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Linear algebra done wrong pdf file
Download linear algebra done wrong solutions librarydoc31 pdf book pdf free download link or read online here in pdf. The author has taken unusual care to motivate concepts and to simplify proofs.
The print and kindle versions of this book are available at. Sheldon axler, linear algebra done right, third edition text videos by author and former mit instructor sheldon axler. Lecture notes math
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Enhancing Decision Quality with Matrix Diagrams
Also known as Matrix Chart and Relationship Matrix.
Variants include Correlation Matrix, Quality Function Deployment and Cause and Effect Matrix.
A matrix diagram is a visual representation that shows the connection between various groups of data. It is a planning tool that essentially displays the existence and strength of relationships
between pairs of items from two or more data sets. It aims to help in problem-solving, decision making, and process improvement efforts. Several advanced problem-solving tools utilize the matrix
diagram concept, such as the Cause-and-Effect Matrix and Quality Function Deployment.
Examples of matrix diagram applications extend across various industries and situations. One of its main applications is in problem-solving scenarios where the relationship between a problem and
potential solutions or causes needs to be explored. Additionally, it helps in identifying the measures most strongly linked to customer or business needs. For example, a marketing team may use a
matrix diagram to identify and select the most effective sales tools among various options.
A matrix diagram displays the existence and strength of relationships between pairs of items from two or more data sets
Types of Matrix Diagrams
Matrix diagrams come in various shapes with the L-shaped matrix being the most basic and commonly used. This type is often represented as a two-dimensional table, with the left-hand column listing
items from the first set, and the top row containing items from the second set. L-shaped matrices are particularly effective for comparing only two lists, for example a problem and its potential
solutions or causes.
Apart from the L-shaped matrix, other shapes like T-shaped, X-shaped, and Y-shaped matrices allow the comparison of more than two lists.
Each shape has a descriptive letter name which indicate its shape
In a matrix diagram, the relationship between any two items is indicated in the cell where they intersect. At each intersection point, the relationship is either present or absent. When it is
present, the strength of the relationship can be indicated using numerical values or symbols placed at the intersection point. A scale from 1 to 5 can be used where 1 indicates a weak relationship
and 5 indicates a strong relationship.
Symbols are preferred over numbers as they enhance clarity and simplify interpretation. Each symbol corresponds to a specific numerical value or level of strength, such as weak, moderate, and strong.
Weighting can be applied in the matrix diagram to assign relative importance to specific items within the data sets. Additional information can also be displayed, including weighted scores, ranks,
and the overall relationship strength score. These weighted scores can later be considered to identify, prioritize, and select the most favorable options.
Constructing and Using a Matrix Diagram
The following steps describe how to construct and use an L-shaped matrix diagram:
• In your team, clearly explain the purpose for building the matrix diagram.
• Select and collect the two sets of data to be compared.
• Agree on symbols and their values.
• Draw a two-dimensional table on a flipchart or table.
• Insert the first data set in the left-hand column and the second data set in the top row.
• Assign weighted scores to show relative importance of items.
• Work through the matrix and discuss the relationships between every pair of items placing the appropriate symbol in the intersecting cell.
• Calculate the final weighted scores for each item in either or both data sets.
• Review the completed matrix with the team to make informed decisions.
Example – Making a Better Cup of Tea
The following is an example of an L-shaped matrix diagram constructed by a coffee shop team. It illustrates the cause-and-effect relationship to enhance the quality and flavor of the tea they
Example – Assigning Human Resources to Multiple Projects
A program manager decided to use a matrix diagram to help allocate human resources to multiple improvement projects. This is an example of a T-shaped matrix diagram that enables the comparison of two
sets of data with a third one.
For example, it is important for those working on the spoilage reduction project to have basic knowledge of statistical process control (SPC).
Example – Application of Improvement Tools
The example below illustrates the workshops attended or conducted by each department in an organization during a change management process.
Note that high scores in columns suggest workshops in their final stages, while high scores in rows suggest departments with significant progress.
Example – Correlation Matrix
It is also possible to compare the same items together by using a triangular half-matrix called the correlation matrix. For example, the following adjacency matrix is created in a facility to analyze
and reorganize rooms and areas in a way that supports workflow efficiency and enhances teams’ collaboration.
There are many tools available to help in creating matrix diagrams. One of the simplest and most straightforward methods is to use these matrix diagram templates.
Wrapping Up
A matrix diagram is one of the seven management and planning tools that are important for making confident and rational decisions. Matrix diagrams are essential tools in decision-making,
problem-solving, and process improvement. The application of matrix diagrams extends to various situations, including cause-and-effect analysis, requirement-specification matching, comparison of
alternative solutions, and identification of improvement opportunities.
Heizer, J., Render, P., & Al-Zu’bi, Z. (2013). Operations Management.
Kubiak, T. M. (2012). The Certified Six Sigma Master Black Belt. Infotech.
Do you want to use the slides in your training courses?
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One to One Functions - Graph, Examples | Horizontal Line Test
One to One Functions - Graph, Examples | Horizontal Line Test
What is a One to One Function?
A one-to-one function is a mathematical function where each input corresponds to just one output. In other words, for each x, there is only one y and vice versa. This implies that the graph of a
one-to-one function will never intersect.
The input value in a one-to-one function is noted as the domain of the function, and the output value is known as the range of the function.
Let's look at the images below:
For f(x), any value in the left circle corresponds to a unique value in the right circle. In conjunction, any value on the right correlates to a unique value in the left circle. In mathematical
jargon, this means that every domain holds a unique range, and every range has a unique domain. Hence, this is an example of a one-to-one function.
Here are some other examples of one-to-one functions:
Now let's examine the second image, which exhibits the values for g(x).
Pay attention to the fact that the inputs in the left circle (domain) do not have unique outputs in the right circle (range). Case in point, the inputs -2 and 2 have the same output, that is, 4.
Similarly, the inputs -4 and 4 have the same output, i.e., 16. We can discern that there are equivalent Y values for many X values. Thus, this is not a one-to-one function.
Here are different examples of non one-to-one functions:
What are the qualities of One to One Functions?
One-to-one functions have these properties:
• The function holds an inverse.
• The graph of the function is a line that does not intersect itself.
• It passes the horizontal line test.
• The graph of a function and its inverse are identical regarding the line y = x.
How to Graph a One to One Function
When trying to graph a one-to-one function, you are required to figure out the domain and range for the function. Let's study a straight-forward example of a function f(x) = x + 1.
Once you have the domain and the range for the function, you ought to graph the domain values on the X-axis and range values on the Y-axis.
How can you tell whether a Function is One to One?
To prove if a function is one-to-one, we can use the horizontal line test. Once you chart the graph of a function, trace horizontal lines over the graph. If a horizontal line moves through the graph
of the function at more than one spot, then the function is not one-to-one.
Due to the fact that the graph of every linear function is a straight line, and a horizontal line does not intersect the graph at more than one spot, we can also reason that all linear functions are
one-to-one functions. Remember that we do not apply the vertical line test for one-to-one functions.
Let's examine the graph for f(x) = x + 1. Immediately after you chart the values for the x-coordinates and y-coordinates, you need to examine whether a horizontal line intersects the graph at more
than one point. In this example, the graph does not intersect any horizontal line more than once. This means that the function is a one-to-one function.
On the other hand, if the function is not a one-to-one function, it will intersect the same horizontal line multiple times. Let's examine the figure for the f(y) = y^2. Here are the domain and the
range values for the function:
Here is the graph for the function:
In this instance, the graph intersects numerous horizontal lines. Case in point, for each domains -1 and 1, the range is 1. Additionally, for either -2 and 2, the range is 4. This signifies that f(x)
= x^2 is not a one-to-one function.
What is the inverse of a One-to-One Function?
Considering the fact that a one-to-one function has just one input value for each output value, the inverse of a one-to-one function is also a one-to-one function. The inverse of the function
basically reverses the function.
For example, in the example of f(x) = x + 1, we add 1 to each value of x in order to get the output, i.e., y. The opposite of this function will deduct 1 from each value of y.
The inverse of the function is denoted as f−1.
What are the properties of the inverse of a One to One Function?
The characteristics of an inverse one-to-one function are identical to any other one-to-one functions. This means that the opposite of a one-to-one function will possess one domain for each range and
pass the horizontal line test.
How do you figure out the inverse of a One-to-One Function?
Finding the inverse of a function is simple. You simply have to swap the x and y values. Case in point, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
As we learned previously, the inverse of a one-to-one function reverses the function. Since the original output value required us to add 5 to each input value, the new output value will require us to
subtract 5 from each input value.
One to One Function Practice Examples
Contemplate these functions:
• f(x) = x + 1
• f(x) = 2x
• f(x) = x2
• f(x) = 3x - 2
• f(x) = |x|
• g(x) = 2x + 1
• h(x) = x/2 - 1
• j(x) = √x
• k(x) = (x + 2)/(x - 2)
• l(x) = 3√x
• m(x) = 5 - x
For any of these functions:
1. Determine whether or not the function is one-to-one.
2. Graph the function and its inverse.
3. Determine the inverse of the function numerically.
4. Specify the domain and range of every function and its inverse.
5. Apply the inverse to find the solution for x in each formula.
Grade Potential Can Help You Learn You Functions
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RareComb is a combinatorial framework that couples the apriori algorithm with binomial tests to systematically analyze patterns of rare event combinations between groups of interest to identify
specific combinations that significantly associate with phenotypes. This generalizable, modular and extensible framework does not depend on apriori knowledge and can detect rare patterns from
high-dimensional genetic data and generate interpretable results, making it readily useful for analyzing cohorts of all size ranges and providing a structured approach to dissect the genetic basis of
complex disorders.
A general framework for identifying rare variant combinations in complex disorders
Vijay Kumar Pounraja, Santhosh Girirajan
Software Requirements
RareComb is an easy-to-use R package with five built-in user-facing functions that take a sparse Boolean dataframe as their input along with multiple input parameters to constrain the execution of
these functions and the results they generate. The R package can be installed using the ‘install.packages(’RareComb’)’ command and loaded into memory using the ‘library(RareComb)’ command. RareComb
depends on the following R packages,
1. arules
2. dplyr
3. pwr
4. reshape2
5. sqldf
6. stats
7. stringr
8. tidyr
The five user-facing functions supported by the package along with their descriptions are as follows,
1) analyze_in_out_simultaneity (Comorbidity Analysis) : Analyze the relationship between input and output variables for all combinations that include at least a single output variable and meet
all the input criteria specified by the user.
2) compare_enrichment (Enrichment in cases + Non-enrichment in controls) : Quantify the enrichment in the observed frequency of cooccurring rare events in combinations that meet all the input
criteria specified by the user compared to their corresponding expectation derived under the assumption of independence between the constituent elements within each combination. The function then
reports the multiple-testing adjusted significant combinations in which enrichment is observed in cases but not in controls.
3) compare_enrichment_depletion (Enrichment in cases + Depletion in controls) : Quantify the enrichment in the observed frequency of cooccurring rare events in combinations that meet all the
input criteria specified by the user compared to their corresponding expectation derived under the assumption of independence between the constituent elements within each combination. The
function then reports the multiple-testing adjusted significant combinations in which enrichment is observed in cases and depletion is observed in controls.
4) compare_enrichment_modifiers (Must include one of the user-supplied input variables in the significant combinations) : Quantify the enrichment in the observed frequency of combinations that
include at least one of the input variables supplied by the user as well as meet other user-specified criteria compared to their corresponding expectation derived under the assumption of
independence between the constituent elements of each combination. The function then reports the combinations in which enrichment is observed in cases but not in controls.
5) compare_expected_vs_observed (Compare observed frequency with the expected frequency within a single group) : Compare the observed frequency of combinations that meet all the user-specified
criteria with their corresponding expectation derived under the assumption of independence between the constituent elements of each combination. Unlike the method ‘compare_enrichment’, this
method does NOT split the groups based on phenotypes. It simply treats the entire input as a single cohort and measures the magnitude of difference between the expected and observed frequencies
of cooccurring events.
Running RareComb
Running RareComb involves the following considerations and steps,
Things to consider: > 1) Make sure the input variables in the input data are prefixed with ‘Input_’ and the output/outcome variables are prefixed with ‘Output_’. If you prefer a different convention,
please use the optional parameters ‘input_format’ and ‘output_format’ to specify the prefix of your choice
2) Prior to invoking the function ‘analyze_in_out_simultaneity’, make sure the input file has more than one output/outcome variables
3) Prior to invoking the functions ‘compare_enrichment’, ‘compare_enrichment_depletion’ and ‘compare_enrichment_modifiers’, make sure the input file has EXACTLY one output/outcome variable
Steps involved in running the functions in RareComb: > Step 1) Generate an input Boolean dataframe with ‘n’ samples (rows) and ‘p’ variables (columns).
Step 2) Invoke the function of interest using the input file along with the additional mandatory and optional parameters.
Step 3) The final output returned by all the functions is a dataframe that the users can choose save to an output comma or tab separated file.
Usage & Examples
1) analyze_in_out_simultaneity(boolean_input_mult_df, combo_length, min_output_count, max_output_count, min_indv_threshold, max_freq_threshold, input_format, output_format, pval_filter_threshold,
Total input parameters : 10 (6 mandatory + 4 optional)
analyze_in_out_simultaneity(boolean_input_mult_df, 3, 1, 2, 5, 0.25, input_format = 'Input_', output_format = 'Output_', pval_filter_threshold = 0.05, adj_pval_type = 'BH')
2) compare_enrichment(boolean_input_df, combo_length, min_indv_threshold, max_freq_threshold, input_format, output_format, pval_filter_threshold, adj_pval_type, min_power_threshold, sample_names_ind)
Total input parameters : 10 (4 mandatory + 6 optional)
compare_enrichment(boolean_input_df, 3, 5, 0.25, input_format = 'Input_', output_format = 'Output_', adj_pval_type = 'bonferroni', sample_names_ind = 'N')
3) compare_enrichment_depletion(boolean_input_df, combo_length, min_indv_threshold, max_freq_threshold, input_format, output_format, pval_filter_threshold, adj_pval_type, min_power_threshold,
Total input parameters : 10 (4 mandatory + 6 optional)
compare_enrichment_depletion(boolean_input_df, 3, 5, 0.25, input_format = 'Input_', output_format = 'Output_', adj_pval_type = 'bonferroni', sample_names_ind = 'N')
4) compare_enrichment_modifiers(boolean_input_df, combo_length, min_indv_threshold, max_freq_threshold, primary_input_entities, input_format, output_format, pval_filter_threshold, adj_pval_type,
min_power_threshold, sample_names_ind)
Total input parameters : 11 (5 mandatory + 6 optional)
compare_enrichment_modifiers(boolean_input_df, 2, 4, 0.25, input_format = 'Input_', output_format = 'Output_', primary_input_entities = input_list, adj_pval_type = 'bonferroni', sample_names_ind =
5) compare_expected_vs_observed(boolean_input_df, combo_length, min_indv_threshold, max_freq_threshold, input_format, pval_filter_threshold, adj_pval_type)
Total input parameters : 7 (4 mandatory + 3 optional)
compare_expected_vs_observed(boolean_input_df, 2, 10, 0.25, 0.05, input_format = 'Input_', adj_pval_type = 'BH')
Further details on the list of parameters applicable to each function can be found in the documentation for the R package in the CRAN website.
Output files and columns
Each function returns a dataframe with the list of statistically significant combinations that meet the user-specified input criteria as the output. Since the definition of ‘statistical significance’
is defined differently for each function, the number and types of columns in the output file will vary for each function depending on the size of the requested combination, number of groups under
analysis, if the supporting sample names are requested or not etc. For example, for functions that analyze the data based on a single binary outcome (Case/Control), the output file will contain
frequency of individual and cooccurring events in each group separately, whereas the output file from analyzing multiple phenotypes together will only contain frequencies from the single group that
is being analyzed. A list of output column names for each of the five functions along with their descriptions are provided below,
Output from ‘analyze_in_out_simultaneity’,
Item_1 Name of the first item in the combination.
Item_2 Name of the second item in the combination.
.. Other items in the combination.
Item_N Name of the ’N’th item in the combination.
Obs_Count_Combo Observed frequency of the cooccurring event within the combination.
Case_Obs_Count_I1 Observed frequency of the individual item ‘Item_1’ in cases.
Case_Obs_Count_I2 Observed frequency of the individual item ‘Item_2’ in cases.
Case_Obs_Count_IN Observed frequency of the individual item ‘Item_N’ in cases.
Output_Count Number of output variables in the combination.
Exp_Prob_Combo Expected probability of events to cooccur.
Obs_Prob_Combo Observed probability of cooccurring events.
pvalue_more p-values from the one-tailed binomial test to evaluate if the observed frequency is greater than the expected frequency of cooccurring events.
input_only_pvalue_more p-values from the one-tailed binomial test considering only the input variables (genotype). This p-value can be used to evaluate if the genotypes in a combination by themselves
are strongly associated with eachother.
Adj_Pval_bonf p-values of the genotype-phenotype combination adjusted for multiple testing using the ‘bonferroni’ method.
Adj_Pval_BH p-values of the genotype-phenotype combination adjusted for multiple testing using the ‘Benjamini-Hochberg’ method.
Output from ‘compare_expected_vs_observed’,
Item_1 Name of the first item in the combination.
Item_2 Name of the second item in the combination.
.. Other items in the combination.
Item_N Name of the ’N’th item in the combination.
Obs_Count_Combo Observed frequency of the cooccurring event within the combination.
Obs_Count_I1 Observed frequency of the individual item ‘Item_1’ in cases.
Obs_Count_I2 Observed frequency of the individual item ‘Item_2’ in cases.
Obs_Count_IN Observed frequency of the individual item ‘Item_N’ in cases.
Exp_Prob_Combo Expected probability of events to cooccur.
Obs_Prob_Combo Observed probability of cooccurring events.
pvalue_more p-values from the one-tailed binomial test to evaluate if the observed frequency is greater than the expected frequency of cooccurring events.
Adj_Pval_bonf p-values of the genotype-phenotype combination adjusted for multiple testing using the ‘bonferroni’ method.
Adj_Pval_BH p-values of the genotype-phenotype combination adjusted for multiple testing using the ‘Benjamini-Hochberg’ method.
Output from ‘compare_enrichment’, ‘compare_enrichment_depletion’ & ‘compare_enrichment_modifiers’,
Item_1 Name of the first item in the combination.
Item_2 Name of the second item in the combination.
Item_N Name of the ’N’th item in the combination.
Case_Obs_Count_I1 Observed frequency of the individual item ‘Item_1’ in cases.
Case_Obs_Count_I2 Observed frequency of the individual item ‘Item_2’ in cases.
Case_Obs_Count_IN Observed frequency of the individual item ‘Item_N’ in cases.
Case_Exp_Prob_Combo Expected probability of the cooccurring event within the combination in cases.
Case_Obs_Prob_Combo Observed probability of the cooccurring event within the combination in cases.
Case_Exp_Count_Combo Expected frequency of the cooccurring event within the combination in cases.
Case_Obs_Count_Combo Observed frequency of the cooccurring event within the combination in cases.
Case_pvalue_more p-values from the one-tailed binomial test to evaluate if the observed frequency is greater than the expected frequency of cooccurring events in cases.
Cont_Obs_Count_I1 Observed frequency of the individual item ‘Item_1’ in controls.
Cont_Obs_Count_I2 Observed frequency of the individual item ‘Item_2’ in controls.
Cont_Obs_Count_IN Observed frequency of the individual item ‘Item_N’ in controls.
Cont_Exp_Prob_Combo Expected probability of the cooccurring event within the combination in controls.
Cont_Obs_Prob_Combo Observed probability of the cooccurring event within the combination in controls.
Cont_Exp_Count_Combo Expected frequency of the cooccurring event within the combination in controls.
Cont_Obs_Count_Combo Observed frequency of the cooccurring event within the combination in controls.
Cont_pvalue_more p-values from the one-tailed binomial test to evaluate if the observed frequency is greater than the expected frequency of cooccurring events in controls.
Control_pvalue_less (applies only to This output column replaces Cont_pvalue_more when the function compare_enrichment_depletion is invoked. This column provides the p-values from the one-tailed
‘compare_enrichment_depletion’) binomial test to evaluate if the observed frequency is lesser than the expected frequency of cooccurring events in controls.
Case_Adj_Pval_bonf p-values of the combination in cases adjusted for multiple testing using the ‘bonferroni’ method.
Case_Adj_Pval_BH p-values of the combination in cases adjusted for multiple testing using the ‘Benjamini-Hochberg’ method.
Effect_Size Effect size measured as Cohen’s d capturing the magnitude of difference in frequency of cooccurring events between cases and controls.
Power_One_Pct Available statistical power for the 2-sample 2-proportion test to compare the frequencies of cooccurring events in cases and controls at 1% significance
Power_Five_Pct Available statistical power for the 2-sample 2-proportion test to compare the frequencies of cooccurring events in cases and controls at 5% significance
Case_Samples A list of sample names from cases that carry the significant combination identified by the function. This column is part of the output only when the function
is invoked with the input parameter ‘sample_names_ind’ set to ‘Y’.
Control_Samples A list of sample names from controls that carry the significant combination identified by the function. This column is part of the output only when the
function is invoked with the input parameter ‘sample_names_ind’ set to ‘Y’.
MIT License
Copyright (c) 2021 Vijay Kumar Pounraja
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction,
including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do
so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
For questions or comments, please contact Vijay Kumar Pounraja (vxm915@psu.edu) or Santhosh Girirajan (sxg47@psu.edu). | {"url":"https://cran.case.edu/web/packages/RareComb/readme/README.html","timestamp":"2024-11-12T06:30:36Z","content_type":"application/xhtml+xml","content_length":"22609","record_id":"<urn:uuid:fe84438d-205e-4446-b40c-1cc370230850>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00700.warc.gz"} |
Create Vectors In R: A Step-by-step Tutorial
This tutorial will show you how to use and create vectors in R. Vectors allow you to work with multiple pieces of data and then assign them into a single object.
A vector in R looks a lot like a range in Excel. However, unlike in Excel, a vector’s elements should all be of the same type.
In the image above, one line represents one vector. You can see that in each vector, all its elements are of the same type. The first line is numerical, followed by string, and then logical.
This tutorial will focus on two things: combining data and manipulating vectors.
Create Vectors In R By Combining Data
Combining data in R is easy. All you need to do is use the c ( ) function.
So, open your RStudio. Assign multiple values to object a using the c ( ) function, similar to what is shown below.
If you print this, you’ll see that the value of a is 1, 2, and 3.
Now remember that vectors should only contain elements of the same type. So, what happens if you mix two different object classes together? Here’s an example:
If you assign two numerical values to b and then assign a string as the third one, you’ll see that all the values in b are converted to a character.
This is called coercion. It’s where R decides the best way to convert the elements into the same object class. In this case, the best way was to convert the numbers to text instead of the other way
Analyze and Manipulate Vectors In R
You can also perform mathematical operations on vectors.
For example, if you multiply a by 2, you’ll see that each numerical element in a was multiplied by 2.
This is similar to multiplying a range in Excel or multiplying a column in Power BI.
Now let’s try out another case.
Let’s create a new object called my_long_vector and assign a range from 5 to 84. So instead of using the c ( ) function, you can use a colon (:) to indicate a range of values.
When you print this, you’ll see that the object my_long_vector contains all the values from 5 to 84.
You can also locate a specific element in a vector. This is called indexing. You can do this by following the object name with square brackets ( [ ] ) and then placing in the position of the element
you want.
For example, you want to find the 3^rd element for my_long_vector. All you need to do is execute my_long_vector [3]. You’ll then arrive with 7 as the answer.
Regardless of how big a vector is, you can still use this with mathematical operations. If you Run the square root of my_long_vector, the Console will show you the square root of each element from 5
to 84.
***** Related Links *****
Power BI With R And RStudio: How To Get Started
Three Ways To Use R Script In Power BI
Objects And Object Classes In R: The Basics
Vectors are one of the building blocks of R. They’re similar to a range in Excel or column in Power BI. R vectors are more advanced compared to basic objects in R. You can perform simultaneous
operations to an array of data in one go.
In the next tutorials, you’ll learn how to work with a whole data frame which will bring you rows and columns of data.
Learn R by working on practical, real-world projects.
Unlock the full potential of R in your data analysis tasks and elevate your skills from proficient to expert.
Learn to master the art of data visualization using R. This guide covers everything from basic plots to complex, interactive visualizations.
This thread explores advanced topics in data analytics, focusing on building data pipelines, comparing SQL and R for data transformation, and applying predictive modeling techniques such as customer
churn analysis and time series forecasting in R.
A hands-on guided project to discover hidden patterns and relationships in retail transaction data using the Apriori algorithm in R.
An in-depth, hands-on course designed to teach the practical application of hierarchical clustering in R, complete with real-world examples, to enhance advanced analytical skills.
This project aims to teach the principles of prescriptive analytics and optimization through hands-on examples using the R programming language.
A comprehensive guide to effectively manipulate and transform data using the dplyr package in R.
Learn how to harness the power of Random Forest models to tackle real-world business challenges.
A comprehensive guide to writing efficient, reusable code and performing analysis using the R language.
A comprehensive guide to predicting stock price trends using Random Forest models in R.
A project aimed at optimizing inventory levels for a manufacturing company through predictive modeling using Random Forests in R.
Mastering R with Practical Projects
Advanced Data Analysis with R: From Proficiency to Mastery
The Ultimate Guide to Visualization in R Programming
Comprehensive Guide to Data Transformation and Prediction with R
Market Basket Insights Using Association Rule Learning in R
Mastering Hierarchical Clustering with R: Dendrograms and Cluster Trees in Action
Mastering Prescriptive Analytics with R: A Practical Guide
Mastering Data Manipulation in R with dplyr
Mastering Random Forest Models for Business Applications
Mastering Reusable Code and Analysis in R
Forecasting Stock Price Movements Using Random Forest in R
Supply Chain Optimization Using Random Forests and R | {"url":"https://blog.enterprisedna.co/create-vectors-in-r-a-step-by-step-tutorial/","timestamp":"2024-11-10T19:26:00Z","content_type":"text/html","content_length":"457973","record_id":"<urn:uuid:8c033592-5144-47af-b2be-1f4fef17d69b>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00193.warc.gz"} |
Average Max Deviation (avg_max_dev)
Average Max Deviation. There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Deviation scenarios function is calculated
by maximizing losses over (Linear Loss) - (Average Linear Loss over scenarios) functions (over M functions for every scenario). Average Max Deviation is calculated by averaging Maximum Deviation
avg_max_dev(matrix_1,matrix_2,...,matrix_M) short call
avg_max_dev_name(matrix_1,matrix_2,...,matrix_M) call with optional name
matrix_m is a Matrix of Scenarios:
where the header row contains names of variables (except scenario_probability, and scenario_benchmark). Other rows contain numerical data. The scenario_probability, and scenario_benchmark columns are
Mathematical Definition
Average Max Deviation function is calculated as follows
M = number of random Loss Functions ,
= vector of random coefficients for m-th Loss Function;
= j-th scenario of the random vector ,
is a random function with scenarios .
Every Loss Function is defined by a separate matrix of scenarios and has an equal number of scenarios J.
Probability of scenario is defined by the first matrix.
is an argument of Average Max Deviation function.
See also
Average Loss, Average Gain, Average Max, Average Max for Gain, Average Max Deviation for Gain | {"url":"https://aorda.com/html/PSG_Help_HTML/avg_max_dev.htm","timestamp":"2024-11-06T12:22:54Z","content_type":"text/html","content_length":"19148","record_id":"<urn:uuid:50bea954-bc11-4c18-a754-0db56fc472df>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00782.warc.gz"} |
Into Math Kindergarten Module 9 Answer Key Use the Count Sequence to Count to 100
We included HMH Into Math Kindergarten Answer Key PDF Module 9 Use the Count Sequence to Count to 100 to make students experts in learning maths.
HMH Into Math Kindergarten Module 9 Answer Key Use the Count Sequence to Count to 100
Find Groups of Five
Count the items on each table. Circle the groups that have five items.
There group of 5 white papers and building blocks
There are 6 pencils on one table
and 3 books on the other table.
Are You Ready?
Represent Numbers to 10
Question 1.
For number 9 used the birds as counters and counted them.
Order Numbers to 10
Question 2.
1, ______, 3,4, ______, 6, 7, 8, 9, _________
1, 2, 3,4, 5, 6, 7, 8, 9, 10
The missing numbers are 2, 5 and 10
Represent Numbers to 10
Question 3.
There are 10 dots in the above picture.
Here the counters used are dots for the number 10
used 10 dots.
1. Read the number. Draw counters to represent the number. 2. Write the unknown numbers. 3. Count the shapes. Write the number.
Lesson 1 Count to 100 by Ones
Listen to the story. How can you use the puzzle pieces to help Anthony count the steps? Anthony is at the beginning of the corn maze. He wants to know how many steps it takes to get to the center of
the corn maze.
Anthony count the steps.
In the middle he forgets to count some numbers
to reach the hundred.
Build Understanding
Question 1.
Point to the row colored yellow. Look at the numbers. Use blue to color another row in the chart. How are the numbers in the blue row the same as the numbers in the yellow row? How are they
In the yellow line there are sequence of numbers from 1 to 10
for every number 50 is added
so, that is the blue line
1 + 50 = 51
2 + 50 = 52 so on
Question 2.
Point to the column colored yellow. Look at the numbers. Use green to color the numbers in the next column in the chart. How are the numbers in the green column the same as the numbers in the yellow
column? How are they different?
In the yellow line there are sequence of numbers skip by 10s
starting from 1 to 91
In the blue line started from 4 and then skip by 10s
4 to 94
On Your Own
Question 3.
Point to each number as you count. What is the last number you counted? Circle that number. Tell a classmate how the chart helps you when counting by ones to 100.
Started counting from 1 so the last number counted is 100.
so, circled the number 100.
I’m in a Learning Mindset!
Directions: Did you know you could learn the math in this lesson? Color the star. Would you like to feel better about your learning? Color the checkmark.
Lesson 2 Count to 100 by Tens
Listen to the story. Circle the games that Jane finds. Jane is walking around at the Fall Festival. Every ten steps there is a different game. How can the games you circled help you count to 100 by
Jane is walking around at the Fall Festival. Every ten steps there is a different game.
skip by 10s are 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
At this steps the game changes.
Build Understanding
Question 1.
Count by tens to 100. Circle the numbers as you say them. What do you notice about each number?
skip by 10s are 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
so , circled them as I counted
Step It Out
Question 2.
Point to the yellow box. Count to 100 by tens. Color each number as you count.
skip by 10s are 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
so , colored them as I counted
On Your Own
Question 3.
Place rows of numbers as you count by tens to 100. Tell a classmate about counting by tens to 100. How are the numbers you count alike? How are the numbers you count different?
Completed the incomplete rows
When I count with my friend the difference come for me is 20
I’m in a Learning Mindset!
Directions: Did you know you could learn the math in this lesson? Color the star. Would you like to feel better about your learning? Color the checkmark.
Lesson 3 Count Forward From a Given Numbers
Question 1.
There are five pumpkins in the crate. Beginning at 5, count forward. Mark an X on each pumpkin as you count. Tell a classmate how many total pumpkins there are.
There are five pumpkins in the crate. Beginning at 5,
so, there are 20 pumpkins.
Step It Out
Question 2.
Sixteen leaves have been raked into the wheel barrow. Beginning at 16, count forward. Mark an X on each leaf as you count. Tell a classmate how many total leaves there are.
Sixteen leaves have been raked into the wheel barrow. Beginning at 16,
so, there are 26 leaves.
On Your Own
Question 3.
There are 18 eggs in the carton. Beginning at 18. Count forward. Mark on X on each egg as you count. Tell a classmate how many total eggs there are.
There are 18 eggs in the carton. Beginning at 18.
so, there are 30 eggs in all.
Question 4.
There are 22 bananas in a crate. Beginning at 22, count forward. Mark an X on each banana as you count. Tell a classmate how many total bananas there are.
There are 22 bananas in a crate. Beginning at 22,
so, there are 37 bananas in all
Module 9 Review
Concepts and Skills
Question 1.
started from number 1
and circled the number 17
Question 2.
Started with number 10
at last I have counted number 30
Question 3.
Started counting from number 24
so, there are 29 marbles in all.
1. Begin at 1 and count. Circle the number 17. 2. Count by tens as you point to the numbers in the shaded boxes. Start with the number 10. Circle the number you end with. 3. There are 24 marbles in
the bag. Beginning with 24, count forward. Mark an X on each marble as you count. Tell a classmate how many marbles there are in all.
Question 4.
(A) 20
(B) 30
(C) 40
Started counting by ten
so, last number that I counted is 40.
Question 5.
23 24 25 26 27 ________
(A) 28
(B) 29
(C) 30
The number after 27 is 28
Question 6.
(A) 20
(B) 22
(C) 23
The number after 21 is 22.
4. Start at 10. Count by tens. What is the last number you count? Mark below the number. 5. Count forward by ones starting at 23. What number is right after 27? Mark below the number. 6. Begin at 1.
Count by ones. What number is right after 21? Mark below the number.
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What is the equation for kinetic energy of a spring?
From calculus, the formula is (0.5)kx^2, where x^2 is the square of the initial displacement of the end of the spring. The kinetic and potential energy at any point will sum to this value. Identify
the spring’s maximum kinetic energy, at the equilibrium point, as equal to the initial potential energy.
What is the formula for work done by a spring?
Let the spring be stretched through a small distance d x dx dx. Then work done in stretching the spring through a distance d x dx dx is d W = F d x , dW=Fdx, dW=Fdx, where F is the force applied to
stretch the spring.
What is the formula for work and kinetic energy?
Relation bewteen KE and W: The work done on an object by a net force equals the change in kinetic energy of the object: W = KEf – KEi. This relationship is called the work-energy theorem. ):
How do you calculate the energy of a spring?
Energy stored in a spring
1. Work is done when a spring is extended or compressed . Elastic potential energy is stored in the spring.
2. The elastic potential energy stored can be calculated using the equation:
3. elastic potential energy = 0.5 × spring constant × (extension) 2
How do you calculate work done in stretching a spring?
We can find the spring constant of the spring from the given data for the 4 kg mass. Then we use x = F/k to find the displacement of a 1.5 kg mass. The work that must be done to stretch spring a
distance x from its equilibrium position is W = ½kx2.
What is relation between kinetic energy and work?
Thus, the work-energy theorem states that: The net work done by the force on an object is equal to change in its kinetic energy.
What is relation between work done and kinetic energy?
The work-energy theorem states that the net work done by the forces on an object equals the change in its kinetic energy.
What is work done formula?
To express this concept mathematically, the work W is equal to the force f times the distance d, or W = fd. If the force is being exerted at an angle θ to the displacement, the work done is W = fd
cos θ.
Is kinetic energy and work done the same?
A moving object has kinetic energy because work has been done on it. When work is done energy in one form is transferred to the kinetic energy of the moving object. To stop the object again, the same
amount of work would have to be done to bring it back to rest.
What is relation between work and energy?
Transferring energy can be in the form of force. This amount of energy transferred by the force to move an object is called work or work done. Thus, the relation between Work and Energy is direct.
That is, the difference in the Kinetic energy of an object is work done by an object.
Is KE equal to PE?
As it falls, its total energy (the sum of the KE and the PE) remains constant and equal to its initial PE.
How do you calculate KE?
To calculate kinetic energy:
1. Find the square of the velocity of the object.
2. Multiply this square by the mass of the object.
3. The product is the kinetic energy of the object.
What is k in Hooke’s Law?
Hooke’s law measures the force exerted by the spring to the object attached to it with the help of the following equation. F = –kx. Where k is the spring constantand measures how stiff and strong the
spring is and x is the distance the spring is stretched or compressed away from its equilibrium or rest position.
What is the equation used to calculate kinetic energy?
– let’s suppose the mass of the object is 55 kg and velocity is 3.87m/s – enter the values in kinetic energy equation: (KE = 0.5 x mv2) = 0.5*55*3.872 = 411.675 J. KE = 411.675 J. – Manual
calculations can be frustrating. To avoid unsatisfying calculations, the best option is to use kinetic energy calculator for calculating kinetic energy of an object.
How to derive the formula for kinetic energy?
The formula for kinetic energy
Derive the formula of kinetic energy
Kinetic energy equation derivation
Kinetic energy derivation calculus
Derivation of kinetic energy using algebra
What is the formula for potential energy of a spring?
Potential Energy ((E)) of a spring is the energy associated with the state of compression or expansion of an elastic spring. This potential energy of the spring can do work that is given by the
formula, (E=W=frac{1}{2} k x^{2}) where (W) is the work done (k) is the constant of the spring and is called spring constant or force constant
How to maximize kinetic energy?
– The first step is to design the vectors of velocity for each of the bodies before and after the collision. – Choose the positive direction, usually toward the right. – Write the conservation of
momentum and kinetic energy principles or the expressions (1) and (2) from above. – Solve the system of equations (1) and (2). | {"url":"https://www.goodgraeff.com/what-is-the-equation-for-kinetic-energy-of-a-spring/","timestamp":"2024-11-05T01:31:01Z","content_type":"text/html","content_length":"56373","record_id":"<urn:uuid:d9430558-3b25-48ff-88ed-c8b57e06535b>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00715.warc.gz"} |
Group of Quantum Optics (GOQ)
The research in the Group Quantum Optics while cmprise the foundations of quantum mechanics as well as the emerging technology applications related to them, such as quantum cryptography, quantum
teleportation and quantum computing. Research is mostly theoretical, but the group has also made inroads in the experimental investigations.
Concepts: quantum mechanics, entanglement and superposition
Quantum mechanics was created in the first quarter of the twentieth century to expand the physical theories at that time, which could not give account of several phenomena involving light, atoms and
molecules. The result not only changed our understanding of the fundamentals of physics, but also our own basic notions about the microscopic universe.
The electron orbitals of the hydrogen atom for various energies. The orbitals indicate the probability of finding an electron at each point (the brighter the region, the more likely to find an
electron): quantum mechanics indicates only probabilities, not exact locations. The energy growths from top to bottom and from left to right. Source: Wikipedia.
In quantum theory, there seems to be an intrinsic indeterminism in nature (although the results based on the probabilistic theory are closer to the experimental measurements than those of classical
physics). In addition, different physical states - a physical state is a "situation" in which an object can be found, as its position, its temperature and so on. - may be somewhat overlapped,
combined to form a third different state. The theory has also shown that the notion of superposition of states can be extended to pairs of bodies, so that everything happens as if it was not possible
to associate a physical state of each of them individually, but only to the pair as a whole - and it was only possible to change the state of the whole pair, not of each object separately (it is the
"entanglement"). For people in general, these anti-intuitive features remained just as curious and fascinating abstractions until, in the 80s, began the developments of the theoretical bases for the
technologies that depend crucially on them. Today, we talk about quantum computers, quantum cryptography, quantum teleportation, etc.
In general, these technologies (most still are in a very early stage) depend on two fundamental quantum phenomena, closely related to the anti-intuitive features described above: quantum
superposition and quantum entanglement (the latter can be considered a kind of superposition involving two or more physical systems).
The superposition -
Here is an example: your computer is now in the physical state "on". The plastic housing is in its physical state "black" (say it is black). Of course he can not be switched on and off at the same
time, or the same part of the carcass being white and black
. Likewise, a subatomic particle can not be in two states simultaneously. On the other hand, it can be in a combination of them or superposition, which produces a
, different state. My computer could, if it was an electron or an atom, be in a state "on + off". This is equivalent to be both on and off: it is a third state, distinct from the other two, but
"formed" by both.
Entanglement –
if the concept is extended to overlap the two systems, for example, two electrons can form superposition states corresponding to two electrons. To illustrate this, suppose that two computers, A and B
(one black and one white) could be placed in an entangled state - or a superposition of the situation (state) Black / B white with black B / A white, i.e., it is not possible to assign a physical
state to each individual "particle" (in this case your computer); it is possible to identify a state just to the whole system. Any change in the state then reaches the two particles simultaneously -
even if they are separated by large distances.
Contrarily to what may look like - a change in the entangled state by itself can not transmit information instantaneously through space.
Understanding the superposition: "particles" are not particles
It's hard to imagine a quantum superposition because we tend to imagine subatomic “particles” as material objects in the traditional sense, like tiny colored billiard balls flying through space or
around atoms. In fact the theory does not allow to assess such a detailed description. Quantum mechanics is a theory that leads to a kind of probabilistic wave description of particles and we know
that two waves can superimpose themselves – e.g., the waves in a pool, in ocean waves that overlap. In the case of "quantum particles", the ability of waves to overlap manifests the possibility of
combining different physical states.
How can the same object manifest itself in two different ways without being any of them? The Wikipedia site has a very interesting idea to explain this, with the figure above, written by
Jean-Christophe Benoist. Here, a cylinder produces a square shadow on a wall and circular shadow on another wall. If we see only the shadow - and mistakenly identify the shadow with the object itself
– it would seem a contradiction: how can something be round and square at the same time?! But the object is not either a circle or a square: it is another entity (a cylinder), we were not looking
directly - but that is manifested as a square or a circle depending on the situation. But beware: this is only an analogy. Both particles and waves as entities "either-no-wave-nor-particles" are
common in three dimensional space. The correspondence is that, in physics, we can only measure the properties of “things”, we do not have access to the “things” themselves. Of course no one ever saw
a computer in such a situation. It happens that this combination is more unstable the greater the mass of the object in question. Very small disturbances can transform them quickly in one of two
superposed states, breaking the combination. The first superposition quantum states of position within an entire atom was only obtained only in 1996. Something of the size of a computer would
"collapse" almost instantaneously to one of the states, on or off, long before it could be observed. This rapid evolution of a superposition of states towards the individual states, due to the
minimum disturbances of the surrounding environment, constitutes what is called "loss of coherence" or “decoherence”.
Now, let's see what you can do with all this!
New quantum technologies
Quantum cryptography – This is actually a type of quantum distribution of secret keys, which can be used to encode and decode messages. The main difference between the quantum key distribution and
the usual encryption systems is that it is possible to know if the communication between the parties is being intercepted, and thus nullify the action of a spy. This is related to the collapse of
quantum states – superposition states or entangled, described above. In this case, the disturbance that causes the collapse is the action of the interceptor.
Note that in quantum cryptography, the generated secret keys are identical – a random sequence of 0s and 1s, which are held by the interlocutors - Alice and Bob. The keys will be available for
encoding the message being sent (by Alice, for example) and decoding the received message (for Bob). This is done by transmitting signals - quantum states of light from Alice to Bob. Despite problems
such as losses caused by the “real world” transmission methods – e.g., via fiber optics, quantum key distribution has been successfully accomplished over increasingly larger distances. The first
protocol for quantum key distribution (BB84) was conceived in 1984 and the first demonstration in the laboratory was done in 1989.
Quantum teleportation – is a technique to transfer information from one place to another. That is, the quantum teleportation transmits no matter, just information. Moreover, it can not be transmitted
at speeds greater than light. However, it can transmit all the information about physical states in a quantum superposition, which until 1993 no one knew how to do it without destroying the
superposition. The teleportation also uses quantum entanglement - is at the time of its collapse that some information is transmitted from one location to another (taking advantage of the fact that
the collapse happened in the state of the pair as a whole, regardless of the distance separating the interlocutors). But in order to turn transmitted information intelligible, additional information
shall be communicated through conventional (classical) channels (so that you can not have data transmission at infinite speeds). Quantum teleportation was proposed in 1993 and has been demonstrated
experimentally in 1997 for the first time.
Quantum computing – For some applications, a quantum computer would make calculations millions of times faster than an ordinary computer. For others, however, there is not much difference. Among the
cases where it is much faster, is decoding messages encrypted by the method currently used in the internet (multiplication of prime numbers). That is, these computers could break most current
encodings. Other cases are the data search in a disordered file and computer simulations of quantum complex systems.
In quantum computers, the information unit is not the bit, as in normal computing, but the qubit (pronounced "quiu-bit"). The difference is that a bit can have only two values - represented by, say,
0 and 1 -, while a qubit can assume the value 0, 1 or any of the infinite quantum superpositions of these two states. In addition, several qubits can be entangled with each other. Recall that an
entangled state changes affect all qubits simultaneously. The two things together - the infinite superpositions with 0 and 1 and entanglement - can be used in some cases to dramatically increase the
speed of processing.
The qubit can be physically implemented in several ways. You can implement qubits with energy states of atoms. Atom with lowest energy is the "0"; atom with higher energy is the "1". They may be
entangled in various ways, e.g., by making them pass together through suitably arranged electromagnetic fields. Another example: the spin of a particle like the electron. Spin is a characteristic of
the micro-particles world which has no counterpart in the macroscopic world, but that is very similar to a common rotation. Spin "spinning" to one side represents the "0" to the other, represents the
The fight against the loss of coherence - Perhaps the biggest challenges for the scientific research in this area are in quantum computers. In this case (and in cryptography and quantum
teleportation), one of the main problems is the extreme instability of superposed and entangled states. They can "collapse" to separable states due to perturbations of the environment – it is the
process of decoherence. It is therefore necessary that quantum computers are able carry their operations before losing coherence, otherwise the computer will suffer the so-called "sudden death"
(stops working during processing).
Moreover, the greater the number of entangled physical systems, in general the more unstable is the entanglement. Only in 1996 it was achieved in laboratory an entangled state of two atoms. Nowadays,
it is possible to do so with trillions of atoms, but the entanglement is short lived and the system is not quite suitable for quantum computing.
To solve this, it is necessary not only to build a fast computer, but also slow down the loss of coherence. In order to do so, we need to know all the possible disturbances that may occur in the
considered physical system and how they influence the loss of coherence.
Atoms in cavities – also must be taken into account that, depending on the way entanglement is generated and how the qubit is implemented - with electrons, photons, or atoms – we may have more or
less robustness against loss of coherence. In our group research in mainly based on atomic qubits, which may be entangled by means of electromagnetic fields confined in highly reflective cavities
(cavity QED). The energy states of the atom contain the information "0" or "1".
Another very interesting system that has been studied in our group is called the optomechanical oscillator - tiny membranes may have its motion controlled so that they become "quantum objects" -
subject to the laws of quantum mechanics, and therefore with the possibility form of quantum superpositions and entangled states with objects containing many atoms, almost macroscopic.
(Figure 1) - Scheme of a trap for atoms using electromagnetic fields to confine ions (atoms without some electrons and therefore electrically charged that can be "trapped" by electric and magnetic
(Figure 3)
In Figure 2 and 3: Scheme of two optical cavities crossed by a flying atom
Sources: J.M. Raimond, T. Meunier, P. Bertet, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune and S. Haroche, Probing a quantum field in a photon box, J. Phys. B: At. Mol. Opt. Phys. 38, S535
(2005); Master thesis of Fernando Luís Semião da Silva, IFGW/Unicamp (2002), p. 10. Doctoral thesis of Fabiano Kenji Nohama, IFGW/Unicamp (2008), p. 41. Master thesis of Ricardo José Missori, IFGW/
Unicamp (2003), p. 46.
The atoms can be entangled after passing through QED cavities arranged in an appropriate manner, as shown below. Several possibilities have been studied, including cavities connected by optical
fibers, since there should be some communication between qubits at different locations in the circuit of a quantum computer.
Scheme of an apparatus to entangle energy states of two atoms. In the left and right are shown two QED cavities. Within each one an atom is enclosed (in blue). The interaction between atoms and the
electromagnetic fields inside the cavities produces two photons in each one, which follow the dashed path until the photon detectors D1 and D2. In the middle of that path there is a semitransparent
mirror (BS) that entangles the two photons. This mirror and the detectors are able to prepare, under certain circumstances, the two atoms and the two fields within the cavities in entangled states.
Note that the two cavities are separated; the entanglement can be obtained even when the atoms are far apart.
Source: Master thesis of Bruno Ferreira de Camargo Yabu-uti, IFGW / UNICAMP (2007), p. 36.
The rate of loss of coherence also depends on the type of generated entanglement. Our Quantum Optics Group, for example, proposed in 2007 a new type of entangled state, called " cluster type
entangled coherent state" (CTECS), whose advantage is just its robustness against the influence of environment. The CTECS fall into the category of entangled states with continuous variables, in
contrast to states with discrete variables. The difference is that the discrete state variables are characterized by only certain specific energies (the energy is quantized), whereas continuous
variables may be in a range of continuously varying energy values. For example, the laser light source represents a continuous variable state easy to implement and handle. It also investigated how to
transfer data from discrete variables entangled states to continuous variables and vice versa, as well as protocols for quantum cryptography using continuous variables.
Experimental apparatus proposed by the Group to make the entangled state CTECS. The scheme uses strategically placed QED cavities so that two atoms cross the cavities and generate entangled CTECS in
Source: PhD thesis of Parmezani Pablo Munhoz, IFGW / UNICAMP (2008), p. 40.
A set of squeezed states of light to be used in an alternative protocol quantum key distribution. Two distinct states are used to encode the bits 0 and 1 while a third state "decoy" is used to
confuse a possible spy.
Source: Master thesis of Douglas Delgado de Souza, IFGW / UNICAMP (2011), p. 100.
It is important to emphasize that our Group investigates not only the practical possibilities of implementing the new quantum technologies, but also the foundations of quantum mechanics, for a better
understanding of the extraordinary nature of quantum states and their relation to the macroscopic world.
History of the group
The Group began with the researche of José Antonio Roversi while he was still in the Group of Phase Transitions in the IFGW, led by prof. Paulo Roberto de Paula e Silva. With the arrival of Antonio
Vidiella Barranco, the Quantum Optics group was formally constituted. In the beginning, it was studied methods of generation of quantum states (entangled or not) from the theoretical point of view.
Later, they began to work also with quantum cryptography, as well as in the investigation of the physical implementation of the entangled states, mainly in atoms in optical cavities, trapped ions and
in optomechanical systems. | {"url":"https://portal.ifi.unicamp.br/en/research/deq-department-of-quantum-electronics/group-of-quantum-optics-goq","timestamp":"2024-11-04T14:50:00Z","content_type":"text/html","content_length":"149239","record_id":"<urn:uuid:e300dcc4-7525-41a4-92f4-8502411f6a9a>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00578.warc.gz"} |
Chris Adolph :: Panel @Essex
Essex 2G Summer 2022
Panel Data Analysisfor Comparative Research Class meets: MTWThF 10:00–13:30 Taught in person at ESS
Offered every Summer at the University of Essex ↤ For my University ofWashington course, click Panel Data again.
Lectures Click on lecture titles to view slides or the buttons to download them as PDFs.
Topic 1
Course Introduction / Review of Linear Regression and Simulation
Students looking to brush up on matrix algebra may want to read through Kevin Quinn’s matrix algebra review. Finally, in conjunction with these slides I discuss examples that can be found in slides
on labor standards in Africa and intimate partner homicide in the US.
Lab 1
Introduction to R
This lab comes with three example scripts. First, there is an R code and data for exploratory data analysis using histograms and boxplots. Next, there is an R code and data for a simple bivariate
linear regression. Finally, there is an R code and data for a multiple regression example. Interested students can find detailed instructions for downloading, installing, and learning my recommended
software for quantitative social science here. Focus on steps 1.1 and 1.3 for now, and then, optionally, step 1.2. (Note: These recommendations may seem dated, as many students prefer to use RStudio
as an integrated design environment in combination with RMarkdown. You are free to follow that model, which minimizes start-up costs. I still prefer a combination of Emacs, the plain R console, and
Latex/XeLatex for my own productivity, with occasional use of Adobe Illustrator for graphics touch-up.)
Topic 2
Basic Concepts for Time Series: Trends, Lags, and Cycles
The univariate time series simulation function for R mentioned in the lecture is available here; this function allows for deterministic trends, stationary and nonstationary ARMA processes, and
additive or multiplicative seasonality. Also available is a simpler but less flexible R script showing simulation and diagnostics of stationary processes using built-in functions.
Topic 3
Models of Stationary Time Series
Example code and csv data for estimating and interpreting ARMA models in R.
Topic 4
Models of Non-stationary Time Series
Example code for estimating and interpreting ARIMA and ECM models in R. You will also need these presidential approval data and this helper function for plotting counterfactual time series using R
base graphics.
Topic 5
Basic Concepts for Panel Data
For the curious, the R script used to construct the example plots in the first half of this lecture is here.
Topic 6
Panel Data Models with Many Time Periods
Panel ARIMA and fixed effects models in R: example code and csv data for estimating and interpreting panel models with a large number of periods and a small number of cross-sectional units.
Topic 7
Panel Data Models with Few Time Periods
Panel GMM models (Arellano-Bond/Blundell-Bond) in R: example code, helper functions, and csv data for estimating and interpreting panel models with a small number of periods and a large number of
cross-sectional units. On Nickell bias: a script to plot the asymptotic results of Nickell (1981) as well as a helper file and two Monte Carlo scripts (large N / large β and large N / small β) to
produce the finite sample results in the lecture slides.
Topic 8
Heteroskedasticity in Panel Data
Topic 9
In-Sample Simulation for Panel Data Models
Example code for simulating in-sample unit-by-unit from a panel data model. Uses the cigarette data and helper functions from Topic 7.
Advanced Topic 1
Missing Data and Multiple Imputation for Panel Data
Advanced Topic 2
Panel Data with Binary Dependent Variables
Student Assignments
Course Problem Set
Problems may be turned in at any time during the course
Data for problems 1, 3, and 4 in comma-separated variable format. Data for problem 4. Data for problem 5. Data for Bonus Problem A. Data for Bonus Problems B and C. Data for Bonus Problem D. | {"url":"http://faculty.washington.edu/cadolph/index.php?page=23&uwphoto=7&banner=6","timestamp":"2024-11-05T06:31:35Z","content_type":"application/xhtml+xml","content_length":"24346","record_id":"<urn:uuid:fcbd7e53-242d-4025-b6fc-8ddd3498a3d1>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00351.warc.gz"} |
Can anyone answer this question? Please!
johnny Registered Posts: 1 New contributor 🐸
Riley Ltd has made the following estimates for the next month:
Selling Price £25 per unit
Variable Cost £10 per unit
Fixed Costs £300,000
Forecast Output 30,000 units
Maximum Output 40,000 units
a. The profit volume ratio
b. The break even point in units
c. The break even point in sales revenue
d. The margin of safety at the forecast level of output expressed both in units and as a percentage of sales
e. The number of units required to generate a profit £100,000
f. Calculate the profit expected at the forecast level of output and at the maximum level of output
Riley Ltd has made the following estimates for the next month:
Selling Price £25 per unit
Variable Cost £10 per unit
Fixed Costs £300,000
Forecast Output 30,000 units
Maximum Output 40,000 units
a. The profit volume ratio
b. The break even point in units
c. The break even point in sales revenue
d. The margin of safety at the forecast level of output expressed both in units and as a percentage of sales
e. The number of units required to generate a profit £100,000
f. Calculate the profit expected at the forecast level of output and at the maximum level of output
These will do most of that I think or give you the figures to do so
Chloe -Contribution = Selling Price per Unit-Variable Costs per Unit
Buys - Breakeven in Units = Fixed Costs/Contribution
Pink - PV Ratio = Contribution/Sales Price
Biscuits-Breakeven In Revenue = Fixed Costs/PV Ratio
Mummy-Margin of Safety In Units = Forecast - Breakeven
Makes -Margin of Safety % = Margin of Safety in Units/Forecast x 100
Them-Target Profit = Fixed Costs + Target Profit/Contribution
• Wow, someones using my Chloe acronym to actually remember the formulas!! Just shown this to my daughter - Chloe who thinks she's famous as her name is "published"!!! Lol
Sorry, had to share!
• please tell me if i got the right answer!
A 06
B 20,000 BEP
C £500,000 breakeven in revue
D 10,000 unit
E 26666 units need for £100,000 proft
F unknown how to work it out
please PM me too
• Hi any2002uk, I mostly agree with your answers.
In A) I assume you meant Profit volume Ratio = 0.6
In D) the margin of safety of 10,000 units is 33.3 % as a percentage of sales
For E) I would have said 26,667 units as 26,666 comes in slightly under the target profit but I expect you would get the mark.
In F) I make the expected profits
i) £15 contribution x 30,000 units - £300,000 fixed O/H = £150,000
ii) £15 x 40,000 - £300,000 = £300,000
Hope this helps! :001_smile:
• where do i and ii from?
how will i know that the quesiton will be asking for CS ratio? can you gve me the example that will appear in the question so i know that i need to use this method
where do i and ii from?
how will i know that the question will be asking for CS ratio? can you give me the example that will appear in the question so i know that i need to use this method
Re: i & ii - Well, its just the way I thought about it. You would get the same result by subtracting the total costs from the total sales revenue if you prefer that method.
I'm not sure I understand your Q about CS ratio - are you asking about part d?
• Please can someone help me on this question, I'd be most great full!
I've been given the information as per below:
Broadsworth has now incurred a fixed cost of £20000
Other than the change in fixed costs you can assume that the sales demand, selling price and all costs remain as budgeted.
Number of units to revise fixed costs £3200
Revised margin of safety % 36
Revised margin of safety in sales revenue 70200
I have to now work out:
The total cost of one unit =
The full absorption cost of one unit =
The marginal cost of one unit =
The full absorption cost of a batch =
The marginal cost of a batch =
• Breakeven is 2400 units and budgeted fix overhead is £60000 | {"url":"https://forums.aat.org.uk/Forum/discussion/20827/can-anyone-answer-this-question-please","timestamp":"2024-11-14T08:28:54Z","content_type":"text/html","content_length":"310077","record_id":"<urn:uuid:dfd1387f-4ae9-470f-9ad6-3d141fec61ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00635.warc.gz"} |
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Two step proxy regression
Title: Two-Step Proxy Regression
Proxy regression is a statistical method commonly used to estimate the relationship between two or more variables. In this article, we introduce and discuss the two-step proxy regression technique,
which can be used to estimate the causal relationship between a dependent variable and one or more independent variables. This method is particularly useful when the proxy variable is not observed
directly, but can be measured using other observed variables. We also discuss the assumptions of this method and consider some limitations and extensions.
Two-step proxy regression is a statistical method which can be used to estimate the relationship between a dependent variable and one or more independent variables. This technique is particularly
useful when the proxy variable is not observed directly, but can be measured using other observed variables. For example, in settings where it is difficult or impossible to directly measure a
variable of interest, such as a disease incidence or educational attainment, researchers may use proxy variables such as income or age to estimate the relationship between this variable and the
outcome of interest. In this scenario, two-step proxy regression can be used to account for the fact that the proxy variable is not directly observed, and to estimate the causal relationship between
the variables of interest. Additionally, researchers may use two-step proxy regression when they are trying to estimate the relationship between a dependent variable and multiple independent
variables, and they want to account for the fact that some of these independent variables may be correlated with the proxy variable. Overall, two-step proxy regression can be a useful technique in a
variety of settings where the relationship between variables is not straightforward and needs to be estimated using indirect measures.
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Pros and Cons of Two step proxy regression
1. Two-step proxy regression can be used in settings where the dependent variable is difficult or impossible to directly measure, allowing researchers to estimate the relationship between the
variable of interest and one or more independent variables.
2. This technique can account for the fact that the proxy variable is not directly observed and may be correlated with the dependent variable.
3. Two-step proxy regression can be used to estimate the relationship between the dependent variable and multiple independent variables, accounting for the fact that some of these independent
variables may be correlated with the proxy variable.
4. The use of a proxy variable in two-step proxy regression, can make it possible to control for confounding variables, which are variables that might be related to both the independent and
dependent variable.
1. Two-step proxy regression assumes that the relationship between the dependent variable and the proxy variable is linear, and this assumption may not always hold in practice.
2. This technique can be sensitive to the choice of the proxy variable. If the proxy variable used is not appropriate or not enough related to the dependent variable, the estimates may be biased or
have low precision.
3. Two-step proxy regression can be computationally intensive, particularly if the model has a large number of independent variables or a large sample size.
4. The use of a proxy variable in two-step proxy regression, can make it harder to interpret the results and to make causal inference, because the relationship between the independent variable and
the proxy variable may not be straightforward.
Overall, two-step proxy regression can be a useful technique, but researchers should be aware of the potential limitations and assumptions present when using this method, and be careful when
interpreting the results.
Future of Two-Step Proxy Regression
The demand for accurate and efficient methods to estimate the relationships between variables is increasing, especially in fields such as economics, public health, and social sciences. Two-step proxy
regression is a statistical method that can be used to estimate the relationship between a dependent variable and one or more independent variables. While this technique has been around for several
years, it is likely to continue to develop and improve in the near future.
One potential development of two-step proxy regression is the integration of machine learning and artificial intelligence techniques, to make the technique more advanced and powerful. This could
allow for more accurate and predictions, and also allow for the incorporation of big data and complex datasets.
Another potential development is the use of more advanced statistical tests, that can account for potential confounding factors and external influences, making the results more robust and reliable.
Additionally, new methods might be developed to handle missing data and outliers in the dataset, which can be common in real-world data.
As well, it is likely that more research will be conducted on the assumptions and limitations of the method, and ways to mitigate them, leading to a more comprehensive understanding of the technique.
Overall, the future of two-step proxy regression is likely to be promising and continues to be a useful tool for researchers who want to estimate the relationship between variables using indirect
measures. And as new technology and techniques develops, it is likely to become more powerful and efficient.
Key Definitions for Better Understanding
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How could Two step proxy regression be improved?
Two-step proxy regression is a statistical method that can be used to estimate the relationship between a dependent variable and one or more independent variables. While this technique has been
around for several years, it is likely to continue to develop and improve with new technology.
One technology that could be used to improve two-step proxy regression is the use of machine learning algorithms, which are computer programs that can learn from data and make predictions. For
example, one machine learning algorithm that could be used for this purpose is the neural network, which is a type of artificial intelligence that can learn and recognize patterns in data. Another
algorithm that could be used is decision tree, random forest, gradient boosting, or other supervised learning algorithms, which can handle missing values, outliers and non-linear relationships.
Another technology that could be used to improve two-step proxy regression is the use of big data, which is the term used to describe very large datasets. As big data becomes more prevalent, it
becomes possible to use more advanced statistical techniques, such as those used in big data analysis, to estimate the relationships between variables. This technology allows researchers to handle
datasets with large number of observations, missing data and outliers, and use more powerful methods to infer causality
Additionally, technology such as cloud computing, parallel processing and distributed computing makes it possible to store, process and analyze large datasets, making it more feasible to conduct
large-scale studies, which would have been too computationally intensive, or data processing would have been time-consuming.
Overall, the use of machine learning algorithms, big data and cloud computing, can greatly improve the accuracy and efficiency of two-step proxy regression, allowing for more complex analyses to be
performed and more robust results to be obtained.
Two step proxy regression: Insights and Analysis
Two-step proxy regression is a statistical method that can be used to estimate the relationship between a dependent variable and one or more independent variables. The method is particularly useful
when the dependent variable is difficult to measure directly, allowing researchers to estimate the relationship between the variable of interest and one or more independent variables. By using a
proxy variable, researchers can control for confounding variables that are correlated with both the independent and dependent variable, making it possible to obtain more robust results.
The analysis of two-step proxy regression involves first identifying the appropriate independent variables that can be used to predict the dependent variable, and then selecting a proxy variable that
is closely related to these independent variables. Once the proxy variable has been chosen, the researcher can estimate the relationship between the proxy variable and the dependent variable using
the method of least squares or other optimization methods. The validity of the results may be checked through diagnostic tests, such as the coefficient of determination (R2) which measures how much
of the variance in the dependent variable is explained by the independent variable and proxy variable collectively.
Furthermore, the model can be checked for assumptions and limitations. For example, one limitation of the method is that if the relationship between the independent variable and the proxy variable is
non-linear, it must be strengthened by transforming these variables, such as logarithmic or polynomial transformations. Another limitation is that the method may not be suitable for datasets where
the missing data is present and need to be handled.
Considering the insider knowledge, the researcher can gain valuable insights by two-step proxy regression. Researchers can find the relationship between different variables, identify the most
important variables that predict the dependent variable, and understand the underlying mechanisms that drive the relationships. This can help to inform policy decisions or design interventions that
aim to improve the outcomes of interest.
Overall, two-step proxy regression is a useful tool for researchers who want to estimate the relationship between variables using indirect measures, it provides insights and allows researchers to
gain a better understanding of the underlying relationships between different variables, which can lead to better informed decision making.
Two step proxy regression: Expert Opinion
The field of statistics and data analysis is constantly evolving, and experts in the field have different opinions on the constant development of two-step proxy regression. Some experts, when
encountered with a new technique it can be improved by using machine learning algorithms, such as neural networks and big data, to make it more powerful, accurate and efficient. They also suggest
ways to handle missing data and outliers, and to account for external influences. They also highlight that this technique is useful for researchers who want to estimate the relationship between
variables using indirect measures.
Other experts, suggest that the best approach to two-step proxy regression is to focus on the validity of the results, pay attention to the assumptions of the method, and be cautious about
overfitting. They also highlight the importance of selecting appropriate independent variables and a proxy variable that is closely related to these independent variables, and that the method should
be used in conjunction with other methods to validate the results.
Some experts also consider that the use of a proxy variable in two-step proxy regression, can make it harder to interpret the results, and to make causal inference, because the relationship between
the independent variable and the proxy variable may not be straightforward. And that the use of a proxy could be a challenge for the researcher, as it requires to understand the underlying mechanisms
that drive the relationship between the independent variable and the proxy variable.
Overall, experts in the field of statistical analysis have different opinions on the development of two-step proxy regression. While some experts suggest using machine learning algorithms, big data
and cloud computing to improve the accuracy and efficiency of the method, others focus on the importance of validating the results, selecting appropriate independent variables and interpreting the
results. Researchers should be aware of the potential challenges and limitations of this method and use it in conjunction with other methods to validate the results.
Final Thoughts
In conclusion, two-step proxy regression is a statistical method that can be used to estimate the relationship between a dependent variable and one or more independent variables. This technique is
particularly useful when the dependent variable is difficult to measure directly, allowing researchers to estimate the relationship between the variable of interest and one or more independent
variables. By using a proxy variable, researchers can control for confounding variables that are correlated with both the independent and dependent variable, making it possible to obtain more robust
While this method has several advantages, it also has limitations and assumptions that researchers should be aware of. Therefore, researchers should be cautious when interpreting the results and use
it in conjunction with other methods to validate the results.
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In summary, two-step proxy regression is a useful tool for researchers who want to estimate the relationship between variables using indirect measures. By understanding the limitations and
assumptions of the method and carefully selecting the appropriate independent variables and proxy variable, researchers can gain valuable insights into the relationships between different variables,
which can lead to better informed decision making.
What is two-step proxy regression?
Two-step proxy regression is a statistical method that can be used to estimate the relationship between a dependent variable and one or more independent variables. This technique is particularly
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How does two-step proxy regression work?
Two-step proxy regression works by using a proxy variable to estimate the relationship between a dependent variable and one or more independent variables. The proxy variable is chosen based on its
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What are the assumptions of two-step proxy regression?
Two-step proxy regression assumes that the relationship between the proxy variable and the dependent variable is linear, and that the relationship between the independent variables and the proxy
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dependent variable, which may not always be true.
What is machine learning and how is it used in two-step proxy regression?
Machine learning is a technique used in two-step proxy regression to improve the accuracy and efficiency of the method. Machine learning algorithms, such as neural networks and big data, are used to
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What is big data and how is it used in two-step proxy regression?
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You might also be interested in... | {"url":"https://cyber-gateway.net/nfo/14799-two-step-proxy-regression","timestamp":"2024-11-03T20:08:20Z","content_type":"text/html","content_length":"79251","record_id":"<urn:uuid:f20b2d70-8559-4217-aeb7-dede0104e2ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00703.warc.gz"} |
The spread or variation in the sampling distribution of means is
called the __________ of the...
The spread or variation in the sampling distribution of means is called the __________ of the...
The spread or variation in the sampling distribution of means is called the __________ of the mean.
Group of answer choices:
Standard error
Standard deviation
Standard dispersion
The Central Limit Theorem tells us that : .
Group of answer choices
- If we keep taking samples from a population, and plot those means on a curve, they will form a bell-shaped, or normal, curve.
- Standard error is a measure of how much the sample means do vary.
- A sampling distribution of means from small samples will have a smaller standard error than means of larger samples do.
- A and B
- All of the above
Standard dispersion
The spread or variation in the sampling distribution of means is called the __________ of the mean.
This is called Standard deviation as it measures the spread about the mean
- If we keep taking samples from a population, and plot those means on a curve, they will form a bell-shaped, or normal, curve.
- Standard error is a measure of how much the sample means do vary.
TRUE .
- A sampling distribution of means from small samples will have a smaller standard error than means of larger samples do.
FASLE ( as sample size increases the standard error decreases )
- A and B is the correct answer | {"url":"https://justaaa.com/statistics-and-probability/66550-the-spread-or-variation-in-the-sampling","timestamp":"2024-11-02T11:42:13Z","content_type":"text/html","content_length":"42551","record_id":"<urn:uuid:1fbe786c-5b57-49bb-ac2a-a16349afa369>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00869.warc.gz"} |
[QSMS-BK21 Seminar 2022-08-29] Symplectic Cohomology, String Topology, and Deformations
• Date : 2022-08-29 10:00 ~ 18:00. (Lunch: 13:00 - 14:00, Break: 15:20 - 15:30)
• Place : 129-406 (SNU)
• Speaker : 김용환 (SNU)
• Title : Symplectic Cohomology, String Topology, and Deformations
• Abstract :
This talk is intended to be a review of Seidel's 2002 ICM talk "Fukaya Categories and Deformations", from the viewpoint of symplectic cohomology. Given an affine variety, one can consider its normal
crossings divisor compactification to view it as a Liouville domain: then it is believed that the differentials in symplectic cohomology should be related to relative Gromov-Witten invariants.
Extrapolating this viewpoint, Borman-Sheridan-Varolgunes has conjectured that quantum cohomology can be viewed as a deformation of the "L-infinity structure" on symplectic cohomology. Though the
existence of a L-infinity structure has not been proven yet, works of Tonkonog and Ganatra-Pomerleano have verified the Borman-Sheridan class for some cases.
In this talk, I will give an overview of these developments, and introduce some key properties that govern the behavior of holomorphic curves. | {"url":"https://qsms.math.snu.ac.kr/index.php?mid=board_sjXR83&order_type=desc&listStyle=list&sort_index=title&page=5&document_srl=2350&l=ko","timestamp":"2024-11-15T03:41:17Z","content_type":"text/html","content_length":"66200","record_id":"<urn:uuid:b65805e7-56a5-4b1b-abc9-524cf68d2d16>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00846.warc.gz"} |
3.3: Equivalence and Implication (2024)
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Consider two propositions generated by \(p\) and \(q\text{:}\) \(\neg (p \land q)\) and \(\neg p \lor \neg q\text{.}\) At first glance, they are different propositions. In form, they are different,
but they have the same meaning. One way to see this is to substitute actual propositions for \(p\) and \(q\text{;}\) such as \(p\text{:}\) I've been to Toronto; and \(q\text{:}\) I've been to
Then \(\neg (p \land q)\) translates to “I haven't been to both Toronto and Chicago,” while \(\neg p \lor \neg q\) is “I haven't been to Toronto or I haven't been to Chicago.” Determine the truth
values of these propositions. Naturally, they will be true for some people and false for others. What is important is that no matter what truth values they have, \(\neg (p \land q)\) and \(\neg p \
lor \neg q\) will have the same truth value. The easiest way to see this is by examining the truth tables of these propositions.
Table \(\PageIndex{1}\): Truth Tables for \(\neg (p\land q)\) and \(\neg p\lor \neg q\)
\(p\) \(q\) \(\neg(p\land q)\0 \(\neg p\lor \neg q\)
\(0\) \(0\) \(1\) \(1\)
\(0\) \(1\) \(1\) \(1\)
\(1\) \(0\) \(1\) \(1\)
\(1\) \(1\) \(0\) \(0\)
In all four cases, \(\neg (p \land q)\) and \(\neg p \lor \neg q\) have the same truth value. Furthermore, when the biconditional operator is applied to them, the result is a value of true in all
cases. A proposition such as this is called a tautology.
Tautologies and Contradictions
Definition \(\PageIndex{1}\): Tautology
An expression involving logical variables that is true in all cases is a tautology. The number 1 is used to symbolize a tautology.
Example \(\PageIndex{1}\): Some Tautologies
All of the following are tautologies because their truth tables consist of a column of 1's.
1. \((\neg (p \land q))\leftrightarrow ( \neg p \lor \neg q)\text{.}\)
2. \(\displaystyle p \lor \neg p\)
3. \(\displaystyle (p \land q)\to p\)
4. \(\displaystyle q\to (p\lor q)\)
5. \(\displaystyle (p \lor q)\leftrightarrow (q \lor p)\)
Definition \(\PageIndex{2}\): Contradiction
An expression involving logical variables that is false for all cases is called a contradiction. The number 0 is used to symbolize a contradiction.
Example \(\PageIndex{2}\): Some Contradictions
\(p \land \neg p\) and \((p\lor q)\land (\neg p) \land (\neg q)\) are contradictions.
Definition \(\PageIndex{3}\): Equivalence
Let \(S\) be a set of propositions and let \(r\) and \(s\) be propositions generated by \(S\text{.}\) \(r\) and \(s\) are equivalent if and only if \(r\leftrightarrow s\) is a tautology. The
equivalence of \(r\) and \(s\) is denoted \(r \iff s\text{.}\)
Equivalence is to logic as equality is to algebra. Just as there are many ways of writing an algebraic expression, the same logical meaning can be expressed in many different ways.
Example \(\PageIndex{3}\): Some Equivalences
The following are all equivalences:
1. \((p \land q)\lor (\neg p \land q)\iff q\text{.}\)
2. \(\displaystyle p \to q \iff \neg q \rightarrow \neg p\)
3. \(p \lor q \iff q \lor p\text{.}\)
All tautologies are equivalent to one another.
Example \(\PageIndex{4}\): An Equivalence to \(1\)
\(p\lor \neg p\iff 1\text{.}\)
All contradictions are equivalent to one another.
Example \(\PageIndex{5}\): An Equivalence to \(0\)
\(p\land \neg p\iff 0\text{.}\)
Consider the two propositions:
Table \(\PageIndex{2}\)
\(x\): The money is behind Door A; and
\(y\): The money is behind Door A or Door B
Imagine that you were told that there is a large sum of money behind one of two doors marked A and B, and that one of the two propositions \(x\) and \(y\) is true and the other is false. Which door
would you choose? All that you need to realize is that if \(x\) is true, then \(y\) will also be true. Since we know that this can't be the case, \(y\) must be the true proposition and the money is
behind Door B.
This is an example of a situation in which the truth of one proposition leads to the truth of another. Certainly, \(y\) can be true when \(x\) is false; but \(x\) can't be true when \(y\) is false.
In this case, we say that \(x\) implies \(y\text{.}\)
Consider the truth table of \(p \to q\text{,}\) Table 3.1.1. If \(p\) implies \(q\text{,}\) then the third case can be ruled out, since it is the case that makes a conditional proposition false.
Definition \(\PageIndex{4}\): Implication
Let \(S\) be a set of propositions and let \(r\) and \(s\) be propositions generated by \(S\text{.}\) We say that \(r\) implies \(s\) if \(r \to s\) is a tautology. We write \(r \Rightarrow s\) to
indicate this implication.
Example \(\PageIndex{6}\): Disjunctive Addition
A commonly used implication called “disjunctive addition” is \(p \Rightarrow (p \lor q)\text{,}\) which is verified by truth table Table \(\PageIndex{3}\).
Table \(\PageIndex{3}\): Truth Table to verify that \(p\Rightarrow (p\lor q)\)
\(p\) \(q\) \(p\lor q\) \(p\to o\lor q\)
\(0\) \(0\) \(0\) \(1\)
\(0\) \(1\) \(1\) \(1\)
\(1\) \(0\) \(1\) \(1\)
\(1\) \(1\) \(1\) \(1\)
If we let \(p\) represent “The money is behind Door A” and \(q\) represent “The money is behind Door B,” \(p \Rightarrow (p \lor q)\) is a formalized version of the reasoning used in Example \(\
PageIndex{6}\). A common name for this implication is disjunctive addition. In the next section we will consider some of the most commonly used implications and equivalences.
When we defined what we mean by a Proposition Generated by a Set, Definition 3.2.1, we didn't include the conditional and biconditional operators. This was because of the two equivalences \(p \to q \
Leftrightarrow \neg p \lor q\) and \(p \leftrightarrow q \Leftrightarrow (p \land q) \lor (\neg p \land \neg q)\text{.}\) Therefore, any proposition that includes the conditional or biconditional
operators can be written in an equivalent way using only conjunction, disjunction, and negation. We could even dispense with disjunction since \(p \lor q\) is equivalent to a proposition that uses
only conjunction and negation.
Universal Operation
We close this section with a final logical operation, the Sheffer Stroke, that has the interesting property that all other logical operations can be created from it. You can explore this operation in
Exercise \(\PageIndex{8}\)
Exercise \(\PageIndex{1}\)
Given the following propositions generated by \(p\text{,}\) \(q\text{,}\) and \(r\text{,}\) which are equivalent to one another?
1. \(\displaystyle (p \land r) \lor q\)
2. \(\displaystyle p\lor (r\lor q)\)
3. \(\displaystyle r \land p\)
4. \(\displaystyle \neg r \lor p\)
5. \(\displaystyle (p\lor q)\land (r\lor q)\)
6. \(\displaystyle r\to p\)
7. \(\displaystyle r \lor \neg p\)
8. \(\displaystyle p\to r\)
\(a\Leftrightarrow e, d\Leftrightarrow f, g\Leftrightarrow h\)
Exercise \(\PageIndex{2}\)
1. Construct the truth table for \(x= (p \land \neg q) \lor (r \land p)\text{.}\)
2. Give an example other than \(x\) itself of a proposition generated by \(p\text{,}\) \(q\text{,}\) and \(r\) that is equivalent to \(x\text{.}\)
3. Give an example of a proposition other than \(x\) that implies \(x\text{.}\)
4. Give an example of a proposition other than \(x\) that is implied by \(x\text{.}\)
Exercise \(\PageIndex{3}\)
Is an implication equivalent to its converse? Verify your answer using a truth table.
No. In symbolic form the question is: Is \((p\to q)\Leftrightarrow (q\to p)\text{?}\)
\(\begin{array}{ccccc} p & q & p\to q & q\to p & (p\to q)\leftrightarrow (q\to p) \\ \hline 0 & 0 & 1 & 1 & 1\\ 0 & 1 & 1 & 0 & 0\\ 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ \end{array}\)
This table indicates that an implication is not always equivalent to its converse.
Exercise \(\PageIndex{4}\)
Suppose that \(x\) is a proposition generated by \(p\text{,}\) \(q\text{,}\) and \(r\) that is equivalent to \(p \lor \neg q\text{.}\) Write out the truth table for \(x\text{.}\)
Exercise \(\PageIndex{5}\)
How large is the largest set of propositions generated by \(p\) and \(q\) with the property that no two elements are equivalent?
Let \(x\) be any proposition generated by \(p\) and \(q\text{.}\) The truth table for \(x\) has 4 rows and there are 2 choices for a truth value for \(x\) for each row, so there are \(2\cdot 2\
cdot 2\cdot 2=2^4\) possible propositions.
Exercise \(\PageIndex{6}\)
Find a proposition that is equivalent to \(p \lor q\) and uses only conjunction and negation.
Exercise \(\PageIndex{7}\)
Explain why a contradiction implies any proposition and any proposition implies a tautology.
\(0\to p\) and \(p\to 1\) are tautologies.
Exercise \(\PageIndex{8}\)
The significance of the Sheffer Stroke is that it is a “universal” operation in that all other logical operations can be built from it.
1. Prove that \(p | q\) is equivalent to \(\neg (p \land q)\text{.}\)
2. Prove that \(\neg p \Leftrightarrow p | p\text{.}\)
3. Build \(\land\) using only the Sheffer Stroke.
4. Build \(\lor\) using only the Sheffer Stroke. | {"url":"https://itconrads.com/article/3-3-equivalence-and-implication","timestamp":"2024-11-02T18:02:02Z","content_type":"text/html","content_length":"79655","record_id":"<urn:uuid:fb97e8eb-4467-473e-b7b7-3b67e852d997>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00096.warc.gz"} |
Concept Based Neighbor Cluster Ensemble Re-Clustering Method
Clustering Ensemble combines the several partitions generated by different clustering algorithm into single clustering solution. The optimization-based method is proposed for the combination of
cluster ensembles for the class of problems with intracluster criteria, such as Minimum-Sum-of-Squares-Clustering (MSSC). To find the solution for MSSC problem we are using simple and efficient
algorithm called improved Exact Method for cluster ensemble re-clustering algorithm which uses similarity measures and distance between the weak clusters. The solution obtained by the single
clustering algorithm does not provide better solution. The solution obtained by this algorithm guarantees better solutions than the ones in the individual cluster. For the MSSC problem in particular,
a prototype implementation of improved Exact Method for cluster ensemble algorithm will produce a new better solution. The algorithm is particularly effective when the number of clusters is large, in
which case it is able to escape the local minima found by K-means type algorithms by recombining the solutions in a Set-Covering context. The stability of the algorithm is also establish by running
this algorithm several times for the same clustering problem instance, produce high-quality solutions. Finally, in experiments utilize external criteria to compute the validity of clustering. The
algorithm is capable of producing high-quality results that are comparable in quality to those of the best known clustering algorithms.
Clustering Ensemble, MSSC, K-Means Algorithm, Set Covering Context.
Ioannis T. Christou ―Coordination of Cluster Ensembles via Exact Methods‖,Vol33,.No2 February 2011.
H. Li, K. Zhang, and T. Jiang, ―Minimum Entropy Clustering and Applications to Gene Expression Analysis,‖ Proc. IEEE Conf. Computational Systems Bioinformatics, pp. 142-151, 2004.
O. du Merle, P. Hansen, B. Jaumard, and N. Mladenovich, ―An Interior Point Algorithm for Minimum Sum of Squares Clustering,‖SIAM J. Scientific Computing, vol. 21, no. 4, pp. 1484-1505,Mar. 2000.
A. Topchy, A.K. Jain, and W. Punch, ―Clustering Ensembles: Models of Consensus and Weak Partitions,‖ IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 27, no. 12, pp. 1866-1881,Dec. 2005.
M.G.C. Resende and R.F. Werneck, ―A Hybrid Heuristic for the PMedian Problem,‖ J. Heuristics, vol. 10, pp. 59-88, 2004.
V. Singh, L. Mukherjee, J. Peng, and J. Xu, ―Ensemble Clustering Using Semidefinite Programming,‖ Advances in Neural Information Processing Systems, J.C. Platt, D. Koller, Y. Singer, and S.
Roweis,eds., pp. 1353-1360, MIT Press, 2008.
A. Strehl and J. Ghosh, ―Cluster Ensembles—A Knowledge Re-Use Framework for Combining Multiple Partitions,‖ J. MachineLearning Research, vol. 3, pp. 583-618, 2002.
A. Asuncion and D.J. Newman, ―UCI Machine Learning Repository,‖School of Information and Computer Science, Univ. of California, http://www.ics.uci.edu/~mlearn/MLRepository.html, 2007.
D. Pelleg and A. Moore, ―X-Means: Extending K-Means with Efficient Estimation of the Number of Clusters,‖ Proc. 17th Int’l Conf. Machine Learning, pp. 727-734, 2000
A.K. Jain and A. Fred, ―Evidence Accumulation Clustering Based on the K-Means Algorithm,‖ Structural, Syntactic, and Statistical Pattern Recognition, pp. 442-451, Springer, 2002.
S.B. KOTSIANTIS, P. E. PINTELAS Recent Advances in Clustering: A Brief Survey
H. Ayad and M. Kamel, ―Cumulative Voting Consensus Method for Partitions with Variable Number of Clusters,‖ IEEE Trans. Jan. 2008. Jan. 2008.
J. Pacheco, ―A Scatter-Search Approach for the Minimum-Sum-of-Squares Clustering Problem,‖ Computers and Operations Research,vol. 32, no. 5, pp. 1325-1335, May 2005.
E. Dimitriadou, A. Weingessel, and K. Hornik, ―A combination Scheme for Fuzzy Clustering,‖ Int’l J. Pattern Recognition and Artificial Intelligence, vol. 16, no. 7, pp. 901-912, 2002.
T. Lange and J.M. Buhmann, ―Combining Partitions by Probabilistic Label Aggregation,‖ Proc. Int’l Conf. Knowledge Discovery in Databases, 2005.
S.Murali Krishna, S. Durga Bhavani, ―An Efficient Approach for Text Clustering based on Frequent Itemsets‖ European Journal of Scientific Research, ISSN 1450-216X, Vol. 42 No. 3, pp. 399-410, 2010.
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Properties of Cyclic Quadrilaterals
Lesson Video: Properties of Cyclic Quadrilaterals Mathematics • Third Year of Preparatory School
In this video, we will learn how to use cyclic quadrilateral properties to find missing angles and identify whether a quadrilateral is cyclic or not.
Video Transcript
In this video, we will learn how to use cyclic quadrilateral properties to find missing angles and identify whether a quadrilateral is cyclic or not. We will begin by recalling what is meant by an
inscribed angle.
An inscribed angle is the angle made when two chords intersect on the circumference of a circle. This means that the vertex of the angle lies on the circumference. We can use our understanding of
inscribed angles to define a cyclic quadrilateral. This is a four-sided polygon whose vertices are inscribed on a circle. If we consider the cyclic quadrilateral π ΄π ΅π Άπ ·, we can join two
vertices π ΄ and π Ά to the center π in order to create two radii π ΄π and π π Ά. We can then label the angle measures created at the center of the circle as π ₯ degrees and π ¦
degrees. Since angles about a point sum to 360 degrees, we have π ₯ degrees plus π ¦ degrees is equal to 360 degrees.
The inscribed angle theorem tells us that an angle π inscribed in a circle is half of the central angle two π that subtends the same arc on the circle, as shown. In other words, the angle of
the circumference is half the angle at the center. This means that the measure of the angle at vertex π ΅ is a half of π ₯ degrees and the measure of the angle at vertex π · is a half of π ¦
degrees. We can then combine these three equations. Firstly, we have the measure of angle π ΅ plus the measure of angle π · is equal to a half of π ₯ degrees plus a half of π ¦ degrees. Factoring
out a half on the right-hand side gives us a half of π ₯ degrees plus π ¦ degrees. And since π ₯ degrees plus π ¦ degrees is 360 degrees, the measure of angle π ΅ plus the measure of angle π ·
is equal to a half of this, which is equal to 180 degrees. This means that the sum of this pair of opposite angles is 180 degrees.
We can complete the same process to demonstrate that the measure of angle π ΄ and the measure of angle π Ά also sum to 180 degrees. And these two equations lead us to the following property
regarding opposite angles in a cyclic quadrilateral. The measures of the opposite angles in a cyclic quadrilateral sum to 180 degrees. This means they are supplementary angles. We can use this to
calculate the measures of missing angles in a cyclic quadrilateral. If the measure of angle π · was 75 degrees, we could calculate the measure of angle π ΅ by subtracting this from 180, giving us
105 degrees. In the same way, if angle π ΄ measured 120 degrees, then the measure of angle π Ά would be 60 degrees as 180 minus 120 is 60. We will now consider an example where the angles in the
cyclic quadrilateral are given algebraically.
Given that the measure of angle π ΄ is equal to π ¦ degrees, the measure of angle π ΅ is equal to four π ₯ minus three degrees, and the measure of angle π Ά is equal to five π ₯ degrees, find
the values of π ₯ and π ¦.
The figure shows a cyclic quadrilateral π ΄π ΅π Άπ · such that the vertices of the quadrilateral lie on the circumference of the circle. We recall that opposite angles in a cyclic quadrilateral
sum to 180 degrees. We are told in the question that the measure of angle π ΅ is four π ₯ minus three degrees. And from the diagram, we see that the measure of angle π · is 115 degrees. This means
that four π ₯ minus three degrees plus 115 degrees is equal to 180. And since the angles are all given in degrees, we can rewrite this as shown. Negative three plus 115 is equal to 112. So the
equation simplifies to four π ₯ plus 112 is equal to 180. We can then subtract 112 from both sides such that four π ₯ is equal to 68. And dividing through by four, we have π ₯ is equal to 17.
We are also told in the question that the measure of angle π ΄ is π ¦ degrees and the measure of angle π Ά is five π ₯ degrees. This means that π ¦ plus five π ₯ equals 180. We have already
calculated that π ₯ is equal to 17, so π ¦ plus five multiplied by 17 is equal to 180. Multiplying five by 17 gives us 85. And subtracting this from both sides, we have π ¦ is equal to 180 minus
85. And this is equal to 95. The two solutions to this question are π ₯ equals 17 and π ¦ is equal to 95.
Whilst it is not required in this question, we could substitute these values back into the expressions for the measures of angles π ΄, π ΅, and π Ά to calculate the missing angles. Angle π ΄ is
equal to 95 degrees. Four multiplied by 17 is 68. And subtracting three from this gives us 65, so the measure of angle π ΅ is 65 degrees. Finally, the measure of angle π Ά is equal to 85 degrees.
At this stage, as with any quadrilateral, it is worth checking that all four angles sum to 360 degrees.
We will now consider how we can extend the property of interior angles of a cyclic quadrilateral to the measure of an exterior angle. Letβ s begin by considering the cyclic quadrilateral π ΄π ΅π
Άπ ·, where the measure of angle π ΄ is π degrees and the measure of angle π Ά is π degrees as shown. We know that π degrees plus π degrees is equal to 180 degrees. And this can be
rewritten as π degrees is equal to 180 degrees minus π degrees. Letβ s now consider an external angle by extending the line segment π ·π Ά to point π Έ in order to create an external angle
π ΅π Άπ Έ. If we label this angle β degrees and since angles π ΅π Άπ · and π ΅π Άπ Έ lie on a straight line, their measures will sum to 180 degrees. This means that π degrees plus β
degrees is equal to 180 degrees.
Once again, we can rewrite this equation as β degrees is equal to 180 degrees minus π degrees. Since the right-hand sides of our two equations are equal, the left-hand sides must be equal and π
degrees is equal to β degrees. Replacing β with π on our diagram, we see that angle π ΅π Άπ Έ is equal to π degrees. And this leads us to a general property. An exterior angle of a
cyclic quadrilateral is equal to the interior angle at the opposite vertex. We will now apply this property to an example.
Find the measure of angle π Έπ Άπ Ή and the measure of angle π ΄π ΅π Ή.
In this question, weβ re asked to find the measure of two angles, firstly the measure of angle π Έπ Άπ Ή and secondly the measure of angle π ΄π ΅π Ή. And in order to find these two measures,
weβ ll use two properties of cyclic quadrilaterals. Firstly, we recall that opposite angles in a cyclic quadrilateral sum to 180 degrees. And secondly, exterior angles of a cyclic quadrilateral are
equal to the interior angle at the opposite vertex. Using the second property, we see that the measure of angle π Έπ Άπ Ή is equal to the measure of the angle at vertex π ΄ and is therefore equal
to 80 degrees.
Using the same property, the measure of the exterior angle π ΄π ΅π Ή is equal to the measure of the interior angle at vertex π ·, that is, the measure of angle π ΄π ·π Ά. Since angles on a
straight line sum to 180 degrees, we can calculate the measure of this angle by subtracting 104 degrees from 180 degrees. This is equal to 76 degrees. The measure of angle π ΄π ΅π Ή is 76 degrees.
And we now have the two solutions as required. Whilst we didnβ t do so in this question, we could have used the first property that opposite angles sum to 180 degrees to find the interior angles at
vertices π ΅ and π Ά first. We could then have used these together with the fact that angles on a straight line sum to 180 degrees to find the measures of π Έπ Άπ Ή and π ΄π ΅π Ή. Either way,
we end up with two answers of 80 degrees and 76 degrees.
We will now consider the converse of these theorems. This states that a quadrilateral is cyclic if we can demonstrate one of the following: either the opposite angles measures are supplementary,
i.e., they sum to 180 degrees, or an exterior angle is equal to the interior angle at the opposite vertex. For example, since the opposite angles in the quadrilateral drawn sum to 180 degrees, this
must be a cyclic quadrilateral. If, on the other hand, the two angles did not sum to 180 degrees, the quadrilateral would not be cyclic and we would not be able to draw a circle through all four
vertices of the quadrilateral. In the second diagram, since the exterior angle is equal to the interior angle at the opposite vertex, the quadrilateral π ΄π ΅π Άπ · is also cyclic. We will now
look at an example where we need to prove whether a quadrilateral is cyclic or not.
Is π ΄π ΅π Άπ · a cyclic quadrilateral?
We begin by recalling that there are two ways we can prove that a quadrilateral is cyclic: firstly, if the opposite angles in the quadrilateral sum to 180 degrees and secondly if an exterior angle is
equal to the interior angle at the opposite vertex. It is the first of these we will use in this question. If we can prove that the measure of angle π ΅ plus the measure of angle π · is equal to
180 degrees, then the quadrilateral is cyclic. We can also do this with angles π ΄ and π Ά. We begin by noticing that triangle π ΄π ·π Ά is isosceles. This means that the measure of angle π Άπ
΄π · is equal to the measure of angle π ΄π Άπ ·, which is equal to 53 degrees. Since angles in a triangle sum to 180 degrees, the measure of angle π ΄π ·π Ά is equal to 180 degrees minus 53
degrees plus 53 degrees. This is therefore equal to 74 degrees.
We now have the measures of two opposite angles in our quadrilateral. 106 plus 74 is equal to 180. So the measure of angle π ΅ and the measure of angle π · do sum to 180 degrees. And we can
therefore conclude that π ΄π ΅π Άπ · is a cyclic quadrilateral, and the correct answer is yes.
We will now summarize the key points from this video. We saw in this video that a cyclic quadrilateral is a four-sided polygon whose vertices are inscribed on a circle. We saw that in a cyclic
quadrilateral the measures of opposite angles are supplementary, i.e., they sum to 180 degrees, and also an exterior angle is equal to the interior angle at the opposite vertex. Finally, we saw that
the opposite is also true. A quadrilateral is cyclic if we can prove one of the following. The opposite angles sum to 180 degrees, or an exterior angle is equal to the interior angle at the opposite | {"url":"https://www.nagwa.com/en/videos/939168969256/","timestamp":"2024-11-11T20:14:07Z","content_type":"text/html","content_length":"271015","record_id":"<urn:uuid:da0a4b39-26f8-4fcb-9aba-adbfd3ce9d0a>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00110.warc.gz"} |
Intuitive Introduction to Quantum Computing [+ Evolution of Computers]
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In this article, we have presented a detailed and intuitive introduction to Quantum Computing which is one of the promising emerging sub-domain of Computer Science.
Table of contents:
1. Introduction: Analog to Quantum Computing
2. What is Quantum Computing?
3. How is it useful to Computer Engineers?
4. Difference between Quantum Computing and Classical Computing
5. Benefits of Quantum Computing over Classical Computing
6. Future
Introduction: Analog to Quantum Computing
In 1901, an artifact was discovered from a shipwreck which is today known as Antikythera mechanism. It is dated to be from around 100 BC. It was an Analog Computer of intense complexity.
Computers have existed from ancient times but its look and form and internal working have been changing.
Analog Computer is a mechanical device that use various physical properties like electrical or hydraulic quantities to solve various problems. Analog Computers were very useful but was prone to error
as it had a continuous range (input and output unlike 0 and 1 of Classical Computing). This was evident by the failure of the major military project Norton Bomb Project. Despite this, Analog
Computers played a major role in World War 1 and 2 by designing military equipment. It was an Analog Computer that was able to predict the tide levels successfully in real time.
Since 1960, Digital Computing or Classical Computing started to take over as the main form of computing. This was possible by the development of transistors and the work of Claude Shannon who showed
2 bits 0 and 1 with 3 operations AND, OR and NOT is enough to do any calculation in a Classical Computer. This made Classical Computers general purpose unlike Analog Computers.
In fact, we can make a Computing device from any physical property. The usefulness, strengths and properties will vary. For example,
• Analog Computers are based on “Mechanical Devices capturing different properties like electrical, hydraulic, pressure”.
• Digital/ Classical Computers -> transistors, binary calculations
• Thermodynamic computing -> various thermodynamic properties of materials
• Molecular computing -> based on DNA and biochemistry
• Gas flow computer -> based on difference in pressure
• Quantum Computer -> based on Superposition principle from Quantum Physics.
Just as Digital/ Classical Computing started to take over Analog Computing in terms of importance since 1960, today Quantum Computing has the potential to take over Classical Computing.
The entire concept and use-case of Quantum Computing is different and is in rapid development which we will understand in this project. Theoretically, Quantum Computer has huge potential and a few
breakthroughs will make it practical and the main form of Computing in near future.
What is Quantum Computing?
Quantum Computing is a form of Computing that is based on the superposition principle and entanglement of Quantum Physics.
It is based on the principle of Qubits which can be seen as a modification of bits which are in the state of both 0 and 1 with some probabilities. Due to this, multiple cases can be checked at once.
These are captured by spin of electrons in a silicon chip (quantum bits). This forms a quantum gate just like AND or NOT gate in a Classical Computer. The basic theoretical concepts are similar.
How is it useful to Computer Engineers?
Help us solve problems which are impossible to solve by Classical Computers. There is a class of problems known as NP-complexity class which has problems which take exponential time to solve using
Classical Computers. One such problem is Travelling Salesman Problem.
Quantum Computer will be a big leap forward to break-through this limitation of Classical Computers. Today, such problems are solved using Machine Learning or Approximation Algorithms on Classical
Computers which take reasonable time but gives a good answer (not the exactly correct answer).
Computer Engineers will get exact answers to challenging problems which are impractical today. Several applications like Google Maps, Uber and others will benefit.
Computer Engineers will work on designing Quantum Algorithms like Grover’s Algorithm which are radically different from algorithms for Classical Computing like Linear and Binary Search.
Quantum Computers will solve problems like Prime Factorization with exponential speedup. This is a danger as it will disrupt the field of Cryptography and the use of passwords will be useless as
anyone can break it with a Quantum Computer instantly. Today, Crypotography is based on the idea that “Use so big numbers and classical computers will not be able to break it in reasonable time,
will take more than 10 years”.
Difference between Quantum Computing and Classical Computing
• Classical uses bits (0 or 1). Quantum uses qubits (0 and 1 both with probability).
• Prime Factorization is a challenging problem for Classical computers. Same is an easy problem for Quantum Computer.
• Classical Computer follow the rules of classical physics whereas Quantum Computer follow the rules of quantum physics.
Benefits of Quantum Computing over Classical Computing
Following are the Benefits of Quantum Computing over Classical Computing:
• Specific set of challenging problems like Prime Factorization will become trivial problems.
• The field of Cryptography will change forever driving the next growth in the field of Computer Science.
The future of Quantum Computing is bright though not a household tool today. Few break-throughs will make it practically.
Big corporations like IBM and Google are investing to make it practical.
The theoretical work of Quantum Computing has a strong foundation and is in good advanced form. Once the physical/ practical form of Quantum Computers is ready, all the theoretical work can be
Just few months back, I started to learn basics of Algorithms and Data Structure. Thanks to the opportunity given by Lalatendu Sir that that I started to explore Quantum Computing in parallel and
understood that the Algorithms for this are completely different. Classical Algorithms will not given any benefit on Quantum Computer. This exploration and preparation of this report opened my mind
and expanded my insights. Today, Computer Science is even more exciting.
All fields will be revolutionized. Recently, I have been interested in Artificial Intelligence and Machine Learning and have been going through the basics like A* algorithm. Even these will change
and form a field known as “Quantum Machine Learning” (already in existence).
This is an exciting time as I am able to witness the growth of Quantum Computers and hopefully, as I enter the industry, this technology will become feasible. I look forward to make a research
contribution in this field. | {"url":"https://iq.opengenus.org/introduction-to-quantum-computing/","timestamp":"2024-11-04T11:09:13Z","content_type":"text/html","content_length":"60148","record_id":"<urn:uuid:12105ee2-a317-41d1-9c25-1b097e6ff1cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00631.warc.gz"} |
Matrix inversion - (Linear Algebra for Data Science) - Vocab, Definition, Explanations | Fiveable
Matrix inversion
from class:
Linear Algebra for Data Science
Matrix inversion is the process of finding a matrix, called the inverse, that when multiplied by the original matrix results in the identity matrix. The identity matrix acts like the number '1' in
matrix multiplication, meaning that if you multiply any matrix by its inverse, you get back to where you started. In many applications, especially in solving systems of equations and optimization
problems, being able to calculate the inverse of a matrix is crucial for efficient computations and understanding relationships between variables.
congrats on reading the definition of matrix inversion. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. A matrix can only be inverted if it is square (same number of rows and columns) and its determinant is non-zero.
2. The inverse of a matrix A is denoted as A^(-1), and the relationship A * A^(-1) = I holds true, where I is the identity matrix.
3. If a matrix is invertible, it means that its rows (or columns) are linearly independent, which ensures a unique solution exists for the corresponding system of equations.
4. Matrix inversion can be computationally expensive, especially for large matrices, which often leads to using alternative methods like LU decomposition or Cholesky decomposition for solving linear
equations efficiently.
5. In data science, matrix inversion is frequently used in algorithms such as linear regression to solve for coefficients that minimize prediction error.
Review Questions
• How does the concept of matrix inversion relate to solving systems of equations in data science?
□ Matrix inversion is essential in solving systems of equations because it allows us to find unique solutions efficiently. When we represent a system of equations in matrix form as Ax = b,
where A is the coefficient matrix and b is the output vector, we can find x by multiplying both sides by A^(-1), yielding x = A^(-1)b. This application highlights how important it is to
understand when a matrix can be inverted and how to compute it.
• Discuss the role of LU decomposition in relation to matrix inversion and solving linear equations.
□ LU decomposition factors a matrix into two simpler matrices: a lower triangular matrix L and an upper triangular matrix U. This approach simplifies the process of solving linear equations
compared to directly calculating the inverse. When we need to solve Ax = b using LU decomposition, we first solve Ly = b for y and then Ux = y for x. This method is often more numerically
stable and efficient than directly computing A^(-1), especially for larger matrices.
• Evaluate how understanding singular matrices can impact the application of matrix inversion in data science problems.
□ Recognizing singular matrices is crucial because they cannot be inverted, leading to challenges in data analysis or predictive modeling. When dealing with datasets, if the feature matrix has
multicollinearity (where some features are linearly dependent), it results in a singular matrix. Understanding this concept helps data scientists select appropriate algorithms or preprocess
data to mitigate these issues, ensuring they can derive meaningful insights and reliable predictions without running into computational problems.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/linear-algebra-for-data-science/matrix-inversion","timestamp":"2024-11-12T15:37:30Z","content_type":"text/html","content_length":"159747","record_id":"<urn:uuid:3ddecb65-2db0-4f00-b071-185297b290b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00454.warc.gz"} |
Heuristics - LD@schoolHeuristics
At its most basic, a heuristic is a shortcut or rule for reducing the cognitive load of information processing. The use of heuristics in mathematics can have a profound impact on a student’s ability
to quickly and accurately solve a math fact or word problem. Students with LDs, whether math-specific or not, will especially benefit from the structure and sequence a heuristic provides (Robinson &
Hutchinson, 2014).
Some examples of effective heuristics for the math class are:
Keyword strategy:
The “keyword” strategy involves associating common words with the operation they represent. For example, they might associate “gave away” to mean the question involves subtraction.
Underlining important information:
One common characteristic of word problems is that they tend to have extraneous information. Students should underline or highlight the important information, allowing for a simplification of the
Heuristics for problem-solving:
Click here to access LD@school’s problem-solving worksheet template including the self-questions for representing and solving word problems. | {"url":"https://www.ldatschool.ca/learning-modules/secondary-school-2/subject-specific-support/math/heuristics/","timestamp":"2024-11-07T19:04:19Z","content_type":"text/html","content_length":"180193","record_id":"<urn:uuid:77268aa3-1ca1-4644-a00f-8ccaaaa0ba5d>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00654.warc.gz"} |
What is: Convergence
What is Convergence in Statistics?
Convergence in statistics refers to the idea that a sequence of random variables approaches a specific value or distribution as the number of observations increases. This concept is fundamental in
the field of statistics and data analysis, as it underpins many theoretical results and practical applications. In essence, convergence helps statisticians understand how sample statistics behave as
the sample size grows, providing insights into the reliability and accuracy of estimates derived from data.
Types of Convergence
There are several types of convergence that statisticians and data scientists commonly encounter, including convergence in distribution, convergence in probability, and almost sure convergence.
Convergence in distribution, also known as weak convergence, occurs when the cumulative distribution functions of a sequence of random variables converge to a limiting distribution. Convergence in
probability, on the other hand, implies that the probability of the random variables deviating from a certain value approaches zero as the sample size increases. Almost sure convergence is a stronger
form of convergence, indicating that the sequence converges to a limit with probability one.
Convergence of Random Variables
When discussing the convergence of random variables, it is essential to understand the implications of each type of convergence on statistical inference. For instance, the Central Limit Theorem (CLT)
is a pivotal result that illustrates convergence in distribution. It states that the sum of a large number of independent and identically distributed random variables will tend to follow a normal
distribution, regardless of the original distribution of the variables. This theorem is crucial for hypothesis testing and confidence interval estimation, as it justifies the use of normal
approximations in many practical scenarios.
Convergence in Statistical Estimation
In the context of statistical estimation, convergence plays a vital role in determining the consistency and efficiency of estimators. An estimator is said to be consistent if it converges in
probability to the true parameter value as the sample size increases. This property is essential for ensuring that the estimates produced by a statistical model become more accurate as more data is
collected. Furthermore, the concept of convergence is closely related to the notion of bias and variance in estimation, where a good estimator should have low bias and low variance, leading to
convergence to the true parameter.
Applications of Convergence in Data Science
In data science, convergence is a critical concept that informs various algorithms and methodologies, particularly in machine learning. For example, many optimization algorithms, such as gradient
descent, rely on the principle of convergence to minimize loss functions. The convergence of these algorithms ensures that as iterations progress, the model parameters stabilize and approach optimal
values. Understanding convergence is essential for data scientists to evaluate the performance and reliability of their models, as well as to diagnose potential issues during training.
Convergence in Bayesian Statistics
Bayesian statistics also incorporates the concept of convergence, particularly in the context of posterior distributions. As more data becomes available, the posterior distribution of a parameter
converges to the true value of that parameter, given the prior distribution. This property is known as posterior consistency. In Bayesian analysis, convergence is crucial for making reliable
inferences and predictions, as it allows practitioners to update their beliefs about parameters as new evidence is obtained.
Convergence and the Law of Large Numbers
The Law of Large Numbers (LLN) is another foundational theorem in probability theory that relates to convergence. It states that as the sample size increases, the sample mean will converge to the
expected value of the population mean. This principle is fundamental in statistics, as it provides a theoretical basis for the reliability of sample estimates. The LLN assures researchers that larger
samples yield more accurate estimates, reinforcing the importance of collecting sufficient data in statistical studies.
Factors Affecting Convergence
Several factors can influence the convergence of random variables and estimators. The underlying distribution of the data, the presence of outliers, and the choice of estimator can all impact the
rate and nature of convergence. For instance, heavy-tailed distributions may lead to slower convergence rates, while robust estimators can mitigate the effects of outliers, promoting faster
convergence. Understanding these factors is crucial for statisticians and data scientists to ensure that their analyses yield valid and reliable results.
Implications of Non-Convergence
Non-convergence can have significant implications in statistical analysis and data science. When estimators do not converge, it may indicate model misspecification, inadequate sample size, or the
presence of biases in the data. Non-convergence can lead to unreliable estimates, erroneous conclusions, and poor decision-making based on flawed analyses. Therefore, it is essential for
practitioners to diagnose and address issues related to convergence to maintain the integrity of their statistical findings and ensure robust data-driven decisions. | {"url":"https://statisticseasily.com/glossario/what-is-convergence/","timestamp":"2024-11-04T07:26:45Z","content_type":"text/html","content_length":"139041","record_id":"<urn:uuid:32303439-e4b2-42ae-9f9e-6f0751768a34>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00752.warc.gz"} |
13.2.3 Statistics (I), PT3 Focus Practice
Question 7:
Diagram below is an incomplete pictograph showing the sales of books for a duration of five months.
(a) The sales in May is ¼ of the total sales in January and February.
Complete the pictograph in the Diagram.
(b) Find the total number of books sold before April.
$\begin{array}{l}\text{Sales in May}\\ =\frac{1}{4}×8\\ =2\end{array}$
Total number of books sold before April
= (5 + 3 + 6) × 15
= 14 × 15
= 210
Question 8:
Diagram below shows an incomplete line graph of the number of eggs sold in four weeks. The number of eggs sold on week 1 is 2000 and 4000 on week 4.
(a) Complete the line graph in the Diagram.
(b) Complete the pie chart in the second Diagram to represent sales from Week 1 to Week 4.
$\begin{array}{l}\text{First week}\\ =\frac{2000}{2000+1500+2500+4000}×{360}^{o}\\ =\frac{2000}{10000}×{360}^{o}\\ ={72}^{o}\\ \\ \text{Fourth week}\\ =\frac{4000}{10000}×{360}^{o}\\ ={144}^{o}\end | {"url":"https://content.myhometuition.com/2017/11/23/13-2-3-statistics-i-pt3-focus-practice/","timestamp":"2024-11-06T13:35:23Z","content_type":"text/html","content_length":"22093","record_id":"<urn:uuid:c1fba240-0d22-4caf-8a10-b23289ee0e54>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00625.warc.gz"} |
Average value of $\pi(t) - li(t)$
Speaker: Daniel Johnston Date: Thu, Jun 20, 2024 Location: PIMS, University of British Columbia Conference: Comparative Prime Number Theory Subject: Mathematics, Number Theory Class: Scientific CRG:
L-Functions in Analytic Number Theory
Central to comparative number theory is the study of the difference $\Delta(t) = \pi(t) − li(t)$, where $\pi(t)$ is the prime counting function and $li(t)$ is the logarithmic integral. Prior to a
celebrated 1914 paper of Littlewood, it was believed that $\Delta < 0$ for all $t > 2$. We now know however that $\Delta(t)$ changes sign infinitely often, with the first sign change occuring before
10320. Despite this, it still appears that $\Delta(t)$ is negative “on average”, in that integrating $\Delta (t)$ from $t = 2$ onwards yields a negative value. In this talk, we will explore this idea
in detail, discussing links with the Riemann hypothesis and also extending such ideas to other differences involving arithmetic functions. | {"url":"https://www.mathtube.org/lecture/video/average-value-pit-lit","timestamp":"2024-11-10T22:07:03Z","content_type":"application/xhtml+xml","content_length":"26653","record_id":"<urn:uuid:a16619bc-3b96-4420-b936-ad7013cab28b>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00398.warc.gz"} |
free calculus lesson plans
247 Pre-Calculus Lesson Plans __ Goals and procedures. - From lessoncorner.com - http://test.lessoncorner.com/search?page=7&q=Pre-Calculus
AP Calculus AB Course Home Page __ Numerous lesson plans, work sheets and other resources. - From apcentral.collegeboard.com - http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/
Calculus Basics - Problems and Worksheets __ You will find problems and instructions designed for beginning students, but helpful to all students to gain the basics of calculus. There are 14
examples. - From awesomelibrary.org - http://www.awesomelibrary.org/Classroom/Mathematics/Middle-High_School_Math/Calculus.html
Calculus lesson plans __ A collection of lesson plans. "Calculus lesson plans. Lessons for Single-variable Calculus, Multi-variable Calculus and much more." - From about.com - http://math.about.com/
Calculus Lesson Plans __ You will find 13 lesson plans and references, some leading to many more references and plans. - From teach-nology.com - http://www.teach-nology.com/teachers/lesson_plans/math
Calculus Lesson Plans __ Almost 400 calculus lesson plans. goals and procedures. - From lessoncorner.com - http://test.lessoncorner.com/Math/Calculus/
Calculus Lesson Plans __ Several lesson plans with goals, procedures - From digitalwish.com - http://www.digitalwish.com/dw/digitalwish/view_lesson_plans?subject=calculus
Calculus Lesson Plans and Work Sheets __ Many lesson plans, worksheets and more. Goals, procedures. - From pleacher.com - http://www.pleacher.com/mp/mlessons/mcalc.html
Calculus Problems and Solutions __ You may want to print these then use them with your students, including problems on beginner Differential and Integral Calculus. - From .ucdavis.edu/D. A. Kouba -
Cool math Lessons - Calculus - What's a limit? __ "This lesson will explain the concept of a limit from various points of view." - illustrated - From coolmath.com - http://www.coolmath.com/
dansmath - lessons - calculus 1 __ "Features step by step lessons to introductory calculus. Included are function and precalculus preview pages." - From Dan Bach - http://home.earthlink.net/~djbach/
Free Calculus Worksheets - Calculus Printables with Answers __ Worksheets cover various Calculus topics including limits, derivatives, integrals, and more. - From tutor-usa.com - http://tutor-usa.com
Free Calculus Worksheets to Download __ Free calculus worksheets with solutions, in PDF format, to download. - From analyzemath.com - http://www.analyzemath.com/calculus_worksheets.html
Homework Help Calculus __ Several calculus resources for student and teacher. - From math.com - http://www.math.com/homeworkhelp/Calculus.html
Lesson Plans 4 Teachers: Calculus Lesson Directory __ Directory of calculus lesson plan sites. - From lessonplans4teachers.com - http://www.lessonplans4teachers.com/calculus.php
Lesson Plans: Differential Calculus __ "Objective: How to find the x-Intercepts and the y-Intercepts...Pre-requisite knowledge: Students should be able to solve a first degree equation and a second
degree equation." Goal and procedure. - From teachers.net - http://teachers.net/lessonplans/posts/1058.html
Lesson Plans: Maxima-Minima (Differential Calculus) (Senior, Mathematics) __ "Assumption : Students have taken the topic prior to this topic. They know how to evaluate functions using
differentiation. Students are in Grade 11 or 12 level." Goals, procedure and materials. - From teachers.net - http://teachers.net/lessonplans/posts/2976.html
Math Forum: Pre-Calculus Lesson Plans __ Half a dozen lesson plans. - From mathforum.org - http://mathforum.org/precalc/precalc.units.html
Pre-Calculus __ Dozens of lesson plans and resources. - From math.utk.edu - http://archives.math.utk.edu/topics/precalculus.html
Problem Sets for Honors Multi Variate Calculus __ Here are problem sets which come in text and PDF format. All of them were written by a math professor. - From Stephen Bennett Maurer - http:// | {"url":"https://ezorigin.archaeolink.com/calculus_lesson_plans.htm","timestamp":"2024-11-12T07:04:16Z","content_type":"text/html","content_length":"29285","record_id":"<urn:uuid:6e6c03d1-4190-4d04-bc27-192225e41edb>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00083.warc.gz"} |
Parameter estimates
Parameter estimates (also called coefficients) are the change in the response associated with a one-unit change of the predictor, all other predictors being held constant.
The unknown model parameters are estimated using least-squares estimation.
A coefficient describes the size of the contribution of that predictor; a near-zero coefficient indicates that variable has little influence on the response. The sign of the coefficient indicates the
direction of the relationship, although the sign can change if more terms are added to the model, so the interpretation is not particularly useful. A confidence interval expresses the uncertainty in
the estimate, under the assumption of normally distributed errors. Due to the central limit theorem, violation of the normality assumption is not a problem if the sample size is moderate.
• For quantitative terms, the coefficient represents the rate of change in the response per 1 unit change of the predictor, assuming all other predictors are held constant. The units of measurement
for the coefficient are the units of response per unit of the predictor.
For example, a coefficient for Height of 0.75, in a simple model for the response Weight (kg) with predictor Height (cm), could be expressed as 0.75 kg per cm which indicates a 0.75 kg weight
increase per 1 cm in height.
When a predictor is a logarithm transformation of the original variable, the coefficient is the rate of change in the response per 1 unit change in the log of the predictor. Commonly base 2 log
and base 10 log are used as transforms. For base 2 log the coefficient can be interpreted as the rate of change in the response when for a doubling of the predictor value. For base 10 log the
coefficient can be interpreted as the rate of change in the response when the predictor is multiplied by 10, or as the % change in the response per % change in the predictor.
• For categorical terms, there is a coefficient for each level:
□ For nominal predictors the coefficients represent the difference between the level mean and the grand mean.
Analyse-it uses effect coding for nominal terms (also known as the mean deviation coding). The sum of the parameter estimates for a categorical term using effect coding is equal to 0.
□ For ordinal predictors, the coefficients represent the difference between the level mean and the baseline mean.
Analyse-it uses reference coding for ordinal terms. The first level is used as the baseline or reference level.
• For the constant term, the coefficient is the response when all predictors are 0, and the units of measurement are the same as the response variable.
A standardized parameter estimate (commonly known as standardized beta coefficient) removes the unit of measurement of predictor and response variables. They represent the change in standard
deviations of the response for 1 standard deviation change of the predictor. You can use them to compare the relative effects of predictors measured on different scales.
VIF, the variance inflation factor, represents the increase in the variance of the parameter estimate due to correlation (collinearity) between predictors. Collinearity between the predictors can
lead to unstable parameter estimates. As a rule of thumb, VIF should be close to the minimum value of 1, indicating no collinearity. When VIF is greater than 5, there is high collinearity between
A t-test formally tests the null hypothesis that the parameter is equal to 0, against the alternative hypothesis that it is not equal to 0. When the p-value is small, you can reject the null
hypothesis and conclude that the parameter is not equal to 0 and it does contribute to the model.
When a parameter is not deemed to contribute statistically to the model, you can consider removing it. However, you should be cautious of removing terms that are known to contribute by some
underlying mechanism, regardless of the statistical significance of a hypothesis test, and recognize that removing a term can alter the effect of other terms. | {"url":"https://analyse-it.com/docs/user-guide/fit-model/linear/parameter-estimates","timestamp":"2024-11-01T19:31:41Z","content_type":"text/html","content_length":"32539","record_id":"<urn:uuid:7836945e-fa23-4e4b-a77b-1a562c1d5894>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00591.warc.gz"} |
Exciton and trion spectral line shape in the presence of an electron gas in GaAs/AlAs quantum wells
We studied the photoluminescence of (e1:hh1)1S excitons (X) and negatively charged excitons (trions, (Formula presented)) in quantum wells (QW’s) having a low-density ((Formula presented)<5×(Formula
presented)) two-dimensional electron gas (2DEG) at T⩽12 K. Mixed type-I-type-II GaAs/AlAs quantum wells are studied in which the 2DEG is photogenerated in the type-I QW’s and (Formula presented) is
determined by the excitation intensity. At a given temperature and for every excitation intensity the photoluminescence spectrum is decomposed into a Lorentzian-shaped (Formula presented) line and a
convoluted Lorentzian-Gaussian X line. Their intensity ratio is analyzed by assuming a thermal equilibrium distribution of X and (Formula presented) that is determined by the chemical potential of
the 2DEG. The (Formula presented) linewidth dependence on (Formula presented) is analyzed as originating from an increased (Formula presented) dephasing rate that is caused by trion-electron
((Formula presented)-e) scattering. We present a model of the elastic ((Formula presented)-e) scattering and calculate its rate as a function of (Formula presented) assuming the 2DEG screening wave
vector ((Formula presented)) to be an adjustable parameter. Although of the same order of magnitude, the fitted (Formula presented) values differ from those calculated for the ideal gas model using
the Thomas-Fermi approximation. Since, to our knowledge, there is no model for calculating (Formula presented) in the low 2DEG density range studied here and T>0, our spectroscopically extracted
(Formula presented)((Formula presented)) values might serve as guidelines for the required theory.
All Science Journal Classification (ASJC) codes
• Electronic, Optical and Magnetic Materials
• Condensed Matter Physics
Dive into the research topics of 'Exciton and trion spectral line shape in the presence of an electron gas in GaAs/AlAs quantum wells'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/exciton-and-trion-spectral-line-shape-in-the-presence-of-an-elect","timestamp":"2024-11-05T03:39:35Z","content_type":"text/html","content_length":"53393","record_id":"<urn:uuid:acb0242f-68d0-4a88-9251-d4d0e4c37e46>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00776.warc.gz"} |
Correlation does not imply causation - biostatsquid.com
Correlation does not imply causation
Simple explanation of what is correlation, positive and negative correlation, and the correlation coefficient r.
In this post I will try to give you a simple and practical explanation of correlation.
Correlation is one of the most used statistical techniques. However, it is very often misinterpreted.
Correlation is actually the basis of many biological studies, from gene expression analysis to analysis of clinical trials data.
I hope this easy explanation helps to get a sense of what is correlation and how to know if two variables are correlated. I will also give you a lot of examples of positive correlation, negative
correlation, and examples of uncorrelated variables.
If you are more of a video-based learner, you can check out my Youtube video, otherwise, just keep reading!
Let’s dive in!
You might notice that when ice cream sales increase, so do shark attacks.
When ice cream sales decrease, so do shark attacks.
Does this mean that eating ice cream causes shark attacks?
Of course not!
In this case, sunny weather is making more people eat ice cream. Also, with good weather, more people swim in the sea, which in turn increases the chances of a shark attack. Ice cream sales and shark
attacks are correlated, which does not mean ice cream sales cause shark attacks or the other way around.
Correlation does not imply causation.
Let’s take a look at the differences between correlation and causation.
Causation means that one variable (often called the predictor variable or independent variable) causes the other (often called the outcome variable or dependent variable).
Correlation measures how two variables are related, so the association between the two variables.
When two variables are correlated, we cannot conclude that one variable causes a change in the other. This relationship could be coincidental, or a third factor may be causing both variables to
For example, you might notice that the more people walk with umbrellas, the more hot cocoa sales there are. When cocoa sales decrease, so does the number of people walking outside with umbrellas.
From this we can only conclude that both variables are correlated. Walking with umbrellas does not cause hot cocoa sales to increase (or vice versa), but rather a third variable (the rain), is
causing both variables to increase.
How do we know if two variables are correlated?
To visualise if 2 variables are correlated, we can plot them in a scatterplot.
If there is a correlation, an overall pattern can be seen when the variables are plotted on a scatterplot.
If this pattern can be approximated by a line, the correlation is linear.
Otherwise, the correlation is non-linear.
So, in summary, a great way to visualise if two variables are correlated is through a scatterplot.
Variables that are correlated form a pattern.
If there is a correlation between two variables, an overall pattern can be seen when the variables are plotted on a scatterplot.
There are three types of correlation
There are three ways to describe correlations between variables.
• Positive correlation: as x increases, y tends to increase, and viceversa. An example of positive correlation would be height and weight. Taller people tend to be heavier.
• Negative correlation: as x increases, y tends to decrease, and viceversa. For example, increased exercise is correlated with less heart disease.
• No correlation (uncorrelated): as x increases, y tends to stay about the same or have no clear pattern. For example, coffee consumption vs. intelligence. The amount of coffee that individuals
consume and their IQ level has a correlation of zero. In other words, knowing how much coffee an individual drinks doesn’t give us an idea of what their IQ level might be.
Let’s look at one last example.
You might notice that places with higher numbers of sunburns have also higher number of cases of skin cancer.
Does this mean that sunburns cause skin cancer?
No! with this data, we cannot say that sunburns cause skin cancer.
But wait a minute.
We can design an experiment to determine causation.
For example, randomised controlled trials can provide good evidence of causal relationships. The goal is to isolate, and manipulate the independent variable to observe its effect on the dependent
variable and control the environment so that other variables are eliminated.
For example, we know that sunburns are caused by UV radiation from the sun. We will find that UV radiation damages DNA in our skin cells. If enough DNA damage build up over time and affects specific
genes à it can cause cells to start growing out of control. This is the start of skin cancer.
So no, sunburn itself does not cause skin cancer. It’s positively correlated to skin cancer. But there are many more factors to take into account.
However, overexposure to the dangerous ultraviolet radiation that damages skin cells, weakening them and creating the opportunity for cancer to form.
Don’t forget to wear sunscreen in the sun!
Variables can be positively correlated, if they change in the same direction, negatively correlated, if they change in opposite directions, or uncorrelated.
How do we measure correlation?
Correlation is measured with the correlation coefficient r.
The correlation coefficient shows both the strength and direction of correlation.
It takes values from -1 to +1.
A positive correlation coefficient (r > 0) means, as you can probably guess, a positive correlation between two variables. That is, they move in the same direction. The higher r is, the stronger the
correlation is. When r = 1 that is a perfect positive correlation, meaning that if 1 variables increases or decreases by 10%, the other will increase and decrease also by 10%.
If the correlation is 0 (r = 0), there is no correlation, no association between the two variables. Remember the scatterplot can either show that Y does not really change even if X changes, or that
there is no clear pattern.
A negative correlation coefficient (r < 0) means a negative correlation. The lower it is, the stronger the negative correlation is. If r = -1, it means there is a perfect negative correlation between
the two variables: if one variable moves by 10%, other variables will also move by 10% in the opposite direction.
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Alim Physics 1st Paper Assignment Answer 2024 (1st Week)
Bangladesh Madrasah Education Board has given 1^st week assignment for the 2024 session of Alim candidates. There are seven assignments in the first week, Madrasah students have to submit four
assignments. Among these, Physics is one. Physics Assignment is quite a tough one for students. Here we will provide you with the details of Alim physics 1^st paper assignment. Here on our website,
you will also get the PDF file of your Physics assignment.
Alim Physics 1^st Paper Assignment
Due to the COVID pandemic, teachers will evaluate the results of every student by this assignment. Physics book is common for all the HSC/Alim students. They study the same Physics board book
approved by NCTB. The assignment is based on chapter 4 of the Physics book. Here in this chapter, Newtonian Mechanism and force are discussed thoroughly. This assignment is based on
• Acceleration
• Force and
• Speed
You need to have a clear concept about these topics to finish your assignment properly. These are the basic Physics knowledge students have to acquire during this course.
Check also: Alim Bangla 1st Paper Assignment Answer
Alim Physics 1st Paper Assignment Answer 1st Week
If you go through this brief discussion about Alim Physics 1^st Paper Assignment, you will see that the assignment is about Newtonian Mechanism and Force. The assignment is on a math problem, where a
diagram is shown with some problems. You have to solve the problems individually.
তুমি 20 ms-1 বেগে একদম খাড়াভাবে একটি 400 gm ভরের ক্রিকেট বল উপরের দিকে ছুড়ে মারলে।
(ক) বলটির বেগ নাম সময়ের গ্রাফ আঁকো।
(খ) গতিপথে সর্বোচ্চ বিন্দুতে বলটির বেগ কত।
(গ) ঐ বিন্দুতে ত্বরণ কত?
(ঘ) ঐখানে ক্রিকেট বলটির উপর ক্রিয়ারত মােট বল কত?
(ঙ) Fig-1 এ 1.5 kg ভরটি একটি টেবিলের উপর স্থির অবস্থানে আছে। 2 kg ভরের আরেকটি ভর একটি অসম্প্রসারণশীল সূতা দিয়ে ঝােলানাে হলাে। টেবিল ও 1.5 kg ভরের মাঝে ঘর্ষণ গুণাঙ্ক 0.2
(১) ভরদ্বয়ের ত্বরণ কত? সুতাটি অসম্প্রসারণশীল না হলে তােমার উত্তরের কী পরিবর্তন হতাে? (২) সূতার টান কত? (৩) 2 kg ভরের সরণ বনাম সময় গ্রাফ আঁকো?
Physics 1st Paper Assignment Answer (1st Week)
The problem is: Suppose you throw a cricket ball, which is 400 gm at the speed of twenty-meter per second. Now
• In the first problem you have to draw a graph between the time and force of the cricket ball.
• The Second problem, you need to find out the ball’s force at the highest point.
• In the third problem, you have to find out the accelerations on that point.
• You need to find out the total force of the ball.
Here is the assignment question for your convenience. Try to read it carefully to solve it in the correct order. The good thing is that we also help you by providing the answers to your assignment
topics. So keep your eyes on our website regularly if you want to get the answer copy.
Read more: Alim Assignment Question & Answer 1st Week
In this current pandemic condition, students must concentrate on their education. Submit your Alim physics 1^st paper assignment accurately. You will be evaluated according to your assignment answer.
This assignment will be the substitution for the exam. Try your best to solve all the Physics questions.
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Blue Jays: What Would a World Series Team Look Like?
TORONTO, ON - SEPTEMBER 28: Bo Bichette #11 and Vladimir Guerrero Jr. #27 of the Toronto Blue Jays sit in the dugout during the ninth inning of their MLB game against the Tampa Bay Rays at Rogers
Centre on September 28, 2019 in Toronto, Canada. (Photo by Cole Burston/Getty Images)
What does a World Series winning team look like, and what would it take for the Blue Jays to get there?
No team has an All-Star at every position, but to win the World Series takes an exceptional amount of talent. Which begs two questions: What does a World Series winning team look like, in terms of
fWAR in the regular season, and how close are the Jays to achieving that level?
World Series winners
If you take the World Series winning teams from 2010-2019 and calculate the total fWAR they received in the preceding regular season, you will find that the median fWAR from position players is 30
and from pitchers is 16. This of course varies from year to year – the 2015 Royals only had 21.8 fWAR from the position players, while the 2016 Cubs had over 37, and the pitching fWAR ranged from
9.4 (San Fran, 2014) to 22.3 (Washington, 2019). But 30 + 16 = 46 is perhaps an appropriate “ballpark”
2022 Blue Jays
So now let’s consider the 2022 Blue Jays in the context of those medians. Start with a few caveats. First, I am looking at the Jays as a whole. So, for example, the Jays might have a below-average
offensive outfield, but make up for that with additional infield offence. Second, I have made several assumptions. On the conservative side, I have assumed no major new signings – no Mookie Betts,
no Francisco Lindor, no “Thor”. I have also assumed no out-of-the-blue breakthroughs by current Jays’ minor leaguers – no Sean Reid-Foley suddenly developing pinpoint control, or Griffin Conine
having more walks than strikeouts. And I have assumed that any players drafted in 2020 are not ready by 2022.
On the optimistic side, I have assumed that existing players continue on their current trajectories with no major injuries or slumps. I also assume that a few current top prospects do make it to the
bigs by that time.
Note that when I show a player at a particular position (such as Vladdy at 1B), the WAR figure I show for them includes any ABs at DH or at other positions.
Catchers: Danny Jansen (2 WAR) + Reese McGuire (1 WAR)
Jansen had 1.4 WAR in 2019 in 384 plate appearances with a very unlucky BABIP of .230. His bat was looking good in 2020 spring training, but alas … I see him as the primary catcher in 2022, and
with 450 PAs combined with average luck he should easily be able to put up 2.0 WAR. McGuire had a 1.2 WAR in 105 PAs in 2019. I’m not sure he can sustain that pace, but he should still contribute
defensively so I have reduced him to 1.0 WAR in 150 PAs in 2022. Note that it is alternatively possible that McGuire is traded by then, and Alejandro Kirk (or Gabriel Moreno) is the backup catcher.
First Base: Vladimir Guerrero Jr. (5 WAR)
I assume that Vladdy does not stick at third (although that *is* possible) but that by 2022 he has reached his full hitting potential. Think Miguel Cabrera, or Carlos Delgado, or Albert Pujols
(without perhaps quite the defensive skills at 1B).
Second base : Cavan Biggio (3.5 WAR)
Biggio earned 2.4 WAR in 2019 in 430 PAs. Extrapolating the same pace to 600 PAs would give 3.35. Assuming further growth, I assume 3.5 in 2022.
Third base: Jordan Groshans (3)
I assume that Groshans is in the majors by 2022 and that he is putting up above-average-but-not-superstar numbers at third, based on a combination of solid defence and a good bat. Think Cavan Biggio
in 2019, but with a full season.
Shortstop: Bo Bichette (4.5 WAR)
Bo earned 1.7 WAR in 212 PAs in 2019. Extrapolating to 600 PAs would give 4.8 WAR. It would not be crazy to assume that, with experience, he improves on the -5.2 UZR/150 that he put up in 2019 and
turns into a Lindor-esque 6 WAR shortstop, but for this exercise assume that he puts up a solid 4.5 fWAR in 2022.
Outfield: Lourdes Gurriel Jr. (3.5), Randal Grichuk (2.5), Teoscar Hernandez (3.5)
Gurriel Jr’s 2019 extrapolates to 3.1 WAR over 600 PAs, despite his slow start. And this was his first year in LF. Assume that he will be a better fielder in 2022, and that he can maintain his 2019
wRC+ of 124, and he should be an above-average but not star 3.5 WAR OF. Think Tommy Pham, or Michael Conforto.
Grichuk averaged 1.93 WAR from 2016-18. Assume that playing centre field (where his defence is well above average) he can return to that level and a bit more for a 2.5 WAR.
Hernandez is very much a wild card. He had a terrible start to 2019, but he earned 1.7 WAR in the second half. And he continued his positive trend in UZR/150, going from -14.6 in 2017 to -12.2 in
2018 to -5.4 in 2019. Assume that he can continue this trend, to become an average defensive right fielder (he has 55 speed and arm, and 50-grade fielding, so the tools are there), and that he can
maintain a 120+ wRC+ (his H2 2019 wRC+ was 142), he becomes a second Gurriel – above average, but not a star. Note that if Teo did not pan out, this position could easily be filled by a Joc Pederson
– or possibly a Travis Shaw.
Designated hitter: Alejandro Kirk (2)
I assume that the DH spot will be used to rest the position players, but that Kirk will have ~100 games there, as well as a few games at catcher (and possibly at 1B to spell Vladdy).
Bench: combined (2)
I assume that the 4-man bench (remember that the roster will increase to 26 players in 2020), not including McGuire/Moreno, will earn 2 WAR.
Starters: Nate Pearson (4), Hyun-Jin Ryu (3.5), Alek Mahoah (2.5), Simeon Woods Richardson (2.5), Anthony Kay (2)
Assume Nate is a solid success (think Sonny Gray in 2019) but not a superstar. I also assume that Ryu, who earned ~5 WAR in 2019, declines by 0.5 WAR per year and is only a 3.5 WAR pitcher in 2022.
And I assume that both Manoah and SWR are up in the bigs and producing in 2022, but not (yet?) at star levels.
More Articles About Blue Jays World Series History:
Note that the 2022 rotation is a particularly fuzzy part of this projection, in that many other scenarios are possible. Could Ryan Borucki, completely recovered from his injury issues, be back and
building on his 1.7 fWAR in only 97 innings in 2018? How about Trent Thornton and his 1.9 fWAR in 2019? Could Adam Kloffenstein, or Eric Pardinho, or Julian Merryweather have forced themselves into
the conversation and onto the roster? Perhaps another way for me to state this prediction is that the 2022 Jays get 7.5 WAR from Nate and Hyun-Jin, and 7 WAR from some combination of three of their
other options.
Also note that these figures include all starters, even though I only show the top five.
Bullpen: combined (2)
The Jays’ bullpen earned a combined 1.9 WAR in 2019. I assume a similar number in 2022
The total: 45.5 WAR
When you add up all of these assumptions, you get 45.5 WAR, or essentially the same figure as the median World Series winning team of the past decade. (And no, I honestly did not jig these numbers
to get that result!)
Next. Blue Jays prospects with the best tools. dark
The bottom line
Any projection three years into the future is nothing more than an educated guess – one scenario among many. And any one of the individual assumptions in this projection could be challenged. But in
aggregate, I believe it to be reasonable – particularly since it does not assume any additions through trade or free agency. And in aggregate, it paints a very attractive picture. | {"url":"https://jaysjournal.com/2020/04/02/blue-jays-world-series-team-look-like/","timestamp":"2024-11-10T08:14:55Z","content_type":"text/html","content_length":"120015","record_id":"<urn:uuid:b60355bf-8177-4648-b737-563c568ca217>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00443.warc.gz"} |
Re: [Gzz] Re: One-time signature possibilities
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Re: [Gzz] Re: One-time signature possibilities
From: Benja Fallenstein
Subject: Re: [Gzz] Re: One-time signature possibilities
Date: Mon, 12 May 2003 15:37:52 +0200
User-agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.3) Gecko/20030430 Debian/1.3-5
Tuomas Lukka wrote:
On Mon, May 12, 2003 at 01:26:33PM +0200, Benja Fallenstein wrote:
I'm starting to be quite convinced that one-time signatures are the way to go with pointers. One-time signatures are signatures where the key can only be used to sign n messages for small n
(so actually they're n-time) (you can get around the problem by signing as one of the n messages the next public key to be used).
How does this BTW impact the chain length? If you need to check
1000 signatures in order to see that a key is valid, that's not really nice...
You make a tree, so than you need to verify the mth signature, you need to verify O(log m) signatures in whole.
If you choose n = 1024, so that you can sign 1024 messages with the same public key, then seed 128 new private keys from there with n = 1024 each, you could sign 128K messages; each of these 128K
messages would require checking two signatures. So you can get to pretty large numbers pretty easily if you need to.
- faster: factoring large integers is very costly compared to hashing.
??? ;)
Um, multiplying. Blah :-)
On the downside, one-time signatures are *large*. Remember that for every save we want to keep, we have to keep the signatures. The storage space per signature is O(b*b) with b the number of
bits in the hash, a problem if we want to use SHA-1 + Tiger. To some degree, we can trade off running time against signature storage space; we need to decide what is reasonable.
Do you have the numbers for e.g. DSA? How many bits more will
we have for this?
I think it's the same length as the key -- something like 1024 bit recommended-- but I'm not sure.
I've benchmarked, and on my machine, a DSA verification takes 30ms, a SHA-1 hash 5/1000 ms and a Tiger hash 6/1000 ms. The time estimates below are based on this.
If we use only SHA-1, not Tiger, some of our options are:
- Store ~3KB, verify ~160 hashes, ~.8 ms
- Store ~1.5KB, verify ~240 hashes, ~1.2ms
- Store ~840 bytes, verify ~600 hashes, ~3ms
- Store ~440 bytes, verify ~5100 hashes, ~25.5ms
Using SHA-1 + Tiger, we have:
- Store ~15KB, verify ~350 hashes, ~4ms
- Store ~8KB, verify ~530 hashes, ~6ms
- Store ~4KB, verify ~1320 hashes, ~15ms
- Store ~2KB, verify ~11000 hashes, ~120ms
I don't quite understand how you get these tradeoffs...
Not surprising since you don't know the algorithm :-)
It was AFAIK invented by Merkle/Winternitz. The description I found is in:
section 4.2, Construction 1, article page 13. (The article explains this as work itself is based on.) The parameter varying in the tradeoff is t: it's {1,2,3,4} respectively.
If, in addition to the above, you store another 20c or 44c bytes (for SHA-1 and SHA-1+Tiger, respectively), you can sign 2^c messages with the same public key. (Generation of the public key
takes 2*(2^c) times the verification time for a single hash.)
How does this work?
Generate 2^c private keys and the public key for each (a single hash); create a c-level binary Merkle hash tree of these public keys; publish the root of the hash tree as the combined public key.
To verify that one leaf is part of a hash tree (given the hash tree's root), you need to give the other hash, at every branch; this needs to be included in the signature; therefore, the signature
needs to contain c more hashes.
Discussion, please: What (if any) of this do you think is reasonable?
This might be reasonable.
Which version?
Note of course that you can sign a group of blocks, you don't need
to sign each block separately.
Of course. Thus, one block per save.
- Benja
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Function to calculate optimal designs
optDesign {DoseFinding} R Documentation
Function to calculate optimal designs
Given a set of models (with full parameter values and model probabilities) the ‘optDesign’ function calculates the optimal design for estimating the dose-response model parameters (D-optimal) or
the design for estimating the target dose (TD-optimal design) (see Dette, Bretz, Pepelyshev and Pinheiro (2008)), or a mixture of these two criteria. The design can be plotted (together with the
candidate models) using ‘plot.design’. ‘calcCrit’ calculates the design criterion for a discrete set of design(s). ‘rndDesign’ provides efficient rounding for the calculated continous design to
a finite sample size.
optDesign(models, probs, doses,
designCrit = c("Dopt", "TD", "Dopt&TD", "userCrit"),
Delta, standDopt = TRUE, weights,
nold = rep(0, length(doses)), n,
optimizer = c("solnp", "Nelder-Mead", "nlminb", "exact"),
lowbnd = rep(0, length(doses)), uppbnd = rep(1, length(doses)),
userCrit, ...)
## S3 method for class 'DRdesign'
plot(x, models, lwdDes = 10, colDes = rgb(0,0,0,0.3), ...)
calcCrit(design, models, probs, doses,
designCrit = c("Dopt", "TD", "Dopt&TD"),
Delta, standDopt = TRUE, weights,
nold = rep(0, length(doses)), n)
rndDesign(design, n, eps = 0.0001)
An object of class ‘c(Mods, fullMod)’, see the Mods function for details. When an TD optimal design should be calculated, the TD needs to exist for all models. If a D-optimal design
models should be calculated, you need at least as many doses as there are parameters in the specified models.
probs Vector of model probabilities for the models specified in ‘models’, assumed in the same order as specified in models
doses Optional argument. If this argument is missing the doses attribute in the ‘c(Mods, fullMod)’ object specified in ‘models’ is used.
designCrit Determines which type of design to calculate. "TD&Dopt" uses both optimality criteria with equal weight.
Delta Target effect needed for calculating "TD" and "TD&Dopt" type designs.
Logical determining, whether the D-optimality criterion (specifically the log-determinant) should be standardized by the number of parameters in the model or not (only of interest if type
standDopt = "Dopt" or type = "TD&Dopt"). This is of interest, when there is more than one model class in the candidate model set (traditionally standardization this is done in the optimal design
Vector of weights associated with the response at the doses. Needs to be of the same length as the ‘doses’. This can be used to calculate designs for heteroscedastic or for generalized
weights linear model situations.
When calculating an optimal design at an interim analysis, ‘nold’ specifies the vector of sample sizes already allocated to the different doses, and ‘n’ gives sample size for the next
nold, n
For ‘optimizer = "exact"’ one always needs to specify the total sample size via ‘n’.
List containing control parameters passed down to numerical optimization algorithms (optim, nlminb or solnp function).
control For ‘type = "exact"’ this should be a list with possible entries ‘maxvls1’ and ‘maxvls2’, determining the maximum number of designs allowed for passing to the criterion function
(default ‘maxvls2=1e5’) and for creating the initial unrestricted matrix of designs (default ‘maxvls1=1e6’). In addition there can be an entry ‘groupSize’ in case the patients are
allocated a minimum group size is required.
Algorithm used for calculating the optimal design. Options "Nelder-Mead" and "nlminb" use the optim and nlminb function and use a trigonometric transformation to turn the constrained
optimization problem into an unconstrained one (see Atkinson, Donev and Tobias, 2007, pages 130,131).
Option "solnp" uses the solnp function from the Rsolnp package, which implements an optimizer for non-linear optimization under general constraints.
Option "exact" tries all given combinations of ‘n’ patients to the given dose groups (subject to the bounds specified via ‘lowbnd’ and ‘uppbnd’) and reports the best design. When
optimizer patients are only allowed to be allocated in groups of a certain ‘groupSize’, this can be adjusted via the control argument. ‘n/groupSize’ and ‘length(doses)’ should be rather small
for this approach to be feasible.
When the number of doses is small (<8) usually ‘"Nelder-Mead"’ and ‘"nlminb"’ are best suited (‘"nlminb"’ is usually a bit faster but less stable than ‘"Nelder-Mead"’). For a
larger number of doses ‘"solnp"’ is the most reliable option (but also slowest) (‘"Nelder-Mead"’ and ‘"nlminb"’ often fail). When the sample size is small ‘"exact"’ provides the
optimal solution rather quickly.
lowbnd, Vectors of the same length as dose vector specifying upper and lower limits for the allocation weights. This option is only available when using the "solnp" and "exact" optimizers.
User defined design criterion, should be a function that given a vector of allocation weights and the doses returns the criterion function. When specified ‘models’ does not need to be
handed over.
The first argument of ‘userCrit’ should be the vector of design weights, while the second argument should be the ‘doses’ argument (see example below). Additional arguments to ‘
userCrit’ can be passed via ...
For function ‘optDesign’ these are additional arguments passed to ‘userCrit’.
For function ‘plot.design’ these are additional parameters passed to plot.Mods.
Argument for ‘rndDesign’ and ‘calcCrit’ functions: Numeric vector (or matrix) of allocation weights for the different doses. The rows of the matrices need to sum to 1. Alternatively
design also an object of class "DRdesign" can be used for ‘rndDesign’. Note that there should be at least as many design points available as there are parameters in the dose-response models
selected in models (otherwise the code returns an NA).
eps Argument for ‘rndDesign’ function: Value under which elements of w will be regarded as 0.
x Object of class ‘DRdesign’ (for ‘plot.design’)
lwdDes, Line width and color of the lines plotted for the design (in ‘plot.design’)
Let M_m denote the Fisher information matrix under model m (up to proportionality). M_m is given by \sum a_i w_i g_i^Tg_i, where a_i is the allocation weight to dose i, w_i the weight for dose i
specified via ‘weights’ and g_i the gradient vector of model m evaluated at dose i.
For ‘designCrit = "Dopt"’ the code minimizes the design criterion
-\sum_{m}p_m/k_m \log(\det(M_m))
where p_m is the probability for model m and k_m is the number of parameters for model m. When ‘standDopt = FALSE’ the k_m are all assumed to be equal to one.
For ‘designCrit = "TD"’ the code minimizes the design criterion
\sum_{m}p_m \log(v_m)
where p_m is the probability for model m and v_m is proportional to the asymptotic variance of the TD estimate and given by b_m'M_m^{-}b_m (see Dette et al. (2008), p. 1227 for details).
For ‘designCrit = "Dopt&TD"’ the code minimizes the design criterion
Again, for ‘standDopt = FALSE’ the k_m are all assumed to be equal to one.
For details on the ‘rndDesign’ function, see Pukelsheim (1993), Chapter 12.
In some cases (particularly when the number of doses is large, e.g. 7 or larger) it might be necessary to allow a larger number of iterations in the algorithm (via the argument ‘control’),
particularly for the Nelder-Mead algorithm. Alternatively one can use the solnp optimizer that is usually the most reliable, but not fastest option.
Bjoern Bornkamp
Atkinson, A.C., Donev, A.N. and Tobias, R.D. (2007). Optimum Experimental Designs, with SAS, Oxford University Press
Dette, H., Bretz, F., Pepelyshev, A. and Pinheiro, J. C. (2008). Optimal Designs for Dose Finding Studies, Journal of the American Statisical Association, 103, 1225–1237
Pinheiro, J.C., Bornkamp, B. (2017) Designing Phase II Dose-Finding Studies: Sample Size, Doses and Dose Allocation Weights, in O'Quigley, J., Iasonos, A. and Bornkamp, B. (eds) Handbook of methods
for designing, monitoring, and analyzing dose-finding trials, CRC press
Pukelsheim, F. (1993). Optimal Design of Experiments, Wiley
See Also
Mods, drmodels
## calculate designs for Emax model
doses <- c(0, 10, 100)
emodel <- Mods(emax = 15, doses=doses, placEff = 0, maxEff = 1)
optDesign(emodel, probs = 1)
## TD-optimal design
optDesign(emodel, probs = 1, designCrit = "TD", Delta=0.5)
## 50-50 mixture of Dopt and TD
optDesign(emodel, probs = 1, designCrit = "Dopt&TD", Delta=0.5)
## use dose levels different from the ones specified in emodel object
des <- optDesign(emodel, probs = 1, doses = c(0, 5, 20, 100))
## plot models overlaid by design
plot(des, emodel)
## round des to a sample size of exactly 90 patients
rndDesign(des, n=90) ## using the round function would lead to 91 patients
## illustrating different optimizers (see Note above for more comparison)
optDesign(emodel, probs=1, optimizer="Nelder-Mead")
optDesign(emodel, probs=1, optimizer="nlminb")
## optimizer solnp (the default) can deal with lower and upper bounds:
optDesign(emodel, probs=1, designCrit = "TD", Delta=0.5,
optimizer="solnp", lowbnd = rep(0.2,3))
## exact design using enumeration of all possibilites
optDesign(emodel, probs=1, optimizer="exact", n = 30)
## also allows to fix minimum groupSize
optDesign(emodel, probs=1, designCrit = "TD", Delta=0.5,
optimizer="exact", n = 30, control = list(groupSize=5))
## optimal design at interim analysis
## assume there are already 10 patients on each dose and there are 30
## left to randomize, this calculates the optimal increment design
optDesign(emodel, 1, designCrit = "TD", Delta=0.5,
nold = c(10, 10, 10), n=30)
## use a larger candidate model set
doses <- c(0, 10, 25, 50, 100, 150)
fmods <- Mods(linear = NULL, emax = 25, exponential = 85,
linlog = NULL, logistic = c(50, 10.8811),
doses = doses, addArgs=list(off=1),
placEff=0, maxEff=0.4)
probs <- rep(1/5, 5) # assume uniform prior
desDopt <- optDesign(fmods, probs, optimizer = "nlminb")
desTD <- optDesign(fmods, probs, designCrit = "TD", Delta = 0.2,
optimizer = "nlminb")
desMix <- optDesign(fmods, probs, designCrit = "Dopt&TD", Delta = 0.2)
## plot design and truth
plot(desMix, fmods)
## illustrate calcCrit function
## calculate optimal design for beta model
doses <- c(0, 0.49, 25.2, 108.07, 150)
models <- Mods(betaMod = c(0.33, 2.31), doses=doses,
placEff=0, maxEff=0.4)
probs <- 1
deswgts <- optDesign(models, probs, designCrit = "Dopt",
## now compare this design to equal allocations on
## 0, 10, 25, 50, 100, 150
doses2 <- c(0, 10, 25, 50, 100, 150)
design2 <- c(1/6, 1/6, 1/6, 1/6, 1/6, 1/6)
crit2 <- calcCrit(design2, models, probs, doses2, designCrit = "Dopt")
## ratio of determinants (returned criterion value is on log scale)
## example for calculating an optimal design for logistic regression
doses <- c(0, 0.35, 0.5, 0.65, 1)
fMod <- Mods(linear = NULL, doses=doses, placEff=-5, maxEff = 10)
## now calculate weights to use in the covariance matrix
mu <- as.numeric(getResp(fMod, doses=doses))
mu <- 1/(1+exp(-mu))
weights <- mu*(1-mu)
des <- optDesign(fMod, 1, doses, weights = weights)
## one can also specify a user defined criterion function
## here D-optimality for cubic polynomial
CubeCrit <- function(w, doses){
X <- cbind(1, doses, doses^2, doses^3)
CVinv <- crossprod(X*w)
optDesign(doses = c(0,0.05,0.2,0.6,1),
designCrit = "userCrit", userCrit = CubeCrit,
optimizer = "nlminb")
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Baccarat Rules and Scheme
Baccarat Chemin de Fer Principles
Baccarat chemin de fer is gambled on with eight decks of cards in a dealing shoe. Cards valued less than ten are valued at face value and with Ten, Jack, Queen, King are zero, and Ace is 1. Wagers
are placed on the ‘bank’, the ‘player’, or on a tie (these aren’t actual people; they simply represent the 2 hands that are dealt).
Two cards are dealt to both the ‘bank’ and ‘gambler’. The score for each hand is the total of the two cards, but the beginning digit is dumped. For instance, a hand of 5 and 6 has a total of 1 (5
plus 6 equals 11; drop the 1st ‘1′).
A third card can be given using the following rules:
- If the player or bank has a value of 8 or nine, both players stay.
- If the gambler has less than five, he takes a card. Players stays otherwise.
- If the gambler holds, the bank takes a card on a value less than 5. If the gambler takes a card, a guide is employed to determine if the bank stays or hits.
Baccarat Chemin de Fer Odds
The greater of the 2 scores wins. Winning wagers on the banker payout nineteen to Twenty (even money minus a 5% commission. The Rake is recorded and paid off once you quit the table so make sure you
still have money left just before you depart). Winning bets on the gambler pays 1:1. Winning bets for tie typically pays out at 8 to 1 but on occasion 9:1. (This is a bad bet as a tie occurs less
than one in every ten rounds. Be cautious of wagering on a tie. However odds are substantially greater for nine to one vs. 8:1)
Gambled on properly baccarat provides generally good odds, aside from the tie bet of course.
Baccarat Chemin de Fer Method
As with all games Baccarat has a few familiar false impressions. One of which is similar to a misconception in roulette. The past is not a harbinger of future actions. Tracking past results at a
table is a poor use of paper and a snub to the tree that gave its life for our paper desires.
The most established and probably the most accomplished plan is the one, three, two, six tactic. This method is employed to build up earnings and minimizing losses.
Start by placing 1 chip. If you succeed, add another to the 2 on the table for a grand total of three units on the second bet. Should you succeed you will have 6 on the game table, take away 4 so you
are left with two on the third round. Should you come away with a win on the 3rd bet, add two to the 4 on the table for a grand total of six on the 4th bet.
If you don’t win on the first bet, you take a loss of 1. A win on the initial bet followed by a hit on the second creates a loss of two. Wins on the 1st two with a defeat on the 3rd provides you with
a gain of two. And wins on the initial 3 with a defeat on the 4th means you balance the books. Succeeding at all 4 wagers leaves you with 12, a gain of ten. This means you are able to squander the
2nd bet 5 times for each successful run of 4 wagers and still break even.
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Spanish Ordinal Numbers Worksheet Pdf - OrdinalNumbers.com
Spanish Numbers Ordinal – There are a myriad of sets that can be enumerated using ordinal numbers as an instrument. They can also be used to generalize ordinal quantities. 1st Ordinal numbers are one
of the most fundamental ideas in mathematics. It is a number that indicates the location of an object within an array. … Read more
Ordinal Numbers Worksheets Pdf
Ordinal Numbers Worksheets Pdf – You can count unlimited sets with ordinal numbers. They can also be utilized as a method to generalize ordinal numbers. 1st Ordinal numbers are among the most
fundamental ideas in math. It is a numerical value that represents the location of an object within the list. The ordinal number is … Read more | {"url":"https://www.ordinalnumbers.com/tag/spanish-ordinal-numbers-worksheet-pdf/","timestamp":"2024-11-06T07:09:18Z","content_type":"text/html","content_length":"53146","record_id":"<urn:uuid:b682c633-a670-41d9-b9bc-d889e165b382>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00509.warc.gz"} |
On the duplication distance of binary strings
We study the tandem duplication distance between binary sequences and their roots. This distance is motivated by genomic tandem duplication mutations and counts the smallest number of tandem
duplication events that are required to take one sequence to another. We consider both exact and approximate tandem duplications, the latter leading to a combined duplication/Hamming distance. The
paper focuses on the maximum value of the duplication distance to the root. For exact duplication, denoting the maximum distance to the root of a sequence of length n by f(n), we prove that f(n) = Θ
(n). For the case of approximate duplication, where a β-fraction of symbols may be duplicated incorrectly, we show using the Plotkin bound that the maximum distance has a sharp transition from linear
to logarithmic in n at β = 1/2.
Publication series
Name IEEE International Symposium on Information Theory - Proceedings
Volume 2016-August
ISSN (Print) 2157-8095
Other 2016 IEEE International Symposium on Information Theory, ISIT 2016
Country/Territory Spain
City Barcelona
Period 7/10/16 → 7/15/16
All Science Journal Classification (ASJC) codes
• Theoretical Computer Science
• Information Systems
• Modeling and Simulation
• Applied Mathematics
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Gaganova N.V.
Superelasticity description based on the combined model of shape memory alloys deformation considering development of the martensitic elements
The article considers a constitutive model for shape memory alloys, which allows to take into account the differences between phase and structural transformation. The model reflects the fact that
hardening effect is typical for structural transformation, but not for phase transformation. Deformation due to structural transformation is described with the use of loading surface by analogue of
the plasticity theory with isotropic hardening. The deformed state is determined by one parameter, which can be changed by phase or structural deformation. Inelastic deformation due to structural
transformation in the active process is subject to the associated flow rule. The article examines the possibilities of the model for describing the phenomenon of superelasticity. The temperature of
the phase transition in shape memory alloys depends on the operating stress, so the phase transition can occur at the constant temperature. In a certain range of stresses, dependence of deformations
on stresses becomes nonlinear. This phenomenon can be explained by a phase-structural transition. In this paper, a proportional monotonic loading at a constant temperature and phase transitions
caused by increasing and decreasing stresses. The model is extended to the case of the development of martensitic elements during the phase and structural transition. Phase-structural and phase
deformations plots are provided. It is shown that the model allows to describe the phenomenon of superelasticity correctly. The obtained results are compared for different material functions that
determine the relationship between the processes of origin and development of martensitic elements. It is shown that under the considered loading conditions, phase deformations increase with
temperature. The values of phase deformations are higher for material functions that take into account the development of martensitic elements.
Pages: 441-454 Elibrary
Khokhlov Andrew V.
Exact solution for strain and stress fields in a multilayer thick-walled tube of non-linear viscoelastic materials under given internal and external pressures
We construct the exact solution of the quasi-static boundary value problem for a multilayer thick-walled tubes made of physically non-linear viscoelastic materials obeying the Rabotnov constitutive
equation with two arbitrary material functions for each layer (a material creep compliance and a function which governs physical non-linearity). We suppose that every layer material is homogeneous,
isotropic and incompressible and that a tube is loaded by time-dependent internal and external pressures (varying slowly enough to neglect inertia terms in the equilibrium equations) and that a plain
strain state is realized, i.e. zero axial displacements are given on the edge cross sections of the tube. We obtained the closed form expressions for displacement, strain and stress fields via the
single unknown function of time and integral operators involving this function, pairs of (arbitrary) material functions of each tube layer, preset pressure values and ratios of tube layers radii and
derived integral equation to determine this unknown function. To derive it we split and solve the set of non-linear integral equations for unknown functions of time governing strain fields of every
tube layer and for unknown interlayer normal stresses (depending on time). Assuming material functions are arbitrary, we proved that the total axial force at a tube cross section doesn’t depend on a
number of layers, their thicknesses and pairs of material functions governing their mechanical behavior and on a history of loading although stresses and strains do. The axial force depends only on a
tube radii and current values of given pressures. It proved to be equal to the axial force calculated for homogeneous linear elastic tube although axial stress depends on radial coordinate in the
case of non-linear viscoelastic materials. Assuming the material functions that govern non-linearity of each layer material coincide with a power function with a positive exponent and assuming their
relaxation moduli are proportional to a single (arbitrary) function of time, we constructed exact solution of the resolving functional equation, calculated all the integrals involved in the general
representation for the tube stress field and reduced it to simple algebraic formulas convenient for analysis.
Pages: 455-476 Elibrary
Blohin V.V., Kulakov V.V., Lisin A.N., Mozalev V.V., Pankov M.I., Sivurova V.A.
Evaluation of capacity for work of carbon/carbon composite materials for aircraft brake disks
An analysis of the effectiveness of using carbon-carbon composite materials for the manufacture of aircraft brake discs is carried out. Materials made using fundamentally different technologies were
considered: materials with a pitch matrix formed by the liquid-phase method, with graphitized, carbonized and with a different ratio of carbonized and graphitized fibers, as well as materials made by
the needle-stitching method, with a pyrocarbon matrix reinforced with tapes and felt. The performance of the brake discs was evaluated by testing the disc on a device that allows you to determine the
strength of the most vulnerable zone of the disc – the spike. The disk loading scheme is close to real conditions, i.e. the groove is engaged with a power finger, which acts as a guide for the drum,
and the load force is directed in the circumferential direction. During the testing process, the current loading parameters are continuously monitored up to failure. In addition, tests of samples of
materials for tensile strength when the interlaminar shear and bending. Samples for testing the bending strength were cut from the disk along the radius with the application of force in the
circumferential direction. The regularity of changes in strength during interlayer shear, bending, and spiking of the disk from the apparent density of the material used is determined. It was found
that the strength of materials during interlayer shear is practically independent of the apparent density of the material, while the Flexural type of material is most preferable for improving the
performance of the brake discs. strength and strength of the force elements of the disk structure correlates with the density of the material. Set the type of material is most preferable for
improving the performance of the brake discs.
Pages: 477-489 Elibrary
Azarov A.V., Babichev A.A., Razin A.F.
Optimal design of an airplane wing composite lattice panel under axial compression
Composite materials are now widely used in aircraft structures. However, the main structural concept for composite wing panels is a traditional stringer structure. Such structure consists of a
load-bearing skin and reinforcing longitudinal ribs-stringers. Due to the special properties of polymer composites, this design usually does not reduce the composite panel’s weight compared to the
metal prototype. An alternative to structures with a load-bearing skin is a composite lattice structure, where the main load-bearing elements are intersecting diagonal and transverse ribs. The ribs
are formed from unidirectional CFRP, which has high specific strength and stiffness. Such a design scheme allows us to realize the high longitudinal properties of the composite material and to ensure
a high weight efficiency of the structure. The paper is concerned with design of a flat rectangular panel consisting of a system of intersecting ribs made of unidirectional composite material by
filament winding or lay-up processes. The panel is a structural element of an airplane wing. The panel is simply supported and compressed in the longitudinal direction. The relations which specify
the optimal structural parameters of the lattice panel, i.e., the panel thickness, the ribs orientation angle, the ribs thickness and spacing are obtained. The optimal parameters provide the panel
minimum mass under strength and buckling constraints. Optimization is undertaken by the method of minimization of the safety factors corresponding to possible modes of the panel failure. The method
allows us to reduce the constrained optimization problem to the problem of conditional minimization. Two types of panels for which the length-to-width ratio is more or less then unity are considered.
Optimization of a carbon-epoxy lattice panel is presented as an example.
Pages: 490-500 Elibrary
Firsanov Vic.V.
A variant of refinement of the classical theory of thin plate bending
The classical theory of bending of thin plates based on Kirchhoff’s hypotheses about the absence of normal stresses in the transverse to the bases direction, invariability of the length of the normal
element to the middle plane of the plate, which means the invariability of the thickness and lack of linear deformation in the transverse direction, the absence of shear strains in planes
perpendicular to the bases of the plate. At the same time, in the equilibrium equations, both normal stresses in the transverse direction and tangential stresses associated with shear deformations by
physical relations remain, but the physical connections are obviously broken. Refinement of the classical theory is usually associated with the rejection of all Kirchhoff hypotheses, which
significantly complicates such a model, or the rejection of one or two kinematic hypotheses. For example, you can set a displacement in the transverse direction as a power series along the transverse
coordinate. In this case, if the degrees are even, the linear deformation in the transverse direction is different from zero, but the normal element connecting the bases of the plate does not change
its length, which is not in contradiction with the Kirchhoff hypothesis. However, this approach may not lead to significant refinements of the classical model, so for a more or less significant
refinement, it is assumed that the most acceptable rejection of the hypothesis of the absence of shear deformations in the planes transverse to the plate bases. In this case, the physical
relationship between shear and stress is restored. Accounting for these shear deformations is especially important for materials with low shear stiffness in the transverse directions. Another reason
for refining the classical model of plate bending is that some boundary conditions are not satisfied accurately enough, which is due to the introduction of a generalized Kirchhoff shear force into
the calculation model, which consists of a purely shear force and an increment along one of the plane coordinates of the torque. With certain refinements, it is possible to solve the problem of three
boundary conditions on the free edges of the plate.
Pages: 501-512 Elibrary
Belov P.A., Lurie S.A.
Variational formulation of coupled of heat/mass transfer and thermoelasticity problem
A variational formulation of a coupled system of equations of thermoelasticity, heat and mass transfer is given. A special case of the gradient model of the Mindlin-Tupin medium is proposed, when the
gradient component of the potential energy depends only on the gradients of the constrained dilation. In general, a generalized variational model is considered, in which the gradient variational
model is expanded by taking into account the potential energy of defective media with dilatational damage, combining two types of free (incompatible) dilations: free dilations associated with a
change in volume due to temperature effects, and free dilations associated with the concentration of impurities due to diffusion processes. As a result, the equations of motion included in the
coupled system of equations are a special case of the gradient theory (dilation model) in the part of the differential operator over displacements. The heat and mass transfer equations have the same
structure and reflect the diffusion-wave mechanism of evolution in a continuous medium of temperature and impurity. It was found that the coupled system of equations decomposes into three independent
boundary value problems with respect to displacements, a free (incompatible) change in volume associated with temperature loading, and a free (incompatible) change in volume associated with the
diffusion (concentration) process, when the tensors of physical properties are spherical and the corresponding connectivity coefficients are equal to zero. The consistency equations obtained from the
generalized equations of Hooke’s law for force factors and their fluxes by eliminating kinematic variables give a whole spectrum of laws of heat conduction, diffusion and thermoelasticity, including
the laws of Fourier, Maxwell-Cattaneo, Soret and Dufour.
Pages: 513-527 Elibrary
Afanasyeva E.A., Dolgova E.V., Goryashnik Yu.S., Krivushina A.A.
Change of epoxy polymer properties after mold fungi influence depending on initial epoxy resin structure
The reaction activity of mixtures based on epoxy resins of bisphenol A as well as 4,4′-methylenedianiline tetraglycidyl ether with an amine hardener were studied. The gelation of the mixtures was
determined. The curing reaction parameters of epoxy compositions are determined by differential scanning calorimetry. The temperature-time regime of specimen curing was selected by means of
mathematical modeling implementing and the experimental data studying of the curing kinetics obtained the thermoanalytical method. Given regime was recommended for polymer matrices with a predictable
high degree of reactive groups conversion obtaining. A mixture of bisphenol A diglycidyl ether epoxy oligomer (ED-22 brand resin) with an amine hardener was chosen as a model sample for mode
selecting. When using the proposed mode, the degree of curing of epoxy matrices of all samples under study exceeded 90%. A high completeness of the reaction was suggested. The fungal resistance of
epoxy polymers has been studied. Microstructural studies of the samples were performed by means of optical microscopy. The predicted damage of the samples surfaces of all epoxy matrices exposed to
micromycetes was found. It could be associated with the frequency of polymer chains cross-linking, which directly depends on the value of the initial resin epoxy number. To assess the change in the
characteristics of polymers after microbiological tests, the mechanical and thermal properties of the cured epoxy matrices were carried out. The research of heat resistance and static bending
strength of samples after exposure to micromycetes was performed. The dependence of the change in the properties of epoxy polymers after tests for resistance to mold fungi on the frequency of
cross-linking of polymer matrices was revealed.
Pages: 528-543 Elibrary
Fedotenkov G.V., Lokteva N.A., Serdyuk D.O., Skopintsev P.D.
Transient stress-strain state of a composite cylindrical shell
This work is devoted to the description of an approach to studying the propagation of non-stationary disturbances, stresses and strains in a thin elastic composite cylindrical shell. The shell is
accepted unlimited, with a constant thickness. An aggregate of non-stationary moving pressures affects along the normal to the outside surface of the shell. The shell is assumed to be unbounded, with
a constant thickness. The outer face of the shell is subjected to a set of non-steady moving loads. It is assumed also that the composite material of the shell is linearly elastic, with a lamination
scheme symmetric with respect to the midsurface. The shell model is based on the Kirchhoff-Lowe hypotheses, while the load instantly applied to the shell are modeled by Dirac functions. The study of
non-steady deformation of the shell is carried out using the transient function, which is a normal displacement that occurs as a response to a single load concentrated in time and coordinates. The
transient function is constructed using exponential Fourier series expansion, Laplace integral transformations in time domain and Fourier transforms with respect to the longitudinal coordinate. The
inverse Laplace transform is performed analytically, whereas the original of the Fourier transform is found by using the numerical method of integrating rapidly oscillating functions. The non-steady
normal deflection of the cylindrical shell is represented as a triple convolution of the transient function with the functions defining the moving concentrated loads with time-varying amplitudes and
coordinates of the impact. The convolution integrals are evaluated using rectangle quadrature formulae. The study of the space-time stress-strain state of an unbounded thin elastic composite
cylindrical shell becomes possible after constructing a non-steady deflection function with further use of constitutive and kinematic relations to obtain the stress state of the shell. In the study
of the non-steady stress-strain state of the composite shell, the given technical constants determined through the generalized stiffness of the material are used. As an example, the space-time
dependences of the non-steady deflection, the distribution of stresses and deformations on the outer surface of the polymer composite shell are constructed. Non-steady impact was considered as a set
of moving concentrated loads.
Pages: 544-559 Elibrary
Yankovskii A.P.
Modeling of viscoelastic-plastic bending behavior of cylindrical shells with spatial reinforcement structures
Based on the method of time steps, a numerical-analytical model of viscoelastic-plastic deformation of circular cylindrical shells with spatial reinforcement has been developed. Instant plastic
deformation of the materials of the components of the composition is described by the theory of flow with isotropic hardening. The viscoelastic deformation of these materials is described by the
governing equations of the Maxwell – Boltzmann model of the body. The geometric nonlinearity of the problem is taken into account in the Karman approximation. The possible weak resistance of
composite shells to transverse shears is taken into account on the basis of the Ambardzumyan theory. The developed model of the mechanical behavior of materials of the components of the composition
is adapted to the use of an explicit numerical “cross” type scheme. The viscoelastic-plastic and elastoplastic dynamic and quasistatic deformation of thin flexible fiberglass cylindrical shells under
the influence of internal pressure, as well as rectangular elongated plates under the action of a uniformly distributed transverse load, are studied. The constructions have traditional reinforcement
structures with orthogonal laying of fibers on equidistant surfaces or have spatial reinforcement structures. It is demonstrated that calculations performed on the theory of elastoplastic and
rigid-plastic deformation of reinforced shells and plates do not give even an approximate idea of the residual states of composite constructions under their dynamic loading. It is shown that, after
viscoelastic-plastic dynamic deformation, relatively thin reinforced constructions acquire corrugated residual forms. With quasistatic transverse loading of composite plates, the residual deflection
has a traditional form, i.e., folds are not formed.
Pages: 560-578 Elibrary | {"url":"https://iampress.ru/en/n4-2020/","timestamp":"2024-11-09T03:16:27Z","content_type":"text/html","content_length":"83127","record_id":"<urn:uuid:f5241d37-5017-4636-b4f6-c629d3fcac35>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00474.warc.gz"} |
bigsparser: sparse matrix format with data on disk
For now, only a few features are implemented:
• convert a dgCMatrix or a dsCMatrix to an SFBM, a Sparse Filebacked Big Matrix
• grow an SFBM using $add_columns() (similar to ‘cbind’ or ‘bdiag’)
• compute the product and crossproduct of an SFBM with a vector
• solve Ax=b, where A is a symmetric SFBM and b is a vector
• access the subset of an SFBM as a dgCMatrix (matrix accessor, since v0.6.1)
A new compact format is available (since v0.5), which is useful when non-zero values in columns are contiguous (or almost). | {"url":"https://cran.gedik.edu.tr/web/packages/bigsparser/readme/README.html","timestamp":"2024-11-06T20:15:30Z","content_type":"application/xhtml+xml","content_length":"6368","record_id":"<urn:uuid:1a2aeeb2-4fd9-4158-b26e-7910856ea077>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00869.warc.gz"} |
AP Board Solutions Class 10 Maths Notes In English & Telugu Mediums - Ap Board Solutions Class 10
AP Board Solutions Class 10 Maths Notes in English & Telugu Mediums
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Optokinetic nystagmus (OKN) is a fundamental oculomotor response to retinal slip generated during natural movement through the environment. The timing and amplitude of the compensatory slow phases
(SPs) alternating with saccadic quick phases (QPs) are remarkably variable, producing a characteristic irregular sawtooth waveform. We have previously found three stochastic processes that underlie
OKN: the processes that determine QP and SP amplitude and the update dynamics of SP velocity. SP and QP parameters are interrelated and dependent on SP velocity such that changes in stimulus speed
can have a seemingly complex effect on the nystagmus waveform. In this study we investigated the effect of stimulus spatial frequency on the stochastic processes of OKN. We found that increasing the
spatial frequency of suprathreshold stimuli resulted in a significant increase in SP velocity with a corresponding reduction in retinal slip. However, retinal slip rarely reached values close to 0,
indicating that the OKN system does not or cannot always minimize retinal slip. We deduce that OKN gain must be less than unity if extraretinal gain is lower than unity (as empirically observed), and
that the difference between retinal and extraretinal gain determines the Markov properties of SP velocity. As retinal gain is reduced with stimuli of lower spatial frequency, the difference between
retinal and extraretinal gain increases and the Markov properties of the system can be observed.
Natural movement through the environment causes the image of the visual scene to drift across the retina and disrupt vision. Optokinetic nystagmus (OKN) is an oculomotor response that is believed to
compensate for this retinal slip during slow head and body movements. OKN consists of an alternating sequence of compensatory slow phases (SPs) made in the direction of retinal slip and saccadic
quick phases (QPs) made predominantly in the opposite direction. Although neural pathways for OKN have been studied for decades, it is a surprisingly complex behavior, and its functional role in
humans still remains uncertain.
In birds and lateral-eyed afoveate mammals, the OKN response is generated predominantly by subcortical pathways that produce a slow buildup of eye velocity referred to as the indirect, delayed, or
velocity storage component of the optokinetic response (OKNd). OKNd appears to supplement the rotational vestibulo-ocular reflex at low temporal frequencies. It is elicited during prolonged
self-rotation in the light by global retinal motion, as the response of the rotational vestibulo-ocular reflex declines (Robinson,
; Schweigart, Mergner, Evdokimidis, Morand, & Becker,
However, in humans and other primates, OKNd is typically dominated by another form of OKN with fast dynamics, referred to as the direct or early optokinetic response. This form shares similarities
with the ocular following response and is thought to be driven by local retinal motion involving similar (if not the same) cortical and cerebellar pathways as smooth-pursuit eye movements (Abadi,
Howard, Ohmi, & Lee,
; Büttner & Büttner-Ennever,
; Cohen, Henn, Raphan, & Dennett,
; Cohen, Matsuo, & Raphan,
; Gellman, Carl, & Miles,
; Miles,
; Miles, Kawano, & Optican,
; Simons & Büttner,
). This has been highlighted in clinical cases where the smooth-pursuit pathway was damaged and the early optokinetic response was lost, while the slow buildup of eye velocity generated from OKNd was
preserved (Harris, Walker, Shawkat, Wilson, & Russell-Eggitt,
; Yee, Baloh, Honrubia, Lau, & Jenkins,
Most studies of OKN have ignored QPs and averaged SP velocity over consecutive cycles to obtain a smooth measure of eye velocity. However, when the details of individual cycles are examined, a
remarkable degree of variability is observed, even when stimulus velocity is held constant (Anastasio,
; Balaban & Ariel,
; Carpenter,
; Cheng & Outerbridge,
; Kolarik, Margrain, & Freeman,
; Shelhamer,
; Trillenberg, Zee, & Shelhamer,
; Waddington & Harris,
). In a recent, study we analyzed the local correlations between cycles using principal-components analysis (PCA) and found evidence for a complex stochastic system of three components (Waddington &
component determines the SP velocity of the
th cycle, which we describe as a linear first-order autoregressive equation with the update dynamics
is a constant and
) is a stochastic process with mean
and standard deviation
. This accounts for the remarkable degree of variability observed in SP velocity from one cycle to the next. The
component determines the amplitude of the
th SP (
) according to
is the start position of the SP,
) is a second stochastic process with mean
and standard deviation
, and
are constants. The
component determines QP amplitude (
) according to
is the start position of the QP (end position of the previous SP),
) is a third stochastic process with mean
and standard deviation
, and
are constants. The stochastic processes
), and
) are uncorrelated. However, because
loads onto multiple components, the variables
, and
are mutually correlated (and autocorrelated) in a complex sequence of cycles (see
Figure 1
Table 1
for a full description of the system).
More recently, we investigated the stochastic nature of the QP switching mechanism (Waddington & Harris,
). Previous models of OKN had typically considered QPs to be regular resetting saccades, which are triggered stochastically after a certain period of time. However, our model indicated that QPs were
triggered primarily after the eyes had moved a certain distance, with some constraints on timing caused by the dependence of the SP amplitude threshold on SP velocity (
Equation 2
). This model gives rise to a complex ratio distribution of SP duration (
) in which the denominator and numerator variables are correlated. We derived the probability density function (pdf) for this model in our previous study and compared it to the pdfs predicted from
five other models of the QP trigger to demonstrate its superior fit to the data.
From these studies we concluded that OKN is controlled by a more complicated system than a simple retinal-slip feedback loop with regular resetting saccades. In particular, SP velocity tends to
wander in a Markov fashion (see
for an explanation of Markov processes) and does not always maintain low retinal slip, leading us to question the long-held assumption that OKN strives to minimize retinal slip per se.
It is not clear why SP velocity should be so variable, although one possibility is that it is the result of variability in retinal gain, which is dependent on visual contrast and presumably contrast
sensitivity, and hence the speed, spatial frequency, and retinal location of the stimulus (Kelly,
). Most previous studies of the effect of spatial frequency on OKN have used stimuli at or near the contrast threshold, with the goal of using eye movements as a behavioral response measure in a
detection task. In these studies, OKN eye-movement parameters are typically not investigated, other than to verify whether or not they occur under given stimulus conditions (Çetinkaya, Oto, Akman, &
; Leguire, Zaff, Freeman, Rogers, & Bremer,
; Wester, Rizzo, Balkwill, & Wall,
). Suprathreshold studies are few and conflicting. Schor and Narayan (
) demonstrated that at low stimulus speeds, increasing spatial frequency had very little effect, but at moderate speeds, it resulted in a decrease in SP velocity. More recently, Sumnall, Freeman, and
Snowden (
) demonstrated that at low stimulus speeds, increasing spatial frequency actually resulted in an increase in SP velocity.
It is not clear how parameters of the OKN stimulus other than stimulus speed affect the OKN waveform. If the purpose of OKN is simply to minimize retinal slip, then it seems appropriate to assume
that the spatial parameters of the stimulus would have no effect. However, there is some evidence to indicate that this is not the case. In this study we decided to investigate the effect of spatial
frequency on OKN. We chose three spatial frequencies—0.05, 0.1, and 0.2 c/°—each presented to eight subjects at two speeds: 10°/s and 30°/s. We employed PCA as previously reported in order to
quantify how (or whether) spatial frequency changed the stochastic
Equations 1
Materials and methods
Eight healthy adults (six male, two female), with a mean age of 26 (SD = 4) years, participated in the study and had no self-reported neurological, ophthalmological, or vestibular impairments. The
visual acuity of each participant was measured using a Snellen chart before consent to participate was elicited, and only participants with uncorrected binocular visual acuity of 6/6 or higher on the
Snellen scale were included in this study. All protocols were approved by the Plymouth University Faculty of Science Human Research Ethics Committee. Participants gave informed written consent and
were made aware of their right to withdraw at any time.
Participants sat in a chair 1 m from the middle of a flat white screen (1.57 × 1.17 m landscape, subtending 76° × 61°) with the tangent error uncorrected. The OKN stimulus was rear projected (Epson
EMP-500; Seiko Epson Corporation, Suwa, Japan) onto the screen at a frame rate of 60 Hz. During the procedure, the room was kept completely dark except for the luminance of the projection onto the
screen. The participant's head was constrained using a chin rest. Eye movements were measured using a binocular head-mounted eye tracker (Skalar IRIS Infrared Light Eye Tracker; Skalar Medical BV,
Breda, Netherlands) that recorded horizontal eye movements with a maximum resolution of 3 arcmin at a sampling rate of 1 kHz. Measurements were recorded on a computer (vsgEyetrace v. 3.0.beta
software for Windows; Cambridge Research Systems, Cambridge, UK) and stored for off-line analysis.
The eye tracker was calibrated for each participant at the start of each procedure and again halfway through the procedure, after participants had a break. Calibration was performed by recording the
voltage output of the eye tracker during fixation of 40 targets placed at different positions on the horizontal midline of the screen. The voltage was linearly proportional to eye position within
±20° of the center of the display, and linear regression was used to generate a calibration scale and offset.
Translational OKN was elicited with a flat vertical square-wave grating, composed of alternating black and white vertical stripes moving horizontally at a fixed tangential speed. Each recording
session comprised a pseudorandom sequence of 12 trials, each with a different spatial frequency (0.05, 0.1, or 0.2 c/°), stimulus speed (10°/s or 30°/s), or direction (leftward or rightward).
For our analysis we required a relatively long time series of OKN data, so each stimulus presentation was recorded for 100 s. To alleviate discomfort from long periods of OKN stimulation we chose to
use stimuli with comparatively low spatial frequency and speed. Participants were instructed to stare at the center of the screen rather than follow the moving lines, to evoke “stare” OKN rather than
“look” OKN (Honrubia, Downey, Mitchell & Ward,
), and attention was maintained by giving brief verbal feedback about the amount of time lapsed during the stimulus presentation approximately every 10 s. Participants were given a break for 1 min
after each trial to alleviate discomfort and tiredness and to minimize the effects of any optokinetic after-nystagmus. Participants were given an additional 5-min break after the sixth presentation,
after which the eye tracker was recalibrated.
Data analysis
Movement in the direction of the stimulus motion was defined as positive, and movement in the opposite direction as negative. Position was defined as positive on the side that the stimulus is
drifting towards relative to the center of the stimulus display, and negative on the opposite side.
All programs and algorithms for analyzing data were developed and created in MATLAB (MathWorks, Natick, MA). Each eye was calibrated separately, and the average was computed to yield a cyclopean eye
position. Eye velocity was derived from the eye position using a central difference algorithm and an 80-Hz Butterworth filter with zero phase. Eye acceleration was derived from the filtered
eye-velocity data using a central difference algorithm.
During a forward pass of the data, possible QPs were detected when eye acceleration became greater than ±1000°/s^2. Peak velocity was then recorded from the time 1 ms before velocity first decreased,
if velocity remained at a lower magnitude for more than 4 ms. The starting and ending point of each QP was determined by respective backward and forward passes of the data from the time of peak
velocity to the time when eye velocity returned to a value between 0°/s and the stimulus speed for a period of 2 ms or more. This procedure allowed us to collect QPs that were made in the direction
of optic flow as well as QPs made in the opposite direction.
All eye movements were reviewed in a customized interactive graphical interface. Blinks were detected manually, and cycles containing blinks were marked and removed from the analysis. After blinks
were extracted, each recorded trial contained
SP–QP cycles, where
ranged from 63 to 342. Six measurements were taken from each OKN cycle:
, and
= 1, … ,
), according to the scheme in
Figure 1
is the SP start position and
is the QP start position). When calculating the mean, median, and standard deviation of variables from each trial, the first 5 s was removed from the analysis to ignore early OKN behavior.
Linearity of slow phases
There are three constraints to consider on the OKN variables we have selected (
Equations 4
Equations 4
must be true by definition, and
Equation 6
is true if SPs have constant SP velocity. We observed that SPs were grossly linear when plotted as position against time, although individual SPs could be nonlinear. We performed linear regression of
eye position against time on all 19,120 recorded SPs and collected the residuals of each linear fit, sampled every 1 ms from the start to the end of each SP. If SPs were grossly nonlinear, we would
expect residuals to depend on when they were sampled from each SP (e.g., performing a linear fit on an accelerating SP would produce predominantly positive residuals at the start and end of each SP,
and negative residuals in the middle).
We plotted the mean value of residuals against the time at which they were sampled during each SP, as a fraction of the total duration of the SP on a scale from 0 (the start of each SP) to 1 (the end
of each SP). We observed a very slight but significant trend in the mean value of residuals (
Figure 2
) that indicated that SPs tended to start slightly faster than the average SP velocity and decelerate over the course of the SP. However, the deviation from expected position was typically less than
3 arcmin (0.05°), indicating that SPs were still grossly linear.
Distribution fitting
In our model of OKN, we expect the distribution of SP duration to be determined by the ratio of two variables: SP amplitude and SP velocity. In a previous study we derived the pdf for the ratio of
two correlated and zero-truncated normal variables (Waddington & Harris,
). We considered the goodness of fit of six different distributions (including the ratio distribution) to the histogram of SP duration from each trial to determine whether the ratio distribution gave
the best fit to the data.
The distributions we tested were predicted from models of the QP trigger that we and others had previously investigated (Waddington & Harris,
). These distributions included: (a) the ratio distribution described; (b) the reciprocal zero-truncated normal distribution predicted by the single-unit LATER model (Carpenter,
), in which a decision signal accumulates linearly in time at a rate that varies between cycles with a normal distribution; (c) the mixture distribution of two reciprocal zero-truncated normal
variables (with one mean rate fixed at zero) predicted by a model in which two LATER units compete in a parallel race to threshold (Carpenter,
); (d) the inverse Gaussian distribution (Anastasio,
); (e) the gamma distribution (Tuckwell,
); and (f) the lognormal distribution (Balaban & Ariel,
). The pdfs of these distributions have been given by Waddington and Harris (
). The principal difference between our model (a) and the others (b–f) is that we assume that QPs are triggered after the eyes have moved over a variable distance threshold rather than after a
regular period of time. In this study, we attempted to replicate our previous findings that the ratio distribution gives a better fit to the data than the other models previously reported in the
We found the maximum-likelihood estimates of the pdf parameters for each distribution, and tested the goodness of fit of all six distributions to the SP duration histogram from each trial using the
chi-square criterion. Maximum-likelihood estimates were obtained using the MATLAB statistics toolbox function mle with the exception of the correlation coefficient, which was estimated using the
sample Pearson's correlation coefficient between SP amplitude and SP velocity from the respective trial. The chi-square test statistic was calculated using the MATLAB function chi2gof, using the
fitted pdf to give the expected frequency and the SP duration histogram to give the observed frequency. Each SP duration histogram was originally covered by 45 bins of equal size, and the expected
and observed frequencies were determined from the midpoint of each bin. When the expected frequency from a bin in either tail of a histogram was less than 5, the chi2gof function automatically merged
neighboring bins until there was a minimum expected frequency of 5 or more in every bin, to maintain the reliability of the test.
Principal-components analysis
Variables from adjacent OKN cycles were grouped to create a vector that was used to generate a correlation matrix for each trial. We defined an OKN cycle as one SP followed by one QP. Each vector
contained the variables x[i], i = 1, … , 5, and S[i], V[i], T[i], y[i], and Q[i], i = 1, … , 4, to generate a 25 × 25 element correlation matrix. Cycles separated by blinks were not included in
generating the correlation matrix. During one trial, blinks were so frequent that a correlation matrix including all variables from four consecutive cycles of OKN could not be created, so only 95 of
96 correlation matrices were analyzed using PCA.
We performed PCA using the MATLAB function
to diagonalize the correlation matrices and yield the underlying eigenvectors and eigenvalues. We discarded the eight components with the smallest eigenvalues from each trial to retain 13 principal
components. The remaining eigenvectors explained over 99.3% of the variability in the data, indicating that the data were well explained by the (linear and stochastic) components. After discarding
redundant components, we performed factor rotation using the varimax strategy to obtain orthogonal rotated components using the MATLAB function
. After factor rotation, similar loading patterns were observed across trials but expressed in a different eigenvalue order. The 13 principal components from each trial were sorted into categories
according to their loading pattern using numerical heuristics. As eigenvectors can be rotated to face in the opposite direction, it was necessary to flip the sign of all loadings in these mirrored
components so that they could be sorted correctly. Each loading pattern was then displayed as a line plot of loading value against OKN variable, where components placed in the same category were
averaged across trials and error bars were used to show the variability between trials. This sorting was exhaustive, and we found that all loading patterns clearly fell into only three qualitatively
different categories (see
Stimulus effects on OKN variables
OKN waveforms were grossly similar to those previously reported in the literature, although we noted that SP velocity was remarkably variable across cycles, as we had observed in our previous
investigation (Waddington & Harris,
). The variability was particularly noticeable at the higher stimulus speed (see
Figure 3
for an example waveform).
We performed a three-way analysis of variance on the mean value of OKN variables, using stimulus spatial frequency, speed, and direction as factors. Increasing spatial frequency resulted in a
significant increase in mean SP velocity,
= 6.8,
= 0.009, and consequently a decrease in retinal slip (
Figure 4A
), although retinal slip did not fall to zero. Increasing stimulus speed also resulted in a significant increase in mean SP velocity as expected,
= 14.5,
= 0.007. Remarkably, we found that at stimulus speeds of 10°/s and 30°/s, retinal slip reached values greater than 3°/s in over 30% and 87% of SPs, respectively.
SPs and QPs tended to compensate for each other over time, so the mean magnitude of SPs and QPs for each trial were typically identical. Spatial frequency and stimulus speed had an interaction effect
on the mean SP magnitude,
= 4.8,
= 0.025, and mean QP magnitude,
= 6.4,
= 0.01. Essentially, increasing spatial frequency resulted in an increase in both mean SP and QP magnitude, but the effect of increasing spatial frequency was greater at 30°/s than at 10°/s (
Figure 4B
). We did not find an effect of spatial frequency or stimulus speed on the mean angle of contraversion, which was held at −3.9° when averaged across all trials.
Increasing spatial frequency did not have a significant effect on the median SP duration, but increasing stimulus speed resulted in a significant decrease,
= 12.5,
= 0.009 (
Figure 4C
Distribution of SP duration
Histograms of SP duration were usually positively skewed, as expected (e.g., Cheng & Outerbridge,
), and 89% were significantly different from Gaussian (Holm–Bonferroni-corrected Lilliefors test,
< 0.004 for 85 trials; Holm,
). Histograms of reciprocal SP duration (QP rate) were also usually positively skewed, and 85% were significantly different from Gaussian (Holm–Bonferroni-corrected Lilliefors test,
< 0.003 for 82 trials).
We found the maximum-likelihood estimates of pdf parameters from each trial and tested the goodness of fit of six defined pdfs to each SP duration histogram using the chi-square criterion (see
Materials and methods
). The ratio distribution (RATIO) gave the best fit to the data and was able to fit 90% of SP duration histograms (
Table 2
). This appeared to verify our model of QPs as saccades triggered after the eyes had moved a certain distance, with the complex distribution of SP duration determined by the ratio of an SP amplitude
threshold and SP velocity that varied between cycles.
The other pdfs that were tested were based on models of QPs as regular resetting saccades that are triggered stochastically after a certain period of time (see
Materials and methods
). The reciprocal normal mixture model (mixRN), the gamma (GAM), and the lognormal (LN) did give reasonably good fits to the data, fitting between 79% and 86% of individual histograms. The inverse
Gaussian (IG) and the reciprocal zero-truncated normal (rectrN) were relatively poor models, as they were only able to give a good fit to 69% and 49% of histograms, respectively.
Autocorrelation of SP velocity
To replicate the observation from our previous investigation (Waddington & Harris,
) that the update dynamics of SP velocity could be explained as a first-order Markov process, we found the sample autocorrelation and partial autocorrelation functions of SP velocity from each trial,
using (respectively) the MATLAB functions
. We plotted the mean autocorrelogram of SP velocity across all trials and found that autocorrelation of SP velocity extended back as far as five or six cycles (
Figure 5A
). However, the mean partial autocorrelogram of SP velocity showed a distinct cutoff at a lag of 1 (
Figure 5B
), implying that the correlation between SP velocity in one cycle and the cycle before could explain all the higher order autocorrelation observed in the autocorrelogram, identifying the time series
as a first-order autoregressive process.
PCA results
We performed PCA on the correlation matrices of OKN variables recorded over four cycles in series (see
Materials and methods
). The analysis revealed that eigenvalues of components varied between trials, but eigenvectors remained predominantly the same and could be sorted into 13 distinct categories, which represented
three groups of similar loading patterns shifted by one, two, or three cycles (
Figure 6
). This indicated that each cycle of OKN constituted three uncorrelated stochastic processes.
These loading patterns appeared to be almost identical to the loading patterns found in our previous study (Waddington & Harris,
). We tested the significance of the linear relationships between OKN variables predicted by our model (
Equations 1
) with ordinary least-squares multiple linear regression using the MATLAB function
and Holm–Bonferroni correction for multiple comparisons. We found that regression of
gave a mean
= 0.34 (
< 0.006 for 91 trials),
gave a mean
= 0.28 (
< 0.005 for 95 trials), and
gave a mean
= 0.27 (
< 0.003 for 78 trials). The similarity of these results to the results from our previous study indicated that our model could extend to account for spatial-frequency effects on the OKN waveform.
Spatial-frequency and stimulus-speed effects on stochastic processes
We estimated the parameters of the proposed stochastic processes for each trial with weighted least-squares multiple linear regression using the MATLAB function robustfit, and assessed their
dependency on stimulus spatial frequency and stimulus speed using repeated-measures analysis of variance.
We found no significant effect of spatial frequency or stimulus speed on the parameters
, or
; the mean values of these parameters were −0.25, 0.12, −0.38, and −0.18, respectively. We noted that these were approximately the same values as found in our previous study (−0.25, 0.16, −0.48 and
−0.17; Waddington and Harris,
Table 3
—which indicated that these parameters may represent fundamentally invariant relationships between OKN variables.
Interestingly, increasing spatial frequency from 0.05 to 0.1 and 0.2 c/° did result in a significant increase in v̂, from 4.6 to 6.1 and 7.7, F = 4.2, p = 0.037, but did not have a significant effect
on e or σ[v]. Conversely, increasing stimulus speed from 10°/s to 30°/s resulted in a significant increase in e from 0.38 to 0.60, F = 19.5, p = 0.003, and an increase in σ[v] from 1.6 to 3.8, F =
95.4, p < 0.001, but did not have a significant effect on v̂. Effectively, increasing the stimulus spatial frequency resulted in the mean SP velocity being shifted by a constant value to be more
positive, without having any other effects on the update dynamics of SP velocity.
Spatial frequency and stimulus speed had a significant interaction effect on q̂, F = 4.2, p = 0.037, but not σ[q]. When the stimulus speed was 30°/s, increasing spatial frequency resulted in q̂
becoming more negative, from −1.9 to −2.4 and −3.4, but there was no clear trend at 10°/s. This may reflect the interaction effect observed between stimulus speed and spatial frequency on the mean SP
and QP amplitude in our initial analysis of the stimulus effects on OKN variables. Spatial frequency did not have a significant effect on ŝ or σ[s]. However, increasing stimulus speed did cause an
increase in both σ[s], F = 18.2, p = 0.004, and σ[q], F = 26.2, p = 0.001.
These results indicated that the spatial frequency of the OKN stimulus had two primary effects on the stochastic processes described in
Equations 1
. Firstly, increasing spatial frequency resulted in an increase in SP velocity by shifting
to be more positive. Additionally, increasing spatial frequency increased the magnitude of QPs and SPs (dependent on stimulus speed) by shifting
to be more negative.
We have found that the spatial frequency of suprathreshold OKN stimuli has a significant effect on eye movements during OKN. Mean values of SP velocity, SP amplitude, and QP amplitude all increased
with increasing spatial frequency, and these effects became more pronounced at the higher stimulus velocity (
Figure 4
). There was also a decrease in SP duration (increase in nystagmus frequency) with stimulus speed. Although increasing spatial frequency resulted in a significant decrease in retinal slip, it did not
fall to zero.
Sequential analysis of OKN cycles using PCA revealed three underlying components, as found in our previous study (Waddington & Harris,
). The
component described how SP velocity is updated on every cycle (
Equation 1
). We found that it is independent of the other processes but dependent on stimulus velocity and spatial frequency (
Equation 12
). We also found that the process that determines QP amplitude (the
component) is dependent on stimulus velocity and spatial frequency (
Equation 14
), and confirmed that both SP and QP amplitudes depend on SP velocity and each other (
Equations 1
). The three underlying stochastic processes that drive the system are uncorrelated, but because individual OKN variables load onto multiple components (and multiple variables load onto each
component), many variables are dependent on more than one of the underlying sources of variance. This shared variance gives rise to a complex sequence of cycles in which OKN variables are mutually
correlated and autocorrelated (
Equations 1
The most important findings to discuss from this investigation are that spatial frequency has a significant effect on steady-state SP velocity (and hence retinal slip) and that the OKN system does
not reduce retinal slip to zero even when stimulus parameters are kept constant and participants are given a substantial period of time (100 s) to adapt.
Spatial frequency and contrast have a significant effect on the OKN waveform
A previous study by Sumnall et al. (
) found a small increase in SP velocity with spatial frequency, although their stimulus differed from ours (drifting Gaussian blobs with differing densities of 1.4 elements/°
and 0.16 elements/°
, at a speed of 4.6°/s). However, they also noted that Schor and Narayan (
) had found a decrease in SP velocity at high spatial frequencies and high speeds (drifting gratings with a spatial frequency of 0.5–8 c/° and speeds of 3°/s–48°/s).
Interestingly, Sumnall et al. (
) found a more pronounced increase in SP velocity with increasing stimulus contrast (for a given speed). Similarly, Spering, Kerzel, Braun, Hawken, and Gegenfurtner (
) have shown a systematic increase in the gain of smooth-pursuit eye movements (SPEM) with increasing contrast. Perceived contrast depends not only on physical contrast but also on contrast
sensitivity to different spatial frequencies, retinal image velocities, and retinal location as determined by the bandpass spatiotemporal contrast sensitivity function (Kelly,
). For central retinal stimulation and very low spatial frequencies (as we have used), we would expect contrast to increase with image velocity, but for high spatial frequencies contrast would
decrease with image velocity. Of course, image velocity and image position are in turn controlled by the oculomotor system, leading to a highly complex nonlinear system (Harris & Waddington,
) that is not fully understood.
It appears that global visual contrast is not being maximized either. Based on the contrast sensitivity curves observed by Burr and Ross (
), when using drifting sinusoidal gratings with very low spatial frequency (0.05 c/°), contrast would be maximized by very high image velocity (≈100°/s). Higher spatial frequencies (0.5 c/°) require
more moderate values of retinal slip (>10°/s) to maximize contrast. We might expect these speeds to be overestimates of our own results because we have used square-wave gratings, which have Fourier
energy at higher harmonics. However, if we treat each of our grating cycles as two bars with widths 10°, 5°, and 2.5°, respectively, then we still expect contrast to be maximized at high image
velocities (>10°/s; Burr & Ross,
). To achieve such retinal slip could even require OKN SPs in the opposite direction of stimulus motion, which we have not observed.
The OKN waveform is clearly dependent on the spatial frequency of the visual stimulus, and the gain of both OKN SPs and SPEM are dependent on visual contrast. It is now well established that retinal
slip alone does not necessarily drive SPEM (Beutter & Stone,
; Stone & Krauzlis,
). Likewise, the traditional view that OKN simply minimizes retinal slip does not seem tenable.
Internal estimates of stimulus speed must be reconstructed from prior retinal and extraretinal signals
Consider the problem of choosing an SP velocity during OKN. We know that SP velocity V[i] depends on stimulus velocity V[S], but V[S] is not known directly and can only be estimated from retinal
information and extraretinal information (efference copy and other proprioceptive cues). At the start of each SP, retinal information can only arise from previous SPs: R[j] = V[S] − V[j] for j < i.
Thus the current estimate of stimulus velocity v̂[S,][i] must depend somehow on actual previous SP velocities V[j][<][i], and hence must be intrinsically Markov when estimates are stochastic. In
principle, dependencies on previous SPs could stretch far back, but our empirical findings indicate a dependency on only the last SP (j = i − 1).
Figure 7
we present a simple scheme where we assume that the estimate of stimulus velocity is given by the sum of the estimates of retinal slip and eye velocity:
. Psychophysical evidence suggests that the gains of
are different (Freeman,
; Freeman & Banks,
) and possibly less than unity (Pola & Wyatt,
), which would lead to an underestimate of stimulus velocity. We denote the retinal slip estimate by the nonlinear equation
is a nonconstant gain that depends on the spatiotemporal contrast sensitivity of the retina (and hence spatial frequency and retinal slip),
is a standard normal noise process, and
is the standard deviation of the noise. We denote the SP velocity estimate by the nonlinear equation
is extraretinal gain and may also depend on eye velocity
is a standard normal noise process, and
is the standard deviation of the noise. If we assume that current SP velocity attempts to match the internal estimate of the current stimulus speed, then
= (
); and substituting
Equations 12
, we obtain the first-order Markov relationship
In this study
are not directly observable, but comparing
Equation 14
Equation 1
, we have
, and
) have proposed that the perceived speed of visual stimuli during SPEM is based on Bayesian estimates of retinal slip velocity and extraretinal eye velocity. Their key assumption is that prior
expectation of stimulus speed is zero, causing shifts to lower velocities for the posterior estimates. Further, because the combined signal is less certain than the retinal signal alone (
The effect of retinal and extraretinal gains less than unity on SP gain
We have found that
. This indicates a relationship between
that is linear if extraretinal gain remains constant:
. Plotting
Figure 8
) revealed an intercept of 0.82 and a slope of −0.92. The nearly (negative) unity slope supports the scheme in
Figure 7
, although we should exercise some caution, as it is slightly lower than expected.
It is possible that extraretinal gain varies or depends on stimulus parameters such as spatial frequency. Extraretinal signals for eye movements are mostly in the form of efference copy, where a copy
of the motor command is used to predict the behavior (Bridgeman,
; Sperry,
). Efference copy cannot be intrinsically veridical, and it must be learned or adapted from its sensory consequences, namely retinal slip (Haarmeier, Bunjes, Lindner, Berret, & Thier,
). If retinal slip is perturbed by noise, sensory consequences cannot be certain. Thus extraretinal gain would be attempting to adapt to a noisy reference, and hence would become noisy itself.
Therefore, noise in the extraretinal pathway is unlikely to be independent of noise in the retinal pathway.
We computed
Table 3
), which showed a significant decrease with stimulus speed and a significant increase with spatial frequency, whereas
only showed a slight decrease with speed and a slight increase with spatial frequency. These results mirror the findings that perceived retinal speed is dependent on spatial frequency (Campbell &
; Diener, Wist, Dichigans, & Brandt,
; Ferrera & Wilson,
; Freeman & Banks,
; Smith & Edgar,
) and support the findings of Sumnall et al. (
) that extraretinal signals do not appear to be significantly affected by spatial frequency.
From the traditional viewpoint of minimizing retinal slip, having an extraretinal gain close to unity seems highly advantageous. From
Equations 14
, the steady-state mean gain for SP velocity is
As can be seen, when
= 1 the mean OKN gain becomes unity regardless of retinal gain. Thus, in principle, it is possible to track the stimulus perfectly at steady state. Indeed, efference copy was introduced implicitly by
Young, Forster, and van Houtte (
) for a discrete-time smooth-pursuit model and explicitly by Robinson, Gordon, and Gordon (
) for a continuous-time smooth-pursuit model to explain how the smooth-pursuit system could track a moving target with large open-loop gain and long loop delays.
has been shown to be considerably less than unity (Pola & Wyatt,
), and we surmise that the report by Spering et al. (
) of low SPEM gain to low contrast stimuli could only occur if
< 1. Our deduction that
is less than unity for discrete-time OKN is consistent with these results (see
Table 3
), although we perhaps find a higher extraretinal gain for OKN SPs than some SPEM estimates.
The difference between retinal and extraretinal gains affects the variance of all OKN variables
An important property of Markov systems is the evolution of variance towards an asymptotic steady state (see
). From
Equations 14
, steady-state variance of SP velocity is given by
Thus, the noise on the estimate of SP velocity g[e] = g[r], variance is at a minimum (
The term
affects the steady-state mean and variance of most OKN variables that we have measured, including QP and SP amplitude, duration, and position (see
Table 4
). It is possible that variance constraints or costs on OKN variables other than SP velocity could lower the optimal
to reduce the difference
and hence reduce
; also see Equation 27 in
Table 4
) and more variability in the end position of QPs (Equations 23 and 26). Too much variability in the end position of QPs could be extremely costly to the visual system, as contrast sensitivity is
dependent on the retinal location of the visual stimulus and visual acuity decreases rapidly outside the central 2° of foveal vision. It is not difficult to imagine a situation where a compromise
needs to be made between minimizing positional variance on QPs and minimizing retinal slip.
The difference between extraretinal and retinal gains also determines the response speed of the velocity update dynamics. Consider a reference cycle labeled as
= 0 with velocity
; then from
Equation A3
, the velocity of the
th cycle is
which clearly takes time to reach mean velocity
. That is, any deviation from the mean takes time to recover depending on
. In natural OKN, stimulus velocity may change rapidly when gaze is shifted from one region of optic flow to another, so it seems plausible that responding quickly to any changes could be important.
→ 1, response time becomes infinite (a random walk) and the system would be trapped by its history and unable to change. On the other hand, when
response is instantaneous, but
, would never reach
< 1. Presumably some compromise is needed, but how speed and accuracy trade off is unknown.
Of course, lowering
to reduce the difference
will also have visual consequences that cannot be directly inferred from Equations 17–28. Recently, Harrison, Freeman, and Sumner (
) have shown that the horizontal component of saccades made to visual targets flashed during ongoing OKN falls well short of the target. That is, the saccades do not compensate for the excursion of
the current SP. This undershoot error was proportional to the distance traveled by the eye during the saccade latency period, which is expected if the error was due to extraretinal underestimation.
Therefore, if we assume that QPs are visually guided, lower extraretinal gain will lead to QPs that undershoot (or overshoot, depending on location) their target, as well as lower SP gain. We should
also recognize that the SP amplitude threshold is likely to be estimated by an extraretinal signal of eye position. If this signal is also an underestimate, then QPs will tend to be triggered before
they reach the target threshold, and any variance in the end position of QPs could not be fully compensated for during the SP. Indeed, this would lead to a partial negative correlation between the
amplitude and start position of SPs (and QPs), and to the first-order Markov properties that we have observed.
It is plausible, therefore, that adapting or optimizing OKN requires controlling g[e] − g[r] (or g[e], if g[r] is unknown). The overall process is complex, however, and at present we have no specific
model of adaptive control of OKN. Indeed, it is even possible that there are three separate adaptive mechanisms each attempting to optimize some visual consequence of OKN, with competing and
nonintuitive effects.
Markov properties may be modified at high spatial frequencies
In this study we have used stimuli with very low spatial frequency, and it is interesting to speculate what would happen with more naturalistic high-contrast and high-spatial-frequency stimuli.
Clearly, we expect retinal gain to increase. Thus the difference g[e] − g[r] would decrease, leading to an overall increase in SP gain and also a decrease in SP velocity variance and a general
reduction in cyclic variability. Should g[e] = g[r], the Markov properties would disappear and zero-order statistics would set in.
However, visual contrast depends on retinal slip, particularly at high spatial frequencies, due to the spatiotemporal contrast sensitivity of the visual system (Kelly,
). There must therefore also be a nonlinearity in the time course of OKN. To illustrate, consider a random fluctuation that happens to reduce SP velocity on the
th cycle. This will increase retinal slip, and hence reduce retinal gain
). This in turn will reduce SP velocity on the next cycle, causing a further increase in retinal slip, and so on in a positive-feedback fashion. Although a steady state may occur, it is also possible
that OKN will simply cease if the random fluctuation is large enough. The opposite effect could occur with a fluctuation that increases SP velocity. A moving high-spatial-frequency grating may have
minimal contrast for a stationary eye and generate no OKN. However, if the eye happened to move spontaneously in the direction of the stimulus, retinal slip would decrease, increasing contrast and
potentially sustaining OKN for a while. This nonlinearity leads to a nonstationary Markov process, but whether it can be detected remains to be explored, although we expect it to be stronger for
high-spatial-frequency stimuli because of the sharp dependency on retinal slip.
Clearly, the OKN system does not simply minimize retinal slip and generate regular resetting saccades. The SPs and QPs of OKN are remarkably variable and must depend on preceding extraretinal and
retinal signals encoding velocity and position, such that they vary in a Markov fashion. In principle, an OKN gain of unity could be maintained if extraretinal gain were also unity, but empirical
studies have demonstrated that it is typically significantly lower. We have deduced that the Markov properties of SP velocity depend only on the difference between retinal and extraretinal gain: When
retinal gain is much lower than extraretinal gain (e.g., when viewing low-spatial-frequency stimuli), the Markov properties of the system can be observed and variability in the whole system
increases. Any noise on the internal estimates of SP velocity (and position) will be amplified by a Markov process. It is therefore possible that extraretinal gain (or the difference between retinal
and extraretinal gain) adapts to control a trade-off between perfect tracking and minimizing the variability of OKN parameters, but this remains to be investigated.
This research was funded with an Engineering and Physical Sciences Research Council Standard Research Student Award (DTG: EP/P502675/1), and a Knowledge Transfer Partnership associate development
grant provided by the UK Technology Strategy Board, Medical Research Council, and WESC Foundation (KTP008989).
Commercial relationships: none.
Corresponding author: Jonathan Waddington.
Abadi R. V., Howard I. P., Ohmi M., Lee E. E. (2005). The effect of central and peripheral field stimulation on the rise time and gain of human optokinetic nystagmus. Perception , 34 , 1015–1024.
Anastasio T. J. (1996). A random walk model of fast-phase timing during optokinetic nystagmus. Biological Cybernetics , 75 , 1–9.
Balaban C. D., Ariel M. (1992). A “beat-to-beat” interval generator for optokinetic nystagmus. Biological Cybernetics , 66 , 203–216.
Beutter B. R., Stone L. S. (2000). Motion coherence affects human perception and pursuit similarly. Visual Neuroscience , 17 , 139–153.
Burr D. C., Ross J. (1982). Contrast sensitivity at high velocities. Vision Research , 22 , 479–484.
Büttner U., Büttner-Ennever J. A. (2006). Present concepts of oculomotor organization. Progress in Brain Research , 151 , 1–42.
Bridgeman B. (1995). A review of the role of efference copy in sensory and oculomotor control systems. Annals of Biomedical Engineering , 4 , 409–422.
Campbell F. W., Maffei L. (1981). The influence of spatial frequency and contrast on the perception of moving patterns. Vision Research , 21 , 713–721.
Carpenter R. H. S. (1993). Distribution of quick-phase intervals in optokinetic nystagmus. Ophthalmic Research , 25 , 91–93.
Carpenter R. H. S. (1994). Express optokinetic nystagmus. In Fuchs A. F. Brandt T. Büttner U. Zee D. (Eds.), Contemporary ocular motor and vestibular research (pp. 185–187). Stuttgart, Germany: Georg
Çetinkaya A., Oto S., Akman A., Akova Y. A. (2008). Relationship between optokinetic nystagmus response and recognition visual acuity. Eye, 22, 77–81.
Cheng M., Outerbridge J. S. (1974). Inter-saccadic interval analysis of optokinetic nystagmus. Vision Research , 14 , 1053–1058.
Cohen B., Henn V., Raphan T., Dennett D. (1981). Velocity storage, nystagmus, and visual-vestibular interactions in humans. Annals of the New York Academy of Sciences , 374 , 421–433.
Cohen B., Matsuo V., Raphan T. (1977). Quantitative analysis of the velocity characteristics of optokinetic nystagmus and optokinetic after-nystagmus. Journal of Physiology , 270 , 321–344.
Diener H. C., Wist E. R., Dichigans J., Brandt T. (1976). The spatial frequency effect on perceived velocity. Vision Research , 16 , 169–176.
Ferrera V. P., Wilson H. R. (1991). Perceived speed of moving 2-dimensional patterns. Vision Research , 31 , 877–893.
Freeman T. C. (2001). Transducer models of head-centered motion perception. Vision Research , 41 , 2741–2755.
Freeman T. C., Banks M. S. (1998). Perceived head-centric speed is affected by both extra-retinal and retinal errors. Vision Research , 38 , 941–945.
Freeman T. C., Champion R. A., Warren P. A. (2010). A Bayesian model of perceived head-centered velocity during smooth pursuit eye movement. Current Biology , 20 , 757–762.
Gellman R. S., Carl J. R., Miles F. A. (1990). Short latency ocular-following responses in man. Visual Neuroscience , 5 , 107–122.
Haarmeier T., Bunjes F., Lindner A., Berret E., Thier P. (2001). Optimizing visual motion perception during eye movements. Neuron , 32 , 527–535.
Harris C. M., Waddington J. W. (2012). On the convergence of time interval moments: Caveat sciscitator. Journal of Neuroscience Methods , 205 , 345–356.
Harris C. M., Waddington J. W. (2013). Optimal control theory of normal and pathological slow eye movements. Journal of Control Engineering and Technology , 3 , 181–188.
Harris C. M., Walker J., Shawkat F., Wilson J., Russell-Eggitt I. (1993). Eye movements in a familial vestibulocerebellar disorder. Neuropediatrics , 24 , 117–122.
Harrison J. J., Freeman T. C. A., Sumner P. (2015). Saccadic compensation for reflexive optokinetic nystagmus just as good as compensation for volitional pursuit.
Journal of Vision
, 15 (1): 24, 1–13, doi:10.1167/15.1.24 [
] [
Holm S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics , 6 , 65–70.
Honrubia V., Downey W. L., Mitchell D. P., Ward P. H. (1968). Experimental studies on optokinetic nystagmus. II: Normal humans. Acta Oto-Laryngologica, 65, 441–448.
Kelly D. H. (1979). Motion and vision. II: Stabilized spatio-temporal threshold surface. Journal of the Optical Society of America, 69, 1340–1349.
Kelly D. H. (1984). Retinal inhomogeneity. I: Spatiotemporal contrast sensitivity. Journal of the Optical Society of America A, 1, 107–113.
Kolarik A. J., Margrain T. H., Freeman T. C. A. (2010). Precision and accuracy of ocular following: Influence of age and type of eye movement. Experimental Brain Research, 201, 271–282.
Leguire L. E., Zaff B. S., Freeman S., Rogers G. L., Bremer D. L. (1991). Contrast sensitivity of optokinetic nystagmus. Vision Research, 31, 89–97.
Miles F. A. (1995). The sensing of optic flow by the primate optokinetic system. In Findlay J. M. Kentridge R. W. Walker R. (Eds.), Eye movement research: Mechanisms, processes and applications (pp.
47–62). Amsterdam: Elsevier.
Miles F. A. (1998). The neural processing of 3-D visual information: Evidence from eye movements. European Journal of Neuroscience, 10, 811–822.
Miles F. A., Kawano K., Optican L. M. (1986). Short-latency ocular following responses of monkey. I: Dependence on temporospatial properties of the visual input. Journal of Neurophysiology, 10,
Pola J., Wyatt H. J. (1989). The perception of target motion during smooth pursuit eye movements in the open-loop condition: Characteristics of retinal and extraretinal signals. Vision Research, 29,
Robinson D. A. (1981). The use of control systems analysis in the neurophysiology of eye movements. Annual Review of Neuroscience , 4 , 463–503.
Robinson D. A., Gordon J. L., Gordon S. E. (1986). A model of the smooth pursuit eye movement system. Biological Cybernetics , 55 , 43–57.
Schor C., Narayan V. (1981). The influence of field size upon the spatial frequency response of optokinetic nystagmus. Vision Research , 21 , 985–994.
Schweigart G., Mergner T., Evdokimidis I., Morand S., Becker W. (1997). Gaze stabilization by optokinetic reflex (OKR) and vestibuloocular reflex (VOR) during active head rotation in man. Vision
Research , 37 , 1643–1652.
Shelhamer M. (1992). Correlation dimension of optokinetic nystagmus as evidence of chaos in the oculomotor system. IEEE Transactions on Biomedical Engineering , 39 , 1319–1321.
Shelhamer M. (1996). On the correlation dimension of optokinetic nystagmus eye movements: Computational variables, filtering, nonstationarity, and surrogate data. Biological Cybernetics , 76 ,
Simons B., Büttner U. (1985). The influence of age on optokinetic nystagmus. European Archives of Psychiatry and Clinical Neurosciences , 234 , 369–373.
Smith A. T., Edgar G. K. (1990). The influence of spatial-frequency on perceived temporal frequency and perceived speed. Vision Research , 30 , 1467–1474.
Spering M., Kerzel D., Braun D. I., Hawken M. J., Gegenfurtner K. R. (2005). Effects of contrast on smooth pursuit eye movements.
Journal of Vision
, 5 (5): 6, 455–465, doi:10.1167/5.5.6. [
] [
Sperry R. W. (1950). Neural basis of the spontaneous optokinetic response produced by visual inversion. Journal of Comparative and Physiological Psychology, 43, 482–489.
Stone L. S., Krauzlis R. J. (2003). Shared motion signals for human perceptual decisions and oculomotor actions.
Journal of Vision
, 3 (11): 7, 725–736, doi:10.1167/3.11.7. [
] [
Sumnall J. H., Freeman T. C., Snowden R. J. (2003). Optokinetic potential and the perception of head-centred speed. Vision Research , 43 , 1709–1718.
Trillenberg P., Zee D., Shelhamer M. (2002). On the distribution of fast-phase intervals in optokinetic and vestibular nystagmus. Biological Cybernetics , 87 , 68–78.
Tuckwell H. (1988). Introduction to theoretical neurobiology: Nonlinear and stochastic theories. Cambridge, UK: Cambridge University Press.
Waddington J., Harris C. M. (2012). Human optokinetic nystagmus: A stochastic analysis.
Journal of Vision
, 12 (12): 5, 1–17, doi:10.1167/12.12.5. [
] [
Waddington J., Harris C. M. (2013). The distribution of quick phase interval durations in human optokinetic nystagmus. Experimental Brain Research , 224 , 179–187.
Wester S. T., Rizzo J. F.,III, Balkwill M. D., Wall C.,III. (2007). Optokinetic nystagmus as a measure of visual function in severely visually impaired children.
Investigative Ophthalmology and Visual Science
, 48, 4542–4548. [
] [
Yee R. D., Baloh R. W., Honrubia V., Lau C. G. Y., Jenkins H. A. (1979). Slow build-up of optokinetic nystagmus associated with downbeat nystagmus.
Investigative Ophthalmology and Visual Science
, 18 , 622–629. [
] [
Young L. R., Forster J. D., Van Houtte N. A. J. (1968). A revised stochastic sampled data model for eye tracking movements. 4th annual NASA-university conference on manual control (pp. 489–509 ). Ann
Arbor, MI: NASA SP-192.
We consider a sequence of random variables
≥ 0, where each
are mutually independent. If the probability distribution of
depends explicitly on the outcome of the previous random variable
but not explicitly on the outcome of earlier random variables, the system is said to be a first-order Markov process. To describe
requires the specification of the infinite dimensional probability transition matrix Pr(
). A special case is given by the stationary Gauss–Markov first-order autoregressive system
is the initial value and may be a random variable,
is a constant, and
) is a sample from a continuous white-noise process that is normally distributed with a nonzero mean
and variance
) ∼
). Samples are mutually independent so that cov(
)) = 0 for
Equations 1
can be written in this form. Expanding
Equation A1
) is a standard normal random variable
) ∼
(0, 1). From
Equation A2
, we have
(1) +
(2) +
(3); etc. Summing the power series yields the
th term
with mean
and variance
. Provided |
| < 1, this series converges with an initial transient that depends on
, followed asymptotically by a steady-state behavior that is independent of
. Thus the steady-state mean is given by
and hence
Similarly, the steady-state variance is | {"url":"https://jov.arvojournals.org/article.aspx?articleid=2440954","timestamp":"2024-11-08T07:30:47Z","content_type":"text/html","content_length":"363612","record_id":"<urn:uuid:aa9b6b38-e1fd-4740-b30c-4b049c647fac>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00787.warc.gz"} |
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Question Text Ratnalise with denominator i) (ii)
Updated On Dec 31, 2022
Topic All topics
Subject Mathematics
Class Class 9
Answer Type Video solution: 1
Upvotes 114
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From Encyclopedia of Mathematics
(Redirected from Non-atomic measure)
2020 Mathematics Subject Classification: Primary: 03E04 [MSN][ZBL] in set theory
2020 Mathematics Subject Classification: Primary: 28A [MSN][ZBL] in measure theory
Ordered sets
A minimal non-zero element of a partially ordered set with a zero $0$, i.e. a covering element of $0$; an element $p > 0$ such that $0<x\leq p$ implies $x=p$.
Measure algebras
For the definition and relevance in the theory of measure algebras we refer to Measure algebra.
Classical measure theory
Let $\mu$ be a (nonnegative) measure on a $\sigma$-algebra $\mathcal{S}$ of subsets of a set $X$. An element $a\in \mathcal{S}$ is called an atom of $\mu$ if
• $\mu (A)>0$;
• For every $B\in \mathcal{S}$ with $B\subset A$ either $\mu (B)=0$ or $\mu (B)=\mu (A)$
(cp. with Section IV.9.8 of [DS] or [Fe]).
Remark If we denote by $\mathcal{N}$ the null sets and consider the standard quotient measure algebra $(\mathcal{S}/\mathcal{N}, \mu)$, then any atom of such quotient measure algebra corresponds to
an equivalence class of atoms of $\mu$.
Atomic measures
A σ-finite measure $\mu$ is called atomic if there is a partition of $X$ into countably many elements of $\mathcal{A}$ which are either atoms or null sets. An atomic probability measure is often
called atomic distribution. Examples of atomic distributions are the discrete distributions.
Nonatomic measures
A measure $\mu$ is called nonatomic if it has no atoms.
Jordan decomposition
If $\mu$ is $\sigma$-finite, it is possible to decompose $\mu$ as $\mu_a+\mu_{na}$, where $\mu_a$ is an atomic measure and $\mu_{na}$ is a nonatomic measure. In case $\mu$ is a probability measure,
this means that $\mu$ can be written as $p \mu_a + (1-p) \mu_{na}$, where $p\in [0,1]$, $\mu_a$ is an atomic probability measure and $\mu_{na}$ a nonatomic probability measure (see [Fe]), which is
sometimes called a continuous distribution. This decomposition is sometimes called Jordan decomposition, although several authors use this name in other contexts, see Jordan decomposition.
Measures in the euclidean space
If $\mu$ is a $\sigma$-finite measure on the Borel $\sigma$-algebra of $\mathbb R^n$, then it is easy to show that, for any atom $B$ of $\mu$ there is a point $x\in B$ with the property that $\mu (B)
= \mu (\{x\})$. Thus such a measure is atomic if and only if it is the countable sum of Dirac deltas, i.e. if there is an (at most) countable set $\{x_i\}\subset \mathbb R^n$ and an (at most)
countable set $\{\alpha_i\}\subset ]0, \infty[$ with the property that \[ \mu (A) = \sum_{x_i\in A} \alpha_i \qquad \mbox{for every Borel set } A\, . \]
Sierpinski's theorem
A nonatomic measure takes a continuum of values. This is a corollary of the following Theorem due to Sierpinski (see [Si]):
Theorem If $\mu$ is a nonatomic measure on a $\sigma$-algebra $\mathcal{A}$ and $A\in \mathcal{A}$ an element such that $\mu (A)>0$, then for every $b\in [0, \mu (A)]$ there is an element $B\in \
mathcal{A}$ with $B\subset A$ and $\mu (B) = b$.
Set theory
In some models of set theory, an atom or urelement is an entity which may be an element of a set, but which itself can have no elements. Zermelo–Fraenkel axiomatic set theory with atoms is denoted
ZFA (see [Je]).
By a natural extension of meaning, the term atom is also used for an object of a category having no subobjects other than itself and the null subobject (cf. Null object of a category).
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory", 1, Interscience (1958). MR0117523 Zbl 0635.47001
[Fe] "An introduction to probability theory and its applications", 2, Wiley (1971).
[Je] T. Jech, "Set theory. The third millennium edition, revised and expanded" Springer Monographs in Mathematics (2003). ISBN 3-540-44085-2 Zbl 1007.03002
[Lo] M. Loève, "Probability theory", Princeton Univ. Press (1963). MR0203748 Zbl 0108.14202
[Si] W. Sierpiński, "Sur les fonctions d’ensemble additives et continues", 3, Fund. Math. (1922) pp. 240-246 Zbl 48.0279.04
How to Cite This Entry:
Non-atomic measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-atomic_measure&oldid=27993
This article was adapted from an original article by N.N. Vorob'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
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निम्न का सरल किजीए
i) (x+4)(x+10)
(ii) 104×35... | Filo
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Question Text निम्न का सरल किजीए i) (ii)
Updated On Aug 20, 2024
Topic All topics
Subject Mathematics
Class Class 9
Answer Type Video solution: 1
Upvotes 80
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Chaos 3 Butterfly Effect
The "Butterfly Effect" is often ascribed to Lorenz. In a paper in 1963 given to the New York Academy of Sciences he remarks:
One meteorologist remarked that if the theory were correct, one flap of a seagull's wings would be enough to alter the course of the weather forever.
By the time of his talk at the December 1972 meeting of the American Association for the Advancement of Science in Washington, D.C. the sea gull had evolved into the more poetic butterfly - the title
of his talk was* :
Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?
Lorenz Definition of Chaos
"When the present determines the future, but the approximate present does not approximately determine the future."
Lorenz and the Butterfly Effect.
Meterologist Ed Lorenz MIT
Created Modern Chaos Theory
Model of weather patterns he developed
1) Dynamical Forecasting in the Spirit of Newton
Inch weather forward on instant at a time..
Too hard
2) Pattern Matching Forecasting - common sense based on previous experiences
3) Statisical Forecasting - numbers adjusting .... curve fitting
Lorenz- Tested the statistical methods.
Used a computer for thought experiments (telescope of the mind).
Computer centers whole rooms
Too simple at first... Either stationary state or oscillating.
Deterministic system of differential equations but non-periodic
Found a system that was deterministic but did not repeat itself (non-periodic).
In studying his artificial weather model what he found was
Butterfly extreme sensitivity of a chaotic system to tiny changes in initial conditions.
Image of butterfly flapping its wings in Brazil creating a tornado in texas.
Goes down the hall gets his cup of coffee and looks nothing like the weather he did the first time.
(he entered the numbers and put them back in the equations).
Really puzzled....
Where in computer program got the
they agreed perfectly at first
Noticed the error growed ... doubled every four days of his simulation...
The initial conditions he used was three digit numbers in printout, but internally computer was used
Round off errors in fourth decimal place...
Lorenz Equation
Butterfly effect is signature of chaos
Any tiny error gets amplified infinitely fast.
Unable to predict long term...
The current state of the atmosphere.... cannot measure precisely.
Non periodic system that was deterministic... keenly vigilant.
Meaning of the Butterfly Effect
Term probably originated from a Lecture Lorenz gave back in 1972
"Predictability: Does the flap of a Butterfly's Wings in Brazil Set off a tornado in texas?"
Jurassic Park ... Jeff Goldbloom puts a drop on her hand and asks which
Too complex, you cannot predict it.
Sliding doors... She plays a character
2 trajectories of a persons life... How whole outcome of your life can be changed
Arcadia - Tom Stoppard
Everyone knows little things can make a big difference.. Ancient idea and concept.
Storm 1941
George Stewart
Meteorologist character
A man sneezing in China can set off a blizzard in NY
Part of the culture of meteorology... Slightest change could affect the weather.
A sound of thunder... A man goes back in time and steps on a butterfly.
Small effects can cascade and create big
John Gower 1390
For Want of a Nail...
For want of a nail the shoe was lost,
for want of a shoe the horse was lost,
for want of a horse the knight was lost,
for want of a knight the battle was lost,
for want of a battle the kingdom was lost.
So a kingdom was lost—all for want of a nail.
There was one nail that led to the loss of a Kingdom.
Small changes can make a big difference in outcomes by a cascading of effects.
So what is so novel.
What is new is that the same kind of sensitivity to tiny changes can afflict even the simplest systems.
Double pendulum for example... Poincares 3 body problem or Lorenzs simple weather problem, systems like that you would not think chaos would manifest.
Because we know ALL the variables and we know the laws that predict their behavior.
So many complexities in our life, but simple deterministic systems also display chaos.
We know we cannot predict our fate, or wars, etc.. Too complex.
Poincare Quote
Poincare - Nightmarish tangle... How two states of a system that are indistinguishably close could lead to different fates for the 3 body problem.
Address the Question how chance can emerge in a deterministic world.
Exactly the Laws and Exactly the situation of the Universe (initial states).
"Even if we know the laws of nature EXACTLY we could still only know the initial situation approximately!"
You can never know the state of the universe exactly.
"But it is not always so; it may happen that the small differences in the initial conditions produce very great ones in the final phenomena."
When a small change blows up into a big change.
"A small error in the former will produce an enormous error in the latter. Prediction becomes impossible and we have a fortuitous phenomenon."
That is how chance arises as a fortuitous one, from small uncertainties cascading into enormous ones.
Poincare really understood the butterfly effect.
Lyapunov exponent - a measure of the butterfly effect.
Lyapunov exponent - a measure of the butterfly effect.
Rate of exponential growth.
Sir James Lighthill
"The recently recognized failure of predictability in Newtonian Dynamics." (1986)
Sir James Lighthill
We thought Newtons laws gave us total predictability.
Why were scientists so slow to recognize the butterfly effect.
Distasteful because threatening to science itself.
Sciences need to put systems in a box.
Scientists hate "everything effects everything else.." its a matter of degree.
The things you thought were negligible can have big effects.
Butterfly effect does not apply to
Does not occur in systems gently relaxing to some equilibrium state (particle rolling down a curved surface).
Two metronomes oscillating and flick it... errors grow linearly.
Errors do not snowball
Tides very predictable - huge fluid mechanical system. Weather is too... yet tides predictable.
One is periodic, and one is not.
Halleys Comet
Timing of Eclipses
All of these things very periodic which means they are predictable.
Tiny errors do not mushroom
It is Not Determinism, it is Determinism PLUS Periodicity that gives predictability.
It is by putting Non-periodicity into the weather that created chaos and the butterfly effect.
Two double pendulums initially are in sync, but then become unpredictable.
***Horizon of predictability is a phrase the Lighthill introduced.
It is the TIME it takes for these tiny errors to double in size.
What is the length of the Horizon of predictability.
Predictibility Horizon demo.. 3 time units yellow and blue not the same (Lorenz computer animate)
Decorrelate and become totally different.
Solar System is Chaotic
But 5 million year Horizon of Predictability.
The solar system is chaotic but predictable because 5 million years is so long on the timescale of a human life.
You cannot predict beyond the predictability Horizon no matter how good your instruments are.
Trying to figure out the position of the planets 4 billion years ago is meaningless.
What if my instruments give me no error, but are perfect.
You cannot do that because it would require an infinite number of decimal places.
There is always some errors somewhere.
Even with a thousand decimal places, because errors grow exponentially the answers will be wrong at some point.
You need exponential accuracy to get a linear increase in approximation time.
10 times to get one unit time, 100 times to get 2 more units of time, 1000 times to get 3 more units of time...etc.
So far we have emphasized the unpredictable and disorderly side of chaos.
Chaos also has a secret order hidden within and the patterns will amaze you. | {"url":"https://www.pemft.net/chaos-3-butterfly-effect.html","timestamp":"2024-11-09T17:07:37Z","content_type":"text/html","content_length":"73458","record_id":"<urn:uuid:48d5d0fa-f932-4cc8-ac44-0fda63c85604>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00614.warc.gz"} |
Syllogism Explained
A syllogism (Greek, Ancient (to 1453);: συλλογισμός, syllogismos, 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two
propositions that are asserted or assumed to be true.
In its earliest form (defined by Aristotle in his 350 BC book Prior Analytics), a deductive syllogism arises when two true premises (propositions or statements) validly imply a conclusion, or the
main point that the argument aims to get across.^[1] For example, knowing that all men are mortal (major premise), and that Socrates is a man (minor premise), we may validly conclude that Socrates is
mortal. Syllogistic arguments are usually represented in a three-line form:
All men are mortal.Socrates is a man.Therefore, Socrates is mortal.^[2]
In antiquity, two rival syllogistic theories existed:
Aristotelian syllogism
Stoic syllogism
. From the
Middle Ages
categorical syllogism
were usually used interchangeably. This article is concerned only with this historical use. The syllogism was at the core of historical deductive reasoning, whereby facts are determined by combining
existing statements, in contrast to
inductive reasoning
, in which facts are predicted by repeated observations.
Within some academic contexts, syllogism has been superseded by first-order predicate logic following the work of Gottlob Frege, in particular his Begriffsschrift (Concept Script; 1879). Syllogism,
being a method of valid logical reasoning, will always be useful in most circumstances, and for general-audience introductions to logic and clear-thinking.^[3] ^[4]
Early history
See main article: History of logic. In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism.^[5]
See main article: Term logic. Aristotle defines the syllogism as "a discourse in which certain (specific) things having been supposed, something different from the things supposed results of
necessity because these things are so."^[6] Despite this very general definition, in Prior Analytics Aristotle limits himself to categorical syllogisms that consist of three categorical propositions,
including categorical modal syllogisms.^[7]
The use of syllogisms as a tool for understanding can be dated back to the logical reasoning discussions of Aristotle. Before the mid-12th century, medieval logicians were only familiar with a
portion of Aristotle's works, including such titles as Categories and On Interpretation, works that contributed heavily to the prevailing Old Logic, or logica vetus. The onset of a New Logic, or
logica nova, arose alongside the reappearance of Prior Analytics, the work in which Aristotle developed his theory of the syllogism.
Prior Analytics, upon rediscovery, was instantly regarded by logicians as "a closed and complete body of doctrine", leaving very little for thinkers of the day to debate, and reorganize. Aristotle's
theory on the syllogism for assertoric sentences was considered especially remarkable, with only small systematic changes occurring to the concept over time. This theory of the syllogism would not
enter the context of the more comprehensive logic of consequence until logic began to be reworked in general in the mid-14th century by the likes of John Buridan.
Aristotle's Prior Analytics did not, however, incorporate such a comprehensive theory on the modal syllogism—a syllogism that has at least one modalized premise, that is, a premise containing the
modal words necessarily, possibly, or contingently. Aristotle's terminology in this aspect of his theory was deemed vague, and in many cases unclear, even contradicting some of his statements from On
Interpretation. His original assertions on this specific component of the theory were left up to a considerable amount of conversation, resulting in a wide array of solutions put forth by
commentators of the day. The system for modal syllogisms laid forth by Aristotle would ultimately be deemed unfit for practical use, and would be replaced by new distinctions and new theories
Medieval syllogism
Boethius (c. 475–526) contributed an effort to make the ancient Aristotelian logic more accessible. While his Latin translation of Prior Analytics went primarily unused before the 12th century, his
textbooks on the categorical syllogism were central to expanding the syllogistic discussion. Rather than in any additions that he personally made to the field, Boethius' logical legacy lies in his
effective transmission of prior theories to later logicians, as well as his clear and primarily accurate presentations of Aristotle's contributions.
Peter Abelard
Another of medieval logic's first contributors from the Latin West, Peter Abelard (1079–1142), gave his own thorough evaluation of the syllogism concept, and accompanying theory in the Dialectica—a
discussion of logic based on Boethius' commentaries and monographs. His perspective on syllogisms can be found in other works as well, such as Logica Ingredientibus. With the help of Abelard's
distinction between de dicto modal sentences and de re modal sentences, medieval logicians began to shape a more coherent concept of Aristotle's modal syllogism model.
Jean Buridan
The French philosopher Jean Buridan (c. 1300 – 1361), whom some consider the foremost logician of the later Middle Ages, contributed two significant works: Treatise on Consequence and Summulae de
Dialectica, in which he discussed the concept of the syllogism, its components and distinctions, and ways to use the tool to expand its logical capability. For 200 years after Buridan's discussions,
little was said about syllogistic logic. Historians of logic have assessed that the primary changes in the post-Middle Age era were changes in respect to the public's awareness of original sources, a
lessening of appreciation for the logic's sophistication and complexity, and an increase in logical ignorance—so that logicians of the early 20th century came to view the whole system as ridiculous.^
Modern history
The Aristotelian syllogism dominated Western philosophical thought for many centuries. Syllogism itself is about drawing valid conclusions from assumptions (axioms), rather than about verifying the
assumptions. However, people over time focused on the logic aspect, forgetting the importance of verifying the assumptions.
In the 17th century, Francis Bacon emphasized that experimental verification of axioms must be carried out rigorously, and cannot take syllogism itself as the best way to draw conclusions in nature.^
[9] Bacon proposed a more inductive approach to the observation of nature, which involves experimentation, and leads to discovering and building on axioms to create a more general conclusion. Yet, a
full method of drawing conclusions in nature is not the scope of logic or syllogism, and the inductive method was covered in Aristotle's subsequent treatise, the Posterior Analytics.
In the 19th century, modifications to syllogism were incorporated to deal with disjunctive ("A or B") and conditional ("if A then B") statements. Immanuel Kant famously claimed, in Logic (1800), that
logic was the one completed science, and that Aristotelian logic more or less included everything about logic that there was to know. (This work is not necessarily representative of Kant's mature
philosophy, which is often regarded as an innovation to logic itself.) Kant's opinion stood unchallenged in the West until 1879, when Gottlob Frege published his Begriffsschrift (Concept Script).
This introduced a calculus, a method of representing categorical statements (and statements that are not provided for in syllogism as well) by the use of quantifiers and variables.
A noteworthy exception is the logic developed in Bernard Bolzano's work Wissenschaftslehre (Theory of Science, 1837), the principles of which were applied as a direct critique of Kant, in the
posthumously published work New Anti-Kant (1850). The work of Bolzano had been largely overlooked until the late 20th century, among other reasons, because of the intellectual environment at the time
in Bohemia, which was then part of the Austrian Empire. In the last 20 years, Bolzano's work has resurfaced and become subject of both translation and contemporary study.
This led to the rapid development of sentential logic and first-order predicate logic, subsuming syllogistic reasoning, which was, therefore, after 2000 years, suddenly considered obsolete by many.
The Aristotelian system is explicated in modern fora of academia primarily in introductory material and historical study.
One notable exception to this modern relegation is the continued application of Aristotelian logic by officials of the Congregation for the Doctrine of the Faith, and the Apostolic Tribunal of the
Roman Rota, which still requires that any arguments crafted by Advocates be presented in syllogistic format.
Boole's acceptance of Aristotle
George Boole's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic John Corcoran in an accessible introduction to Laws of Thought.^[10] ^[11] Corcoran also wrote a
point-by-point comparison of Prior Analytics and Laws of Thought.^[12] According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were "to go under, over, and beyond"
Aristotle's logic by:
1. providing it with mathematical foundations involving equations;
2. extending the class of problems it could treat, as solving equations was added to assessing validity; and
3. expanding the range of applications it could handle, such as expanding propositions of only two terms to those having arbitrarily many.
More specifically, Boole agreed with what Aristotle said; Boole's 'disagreements', if they might be called that, concern what Aristotle did not say. First, in the realm of foundations, Boole reduced
Aristotle's four propositional forms to one form, the form of equations, which by itself was a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation solving to
logic—another revolutionary idea—involved Boole's doctrine that Aristotle's rules of inference (the "perfect syllogisms") must be supplemented by rules for equation solving. Third, in the realm of
applications, Boole's system could handle multi-term propositions and arguments, whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's
system could not deduce: "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is a
rectangle is a square that is a quadrangle."
Basic structure
A categorical syllogism consists of three parts:
1. Major premise
2. Minor premise
3. Conclusion/Consequent
Each part is a categorical proposition, and each categorical proposition contains two categorical terms.^[13] In Aristotle, each of the premises is in the form "All S are P," "Some S are P", "No S
are P" or "Some S are not P", where "S" is the subject-term and "P" is the predicate-term:
More modern logicians allow some variation. Each of the premises has one term in common with the conclusion: in a major premise, this is the major term (i.e., the predicate of the conclusion); in a
minor premise, this is the minor term (i.e., the subject of the conclusion). For example:
Major premise: All humans are mortal.
Minor premise: All Greeks are humans.
Conclusion/Consequent: All Greeks are mortal.
Each of the three distinct terms represents a category. From the example above, humans, mortal, and Greeks: mortal is the major term, and Greeks the minor term. The premises also have one term in
common with each other, which is known as the middle term; in this example, humans. Both of the premises are universal, as is the conclusion.
Major premise: All mortals die.
Minor premise: All men are mortals.
Conclusion/Consequent: All men die.
Here, the major term is die, the minor term is men, and the middle term is mortals. Again, both premises are universal, hence so is the conclusion.
See main article: Polysyllogism. A polysyllogism, or a sorites, is a form of argument in which a series of incomplete syllogisms is so arranged that the predicate of each premise forms the subject of
the next until the subject of the first is joined with the predicate of the last in the conclusion. For example, one might argue that all lions are big cats, all big cats are predators, and all
predators are carnivores. To conclude that therefore all lions are carnivores is to construct a sorites argument.
There are infinitely many possible syllogisms, but only 256 logically distinct types and only 24 valid types (enumerated below). A syllogism takes the form (note: M – Middle, S – subject, P –
Major premise: All M are P.
Minor premise: All S are M.
Conclusion/Consequent: All S are P.
The premises and conclusion of a syllogism can be any of four types, which are labeled by letters^[14] as follows. The meaning of the letters is given by the table:
code quantifier subject copula predicate type example
A All S are P universal affirmative All humans are mortal.
E No S are P universal negative No humans are perfect.
I Some S are P particular affirmative Some humans are healthy.
O Some S are not P particular negative Some humans are not old.
In Prior Analytics, Aristotle uses mostly the letters A, B, and C (Greek letters alpha, beta, and gamma) as term place holders, rather than giving concrete examples. It is traditional to use is
rather than are as the copula, hence All A is B rather than All As are Bs. It is traditional and convenient practice to use a, e, i, o as infix operators so the categorical statements can be written
succinctly. The following table shows the longer form, the succinct shorthand, and equivalent expressions in predicate logic:
Form Shorthand Predicate logic
\forallx(A(x) → B(x))
All A is B AaB or
No A is B AeB or
\forallx(A(x) → \negB(x))
Some A is B AiB \existx(A(x)\landB(x))
Some A is not B AoB \existx(A(x)\land\negB(x))
The convention here is that the letter S is the subject of the conclusion, P is the predicate of the conclusion, and M is the middle term. The major premise links M with P and the minor premise links
M with S. However, the middle term can be either the subject or the predicate of each premise where it appears. The differing positions of the major, minor, and middle terms gives rise to another
classification of syllogisms known as the figure. Given that in each case the conclusion is S-P, the four figures are:
Figure 1 Figure 2 Figure 3 Figure 4
Major premise M–P P–M M–P P–M
Minor premise S–M S–M M–S M–S
(Note, however, that, following Aristotle's treatment of the figures, some logicians—e.g., Peter Abelard and Jean Buridan—reject the fourth figure as a figure distinct from the first.)
Putting it all together, there are 256 possible types of syllogisms (or 512 if the order of the major and minor premises is changed, though this makes no difference logically). Each premise and the
conclusion can be of type A, E, I or O, and the syllogism can be any of the four figures. A syllogism can be described briefly by giving the letters for the premises and conclusion followed by the
number for the figure. For example, the syllogism BARBARA below is AAA-1, or "A-A-A in the first figure".
The vast majority of the 256 possible forms of syllogism are invalid (the conclusion does not follow logically from the premises). The table below shows the valid forms. Even some of these are
sometimes considered to commit the existential fallacy, meaning they are invalid if they mention an empty category. These controversial patterns are marked in italics. All but four of the patterns in
italics (felapton, darapti, fesapo and bamalip) are weakened moods, i.e. it is possible to draw a stronger conclusion from the premises.
Figure 1 Figure 2 Figure 3 Figure 4
Barbara Cesare Datisi Calemes
Celarent Camestres Disamis Dimatis
Darii Festino Ferison Fresison
Ferio Baroco Bocardo Calemos
Barbari Cesaro Felapton Fesapo
Celaront Camestros Darapti Bamalip
The letters A, E, I, and O have been used since the medieval Schools to form mnemonic names for the forms as follows: 'Barbara' stands for AAA, 'Celarent' for EAE, etc.
Next to each premise and conclusion is a shorthand description of the sentence. So in AAI-3, the premise "All squares are rectangles" becomes "MaP"; the symbols mean that the first term ("square") is
the middle term, the second term ("rectangle") is the predicate of the conclusion, and the relationship between the two terms is labeled "a" (All M are P).
The following table shows all syllogisms that are essentially different. The similar syllogisms share the same premises, just written in a different way. For example "Some pets are kittens" (SiM in
Darii) could also be written as "Some kittens are pets" (MiS in Datisi).
In the Venn diagrams, the black areas indicate no elements, and the red areas indicate at least one element. In the predicate logic expressions, a horizontal bar over an expression means to negate
("logical not") the result of that expression.
It is also possible to use graphs (consisting of vertices and edges) to evaluate syllogisms.^[15]
Celarent (EAE-1)
Similar: Cesare (EAE-2)
Darii (AII-1)
Similar: Datisi (AII-3)
Ferio (EIO-1)
Similar: Festino (EIO-2), Ferison (EIO-3), Fresison (EIO-4)
Bocardo (OAO-3)
Barbari (AAI-1)
Celaront (EAO-1)
Similar: Cesaro (EAO-2)
Camestros (AEO-2)
Similar: Calemos (AEO-4)
Felapton (EAO-3)
Similar: Fesapo (EAO-4)
Darapti (AAI-3)
Table of all syllogisms
This table shows all 24 valid syllogisms, represented by Venn diagrams. Columns indicate similarity, and are grouped by combinations of premises. Borders correspond to conclusions. Those with an
existential assumption are dashed.
Terms in syllogism
With Aristotle, we may distinguish singular terms, such as Socrates, and general terms, such as Greeks. Aristotle further distinguished types (a) and (b):
Such a predication is known as a distributive, as opposed to non-distributive as in Greeks are numerous. It is clear that Aristotle's syllogism works only for distributive predication, since we
cannot reason All Greeks are animals, animals are numerous, therefore all Greeks are numerous. In Aristotle's view singular terms were of type (a), and general terms of type (b). Thus, Men can be
predicated of Socrates but Socrates cannot be predicated of anything. Therefore, for a term to be interchangeable—to be either in the subject or predicate position of a proposition in a syllogism—the
terms must be general terms, or categorical terms as they came to be called. Consequently, the propositions of a syllogism should be categorical propositions (both terms general) and syllogisms that
employ only categorical terms came to be called categorical syllogisms.
It is clear that nothing would prevent a singular term occurring in a syllogism—so long as it was always in the subject position—however, such a syllogism, even if valid, is not a categorical
syllogism. An example is Socrates is a man, all men are mortal, therefore Socrates is mortal. Intuitively this is as valid as All Greeks are men, all men are mortal therefore all Greeks are mortals.
To argue that its validity can be explained by the theory of syllogism would require that we show that Socrates is a man is the equivalent of a categorical proposition. It can be argued Socrates is a
man is equivalent to All that are identical to Socrates are men, so our non-categorical syllogism can be justified by use of the equivalence above and then citing BARBARA.
Existential import
If a statement includes a term such that the statement is false if the term has no instances, then the statement is said to have existential import with respect to that term. It is ambiguous whether
or not a universal statement of the form All A is B is to be considered as true, false, or even meaningless if there are no As. If it is considered as false in such cases, then the statement All A is
B has existential import with respect to A.
It is claimed Aristotle's logic system does not cover cases where there are no instances. Aristotle's goal was to develop a logic for science. He relegates fictions, such as mermaids and unicorns, to
the realms of poetry and literature. In his mind, they exist outside the ambit of science, which is why he leaves no room for such non-existent entities in his logic. This is a thoughtful choice, not
an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is "a phrase signifying a thing's essence." Because non-existent entities cannot be
anything, they do not, in Aristotle's mind, possess an essence. This is why he leaves no place for fictional entities like goat-stags (or unicorns).^[16]
However, many logic systems developed since do consider the case where there may be no instances. Medieval logicians were aware of the problem of existential import and maintained that negative
propositions do not carry existential import, and that positive propositions with subjects that do not supposit are false.
The following problems arise:
For example, if it is accepted that AiB is false if there are no As and AaB entails AiB, then AiB has existential import with respect to A, and so does AaB. Further, if it is accepted that AiB
entails BiA, then AiB and AaB have existential import with respect to B as well. Similarly, if AoB is false if there are no As, and AeB entails AoB, and AeB entails BeA (which in turn entails BoA)
then both AeB and AoB have existential import with respect to both A and B. It follows immediately that all universal categorical statements have existential import with respect to both terms. If AaB
and AeB is a fair representation of the use of statements in normal natural language of All A is B and No A is B respectively, then the following example consequences arise:
"All flying horses are mythical" is false if there are no flying horses.
If "No men are fire-eating rabbits" is true, then "There are fire-eating rabbits" is true; and so on.
If it is ruled that no universal statement has existential import then the square of opposition fails in several respects (e.g. AaB does not entail AiB) and a number of syllogisms are no longer valid
(e.g. BaC, AaB->AiC).
These problems and paradoxes arise in both natural language statements and statements in syllogism form because of ambiguity, in particular ambiguity with respect to All. If "Fred claims all his
books were Pulitzer Prize winners", is Fred claiming that he wrote any books? If not, then is what he claims true? Suppose Jane says none of her friends are poor; is that true if she has no friends?
The first-order predicate calculus avoids such ambiguity by using formulae that carry no existential import with respect to universal statements. Existential claims must be explicitly stated. Thus,
natural language statements—of the forms All A is B, No A is B, Some A is B, and Some A is not B—can be represented in first order predicate calculus in which any existential import with respect to
terms A and/or B is either explicit or not made at all. Consequently, the four forms AaB, AeB, AiB, and AoB can be represented in first order predicate in every combination of existential import—so
it can establish which construal, if any, preserves the square of opposition and the validity of the traditionally valid syllogism. Strawson claims such a construal is possible, but the results are
such that, in his view, the answer to question (e) above is no.
Syllogistic fallacies
People often make mistakes when reasoning syllogistically.^[17]
For instance, from the premises some A are B, some B are C, people tend to come to a definitive conclusion that therefore some A are C.^[18] ^[19] However, this does not follow according to the rules
of classical logic. For instance, while some cats (A) are black things (B), and some black things (B) are televisions (C), it does not follow from the parameters that some cats (A) are televisions
(C). This is because in the structure of the syllogism invoked (i.e. III-1) the middle term is not distributed in either the major premise or in the minor premise, a pattern called the "fallacy of
the undistributed middle". Because of this, it can be hard to follow formal logic, and a closer eye is needed in order to ensure that an argument is, in fact, valid.^[20]
Determining the validity of a syllogism involves determining the distribution of each term in each statement, meaning whether all members of that term are accounted for.
In simple syllogistic patterns, the fallacies of invalid patterns are:
Neither of the premises accounts for all members of the middle term, which consequently fails to link the major and minor term.
The conclusion implicates all members of the major term (P – meaning the proposition is negative); however, the major premise does not account for them all (i.e., P is either an affirmative predicate
or a particular subject there).
Same as above, but for the minor term (S – meaning the proposition is universal) and minor premise (where S is either a particular subject or an affirmative predicate).
Both premises are negative, meaning no link is established between the major and minor terms.
If either premise is negative, the conclusion must also be.
If both premises are affirmative, the conclusion must also be.
Other types
See also
External links
Notes and References | {"url":"https://everything.explained.today/Syllogism/","timestamp":"2024-11-12T10:07:25Z","content_type":"text/html","content_length":"59513","record_id":"<urn:uuid:2c8012e4-4f61-4ff3-b8c1-eb640c9005ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00743.warc.gz"} |
A First Course in Linear Algebra
Textbook Title: A First Course in Linear Algebra
Textbook Description:
A First Course in Linear Algebra is an introductory textbook designed for university sophomores and juniors. Typically such a student will have taken calculus, but this is not a prerequisite. The
textbook begins with systems of linear equations, then covers matrix algebra, before taking up finite-dimensional vector spaces in full generality.
Author: Ben Crowell
Subjects: Mathematics
Key words: Mathematics, Algebra
Download URL: http://linear.pugetsound.edu/
Only Registered Users Can Save eTextbooks to Their TextBookGo Account.
Sign Up For Free! | {"url":"https://textbookgo.com/a-first-course-in-linear-algebra/","timestamp":"2024-11-09T00:25:29Z","content_type":"text/html","content_length":"13354","record_id":"<urn:uuid:2e083601-09a3-443a-af04-59227d223483>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00182.warc.gz"} |
Math Games
• Personal Financial Literacy
□ 5.PFL.111.7.b.10A
Define income tax, payroll tax, sales tax, and property tax.
□ 5.PFL.111.7.b.10B
Explain the difference between gross income and net income.
□ 5.PFL.111.7.b.10C
Identify the advantages and disadvantages of different methods of payment, including check, credit card, debit card, and electronic payments.
□ 5.PFL.111.7.b.10D
Develop a system for keeping and using financial records.
□ 5.PFL.111.7.b.10E
Describe actions that might be taken to balance a budget when expenses exceed income.
□ 5.PFL.111.7.b.10F
Balance a simple budget.
• Mathematical Process Standards
□ 5.MPS.111.7.b.1A
Apply mathematics to problems arising in everyday life, society, and the workplace.
□ 5.MPS.111.7.b.1B
Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving
process and the reasonableness of the solution.
□ 5.MPS.111.7.b.1C
Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to
solve problems.
□ 5.MPS.111.7.b.1D
Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
□ 5.MPS.111.7.b.1E
Create and use representations to organize, record, and communicate mathematical ideas.
□ 5.MPS.111.7.b.1F
Analyze mathematical relationships to connect and communicate mathematical ideas.
□ 5.MPS.111.7.b.1G
Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
□ 5.NO.111.7.b.2
Represent the value of the digit in decimals through the thousandths using expanded notation and numerals.
□ 5.NO.111.7.b.2B
Compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =.
□ 5.NO.111.7.b.2C
Round decimals to tenths or hundredths.
□ 5.NO.111.7.b.3A
Estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division.
□ 5.NO.111.7.b.3B
Multiply with fluency a three-digit number by a two-digit number using the standard algorithm.
□ 5.NO.111.7.b.3C
Solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm.
□ 5.NO.111.7.b.3D
Represent multiplication of decimals with products to the hundredths using objects and pictorial models, including area models.
□ 5.NO.111.7.b.3E
Solve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationship to
the multiplication of whole numbers.
□ 5.NO.111.7.b.3F
Represent quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using objects and pictorial models, including area models.
□ 5.NO.111.7.b.3G
Solve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm.
□ 5.NO.111.7.b.3H
Represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations.
□ 5.NO.111.7.b.3I
Represent and solve multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models.
□ 5.NO.111.7.b.3J
Represent division of a unit fraction by a whole number and the division of a whole number by a unit fraction such as 1/3 � 7 and 7 � 1/3 using objects and pictorial models, including area
□ 5.NO.111.7.b.3K
Add and subtract positive rational numbers fluently.
□ 5.NO.111.7.b.3L
Divide whole numbers by unit fractions and unit fractions by whole numbers.
□ 5.AR.111.7.b.4A
Identify prime and composite numbers.
□ 5.AR.111.7.b.4B
Represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity.
□ 5.AR.111.7.b.4C
Generate a numerical pattern when given a rule in the form y = ax or y = x + a and graph.
□ 5.AR.111.7.b.4D
Recognize the difference between additive and multiplicative numerical patterns given in a table or graph.
□ 5.AR.111.7.b.4E
Describe the meaning of parentheses and brackets in a numeric expression.
□ 5.AR.111.7.b.4F
Simplify numerical expressions that do not involve exponents, including up to two levels of grouping.
□ 5.AR.111.7.b.4G
Use concrete objects and pictorial models to develop the formulas for the volume of a rectangular prism, including the special form for a cube (V = l x w x h, V = s x s x s, and V = Bh).
□ 5.AR.111.7.b.4H
Represent and solve problems related to perimeter and/or area and related to volume.
□ 5.GM.111.7.b.5
The student applies mathematical process standards to classify two-dimensional figures by attributes and properties. The student is expected to classify two-dimensional figures in a hierarchy
of sets and subsets using graphic organizers based on their attributes and properties.
□ 5.GM.111.7.b.6A
Recognize a cube with side length of one unit as a unit cube having one cubic unit of volume and the volume of a three-dimensional figure as the number of unit cubes (n cubic units) needed to
fill it with no gaps or overlaps if possible.
□ 5.GM.111.7.b.6B
Determine the volume of a rectangular prism with whole number side lengths in problems related to the number of layers times the number of unit cubes in the area of the base.
□ 5.GM.111.7.b.7
The student applies mathematical process standards to select appropriate units, strategies, and tools to solve problems involving measurement. The student is expected to solve problems by
calculating conversions within a measurement system, customary or metric.
□ 5.GM.111.7.b.8A
Describe the key attributes of the coordinate plane, including perpendicular number lines (axes) where the intersection (origin) of the two lines coincides with zero on each number line and
the given point (0, 0); the x-coordinate, the first number in an ordered pair, indicates movement parallel to the x-axis starting at the origin; and the y-coordinate, the second number,
indicates movement parallel to the y-axis starting at the origin.
□ 5.GM.111.7.b.8B
Describe the process for graphing ordered pairs of numbers in the first quadrant of the coordinate plane.
□ 5.GM.111.7.b.8C
Graph in the first quadrant of the coordinate plane ordered pairs of numbers arising from mathematical and real-world problems, including those generated by number patterns or found in an
input-output table.
□ 5.DA.111.7.b.9A
Represent categorical data with bar graphs or frequency tables and numerical data, including data sets of measurements in fractions or decimals, with dot plots or stem-and-leaf plots.
□ 5.DA.111.7.b.9B
Represent discrete paired data on a scatterplot.
□ 5.DA.111.7.b.9C
Solve one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot. | {"url":"https://teks.mathgames.com/standards/grade5","timestamp":"2024-11-02T14:43:34Z","content_type":"text/html","content_length":"530467","record_id":"<urn:uuid:ef450cb2-6922-4c3e-b8ed-f7fc9fb3872e>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00019.warc.gz"} |
Compound Interest Calculator - Open Office Templates
Compound Interest Calculator
Compound interest is an interest that arises when interest is added to the principal everytime the interest is due, so the total amount will be calculated together to earn next interest. The addition
of interest to the principal to get another interest is called compounding. Banks usually apply this kind of interest rate where your savings amount will grow based on that interest period. For
example, a saving with $1000 initial principal and 1% interest per month would have a balance of $1010 at the end of the first month, $1020.1 at the end of the second month, and so on.
This compound interest also used as a loan interest. This compound interest calculator is a simple calculator that will calculate the future value of your savings or loan amount based on daily,
weekly, quarterly, semi-annually and annually compounded period. And you can also see how the total interest applied to your savings or loans based on that compounded period. | {"url":"https://openofficetemplates.net/calc/compound-interest-calculator/","timestamp":"2024-11-10T11:11:59Z","content_type":"text/html","content_length":"52334","record_id":"<urn:uuid:685a8a82-77d8-4343-981c-1a0b9fd042fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00315.warc.gz"} |
Let's Do Math! - Guidance Control
Down below you'll find some discussion of doing some "physics-based guidance control," but first the daily update:
Particle Event Components
I basically have two separate particle systems:
One is a pretty typical particle system, it just emits and renders mostly ballistic sprite particles, for explosions or smoke, that sort of thing.
The other is really more of a batch model-rendering system - it takes a single model and spawns multiple rigid bodies, so it has a physical presence in the physics simulation - good for bullets,
swarms, that sort of thing.
So, what if you have a bullet, and you want a small explosion when it hits something?
I put together a sort of mediation component that waits for notifications from the physics particle system that a particle is being destroyed, and then it emits a particle on a an explosion particle
Which is fine, except now I have a cSpawnParticleOnPhysicsCollideComponent class, which is getting a bit wordy. There's also a cSpawnParticleOnPhysicsTimeoutComponent class. It also doesn't trigger
any audio yet. This is going to grow out of control quickly, seems like it needs a rethink.
Now, Let's Do Math!
I wanted to work out the math required to solve this problem:
• Under a maximum acceleration, how do we turn to face a particular direction as quickly as possible without overshooting the target?
So, you want to start turning in a particular direction, then slow down and come to a stop directly facing your target. I'm assuming that your target is stationary, otherwise this gets
First, your velocity under maximum deceleration will look something like this:
where the y-axis is your current speed, x-axis is the amount of time it will take you to stop, and the area under the curve is the distance you will travel.
• If you are on the line, you just need to decelerate at the maximum rate, and when you hit zero angular velocity, you should be facing the right direction.
• If you are below the line, you basically want to get to the line as quickly as possible.
• If you are above the line, you can't slow down fast enough to avoid overshooting the target, but let's do our best...
So what do we need to calculate?
Converting between angular acceleration and torque is pretty easy:
Then you just need to compute the optimal acceleration, and make sure you don't exceed it:
Which looks a bit more sensible in code:
VECTOR4 axis, angle; //Desired rotation
turn_axis_angle(from, to, &axis, &angle);
VECTOR4 w_opt = axis * VECTOR4::sqrt2() * sqrt(angle * max_ang_accel); //How fast we should be turning right now (w0) to decelerate to w=0 after turning angle radians
VECTOR4 w_delta = w_opt - w0; //Difference between optimal and current angular velocities
VECTOR4 w_delta_accel = length3(w_delta) / VECTOR4(GameLib::physics_dt); //Acceleration required to get to optimal by next frame
VECTOR4 w_cmp = w_delta_accel > max_ang_accel; //Choose lesser of optimal and maximum acceleration
VECTOR4 w_accel = sel(w_cmp, safe_normalize3(w_delta) * max_ang_accel, axis * w_delta_accel);
VECTOR4 torque = I * w_accel; //Compute torque from acceleration
OK, I'm not 100% certain that is all correct, but it seems to work, which is a good sign. :)
All of the math looks very similar for doing the same thing with linear acceleration. | {"url":"http://blog.basemetalgames.com/2013/08/lets-do-math-guidance-control.html","timestamp":"2024-11-13T18:25:30Z","content_type":"text/html","content_length":"54208","record_id":"<urn:uuid:cf687165-281a-4854-873b-ab751de0df94>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00039.warc.gz"} |
5 Machine Learning BEGINNER Projects (+ Datasets & Solutions)
I this tutorial I share 5 Beginner Machine Learning projects with you and give you tips how to solve all of them. These projects are for complete beginners and should teach you some basic machine
learning concepts. With each project the difficulty increases a little bit and you'll learn a new algorithm.
For each project I give you an algorithm that you can use and include the links to the datasets, so you can start right away!
For all those projects I recommend to use the scikit-learn library. This is the go-to library in Python when it comes to machine learning. It's incredibly easy to get started with this library and to
implement your own Machine Learning algorithms with it.
Regression vs. Classification
Before we go over the projects you should know about the 2 basic types of machine learning tasks: Regression vs. Classification.
Fundamentally, classification is about predicting a label, so a concrete class value while regression is about predicting a quantity, so a continuous value.
Project 1
As first project I recommend to start with a regression problem. For this problem I recommend to do actually 2 projects. One is a super simple project to predict the salary based on the number of
years of experience. This only contains 2 variables, so you stay in 2 dimensions and this should give you a good understanding of how the model works. After that I recommend to do the Boston Housing
dataset. Here you should predict the price of a home based on multiple different variables. The algorithm you should use here is the so-called Linear Regression model. This is one of the easiest
algorithms and shouldn't be too hard to understand.
Project 2
After that I recommend to tackle your first classification problem. The dataset is the Iris dataset. This is probably the most famous dataset in the world of machine learning, and everyone should
have solved it at least once. Here we have samples from 3 different flower species, and for each sample we have 4 different features that describe the flower. With this information we want to predict
the species of the flower then. As algorithm I recommend to use the K Nearest Neighbor (KNN) algorithm. This is one of the simplest classification algorithms but works pretty well here. The species
are very clearly distinguishable, so you should be able to train a good KNN model and reach 100% correct predictions.
I know everyone is using the Iris dataset as first example, so if you cannot see it anymore and want to have an alternative then you can check out the Penguin dataset where we want to predict the
species of a penguin based on certain features.
Project 3
Next, I recommend to use the Breast Cancer dataset. This is another famous dataset with the interesting task to predict if a cancer cell is good or bad (or in medical terms: malignant or benign).
Here we have 30 different features for each cancer cell that have been computed from medical images. This is certainly more complex and more difficult than the project before, but still you should be
able to reach an accuracy of 95% here. As algorithm I recommend to try out the Logistic Regression model. This is similar to the Linear Regression model in the beginning. Don't be confused by the
name, because even though it has Regression in its name, it is actually used for a classification task. The Logistic Regression algorithm also models a continuous value, but this is a probability
value between 0 and 1 and can therefore be used for classification. I also recommend to have a look at another new technique that is called feature standardization. Because the 30 different features
may have values in different ranges, and this might confuse the model a little bit. So play around with feature standardization here and see if you can improve your model even further with that.
(Note: Feature standardization is not required for Logistic Regression, but it's still an important technique and can be important for other classifier here.)
Project 4
The fourth project is interesting because it is implemented in everyones email client. Here we want to create a spam filter based on the Spambase dataset. In this dataset we have the frequency of
different words and characters, so we calculate the total number of appearances of each word and divide it by the total number of words in the email. Spam emails clearly show certain key words more
often than normal mails, so with this information we are able to create a spam classifier. As algorithm I recommend to have a look at the Naive Bayes algorithm here. The new challenge here is then
not only to use this dataset and evaluate your model, but then after you have trained your classifier also apply it to a real application. So what do you do with a new email? What do you have to do
before you pass it to the classifier? Here you somehow have to find out how to transform the text from the email to the same format that your classifier expects. This should give you a better
understanding of how datasets are shaped and created.
Project 5
The last project I recommend is the Titanic dataset. This is the first beginner project that Kaggle recommends on their site in the Getting Started section. Here we have a list of all Titanic
passengers with certain features like the age, the name, or the sex of the person, and we want to predict if this passenger survived or not. The Titanic dataset requires a little more work before we
can use it, because not all information in this dataset are useful and we even have missing values. So here you should learn some preprocessing techniques and how to visualise, analyze, and clean the
data. Up to this point we could use the datasets right away, but in real world applications this is actually almost never the case, so you should definitely learn how to analyze datasets. As
algorithm I recommend to have a look at Decision Trees, and also at a second algorithm, the Random Forest algorithm, which extends decision trees. As another tip i recommend to have a look at the
pandas library here. This makes your life a lot easier when it comes to data visualisation and processing the data beforehand.
If you complete all projects you should have a good understanding of 6 popular machine learning algorithms, and you should also have a feeling for different datasets and some knowledge of how to
analyze and process the data.
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For further actions, you may consider blocking this person and/or reporting abuse | {"url":"https://dev.to/pat_loeber/5-machine-learning-beginner-projects-datasets-solutions-4pjo","timestamp":"2024-11-13T12:47:22Z","content_type":"text/html","content_length":"73681","record_id":"<urn:uuid:5e92ecfb-8f4d-4f64-b66b-2c88a67f447b>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00404.warc.gz"} |
MathsCity Leeds opening weekend
You're reading: Travels in a Mathematical World
MathsCity Leeds opening weekend
One day in February 2014, I was fortunate enough to battle through London during a Tube strike to attend a reception at the House of Commons for MathsWorldUK – an initiative then just two years in
development which aimed “to establish a national Mathematics Exploratorium in the United Kingdom … an interactive centre full of hands-on activities showcasing mathematics in all its aspects for
people of all ages and backgrounds”.
That initiative took a huge leap forwards last week with the launch of MathsCity Leeds, which my son and I visited on its opening weekend.
MathsCity is a play and discovery exhibit in a space within Trinity Shopping Centre in Leeds. It consists of a large room on two levels with perhaps three dozen different activity stations each
containing some kit and a brief instruction of what to do with it.
MathsCity Leeds
The exhibits were all well made, usually out of wood or plastic, with nice clear instructions – though there were also plenty of staff on hand to help if explanations were needed. Here are some of
the things we played with during our two-hour visit.
A selection of the exhibits we played with.
With this sort of thing, I always remember a time when I introduced a group of first-year maths undergraduates to a game at our Maths Arcade. After they had played for a bit, I asked if they were
getting on okay. “Fine,” one of them said, “but where’s the maths?” “What do you think maths is?” I asked. “Well, there are no numbers.” Here, there is a definite attempt to present maths as more
than just arithmetic and calculation with a single correct answer. Some exhibits were puzzles with an objective, lots were something to play with or explore goal-free, which is nice to see. There
were lots of shapes, symmetry and logic on display, and lots of mirrors!
Self-portrait; surrounded by the three mirrors of the kaleidoscope, can you tell I’m thinking “better take this quick then work out what trouble my son is getting into now”?
We had a great time and both enjoyed ourselves. We easily passed two hours and could have stayed longer – a little tantrum when I said it was time to leave is usually a sign an activity is going
Find \(x\)
I should say my son is six and a totally receptive audience. He was happy enough to learn that we were going on a long train journey – when I told him we were going to a ‘maths museum’ he boiled over
with enthusiasm (and when his mum promised a roast dinner on our return he declared the day “just like Christmas!”).
I called it a ‘maths museum’ as a deliberate error using a word he’d understand. Nomenclature is important, though. I notice the early MathsWorldUK material talks of an “exploratorium”, while the
current material uses “discovery centre”. My invitation to the House of Commons says its imagined exploratorium will be an interactive activity centre and will also
convey a feeling for the rich history of mathematics and some of the outstanding personalities whose ideas have shaped its development, but it will also show how much it contributes to modern-day
When I try to think about what a national maths centre might look like (contextualised, say, by comparison with the National Space Centre at Leicester) I might imagine something like MathsCity as the
playful heart of a broader experience that also attempts to showcase modern applications, illuminate the history of mathematics and explain a little of the technical detail for a general audience.
Here the focus – quite rightly – is on playful discovery and presenting a positive view of mathematics, and MathsCity does a good job of these aspects.
Playful misuse of the mirror letters.
Not knowing how far we’d come, the lady who greeted us on arrival told us proudly how lucky we are to have the one-of-a-kind MathsCity in Leeds, and when we said goodbye she told my son to tell all
his friends to come down. I think MathsCity does an amazing job at its intended mission and provides an entertaining space with a positive view of maths for people coming into Leeds to shop, though I
can’t quite imagine pitching it to non-mathematician parents at the school gate here in Nottingham as the focus of a long day trip.
Is it trying to be a national maths centre? Not yet. Does it work well as a playful discovery centre? Definitely. My interest in mathematics communication and my son’s general enthusiasm for train
rides and maths carried us though a happy day trip to Leeds. I feel I should acknowledge how hard it is to try to launch something like this during the COVID pandemic, where the need to minimise the
spread of the virus and changing Government restrictions interact with the desire to get people through the door and demonstrate the viability of the idea. If you’re a reader of this blog, you might
also go the extra mile to visit, or certainly please do your best to support this fledgling initiative by telling people local to Leeds about it!
All information about MathsCity and the place to book tickets is mathscity.co.uk.
3 Responses to “MathsCity Leeds opening weekend”
1. Jack Abramsky
Dear Peter
As a Trustee of MathsWorldUK I want to say how much we at MWUK appreciated your article on your visit with your son to MathsCity in Leeds. I am writing to get permission to use some of your
photographs of MathsCity in our forthcoming Newsletter. I cannot guarantee to include them but it would be good to get your permission ahead of time. Do you have others that were not included in
your article?
Best wishes
Jack Abramsky | {"url":"https://aperiodical.com/2021/10/mathscity-leeds-opening-weekend/","timestamp":"2024-11-11T03:27:59Z","content_type":"text/html","content_length":"62215","record_id":"<urn:uuid:e27dc833-8930-474f-84f7-29c0b5e8760e>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00769.warc.gz"} |
Assessing Instructor Effectiveness Based on Future Student Performance
Educational institutions rely on instructor assessment to determine course assignments, which instructors to retain or promote, and whom to provide with additional assistance or training. Instructor
assessment is most commonly based on student surveys or peer evaluation—which are both subject to the evaluator’s personal biases. This study describes an assessment method based on future student
grade performance, which has the potential to avoid these biases. This study is based on eight years of undergraduate course-grade data from over 24,000 students in a large metropolitan university.
The methodology introduced in this paper accounts for confounding factors, such as diverse instructor grading policies and varying student abilities. Top and bottom performing instructors are
identified for each course.
Instructor effectiveness, instructor evaluation, data mining, educational data mining, grade analysis, data analysis.
Assessing instructor effectiveness is important for determining which instructors to retain or promote, optimal assignment of courses, and providing additional mentorship or training to weak
instructors [3, 8]. It is also often a key factor in tenure decisions. In a university, these assessments are typically done through student surveys or peer evaluations based on classroom
observations [8]. Both of these methods are subject to the biases of the evaluators, which may be impacted by instructor gender and race, and may not measure student learning [1, 2, 5, 7]. The
justification for using student surveys is derived from several studies in which positive correlations are found between student evaluations and instructor effectiveness as measured through exams at
the end of each course. However, a recent meta-analysis conducted on thirty-two of these studies shows that there is no such positive correlation for the studies containing the most course sections,
indicating earlier conclusions were due to a lack of data and providing argument against the use of student evaluations to measure instructor effectiveness [7]. These studies also measure instructor
effectiveness using the grades for the course being taught. Given that the exams and grades are usually designed by the instructor, this yields another potential source of bias; our methods avoid
this bias by relying on students’ performance in future courses. Peer evaluations are most likely subject to similar biases.
The method introduced in this paper assesses instructors by quantifying their impact on future student grade performance. If students with a given instructor perform better (worse) in future courses
than students who have a different instructor, then the instructor is ranked favorably (unfavorably). Our study only assumes basic course-grade data is available, and does not account for all
potential confounding factors, such as the time of day of the class or class size [9]. However, we do account for grade‑related factors such as instructor grading leniency and student ability as
measured by grade-point average. We have developed a publicly available Python-based software tool that implements our methodology and generates the instructor effectiveness metrics [6].
Our study identifies instructors who appear to be much more or less effective than other instructors based on their student’s future performance. Our analysis first focuses on two case studies,
assessing instructors of “Spanish 1” and “Computer Science 2” based on future performance in “Spanish 2” and “Data Structures,” respectively. We then identify the best instructors in the university
based on the teaching of a single course, and then identify the top‑10 and bottom-10 instructors based on the performance over all courses each instructor teaches, with future student performance
measured on all future courses taken within a single department. We discuss several interesting patterns across these results.
Student Course-Grade Data Set
The work presented in this paper is based on eight years of undergraduate course-grade data from Fordham University. Each record in the dataset represents a student earning a grade in a specific
course section and includes the following fields: student identifier, instructor ID, course name, course number, course department, course term (semester and year), and student grade using a 0.0 (F)
- 4.0 (A) scale. Table 1 provides key dataset statistics. In order to enhance privacy, student identifiers were remapped and data for course sections with fewer than five students were omitted. Even
with such measures, due to federal regulations we are not permitted to publicly share the dataset.
Table 1. Summary Dataset Statistics
Feature Unique Values
Record Number 442,230
Student ID (SID) 24,654
Instructor ID (IID) 2,195
Course Name & Number 2,505
Course Section 21,504
Demographic information was not included in the data set as it could facilitate de‑anonymization. We therefore characterize the population using university statistics for the middle year of the data:
gender distribution is 60% female and 40% male, and the racial/ethnic breakdown is 55% White, 14% Hispanic, 11% Asian, 7% International, 4% Black, and 9% other. The majority of students are between
the ages of 17 and 22.
Measuring Instructor Benefit
This section describes the methodology used to calculate instructor benefit and introduces our three instructor benefit metrics. This methodology is implemented in a publicly available Python‑based
software tool developed by our research group [6], which enables other researchers to apply our research to other student grade datasets. The steps used to generate our results are summarized in
Figure 1 and described in subsequent subsections.
Figure 1. Overview of Data Processing Steps
Remove Sections with Too Similar Grades
Differences in instructor effectiveness can only be measured if there is reasonable variance in the grades assigned to students. Thus we remove the data associated with course sections with very low
grade variance. Step 1 computes the standard deviation of the grades in every course section and eliminates sections where the standard deviation is below MinSD. The distribution of grade
standard-deviation values at the section level is provided in the Appendix, and Section 4 specifies the MinSD value of 0.2 that is used throughout this study.
Normalize Grades in each Section
To account for different instructor grading schemes, we employ z‑score normalization for each course section. The Level-1 normalized score, defined below, tells us how many standard deviations away
from the course section mean a student scores.
${L1}_{\text{SID}}^{\text{CrsSec}} = \frac{G_{\text{SID}}^{\text{CrsSec}} - \mu_{\text{CrsSec}}^{}}{\sigma_{\text{CrsSec}}^{}}$
In this formula, ${L1}_{\text{SID}}^{\text{CrsSec}}$ is the normalized course grade for student SID in the specified course section CrsSec, $G_{\text{SID}}^{\text{CrsSec}}$ is the original grade for
student SID in CrsSec, and $\mu_{\text{CrsSec}}$ and $\sigma_{\text{CrsSec}}$ are the mean and standard deviation of the grades in CrsSec. Instructor benefits calculated using this Level-1 normalized
grade are referred to as L1 Instructor Benefits. Instructor benefits calculated using the unnormalized grades are referred to as Grade Benefits (although they can be viewed as Level-0 Instructor
Normalize Grades by Overall Performance
Student performance is not just dependent on the effectiveness of the instructor but also depends on a student’s abilities. We therefore employ a second level of grade normalization that is based on
the student’s overall performance in all of their courses (i.e., GPA). Without this normalization, an instructor that coincidentally is assigned high-performing students will appear to perform better
than an instructor that is assigned weaker students. This Level-2 normalization is defined by the formula below and L2 Instructor Benefits are calculated using these values.
${L2}_{\text{SID}}^{\text{CrsSec}} = \frac{{L1}_{\text{SID}}^{\text{CrsSec}} - \mu_{\text{SID}}^{\text{norm}}}{\sigma_{\text{SID}}^{\text{norm}}}$
In this formula, ${L2}_{\text{SID}}^{\text{CrsSec}}$ is the Level-2 normalized grade of student SID in course section CrsSec, ${L1}_{\text{SID}}^{\text{CrsSec}}$ is the Level-1 normalized score from
the prior step, and $\mu_{\text{SID}}^{\text{norm}}$ and $\sigma_{\text{SID}}^{\text{norm}}$ are the mean and standard deviation of student SID’s L1-normalized grades across all courses.
Find Instructor Benefit by Course Pair
We next consider every ordered course pair, C1 → C2, where course C1 is taken prior to course C2. Assume that the instructor for C1 has an instructor ID, IID. The instructor benefit associated with
instructor IID teaching C1 based on C2 performance is computed using the C2 grades for those students who previously had instructor IID for C1. The type of C2 grade (unnormalized, L1-normalized,
L2‑normalized) determines the type of Instructor Benefit (Grade, L1, L2). The calculations just described are aggregated over all sections for a given course. More formally, $IB_{\text{IID}}^{C1 \
rightarrow C2}$ is the instructor benefit (IB) for students taking C2 after taking C1 with instructor IID.
$IB_{\text{IID}}^{C1 \rightarrow C2} = \left\langle \text{GRADE}_{SID \in \left( C1,IID \right)}^{CrsSec \in \left( C2 \right)} \right\rangle$
In this formula, $CrsSec \in \left( C2 \right)$ is any course section of course C2, SID $\in \left( C1,IID \right)$ is every student who took course C1 with instructor IID, and $\left\langle \mathbf
{x} \right\rangle$ is the average of all values in x.
$IB_{\text{IID}}^{C1 \rightarrow C2}$ is computed for all ordered course pairs and for every C1 instructor, as long as at least 80% of the students who complete both courses take C1 first. This
restriction ensures that we only evaluate instructor effectiveness between courses that are taken in the expected order. In Section 3.6, we discuss aggregating the instructor effectiveness metrics
for each instructor teaching C1 over all C2 courses, and then again over all C1 courses for each instructor. However, we believe that these higher level metrics are less meaningful, since each C1
course is often best evaluated on a single C2 course (e.g., Spanish 1 is best evaluated using performance in Spanish 2). For this reason most of the results in Section 5 are at course-pair level.
When the most appropriate future course is not clear based on domain knowledge, the choice can be guided by looking at the course pairs with the highest pairwise grade correlation, as described in
one of our recent studies [4].
Remove Course Pairs with Few Students
Instructor effectiveness is computed for pairs C1 → C2. For the resulting instructor effectiveness metrics to be reliable, the instructor will need to have taught at least MinStudents in C1 who
subsequently completed C2. However, since this is a comparative statistic, we also require that there are MinStudents who completed C1 with other instructors and then completed C2 (hence if
MinStudents = 50 there must be at least 100 students that took C1 and then completed C2). Section 4 explores how the MinStudents threshold impacts the number of available course pairs.
Aggregate Average Instructor Benefits
Course-pair level instructor benefit metrics are aggregated to yield higher level views of instructor performance. The instructor benefit values for each course pair are first aggregated over the set
of C2; restrictions on C1 and C2 may be applied. The results in Section 5.3 are at the instructor level but are aggregated over all C1 courses taught by instructor IID in a single department and
measured on future performance over classes in another department (i.e., $\text{AI}B_{\text{IID}}^{D1 \rightarrow D2})$. This aggregation formula, which is provided below, also weights the courses by
the number of students.
$\text{AI}B_{\text{IID}}^{D1 \rightarrow D2} = \frac{1}{\sum_{CX_{\text{IID}}}^{}\left| \left\{ S \in CX_{\text{IID}} \right\} \right|}\sum_{\begin{pmatrix} C1 \in D1, \\ C2 \in D2 \\ \end{pmatrix}}^
{}{IB_{\text{IID}}^{C1 \rightarrow C2}}\left| \left\{ S \in IB_{\text{IID}}^{C1 \rightarrow C2} \right\} \right|$
The computation is performed across all relevant course pairs in the department $C1 \rightarrow C2$, where $IB_{\text{IID}}^{C1 \rightarrow C2}$ is an instructor benefit from C1 to C2, $|\{ S \in IB_
{\text{IID}}^{C1 \rightarrow C2}\}|$ is the number of students who took the instructor in that particular course pairing, and $\sum_{CX_{\text{IID}}}^{}\left| \left\{ S \in CX_{\text{IID}} \right\} \
right|$ is the number of students who took a course with instructor IID (summed over all possible courses CX). The formula above can be used to find, amongst other things, the instructor
effectiveness scores for a Computer Science instructor when evaluated on future Math classes, as well as when evaluated on future Computer Science classes (i.e., when D1 = D2).
Threshold Sensitivity Analysis
We selected appropriate thresholds for MinSD, the minimum standard deviation of section grades, and MinStudents, the minimum number of students in each course pair. The Appendix provides additional
relevant information related to the selection of these thresholds. We select MinSD=0.2, retaining most (20,904 of 21,504) course sections while ensuring some variability in student grades, and
MinStudents=50, to keep a large number of course pairs while maintaining reliability of the instructor benefit metrics.
This section provides our instructor benefit results. Section 5.1 and Section 5.2 provide results at the course-pair level, while Section 5.3 identifies the top and bottom performing instructors
based on their performance across all courses in a department. Grade Benefit, Level-1 Benefit, and Level-2 Benefit metrics are all provided, with a focus on Level-2 Instructor Benefits.
Section Level Course-Pair Results
This section provides results at the course-pair level. Due to space limitations we can only provide instructor benefit results for instructors teaching Spanish 1 based on future student performance
in Spanish 2 and for instructors teaching Computer Science 2 (CS2) based on future performance in Data Structures. We include the Spanish courses because they are popular and Computer Science courses
because they are offered by our home department and there is great interest in Computer Science education. In both cases, the course pairs are part of a common introductory sequence, and the second
course directly follows the first course.
Table 2. Instructor Benefit for Spanish 1 → Spanish 2
Instructor Sections Total # Students Instructor Benefit
ID Taught Spanish 1 Spanish 2 Grade Level 1 Level 2
F980 22 381 325 -0.024 0.016 0.050
F494 17 367 295 -0.189 -0.091 -0.090
F787 12 217 166 -0.154 -0.079 -0.210
F424 11 231 191 -0.278 -0.259 -0.146
F425 11 213 171 0.034 -0.076 -0.097
F819 10 201 176 0.050 0.124 0.233
F883 9 154 129 0.030 -0.039 0.065
F719 8 179 134 0.090 0.076 0.147
F890 7 172 138 -0.097 -0.009 -0.282
F541 7 86 67 0.061 0.045 -0.219
All 189 3485 2644 -0.088 -0.092 -0.073
The results for the Spanish classes are summarized in Table 2. Sections where fewer than five students continue to Spanish 2 are excluded to allow section-level instructor benefit values to be
reliable for statistical analyses. Due to space limitations, instructors with fewer than seven sections are not listed but are included in the summary statistics in the last row.
Our analysis focuses on Level 2 instructor benefit because it accounts for the two confounding factors discussed earlier, but Figure 2 shows that the three metrics are generally correlated, and the
Pearson correlation coefficient (ρ) using the 189 section level values confirms this with ρ=0.74 between the Grade and Level-1 metric and ρ=0.76 between the Level-1 and Level-2 metric.
Figure 2. Instructor Benefit Metrics for Spanish 1 Instructors
Table 2 and Figure 2 show that there are substantial differences in the Level-2 values since they vary from +0.233 to -0.282. Small differences in instructor ability may be hard to distinguish, so in
this initial study, we focus on cases with the largest differences and where the values are based on many students. Given this, we conclude that instructor F819 is highly effective, while F890, F541,
and F787 are least effective. A two-sample unequal variance t-test at the section level for Instructor F819 (+0.283) and F787 (‑0.210) yields p=0.0156 for the one-tailed distribution and p=0.0312 for
the two-tailed distribution. These p-values suggest that the differences are statistically significant, although this is partially due to comparing instructors at the two extremes. More students
would make more refined assessments possible.
Table 3 provides analogous results for the course pair. Instructor data is limited because the CS major was not heavily populated in the timeframe considered (2010 to 2018). The table includes all
instructors that taught three or more sections of CS2 (the bottom two instructors did not meet our preferred MinStudents threshold of 50). Based on the level 2 instructor benefit, two instructors are
strongly positive and two moderately negative. A t-test on the first two instructors in Table 3 yields p‑values of 0.00030 (1-tail) and 0.0003 (2-tail).
Table 3. Instructor Benefit for CS2 → Data Structures
Instructor Sections Total # Students Instructor Benefit
ID Taught CS2 DataStr. Grade Level 1 Level 2
F212 12 293 158 -0.304 -0.226 -0.189
F177 4 92 62 0.237 0.151 0.396
F589 3 56 36 -0.329 -0.177 -0.228
F653 3 35 33 -0.385 -0.042 0.400
All 32 697 410 -0.227 -0.145 -0.054
The section level results suggest that it is possible to distinguish between high and low-performing instructors when using future student performance in a single highly related course. It may be
difficult to reliably assess less extreme differences, but universities with larger classes or higher teaching loads should be better able to perform more refined assessments.
Global Course-Pair Instructor Results
Table 4 provides the best Level 2 Instructor Benefit results at the course-pair level. The course pairs are restricted to the same department or between departments that share major requirements,
since it is best to measure instructor effectiveness using related courses. The results are based on MinSD=0.2 and MinStudents=50. Each entry in Table 4 corresponds to a single instructor. Course
names are abbreviated using department codes and course numbers, but the full names are provided in our discussion.
Table 4. Top 6 Instructor Course Pairs by Level 2 Benefit
Course1 Course2 Grade Benefit Level 1 Benefit Level 2 Benefit
Chem 211 Bio 342 0.11 0.09 1.50
Econ 220 Econ 332 0.53 0.50 1.12
Phys 140 Chem 121 0.24 0.43 0.79
Chem 121 Chem 212 0.14 0.22 0.76
NatSci 304 NatSci 321 0.44 0.48 0.75
Comm 112 Comm 242 0.41 0.38 0.72
The courses pairs that appear in Table 4 exhibit a strong relationship between Course 1 and Course 2. For example, the first row involves “Organic Chemistry Lab” and “Biochemistry,” while the two
Natural Science courses correspond to “Organic Chemistry I Lab” and “Organic Chemistry II.” Our belief is that the instructor associated with each entry is a very effective instructor, although, as
discussed in Section 6, we cannot validate this. The Grade and L1 Benefit values are rarely as high as the L2 Benefit values, which indicates that the second round of normalization has a substantial
impact. Note that students who take Econ 220, with the instructor represented by that entry, obtain a grade that is, on average, 0.53 higher than otherwise expected; this corresponds to a difference
of more than a half letter grade.
Department Level Instructor Results
This section describes the aggregate effectiveness of an instructor ($\text{AIB}_{\text{IID}}^{D1 \rightarrow D2}$, as defined in Section 3.6.) based on all courses that instructor teaches in one
department, measured by student success in future courses in one (potentially different) department. We consider these results to be less meaningful than results on course-pairs selected for mutual
relevance. Nonetheless, some interesting high level observations arise. The results in Table 5 include the top and bottom performing instructors using MinStudents=50 and MinSD=0.2. Each entry
corresponds to a single instructor.
Table 5. Top and Bottom 10 Instructors by Level 2 Benefit
Course 1 Course 2 Instructor Benefit
Department Department Grade Level 1 Level 2
Top 10 Instructors
Political Sci. Natural Science 0.317^(8) 0.340^(3) 0.588
Mathematics Mathematics 0.033 0.242^(8) 0.570
Economics Theology 0.140 0.136 0.546
Economics Theology -0.127 -0.035 0.525
Italian Physics 0.131 0.225 0.510
Art History Spanish 0.314^(9) 0.260^(6) 0.503
Philosophy Communications 0.296 0.350^(2) 0.502
Mathematics Chemistry 0.063 0.103 0.497
Natural Science English 0.267 0.237^(9) 0.488
Physics Political Science 0.233 0.009 0.465
Bottom 10 Instructors
Mathematics Mathematics 0.163 -0.033 -0.585
Mathematics Mathematics -0.182 -0.131 -0.563
Physics Physics 0.154 -0.075 -0.511
Sociology Physics 0.107 -0.066 -0.476
Natural Science English -0.228 -0.203 -0.463
Visual Arts Anthropology -0.209 -0.072 -0.422
Natural Science Natural Science -0.475^(1) -0.372^(3) -0.416
Chemistry Biology -0.342^(8) -0.443^(1) -0.415
Comp. Science Physics 0.304 -0.185 -0.411
Mathematics Natural Science 0.085 -0.042 -0.406
The instructor benefits in Table 5 indicate that there is a substantial difference between the top and bottom performing instructors. Many unnormalized grade differences are about 0.3, representing
one-third of a letter grade difference. The values for the three effectiveness metrics appear correlated. To verify this we computed the Pearson correlation ρ between all three pairs of metrics, with
the following results: ρ(Grade, Level 1)[ ]= 0.977, ρ(Grade, Level 2) = 0.990, ρ[ ](Level 1, Level 2) = 0.989. These correlations are higher than in the prior section, showing a difference through
aggregation over many sections and courses. For comparison, when the Grade or Level 1 benefit metric appears in the top-10 or bottom-10 for the listed instructor entry, we provide the rank in
parentheses as a superscript (e.g., the instructor in the first row of data has the third highest Level‑1 Benefit). The ranks are not needed for Level 2 Benefits since they are already in rank order.
We generally would expect that instructor effectiveness in a course will have the biggest impact on other courses in the same discipline. Many of the entries do involve the same department or related
departments, but quite a few do not. There is much more agreement between the departments for the bottom 10 instructors, seeming to indicate weak instructors fail to convey field-specific concepts
for future use, while strong instructors may convey broader skills useful across disciplines. We note that STEM (Science, Technology, Engineering, and Math) instructors account for 80% of the bottom
performing instructors but only 40% of the top performing instructors, which is plausible given that STEM graduate programs generally provide little pedagogy instruction.
Bias in student and peer evaluations of instructor effectiveness have been widely observed [1, 2, 5, 7], supporting the need for more objective assessment methods. This study presents an alternative
method for instructor evaluation based on student performance in future courses. Our study accounts for instructor grading leniency and overall student ability; these factors impact assessment, but
all three metrics are nonetheless highly correlated.
Instructor assessment appears most appropriate at the course level and provides most insight when considering future performance in a single, highly related course. We focused on instructor
performance for Spanish 1 and CS2, based on future student performance in Spanish 2 and Data Structures, respectively. In both cases, instructor benefit varied substantially and the Level 2
instructor benefit for instructors at these two extremes differed with reasonable levels of statistical confidence. Our methodology distinguishes between instructors and identifies high and low
performing instructors. Evaluation of single instructors across courses was less clear, but revealed patterns, such as the weakest instructors often being associated with STEM disciplines.
The methodology and metrics described in this paper are calculated from traditional student course-grade data using a publicly available Python-based tool developed by our research group [6]. This
tool can be used by other researchers and practitioners to extend our analysis to other educational institutions. We plan to improve the tool’s documentation and usability in the near future.
There are numerous areas for further work. Increasing the size of our data set would substantially strengthen future analysis, especially within our Computer Science department. Effects of instructor
rank, title, years of experience, gender, and race also would be valuable to study. Furthermore, we aim to identify further discipline-based patterns, such as differences in instructor effectiveness
distributions across departments.
The most fundamental limitation of this work relates to validation. Currently we only perform limited validation across course sections. Additional validation will inherently be limited since there
is no way to assess the “ground truth.” Still, we aim to measure the relationship with weaker metrics like student survey results.
1. Boring, A., Ottoboni, K., and Stark, P.B. 2016. Student evaluations of teaching (mostly) do not measure teaching effectiveness. Science Open Research. DOI= https://doi.org/10.14293/
S2199-1006.1.SOR-EDU.AETBZC.v1 .
2. Chávez, K., Mitchell, K. 2020. Exploring bias in student evaluations: Gender, race, and ethnicity. PS: Political Science & Politics, 53(2), 270-274. DOI= https://doi.org/10.1017/S1049096519001744
3. Goldhaber, Dan. 2015. Teacher effectiveness research and the evolution of U.S. teacher policy. The George W. Bush Institute. https://files.eric.ed.gov/fulltext/ED560206.pdf.
4. Leeds, D.D. Zhang, T., and Weiss, G.M. 2021.Mining course groupings using academic performance. In Proceedings of The 14th International Conference on Educational Data Mining, International
Educational Data Mining Society, Paris France, June 29-July 2, 804-808.
5. Lilienfeld, E. 2016. How student evaluations are skewed against women and minority professors. The Century Foundation. https://tcf.org/content/commentary/
6. Riad-Zaky, M., Weiss, G.M., and Leeds, D.D. 2022. Course Grade Analytics with Networks (CGAN) [computer software], available https://www.cis.fordham.edu/edmlab/software.
7. Uttl, B., White, A. C., and Gonzalez, D. W. 2017. Meta-analysis of faculty’s teaching effectiveness: Student Evaluation of teaching ratings and student learning are not related. Studies in
Educational Evaluation., 54, 22-42. DOI= https://doi.org/10.1016/j.stueduc.2016.08.007.
8. Vlieger, P., Jacob, B., Stange, K. 2016. Measuring instructor effectiveness in higher education. National Bureau of Economic Research. https://www.nber.org/papers/w22998.
9. Wachtel, H.K. 1998. Student evaluation of college teaching effectiveness: a brief review, Assessment & Evaluation in Higher Education, 23:2, 191-212. DOI= https://doi.org/10.1080/0260293980230207
This appendix provides additional information related to the two thresholds that were discussed in Section 4. The relevant underlying data distributions are shown, which inform the choice of specific
threshold values.
Figure 3 shows the distribution of the standard deviation values for grades at the course section level. There are quite a few sections with grade standard deviation near zero, most likely due to
small project-based courses, where instructors often assign grades of “A”. As discussed in Section 3.1, sections with grade standard deviations below MinSD are removed since student performance
cannot be effectively measured in such cases. Threshold values of 0.1, 0.2, 0.3, 0.4, and 0.5, were evaluated before a MinSD value of 0.2 was selected; that value was selected because it ensures a
reasonable level of variance in the course grades while retaining most of the course sections.
Figure 3. Distribution of Section Grade Standard Deviations
Figure 4 shows how the number of course pairs vary, in log scale, based on the number of students in the course pair. This number is based on the students in the first course in the pair taught by a
particular instructor. The figure is used to help select the MinStudents threshold defined in Section 3.5, which removes course pairs with too few students. Most pairings have less than 100 students,
even though course pairs are aggregated over all relevant course sections; this occurs because many course pairs involve disparate courses in different disciplines. Our results are based on
MinStudents=50. A larger threshold would increase the reliability of our instructor benefit scores but eliminate too many course pairs.
Figure 4. Number of Students in Course Pairs by Instructor
© 2022 Copyright is held by the author(s). This work is distributed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license. | {"url":"https://educationaldatamining.org/EDM2022/proceedings/2022.EDM-posters.70/index.html","timestamp":"2024-11-02T17:02:05Z","content_type":"text/html","content_length":"56453","record_id":"<urn:uuid:108366f4-edc1-4007-b120-682cfcfc6a3f>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00841.warc.gz"} |
Density in context of speed of motor formula
30 Aug 2024
Title: The Role of Density in the Context of Speed: A Study on Motor Formulae
This article delves into the significance of density in the calculation of speed, particularly in the context of motor formulae. We will explore the relationship between density, mass, and volume,
and demonstrate how these concepts are intertwined in the calculation of speed. The article will also provide a comprehensive overview of the most commonly used motor formulae, including the BODMAS
(Brackets, Orders, Division, Multiplication, Addition, Subtraction) and ASCII formats.
The concept of density is crucial in understanding various physical phenomena, including the behavior of motors. Density is defined as the mass per unit volume of a substance, typically measured in
units such as grams per cubic centimeter (g/cm³). In the context of motor formulae, density plays a vital role in calculating speed.
The Relationship Between Density, Mass, and Volume:
The relationship between density, mass, and volume can be expressed mathematically using the following equation:
ρ = m / V
Where ρ is the density (in units such as g/cm³), m is the mass (in units such as grams), and V is the volume (in units such as cubic centimeters).
Motor Formulae:
Several motor formulae rely on the concept of density to calculate speed. One of the most commonly used formulae is:
v = √(2gh)
Where v is the velocity (speed) in meters per second, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height (in meters).
Using BODMAS format, this equation can be rewritten as:
v = (√(2 × g × h))
In ASCII format, this equation would appear as follows:
v = (√(2gh))
The Role of Density in Motor Formulae:
Density plays a crucial role in calculating speed using motor formulae. For example, the following formula is used to calculate the torque (τ) of a motor:
τ = k × ρ × V
Where τ is the torque (in units such as newton-meters), k is a constant dependent on the motor design, ρ is the density of the motor material, and V is the volume of the motor.
Using BODMAS format, this equation can be rewritten as:
τ = (k × ρ × V)
In ASCII format, this equation would appear as follows:
τ = (kρV)
Density plays a vital role in calculating speed using motor formulae. By understanding the relationship between density, mass, and volume, engineers can design more efficient and effective motors.
This article has demonstrated the importance of density in various motor formulae, including the calculation of velocity and torque.
1. Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
2. Serway, R. A., & Jewett, J. W. (2014). Physics for Scientists and Engineers. Cengage Learning.
ASCII Format:
v = (√(2gh)) τ = (kρV)
Note: The ASCII format is used to represent the mathematical equations in a plain text format, using symbols such as * for multiplication and ^ for exponentiation.
Related articles for ‘speed of motor formula’ :
• Reading: Density in context of speed of motor formula
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Identifying an Electron Energy Level Transition Given the Wavelength of an Absorbed Photon
Question Video: Identifying an Electron Energy Level Transition Given the Wavelength of an Absorbed Photon Physics • Third Year of Secondary School
The diagram shows the binding energy of each energy level of a hydrogen atom. If an electron is in the ground state, what energy level would it transition to if it absorbed a photon with a wavelength
of 97.4 nm? Use a value of 4.14 × 10⁻¹⁵ eV.s for the value of the Planck constant.
Video Transcript
The diagram shows the binding energy of each energy level of a hydrogen atom. If an electron is in the ground state, what energy level would it transition to if it absorbed a photon with a wavelength
of 97.4 nanometers? Use a value of 4.14 times 10 to the negative 15 electron volt seconds for the value of the Planck constant.
Here, we see several energy levels available to an electron. And we know the electron begins in the ground state or energy level one. Say it absorbs a photon. This transfers the energy of the photon
to the electron. And that amount of energy must match the energy difference between the electron’s initial level and some other level, causing the electron to transition to that other level. In other
words, when this electron transitions, the difference in binding energy between the electron’s final and initial levels must be accounted for by the energy of the absorbed photon. We call this energy
difference Δ𝐸, and it equals 𝐸 final minus 𝐸 initial.
Now, we already know 𝐸 initial from the diagram. So, if we also know the energy of the photon which accounts for Δ𝐸, we can calculate the binding energy of the level that the electron will transition
to. Then, we can match it to one of these levels, and we’ll have our answer. Put mathematically, we can solve this formula for 𝐸 final by adding 𝐸 initial to both sides. So 𝐸 initial cancels out of
the right-hand side. And writing the formula a bit more neatly, we have that Δ𝐸 plus 𝐸 initial equals 𝐸 final.
Now, we already know 𝐸 initial. And it turns out we also have all the info we need to calculate the energy of the photon, which, remember, acts as Δ𝐸. The photon has a wavelength of 97.4 nanometers.
And recall that we can relate the wavelength, 𝜆, of a photon to its energy, 𝐸, using the formula 𝐸 equals ℎ𝑐 over 𝜆, where ℎ is the Planck constant, whose value was given to us in the question
statement, and 𝑐 is the speed of light, 3.0 times 10 to the eight meters per second.
Now, substituting in the values on the right-hand side, we have that the photon’s energy equals the Planck constant times the speed of light divided by the photon’s wavelength. But notice that the
wavelength is currently expressed in nanometers. And to calculate, it should be written in plain meters. So recall that one nanometer equals 10 to the negative nine meters. And we can make a
substitution to write 𝜆 as 97.4 times 10 to the negative nine meters.
Now, calculating, we found that the energy of the photon is 12.75 electron volts, which will act as our Δ𝐸 value. So, now that we know Δ𝐸 and 𝐸 initial, we can substitute them into this formula and
find the binding energy at the electron’s final level. Thus, we have 12.75 electron volts plus negative 13.6 electron volts, which equals negative 0.85 electron volts. According to the diagram, this
value matches the binding energy at energy level four. Thus, if the electron in the ground state absorbed a photon with a wavelength of 97.4 nanometers, it would transition to energy level four. | {"url":"https://www.nagwa.com/en/videos/725160536819/","timestamp":"2024-11-03T03:38:37Z","content_type":"text/html","content_length":"255226","record_id":"<urn:uuid:f5ee717b-bc31-4c87-bbfe-0963b016778e>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00108.warc.gz"} |
Trace Training Priors
Jun 13, 2022 13:14 · 1255 words · 6 minute read
(Crossposted from the AI Alignment Forums)
I’m worried about scenarios involving deceptive models. We’ve failed at inner alignment so the model has goals that are not aligned with ours. It can somehow detect when it’s in training, and during
training it pretends to share our goals. During deployment, surprise! The model paperclips the universe.
In this story deception is all about the model having hidden behaviors that never get triggered during training. There are many inputs where the model does what we want. These correspond to inputs
where it ‘plays along’. Then on some rare input or an input outside of the training distribution it defects, triggering some catastrophic behavior that corresponds to maximizing an ulterior motive.
This suggests that we should be wary of models that have logic/subcircuits/states that are never used during training, and we can penalize models for having such hidden behaviors.
In the rest of this post I’ll flesh out a family of priors that penalize hidden behaviors, which I think provide at least a few bits of optimization away from deception. These are very much working
notes, and feedback is very welcome.
One-Gate Trace Prior: Entropy
Consider a boolean circuit. The output of each gate is 0/1. One way to implement the entropy prior is to record traces of these outputs during training, and penalize the model for having strong bias
between 0/1:
$$ p \propto e^{-\sum_i \sum_{j\in [0,1]} p_i(j) \ln p_i(j)} $$
Here $p_i$ is the observed fraction of the time gate $i$ produced output $j$ during training. I call this the one-gate trace prior because it only relies on probabilities defined over single gates.
This prior favors models where each gate has even odds of outputting 0/1.
The intuition for why this might work is that deceptive models need some machinery specifying when and how they defect, and this machinery is never active during training. This is at the core of what
makes them bad! If the defecting machinery activated during training we’d notice because the model would stop optimizing the outer goal, so we’d assign it a high loss and move on to better models.
Another framing is: It is inherently suspicious if a model has machinery that is rarely activated. That machinery wasn’t vetted during training, so we have no reason to trust it. If the model has a
hidden love of paperclips, this is the natural place to hide it.
As a technical detail, in order to do useful computation the model may need access to some constant input wires, and these should be excluded from the prior.
The one-gate trace prior isn’t normalized, and can’t be normalized without an additional prior over e.g. complexity. So in reasoning about it we need to remember that there will always be another
piece to the prior setting the distribution over circuits. I’m agnostic to the details of that prior, and view the one-gate trace prior as a regularizer we apply to other (normalized) priors to make
them safer.
Objection: Model Performance
This prior could cause a performance hit. The intuition is that sometimes a model will need to respond to unusual events even in the training distribution, so forcing the distribution over gate
outputs to be as uniform as possible could make it hard to act in environments containing rare events.
In particular, this prior disfavors modularity. Suppose the model is meant to classify images. It might be natural have separate sub-circuits for recognizing cats and dogs. If cats are much more
likely than dogs during training, our prior will disfavor models that separate that machinery out, because the ‘dog’ sub-circuit will rarely be active and the ‘cat’ sub-circuit will often be active.
If the pressure to perform well during training is strong enough this could push models towards obfuscating their calculations just to hide from the prior, which is bad.
Patch: Don’t use entropy
We can make our prior more forgiving of modularity and biased inputs by just disfavoring really rarely activations, e.g.
$$ p \propto e^{-\sum_i \sum_{j\in [0,1]} p_i(j)^m} $$
Here $m$ is a positive number, and as we make $m$ large the prior becomes indifferent to the middle of the probability distribution. For example, below I’ve plotted $-(p^{m}+(1-p)^{m})$ over [0,1]
for $m=20$.
How should we pick $m$? We just want assurances that all the gears in the model got tried out at least a few times during training. So we’re looking to see that each gate produced each output at
least $q$ times during training, where $q$ is fixed.
Suppose we have $N$ training samples. We want our prior to be flat over $[q/N,1-q/N]$ and to penalize models outside that window, which means $m \sim \log N/q$. That is, as we get more data we can
make our prior indifferent to larger windows because we’re able to observe rarer behavior.
In this framing, $q$ is a sort of safety parameter: larger $q$ means the prior is more suspicious of uneven distributions and requires that we see more examples before we trust the model.
Relation to Performance
This patch resolves the performance issues we saw using entropy. Modular circuits are totally fine under this prior so long as they all get checked out during training. Even if dog images come up
less often than cat images, they still have to come up a decent number of times for us to be confident the model has learned what a dog is, and if that number is at least $q$ then we won’t penalize
the model for having separate dog and cat circuits.
A naive way to implement this prior is to run each circuit $N$ extra times per training evaluation, and to use traces from those extra evaluations to compute the prior. This is obviously expensive if
we make $N$ large, though only linearly so.
If we’re dealing with a neural network instead of a boolean circuit we can reinforce pathways taken during training. Specifically, the first $q$ times a neuron is active (has a positive sum), and the
first $q$ times it is inactive, we reinforce all weights emerging from that neuron.
As a technical detail, activations early in training probably shouldn’t ‘count’ as exploring the model’s behavior space. That can probably be addressed by only tracking the most recent $N$ training
samples when evaluating if a given neuron has activated $q$ times.
Objection: Moving Computation
One way that deception can hide from this prior is by changing the location on the circuit where the deceptive conditional (”Am I in training or deployment?”) gets evaluated. For instance if the
model can make this conditional get evaluated in different places in the circuit depending on the inputs it receives, it is no longer helpful to compare the behavior of any single gate across
This actually seems like a very big problem, because only takes a small number of bits of model complexity to make the relevant evaluation shift around on the circuit. In a future post I’ll flesh out
this failure mode further, and consider additional patches.
Training trace priors provide a way of looking for hidden behaviors, which are signs of deception. This makes them very closely related to methods of fuzzing, where automated systems try to explore
the space of program behaviors.
The specific priors I’ve constructed above are probably not too helpful on their own, but my hope is that there is a version of a trace prior that more strongly pushes away from deception.
(Thanks to Evan Hubinger and Nicholas Schiefer for suggestions and discussions around these ideas)
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What are Bourbon Legs I What Causes Whiskey Legs
how do whiskey legs form?
What Are Bourbon Legs?
What do bourbon legs tell us about a whiskey?
What are Bourbon Legs and What do they Tell us about a Bourbon or Whiskey?
What are bourbon legs (whiskey legs, wine legs…also called tears), how do they form, what if anything, do they tell you about the whiskey before you even taste it? This question led to the following
research, tests, and conclusions. First, the questions that I had:
1. What creates bourbon legs? There are 2 theories:
□ Only the alcohol affects the legs.
□ Alcohol proof plus congeners (fatty acids, tannins, and other components from the barrel aging process) can also impact the legs, increasing the time for the legs to form and fall.
2. Does the proof of a whiskey impact the thickness, time to form, or the length of the legs?
3. Does a whiskey’s age/time in the barrel impact the legs?
4. Does a higher proof whiskey form MORE legs, and/or, do the legs from higher proof whiskeys fall more slowly than lower proof whiskeys?
So, I created a variety of tests to try to prove/disprove these questions and I have video recorded the tests so you can assess for yourself if my conclusions are correct, or horseshit! The tests are
further down the page.
What are Bourbon Legs and How do they Form?
When you swirl a glass of bourbon (or whiskey or wine), you will notice a film around the inner edge of the glass, and if you wait and watch for a bit, you will see streaming legs (sometimes called
tears) form and drip down the sides of the glass. At first, you will see a thin “crown” form at the top of the swirl. After a bit, small beads will form, and eventually those beads will stream down
the sides of the glass as legs or tears.
This is scientifically known as the Marangoni effect. Whiskey is essentially made of two components, alcohol and water. While in the bowl of the glass, they are homogenously mixed together. But when
you swirl the glass, now you have two versions of the whiskey, one with the higher alcohol content in the base, and the other with a thin layer of whiskey on the sides of the glass. Since alcohol
evaporates much faster than water, this thin layer of whiskey on the sides of the glass quickly becomes higher in water content than alcohol.
The water molecules are drawn to other water and since water has a higher surface tension than alcohol does, the water draws more strongly on the surrounding whiskey, both on the sides of the glass
and in the bowl. The Marangoni effect means that the water molecules from the base and sides of the glass will be pulled up the sides of the glass to the water at the crown (i.e. is attracted to the
higher water density at the top of the crown and away from the lower water content in the bowl).
The effect defies gravity because the water is being drawn upwards, until it becomes heavy enough to form tears (legs) and streams back down the sides of the glass.
The Tests
So, here are the tests I devised:
There is a lot of debate and differing opinions on what affects the length of time for the legs to form and how long it takes for them to begin falling from the crown. Some say that only the alcohol
proof affects the legs and others say the proof AND congeners from the barrel (in the case of whiskey/bourbon) impact the forming of the legs.
1. What Creates Bourbon Legs?
There are 2 theories:
• Only the alcohol proof affects the legs.
• Alcohol proof plus congeners (fatty acids, tannins, and other components from the barrel aging process) can also impact the legs, increasing the time for the legs to form and fall.
To evaluate these two theories I’m using the following products for the test:
Test #1: Smirnoff 100 proof Vodka -vs- Knob Creek 12-year 100 proof
This test uses two alcohol products, both with the same proof but one is highly distilled (the vodka) and the other is aged in oak barrels for 12 years. If only the proof affects the formation of the
legs, then both products should develop legs in a very similar timeframe. But if time in the barrel also affects the legs (by adding congeners) then the Knob Creek 12 year will develop legs at a
noticeably different rate.
In the video, remember to adjust the time for the vodka legs to form by about minus 3 seconds because of the time it took to swirl the glass after the bourbon. You can see beads begin to form on the
Smirnoff after about 7 seconds, and on the Knob Creek 12-year after about 17 seconds.
Do Congeners Affect Legs or is Alcohol Proof the only Factor?
Smirnoff Vodka 100 proof Knob Creek 12 yr Old 100 proof
Beads to form on Crown 7 seconds 17 seconds
Legs to Begin Falling 14 seconds 23 seconds
Legs to Fall to Bottom 32 seconds 60+ seconds
This test seems to prove that congeners (fatty acids, tannins, and other components from the barrel aging process) and time in the barrel DO have an impact on the time it takes for the legs to form
and fall. And it seems to disprove that alcohol proof is the only determining factor in the formation of bourbon legs. But you can watch the video test and decide for yourself. Please leave your
constructive observations in the comments.
2. Does the proof of a whiskey impact the thickness, time to form, or the length of the legs?
Need two whiskeys that are the same age but different proofs.
Test #2: Calumet Farms 15-year 105 proof -vs- George T Stagg 15-year 135 proof
Does Alcohol Proof Affect Formation of Bourbon Legs?
Calumet Farms 15-year 105p George T Stagg 15-year 135p
Beads to form on Crown 7 seconds 17 seconds
Legs to Begin Falling 14 seconds 23 seconds
Legs to Fall to Bottom 32 seconds 60+ seconds
Both of these bourbons are 15 years old but are separated by 30 points in terms of the proof (105 -vs- 135). This test reveals that in a higher proof bourbon the legs will definitely take longer to
form and longer to fall than in a lower proof bourbon (or whiskey).
So now we can see that both time in the barrel (and the congeners that develop) as well as the proof level of a bourbon can affect the development of the legs.
3. Does a whiskey’s age/time in the barrel impact the legs?
Need two whiskeys with the same proof but different ages.
Test #3: Jack Daniel’s Bonded 100 proof (about 4 years) -vs- Knob Creek 12-year 100 proof
Does Age (Time in the Barrel) Affect Legs?
Jack Daniel’s Bonded 100 proof Knob Creek 12 yr Old 100 proof
Beads to form on Crown 8 seconds 20 seconds
Legs to Begin Falling 14 seconds 33 seconds
Legs to Fall to Bottom 30 Seconds (left side) 50+ seconds
This test shows that time in the barrel (age) definitely impacts the legs. They are both the same proof but the 12-year aged whiskey took noticeably longer to develop the legs than the roughly 4-year
old whiskey. This is a second validation of Test #1 which strongly supports the idea that congeners do impact the formation of bourbon legs, and that it is not just the alcohol proof.
4. Does a higher proof whiskey form MORE legs, and/or, do the legs from higher proof whiskeys fall more slowly than lower proof whiskeys?
This video test is the same as #2 but with a different title screen, because the whiskeys in test #2 can answer this question as well.
Test #4: Calumet Farms 15-year 105 proof -vs- George T Stagg 15-year 135 proof (same as Test 2)
Does Alcohol Proof Affect Quantity & Falling-Speed of Whiskey Legs?
Calumet Farms 15-year 105p George T Stagg 15-year 135p
Number of Legs 10ish (1:03 timestamp) 17ish (1:36 timestamp)
Closeness of Legs Fairly spaced Tightly spaced
Legs to Begin Falling 14 seconds 23 seconds
Legs to Fall to Bottom 32 seconds 60+ seconds
By having two bourbons of the same age but with different proofs this test will determine if a higher proof bourbon’s legs develop differently than a lower proof whiskey of the same age.
The results of the test shows that higher proof whiskey does produce more tightly spaced legs that fall more slowly than lower proof whiskey.
So, going back to my original questions, the answers are:
1. What creates bourbon legs? Alcohol proof plus congeners (fatty acids, tannins, and other components from the barrel aging process) can also impact the legs, increasing the time for the legs to
form and fall.
□ The theory that only the alcohol proof affects the legs seems to have been debunked.
2. Does the proof of a whiskey impact the thickness, time to form, or the length of the legs? Yes
3. Does a whiskey’s age/time in the barrel impact the legs? Yes
4. Does a higher proof whiskey form MORE legs, and/or, do the legs from higher proof whiskeys fall more slowly than lower proof whiskeys? Yes
What do Bourbon and Whiskey Legs Tell Us?
• Quickly forming legs can indicate:
□ a whiskey with less flavor and/or mouthfeel (due to congeners or lack thereof)
• Legs that take a long time to fall can indicate (keyword here is CAN indicate, not necessarily WILL indicate!):
□ a non-chill filtered whiskey
□ perhaps a more flavorable and/or better mouthfeel whiskey
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2 hours ago
Lots of great info-thanks for sharing!
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Megaelectron-volts to Joules
Note: Fractional results are rounded to the nearest 1/64. For a more accurate answer please select 'decimal' from the options above the result.
Note: You can increase or decrease the accuracy of this answer by selecting the number of significant figures required from the options above the result.
Note: For a pure decimal result please select 'decimal' from the options above the result.
One megaelectron volt is the energy acquired by an electron when accelerated through a potential difference of 1,000,000 volts.It is equal to 1.602 × 10−13 joules.
One Joule is 1 Newton Metre, ie the work done or energy transfered to an object when a one Newton force acts on it over one metre. It can also be defined as the heat energy dissipated by a current of
one ampere passing through a one Ohm resistor for one second
Megaelectron-volts Joules
0MeV 0.00J
1MeV 0.00J
2MeV 0.00J
3MeV 0.00J
4MeV 0.00J
5MeV 0.00J
6MeV 0.00J
7MeV 0.00J
8MeV 0.00J
9MeV 0.00J
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Does PLECS have a matrix operation module? If so, can these modules be deployed to RT box2?
I want to implement the recursive least square algorithm on PLECS, which requires matrix operation, but I can’t find the block of matrix generation and operation. I hope to get a reply
If the matrix coefficients are constants, you can use the Gain block for multiplying an incoming vector signal with a matrix (the ‘Mulitplication’ parameter needs to be set to ‘Matrix (K*u)’).
PLECS does not support matrices as signal values, it only supports vectors, so therefore there is no C=A*B operation where A,B, and C are matrices.
If the coefficients are variable over time or you need to determine the product of two matrices, then you need to unwrap the matrix multiplication using Sum and Product blocks.
Another alternative is to implement matrix multiplication in a C-Script or DLL Block. The DLL Block is not supported when generating code for the RT Box, so therefore the C-Script block would be the
best choice. You will also likely need to go this route if you require computations beyond basic operations like addition/multiplication.
A good reference for matrix operations in a C-Script is the “Multistep Model Predictive Control for NPC Inverter Driving an Induction Machine” demo model in the RT Box demo model library. | {"url":"https://forum.plexim.com/t/does-plecs-have-a-matrix-operation-module-if-so-can-these-modules-be-deployed-to-rt-box2/1145","timestamp":"2024-11-02T15:42:10Z","content_type":"text/html","content_length":"24369","record_id":"<urn:uuid:18bceaa0-0d36-4e79-af0a-26abe4bd546f>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00250.warc.gz"} |
WRF Microphysics Driver: Arithmetic Intensity
The graph relates to the computation of the WRF Microphysics Driver, using single precision numbers (less data traffic across the processor-memory interface).
The arithmetic intensity of the computation (the flop/byte ratio - F/B) is shown as a function of the cache size (in single precision numbers); a virtual computer with variable cache size was used.
The y-axis also shows the roof-line extrapolated Summit performance corresponding to the given F/B ratio (note that the Summit's single precision peak is 400 PFlop/s).
The upper bundle of curves corresponds to the assumption that one EXP function computation takes the same time as 15 MPY (a realistic assumption for single precision).
The horizontal magenta line (SP SpMV) is an estimation of the F/B of a single precision of a sparse matrix - vector product, the basic part of the HPCG. The Driver data for 1 EXP = 15 MPY are not
much higher.
The left vertical Volta-100 line shows the size of cache of Nvidia Volta-100 per one core. The right vertical line is the size of all caches of Volta-100 in one package. Note that the grid sizes used
in the experiment are quite small for recent standards, larger grids would shift the increase of the F/B to much larger cache sizes.
Consequently, the F/B of the Microphysics driver, using caches of realistic sizes, is almost as low as the F/B of the HPCG, thus making LOWAIN a promising architecture for WRF (and NWP) computations. | {"url":"https://lowain.org/WRF/WRFMicrophysics.html","timestamp":"2024-11-08T01:53:28Z","content_type":"text/html","content_length":"2336","record_id":"<urn:uuid:53aaf63f-6ce8-4cbe-9136-def477b5d163>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00146.warc.gz"} |
USU Internal Contest March'2001
The sergeant ordered that all the recruits stand in rows. The recruits have formed K rows with N people in each, but failed to stand according to their height. The right way to stand in a row is as
following: the first soldier must be the highest, the second must be the second highest and so on; the last soldier in a row must be the shortest. In order to teach the young people how to form rows,
the sergeant ordered that each of the recruits jump as many times as there are recruits before him in his row who are shorter than he. Note that there are no two recruits of the same height.
The sergeant wants to find which of the rows will jump the greatest total number of times in order to send this row to work in the kitchen. Help the sergeant to find this row.
The first line contains integers N and K (2 ≤ N ≤ 10000; 1 ≤ K ≤ 20). Each of the following K lines contains N different integers from 1 to N. The recruits in each row are numbered according to their
height (1 — the highest, N — the shortest). Each line shows the order in which the recruits stand in the corresponding row. The first integer in a line is the number of the first recruit in a row and
so on. Therefore a recruit jumps as many times as there are numbers which are greater than his number in the line before this number.
You should output the number of the row in which the total amount of jumps is the greatest. If there are several rows with the maximal total amount of jumps you should output the minimal of their
input output
Problem Author: Nikita Shamgunov
Problem Source: USU Open Collegiate Programming Contest March'2001 Senior Session | {"url":"https://timus.online/problem.aspx?space=25&num=1","timestamp":"2024-11-14T17:42:55Z","content_type":"text/html","content_length":"6844","record_id":"<urn:uuid:53c73117-fb46-411c-9da6-d98879167b2a>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00276.warc.gz"} |
How do you graph xy = -8? | Socratic
How do you graph #xy = -8#?
1 Answer
The graph is a rectangular hyperbola, with the two branches in Q2 and Q4. The asymptotes of the hyperbola are the axes of coordinates.
This rectangular hyperbola can be obtained by rotating the
rectangular hyperbola $x y = 8$ about the origin through a right
In Q2 and Q4, xy < 0. So the branches of the given
rectangular hyperbola lie in Q2 and Q4,
The asymptotes are x = 0 and y =0, so that,
when $x \to 0 , y \to \pm \infty$ and when $y \to 0 , x \to \pm \infty$.
The points that are closest to the center C at the origin are the
vertices $V \left(2 \sqrt{2} , - 2 \sqrt{2}\right) \mathmr{and} V ' \left(- 2 \sqrt{2} , 2 \sqrt{2}\right)$.
The eccentricity e = sqrt 2, for a rectangular hyperbola.
The transverse axis length 2a = 8.
The foci are at $S \left(4 , - 4\right) \mathmr{and} S ' \left(- 4 , 4\right)$. .
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If the temperature a substance increases from 20 to 35 Celsius, what is the total temperature change on the Kelvin scale? | Socratic
If the temperature a substance increases from 20 to 35 Celsius, what is the total temperature change on the Kelvin scale?
1 Answer
The change is an increase of 15 Kelvin.
Calculating changes in Celsius and Kelvin will reveal the same magnitude. If the change is +15 degrees Celsius, the change is +15 Kelvin (note that it's not degrees Kelvin, just simply Kelvin).
You can use the conversion, which is 273.15 Kelvin = zero degrees Celsius. Since it's a relative scale, ${1}^{0}$ C = 274.15 K, and ${2}^{0}$ C = 275.15 and so forth. This isn't really useful in your
calculation, because you're just trying to evaluate the total temperature change, which will always be equal when comparing Celsius and Kelvin. It gets a little trickier when involving Fahrenheit
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The Popper-Miller argument against probabilistic induction
Let us define the support that the hypothesis receives from the evidence as the increase in its probability: . Popper & Miller [1987, p. 574] prove that this support function may be split into two
$$\label{one}s(h|e)=s(h\lor e|e)+s(h\leftarrow e | e).$$
It is easy to evaluate these terms, which shows that the first term is always positive (or 0):
$$\label{dedsup} s(h\lor e|e)= 1-p(h \lor e),$$
while the second term is always negative (or 0):
$$\label{indneg} s(h\leftarrow e | e)=- \left ( 1-p(e) \right)\left ( 1-p(h|e) \right).$$
From here Gillies [1986, p. 111] is lured into concluding the refutation of induction as follows:
"Since follows logically from , must represent purely deductive support. So, if there is such a thing as inductive support in the Bayesian sense, it must be contained in the term . However ... this
term is always negative. It therefore follows that there cannot be inductive support of the kind that the Bayesians postulate."
But to say that " must represent purely deductive support" is naive. As Townsend [1989, pp. 493f.] points out, it is clear from (2) that such a stipulation has absurd consequences, namely that any
hypothesis is deductively supported by any evidence (unless ). In particular, a contradictory hypothesis receives deductive support, as does a hypothesis which is directly contradicted by the
Popper and Miller are not as naive as Gillies. They never explicitly identify with deductive support. How, then, do they conclude their proof that there can be no inductive support? They are actually
alluding to two different ways of completing the proof. Let me begin with the first of these, which is related to Gillies'. In fact, while this approach appears more sophisticated, I shall argue that
it is still susceptible to the same criticism.
Note first that "unless happens to be deductively independent from , the value of is deductively contaminated." (This and the following quotations are all from Popper & Miller [1987, p. 574]. They
are insubstantially edited.) That is to say: if and have common consequences, then will be supported by having these consequences confirmed. But this is deduction, not induction: the evidence does
not point beyond itself. But is presumably inductively contaminated as well. How are we to disentangle one contaminant from the other? Gillies' simplistic approach will not do, as we know.
Nevertheless, one feels that is the key. "If there is such a thing as pure inductive dependence at all, there seems nothing for it but to measure it by something like ," which is not deductively
contaminated because nothing (nontautological) can be derived from both and , so there can be no deductive support in this case. We already know that is negative by (\ref{indneg}), so it is tempting
to jump to the conclusion that inductive support is negative. But this would be too hasty: how do we know that the inductive contamination of is negative simply because is negative? To solve this
problem Popper and Miller make this very convenient assumption:
(A) "If there were to be some genuinely inductive dependence [i.e. inductive contribution to ] between a hypothesis and some evidence , it could hardly change if were replaced by some hypothesis
equivalent to (given , or equivalent to in the presence of )."
Indeed, is equivalent to in the presence of , so we have the desired result: "Inductive dependence is counterdependence," because it is so in the pure case and thus also in all other cases by (A).
But isn't (A) just Gillies in disguise? Let us investigate the case where the evidence refutes the hypothesis, which was so embarrassing to Gillies. If we take for our hypothesis we find that and .
Since in general , this means that . According to (A) the inductive support should be the same in both cases, so apparently the difference must be due to some positive deductive support for , which
is absurd. This refutation of (A) is simpler than the only other refutation of (A) that I have found, namely that of Rodriguez [1987, pp. 356f.]. It seems so simple, indeed, that I cannot help but
suspect that I am missing something in my interpretation of (A). But I can find no indication of this in Popper and Miller. They also follow the statement of (A) by saying "This much has recently
been argued at length by Levi [1986]." This is an extremely leisurely piece without a single formula in it which indeed seems to embrace the simple interpretation of (A) that I have used (esp. p.
It seems that Popper and Miller were themselves aware of the dubious status of (A), for later they offered a second (more famous) proof which avoids it. The idea of this proof is:
(B) is the part of that goes beyond .
Here I am using the shorthand of identifying with its content, i.e., the set of all its (nontautological) logical consequences. From (B) Popper and Miller claim to prove their theorem roughly as
follows: inductive support means support that goes beyond , but the part that goes beyond always has negative support by (3). QED.
Popper and Miller justify (B) thus: " is just what needs to be added to , without duplicating anything already there, in order to yield the system ." The problem is that in general most of is in
neither nor : such propositions can be derived only when the two are pooled. And yet such propositions clearly seem to go beyond , so it seems that (B) patently fails to capture the part that goes
beyond . Indeed, (B) "has been almost uniformly rejected" (Howson & Franklin [1994, p. 452]), primarily for this reason, and a number of convincing refutations of (B) using such propositions have
been given. These examples span the range from simplistic logical ones ( itself), to ravens, to actual scientific ones.
Such examples are summarised in Popper & Miller [1987, p. 580]. Their reply is revealing. They begin by admitting that "the facts are of course as stated." But then they insist that "the objection is
devoid of merit" (p. 581). And why? Because "the objection misses its mark: it fails to show that positive probabilistic dependence can be achieved in the absence of some degree of deductive
dependence"! Of course there can be no positive probabilistic dependence without some degree of deductive dependence. That is an immediate corollary of (1). Popper and Miller are trying to defuse a
correct rebuttal of (B) by claiming that "its mark" was not (B) at all but rather a completely different theorem, and one with an elementary proof.
A second diversion they employ is to try to shift the burden of proof onto their critics by challenging them to characterise the excess content of in a better way: "that, we feel, is a problem for
those who believe in induction, not for us." (Miller [1990, p. 151]; cf. Popper & Miller [1987, p. 581]. This challenge has been taken up by Mura [1990] and Elby [1994].) Perhaps, but this does not
change the fact that Popper and Miller's "proof" that deductive support is countersupport is only as good as their characterisation of excess content. One must conclude that they use these rather
dishonest tricks because they have no convincing arguments in defence of (B). (Even their own disciple, Gillies [1986], apparently wished to avoid (B), which was his motivation for giving the foolish
argument discussed above.)
It may seem obvious that (B) is the weak point in the Popper-Miller argument, but actually quite a few attempted refutations mistakenly charge at other parts of it. I want to discuss the attempt of
Good [1990]. Note first that it is not necessary to identify with the part of that does not go beyond . To do so is a natural complement to (B), especially in light of (1), and Popper and Miller do
indeed use this mode of expression. But as I outlined above one may prove that inductive support is countersupport from (B) alone. Thus it is not quite fair to formulate the Popper-Miller argument as
if it depends crucially on this additional assumption, and then go on to reject it for this reason. And yet this is what Good [1990] does, for example. Here is what he calls a "suspicious feature" of
the Popper-Miller argument:
"Whatever may be, whether it is supported or completely refuted by , or if it has nothing at all to do with , it remains true that a 'part' of , namely , is deducible from ."
Besides missing the core of Popper-Miller, this argument does not appear to me to be as "suspicious" as it first looks. If and are unrelated, then nothing nontrivial can be derived from both and
separately, which means that nothing can be derived from , which means that the content of is in effect empty. Thus there is nothing suspicious about this case. And why is it suspicious that part of
is deducible from even when refutes ? Take the simple case and . The part of deducible from is of course . In terms of content: . Hardly very suspicious.
Good [1990] continues:
"[If consists of these two parts,] Does this then put us under an obligation, whenever refutes , to express this refutation by saying that refutes , thus removing from the part that supports? I don't
believe that Popper and Miller would use such an absurd mode of expression."
But this mode of expression is not absurd. It would in fact be perfectly sensible if (B) was correct. Consider an example: my hypothesis is Euclidean geometry , where I have divided its assumptions
into the parallel postulate and the other axioms . Suppose the evidence tells us that space is non-Euclidean, that is: . Now is refuted, but this does not mean that we should throw out all of . On
the contrary, remains intact. To say that what is falsified is the part that goes beyond is to say that what is falsified is precisely the part of Euclidean geometry that can only be proved using the
parallel postulate . The remaining part , which supports, includes the first handful propositions of Euclid's Elements. It should indeed be preserved. The problem for Popper-Miller is again (B):
Pythagoras' theorem, for example, is a consequence of that clearly goes beyond ; and yet it is obviously not in . Thus Good should have attacked (B) instead of this "mode of expression," which would
in fact be correct if (B) was correct.
Chihara, C. S. & Gillies, D. A. (1988). An Interchange on the Popper–Miller Argument. Philosophical Studies, 54, 1–8.
Elby, A. (1994). Contentious Contents: For Inductive Probability. The British Journal for the Philosophy of Science, 45(1), 193–200.
Gillies, D. (1986). In Defense of the Popper–Miller Argument. Philosophy of Science, 53(1), 110–113.
Good, I. J. (1987). A Reinstatement, in Response to Gillies, of Redhead’s Argument in Support of Induction. Philosophy of Science, 54(3), 470–472.
Good, I. J. (1990). Discussion: A Suspicious Feature of the Popper/Miller Argument. Philosophy of Science, 57, 535–536.
Howson, C. & Franklin, A. (1994). Bayesian Conditionalization and Probability Kinematics. The British Journal for the Philosophy of Science, 45(2), 451–466.
Levi, I. (1986). Probabilistic pettifoggery. Erkenntnis, 25(2), 133–140.
Miller, D. W. (1990). Reply to Zwirn & Zwirn. Cahiers du CREA, 14, 149–153.
Mura, A. (1990). When Probabilistic Support Is Inductive. Philosophy of Science, 57(2), 278–289.
Popper, K. R. & Miller, D. W. (1987). Why Probabilistic Support is Not Inductive. Philosophical Transactions of the Royal Society of London, A321, 569–596.
Rodriguez, A. R. (1987). On Popper–Miller’s proof of the impossibility of inductive probability. Erkenntnis, 27, 353–357.
Townsend, B. (1989). Partly Deductive Support in the Popper-Miller Argument. Philosophy of Science, 56(3), 490–496. | {"url":"https://intellectualmathematics.com/blog/the-popper-miller-argument-against-probabilistic-induction/","timestamp":"2024-11-12T23:40:11Z","content_type":"text/html","content_length":"80182","record_id":"<urn:uuid:20255698-1b88-45d5-b3e6-938baaf6f4ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00298.warc.gz"} |
Fixed Asset Turnover Overview, Formula, Ratio and Examples
A higher ratio indicates that the company is utilizing its assets efficiently to generate sales, which is generally seen as a positive sign. The Net Asset Turnover Ratio measures how effectively a
company generates sales from its net assets. Net assets refer to total assets minus total liabilities, representing the shareholders’ equity or the portion of assets owned by shareholders. This ratio
provides a broader view of asset utilization since it considers both fixed assets and current assets. The asset turnover ratio, also known as the total asset turnover ratio, measures the efficiency
with which a company uses its assets to produce sales. The asset turnover ratio formula is equal to net sales divided by the total or average assets of a company.
1. A higher ratio indicates efficient utilization of fixed and current assets to generate sales.
2. Alternatively, a company can gain insight into their competitors by evaluating how their fixed asset ratio compares to others.
3. It suggests that the company is effectively deploying its long-term assets to drive revenue generation.
4. All of these are depreciated from the initial asset value periodically until they reach the end of their usefulness or are retired.
5. In contrast, the total asset version encompasses all assets employed by the company, including both fixed and current assets.
The fixed asset turnover ratio (FAT) is, in general, used by analysts to measure operating performance. The asset turnover ratio measures how effectively a company uses its assets to generate revenue
or sales. The ratio compares the dollar amount of sales or revenues to the company’s total assets to measure the efficiency of the company’s operations. The fixed asset turnover ratio is useful in
determining whether a company is efficiently using its fixed assets to drive net sales.
Check out our debt to asset ratio calculator and fixed asset turnover ratio calculator to understand more on this topic. By adding the two asset values and then dividing by 2, https://
intuit-payroll.org/ you get the average value of the assets over the course of the year. This is then compared to the total annual sales or revenue, which can be found on the income statement.
To reiterate from earlier, the average turnover ratio varies significantly across different sectors, so it makes the most sense for only ratios of companies in the same or comparable sectors to be
benchmarked. Hence, it is often used as a proxy for how efficiently a company has invested in long-term assets. The turnover metric falls short, however, in being distorted by significant one-time
capital expenditures (Capex) and asset sales. All of these categories should be closely managed to improve the asset turnover ratio.
The Asset Turnover Ratio is calculated by dividing the company’s revenue by its average total assets during a certain period. The asset turnover ratio gauges a company’s asset efficiency in
generating revenue, comparing sales to total assets annually. A variation, the Fixed Asset Turnover (FAT) ratio, considers only a company’s fixed assets. Instead of dividing net sales by total
assets, the fixed asset turnover divides net sales by only fixed assets.
Asset Turnover Ratio Calculator
A higher ATR signifies a company’s exceptional ability to generate significant revenue using a relatively smaller pool of assets. For optimal use, it is best employed for comparing companies within
the same industry, providing valuable insights into their operational efficiency and revenue generation capabilities. Therefore, the ratio fails to tell analysts whether or not a company is even
Is It Better to Have a High or Low Asset Turnover?
The Asset Turnover Ratio is a performance measure used to understand the efficiency of a company in using its assets to generate revenue. It measures how effectively a company is managing its assets
to produce sales and is a key indicator of operational efficiency. A higher ratio suggests that the company is using its assets more effectively to generate revenue.
Comparisons of Ratios
The asset turnover ratio provides valuable insights into how effectively a company utilizes its assets to generate revenue. Therefore, comprehending and interpreting this ratio is crucial for
students interested in corporate finance. This article will delve into the asset turnover ratio, its calculation, interpretation, and significance in financial analysis. Companies with higher fixed
asset turnover ratios earn more money for every dollar they’ve invested in fixed assets. Therefore, there is no single benchmark all companies can use as their target fixed asset turnover ratio.
Instead, companies should evaluate what the industry average is and what their competitor’s fixed asset turnover ratios are.
On the other hand, a low asset turnover ratio could indicate inefficiency in using assets, suggesting problems with the company’s inventory management, sales generation, or asset acquisition
strategies. It could also mean that the company is asset-heavy and may not be generating adequate revenue relative to the assets it owns. The asset turnover ratio assesses a company’s efficiency in
using assets for sales generation, while return on assets (ROA) gauges its efficiency in generating profits with assets. ATR focuses on operational efficiency, whereas ROA encompasses both
operational efficiency and profitability. Lastly, by combining the asset turnover ratio with DuPont analysis, investors and analysts can gain a comprehensive understanding of a company’s financial
The asset turnover ratio for each company is calculated as net sales divided by average total assets. Generally, a higher ratio is favored because it implies that the company is efficient in
generating sales or revenues from its asset base. A lower ratio indicates that a company is not using its assets efficiently and may have internal problems. The asset turnover ratio tends to be
higher for companies in certain sectors than in others. Retail and consumer staples, for example, have relatively small asset bases but have high sales volume—thus, they have the highest average
asset turnover ratio. Conversely, firms in sectors such as utilities and real estate have large asset bases and low asset turnover.
One of the key metrics used to measure this efficiency is the Asset Turnover Ratio. This financial ratio gives an insight into how well a company is using its assets to generate revenue. It serves as
an indicator of the company’s operational efficiency and can be particularly telling in comparison i9 processor list with competitors within the same industry. The asset turnover ratio measures the
value of a company’s sales or revenues relative to the value of its assets. The asset turnover ratio can be used as an indicator of the efficiency with which a company is using its assets to generate
It’s crucial to be consistent with the time periods for both net sales and total assets when calculating this ratio. If you’re looking at net sales for the year, make sure to use the total assets at
the start and end of the same year to calculate the average. For example, retail companies have high sales and low assets, hence will have a high total asset turnover. On the other hand,
Telecommunications, Media & Technology (TMT) may have a low total asset turnover due to their high asset base. Thus, it is important to compare the total asset turnover against a company’s peers.
However, a very high ratio could also indicate underinvestment in fixed assets, which may impact future growth prospects or operational capacity. To do so, divide the company’s net sales (or total
revenue) by its average total assets formula during a specific period. Net sales represent a company’s total sales revenue after deducting returns, discounts, and allowances. Average total assets are
the average value of a company’s total assets over a specific period, usually calculated by taking the average of the beginning and ending asset balances.
This variation isolates how efficiently a company is using its capital expenditures, machinery, and heavy equipment to generate revenue. The fixed asset turnover ratio focuses on the long-term
outlook of a company as it focuses on how well long-term investments in operations are performing. Investors, analysts, lenders, management, industry peers, financial consultants, and regulators use
this metric to gain insight into a company’s operational efficiency and asset utilization. Asset Turnover Ratio is a fundamental metric that plays a crucial role in assessing a company’s operational
efficiency and overall financial health. It measures how effectively a company utilizes its assets to generate sales revenue. The total asset turnover formula ratio measures a company’s ability to
generate revenue or sales in relation to its total assets.
Overall, investments in fixed assets tend to represent the largest component of the company’s total assets. The FAT ratio, calculated annually, is constructed to reflect how efficiently a company, or
more specifically, the company’s management team, has used these substantial assets to generate revenue for the firm. The asset turnover ratio is calculated by dividing net sales or revenue by
average total assets. In the financial world, understanding a company’s efficiency in utilizing its assets is crucial for investors, analysts, and the company’s management.
It can be used to compare how a company is performing compared to its competitors, the rest of the industry, or its past performance. Average total assets are found by taking the average of the
beginning and ending assets of the period being analyzed. The standard asset turnover ratio considers all asset classes including current assets, long-term assets, and other assets. Like other
financial ratios, the fixed ratio turnover ratio is only useful as a comparative tool. For instance, a company will gain the most insight when the fixed asset ratio is compared over time to see the
trend of how the company is doing. | {"url":"https://dutadamainusatenggarabarat.id/fixed-asset-turnover-overview-formula-ratio-and/","timestamp":"2024-11-05T22:44:33Z","content_type":"text/html","content_length":"102817","record_id":"<urn:uuid:faa614b3-442e-4749-9274-4ece11348666>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00289.warc.gz"} |
2318: Dynamic Entropy
Dynamic Entropy
Title text: Despite years of effort by my physics professors to normalize it, deep down I remain convinced that 'dynamical' is not really a word.
This is another one of Randall's Tips, this time a Science Tip. This time it is a bit special since it came less than three weeks after another Science Tip: 2311: Confidence Interval (which was
itself the first time that a non-Protip Tip type has been re-used). This is the first time a type of tip (that was not a Protip) has been used for two "tips comics" in a row.
This Science Tip suggests that if you have a cool new concept, you should call it dynamic entropy.
Dynamic programming is a mathematical optimization method and computer programming method developed by Richard Bellman in the 1950s. The History section of the Wikipedia article contains the full
paragraph from Bellman's autobiography that contains the quote that is in the comic strip. Bellman describes how he was doing mathematical research funded by the military at a time when the Secretary
of Defense had a literal pathological fear of the word "research", and by extension, "mathematical". Bellman borrowed the word "dynamic" from physics as being both accurate for his work and as a word
that in plain English has positive connotations and is never used in a pejorative sense (expressing contempt or disapproval). The word "dynamic" itself comes from the Greek dynamikos, "powerful",
which is a positive meaning in itself, and has been applied to topics in physics that are related to motion and forces and used in ordinary English to refer to things that exert power, force, growth,
and change (dynamo, dynamite, and as an adjective). Even though those things aren't always good, when they're bad, we use other words instead (e.g. cancer undergoes metastasis, not "dynamism").
Entropy is a term from physics, specifically statistical mechanics, describing a property of a thermodynamic system. When Claude Shannon developed a mathematical framework for studying signal
processing and communications systems, which became known as Information theory, he struggled to come up with a proper name for one mathematical concept in his theory that quantified amount of noise
or uncertainty in a signal. Computer scientist John von Neumann noticed the similarity of the equations with some in thermodynamics and suggested, "You should call it entropy, for two reasons. In the
first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy
really is, so in a debate you will always have the advantage." (see History of information theory). The following is an excerpt from the explanation of 1862: Particle Properties:
The term "entropy", which began as a thermodynamic measure, has since been adopted by analogy into multiple seemingly unrelated domains including, for example, information theory. The table
allows that the term "entropy" must mean something in the context of particle physics, but isn't certain whether it's the classical, Gibbs' modern statistical mechanics, Von Neumann's quantum
entropy, or some other meaning.
In classical thermodynamics, entropy is a macroscopic property describing the disorder or randomness of a system with many particles. However, in statistical mechanics and quantum mechanics, the
concept of entropy can also be applied to single particles under certain conditions. If the particle's position is not precisely known and can be described by a probability distribution, this
contributes to entropy. Similarly, if the particle's momentum is uncertain and described probabilistically, this also contributes to entropy. A single quantum particle in a pure state (e.g., an
electron in a specific atomic orbital) has zero entropy. This is because there is no uncertainty about the state of the system. If the single particle's state is described by a density matrix
representing a mixed state (a probabilistic mixture of several possible states), the Von Neumann entropy can quantify the degree of uncertainty or mixedness of the state.
Imagine two identical balloons filled with the same gas and heated from two opposite sides with identical heat sources, creating symmetric temperature gradients in both; because the distribution
of temperatures is the same, the Gibbs statistical thermodynamic entropy ๐ of the gas molecule particles in each balloon will be the same. In contrast, if one balloon is heated by a low-power
heat source and another by an otherwise identical high-power heat source, the balloon next to the high-power heat source will have a steeper temperature gradient, increasing the number of
accessible microstates, so the Gibbs entropy ๐ [low_power] < ๐ [high_power]. Now consider electrons in two atoms excited by absorbing identical photons to a mixed state; if the mixed states
have the same probabilities for different energy levels, their Von Neumann quantum entropy ๐ values will be the same. Conversely, if one atom has electrons excited to a pure state and another
to a mixed state by photons of different energies, the mixed state will have higher entropy due to greater uncertainty, i.e., ๐ [pure] = 0 and 0 < ๐ [mixed] โ ค ln(2).
The naming of dynamic programming and of entropy in information theory are both examples of scientists choosing a name for what were at least partially very non-scientific seeming reasons. In one
case because it has only positive and no negative connotations in plain English. In the other case because there is much confusion over the meaning of the word so Shannon would be free to adopt it in
a new context. Randall is claiming that would make them great to put together to name some new concept; the combination will mean whatever the creator wants it to mean (even able to change
mid-debate), and never sound bad the way that e.g. cold fusion has come to be.
Even though the caption implies that "dynamic entropy" would be available as a new name, it has actually been used in physics^[1], probability^[2], computer science^[3], and even the term "dynamical
entropy" in physics^[4]^[5] and bioscience^[6].
In the title text Randall mentions that, even though his physics professors have continued to use the word "dynamical", "trying to normalize it" by repetitive usage, he remains convinced that it is
not really a word. Presumably he doesn't like that it has two suffixes used to make words into adjectives, -ic and -al, as if "dynamic" wasn't already positive enough. The Free Dictionary discusses
how -ic and -ical suffixes are confused in many common words and explains their different uses.
The term "dynamical" in physics generally is used in "Dynamical system" or as an adjective to name a concept as applied to dynamical systems such as "dynamical entropy"^[7].
[One panel only with text and a few lines and arrows. There are two columns each with a heading. Beneath each heading is a quote written on four lines. Below the quote, in grey font, and
indented, starting with a hyphen, with the text aligned to the right of this are five lines of text. This explains who the quote belongs to and where it was stated (in brackets at the end). From
the bottom of each of these two gray text paragraphs gray curved arrows goes down to two gray lines. Below each of these two lines are one large word per line. They are again in black text.]
"It's impossible to use the word 'dynamic' in the pejorative sense... Thus, I thought 'Dynamic Programming' was a good name."
- Richard Bellman, explaining how he picked a name for his math research to try to protect it from criticism (Eye of the Hurricane, 1984)
"You should call it 'Entropy'... No one knows what entropy really is, so in a debate you will always have the advantage."
- John von Neumann, to Claude Shannon, on why he should borrow the physics term in information theory (as told to Myron Tribus)
Dynamic Entropy
[Caption below the panel:]
Science Tip: If you have a cool concept you need a name for, try "Dynamic Entropy."
Many of Buckminster Fuller's designs and works were associated with the word "dymaxion", a combination of the words "dynamic", "maximum", and "tension", all words that Fuller himself used a lot in
talking about his work, and which are words that simultaneously have use in science and positive connotations in lay English.
add a comment! ⋅ add a topic (use sparingly)! ⋅ refresh comments!
Can confirm, have never lost an argument. Dynamic Entropy (talk) 00:45, 11 June 2020 (UTC)
You came here for an argument? No, this here is abuse. Argument's next door. Kev (talk) 10:26, 13 June 2020 (UTC)
Allegrini, P., Douglas, J. F., & Glotzer, S. C. (1999). Dynamic entropy as a measure of caging and persistent particle motion in supercooled liquids. Physical Review E, 60(5), 5714, doi: 10.1103/
Asadi, M., Ebrahimi, N., Hamedani, G., & Soofi, E. (2004). Maximum Dynamic Entropy Models. Journal of Applied Probability, 41(2), 379-390. Retrieved June 11, 2020, from www.jstor.org/stable/
Green, J. R., Costa, A. B., Grzybowski, B. A., & Szleifer, I. (2013). Relationship between dynamical entropy and energy dissipation far from thermodynamic equilibrium. Proceedings of the National
Academy of Sciences, 110(41), 16339-16343.
S. Satpathy et al., "An All-Digital Unified Static/Dynamic Entropy Generator Featuring Self-Calibrating Hierarchical Von Neumann Extraction for Secure Privacy-Preserving Mutual Authentication in
IoT Mote Platforms," 2018 IEEE Symposium on VLSI Circuits, Honolulu, HI, 2018, pp. 169-170, doi: 10.1109/VLSIC.2018.8502369.
Bugstomper (talk) 01:28, 11 June 2020 (UTC)
Can someone with knowledge of the reference system in a wiki make the reference appear above the discussion, maybe in a section named References?--Kynde (talk) 07:06, 11 June 2020 (UTC)
Done Bugstomper (talk) 09:13, 11 June 2020 (UTC)
Thanks. Could not find the correct way to do it, so nice my call was answered quickly. --Kynde (talk) 11:42, 12 June 2020 (UTC)
Well bugger me (METAPHOR! METAPHOR!) but my current Master thesis in Computer Science could use that term without much shoehorning. (tl;dr: Binary search trees that adapt, =dynamic, can serve a query
series faster than static, and the gain depends on the structure of the query series, =entropy. I prefer the good old "instance optimality", though...) 162.158.159.122 08:58, 11 June 2020 (UTC)
This seems to tie in with the recent comic 2315: Eventual Consistency, which is also about entropy (in a thermodynamic(al) sense), but I guess that like the rest of the world I don't know what
entropy really is, because if entropy is a measure of how "surprising" a variable is, why is everything being flat and spread out evenly called a state of maximum entropy? Everything being the same
doesn't sound very surprising to me... --IByte (talk) 09:08, 11 June 2020 (UTC)
Because entropy is the inverse of how "surprising" or organized or full of information a system is. Bugstomper (talk) 09:27, 11 June 2020 (UTC)
It's not about "surprising" but about an even spread of probability, so for matter a complex molecule has less entropy than a smaller molecule because the atoms are held in place, and if the
quarks in the atoms aren't even held together in subatomic particles then that is ultimate entropy. For information having more possible choices for the message or password spreads the
probability of any one occurrence around more possibilities. If it is narrowly defined it has low entropy because the probability is concentrated in a few items.162.158.75.104 13:53, 11 June
2020 (UTC)
I can think of a number of cases where "dynamic" would be a bad thing, but not necessarily pejorative. The structure of a building had better not be dynamic (think "sudden energetic disassembly"),
and when my (salaried, should be steady) paycheck becomes dynamic, I have to talk to HR. Can someone come up with a pejorative? 162.158.78.212 11:16, 11 June 2020 (UTC)
2050 retro slang: Dynamic, variant of Dienamic, portmanteau of Die and (Viet) Nam + adjectival suffix // Dienam (n): a long, brutal slog with an unsatisfying ending, possibly with unintended
consequences arising some time after the conclusion (cv Agent Orange) "Just like the soldiers a century ago, they knew the project was a dienam -- resources would be wasted, careers ended, and
goals unmet -- but they couldn't convince the executives to abandon the project." // Some[thing] that is a failure, many years in the making, from which no success can be extricated. "[The
development team] was working on a dynamic project for the studio when [the studio] finally was forced to declare bankruptcy and shutter due to literally years of overcomitting and
underdelivering." 162.158.75.104 (talk) (please sign your comments with ~~~~)
As there exists such a thing as resilience by explicitness, when it comes to type system discussions in computer programming language design the term 'dynamic' can be pretty condemning.
141.101.76.204 (talk) (please sign your comments with ~~~~)
(Looks like people aren't ~~~~ing. My contribution starts here.) I can just see a reporter, at the scene of some breaking news, saying "It's very much a dynamic situation here", while dodging various
rioter/police missiles, hurricane debris, moving away from a wildfire front or in the middle of a rescue situation in post-earthquake aftermath yet still amidst aftershocks. 141.101.107.164 05:28, 12
June 2020 (UTC)
The 'Use โ -icโ with nouns ending in โ -icsโ (usually)' section of that Free Dictionary article is relevant here, since dynamics is a thing. 172.69.62.148 15:28, 12 June 2020 (UTC)
Dynamic gravity, dynamic oxygen level, dynamic surgical compentency, dynamic astronomical unit ... none of these are desireable. These Are Not The Comments You Are Looking For (talk) 20:18, 15 June
2020 (UTC) | {"url":"https://explainxkcd.com/wiki/index.php/2318:_Dynamic_Entropy","timestamp":"2024-11-03T13:24:28Z","content_type":"text/html","content_length":"50347","record_id":"<urn:uuid:704e81f3-ea2c-4d53-80b7-96d3c8fc5d4f>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00235.warc.gz"} |
Are tensors just matrices?
Asked by: Estella Kunde II Score: 5/5
71 votes
A tensor is often thought of as a generalized matrix. ... Any rank-2 tensor can be represented as a matrix, but not every matrix is really a rank-2 tensor. The numerical values of a tensor's matrix
representation depend on what transformation rules have been applied to the entire system.
Why are tensors not matrices?
The tensor is not that matrix, because different types of tensors can correspond to the same matrix. The differences between those tensor types are uncovered by the basis transformations (hence the
physicist's definition: "A tensor is what transforms like a tensor").
What exactly is a tensor?
In simple terms, a tensor is a dimensional data structure. Vectors are one-dimensional data structures and matrices are two-dimensional data structures. ... For instance, we can represent second-rank
tensors as matrices. This stress on "can be" is important because tensors have properties that not all matrices will have.
Is a matrix A 2d tensor?
A matrix is a two dimensional array of numbers (or values from some field or ring). A 2-rank tensor is a linear map from two vector spaces, over some field such as the real numbers, to that field.
What are tensors matrices?
A tensor is a container which can house data in N dimensions. Often and erroneously used interchangeably with the matrix (which is specifically a 2-dimensional tensor), tensors are generalizations of
matrices to N-dimensional space. Mathematically speaking, tensors are more than simply a data container, however.
What’s the difference between a TENSOR and a MATRIX?
28 related questions found
Who invented tensors?
Born on 12 January 1853 in Lugo in what is now Italy, Gregorio Ricci-Curbastro was a mathematician best known as the inventor of tensor calculus.
What is a tensor vs a matrix?
In a defined system, a matrix is just a container for entries and it doesn't change if any change occurs in the system, whereas a tensor is an entity in the system that interacts with other entities
in a system and changes its values when other values change.
Are all rank 2 tensors matrices?
Any rank-2 tensor can be represented as a matrix, but not every matrix is really a rank-2 tensor. The numerical values of a tensor's matrix representation depend on what transformation rules have
been applied to the entire system.
Are all scalars tensors?
However, all scalars are not tensors and all tensors of rank 0 are scalars. The same applies to vectors, dyads or triads. A vector dyad product results in a dyad which would make it possible to alter
the direction of a vector and also an increase from rank1 to rank2 tensor.
What is a 3 dimensional tensor?
A tensor with one dimension can be thought of as a vector, a tensor with two dimensions as a matrix and a tensor with three dimensions can be thought of as a cuboid. The number of dimensions a tensor
has is called its rank and the length in each dimension describes its shape .
What is tensor example?
A tensor field has a tensor corresponding to each point space. An example is the stress on a material, such as a construction beam in a bridge. Other examples of tensors include the strain tensor,
the conductivity tensor, and the inertia tensor.
Why do we use tensor?
Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid
mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic ...
Are vectors tensors?
In fact tensors are merely a generalisation of scalars and vectors; a scalar is a zero rank tensor, and a vector is a first rank tensor. The rank (or order) of a tensor is defined by the number of
directions (and hence the dimensionality of the array) required to describe it.
Are tensors part of linear algebra?
Tensors are a type of data structure used in linear algebra, and like vectors and matrices, you can calculate arithmetic operations with tensors. ... That tensors are a generalization of matrices and
are represented using n-dimensional arrays.
What is the difference between matrix and vector?
A vector is a list of numbers (can be in a row or column), A matrix is an array of numbers (one or more rows, one or more columns).
What is 2nd rank tensor?
SECOND RANK TENSOR PROPERTIES. Many properties are tensors that relate one vector to another or relate a scalar to a tensor. If the driving force and the response are collinear the property can be
expressed as a scalar, but when that are not, the property must be expressed as a second rank tensor.
Are Spinors tensors?
Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation. ... Unlike vectors and tensors, a spinor
transforms to its negative when the space is continuously rotated through a complete turn from 0° to 360° (see picture).
Why stress is a tensor?
Stress has both magnitude and direction but it does not follow the vector law of addition thus, it is not a vector quantity. Instead, stress follows the coordinate transformation law of addition, and
hence, stress is considered as a tensor quantity. ... Therefore, stress is a tensor quantity, and (C) is the correct option.
Is torque a tensor quantity?
EXPLAIN GIVING SOME EXAMPLES . IS TORQUE IS A TENSOR QUANTITY AND WHAT ABOUT MOMENT OF INERTIA. The moment of inertia is a tensor because it involves two directions- the axis of rotation, and the
position of the center of mass (with respect to the rotation axis). ...
What is not a tensor?
Area, Volume, and Tensor Densities. ... But by the tensor transformation laws, a rank-0 tensor is supposed to be invariant under a change of coordinates. We therefore conclude that quantities like
area and volume are not tensors.
What is the difference between a matrix and a sparse matrix?
Sparse matrices are distinct from matrices with mostly non-zero values, which are referred to as dense matrices. A matrix is sparse if many of its coefficients are zero. ... The example has 13 zero
values of the 18 elements in the matrix, giving this matrix a sparsity score of 0.722 or about 72%.
Do tensors have to be square?
Firstly, a tensor is simply an element of the tensor product of some vector spaces or bimodules or something. In this sense, of course there are non-square tensors. For example an element of V⊗kW
would be called a tensor, for any k-vector spaces V and W.
What are the different types of tensors?
There are four main tensor type you can create:
• Variable.
• constant.
• placeholder.
• SparseTensor.
What is difference between vector and tensor?
The tensor is a more generalized form of scalar and vector. ... If a tensor has only magnitude and no direction (i.e., rank 0 tensor), then it is called scalar. If a tensor has magnitude and one
direction (i.e., rank 1 tensor), then it is called vector.
Who discovered tensor analysis?
Gregorio Ricci-Curbastro, (born January 12, 1853, Lugo, Papal States [Italy]—died August 6, 1925, Bologna), Italian mathematician instrumental in the development of absolute differential calculus,
formerly also called the Ricci calculus but now known as tensor analysis. | {"url":"https://moviecultists.com/are-tensors-just-matrices","timestamp":"2024-11-06T09:06:51Z","content_type":"text/html","content_length":"42871","record_id":"<urn:uuid:1326ae86-442e-4f4c-8121-b16544f995af>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00466.warc.gz"} |
Importance of Using Elimination | Sets & Statistics | ANA Prep
Process of elimination is only next to number plugging in popularity as a strategy for solving Quant questions on the GMAT. I am not a fan of either method. Yes, they are useful sometimes, and even
necessary in some questions but for most questions, I like to use logic/reasoning.
That said, there is a set of questions in which we should think of these strategies. Number plugging is very useful when you have one or two variables in the options. Algebra can be time consuming in
these cases because of equation manipulation required.
Similarly, some questions beg you to use the process of elimination. Their question stem goes something like ”which of the following options can be the value of
Question: A list of numbers has six positive integers. Three of those integers are known – 4, 5 and 24 and three of those are unknown – Which of the following CANNOT be the value of any one of the
Solution: The question gives us concrete information about mean – it is 10 – but not about median – it is between 7 and 8 (exclusive). What can we say about median from this? That it cannot be 7 or 8
but anything in between. But we know that the list has all integers. When we have even number of integers, we know that the median is the average of the middle two numbers – when all are placed in
increasing order. So can the average of the two middle numbers be, say, 7.1? Which two positive integers can average to give 7.1? None! Note that if the average of two integers is a decimal, the
decimal must be (some number).5 such as 7.5 or 9.5 or 22.5 etc. This happens in case one number is odd and the other is even. In all other cases, the average would be an integer.
Since the median is given to be between 7 and 8, the median of the list of the six positive integers must be 7.5 only.
Now we know that the mean = 10 and median = 7.5
There are 6 elements in the list. The average of the list is 10 which means the sum of all 6 elements =
The sum of the third and fourth elements of the list is 15 so
So, two numbers whose sum is 15 such that one is less than 7.5 and the other greater than 7.5 could be
Case 2: The known 5 could be the third number in which case one of the unknown numbers is less than 5 and two of the unknown numbers would be more than 7.5.
If the third number is 5, the fourth number has to be 10 to get a median of 7.5. Hence, 10 must be one of the unknown numbers.
The sum of the other two unknown numbers would be 27 – 10 = 17.
One of them must be less than 5 and the other greater than 10. So possible options are
Let’s now try to look at the process of elimination here and see if we can find an easier way.
Two of the given options are 5 and 10. They have a median of 7.5 so lets assume that two of the unknown numbers are 5 and 10 (5 can be one of the unknowns since we are not given that all six integers
need to be distinct). If two unknowns make up third and fourth numbers in the list and have a median of 7.5, their sum would be 15 and the third unknown will be 12 (to get the mean of 10). This case
(5, 10, 12) satisfies all conditions so options (B), (D) and (E) are out of play.
Now we are left with two options 13 and 11. Check any one of them and you will know which one is not possible. Let’s check 13.
From the given options, any number greater than 7.5 must be either the fourth number or the fifth number. 13 cannot be the fourth number since the third number would need to be 2 in that case to get
median 7.5. But we have 4 and 5 more than 2 so it cannot be the third number. So 13 must be the fifth number of the list. We saw in the case above that if two unknowns are third and fourth numbers
then the fifth number HAS TO BE 12. So the already present 5 must be the third number and the fourth number must be 10. In that case, the leftover unknown would be 4 (to get a sum of 27). So the
three unknowns would be 4, 10 and 13. This satisfies all conditions and is possible. Hence answer must be (C). 11 will not be possible.
Let’s see what would have happened had you picked 11 to try out. If 11 were the fourth number, to get a median of 7.5, we would need 4 as the third number. That is not possible since we already have
a 5 given. So 11 must have been the fifth number. This would mean that the already present 5 and one unknown 10 would make the median of 7.5. So the third unknown in this case would be 6 (to get a
sum of 27). But 6 would be the third number and the median in this case would be | {"url":"https://anaprep.com/sets-statistics-importance-of-using-elimination/","timestamp":"2024-11-14T22:04:55Z","content_type":"text/html","content_length":"148510","record_id":"<urn:uuid:6008a398-fbbc-4e34-a9cf-02d9bda89d86>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00025.warc.gz"} |
Divison of decimal numbers - I
In our daily life, we use decimal number division in plenty of situations. For example, Ram went to buy eggs that cost \(Rs. 5.50\). If he gave \(Rs. 55\) to shop keeper, can you tell me how many
eggs he bought?
We can simply tell the answer, by dividing the total cost by the cost of the egg. That is $\frac{55}{5.5}=10$. So, Ram has bought \(10\) eggs.
Now, we got a brief intro about decimal division, so, we dive to the varies methods involve in the decimal division.
1. Decimal Division through models.
2. Division of a decimal number by a whole number.
3. Division by \(10\), \(100\) and \(1000\).
4. Division of a decimal number by another decimal number.
1. Decimal Division through models
Let us divide \(3.26\) by \(2\) using
grid mode
• Shade grids to show the decimal number \(3.26\) by \(2\).
• Separate the shaded region into \(2\) equal parts, as shown below.
• Each colour represents an equal portion of \(1.63\).
• Therefore, \(3.26\) \(÷\) \(2\) \(=\) \(1\) whole \(+\) \(6\) tenth \(+\) \(3\) Hundredth | {"url":"https://www.yaclass.in/p/mathematics-state-board/class-7/number-system-2789/division-of-decimal-numbers-17807/re-5154a7ff-46d5-4f82-aeea-2c61aa60f5e5","timestamp":"2024-11-06T11:34:39Z","content_type":"text/html","content_length":"50700","record_id":"<urn:uuid:19fc2828-a792-4b67-90da-b30afe8aa58c>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00364.warc.gz"} |
old Permuations and Combinations | Path2TJ
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To enable every aspiring TJ student to present their best self
Check out our Feature in WTOP!
Welcome to the guide on Permutations and Combinations
Please note this topic is a big differentiator in getting a high score.
*PATH2TJ approved use only*
The prerequisites are a thorough understanding of Factorials. If you do not have a good handle on these concepts, use the links below to explore these concepts and skills first. Though this topic is
often taught in schools, it is not always covered to the depth necessary to quickly answer questions. This lesson will focus on the specifics of permutations and combinations to the level needed for
the test.
Choose one or more of the following links to come up to the speed with the concepts:
https://betterexplained.com/articles/easy-permutations-and-combinations/ (The first site to visit giving a basic introduction)
https://youtu.be/6B-dvOMTeV8?t=44 (A short video to bring you up to speed) https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:prob-comb#x9e81a4f9838 9efdf:combinatorics-precalc (A classic
site to learn and test yourself)
https://youtu.be/wJDKqEYq7SY (A short video on combinations)
https://youtu.be/p8vIcmr_Pqo (A longer, in-depth video on combinations) https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/index.html (Use tutorials 55,56, and 57 to attain a
college-level understanding)
Practice and check your understanding with the practice sheets available here:
https://leagueland.typepad.com/files/ws-1---counting-combinations-permutations-key.pdf (A question bank with any questions, do enough to make you feel comfortable) https://mi01000971.schoolwires.net/
cms/lib/MI01000971/Centricity/Domain/429/Permutations% 20and%20Combinations%20Worksheet.docx (Practice problems to learn Permutations and Combinations. HIGHLY RECOMMENDED THAT YOU DO ALL OF THESE)
https://mi01000971.schoolwires.net/cms/lib/MI01000971/Centricity/Domain/429/Permutations% 20and%20Combinations%20Worksheet%20KEY.pdf (Answers to the questions above)
Now check if you are battle-ready by answering the following questions:
1. A letter lock Consists of four rings, each ring contains 9 digits. The lock can be opened by setting a four-digit code with the correct combination of the four rings. How many unsuccessful
attempts are possible in which the lock cannot be opened?
a. 4 9 -1
b. 9 4 -1
c. 9C4
d. None of the above
2. In how many ways can 11 identical books on English and 9 identical books on Math be placed in a row on a shelf so that two books on Math are not together?
a. 110
b. 220
c. 330
d. 440
3. There are 6 numbered chairs placed around a circular table. 3 boys and 3 girls want to sit on them in a way that no two boys nor girls sit next to each other. How many such arrangements are
a. 36
b. 58
c. 72
d. None of the above
4. A committee is to be formed in which 5 people are chosen from 6 men and 4 women is to be chosen. In how many ways can this committee be made if there can be, at most, 2 women? a. 186 b. 168 c. 136
d. 169 5. Allen and Mary host a TV show together in which one day N number of guests attend the show. Each guest shakes hands with every other guest and each guest also shakes hands with each host.
If there happens to be a total of 65 handshakes, find the number of guests that attended the show.
a. 13
b. 14
c. 10
d. 9
6. If 10 objects are arranged in a row, then the number of ways of selecting three of these objects such that no two of them are adjacent is:
a. 56
b. 65
c. 28
d. 13
7. 8 people are to be seated at Melville's Restaurant. The only table available is a round table. However, Jack and Jill insist on sitting next to each other. In how many ways can the 8 people be
a. 8!
b. 8 + 3!
c. 7!
d. 6! + 2
8. Suppose the Lincoln-Douglas debate team consists of 10 seniors. Two of which will be randomly selected to become captains. However, one of the seniors is already captain of another club and is
ineligible. In how many ways can the two captains be chosen?
a. 36
b. 72
c. 45
d. 90
9. Oliver is applying for an internship and says, “I have a 15% chance of being accepted to Internship A and I have a 5% chance of getting accepted into both A and B. Assuming Oliver will be accepted
by at least one internship, what is the chance he is accepted for internship B but not A?
a. 15%
b. 65%
c. 80%
d. 20%
Answer key:
1. B
2. B
3. C
4. A
5. C
6. A
7. D
8. A
9. C
Congrats on reaching the end of this lesson. If you did not understand any concept be sure to go back through and review, so that you can be fully prepared. We learn the most from our mistakes so
don’t be disheartened by them. When you feel that you are ready, go ahead and take a little break. Then come back and move onto another section!
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Project: rdf-n3 - The Ruby Toolbox
RDF::N3 reader/writer and reasoner
Notation-3 reader/writer for RDF.rb .
RDF::N3 is an Notation-3 parser and reasoner for Ruby using the RDF.rb library suite.
Reader inspired from TimBL predictiveParser and Python librdf implementation.
Uses CG Specification
This version tracks the W3C N3 Community Group Specification which has incompatibilities with the Team Submission version. Notably:
• The @keywords declaration is removed, and most form of @ keywords (e.g., @is, @has, @true) are no longer supported.
• Terminals adhere closely to their counterparts in Turtle.
• The modifier <- is introduced as a synonym for is ... of.
• The SPARQL BASE and PREFIX declarations are supported.
• Implicit universal variables are defined at the top-level, rather than in the parent formula of the one in which they are defined.
• Support for explicit existential and universal variables (@forAll and @forSome) has been removed. Quick variables are the standard for universal quantification and blank nodes for existential,
but scoping rules are different: Quickvars have top-level scope, and blank nodes formula scope.
This brings N3 closer to compatibility with Turtle.
RDF::N3 parses Notation-3, Turtle and N-Triples into statements or quads. It also performs reasoning and serializes to N3.
Install with gem install rdf-n3
• Support for Variables in Formulae. Existential variables are quantified to RDF::Node instances, Universals to RDF::Query::Variable, with the URI of the variable target used as the variable name.
Instantiate a reader from a local file:
RDF::N3::Reader.open("etc/foaf.n3") do |reader|
reader.each_statement do |statement|
puts statement.inspect
Define @base and @prefix definitions, and use for serialization using :base_uri an :prefixes options
Write a graph to a file:
RDF::N3::Writer.open("etc/test.n3") do |writer|
writer << graph
Experimental N3 reasoning is supported. Instantiate a reasoner from a dataset:
RDF::N3::Reasoner.new do |reasoner|
RDF::N3::Reader.open("etc/foaf.n3") {|reader| reasoner << reader}
reader.each_statement do |statement|
puts statement.inspect
Reasoning is performed by turning a repository containing formula and predicate operators into an executable set of operators (similar to the executable SPARQL Algebra). Reasoning adds statements to
the base dataset, marked with :inferred (e.g. statement.inferred?). Predicate operators are defined from the following vocabularies.
When dispatching built-in operators, precedence is given to operators whos operands are fully evaluated, followed by those having only variable output operands, followed by those having the fewest
operands. Operators are evaluated until either no solutions are derived, or all operators have been completed.
Reasoning is discussed in the Design Issues document.
• list:append (See {RDF::N3::Algebra::List::Append})
• list:first (See {RDF::N3::Algebra::List::First})
• list:in (See {RDF::N3::Algebra::List::In})
• list:iterate (See {RDF::N3::Algebra::List::Iterate})
• list:last (See {RDF::N3::Algebra::List::Last})
• list:length (See {RDF::N3::Algebra::List::Length})
• list:member (See {RDF::N3::Algebra::List::Member})
• log:conclusion (See {RDF::N3::Algebra::Log::Conclusion})
• log:conjunction (See {RDF::N3::Algebra::Log::Conjunction})
• log:content (See {RDF::N3::Algebra::Log::Content})
• log:dtlit (See {RDF::N3::Algebra::Log::DtLit})
• log:equalTo (See {RDF::N3::Algebra::Log::EqualTo})
• log:implies (See {RDF::N3::Algebra::Log::Implies})
• log:includes (See {RDF::N3::Algebra::Log::Includes})
• log:langlit (See {RDF::N3::Algebra::Log::LangLit})
• log:n3String (See {RDF::N3::Algebra::Log::N3String})
• log:notEqualTo (See {RDF::N3::Algebra::Log::NotEqualTo})
• log:notIncludes (See {RDF::N3::Algebra::Log::NotIncludes})
• log:outputString See {RDF::N3::Algebra::Log::OutputString})
• log:parsedAsN3 (See {RDF::N3::Algebra::Log::ParsedAsN3})
• log:semantics (See {RDF::N3::Algebra::Log::Semantics})
• math:absoluteValue (See {RDF::N3::Algebra::Math::AbsoluteValue})
• math:acos (See {RDF::N3::Algebra::Math::ACos})
• math:asin (See {RDF::N3::Algebra::Math::ASin})
• math:atan (See {RDF::N3::Algebra::Math::ATan})
• math:acosh (See {RDF::N3::Algebra::Math::ACosH})
• math:asinh (See {RDF::N3::Algebra::Math::ASinH})
• math:atanh (See {RDF::N3::Algebra::Math::ATanH})
• math:ceiling (See {RDF::N3::Algebra::Math::Ceiling})
• math:cosh (See {RDF::N3::Algebra::Math::CosH})
• math:cos (See {RDF::N3::Algebra::Math::Cos})
• math:difference (See {RDF::N3::Algebra::Math::Difference})
• math:equalTo (See {RDF::N3::Algebra::Math::EqualTo})
• math:exponentiation (See {RDF::N3::Algebra::Math::Exponentiation})
• math:floor (See {RDF::N3::Algebra::Math::Floor})
• math:greaterThan (See {RDF::N3::Algebra::Math::GreaterThan})
• math:lessThan (See {RDF::N3::Algebra::Math::LessThan})
• math:negation (See {RDF::N3::Algebra::Math::Negation})
• math:notEqualTo (See {RDF::N3::Algebra::Math::NotEqualTo})
• math:notGreaterThan (See {RDF::N3::Algebra::Math::NotGreaterThan})
• math:notLessThan (See {RDF::N3::Algebra::Math::NotLessThan})
• math:product (See {RDF::N3::Algebra::Math::Product})
• math:quotient (See {RDF::N3::Algebra::Math::Quotient})
• math:remainder (See {RDF::N3::Algebra::Math::Remainder})
• math:rounded (See {RDF::N3::Algebra::Math::Rounded})
• math:sinh (See {RDF::N3::Algebra::Math::SinH})
• math:sin (See {RDF::N3::Algebra::Math::Sin})
• math:sum (See {RDF::N3::Algebra::Math::Sum})
• math:tanh (See {RDF::N3::Algebra::Math::TanH})
• math:tan (See {RDF::N3::Algebra::Math::Tan})
• string:concatenation (See {RDF::N3::Algebra::Str::Concatenation})
• string:contains (See {RDF::N3::Algebra::Str::Contains})
• string:containsIgnoringCase (See {RDF::N3::Algebra::Str::ContainsIgnoringCase})
• string:endsWith (See {RDF::N3::Algebra::Str::EndsWith})
• string:equalIgnoringCase (See {RDF::N3::Algebra::Str::EqualIgnoringCase})
• string:format (See {RDF::N3::Algebra::Str::Format})
• string:greaterThan (See {RDF::N3::Algebra::Str::GreaterThan})
• string:lessThan (See {RDF::N3::Algebra::Str::LessThan})
• string:matches (See {RDF::N3::Algebra::Str::Matches})
• string:notEqualIgnoringCase (See {RDF::N3::Algebra::Str::NotEqualIgnoringCase})
• string:notGreaterThan (See {RDF::N3::Algebra::Str::NotGreaterThan})
• string:notLessThan (See {RDF::N3::Algebra::Str::NotLessThan})
• string:notMatches (See {RDF::N3::Algebra::Str::NotMatches})
• string:replace (See {RDF::N3::Algebra::Str::Replace})
• string:scrape (See {RDF::N3::Algebra::Str::Scrape})
• string:startsWith (See {RDF::N3::Algebra::Str::StartsWith})
• time:dayOfWeek (See {RDF::N3::Algebra::Time::DayOfWeek})
• time:day (See {RDF::N3::Algebra::Time::Day})
• time:gmTime (See {RDF::N3::Algebra::Time::GmTime})
• time:hour (See {RDF::N3::Algebra::Time::Hour})
• time:inSeconds (See {RDF::N3::Algebra::Time::InSeconds})
• time:localTime (See {RDF::N3::Algebra::Time::LocalTime})
• time:minute (See {RDF::N3::Algebra::Time::Minute})
• time:month (See {RDF::N3::Algebra::Time::Month})
• time:second (See {RDF::N3::Algebra::Time::Second})
• time:timeZone (See {RDF::N3::Algebra::Time::Timezone})
• time:year (See {RDF::N3::Algebra::Time::Year})
Parser features
Chaining with iriPropertyList
Adds a proposed syntactic extension for subject embedding similar to a blankNodePropertyList. An iriPropertyList begins with [ id _id_, instead of a simple [. This sets id as the subject to be used
for the following propertyList. This provides a mechanisms similar to JSON-LD Embedding.
@prefix dc: <http://purl.org/dc/terms/>.
@prefix : <http://example.org/nd#>.
:SummerReadingList a :OrderedListOfBooks ;
:toRead (
[id :mobyDick dc:title "Moby Dick"; :setting :WhaleIntestines ]
id :jaws
dc:title "Jaws";
:setting :Beach
Note that the id used in the iriPropertyList is not delimited by a ;
Formulae / Quoted Graphs
N3 Formulae are introduced with the { statement-list } syntax. A given formula is assigned an RDF::Node instance, which is also used as the graph_name for RDF::Statement instances provided to
RDF::N3::Reader#each_statement. For example, the following N3 generates the associated statements:
@prefix x: <http://example.org/x-ns/#> .
@prefix log: <http://www.w3.org/2000/10/swap/log#> .
@prefix dc: <http://purl.org/dc/elements/1.1/#> .
{ [ x:firstname "Ora" ] dc:wrote [ dc:title "Moby Dick" ] } a log:falsehood .
when turned into an RDF Repository results in the following quads
_:form <http://www.w3.org/1999/02/22-rdf-syntax-ns#type> <http://www.w3.org/2000/10/swap/log#falsehood> .
_:moby <http://purl.org/dc/elements/1.1/#title> "Moby Dick" _:form .
_:ora <http://purl.org/dc/elements/1.1/#wrote> _:moby _:form .
_:ora <http://example.org/x-ns/#firstname> "Ora" _:form .
Reasoning uses a Notation3 Algebra, similar to SPARQL S-Expressions. This implementation considers formulae to be patterns, which may be asserted on statements made in the default graph, possibly
loaded from a separate file. The logical relationships are reduced to algebraic operators.
The latest version of N3 supports only quickVars (e.g., ?x). THe former explicit @forAll and @forSome of been removed. Existential variables are replaced with an allocated RDF::Node instance.
Note that the behavior of both existential and universal variables is not entirely in keeping with the Team Submission, and neither work quite like SPARQL variables. When used in the antecedent part
of an implication, universal variables should behave much like SPARQL variables. This area is subject to a fair amount of change.
• Variables, themselves, cannot be part of a solution, which limits the ability to generate updated rules for reasoning.
• Both Existentials and Universals are treated as simple variables, and there is really no preference given based on the order in which they are introduced.
Formulae are typically used to query the knowledge-base, which is set from the base-formula/default graph. A formula is composed of both constant statements, and variable statements. When running as
a query, such as for the antecedent formula in log:implies, statements including either explicit variables or blank nodes are treated as query patterns and are used to query the knowledge base to
create a solution set, which is used either prove the formula correct, or create bindings passed to the consequent formula.
Blank nodes associated with rdf:List statements used as part of a built-in are made non-distinguished existential variables, and patters containing these variables become optional. If they are not
bound as part of the query, the implicitly are bound as the original blank nodes defined within the formula, which allows for both constant list arguments, list arguments that contain variables, or
arguments which are variables expanding to lists.
Full documentation available on RubyDoc.info
Principle Classes
• {RDF::N3}
• {RDF::N3::Format}
• {RDF::N3::Reader}
• {RDF::N3::Reasoner}
• {RDF::N3::Writer}
• {RDF::N3::Algebra}
□ {RDF::N3::Algebra::Formula}
Additional vocabularies
• {RDF::N3::Rei}
• {RDF::N3::Crypto}
• {RDF::N3::Math}
• {RDF::N3::Time}
Change Log
This repository uses Git Flow to mange development and release activity. All submissions must be on a feature branch based on the develop branch to ease staging and integration.
• Do your best to adhere to the existing coding conventions and idioms.
• Don't use hard tabs, and don't leave trailing whitespace on any line.
• Do document every method you add using YARD annotations. Read the tutorial or just look at the existing code for examples.
• Don't touch the .gemspec, VERSION or AUTHORS files. If you need to change them, do so on your private branch only.
• Do feel free to add yourself to the CREDITS file and the corresponding list in the the README. Alphabetical order applies.
• Do note that in order for us to merge any non-trivial changes (as a rule of thumb, additions larger than about 15 lines of code), we need an explicit public domain dedication on record from you,
which you will be asked to agree to on the first commit to a repo within the organization. Note that the agreement applies to all repos in the Ruby RDF organization.
This is free and unencumbered public domain software. For more information, see https://unlicense.org/ or the accompanying {file:UNLICENSE} file. | {"url":"https://www.ruby-toolbox.com/projects/rdf-n3","timestamp":"2024-11-08T11:31:17Z","content_type":"text/html","content_length":"73546","record_id":"<urn:uuid:65de44c7-2b40-4351-acca-d03b1ac6f89a>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00618.warc.gz"} |
Understanding Arc Length: A Comprehensive Guide
Introduction to Arc Length
What is Arc Length?
Arc length is a fundamental concept in geometry, representing the distance along a curved line, or arc, between two points. Unlike the straight-line distance between two points, arc length considers
the curve's nature, making it a more complex and intriguing measurement. This concept is particularly important in various fields, including mathematics, engineering, and architecture, where precise
measurements of curves are crucial.
Importance of Calculating Arc Length
The ability to calculate arc length is essential in many practical applications. Whether designing a curved structure, mapping a circular path, or analyzing the trajectory of a moving object,
understanding the arc length allows for accurate planning and execution. It also plays a significant role in calculus and trigonometry, where it serves as a foundation for more advanced mathematical
The Arc Length Formula: A Closer Look
The Concept Behind the Formula
The formula for calculating arc length is rooted in the relationship between the radius of a circle and the angle subtended by the arc at the circle's center. By understanding this relationship, one
can determine the exact length of the arc without needing to physically measure it. This formula is a powerful tool that simplifies the process of working with curves in both theoretical and
practical scenarios.
Limitations and Challenges
While the arc length formula is a valuable tool, it does have its limitations. Calculating arc length for more complex curves, such as parabolas or ellipses, can be challenging and may require
advanced mathematical techniques. Additionally, the accuracy of the calculation can be affected by factors such as rounding errors and the precision of the measurements used.
Arc Length Calculators: Simplifying the Process
Introduction to Arc Length Calculators
Arc length calculators are digital tools designed to simplify the process of determining the length of an arc. By inputting the necessary parameters, such as the radius and angle, users can quickly
and accurately calculate the arc length without needing to manually apply the formula. These calculators are widely used in various fields, offering a convenient solution for both professionals and
How Arc Length Calculators Work
Arc length calculators are typically user-friendly, requiring only basic information to perform the calculation. Most calculators ask for the radius of the circle and the angle in degrees or radians.
Once these values are entered, the calculator applies the arc length formula to provide the result. Some advanced calculators may also offer additional features, such as converting between units or
handling more complex curves.
Benefits of Using an Arc Length Calculator
Using an arc length calculator offers several benefits, including:
1. Speed and Efficiency: Calculating arc length manually can be time-consuming, especially for complex curves. An arc length calculator provides quick results, allowing users to focus on other
2. Accuracy: Digital calculators minimize the risk of human error, ensuring more precise measurements.
3. Accessibility: Arc length calculators are widely available online and often free to use, making them accessible to anyone needing to calculate arc length.
4. Versatility: Many calculators are designed to handle various types of curves, making them versatile tools for different applications.
Practical Applications of Arc Length Calculation
Engineering and Architecture
In engineering and architecture, accurate measurements of curves are crucial for designing structures that are both functional and aesthetically pleasing. From bridges and tunnels to curved facades
and arches, understanding the arc length ensures that these designs are executed precisely and safely.
Astronomy and Space Science
Arc length calculation is also vital in astronomy and space science, where it helps in determining the paths of celestial bodies, such as the orbits of planets and the trajectories of spacecraft. By
understanding the arc length of these paths, scientists can make more accurate predictions and adjustments in their observations and missions.
Cartography and Geography
In cartography and geography, arc length plays a key role in mapping curved surfaces, such as the Earth's surface. By calculating the arc length of geographical features, cartographers can create
more accurate maps and models, aiding in navigation, exploration, and analysis.
Advanced Topics in Arc Length Calculation
Calculating Arc Length for Different Curves
While the basic arc length formula applies to circular arcs, calculating arc length for other types of curves, such as parabolas, ellipses, and hyperbolas, requires more advanced mathematical
techniques. These calculations often involve integral calculus and may require numerical methods to approximate the arc length accurately.
Numerical Methods for Arc Length Calculation
Numerical methods, such as the trapezoidal rule and Simpson's rule, are often used to approximate the arc length of complex curves. These methods break down the curve into smaller segments, calculate
the arc length for each segment, and then sum the results to obtain the total arc length. While these methods can be computationally intensive, they provide a way to calculate arc length when an
exact formula is not available.
The Role of Arc Length in Calculus
In calculus, arc length is closely related to the concept of the definite integral. The arc length of a curve can be represented as the integral of the square root of the sum of the squares of the
derivatives of the curve's parametric equations. This relationship forms the basis for many advanced mathematical techniques used in both theoretical and applied mathematics.
The Future of Arc Length Calculation
Technological Advancements
As technology continues to advance, arc length calculators are becoming more sophisticated, offering enhanced features and greater accuracy. With the integration of artificial intelligence and
machine learning, future calculators may be able to handle even more complex curves and provide real-time results for dynamic systems.
Potential Innovations
In the future, we may see innovations in arc length calculation that go beyond traditional methods. For example, new mathematical models and algorithms could be developed to simplify the process of
calculating arc length for non-standard curves. Additionally, advances in virtual and augmented reality could enable users to visualize and interact with curves in three-dimensional space, providing
a more intuitive understanding of arc length.
The Growing Importance of Arc Length in Emerging Fields
As emerging fields, such as robotics, autonomous vehicles, and virtual reality, continue to evolve, the importance of accurate arc length calculation is likely to grow. In these fields, precise
measurements of curves are essential for tasks such as path planning, motion control, and environment modeling. As a result, arc length calculation will continue to play a critical role in the
development and implementation of these technologies.
The Essential Nature of Arc Length
Arc length is a fundamental concept in geometry and mathematics, with wide-ranging applications in various fields. Whether in engineering, astronomy, or cartography, the ability to calculate arc
length accurately is crucial for success. As technology continues to advance, arc length calculators will play an increasingly important role in simplifying this process and ensuring precise
Arc length is the distance along a curved line or arc between two points, as opposed to a straight line distance.
Arc length calculators use input parameters like the radius and angle to quickly compute the length of a curve based on established formulas.
Arc length calculators offer speed, accuracy, and convenience, reducing manual calculation time and minimizing errors.
Many calculators are designed for circular arcs, but advanced versions can approximate lengths for more complex curves using numerical methods.
Arc length calculations are crucial in fields like engineering, architecture, astronomy, and cartography for designing curves and mapping curved paths. | {"url":"https://hightechtools.online/arc-length-calculator","timestamp":"2024-11-12T10:09:25Z","content_type":"text/html","content_length":"79989","record_id":"<urn:uuid:34bc4261-aa6a-49d7-9444-cd541937ce87>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00165.warc.gz"} |
Statement 1:oxidation number of ni in is zero. statement 2:nick-Turito
Are you sure you want to logout?
Statement 1:Oxidation number of Ni in is zero.
Statement 2:Nickel is bonded to neutral ligand carbonyl.
A. Statement 1 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
B. Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
C. Statement 1 is True, Statement 2 is False
D. Statement 1 is False, Statement 2 is True
The correct answer is: Statement 1 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/statement-1-oxidation-number-of-ni-in-is-zero-statement-2-nickel-is-bonded-to-neutral-ligand-carbonyl-statement-1-i-q26065f","timestamp":"2024-11-01T19:43:38Z","content_type":"application/xhtml+xml","content_length":"388287","record_id":"<urn:uuid:c2169a4f-1262-4a54-8ab3-85c569a38652>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00214.warc.gz"} |
Extended Isolation Forest for anomaly detection
Project description
Table of contents
Extended Isolation Forest
This is a simple package implementation for the Extended Isolation Forest method described in this paper. It is an improvement on the original algorithm Isolation Forest which is described (among
other places) in this paper for detecting anomalies and outliers for multidimensional data point distributions.
The problem of anomaly detection has wide range of applications in various fields and scientific applications. Anomalous data can have as much scientific value as normal data or in some cases even
more, and it is of vital importance to have robust, fast and reliable algorithms to detect and flag such anomalies. Here, we present an extension to the model-free anomaly detection algorithm,
Isolation Forest Liu2008. This extension, named Extended Isolation Forest (EIF), improves the consistency and reliability of the anomaly score produced by standard methods for a given data point. We
show that the standard Isolation Forest produces inconsistent anomaly score maps, and that these score maps suffer from an artifact produced as a result of how the criteria for branching operation of
the binary tree is selected.
Our method allows for the slicing of the data to be done using hyperplanes with random slopes which results in improved score maps. The consistency and reliability of the algorithm is much improved
using this extension. Here we show the need for an improvement on the source algorithm to improve the scoring of anomalies and the robustness of the score maps especially around edges of nominal
data. We discuss the sources of the problem, and we present an efficient way for choosing these hyperplanes which give way to multiple extension levels in the case of higher dimensional data. The
standard Isolation Forest is therefore a special case of the Extended Isolation Forest as presented it here. For an N dimensional dataset, Extended Isolation Forest has N levels of extension, with 0
being identical to the case of standard Isolation Forest, and N-1 being the fully extended version.
Figure 1: Example training data. a) Normally distributed cluster. b) Two normally distributed clusters. c) Sinusoidal data points with Gaussian noise.
While various techniques exist for approaching anomaly detection, Isolation Forest Liu2008 is one with unique capabilities. This algorithm can readily work on high dimensional data, it is model free,
and it scales well. It is therefore highly desirable and easy to use. However, looking at score maps for some basic example, we can see that the anomaly scores produced by the standard Isolation
Forest are inconsistent, . To see this we look at the three examples shown in Figure 1.
In each case, we use the data to train our Isolation Forest. We then use the trained models to score a square grid of uniformly distributed data points, which results in score maps shown in Figure 2.
Through the simplicity of the example data, we have an intuition about what the score maps should look like. For example, for the data shown in Figure 1a, we expect to see low anomaly scores in the
center of the map, while the anomaly score should increase as we move radially away from the center. Similarly for the other figures.
Looking at the score maps produced by the standard Isolation Forest shown in Figure 2, we can clearly see the inconsistencies in the scores. While we can clearly see a region of low anomaly score in
the center in Figure 2a, we can also see regions aligned with x and y axes passing through the origin that have lower anomaly scores compared to the four corners of the region. Based on our intuitive
understanding of the data, this cannot be correct. A similar phenomenon is observed in Figure 2b. In this case, the problem is amplified. Since there are two clusters, the artificially low anomaly
score regions intersect close to points (0,0) and (10,10), and create low anomaly score regions where there is no data. It is immediately obvious how this can be problematic. As for the third
example, figure 2c shows that the structure of the data is completely lost. The sinusoidal shape is essentially treated as one rectangular blob.
Figure 2: Score maps using the Standard Isolation Forest for the points from Figure 1. We can see the bands and artifacts on these maps
Isolation Forest
Given a dataset of dimension N, the algorithm chooses a random sub-sample of data to construct a binary tree. The branching process of the tree occurs by selecting a random dimension x_i with i in
{1,2,...,N} of the data (a single variable). It then selects a random value v within the minimum and maximum values in that dimension. If a given data point possesses a value smaller than v for
dimension x_i, then that point is sent to the left branch, otherwise it is sent to the right branch. In this manner the data on the current node of the tree is split in two. This process of branching
is performed recursively over the dataset until a single point is isolated, or a predetermined depth limit is reached. The process begins again with a new random sub-sample to build another
randomized tree. After building a large ensemble of trees, i.e. a forest, the training is complete.
During the scoring step, a new candidate data point (or one chosen from the data used to create the trees) is run through all the trees, and an ensemble anomaly score is assigned based on the depth
the point reaches in each tree. Figure 3 shows an schematic example of a tree and a forest plotted radially.
Figure 3: a) Shows an example tree formed from the example data while b) shows the forest generated where each tree is represented by a radial line from the center to the outer circle. Anomalous
points (shown in red) are isolated very quickly,which means they reach shallower depths than nominal points (shown in blue).
It turns out the splitting process described above is the main source of the bias observed in the score maps. Figure 4 shows the process described above for each one of the examples considered thus
far. The branch cuts are always parallel to the axes, and as a result over construction of many trees, regions in the domain that don't occupy any data points receive superfluous branch cuts.
Figure 4: Splitting of data in the domain during the process of construction of one tree.
The Extended Isolation Forest remedies this problem by allowing the branching process to occur in every direction. The process of choosing branch cuts is altered so that at each node, instead of
choosing a random feature along with a random value, we choose a random normal vector along with a random intercept point.
Figure 5 shows the resulting branch cuts int he domain for each of our examples.
Figure 5: Same as Figure 4 but using Extended Isolation Forest
We can see that the region is divided much more uniformly, and without the bias introducing effects of the coordinate system. As in the case of the standard Isolation Forest, the anomaly score is
computed by the aggregated depth that a given point reaches on each iTree.
As we see in Figure 6, these modifications completely fix the issue with the score maps that we saw before and produce reliable results. Clearly, these score maps are a much better representation of
anomaly score distributions.
Figure 6: Score maps using the Extended Isolation Forest.
Figure 7 shows a very simple example of anomalies and nominal points from a Single blob example as shown in Figure 1a. It also shows the distribution of the anomaly scores which can be used to make
hard cuts on the definition of anomalies or even assign probabilities to each point.
Figure 7: a) Shows the dataset used, some sample anomalous data points discovered using the algorithm are highlighted in black. We also highlight some nominal points in red. In b), we have the
distribution of anomaly scores obtained by the algorithm.
The Code
Here we provide the source code for the algorithm as well as documented example notebooks to help get started. Various visualizations are provided such as score distributions, score maps, aggregate
slicing of the domain, and tree and whole forest visualizations. Most examples are in 2D. We present one 3D example. However, the algorithm works readily with higher dimensional data.
pip install git+https://github.com/sahandha/eif.git
No extra requirements are needed. In addition, it also contains means to draw the trees created using the igraph library. See the example for tree visualizations.
See these notebooks for examples on how to use it
If you use this code and method, please considering using the following reference:
A link to the paper can be found here
author={S. {Hariri} and M. {Carrasco Kind} and R. J. {Brunner}},
journal={IEEE Transactions on Knowledge and Data Engineering},
title={Extended Isolation Forest},
keywords={Forestry;Vegetation;Distributed databases;Anomaly detection;Standards;Clustering algorithms;Heating systems;Anomaly Detection;Isolation Forest},
• Convert code into C++ with using cython. Much faster and efficient forest generation and scoring procedures
• Added documentation, examples and software paper
• Bugfix for multidimensional data
Project details
Download files
Download the file for your platform. If you're not sure which to choose, learn more about installing packages.
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Halton quasirandom point set
haltonset is a quasirandom point set object that produces points from the Halton sequence. The Halton sequence uses different prime bases in each dimension to fill space in a highly uniform manner.
p = haltonset(d) constructs a d-dimensional point set p, which is a haltonset object with default property settings. The input argument d corresponds to the Dimensions property of p.
p = haltonset(d,Name,Value) sets properties of p using one or more name-value pair arguments. Enclose each property name in quotes. For example, haltonset(5,'Leap',2) creates a five-dimensional point
set from the first point, fourth point, seventh point, tenth point, and so on.
The returned object p encapsulates properties of a Halton quasirandom sequence. The point set is finite, with a length determined by the Skip and Leap properties and by limits on the size of the
point set indices (maximum value of 2^53). Values of the point set are generated whenever you access p using net or parenthesis indexing. Values are not stored within p.
Dimensions — Number of dimensions
positive integer scalar
This property is read-only.
Number of dimensions of the points in the point set, specified as a positive integer scalar. For example, each point in the point set p with p.Dimensions = 5 has five values.
Use the d input argument to specify the number of dimensions when you create a point set using the haltonset function.
Leap — Interval between points
0 (default) | positive integer scalar
Interval between points in the sequence, specified as a positive integer scalar. In other words, the Leap property of a point set specifies the number of points in the sequence to leap over and omit
for every point taken. The default Leap value is 0, which corresponds to taking every point from the sequence.
Leaping is a technique used to improve the quality of a point set. However, you must choose the Leap values with care. Many Leap values create sequences that fail to touch on large
sub-hyper-rectangles of the unit hypercube and, therefore, fail to be a uniform quasirandom point set. For more information, see [1].
One rule for choosing Leap values for Halton sets is to set the value to (n–1), where n is a prime number that has not been used to generate one of the dimensions. For example, for a d-dimensional
point set, specify the (d+1)th or greater prime number for n.
Example: p = haltonset(2,'Leap',4); (where d = 2 and n = 5)
Example: p.Leap = 100;
ScrambleMethod — Settings that control scrambling
0x0 structure (default) | structure with Type and Options fields
Settings that control the scrambling of the sequence, specified as a structure with these fields:
• Type — A character vector containing the name of the scramble
• Options — A cell array of parameter values for the scramble
Use the scramble object function to set scrambles. For a list of valid scramble types, see the type input argument of scramble. An error occurs if you set an invalid scramble type for a given point
The ScrambleMethod property also accepts an empty matrix as a value. The software then clears all scrambling and sets the property to contain a 0x0 structure.
Skip — Number of initial points in sequence to omit
0 (default) | positive integer scalar
Number of initial points in the sequence to omit from the point set, specified as a positive integer scalar.
Initial points of a sequence sometimes exhibit undesirable properties. For example, the first point is often (0,0,0,...), which can cause the sequence to be unbalanced because the counterpart of the
point, (1,1,1,...), never appears. Also, initial points often exhibit correlations among different dimensions, and these correlations disappear later in the sequence.
Example: p = haltonset(__,'Skip',2e3);
Example: p.Skip = 1e3;
Type — Sequence type
'Halton' (default)
This property is read-only.
Sequence type on which the quasirandom point set p is based, specified as 'Halton'.
Object Functions
net Generate quasirandom point set
scramble Scramble quasirandom point set
You can also use the following MATLAB^® functions with a haltonset object. The software treats the point set object like a matrix of multidimensional points.
length Length of largest array dimension
size Array size
Create Halton Point Set
Generate a three-dimensional Halton point set, skip the first 1000 values, and then retain every 101st point.
p = haltonset(3,'Skip',1e3,'Leap',1e2)
p =
Halton point set in 3 dimensions (89180190640991 points)
Skip : 1000
Leap : 100
ScrambleMethod : none
Apply reverse-radix scrambling by using scramble.
p =
Halton point set in 3 dimensions (89180190640991 points)
Skip : 1000
Leap : 100
ScrambleMethod : RR2
Generate the first four points by using net.
X0 = 4×3
0.0928 0.6950 0.0029
0.6958 0.2958 0.8269
0.3013 0.6497 0.4141
0.9087 0.7883 0.2166
Generate every third point, up to the eleventh point, by using parenthesis indexing.
X = 4×3
0.0928 0.6950 0.0029
0.9087 0.7883 0.2166
0.3843 0.9840 0.9878
0.6831 0.7357 0.7923
• The Skip and Leap properties are useful for parallel applications. For example, if you have a Parallel Computing Toolbox™ license, you can partition a sequence of points across N different
workers by using the function spmdIndex (Parallel Computing Toolbox). On each nth worker, set the Skip property of the point set to n – 1 and the Leap property to N – 1. The following code shows
how to partition a sequence across three workers.
Nworkers = 3;
p = haltonset(10,'Leap',Nworkers-1);
p.Skip = spmdIndex - 1;
% Compute something using points 1,4,7...
% or points 2,5,8... or points 3,6,9...
Halton Sequence Generation
Consider a default haltonset object p that contains d-dimensional points. Each p(i,:) is a point in a Halton sequence. The jth coordinate of the point, p(i,j), is equal to
$\sum _{k}{a}_{ij}\left(k\right){b}_{j}{}^{-k-1}.$
• b[j] is the jth prime.
• The ${a}_{ij}\left(k\right)$ coefficients are nonnegative integers less than b[j] such that
$i-1=\sum _{k=0}{a}_{ij}\left(k\right){b}_{j}{}^{k}.$
In other words, the ${a}_{ij}\left(k\right)$ values are the base b[j] digits of the integer i – 1.
For more information, see [1].
[1] Kocis, L., and W. J. Whiten. “Computational Investigations of Low-Discrepancy Sequences.” ACM Transactions on Mathematical Software. Vol. 23, No. 2, 1997, pp. 266–294.
Version History
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How compilers are structured.
Compilers are programs that translate source code from one language (the input language) to another (the output language). Usually, the input language is the language the programmer writes in, and
the output language is either assembly code or machine code.
Compilers make their job easier by breaking it into parts.
The first part is lexical analysis. This is done by a lexical analyzer, also known as a lexer. This is breaks up the input stream into tokens. Tokens are lumps of text from the input stream,
annotated with a type. So, given the expression:
42 + 4
The following tokens would be produced:
1. INTEGER "42"
2. PLUS "+"
3. INTEGER "4"
4. EOF
The last element is important: this is the End Of File marker, and tells the grammar when no more source code is to be expected.
The second part is syntactic analysis, using a grammar. A grammar is a list of rules describing the recursive structure of the language.
(I'm going to use recursive descent in my parser since it's the easiest one to write and read without using a parser generator.) A recursive descent grammar for simple expressions might look like:
expression : factor ( PLUS factor )* EOF ;
This rule means that an 'expression' is expected to be made up of a factor, followed by zero or more reptitions of the part in the ()s: that is, zero or more of "PLUS token immediately followed by a
factor". The '*' means zero or more of the element just before it. A '+' means one or more of the element just before it. Finally, the expression ends in EOF, so the source code should finish after
the expression rule has been matched.
factor : INTEGER ;
Here, a 'factor' is in turn made up of an INTEGER.
The job of a parser is to eat tokens (skip over them, causing the next token to come to the front) from the lexer, but only following rules from the grammar. The parser drives the whole process. It
follows the grammar as if each rule in the grammar was a function, and it pulls and eats tokens from the lexer.
Consider the expression "42 + 4" above. A parser using the 'expression' rule as its starting rule and given the stream of tokens INTEGER, PLUS, INTEGER from the lexer (from 42 + 4), it's runtime
perspective looks like this:
1. Parser starts with the expression rule. First thing is supposed to be a factor, and all factors start with INTEGER, so check that the lexer's current (i.e. head of the stream) token is an
2. The lexer's current token is an INTEGER, so recurse into the factor rule.
3. The factor rule eats the INTEGER token from the lexer, causing the lexer to move on to the next token (PLUS), and returns to the expression rule.
4. The expression rule is now positioned just after the first 'factor'; it now has a choice to make. Either enter the loop of 'PLUS factor' (choice PLUS), or skip the loop (choice EOF). It uses the
current token from the lexer (PLUS) to make a choice, so it enters the loop.
5. The loop has been entered, and the PLUS token from the stream is eaten. The next token from the lexer is INTEGER "4". The parser recurses into the factor rule.
6. The factor rule eats the INTEGER, leaving EOF as the current token in the lexer, and returns to the expression rule.
7. The expression rule has reached the end of the loop, and must decide if it is going to go around again. The start of the loop is a PLUS, while breaking out of the loop causes an EOF to be
matched. The current token (EOF) is used to make the decision, so the loop is exited.
8. The EOF is matched, and expression returns to whoever called into the parser.
That's the basic process of a parser. Of course, if it just follows the grammar and eats tokens, it doesn't do anything other than match its language and throw errors on invalid input. That isn't
terribly useful, so usually the parser does more work as it matches elements. However, that's enough on parsers for the moment.
Next, I'll write a very simple little lexer that breaks text up into the tokens described above.
No comments: | {"url":"http://blog.barrkel.com/2005/06/how-compilers-are-structured.html","timestamp":"2024-11-05T09:41:08Z","content_type":"application/xhtml+xml","content_length":"32235","record_id":"<urn:uuid:209607b7-0061-42ad-b3b9-e7d1177692a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00004.warc.gz"} |
Law of Large Numbers - (Probabilistic Decision-Making) - Vocab, Definition, Explanations | Fiveable
Law of Large Numbers
from class:
Probabilistic Decision-Making
The law of large numbers states that as the size of a sample increases, the sample mean will get closer to the expected value or population mean. This principle highlights the idea that larger
samples provide more accurate estimates of the true characteristics of a population, reducing variability and error.
congrats on reading the definition of Law of Large Numbers. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The law of large numbers applies to independent random variables, meaning that each observation does not influence others.
2. This law assures that as more observations are collected, the likelihood of the sample mean deviating significantly from the population mean decreases.
3. There are two forms of the law: the weak law, which deals with convergence in probability, and the strong law, which deals with almost sure convergence.
4. It is fundamental in statistics because it justifies using sample means for inference about population parameters.
5. The law provides a foundation for the Central Limit Theorem, which states that the distribution of sample means approaches a normal distribution as the sample size grows.
Review Questions
• How does the law of large numbers support decision-making in management?
□ The law of large numbers helps management make better decisions by ensuring that larger samples provide estimates closer to the true population parameters. By relying on larger datasets when
analyzing market trends or customer preferences, managers can reduce uncertainty and improve their forecasts. This leads to more informed strategic decisions, minimizing risk and enhancing
• Discuss how the weak and strong forms of the law of large numbers differ and their implications for statistical analysis.
□ The weak form of the law of large numbers focuses on convergence in probability, suggesting that as sample size increases, the probability that the sample mean deviates from the population
mean approaches zero. The strong form guarantees almost sure convergence, meaning that with increasing sample sizes, the sample mean will almost certainly converge to the population mean.
This distinction is important in statistical analysis as it affects how we interpret data reliability and make predictions based on varying sample sizes.
• Evaluate how understanding the law of large numbers can impact interpretations of sampling distributions in practice.
□ Understanding the law of large numbers allows analysts to critically assess how sampling distributions behave with increasing sample sizes. It highlights that smaller samples can lead to
significant variability and potential misinterpretations, while larger samples yield more stable estimates aligned with true population parameters. This knowledge enables analysts to select
appropriate sample sizes and enhances confidence in statistical conclusions drawn from empirical data, ultimately improving research quality and decision-making.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/probabilistic-and-statistical-decision-making-for-management/law-of-large-numbers","timestamp":"2024-11-07T00:49:04Z","content_type":"text/html","content_length":"175323","record_id":"<urn:uuid:1aa5e218-7247-44b0-9654-2b81515e51e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00068.warc.gz"} |
Bartels, R. (1971), “A Stabilization of the Simplex Method,” Numerical Mathematics, 16, 414–434.
Bland, R. G. (1977), “New Finite Pivoting Rules for the Simplex Method,” Mathematics of Operations Research, 2, 103–107.
Breau, R. and Burdet, C. A. (1974), “Branch and Bound Experiments in Zero-One Programming,” Mathematical Programming Study, 2, 1–50.
Crowder, H., Johnson, E. L., and Padberg, M. W. (1983), “Solving Large-Scale Zero-One Linear Programming Problems,” Operations Research, 31, 803–834.
Dantzig, G. B. (1963), Linear Programming and Extensions, Princeton, NJ: Princeton University Press.
Garfinkel, R. S. and Nemhauser, G. L. (1972), Integer Programming, New York: John Wiley & Sons.
Greenberg, H. J. (1978), “Pivot Selection Tactics,” in H. J. Greenberg, ed., Design and Implementation of Optimization Software, 143–174, Netherlands: Sijthoff & Noordhoff.
Hadley, G. (1962), Linear Programming, Reading, MA: Addison-Wesley.
Harris, P. (1975), “Pivot Selection Methods of the Devex LP Code,” Mathematical Programming Study, 4, 30–57.
Ignizio, J. P. (1976), Goal Programming and Extensions, Lexington, MA: D.C. Heath and Company.
Murtagh, B. A. (1981), Advanced Linear Programming, Computation and Practice, New York: McGraw-Hill.
Nelson, M. (1992), The Data Compression Book, M&T Books.
Reid, J. K. (1975), “A Sparsity-Exploiting Variant of the Bartels-Golub Decomposition for Linear Programming Bases,” Harwell Report CSS 20.
Reid, J. K. (1976), “Fortran Subroutines for Handling Sparse Linear Programming Bases,” Harwell Report R 8269.
Savelsbergh, M. W. P. (1994), “Preprocessing and Probing Techniques for Mixed Integer Programming Problems,” ORSA J. on Computing, 6, 445–454.
Taha, H. A. (1975), Integer Programming, New York: Academic Press. | {"url":"http://support.sas.com/documentation/cdl/en/ormpug/63352/HTML/default/ormpug_lp_sect064.htm","timestamp":"2024-11-14T22:09:43Z","content_type":"application/xhtml+xml","content_length":"11412","record_id":"<urn:uuid:182c16f1-5b0d-48cf-addb-9a5c61ee11a6>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00721.warc.gz"} |
Degrees of freedom
Degrees of freedom, according to my understanding should be n-1. But as I am observing in most of the questions (Schweser question bank), they have taken them as n-2. Any comment or clarification,
would help me.
The number of degrees of freedom depends on how many parameters you’ve estimated.
In a multiple regression, for example, in which you have n data points and k independent variables and you’re estimating k slope coefficients and one intercept, dof = n − k − 1.
What are types of problems in which you’re seeing dof = n − 2?
These are the kinds of things I can’t believe the prep providers and the CFAI fail to explain or even give the first sentence that s2000 provided. From the whacky misconception that multicollinearity
inflates R2 or the F-statistic to the “mysteriously new” t-statistic for Pearson’s correlation coefficient… they look like a deer on ice. | {"url":"https://www.analystforum.com/t/degrees-of-freedom/125325","timestamp":"2024-11-03T19:58:40Z","content_type":"text/html","content_length":"20645","record_id":"<urn:uuid:578eaad2-91e1-4366-81ee-0622a962105d>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00634.warc.gz"} |
ball mill particle size distribution
The crushed particles were ground in a porcelain ball mill in order to obtain particle sizes between 75 and 250 μm. After homogenization, 25 aliquots of 40 ml (~ 60 g) were obtained in each round
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A single pass through the mill would create a very wide Particle Size Distribution. In order to narrow the PSD, jet mills are equipped with a classifier. ... Mills with size reduction media: Ball
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Criticalsized particles are those where the product of the mill feedsize distribution and the mill breakage rates result in a buildup of a size range of material in the mill load, the
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Ball size distribution in tumbling mills 37 Milling performance of a ball size distribution 40 Summary 41 ... Grinding rate versus particle size for a given ball diameter 25 Cumulative breakage
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The discharge end design of a ball mill plays an important role in discharging the desired particle sizes (−150 + 10 µm) and the percentage of recirculating load from the discharge end of the
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Six different particle size distribution (GatesGaudinSchuhmann (GGS), RosinRammler (RR), Lognormal, Normal, Gamma, and Swebrec) models were compared under different metallurgical coke grinding
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As a result, ball mills produce a rather wide particlesize distribution (PSD), requiring additional work downstream, while rod mills achieve a lower PSD but produce a coarser product. By
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Mg/h, assuming that the 80percent of particle size of the hydrocyclone overflow will be approximately of diameter 140 m. In 1991, two selfgrinding mills were installed to replace two crushing
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Liu et al. demonstrated that the specific surface area of corn stover powder increased from m 2 /g (untreated) to m 2 /g in a 30min vibratory ball milling time due to the reduction of the
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costs. Despite this, the variability of wi when performing the ball mill Bond's standard test is not always understood enough. This study shows the results of a variability analysis (a 33
factorial design) performed to elucidate the influence on wi of several parameters obtained from the particle size distribution (PSD) in feed and product.
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The planetary ball mill is promising in that it makes grinding to submicron sizes possible by imparting high energy to the ground powder. ... The experiments were conducted on a laboratory scale
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The optimization of processing plants is one of the main concerns in the mining industry, since the comminution stage, a fundamental operation, accounts for up to 70% of total energy consumption.
The aim of this study was to determine the effects that ball size and mill speed exert on the milling kinetics over a wide range of particle sizes. This was done through dry milling and batch
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Breakage rate and particle size have a maximum for each ball size distribution using a pilotscale ball mill on size at maximum breakage (X m) is strongly related to top ball size (D b) in terms
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He concluded that the size distribution from a laboratory rod mill gave a similarshaped size distribution to that of a closed circuit laboratory ball mill. He also demonstrated how a laboratory
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The tests conducted for ball mills [14], cylindrical mills [15,16], and hammer mills [1,17] showed that the particle size distribution curves vary depending on the process and design parameters
of ...
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A section cutthrough of ball mills. A ball mill is a type of grinder used to grind or blend materials for use in mineral dressing processes, paints, pyrotechnics, ceramics, and selective laser
works on the principle of impact and attrition: size reduction is done by impact as the balls drop from near the top of the shell.
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Now, the recent development of highenergy ball mills that reach rotational speeds of up to 2000 rpm provides the opportunity to acquire materials with novel characteristics, ... E. Effect of ball
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As it is difficult to detect the particle size distribution of ball milling process on line, a prediction model of particle size distribution in bauxite continuous ballmilling...
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This paper presents a prediction of the discharge particle size distribution using three components, mass balance model (sometimes referred to as a population balance model, PBM), impact energy
distribution of the mill obtained from the simulation of the charge motion using DEM and breakage characteristics of particles determined from dropball ...
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In this research, ball size distribution which is a function of makeup ball sizes was investigated to optimise the milling stage of a grinding circuit in order to maximise the production of the
narrowlysized mill product for floatation, in this case − 75 + 9 μ m. Section snippets
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Abstract. This study investigates the evolution of dimensional properties of grinding products, namely, the mass, the surface area, the length, and the number of particle distributions with the
energy input in a ball mill. The size analysis of the mill products enables the calculation of the mass distribution of each material at predetermined ...
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Figure Measured and predicted product size distribution for a 297 +210μm feed sample .. 96 Figure Measured and predicted product size distribution for a 210 +150μm feed sample .. 97 Figure
Measured and predicted product size distribution for a 150 +105μm feed sample .. 97
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The Effect of Ball Size on the Energy Efficiency of Hybrid HighPressure Roll Mill/Ball Mill Grinding. Powder Technology, Vol. 105, 1999, 199204. /S(99)001382 Search in Google Scholar
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Volume 25, June 2018, Pages 7984 Effect of ball and feed particle size distribution on the milling efficiency of a ball mill: An attainable region approach N. Hlabangana a, G. Danha, E. Muzenda
Add to Mendeley https://// Get rights and content Under a Creative Commons license open access •
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The specific rates of breakage of particles in a tumbling ball mill are described by the equation S i = ax α i (Q(z), where Q(z) is the probability function which ranges from 1 to 0 as particle
size equation produces a maximum in S, and the particle size of the maximum is related to ball diameter by x m = k 1 d variation of a with ball diameter was found to be of the form ...
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The influence of grinding conditions on the production of fine particles and the width of the particle size distribution produced during ball mill grinding was investigated. The grinding
experiments were carried out varying the grinding ball diameter under dry and wet conditions.
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Digital Resonator
A digital resonator is a special two pole bandpass filter with a pair of complex conjugate poles located near the unit circle. The name resonator refers to the fact that the filter has a larger
magnitude response in the vicinity of the pole locations. Digital resonators are useful in many applications, including simple bandpass filtering and speech generations.
IDEAL FILTERS ARE NOT PHYSICALLY REALIZABLE. Why?
Ideal filters are not physically realizable because Ideal filters are anti-causal and as only causal systems are physically realizable.
Let take example of ideal lowpass filter.
H(ω) = 1 for - ωc ≤ ω ≤ ωc
= 0 elsewhere
The unit sample response of this ideal LPF can be obtained by taking IFT of H(ω).
LSI system is causal if its unit sample response satisfies following condition. h(n) = 0
for n<0
In above figure h(n) extends -∞ to ∞ . Hence h(n) ≠0 for n<0. This means causality condition is not satisfied by the ideal low pass filter. Hence ideal low pass filter is non causal and it is not
physically realizable.
The following examples illustrate the essential features of digital filters.
1. UNITY GAIN FILTER: y[n] = x[n]
Each output value y[n] is exactly the same as the corresponding input value x[n]:
2. SIMPLE GAIN FILTER: y[n] = Kx[n] (K = constant) Amplifier or attenuator) This simply applies a gain factor K to each input value:
3. PURE DELAY FILTER: y[n] = x[n-1]
The output value at time t = nh is simply the input at time t = (n-1)h, i.e. the signal is delayed by time h:
4. TWO-TERM DIFFERENCE FILTER: y[n] = x[n] - x[n-1]
The output value at t = nh is equal to the difference between the current input xn and the previous input x[n-1]:
5. TWO-TERM AVERAGE FILTER: y[n] = (x[n] + x[n-1]) / 2
The output is the average (arithmetic mean) of the current and previous input:
6. THREE-TERM AVERAGE FILTER: yn = (xn + xn-1 + xn-2) / 3
This is similar to the previous example, with the average being taken of the current and two previous inputs.
7. CENTRAL DIFFERENCE FILTER: y[n] = (x[n] - x[n-2]) / 2
This is similar in its effect to example (4). The output is equal to half the change in the input signal over the previous two sampling intervals:
The order of a digital filter can be defined as the number of previous inputs (stored in the processor's memory) used to calculate the current output.
This is illustrated by the filters given as examples in the previous section.
Example (1): y[n] = x[n]
This is a zero order filter, since the current output yn depends only on the current input xn and not on any previous inputs.
Example (2): y[n] = Kx[n]
The order of this filter is again zero, since no previous outputs are required to give the current output value.
Example (3): y[n] = x[n-1]
This is a first order filter, as one previous input (xn-1) is required to calculate y n. (Note that this filter is classed as first-order because it uses one previous input, even though the current
input is not used).
Example (4): y[n] = x[n] - x[n-1]
This is again a first order filter, since one previous input value is required to give the current output.
Example (5): y[n] = (x[n] + x[n-1]) / 2
The order of this filter is again equal to 1 since it uses just one previous input value.
Example (6): y[n] = (x[n] + x[n-1] + x[n-2]) / 3
To compute the current output yn, two previous inputs (xn-1 and xn-2) are needed; this is therefore a second-order filter.
Example (7): y[n] = (x[n] - x[n-2]) / 2
The filter order is again 2, since the processor must store two previous inputs in
order to compute the current output. This is unaffected by the absence of an explicit xn-1 term in the filter expression.
Q) For each of the following filters, state the order of the filter and identify the values of its coefficients:
(a) yn = 2xn - xn-1 A) Order = 1: a0 = 2, a1 = -1
(b) yn = xn-2 + 2xn-2 + xn-3 B) Order = 2: a0 = 0, a1 = 0, a2 = 1
(c) yn = xn - 2xn-1 C) Order = 3: a0 = 1, a1 = -2, a2 = 2, a3 = 1
Number Representation
In digital signal processing, (B + 1)-bit fixed-point numbers are usually represented as two’s- complement signed fractions in the format
bo b-ib-2 …… b-B
The number represented is then
X = -bo + b-i2^- 1 + b-22^- 2 + ••• + b-B 2^-B (3.1)
where bo is the sign bit and the number range is —1 <X < 1. The advantage of this representation is that the product of two numbers in the range from — 1 to 1 is another number in the same range.
Floating-point numbers are represented as
X = (-1)^s m2^c (3.2)
where s is the sign bit, m is the mantissa, and c is the characteristic or exponent. To make the representation of a number unique, the mantissa is normalized so that 0.5 <m < 1.
Although floating-point numbers are always represented in the form of (3.2), the way in which this representation is actually stored in a machine may differ. Since m > 0.5, it is not necessary to
store the 2^- 1 -weight bit of m, which is always set.
Therefore, in practice numbers are usually stored as
X = (-1)^s(0.5 + f)2^c (3.3)
where f is an unsigned fraction, 0 <f < 0.5.
Most floating- point processors now use the IEEE Standard 754 32-bit floating-point format for storing numbers. According to this standard the exponent is stored as an unsigned integer p where
p = c + 126 (3.4)
Therefore, a number is stored as
X = ( -1)^s(0.5 + f )2^p - 1 2 6 (3.5) where s is the sign bit, f is a 23-b unsigned fraction in the range 0 <f < 0.5, and p is an 8 -b unsigned integer in the range 0 <p < 255. The total number of
bits is 1 + 23 + 8 = 32. For example, in IEEE format 3/4 is written (-1)^0 (0.5 + 0.25)2° so s = 0, p = 126, and f = 0.25. The value X = 0 is a unique case and is represented by all bits zero (i.e.,
s = 0, f = 0, and p = 0). Although the 2^-1-weight mantissa bit is not actually stored, it does exist so the mantissa has 24 b plus a sign bit.
1. Fixed-Point Quantization Errors
In fixed- point arithmetic, a multiply doubles the number of significant bits. For example, the product of the two 5-b numbers 0.0011 and 0.1001 is the 10 -b number 00.000 110 11. The extra bit to
the left of the decimal point can be discarded without introducing any error. However, the least significant four of the remaining bits must ultimately be discarded by some form of quantization so
that the result can be stored to 5 b for use in other calculations. In the example above this results in 0.0010 (quantization by rounding) or 0.0001 (quantization by truncating). When a sum of
products calculation is performed, the quantization can be performed either after each multiply or after all products have been summed with double-length precision.
We will examine three types of fixed-point quantization—rounding, truncation, and magnitude truncation. If X is an exact value, then the rounded value will be denoted Q[r] (X), the truncated value Q
[t] (X), and the magnitude truncated value Q[m t] (X). If the quantized value has B bits to the right of the decimal point, the quantization step size is
A = 2^-B (3.6)
Since rounding selects the quantized value nearest the unquantized value, it gives a value which is never more than ± A /2 away from the exact value. If we denote the rounding error by
fr = Qr(X) – X (3.7)
Truncation simply discards the low-order bits, giving a quantized value that is always less than or equal to the exact value so
- A < ft < 0 (3.9)
Magnitude truncation chooses the nearest quantized value that has a magnitude less than or equal to the exact value so
— A <fmt <A (3.10)
The error resulting from quantization can be modeled as a random variable uniformly distributed over the appropriate error range. Therefore, calculations with roundoff error can be considered
error-free calculations that have been corrupted by additive white noise.
2. Floating-Point Quantization Errors
With floating-point arithmetic it is necessary to quantize after both multiplications and additions. The addition quantization arises because, prior to addition, the mantissa of the smaller number in
the sum is shifted right until the exponent of both numbers is the same. In general, this gives a sum mantissa that is too long and so must be quantized.
We will assume that quantization in floating-point arithmetic is performed by rounding. Because of the exponent in floating-point arithmetic, it is the relative error that is important. | {"url":"https://www.brainkart.com/article/Digital-Resonator_13054/","timestamp":"2024-11-03T06:43:33Z","content_type":"text/html","content_length":"80838","record_id":"<urn:uuid:2dbeb27c-d8b4-4be5-bf6c-2253d0b2397c>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00607.warc.gz"} |
Free online literal equation calculator
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Multiply, Dividing;
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and Solving Equations
LinearEquations sxodasos Posted: Saturday 30th of Dec 07:29
Solving a Quadratic I am in a real bad state of mind. Somebody help me please. I experience a lot of issues with interval notation, converting decimals and
Equation inequalities and especially with free online literal equation calculator. I have to show some immediate growth in my math. I came to know
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Equations Introduction tool which helps me with my studies. Any link is greatly appreciated. Thanks.
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System of Equations Vofj Timidrov Posted: Monday 01st of Jan 07:26
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Functions Zeros, and
Applications Registered:
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Functions From: Bulgaria
Linear equations in two
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Comparing and Ordering alhatec16 Posted: Wednesday 03rd of Jan 09:20
Rational Numbers It would really be nice if you could let us know about a software that can provide both. If you could get us a resource that would give a
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Solving Exponential and
Logarithmic Equations
Solving Systems of Dnexiam Posted: Wednesday 03rd of Jan 16:10
Linear Equations I would suggest using Algebrator. It not only assists you with your math problems, but also provides all the required steps in detail so
DISTANCE,CIRCLES,AND that you can enhance the understanding of the subject.
Solving Quadratic
Equations Registered:
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Inequalit From: City 17
Applications of Systems
of Linear Equations in
Two Variables
Systems of Linear
Test Description for
RATIONAL EX
Exponential and
Logarithmic Equations
Systems of Linear
Equations: Cramer's Rule
Introduction to Systems
of Linear Equations
Literal Equations &
Equations and
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SOLVING LINEAR AND
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Linear equations in two
free online literal equation calculator
Related topics:
Home online free algebra solver working | solving algebra homework step by step | multivariable algebra | glencoe biology worksheet answers | 7th grade exponent
Rational Expressions worksheet | free 8th grade algebra study material | middle school physics calculation worksheets | differential equations derivative squared | "prime factored
Graphs of Rational form" | simplify calculator division, square root, radicals, and fractions | matlab divide ellipse | algebra 2 10th grade rinehart | 10th maths sample question |
Functions non linear differential equation matlab
Solve Two-Step Equations
Multiply, Dividing;
Exponents; Square Roots; Author Message
and Solving Equations
LinearEquations sxodasos Posted: Saturday 30th of Dec 07:29
Solving a Quadratic I am in a real bad state of mind. Somebody help me please. I experience a lot of issues with interval notation, converting decimals and
Equation inequalities and especially with free online literal equation calculator. I have to show some immediate growth in my math. I came to know
Systems of Linear there are many Software Tools available online which can help you in algebra. I can spend some money too for an effective and inexpensive
Equations Introduction tool which helps me with my studies. Any link is greatly appreciated. Thanks.
Equations and Registered:
Inequalities 15.05.2002
Solving 2nd Degree From: Paralia
Review Solving Quadratic
System of Equations Vofj Timidrov Posted: Monday 01st of Jan 07:26
Solving Equations & The perspective you’ve adopted towards the free online literal equation calculator is not the a good one. I do understand that one can’t
Inequalities really think of anything else in such a scenario . Its good that you still want to try. My key to successful equation solving is Algebrator
Linear Equations I would advise you to give it a try at least once.
Functions Zeros, and
Applications Registered:
Rational Expressions and 06.07.2001
Functions From: Bulgaria
Linear equations in two
Lesson Plan for
Comparing and Ordering alhatec16 Posted: Wednesday 03rd of Jan 09:20
Rational Numbers It would really be nice if you could let us know about a software that can provide both. If you could get us a resource that would give a
LinearEquations step-by-step solution to our problem, it would really be great . Please let us know the reliable links from where we can get the tool .
Solving Equations
Radicals and Rational
Exponents Registered:
Solving Linear Equations 10.03.2002
Systems of Linear From: Notts, UK.
Solving Exponential and
Logarithmic Equations
Solving Systems of Dnexiam Posted: Wednesday 03rd of Jan 16:10
Linear Equations I would suggest using Algebrator. It not only assists you with your math problems, but also provides all the required steps in detail so
DISTANCE,CIRCLES,AND that you can enhance the understanding of the subject.
Solving Quadratic
Equations Registered:
Quadratic and Rational 25.01.2003
Inequalit From: City 17
Applications of Systems
of Linear Equations in
Two Variables
Systems of Linear
Test Description for
RATIONAL EX
Exponential and
Logarithmic Equations
Systems of Linear
Equations: Cramer's Rule
Introduction to Systems
of Linear Equations
Literal Equations &
Equations and
Inequalities with
Absolute Value
Rational Expressions
SOLVING LINEAR AND
Steepest Descent for
Solving Linear Equations
The Quadratic Equation
Linear equations in two
Rational Expressions
Graphs of Rational
Solve Two-Step Equations
Multiply, Dividing;
Exponents; Square Roots;
and Solving Equations
Solving a Quadratic
Systems of Linear
Equations Introduction
Equations and
Solving 2nd Degree
Review Solving Quadratic
System of Equations
Solving Equations &
Linear Equations
Functions Zeros, and
Rational Expressions and
Linear equations in two
Lesson Plan for
Comparing and Ordering
Rational Numbers
Solving Equations
Radicals and Rational
Solving Linear Equations
Systems of Linear
Solving Exponential and
Logarithmic Equations
Solving Systems of
Linear Equations
Solving Quadratic
Quadratic and Rational
Applications of Systems
of Linear Equations in
Two Variables
Systems of Linear
Test Description for
RATIONAL EX
Exponential and
Logarithmic Equations
Systems of Linear
Equations: Cramer's Rule
Introduction to Systems
of Linear Equations
Literal Equations &
Equations and
Inequalities with
Absolute Value
Rational Expressions
SOLVING LINEAR AND
Steepest Descent for
Solving Linear Equations
The Quadratic Equation
Linear equations in two
free online literal equation calculator
Related topics:
online free algebra solver working | solving algebra homework step by step | multivariable algebra | glencoe biology worksheet answers | 7th grade exponent worksheet | free 8th grade
algebra study material | middle school physics calculation worksheets | differential equations derivative squared | "prime factored form" | simplify calculator division, square root,
radicals, and fractions | matlab divide ellipse | algebra 2 10th grade rinehart | 10th maths sample question | non linear differential equation matlab
Author Message
sxodasos Posted: Saturday 30th of Dec 07:29
I am in a real bad state of mind. Somebody help me please. I experience a lot of issues with interval notation, converting decimals and inequalities and especially
with free online literal equation calculator. I have to show some immediate growth in my math. I came to know there are many Software Tools available online which can
help you in algebra. I can spend some money too for an effective and inexpensive tool which helps me with my studies. Any link is greatly appreciated. Thanks.
From: Paralia
Vofj Timidrov Posted: Monday 01st of Jan 07:26
The perspective you’ve adopted towards the free online literal equation calculator is not the a good one. I do understand that one can’t really think of anything else
in such a scenario . Its good that you still want to try. My key to successful equation solving is Algebrator I would advise you to give it a try at least once.
From: Bulgaria
alhatec16 Posted: Wednesday 03rd of Jan 09:20
It would really be nice if you could let us know about a software that can provide both. If you could get us a resource that would give a step-by-step solution to our
problem, it would really be great . Please let us know the reliable links from where we can get the tool .
From: Notts, UK.
Dnexiam Posted: Wednesday 03rd of Jan 16:10
I would suggest using Algebrator. It not only assists you with your math problems, but also provides all the required steps in detail so that you can enhance the
understanding of the subject.
From: City 17
Author Message
sxodasos Posted: Saturday 30th of Dec 07:29
I am in a real bad state of mind. Somebody help me please. I experience a lot of issues with interval notation, converting decimals and inequalities and especially with free
online literal equation calculator. I have to show some immediate growth in my math. I came to know there are many Software Tools available online which can help you in algebra.
I can spend some money too for an effective and inexpensive tool which helps me with my studies. Any link is greatly appreciated. Thanks.
From: Paralia
Vofj Timidrov Posted: Monday 01st of Jan 07:26
The perspective you’ve adopted towards the free online literal equation calculator is not the a good one. I do understand that one can’t really think of anything else in such a
scenario . Its good that you still want to try. My key to successful equation solving is Algebrator I would advise you to give it a try at least once.
From: Bulgaria
alhatec16 Posted: Wednesday 03rd of Jan 09:20
It would really be nice if you could let us know about a software that can provide both. If you could get us a resource that would give a step-by-step solution to our problem,
it would really be great . Please let us know the reliable links from where we can get the tool .
From: Notts, UK.
Dnexiam Posted: Wednesday 03rd of Jan 16:10
I would suggest using Algebrator. It not only assists you with your math problems, but also provides all the required steps in detail so that you can enhance the understanding
of the subject.
From: City 17
Posted: Saturday 30th of Dec 07:29
I am in a real bad state of mind. Somebody help me please. I experience a lot of issues with interval notation, converting decimals and inequalities and especially with free online literal equation
calculator. I have to show some immediate growth in my math. I came to know there are many Software Tools available online which can help you in algebra. I can spend some money too for an effective
and inexpensive tool which helps me with my studies. Any link is greatly appreciated. Thanks.
Posted: Monday 01st of Jan 07:26
The perspective you’ve adopted towards the free online literal equation calculator is not the a good one. I do understand that one can’t really think of anything else in such a scenario . Its good
that you still want to try. My key to successful equation solving is Algebrator I would advise you to give it a try at least once.
Posted: Wednesday 03rd of Jan 09:20
It would really be nice if you could let us know about a software that can provide both. If you could get us a resource that would give a step-by-step solution to our problem, it would really be
great . Please let us know the reliable links from where we can get the tool .
Posted: Wednesday 03rd of Jan 16:10
I would suggest using Algebrator. It not only assists you with your math problems, but also provides all the required steps in detail so that you can enhance the understanding of the subject. | {"url":"https://rational-equations.com/in-rational-equations/graphing-equations/free-online-literal-equation.html","timestamp":"2024-11-06T05:21:42Z","content_type":"text/html","content_length":"96952","record_id":"<urn:uuid:3d3b5ff6-7562-439a-9350-fbb0a17377a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00760.warc.gz"} |
Inverse problem for the Forgotten and the hyper Zagreb indices of trees
[1] H. Aram and N. Dehgardi, Reformulated F-index of graph operations, Commun. Comb. Optim. 2 (2017), no. 2, 87–98.
[2] M. Eliasi and A. Ghalavand, On trees and the multiplicative sum Zagreb index, Commun. Comb. Optim. 1 (2016), no. 2, 137–148.
[3] W. Gao, M.K. Siddiqui, N.A. Rehman, and M.H. Muhammad, Topological characterization of dendrimer, benzenoid, and nanocone, Zeitschrift für Naturforschung C 74 (2018), no. 1-2, 35–43.
[4] W. Gao, W. Wang, and M.R. Farahani, Topological indices study of molecular structure in anticancer drugs, J. Chemistry 2016 (2016), Article ID 3216327.
[5] I. Gutman, M. Togan, A. Yurttas, A.S. Cevik, and I.N. Cangul, Inverse problem for sigma index, MATCH Commun. Math. Comput. Chem. 79 (2018), no. 2, 491–508.
[6] X. Li, Z. Li, and L. Wang, The inverse problems for some topological indices in combinatorial chemistry, J. Comput. Biology 10 (2003), no. 1, 47–55.
[7] T. Réti, R. Sharafdini, A. Drégelyi-Kiss, and H. Haghbin, Graph irregularity indices used as molecular descriptors in QSPR studies, MATCH Commun. Math. Comput. Chem. 79 (2018), 509–524.
[8] D. Vukičević and M. Gašperov, Bond additive modeling 1. Adriatic indices, Croatica Chemica Acta 83 (2010), no. 3, 243–260.
[9] A. Yurtas, M. Togan, V. Lokesha, I.N. Cangul, and I. Gutman, Inverse problem for Zagreb indices, J. Math. Chemistry 57 (2019), no. 2, 609–615. | {"url":"https://comb-opt.azaruniv.ac.ir/article_14266.html","timestamp":"2024-11-09T01:53:42Z","content_type":"text/html","content_length":"48098","record_id":"<urn:uuid:cff8f48f-b415-423c-a445-04014110cb56>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00472.warc.gz"} |
[Haskell-cafe] Proving correctness
Pavel Perikov perikov at gmail.com
Mon Feb 14 21:03:26 CET 2011
Sorely, Haskell can't prove logic with it. No predicates on values, guarantee that proof is not _|_. Haskell makes bug free software affordable, that's true. But it's not a proof assistant.
On 14.02.2011, at 22:57, Albert Y. C. Lai wrote:
> On 11-02-12 09:40 PM, Brandon S Allbery KF8NH wrote:
>> Only up to a point. While most of the responses so far focus on the
>> question from one direction, the other is epitomized by a Knuth quote:
>> "Beware of bugs in the above code; I have only proved it correct, not tried it."
> Knuth's definition of "proof" is of the sketchy kind of the mathematics community, not remotely close to the Coq kind. Even Dijsktra's and Bird's kind offers higher assurance than the traditional mathematician's sketchy kind.
> There are still gaps, but drastically narrower than Knuth's gaps, and bridgeable with probability arbitrarily close to 1:
> Possible defects in theorem provers: Use several theorem provers and/or several independent alternative implementations (both alternative software and alternative hardware).
> Possible deviation of Haskell compilers from your assumed formal semantics of Haskell: Verify the compilers too, or modify the compilers to emit some sort of proof-carrying code.
> Possible defects in target hardware: The hardware people are way ahead in improving both formal verification and manufacturing processes to reduce defects.
> When John Harrison ( http://www.cl.cam.ac.uk/~jrh13/ ) has a proof for a floating-point algorithm, I would not dare to apply the Knuth quote on it.
> _______________________________________________
> Haskell-Cafe mailing list
> Haskell-Cafe at haskell.org
> http://www.haskell.org/mailman/listinfo/haskell-cafe
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QUATERNIONS, in mathematics. The word " quaternion " properly means " a set of four." In employing such a word to denote a new mathematical method, Sir W. R. Hamilton was probably influenced by the
recollection of its Greek equivalent, the Pythagorean Tetractys (TerpaKriis, the number four), the mystic source of all things. Quaternions (as a mathematical method) is an extension, or improvement,
of Cartesian geometry, in which the artifices of co-ordinate axes, etc., are got rid of, all directions in space being treated on precisely the same terms. It is therefore, except in some of its
degraded forms, possessed of the perfect isotropy of Euclidian space. From the purely geometrical point of view, a quaternion may be regarded as the quotient of two directed lines in space or, what
comes to the same thing, as the factor, or operator, which changes one directed line into another. Its analytical definition will appear later.
History. The evolution of quaternions belongs in part to each of two weighty branches of mathematical history the interpretation of the imaginary (or impossible) quantity of common algebra, and the
Cartesian application of algebra to geometry. Sir W. R. Hamilton was led to his great invention by keeping geometrical applications constantly before him while he endeavoured to give a real
significance to V-L We will therefore confine ourselves, so far as his predecessors are concerned, to attempts at interpretation which had geometrical applications in view.
One geometrical interpretation of the negative sign of algebra was early seen to be mere reversal of direction along a line. Thus, when an image is formed by a plane mirror, the distance of any point
in it from the mirror is simply the negative of that of the corresponding point of the object. Or if motion in one direction along a line be treated as positive, motion in the opposite direction
along the same line is negative. In the case ot time, measured from the Christian era, this distinction is at once given.by the letters A.D. or B.C., prefixed to the date. And to find the position,
in time, of one event relatively to another, we have only to subtract the date of the second (taking account of its sign) from that of the first. Thus 1 to find the interval between the battles of
Marathon (4QO B.C.) and Waterloo (A.D. 1815) we have + l8l5-(-49o)=2305 years.
And it is obvious that the same process applies in all cases in which we deal with quantities which may be regarded as of one directed dimension only, such as distances along a line, rotations about
an axis, etc. But it is essential to notice that this is by no means necessarily true of operators. To turn a line through a certain angle in a given plane, a certain operator is required; but when
we wish to turn it through an equal negative angle we must not, in general, employ the negative of the former operator. For the negative of the operator which turns a line through a given angle in a
given plane will in all cases produce the negative of the original result, which is not the result of the reverse operator, unless the angle involved be an odd multiple of a right angle. This is, of
course, on the usual assumption that the sign of a product is changed when that of any one of its factors is changed, which merely means that-i is commutative with all other quantities.
John Wallis seems to have been the first to push this idea further. In his Treatise of Algebra (1685) he distinctly proposes to construct the imaginary roots of a quadratic equation by going out of
the line on which the roots, if real, would have been constructed.
In 1804 the Abbe Buee (Phil. Trans., 1806), apparently without any knowledge of Wallis's work, developed this idea so far as to make it useful in geometrical applications. He gave, in fact, the
theory of what in Hamilton's system is called Composition of Vectors in one plane i.e. the combination, by + and , of complanar directed lines. His constructions are based on the idea' that the
imaginaries =t V i represent a unit line, and its reverse, perpendicular to the line on which the real units =*= i are measured. In this sense the imaginary expression a + b V i is constructed by
measuring a length a along the fundamental line (for real quantities), and from its extremity a line of length b in some direction perpendicular to the fundamental line. But he did not attack the
question of the representation of products or quotients of directed lines. The step he took is really nothing more than the kinematical principle of the composition of linear velocities, but
expressed in terms of the algebraic imaginary.
In 1806 (the year of publication of BueVs paper) Jean Robert Argand published a pamphlet 2 in which precisely the same ideas are developed, but to a considerably greater extent. For an interpretation
is assigned to the product of two directed lines in one plane, when each is expressed as the sum of a real and an imaginary part. This product is interpreted as another directed line, forming the
fourth term of a proportion, of which the first 1 Strictly speaking, this illustration of Tait's is in error by unity because In our calendar there is no year denominated zero. Thus the interval
between June the first of i B.C. and June the first of i A.D. is one year, and not two years as the text implies. (A.McA.)
1 Essai sur une maniere He rcprisenter let Quantites Imaginaires dans les Constructions Geometriquts. A second edition was published by J. Houel (Paris, 1874). There is added an important Appendix,
consisting of the papers from Gergonne's Annales which are referred to in the text above. Almost nothing can, it seems, be learned of Argand's private life, except that in all probability he was born
at Geneva in 1768.
term is the real (positive) unit-line, and the other two are the factor-lines. Argand's work remained unnoticed until the question was again raised in Gergonne's Annales, 1813, by J. F. Franjais.
This writer stated that he had found the germ of his remarks among the papers of his deceased brother, and that they had come from Legendre, who had himself received them from some one unnamed. This
led to a letter from Argand, in which he stated his communications with Legendre, and gave a resume of the contents of his pamphlet. In a further communication to the Annales, Argand pushed on the
applications of his theory. He has given by means of it a simple proof of the existence of roots, and no more, in every rational algebraic equation of the th order with real coefficients. About 1828
John Warren (1796-1852) in England, and C. V. Mourey in France, independently of one another and of Argand, reinvented these modes of interpretation; and still later, in the writings of Cauchy, Gauss
and others, the properties of the expression a + b V i were developed into the immense and most important subject now called the theory of complex numbers (see NUMBER). From the more purely
symbolical view it was developed by Peacock, De Morgan, etc., as double algebra.
Argand's method may be put, for reference, in the following form. The directed line whose length is a, and which makes an anglefl with the real (positive) unit line, is expressed by a(cps0+ sin 6),
where i is regarded as + V i - The sum of two such lines (formed by adding together the real and the imaginary parts of two such expressions) can, of course, be expressed as a third directed line)
the diagonal of the parallelogram of which they are conterminous sides. The product, P, of two such lines is, as we have seen, given by i:a(cose + sin0) : : a'(cos0' + t sin 6'): P, or P = aa'|cos(0+
') + ' sin (+')}.
Its length is, therefore, the product of the lengths of the factors, and its inclination to the real unit is the sum of those of the factors. If we write the expressions for the two lines in the form
A+B*, A'+B'i, the product is AA'-BB'+t(AB'+BA'); and the fact that the length of the product line is the product of those of the factors is seen in the form (A'+B)(A' 2 +B' 2 ) = (AA'-BB') 2 +
(AB'+BA')*. In the modern theory of complex numbers this is expressed by saying that the Norm of a product is equal to the product of the norms of the factors.
Argand's attempts to extend his method to space generally were fruitless. The reasons will be obvious later; but we mention them just now because they called forth from F. J. Servois (Gergonne's
Annales, 1813) a very remarkable comment, in which was contained the only yet discovered trace of an anticipation of the method of Hamilton. Argand had been led to deny that such an expression as i*
could be expressed in the form A+Bj, although, as is well known, Euler showed that one of its values is a real quantity, the exponential function of ir/2. Servois says, with reference to the general
representation of a directed line in space:
" L'analogie semblerait exiger que le trinome fQt de la forme p cos a+q cos ft+r cos 7: o, 0, y 6tant les angles d'une droite avec trois axes rectangulaires; et qu'on edt (p cos a. + q cos (3 + r cos
y)(p' cos a + q' cos ft + r' cos 7) = cos*a-f-cos 2 /3+cos 2 7 = i. Les valeurs de p, q, r, p', q,' r', qui satisferaient a cette condition seraient absurdes; mais seraient-elies imaginaires,
reductibles a la forme ge'ndrale A+BV I? Voila une question d'analyse fort singuliere que je soumets a vos lumieres. La simple proposition que je vous en fais suffit pour vous faire voir que je ne
crois point que toute fonction analytique non reelle soit vraiment reductible a la forme A+B V I."
As will be seen later, the fundamental *', j, k of quarternions, with their reciprocals, furnish a set of six quantities which satisfy the conditions imposed by Servois. And it is quite certain that
they cannot be represented by ordinary imaginaries. Something far more closely analogous to quaternions than anything in Argand's work ought to have been suggested by De Moivre's theorem (1730).
Instead of regarding, as Bu6e and Argand had done, the expression o(cos 9 + i sin 6) as a directed line, let us suppose it to represent the operator which, when applied to any line in the plane in
which 6 is measured, turns it in that plane through the angle 6, and at the same time increases its length in the ratio a : i. From the new point of view we see at once, as it were, why it is true
that (cos 0+ ' sin )" =cos m8+ i sin W0.
For this equation merely states that m turnings of a line through successive equal angles, in one plane, give the same result as a single turning through m times the common angle. To make this
process applicable to any plane in space, it is clear that we must have a special value of i for each such plane. In other words, a unit line, drawn in any direction whatever, must have i for its
square. In such a system there will be no line in space specially distinguished as the real unit line: all will be alike imaginary, or rather alike real. We may state, in passing, that every
quaternion can be represented as a (cos 0+7T sin 6), where a is a real number, 6 a real angle, and ir a directed unit line whose square is i. Hamilton took this grand step, but, as we have already
said, without any help from the previous work of De Moivre. The course of his investigations is minutely described in the preface to his first great work (Lectures on Quaternions, 1853) on the
subject. Hamilton, like most of the many inquirers who endeavoured to give a real interpretation to the imaginary of common algebra, found that at least two kinds, orders or ranks of quantities were
necessary for the purpose. But, instead of dealing with points on a line, and then wandering out at right angles to it, as Buee and Argand had done, he chose to look on algebra as the science of "
pure time," 1 and to investigate the properties of " sets " of time-steps. In its essential nature a set is a linear function of any number of " distinct " units of the same species. Hence the
simplest form of a set is a " couple "; and it was to the possible laws of combination of couples that Hamilton first directed his attention. It is obvious that the way in which the two separate
time-steps are involved in the couple will determine these laws of combination. But Hamilton's special object required that these laws should be such as to lead to certain assumed results; and he
therefore commenced by assuming these, and from the assumption determined how the separate time-steps must be involved in the couple. It we use Roman letters for mere numbers, capitals for instants
of time, Greek letters for time-steps, and a parenthesis to denote a couple, the laws assumed by Hamilton as the basis of a system were as follows:
(B,, B 2 )-(A,, A 2 ) = (B,-A 1 , B 2 -A 2 ) = (o, 0); (a, b) (a,/3) = (aa-b/J, ba+a/3).' To show how we give, by such assumptions, a real interpretation to the ordinary algebraic imaginary, take the
simple case a=o, b= i, and the second of the above formulae gives (o, i)(a,/J) = (-/3, a). Multiply once more by the number-couple (o, i), and we have Thus the number-couple (o, i), when twice
applied to a step-couple, simply changes its sign. That we have here a perfectly real and intelligible interpretation of the ordinary algebraic imaginary is easily seen by an illustration, even if it
be a somewhat extravagant one. Some Eastern potentate, possessed of absolute power, covets the vast possessions of his vizier and of his barber. He determines to rob them both (an operation which may
be very satisfactorily expressed by i); but, being a wag, he chooses his own way of doing it. He degrades his vizier to the office of barber, taking all his goods in the process; and makes the barber
his vizier. Next day he repeats the operation. Each of the victims has been restored to his former rank, but the operator i has been applied to both.
Hamilton, still keeping prominently before him as his great object the invention of a method applicable to space of three dimensions, proceeded to study the properties of triplets of the form x+iy-\
-jz, by which he proposed to represent the directed line in space whose projections on the co-ordinate axes are x, y, z. The composition of two such lines by the algebraic 1 Theory of Conjugate
Functions, or Algebraic Couples, with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time, read in 1833 and 1835, and published in Trans. R. I. A. xvii. ii. (1835).
1 Compare these with the long-subsequent ideas of Grassmann.
addition of their several projections agreed with the assumption of Buee and Argand for the case of coplanar lines. But, assuming the distributive principle, the product of two lines appeared to give
the expression xx'-yy'-zz'+i(yx'+xy")+j(xz'+zx')+ij(yz'+z y '-). For the square of j, like that of i, was assumed to be negative unity. But the interpretation of ij presented a difficulty in fact the
main difficulty of the whole investigation and it is specially interesting to see how Hamilton attacked it. He saw that he could get a hint from the simpler case, already thoroughly discussed,
provided the two factor lines were in one plane through the real unit line. This requires merely that y : z :: y' : z' ; or yz'zy'=o; but then the product should be of the same form as the separate
factors. Thus, in this special case, the term in ij ought to vanish. But the numerical factor appears to be yz'+zy', while it is the quantity yz'zy' which really vanishes. Hence Hamilton was at first
inclined to think that ij must be treated as nil. But he soon saw that " a less harsh supposition " would suit the simple case. For his speculations on sets had already familiarized him with the idea
that multiplication might in certain cases not be commutative; so that, as the last term in the above product is made up of the two separate terms ijyz' a.ndjizy', the term would vanish of itself
when the factorlines are coplanar provided ij= ji, for it would then assume the form ij(yz' zy'). He had now the following expression for the product of any two directed lines:
xx' -yy'-zz'+i(yx'+ xy") +j(xz'+zx") +ij (yz r -zy'). But his result had to be submitted to another test, the Law of the Norms. As soon as he found, by trial, that this law was satisfied, he took the
final step. " This led me," he says, " to conceive that perhaps, instead of seeking to confine ourselves to triplets, ... we ought to regard these as only imperfect forms of Quaternions, . . . and
that thus my old conception of sets might receive a new and useful application." In a very short time he settled his fundamental assumptions. He had now three distinct space-units, i, j, k; and the
following conditions regulated their combination by multiplication:
&=]* = & i, ij= ji = k,jk = kj=i, ki= ik=j.' And now the product of two quaternions could be at once expressed as a third quaternion, thus (o +ib +jc +kd) (a'+ib'+jc'+kd") = A -HB +JC +kD, where A =
aa'-bb'-cc'-dd', B=ab'+ba'+cd'dc', C=ac'+ca'+db'-bd', D=ad'+da'+bc'-cb'.
Hamilton at once found that the Law of the Norms holds, not being aware that Euler had long before decomposed the product of two sums of four squares into this very set of four squares. And now a
directed line in space came to be represented as ix+jy+kz, while the product of two lines is the quaternion -(xx'+yy'+zz r )+i(yz'-zy r )+j(zx'-xz r )+k(xy'- y x"). To any one acquainted, even to a
slight extent, with the elements of Cartesian geometry of three dimensions, a glance at the extremely suggestive constituents of this expression shows how justly Hamilton was entitled to say: " When
the conception . . . had been so far unfolded and fixed in my mind, I felt that the new instrument for applying calculation to geometry, for which I had so long sought, was now, at least in part,
attained." The date of this memorable discovery is October 16, 1843.
Suppose, for simplicity, the factor-lines to be each of unit length. Then x, y, z, x 1 , y', z' express their direction-cosines. Also, if be the angle between them, and x", y", z* the
direction-cosines of a line perpendicular to each of them, we have xx'+yy'+zz' = cos 9, yz'zy* = x* sin 8, etc., so that the product of two unit lines is now expressed as cos9+(ix"-\-jy"+kz") sin 0.
Thus, when the factors ' It will be easy to see that, instead of the last three of these, we may write the single one ijk = i.
are parallel, or 8 = 0, the product, which is now the square of any (unit) line is I. And when the two factor lines are at right angles to one another, or = ir/2, the product is simply ix* + iy'+kz',
the perpendicular to both. Hence, and in this lies the main element of the symmetry and simplicity of the quaternion calculus, all systems of three mutually rectangular unit lines in space have the
same properties as the fundamental system i, j, k. In other words, if the system (considered as rigid) be made to turn about till the first factor coincides with * and the second with j, the product
will coincide with k. This fundamental system, therefore, becomes unnecessary; and the quaternion method, in every case, takes its reference lines solely from the problem to which it is applied. It
has therefore, as it were, a unique internal character of its own.
Hamilton, having gone thus far, proceeded to evolve these results from a characteristic train of a priori or metaphysical reasoning.
Let it be supposed that the product of two directed lines is something which has quantity; i.e. it may be halved, or doubled, for instance. Also let us assume (a) space to have the same properties in
all directions, and make the convention (6) that to change the sign of any one factor changes the sign of a product. Then the product of two lines which have the same direction cannot be, even in
part, a directed quantity. For, if the directed part have the same direction as the factors, (ft) -shows that it will be reversed by reversing either, and therefore will recover its original
direction when both are reversed. But this would obviously be inconsistent with (a). If it be perpendicular to the factor lines, (a) shows that it must have simultaneously every such direction. Hence
it must be a mere number.
Again, the product of two lines at right angles to one another cannot, even in part, be a number. For the reversal of either factor must, by (6), change its sign. But, if we look at the two factors
in their new position by the light of (a), we see that the sign must not change. But there is nothing to prevent its being represented by a directed line if, as further applications of (a) and (6)
show we must do, we take it perpendicular to each of the factor lines. Hamilton seems never to have been quite satisfied with the apparent heterogeneity of a quaternion, depending as it does on a
numerical and a directed part. He indulged in a great deal of speculation as to the existence of an extra-spatial unit, which was to furnish the ration d'itre of the numerical part, and render the
quaternion homogeneous as well as linear. But for this we must refer to his own works.
Hamilton was not the only worker at the theory of sets. The year after the first publication of the quaternion method, there appeared a work of great originality, by Grassmann, 1 in which results
closely analogous to some of those of Hamilton were given. In particular, two species of multiplication (" inner " and " outer ") of directed lines in one plane were given. The results of these two
kinds of multiplication correspond respectively to the numerical and the directed parts of Hamilton's quaternion product. But Grassmann distinctly states in his preface that he had not had leisure to
extend his method to angles in space. Hamilton and Grassmann, while their earlier work had much in common, had very different objects in view. Hamilton had geometrical application as his main object;
when he realized the quaternion system, he felt that his object was gained, and thenceforth confined himself to the development of his methqd. Grassmann's object seems to have been, all along, of a
much more ambitious character, viz. to discover, if possible, a system or systems in which every conceivable mode of dealing with sets should be included. That he made very great advances towards the
attainment of this object all will allow; that his method, even as completed in 1862, fully attains it is not so certain. But his claims, however great they may be, can in no way conflict with those
of Hamilton, whose mode of multiplying couples (in which the " inner " and " outer " multiplication are essentially involved) was produced in 1833, and whose quaternion system was completed and
published before Grassmann had elaborated for press even the rudimentary portions of his own system, in which the veritable difficulty of the whole subject, the application to angles in space, had
not even been attacked. Grassmann made in 1854 a somewhat savage onslaught on Cauchy and De St Venant, the former of whom had invented, while the latter had exemplified in application, the system of
" clefs algfbriques," which is almost precisely 1 Die Ausdehnungslehre, Leipsic, 1844; 2nd ed., vollstandig und in stronger Form bearbeitet, Berlin, 1862. See also the collected works of Mobius, and
those of Clifford, for a general explanation of Grassmann's method.
that of Grassmann. But it is to be observed that Grassmann, though he virtually accused Cauchy of plagiarism, does not appear to have preferred any such charge against Hamilton. He does not allude to
Hamilton in the second edition of his work. But in 1877, in the Matttematische Annalen, xii., he gave a paper "On the Place of Quaternions in the Ausdehnungslehre," in which he condemns, as far as he
can, the nomenclature and methods of Hamilton.
There are many other systems, based on various principles, which have been given for application to geometry of directed lines, but those which deal with products of lines are all of such complexity
as to be practically useless in application. Others, such as the Barycentrische Calciil of Mobius, and the Methodc des equipollences of Bellavitis, give elegant modes of treating space problems, so
long as we confine ourselves to prpjective geometry and matters of that order; but they are limited in their field, and therefore need not be discussed here. More general systems, having close
analogies to quaternions, have been given since Hamilton's discovery was published. As instances we may take Goodwin's and O'Brien's papers in the Cambridge Philosophical Transactions for 1849. (See
also ALGEBRA: special kinds.)
Relations to other Branches of Science. The above narrative shows how close is the connexion between quaternions and the ordinary Cartesian space-geometry. Were this all, the gain by their
introduction would consist mainly in a clearer insight into the mechanism of co-ordinate systems, rectangular or not a very important addition to theory, but little advance so far as practical
application is concerned. But, as yet, we have not taken advantage of the perfect symmetry of the method. When that is done, the full value of Hamilton's grand step becomes evident, and the gain is
quite as extensive from the practical as from the theoretical point of view. Hamilton, in fact, remarks, 2 " I regard it as an inelegance and imperfection in this calculus, or rather in the state to
which it has hitherto been unfolded, whenever it becomes, or seems to become, necessary to have recourse ... to the resources of ordinary algebra, for the solution of equations in quaternions." This
refers to the use of the x, y, z co-ordinates, associated, of course, with i,j, k. But when, instead of the highly artificial expression ix-\-jy-\-kz, to denote a finite directed line, we employ a
single letter, a (Hamilton uses the Greek alphabet for this purpose), and find that we are permitted to deal with it exactly as we should have dealt with the more complex expression, the immense gain
is at least in part obvious. Any quaternion may now be expressed in numerous simple forms. Thus we may regard it as the sum of a number and a line, a+a, or as the product, j3-y, or the quotient, 5-',
of two directed lines, etc., while, in many cases, we may represent it, so far as it is required, by a single letter such as q, r, etc.
Perhaps to the student there is no part of elementary mathematics so repulsive as is spherical trigonometry. Also, everything relating to change of systems of axes, as for instance in the kinematics
of a rigid system, where we have constantly to consider one set of rotations with regard to axes fixed in space, and another set with regard to axes fixed in the system, is a matter of troublesome
complexity by the usual methods. But every quaternion formula is a proposition in spherical (sometimes degrading to plane) trigonometry, and has the full advantage of the symmetry of the method. And
one of Hamilton's earliest advances in the study of his system (an advance independently made, only a few months later, by Arthur Cayley) was the interpretation of the singular operator q( )q- 1 ,
where q is a quaternion. Applied to any directed line, this operator at once turns it, conically, through a definite angle, about a definite axis. Thus rotation is now expressed in symbols at least
as simply as it can be exhibited by means of a model. Had quaternions effected nothing more than this, they would still have inaugurated one of the most necessary, and apparently impracticable, of
The physical properties of a heterogeneous body (provided they vary continuously from point to point) are known to depend, in the neighbourhood of any one point of the body, on a quadric function of
the co-ordinates with reference to that point. The 1 Lectures on Quaternions, 513.
same is true of physical quantities such as potential, temperature, etc., throughout small regions in which their variations are continuous; and also, without restriction of dimensions, of moments of
inertia, etc. Hence, in addition to its geometrical applications to surfaces of the second order, the theory of quadric functions of position is of fundamental importance in physics. Here the
symmetry points at once to the selection of the three principal axes as the directions for *, /, k; and it would appear at first sight as if quaternions could not simplify, though they might improve
in elegance, the solution of questions of this kind. But it is not so. Even in Hamilton's earlier work it was shown that all such questions were reducible to the solution of linear equations in
quaternions; and he proved that this, in turn, depended on the determination of a certain operator, which could be represented for purposes of calculation by a single symbol. The method is
essentially the same as that developed, under the name of " matrices," by Cayley in 1858; but it has the peculiar advantage of the simplicity which is the natural consequence of entire freedom from
conventional reference lines.
Sufficient has already been said to show the close connexion between quaternions and the theory of numbers. But one most important connexion with modern physics must be pointed out. In the theory of
surfaces, in hydrokinetics, heat- conduction, potentials, etc., we constantly meet with what is called d? d? d? " Laplace's operator," viz. ^" r "^" r "T2- We know that this is an invariant; i.e. it
is independent of the particular directions chosen for the rectangular co-ordinate axes. Here, then, is a case specially adapted to the isotropy of the quaternion system; and Hamilton easily saw that
the 'expression *j~+Jj~ + *J" could be, like ix+jy+kz, effectively expressed by a single letter. He chose for this purpose V. And we now see that the square of V is the negative of Laplace's
operator; while V itself, when applied to any numerical quantity conceived as having a definite value at each point of space, gives the direction and the rate of most rapid change of that quantity.
Thus, applied to a potential, it gives the direction and magnitude of the force; to a distribution of temperature in a conducting solid, it gives (when multiplied by the conductivity) the flux of
heat, etc.
No better testimony to the value of the quaternion method could be desired than the constant use made of its notation by mathematicians like Clifford (in his Kinematic) and by physicists like
ClerkMaxwell (in his Electricity and Magnetism). Neither of these men professed to employ the calculus itself, but they recognized fully the extraordinary clearness of insight which is gained even by
merely translating the unwieldy Cartesian expressions met with in hydrokinetics and in electrodynamics into the pregnant language of quaternions. (P. G. T.)
Supplementary Considerations. There are three fairly wellmarked stages of development in quaternions as a geometrical method, (i) Generation of the concept through imaginaries and development into a
method applicable to Euclidean geometry. This was the work of Hamilton himself, and the above account (contributed to the gth ed. of the Ency. Brit, by Professor P. G. Tait, who was Hamilton's pupil
and after him the leading exponent of the subject) is a brief resume of this first, and by far the most important and most difficult, of the three stages. (2) Physical applications. Tait himself may
be regarded as the chief contributor to this stage. (3) Geometrical applications, different in kind from, though more or less allied to, those in connexion with which the method was originated. These
last include (a) C. J. Joly's projective geometrical applications starting from the interpretation of the quaternion as a point-symbol; 1 these applications may be said to require no addition to the
quaternion algebra; (b) W. K. Clifford's biquaternions and G. Combebiac's tri-quaternions, which require the addition of quasi-scalars, independent of one another and of true scalars, and analogous
to true scalars. As an algebraic 1 It appears from J[oly's and Macfarlane's references that J. B. Shaw, in America, independently of Joly, has interpreted the quaternion as a point-symbol.
method quaternions have from the beginning received much attention from mathematicians. An attempt has recently been made under the name of multenions to systematize this algebra.
We select for description stage (3) above, as the most characteristic development of quaternions in recent years. For (3) (a) we are constrained to refer the reader to Joly's own Manual of
Quaternions (1905).
The impulse of W. K. Clifford in his paper of 1873 (" Preliminary Sketch of Bi-Quaternions," Mathematical Papers, p. 181) seems to have come from Sir R. S. Ball's paper on the Theory of Screws,
published in 1872. Clifford makes use of a quasi-scalar co, commutative with quaternions, and such that if p,q,8tc., are quaternions, when p+uq = p' -\-aiq', then necessarily P = P'> <? = ?' He
considers two cases, viz. co 2 =i suitable for non-Euclidean space, and co 2 =o suitable for Euclidean space; we confine ourselves to the second, and will call the indicated bi-quaternion p+uq an
octonion. In octonions the analogue of Hamilton's vector is localized to the extent of being confined to an indefinitely long axis parallel to itself, and is called a rotor; if p is a rotor then cop
is parallel and equal to p, and, like Hamilton's vector, cop is not localized; cop is therefore called a vector, though it differs from Hamilton's vector in that the product of any two such vectors
cop and cocr is zero because co 2 = o . p+coo- where p, er are rotors (i.e. p is a rotor and coer a vector), is called a motor, and has the geometrical significance of Ball's wrench upon, or twist
about, a screw. Clifford considers an octonion p+wq as the quotient of two motors p+coo- , p'+coa'. This is the basis of a method parallel throughout to the quaternion method; in the specification of
rotors and motors it is independent of the origin which for these purposes the quaternion method, pure and simple, requires.
Combebiac is not content with getting rid of the origin in these limited circumstances. The fundamental geometrical conceptions are the point, line and plane. Lines and complexes thereof are
sufficiently treated as rotors and motors, but points and planes cannot be so treated. He glances at Grassmann's methods, but is repelled because he is seeking a unifying principle, and he finds that
Grassmann offers him not one but many principles. He arrives at the tri-quaternion as the suitable fundamental concept.
We believe that this tri-quaternion solution of the very interesting problem proposed by Combebiac is the best one. But the first thing that strikes one is that it seems unduly complicated. A point
and a plane fix a line or axis, viz. that of the perpendicular from point to plane, and therefore a calculus of points and planes is ipso facto a calculus of lines also. To fix a weighted point and a
weighted plane in Euclidean space we require 8 scalars, and not the 12 scalars of a tri-quaternion. We should expect some species of biquaternion to suffice. And this is the case. Let i\, co be two
quasi-scalars such that if=ii, cor/=co, j)co=co 2 =o. Then the biquaternion ijq+ur suffices. The plane is of vector magnitude JVcj, its equation is %Spq=Sr, and its expression is the bi-quaternion ^
Vtr+coSr; the point is of scalar magnitude Scj, and its position vector is 0, where %Vflq=Vr (or what is the same, fl = [Vr+q.Vr. g^l/Sj^anditS expression is ijSq+uVr. (Note that the here occurring
is only required to ensure harmony with tri-quaternions of which our present biquaternions, as also octonions, are particular cases.) The point whose position vector is Vrq' 1 is on the axis and may
be called the centre of the bi-quaternion; it is the centre of a Sphere of radius Srq~ l with reference to which the point and plane are in the proper quaternion sense polar reciprocals, that is, the
position vector of the point relative to the centre is Srq~ l . Vq/Sq, and that of the foot of perpendicular from centre on plane is Srcf 1 . Sq/Vq, the product being the (radius) 2 , that is (Srq~ 1
) 2 . The axis of the member zQ+a/Q' of the second-order complex Q, Q' (where Q=t}q+a>r, Q^ij^+co/ and x, y! are scalars) is parallel to a fixed plane and intersects a fixed transversal, viz. the
line parallel to q'q- 1 which intersects the axes of Q and Q'; the plane of the member contains a fixed line; the centre is on a fixed ellipse which intersects the transversal; the axis is on a fixed
ruled surface to which the plane of the ellipse is a tangent plane, the ellipse being the section of the ruled surface by the plane; the ruled surface is a cylindroid deformed by a simple shear
parallel to the transversal. In the third-order complex the centre locus becomes a finite closed quartic surface, with three (one always real) intersecting nodal axes, every plane section of which is
a trinodal quartic. The chief defect of the geometrica properties of these bi-quaternions is that the ordinary algebraic scalar finds no place among them, and in consequence Q" 1 is meaningless.
Putting in= we get Combebiac's tri-quaternion under the form Q= p+ijq+wr. This has a reciprocal Q~>= />-i= -<ap- l rq~ l , and a conjugate KQ (such that K[QQ'] KQ'KQ, K[KQ] = Q) given by KQ=
tKq+r,Kp+uKr; the product QQ' of Q and Q' is pp'+wq'+u(pr'+rq'); the quasi-vector j(i K)Q is Combebiac's linear element and may be regarded as a point on a line; the quasi-scalar (in a different
sense from the rest of this article) i(i-f-K)Q is Combebiac scalar (Sp+Sq) + Combebiac's plane. Combebiac does not use K; and in place of {, t\ he uses V.=T\ , so that fj?= i,w/j= =, w 2 = o.
Combebiac's tri-quaternion may be regarded from many simplifying points of view. Thus, in place of his general tri-quaternion we might deal with products of an odd number of point-plane-scalars (of
form pq+ur) which are themselves point-plane-scalars; and products of an even number which are octonions; the quotient of two point-plane-scalars would be an octonion, of two octonions an octonion,
of an octonion by a point-plane-scalar or the inverse a point-plane-scalar. Again a unit point n may be regarded as by multiplication changing (a) from octonion to point-plane-scalar, (6) from
point-plane-scalar to octonion, (c) from plane-scalar to linear element, (d) from linear element to plane-scalar.
If Q=/>+77<?+wr and we put Q=(i+|to<) (/>+??) X (i+fat)- 1 we find that the quaternion / must be 2/(r) //(?/>), where f(r)=rq-Kpr. The point p=V/ may be called the centre of Q and the length St may
be called the radius. If Q and Q' are commutative, that is, if QQ' = Q'Q, then Q and Q' have the same centre and the same radius. Thus Q" 1 , Q Q 2 . Q 3 - have a common centre and common radius. Q
and KQ have a common centre and equal and opposite radii; that is, the t of KQ is the negative conjugate of that of Q. When S = o, (i+fau) ( ) (i+Jww)" 1 is an operator which shifts (without further
change) the tri-quaternion operand an amount given by u in direction and distance.
BIBLIOGRAPHY. In 1904 Alexander Macfarlane published a Bibliography of Quaternions and allied systems of Mathematics for the I nternational Association for promoting the study of Quaternions and
allied systems of Mathematics (Dublin University Press); the pamphlet contains 86 pages. In 1899 and 1901 Sir W. R. Hamilton's classical Elements of Quaternions of 1866 was republished under C. J.
Joly's editorship, in two volumes (London). Joly adds valuable notes and thirteen important apjpendices. In 1890 the 3rd edition of P. G. Tail's Elementary Treatise on Quaternions appeared
(Cambridge). In 1905 C. J. Joly published his Manual of Quaternions (London) ; the valuable contents of this are doubled by copious so-called examples; every earnest student should take these as part
of the main treatise. The above three treatises may be regarded as the great storehouses; the handling of the subject s very different in the three. The following should also be mentioned: A.
McAulay, Octonions, a development of Clifford's Bi-quaternions (Cambridge, 1898); G. Combebiac, Cakul des trtquaternions (Paris, 1002); Don Francisco Perez de Munoz, Introduccion al e studio del
cdlculo de Cuaterniones y otras Algebras especiales (Madrid, 1905) ; A. McAulay, Algebra after Hamilton, or Multentons (Edinburgh, 1908). (A. McA.)
Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)
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Finding Galaxy Clusters with the Sunyaev-Zel'dovich Effect
Finding galaxy clusters is an important task in cosmology. By finding them and mapping them, we've determined the filamentary structure on the largest of scales in our universe. Their properties
constrain cosmological models.
But finding them isn't an easy proposition. We can do it fairly easily in the local universe, but since light follows the inverse square law, they become very faint very quickly. As such, finding
them by looking for their light isn't terribly easy. Often distant galaxies are found when something special happens: like a GRB or a supernova.
The most recent issue of the ApJ has a
cool article
that adds a powerful new trick to the mix; It employs the Sunyaev-Zel'dovich (SZ) effect.
First, a science lesson:
The SZ effect is essentially reverse Compton scattering. Compton scattering is what happens all the time in our atmosphere: (relatively) high energy photons come in, bounce off (relatively) low
energy electrons and atoms and molecules and lose a bit of that energy (which is why the sky is blue) as well as bouncing off in a new direction (which is why we see light coming from the entire sky
and not just directly from the Sun).
Of course, as with most physics, if it can go one way, why not posit the reverse and slap your name on it.
The SZ effect occurs when a relatively low energy photon passes through a hot gas, and picks up some of that energy. In other terms you could say the photon gets blueshifted or gains a shorter
Now if you're galaxy savvy, you probably realize that clusters of galaxies are (usually) full of hot, ionized gas. (I say usually since galactic collisions can strip them of their gas, but this is
more on a individual galactic scale as opposed to a cluster scale.) Propagating though this gas is low energy microwave photons from the Cosmic Microwave Background.
There's been several observations of known clusters showing the SZ effect, but this paper is the first to go the other way around: Using the SZ effect to find the clusters.
With this technique, the group found 4 clusters of galaxies in a ~40º section of the sky, three of which were previously unknown. And looking at the optical images of them, it's not hard to see why
they were missed! The green circles show the cluster and the blue ones note galaxies that are gravitationally lensed.
Staniszewski, Z., Ade, P., Aird, K., Benson, B., Bleem, L., Carlstrom, J., Chang, C., Cho, H., Crawford, T., Crites, A., de Haan, T., Dobbs, M., Halverson, N., Holder, G., Holzapfel, W., Hrubes, J.,
Joy, M., Keisler, R., Lanting, T., Lee, A., Leitch, E., Loehr, A., Lueker, M., McMahon, J., Mehl, J., Meyer, S., Mohr, J., Montroy, T., Ngeow, C., Padin, S., Plagge, T., Pryke, C., Reichardt, C.,
Ruhl, J., Schaffer, K., Shaw, L., Shirokoff, E., Spieler, H., Stalder, B., Stark, A., Vanderlinde, K., Vieira, J., Zahn, O., & Zenteno, A. (2009). GALAXY CLUSTERS DISCOVERED WITH A SUNYAEV-ZEL'DOVICH
EFFECT SURVEY The Astrophysical Journal, 701 (1), 32-41 DOI: 10.1088/0004-637X/701/1/32
3 comments:
Torbjörn Larsson said...
Both scattering as such and old physics models in general trips me up. Which is why I reacted, since I've once learned that the main reason Earth atmosphere is blue is presumably that higher
energy photons are diffused more by elastic scattering, not by wavelength shift. That is, Rayleigh, not Compton, scattering is the main mechanism.
FWIW, Wikipedia seems to agree, so maybe that old model is still considered the best.
Alex said...
I’m looking for somebody who knows some astronomical computer programs. I recovered the stars from 30 NASA images and I would like to determine where they were taken, on the Moon or on the
Anybody knows such computer program?
Anybody could generate the picture of stars above the horizon from the Moon at a given time (1969-1972) and give location (Apollo landing sites)?
If you do or can help in any way, please email me: Alex22@easy.com
Alex said...
I’m looking for somebody who knows some astronomical computer programs. I recovered the stars from 30 NASA images and I would like to determine where they were taken, on the Moon or on the
Anybody knows such computer program?
Anybody could generate the picture of stars above the horizon from the Moon at a given time (1969-1972) and give location (Apollo landing sites)?
If you do or can help in any way, please email me: Alex22@easy.com | {"url":"https://angryastronomer.blogspot.com/2009/08/finding-galaxy-clusters-with-sunyaev.html","timestamp":"2024-11-13T02:04:05Z","content_type":"application/xhtml+xml","content_length":"94071","record_id":"<urn:uuid:a22f3ed7-b53e-4b84-a3de-cf2553a46158>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00684.warc.gz"} |
Inflation in warped geometries
We argue that brane anti-brane inflation in string theory de-Sitter vacua of Kachru-Kallosh-Linde-Trivedi (KKLT) is captured by the dynamics of a D3-brane probe in the local KKLT model constructed in
hep-th/0203041. This provides a framework to study in a controllable way corrections to the inflationary slow roll parameter η due to conformal symmetry breaking in a warped geometry throat. We
compute the leading correction to η for the inflation in the Klebanov-Tseytlin throat geometry. We find that in certain regime this correction tends to decrease η. Computations in a different regime
suggest however that it is unlikely that η≪1 can be achieved with the D3-brane throat inflation.
All Science Journal Classification (ASJC) codes
• Nuclear and High Energy Physics
Dive into the research topics of 'Inflation in warped geometries'. Together they form a unique fingerprint. | {"url":"https://pure.psu.edu/en/publications/inflation-in-warped-geometries","timestamp":"2024-11-12T14:31:27Z","content_type":"text/html","content_length":"46503","record_id":"<urn:uuid:73738150-7170-4dbc-8f3a-91d0b19ced7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00154.warc.gz"} |
Ap Human Calculator - Calculator City
Ap Human Calculator
Enter the number of multiple-choice questions, correct answers, and free response score into the calculator to determine your final AP Human Geography score.
AP Human Geography Score Calculation Formula
The following formula is used to calculate the final AP score from your multiple-choice and free response scores.
Final Score = (MC Percentage * 0.5) + (FRQ Score * 0.5)
• Final Score is the combined AP score
• MC Percentage is the percentage of correct answers in multiple-choice questions
• FRQ Score is the score obtained in free response questions
To calculate the final AP score, multiply the percentage of correct multiple-choice answers by 0.5 and add it to the free response score multiplied by 0.5.
What is AP Human Geography Score Calculation?
AP Human Geography score calculation refers to the process of determining the final score you achieve on the AP Human Geography exam. This involves understanding the weightage of multiple-choice
questions and free response questions, as well as calculating the respective scores accurately. Accurate AP score calculation ensures that students can predict their performance and prepare better
for their exams.
How to Calculate Your AP Human Geography Score?
The following steps outline how to calculate your final AP Human Geography score using the given formula.
1. First, determine the number of multiple-choice questions and the number of correct answers.
2. Next, determine the score obtained in the free response section.
3. Use the formula from above: Final Score = (MC Percentage * 0.5) + (FRQ Score * 0.5).
4. Calculate the percentage of correct multiple-choice answers.
5. Finally, calculate the final AP score by plugging in the values.
6. After inserting the variables and calculating the result, check your answer with the calculator above.
Example Problem:
Use the following variables as an example problem to test your knowledge.
Number of Multiple-Choice Questions = 60
Number of Correct Answers = 45
Free Response Score = 3
1. What is the multiple-choice percentage?
The multiple-choice percentage is the ratio of the number of correct answers to the total number of multiple-choice questions, expressed as a percentage.
2. How is the final AP score calculated?
The final AP score is calculated by combining the weighted scores of the multiple-choice and free response sections, using the formula provided above.
3. Why is it important to use the AP Human Geography calculator?
Using the AP Human Geography calculator helps students accurately predict their scores and understand their performance, which can aid in better exam preparation.
4. Can this calculator be used for other AP subjects?
While the specific weights may vary, the general approach to score calculation can be adapted for other AP subjects by adjusting the weightages accordingly.
5. How often should I use the AP score calculator?
It’s helpful to use the AP score calculator after taking practice exams or completing test sections to gauge your performance and adjust your study plan accordingly. | {"url":"https://calculator.city/ap-human-calculator/","timestamp":"2024-11-05T23:35:37Z","content_type":"text/html","content_length":"76197","record_id":"<urn:uuid:dbb4274e-c59c-4b26-8763-08dd35c1b696>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00723.warc.gz"} |
Finding the Components of the Sum of Two Vectors in Component Form
Question Video: Finding the Components of the Sum of Two Vectors in Component Form Mathematics • First Year of Secondary School
Given that 𝐮 = ⟨2, −4⟩ and 𝐯 = ⟨−2, 4⟩, find the components of 𝐮 + 𝐯.
Video Transcript
Given that vector 𝐮 is equal to two, negative four and vector 𝐯 is equal to negative two, four, find the components of vector 𝐮 plus vector 𝐯.
We know that in order to add two vectors, we simply add their corresponding components. In this question, we need to add two and negative two and then negative four and four. Two plus negative two is
the same as two minus two, which gives us an answer of zero. Negative four plus four is also equal to zero.
If vector 𝐮 is equal to two, negative four and vector 𝐯 is equal to negative two, four, then vector 𝐮 plus vector 𝐯 is equal to zero, zero, otherwise known as the zero vector. | {"url":"https://www.nagwa.com/en/videos/728160956708/","timestamp":"2024-11-04T09:14:32Z","content_type":"text/html","content_length":"247280","record_id":"<urn:uuid:26925b2c-77fb-43cf-bfc3-ba0727252f7b>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00122.warc.gz"} |
Preservice Teachers as Document Detectives
Source: Australian Journal of Teacher Education, 45(6)
(Reviewed by the Portal Team)
This study examines the preservice teachers’ understanding of critical analysis practices.
The paper argues for the importance of preservice teachers being encouraged to collaboratively construct knowledge of critical analysis practices, through interrogation of media text containing
graphs and representations, the process of analysis dependent on ideas from mathematics.
Further, the paper describes the value of using items from the media as resources for enhancing knowledge of critical analysis practices.
This study focuses on preservice teachers’ ability to comprehend text from the media, in particular text that included mathematical representations.
Interrogation of such text has the potential to deepen students’ understanding of text that they may encounter in society.
The long term aim of interrogating such text in preservice classrooms would be to encourage the preservice teachers to carry such skills into the classroom.
Research process
During the research, a variety of media text that included graphs and representations was investigated and critically analysed by the preservice teachers.
The research questions were:
1. Why is it important to refer to ideas from mathematics, in the process of critically analysing media text containing visual representations?
2. What resources can be used in the process of scaffolding critical analysis competencies?
The case study research involved 23 preservice teacher participants who were enrolled in the Primary Graduate Diploma program at one Australian university.
The participants were involved in a series of learning sessions, the learning taking place once weekly for four weeks in a university classroom.
A selection of six samples of media text containing tables, graphs, and varied visuals were collaboratively discussed and critically analysed by small groups of five to six preservice teacher
The process was guided by a mathematics education lecturer, who fulfilled the role of the facilitator of learning, encouraging and guiding the learners to participate in extensive critical discourse
relating to the content and context of the media items.
During the process, data presented in the tables and graphs were used to justify, confirm, disconfirm, and summarise ideas, and make decisions, thereby aiding collaborative construction of meaning.
The preservice teachers’ critical discussions about the media items were recorded and data collated.
Results and discussion
Students require opportunities to engage with everyday text that reflects authentic cultural and social practices (Lemke, 2003), to aid them to efficiently utilise text that they encounter in
everyday situations (Monteiro & Ainley, 2010).
Engaging everyday examples can aid preservice teachers (and other students) to “become media detectives themselves, looking for cases of poor practice or reporting” (Watson, 2000, p. 5).
Use of such text can aid them to uncover cases where text is written to influence ideas through presenting certain views and silencing others, for instance (Luke & Freebody, 1999).
In the current study, dialogue about the above-mentioned text depended on a deep engagement with the subtle meanings in the visual text by the preservice teachers.
Elements of the dialogue reflected critical analysis, referring to the views portrayed in the text, the purposes, bias and misrepresentations, disguised information and specific emphasis, the
author’s background, and choices made when presenting the data.
However, in general, the preservice teachers struggled to comprehensively critically analyse the text.
As evident in the examples, with focus on such media visuals and scaffolding from other students and the facilitator, the preservice teachers were encouraged to collaboratively construct knowledge of
critical analysis practices relating to tables, graphs, and varied visuals.
As illustrated in the examples, mathematical knowledge is often crucial in terms of drawing deep understanding and critically analysing and describing such visuals and representations.
This indicates the importance of extending preservice teachers’ knowledge of mathematical ideas in order to scaffold their ability to critically analyse media visuals (as a first step towards them
conveying such skills to their students).
In this study, relying on mathematical ideas to analyse the text at times posed challenges to the preservice teachers.
For example, an understanding of the importance of random sampling when collecting data is crucial in terms of critically analysing the messages in such presentations.
Further, an understanding of the impact that the presentation of the data can have on the perceived meaning is also crucial in the critical analysis process.
With input from others and guidance from the facilitator, discussion can be extended to factors including mathematical factors that can impact on the authenticity of messages presented in graphs and
visuals, such as the factors discussed in the above-mentioned examples.
The media resources selected in this study proved to be useful resources for the preservice teachers, providing opportunities for them to develop their ideas about critical analysis.
This concurs with other studies that found such items to be useful resources for scaffolding preservice teachers’ critical analysis competencies (Monteiro & Ainley, 2003, 2004, 2007; Robertson &
Hughes, 2011).
Notably, these examples illustrate the significance of mastery of critical competencies in terms of constructing a deep understanding of mathematical information presented in media text.
Investigation of such text in instruction has the potential to enhance understanding of the mathematical messages depicted in visuals and of claims made about the visuals (see Watson, 2015), and
place learners in a better position to draw deep understanding of both the mathematical content and its representation in text.
In answer to the research questions, the examples illustrate the importance of a focus on mathematical ideas and language when scaffolding critical analysis competencies, a deep understanding of
media visuals and representations often being dependent on mathematical knowledge.
In the examples, mathematical ideas were clearly crucial in terms of the preservice teachers’ ability to navigate, make deep meaning of, and describe the given text.
In many instances, the preservice teachers in this study required scaffolding to expand their critical competencies based on mathematical ideas.
This indicates the importance of preservice educators endeavouring to aid their students to navigate such text, similar text commonly found in the media rich world.
Although the study addressed the research questions, the outcomes of the study are limited by the small number of participants and small amount of media text investigated, and the fact that the study
was non-longitudinal.
The results, do however, have the potential to aid understanding of the challenges faced by preservice teachers in the critical analysis of graphs and representations and the value of using media
text in this regard.
These results can be extended by further comprehensive studies on the topic.
Lemke, J. L. (2003). Mathematics in the Middle: Measure, picture, gesture, sign, and word. In M. Anderson, A. Sáenz-Ludlow, S. Zellweger & V. V. Cifarelli (Eds.), Educational perspectives on
mathematics as semiosis: From thinking to interpreting to knowing (pp. 215-234). Brooklyn, NY, and Ottawa, Ontario: Legas.
Luke, A., & Freebody, P. (1999). Further notes on the four resources model. Retrieved October 19, 2018, from https://www.semanticscholar.org/paper/Further-notes-on-the-four-resource...
Monteiro, C., & Ainley, J. (2003). Developing critical sense in graphing. CERME 3: Proceedings of Third Conference of the European Society for Research in Mathematics Education, Bellaria, Italy (pp.
1-10). Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.542.8950&rep=re... pdf
Monteiro, C., & Ainley, J. (2004). Critical sense in interpretations of media graphs. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3
(pp. 361-368).
Monteiro, C., & Ainley, J. (2007). Investigating the interpretation of media graphs among student teachers. International Electronic Journal of Mathematics Education, 2(3), 187-207.
Monteiro, C. E. F., & Ainley, J. M. (2010). The interpretation of graphs: Reflecting on contextual aspects. Alexandria (UFSC), 3(2), 17-30.
Robertson, L., & Hughes, J. M. (2011). Investigating pre-service teachers’ understanding of critical media literacy. Language and Literacy, 13(2), 37-53.
Watson, J. M. (2000). Lessons from chance and data research for the classroom. Teaching Mathematics, 25(4), 3-9.
Watson, J. (2015). Statistical literacy in action: Should all graphs start at zero? Australian Primary Mathematics Classroom, 20(4), 26-30. | {"url":"https://education.eng.macam.ac.il/article/5169","timestamp":"2024-11-03T09:46:41Z","content_type":"text/html","content_length":"42912","record_id":"<urn:uuid:6994575c-e230-4952-862f-bef19ceb2fe2>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00493.warc.gz"} |
[Haskell-cafe] Problems with truncate big float values in Haskell
Henson jfsihenson at gmail.com
Thu Jan 5 18:16:57 UTC 2017
Firstly I desire one year brilliant for all on group.
I have one simple problem but not solved for me for some mistake mine.
I have one Rational number (3 + sqrt 5) and I need get the last three
digits from
integral part of that number.
The question is the number is arbitrary and that number always is power
of n and n is between 2(Two) and 2000000000(Two billions).
I have attempted that on ghci for various values and get same result
with different powers:
let x = (truncate ((3 + sqrt 5) ^ 2000000)) `mod` 1000 and result is 216.
let y = (truncate ((3 + sqrt 5) ^ 20000000)) `mod` 1000 and result is 216.
let w = (truncate ((3 + sqrt 5) ^ 200000000)) `mod` 1000 and result is 216.
let z = (truncate ((3 + sqrt 5) ^ 2000000000)) `mod` 1000 and result is 216.
The result is same for all:
216, and is incorrect because any that expressions have one exclusive value.
What I do for complete that task correctly and what I'm mistaken here?!
Thank you,
Josenildo Silva
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Convert Date Format in Excel using Python
In this tutorial, we will learn how to convert a date format in Excel using a Python formula. Specifically, we will focus on converting the date format of '20230801' into '2023-08-01' so that it can
be recognized as a date by Excel.
To achieve this, we will use a combination of Excel functions including the DATEVALUE function, LEFT function, MID function, and RIGHT function. These functions will allow us to extract the year,
month, and day from the given date format and concatenate them together with hyphens to form the desired date format.
By using the formula =DATEVALUE(LEFT(A1,4)&"-"&MID(A1,5,2)&"-"&RIGHT(A1,2)), where A1 is the cell containing the date in the format '20230801', we can convert the date format into '2023-08-01' which
Excel can recognize as a date.
Let's walk through the steps of the formula:
1. The LEFT function extracts the first 4 characters from the cell A1, representing the year.
2. The MID function extracts the characters from position 5 to position 6 from the cell A1, representing the month.
3. The RIGHT function extracts the last 2 characters from the cell A1, representing the day.
4. The '&' operator concatenates the extracted year, month, and day together with hyphens to form the desired date format.
5. The DATEVALUE function converts the concatenated string into a date value that Excel can recognize.
By following these steps and using the provided formula, you can easily convert the date format of '20230801' into '2023-08-01' so that it can be recognized as a date by Excel. This tutorial will
provide you with a clear understanding of the process and enable you to apply it to other date formats as well.
An Excel formula
Formula Explanation
This formula uses the DATEVALUE function in combination with the LEFT, MID, and RIGHT functions to convert the date format of "20230801" into "2023-08-01" so that it can be recognized as a date.
Step-by-step explanation
1. The LEFT function is used to extract the first 4 characters from the cell A1, which represents the year.
2. The MID function is used to extract the characters from position 5 to position 6 from the cell A1, which represents the month.
3. The RIGHT function is used to extract the last 2 characters from the cell A1, which represents the day.
4. The "&" operator is used to concatenate the extracted year, month, and day together with hyphens ("-") to form the desired date format.
5. The DATEVALUE function is used to convert the concatenated string into a date value that Excel can recognize.
6. The formula is entered into the desired cell, such as B1, and it will display the converted date format.
For example, if we have the date "20230801" in cell A1, the formula =DATEVALUE(LEFT(A1,4)&"-"&MID(A1,5,2)&"-"&RIGHT(A1,2)) would return the date value "8/1/2023" in cell B1, which is the converted
date format that Excel recognizes.
Similarly, if we have the date "20211225" in cell A2, the formula =DATEVALUE(LEFT(A2,4)&"-"&MID(A2,5,2)&"-"&RIGHT(A2,2)) would return the date value "12/25/2021" in cell B2.
By using this formula, you can easily convert the date format of "20230801" into "2023-08-01" so that it can be recognized as a date by Excel. | {"url":"https://codepal.ai/excel-formula-generator/query/lGLfNUqJ/excel-formula-convert-date-format","timestamp":"2024-11-03T15:46:10Z","content_type":"text/html","content_length":"92543","record_id":"<urn:uuid:b0244683-878d-4adc-95a6-7565be6cf563>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00200.warc.gz"} |
INFIQ | GRADE 1 Math Worksheets- Place Value - Groups of Ten
The grade 1 worksheets on this page will help the students practice & sharpen the concepts for ‘Place Value – Groups of Ten.’ With these worksheets, students will improve their understanding of the
tens place. They will convert groups of ten to a number and conversely, break a number that is multiple of 10 into a bundle of ten. The worksheets use different variations to reinforce student’s
understanding of the concept, while keeping them engaged and challenged.
Tired of managing the worksheets? Download the INFIQ app & let the app do the heavy lifting for you.
You can create 'infinite' topic-level and lesson-level quizzes with the INFIQ app.
With the INFIQ app, you can even create your own 'infinite custom' quizzes by combining multiple different lessons.
The INFIQ app comprehensively covers the Common Core curriculum and multiple state curriculums. Check out the Features Page for all the features.
Groups of Ten Worksheet Booklet
While these worksheets are great, nothing beats Going Digital Saving Trees Unlimited Infinite Download INFIQ workbook for Infinite Quizzes
1. Unlimited worksheets or quizzes for any grade (grades 1-5).
2. Create unlimited Topic-level quizzes
3. Create unlimited Lesson-level quizzes.
4. Create unlimited custom quizzes by combining lessons from different topics.
5. Parents can review completed quizzes from anywhere, anytime.
6. Question-level scratch pad.
7. Comprehensively covers Common Core Curriculum.
8. Practice Mode provides instant feedback.
9. Countless word problems.
10. Parents can create ‘Goals & Rewards.’
11. Detailed Progress Report.
12. Daily, Weekly and Monthly Activity Report.
13. Offline mode.
14. Multiple student profiles
15. Homework Reminder
** The INFIQ app contains all the questions shown here and a lot more question types. The questions may have have been digitally adapted for the mobile form factor. | {"url":"https://www.getinfiq.education/worksheets/g1-groups-of-ten/","timestamp":"2024-11-12T09:17:48Z","content_type":"text/html","content_length":"335476","record_id":"<urn:uuid:07b0bc03-0d92-476f-80e1-8717882584e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00015.warc.gz"} |
Bogdan and Wally :: Transum Newsletter
Bogdan and Wally
Saturday 1st August 2020
This is the Transum Newsletter for the month of August 2020. It begins, as usual with the puzzle of the month.
Would you like to win the Ultimate Teacher Trophy? If so you need to win two games of Ultimate Noughts and Crosses in a row. You will play three games in total.
Your opponents will be the super-intelligent Professor Bogdan Plopmush and the enthusiastic but absent-minded Wally Pip.
• You have a 30% chance of beating Bogdan;
• You have an 85% chance of beating Wally.
You can choose the order of your games from these two options:
Bogdan, Wally, Bogdan
Wally, Bogdan, Wally
In order to have the greatest chance of winning the Ultimate Teacher Trophy, which option should you choose?
The answer will be at the end of this newsletter and while you are thinking about it I will update you on some developments.
I am currently lucky enough to be in a country where there have been no locally transmitted case of you-know-what for months so I was able to have a summer beach holiday. Lucky me. I justified it by
knowing that I was helping, albeit in a very small way, the tourist industry here. But while sun, sea and sand were perfect for relaxing they did tend to limit the number of new additions to the
Transum website for the month of July.
HCF and LCM worded questions did however flow from my mind (and my deckchair) and they can be seen as the new level 6 of the existing online exercise. The last three questions should hopefully
challenge even your brightest sparks.
Factor Pairs was next to be created. I decided on presenting the resource as nine puzzles with the first coming with lots of help. This is a drag and drop activity but designed to help pupils see the
patterns in the ordered lists of factors arranged in pairs.
Two new help videos were made at the beginning of July. They feature the cartoon character Tran Sum (watch out Disney!) and cover the topics of Addition and Subtraction. They are not designed to be
watched all the way from beginning to end. They can be found in the help tabs of the Addition and Subtraction online exercises and when a pupil clicks on the help tab the video will be cued to the
section matching the level of the exercise they are viewed from.
On the last day of July I made a video on the order of operations commonly known as BIDMAS (or is it BODMAS or PEMDAS?)
Did you know that you can work out the number of factors a number has by multiplying together the indices (each increased by one) seen in the prime factorisation of that number? You can apply that
knowledge using this next new resource called How Many Factors? It is a bit of drill and practice and provides plentiful mental methods practice. The instructions for calculating prime factorisations
using a calculator are in the help tab, but only appear when subscribers are signed in. I’ll leave it to you to figure out the best time to share those instructions with your pupils.
Connect 4 Factors is one of the Transum classics with many people playing the game or attempting the challenge each day. It has had an upgrade (frequent flyer points) making it even more robust and
easy to use. It you haven’t seen it already go and have a peep.
HELP! How do you convert a fraction to a percentage? Do you multiply by 100 or do you multiply by 100% ? Textbooks differ and I have been having a conversation with a wonderful subscriber which is
leaving me undecided. You can see the discussion on the Percentages Debate page. What do you think?
And now the answer to the puzzle of the month that you have been thinking about since the beginning of this newsletter:
The puzzle is adapted from a puzzle I saw in Alex Bellos’ book So You Think You’ve Got Problems. It’s one of those puzzles that most people get wrong as the correct answer is not immediately obvious.
Most people think that the option of playing Wally twice is going to give the best chance of winning two games. However these two wins will not be ‘in a row’ so won’t win you the trophy.
The correct answer is Bogdan, Wally, Bogdan. You are quite likely to win the middle game and you then have two chances of making it a double with your two games against Bogdan.
If you are not convinced, draw two tree diagrams, one for each option, then you will see why the first option is best. Alternatively the combined events probability formula could be used.
That's all for now,
PS. I just asked Alexa to tell me a maths joke. She said "What happened when the Maths teacher gave out extra homework? The addition cause the division to multiply!" - I think I get it but not sure.
Do you have any comments? It is always useful to receive feedback on this newsletter and the resources on this website so that they can be made even more useful for those learning Mathematics
anywhere in the world. Click here to enter your comments. | {"url":"https://transum.org/Newsletter/?p=501","timestamp":"2024-11-14T17:36:58Z","content_type":"text/html","content_length":"21753","record_id":"<urn:uuid:d189518d-91b3-4ce1-b632-922a5ea41293>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00499.warc.gz"} |
Trigonometry 5th Edition Mark Dugopolski-Test Bank - Test Bank Goo
Trigonometry 5th Edition Mark Dugopolski-Test Bank
Format: Downloadable ZIP File
Resource Type: Test bank
Duration: Unlimited downloads
Delivery: Instant Download
Soltion Handbook For Trigonometry 5th Edition Mark Dugopolski
ISBN-13: 9780136880820
Trigonometry with Built-in Evaluate prepares you to reach trigonometry and past with an emphasis on downside fixing and important pondering. It helps you develop the comprehension and confidence
wanted to reach and out of the classroom. Studying aids are strategically positioned all through, supplying you with steering proper the place you want it. Ample chapter evaluate materials provides
options corresponding to Highlights, Chapter Evaluate Workout routines and Chapter Checks to assist evaluate and synthesize the fabric as you put together for the highway forward. Together with many
different content material updates and enhancements, the 5th Edition rewrites quite a few explanations, examples, and workouts in response to suggestions from customers.
Desk of Contents
The desk of contents under reveals the evaluate matters (bulleted textual content) which are built-in into the MyLab Math course. The Built-in Evaluate consists of worksheets, movies, and different
supplies designed to assist college students grasp these essential matters.
P. Algebraic Conditions
□ P.1 The Cartesian Coordinate System
□ P.2 Features
□ P.3 Households of Features, Transformations, and Symmetry
□ P.4 Compositions and Inverses
1. Angles and the Trigonometric Features
□ 1.1 Angles and Diploma Measure
□ 1.2 Radian Measure, Arc Size, and Space
□ 1.3 Angular and Linear Velocity
□ 1.4 The Trigonometric Features
□ 1.5 Proper Triangle Trigonometry
□ 1.6 The Basic Identification and Reference Angles
□ Chapter 1 Built-in Evaluate Matters:
☆ Multiplying easy rational expressions
☆ Utilizing Rational Expressions in Conversions
☆ Discovering space and circumference of a circle utilizing the usual system.
☆ Simplifying sq. roots.
☆ Performing operations with sq. roots.
☆ Utilizing the Pythagorean theorem to seek out lacking sides of a proper triangle.
☆ Discovering middle and radius of a circle given the equation for the circle.
☆ Discovering the inverse of a perform.
2. Graphs of the Trigonometric Features
□ 2.1 The Unit Circle and Graphing
□ 2.2 The Basic Sine Wave
□ 2.3 Graphs of the Secant and Cosecant Features
□ 2.4 Graphs of the Tangent and Cotangent Features
□ 2.5 Combining Features
□ Chapter 2 Built-in Evaluate Matters:
☆ Shifting graphs of algebraic features horizontally and vertically.
☆ Discovering area and vary of algebraic features; Reflecting, stretching, and shrinking of algebraic features.
☆ Writing equations of horizontal and vertical strains.
☆ Performing arithmetic with fractions involving pi.
☆ Discovering horizontal and vertical asymptotes for rational features.
☆ Figuring out area and vary of rational features.
3. Trigonometric Identities
□ 3.1 Primary Identities
□ 3.2 Verifying Identities
□ 3.3 Sum and Distinction Identities for Cosine
□ 3.4 Sum and Distinction Identities for Sine and Tangent
□ 3.5 Double-Angle and Half-Angle Identities
□ 3.6 Product and Sum Identities
□ Chapter 3 Built-in Evaluate Matters:
☆ Recognizing identities in algebra.
☆ Utilizing the Basic Identification from Trigonometry to simplify expressions.
☆ Utilizing reciprocal identities to simplify expressions.
☆ Multiplying binomials.
☆ Squaring a binomial.
☆ Factoring expressions right into a product of two binomials.
☆ Discovering compositions of algebraic features.
☆ Proving that an equation isn’t an id.
☆ Operations with rational expressions in algebra.
4. Fixing Conditional Trigonometric Equations
□ 4.1 The Inverse Trigonometric Features
□ 4.2 Primary Sine, Cosine, and Tangent Equations
□ 4.3 Equations Involving Compositions
□ 4.4 Trigonometric Equations of Quadratic Kind
□ Chapter 4 Built-in Evaluate Matters:
☆ Evaluating a composition of algebraic features.
☆ Figuring out identities in trigonometry.
☆ Fixing proportions for a variable.
☆ Fixing for a variable in an algebraic equation.
☆ Fixing quadratic equations by factoring.
☆ Remedy quadratic equations by utilizing the sq. root property.
☆ Remedy quadratic equations by utilizing the quadratic system.
☆ Squaring all sides of an equation and getting extraneous roots.
☆ Area and vary of the trig features.
☆ Discovering the precise values of the sine perform.
☆ Discovering the precise values of secant, cosecant, and cotangent.
5. Purposes of Trigonometry
□ 5.1 The Regulation of Sines
□ 5.2 The Regulation of Cosines
□ 5.3 Space of a Triangle
□ 5.4 Vectors
□ 5.5 Purposes of Vectors
□ Chapter 5 Built-in Evaluate Matters:
☆ Fixing proportions for x.
☆ Fixing proportions utilizing the inverse sine perform.
☆ Discovering the realm of a triangle utilizing the usual system.
☆ Fixing proper triangles.
☆ Discovering the gap between two factors with the gap system.
6. Advanced Numbers, Polar Coordinates, and Parametric Equations
□ 6.1 Advanced Numbers
□ 6.2 Trigonometric Type of Advanced Numbers
□ 6.3 Powers of Roots of Advanced Numbers
□ 6.4 Polar Equations
□ 6.5 Parametric Equations
□ New – 6.6 Enjoyable with Polar and Parametric Equations
□ Chapter 6 Built-in Evaluate Matters:
☆ Simplifying sq. roots.
☆ Including and subtracting binomials.
☆ Multiplying binomials.
☆ Discovering nth roots with 1/n notation.
☆ Fixing cubic equations.
☆ Discovering sine and cosine of huge angles.
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Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groupssearch
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Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groups
Hardcover ISBN: 978-0-8218-4593-6
Product Code: MMONO/116
List Price: $165.00
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Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groups
Hardcover ISBN: 978-0-8218-4593-6
Product Code: MMONO/116
List Price: $165.00
MAA Member Price: $148.50
AMS Member Price: $132.00
eBook ISBN: 978-1-4704-4527-0
Product Code: MMONO/116.E
List Price: $155.00
MAA Member Price: $139.50
AMS Member Price: $124.00
Hardcover ISBN: 978-0-8218-4593-6
eBook ISBN: 978-1-4704-4527-0
Product Code: MMONO/116.B
List Price: $320.00 $242.50
MAA Member Price: $288.00 $218.25
AMS Member Price: $256.00 $194.00
• Translations of Mathematical Monographs
Volume: 116; 1993; 223 pp
MSC: Primary 60; Secondary 22; 43
This book studies the problem of the decomposition of a given random variable into a sum of independent random variables (components). Starting from the famous Cramér theorem, which says that all
components of a normal random variable are also normal random variables, the central feature of the book is Fel′dman's use of powerful analytical techniques. In the algebraic case, one cannot
directly use analytic methods because of the absence of a natural analytic structure on the dual group, which is the domain of characteristic functions. Nevertheless, the methods developed in
this book allow one to apply analytic techniques in the algebraic setting. The first part of the book presents results on the arithmetic of probability distributions of random variables with
values in a locally compact abelian group. The second part studies problems of characterization of a Gaussian distribution of a locally compact abelian group by the independence or identical
distribution of its linear statistics.
Specialists in probability theory, mathematical statistics and functional analysis.
□ Chapters
□ Introduction
□ Chapter I. Auxiliary results
□ Chapter II. Arithmetic of distributions
□ Chapter III. Characterization problems
□ There is no question that it was a wise decision by the editors of the Mathematical Monographs Translations series to accept this piece of work for publication and make it accessible to a
broad community of mathematicians working in structural probability theory. Moreover, the handy monograph of a few more than 200 pages advertises a most interesting aspect of probability
theory to all analysts who want to see abstract harmonic analysis at work.
Bulletin of the AMS
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Reviews
• Requests
Volume: 116; 1993; 223 pp
MSC: Primary 60; Secondary 22; 43
This book studies the problem of the decomposition of a given random variable into a sum of independent random variables (components). Starting from the famous Cramér theorem, which says that all
components of a normal random variable are also normal random variables, the central feature of the book is Fel′dman's use of powerful analytical techniques. In the algebraic case, one cannot
directly use analytic methods because of the absence of a natural analytic structure on the dual group, which is the domain of characteristic functions. Nevertheless, the methods developed in this
book allow one to apply analytic techniques in the algebraic setting. The first part of the book presents results on the arithmetic of probability distributions of random variables with values in a
locally compact abelian group. The second part studies problems of characterization of a Gaussian distribution of a locally compact abelian group by the independence or identical distribution of its
linear statistics.
Specialists in probability theory, mathematical statistics and functional analysis.
• Chapters
• Introduction
• Chapter I. Auxiliary results
• Chapter II. Arithmetic of distributions
• Chapter III. Characterization problems
• There is no question that it was a wise decision by the editors of the Mathematical Monographs Translations series to accept this piece of work for publication and make it accessible to a broad
community of mathematicians working in structural probability theory. Moreover, the handy monograph of a few more than 200 pages advertises a most interesting aspect of probability theory to all
analysts who want to see abstract harmonic analysis at work.
Bulletin of the AMS
Permission – for use of book, eBook, or Journal content
Please select which format for which you are requesting permissions. | {"url":"https://bookstore.ams.org/MMONO/116","timestamp":"2024-11-10T05:32:48Z","content_type":"text/html","content_length":"87914","record_id":"<urn:uuid:414b83e1-1c27-43ee-a458-6300e1657991>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00843.warc.gz"} |
OEF vector space definition
OEF vector space definition --- Introduction ---
This module actually contains 13 exercises on the definition of vector spaces. Different structures are proposed in each case; up to you to determine whether it is really a vector space.
See also the collections of exercises on vector spaces in general or definition of subspaces.
Let S be the set of all circles on the (cartesian) plane, with rules of addition and multiplication by scalars defined as follows.
• If C[1] (resp. C[2]) is a circle of center (x[1],y[1]) (resp. (x[2],y[2]) ) and radius , C[1] + C[2] will be the circle of center (x[1]+x[2],y[1]+y[2]) and radius .
• If C is a circle of center (x,y) and radius , and if a is a real number, then aC is a circle of center (ax,ay) and radius .
Is S with the addition and multiplication by scalar defined above is a vector space over the field of real numbers?
Space of maps
Let S be the set of maps
f: ---> ,
(i.e., from the set of to the set of ) with rules of addition and multiplication by scalar as follows:
• If f[1] and f[2] are two maps in S, f[1]+f[2] is a map f: : -> such that f(x)=f[1](x)+f[2](x) for all x belonging to .
• If f is a map in S and if a is a real number, af is a map from to such that (af)(x)=a·f(x) for all x belonging to .
Is S with the structure defined above is a vector space over R ?
Absolute value
Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows:
• For any (x,y) and (x,y) belonging to S, we define (x,y)+(x,y) = (x+x,y+y).
• For any (x,y) belonging to S and any real number a, we define a(x,y) = (|a|x,|a|y).
Is S with the structure defined above is a vector space over R?
Affine line
Let L be a line in the cartesian plane, defined by an equation c[1]x+c[2]y=c[3], and let =(x,y) be a fixed point on L.
We take S to be the set of points on L. On S, we define addition and multiplication by scalar as follows.
• If =(x,y) and =(x,y) are two elements of S, we define + = .
• If =(x,y) is an element of S and if is a real number, we define = .
Is S with the structure defined above is a vector space over R?
Alternated addition
Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows:
• For any (x,y) and (x,y) belonging to S, (x,y)+(x,y) = (x+y,y+x).
• For any (x,y) belonging to S and any real number a, a(x,y) = (ax,ay).
Is S with the structure defined above is a vector space over R?
The set of all , together with the usual addition and multiplication, is it a vector space over the field of ?
Let be the set of real matrices. On , we define the multiplicatin by scalar as follows. If is a matrix in , and if is a real number, the multiplication of by the scalar is defined to be the matrix ,
where .
Is together with the usual addition and the above multiplication by scalar a vector space over ?
Matrices II
The set of matrices of elements and of , together with the usual addition and multiplication, is it a vector space over the field of ?
Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows:
• For any (x,y) and (x,y) belonging to S, we define (x,y)+(x,y) = (x+x,y+y).
• For any (x,y) belonging to S and any real number a, we define a(x,y) = (x/a , y/a) if a is non-zero, and 0(x,y)=(0,0).
Is S with the structure defined above is a vector space over R?
Non-zero numbers
Let S be the set of real numbers. We define addition and multiplication by scalare on S as follows:
• If x and y are two elements of S, the sum of x and y in S is defined to be xy.
• If x is an element of S and if a is a real number, the multiplication of x by the scalare a is defined to be x^a.
Is S with the structure defined above is a vector space over R?
Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows:
• If (x,y) and (x,y) are two elements of S, their sum in S is defined to be the couple (x+x,y+y).
• If (x,y) is an element of S, and if a is a real number, the multiplication of (x,y) by the scalar a in S is defined to be the couple (ax(),ay()).
Is S with the structure defined above is a vector space over R?
Let S be the set of couples (x,y) of real numbers. We define the addition and multiplication by scalar on S as follows:
• For any (x,y) and (x,y) belonging to S, (x,y)+(x,y) = (x+x,y+y).
• For any (x,y) belonging to S and any real number a, a(x,y) = (ax,ay()^2).
Is S with the structure defined above is a vector space over R?
Unit circle
Let S be the set of points on the circle x^2+y^2=1 in the cartesian plane. For any point (x,y) in S, there is a real number t such that x=cos(t), y=sin(t).
We define the addition and multiplication by scalare on S as follows:
• If (cos(t[1]),sin(t[1])) and (cos(t[2]),sin(t[2])) are two points in S, their sum is defined to be (cos(t[1]+t[2]),sin(t[1]+t[2])).
• If p=(cos(t), sin(t)) is a point in S and if a is a real number, the multiplication of p by the scalar a is defined to be (cos(at), sin(at)).
Is S with the structure defined above is a vector space over R?
Other exercises on: vector spaces linear algebra
The most recent version
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The Mathematics of Catenary - Alan Zucconi
Many modern games feature hanging wires, cables and chains; this series of tutorials will explore the mathematics behind their shape, which is known as catenary.
You can find the Unity package to create catenaries in Unity at the end of the post.
An Introduction to Catenaries
Out of the many mathematical objects that have been studied and described, there is one that is very dear to many game developers. And yet, only a small number of them actually know its name:
A catenary is the shape that a rope or chain will naturally converge to, when suspended at its ends. It is not a coincidence that the name catenary itself comes from the Latin catenaria—which indeed
means chain.
Modern games feature an increasing number of run down facilities and destroyed environments. And most comes with their fair share of hanging wires. Such as the ones seen in GLaDOS’ room in “Portal”,
or “Half-Life: Alyx”, just to name a couple.
Because catenaries occurs everywhere around us, we have grown accustomed to their shape. This also means that it is very easy to spot when something is not hanging the right way. Like skin-complexion
and cloth-physics, a wrong catenaries are hanging over an uncanny valley of their own.
And yet, so many games are getting catenaries wrong! The reason, however, is not surprising. While they are so easy to generate in the real world, their mathematical definition is made of the same
substance of nightmares. Exception made for a few special cases, there is no “easy” equation to generate a catenary; at least not in the form that we need to properly decorate a level.
One common ways of creating physically-based catenaries “for free” is to use rigid bodies and hinge joints to create chains and ropes. This has the extra benefit of making them reactive to the
player’s interaction, but at the cost of being computationally expensive. Most hanging wires and cables are part of the background, and using physics to create them would be too expensive.
Consequently, being able to place static catenaries with no run-time cost is pretty critical.
On top of that, drawing catenaries comes with an additional benefit. Let’s imagine that you want to create an actual, physically-driven hanging wire for your game. How do you place the wire segments,
when instantiating them? Many developers would simply place them along a line, letting the physics engine do the work for them by finding an equilibrium state. Drawing catenaries allows you to
initialise physically-driven wires and cables in their equilibrium state, without having to wait for them to settle into position by themselves.
It is worth noticing that while Unity does not offer any build-in tools for cables and chains, Unreal Engine comes with a Cable Component that solves exactly this problem through a technique called
Verlet Integration (which, incidentally, will be the topic of a future series). And in case you are into shaders, Ross Beardsall recently came up with an ingenious solution to simulate coiled cables
in Unreal Engine 4…
A Formal Definition
If we want to get physically-accurate catenaries, perhaps is best to start from the beginning. The simplest catenary one can image has a well-defined equation, which almost entirely relies on
hyperbolic cosine:
The catenary equation has a parameter,
❓ What is the hyperbolic cosine?
Many of you might be familiar with the more “traditional” cosine function. While sine and cosine are defined over a circle, their hyperbolic partners are defined over an hyperbola (see animation
Their natural use is to study hyperbolic geometries. Incidentally, they also occur in the solutions of many linear differential equations. In fact,
which is:
They are also strongly related to
❓ Show me the derivation!
The derivation of the catenary equation is a tricky one, and it requires some pretty advanced calculus. If you are interested, Math24 has a very detailed article, Equation of Catenary, showing how
that can be done step by step.
The derivation starts by assuming that for each small segment of the chain, the forces of gravity are in perfect balance with the tension forces from its neighbouring segments. This leads to a set of
equations then, once solved, result in (1):
The derivation also gives some intuition into what the parameter
❓ What are catenaries used for?
Parametrising the catenary
If we want to draw physically sound catenaries, (1) might not be the best way to do it. The reason is simple: asides from
A more “customisable” equation is (2), which allows to move the curve horizontally and vertically using two additional parameters,
Ideally, however, we would like the equation of a catenary that passes by two anchored points,
The parameter
This is actually perfect if both
Solving the catenary problem
The following section will show the equations of a physically correct catenary that represents a rope anchored at two points in space,
First, let’s define two auxiliary parameters,
For this derivation, we are assuming that
The resulting values for
With the interactive chart below you can move the second anchor point around (
Finding a
The section above failed to provide an equation for the first parameter of the catenary: transcendental equation:
To put it simply, this means that we cannot rearrange this equation in a simple form such as
When this happens, it means that we need to a different approach to calculate the value of analytical tools fails, we resort to numerical ones. Which means that we need to use an algorithm to find an
approximated solution.
The next post in the series will explore how we can do that.
What’s Next…
This first post introduced catenaries, the mathematical objects used to model hanging chains. The next one in the series will explore how to implement them in a game engine such as Unity.
Download Unity Package
Become a Patron!
There are two different Unity packages available for this tutorial. They contain a simple library to draw efficiently catenaries, which you can use in your games. Both packages are available through
The Standard package one contains the scripts to draw catenaries in 3D and 3D, along with a test scene. The Advanced package contains support for rigged models (such as corded cables or chains),
along with some advanced code to sample catenaries uniformly.
Feature Standard Advanced
Catenary 2D ✅ ✅
Catenary 3D ✅ ✅
Rigged Model Support ❌ ✅
Cable Shader ❌ ✅
3 responses to “The Mathematics of Catenary”
Hi Alan,
I am a student in Year 13 interested in the derivation and parametrisation of the catenary curve formula. You stated that the derivation to solve the catenary problem is rather convoluted but I
would love to read about it and I can’t seem to find the information on the internet. Would you be able to share the derivation with me, or the source you used to find p and q?
Thank you so much for your epic article!
– Mia 😀
Hi Mia!
These are two links that might help you!
– https://proofwiki.org/wiki/Equation_of_Catenary
– https://math24.net/equation-catenary.html
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Reply To: Einstein
April 29, 2016 at 10:03 am #564
Unfortunately, the editors and reviewers of the physical journals, Phys. Rev., EJPC, ZNA and Foundation of Physics could not begin somewhat physically with my new quantum field theory and rejected
all my emitted articles. Therefore, the new theory came not in the circulation of physical science.
A comparison of quantum field theories
The currently used quantum field theories quantize the energy and the interaction with the Planck constant h and the special relativity theory states the energy-mass equivalence relation, E = m∙c^2.
I have defined another kind of quantum field theory in which only the sources of the interactions are quantized. The energy and the interactions are not quantized and the energy is not equivalent the
mass. We have obviously to do with two physically quite different kinds of quantum field theories.
I collect the key features of the new quantum field theory:
– The sources of the interactions are the conserved elementary charges. We know two kinds of elementary charges, they are the elementary electric charges qi = {± e} and the elementary gravitational
charges gi = {± g∙mi}. Assumption: Physically no more elementary charges exist.
– The elementary charges generate two kinds of continuous fundamental interaction fields which propagate with c. These fields are at the presence of charges non-conservative and the two fields are
independent from each other, they do not influence each other. The universal gravitational constant is G = g^2/4π.
– The elementary charges are distributed on four stable elementary particles, e, p, P and E. Each particle has two kinds of elementary charges. The elementary pariticles are the electron (e), the
positron (p), the proton (P) and the elton (E). The elton is usually called as “antiproton”. Assumption: Physically no more stable elementary particles exist.
– The masses me and mP are the elementary masses of electron (e) and proton (P). The inertial and the gravitational masses of the elementary particles, e,p,P,E, are in each case equal and only for
the stable elementary particles are the inertial and the masses equal. Assumption: These stable particles are not composed of any other particles.
– There exists a general uncertainty: neither the positions, nor the velocities of the elementary particles can be ever exactly observed. Infinitely large and infinitely small relative distances
between particles do not belong to a physical description.
– Time and space are homogeneous and the space is isotropic. Because the interaction propagation with c the time and space are connected; the Minkowski space describes the connectivity of space and
Results of my new quantum field theory:
An action integral
I = ∫Ω L(x) (dx)^4 (1)
can be constructed in finite ranges {x}εΩ of Minkowski space with the key features. The elementary charges, qi and gi, generate the vector fields A(em)ν(x) and A(g)ν(x). The Lagrange density L(x) is
the sum of the free particles parts, of the free fields parts constructed with the Faraday tensors, F(em)νμ, F(g)νμ, and of the interaction parts
L(int)(x) = + j(em)ν A(em)ν(x) – j(g)νA(g)ν(x), (2)
whereby j(em)ν and j(g)ν are the probability density currents of the charges. All parts of L(x) are constructed Lorentz-invariantly. Since all parts of L(x) are caused by the conserved elementary
charges the action integral does not depend on the boundary conditions of the surface of Ω containing some numbers of particles, Ni. The action integral is not an expression of energy. Within Ω there
exist different kinds of subsidiary conditions, one kind for the fields and another kind for j(em)ν and j(g)ν. The subsidiary conditions of the fields are known as Lorenz conditions. The Lorentz
conditions express that the fields propagate with the constant velocity c within Ω. With these subsidiary conditions for the fields and applying the Hamilton principle on the action integral, I, we
get the Maxwell equations for the fields. The Maxwell equations for the electromagnetic field and the gravitation field differ only on the sign of the probability density currents of the charges.
The determination of the equations for the particle motions
In order to get a Lorentz-invariant action integral, we put for the particles parts in the Lagrange density, L(x), the continuity equations of particle numbers, ji(n)ν(x), for i=e,p,P,E, multiplied
by the constants mi∙c
L(p)(x) = Σi=e,p,P,E mi∙c∙∂ν ji(n)ν(x). (3)
Additionally, the subsidiary conditions of the particles, Gi, are to be considered at the variation
Gi = ∫Ω ∂ν ji(n)ν(x) (dx) 4 = 0, i=e,p,P,E. (4)
Before we perform the variation we must express the j(n)ν (x) in a quadratic form, with the Dirac spinors and with the γν matrices
j(n)ν (x) = c ⋅ Ψi(x) γν Ψi(x). (5)
Ψi(x) = Ψi(x) γ0 are the adjoin spinors to the spinors Ψi(x) and the expression Eq. (5) has the correct transformation behavior under Lorentz transformation. γ0 γ0 = 1 is the unit four matrix. We
have to use the spinors because neither the positions, nor the velocities of the particles are ever exactly known. Furthermore, the Noether charges
∫V Ψi(r,t) γ0 Ψi(r,t) d^3x = ∫V Σj=1,4 Ψ*i,j(r,t) Ψi,j(r,t) d^3x = Ni, i=e,p,P,E (6)
are used for the normalization of the spinors for each volume V and at each time t. The sum j is taken about the four components of the four dimensional spinor. Applying the Hamilton principle we get
the equations of motions for the particles expressed with the spinors. The stationary of the variation of I, considering the subsidiary condition, Gi,
δI + Σj Σi=e,p,P,E λj∙δGi = 0, (7)
cause the appearance of Lagrange multipliers, λj, in the Euler-Lagrange equation of the spinors in using independent variations of the adjoint spinors Ψi(x) and Ψi(x) for each particle i=e,p,P,E.
Bound states of elementary particles
In order to get temporary stationary bound state of elementary particles, we have to consider conditional probabilities for particle density currents relative to center of mass system (COM) in the
mutual fields of the particles: we are looking for the temporary stationary solutions for the particles and the mutual fields. The conditional probabilities depend on the relative distances of
particles. Since the particle number conservations are further valid, the appearance of Lagrange multipliers, λi, is expected. Thus, the temporal stationary conditions are connected with the Lagrange
multipliers in such a way that the probabilities to find particles in distances relative to COM are temporally stationary and the relative currents vanish. Simultaneously, the mutual interacting
field is also temporary stationary. On this point enter the relativity in the physical description. We have to express the action integral, I, and the subsidiary conditions, Gi, relatively to COM.
Thus, we have to use the spinors as conditional probabilities of particle density currents with relative coordinates. The finite range Ω must contains the unique point of COM. Generally, temporary
stationary solutions of the variation for a λi are given with different values of an additional positive real parameter E. The largest possible discrete value of E belonging to a λi are labeled as
the ground state with the bound energy E0(λi). The bound energies are always negative, therefore, we label with the positive E0(λi) the negative of the bound state energies. Generally, a set of
discrete values of the parameter E exists with
E0(λi) > E1(λi) > E2(λi) > E3(λi) …. > 0 (8)
for each λi. A bound state is generally a superposition of temporary stationary solutions. Nevertheless, we don’t speak about energy quantization at the relation (8) because this superposition is
connected not only to temporary stationary mutual fields, but also to the simultaneous presence of radiation component of the field. Bound states emit always radiation with continuous submission of
radiation energy, until the energy E0(λi) is reached.
The physical interpretation of the bound state problem
We have to describe the problem of the capture of an electric charged particle, i, with the elementary mass, mi, in the electric field of other moving particles with elementary charges qj = { ± e}
with the elementary masses {me, mP}. Thereby we know neither the initial positions, not the initial velocities of the particles (for instance for the captured electron by atoms). Generally, electric
charged particles moving in an electromagnetic field radiate always electromagnetic waves and the waves (the fields) propagate with c. According to the subsidiary conditions of the particles, Eq.
(4), Lagrange multipliers, λi, are appearing in the equations of particles motions and these constants ensure temporally stationary solutions at some real values Ei(λi) of the whole problem.
Mathematically, we have to solve four coupled differential equations of the first order in space with the normalization conditions, Eq. (6), for the particles. Since the spinors describe the
probability density currents of particles, in the four components of the spinors, Ψi,j(r), j=1,4, the three components of the velocities are coded. The result is finally the determination of the
bound energies Ei(λi) of the particle system for the temporary stationary case. In the following we shall write for simplicity E0(λi) = E(bound) for the ground state belonging for one λi. and rename
λi. with h.
Fortunately, in the case of two-particle systems we have the possibility to say something about the relative velocity and about the relative distance of the ground state of particle systems in the
mutual temporary stationary interaction as a function of E(bound) and of the reduced mass m’ = mi∙mj/(mi+mj), without solving explicitly the variation problem.
For the Planck constant h, which describe the atomic shells, there exists an old expression set up by Sommerfeld
h = e^2/2c∙(m’∙c^2/2∙E(bound))^1/2 = e^2/2c∙1/α, (9)
for the hydrogen atom, where m‘ = me∙mP/(me+mP) ≈ me is the reduced mass of electron and proton and the bound energy of the ground state energy is E(bound) = 13.8 eV. However, in the current quantum
mechanics is not understood why α = v/c has a value of α = 1/137.036. v is the relative velocity of the electron. With the same relation for h, Eq. (9), the positronium problem can also be determined
with the energy of the ground state, E(bound; positronium) = 13.8/2 eV = 6.9 eV, since the reduced mass is m’ = me/2. We shall use the relation, Eq. (9), for each two-particle-systems with the
opposite signs of electric charges in order to get other values for h, if we have different reduced masses m’ and different ground state energies, E(bound). With other words, we have with Eq. (9) a
simple way to get expressions of the values of the constant h in cases of two-particle-systems. For two-particle-system we have the expression for the inertial mass
mi = mi + mj – E(bound)/c^2.
In case of an electron-positron system, (e,p), if we use the condition for the bound state that the inertial mass is zero, mi = 0, we have the condition
2∙me = E(bound)/c^2.
Since the inertial mass, mi, cannot be negative, this states the lowermost bound energy of the (e,p)-system. Setting this equation and the reduced mass m‘(e,p) = me/2 in the relation, Eq. (9) for h,
we get another value
h0 = e^2/2c∙(1/8)^1/2 = h/387. (10)
Thus, we get a much smaller value for h0 as the Planck constant h. The relative velocities of the bound particle (e,p) can be calculated for instance according
(v/c)/(1-(v/c)2) = (2∙E(bound)/(me∙c2).
Since in the case of (e,p), we have two particles with the same mass, we must set in the half of the ground state energy, E(bound), in order to get (v/c) = 0.894. The particles move with the velocity
89.4% of c. We label this state of the electron-positron system as the electron-neutrino, νe = (e,p); the neutrino has as well the inertial as the gravitational mass zero.
We have also a simple relation for the calculation of the relative distance of particles, of the size of the ground states radii,
r = h^2/(4∙π2∙m’∙e^2), (11)
For the electron-neutrino, νe, we get r(νe) = 0.703∙10^-13cm. Besides the (e,p)-system, we can also calculate with h0, Eq. (10), for the (P,e), (p,E) and (P,E)-systems the ground state energies, the
relative velocities and the sizes. The ground state energies of the (P,e), (p,E) are 2.04 MeV. The ground state energy of the (P,E)-system is E(bound) = 2∙mP∙c^2. The size are d(P,e) = 2∙r(P,e) =
0.702∙10^-13cm and the same size appear also for the (p,E)-system. With the h0 calculated ground state of (P,E) we get a size of r(P,E) = 0.383∙10^-16cm: we label this state as the proton-neutrino,
νP = (P,E). According to the finite sizes of two-particle ground states, we can state: in the interaction of elementary particles no singularities can occur and particles with the same mass don’t
In opposition to Einstein’s energy-mass-equivalence, E = m∙c^2, the electron-positron and the proton-elton pairs do not annihilate at their merging in our theory. The energies of particle systems and
the electromagnetic interaction are not quantized with E = h ν in our new quantum field theory. The new quantum field theory is an atomistic theory of matter based on the four kind of stable
particles e, p, P and E with the conserved charges qi and gi.
Unfortunately, the transmission of upper and lower indexes does not work. Double appearing indexes are summed.
Gyula I. Szász
• This reply was modified 8 years, 6 months ago by Gyula Szász.
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s about STACK
Documentation home
Category index
Site map
STACK is a very popular online assessment system, used by many groups in a variety of languages. STACK has been the subject of research, and has itself enabled research projects to take place. This
page contains a selection of publications.
Publicity materials.
We have a booklet of case studies, and a PDF flyer about the STACK project.
We have a "Getting started with STACK" guide (Spanish).
Computer aided assessment of mathematics
Computer Aided Assessment of Mathematics, Chris Sangwin, Oxford University Press, 2013.
This book provides an Introduction to Computer Aided Assessment using STACK as the main working example.
Nakamura (2010)
Y. Nakamura, The STACK e-Learning and Assessment System for mathematics, science and engineering education through Moodle, Tokyo Denki University Press, 2010, (In Japanese) ISBN 978-4-501-54820-9.
Papers about STACK
Proceedings of the 1st International STACK Conference can be found can be found on the open access publication server Zenodo here: https://zenodo.org/communities/stack
These recent papers about STACK are a good place to start:
A comprehensive bibliography is available here: STACK bibliography, with the entries in BiBTeX format.
Documentation home
Category index
Site map
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Square Root Function In Dev C++
Square Root Function In Dev C++ Rating: 3,1/5 2625 reviews
Nov 25, 2012 Check out for more free engineering tutorials and math lessons! C Programming Tutorial: Square root and power functions Please. Returns the square root of x. Header provides a
type-generic macro version of this function. This function is overloaded in and (see complex sqrt and valarray sqrt ). However, teachers at universities don't like to let the things easy for
students, that's why in programming classes you may need to find a way to find the square root of a number without using this library in C! As homeworks or tasks aren't optional, we'll show you how
you can easily achieve this goal without using the sqrt function in C.
• C Standard Library Resources
The math.h header defines various mathematical functions and one macro. All the functions available in this library take double as an argument and return double as the result.
Library Macros
There is only one macro defined in this library −
Sr.No. Macro & Description
1 This macro is used when the result of a function may not be representable as a floating point number. If magnitude of the correct result is too large to be represented, the function sets errno
to ERANGE to indicate a range error, and returns a particular, very large value named by the macro HUGE_VAL or its negation (- HUGE_VAL).
If the magnitude of the result is too small, a value of zero is returned instead. In this case, errno might or might not be set to ERANGE.
Library Functions
Following are the functions defined in the header math.h −
Square Root Function In Dev C 5
Sr.No. Function & Description
double acos(double x)
Returns the arc cosine of x in radians.
double asin(double x)
Returns the arc sine of x in radians.
double atan(double x)
Returns the arc tangent of x in radians.
double atan2(double y, double x)
Returns the arc tangent in radians of y/x based on the signs of both values to determine the correct quadrant.
double cos(double x)
Returns the cosine of a radian angle x.
double cosh(double x)
Returns the hyperbolic cosine of x.
double sin(double x)
Returns the sine of a radian angle x.
double sinh(double x)
Returns the hyperbolic sine of x.
double tanh(double x)
Returns the hyperbolic tangent of x.
double exp(double x)
Returns the value of e raised to the xth power.
double frexp(double x, int *exponent)
The returned value is the mantissa and the integer pointed to by exponent is the exponent. The resultant value is x = mantissa * 2 ^ exponent.
double ldexp(double x, int exponent)
Returns x multiplied by 2 raised to the power of exponent.
double log(double x)
Returns the natural logarithm (base-e logarithm) of x.
double log10(double x)
Returns the common logarithm (base-10 logarithm) of x.
double modf(double x, double *integer)
The returned value is the fraction component (part after the decimal), and sets integer to the integer component.
double pow(double x, double y)
Returns x raised to the power of y.
double sqrt(double x)
Returns the square root of x.
double ceil(double x)
Returns the smallest integer value greater than or equal to x.
double fabs(double x)
Returns the absolute value of x.
double floor(double x)
Returns the largest integer value less than or equal to x.
double fmod(double x, double y)
Returns the remainder of x divided by y.
In this Example we will learn how to find the square root of a given number using C++. Dev c does not name a type pe error.
Square Root Function In Dev C System
In the first example we are going to use std::pow function to calculate the square root.
#include <cmath>
cout<<'nSqure of '<<number<<' is: '<<result<<endl;
Squre of 9 is: 3
Now In the next example we will learn how to find the square root of a given number using std::sqrt function.
#include <cmath>
cout<<'nSqure of '<<number<<' is: '<<result;
Squre of 36 is: 6
• C++ Simple Programs And Examples
Square Root Function In Dev C Pdf
• C++ Sorting algorithms & Techniques
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Andrew D. Smith
will speak on
Spirals in Spaces of Holomorphic Functions
Tue 17th September 2024
E0.32 (beside Pi restaurant)
[map] Further informationAbstract:
Functions $W(t,z)$ of real time $t \ge 0$ and $z \in \mathbb C$ satisfy the spiral relation:
W(2t, z) = (1+e^z) W(t, z)
For fixed $t$, these are holomorphic functions of $z$ in the region:
\vert \Im z \vert < \cos^{-1} \left[ - \tfrac 1 2 e^{ - \vert \Re z \vert} \right]
Viewed as functions of $t$, for fixed $z$, the functions $W(t,z)$ are H\'{o}lder continuous and nowhere differentiable. They have a time-homogeneity property if $\Re z = 0$, while for $\Im z = \pm \
frac 1 2 \pi $ the paths have finite quadratic variation; a property also associated with semi-martingale paths in the theory of stochastic processes.
The $W$ functions can produce beautiful images. Familiar fractal sets: L\'{e}vy's C-curve, Heighway's dragon curve and van Roy's unicorn curve arise as the loci of $W(t,z)$ when $0 \le t \le 1$ and
$z = \pm \frac 1 2 i\pi$, that is, functions of $t$ that satisfy both the time-homogeneity and quadratic variation criteria.
(This talk is part of the Analysis series.) | {"url":"https://maths.ucd.ie/seminars/1916","timestamp":"2024-11-03T21:59:03Z","content_type":"application/xhtml+xml","content_length":"13894","record_id":"<urn:uuid:2955c42d-ac33-434a-9d6e-187f4947a6a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00054.warc.gz"} |