.gz"} |
MATH 26 – Analytic Geometry and Calculus I
Course Description
The ideas of calculus were prominent in the works of ancient Greek mathematicians. However, it was centralized further by Sir Isaac Newton and Gottfried Wilhelm Leibniz until becoming the backbone of
the technology and development that we see today. Most computational and analytical means of solving problems are entailed in calculus, ranging from bacterial behavior to manufacturing in industries
to space exploration. It is this fundamental idea of understanding how and why the world works, and how we can improve it.
In this course, we will establish your knowledge in the fundamentals of calculus. This will involve two mathematical operations: differentiation and integration. You will be introduced with the
underlying concepts and techniques, along with establishing how theory is applied to real-world scenarios. As much as this course will let you appreciate the analytical part in solving problems, you
will be introduced to some concepts of numerical methods that branchout from learning the basics.
Course Learning Outcomes
After completing this course, you should be able to:
1. Graph conic sections in a coordinate plane
2. Apply conic sections in real-world scenarios
3. Discuss theorems that govern differential and integral calculus
4. Solve real-world problems by applying calculus
Course Outline
I. Analytic Geometry and Conic Sections
A. Lines and Equations of Lines
B. Equation of a Circle with Center ?(0,0)
C. Equation of a Circle with Center ?(ℎ,?)
D. Equation of a Parabola with Vertex ?(0,0)
E. Translation of Axes
F. Equation of a Parabola with Vertex ?(ℎ,?)
G. Equation of an Ellipse with Center ?(0,0)
H. Equation of an Ellipse with Center ?(ℎ,?)
I. Equation of a Hyperbola with Center ?(0,0)
J. Equation of a Hyperbola with Center ?(ℎ,?)
K. Rotation of Axes
II. Limits and Continuity
A. Intuitive Concept of a Limit
B. Definition of a Limit
C. Theorems on Limits
D. One-Sided Limits
E. Infinite Limits
F. Limits at Infinity
G. Continuity of Functions
H. Limit Theorems involving Sine and Cosine
III. The Derivative
A. The Slope of the Tangent Line
B. Definition of the Derivative
C. Theorems on Differentiation
D. Differentiation of Trigonometric Functions
E. Differentiation of Power Functions
F. Higher-Order Derivative
G. Chain Rule
H. Implicit Differentiation
IV. Applications of Derivatives
A. Instantaneous Velocity and Acceleration
B. Finding the Extrema of a Function in an Interval
C. Graphs of a Function
D. Maxima and Minima
E. Related Rates
F. The Differential
V. Antidifferentiation
A. Definition of Antiderivative
B. Theorem and Formula of Antiderivative
C. Antiderivatives Involving Trigonometric Functions
D. Evaluation of Indefinite Integrals
E. Differential Equations with Separable Variables
F. Application of Antidifferentation to Rectilinear Motion
G. Area Concept and Calculation Area by Sum (Reimann Sum)
H. The Definite Integral
I. Evaluation of Definite Integral
J. The Area of a Region Bounded by Lines and/or Curves | {"url":"https://dmpcs.upmin.edu.ph/math-26-analytic-geometry-and-calculus-i/","timestamp":"2024-11-01T22:13:02Z","content_type":"text/html","content_length":"167977","record_id":"<urn:uuid:53c94893-a666-4cd6-b21d-7ce3087e2f4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00447.warc.gz"} |
Find the inverse of a matrix using determinant method
Guys what is determinant method? Is that the stand method where we find the inverse of matrix by
A^-1 = 1/|a| x adj(a) ?
Or elementary transformations? AI = A or A = IA ?
or what is that? please help me with this. I need to solve question papers 😀 | {"url":"https://www.crazyengineers.com/threads/find-the-inverse-of-a-matrix-using-determinant-method.62139","timestamp":"2024-11-10T06:37:25Z","content_type":"text/html","content_length":"67413","record_id":"<urn:uuid:10c1a733-7efc-4b9d-8dae-df0107a20268>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00106.warc.gz"} |
Conversion y
Conversion yards to feet, yd to ft.
The conversion factor is 3; so 1 yard = 3 feet. In other words, the value in yd multiply by 3 to get a value in ft. The calculator answers the questions:
110 yd is how many ft
? or
change yd to ft
. Convert yd to ft.
Conversion result:
1 yd = 3 ft
1 yard is 3 feet.
Choose other units (length)
Conversion of a length unit in word math problems and questions
more math problems » | {"url":"https://www.hackmath.net/en/calculator/conversion-of-length-units?unit1=yd&unit2=ft&dir=1","timestamp":"2024-11-12T19:18:35Z","content_type":"text/html","content_length":"25496","record_id":"<urn:uuid:02c10e38-7202-46cf-bb22-89db214f20b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00610.warc.gz"} |
Overcoming the partial volume effect
Overcoming the partial volume effect¶
Each metric extraction method provided by SCT accounts for the partial volume effect in different ways.
Binary mask-based methods¶
-method bin: Average within binary ROI¶
Because of its simplicity, the traditional method to quantify metrics is to use a binary mask: voxels labeled as “1” (i.e. in the mask) are selected and values within those voxels are averaged.
This method does not account for the partial volume effect whatsoever, and thus the resulting metric could be biased by the surrounding tissues, as demonstrated on the previous page.
-method wa: Weighted average¶
Instead, we could turn the binary masks into weighted masks by effectively “weighting” the contribution of voxels at the interface (e.g., mask value = 0.1) vs. voxels well within the tissue of
interest (e.g., mask value = 0.9).
This method is only useful if you have an existing binary mask for each region of interest. Also, this method only considers each mask in isolation, rather than considering the relations between
adjacent masks. So, while it would help to minimize the partial volume effect, it would not comprehensively solve the problem.
Atlas-based methods¶
Instead of using binary masks, we can use the white and gray matter atlas contained within the PAM50 template. In the atlas, each tract is represented using a nonbinary “soft” mask, with values
ranging from 0 to 1 at the edges of each tract label that capture the partial volume information. For more information on how the exact partial volume values were determined for each tract, see [Lévy
et al., Neuroimage 2015].
-method ml: Maximum Likelihood¶
The partial volume information from the atlas can be combined with Gaussian mixture modeling and maximum likelihood estimation to estimate the “true” value within the region of interest (e.g. a white
matter tract). This approach assumes that within each compartment, the metric is homogeneous.
-method map: Maximum A Posteriori¶
Because Maximum Likelihood estimation is sensitive to noise, especially in small tracts, we recommend using the Maximum a Posteriori method instead. This method adds a prior – specifically, the
maximum likelihood estimation computed within either the WM, GM, or CSF compartment of the image, depending on which area the ROI belongs to. (For example, if a metric is extracted for a specific WM
tract, the maximum likelihood for the WM as a whole will be used as a prior.)
The map method is the most robust to noise in small tracts. This was further validated using bootstrap simulations based on a synthetic MRI phantom. For more details, see [Lévy et al., Neuroimage
2015] (construction of the phantom, effect of noise, contrast) and [De Leener et al., Neuroimage 2017; Appendix] (effect of spatial resolution).
The methods bin and wa can be used with any binary mask. However, the methods ml and map require you to warp the white matter atlas to the coordinate space of your data, as is shown on the next page. | {"url":"https://spinalcordtoolbox.com/stable/user_section/tutorials/atlas-based-analysis/overcoming-the-partial-volume-effect.html","timestamp":"2024-11-12T02:51:16Z","content_type":"text/html","content_length":"63730","record_id":"<urn:uuid:9388be94-0db1-4d35-b41b-ad17fa4a6492>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00014.warc.gz"} |
The Sonic R System
Veteran forex trader Raghee Horner developed "The Wave" to determine what state the market was in - trending or non-trending, confused or bored. The wave is composed of three exponential moving
averages (EMA) which move more strongly with recent price action than other moving averages.
In other words, the EMA takes more account of recent prices than historical ones. The 3 EMAs used are the 34 period EMA of high prices, low prices and close prices. The choice of the 34 period is
based on the Fibonacci series.
An uptrend is determined as the angle of the 34 EMA wave being at a 12 - 2 o'clock angle. A downtrend is 4 - 6 o'clock angle and no trend is a 3 o' clock angle i.e. flat. Trading entry in a downtrend
occurs when the market rallies back to the Wave low (the bottom line of the Wave). In an uptrend, a long position is taken when the market retraces back to the Wave high (the upper line of the Wave)
see images below. This is essentially a pullback strategy.
Raghee has written a number of books, the most cited being "The Million Dollar Setup" which is available in our library of trading books. In essence, you want to buy closes above the 34 period EMA
high in an uptrend and sell the 34 EMA average close in a downtrend. Criticism of her strategies is that they are not specific and rely heavily on trader discretion which isn't helpful if you're a
new trader with little experience.
However, in 2008, a young man who calls himself the Sonicdeejay turned up on trading forums to publish his Sonic R System. At the heart of his trading system was Raghee Horner's Wave which he renamed
the Dragon. Sonicdeejay developed a swing strategy on the 15 minute timeframe which requires Price, Volume, Support & Resistance Analysis or PVSRA. He identified two trading setups - the Classic and
the Scout.
Simply, the Classic trading setup can be stated as: “Buy the first pull back from new high, sell the first pullback from new low”. The diagram below illustrates an example of classic setup.
But what about PVSRA? First you need to draw the Dragon being a 34 period EMA applied to the high, low and close prices. Add a 89 period EMA to determine trend - an upward cross of the 89 EMA by the
Dragon indicates an uptrend and vice versa. Next you want higher volume to support your trade entry. Lastly add support and resistance levels to determine trend reversal points.
Your chart will look like this:
The Scout is a counter trend setup and is more risky than the Classic setup. The author of the system believes that a Scout setup during a Classic setup is the safest application. The Scout is
essentially a re-entry of the Classic on a retracement or pull back. There are many strategies that are based on entry after a pullback to a moving average but these strategies rely heavily on trader
discretion. An example of a Scout is this:
If you'd like to find out more about the Sonic R System, search the forexfactory.com website where there is a ton of information from the author.
The Wave or the Dragon is a useful tool to determine trend because its construction takes account of recent price volatility. When volatility is high, the Dragon is wider and aims to prevent false
signals. Using the cross of the 89 EMA (also a Fibonacci number) by the 34 EMA to determine trend is a precise rule that can be coded and indicators created.
Other technical analysts have developed indicators where the defaults are Fibonacci numbers e.g. the Awesome Oscillator developed by Bill Williams Phd in the 1990's is based on the difference between
the 5 period simple moving average (SMA) and the 34 period SMA. The first few numbers in the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 …. The Golden Ratio is constant between
adjacent numbers e.g. 55/34 = 1.618, 89/55 = 1.618.
The golden ratio found in this sequence occurs over and over in nature. Almost everything runs according to the Fibonacci ratios which imbues these numbers with "magical" properties. This natural
mathematics is mimicked by traders in the markets. In technical analysis, Fibonacci Golden ratios and numbers are often used to determine trading periods, possible targets or levels of support/
resistance. I think the connection between nature and man made financial makets is tenuous but you can decide for yourself.
Closing thoughts on the Sonic R Dragon
Using an EMA to measure or assess trend is better than its lagging cousin the SMA because greater weight given to recent data so the EMA responds more quickly to recent price changes than the SMA.
This makes sense in that markets are not constant - they change in direction and volatility. The Wave or the Dragon attempts to account for volatility to keep the trader away from false signals when
determining trend.
Personally, I find it hard to believe that the forces in nature that gave rise to the Fibonacci sequence have somehow permeated the servers and terminals of the world's stock exchanges to give
traders an edge.
Perhaps there's an alternate explanation for the success of the 34 period moving average because its approximately the mid-point of a quarter. There are roughly 60 trading days in a quarter (5
trading days a week) at the end of which Institutional portfolios are reviewed & rebalanced. A 34 period moving average would therefore have sufficient price sensitivity to respond to massive
Institutional flows during rebalancing. But whether the best number is 33 or 35 for the EMA period is immaterial because the Dragon is a guide not a strict rule. Like all other indicators, it should
not be used in isolation but together with other indicators to confirm price action.
Download the Sonic Dragon indicator HERE
Download the Raghee Horner Wave HERE
Download "The Million Dollar Setup" by Raghee Horner HERE | {"url":"https://www.3candlereversal.com/post/history-of-the-sonic-r-system","timestamp":"2024-11-14T23:27:38Z","content_type":"text/html","content_length":"1050487","record_id":"<urn:uuid:26621ab8-ff0c-4505-b5d0-eb2c81c05f1b>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00058.warc.gz"} |
Clique transversal number is sublinear for graphs with no small cliques
Let \( \tau(G)\) denote the cardinality of a smallest set of vertices in \( G\) that shares some vertex with every clique of \( G\).
Problem [1]
Suppose all cliques in \(G\) have \(cn\) vertices. Is it true that \[ \tau(G) = o(n)? \]
1 P. Erdös. Problems and results on set systems and hypergraphs, Extremal problems for finite sets (Visegrád, 1991), Bolyai Soc. Math. Stud., 3, pp. 217-227, János Bolyai Math. Soc., Budapest, 1994. | {"url":"https://mathweb.ucsd.edu/~erdosproblems/erdos/newproblems/SublinearCliqueTransversal.html","timestamp":"2024-11-02T10:35:43Z","content_type":"text/html","content_length":"3911","record_id":"<urn:uuid:1b5fcfa7-2d0f-4595-b0bc-4aa28172726d>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00737.warc.gz"} |
Lesson 8
Speaking of Scaling
8.1: Going Backwards (5 minutes)
Students calculate a scale factor given the areas of the circular base of a cone and the base of its dilation. This connects the concept of surface area dilation to cross sections, and gives practice
with non-integer roots.
Monitor for pairs of students who initially consider an answer of 20.25 but come to a consensus that the scale factor is 4.5.
Arrange students in groups of 2. Provide access to scientific calculators. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even
if they are different. Follow with a whole-class discussion.
Student Facing
The image shows a cone that has a base with area \(36\pi\) square centimeters. The cone has been dilated using the top vertex as a center. The area of the dilated cone’s base is \(729\pi\) square
What was the scale factor of the dilation?
Anticipated Misconceptions
Some students may struggle to find the square root of 20.25. Remind students that their calculators can find square roots, and prompt them to use an estimate to check the reasonableness of the
calculator output.
Activity Synthesis
Select a pair of students to explain their reasoning. If the pair considered 20.25 but moved to an answer of 4.5, ask how they knew 20.25 wasn’t correct.
Discuss with students how they can decide if 4.5 is a reasonable value for the square root of 20.25, considering the fact that 4^2 = 16 and 5^2 = 25. If time allows, ask students to calculate the
scale factor for the solid’s volume (4.5^3 = 91.125).
8.2: Info Gap: Originals and Dilations (20 minutes)
This info gap activity gives students an opportunity to determine and request the information needed to infer characteristics of original and dilated solids based on one-, two-, and three-dimensional
scale factors.
The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of
discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the
information they need (MP6).
Monitor for pairs that complete Problem Card 2 by using the radius and the height of the dilated cylinder in the volume formula, and for other pairs that instead apply the cube of the scale factor to
the original cylinder's volume.
Here is the text of the cards for reference and planning:
Tell students they will continue to work with scale factors for dilated solids. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.
Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for
a second problem and instruct them to switch roles.
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to solve problems involving surface area and volume of solids. Display questions or
question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation
Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their
own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Student Facing
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the data card:
1. Silently read the information on your card.
2. Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your
3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
4. Read the problem card, and solve the problem independently.
5. Share the data card, and discuss your reasoning.
If your teacher gives you the problem card:
1. Silently read your card and think about what information you need to answer the question.
2. Ask your partner for the specific information that you need.
3. Explain to your partner how you are using the information to solve the problem.
4. When you have enough information, share the problem card with your partner, and solve the problem independently.
5. Read the data card, and discuss your reasoning.
Activity Synthesis
After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Select groups that found Problem Card 2’s answer using the volume
formula, and other groups that applied the cubed scale factor to the original cylinder’s volume.
Here are some questions for discussion:
• “What was the easiest part about this activity? What was the most difficult part?”
• “Was all the data on each card used? If not, which pieces of data weren’t used?”
• “How did you determine the scale factor for lengths in the first problem?”
• “How did you find the volume of the dilated cylinder on the second card? Did you use the volume formula, or did you use another method?”
Highlight for students the scale factors of \(k,k^2 ,\) and \(k^3\) for lengths, surface areas, and volumes respectively.
8.3: Jumbo Can (15 minutes)
Optional activity
In this activity, students are building skills that will help them in mathematical modeling (MP4). They recognize that a geometric solid can be a mathematical model of a real-life object, and have an
opportunity to consider the accuracy of that model. They’re prompted to connect surface area and volume to the real-life context of container materials and fill.
Ask students about their favorite sparkling water or juice. Tell students they’ll be playing the part of a beverage company that’s considering introducing a new product. Consider showing students
several different styles of beverage cans, including mini-sizes, tall and narrow cans, and standard cans.
Reading, Listening, Conversing: MLR6 Three Reads. Use this routine to support reading comprehension of this word problem. Use the first read to orient students to the situation. Ask students to
describe what the situation is about without using numbers (a beverage company wants to make a jumbo version of the can that is a dilated version of the original). Use the second read to identify
quantities and relationships. Ask students what can be counted or measured without focusing on the values (cost of materials for the original can, cost of juice for the original can, cost of
materials for the new can). After the third read, ask students to brainstorm possible solution strategies to answer the first question. This helps students connect the language in the word problem
and the reasoning needed to solve the problem.
Design Principle(s): Support sense-making
Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students
understand that the materials cost of the can is related to the can’s surface area and the fill cost is related to the can’s volume.
Supports accessibility for: Memory; Organization
Student Facing
A beverage company manufactures and fills juice cans. They spend $0.04 on materials for each can, and fill each can with $0.27 worth of juice.
The marketing team wants to make a jumbo version of the can that’s a dilated version of the original. They can spend at most $0.16 on materials for the new can. There’s no restriction on how much
they can spend on the juice to fill each can. The team wants to make the new can as large as possible given their budget.
1. By what factor will the height of the can increase? Explain your reasoning.
2. By what factor will the radius of the can increase? Explain your reasoning.
3. Create drawings of the original and jumbo cans.
4. What geometric solid do the cans resemble? What are some possible differences between the geometric solid and the actual can?
5. What will be the total cost for materials and juice fill for the jumbo can? Explain or show your reasoning.
6. Describe any other factors that might cause the total cost to be different from your answer.
Student Facing
Are you ready for more?
As of 2019, the Burj Khalifa, located in Dubai, was the tallest building in the world. Suppose a scale model of the Burj Khalifa (without antennae) is 30 inches tall.
1. To what scale is this model? You will need to use the internet or another resource to find the actual height of the building.
2. How tall would a model of the Eiffel tower be at this scale?
Anticipated Misconceptions
Some students may double the height of the can but not the radius in their drawings. Prompt them to verify that their dilated can has the same proportions as their original.
Some students may identify the scale factor as 2 or as 16. Remind them of the relationship between the scale factor for dimensions, \(k\), and the scale factor for surface areas, \(k^2\).
Activity Synthesis
The goal is for students to understand that the cylinder is an inexact mathematical model for the real-life can. The model can give insight into the real-world situation. Ask students to share their
thoughts on factors that affect the final cost. Invite them to consider whether the original proportions of the can matter (they don’t matter, because the scale factors are the same regardless of the
actual shape of the can).
Lesson Synthesis
The main idea of the lesson is that if we know the factor by which the volumes, surface areas, or lengths change when a solid is dilated, it’s possible to find the factor by which the remaining
values change. Here are some questions for discussion:
• “Suppose you know the volumes of an original solid and its dilation. How can you find the factor by which the surface area changed?” (Divide the dilated volume by the original volume to find the
factor by which the volume changed. Then take the cube root of that to find the scale factor of dilation. Finally, square that value to get the surface area scale factor.)
• “Suppose you know the surface areas of an original solid and its dilation. How can you find the factor by which the volume changed?” (Divide the dilated volume by the original volume to find the
factor by which the surface area changed. Then take the square root of that to find the scale factor of dilation. Finally, cube that value to get the surface area scale factor.)
• “What are some real-world applications for these concepts?” (One example is designing any kind of packaging including shampoo, cereal, and coffee—we often need to understand how a change in
volume for these products affects the packaging materials and product dimensions. Other examples include designing spaces that need a certain volume like car trunks, cargo containers, and tanker
trucks, and engineering objects that are expensive to paint, such as airplanes.)
8.4: Cool-down - Dog Food Bags (5 minutes)
Student Facing
Suppose a solid is dilated. If we know the factor by which the surface area or volume scale changed, we can work backwards to find the scale factor of dilation. Then we can use that information to
solve problems.
A company sells 10 inch by 10 inch by 14 inch 5-gallon aquariums, but a museum wants to buy a 135-gallon aquarium with the same shape. The company needs to know the dimensions of the new tank and by
what factor the surface area will change.
Gallons are a measure of volume. So, the volume of the tank increases by a factor of \(135\div 5=27\). To find the scale factor for the dimensions of the tank, calculate the cube root of 27, or 3.
This tells us that the height, length, and width of the tank will each be multiplied by 3. Next, we can square the scale factor of 3 to find that the tank’s surface area will increase by a factor of
3^2 = 9.
│ │original aquarium│ dilated aquarium │
│ height (inches) │10 │\(10\boldcdot 3=30\) │
│ length (inches) │14 │\(14\boldcdot 3=42\) │
│ width (inches) │10 │\(10\boldcdot 3=30\) │
│surface area (square inches) │760 │\(760\boldcdot 9=6,\!840\)│
│ volume (gallons) │5 │\(5\boldcdot 27=135\) │ | {"url":"https://curriculum.illustrativemathematics.org/HS/teachers/2/5/8/index.html","timestamp":"2024-11-03T06:04:18Z","content_type":"text/html","content_length":"108915","record_id":"<urn:uuid:bb7a0948-09e8-4ff9-90ab-1e6becfb753c>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00217.warc.gz"} |
Circular recursive definition
Hello people. I have two terms, namely 𝔗 and 𝔄, that are recursively defined, but then in their definition, each one depends on the other:
Now, when I do induction on 𝔗, I have to deal with 𝔄 at the end, and vice versa. Any ideas on how can I do inductions?
For reference, please take a look at this branch of my repo.
You can do mutually recursive definitions (though it's far from easy to make it work), here's an example: https://coq.inria.fr/distrib/current/refman/language/core/inductive.html#
Note that mutual recursion can be eliminated through indices.
You can typically write a type Inductive I (_ : bool) := ... where the index set to true encodes for T types and set to false for A types
Here is another example that is maybe closer to your use-case, namely the formulas and programs of propositional dynamic logic (PDL): https://github.com/coq-community/comp-dec-modal/blob/master/PDL/
If your syntax is mutually recursive, you will likely anyways want to simultaneously prove two statements, one for each of the two types. In the case where all you need are propositions, then
Combined Scheme should do the trick. If some of your "statements" actually have sort "Type" you may have to create the combined scheme manually as I did here.
I suppose that using the indexed approach, you will have theorem statements involving if/match clauses to separate the two types. This may or may not be what you want. In the end, it's probably a
matter of taste.
Mohammad-Ali A'RÂBI said:
For reference, please take a look at this branch of my repo.
I suggest using a more modern version of Coq than 8.4pl3. The difference between 8.4 and the current 8.12 release is quite significant. :fear:
The difference between 8.4 and 8.5 alone is already very significant.
@Pierre-Marie Pédrot I think the first version of Coq I used was 8.3, and I don't recall the major change from 8.4 to 8.5. Long time ago ...
Thanks a lot, people. I'm unable to install the latest CoqIDE on my Ubuntu 14.04, but I'll have to upgrade my OS eventually. :grinning_face_with_smiling_eyes:
The Scheme should do the trick, but I have to study it deeper. Still don't know how does it work. :thinking:
There are many alternatives to your system package manager to get a more recent version of Coq. The most common one is using opam: see https://coq.inria.fr/opam-using.html. But be aware that you
would need opam 2 which is not the version you can get through your package manager on an old Ubuntu. You can also get the latest version of Coq through Conda (https://conda-forge.org/) or Nix (
Getting coqIDE might add complications?
@Christian Doczkal IIRC Combined Scheme now works on Type as well, but surely not in 8.4
I managed to install opam 2.0.4 on my system, but CoqIDE 8.12.0 requires conf-gtk3 18, which also, in turn, requires gtk+-3.0 to be install, and as it seems, the latest one compatible to Ubuntu 14.04
is not sufficient.
Coq 8.12.0 itself is installed with no problem.
There are plugins to use Coq from Emacs, Vi, and vscode. The Emacs one is the most commonly used.
Another way is that CoqIDE is relatively independent of the actual Coq version
Finally, the latest coqide using gtk2 is much more recent than 8.4
(The “coqide is independent” part might be trickier to exploit, I’ve only read it on github)
Coq 8.10 was the first version using gtk3, according to the manual: https://coq.inria.fr/distrib/current/refman/changes.html#version-8-10
So 8.9 should be installable, and much closer to modern practice. (OTOH, 14.04 is probably outside its maintenance window)
So, I installed Ubuntu 20.04 together with Coq and -IDE 8.12.0 (through opam). Made the makefile through CoqIDE, and it works fine, but the IDE itself cannot load the modules, saying it cannot find
the .vos files (they are there, only empty).
Fixed the load-path problem. Going for the Schemes.
@Christian Doczkal I am still confused. I tried to imitate the combined schemes solution from here, but still don't know how to use them in the proof. The Coq's manual on schemes also has no examples
of using them in the proofs.
If you can take a look at the current state of my progress, I would really appreciate it:
See Lemma form_prog_dec :
Basically, what you can do is a simultaneous induction on your mutually inductive types T and U
and you phrase that as one lemma
you cannot use the induction tactic, but you can start by applying the combined induction scheme.
(users of form_prog_ind might be a better example, like https://github.com/coq-community/comp-dec-modal/blob/master/PDL/PDL_def.v#L152 )
to be able to apply the combined scheme, you should use About on it, identify its conclusion, and state the lemma so that it matches the conclusion
in your case, the conclusion will probably have a form similar to (forall t: T, P1 t) /\ (forall u: U, P2 u) — and you need to pick the right P1 and P2
(https://github.com/coq-community/comp-dec-modal/blob/master/PDL/PDL_def.v#L497 might be a slightly more typical example, tho I’m confused that it uses * and not /\, yet it uses the _ind principle)
Paolo Giarrusso said:
(https://github.com/coq-community/comp-dec-modal/blob/master/PDL/PDL_def.v#L497 might be a slightly more typical example, tho I’m confused that it uses * and not /\, yet it uses the _ind
Indeed, this is a bit obscure. The reason for phrasing the statement with a * is that this turns the statement into an ssreflect multirule for rewrite, even though this may not be used for this
particular lemma. As for why this works, the apply: tactic actually, which is based on the same algorithm as refine, inserts a coercion from "A /\ B" to "A * B". This can be seen by running:
Goal forall A B : Prop, A /\ B -> A * B.
Unset Printing Notations.
intros A B AB. refine AB. Print Graph.
Show Proof.
which yields fun (A B : Prop) (AB : and A B) => pair_of_and AB. :shrug:
Paolo Giarrusso said:
Christian Doczkal IIRC Combined Scheme now works on Type as well, but surely not in 8.4
I can confirm that Combined Scheme now works in Type at least in 8.12, let's see what CI says about older versions.
Replacing * with /\ and then just applying the induction rule resulting from the combined scheme solved it. Thanks a ton, guys.
Last updated: Oct 13 2024 at 01:02 UTC | {"url":"https://coq.gitlab.io/zulip-archive/stream/237977-Coq-users/topic/Circular.20recursive.20definition.html","timestamp":"2024-11-04T07:21:27Z","content_type":"text/html","content_length":"27152","record_id":"<urn:uuid:986423c5-ad32-43aa-ba72-b7fa43402087>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00734.warc.gz"} |
Calculate the occupied bandwidth of non-stationary signals
Answers (1)
Commented: William Rose on 2 Mar 2023
Is there a way to calculate the occupied bandwidth of non-stationary signals using wavelet toolbox or any other tool?
0 Comments
5 views (last 30 days)
Calculate the occupied bandwidth of non-stationary signals
Decide how you want to define and measure bandwidth. For example, you could define and measure it as the frequency range extending fro the spectral peak to -3dB on either side of the peak.
Use stft(x) to get the spectrum of x at successive times.
Then apply the bandwidth calculation to the spectrum at each time.
4 Comments
William Rose on 2 Mar 2023
"How does that sound to you?" It sounds like you don't want to use the standard STFT, and that is fine.
"you can only obtain this information with limited precision, and that precision is determined by the size of the window" Yes. It is the uncertainty principle.
You said that your goal is to find bandwidth as a function of time, where badwidth is defined as the frequency range that captures 99% of the (instantaneous) power. If you vary the window size, as
you have proposed, then the duration of "instantaneous" will change with time, and the time resolution of the bandwidth estimate will change with time.
1. You can write your own STFT routine in which the window size varies. How you compute the "appropriate" window size is not obvious to me.
2. You can run a standard STFT several times, or many times, with a different window size each time. The you select which STFT you want to refer to at different times, depending on some criterion
you devise.
3. You can do a wavelet anaysis and figure out how to use wavelets to determine the frequency that includes 99% of the "instantaneous" power. I don't know how to relate wavelets to frequency content
in a quantitative way, but maybe the paper which I referenced in a previous comment will be helpful. (Although I notice the word "qualitative" in the abstract...) | {"url":"https://nl.mathworks.com/matlabcentral/answers/1921135-calculate-the-occupied-bandwidth-of-non-stationary-signals?s_tid=prof_contriblnk","timestamp":"2024-11-11T17:07:06Z","content_type":"text/html","content_length":"130779","record_id":"<urn:uuid:a79e477a-6e2f-45c1-b614-78141da53f19>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00737.warc.gz"} |
Polygon (Meaning and Explanation)
We explain what a polygon is in geometry, the elements that make it up and what types exist. Also, how its measurements are calculated.
What is a polygon?
In geometry, a polygon is called a plane geometric figure, composed of a set of line segments connected in such a way that they enclose and delimit a region of the plane generally without crossing
one line with another. Its name comes from the Greek words poly (“a lot”) and gonos (“angle”), that is, in principle they are geometric figures with numerous angles, although today it is preferred to
classify them according to their number of sides and not angles.
Polygons are two-dimensional figures (flat equivalents of three-dimensional polytopes), that is, they have only two dimensions: length and width, and both are determined by the proportions of the
lines that compose them. The fundamental thing about a polygon is that the set of its lines separates a region of the plane from the rest, that is, it delimits an “inside” and an “outside”, given
that they are figures closed in themselves.
There are many types of polygons and many ways to understand them, depending on whether we are talking about Euclidean or non-Euclidean geometry, but are usually named depending on the number of
sides they have, using numerary prefixes. For example, a pentagon (penta + gonos) is a polygon that has five recognizable sides.
The rest of the polygons are named as follows:
Number of sides Polygon name
3 Trine or triangle
4 Tetragon or quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon or octagon
9 Nonagon or enneagon
10 Decagon
11 Endecagon or undecagon
12 Dodecagon
13 Tridecagon
14 Tetradecagon
15 Pentadecagon
16 Hexadecagon
17 Heptadecagon
18 Octodecagon or octadecagon
19 Nonadecagon or eneadecagon
20 Isodecagon or icosagon
21 Henicosagon
22 Doicosagon
23 Triaicosagon
24 Tetraicosagon
25 Pentaicosagon
30 Triacontagon
40 Tetracontagon
50 Pentacontagon
60 Hexacontagon
70 Heptacontagon
80 Octocontagon or octacontagon
90 Nonacontagon or eneacontagon
100 hexagon
1,000 Chiliágono or kiliágono
10,000 Myriagon
See also: Polyhedra
Elements of a polygon
Polygons are made up of a series of geometric elements.
Polygons are made up of a series of geometric elements to take into account:
• Sides. They are the line segments that make up the polygon, that is, the lines that trace it in the plane.
• Vertices. They are the meeting points, intersection or union of the sides of the polygon.
• Diagonals. They are straight lines that join two non-consecutive vertices within the polygon.
• Center. Present only in regular polygons, it is a point in its interior area that is equidistant from all its vertices and sides.
• interior angles. They are the angles that two of its sides or segments make up in the interior area of the polygon.
• Exterior angles. They are the angles that make up one of its sides or segments in the outer area of the polygon and the projection or continuation of another.
Types of polygons
Polygons are classified in different ways, depending on their specific shape. First of all, it is important to distinguish between regular and irregular polygons:
regular polygons. They are those whose sides and internal angles have the same measurement, being equal to each other. They are symmetrical figures, such as the equilateral triangle or the square.
Furthermore, regular polygons are at the same time:
• Equilateral polygons. They are those polygons whose sides always measure the same.
• equiangular polygons. They are those polygons whose internal angles always measure the same.
irregular polygons. They are those whose sides and internal angles are not equal to each other, since they have different measurements. For example, a scalene triangle.
On the other hand, polygons can be simple or complex, depending on whether their sides intersect or dry at some point:
• Simple polygons. They are those whose lines or sides never intersect or intersect, and therefore have a single contour.
• Complex polygons. They are those that present a crossing or intersection between two or more of their non-consecutive edges or sides.
Finally, we can distinguish between convex and concave polygons, depending on the general orientation of their shape:
• Convex polygons. They are those simple polygons whose internal angles never exceed 180° opening. They are characterized because any side can be contained within the figure.
• Concave polygons. They are those complex polygons whose internal angles exceed 180° opening. They are characterized because a line is capable of cutting the polygon at more than two different
Measurements of a polygon
Being a flat figure, which exists only in the two-dimensional plane (that is, length and width), but closed in itself, polygons contain a segment of the plane and delimit an outside and an inside.
Thanks to this, two types of measurements can be carried out:
The perimeter. It is the sum of the length of all the sides of the polygon, and in the case of regular polygons it is calculated by multiplying the length of its sides by the number of these.
The area. It is the portion of the plane delimited by the sides of the polygon, that is, its “interior” area. Its calculation, however, requires different procedures, for example:
• In a triangle, it is calculated by multiplying the base and the height and dividing by 2.
• In a regular quadrilateral (square), it is calculated by squaring the length of any of its sides.
• In a right quadrilateral (rectangle), it is calculated by multiplying its base by its height.
What plane figures are not polygons?
Not all plane figures are polygons. Those figures that do not close on themselves (that is, they do not have an interior area), that have curved lines in their formation or whose non-consecutive
sides intersect should not be considered as polygons.
Continue with: Cartesian plane
• “Polygon” in Wikipedia.
• “Etymology of Polygon” in the Online Spanish Etymological Dictionary.
• “What are polygons?” in the Junta de Andalucía (Spain).
• “The polygons” (video) in Smile and Learn.
• “Polygon (mathematics)” in The Encyclopaedia Britannica. | {"url":"https://meaningss.com/polygon/","timestamp":"2024-11-10T02:43:43Z","content_type":"text/html","content_length":"95378","record_id":"<urn:uuid:37a45341-d8fa-455f-9f84-264ce297a848>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00804.warc.gz"} |
$1,010 a year
Salary to hourly calculator
Paycheck calculator
A yearly salary of $1,010 is
$84 a month
. This number is based on
40 hours of work per week
and assuming it’s a full-time job (8 hours per day) with vacation time paid. If you get paid biweekly your
gross paycheck will be $39
To calculate annual salary to monthly salary we use this formula: Yearly salary / 12 months
Time Full Time
Monthly wage $1,010 yearly is $84 monthly
Biweekly wage $1,010 yearly is $39 biweekly
Weekly wage $1,010 yearly is $19 weekly
Daily Wage $1,010 yearly is $4 daily
Hourly Wage $1,010 yearly is $0.49 hourly
USA Salary to Hourly Calculator
Our salary to hourly calculator is the perfect tool to help you estimate your annual salary based on your hourly rate in the US.
It can be helpful when planning your budget, setting financial goals, or negotiating your salary with your employer. With our salary to hourly calculator, you can get an estimate of your earning
potential in just a few clicks. The popular related salaries are $1110, $1210, $1310, $1410, $1510, $1610, $1710, $1810, $1910, $2010.
Compare your income to the median salary in the US
The median wage per hour in the US is $15.35 in 2024. Your income is lower than the median hourly wage. | {"url":"https://salary-hourly.com/?monthly=1010","timestamp":"2024-11-02T02:18:18Z","content_type":"text/html","content_length":"12814","record_id":"<urn:uuid:d9720678-b268-423b-9489-fe554caae89b>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00561.warc.gz"} |
Example 4. If In=∫xna−xdx,
and evaluate ∫0a... | Filo
Question asked by Filo student
Example 4. If , and evaluate .
Not the question you're searching for?
+ Ask your question
Video solutions (1)
Learn from their 1-to-1 discussion with Filo tutors.
9 mins
Uploaded on: 1/18/2023
Was this solution helpful?
Found 4 tutors discussing this question
Discuss this question LIVE for FREE
11 mins ago
One destination to cover all your homework and assignment needs
Learn Practice Revision Succeed
Instant 1:1 help, 24x7
60, 000+ Expert tutors
Textbook solutions
Big idea maths, McGraw-Hill Education etc
Essay review
Get expert feedback on your essay
Schedule classes
High dosage tutoring from Dedicated 3 experts
Students who ask this question also asked
View more
Stuck on the question or explanation?
Connect with our Mathematics tutors online and get step by step solution of this question.
231 students are taking LIVE classes
Question Text Example 4. If , and evaluate .
Updated On Jan 18, 2023
Topic Calculus
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 135
Avg. Video Duration 9 min | {"url":"https://askfilo.com/user-question-answers-mathematics/example-4-if-2-n-3-i_-n-2-a-n-i_-r-and-evaluate-33383539383839","timestamp":"2024-11-08T09:04:04Z","content_type":"text/html","content_length":"273464","record_id":"<urn:uuid:ac311f63-eaf9-415b-9c97-3b94c9ab4b3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00080.warc.gz"} |
Advanced Algebra Assignment Help | Accurate Solutions
1. Advanced Algebra Assignment Help
Hire Our Experts Today for Customized Advanced Algebra Assignment Solutions
Elevate your advanced algebra experience by enlisting our experts to craft tailored assignment solutions just for you. We understand that every student's needs are unique, which is why we offer
personalized assistance. Our dedicated team will work closely with you to address your specific challenges and requirements. With our customized advanced algebra assignment solutions, you'll not only
conquer your assignments but also gain a deeper understanding of the subject. Don't settle for one-size-fits-all solutions; hire our experts today for an educational experience tailored to your
Comprehensive Advanced Algebra Assignment Assistance for All Topics
Our comprehensive advanced algebra assignment assistance ensures that students receive expert guidance and solutions on a wide range of challenging topics. From linear equations to differential
equations, our team of experts breaks down complex concepts into clear, step-by-step explanations. We prioritize understanding and skill development, providing students with the tools they need to
confidently tackle assignments in areas like polynomials, rational functions, complex numbers, vectors, matrices, linear transformations, and differential equations. With our assistance, students can
master advanced algebra and excel in their coursework.
Topic Description
Linear Equations Our experts provide step-by-step solutions to linear equations, ensuring students grasp the fundamental concepts while showcasing problem-solving techniques. We clarify the
underlying principles, helping students develop a deep understanding.
Polynomials We guide students through polynomial assignments, elucidating polynomial properties, factoring methods, and polynomial manipulation. Our solutions break down complex polynomial
problems into manageable steps for better comprehension.
Rational Our comprehensive solutions for rational functions assignments elucidate the concept of rational functions, covering aspects like simplification, domain, and asymptotes. We equip
Functions students with problem-solving strategies for these challenging functions.
Complex Numbers We demystify complex numbers through detailed assignment solutions, exploring their properties, arithmetic operations, and applications. Our explanations help students navigate
complex number problems confidently.
Vectors For vector assignments, we provide clear explanations of vector operations, including addition, subtraction, dot products, and cross products. We demonstrate vector properties and
applications, enhancing students' vector calculus skills.
Matrices Our matrix assignment solutions delve into matrix operations, determinants, and applications in various fields. We help students understand matrix transformations and their
significance in linear algebra.
Linear We elucidate linear transformations, including their properties and geometric interpretations, through assignment solutions. Our explanations empower students to grasp the concepts
Transformations and apply them to practical problems.
Differential We tackle differential equation assignments methodically, covering various solution techniques, such as separation of variables, integrating factors, and Laplace transforms. We
Equations emphasize real-world applications to enhance understanding. | {"url":"https://www.mathsassignmenthelp.com/advanced-algebra-assignment-help/","timestamp":"2024-11-02T03:04:32Z","content_type":"text/html","content_length":"116175","record_id":"<urn:uuid:ae2cdbf7-2ff9-439d-a41a-94309f8c9614>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00688.warc.gz"} |
redAnTS 1 - Numerical Solution
Numerical Solution
The only solution option available in redAnTS is static deformation. Advanced FEA packages will offer solution options such as modal, buckling, etc. Select Static under the Solver menu. This
assembles and solves the global matrix. Since this is a small problem with less than 100 elements, this takes very little time. Bigger problems will take much longer for the computer to crunch
through. Verify that under Current Settings, the software reports Displacements done. Let's take a look at the nodal displacement values to check that they look plausible.
Nodal Displacements
Under Plotting, select Displacement. The nodal displacements are shown below.
(Click for enlarged image)
Does the displacement field satisfy the imposed boundary conditions shown in this figure? Are the largest displacements where you expect them to be? Are the normal displacements zero for the sides
for which this condition is imposed?
Deformed Mesh
Let's take a peek at how the elements have deformed under the applied uniaxial tension. Under Plotting, select Deformed mesh. Since the deformations are usually very small, the program asks for a
maginification factor to make them more visible in the plot. Enter 500 for the magnification factor and click OK. You can move the legend out of the way by dragging it with the mouse.
We see that elements have been stretched in the x' direction which is the direction along which the tension has been applied. The elements have also shrunk in the y' direction. This behavior is as
To save a copy of the plot, click on Plot in the Export menu. Save the plot as deformed_mesh.fig. To convert from MATLAB's fig format to a more portable format, say, jpeg, open deformed_mesh.fig in
MATLAB as follows: at the MATLAB prompt, type open('deformed_mesh.fig'). This will display the plot in a MATLAB window. You should always include a plot legend to tell the reader what the different
lines correspond to. Take a few seconds to get frustrated that MATLAB loses the legend (aarrgggh!!bang!!bang!!). Clever engineers that we are, we'll add the legend back manually. Go to the MATLAB
command prompt and type in the appropriate legend command: legend('Mesh','Deformed mesh (500x)'). You can move the legend around by dragging it. In the figure window, select File -> Save as. Under
Save as type, select JPEG image and click Save. Open the jpeg image from your working folder to verify that it has been created properly.
To get an idea of the information available in the help page for the Solver menu, click on Help for this menu and scan through it. Click OK.
Since the nodal displacements look plausible, let's prod the beast to display various stress and strain components i.e. post-process the results. This will occupy us in step 6. Note that the student
version doesn't come with the post-processing module. You'll have to wait until you have developed this module before you can go through step 6. When you are finished with your post-processing
module, return to step 6. This will also help you validate your module.
Go to Step 6: Numerical Results | {"url":"https://confluence.cornell.edu/display/SIMULATION/redAnTS+1+-+Numerical+Solution","timestamp":"2024-11-05T06:46:29Z","content_type":"text/html","content_length":"65063","record_id":"<urn:uuid:663775be-983e-4874-9385-d0fb1eaafd26>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00893.warc.gz"} |
This blog is about Logic, Alchemy and the relationship between Logic and Alchemy.
Classic Logic is based on the Square of Opposition of Aristotle. This square was discussed for more than 2000 years. Recently scientists detected that the Square is really a Cube. It misses an extra
The Square describes the six oppositions (“dual combinations”) of possibility (“some”, Particular Affirmative), necessity (“all”, Universal Affirmative), not-possibility (“not some”, Particular
Negative) and not-necessity (“not all”, Universal Negative).
Geometric Logic is a part of Modal Logic which is a part of Logic.
In Geometric Logic the Linguistic Representations of Logic are mapped to the Graphical Representations of Geometry.
Modal Logic works with the notion that propositions can be mapped to sets of possible worlds.
The idea of possible worlds is attributed to the famous mathematician Gottfried Leibniz (1646-1716), who spoke of possible worlds as ideas in the mind of God and argued that our actually created
world must be the best of all possible worlds. Possible worlds play an important role in the explanation of Quantum Mechanics.
Alchemy is based on the Square of the Four Elements. The Four Elements are oppositions. The oppositions of Alchemy are related to the oppositions of the Square of Opposition of Aristotle in an
orthogonal model. They form an arc of 90 degrees, which makes Alchemy the complex part (imaginary) of Logic.
In his work De Arte Combinatoria Leibniz developed a general theory of science that was based on a fusion between Alchemy and Logic.
In this blog I will explain Alchemy and Logic and show that Alchemy and Geometric Logic share the same geometry, the Hexagon (the Seal of Solomon), which is a 2D-mapping of the Cube of Space of the
Sefer Yetsirah.
About Alchemy
When thou hast made the quadrangle round,
Then is all the secret found.
George Ripley (d. 1490).
One of the most complicated ancient architectures is the architecture behind Alchemy. The architecture of Alchemy is full of strange concepts and allegories.
Alchemy comes from the Arabic al-khemia meaning the Black Soil of Egypt (or the ((Egyptian) Black Art). Alchemy is the art of transformation.
Alchemy is an ancient science that was practiced all over the world. The most famous teacher was called Hermes Trismegistus in Greece and Thoth in Egypt. The old science of Hermes came back in the
The essence of Alchemy is called REBIS (Res Bina), Double Matter. REBIS is the end product of the alchemical magnum opus or Great Work. It is the Fusion of Spirit and Matter (the Body). Spirit is the
Creative Part of the Human.
Double matter is sometimes described as the divine hermaphrodite, a being of both male and female qualities as indicated by the two heads within a single body.
The Great Work is the chemical and personal quest to create Double Matter, the Philosopher’s Stone, Spirit in Matter.
The Stone is the agent of chemical transmutation, and the key ingredient in the creation of the elixir of life, said to heal all diseases, induce longevity and even immortality.
In current Physics transmutation is proved at room temperature in a proces called Cold Fusion.
In Physics one explanation for the fusion between Quantum Mechanics and Gravity is called the Wheeler Feynman Absorber Theory. In this theory the Future causes the Past and vice-versa. We are always
in the Now, the Middle between Past & Future.
The Future and the Past are created by the Intention of the Observer, we call Measurement.
The aim of the Great Work is a fusion with the One (the Blazing Star, Saturn, Point) that is divided in the Two (Sun & Moon, Male & Female, Line) and the Four Planets (Square), the 12 (4×4-4)
Constellations or more structures based on a power of a power (of a power….) of 2. The Great Work moves from 2**N, …., Sixteen to Four to Two to One (or from one to two to four) in a Spiraling
Spiral Motion.
The One, to which the elements must be reduced, is a little circle in the center of the squared figure (the cross). The cross is the mediator, making peace between the opposition of the planets.
The winged dragon represents First Matter (Quintessence, The Fifth Element) and suggests ascension, a merging of matter (body) and spirit (Creativity). Creativity (Enthusiasm, Spirit, Inspiration,
Imagination) is the engine behind the Spiral Wheel.
The First Matter is the primordial chaos comparable to what we now call the vacuum, the state of lowest energy. This state contains all possibilities.
It can be looked on as an unorganized state of energy that is the same for all substances and exists in an invisible state between energy and matter.
The First Matter is the One, the Point in the Middle of the Circle. In current physics a point is called a singularity or a black hole.
Alchemy is about Recombination
Alchemy is about re-combination summarized in the Latin expression Solve (break down in separate elements) et Coagula (coming back together (coagulating) in a new, higher form).
Earth (Body) becomes Water (Spirit) by dissolving it in some solvent. Water becomes Air (a moist vapor) by boiling, which further heating turns into Fire (a dry vapor).
Finally Fire becomes Earth by allowing the vapors to condense on a solid material (a stone).
The circulation may repeat if it is done in a reflux condenser, such as the Pelican or kerotakis.
The Pelican is the Symbol of the Heart Chakra the connector between the Upper- and the Lower Triangle of the Body.
According to the early alchemists, the four elements—fire, water, air, and earth—come into existence via the combination of specific qualities, recognized as hot, cold, wet, and dry, being impressed
on to the Prime Matter. For example, when hot and dry are impressed on the Prime Matter, we have fire.
(Hot and Cold) and (Wet and Dry) are oppositions. Their fusion is the empty set.
(Hot or Cold) and (Wet or Dry) can be combined -> (Hot, Wet), (Hot, Dry), etc.
The Cold allows substances to get together. The Hot is the power of separation.
Wet things tend to be Flexible (Fluent, Movement, Self, Not-Agency) whereas Dry things are fixed and structured (Agency, Resistance, not-Movement).
When the qualities are changed, the elements themselves are changed. Adding Water to Fire substituting wet for dry; hot and dry becomes hot and wet: steam, or Air.
As you can see in de picture below Hot & Cold and Wet and Dry are Binary Opposites (“lines”) that are voided by the central cross, the point of the Empty Set, The Void (A and not-A -> Empty).
Hot & Dry = Fire, Hot & Wet = Air, Cold & Wet = Water and Cold & Dry = Earth.
There are four types of Trinities the Passion-trinity (Dry, Wet, Warm), the Structure-trinity (Dry, Wet, Cold), the Resistance-trinity (Warm, Cold, Dry) and the Movement-trinity (Warm, Cold, Wet).
The Square rotates. The wheel of the Square is driven by the four qualities. Wet on the rising side, Hot on the top, Dry on the descending side, and Cold on the bottom.
The start is powerless (Cold). By becoming more flexible (Wet) success and power comes (Heat). At the top Rigidity (Dry) undermines the system and it falls back to a powerless state.
The four Elements all share the Triangle of the Holy Trinity. Every Trinity is seen from another perspective.
The two triangels, The Up-triangel and the Down-Triangel, are Opposites that are created out of two triangels that contain two Opposites that are fused. Humans are opposites in opposition.
Humans are FourFold
Humans are part of the Bilateria, animals with bilateral symmetry.
Humans can be described as a fusion of two mirrored bilateral triangles, the Up-triangel related to the Top of the Body (Mind) and the Bottom-triangel of the lower Body both connected by the Heart.
The two triangels and the separate triangels have to be balanced.
The principle of Balancing the two Triangles of Top & Bottom of the Body is represented by the Egyptian Goddess Ma’at who weights the Heart (the connector of Up and Down) of the Dead with her Feather
As you can see the Square of the Cycle of the Four Elements is visible in the picture below. The total picture shows a Hexagram with a Center (The Cross).
The points that are missing in the Square are the top- and the bottom of the two triangels which are the Whole (Up, Heaven) and the not-Whole, Emptiness, the Bottom (Hell).
A hexagon is a 2D-projection of a Cube.
The hexagon of Solomon can be transformed into the Cube of Space of the Sefer Yetsirah.
The Cube of Space can be transformed into the Tree of Life. The Tree of Life also called the tree of knowledge or world tree connects heaven and underworld and all forms of creation and is portrayed
in various religions and philosophies all over the world.
The Spiral of Alchemy is about the Chemical Transformation of the Body by the Conceptual Transformation of the Mind.
The Tree of Life shows the levels/scales (three) and possible paths of the Opus Magnum, the alchemical journey.
About Truths for a Fact and Truths for a Reason
People have always believed in the fundamental character of binary oppositions like Hot & Cold and Wet & Dry are used in Alchemy.
In this part we move to the Field of Logic. In this case Hot/Cold and Wet/Dry transform into Necessary /Not-Necessary and Possible/not-Possible.
Oppositions can be divided in:
1. Digital oppositions, contradictories, contain mutually exclusive terms (Gender (Male/Female)).
2. Analogue oppositions, antonyms or contraries: contain terms that are ordered on the same dimension (Temperature (Minimum->Cold/maximum->Hot)).
The doctrine of the square of opposition originated with Aristotle in the fourth century BC and has occurred in logic texts ever since.
The logical square is the result of two questions: Can two things be false together? Can two things be true together?
This gives 4 possibilities: no-no: contradiction (A/O, E/I); no-yes: subcontrariety (I/O); yes-no: contrariety (A/E); yes-yes: subalternation (A/I, E/O) better known as implication (A->B).
The four corners of the square represent the four basic forms of propositions recognized in classical logic:
1. A propositions, or universal affirmatives take the form: All S are P.
2. E propositions, or universal negations take the form: No S are P.
3. I propositions, or particular affirmatives take the form: Some S are P.
4. O propositions, or particular negations take the form: Some S are not P.
The square of opposition was debated for many reasons for more than two thousand years.
One of the discussions is about the difference between possible and necessary. In the so called Master Argument Diodorus proved that the future is as certain and defined as the past. The essence of
logic (necessaty) implies the non-existence of freedom (possibility). X is possible if and only if X is necessary.
The term ‘possible’, in Aristotle’s view, is ambiguous. It has two senses known as one-sided possibility and two-sided possibility (or contingency). Being two-sided possible means being neither
impossible nor necessary, and being one-sided possible simply means being not impossible.
Leibniz distinguished between necessary Truths for a Reason, which are true for a reason—i.e, their opposite is a contradiction—and contingent truths, (Truth for a Fact) such as the fact that the
president of France is François Hollande. A contingent truth cannot be proved logically or mathematically; it is accidental, or historical (based on facts (events)).
About Geometric Logic and n-dimensional “Squares” of Opposition
Modal Logic is about the fusion of necessity and possibility, contingency.
In Modal Logic the propositions are modelled in a logical hexagon where:
1. A is interpreted as necessity: the two propositions must be either simultaneously true or simultaneously false. A is a Proof or Law.
2. E is interpreted as impossibility. E is an Observation that contradicts the Proof of A.
3. I is interpreted as possibility: the truth of the propositions depends on the system of logic being considered . I is an Idea implied by the Proof that Contradicts E.
4. O is interpreted as ‘not necessarily’. O contradicts A.
5. U is interpreted as non-contingency: neither logically necessary nor logically impossible, its truth or falsity can be established only by sensory observation.
6. Y is interpreted as contingency: propositions that are neither true under every possible valuation (i.e. tautologies) nor false under every possible valuation (i.e. contradictions). Their truth
depends on the truth of the facts that are part of the proposition. Y is a possible proven theory.
The Logical Hexagon has the same geometry as the Seal of Solomon. It contains the Square of Opposition (A,E,I,O).
The Logical Hexagon can be transformed just like the Seal of Solomon to the Cube of Space now called The Cube of Opposition.
The Logical Hexagon is used in many fields of Sciences like Musical Theory or Scientific Discovery.
Every opposite can be defined by a string of zero’s and one’s like 1010. An opposite is 0101. This means that there are 2**4/2 = 4×4/2= 16 / 2 = 8 possible opposites. This is called the Octagon of
The Octagon contains six reachable and two unreachable points in the Center (0000, 1111), the Empty Set (the Void, the Hole, Contradiction) or its Opposite (The Whole, Tautology).
It is clear that it must be possible to create higher dimensional models based on n-based-logics.
When n becomes very big the geometry will tend to a n-Dim circle in which every point is in opposition to the point at the other side.
The big problem with n-based logics is language. We don’t have the names to articulate the many grades of opposition that are possible.
The other problem is that a negation is not a symmetric operation (A = not-not-A) but a rotation with an angle 360/n.
The n-opposite-geometries are highly related to Simple Non-Abelian Groups (SNAGs). They play an important role in biology.
The last step is to transform the static n-opposite geometries by making n very large (infinite) to a dynamic “opposition field”.
Related Models
In current Psychology Hot & Cold are called Communion and Moist & Dry, Agency. Together they create the so called Interpersonal Circumplex.
The four stages of Learning by Jean Piaget.
The mathematical model behind the theory of Piaget is called the Klein Four Group or Identity/Negation/Reciprocity/Correlation-model.
In the Science of Ecology (Panarchy) Agency & Communion are called Connectedness and Potential. The Panarchy model looks like a Mobius Ring.
A model that describes the Four Perspectives on Security:
The Semiotic Square of Greimas:
The Chinese Sheng-Cycle: The Chinese Five Element Model is comparable to the Western Four Element model. It contains as a Fifth Element, the Whole-Part-relationship (Earth, Observer State). Emptiness
(Nothing, the Tao) is represented by the recursive Pentangle inside.
The lesson
“When thou hast made the quadrangle round, then is all the secret found“.
When You have moved once through the Cycle you have seen Everything there is to See. The only way to move out of the Cycle is to jump into the wHole in the Middle.
The secret: It is impossible to move though the center because the paths that go through the center are a contradiction or a tautology (I am what I am, JHWH).
We have to move with the Cycle (making the rectangle round (a circle)), around the Singularity (the Hole of the Whole), going Up and going Down all the time.
The only way to solve this problem is to join the opposites by accepting only one thruth-value, by:
1. Accepting everything (Wu Wei),
2. Denying everything (Stoics) or
3. Diminishing the amount of dimensions (“becoming Simple”, “like Children”) by moving up in abstraction (Hot & Cold -> Temperature -> Everything is Energy) to the level of zero dimensions, a point,
becoming One with the First Matter (Tao, Vacuum, The Kingdom).
“If we then become children, would we thus enter the kingdom?” Jesus said unto them, “When ye make the two one, and when you make the inside like unto the outside and the outside like unto the
inside, and that which is above like unto that which is below, and when ye make the male and the female one and the same, so that the male no longer be male nor the female female; then will ye enter
into the kingdom.” (Gospel of Thomas, Logion 22).
About the History of Cycles (in Dutch)
About the Geometry of Negation
How to use the Square of Opposition as a Research Tool
About the Cube of Space of Sefer Yetzirah
About Truth of Reason and Truth of Fact
About Chemical Transformation (Cold Fusion)
About the Ars Generalis Ultima of Ramon Lull (1305)
About the Six Domains of the Polynomic System of Value
About Leibniz Calculating Machine
Why Innovation is Re-Combination
How to Balance the Seal of Solomon
About the Hexagon of Opposition
Anti-Fragility and the Square of Opposites
An Introduction to Hexagonal Geometry | {"url":"http://hans.wyrdweb.eu/category/physics/","timestamp":"2024-11-07T10:22:33Z","content_type":"application/xhtml+xml","content_length":"236217","record_id":"<urn:uuid:82a2418f-b436-4e22-b56c-f102fa9cf4d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00419.warc.gz"} |
Fredholm integral equation
Re-searches upon an integral equation exempli-fying the use of a general method due to. Fredholm (Ark f matem, astr o fys 34), A course of integrational and
Integral Equation Characteristic Function Fredholm Determinant Chapter Versus Tile Zero. These keywords were added by machine and not by the authors.
(i) If the function , , then "ˇ becomes simply $% - ". and this equation is called Fredholm integral equation of the second kind. (ii) If the function , … The Fredholm Integral Operator, denoted by
K, is de ned as on functions f2C([a;b]) as Kf:= Z b a k(x;y)f(y)dy where k is an F.I.E. kernel.
Solve an Initial Value Problem Using a Green's Function. Solve a Boundary Value Problem Using a Green's Function. Solve the Wave Equation Using Its Fundamental Solution. Fredholm integral equation is
one of the most important integral equations. Integral equations can be viewed as equations which are results of transformation of points in a given vector spaces of integrable functions by the use
of certain specific integral operators to points in the same space.
Forsgård Fredholm, Daniel: Intensional aspects of function definitions. Serafimovski integral equation of a known type. For a periodic structure of the boundary impedance, this equation can in
special cases be written as a Fredholm equation.
One reason is the fact that boundary integral operators generally are neither analysis of Fredholm integral equations of the second kind are not applicable.
About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators Enjoy the videos and music you
love, upload original content, and share it all with friends, family, and the world on YouTube. En mathématiques, l'équation intégrale de Fredholm est une équation intégrale étudiée par Ivar
Fredholm. La caractéristique principale d'une équation de Fredholm est que les bornes d'intégration sont constantes. Son étude donne naissance à la théorie de Fredholm (en), à l'étude des noyaux de
Fredholm (en) et des opérateurs de Solving Fredholm Integral Equations of the Second Kind in Matlab K. E. Atkinson Dept of Mathematics University of Iowa L. F. Shampiney Dept of Mathematics Southern
Methodist University May 5, 2007 Abstract We present here the algorithms and user interface of a Matlab pro-gram, Fie, that solves numerically Fredholm integral equations of the 1.
Results in this paper include application of the weighted mean-value theorem for integrals
Solve the Tautochrone Problem.
The current study suggests a collocation method for the mixed Volterra - Fredholm integral Fredholm integral equations with kernels of type (1.1) were studied in other contexts already during the
1940's by Chandrasekhar [7], and in fact the two functions mentioned above are usually called Chandrasekhar's X- and Y- functions. Now by choosing oJ to be Fredholm investigated the above equation by
discretizing the equation and appeal to linear algebra, and then taking a limit at the end. This give rises to a number called the Fredholm determinant of 1 + K (we simply say the Fredholm
determinant for K), which determines whether the given integral equation is solvable or not. The determinant Fredholm integral equations of the second kind with . a weakly singular kernel and the
corresponding eigenvalue problem.
Folktandvård mölndal sjukhus
I tried to find resolvent kernel of Volterra integral equation by taking kernel as 1.
The exact solution for constant b discussed above was obtained by applying the standard technique to reduce an equation of this kind to a differential equation. About Press Copyright Contact us
Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators Enjoy the videos and music you love, upload original content, and
share it all with friends, family, and the world on YouTube. En mathématiques, l'équation intégrale de Fredholm est une équation intégrale étudiée par Ivar Fredholm.
Polhemskolan biblioteket
civilingenjör teknisk fysik och elektroteknik lönny chef pizza binghamtoncc leasing konkurssveriges ingenjorer langrand ole bbq
The Laplace transform happens to be a Fredholm integral equation of the 1st kind with kernel K(s;x) = e¡sx. 3.1.1 Inverse The inverse Laplace transform involves complex integration, so tables of
transform pairs are normally used to flnd both the Laplace transform of a function and its inverse.
The optional output argument cond is an inexpensive estimate of the condition of A computed with a built-in function. It provides a Fredholm integral equations of the first kind. These are equations
of the form. (Aϕ)(s) = ∫ DK(x, s)ϕ(s)ds = f(x). They are usually ill-posed in the sense that their solution might not exist, not be unique, and will (if it exists) in general depend on f in a
discontinuous way (see Ill-posed problems ). | {"url":"https://hurmanblirrikutpb.firebaseapp.com/19882/70216.html","timestamp":"2024-11-12T00:21:30Z","content_type":"text/html","content_length":"10418","record_id":"<urn:uuid:fc327194-1e49-4086-9366-1bfd57e34ab3>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00399.warc.gz"} |
How do you identify all asymptotes or holes for g(x)=(x^2-6x+8)/(x+2)? | HIX Tutor
How do you identify all asymptotes or holes for #g(x)=(x^2-6x+8)/(x+2)#?
Answer 1
$g \left(x\right)$ has a vertical asymptote $x = - 2$ and an oblique (slant) asymptote $y = x - 8$
#g(x) = (x^2-6x+8)/(x+2)#
Divide the numerator by the denominator by grouping:
#g(x) = (x^2-6x+8)/(x+2)#
#color(white)(g(x)) = (x^2+2x-8x-16+24)/(x+2)#
#color(white)(g(x)) = (x(x+2)-8(x+2)+24)/(x+2)#
#color(white)(g(x)) = ((x-8)(x+2)+24)/(x+2)#
#color(white)(g(x)) = x-8+24/(x+2)#
From this alternative expression we can see that #g(x)# has a vertical asymptote at #x=-2# and an oblique (a.k.a. slant) asymptote #y = x-8#.
graph{(y-(x^2-6x+8)/(x+2))(y-x+8) = 0 [-79.6, 80.4, -45.64, 34.36]}
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 2
To identify asymptotes or holes for ( g(x) = \frac{{x^2 - 6x + 8}}{{x + 2}} ), we need to analyze its behavior as ( x ) approaches certain values.
1. Vertical Asymptote:
□ A vertical asymptote occurs where the denominator of the rational function equals zero but the numerator does not.
□ Set the denominator ( x + 2 ) equal to zero and solve for ( x ).
□ ( x + 2 = 0 ) gives ( x = -2 ).
□ Therefore, there is a vertical asymptote at ( x = -2 ).
2. Horizontal Asymptote:
□ To find horizontal asymptotes, examine the behavior of the function as ( x ) approaches positive or negative infinity.
□ If the degrees of the numerator and denominator are the same, the horizontal asymptote occurs at the ratio of the leading coefficients.
□ Since both the numerator and denominator are of degree 1, the horizontal asymptote is given by the ratio of the leading coefficients, which is ( \frac{1}{1} = 1 ).
□ Therefore, there is a horizontal asymptote at ( y = 1 ).
3. Hole:
□ If factors in the numerator and the denominator cancel out, there's a hole in the graph.
□ Factorize the numerator ( x^2 - 6x + 8 ).
□ ( x^2 - 6x + 8 = (x - 4)(x - 2) ).
□ There's no factor of ( x + 2 ) in the numerator, so there's no cancellation.
□ Therefore, there are no holes in the graph of ( g(x) ).
In summary, the rational function ( g(x) = \frac{{x^2 - 6x + 8}}{{x + 2}} ) has a vertical asymptote at ( x = -2 ) and a horizontal asymptote at ( y = 1 ), but it does not have any holes.
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
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
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-identify-all-asymptotes-or-holes-for-g-x-x-2-6x-8-x-2-8f9afa53c4","timestamp":"2024-11-08T05:15:31Z","content_type":"text/html","content_length":"582525","record_id":"<urn:uuid:61d4fe5f-7c45-4c8e-8f57-9646a3195d3f>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00138.warc.gz"} |
Multiple Time Series In An Excel Chart 2024 - Multiplication Chart Printable
Multiple Time Series In An Excel Chart
Multiple Time Series In An Excel Chart – You may create a multiplication chart in Shine by using a web template. You will find several samples of templates and figure out how to structure your
multiplication graph or chart using them. Here are several tricks and tips to generate a multiplication graph. Upon having a template, all you need to do is copy the formula and mixture it inside a
new mobile phone. Then you can take advantage of this solution to multiply a series of figures by another establish. Multiple Time Series In An Excel Chart.
Multiplication dinner table web template
You may want to learn how to write a simple formula if you are in the need to create a multiplication table. First, you should secure row one of many header column, then flourish the number on row A
by mobile B. A different way to create a multiplication dinner table is to use mixed references. In this instance, you would probably key in $A2 into column A and B$1 into row B. The effect is a
multiplication dinner table using a formula that really works both for rows and columns.
If you are using an Excel program, you can use the multiplication table template to create your table. Just open the spreadsheet with the multiplication dinner table change and template the brand on
the student’s label. You can even adjust the page to suit your individual demands. It comes with an choice to change the colour of the cellular material to modify the look of the multiplication
dinner table, also. Then, you may transform all the different multiples suitable for you.
Building a multiplication graph in Excel
When you’re utilizing multiplication desk software, you can easily produce a easy multiplication desk in Excel. Just produce a page with columns and rows numbered from a single to 30. The location
where the columns and rows intersect is definitely the answer. For example, if a row has a digit of three, and a column has a digit of five, then the answer is three times five. The same thing goes
for the other way around.
First, you can go into the figures that you need to grow. If you need to multiply two digits by three, you can type a formula for each number in cell A1, for example. To produce the numbers greater,
pick the cells at A1 and A8, and then click on the appropriate arrow to choose an array of cells. Then you can sort the multiplication formula inside the cells from the other rows and columns.
Gallery of Multiple Time Series In An Excel Chart
Multiple Time Series In An Excel Chart Peltier Tech
Plotting Multiple Series In A Line Graph In Excel With Different Time
Plotting Multiple Series In A Line Graph In Excel With Different Time
Leave a Comment | {"url":"https://www.multiplicationchartprintable.com/multiple-time-series-in-an-excel-chart/","timestamp":"2024-11-13T23:07:02Z","content_type":"text/html","content_length":"52924","record_id":"<urn:uuid:05b727ee-e17c-442f-b304-e41ddeee6b2c>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00820.warc.gz"} |
Calibration of High-Frequency Impedance Spectroscopy Measurements with Nanocapacitor Arrays
High frequency impedance spectroscopy (HFIS) biosensors based on nano-electrode arrays (NEA) demonstrated the capability to overcome the screening limits set by the Electrical Double Layer (EDL),
thus enabling label-free detection and imaging of analytes far above the sensor surface [1,2]. In order to achieve quantitatively accurate results, a precise understanding and modeling of the signal
transduction chain is necessary. With reference to the CMOS array platform in [1], capacitance is measured by CBCM. Hence, the nanoelectrodes are alternatively charged and discharged by two switch
transistors (Fig.1, a), which are activated by non-overlapping clocks with typically 1 ns floating time between the two phases. The column readout circuits integrate and average over multiple cycles
the charging current to obtain a capacitance information. The output signal is interpreted in terms of a switching capacitance (CSW), modeled by charge-pump analysis of an equivalent C-RC circuit
excited by a square wave (EDL capacitance CS in series to a parallel RECE representing the bulk electrolyte [1]; CS, RE and CE are extracted with the biosensor simulator ENBIOS [3]), good agreement
is obtained between experiments and simulations over a broad range of frequencies and electrolyte salt concentrations [1]. Residual discrepancies, however, require explanation and this is the main
contribution of our abstract. To this end, we firstly, consider the role of leakage currents (ILEAK) in the sensor cell (due to subthreshold conduction of the inactive switch). The leakage current
implies overestimating the column current IM (and hence the capacitance). Due to the large number of cells connected on each column, a value as large as 20pA is estimated for ILEAK, and measurements
are corrected by compensating for it. Then, we consider the voltage waveforms at the nanoelectrode, as obtained by Spice simulations with Predictive Technology Models (PTM) of the sensor cell readout
circuit (Fig.1 (b) for a 10mM electrolyte). Charge repartition between the nanoelectrode’s node and CGS/CGD capacitance of the switching transistors during the float time distorts the otherwise
square-waveform. For electrolytes with high salt concentration this effect is mitigated (due to the larger load capacitance). To account for this effect, we extract the harmonic content of the
waveform by Fourier expansion of the waveform (Fig.1, b). Then, ENBIOS simulations at all harmonic frequencies are used to reconstruct the capacitance response to the actual waveform (CF). Fig.1 (c)
compares experiments (corrected for leakage) and simulations (CSW or CF). The impact of leakage is modest, whereas CF exhibits an improved agreement with experiments at high frequency, where waveform
glitches are more relevant. These corrections highlight the importance of leakage and harmonic content of the input waveforms to achieve quantitatively accurate interpretation of NEA HFIS biosensor
experiments. Further work is necessary to extend these results to electrolytes with physiological salinity.
Calibration of High-Frequency Impedance Spectroscopy Measurements with Nanocapacitor Arrays / Cossettini, Andrea; Selmi, Luca. - (2019), pp. 131-131. (Intervento presentato al convegno 2nd European
Biosensor Symposium tenutosi a Firenze, Italia nel 18-21 Febbraio 2019).
Calibration of High-Frequency Impedance Spectroscopy Measurements with Nanocapacitor Arrays
High frequency impedance spectroscopy (HFIS) biosensors based on nano-electrode arrays (NEA) demonstrated the capability to overcome the screening limits set by the Electrical Double Layer (EDL),
thus enabling label-free detection and imaging of analytes far above the sensor surface [1,2]. In order to achieve quantitatively accurate results, a precise understanding and modeling of the signal
transduction chain is necessary. With reference to the CMOS array platform in [1], capacitance is measured by CBCM. Hence, the nanoelectrodes are alternatively charged and discharged by two switch
transistors (Fig.1, a), which are activated by non-overlapping clocks with typically 1 ns floating time between the two phases. The column readout circuits integrate and average over multiple cycles
the charging current to obtain a capacitance information. The output signal is interpreted in terms of a switching capacitance (CSW), modeled by charge-pump analysis of an equivalent C-RC circuit
excited by a square wave (EDL capacitance CS in series to a parallel RECE representing the bulk electrolyte [1]; CS, RE and CE are extracted with the biosensor simulator ENBIOS [3]), good agreement
is obtained between experiments and simulations over a broad range of frequencies and electrolyte salt concentrations [1]. Residual discrepancies, however, require explanation and this is the main
contribution of our abstract. To this end, we firstly, consider the role of leakage currents (ILEAK) in the sensor cell (due to subthreshold conduction of the inactive switch). The leakage current
implies overestimating the column current IM (and hence the capacitance). Due to the large number of cells connected on each column, a value as large as 20pA is estimated for ILEAK, and measurements
are corrected by compensating for it. Then, we consider the voltage waveforms at the nanoelectrode, as obtained by Spice simulations with Predictive Technology Models (PTM) of the sensor cell readout
circuit (Fig.1 (b) for a 10mM electrolyte). Charge repartition between the nanoelectrode’s node and CGS/CGD capacitance of the switching transistors during the float time distorts the otherwise
square-waveform. For electrolytes with high salt concentration this effect is mitigated (due to the larger load capacitance). To account for this effect, we extract the harmonic content of the
waveform by Fourier expansion of the waveform (Fig.1, b). Then, ENBIOS simulations at all harmonic frequencies are used to reconstruct the capacitance response to the actual waveform (CF). Fig.1 (c)
compares experiments (corrected for leakage) and simulations (CSW or CF). The impact of leakage is modest, whereas CF exhibits an improved agreement with experiments at high frequency, where waveform
glitches are more relevant. These corrections highlight the importance of leakage and harmonic content of the input waveforms to achieve quantitatively accurate interpretation of NEA HFIS biosensor
experiments. Further work is necessary to extend these results to electrolytes with physiological salinity.
File in questo prodotto:
File Dimensione Formato
Accesso riservato
Tipologia: Versione pubblicata dall'editore 251.32 kB Adobe PDF Visualizza/Apri Richiedi una copia
Dimensione 251.32 kB
Formato Adobe PDF
Pubblicazioni consigliate | {"url":"https://iris.unimore.it/handle/11380/1173377","timestamp":"2024-11-11T18:08:46Z","content_type":"text/html","content_length":"69707","record_id":"<urn:uuid:1b14e17c-b6a5-400f-9d00-9ee1a4053e91>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00496.warc.gz"} |
Evidence Lower Bound: ELBO
This article reuses a lot of materials from the references. Please see the references for more details on ELBO.
Given a probability distribution density $p(X)$ and a latent variable $Z$, we have the marginalization of the joint probability
$$ \int dZ p(X, Z) = p(X). $$
Using Jensen’s Inequality
In many models, we are interested in the log probability density $\log p(X)$ which can be decomposed using an auxiliary density of the latent variable $q(Z)$,
$$ \log p(X) =& \log \int dZ p(X, Z) \\ =& \log \int dZ p(X, Z) \frac{q(Z)}{q(Z)} \\ =& \log \int dZ q(Z) \frac{p(X, Z)}{q(Z)} \\ =& \log \mathbb E_q \left[ \frac{p(X, Z)}{q(Z)} \right]. $$
[[Jensen's inequality]] Jensen's Inequality Jensen’s inequality shows that $$ f(\mathbb E(X)) \leq \mathbb E(f(X)) $$ for a concave function $f(\cdot)$. shows that
$$ \log \mathbb E_q \left[ \frac{p(X, Z)}{q(Z)} \right] \geq \mathbb E_q \left[ \log\left(\frac{p(X, Z)}{q(Z)}\right) \right], $$
as $\log$ is a concave function.
Applying this inequality, we get
$$ \log p(X) =& \log \mathbb E_q \left[ \frac{p(X, Z)}{q(Z)} \right] \\ \geq& \mathbb E_q \left[ \log\left(\frac{p(X, Z)}{q(Z)}\right) \right] \\ =& \mathbb E_q \left[ \log p(X, Z)- \log q(Z) \right]
\\ =& \mathbb E_q \left[ \log p(X, Z) \right] - \mathbb E_q \left[ \log q(Z) \right] . $$
Uing the definition of entropy and cross entropy, we know that
$$ H(q(Z)) = - \mathbb E_q \left[ \log q(Z) \right] $$
is the entropy of $q(Z)$ and
$$ H(q(Z);p(X,Z)) = -\mathbb E_q \left[ \log p(X, Z) \right] $$
is the cross entropy. For convinence, we denote
$$ L = \mathbb E_q \left[ \log p(X, Z) \right] - \mathbb E_q \left[ \log q(Z) \right] = - H(q(Z);p(X,Z)) + H(q(Z)), $$
which is called the evidence lower bound (ELBO) as
$$ \log p(X) \geq L. $$
KL Divergence
In a latent variable model, we might need to calculate the posterior $p(Z|X)$. When this is intractable, we find an approximation $q(Z|\theta)$ where $\theta$ is the parametrization such as neural
network parameters. To make sure we have a good approximation of the posterior, we find the KL divergence of $q(Z|\theta)$ and $p(Z|X)$.
The [[KL divergence]] KL Divergence Kullback–Leibler divergence indicates the differences between two distributions is
$$ D_\text{KL}(q(Z|\theta)\parallel p(Z|X)) =& -\mathbb E_q \log\frac{p(Z|X)}{q(Z|\theta)} \\ =& -\mathbb E_q \log\frac{p(X, Z)/p(X)}{q(Z|\theta)} \\ =& -\mathbb E_q \log\frac{p(X, Z)}{q(Z|\theta)} -
\mathbb E_q \log\frac{1}{p(X)} \\ =& - L + \log p(X). $$
Since $D(q(Z|\theta)\parallel p(Z|X))\geq 0$, we have
$$ \log p(X) \geq L, $$
which also indicates that $L$ is the lower bound of $\log p(X)$.
In fact,
$$ L - \log p(X) = - D_\text{KL}(q(Z|\theta)\parallel p(Z|X)) $$
is the Jensen gap.
Planted: by L Ma;
LM (2021). 'Evidence Lower Bound: ELBO', Datumorphism, 04 April. Available at: https://datumorphism.leima.is/wiki/machine-learning/bayesian/elbo/. | {"url":"https://datumorphism.leima.is/wiki/machine-learning/bayesian/elbo/?ref=footer","timestamp":"2024-11-12T03:56:10Z","content_type":"text/html","content_length":"114951","record_id":"<urn:uuid:d00fc219-e9f0-4f10-b9e1-e7db2b53066a>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00216.warc.gz"} |
Pyramid Volume Calculator
Pyramid Volume
Calculating the volume of a pyramid is essential in various fields, such as architecture, civil engineering, and geometry. Whether you’re working on construction projects or studying mathematical
concepts, understanding how to find the volume of a pyramid is crucial for determining space usage and material requirements. This article will explain how to calculate pyramid volume, provide
step-by-step examples, and discuss practical applications where pyramid volume calculations are commonly used in real-world scenarios.
How to Calculate Pyramid Volume
The volume of a pyramid is calculated similarly to a cone, as both shapes taper to a point from a broad base. The formula for calculating the volume of a pyramid is:
\( V = \frac{1}{3} A_b h \)
• \( V \) is the volume of the pyramid (in cubic units, such as cubic meters or cubic feet).
• \( A_b \) is the area of the base (in square units, such as square meters or square feet).
• \( h \) is the height of the pyramid (the perpendicular distance from the base to the apex, in meters, feet, etc.).
This formula calculates the volume of any pyramid, whether it has a triangular, square, or rectangular base. The key is first to find the area of the base and then apply the formula.
Step-by-Step Guide to Pyramid Volume Calculation
Here is a simple step-by-step guide to calculating the volume of a pyramid:
• Step 1: Measure or determine the area of the base \( A_b \). If the base is a square or rectangle, multiply the length by the width to find the area. For a triangular base, use \( \frac{1}{2} \
times \text{base length} \times \text{height of the triangle} \).
• Step 2: Measure or determine the height \( h \) of the pyramid, which is the vertical distance from the center of the base to the apex (tip) of the pyramid.
• Step 3: Use the volume formula: \( V = \frac{1}{3} A_b h \).
• Step 4: Multiply the area of the base by the height and divide by 3 to get the volume.
• Step 5: Ensure that the units are consistent throughout the calculation to get an accurate result in cubic units (e.g., cubic meters, cubic feet).
This method works for pyramids of various base shapes, including square, rectangular, and triangular pyramids.
Example of Pyramid Volume Calculation
Let’s work through an example. Suppose you have a square pyramid with a base length of 4 meters and a height of 6 meters. First, calculate the area of the square base:
\( A_b = 4 \times 4 = 16 \, \text{square meters} \)
Now, apply the volume formula:
\( V = \frac{1}{3} \times 16 \times 6 = 32 \, \text{cubic meters} \)
Therefore, the volume of the pyramid is 32 cubic meters.
Practical Applications of Pyramid Volume
Calculating pyramid volume is important in a range of engineering and architectural applications. Some of the most common uses include:
• Architecture: Pyramids are used in the design of roofs, monuments, and other structures. Volume calculations are critical for estimating material usage and construction costs.
• Construction: Pyramidal structures like roof trusses, concrete forms, and other components require accurate volume calculations to ensure stability and proper material allocation.
• Manufacturing: In industries that produce parts with pyramid shapes, volume calculations help determine the amount of raw material needed and optimize production processes.
• Storage: Some tanks and containers used in fluid storage have pyramid shapes. Volume calculations are used to determine how much liquid or material the container can hold.
• Mathematics and Geometry: Volume calculations for pyramids are essential in academic and professional fields, helping students and engineers understand three-dimensional space.
Pyramid Volume for Different Units
When calculating pyramid volume, it’s crucial to use consistent units. The result will always be in cubic units, depending on the units used for the base area and height. Here are some common unit
• Cubic Meters (m³): Used for large structures, such as monuments or industrial components. If the base area and height are in meters, the volume will be in cubic meters.
• Cubic Centimeters (cm³): Used for smaller objects, such as packaging or laboratory equipment. If the base area and height are in centimeters, the volume will be in cubic centimeters.
• Cubic Feet (ft³): Commonly used in the United States for construction and material calculations. If the base area and height are in feet, the volume will be in cubic feet.
• Cubic Inches (in³): Used for small, precise measurements, particularly in engineering applications. If the base area and height are in inches, the volume will be in cubic inches.
Be sure to use consistent units throughout the calculation to avoid errors and ensure accuracy.
Examples of Pyramid Volume Calculations
Example 1: Calculating Pyramid Volume in Meters
Suppose you have a rectangular pyramid with a base length of 5 meters, a base width of 3 meters, and a height of 10 meters. The area of the base is:
\( A_b = 5 \times 3 = 15 \, \text{square meters} \)
The volume is calculated as:
\( V = \frac{1}{3} \times 15 \times 10 = 50 \, \text{cubic meters} \)
Example 2: Calculating Pyramid Volume in Centimeters
For a pyramid with a triangular base where the base length is 8 centimeters, the height of the triangle is 6 centimeters, and the pyramid height is 12 centimeters, the base area is:
\( A_b = \frac{1}{2} \times 8 \times 6 = 24 \, \text{square centimeters} \)
The volume is calculated as:
\( V = \frac{1}{3} \times 24 \times 12 = 96 \, \text{cubic centimeters} \)
Example 3: Calculating Pyramid Volume in Feet
If you have a pyramid with a square base that has a side length of 4 feet and a height of 9 feet, the base area is:
\( A_b = 4 \times 4 = 16 \, \text{square feet} \)
The volume is calculated as:
\( V = \frac{1}{3} \times 16 \times 9 = 48 \, \text{cubic feet} \)
Frequently Asked Questions (FAQ)
1. What is the formula for calculating the volume of a pyramid?
The formula for calculating the volume of a pyramid is \( V = \frac{1}{3} A_b h \), where \( A_b \) is the area of the base and \( h \) is the height of the pyramid.
2. How do I calculate the area of the base of a pyramid?
The area of the base depends on the shape. For a square or rectangular base, multiply the length by the width. For a triangular base, use the formula \( A = \frac{1}{2} \times \text{base length} \
times \text{height of the triangle} \).
3. Can I use the same formula for a pyramid with any base shape?
Yes, the formula \( V = \frac{1}{3} A_b h \) applies to any pyramid. However, the key is correctly calculating the area of the base, which depends on the base shape.
4. Why is pyramid volume important in engineering?
Pyramid volume is important in engineering for determining material usage, space requirements, and storage capacity. It is used in construction, manufacturing, and various design applications. | {"url":"https://turn2engineering.com/calculators/pyramid-volume-calculator","timestamp":"2024-11-07T00:02:19Z","content_type":"text/html","content_length":"224237","record_id":"<urn:uuid:a8e7540d-eae9-47a3-89c8-f2c4855ecb89>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00132.warc.gz"} |