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The Order of Operation is the single most important thing you need to understand as you progress in your journey. It is the root cause of most mistakes I see in questions posted on the Forum – So
what is it –
It is the sequence that Tableau goes through each time you create a viz. There are several models – the one I use consistently is 10 steps in 3 groups:
Each worksheet in Tableau has its own data table – a subset of the total data set that was uploaded. That subset is created first by filtering out unneeded data at the workbook level (Steps 1-2)
then creating and filtering a table structure – think like a spreadsheet even though it is a tall narrow structure internal to Tableau) In Steps 3-5 and the remaining steps load values into the
table, do some calculations, and create the viz.
The first 2 steps operate at the “Workbook Level” and filter out data as files are uploaded into Tableau reducing the data volume and improving performance. They are very easy to use – on the data
source tab – Add a filter – select a dimension or measure and then the values you want to filter out:
It’s no more difficult than that!
The use of the Context filter in Step 3 and the effect on Fixed LODs, Top N, and Set formation causes much confusion for new and more experienced users.
It is not difficult – just follow a simple rule –
If you want the filter to affect the results of the Fixed LOD, the Top N or the creation of a set then place the filter in Context – If not then use Dimension filters that are applied in Step 5.
In this example, I’ve written a LOD to total the Sales at the Category level:
{ FIXED [Category]: sum([Sales])}
And used it to calculate the percent of Sales by category
sum([Sales])/sum([Fixed LOD Category Sales])
Category and Sub-categories are on rows and the Fixed Category, Sales, and % to Category measures make up the chart – A filter will be applied to Sub-Category
First Sub-Category filter is applied in Context before the LOD is calculated so the Fixed Category totals are affected and the result changes with the filter –
Now the same filter is applied as a Dimension filter After the Fixed Category LOD is calculated so the totals are not affected by the filter – (Note the filter is NOT ignored – you as the developer
used a dimension filter (Step 5) not a context filter Step 3 and filtering is done after the LOD is calculated Step 4))
Data from Blended data sets is loaded in Step 6 of the Order of Operations – After all, filtering has been applied and after Fixed LODs, Top N and Set are created – Remember when we talked about
connecting data tables and specifically about some of the limitations of Blended? The Order of Operation is immutable which is why you can’t use data from Blended data sets in Fixed LODs, Top N, or
Set, and why filtering is limited to the level of the link between the data sets (More on that in the next Session)
Include and Exclude were 2 of the most difficult functions for me to understand when I first started using Tableau. It took some work but I think this explains the concept –
Think of Include as setting a limit in the dimensions in the view – below that dimension, the detail and the Include expression return the same value – above the limit, the Include expression works
like “Fixed” and returns the value of the Fixed dimension
In this example "Ship Mode" is the Dimension used as the base
{ INCLUDE [Ship Mode]:avg([Sales])}
Below Ship Mode, the Tableau Average and Include Return the same value – the average is the total sales divided by the record count at the detail level (Region-Ship mode – Segment) – Include
returns the same values at the level
When evaluated at the level of the Include statement (Ship Mode) the individual row level values are the same as the Tableau calculated averages – the sub-total of the Include is the average of the 4
regions where the Tableau values are the average value of all records in the Region
At the Region level – above the level of Include we see only 4 rows – the Include statement averages are the average of the 4 Ship Modes in the regions where the Tableau average is the average of all
the records in the region.
In contrast, the Exclude function performs the aggregation at the next level above the dimension level in the Exclude statement
{ EXCLUDE [State/Province]:sum([Sales])}
Will aggregate the Sales at the next highest level above State in the map – the Region
For the advanced user, a more detailed explanation can be found at the Link to Include and Exclude on my blog
In the previous steps, we have looked at filters that affect the structure of the data table. In Step 8 we look at Measure Filters that operate on the Values in the data table. They can be applied
at individual record level in the underlying table for the worksheet or at the aggregate level:
Using Superstore data we first find the number of orders that have an individual record (think row on the invoice) that have Sales $ greater than $10K then we find the total number of Orders with
Sales $ greater than 10K – There will be more Orders with TOTALS greater than $10K than individual rows greater than $10k
Create the viz and drag Sales to the Filter Shelf – and a window opens – Selecting the “All Values” option applies the filter at the record level in the underlying (detail) level of the data table.
Any other options apply to the aggregate as created by the Dimensions in the viz – The default aggregation is Sum() but many others are available
Here the All Values option is selected and the Sales row level is filtered to greater than 10,000 – the result is 5 orders have individual rows that meet the criteria
In the same viz, if we filter at the aggregate level for orders with a total value greater than 10,000 there are more than 5000 orders
The 9 Step – Grand Totals and Sub-Totals is a broad topic
Here we cover the basics which are cool in themself.
Totals and subtotals are calculated in a separate module in Tableau. There the individual detailed records in the underlying data table are aggregated to get the totals – that differs from
spreadsheet calculators where columns or rows are summed from the sheet itself.
The Grand Totals function can be reached through “Analytics” on the top ribbon or (my fave) can be Dragged and Dropped on the canvas from the Analytics tab in the Date Frame
The default aggregation is Sum()
But that can be easily changed by opening the Measure green pill, picking Total Using, and then the option you want – The aggregation will be used on the Totals (Rows and columns) – the table cell
values will be aggregated by the selection made when they were brought to the viz.
Sub-totals can be applied in the same way- Drag and drop the total to the Subtotals Icon
And Subtotals are applied at the Segment level:
Grand totals and subtotals can get very complicated – nested subtotals, totaling table calculations, and custom totals are a few – When you are ready, I encourage you to read Link to GT and Sub T
which provide detailed use case examples
Table Calculations are evaluated in the 10^th and last step of the Order of Operation – They are an extensive topic that will be covered in detail as a “Special Calculation” in a later section.
As a newbie Table Calculations were in my comfort zone. They are similar in concept to the way calculations work on spreadsheets and they are primarily executed on the table that you see in the viz
With Table Calculations, we talk about Scope and Direction, and position within the table. Scope refers to how much of the table is used in the calculation, Direction – Across or Down – is the
direction in the calculation is executed in the table (In Excel it is equivalent to the direction you copy and paste a formula) and position is a table is the cell location with respect the other
cells around it (similar to Offset in Excel).
In this basic example, the total sales is each Subcategory are added to the Running total going down the table
To get you started, Tableau has a group of commonly used Table Calculations that can be applied directly from a dropdown menu
Table calculations are also used to navigate the table – there are 5 functions – First, Last, Index, Previous Value, and Lookup which can be used to return a value from a relative position in the
For example:
Will look in the column “Sum(sales)” and look back 1 cell (upward) and return the value from that cell.
Will return the first value from the sum(sales) column
The best way to learn table calculations is to use them – practice and see what they return. With practice, you will be able to write custom and nested table calculations and use them in
combinations. When you are ready see the post at Link to Table Calculations where I go into much more depth and include use case examples using advanced table calculations. | {"url":"https://jimdehner.com/2023/09/06/bgs-session-4-the-order-of-operation/","timestamp":"2024-11-09T20:13:54Z","content_type":"text/html","content_length":"95583","record_id":"<urn:uuid:2bde5f06-2749-495a-adf0-3b9125b39540>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00751.warc.gz"} |
Risk-Adjusted Returns: Understanding Metrics and Their Importance in Investment Analysis
6.4.3 Risk-Adjusted Returns
In the world of investments, understanding the relationship between risk and return is crucial for making informed decisions. Risk-adjusted returns are metrics that help investors evaluate the
performance of an investment by considering the amount of risk involved in achieving those returns. This section will delve into the concept of risk-adjusted returns, explain key metrics such as the
Sharpe Ratio and Alpha, and illustrate how these metrics are calculated and interpreted. By the end of this section, you will appreciate the importance of incorporating risk considerations into
investment performance evaluation.
Understanding Risk-Adjusted Returns
Risk-adjusted returns are measures that account for the risk taken to achieve returns, enabling comparison between funds with different risk profiles. They provide a more comprehensive view of an
investment’s performance by considering both the return it generates and the risk it incurs. This approach allows investors to compare investments on a level playing field, regardless of their
inherent risk levels.
Key Metrics for Risk-Adjusted Returns
Sharpe Ratio
The Sharpe Ratio is a widely used metric that measures the excess return per unit of risk. It helps investors understand how much additional return they are receiving for the extra volatility
endured. The formula for the Sharpe Ratio is:
$$ \text{Sharpe Ratio} = \frac{\text{Average Fund Return} - \text{Risk-Free Rate}}{\text{Standard Deviation of Fund Returns}} $$
• Average Fund Return: The average return generated by the fund over a specific period.
• Risk-Free Rate: The return on a risk-free investment, typically represented by government bonds.
• Standard Deviation of Fund Returns: A measure of the volatility or risk associated with the fund’s returns.
A higher Sharpe Ratio indicates better risk-adjusted performance, as it suggests that the fund is generating more return per unit of risk.
Alpha measures the excess return of a fund relative to the return of its benchmark index. It indicates the value added by the fund manager after adjusting for the market risk taken. A positive alpha
suggests that the fund has outperformed its benchmark on a risk-adjusted basis, while a negative alpha indicates underperformance.
• Fund Return: The actual return of the fund.
• Benchmark Return: The return of a comparable market index.
• Beta: A measure of the fund’s sensitivity to market movements.
• Market Return: The return of the overall market.
Interpreting Risk-Adjusted Performance Measures
When interpreting risk-adjusted performance measures, it’s important to consider the context and compare the metrics to similar funds or benchmarks. Here are some key points to keep in mind:
• Sharpe Ratio: A higher Sharpe Ratio is generally preferred, as it indicates that the fund is delivering more return for each unit of risk. However, it’s essential to compare the Sharpe Ratio to
those of similar funds to determine its relative performance.
• Alpha: A positive alpha indicates that the fund manager has added value by outperforming the benchmark on a risk-adjusted basis. Conversely, a negative alpha suggests underperformance. Investors
should consider the consistency of alpha over time to assess the fund manager’s skill.
Illustrating Calculations of Risk-Adjusted Returns
Let’s illustrate the calculation of risk-adjusted returns with a hypothetical example:
• Fund Return: 8%
• Risk-Free Rate: 2%
• Standard Deviation: 10%
Using the Sharpe Ratio formula:
$$ \text{Sharpe Ratio} = \frac{8\% - 2\%}{10\%} = 0.6 $$
A Sharpe Ratio of 0.6 is considered reasonable, but it’s important to compare this ratio to those of similar funds to assess its relative performance.
For Alpha, assume the following:
• Benchmark Return: 6%
• Beta: 1.2
• Market Return: 7%
Using the Alpha formula:
$$ \text{Alpha} = 8\% - (6\% + 1.2 \times (7\% - 2\%)) = 8\% - (6\% + 6\%) = 8\% - 12\% = -4\% $$
In this example, the fund has a negative alpha of -4%, indicating underperformance relative to its benchmark on a risk-adjusted basis.
The Importance of Considering Risk in Fund Performance
Focusing solely on returns may overlook the risks taken to achieve them. Risk-adjusted metrics provide a more complete picture of a fund’s performance by considering both the return generated and the
risk incurred. This approach helps investors select funds that align with their risk tolerance and investment objectives.
By evaluating both returns and risk, investors can make more informed decisions and construct portfolios that balance potential rewards with acceptable levels of risk. This is particularly important
in volatile markets, where understanding the risk-return trade-off is crucial for achieving long-term investment success.
Risk-adjusted returns are essential tools for evaluating investment performance. By considering both the returns generated and the risks incurred, these metrics provide a more comprehensive view of a
fund’s performance. The Sharpe Ratio and Alpha are key metrics that help investors assess risk-adjusted returns and make informed investment decisions. By incorporating risk considerations into
performance evaluation, investors can select funds that align with their risk tolerance and investment objectives, ultimately enhancing their investment outcomes.
Quiz Time!
📚✨ Quiz Time! ✨📚
### What is the primary purpose of risk-adjusted returns? - [x] To account for the risk taken to achieve returns - [ ] To maximize returns regardless of risk - [ ] To minimize risk without
considering returns - [ ] To compare funds based solely on returns > **Explanation:** Risk-adjusted returns measure the performance of an investment by considering the amount of risk involved in
achieving those returns, enabling comparison between funds with different risk profiles. ### Which formula represents the Sharpe Ratio? - [x] (Average Fund Return - Risk-Free Rate) / Standard
Deviation of Fund Returns - [ ] (Fund Return - Benchmark Return) / Beta - [ ] (Market Return - Risk-Free Rate) / Standard Deviation of Market Returns - [ ] (Fund Return - Risk-Free Rate) / Alpha >
**Explanation:** The Sharpe Ratio formula is (Average Fund Return - Risk-Free Rate) / Standard Deviation of Fund Returns, indicating how much excess return is received for the extra volatility
endured. ### What does a positive alpha indicate? - [x] Outperformance on a risk-adjusted basis - [ ] Underperformance relative to the benchmark - [ ] Higher volatility than the market - [ ] Lower
returns than the risk-free rate > **Explanation:** A positive alpha suggests that the fund has outperformed its benchmark on a risk-adjusted basis, indicating value added by the fund manager. ### How
is the Sharpe Ratio interpreted? - [x] A higher Sharpe Ratio indicates better risk-adjusted performance - [ ] A lower Sharpe Ratio indicates better risk-adjusted performance - [ ] A higher Sharpe
Ratio indicates higher risk - [ ] A lower Sharpe Ratio indicates lower returns > **Explanation:** A higher Sharpe Ratio indicates better risk-adjusted performance, as it suggests that the fund is
generating more return per unit of risk. ### What is the significance of comparing the Sharpe Ratio to similar funds? - [x] To assess relative performance - [ ] To determine absolute performance - [
] To calculate risk-free returns - [ ] To measure market volatility > **Explanation:** Comparing the Sharpe Ratio to similar funds helps assess relative performance and determine how well a fund is
performing compared to its peers. ### What does a negative alpha suggest? - [x] Underperformance relative to the benchmark - [ ] Outperformance on a risk-adjusted basis - [ ] Higher returns than the
market - [ ] Lower volatility than the benchmark > **Explanation:** A negative alpha indicates underperformance relative to the benchmark on a risk-adjusted basis, suggesting that the fund manager
has not added value. ### Why is it important to consider both returns and risk? - [x] To make informed investment decisions - [ ] To maximize returns regardless of risk - [ ] To minimize risk without
considering returns - [ ] To focus solely on short-term gains > **Explanation:** Considering both returns and risk helps investors make informed decisions and construct portfolios that balance
potential rewards with acceptable levels of risk. ### What is the role of the risk-free rate in the Sharpe Ratio? - [x] It represents the return on a risk-free investment - [ ] It measures the
volatility of the market - [ ] It indicates the fund's sensitivity to market movements - [ ] It calculates the excess return of the fund > **Explanation:** The risk-free rate represents the return on
a risk-free investment, typically government bonds, and is used in the Sharpe Ratio formula to calculate excess return per unit of risk. ### How does Alpha measure a fund's performance? - [x] By
comparing the fund's return to its benchmark on a risk-adjusted basis - [ ] By calculating the fund's volatility - [ ] By measuring the fund's sensitivity to market movements - [ ] By determining the
fund's absolute return > **Explanation:** Alpha measures a fund's performance by comparing its return to its benchmark on a risk-adjusted basis, indicating the value added by the fund manager. ###
True or False: Risk-adjusted returns provide a more complete picture of a fund's performance. - [x] True - [ ] False > **Explanation:** True. Risk-adjusted returns consider both the return generated
and the risk incurred, providing a more comprehensive view of a fund's performance. | {"url":"https://csccourse.ca/6/4/3/","timestamp":"2024-11-09T13:17:59Z","content_type":"text/html","content_length":"92673","record_id":"<urn:uuid:ff1c411f-77fe-4b55-8e20-e7fb99b80a18>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00069.warc.gz"} |
Department of Mathematics - MSc. Mathematics
The Department of Mathematics offers a graduate program that leads to the degree of Master of Science in Mathematics. The program features a thesis and a non-thesis options. The thesis option
requires a successful completion of a thesis, and the non-thesis option requires the completion of a project. Both part-time and full-time students can be admitted to the program. Current research
interests of the faculty include:
Differential Equations, Numerical Analysis, Integral Transforms, Fractional Calculus, Algebra (Ring Theory); Analysis (Functional Analysis, Operator Theory; Approximation Theory and Fourier
Analysis), Geometry (General Topology, Differential Geometry, and Algebraic Topology), Discrete Mathematics, Combinatorics, Dynamical Systems, and Control.
To be recognized as one of the top mathematics department in the region for offering strong M.Sc. in Mathematics, and to be valued internationally as a significant contributor to the continuing
graduate education in these fields.
To train students to apply and disseminate mathematical knowledge and understanding.
• Program Educational Objectives (PEOs):
1. To provide students with a comprehensive mathematical education at Kuwait University, so that they can either continue to pursue a Ph. D program at top institutions around the world or to pursue
Master and Ph. D programs abroad.
2. To prepare students to compete successfully for employment and higher positions in government, industry, and non-profit organizations.
3. To attract students with solid background in mathematics and to engage them in mathematical research.
• Student Learning Outcomes (SLOs):
1. To solve problems in the advanced areas of mathematics including (a) differential equations, (b) numerical analysis, (c) algebra, and (d) real analysis.
2. To identify, formulate, and solve complex mathematical problems by applying advanced mathematics.
3. To communicate mathematical ideas with clarity and coherence, both written and verbally.
4. To perform and disseminate research individually and in collaboration with other researchers.
- Excellence
- Creativity
- Integrity and Professionalism
- Service to society
• Comprehensive Examination | {"url":"https://math.sci.kuniv.edu.kw/graduate/msc-mathematics","timestamp":"2024-11-14T07:00:36Z","content_type":"text/html","content_length":"192616","record_id":"<urn:uuid:803a86d3-d295-4f0d-8978-653d7fff92b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00837.warc.gz"} |
running trimmomatic inside Trinity
Hi, I am trying to run trimmomatic directly in the Trinity command, I am using the following:
trinity --seqType fq --max_memory 100G --left ~/Dp_RNAseq/raw_data/All_reads_R1.fastq --right ~/Dp_RNAseq/raw_data/All_reads_R2.fastq --CPU 8 --output ~/Dp_RNAseq/Trinity/ --quality_trimming_params "LEADING:2 TRAILING:2 MINLEN:25"
Doesn't matter how I type it, or the actual options I put in. Trinity doesn't seem to be able to read the parameters in the --quality_trimmin_params. I see getting the error:
Error, do not understand options: TRAILING:2 MINLEN:25
What I am doing wrong?. I checked with several examples and my command seems identical to them.
From Trinity wiki
[> --quality_trimming_params <string> defaults to: \ #
"ILLUMINACLIP:$TRIMMOMATIC_DIR/adapters/TruSeq3-PE.fa:2:30:10 SLIDINGWINDOW:4:5 LEADING:5 TRAILING:5 MINLEN:25"]
If you want to change parameters follow the command structure above.
Explanation for trimmomatic parameters is here.
thanks, I don't know how that is different from what I did. Using the ILLUMINACLIP and SLIDINGWINDOWS options doesn't change anything, it still give me the same error.
It appears that you have forgotten to include --trimmomatic option in your trinity command to indicate to trinity that you want to run that program.
I did run with the --trimmomatic option in the beginning and it didn't work, trying things I deleted it.
Can you try with following options? Replace $TRIMMOMATIC_DIR with real path to the adapters file on your system.
--trimmomatic --quality_trimming_params "ILLUMINACLIP:$TRIMMOMATIC_DIR/adapters/TruSeq3-PE.fa:2:30:10 TRAILING:2 MINLEN:25"
Thank you so much for your help @genomax2. I did tried that, I don't really need ILLUMINACLIP option because I don't want to use it, my reads were trimmed for adapters at the sequencing center, but
seeing that wiring only the two other options wasn't working, I tried writing the command exactly like in most examples out there (with the ILLUMINACLIP option), and yes, I made sure the path to
adapters was correct. Didn't work. I decided to run trimmomatic outside trinity and it worked fine, now I am running the assembly, still not knowing what was wrong.
I guess we won't know what the problem was then .. but as long as you got it to work. | {"url":"https://www.biostars.org/p/185084/#185100","timestamp":"2024-11-13T05:12:03Z","content_type":"text/html","content_length":"29658","record_id":"<urn:uuid:644f2a06-837e-47aa-8ac1-76fc482bb5f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00489.warc.gz"} |
the median is a measure of quizlet
2nd stat quiz Flashcards | Quizlet A distribution with a very large positive skew. PDF Measures of Central Tendency & Dispersion - MWSU Intranet The 3 most common measures of central tendency are the
mean, median and mode. You can calculate percentiles using calculators and computers. It means that 90 percent of test scores are the same as or less than your score and that 10 percent of the test
scores are the same as or greater than your test score. For example: The goal of descriptive statistics is to summarize and organize large amounts of data and measures of central tendency tell us
about the middle of a distribution but we need to select the measure that is most representative of the distribution. All measures of central tendency reflect something about the middle of a
distribution; but each of the three most common measures of central tendency represents a different concept: Mean: average, where is for the population and or M is for the sample (both same
equation). In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median. For each, starting value = ________ and ending value = ________.
9-3Measures of Central Tendency. Compute the weighted mean for the following data. The mode is the point on the x-axis that falls directly below the tallest point on the distribution. Test Scores for
Class A: There are 14 values less than the 28th percentile. Day Stock Price Today your instructor is walking around the room, handing back the quizzes. . 1 Find the median, first quartile, and third
quartile. 4 88 Find the median. The steps for finding the median differ depending on whether you have an odd or an even number of data points. ) will be the middle value, or 2. The variance of a
sample of 169 observations equals 576. Laerd Statistics. 5 85 Find the interquartile range for the following two data sets and compare them. The large skew results in very different values for these
measures. 300 seconds. mode = 165 variance = 324 What is the MEDIAN price of the items Mary bought? Chapter 4: Measures of Central Tendency, 6. ii. A mean refers to a ratio of the sum of the total
number in a data set to the frequency of the data set. If we consider the normal distribution - as this is the most frequently assessed in statistics - when the data is perfectly normal, the mean,
median and mode are identical. The mean of the sample is 5. Measures of Central Tendency The mean, median and mode are all valid measures of central tendency, but under different conditions, some
measures of central tendency become more appropriate to use than others. The third quartile is the same as the 75th percentile. Thus, the median of the lower half, or the first quartile ( Figure 2.
The arithmetic mean is the most common measure of central tendency. Therefore the mode of continuous data is normally computed from a grouped frequency distribution. Figure 2 shows the results of an
experiment on memory for chess positions. These are the three measures of central tendency: One definition of central tendency is the point at which the distribution is in balance. data into two
halves. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each
other. 12 - 16 4 B 25 The median is seven. How do you decide? The fulcrum or balancing point is calculated as the arithmetic mean or mean. 4 Chapter 4: Measures of Central Tendency - Maricopa Fifty
percent of 50 is 25. To calculate the mean, you first add all the numbers together (3 + 11 + 4 + 6 + 8 + 9 + 6 = 47). Are you looking for the average (the mean), do you want to identify the middle
score (the median), or are you looking for the score that appears most often (the mode)? Differences among the measures occur with skewed distributions. For the above sample, which is correct? We
calculated the mean as 6.8. n It can be used with both discrete and continuous data, although its use is most often with continuous data (see our Types of Variable guide for data types). Interpret
the 70th percentile in the context of this situation. Creative Commons Attribution License The common measures of location are quartiles and percentiles.. Quartiles are special percentiles. Figure 5.
Mean, median, and mode all serve a valuable purpose in analyzing psychological data. Therefore, a measure of central tendency is a way to summarize a large set of numbers using one single score.
Compare the mean, median, and mode in terms of their sensitivity to extreme scores. If no number in a set occurs more than once, there is no mode for that set of data. Make up three data sets with 5
numbers each that have: the same mean but different standard deviations. Since you have an odd number of scores, the number in the third position of the data set is the median which, in this case, is
9 (5, 7, 9, 9, 11). Compute Jason's semester grade point average. Which of the following statements about the median is Counting from the bottom of the list, there are three data values less than 25.
In order to find the mode, create a frequency table. To acknowledge that we are calculating the population mean and not the sample mean, we use the Greek lower case letter "mu", denoted as \( \mu \):
The mean is essentially a model of your data set. A 27 You have data measured on an ordinal scale. Subjects were shown a chess position and then asked to reconstruct it on an empty chessboard. Doing
reproducible research. The 75th percentile, then, must be an eight. Take these two steps to calculate the mean: Step 1: Add all the scores together. F 14 If you take too long, you might not be able
to finish. 7 91 Kendra Cherry, MS,is the author of the "Everything Psychology Book (2nd Edition)"and has written thousands of articles on diverse psychology topics. x+.5y Here are a few to consider.
Mean, Median, and Mode - Measures of Central Tendency - ThoughtCo Solved Which of the following statements about the median - Chegg However, as the data becomes skewed the mean loses its ability to
provide the best central location for the data because the skewed data is dragging it away from the typical value. After reading this lesson you should know that there are quite a few options when
one wants to describe central tendency. 6 terms. A measure of dispersion is a number which indicates how far each individual score (in the raw data set) is from the mean, (i.e. If you were the
principal, would you be justified in purchasing new fitness equipment? You can, therefore, sometimes consider the mode as being the most popular option. In computing the mean of a sample, the value
of xi is divided by, is computed by summing all the data values and dividing the sum by the number of items. b. Jan 18, 2023 Texas Education Agency (TEA). Table 3 shows the number of touchdown (TD)
passes thrown by each of the 31 teams in the National Football League in the 2000 season. Remember that measures of central tendency summarize and organize large sets of data that allow researchers
to communicate information with just a few numbers. Say IQ scores have a bell shaped distribution with a mean = 100, and a standard deviation = 15. 2. Verywell Mind's content is for informational and
educational purposes only. In a perfectly symmetrical (normal) distribution, all three measures of central tendency are located at the same value. However, 15 students is a small sample, and the
principal should survey more students to be sure of his survey results. The mode is the most frequent value. What if you had only 10 scores? Q 5 85 For the 11 salaries, calculate the IQR and
determine if any salaries are outliers. There are two important reasons that we must pay attention to the scale of measurement of a variable. Ordinal scale. This is particularly problematic when we
have continuous data because we are more likely not to have any one value that is more frequent than the other. For this reason, universities and colleges use percentiles extensively. In cases where
you have a large number of scores, creating a frequency distribution can be helpful in determining the mode. 2 Imagine this situation: You are in a class with just four other students, and the five
of you took a 5-point pop quiz. Introduction to Mathematical Statistics. 29 Verywell Mind uses only high-quality sources, including peer-reviewed studies, to support the facts within our articles.
The mode, median, and mean are all measures of central tendency. Find the third quartile. Which of the following statements about the median is The symbol (pronounced mew) is used for the mean of a
population. To calculate the mean, you first add all the numbers together (3 + 11 + 4 + 6 + 8 + 9 + 6 = 47). Finally, lets look at Dataset C. This is more like it! Therefore, they are both modes. Of
the three statistics, the mean is the largest, while the mode is the smallest. Germany 36 3 84 In fact, well offer you three definitions! The mean has one main disadvantage: it is particularly
susceptible to the influence of outliers. An example of a normally distributed set of data is presented below: When you have a normally distributed sample you can legitimately use both the mean or
the median as your measure of central tendency. If N or n is even then the median is the average of the middle two numbers, Mean is preferred when using ratio level data unless distribution includes
outliers, Median is the preferred when using ordinal data, Median is preferred when data include outliers, Mode is preferred when using nominal data, explain the purpose of measuring central
tendency, define and compute the three measures of central tendency (mean, median, mode), list the circumstances where each of the three measures of central tendency are appropriate, explain how the
three measures of central tendency are related to distribution (positive skew, negative skew, normal), If the mean time to respond to a stimulus is much. The median is defined as the value with 50%
of scores above it and 50% of scores below it; therefore, 60% of score cannot fall above the median. Solved Which of the following is a measure of central - Chegg 2.3 Measures of the Location of the
Data - Statistics - OpenStax Other statistics are based on ordering or ranking of values (such as the median, which is the middle value when all of the values are ordered by their magnitude), and
these require that the value at least be on an ordinal scale. Today your instructor is walking around the room, handing back the quizzes. The other two values are the minimum value (or min) and the
maximum value (or max). In this set, both 20 and 23 occur twice (13, 17, 20, 20, 21, 23, 23, 26, 29, 30). Changes were made to the original material, including updates to art, structure, and other
content updates. 4 88 90, 72, 80, 92, 90, 97, 92, 75, 79, 68, 70, 80, 99, 95, 78, 73, 71, 68, 95, 100. Again, the mean reflects the skewing the most. The midpoint is the middle score ranging from
lowest to highest values. She stops at your desk and hands you your paper. 20 terms . Which statement is NOT true, If the value of x equals the mean, the z score is 1, Chapter 3 test bank questions
(plus printed o, Statistical Techniques in Business and Economics, Douglas A. Lind, Samuel A. Wathen, William G. Marchal, Daniel S. Yates, Daren S. Starnes, David Moore, Betty Thorne, Paul Newbold,
William Carlson, Spanish Test Review - Di Algo, Poema 20, Soni. 2 A classic example of the above right-skewed distribution is income (salary), where higher-earners provide a false representation of
the typical income if expressed as a mean and not a median. If you answered with the mode of $250,000 or the median of $500,000, you would not be giving any indication that some players make many
millions of dollars. 1) MEAN (AO1) This is calculated by adding up all the scores in a group/ in the raw . In future lessons, we talk about mainly about the mean. 3. c. There is an open ended
distribution (For example, if you have a data field which measures number of children and your options are [latex]0[/latex], [latex]1[/latex], [latex]2[/latex], [latex]3[/latex], [latex]4[/latex],
[latex]5[/latex] or [latex]6[/latex] or more, than the [latex]6[/latex] or more field is open ended and makes calculating the mean impossible, since we do not know exact values for this field).
Chapter 3: Describing Data using Distributions and Graphs, 4. 9 82 This is not just generosity on our part. If you are redistributing all or part of this book in a print format, There are 25 values
less than the median. Measures of Central Tendency. Thus, the mean is more sensitive to skew than the median or mode, and in cases of extreme skew, the mean may no longer be appropriate to use. Hogg
RV, McKean JW, Craig AT. For the 100-meter dash, the third quartile for times for finishing the race was 11.5 seconds. Therefore, if you were to say that 90 percent of the test scores are less, and
not the same or less, than your score, it would be acceptable because removing one particular data value is not significant. There are three main considerations when determining which measure of
central tendency to use: Before deciding to report a mean, median or mode ask yourself what the data are trying to convey, what is the shape of the distribution (e.g., normal or skewed) and the level
of measurement for the data. 4 Levels of Measurement: Nominal, Ordinal, Interval & Ratio - CareerFoundry The problem is that the other four students had higher grades, putting yours below the center
of the distribution. The symbol used for standard deviation of a population is: The symbol used for standard deviation of a sample is: s Median is the preferred measure of central tendency when:
There are a few extreme scores in the distribution of the data. Interpret the 30th percentile in the context of this situation. Step 2: Divide the sum by the number of scores used. The median, M, is
called both the second quartile and the 50 th percentile. Nominal scale. All of your classmates score lower than you so your score is above the center of the distribution. 230,500+387,000 50 - 54 2
$120,000; 6 90 An (equal) interval scale has all of the features of an ordinal scale, but in addition, the intervals between units on the measurement scale can be treated as equal. . Seventy percent
of students study seven or more hours per week. If there are no outliers in your data set, the mean may be the best choice in terms of accuracy since it takes into account each individual score and
finds the average. So, percentiles mean the data is divided into 100 sections. The hourly wages of a sample of 130 system analysts are given below. 639,000+659,000 Please use the following summary
table to know what the best measure of central tendency is with respect to the different types of variable. Share Share by Rosie. Our mission is to improve educational access and learning for
everyone. This is a depressing outcome even though your score is no different than the one in Dataset A. Recall that a percent means one-hundredth. 2. The mode of these numbers would be 3 since this
is the most frequently occurring number (2, 3, 6, 3, 7, 5, 1, 2, 3, 9). Our website is not intended to be a substitute for professional medical advice, diagnosis, or treatment. The results were as
follows: Find the 28th percentile. This works fine when you have an odd number of scores, but what happens when you have an even number of scores? To find the median, add the two values together and
divide by two. Figure 9. $68,500; The first quartile is the median of the lower half of the scores and does not include the median. The mean is the arithmetic average of the scores, the median is the
midpoint of the ordered scores, and the mode is the score with the greatest frequency. Differentiate the function. The local university uses a 4 point grading system, i.e., A = 4, B = 3, C = 2, D =
1, F = 0. A standard example is physical temperature measured in Celsius or Fahrenheit; the physical difference between 10 and 20 degrees is the same as the physical difference between 90 and 100
degrees, but each scale can also take on negative values. The grade point average of the students at UTC is 2.80 with a standard deviation of 0.84. How likely is it that we will find two or more
people with exactly the same weight (e.g., 67.4 kg)? In the remainder of this section, we will give statistical measures for these concepts of central tendency. Figure 3. Answer the following
questions: If you were to do a little research, you would find several formulas for calculating the kth percentile. equal to it, while all the observations above the median are equal You can think of
the median as the "middle value," but it does not actually have to be one of the observed values. How to Find the Median | Definition, Examples & Calculator - Scribbr She stops at your desk and hands
you your paper. 2 Find the third quartile. a) 102 b) 103 c) 150 d) 120 . correct? The difference between a ratio scale variable and an interval scale variable is that the ratio scale variable has a
true zero point. His grade report is presented below. These constraints also imply that there are certain kinds of statistics that we can compute on each type of variable. P.E. So, if we look at the
example below: We again rearrange that data into order of magnitude (smallest first): Only now we have to take the 5th and 6th score in our data set and average them to get a median of 55.5. Since
there are 14 observations (an even number of data values), the median is between the seventh value, 6.8, and the eighth value, 7.2. As an example, imagine that your psychology experiment returned the
following number set: 3, 11, 4, 6, 8, 9, 6. to it or larger. Fifty statistics students were asked how much sleep they get per school night (rounded to the nearest hour). Sweden 46 These are values
that are unusual compared to the rest of the data set by being especially small or large in numerical value. Sets found in the same folder. The left side shows the memory scores of the non-players.
10 86 Counting from the bottom of the list, there are 18 data values less than 58. By the way, although the arithmetic mean is not the only mean (there is also a geometric mean, a harmonic mean, and
many others that are all beyond the scope of this course), it is by far the most commonly used. The mean = 11, the variance = 21, the standard deviation = 4.58. For example, 15 percent of data values
are less than or equal to the 15th percentile. Median is the preferred measure of central tendency when: There are a few extreme scores in the distribution of the data. n While simple to explain, the
median is harder to compute than the mean. Lets consider another example. We can clearly see, however, that the mode is not representative of the data, which is mostly concentrated around the 20 to
30 value range. In figure 5, the median is in the geometric middle as there is a similar distribution of higher and lower scores. The median is the middle score for a set of data that has been
arranged in order of magnitude. 3 The mode is the most frequently occurring value; the median is the middle value (refer back to the section on ordinal data for more information), and the mean is an
average of all values. a) 6 b) 12 c) 11 d) 4 10) If the mean of these numbers is 17 then the sum of these numbers is.? English 3 D n Other Quizlet sets. The term central tendency dates from the late
1920s.. All of the following statements are true of large data sets with negatively skewed distributions except: The mean is less than the mode. When there is an even number of numbers, the median is
the mean of the two middle numbers. Show More. Three possible datasets for the 5-point make-up quiz. The median The highest value The mean The mode The median is The equation f the midpoint of the
ordered scores the arithmetic average of all of the scores the score with the greatest frequency A the average of the lowest and the highest scores A B |< The equation for a sample Show transcribed
image text Expert Answer Interval scale. Mode: most common, or most frequent value, where there can be a tie or there can be no mode. Each measure of central tendency has its own strengths and
weaknesses. (100) = 63.80. Experts are tested by Chegg as specialists in their subject area. We need a formal definition of the center of a distribution. Verywell Mind content is rigorously reviewed
by a team of qualified and experienced fact checkers. Another problem with the mode is that it will not provide us with a very good measure of central tendency when the most common mark is far away
from the rest of the data in the data set, as depicted in the diagram below: In the above diagram the mode has a value of 2. Here are some useful tips to help you distinguish between these measures,
as well as how to calculate mean, median, and mode. $69,000; 114,950; 158,000; 230,500; 387,000; 389,950; 479,000; 488,800; 529,000; 575,000; 639,000; 659,000; 1,095,000; 5,500,000, Q1 = We often
test whether our data is normally distributed because this is a common assumption underlying many statistical tests. Percentiles are useful for comparing values. The median is also a frequently used
measure of central tendency. and you must attribute Texas Education Agency (TEA). Coefficient of variation = 64% What if he had said 60% of the students scored above the mean?. The median, a
different measure of central tendency, is the halfway point. Generally, if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. For
example, we might ask people for their political party affiliation, and then code those as numbers: 1 = Republican, 2 = Democrat, 3 = Libertarian, and so on. They each give us a measure of Central
Tendency (i.e. First, all X values were added up, then divided by the total number of teams. This puts your score at the exact center of the distribution. The median is the middle score in the set.
Solved 1.Which of the following are measures of central - Chegg Finding the Median The median of a set of data is the "middle element" when the data is arranged in ascending order. On the right are
people who play a great deal (tournament players). Determine whether each of the functions is exponential or not. You will hear about median salaries and median prices of houses sold, etc. A
distribution is a graph that shows how scores are distributed along a measurement scale. Since the median is the middle value of a data set it, The relative frequency of a class is computed by,
dividing the frequency of the class by the sample size, If the coefficient of variation is 40% and the mean is 70, then the variance is, Given the following information:
Toastmaster Pressure Cooker Troubleshooting, Articles T | {"url":"https://amandaelisek.com/shands-hospital/the-median-is-a-measure-of-quizlet","timestamp":"2024-11-06T09:07:34Z","content_type":"text/html","content_length":"29418","record_id":"<urn:uuid:ecceaf9b-32a2-4c09-bcca-87c5724cdd10>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00115.warc.gz"} |
Semantics of logic
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In logic, the semantics of logic or formal semantics is the study of the semantics, or interpretations, of formal languages and (idealizations of) natural languages usually trying to capture the
pre-theoretic notion of logical consequence.
The truth conditions of various sentences we may encounter in arguments will depend upon their meaning, and so logicians cannot completely avoid the need to provide some treatment of the meaning of
these sentences. The semantics of logic refers to the approaches that logicians have introduced to understand and determine that part of meaning in which they are interested; the logician
traditionally is not interested in the sentence as uttered but in the proposition, an idealised sentence suitable for logical manipulation.
Until the advent of modern logic,
, but with the generality of modern logics based on the quantifier.
The main modern approaches to semantics for formal languages are the following:
• The archetype of model-theoretic semantics is
first-order predicate logic is given by a mapping from terms to a universe of
, and a mapping from propositions to the truth values "true" and "false". Model-theoretic semantics provides the foundations for an approach to the theory of meaning known as
truth-conditional semantics
, which was pioneered by
Donald Davidson
Kripke semantics
introduces innovations, but is broadly in the Tarskian mold.
• Proof-theoretic semantics associates the meaning of propositions with the roles that they can play in inferences. Gerhard Gentzen, Dag Prawitz and Michael Dummett are generally seen as the
founders of this approach; it is heavily influenced by Ludwig Wittgenstein's later philosophy, especially his aphorism "meaning is use".
• Truth-value semantics (also commonly referred to as substitutional quantification) was advocated by Ruth Barcan Marcus for modal logics in the early 1960s and later championed by J. Michael Dunn,
Nuel Belnap, and Hugues Leblanc for standard first-order logic. James Garson has given some results in the areas of adequacy for intensional logics outfitted with such a semantics. The truth
conditions for quantified formulas are given purely in terms of truth with no appeal to domains whatsoever (and hence its name truth-value semantics).
• Henkin quantifiers
• Probabilistic semantics originated from Hartry Field and has been shown equivalent to and a natural generalization of truth-value semantics. Like truth-value semantics, it is also non-referential
in nature.
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Distribution Using
Fitting a Univariate Distribution Using Cumulative Probabilities
This example shows how to fit univariate distributions using least squares estimates of the cumulative distribution functions. This is a generally-applicable method that can be useful in cases when
maximum likelihood fails, for instance some models that include a threshold parameter.
The most common method for fitting a univariate distribution to data is maximum likelihood. But maximum likelihood does not work in all cases, and other estimation methods, such as the Method of
Moments, are sometimes needed. When applicable, maximum likelihood is probably the better choice of methods, because it is often more efficient. But the method described here provides another tool
that can be used when needed.
Fitting an Exponential Distribution Using Least Squares
The term "least squares" is most commonly used in the context of fitting a regression line or surface to model a response variable as a function of one or more predictor variables. The method
described here is a very different application of least squares: univariate distribution fitting, with only a single variable.
To begin, first simulate some sample data. We'll use an exponential distribution to generate the data. For the purposes of this example, as in practice, we'll assume that the data are not known to
have come from a particular model.
n = 100;
x = exprnd(2,n,1);
Next, compute the empirical cumulative distribution function (ECDF) of the data. This is simply a step function with a jump in cumulative probability, p, of 1/n at each data point, x.
x = sort(x);
p = ((1:n)-0.5)' ./ n;
ylabel('Cumulative probability (p)');
We'll fit an exponential distribution to these data. One way to do that is to find the exponential distribution whose cumulative distribution function (CDF) best approximates (in a sense to be
explained below) the ECDF of the data. The exponential CDF is p = Pr{X <= x} = 1 - exp(-x/mu). Transforming that to -log(1-p)*mu = x gives a linear relationship between -log(1-p) and x. If the data
do come from an exponential, we ought to see, at least approximately, a linear relationship if we plug the computed x and p values from the ECDF into that equation. If we use least squares to fit a
straight line through the origin to x vs. -log(1-p), then that fitted line represents the exponential distribution that is "closest" to the data. The slope of the line is an estimate of the parameter
Equivalently, we can think of y = -log(1-p) as an "idealized sample" from a standard (mean 1) exponential distribution. These idealized values are exactly equally spaced on the probability scale. A
Q-Q plot of x and y ought to be approximately linear if the data come from an exponential distribution, and we'll fit the least squares line through the origin to x vs. y.
y = -log(1 - p);
muHat = y \ x
Plot the data and the fitted line.
plot(x,y,'+', y*muHat,y,'r--');
ylabel('y = -log(1-p)');
Notice that the linear fit we've made minimizes the sum of squared errors in the horizontal, or "x", direction. That's because the values for y = -log(1-p) are deterministic, and it's the x values
that are random. It's also possible to regress y vs. x, or to use other types of linear fits, for example, weighted regression, orthogonal regression, or even robust regression. We will not explore
those possibilities here.
For comparison, fit the data by maximum likelihood.
Now plot the two estimated distributions on the untransformed cumulative probability scale.
hold on
xgrid = linspace(0,1.1*max(x),100)';
plot(xgrid,expcdf(xgrid,muHat),'r--', xgrid,expcdf(xgrid,muMLE),'b--');
hold off
xlabel('x'); ylabel('Cumulative Probability (p)');
legend({'Data','LS Fit','ML Fit'},'location','southeast');
The two methods give very similar fitted distributions, although the LS fit has been influenced more by observations in the tail of the distribution.
Fitting a Weibull Distribution
For a slightly more complex example, simulate some sample data from a Weibull distribution, and compute the ECDF of x.
n = 100;
x = wblrnd(2,1,n,1);
x = sort(x);
p = ((1:n)-0.5)' ./ n;
To fit a Weibull distribution to these data, notice that the CDF for the Weibull is p = Pr{X <= x} = 1 - exp(-(x/a)^b). Transforming that to log(a) + log(-log(1-p))*(1/b) = log(x) again gives a
linear relationship, this time between log(-log(1-p)) and log(x). We can use least squares to fit a straight line on the transformed scale using p and x from the ECDF, and the slope and intercept of
that line lead to estimates of a and b.
logx = log(x);
logy = log(-log(1 - p));
poly = polyfit(logy,logx,1);
paramHat = [exp(poly(2)) 1/poly(1)]
paramHat = 1×2
2.1420 1.0843
Plot the data and the fitted line on the transformed scale.
plot(logx,logy,'+', log(paramHat(1)) + logy/paramHat(2),logy,'r--');
For comparison, fit the data by maximum likelihood, and plot the two estimated distributions on the untransformed scale.
paramMLE = 1×2
2.1685 1.0372
hold on
xgrid = linspace(0,1.1*max(x),100)';
plot(xgrid,wblcdf(xgrid,paramHat(1),paramHat(2)),'r--', ...
hold off
xlabel('x'); ylabel('Cumulative Probability (p)');
legend({'Data','LS Fit','ML Fit'},'location','southeast');
A Threshold Parameter Example
It's sometimes necessary to fit positive distributions like the Weibull or lognormal with a threshold parameter. For example, a Weibull random variable takes values over (0,Inf), and a threshold
parameter, c, shifts that range to (c,Inf). If the threshold parameter is known, then there is no difficulty. But if the threshold parameter is not known, it must instead be estimated. These models
are difficult to fit with maximum likelihood -- the likelihood can have multiple modes, or even become infinite for parameter values that are not reasonable for the data, and so maximum likelihood is
often not a good method. But with a small addition to the least squares procedure, we can get stable estimates.
To illustrate, we'll simulate some data from a three-parameter Weibull distribution, with a threshold value. As above, we'll assume for the purposes of the example that the data are not known to have
come from a particular model, and that the threshold is not known.
n = 100;
x = wblrnd(4,2,n,1) + 4;
hist(x,20); xlim([0 16]);
How can we fit a three-parameter Weibull distribution to these data? If we knew what the threshold value was, 1 for example, we could subtract that value from the data and then use the least squares
procedure to estimate the Weibull shape and scale parameters.
x = sort(x);
p = ((1:n)-0.5)' ./ n;
logy = log(-log(1-p));
logxm1 = log(x-1);
poly1 = polyfit(log(-log(1-p)),log(x-1),1);
paramHat1 = [exp(poly1(2)) 1/poly1(1)]
paramHat1 = 1×2
7.4305 4.5574
plot(logxm1,logy,'b+', log(paramHat1(1)) + logy/paramHat1(2),logy,'r--');
That's not a very good fit -- log(x-1) and log(-log(1-p)) do not have a linear relationship. Of course, that's because we don't know the correct threshold value. If we try subtracting different
threshold values, we get different plots and different parameter estimates.
logxm2 = log(x-2);
poly2 = polyfit(log(-log(1-p)),log(x-2),1);
paramHat2 = [exp(poly2(2)) 1/poly2(1)]
paramHat2 = 1×2
6.4046 3.7690
logxm4 = log(x-4);
poly4 = polyfit(log(-log(1-p)),log(x-4),1);
paramHat4 = [exp(poly4(2)) 1/poly4(1)]
paramHat4 = 1×2
4.3530 1.9130
plot(logxm1,logy,'b+', logxm2,logy,'r+', logxm4,logy,'g+', ...
log(paramHat1(1)) + logy/paramHat1(2),logy,'b--', ...
log(paramHat2(1)) + logy/paramHat2(2),logy,'r--', ...
log(paramHat4(1)) + logy/paramHat4(2),logy,'g--');
xlabel('log(x - c)');
ylabel('log(-log(1 - p))');
legend({'Threshold = 1' 'Threshold = 2' 'Threshold = 4'}, 'location','northwest');
The relationship between log(x-4) and log(-log(1-p)) appears approximately linear. Since we'd expect to see an approximately linear plot if we subtracted the true threshold parameter, this is
evidence that 4 might be a reasonable value for the threshold. On the other hand, the plots for 2 and 3 differ more systematically from linear, which is evidence that those values are not consistent
with the data.
This argument can be formalized. For each provisional value of the threshold parameter, the corresponding provisional Weibull fit can be characterized as the parameter values that maximize the R^2
value of a linear regression on the transformed variables log(x-c) and log(-log(1-p)). To estimate the threshold parameter, we can carry that one step further, and maximize the R^2 value over all
possible threshold values.
r2 = @(x,y) 1 - norm(y - polyval(polyfit(x,y,1),x)).^2 / norm(y - mean(y)).^2;
threshObj = @(c) -r2(log(-log(1-p)),log(x-c));
cHat = fminbnd(threshObj,.75*min(x), .9999*min(x));
poly = polyfit(log(-log(1-p)),log(x-cHat),1);
paramHat = [exp(poly(2)) 1/poly(1) cHat]
paramHat = 1×3
4.7448 2.3839 3.6029
logx = log(x-cHat);
logy = log(-log(1-p));
plot(logx,logy,'b+', log(paramHat(1)) + logy/paramHat(2),logy,'r--');
xlabel('log(x - cHat)');
ylabel('log(-log(1 - p))');
Non-Location-Scale Families
The exponential distribution is a scale family, and on the log scale, the Weibull distribution is a location-scale family, so this least squares method was straightforward in those two cases. The
general procedure to fit a location-scale distribution is
• Compute the ECDF of the observed data.
• Transform the distribution's CDF to get a linear relationship between some function of the data and some function of the cumulative probability. These two functions do not involve the
distribution parameters, but the slope and intercept of the line do.
• Plug the values of x and p from the ECDF into that transformed CDF, and fit a straight line using least squares.
• Solve for the distribution parameters in terms of the slope and intercept of the line.
We also saw that fitting a distribution that is a location-scale family with an additional a threshold parameter is only slightly more difficult.
But other distributions that are not location-scale families, like the gamma, are a bit trickier. There's no transformation of the CDF that will give a relationship that is linear. However, we can
use a similar idea, only this time working on the untransformed cumulative probability scale. A P-P plot is the appropriate way to visualize that fitting procedure.
If the empirical probabilities from the ECDF are plotted against fitted probabilities from a parametric model, a tight scatter along the 1:1 line from zero to one indicates that the parameter values
define a distribution that explains the observed data well, because the fitted CDF approximates the empirical CDF well. The idea is to find parameter values that make the probability plot as close to
the 1:1 line as possible. That may not even be possible, if the distribution is not a good model for the data. If the P-P plot shows a systematic departure from the 1:1 line, then the model may be
questionable. However, it's important to remember that since the points in these plots are not independent, interpretation is not exactly the same as a regression residual plot.
For example, we'll simulate some data and fit a gamma distribution.
n = 100;
x = gamrnd(2,1,n,1);
Compute the ECDF of x.
x = sort(x);
pEmp = ((1:n)-0.5)' ./ n;
We can make a probability plot using any initial guess for the gamma distribution's parameters, a=1 and b=1, say. That guess is not very good -- the probabilities from the parametric CDF are not
close to the probabilities from the ECDF. If we tried a different a and b, we'd get a different scatter on the P-P plot, with a different discrepancy from the 1:1 line. Since we know the true a and b
in this example, we'll try those values.
a0 = 1; b0 = 1;
p0Fit = gamcdf(x,a0,b0);
a1 = 2; b1 = 1;
p1Fit = gamcdf(x,a1,b1);
plot([0 1],[0 1],'k--', pEmp,p0Fit,'b+', pEmp,p1Fit,'r+');
xlabel('Empirical Probabilities');
ylabel('(Provisionally) Fitted Gamma Probabilities');
legend({'1:1 Line','a=1, b=1', 'a=2, b=1'}, 'location','southeast');
The second set of values for a and b make for a much better plot, and thus are more compatible with the data, if you are measuring "compatible" by how straight you can make the P-P plot.
To make the scatter match the 1:1 line as closely possible, we can find the values of a and b that minimize a weighted sum of the squared distances to the 1:1 line. The weights are defined in terms
of the empirical probabilities, and are lowest in the center of the plot and highest at the extremes. These weights compensate for the variance of the fitted probabilities, which is highest near the
median and lowest in the tails. This weighted least squares procedure defines the estimator for a and b.
wgt = 1 ./ sqrt(pEmp.*(1-pEmp));
gammaObj = @(params) sum(wgt.*(gamcdf(x,exp(params(1)),exp(params(2)))-pEmp).^2);
paramHat = fminsearch(gammaObj,[log(a1),log(b1)]);
paramHat = exp(paramHat)
paramHat = 1×2
2.2759 0.9059
pFit = gamcdf(x,paramHat(1),paramHat(2));
plot([0 1],[0 1],'k--', pEmp,pFit,'b+');
xlabel('Empirical Probabilities');
ylabel('Fitted Gamma Probabilities');
Notice that in the location-scale cases considered earlier, we could fit the distribution with a single straight line fit. Here, as with the threshold parameter example, we had to iteratively find
the best-fit parameter values.
Model Misspecification
The P-P plot can also be useful for comparing fits from different distribution families. What happens if we try to fit a lognormal distribution to these data?
wgt = 1 ./ sqrt(pEmp.*(1-pEmp));
LNobj = @(params) sum(wgt.*(logncdf(x,params(1),exp(params(2)))-pEmp).^2);
mu0 = mean(log(x)); sigma0 = std(log(x));
paramHatLN = fminsearch(LNobj,[mu0,log(sigma0)]);
paramHatLN(2) = exp(paramHatLN(2))
paramHatLN = 1×2
0.5331 0.7038
pFitLN = logncdf(x,paramHatLN(1),paramHatLN(2));
hold on
hold off
ylabel('Fitted Probabilities');
legend({'1:1 Line', 'Fitted Gamma', 'Fitted Lognormal'},'location','southeast');
Notice how the lognormal fit differs systematically from the gamma fit in the tails. It grows more slowly in the left tail, and dies more slowly in the right tail. The gamma seems to be a slightly
better fit to the data.
A Lognormal Threshold Parameter Example
The lognormal distribution is simple to fit by maximum likelihood, because once the log transformation is applied to the data, maximum likelihood is identical to fitting a normal. But it is sometimes
necessary to estimate a threshold parameter in a lognormal model. The likelihood for such a model is unbounded, and so maximum likelihood does not work. However, the least squares method provides a
way to make estimates. Since the two-parameter lognormal distribution can be log-transformed to a location-scale family, we could follow the same steps as in the earlier example that showed fitting a
Weibull distribution with threshold parameter. Here, however, we'll do the estimation on the cumulative probability scale, as in the previous example showing a fit with the gamma distribution.
To illustrate, we'll simulate some data from a three-parameter lognormal distribution, with a threshold.
n = 200;
x = lognrnd(0,.5,n,1) + 10;
hist(x,20); xlim([8 15]);
Compute the ECDF of x, and find the parameters for the best-fit three-parameter lognormal distribution.
x = sort(x);
pEmp = ((1:n)-0.5)' ./ n;
wgt = 1 ./ sqrt(pEmp.*(1-pEmp));
LN3obj = @(params) sum(wgt.*(logncdf(x-params(3),params(1),exp(params(2)))-pEmp).^2);
c0 = .99*min(x);
mu0 = mean(log(x-c0)); sigma0 = std(log(x-c0));
paramHat = fminsearch(LN3obj,[mu0,log(sigma0),c0]);
paramHat(2) = exp(paramHat(2))
paramHat = 1×3
-0.0698 0.5930 10.1045
pFit = logncdf(x-paramHat(3),paramHat(1),paramHat(2));
plot(pEmp,pFit,'b+', [0 1],[0 1],'k--');
xlabel('Empirical Probabilities');
ylabel('Fitted 3-param Lognormal Probabilities');
Measures of Precision
Parameter estimates are only part of the story -- a model fit also needs some measure of how precise the estimates are, typically standard errors. With maximum likelihood, the usual method is to use
the information matrix and a large-sample asymptotic argument to approximate the covariance matrix of the estimator over repeated sampling. No such theory exists for these least squares estimators.
However, Monte-Carlo simulation provides another way to estimate standard errors. If we use the fitted model to generate a large number of datasets, we can approximate the standard error of the
estimators with the Monte-Carlo standard deviation. For simplicity, use the helper fitting function, logn3fit.m.
estsSim = zeros(1000,3);
for i = 1:size(estsSim,1)
xSim = lognrnd(paramHat(1),paramHat(2),n,1) + paramHat(3);
estsSim(i,:) = logn3fit(xSim);
ans = 1×3
0.1542 0.0908 0.1303
It might also be useful to look at the distribution of the estimates, to check if the assumption of approximate normality is reasonable for this sample size, or to check for bias.
subplot(3,1,1), hist(estsSim(:,1),20);
title('Log-Location Parameter Bootstrap Estimates');
subplot(3,1,2), hist(estsSim(:,2),20);
title('Log-Scale Parameter Bootstrap Estimates');
subplot(3,1,3), hist(estsSim(:,3),20);
title('Threshold Parameter Bootstrap Estimates');
Clearly, the estimator for the threshold parameter is skewed. This is to be expected, since it is bounded above by the minimum data value. The other two histograms indicate that approximate normality
might be a questionable assumption for the log-location parameter (the first histogram) as well. The standard errors computed above must be interpreted with that in mind, and the usual construction
for confidence intervals might not be appropriate for the log-location and threshold parameters.
The means of the simulated estimates are close to the parameter values used to generate simulated data, indicating that the procedure is approximately unbiased at this sample size, at least for
parameter values near the estimates.
[paramHat; mean(estsSim)]
ans = 2×3
-0.0698 0.5930 10.1045
-0.0690 0.5926 10.0905
Finally, we could also have used the function bootstrp to compute bootstrap standard error estimates. These do not make any parametric assumptions about the data.
estsBoot = bootstrp(1000,@logn3fit,x);
ans = 1×3
0.1490 0.0785 0.1180
The bootstrap standard errors are not far off from the Monte-Carlo calculations. That's not surprising, since the fitted model is the same one from which the example data were generated.
The fitting method described here is an alternative to maximum likelihood that can be used to fit univariate distributions when maximum likelihood fails to provide useful parameter estimates. One
important application is in fitting distributions involving a threshold parameter, such as the three-parameter lognormal. Standard errors are more difficult to compute than for maximum likelihood
estimates, because analytic approximations do not exist, but simulation provides a feasible alternative.
The P-P plots used here to illustrate the fitting method are useful in their own right, as a visual indication of lack of fit when fitting a univariate distribution. | {"url":"https://kr.mathworks.com/help/stats/fitting-a-univariate-distribution-using-cumulative-probabilities.html","timestamp":"2024-11-02T04:55:20Z","content_type":"text/html","content_length":"102204","record_id":"<urn:uuid:b1f86c76-aa99-4ab0-b039-6657341bb1e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00732.warc.gz"} |
cm – Computer Modern fonts
Knuth's final iteration of his re-interpretation of a c.19 Modern-style font from Monotype. The family is comprehensive, offering both sans and roman styles, and a monospaced font, together with
mathematics fonts closely integrated with the mathematical facilities of TeX itself.
The base fonts are distributed as METAFONT source, but autotraced PostScript Type 1 versions are available (one version in the AMS fonts distribution, and also the BaKoMa distribution).
The Computer Modern fonts have inspired many later families, notably the European Computer Modern and the Latin Modern families.
Sources /fonts/cm
Licenses Knuth License
Maintainer Donald E. Knuth
Contained in TeXLive as cm
MiKTeX as cm
MF Font
Topics Monospaced Font
Proportional Font
CM Font
Community Comments
Maybe you are interested in the following packages as well. | {"url":"https://ctan.org/pkg/cm","timestamp":"2024-11-11T02:01:09Z","content_type":"text/html","content_length":"15165","record_id":"<urn:uuid:0e390dd7-0198-4dc5-9005-b1bf4a739b6d>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00460.warc.gz"} |
Vertical Spread Checklist - A Guide to Profitability - OptionBoxer
Vertical Spread Checklist – A Guide to Profitability
by OptionBoxer | Apr 6, 2020 | "Secrets" to Options Trading Success, OptionBoxer Options Locker, Options Strategies, Tactical Options Trading Techniques | 0 comments
With so many possible vertical spread combinations singling out the best one is mind numbing, to say the least. Using these steps can help simplify the process and create confidence that major
factors haven’t been ignored.
Vertical spreads are straightforward enough for those initiated to options trading. Let’s say a trader is bullish on an underlying. The Bull Call vertical or Bull Put vertical would be ideal.
Conversely, the trader may expect poor performance in the coming days. Therefore, the Bear Call Vertical or the Bear Put Vertical would be more appropriate.
Simple enough, Yes? But there’s still questions that must be answered and I’ll attempt to uncover the lions share of them below.
Vertical Spread Step-by-Step
Step 1 – Check IV Percentile or IV Rank
This metric will answer a very important question. Should I buy or sell a vertical spread? If IV is “relatively” high the better option is to sell the vertical spread. Conversely, when IV is
“relatively” low the better option is to buy the vertical spread.
This metric will also benefit the astute trader by answering another question. Are the option premiums dense or light? Dense meaning, higher premiums and light representing lower premiums. With an
understanding of IV Percentile or IV Rank you won’t, generally, have to worry about overpaying or under receiving on the premiums.
Word of Caution – Understand, IV can stay high or stay low.
Step 2 – Determine Underlying Direction
This step will require the trader to determine a directional bias for the desired underlying. Honestly, this section could and does comprise entire books. Therefore, I’ll simply defer to you to
determine the best method. I have my own but that isn’t the purpose of this post. The easiest method would be simply to use a moving average crossover strategy. Possibly, the 10 SMA crossing the 30
EMA for short term trends.
10 SMA(Red) / 30 EMA(Blue) Crossover
Step 3 – Days to Expiration
Now the question is…How Long? Honestly, the only way to answer this would be to have a crystal ball but the reality is just as simple, choose enough time for the position to have a reasonable chance.
To do this, consider adding the average true range indicator to your charts. This tool can help improve the odds of success via some basic arithmatic and some rudimentary forecasting.
Average True Range (ATR) Indicator
I use the ATR over a 200 period average. To add and adjust ATR in ThinkorSwim, click the beaker icon > Search ‘ATR’ > Double click. This will add it to your subcharts. Click the associated gear icon
to open the indicator settings > Change the length field to 200 instead of 14. This uses a larger number of days to determine a more normal average by smoothing some of the extreme moves.
As you can see in the image above, ATR is currently at $4.74. That suggests that the daily range for each day over the last 200 days is $4.74. With this information we can guesstimate a reasonable
amount of time.
For instance, if a trader selected the expiration month ending in 30 Days. Simply Calculate.
$4.74 * 30 = (+/-)$142.2
So if price moved entirely up or down everyday for the next 30 days the underlying price would move $142.2 points. However, this isn’t likely to happen. Therefore, to account for some arbitrary level
of variance we need to multiply by our assumption price will maintain a directional bias. In this case, I’ve suggested price will move directionally only 25% of the time over the next 30 days.
$142.2 * .25 = (+/-)$35.5
This solves the problem of price going straight up or down using a random blind assumption.
However, I’m certain more scientific approaches exist so if possible refer to that. This suits me well and its quick to uncover.
With this information a trader can now determine with some confidence if 30 days will be sufficient. How? By comparing the amount of our hypothetical expected move to the desired or non-desired move
of our selected spread.
I’ll put this, along with all the vertical spread pieces, together again at the end of this post.
Word of Caution – If in doubt allow for more time to expiration.
Step 4 – Vertical Spread Strike Selection
Vertical Spreads are comprised of 2 different strike contracts. One long, one short. At what level each falls rests on the directional assumption and IV.
For example, lets assume we’re bullish on SPY over the next 30 days and IV is relatively high. Thus, a trader may wish to sell the bull put vertical spread and for this trade to be profitable it will
need to expire out of the money.
But at what level should the short strike fall? This can be selected using the Probability out of the money(OTM) metric in ThinkorSwim. This mythical number can be used to determine, mathmatically,
just how likely a strike is to finish OTM.
Word of Caution – This number is theoretical in nature and reality is often much different.
SPY Option Chain – Probability OTM
Then, using the Probability OTM highlighted above a trader is able to see the mathmatical likelihood an option strike will finish OTM. Therefore, it makes sense to desire a strike price further away
with a higher chance of success.
Step 5 – Vertical Spread Risk
The final piece to the vertical spread puzzle is determining an appropriate amount of risk. Of course, to do this, the trader will want to consider his/her account size, risk tolerance, and
confidence level the directional bias is accurate. Moreover, its that confidence in direction that can be problematic ;).
Once those questions are answered simply select the long portion of the vertical spread trade and execute the order.
Hypothetical Vertical Spread Example
Together with the examples discussed above. Suppose a trader is bullish SPY. Price is currently $264.86. IV Rank is 48, which is relatively high.
30 Day possible price movement = (+/-) $35.50
Short strike = $243 put (70% Prob. OTM)
Long strike = $242 put
By subtracting the current market price of SPY with our short strike price we can compare with the 30 Day move.
$264.86 – $243 = $21.86
We’re expecting a directional move of approximately $35.50 but our current trade only allows for $21.86 before breaking our short strike. Its at this point, the trader would have to rely on their
directional bias to determine the viablity of the trade.
If confidence is high price will remain higher then all systems are a go to execute. Should there be any doubt, a trader would be well served to move further out in time or forgo the position all
Great! More Math…
$21.86 / $35.5 = .61 or 61%
In this instance, we can see there is an arbitrary 61% chance this trade will become profitable. Therefore, signaling a potentially positive trade. However, it could also make sense to look at future
expiration months or alternative strikes to find a higher probability vertical spread.
May God bless and keep your trading profitable,
Submit a Comment Cancel reply
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From Encyclopedia of Mathematics
(in logic)
A notion introduced by S. Shelah [a8]. The general theory of forking is also known as stability theory, but more commonly, non-forking (the negation of forking) is defined as a certain well-behaved
relation between a type and its extension (cf. Types, theory of).
Let Formal language; Model (in logic); Model theory). Given an
Given a type
i) for
Assume that
2) if
3) for any
4) for any
5) for any
The ultrapower construction (cf. also Ultrafilter) gives a systematic way of building non-forking extensions [a4].
For a comprehensive introduction of forking see [a1], [a2], [a4], [a5], and [a9]. For applications in algebra, see [a7] and [a6].
The techniques of forking have been extended to unstable theories. In [a2], this is done by considering only types that satisfy stable conditions. In [a3], types are viewed as probability measures
and forking is treated as a special kind of measure extension. The stability assumption is then weakened to theories that do not have the independence property.
[a1] J.T. Baldwin, "Fundamentals of stability theory" , Springer (1987)
[a2] V. Harnik, L. Harrington, "Fundamentals of forking" Ann. Pure and Applied Logic , 26 (1984) pp. 245–286
[a3] H.J. Keisler, "Measures and forking" Ann. Pure and Applied Logic , 34 (1987) pp. 119–169
[a4] D. Lascar, B. Poizat, "An introduction to forking" J. Symb. Logic , 44 (1979) pp. 330–350
[a5] A. Pillay, "Introduction to stability theory" , Oxford Univ. Press (1983)
[a6] A. Pillay, "The geometry of forking and groups of finite Morley rank" J. Symb. Logic , 60 (1995) pp. 1251–1259
[a7] M. Prest, "Model theory and modules" , Cambridge Univ. Press (1988)
[a8] S. Shelah, "Classification theory and the number of non-isomorphic models" , North-Holland (1990) (Edition: Revised)
[a9] M. Makkai, "A survey of basic stability theory" Israel J. Math. , 49 (1984) pp. 181–238
How to Cite This Entry:
Forking. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Forking&oldid=19231
This article was adapted from an original article by Siu-Ah Ng (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
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Section 4 - Polynomial Functions (Edition 1)
Category 22 Example 1 - no calculation required
The values can be directly read from the table. The f(x) value for a certain value of x can be read from the corresponding f(x) column. The x value for a certain value of f(x) can be read from the
corresponding x column.
Category 22 Example 2 - no calculation required
It can be observed from the graph that for x = -1, the value of y = 2. Count the number of times the graph intersects the x-axis at y = -2.
Category 23 Example 2 - no calculation required.
The value of g(4) can be read from the table as -6. The value of f(-6) can be read from the table as 12.
Category 24 Example 1 - no calculation required
The points where the graph passes through the x-axis or touches the x-axis can be directly read from the graph.
Tip: Passes through = 1 factor. Touches and curves back = 2 identical factors.
Category 24 Example 2 - no calculation required
Each factor is a distinct zero. Count them.
Category 24 Example 3 - no calculation required
From the x column read the values where h(x) = 0
Tip: y = 0 at x-intercept. | {"url":"https://www.tutorhubllc.com/post/section-4-mental-math","timestamp":"2024-11-11T22:47:13Z","content_type":"text/html","content_length":"1050409","record_id":"<urn:uuid:2aca71ef-6c9f-4017-a259-0a97459407e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00479.warc.gz"} |
Room 509, Cosmology Building, NTU
Gi-Ren Liu (National Cheng Kung University)
Chun-Hsiung Hsia (National Taiwan University)
1. Introduction and Contents
Considering a parameterized nonlinear transformation, which may arise from partial differential equations or time-frequency analysis, when subjected to random input, yields an output that can
manifest as a temporal-spatial or time-scale random field. The primary objective of this course is to determine the large-scale limit of this output field. Additionally, the course aims to quantify
the discrepancy between the output and its large-scale limit. We expect students to gain a deeper understanding of the convergence speed of standardized partial sums of correlated random variables
through this training. To achieve this goal, the content of this course will contain the spectral representation of random processes, the Stein method, the non-Stein method, and stochastic calculus.
2. Course Outline
We will guide the students in understanding stochastic analysis tools used to determine the convergence speed of the distributions of normalized partial sums/integrals of correlated
random variables/fields. This includes exploring the spectral representation of random processes, the Stein method, the non-Stein method, and fundamental formulas in stochastic calculus. Once
students have grasped these basics, we will compare the convergence speeds obtained through the Stein and non-Stein methods. We expect that students will apply this acquired knowledge to investigate
corresponding convergence rate problems in various research fields.
3. Prerequisites
linear algebra, undergraduate probability theory
4. Grading Scheme
Student presentation
5. Course Goal
(1) Understand the concept of Stein’s method.
(2) Apply Stein’s method to get the rate of convergence of the classical Central Limit Theorem (CLT).
(3) Prove the CLT for weakly dependent random variables.
(4) Introduce the multidimensional Stein method.
(5) Explore the CLT and its convergence speed for random vectors.
6. Reference material (textbooks)
(1) Chatterjee, S. A short survey of Stein's method. arXiv preprint arXiv:1404.1392 (2014).
(2) Chen, L.H., Goldstein, L. and Shao, Q.M., 2011. Normal approximation by Stein's method (Vol. 2). Berlin: Springer.
(3) Krylov, N. V. (2002). Introduction to the theory of random processes, volume 43. American Mathematical Soc.
(4) Nourdin, I. and Peccati, G. (2012). Normal approximations with Malliavin calculus: from Stein’s method to universality. Number 192. Cambridge University Press.
7. Credit: 2
8. Course Number/ ID
No.: NCTS 5056 (三校聯盟之學生於課程網選課適用)
ID: V41 U4110
Contact: Murphy Yu (murphyyu@ncts.tw) | {"url":"https://ncts.ntu.edu.tw/events_3_detail.php?nid=345","timestamp":"2024-11-05T22:50:22Z","content_type":"application/xhtml+xml","content_length":"48204","record_id":"<urn:uuid:cbdaa09c-4271-469e-a274-96fab9034baf>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00895.warc.gz"} |
This repository contains an R package for generating synthetic alpha shapes by either (i) empirical sampling based on an existing dataset with reference shapes, or (ii) probabilistic sampling from a
known distribution function on shapes.
Understanding morphological variation is an important task in many applications. Recent studies in computational biology have focused on developing computational tools for the task of sub-image
selection which aims at identifying structural features that best describe the variation between classes of shapes. A major part in assessing the utility of these approaches is to demonstrate their
performance on both simulated and real datasets. However, when creating a model for shape statistics, real data can be difficult to access and the sample sizes for these data are often small due to
them being expensive to collect. Meanwhile, the landscape of current shape simulation methods has been mostly limited to approaches that use black-box inference—making it difficult to systematically
assess the power and calibration of sub-image models.
In this R package, we introduce the \(\alpha\)-shape sampler: a probabilistic framework for simulating realistic 2D and 3D biological shapes and images based on probability distributions which can be
learned from real data or explicitly stated by the user.
The Method
The ashapesampler package supports two mechanisms for sampling shapes in two and three-dimensions, which we outline below. The first strategy empirically samples new shapes based on an existing
dataset — this was highlighted in the main text of Winn-Nuñez et al. The second strategy probabalistically samples new shapes from a known distrubtion — this approach is also implementated in this
software package with the corresponding theory being derived in the Supporting Information of Winn-Nuñez et al.
Generating New Shapes from an Existing Dataset
The \(\alpha\)-shape sampler consists of four key steps: 1. Input aligned reference shapes as simplicial complexes. A simplicial complex object in this case is a list containing (a) the Euclidean
coordinates of the vertices and (b) a list of all vertices, edges, faces, and tetrahedra. Functions are available to read OFF files into R in the correct format and to extract the simplical complex
information from a generated alpha complex. A method to convert a binary mask to a 2D simplicial complex for use in the algorithm can be found in the vignettes. 2. Calculate the reach for each shape
in the dataset. The reach is estimated based on boundary points of the simplicial complex. Users can choose the summary statistic used for the estimated reach for a reference shape to be either the
mean, median, or minimum across points. Default is mean. Once we have the reach for each shape, users can take some summary statistic (usually the minimum) over a J subset of randomly selected
reference shapes to produce new shapes. 3. Sample new points, using the combined point cloud of the randomly selected J shapes and the estimated reach tau derived from the J reference shapes.
Parameters for rejection sampling can be adjusted by the users and are discussed further in the vignettes. Note that this step is generally the longest computationally—if the user reaches a
computational bottleneck, check to the value of tau relative to the area/volume of the combined point cloud. Parallelizing also speeds up the algorithm. 4. Output newly generated shape as an alpha
shape object.
Users should note that it is critical to align shapes to maximize the pipeline’s success and that there may be some manual parameter tuning for the best results.
Demonstrations for pipeline implementation are in the vignettes. Functions are broken into parts instead of integrated altogether so that users can troubleshoot the pipeline at different stages.
Generating New Shapes from Probability Distributions
Users an also use the ashapesampler package to generate shapes in two and three dimensions from probability distributions. This approach can prove particularly useful for simulating shapes and
benchmarking the performance of different statistical methods. Here, we list the parameters for generating new shapes in two and three dimensions. Options for user-adjusted parameters and defaults
can be found in the vignettes. Users should keep a few key points in mind when generating shapes this way: * The bound parameter is the manifold from which points are sampled. At this time, the
package only supports a square, a circle (i.e., a disk where the function assumes it is filled in), and an annulus in two dimensions. In three-dimensions, it supports a cube, sphere (i.e., a ball
where the function assumes it is filled in), and torus. The size of these manifolds can be specified using the rmax and rmin parameters, where applicable. Adjusting the size may affect computational
time if the reach tau is not adjusted with it. * The reach tau needs to be specified as a finite value in advance, as this hyperparameter affects the choice of alpha. Default of tau is 1, but it can
be any finite value. Keep in mind that the smaller that tau is relative to the area or volume of the manifold, the more detail in the shapes produced and the more time it will take to produce a new
generate opbject shape. * By default, alpha will be as large as theoretically allowed. The smaller alpha is relative to tau, the more points will need to be sampled and the more time it will take to
produce a new generate shape. This is particularly true when the goal is to have shapes to have both full connectivity/no isolated points as well as preserve the homology. * At this time, the package
only supports the truncated normal distribution for randomly selecting alpha. Bounds of this truncated normal can be adjusted by the user up to what is theoretically allowed. Keep in mind that the
general bounds of this distribution should keep alpha as large as possible for best computational performance.
R Packages for ashapesampler and Tutorials
The ashapesampler software requires the installation of the following R libraries:
Unless stated otherwise, the easiest way to install many of these packages is with the following example command entered in an R shell:
install.packages("alphahull", dependecies = TRUE)
Alternatively, one can also install R packages from the command line.
C++ Packages for ashapesampler and Tutorials
The code in this repository assumes that basic C++ functions and applications are already set up on the running personal computer or cluster. If not, some of the packages (e.g., TDA and alphashape3d)
needed to build alpha complexes and alpha shapes in three dimensions will not work properly. A simple option is to use gcc. macOS users may use this collection by installing the Homebrew package
manager and then typing the following into the terminal:
brew install gcc
For macOS users, the Xcode Command Line Tools include a GCC compiler. Instructions on how to install Xcode may be found here. Additional installs for macOS users are automake, curl, glfw3, glew,
xquartz, and qpdf. For extra tips on how to run C++ on macOS, please visit here. For tips on how to avoid errors dealing with “-lgfortran” or “-lquadmath”, please visit here.
R Package Installation
Package will eventually appear on CRAN, at which time one can download the package there.
To install the package from GitHub, we recommend using the remotes package by running the command:
To then load the package in R, use the command
Other common installation procedures may apply.
Code Usage
The vignettes folder contains the following demonstrations for running and analyzing results in the ashapesampler:
• Sampling alpha shapes from a probability distribution in two-dimensions.
• Sampling alpha shapes from a probability distribution in three-dimensions.
• Generating new 2D annuli from a simulated set of annuli.
• Generating new 3D tori from a simulated set of tori.
Additional vignettes and source code can be found in the corresponding results repository.
The auto3dgm paradigm for assigning landmarks via unsupervised learning can be found here.
Primate manibular molar data and neutrophil binary masks can be accessed and downloaded here.
Relevant Citations
E.T. Winn-Nuñez, H. Witt, D. Bhaskar, R.Y. Huang, I.Y. Wong, J.S. Reichner, and L. Crawford. Generative modeling of biological shapes and images using a probabilistic \(\alpha\)-shape sampler.
Questions and Feedback
Please send any questions or feedback to the corresponding authors Emily Winn-Nuñez or Lorin Crawford.
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The Best Online Math Homework Help for College Students
I have basically researched and tested each and every online math homework help site to find the best one for my college needs. Both free & paid.
Read more on the college math homework help site of my choice here.
And for reference I’ve listed the rest of the help sites below.
WebMath — Solve your math problem
An outdated site for solving various math problems from algebra to geometry and calculus. Solutions are “auto-generated”, with no mobile support.
Math.com — Homework help
Another old non-mobile website with a large but somewhat unorganized collection of math answers with somewhat vague examples. Not very useful.
Varsity Tutors — Answers to math homework problems
Large and organized database of subjects for textbook and homework assistance for college students. Supports mobiles and provides expensive math tutoring.
Tutor.com — Online math tutors
Expert tutors around the clock ready to answer any questions on math, algebra, geometry, trigonometry, calculus and statistics for a fee. Not good for anything else.
Mathway — Algebra problem solver
Simple calculator that answers most questions on basic math, algebra, trigonometry, calculus, statistics, finite math, chemistry and graphing.
Slader — Homework help and answers
One of the most popular resources for completely free textbook answers and help to math problems. Site is not so easy to navigate but has easy, guided solutions.
Assignment Geek — Expert math homework help
On this site you can actually pay someone to do your assignments for you, something that I do not recommend. However they have proved out somewhat reliable.
Khan Academy — Free online courses, and lessons
An impressive collection of free video tutorials that explain various math problems and subjects. Best for general grasp of mathematics.
Skooli — Online math tutoring
Access to licensed tutoring that can help with math homework, exams or any problems for a decent price tag. Not that popular though.
Any questions about college homework welcome?
Or what is your experience with math help in college, let us know!
Much thanks for the visit, Tim.
Leave a Comment | {"url":"https://www.kachi.jp/the-best-online-math-homework-help-for-college-students/","timestamp":"2024-11-07T07:48:45Z","content_type":"text/html","content_length":"59967","record_id":"<urn:uuid:f6c6aa7c-bc1d-46ef-adfc-3cd79691fd6f>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00344.warc.gz"} |
During Which Process Of The Water Cycle Does Water Change From A Gas To Liquad
Condensation is the process by which water vapor in the air is changed into liquid water. Condensation is crucial to the water cycle because it is responsible for the formation of clouds.
* when L → H chord too long
in this case we see that the speed to cross the well grows a lot (it goes towards infinity) therefore we do not have enough speed in the movement to cross
* when L → 0 very short string
the speed of the platform is very small, so we do not have the minimum required value
vox = √ (g / (2 (H)) D
For this exercise we are going to solve it using conservation of energy to find the velocity of the body and the launch of projectiles to find the velocity to cross the well.
Let's start with the projectile launch
as the body leaves the vertical its velocity must be horizontal
x = v₀ₓ t
y = y₀ + [tex]v_{oy}[/tex] t - ½ g t²
when reaching the ground its height of zero (y = 0) and the initial vertical velocity is zero
t = √ 2 y₀ / g
we substitute
x = vox √2y₀ / g
v₀ₓ = √(g / 2y₀) x
In the exercise, it tells us that the width of the well is D (x = D) and the initial height is the height of the platform minus the length of the rope (I = H - L)
v₀ₓ = √(g /(2 (H -L)) D
this is the minimum speed to cross the well.
Now let's use conservation of energy
starting point. On the platform
[tex]Em_{o}[/tex] = U = m g H
final point. At the bottom of the swing
Em_{f} = K + U = 1 / 2m v² + m g (H -L)
as there is no friction the mechanical energy is conserved
Em_{o} = Em_{f}
m g H = 1 / 2m v² + m g (H -L)
v = √ (2gL)
let's write our two equations
the minimum speed to cross the well
v₀ₓ = √ (g /(2 (H -L)) D
the speed at the bottom of the oscillatory motion
v = √ (2g L)
we analyze the extreme cases
* when L → H chord too long
in this case we see that the speed to cross the well grows a lot (it goes towards infinity) therefore we do not have enough speed in the movement to cross
* when L → 0 very short string
the speed of the platform is very small, so we do not have the minimum required value
vox = √ (g / (2 (H)) D
From this analysis we see that there is a range of lengths that allows us to have the necessary speeds to cross the well
V₀ₓ = v
g / (2 (H -L) D² = 2g L
4 L (H- L) = D²
4 H L - 4 L2 - D² = 0
L² - H L - D² / 4 = 0
let's solve the quadratic equation
L = [H ± √ (H2-D2)] / 2
we assume that H> D
L = ½ H [1 + - RA (1 - (D / H) 2)]
The two values of La give the range of values for which the two speeds are equal | {"url":"https://smart.gov.qa/question-answers/during-which-process-of-the-water-cycle-does-water-change-fr-zfgb","timestamp":"2024-11-09T17:23:27Z","content_type":"text/html","content_length":"86661","record_id":"<urn:uuid:e0c4b11a-a3c2-428c-82e3-d8c0972b4329>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00205.warc.gz"} |
These are Mathematics Department course numbers; some of these courses are also cross-listed under Computer Science numbers. (Unfortunately, the University has changed its system of catalog
description pages in a way that requires updating links each semester. Therefore, we only direct you to the UI main course page and let you click on from there to find the official catalog
announcements. The “course announcement” links here point to more complete descriptions of the most recent offerings of the courses.)
We have a one-semester introduction to combinatorics at the graduate level (Combinatorial Mathematics, Math 580/CS 571). This course is suitable for students in other areas seeking an overview of the
fundamentals of the area and for students preparing to study more advanced courses in combinatorics.
We also rotate several regular graduate courses. These assume some experience in combinatorics; Math 580 suffices for each. Normally, each of these courses is offered once every two years.
• Combinatorics (Math 580) (course web page)
• Extremal Graph Theory (Math 581/CS 572) |
• Structure of Graphs (Math 582)
• Partial Orders & Combinatorial Optimization (Math 583) (course announcement) (course web page)
• Methods of Combinatorics (Math 584/CS 575) (course announcement) (course web page)
• Probabilistic Methods in Discrete Mathematics (Math 585) (course announcement)
• Algebraic Combinatorics (Math 586)
Semester Course No. Course Title Instructor
Fall 2020 Math 586 Algebraic Combinatorics Yong
Fall 2020 Math 581/CS 572 Extremal Graph Theory Kostochka
Spring 2021 Math 580/CS 571 Combinatorial Mathematics Balogh
Fall 2021 Math 580/CS 571 Combinatorial Mathematics Balogh
Fall 2021 Math 582 Structure of Graphs Kostochka
Fall 2021 Math 585 Probabilistic Methods in Discrete Mathematics Balogh
Spring 2022 Math 584/CS 575 Methods of Combinatorics Balogh
Fall 2022 Math 580/CS571 Combinatorial Mathematics Balogh
Fall 2023 Math 581 Extremal Graph Theory Kostochka
Spring 2023 Math 586 Algebraic Combinatorics Yong
Spring 2023 Math 583 Partial Orders & Combinatorial Optimization Balogh
Spring 2024 Math 580 Combinatorial Mathematics Balogh
Fall 2024 Math 580 Combinatorial Mathematics Methuku
Spring 2025 Math 585 Probabilistic Methods in Discrete Mathematics Methuku | {"url":"https://combinatorics.math.illinois.edu/courses/","timestamp":"2024-11-12T07:09:43Z","content_type":"text/html","content_length":"29766","record_id":"<urn:uuid:5ddf7392-552a-4575-8281-cd8abfdbcf06>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00038.warc.gz"} |
Universe normalization in `make_local_hint_db`
I've been looking at perf profiles of proofs and I saw one suspicious item today that I'd like to understand better. It's the call to Evd.nf_univ_variables sigma in prepare_hint called via
make_local_hint_db. IIUC it is called once for every lemma passed to eauto via using (probably typeclasses eauto, too, but I am not sure). In a big proof I've been looking at today this call to
universe normalization accounts for a total of 5% of the entire runtime. My first impulse was to try to perform the normalization only once for every call to eauto but it turns out that the sigma in
question is the result of "instantiating" the provided lemmas, i.e. sigma from the goal itself goes in and out comes a, presumably, slight larger sigma which is at least in principle different for
every provided lemma. Is there any way the universe normalizaton work could still be shared? Does it make sense to normalize the input sigma once in order to save cost on normalizing the derived
The code also has this comment which I don't fully understand. Is it possible that the normalization could be skipped entirely?
(* We re-abstract over uninstantiated evars and universes.
It is actually a bit stupid to generalize over evars since the first
thing make_resolves will do is to re-instantiate the products *)
let sigma = Evd.nf_univ_variables sigma in
I don't think it's actually needed at all ;)
it was introduced in coq/coq@1e389def84cc3eafc8aa5d1a1505f078a58234bd which did
- let c = drop_extra_implicit_args (Evarutil.nf_evar sigma c) in
+ let sigma, subst = Evd.nf_univ_variables sigma in
+ let c = Vars.subst_univs_constr subst (Evarutil.nf_evar sigma c) in
+ let c = drop_extra_implicit_args c in
maybe nf_evar didn't normalize universe variables back then? IDK
Well, that would be a nice speedup.. at least in our proofs
@Janno does that PR give a speedup in your code? our bench doesn't show any perf change AFAICT
I haven't gotten around to patching everything and rebuilding from scratch. I don't really see how it could not give a speedup, though, since the result of the normalization is discarded. There
shouldn't be any slowdown in other places from not doing it.
Last updated: Oct 13 2024 at 01:02 UTC | {"url":"https://coq.gitlab.io/zulip-archive/stream/237656-Coq-devs-.26-plugin-devs/topic/Universe.20normalization.20in.20.60make_local_hint_db.60.html","timestamp":"2024-11-05T16:44:39Z","content_type":"text/html","content_length":"8901","record_id":"<urn:uuid:5cdf5772-1a36-41b5-9c57-cd3e0a27df9e>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00858.warc.gz"} |
A New Proof of the Existence of Suitable Weak Solutions and Other Remarks for the Navier-Stokes Equations
A New Proof of the Existence of Suitable Weak Solutions and Other Remarks for the Navier-Stokes Equations ()
1. Introduction
The main objective of this paper is to provide a new proof of the existence of suitable weak solutions to the Navier-Stokes equations. Specifically, we show that the semi-discrete and the completely
discrete semi-implicit Euler schemes lead to families of approximate solutions that converge to a weak solution that is suitable in the sense of Caffarelli, Kohn and Nirenberg [2] .
We will be concerned with the 3D Navier-Stokes equations completed with initial and Dirichlet boundary conditions in bounded domains $\Omega ×\left(0,T\right)$ (as usual, Ω is the spatial domain, a
regular, bounded and connected open set in ${ℝ}^{3}$ “filled” by the fluid particles; $\left(0,T\right)$ is the time observation interval).
The key concept of suitable weak solution was introduced in [2] . In few words, this is a weak solution satisfying a local energy inequality. Due to its definition, it is expected that suitable
solutions are regular (and unique). However, up to now, this is unknown. The best we can prove is that the set of singular points of a suitable solution is small, in the sense that has Hausdorff
dimension ≤1. As shown below, in order to improve this result, we would need an estimate that we do not have at hand at present. A similar analysis can be performed in the context of the Boussinesq
system; see [4] .
The estimate in [2] of the Hausdorff dimension relies on some technical results asserting that adequate criteria, applied to suitable solutions in a given space-time region, imply the regularity of
the points in a subregion. During the last years, several authors have tried to improve or weaken these criteria and some achievements have been obtained:
・ In Seregin [5] , a family of sufficient conditions that contains the Caffarelli-Kohn-Nirenberg condition as a particular case is introduced. Its formulation is given in terms of functionals
invariant with respect to scale transformations.
・ In Vasseur [6] , an interesting criterion appears: if we normalize the solution and the sum of the associated kinetic and viscous energies and the L^p norm of the pressure is small enough, we get
regularity. The proof of this assertion is inspired by a method of De Giorgi designed to prove the regularity of elliptic equations, see [7] .
・ In Wolf [8] , the author provides a notion of local pressure. It permits to estimate the integrals involving the pressure in terms of the velocity and, again, deduce regularity. The method is
interesting and can be adapted to get partial regularity results for other systems, such as the equations of quasi-Newtonian or Boussinesq (heat conducting) fluids.
・ Finally, in Choe, Wolf and Yang [9] , an improved version of the Caffarelli-Kohn-Nirenberg criterion is furnished, using ideas from [5] .
Note that, in a recent paper, Buckmaster and Vicol [10] have proved that, for a very weak class of distributional solutions in spatially periodic domains, non-uniqueness occurs.
The techniques employed in this paper can be applied to many other approximation schemes that lead to energy inequalities, as those in [11] [12] [13] [14] . More precisely, we first use the well
known energy estimates, together with appropriate interpolation results and recall that the approximate solutions converge to a weak solution $\left(u,p\right)$ . Then, we analyze the role of the
pressure p; this reduces in fact to a detailed study of the behavior of the time derivate of the velocity field. This way, we are able to take (lower) limits in the local energy identities satisfied
by the approximate solutions and deduce that $\left(u,p\right)$ is suitable.
Our results can be compared to other previous proofs of existence: the one in the Appendix in [2] (based on the construction of a family of time delayed linear approximations), the main result in Da
Veiga [15] (relying on regularization with vanishing fourth-order terms), the main result in Guermond [3] (where Faedo-Galerkin techniques are employed) and, also, the results by Berselli and Spirito
[16] [17] , where the Voigt approximations and the artificial compressibility method are shown to converge.
We think that our results can be useful at least from two points of view. First, a new (relatively simple) constructive argument is used to prove the existence of suitable solutions. Then, there is
some practical interest: by inspection of the behavior of the computed approximations in a prescribed region, we may try to deduce if the related points are regular. In other words, checking whether
or not the Caffarelli-Kohn-Nirenberg criteria are satisfied on the computed numerical solutions can serve to identify or discard singular points. Based on this idea, we will present in a forthcoming
paper several numerical experiments for which interesting conclusions can be obtained.
The plan of the paper is the following:
・ In Section 2, we review the main results in the papers [2] and [18] . In particular, we explain why suitable solutions are relevant in the context of the regularity problem.
・ In Section 3, we recall the Euler approximation schemes and we establish the convergence to a suitable solution of the Navier-Stokes equations.
・ Finally, Section 4 is devoted to some additional comments and open questions.
2. Background: The Basic Results by Caffarelli, Kohn and Nirenberg
In the sequel, we denote by $|\text{ }\cdot \text{ }|$ and $\left(\cdot ,\cdot \right)$ the usual L^2 norm and scalar product, respectively. The symbol C will be used to denote a generic positive
2.1. The Main Properties of Suitable Solutions
In this section, we will recall the main contributions of Caffarelli, Kohn and Nirenberg, see [2] . In this reference, the best results known to date in relation to the regularity of the
Navier-Stokes equations are established.
Let $\Omega \subset {ℝ}^{3}$ be a nonempty, regular, bounded and connected open set and assume that $T>0$ . Let us set $Q:=\Omega ×\left(0,T\right)$ and $\Sigma :=\partial \Omega ×\left(0,T\right)$ .
We will consider local and global solutions to the Navier-Stokes equations in three dimensions
$\left\{\begin{array}{l}{u}_{t}+u\cdot abla u-\Delta u+abla p=f\left(x,t\right),\\ abla \cdot u=0,\end{array}$(1)
where $f=\left({f}^{1},{f}^{2},{f}^{3}\right)$ verifies $f\in {L}^{q}{\left(\Omega ×\left(0,T\right)\right)}^{3}$ with $q\ge 2$ and $abla \cdot f=0$ .
At a local level, we will consider solutions in sets of the form $G×\left(a,b\right)$ , where $G\subset \Omega$ is open and connected and $\left(a,b\right)\subset \left(0,T\right)$ :
Definition 2.1 Let the open set $D:=G×\left(a,b\right)$ be given. It will be said that the couple $\left(u,p\right)$ is a weak solution to the Navier-Stokes equations, Equation (1) in D if the
following holds:
・ $u\in {L}^{2}\left(0,T;{H}^{1}{\left(G\right)}^{3}\right)\cap {L}^{\infty }\left(0,T;{L}^{2}{\left(G\right)}^{3}\right)$ and $p\in {L}^{5/3}\left(G×\left(a,b\right)\right)$ .
・ u and p satisfy the Navier-Stokes equations in Equation (1) in the distributional sense in D.
On the other hand, for the definition of a global solution, it will be convenient to use the spaces H and V, with
$H:=\left\{v\in {L}^{2}{\left(\Omega \right)}^{3}:abla \cdot v=0\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\Omega ,v\cdot n=0\text{\hspace{0.17em}}\text{on}\text{\hspace{0.17em}}\partial \
Omega \right\},$
$V:=\left\{v\in {H}_{0}^{1}{\left(\Omega \right)}^{3}:abla \cdot v=0\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\Omega \right\}.$
Let us assume that ${u}_{0}\in H$ and let us consider the initial-boundary value problem
$\left\{\begin{array}{l}{u}_{t}+u\cdot abla u-\Delta u+abla p=f\left(x,t\right),\left(x,t\right)\in Q,\\ abla \cdot u=0,\left(x,t\right)\in Q,\\ u\left(x,t\right)=0,\left(x,t\right)\in \Sigma ,\\ u\
left(x,0\right)={u}_{0}\left(x\right),x\in \Omega .\end{array}$(2)
Definition 2.2 It will be said that the couple $\left(u,p\right)$ is a (global) weak solution to Equation (2) if the following holds:
・ $u\in {L}^{2}\left(0,T;V\right)\cap {L}^{\infty }\left(0,T;H\right)$ and $p\in {L}^{5/3}\left(Q\right)$ .
・ u and p satisfy the Navier-Stokes equations in Equation (2) in the distributional sense in Q
・ $u\left(x,0\right)={u}_{0}\left(x\right)$ a.e. in Ω.
It is known that any couple $\left(u,p\right)$ satisfying the previous first and second points also verifies
$u\in {L}^{10/3}{\left(Q\right)}^{3}\cap {C}_{w}^{0}\left(\left[0,T\right];H\right),\text{\hspace{0.17em}}{u}_{t}\in {L}^{4/3}\left(0,T;{V}^{\prime }\right).$
In particular, u can be viewed as a well-defined H-valued function and the third assertion in Definition 2.2 has a sense as an equality in H.
In order to understand the role and relevance of the terms in the estimates that follow, it is convenient to associate a dimension to each variable in Equation (2). Note that, if the pair $\left(u,p\
right)$ is a weak solution to the Navier-Stokes equations in $D=G×\left(a,b\right)$ , then for each $\lambda >0$ the functions
${u}_{\lambda }\left(x,t\right):=\lambda u\left(\lambda x,{\lambda }^{2}t\right)$ and ${p}_{\lambda }\left(x,t\right):={\lambda }^{2}p\left(\lambda x,{\lambda }^{2}t\right)$
solve a similar problem in ${D}_{\lambda }:={\lambda }^{-1}G×\left({\lambda }^{-2}a,{\lambda }^{-2}b\right)$ , with force ${f}_{\lambda }:={\lambda }^{3}f\left(\lambda x,{\lambda }^{2}t\right)$ .
Thus, for any integer k, we say that a variable or a linear differential operator is of dimension k if it is non-dimensionalized when it is multiplied by ${\lambda }^{-k}$ , where λ is a
characteristic length. We can affirm that
・ ${x}_{i}$ has dimension 1 and t is of dimension 2,
・ ${u}^{i}$ has dimension −1 and p has dimension −2,
・ f has dimension −3,
・ ${\partial }_{i}$ has dimension −1 and ${\partial }_{t}$ has dimension 2,
so that all the terms of the motion equation in Equation (2) have dimension −3.
The analysis of the existence of a weak solution to Equation (2) can be found for instance in [19] and [20] . Now, we will speak of the regularity problem, that is, the possible regularity properties
of the weak solution. To this purpose, let us consider the following definition:
Definition 2.3 Let $\left(u,p\right)$ be a weak solution to Equation (1) in $D=G×\left(a,b\right)$ and let $\left({x}_{0},{t}_{0}\right)\in D$ be given. It will be said that $\left({x}_{0},{t}_{0}\
right)$ is a singular point if u is not ${L}^{\infty }$ in any neighborhood of $\left({x}_{0},{t}_{0}\right)$ , that is, there are no r and C such that $|u\left(x,t\right)|\le C$ for $\left(x,t\
right)$ a.e. in $B\left(\left({x}_{0},{t}_{0}\right);r\right)$ . The remaining points, those where u is locally bounded, will be called regular points.
According to a result by Serrin [21] , it is known that, if $\left(u,p\right)$ is a weak solution to Equation (2) and $\left({x}_{0},{t}_{0}\right)\in D$ is a regular point for u, then u coincides
a.e. with a ${C}^{\infty }$ function in a neighborhood of $\left({x}_{0},{t}_{0}\right)$ . This gives an idea of how interesting can be to get a description of the set S of singular points.
In fact, Serrin proved that, in order to have u of class ${C}^{\infty }$ near $\left({x}_{0},{t}_{0}\right)$ , one just needs an estimate of the kind ${L}^{r}$ in time and ${L}^{s}$ in space, with
sufficiently large r and s. Note that, in [22] , it is shown that a weaker condition is sufficient for ${C}^{\infty }$ regularity. Note also that, in accordance with the results in [23] , if one
component of the velocity field is essentially bounded in a region, there is no singular point in a subregion.
The first papers devoted to describe S are due to Scheffer [1] [24] [25] . There, some estimates of the size of the set were given in terms of appropriate Hausdorff measures. Actually, the main
result in [1] is the following:
Theorem 2.4 Assume that $f\equiv 0$ . There exists a weak solution to Equation (2) whose associated singular set S satisfies:
${\mathcal{H}}^{5/3}\left(S\right)<+\infty$ and ${\mathcal{H}}^{1}\left(S\cap \left(\Omega ×\left\{t\right\}\right)\right)<+\infty$ uniformly in t.
Here, ${\mathcal{H}}^{k}$ denotes the usual Hausdorff k-dimensional measure in ${ℝ}^{4}$ .
This result was improved by Caffarelli, Kohn and Nirenberg in [2] in several directions. There, the authors used a particular class of weak solution, denoted suitable weak solution or simply suitable
solution, according to the following definition:
Definition 2.5 Let $D=G×\left(a,b\right)$ be a cylinder in ${ℝ}^{3}×ℝ$ . It is said that $\left(u,p\right)$ is a suitable weak solution to the Navier-Stokes equations in D if it satisfies points 1
and 2 of Definition 2.1 and, furthermore, the following generalized energy inequality property: for any $\varphi \in {C}_{0}^{\infty }\left(D\right)$ with $\varphi \ge 0$ ,
$2{\iint }_{D}{|abla u|}^{2}\varphi \le {\iint }_{D}\left({|u|}^{2}\left({\varphi }_{t}+\Delta \varphi \right)+\left({|u|}^{2}+2p\right)u\cdot abla \varphi +2\left(u\cdot f\right)\varphi \right)\text
{ }.$
Then, the authors of [2] introduced the so called “parabolic” Hausdorff measure ${\mathcal{P}}^{1}$ , as follows:
・ First, for any small $\delta >0$ and any $X\subset {ℝ}^{4}$ , they set
${\mathcal{P}}_{\delta }^{1}\left(X\right):=\mathrm{inf}\left\{\underset{i\ge 1}{\sum }\text{ }\text{ }{r}_{i}:X\subset \underset{i\ge 1}{\cup }{Q}_{{r}_{i}},{r}_{i}<\delta \right\}.$
Here, each ${Q}_{{r}_{i}}$ is a parabolic cylinder, that is, a set of the form
${Q}_{r}:=\left\{\left(x,t\right)\in {ℝ}^{4}:|x-\stackrel{¯}{x}|\le r,|t-\stackrel{¯}{t}|\ge {r}^{2}\right\}$
for some $\left(\stackrel{¯}{x},\stackrel{¯}{t}\right)\in Q$ .
・ Then, for any $X\subset {ℝ}^{4}$ , they set
${\mathcal{P}}^{1}\left(X\right):=\underset{\delta \to {0}^{+}}{\mathrm{lim}}{\mathcal{P}}_{\delta }^{1}\left(X\right)$
With the help of ${\mathcal{P}}^{1}$ , a local partial regularity result can be established for any suitable solution:
Theorem 2.6 Let $\left(u,p\right)$ be a suitable solution to Equation (1) in D. Then the associated singular set satisfies ${\mathcal{P}}^{1}\left(S\right)=0$ .
This result improves Theorem 2.4 in several aspects: first, it has local character; then, it allows a rather general force term f; finally, it gives a better estimate of the Hausdorff dimension of S,
since one has
${\mathcal{H}}^{1}\left(X\right)\le C{\mathcal{P}}^{1}\left(X\right),\text{\hspace{0.17em}}\forall X\subset {ℝ}^{4}$
for some $C>0$ .
In the sequel, for any $\left(x,t\right)$ and $r>0$ , we will denote by ${Q}_{r}\left(x,t\right)$ the following parabolic cylinder:
${Q}_{r}\left(x,t\right):=\left\{\left(y,\tau \right):|y-x|<r,t-{r}^{2}<\tau <t\right\}.$
For the proof of Theorem 2.6, we need two results. The first one is the following:
Proposition 2.7 Suppose that $\left(u,p\right)$ is a suitable weak solution to Equation (1) in ${Q}_{1}:={Q}_{1}\left(0,0\right)$ and $f\in {L}^{q}{\left({Q}_{1}\right)}^{3}$ with $q>5/2$ . There
exist ${ϵ}_{1}>0$ , ${C}_{1}>0$ and ${ϵ}_{2}={ϵ}_{2}\left(q\right)>0$ such that, if
${\iint }_{{Q}_{1}}\left({|u|}^{3}+|u||p|\right)+{\int }_{-1}^{0}{\left({\int }_{|x|<1}|p|\text{d}x\right)}^{5/4}\text{d}t\le {ϵ}_{1}$(3a)
${\iint }_{{Q}_{1}}{|f|}^{q}\le {ϵ}_{2}$(3b)
$|u\left(x,t\right)|\le {C}_{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}a.e.\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{Q}_{1/2}:={Q}_{1/2}\left(0,0\right).$(3c)
In particular, $\left(0,0\right)$ is a regular point.
Proposition 2.7 shows that the sizes of the data have an influence on the regularity of suitable solutions. Now, if we introduce
$M\left(r\right):=\frac{1}{{r}^{2}}{\iint }_{{Q}_{r}}\left({|u|}^{3}+|u||p|\right)+{r}^{-13/4}{\int }_{t-{r}^{2}}^{t}{\left({\int }_{|y-x|<r}|p|\text{d}y\right)}^{5/4}\text{d}\tau$(4)
${F}_{q}\left(r\right):={r}^{3q-5}{\iint }_{{Q}_{r}}{|f|}^{q},$(5)
taking into account the dimensions of these quantities, we can easily deduce the following:
Corollary 2.8 Suppose that $\left(u,p\right)$ is a suitable solution to Equation (1) in the cylinder ${Q}_{r}\left(x,t\right)$ and $f\in {L}^{q}{\left({Q}_{r}\left(x,t\right)\right)}^{3}$ , with $q>5
/2$ . Then, if $M\left(r\right)\le {ϵ}_{1}$ and ${F}_{q}\left(r\right)\le {ϵ}_{2}$ , one has
$|u|\le {C}_{1}{r}^{-1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}a.e.\text{\hspace{0.17em}}\text{ }\text{in}\text{\hspace{0.17em}}{Q}_{r/2}\left(x,t\right)$
and, consequently, every point in ${Q}_{r/2}\left(x,t\right)$ is regular.
Let us now set
${Q}_{r}^{*}\left(x,t\right):=\left\{\left(y,\tau \right):|y-x|<r,t-\frac{7}{8}{r}^{2}<\tau <t+\frac{1}{8}{r}^{2}\right\}.$
The second fundamental result used in the proof of Theorem 2.6 is the following:
Proposition 2.9 Let $\left(u,p\right)$ be a suitable solution to Equation (1) in a neighborhood of $\left(x,t\right)$ . There exists ${ϵ}_{3}>0$ such that, if
$\underset{r\to 0}{\mathrm{lim}\mathrm{sup}}\frac{1}{r}{\iint }_{{Q}_{r}^{*}\left(x,t\right)}{|abla u|}^{2}\le {ϵ}_{3},$
then $\left(x,t\right)$ is a regular point.
For the proofs of Propositions 2.7 and 2.9, Caffarelli, Kohn and Nirenberg used the generalized energy inequality with well chosen test functions ϕ; a simpler proof is given in [18] . Then, Theorem
2.6 is deduced from these results by contradiction using a covering lemma and the usual energy estimates.
2.2. Sketch of the Proofs of Theorems 2.4 and 2.6
Theorem 2.6 is a consequence of Proposition 2.9. The argument is explained below.
Consider first the proof of the fact that S has Hausdorff dimension less than or equal to 5/3, that is, Theorem 2.4. Using Corollary 2.8 and a covering lemma, we can easily see that, for each $\delta
>0$ , S can be covered by a family of parabolic cylinders $\left\{{Q}_{{r}_{i}}^{*}\left({x}_{i},{t}_{i}\right)\right\}$ such that ${r}_{i}<\delta$ , the ${Q}_{{r}_{i}/5}^{*}\left({x}_{i},{t}_{i}\
right)$ are mutually disjoint and
$\frac{1}{{r}^{2}}{\iint }_{{Q}_{{r}_{i}/5}^{*}\left({x}_{i},{t}_{i}\right)}\left({|u|}^{3}+|u||p|\right)+{r}^{-13/4}{\int }_{{t}_{i}-7{r}_{i}^{2}/8}^{{t}_{i}+{r}_{i}^{2}/8}{\left({\int }_{|x-{x}_{i}
|<{r}_{i}}|p|\text{d}y\right)}^{5/4}\text{d}\tau >C{ϵ}_{1}$(6)
for all i. Using Hölder’s inequality, we deduce that
${r}_{i}^{-5/3}{\iint }_{{Q}_{{r}_{i}/5}^{*}\left({x}_{i},{t}_{i}\right)}\left({|u|}^{10/3}+{|p|}^{5/3}\right)\ge C\left(ϵ1\right)$
and, therefore,
${\sum }^{\text{}}\text{ }\text{ }{r}_{i}^{5/3}\le {\iint }_{\cup {Q}_{{r}_{i}/5}^{*}\left({x}_{i},{t}_{i}\right)}\left({|u|}^{10/3}+{|p|}^{5/3}\right)\le C.$
Taking $\delta \to 0$ , we find that ${\mathcal{P}}^{5/3}\left(S\right)=0$ , whence we see in particular that the Hausdorff dimension of S is at most 5/3.
To show that ${\mathcal{P}}^{1}\left(S\right)=0$ by a similar method, instead of the integral of $\left({|u|}^{10/3}+{|p|}^{5/3}\right)$ , we need a global quantity of dimension 1. This is furnished
by Proposition 2.9. Indeed, this result allows to replace Equation (6) by
$\frac{1}{{r}_{i}}{\iint }_{{Q}_{{r}_{i}/5}^{*}\left({x}_{i},{t}_{i}\right)}{|abla u|}^{2}>C\left({ϵ}_{3}\right)$(7)
and, this way, we are led to the estimate
${\sum }^{\text{}}\text{ }\text{ }{r}_{i}\le C{\iint }_{\cup {Q}_{{r}_{i}/5}^{*}\left({x}_{i},{t}_{i}\right)}{|abla u|}^{2},$
whence we conclude that ${\mathcal{P}}^{1}\left(S\right)=0$ .
It is natural to ask if we can get a better estimate of the dimension of S. In other words, can we find $k<1$ such that ${\mathcal{P}}^{k}\left(S\right)=0$ ? Unfortunately, this question has not been
answered up to now. Actually, the answer does not seem simple and is related to the possibility of demonstrating an additional estimate of the (suitable) weak solutions of order less than 1.
It is important to note that the assumption $f\in {L}^{q}{\left(Q\right)}^{3}$ with $q>5/2$ is mainly needed to prove Proposition 2.7. On the other hand, note that, in Theorem 2.6, Caffarelli, Kohn
and Nirenberg chose to estimate the measure ${\mathcal{P}}^{1}$ of the set S, instead of the standard measure ${\mathcal{H}}^{1}$ . Both definitions are special cases of a construction made by
Carathéodory that is detailed in [26] .
The argument used by Caffarelli, Kohn and Nirenberg is valid for any suitable solution. In the Appendix of [2] , they prove the existence of such a solution. Thus, the following holds:
Theorem 2.10 Suppose that ${u}_{0}\in V$ , $f\in {L}^{q}{\left(Q\right)}^{3}$ with $q>5/2$ and $abla \cdot f=0$ in Q. Then, there exists at least one suitable weak solution $\left(u,p\right)$ to the
Navier-Stokes equations in Q satisfying
$u\left(t\right)\to {u}_{0}$ weakly in H as $t\to 0$ .
In addition, one has:
$\begin{array}{l}{\int }_{\Omega ×\left\{t\right\}}{|u|}^{2}\varphi +2{\int }_{0}^{t}{\int }_{\Omega }{|abla u|}^{2}\varphi \\ \le {\int }_{\Omega }{|{u}_{0}|}^{2}\varphi \left(x,0\right)+{\int }_{0}
^{t}{\int }_{\Omega }\left({|u|}^{2}\left({\varphi }_{t}+\Delta \varphi \right)+\left({|u|}^{2}+2p\right)u\cdot abla \varphi +2\left(u\cdot f\right)\varphi \right)\end{array}$(8)
for all functions $\varphi \in \mathcal{D}\left(\Omega ×\left[0,T\right]\right)$ with $\varphi \ge 0$ and $\varphi =0$ near $\partial \Omega ×\left(0,T\right)$ .
2.3. On the Existence of Suitable Weak Solutions
The existence of a suitable weak solution to Equation (2) is established in [2] by introducing a family of linear approximated problems and checking that the generalized energy inequalities are
satisfied in the limit. A second proof is given in [3] , using Faedo-Galerkin approximations. In both cases, the main difficult point is passing to the limit in the term $pu\cdot abla \varphi$ in the
right-hand side of the inequality. This requires nontrivial estimates on the pressure. In particular, Guermond [3] is able to achieve by reproducing for the discrete pressure some a priori estimates
similar to the estimates of Sohr and Von Wahl in [27] .
3. Some Convergence Results
3.1. The Convergence of the Semi-Approximate Problems
In this section, we will give a new proof of Theorem 2.10. To do this, we will apply the semi-implicit Euler scheme to produce a family of approximations to the Navier-Stokes problem Equation (2). We
will see that, at least for a subsequence, we have convergence to a suitable weak solution.
The scheme is the following. We take N large enough (the number of time steps) and we define the time step size $\tau :=T/N$ , the instants ${t}^{m}:=m\tau$ and the approximations
${f}^{m}:=\frac{1}{\tau }{\int }_{{t}^{m-1}}^{{t}^{m}}f\left(x,t\right)\text{d}t$ , ${u}^{m}\approx u\left(\cdot ,{t}^{m}\right)$ and ${p}^{m}\approx p\left(\cdot ,{t}^{m}\right)$ , with ${u}^{0}={u}
_{0}$ and
$\left\{\begin{array}{l}\frac{{u}^{m+1}-{u}^{m}}{\tau }+\left({u}^{m}\cdot abla \right){u}^{m+1}-\Delta {u}^{m+1}+abla {p}^{m+1}={f}^{m+1},x\in \Omega \\ abla \cdot {u}^{m+1}=0,x\in \Omega ;{\int }_
{\Omega }{p}^{m+1}\text{d}x=0,\\ {u}^{m+1}=0,x\in \partial \Omega ,\end{array}$(9)
for $m=0,1,\cdots ,N-1$ .
First of all, let us check that the ${u}^{m}$ are well defined:
Lemma 3.1 The Euler scheme in Equation (9) is well defined. In other words, for every $m\ge 0$ , there exists a unique solution $\left({u}^{m+1},{p}^{m+1}\right)$ to Equation (9).
The proof is immediate by induction. We only need to note that for each Equation (9) is a Dirichlet problem for a linear PDE system that can be written in the form
Find $u\in V$ such that $1/\tau \left(u,v\right)+\left(\left(w\cdot abla \right)u,v\right)+\left(abla u,abla v\right)=\left(g,v\right)\text{\hspace{0.17em}}\forall v\in V$ , Now, let us see that the
${u}^{m}$ are uniformly bounded in the L^2 norm. We have:
$\left(\frac{{u}^{m+1}-{u}^{m}}{\tau },{u}^{m+1}\right)+\left(\left({u}^{m}\cdot abla \right){u}^{m+1},{u}^{m+1}\right)+\left(abla {u}^{m+1},abla {u}^{m+1}\right)=\left({f}^{m+1},{u}^{m+1}\right),$
which can be rewritten in the form
$\frac{1}{2}\left({|{u}^{m+1}|}^{2}-{|{u}^{m}|}^{2}\right)+\frac{1}{2}{|{u}^{m+1}-{u}^{m}|}^{2}+\tau {|abla {u}^{m+1}|}^{2}=\tau \left({f}^{m+1},{u}^{m+1}\right).$(11)
Using the Cauchy-Schwarz and Young inequalities, we easily get that
$\frac{1}{2}\left({|{u}^{m+1}|}^{2}-{|{u}^{m}|}^{2}\right)+\tau {|abla {u}^{m+1}|}^{2}\le C{|{f}^{m+1}|}^{2}+\frac{\tau }{2}{|abla {u}^{m+1}|}^{2}.$(12)
${|{u}^{n+1}|}^{2}+\tau {\sum }_{m=0}^{n}{|abla {u}^{m+1}|}^{2}\le C{|{f}^{m+1}|}^{2}+{|{u}^{0}|}^{2}\le C$(13)
for all n and, certainly, ${u}^{m}$ is uniformly bounded in H.
Using this Euler scheme, we can construct the approximate solutions of the Navier-Stokes system. More precisely, let us introduce the functions ${u}_{N}$ and ${u}_{N}^{*}$ as follows:
・ ${u}_{N}:\left[0,T\right]↦V$ is the unique continuous piecewise linear function satisfying
${u}_{N}\left({t}^{m}\right)={u}^{m}$ for $m=0,1,\cdots ,N$ .
・ ${u}_{N}^{*}:\left[0,T\right]↦V$ is the piecewise constant function characterized by
${u}_{N}^{*}\left(t\right)={u}^{m+1}$ in $\left({t}^{m},{t}^{m+1}\right]$ for $m=0,1,\cdots ,N-1$ .
In a similar way, we can introduce the approximate pressures ${p}_{N}^{*}$ and forces ${f}_{N}^{*}$ (again piecewise constant). The following holds:
Lemma 3.2 For any N and almost every $t\in \left(0,T\right)$ , one has
$\left\{\begin{array}{l}{u}_{N,t}+\left({u}_{N}^{*}\left(t-\tau \right)\cdot abla \right){u}_{N}^{*}-\Delta {u}_{N}^{*}+abla {p}_{N}^{*}={f}_{N}^{*},\\ abla \cdot {u}_{N}^{*}=0.\end{array}$(14)
We can now present the main result of this section. It is related to the convergence of ${u}_{N}$ and ${u}_{N}^{*}$ towards a suitable weak solution to the Navier-Stokes equation:
Theorem 3.3 After eventual extraction of a subsequence, the functions ${u}_{N}^{*}$ converge weakly in ${L}^{2}\left(0,T;V\right)$ , weakly-* in ${L}^{\infty }\left(0,T;H\right)$ , strongly in ${L}^
{2}{\left(Q\right)}^{3}$ and a.e. in Q towards a suitable weak solution to Equation (2) as $N\to +\infty$ .
For the proof of Theorem 3.3, it will be convenient to recall the following well known lemma (for instance, see the proof in [19] ):
Lemma 3.4 Let u be a function satisfying $u\in {L}^{2}\left(0,T;V\right)$ and ${u}_{t}\in {L}^{2}\left(0,T;{V}^{\prime }\right)$ . Then, u is a.e. equal to a continuous function from $\left[0,T\
right]$ into H. In addition, the function $t↦{|u\left(t\right)|}^{2}$ is absolutely continuous and
$\frac{\text{d}}{\text{d}t}{|u\left(t\right)|}^{2}=2〈{u}_{t}\left(t\right),u\left(t\right)〉$ a.e. in $\left(0,T\right)$ ,
where $〈\cdot ,\cdot 〉$ denotes the duality product in ${V}^{\prime }×V$ .
Proof of Theorem 3.3:
Let us first try to find the spaces where the functions ${u}_{N}^{*}$ , ${u}_{N}$ and ${p}_{N}^{*}$ are uniformly bounded. This is classical and very well known, but we will give the details for
Consider the Equation (11). Let us fix N and n with $0\le n\le N-1$ and let us carry out summation in m, from 0 to n. The following is obtained:
$\begin{array}{l}\frac{1}{2}{|{u}^{n+1}|}^{2}+\frac{1}{2}\underset{m=0}{\overset{n}{\sum }}{|{u}^{m+1}-{u}^{m}|}^{2}+\tau \underset{m=0}{\overset{n}{\sum }}{|abla {u}^{m+1}|}^{2}\\ =\tau \underset{m=
0}{\overset{n}{\sum }}\left({f}^{m+1},{u}^{m+1}\right)+\frac{1}{2}{|{u}^{0}|}^{2}.\end{array}$
Obviously, this can also be written in the form
$\begin{array}{l}\frac{1}{2}{|{u}_{N}^{*}\left(t\right)|}^{2}+\frac{1}{2}\underset{m=0}{\overset{n}{\sum }}{|{u}^{m+1}-{u}^{m}|}^{2}+\underset{m=0}{\overset{n}{\sum }}\text{ }\text{ }{\int }_{{t}^
{m}}^{{t}^{m+1}}{|abla {u}_{N}^{*}\left(t\right)|}^{2}\text{d}t\\ =\underset{m=0}{\overset{n}{\sum }}\text{ }\text{ }{\int }_{{t}^{m}}^{{t}^{m+1}}\left({f}_{N}^{*}\left(t\right),{u}_{N}^{*}\left(t\
right)\right)\text{d}t+\frac{1}{2}{|{u}^{0}|}^{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall t\in \left({t}^{m},{t}^{m+1}\right].\end{array}$
$\frac{1}{2}{|{u}_{N}^{*}\left(t\right)|}^{2}+{\int }_{0}^{{t}^{n+1}}{|abla {u}_{N}^{*}\left(t\right)|}^{2}\text{d}t\le {\int }_{0}^{{t}^{n+1}}\left({f}_{N}^{*}\left(t\right),{u}_{N}^{*}\left(t\
and, from the Cauchy-Schwarz and Young inequalities, we easily see that
${|{u}_{N}^{*}\left(t\right)|}^{2}+{\int }_{0}^{T}{|abla {u}_{N}^{*}\left(t\right)|}^{2}\text{d}t\le {|{u}^{0}|}^{2}+C{‖{f}_{N}^{*}‖}_{{L}^{2}\left(Q\right)}^{2}\text{\hspace{0.17em}}\text{\hspace
{0.17em}}\text{\hspace{0.17em}}\forall t\in \left[0,T\right].$
This means that
${u}_{N}^{*}$ is uniformly bounded in ${L}^{2}\left(0,T;V\right)$ and ${L}^{\infty }\left(0,T;H\right)$ . (16)
On the other hand, it can also be deduced from Equation (9) that
${\int }_{0}^{T}{|{u}_{N}\left(t\right)-{u}_{N}^{*}\left(t\right)|}^{2}\text{d}t\le \tau \underset{m=0}{\overset{N-1}{\sum }}{|{u}^{m+1}-{u}^{m}|}^{2}\le C\tau ,$
whence ${‖{u}_{N}^{*}-{u}_{N}‖}_{{L}^{2}\left(Q\right)}^{2}\le C\tau$ .
To estimate ${u}_{N}$ , we use its definition and the fact that, for any $t\in \left({t}^{m},{t}^{m+1}\right)$ ,
$|{u}_{N}\left(t\right)|\le |{u}^{m}|+|{u}^{m+1}|$ and $|abla {u}_{N}\left(t\right)|\le |abla {u}^{m}|+|abla {u}^{m+1}|$ .
Accordingly, we also have that
${u}_{N}$ is uniformly bounded in ${L}^{2}\left(0,T;V\right)$ and ${L}^{\infty }\left(0,T;H\right)$ . (17)
Now, from classical interpolation results, we deduce that ${u}_{N}^{*}$ and ${u}_{N}$ is uniformly bounded in
${L}^{r}\left(0,T;{L}^{\frac{6r}{3r-4}}{\left(\Omega \right)}^{3}\right)$$\forall r\in \left[2,+\infty \right]$ . (18)
It is well known that the estimates of Equation (17) and Equation (18) allow us to prove that the ${u}_{N}$ belong to and are uniformly bounded in the Sobolev spaces of fractional order ${H}^{\gamma
}\left(0,T;H\right)$ for $0<\gamma <1/4$ ; see for example [19] . Therefore, as a consequence of Aubin-Lions’ Theorem, the ${u}_{N}$ belong to a compact set of ${L}^{2}\left(Q\right)$ .
As a consequence, at least for a subsequence (again indexed by N), we must have:
$\left\{\begin{array}{l}{u}_{N}\to u\text{\hspace{0.17em}}\text{weakly}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{L}^{2}\left(0,T;V\right)\text{\hspace{0.17em}}\text{and}\text{\hspace
{0.17em}}\text{weakly-}\ast \text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{L}^{\infty }\left(0,T;H\right),\\ {u}_{N}\to u\text{\hspace{0.17em}}\text{strongly}\text{\hspace{0.17em}}\text{in}\
This is enough to pass to the limit in Equation (14) and deduce that u is a weak solution of Equation (2). Note that it can also be assumed that
${u}_{N}\to u$ strongly in ${L}^{r}\left(0,T;{L}^{q}{\left(\Omega \right)}^{3}\right)$ for all $2<r<+\infty$ , $1\le q<6r/\left(3r-4\right)$ . (20)
To show that u is suitable, we have to give new estimates. To this purpose, we will use some regularity results that, as those in [3] , play the role of the Sohr and Wahl’s estimates in [27] .
For $0<s<1$ , the space ${H}^{s}\left(\Omega \right):={\left[{H}^{1}\left(\Omega \right),{L}^{2}\left(\Omega \right)\right]}_{s}$ can be defined by the method of real interpolation between ${H}^{1}\
left(\Omega \right)$ and ${L}^{2}\left(\Omega \right)$ , i.e. the so-called K-method of Lions and Peetre [28] ; see also [29] and [30] . We will denote by ${H}_{0}^{s}$ the closure of $\mathcal{D}\
left(\Omega \right)$ in ${H}^{s}\left(\Omega \right)$ . For any $s<0$ , the space ${H}^{-s}\left(\Omega \right)$ and the corresponding norm ${‖\text{ }\cdot \text{ }‖}_{{H}^{-s}}$ are defined by
duality and, in particular,
${‖v‖}_{{H}^{-s}}:=\underset{w\in \mathcal{D}\left(\Omega \right)\\left\{0\right\}}{\mathrm{sup}}\frac{\left(v,w\right)}{{w}_{{H}^{s}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall v\in {L}^{2}
\left(\Omega \right).$
We will look for a uniform estimate of ${u}_{N,t}$ in a space of the form ${L}^{a}\left(0,T;{H}^{-\sigma }{\left(\Omega \right)}^{3}\right)$ . This way, by applying De-Rham’s Lemma (see [31] ), we
will get a bound of ${p}_{N}^{*}$ in ${L}^{a}\left(0,T;{H}^{1-\sigma }{\left(\Omega \right)}^{3}\right)$ and we will be able to take limits in the generalized energy inequality.
Note that, for all m, one has ${u}^{m}={w}^{m}+{z}^{m}$ , where the ${w}^{m}$ and the ${z}^{m}$ are respectively given by
$\left\{\begin{array}{l}\frac{1}{\tau }\left({w}^{m+1}-{w}^{m}\right)+A{w}^{m+1}=0,\\ {w}^{0}={u}^{0}\end{array}$(21)
$\left\{\begin{array}{l}\frac{1}{\tau }\left({z}^{m+1}-{z}^{m}\right)+A{z}^{m+1}={F}^{m+1},\\ {z}^{0}=0,\end{array}$(22)
where ${F}^{m+1}={f}^{m+1}-\left({u}^{m}\cdot abla \right){u}^{m+1}$ and A is the Stokes operator. Recall that $A:D\left(A\right)\subset H↦H$ , with
$D\left(A\right)={H}^{2}{\left(\Omega \right)}^{3}\cap V,Av=P\left(-\Delta v\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall v\in D\left(A\right)$
(here, $P:{L}^{2}{\left(\Omega \right)}^{3}↦H$ is the orthogonal projector). Also, recall that there exists an orthogonal basis of V formed by eigenfunctions ${\xi }_{j}$ ,
$A{\xi }_{j}={\lambda }_{j}{\xi }_{j},{\xi }_{j}\in V,|{\xi }_{j}|=1,{\lambda }_{j}↗\infty$
$D\left({A}^{r}\right)=\left\{v\in H:\underset{j\ge 1}{\sum }\text{ }\text{ }{\lambda }_{j}^{2r}{|\left(v,{\xi }_{j}\right)|}^{2}<+\infty \right\}$
for all $r\ge 0$ .
In the sequel, we will consider the functions ${w}_{N},{w}_{N}^{*},{z}_{N}$ and ${z}_{N}^{*}$ , whose definitions can be obtained from the ${z}^{m}$ and the ${w}^{m}$ in a way similar to ${u}_{N}$
and ${u}_{N}^{*}$ .
First, note that
${w}^{m}={\left(Id.+\tau A\right)}^{-m}{u}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall m=0,1,\cdots ,N,$
$\begin{array}{c}{‖{w}_{N,t}‖}_{{L}^{2}\left(Q\right)}^{2}=\tau \underset{m=0}{\overset{N-1}{\sum }}{|A{\left(Id.+\tau A\right)}^{-\left(m+1\right)}{u}_{0}|}^{2}\\ =\tau \underset{m=0}{\overset{N-1}
{\sum }}\text{\hspace{0.17em}}\underset{j\ge 1}{\sum }\frac{{\lambda }_{j}^{2}}{{\left(1+\tau {\lambda }_{j}\right)}^{2\left(m+1\right)}}{|\left({u}_{0},{\xi }_{j}\right)|}^{2}\\ =\tau \underset{j\ge
1}{\sum }\left(\underset{m=0}{\overset{N-1}{\sum }}\frac{1}{{\left(1+\tau {\lambda }_{j}\right)}^{2\left(m+1\right)}}\right){\lambda }_{j}^{2}{|\left({u}_{0},{\xi }_{j}\right)|}^{2}\\ \le {\sum }_{j\
ge 1}\frac{{\lambda }_{j}^{2}}{{\left(1+\tau {\lambda }_{j}\right)}^{2\left(m+1\right)}}{|\left({u}_{0},{\xi }_{j}\right)|}^{2}={|{u}_{0}|}^{2}.\end{array}$(23)
${w}_{N,t}$ and $A{w}_{N}^{*}$ are uniformly bounded in ${L}^{2}{\left(Q\right)}^{3}$ . (24)
Let us now see what can be said of ${z}_{N,t}$ and $A{z}_{N}^{*}$ . For all $m\ge 1$ , we have
${z}^{m}=\underset{l=1}{\overset{m}{\sum }}{\left(Id.+\tau A\right)}^{-\left(m+1-l\right)}{F}^{l}$
Let $s,\sigma \in \left(0,1\right)$ be such that with $\sigma >s$ . Then
${‖A{z}^{m+1}‖}_{{H}^{-\sigma }}\le \tau \underset{l=1}{\overset{m+1}{\sum }}{‖A{\left(Id.+\tau A\right)}^{-\left(m+2-l\right)}‖}_{\mathcal{L}\left({H}^{-s};{H}^{-\sigma }\right)}{‖{F}^{l}‖}_{{H}^
{-s}}=\tau \underset{l=1}{\overset{m}{\sum }}\text{ }\text{ }{a}_{m-l}{b}_{l},$
where the ${a}_{n}$ and the ${b}_{l}$ are given by
${a}_{n}={‖A{\left(Id.+\tau A\right)}^{-\left(m+2-l\right)}‖}_{\mathcal{L}\left({H}^{-s};{H}^{-\sigma }\right)},{b}_{l}={‖{F}^{l}‖}_{{H}^{-s}}.$
We will apply the following result, that must be viewed as a discrete version of the well known Young inequality for convolution products:
Lemma 3.5 Let us assume that $k\ge 1$ , $a\in {l}^{p}$ and $b\in {l}^{q}$ . Then, if $r\in \left[1,+\infty \right]$ and
one has
${\left({\sum }_{n=1}^{k}{|{\sum }_{l=1}^{n}{a}_{n-l}{b}_{l}|}^{r}\right)}^{1/r}\le {\left({\sum }_{n=1}^{k}{|{a}_{n}|}^{p}\right)}^{\frac{1}{p}}{\left({\sum }_{n=1}^{k}{|{b}_{n}|}^{q}\right)}^{1/q}$
for all $k\ge 1$ .
The proof of this result can be found in [32] . Using Lemma 2.5 with $r=a$ , $p=1$ and $q=a$ , we find that
${\left(\tau \underset{n=1}{\overset{N}{\sum }}{‖A{z}^{n+1}‖}_{{H}^{-\sigma }}^{a}\right)}^{1/a}\le {\left({\tau }^{1+a}\underset{n=1}{\overset{N}{\sum }}{|\underset{l=1}{\overset{n}{\sum }}\text{ }\
text{ }{a}_{n-l}{b}_{l}|}^{a}\right)}^{\frac{1}{a}}\le \left(\underset{n=1}{\overset{N}{\sum }}\text{ }\text{ }{a}_{n}\right){\left(\underset{n=1}{\overset{N}{\sum }}\text{ }\text{ }{b}_{n}^{a}\
From the estimates in Equation (16) already obtained for ${u}_{N}^{*}$ , it is immediate that, for any $a\in \left[1,2\right]$ , ${F}_{N}^{*}$ is uniformly bounded in ${L}^{a}\left(0,T;{L}^{3a/\left
(4a-2\right)}{\left(\Omega \right)}^{3}\right)$ and, consequently, also in ${L}^{a}\left(0,T;{H}^{-\left(5a-4\right)/\left(2a\right)}{\left(\Omega \right)}^{3}\right)$ . Thus, choosing in $\left[1,2\
right]$ and taking $s=\left(5a-4\right)/\left(2a\right)$ , we get:
${‖{F}_{N}^{*}‖}_{{L}^{a}\left(0,T;{H}^{-s}\right)}={\left(\tau \underset{m=1}{\overset{N}{\sum }}\text{ }\text{ }{b}_{m}^{a}\right)}^{1/a}\le C\left(a\right).$
On the other hand, for any smooth z, one has
${‖A{\left(Id.+\tau A\right)}^{-\left(n+2\right)}z‖}_{{H}^{-\sigma }}^{2}=\underset{j\ge 1}{\sum }\text{ }\text{ }{\lambda }_{j}^{-\sigma }\frac{{\lambda }_{j}^{2}}{{\left(1+\tau {\lambda }_{j}\
right)}^{2\left(n+2\right)}}{|z,{\xi }_{j}|}^{2}\le \left[\underset{j}{\mathrm{sup}}\frac{{\lambda }_{j}^{2\left(1-ϵ\right)}}{{\left(1+\tau {\lambda }_{j}\right)}^{2\left(n+2\right)}}\right]{‖z‖}_
where $ϵ=\left(\sigma -s\right)/2$ . Therefore, recalling the definition of the ${a}_{n}$ , we deduce that
${a}_{n}\le \frac{C\left(ϵ\right)}{{\left(n\tau \right)}^{1-ϵ}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\tau \underset{n=1}{\overset{m}{\sum }}\text{ }\text{ }{a}_{n}\le C\left(ϵ\right){\int }_
{0}^{T}\frac{\text{d}s}{{s}^{1-ϵ}}\le C\left(ϵ\right)$
and, finally,
${‖A{z}_{N}^{*}‖}_{{L}^{a}\left(0,T;{H}^{-\sigma }\right)}\le C\left({\sum }_{m=1}^{N}\tau {a}_{m}\right){\left(\tau {\sum }_{m=1}^{N}{‖{F}^{m}‖}_{{H}^{-s}}^{a}\right)}^{\frac{1}{a}}\le {‖C{F}_{N}^
Note that this estimate is valid for all $a\in \left[1,2\right]$ , with $s=\left(5a-4\right)/\left(2a\right)$ and $\sigma \in \left(0,1\right)$ , $\sigma >s$ .
Obviously, the same estimate is valid for ${‖{z}_{N,t}‖}_{{L}^{a}\left(0,T;{H}^{-\sigma }\right)}$ . This proves that
$\begin{array}{l}{z}_{N,t}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}A{z}_{N}^{*}\text{\hspace{0.17em}}\text{areuniformlyboundedin}\text{\hspace{0.17em}}{L}^{a}\left(0,T;{H}^{-\sigma }{\
left(\Omega \right)}^{3}\right)\\ \forall a\in \left[0,1\right],\forall \sigma >s=\left(5a-4\right)/\left(2a\right).\end{array}$(27)
In view of Equation (24) and Equation (27) and recalling that ${u}_{N}^{*}={w}_{N}^{*}+{z}_{N}^{*}$ , it follows that $A{u}_{N}^{*}$ and ${u}_{N,t}^{*}$ are also uniformly bounded in ${L}^{a}\left
(0,T;{H}^{-\sigma }{\left(\Omega \right)}^{3}\right)$ .
Now, from De-Rham’s Lemma [31] , we see that ${p}_{N}^{*}$ is uniformly bounded in ${L}^{a}\left(0,T;{H}^{1-\sigma }\left(\Omega \right)\right)$ , which is continuously embedded in ${L}^{a}\left(0,T;
{L}^{6/\left(1+2\sigma \right)}\left(\Omega \right)\right)$ . In particular, for $a=5/3$ , we have $s=13/10$ and we can take σ as close as desired to s, which gives $6/\left(1+2\sigma \right)$ as
close as desired to 5/3.
As a consequence of these estimates, after extracting a new sequence (if this is needed), we see that
${p}_{N}^{*}$ converges weakly in ${L}^{5/3}\left(0,T;{L}^{\beta }\left(\Omega \right)\right)$$\forall \beta <5/3$ . (28)
Let us check that the local energy inequality holds for u and p.
If we multiply the Equation (14) by the function ${u}_{N}^{*}\varphi$ , where $\varphi \in {C}_{0}^{\infty }\left(\Omega ×\left[0,T\right]\right)$ is non-negative and we integrate in space, we have:
$\begin{array}{l}{\int }_{\Omega }{u}_{N,t}\cdot {u}_{N}^{*}\varphi +{\int }_{\Omega }\left({u}_{N}^{*}\left(t-\tau \right)\cdot abla \right){|{u}_{N}^{*}|}^{2}\varphi +{\int }_{\Omega }\left(-\Delta
{u}_{N}^{*}\right)\cdot {u}_{N}^{*}\varphi +{\int }_{\Omega }abla {p}_{N}^{*}\cdot {u}_{N}^{*}\varphi \\ ={\int }_{\Omega }{f}_{N}^{*}\cdot {u}_{N}^{*}\varphi .\end{array}$(29)
If $t\in \left[0,T\right]$ , there exists n such that $t\in \left({t}^{n},{t}^{n+1}\right]$ and then, using Lemma 3.4, one has:
$\begin{array}{c}{\int }_{\Omega }{u}_{N,t}\cdot {u}_{N}^{*}\varphi ={\int }_{\Omega }{u}_{N,t}\cdot {u}_{N}\varphi +{\int }_{\Omega }{u}_{N,t}\cdot \left({u}_{N}^{*}-{u}_{N}\right)\varphi \\ =\frac
{1}{2}\frac{\text{d}}{\text{d}t}{\int }_{\Omega }{|{u}_{N}|}^{2}\varphi -\frac{1}{2}{\int }_{\Omega }{|{u}_{N}|}^{2}{\varphi }_{t}+{\int }_{\Omega }{u}_{N,t}\cdot \left({u}_{N}^{\text{*}}-{u}_{N}\
right)\varphi \text{ }\text{.}\end{array}$
Note moreover that ${\int }_{\Omega }{u}_{N,t}\cdot \left({u}_{N}^{*}-{u}_{N}\right)\varphi \ge 0$ , because ${u}_{N}^{*}-{u}_{N}$ is by definition equal to $\left({t}^{n+1}-t\right){u}_{N,t}$ .
• Also,
${\int }_{\Omega }\left({u}_{N}^{*}\left(t-\tau \right)\cdot abla \right){|{u}_{N}^{*}|}^{2}\varphi =-\frac{1}{2}{\int }_{\Omega }{u}_{N}^{*}\left(t-\tau \right){|{u}_{N}^{*}|}^{2}\cdot abla \varphi
• On the other hand,
${\int }_{\Omega }\left(-\Delta {u}_{N}^{*}\right)\cdot {u}_{N}^{*}\varphi ={\int }_{\Omega }{|abla {u}_{N}^{*}|}^{2}\varphi -\frac{1}{2}{\int }_{\Omega }{|{u}_{N}^{*}|}^{2}\Delta \varphi .$
• Finally,
${\int }_{\Omega }abla {p}_{N}^{*}\cdot {u}_{N}^{*}\varphi =-{\int }_{\Omega }{p}_{N}^{*}\cdot {u}_{N}^{*}abla \varphi .$
Consequently, we see from Equation (29) that
$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\int }_{\Omega }{|{u}_{N}|}^{2}\varphi +{\int }_{\Omega }{|abla {u}_{N}^{*}|}^{2}\varphi \le \frac{1}{2}{\int }_{\Omega }{|{u}_{0}|}^{2}\varphi
+{\iint }_{\Omega ×\left(0,t\right)}{|{u}_{N}|}^{2}{\varphi }_{t}\\ \text{ }+{\iint }_{\Omega ×\left(0,t\right)}\left[\left(\frac{1}{2}{u}_{N}^{*}\left(t-\tau \right){|{u}_{N}^{*}|}^{2}+{p}_{N}^{*}
{u}_{N}^{*}\right)\cdot abla \varphi +\frac{1}{2}{|{u}_{N}^{*}|}^{2}\Delta \varphi +{f}_{N}^{*}\cdot {u}_{N}^{*}\varphi \right]\end{array}$
If we integrate in time, we find that
$\begin{array}{l}{\int }_{\Omega }{|{u}_{N}|}^{2}\varphi +2{\iint }_{\Omega ×\left(0,t\right)}{|abla {u}_{N}^{*}|}^{2}\varphi \le {\int }_{\Omega }{|{u}_{0}|}^{2}\varphi +{\iint }_{\Omega ×\left(0,t\
right)}{|{u}_{N}|}^{2}{\varphi }_{t}\\ \text{ }+{\iint }_{\Omega ×\left(0,t\right)}\left[\left({u}_{N}^{*}\left(t-\tau \right){|{u}_{N}^{*}|}^{2}+2{p}_{N}^{*}{u}_{N}^{*}\right)\cdot abla \varphi +{|
{u}_{N}^{*}|}^{2}\Delta \varphi +2{f}_{N}^{*}\cdot {u}_{N}^{*}\varphi \right]\end{array}$
Thanks to the energy estimates Equation (16), we can take the lower limit in the left-hand side. On the other hand, thanks to Equation (19), Equation (20) and Equation (28), we can take limits in all
the terms in the right; for example, since ${u}_{N}^{*}$ converges strongly in ${L}^{5/2}\left(0,T;{L}^{19/5}{\left(\Omega \right)}^{3}\right)$ and ${p}_{N}^{*}$ converges weakly in ${L}^{5/3}\left
(0,T;{L}^{14/9}\left(\Omega \right)\right)$ , we see that ${p}_{N}^{*}{u}_{N}^{*}\cdot abla \varphi$ converges weakly in ${L}^{1}\left(Q\right)$ towards $pu\cdot abla \varphi$ .
The final result is that
$\begin{array}{l}2{\iint }_{\Omega ×\left(0,t\right)}{|abla u|}^{2}\varphi \\ \le \underset{\Omega }{\int }{|{u}_{0}|}^{2}\varphi +{\iint }_{\Omega ×\left(0,t\right)}\left({|u|}^{2}\left({\varphi }_
{t}+\Delta \varphi \right)+\left({|u|}^{2}+2p\right)u\cdot abla \varphi +2\left(u\cdot f\right)\varphi \right),\end{array}$(30)
as desired.
3.2. The Convergence of the Fully Discretized Problems
In this section, we will argue as in [3] and we will check that the approximate solutions obtained via the semi-implicit Euler discrete scheme, used together with an appropriate approximation in
space, converge to a suitable solution to Equation (2).
As before, let us introduce N, $\tau :=T/N$ and the ${t}^{m}:=m\tau$ . We will also consider two families of finite dimensional spaces ${\left\{{X}_{h}\right\}}_{h>0}$ and ${\left\{{P}_{h}\right\}}_
{h>0}$ with the ${X}_{h}\subset {H}_{0}^{1}{\left(\Omega \right)}^{3}$ and the ${P}_{h}\subset {L}^{2}\left(\Omega \right)$ such that
$\left\{\begin{array}{l}{\mathrm{inf}}_{{v}_{h}\in {X}_{h}}{‖v-{v}_{h}‖}_{{H}^{1}}\to 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall v\in {H}_{0}^{1}{\left(\Omega \right)}^{3},\\ {\mathrm{inf}}
_{{q}_{h}\in {P}_{h}}{‖q-{q}_{h}‖}_{{L}^{2}}\to 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall q\in {L}^{2}\left(\Omega \right),\end{array}$(31)
and the $\left({X}_{h},{P}_{h}\right)$ are uniformly compatible, in the sense that there exists a constant $\mu >0$ independent of h such that the following inf-sup conditions are satisfied:
${\mathrm{inf}}_{{q}_{h}\in {P}_{h}\\left\{0\right\}}{\mathrm{sup}}_{{v}_{h}\in {X}_{h}\\left\{0\right\}}\frac{\left(abla {q}_{h},{v}_{h}\right)}{{‖{v}_{h}‖}_{{H}^{-1-s}}{‖{q}_{h}‖}_{{H}^{s}}}\ge \mu
Now, we consider the approximations
${u}_{h}^{m}=u\left(\cdot ,{t}^{m}\right)\in {X}_{h},{p}_{h}^{m}=p\left(\cdot ,{t}^{m}\right)\in {P}_{h}$ ,
with ${u}_{h}^{0}={u}_{0h}$ (the orthogonal projection of ${u}_{0}$ on ${X}_{h}$ ) and
$\left\{\begin{array}{l}\left(\frac{{u}_{h}^{m+1}-{u}_{h}^{m}}{\tau },{v}_{h}\right)+\left(\left({u}_{h}^{m}\cdot abla \right){u}_{h}^{m+1},{v}_{h}\right)+\left(abla {u}_{h}^{m+1},abla {v}_{h}\right)
+\left(abla {p}_{h}^{m+1},{v}_{h}\right)\\ =\left({f}_{h}^{m+1},{v}_{h}\right),\forall {v}_{h}\in {X}_{h},\\ \left({q}_{h},abla \cdot {u}_{h}^{m+1}\right)=0,\\ \left({u}_{h}^{m+1},{p}_{h}^{m+1}\
right)\in \left({X}_{h},{P}_{h}\right),\end{array}$(33)
for $m=0,1,\cdots ,N-1$ . The following result, which is a consequence of Equation (32), gives coherence to our scheme:
As before, the ${u}_{h}^{m}$ and ${p}_{h}^{m}$ serve to construct approximate solutions to the Navier-Stokes system. Thus, we define the functions ${u}_{N,h},{u}_{N,h}^{*},{p}_{N,h}^{*}$ etc.
similarly to ${u}_{N},{u}_{N}^{*},{p}_{N}^{*}$ etc. The main result of this section is the following:
Theorem 3.7 After eventual extraction of a subsequence, the functions ${u}_{N,h}^{*}$ converge weakly in ${L}^{2}\left(0,T;V\right)$ , weakly-* in ${L}^{\infty }\left(0,T;H\right)$ , strongly in ${L}
^{2}{\left(Q\right)}^{3}$ and a.e. in Q towards a suitable weak solution to Equation (2) as $N\to +\infty$ and $h\to 0$ .
Sketch of the proof:
Arguing as in the proof of the Theorem 3.3, it can be seen that the ${u}_{N,h}$ and the ${u}_{N,h}^{*}$ are uniformly bounded in ${L}^{2}\left(0,T;V\right)$ and ${L}^{\infty }\left(0,T;H\right)$ and,
furthermore, ${‖{u}_{N,h}-{u}_{N,h}^{*}‖}_{{L}^{2}\left(Q\right)}^{2}\le C\tau$ . As in [19] , we can also prove that the ${u}_{N,h}$ are uniformly bounded in ${H}^{\gamma }\left(0,T;H\right)$ for
any $\gamma \in \left(0,1/4\right)$ . Consequently, at least for a subsequence (still indexed with N and h), one has:
$\left\{\begin{array}{l}{u}_{N,h}\to u\text{\hspace{0.17em}}\text{weakly}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{L}^{2}\left(0,T;V\right)\text{\hspace{0.17em}}\text{andweakly-}\ast \
text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{L}^{\infty }\left(0,T;H\right),\\ {u}_{N,h}\to u\text{\hspace{0.17em}}\text{strongly}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{L}^{2}
${u}_{N,h}\to u$ strongly in ${L}^{r}\left(0,T;{L}^{q}{\left(\Omega \right)}^{3}\right)$ for all $2<r<+\infty$ , $1\le q<6r/\left(3r-4\right)$ . (35)
As before, this is enough to pass to the limit and deduce that u is a weak solution of Equation (2).
In order to prove that u is suitable, we can argue as in Guermond [3] . Here, we need the spaces ${\stackrel{˜}{H}}_{0}^{s}\left(\Omega \right):={\left[{L}^{2}\left(\Omega \right),{H}_{0}^{1}\left(\
Omega \right)\right]}_{s}$ for $s\in \left(0,1\right)$ and their dual spaces ${\stackrel{˜}{H}}_{0}^{-s}\left(\Omega \right)$ .
The following estimates are established in [3] :
・ For any $\alpha \in \left[1/4,1/2\right]$ and any $\delta <\stackrel{¯}{\delta }\left(\alpha \right)=2\left(1+\alpha \right)/5$ , one has
${‖{u}_{N,h,t}‖}_{{H}^{\delta -1}\left(0,T;{\stackrel{˜}{H}}^{-\alpha }\right)}+{‖{u}_{N,h}^{*}‖}_{{H}^{\delta }\left(0,T;{\stackrel{˜}{H}}^{-\alpha }\right)}\le C\left(\alpha \right),$
・ For any and $s\in \left[1/2,7/10\right]$ any, one also has
${‖{u}_{N,h,t}‖}_{{H}^{-r}\left(0,T;{\stackrel{˜}{H}}^{-s}\right)}+{‖{p}_{N,h}^{*}‖}_{{H}^{-r}\left(0,T;{\stackrel{˜}{H}}^{1-s}\right)}\le C\left(s\right),$
As a consequence, it can be assumed that the ${p}_{N,h}^{*}$ converge weakly (for instance) in ${H}^{-r}\left(0,T;{H}^{3/8}\left(\Omega \right)\right)$ for all $r>7/16$ and the ${u}_{N,h}^{*}$
converge strongly in ${H}^{\delta }\left(0,T;{\stackrel{˜}{H}}^{-\alpha }{\left(\Omega \right)}^{3}\right)$ for all $\alpha <3/8$ and $\delta <11/20$ . This is sufficient to ensure that ${p}_{N,h}^
{*}{u}_{N,h}^{*}\cdot abla \varphi$ converges weakly in ${L}^{1}{\left(Q\right)}^{3}$ towards $pu\cdot abla \varphi$ .
Hence, arguing as in the final part of the proof of Theorem 3.3, it is not difficult to check that the limit $\left(u,p\right)$ of the $\left({u}_{N,h}^{*},{p}_{N,h}^{*}\right)$ is a suitable weak
solution to Equation (2).
This ends the proof.
4. Some Additional Comments and Questions
4.1. The Same Results Hold for the Boussinesq System
The Boussinesq system is the following:
$\left\{\begin{array}{l}{u}_{t}-\Delta u+\left(u\cdot abla \right)u+abla p=f+\theta k,\left(x,t\right)\in Q,\\ abla \cdot u=0,\left(x,t\right)\in Q,\\ {\theta }_{t}+u\cdot abla \theta -\Delta \theta
=g,\left(x,t\right)\in Q,\\ u\left(x,t\right)=0,\theta \left(x,t\right)=0,\left(x,t\right)\in \Sigma ,\\ u\left(x,0\right)={u}_{0}\left(x\right),\theta \left(x,0\right)={\theta }_{0}\left(x\right),x\
in \Omega .\end{array}$(36)
We assume here that
${u}_{0}\in V,{\theta }_{0}\in {H}_{0}^{1}\left(\Omega \right),f\in {L}^{2}\left(0,T;{H}^{-1}{\left(\Omega \right)}^{3}\right),k\in {ℝ}^{3}$ and $g\in {L}^{2}\left(0,T;{L}^{2}\left(\Omega \right)\
right)$ .(37)
As in [4] , we can speak of weak solutions to Equation (36) and, also, of suitable weak solutions to the previous equations in any set of the form $D=G×\left(a,b\right)$ , with $G\subset {ℝ}^{3}$ a
connected open set.
The results in Section 3 can be extended to this framework. Thus, we can for instance consider the semi-implicit Euler scheme
$\left\{\begin{array}{l}\frac{{u}^{m+1}-{u}^{m}}{\tau }+\left({u}^{m}\cdot abla \right){u}^{m+1}-\Delta {u}^{m+1}+abla {p}^{m+1}={f}^{m+1}+{\theta }^{m+1}k,\\ abla \cdot {u}^{m+1}=0,\\ \frac{{\theta
}^{m+1}-{\theta }^{m}}{\tau }+{u}^{m}\cdot abla {\theta }^{m+1}-\Delta {\theta }^{m+1}={g}^{m+1}\end{array}$(38)
and prove that, at least for a subsequence, the corresponding $\left({u}_{N},{p}_{N},{\theta }_{N}\right)$ and $\left({u}_{N}^{*},{p}_{N}^{*},{\theta }_{N}^{*}\right)$ converge, in an appropiate
sense, to a suitable weak solution $\left(u,p,\theta \right)$ .
4.2. Possible Extensions to Other Systems
It would be interesting to prove similar results to Theorems 3.3 and 3.7 for the solutions to the variable-density Navier-Stokes equations
$\left\{\begin{array}{l}{\rho }_{t}+abla \cdot \left(\rho u\right)=0,\left(x,t\right)\in Q,\\ \rho \left({u}_{t}+\left(u\cdot abla \right)u\right)-\Delta u+abla p=\rho f,\left(x,t\right)\in Q\\ abla
\cdot u=0,\left(x,t\right)\in Q,\\ u\left(x,t\right)=0,\left(x,t\right)\in \Sigma ,\\ u\left(x,0\right)={u}_{0}\left(x\right),\rho \left(x,0\right)={\rho }_{0}\left(x\right),x\in \Omega .\end{array}$
However, this is not clear at present. Note that the “reasonable” definition of a suitable weak solution should involve the following property: for any $\varphi \in D\left(Q\right)$ with $\varphi \ge
0$ ,
$2{\iint }_{D}{|abla u|}^{2}\varphi \le {\iint }_{D}\left({|u|}^{2}\left(\rho {\varphi }_{t}+\Delta \varphi \right)+\left(\rho {|u|}^{2}+2p\right)\left(u\cdot abla \varphi \right)+2\rho \left(u\cdot
f\right)\varphi \right).$
But, unfortunately, the apparent lack of regularity of p makes it difficult to prove this.
4.3. Extensions to Other Approximation Schemes for the Navier-Stokes Equations
As we already said, Theorems 3.3 and 3.7 can be adapted to many other time approximation schemes. Among then, let us simply recall the following:
・ Crank-Nicholson scheme:
$\frac{{u}^{m+1}-{u}^{m}}{\tau }+\left({u}^{m}\cdot abla \right){u}^{m+1}-\Delta \left(\frac{{u}^{m+1}+{u}^{m}}{2}\right)+abla {p}^{m+1}={f}^{m+1},abla \cdot {u}^{m+1}=0.$
・ Gear scheme:
$\frac{3{u}^{m+1}-4{u}^{m}+{u}^{m-1}}{2\tau }+\left({u}^{m}\cdot abla \right){u}^{m+1}-\Delta {u}^{m+1}+abla {p}^{m+1}={f}^{m+1},abla \cdot {u}^{m+1}=0.$
θ-scheme: For α and β such that $0<\alpha ,\beta <1$ and $\alpha +\beta =1$ , we compute $\left({u}^{n+\theta },{p}^{n+\theta }\right)$ , then ${u}^{n+1-\theta }$ and finally $\left({u}^{n+1},{p}^
{n+1}\right)$ as follows:
$\frac{{u}^{n+\theta }-{u}^{n}}{\theta \Delta t}-\alpha u \Delta {u}^{n+\theta }+abla {p}^{n+\theta }={f}^{n+\theta }+\beta u \Delta {u}^{n}-\left({u}^{n}\cdot abla \right){u}^{n},abla \cdot {u}^{n+\
theta }=0,$
$\frac{{u}^{n+1-\theta }-{u}^{n+\theta }}{\left(1-2\theta \right)\Delta t}-\beta u \Delta {u}^{n+1-\theta }+\left({u}^{n+1-\theta }\cdot abla \right){u}^{n+1-\theta }={f}^{n+\theta }+\alpha u \Delta
{u}^{n+\theta }-abla {p}^{n+\theta }.$
$\frac{{u}^{n+1}-{u}^{n+1-\theta }}{\theta \Delta t}-\alpha u \Delta {u}^{n+1}+abla {p}^{n+1}={f}^{n+1}+\beta u \Delta {u}^{n+1-\theta }-\left({u}^{n+1-\theta }\cdot abla \right){u}^{n+1-\theta
},abla \cdot {u}^{n+1}=0$ .
As we mentioned in Section 1, it would be interesting to establish an analog of Propositions 2.7 and 2.9 for a family of approximated solutions. This should help to detect or discard the occurrence
of singular points just observing the results of appropriate numerical experiments.
E.F.-C. was partially supported by MINECO (Spain), Grant MTM2016-76990-P. I.M.-G. was partially supported by CEI (Junta de Andalucía, Spain), Grant FQM-131, VI PPIT (Universidad de Sevilla, Spain). | {"url":"https://scirp.org/journal/paperinformation?paperid=84243","timestamp":"2024-11-06T17:08:55Z","content_type":"application/xhtml+xml","content_length":"536936","record_id":"<urn:uuid:0ea10733-ebf2-422b-b151-3f65db7e7d12>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00738.warc.gz"} |
Optimization of perforation orientation for achieving uniform proppant distribution between clusters - ResFrac Corporation
Equal proppant distribution between perforation clusters within a stage is one of the goals of hydraulic fracturing design. However, as we know from multiple experiments and field observations, the
proppant distribution is not always uniform. Therefore, the natural question is whether it is possible to use a specific design of perforations to achieve a more uniform proppant distribution. This
blog post attempts to answer this question using a recently developed model for slurry flow in a perforated wellbore. The primary findings are surprising and demonstrate that a variable perforation
phasing transitioning gradually from 180$^\circ$ (bottom of the well) at the heel to approximately 100$^\circ$ at the toe leads to optimal proppant placement. Under some circumstances, such as the
reduced injection rate and/or increased perforation diameter, it is possible to have precisely uniform distribution of proppant for all perforation holes, according to the model. If the same
perforation orientation is required for all clusters, then the optimal orientation is in the range from 110$^\circ$ to 120$^\circ$, i.e. slightly below the middle of the well. The primary reason for
such result lies in the balancing of two opposing phenomena, namely, particle settling in the wellbore whereby there is always higher particle concentration in the lower part of the well, and
proppant inertia that causes some particles to miss the perforation. When the aforementioned phenomena are perfectly balanced, then the amount of proppant that goes into the perforation is equal to
the average concentration in the wellbore, which, once applied for all perforations, leads to uniform proppant intake for every hole. Despite the mechanisms and trends are clearly captured by the
model, the results should also be tested in future laboratory studies for quantitative confirmation. In addition, we are implementing the model into ResFrac to explore potential value of the result
for practical cases.
Problem statement
This blog post is a follow-up to my previous effort and summarizes the findings from the upcoming paper [1]. Previously, a mathematical model for the problem of slurry flow in a perforated wellbore
was described and the underlying physical mechanisms were discussed. The purpose of this blog post, on the other hand, is to couple the model with an optimization algorithm to investigate optimal
perforation orientations that lead to the desired uniform proppant distribution between perforations. A brief description of the model is added at the beginning to cater for readers who are not
familiar with the previous blog post.
Fig. 1 shows schematics for the problem and specifies the two sub-problems that are relevant for the optimization. Fig. 1$(a)$ shows the flow of suspension in a horizontal well and variation of
particle volume fraction inside the well is schematically shown. In the heel or upstream part of the stage the flow velocity is sufficiently high to fully suspend particles. However, as the slurry
slows down towards the heel of the stage, particles settle, which leads to higher particle concentration in the lower part of the well. Fig. 1$(b)$ shows schematics for the second sub-problem related
to proppant turning. Fluid streamlines located within distance $l_f$ from the perforation enter the hole. Due to higher mass density of particles, only particles that are located within a smaller
distance $l_p$ are able to enter the perforation.
Figure 1: Schematics of slurry flow and particle settling in a perforated wellbore (a). Illustration for the proppant
turning problem (b).
As follows from [2], the first sub-problem is characterized primarily by the following two parameters
G = \dfrac{8\phi_m(\rho_p\!-\!\rho_f)g d_w}{f_D\rho_f v_w^2},\qquad t_0 = \dfrac{9\mu_a d_w}{2(\rho_p\!-\!\rho_f)g a^2}.
The parameter $G$ is dimensionless gravity and it quantifies the degree of asymmetry of the particle distribution in the wellbore. At the same time, $t_0$ is the characteristic settling time that
provides the time scale for reaching steady-state particle distribution. In the above equation $\phi_m=0.585$ is the maximum volume fraction of particles, $\rho_p$ is particle mass density, $\rho_f$
is fluid mass density, $g=9.8$ m/s$^2$ is gravitational constant, $d_w$ is wellbore diameter, $f_D=0.04$ is fitting parameter that can also be interpreted as a friction factor in the pipe, $v_w$ is
the average wellbore velocity, $a$ is particle radius, and $\mu_a$ is apparent viscosity. Note that the parameter $G$ is inversely proportional to Shields number that is commonly used to describe
sediment flow. The apparent viscosity accounts for the small scale motion caused by turbulent flow, which in turn triggers non-linear turbulent drag and effectively increases viscosity, see [2] for
more details.
To illustrate the variability of particle volume fraction inside the wellbore for various values of $G$, Fig. 2 plots the solution for $G=\{0.1,1,10,10^2,10^3\}$ for the average normalized volume
fraction $\langle \bar\phi\rangle = \langle \phi\rangle/\phi_m=0.1$. Such concentration corresponds to approximately 1.4 ppg. Panel $(a)$ shows the variation of the normalized particle volume
fraction $\bar\phi=\phi/\phi_m$ along the vertical line passing through the center of the wellbore. Panel $(b)$ shows the variation of the normalized flow velocity $\bar v_x=v_x/v_w$, also along the
vertical line passing through the center of the wellbore. Panel $(c)$ shows the variation of the normalized particle volume fraction (the same as in panel $(a)$) within the wellbore’s cross-section.
The black color corresponds to the maximum allowable concentration, while the while color corresponds to no particles. Results clearly show that the parameter $G$ significantly affects the particle
distribution inside the wellbore. The particles are suspended for small $G$ (or high velocity) and settle at the bottom of the well for large $G$ (or low velocity). The lower left panel also
schematically shows potential perforation orientations that illustrate how the orientation can lead to accessing various particle concentrations, especially for large values of $G$.
Figure 2: Variation of the normalized particle volume fraction ((a) and (c)) and slurry velocity (b) for different values
of the dimensionless gravity $G$.
The proppant turning model can be summarized as follows. First, the value for $l_f$ is calculated by requiring that all the fluid streamlines that are located within $l_f$ enter the perforation, see
Fig. 1 $(b)$. The total flow rate in these streamlines is made equal to the flow rate through the perforation. As the fluid slows down in order to enter the perforation, particles start to slip and
accumulate the total slip $s$ by the time the horizontal fluid velocity is zero, i.e. by the time the particle reaches the perforation. The magnitude of this slip depends on many parameters, but is
driven by the density mismatch between the phases and the drag force that can be either laminar or turbulent. The exact expression for the slip is lengthy and is omitted here, see [2] for more
details. What is important is the dimensionless slip, defined as $\bar s = s/d_p$, i.e. the slip normalized by the perforation diameter. Since $s$ does not depend on the perforation diameter, this
shows that bigger perforations lead to smaller dimensionless slip and better ability of particles to turn into the perforation. Finally, the value of the dimensionless slip is used to find $l_p$ by
requiring that all the streamlines that are located within $l_p$ can have a normalized horizontal slip of up to $\bar s$. The value of $l_p$ is then used to calculate the proppant flow rate through
the perforation.
It is instructive to introduce a single dimensionless parameter that quantifies the degree of particle slippage past perforation. If the particle volume fraction $\phi$ is approximately constant
within the zone outlined by $l_p$, then the turning efficiency can be defined as
In the above equation, the quantity $\langle\phi\rangle_p$ denotes the average volume fraction of slurry that enters the perforation, while $\langle\phi\rangle_{l_p}$ is the average particle volume
fraction in the zone outlined by $l_p$. Therefore, the ratio $\eta={\langle\phi\rangle_p}/{\langle\phi\rangle_{l_p}}$ represents the apparent decrease of the particle volume fraction or, in other
words, turning efficiency. The smaller the dimensionless slip, the higher the turning efficiency.
Optimization results
What is interesting about the two sub-problems is that there are situations in which they can compensate each other. In particular, the turning efficiency is always $\eta<1$, which reduces the amount
of proppant that enters the perforation. At the same time, with the reference to Fig. 2, particle settling can increase the amount of proppant that enters the perforation if the latter is located in
the lower part of the well. This observation can be used to define the “uniformity” lines as
\eta \phi(\theta)=\langle \phi\rangle_w.
In simple words, this equation states that the volume fraction of particles entering the perforation with orientation $\theta$ should be equal to the average volume fraction in the well. In this
case, the volume fraction of particles after the perforation is not going to change, which is the required step towards achieving the uniform proppant distribution. Note the effect of finite $l_p$ is
ignored, i.e. it is assumed that $l_p\ll d_w$.
The parametric space for the problem consists of two dimensionless parameters, the turning efficiency $\eta$ defined in (2) and the dimensionless gravity $G$ defined in (1). As was discussed in
detail in [cite previous blog post], there are three limits that correspond to the guaranteed uniform distribution $U$, the situation when the last cluster takes all proppant $L$, and the situation
when the sensitivity to perforation phasing is the strongest $P$. The uniformity lines are now added and are shown by the grey lines in Fig.3 for different values of $\theta$ and $\langle \bar\phi\
rangle=0.1$ (the result depends on the average volume fraction). The most left line corresponds to the perforation oriented downwards or $\theta=180^\circ$. If the data points are located on the left
from this line, then it becomes impossible to balance the loss due to turning efficiency by the perforation orientation. At the same time, on the right form this line, it is always possible to find
an orientation that would lead to the preservation of volume fraction in the wellbore. This plot clearly demonstrates that perforations located in the lower part of the well can lead to uniform
proppant placement. Also, comparisons in [2] indicate that $\eta$ varies from approximately 0.6 to 0.9, while $G$ from under 1 to up to 100. This allows to conclude right away that the optimal
perforation orientations are from approximately $100^\circ$ to $180^\circ$ with the majority of points likely falling between $110^\circ$ and $120^\circ$.
Variable phasing
To demonstrate the application of optimal phasing, two test cases are considered. The first case is taken from the field scale experiments [3,4], while the second case is taken from the laboratory
scale experiments from [5]. The field scale case is called PTST2 and used 100 mesh proppant. The stage consisted of 13 perforation clusters, each having 3 holes with $120^\circ$ phasing. The
injection rate is 90 bpm, while the perforation diameter is $d_p=0.33$ in. High Viscosity Friction Reducer (HVFR) was used in the carrier fluid, see complete set of input parameters in [3,4,2].
Fig. 3$(a)$ shows the comparison of proppant volume fraction for each cluster. The red lines with circular markers correspond to the results of experiments, while the black solid lines correspond to
model prediction. The amount of proppant for each individual perforation is shown by the square markers. Note that different shades of grey are used to distinguish between odd and even clusters for
visualization purposes. Finally, the asterisk markers indicate the average proppant volume fraction in the wellbore before a perforation. Panel $(b)$ shows the azimuth of every perforation. Note that
orientations with angles exceeding $180^\circ$ are shown as $360^\circ-\theta$ to have a smaller vertical axis. What is interesting is that the amount of proppant received by each perforation varies
drastically towards the end of the stage. This is because the particle settle in that part of the wellbore and, as a result, the perforations located above receive much less proppant than that
located in the lower part of the well. Panel $(c)$ shows $(G,\eta)$ parametric space for the field case and markers indicate the trajectories for three different scenarios. The crosses correspond to
the original case, the triangles correspond to the case with the reduced rate of 70 bpm, while the circular markers indicate the case with 2 perforations per cluster with an increased diameter $d_p$.
Note that the first or heel cluster has smaller $G$, while the toe clusters have larger $G$, therefore the trajectories follow the path from left to right as the observer moves downstream.
Figure 3: Comparison between the field scale PTST2 case [3,4] and the model. Panel $(a)$ shows particle volume fraction (red lines with circular markers – measurement, solid black lines – cluster
average model, square markers – proppant per perforation, asterisk markers – concentration in the wellbore), panel $(b)$ shows perforation orientation, while panel $(c)$ shows the parametric space
with the trajectories for three different cases: original (crosses), reduced rate (triangles), and increased perforation diameter (circles).
The reasoning behind choosing these three cases is the following. The original case is slightly outside of the $180^\circ$ uniformity line. As a result, it will not be possible to achieve completely
uniform proppant distribution. Two practical ways to move the first cluster on the right from the $180^\circ$ uniformity line is to either increase the dimensionless gravity $G$, or to increase the
turning efficiency $\eta$. By looking at the expression for $G$(1), one simple way is to reduce the average flow velocity by reducing the rate. That’s why the first alternative case has the rate of
70 bpm and the corresponding triangles in Fig. 4 are now all on the right from the $180^\circ$ uniformity line. One of the easiest ways to increase the turning efficiency is to increase perforation
diameter. Recall from above that the dimensionless slip is inversely proportional to perforation diameter and thus increasing the diameter reduces the normalized slip and increases the turning
efficiency. Changing the particle size can also help the efficiency, but this case already has 100 mesh proppant and therefore further reduction of particle size can negatively impact hydraulic
conductivity of fractures and may reduce production. Using lightweight proppant can also be useful. But, as can be seen from (1), the value of $G$ actually decreases, even though the turning
efficiency increases. Thus, the movement in the parametric space is in the up-left direction, which approximately follows the $180^\circ$ uniformity line and thus is less efficient. Of course, it is
possible to use very lightweight proppant and end up in the $U$ limit, in which the result is always uniform, but this will probably not be economically profitable.
Fig. 4 shows various optimization results for the field scale case: $(a)$ the original case; $(b)$ the same original case, but with $10^\circ$ uncertainty added to the optimal perforation
orientation; $(c)$ the case with the reduced rate; $(d)$ the case with an increased perforation diameter. The optimization results for the original case demonstrate that the optimal orientation for
the first half of the stage is downwards, while after that the optimal azimuth gradually declines towards approximately $100^\circ$. The early behavior cannot be perfectly uniform since the first
several clusters are located outside of the $180^\circ$ uniformity line, but nevertheless, the overall result is more uniform than the original design. Also, which is probably even more important,
there is no strong variation of the amount of proppant per individual perforation, which ensures that each individual perforation is used more effectively. The addition of randomness is made to
investigate the sensitivity of the result to uncertainties. The proppant distribution is still fairly uniform, but it deteriorates for the last several clusters, for which the value of $G$ is large.
The reduction of rate allows to have a perfectly uniform proppant distribution, according to the model and without uncertainties. The optimal orientation gradually descends from $180^\circ$ to
approximately $100^\circ$. This is of course possible because the whole trajectory in the parametric space shown in Fig. 3 is on the right from the $180^\circ$ uniformity line. The increase of
perforation diameter also significantly improves the optimal result, even though the optimal proppant distribution is not perfectly uniform.
Figure 4: Results of simulations for optimal perforation design for the original field scale case $(a)$, for the field scale case with $10^\circ$ uncertainty added to perforation orientation $(b)$,
for the field scale case with the reduced rate 70 bpm $(c)$, and for the field scale case with the increased perforation diameter 0.45 in $(d)$.
It is also interesting to examine the laboratory scale experiment, available in [5]. This is one of the rare cases, for which the amount of proppant was measured for each individual perforation.
There were 3 perforation clusters with 4 holes per cluster with $90^\circ$ phasing. Tap water with 40/70 mesh proppant are pumped with the rate of 79 gal/min, see the complete list of parameters in
[5]. Fig. 5 shows the results of comparisons with the model. The results for each individual perforation are compared, as well as the results per cluster are also compared. There is an overall good
agreement between the model and the measurement. But what is remarkable, is that there is a strong variability of the amount of proppant received for each individual perforation, which can be
sub-optimal. Fig. 5$(c)$ shows the parametric space for this laboratory scale case and crosses show the trajectory inside the parametric space. Clearly, all the points are on the right from the $180^
\circ$ uniformity line and therefore it is possible to find optimal perforation orientations to have perfectly uniform proppant distribution.
Figure 5: Comparison between the laboratory scale case [5] and the model. Panel (a) shows particle volume fraction (red lines with circular markers – measurement, solid black lines – cluster average
model, square markers – proppant per perforation, asterisk markers – concentration in the wellbore), panel (b) shows perforation orientation, while panel (c) shows the parametric space with the
trajectory for the considered case.
Fig. 6$(a)$ shows the optimal result when each individual perforation is optimized and the distribution is indeed uniform. The optimal angle varies in a narrow range from $113^\circ$ to $122^\circ$,
which is consistent with the location of the points inside the parametric space between $110^\circ$ and $120^\circ$ lines. Recall that the position of these lines shown in Fig. 5 depend on particle
volume fraction and are therefore not perfectly universal. Fig. 6$(b)$ shows the effect of $10^\circ$ uncertainty. It is noticeable, especially for the last cluster.
Figure 6: Results of simulations for optimal perforation design for the original laboratory scale case $(a)$, for the same case with $10^\circ$ uncertainty adde to perforation orientation $(b)$.
Constant phasing
While the optimization of each individual perforation provides the best results in terms of the uniformity of proppant distribution, there can be operational limitations or perhaps other
considerations why this may not be the best solution overall. For instance, fracture initiation pressure can be very different for different perforation orientations, or near wellbore pressure drop
can be very different for various perforation orientations. It is therefore instructive to consider the optimization that is restricted to having the same perforation orientation for all clusters.
Fig. 7 shows such results for the field scale example. The same four cases are considered, original, original with uncertainty, reduced rate, and increased perforation diameter. Results demonstrate
that the optimal perforation orientation is approximately $110^\circ$ for all cases. There is a noticeable variability caused by using the same orientation. The result is much less sensitive to the
uncertainty and becomes only slightly better if the rate is reduced or the perforation diameter is increased. The actual proppant distribution is quantitatively similar to the original result shown
in Fig. 3. The average perforation orientation within the cluster is $100^\circ$ for the $120^\circ$ phasing shown in Fig.3, which is close to the optimal value of approximately $110^\circ$. The
major difference, however, is the distribution of proppant for each individual perforation. In the original case, there is a very strong variation per hole, while the optimal results show a much
milder variation of the amount of proppant received by each perforation.
Figure 7: Results of simulations for optimal perforation design for the field case when all perforations are required to have the same orientation: the original field scale case $(a)$, the case with
$10^\circ$ uncertainty added to perforation orientation $(b)$, the case with the reduced rate 70 bpm $(c)$, and for case with the increased perforation diameter 0.45 in $(d)$.
Fig. 8 considers the laboratory scale example and employs constant phasing optimization. Fig. 8$(a)$ shows the optimal proppant distribution for the case of optimal azimuth is $118^\circ$. The use of
constant phasing for all clusters introduces some variability of the resultant proppant distribution, but it is much more mild compared to the field scale case. Finally, Fig. 8$(b)$ adds $10^\circ$
uncertainty to the latter case, which provides an estimate of the sensitivity of the result to perturbations. These cases as well as the ones shown in Fig. 6 are very close to each other and the
optimal perforation orientation is also almost the same. Based on the parametric space shown in Fig. 5$(c)$, this three-cluster laboratory design has relatively large values of $G$, which closely
corresponds to the last three clusters within the stage. Recall that for the field case the last clusters also have the optimal perforation azimuth between $110^\circ$ and $120^\circ$.
Figure 8: Results of simulations for optimal perforation design for the laboratory scale case when all perforations are required to have the same orientation $(a)$, and for the case when all
perforations are required to have the same orientation and $10^\circ$ uncertainty is added for perforation orientation $(b)$.
This blog post addresses the problem of proppant distribution between perforation clusters for hydraulic fracturing applications. An optimization algorithm is developed based on the recently
developed model for slurry flow in a perforated wellbore. It is shown that it is possible to adjust orientation of each individual perforation to achieve more uniform proppant distribution between
the clusters. Under some conditions, it is even possible to reach a fully uniform distribution, according to the model. In general, the optimal perforation placement is in the lower part of the well
from the bottom at $180^\circ$ (heel clusters) to approximately middle at $100^\circ$ (toe clusters). If the same orientation is required for all perforations, then the range becomes from $110^\circ$
to $120^\circ$, at least for the examples considered. Optimization for the laboratory scale experiment shows a similar trend that the optimal perforation azimuth is between $110^\circ$ and $120^\
While the obtained result seems almost universal, care must be taken before applying it. First of all, the field scale cases can have quite different parameters, which can shift the optimal azimuths.
The second reason is much more significant. The model does not consider the effect of perforation erosion and stress shadow from the previous stage. These effects can significantly change the slurry
distribution between the clusters and thus affect the resultant proppant distribution. Nevertheless, the developed model significantly enhances the understanding of the processes occurring in the
wellbore and can be used as a building block towards solving the problem with erosion and fractures.
Note that all the perforation angles discussed here are calculated relative to the wellbore center, rather than the center of the perforation gun (located below the wellbore center). The conversion
is not difficult and requires the ratio between the perforation gun diameter and the wellbore diameter.
[1] E.V. Dontsov, C.W. Hewson, and M.W. McClure. A model for optimizing proppant distribution between perforations. In American Rock Mechanics Association, Atlanta, GA, 2023.
[2] E. Dontsov. A model for proppant dynamics in a perforated wellbore. arXiv:2301.10855, 2023.
[3] P. Snider, S. Baumgartner, M. Mayerhofer, and M. Woltz. Execution and learnings from the first two surface tests replicating unconventional fracturing and proppant transport. In Proceedings of
Hydraulic Fracturing Technology Resources Conference, 1-3 February 2022, Houston, Texas, USA, SPE-209141-MS, 2022.
[4] J. Kolle, A. Mueller, S. Baumgartner, and D. Cuthill. Modeling proppant transport in casing and perforations based on proppant transport surface tests. In Proceedings of Hydraulic Fracturing
Technology Resources Conference, 1-3 February 2022, Houston, Texas, USA, SPE-209178-MS, 2022.
[5] X. Liu, J. Wang, A. Singh, M. Rijken, D. Wehunt, L. Chrusch, F. Ahmad, and J. Miskimins. Achieving near-uniform fluid and proppant placement in multistage fractured horizontal wells: A
computational fluid dynamics modeling approach. SPE Production & Operations, 36:926–945, 2021. | {"url":"https://www.resfrac.com/blog/optimization-of-perforation-orientation-for-achieving-uniform-proppant-distribution-between-clusters","timestamp":"2024-11-07T14:15:48Z","content_type":"text/html","content_length":"250625","record_id":"<urn:uuid:8b08b078-0ebf-452c-8ee8-41a8fa6e5fa5>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00647.warc.gz"} |
What is the main purpose of the elbow method in K-means clustering?
To determine the optimal number of clusters
To speed up clustering
To visualize clusters
To perform feature selection
What is the difference between bias and fairness in machine learning?
Bias refers to systematic errors in a model, fairness refers to treating all individuals equally
Bias is used for classification, fairness is used for regression
Bias is faster than fairness
Bias requires more data than fairness
What is the main challenge posed by the curse of dimensionality?
Lack of data
Computational complexity in high-dimensional spaces
Difficulty in data collection
Ease of visualization
What does GAN stand for in deep learning?
General Adversarial Network
Generative Adversarial Network
Grouped Analytical Nodes
Guided Attention Normalization
What is the curse of dimensionality?
Having too few dimensions
Problems that arise with high-dimensional data
The tendency of all data to be high-dimensional
The difficulty of visualizing 3D data
What is the primary goal of time series analysis in machine learning?
To classify data
To predict future values based on past observations
To cluster data points
To reduce dimensionality
What distinguishes time series analysis from cross-sectional data analysis?
Time series data is always categorical
Time series data has a temporal order
Time series data cannot have missing values
Time series data is always normally distributed
What does SVM stand for in machine learning?
Support Vector Machine
Supervised Vector Method
Semi-supervised Vision Model
Structured Variable Method
Which is NOT a common feature selection method?
Filter methods
Wrapper methods
Embedded methods
Random methods
Define supervised/unsupervised learning
Supervised: uses labeled data
Unsupervised: uses unlabeled data
Supervised: predicts output
Unsupervised: finds patterns
Score: 0/10
What is the primary goal of autoencoders?
To automatically encode data
To learn efficient data representations
To increase data dimensions
To encrypt encoded data
What is the main advantage of using multiple agents in LangChain?
To increase processing speed
To handle complex tasks requiring different expertise
To reduce model size
To encrypt communication between models
What is the main difference between Type I and Type II errors?
Type I rejects true null hypothesis
Type II fails to reject false null hypothesis
Type I is always more serious
Type II only occurs in large samples
What is the purpose of cross-validation?
To increase model complexity
To assess model performance on unseen data
To speed up model training
To create more training data
What is the main challenge in few-shot learning?
Large amount of labeled data required
Learning from very few examples
Overfitting on training data
Slow inference time
Which of these is not a common activation function in deep learning?
What is the main goal of hierarchical clustering?
To create flat clusters
To build a hierarchy of clusters
To reduce cluster size
To generate new clusters
In supervised machine learning, what's the difference between training data and test data?
Training data is used to build the model, test data is used to evaluate it
Both are used interchangeably
Training data is labeled, test data is not
Test data improves the model
What is the primary advantage of using deep learning for chatbots?
They never make mistakes
They can understand and generate more natural language
They don't require any training data
They are always faster than rule-based chatbots
What is the process of identifying and removing errors and inconsistencies from a dataset?
Data cleaning
Data splitting
Data transformation
Data aggregation
Score: 0/10
What is the main purpose of data marts?
To shop for data
To provide a subset of data warehouse
To increase data complexity
To encrypt departmental data
What is the diff. between model-based/model-free planner?
Model-based: uses environment model
Model-free: directly acts
Model-based: plans
Model-free: reacts
What is a key advantage of the Isolation Forest algorithm for anomaly detection?
It requires labeled anomalies
It works well with high-dimensional data
It's computationally expensive
It only works with numerical data
Which type of problem is logistic regression typically used for?
Dimensionality reduction
Define Jaccard similarity
Similarity measure for sets
Calculates intersection over union
Range from 0 to 1
All of the above
What is the primary goal of text summarization?
To count words
To generate short summaries
To classify documents
To increase text length
What is the primary goal of decision trees?
To predict continuous outcomes
To classify data points
To group similar data points
To estimate probabilities
Which of these is not a type of big data processing framework?
Apache Hadoop
Apache Spark
Apache Flink
Apache Quantum
What does DBSCAN stand for?
Density-Based Spatial Clustering of Applications with Noise
Distance-Based Scanning Clusters and Nodes
Distributed Bayesian Sampling Clusters and Networks
Data-Based Systematic Clustering Analysis
What is the difference between object detection and image segmentation?
Object detection locates objects
Image segmentation labels pixels
Object detection used for localization
Image segmentation used for analysis
Score: 0/10
What is the purpose of long short-term memory (LSTM) networks?
To reduce memory usage
To handle long-term dependencies in sequential data
To perform clustering
To reduce dimensionality
Which NLP technique does Word2Vec primarily use?
Recurrent Neural Networks
Convolutional Neural Networks
Shallow Neural Networks
Transformer architecture
What is the diff. between L1 and L2 reg.?
L1: adds penalty, creates sparse model
L2: adds penalty, shrinks coefficients
Both prevent overfitting
L1: non-differentiable
In statistics, the p-value represents:
Probability of the null hypothesis being true
Probability of observing data as extreme as the sample, assuming null hypothesis is true
Percentage of data points in the sample
Power of the statistical test
What is a key characteristic of Reinforcement Learning compared to supervised learning?
It always requires less data
It learns from interaction with an environment
It always produces deterministic results
It only works for classification tasks
What is the primary purpose of feature scaling in machine learning?
To increase feature count
To normalize feature ranges
To create new features
To remove irrelevant features
What is the main advantage of deep learning over traditional machine learning?
Reduced training time
Ability to learn hierarchical features
Simplified model interpretation
Reduced need for data
What is the main difference between Pearson and Spearman correlation?
Sample size requirement
Assumption about relationship linearity
Type of data used
Significance level
What does GLOVE stand for in word embeddings?
Global Vectors
Global Vectors for Word Representation
Generalized Linear Optional Vector Embedding
Grouped Lexical Order Validation Engine
What does NLP stand for in artificial intelligence?
Numerical Linear Programming
Natural Language Processing
Network Layer Protocol
Normalized Learning Procedure
Score: 0/10 | {"url":"https://coolgenerativeai.com/data-science-test-online-2/","timestamp":"2024-11-12T09:46:11Z","content_type":"text/html","content_length":"178973","record_id":"<urn:uuid:3005f2a1-174b-442e-9263-594fc359264a>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00701.warc.gz"} |
Probability of No Rain
From: Brooklyn, NY
Registered: 2024-09-07
Posts: 52
Probability of No Rain
There is a 30% chance of rain tomorrow. What is the probability of no rain tomorrow?
P(no rain tomorrow) = 1 - P(rain tomorrow).
This yields the answer of something different happening next time.
P(something not happening tomorrow or next time) = 1 - P(something happening tomorrow or next).
Pretty cool, right?
Focus on the journey for there will be those who would love to see you fall.
Registered: 2010-06-20
Posts: 10,610
Re: Probability of No Rain
Yes. It follows from the following
If one of n events must occur then
P(event 1) + P(event 2) + ... + P(event n) = 1
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
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A closure scheme for chemical master equations
Probability reigns in biology, with random molecular events dictating the fate of individual organisms, and propelling populations of species through evolution. In principle, the master probability
equation provides the most complete model of probabilistic behavior in biomolecular networks. In practice, master equations describing complex reaction networks have remained unsolved for over 70
years. This practical challenge is a reason why master equations, for all their potential, have not inspired biological discovery. Herein, we present a closure scheme that solves the master
probability equation of networks of chemical or biochemical reactions. We cast the master equation in terms of ordinary differential equations that describe the time evolution of probability
distribution moments. We postulate that a finite number of moments capture all of the necessary information, and compute the probability distribution and higherorder moments by maximizing the
information entropy of the system. An accurate order closure is selected, and the dynamic evolution of molecular populations is simulated. Comparison with kineticMonte Carlo simulations, which merely
sample the probability distribution, demonstrates this closure scheme is accurate for several small reaction networks. The importance of this result notwithstanding, a most striking finding is that
the steady state of stochastic reaction networks can now be readily computed in a single-step calculation, without the need to simulate the evolution of the probability distribution in time.
• Entropy maximization
• Information theory
• Statistical mechanics
• Stochastic models
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EViews Help: Granger Causality
Granger Causality
Correlation does not necessarily imply causation in any meaningful sense of that word. The econometric graveyard is full of magnificent correlations, which are simply spurious or meaningless.
Interesting examples include a positive correlation between teachers’ salaries and the consumption of alcohol and a superb positive correlation between the death rate in the UK and the proportion of
marriages solemnized in the Church of England. Economists debate correlations which are less obviously meaningless.
The Granger (1969) approach to the question of whether
It is important to note that the statement “
When you select the view, you will first see a dialog box asking for the number of lags to use in the test regressions. In general, it is better to use more rather than fewer lags, since the theory
is couched in terms of the relevance of all past information. You should pick a lag length,
EViews runs bivariate regressions of the form:
for all possible pairs of F-statistics are the Wald statistics for the joint hypothesis:
for each equation. The null hypothesis is that not Granger-cause not Granger-cause
We illustrate using data on consumption and GDP using the data in the workfile “Chow_var.WF1”. The test results are given by:
For this example, we cannot reject the hypothesis that GDP does not Granger cause CS but we do reject the hypothesis that CS does not Granger cause GDP. Therefore it appears that Granger causality
runs one-way from CS to GDP and not the other way.
If you want to run Granger causality tests with other exogenous variables (e.g. seasonal dummy variables or linear trends) or if you want to carry out likelihood ratio (LR) tests, run the test
regressions directly using equation objects.
Panel causality tests are described in
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Ten slips of paper labeled from 1 to 10 are placed in a hat. The first slip of paper is not replaced before selecting the second slip of paper. What is the probability of selecting an odd number followed by an even number? A. 1/4 B. 1/5 C. 5/18 D.
Ten slips of paper labeled from 1 to 10 are placed in a hat. The first slip of paper is not replaced before selecting the second slip of paper.
What is the probability of selecting an odd number followed by an even number?
A. 1/4
B. 1/5
C. 5/18
D. 2/9
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What is the Difference Between Compressible and Incompressible Flow - Pediaa.Com
What is the Difference Between Compressible and Incompressible Flow
The main difference between compressible and incompressible flow is their change in density. Compressible flow occurs when the density of the fluid changes significantly as it moves through a system,
and changes in pressure and temperature usually accompany this change in density. In contrast, incompressible flow is characterized by the fact that the density of the fluid remains approximately
constant as it flows. It does not change significantly in response to changes in pressure or temperature.
Understanding fluid dynamics is vital in various fields, from engineering to meteorology. Compressible and incompressible flow are fundamental concepts in fluid dynamics.
Key Areas Covered
1. What is Compressible Flow
– Definition, Features, Applications
2. What is Incompressible Flow
– Definition, Features, Applications
3. Similarities Between Compressible and Incompressible Flow
– Outline of Common Features
4. Difference Between Compressible and Incompressible Flow
– Comparison of Key Differences
5. FAQ: Compressible and Incompressible Flow
– Frequently Asked Questions
Key Terms
Compressible Flow, Incompressible Flow
What is Compressible Flow
Compressible flow occurs when the density of the fluid changes significantly as it moves through a system. Changes in pressure and temperature usually accompany this change in density. Unlike
incompressible flow, where density remains nearly constant, compressible flow sees substantial changes in gas density. Compressible flow is often associated with high speeds. When a gas is forced to
move at high velocities, it can experience compressibility effects. These effects become more pronounced as the gas approaches or exceeds the speed of sound.
The ideal gas law governs compressible flow (P = ρRT), where P is pressure, ρ is density, R is the specific gas constant, and T is temperature. This equation relates pressure, density, and
temperature, and it is a fundamental component of compressible flow analysis. The Mach number (Ma) is a dimensionless parameter used to characterize compressible flow. It is defined as the ratio of
the flow velocity to the local speed of sound (Ma = V / a, where V is velocity and a is the speed of sound). A low Mach number indicates that the flow is nearly incompressible, while a high Mach
number indicates compressible flow.
What are the Applications of Compressible Flow
Compressible flow has numerous real-world applications in various fields of engineering and science. Compressible flow analysis is essential in aerospace engineering, particularly in the design and
analysis of aircraft and spacecraft. Understanding the behavior of air at high speeds and altitudes is crucial for aerodynamic and propulsion systems. Compressible flow is prevalent in
turbomachinery, including gas turbines, jet engines, and compressors. Designing efficient and reliable machinery relies on a deep understanding of how gases behave in compressible flow conditions.
These components are useful in various applications, from rocket engines to industrial processes.
Compressible flow analysis is vital for optimizing the performance of nozzles and diffusers. In internal combustion engines, the behavior of the fuel-air mixture in the combustion chamber is highly
compressible. Analyzing compressible flow is critical for optimizing combustion efficiency and emissions. Compressible flow analysis is also essential in chemical and process engineering, where gases
undergo significant pressure and temperature changes in reactors and pipelines.
What is Incompressible Flow
Incompressible flow is characterized by the fact that the density of the fluid remains approximately constant as it flows, and it does not change significantly in response to changes in pressure or
temperature. This means that the volume of the fluid elements remains nearly unchanged. Incompressible flow is typically associated with low fluid velocities. At these low speeds, the effects of
compressibility on density and pressure are negligible. In practice, this corresponds to Mach numbers (Ma) much less than 1 (Ma << 1).
Incompressible flow is often approximated as “ideal” or “perfect” flow, where there are no energy losses due to friction or heat transfer. This simplifies the analysis of fluid dynamics problems,
making it a fundamental concept in many engineering applications. In incompressible flow, the principle of mass conservation, expressed by the continuity equation, is particularly straightforward. It
states that the mass flow rate into a control volume must equal the mass flow rate out of that volume.
What are the Applications of Incompressible Flow
In civil and environmental engineering, incompressible flow principles help to analyze water distribution systems, stormwater management, and the behavior of fluids in pipelines and channels. The
design of ships, submarines, and offshore structures relies on the principles of incompressible flow to understand the behavior of water at low speeds. In the design of heat exchangers, which are
used in HVAC systems, power plants, and refrigeration systems, incompressible flow analysis is essential to optimize heat transfer efficiency.
Incompressible flow principles also help to analyze and design systems for the transport of liquids and low-speed gases in chemical and process engineering applications.
Similarities Between Compressible and Incompressible Flow
• Compressible flow and incompressible flow obey the principle of mass conservation.
• Both flows obey the principle of energy conservation.
Difference Between Compressible and Incompressible Flow
Compressible flow occurs when the density of the fluid changes significantly as it moves through a system, and changes in pressure and temperature usually accompany this change in density.
Incompressible flow is characterized by the fact that the density of the fluid remains approximately constant as it flows, and it does not change significantly in response to changes in pressure or
In compressible flow, density changes with variations in pressure and temperature, whereas in incompressible flow, density remains nearly constant.
Compressible flow is often associated with high speeds, whereas incompressible flow is typically observed at low speeds.
FAQ: Compressible and Incompressible Flow
What is the difference between compressible and incompressible materials?
In compressible materials, the volume changes. If a material is incompressible, the material will only be pushed aside to establish a volume-preserving state.
Is water compressible or incompressible?
Water is considered incompressible because it resists changes in volume when subjected to pressure. This is due to the strong hydrogen bonds between water molecules, which hold them in a relatively
fixed arrangement. When pressure is applied, these bonds prevent the water molecules from coming closer together and significantly reducing the volume.
What are examples of incompressible flow?
Some examples of incompressible flow include water in pipes, blood in blood vessels, airflow at low speeds, hydraulic systems, ship and submarine hydrodynamics, and sewer systems.
The main difference between compressible and incompressible flow is that incompressible flow assumes a constant density, while compressible flow acknowledges changes in density due to variations in
pressure and temperature.
1. “Fluid.” Byju’s.
2. “Equations of Compressible and Incompressible Flow in Fluid Dynamics.” Cadence – Resources.
Image Courtesy:
1. “Compressible Airflow Airplane” By SimScale GmbH – SimScale cloud-based CAE platform. (CC BY-SA 4.0) via Commons Wikimedia
2. “Incompressible Flow in a Venturi Injector” By SimScale GmbH – SimScale cloud-based CAE platform. (CC BY-SA 4.0) via Commons Wikimedia | {"url":"https://pediaa.com/what-is-the-difference-between-compressible-and-incompressible-flow/?noamp=mobile","timestamp":"2024-11-09T19:02:19Z","content_type":"text/html","content_length":"81566","record_id":"<urn:uuid:892812a1-7d83-4021-bb53-f7def0463da4>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00140.warc.gz"} |
Ten categories of statistical errors: a guide for research in endocrinology and metabolism | American Journal of Physiology-Endocrinology and Metabolism
Ten categories of statistical errors: a guide for research in endocrinology and metabolism
A simple framework is introduced that defines ten categories of statistical errors on the basis of type of error, bias or imprecision, and source: sampling, measurement, estimation, hypothesis
testing, and reporting. Each of these ten categories is illustrated with examples pertinent to research and publication in the disciplines of endocrinology and metabolism. Some suggested remedies are
discussed, where appropriate. A review of recent issues of American Journal of Physiology: Endocrinology and Metabolism and of Endocrinology finds that very small sample sizes may be the most
prevalent cause of statistical error in this literature.
statistics offers a widening range of powerful tools to help medical researchers attain full and accurate understanding of biological structure within their data. As research methodologies continue
to advance in sophistication, these data are becoming increasingly rich and complex, thereby requiring increasingly thoughtful analysis. In response to these developments, funding sources, regulatory
agencies, and journal editors are tightening their scrutiny of studies' statistical designs, analyses, and reporting. Compared with the past, statistical errors of any magnitude carry greater weight
today in the competition among scientists for research grants and in the publication of research findings.
This essay presents a framework to help researchers and reviewers identify ten categories of statistical errors. The framework has two axes. The first axis recognizes two canonical types of
statistical error: bias and imprecision. The second axis distinguishes five fundamental sources of statistical error: sampling, measurement, estimation, hypothesis testing, and reporting.
Bias is error of consistent tendency in direction. For example, an assay that consistently tends to underestimate concentrations of a metabolite is a biased assay. In contrast, imprecision is
nondirectional noise. An imprecise assay may give true readings on average, but those readings vary in value among repeats.
These two types of error arise from at least five sources. Sampling error is error introduced by the process of selecting units (e.g., human subjects, mice, and so forth) from a population of
interest. Once a unit has been drawn into the sample, error can arise in measurements made on that unit. The researcher often wishes to use these measured data to estimate parameters (e.g., mean,
median) or test hypotheses pertaining to the population from which units were drawn. In doing so, statistical methods should be chosen to minimize statistical error in these estimates and tests.
Finally, with the study completed, results need to be reported in a manner that ensures that readers are as fully and accurately informed as possible. These five sources are conceptually
distinguishable but interrelated in practice. As will be illustrated, discussion of one source often has implications for others.
The objective of this essay is to illustrate each of these ten categories of statistical error and also provide some constructive suggestions as to how these errors can be minimized or corrected.
This essay is not intended to provide detailed training in statistical methods; rather, its purpose is to alert researchers to potential statistical pitfalls. Details on statistical methods that are
pertinent to a researcher's particular application can be found in texts (some of which are cited in this essay) or from consultation with a biostatistician. In this essay, formal mathematical
presentations have been kept to a minimum. Illustrative examples are drawn from review of a variety of original-research articles published in this journal, American Journal of
Physiology-Endocrinology and Metabolism, and in Endocrinology. Specific articles in the literature are not used as examples, because the purpose of this essay is not to critique particular authors
but instead to identify where improvement may be possible in this literature as a whole.
Sampling begins by identifying a specific population of interest for study. Key to minimizing sampling bias is specificity, because declaring interest in a broad study population (e.g., humans with
type 2 diabetes) essentially guarantees sampling bias. To illustrate, if one truly wished to characterize average peripheral-blood concentrations of some metabolite in humans with type 2 diabetes,
then all such humans would need to be available for sampling and each assigned a known probability of being selected for study. Not only is this impractical, because such a list could never be
compiled, but it is also unethical, because human subjects must be allowed to volunteer for potential entry into research rather than being randomly selected. In fact, because voluntary participation
is the foundation of ethical research in human subjects, research on human subjects invariably contains self-selection bias. Although self-selection bias may not be eliminated, it can be minimized.
During study planning, the researcher simply declares interest in subjects of that type who are likely to volunteer and meet entry criteria; and, during study implementation, the researcher keeps
dropout (and noncompliance) to a minimum. In contrast to research in human subjects, with animal models samples are often drawn from bred populations, so that care must be taken, and apparently often
is in the endocrinology and metabolism (E and M) literature, to designate during study planning the specific breeding line from which a sampling will be obtained.
Sampling need not be from one population. A researcher may wish to study several populations simultaneously. This requires planning to avoid errors in parameter estimation and hypothesis testing. In
particular, if more than one population is of interest, sampling should be stratified by design (13). Here we consider two potentially problematic types of mixed-population sampling that are found in
the E and M literature: sampling from extremes and changes in protocol.
The E and M literature contains examples of deliberate sampling from the extremes. For example, a study in mice may compare the 10% lightest and the 10% heaviest in a batch. The limitation of this
form of stratified sampling is that results obtained and conclusions drawn apply only to those extremes. Any interpolation may be biased if the extremes behave differently from intermediate levels,
which can arise if, say, lightest mice possess medical complications that are uncharacteristic over the remainder of the weight range. A better basic design would be to sample at equal intervals over
the entire weight range. Atop this, one may concentrate sampling around any suspected within-range change points (e.g., a weight at which the slope on weight changes sharply or, say, average response
shifts abruptly up or down).
Mixed-population samples may also result from in-course changes in study protocol. For example, an injection schedule may be changed between subjects enrolled early during a study and those enrolled
later. Post hoc statistical analysis (e.g., regression analysis using covariates) may be attempted to disentangle the effects of the protocol change from any treatment differences, time trends, and
the like that were of primary interest in the original design of the study. However, no guarantee can be made that statistical “correction” for in-course changes in protocol will be fully effective,
especially when in-course protocol changes are strongly confounded with any effects of interest. The best strategy is to avoid in-course protocol changes altogether. Any adjustments in protocol
should be made before the conduct of the full study, perhaps on the basis of results of preliminary studies.
Sampling bias can also arise in experimental studies in the form of an “inadequate control.” This is an example of bias due to sampling from the wrong population. An ideal control possesses all
characteristics of the treatment condition that impact response except for the putative mechanism under study. Such ideals can be very difficult to obtain in practice (e.g., with assays in the
presence of cross-reactivity or in attempts to find “matching” controls in nonrandomized studies). One important safeguard against an inadequate control, which is not always followed in the E and M
literature, is to observe treatment and control samples contemporaneously, not in sequence. For example, in a two-arm randomized trial, both arms should be running between the same start and end
dates, rather than running one arm first and the other arm later. Contemporaneous implementation can control for a number of time-varying factors that may impact the outcome of interest, such as
turnover in lab technicians and drift in instrument calibration.
Inadequate randomizations are a form of sampling bias, because the goal of randomization is to generate two (or more) samples at baseline that are, on average, as homogenous as possible on all
factors that may influence outcome. Simple randomizations alone do not guarantee this homogenization, especially when sample sizes are small, as is common in the E and M literature (see Category II,
for example). The best corrective action is avoidance through use of more sophisticated randomization methods, such as stratified or adaptive designs (14). These include “permuted block” designs, in
which the order of treatment assignments is randomly permuted within small to moderately sized blocks of subjects of shared baseline characteristics. A much less desirable correction, because it is
remedial and not preventive, is to “statistically adjust” for baseline differences during parameter estimation by using, for example, appropriate multiple-regression methods. The formulation of such
regression models should be specified in advance of any data review to avoid “data snooping” (Category VII). Despite their ease of use, these mathematical adjustments to reduce estimation bias can
never fully substitute for starting all arms of an experiment at approximately equivalent compositions.
Sampling error of consistent tendency in direction not only can lead to biased estimates of “location” parameters (e.g., means) but also can result in bias in parameters of any type, including
parameters of “dispersion” such as the variance. Sampling bias in dispersion can mislead understanding of how heterogeneous a population truly is. Sampling bias in the variance from a pilot study can
yield sample-size estimates that are too large or small for the confirmatory study. Overestimates of variances result in unnecessary losses in power, whereas underestimates increase chances that one
will falsely conclude that a treatment is actually effective, that a mean response truly increases over time, and the like. A fundamental tenet to minimizing bias in variance estimates is to sample
across those particular units to which conclusions are to apply. For instance, suppose one wished to test for the response of patients to a possible medical therapy that acts on pancreatic cells. One
could conduct this test on multiple pancreatic cells drawn from a single donor, but the resulting variance estimate would be an estimate of within-subject variance (wrong parameter) from a single
subject (potentially idiosyncratic), so that one could not use these results to draw conclusions of general use to therapists. Instead, one would need to draw a sampling of cells from each of several
donors, calculate mean cell response for each donor, and then estimate the among-subject variance from these subjects' means.
Compared with other medical and scientific disciplines, many E and M research studies tend to use very small sample sizes, with sample sizes <10 being commonplace. Doubtless practitioners use small
samples because observations are difficult and/or expensive to obtain. Some may also argue that physiological characteristics are less variable than, say, clinical or epidemiological characteristics
and that, therefore, smaller sample sizes are permissible in most laboratory E and M studies. Although these arguments carry some weight, they cannot justify the extremely small sample sizes that are
so common in this literature. Just how extremely small these samples are can be illustrated as follows.
Suppose we randomly draw a sample of size n from a population and calculate the sample arithmetic mean. Then we repeat this process many times. Our estimate of the sample mean will vary to some
degree among our different samples. Traditionally, one way to characterize this sampling imprecision in the mean is to estimate the standard deviation (i.e., standard error) of the samples' means.
Fortunately, one can estimate the standard error from a single sample as , where s is the sample standard deviation. Notice how the standard error depends inversely on sample size n, which makes
sense intuitively, as one would expect precision of the sample mean to increase as sample size increases. However, this relationship is not linear, in that the standard error is proportional to the
inverse of the square root of sample size. As illustrated in Fig. 1, sample sizes <10 tend to have large relative imprecision () and thus yield unquestionably imprecise estimates of the mean.
The degree of sampling imprecision in the E and M literature could see major reductions if samples of size 15-30 were more widely employed or required. For example, Fig. 1 shows that a sample size of
5 has relative imprecision of ∼0.45, whereas a sample size of 20 has relative imprecision of ∼0.22, which is a reduction in imprecision of >50%.
Imprecision can also result if sampling is from more than one source population but not stratified by design (13). Mixed-population sampling can arise during recruitment in a hemodynamic study in
which, for instance, some subjects present with arterial hypertension and others do not. Because hypertension may affect hemodynamic response, an efficient design would deliberately stratify sampling
on hypertension status. If stratified sampling were not designed into the study, some type of poststratification could be performed after data collection; however, this remedial approach risks large
disparities in sample size among strata, so that some strata may have small sample sizes, which reduces precision and statistical power. Instead, in advance of implementation of a study, the
researcher could identify any factors that may have a significant impact on outcome, form a parsimonious quantity of strata from these factors, and then sample from each stratum. If the researcher
wishes to test statistically for differences among these strata, then sample size could be controlled by study administrators to be nearly equal across all strata.
In contrast to sampling errors, E and M research is clearly devoted to minimizing errors of measurement, as evidenced by the large proportion of published methods sections that address issues of
measurement (e.g., specimen handling and storage, preparation of solutions and cells, measurement of fluxes, and the like). Seemingly, enough detail is typically given to permit readers to reproduce
all or most all of the reported measurement methods.
Nevertheless, measurement bias can creep into E and M research data in subtle ways, often through some alteration that takes place over the course of the study. Changes in reagent manufacturers and
turnover in device operators are just a couple of possible examples. All such alterations should be identified and tested to see if they introduce bias into measured outcomes. For instance, a switch
to a new source of reagent should never be sequential; rather, if a switch is identified in advance, a series of test assays should be run simultaneously on both reagents, and results compared. Key
to testing an alteration is to recognize that one's intent is to prove that the two conditions (e.g., reagents, operators) yield equivalent, not different, results. As such, instead of standard
two-sample testing (e.g., with Student's t-test), bioequivalency testing should be performed in consultation with a statistician. In these discussions, the researcher should come prepared to provide
the statistician with quantitative upper and lower bounds that define the range of biological equivalence (e.g., ±1% difference between means of old and new reagents).
Measurement imprecision is straightforward to characterize through use of “technical repeats.” Technical repeats result from taking multiples of the same measurement (e.g., an assay) on the same
specimen at the same time. Technical error, sometimes termed “intra-assay variation,” is not commonly reported in the E and M literature. Whether technical repeats are rarely performed or this simply
represents a failure to report technical error is unclear. The coefficient of variation (CV) provides a unitless measure of technical error. Ideally, one hopes to see technical-error CVs of <5-10% at
most. Typically, the CV is estimated by the ratio of the sample standard deviation to the sample mean. This estimator is biased, and in small samples this bias is large enough that a corrected
formulation should be employed (12).
Estimation bias is error of consistent tendency in direction for a sample-based estimate of the value of a parameter, where a parameter is a characteristic of the population of interest (e.g., mean
or variance). Estimation bias is distinct from sampling bias and measurement bias in that we are not concerned with bias arising from the collection of data per se. Rather our focus is limited to
bias that arises in the computational process by which we formulate an estimate of a parameter from data that have already been collected. Formally, any such formulation is termed an “estimator” of a
This is not to say that parameter estimation and sampling bias are wholly separable. A clear example of their overlap is with missing data. Missing data do occur in the E and M literature, perhaps
more often than can be detected because of spotty reporting of sample sizes in figures and tables. Possible signs of missing data are when an article reports that all “available” data were analyzed
or when sample sizes differ between methods and results. In addition, the cause of missing data is often not reported. This cause should not only be given but detailed, as details are all-important
to understanding the scale and character of any effect of missing data on estimation bias. Roughly speaking, missing data consist of two types: those which are missing because of the value they would
take if observed (“informative”), and those that are missing for reasons unrelated to the value that they would take if observed (“noninformative”) (7). A specimen that is accidentally dropped is an
example of noninformative missing data. In contrast, informative missing data arise when specimens of a particular type of response (e.g., high readings) are more likely to be lost. Informative
missing data have the potential for making unbiased estimation difficult to impossible, particularly when the precise reason for why data are missing is unknown or unclear. Unbiased estimation in the
presence of missing data is most complex when data consist of repeated measurements on each subject and some subjects are missing some of their repeated measurements. Of course, the quantity of
missing data should be minimized whenever possible, even if this requires preparation of more reagent, buffer, collection tubes, and the like. A complete analysis data set includes information on the
precise cause for each missing datum. For example, separate codes could be used for measurements not obtained due to 1) specimen lost, 2) reading above instrument range, 3) reading below instrument
range, and so forth. Suppose a research project generated data of which one-fourth measured above instrument range, one-fourth measured below instrument range, and the remainder were nonmissing.
These data could be salvaged by transformation to an ordinal scale (below range, in-range, above range) and analyzed, albeit with a potential loss of power compared with an analysis on the original
interval or ratio scale with all data nonmissing.
Another situation in which estimation bias can arise is when estimation fails to recognize the presence of an “incomplete design.” Suppose the effect of compound Y on intracellular pH has been
previously studied, and primary interest is determining how compound X affects intracellular pH. An experiment is designed in which cell specimens are randomized to being cultured in the absence of X
and Y, with Y alone, and with X and Y together (X + Y). This design does not permit unbiased estimation of the effects of X alone, because the difference in performance between Y alone and X + Y
provides an estimate of the added effect of X in the presence of Y. A “complete” design would also include randomization to X alone, so that this response could be compared with the response obtained
in the absence of X and Y, allowing a separate effect of X to be estimated.
A special category of designs known as “crossover designs” appears on occasion in the E and M literature but may not be identified as such, so that estimation of means and variances may be biased. In
the simplest of these designs, all treatment conditions are applied sequentially, and in some random order, to each subject. For example, one-half of the subjects may be randomly assigned to receive
an infusion of compound A first and compound B second, with the remainder randomly assigned to receive the two infusions in the order of B and then A. Crossover designs are rife with opportunities to
introduce estimation bias into estimates of treatment effects and experimental error. The possibilities of carryover effects from one treatment application to the next within a subject and for change
in outcome over time within subjects, regardless of treatment ordering, are among the factors that should be considered in estimation. For more information see Ref. 8.
Like estimation bias, estimation imprecision is error in the estimation of the value of a parameter; however, unlike estimation bias, estimation imprecision lacks consistent tendency in direction and
is therefore sometimes referred to as estimation “noise,” “instability,” or “unreliability.” Of course, small sample sizes are a common reason for imprecise estimates in the E and M literature
(discussed in categories I and II). However, one can treat that source of imprecision as distinct from estimation imprecision, because it is introduced during the sampling process. Instead, in this
section, we consider only imprecision that is introduced during estimation.
Estimation imprecision can potentially be reduced through careful choice of estimation methods. Statisticians refer to such reductions as improvements in estimation “efficiency,” because greater
precision is obtained from a data set of a given sample size. Common situations in which reductions arise are when only a portion of available data is analyzed (e.g., the last 10 min of recordings),
or some form of data reduction is performed before estimation. To illustrate the latter, suppose one is investigating the impacts of body weight on concentrations of a circulating hormone. Ten
subjects are sampled within each of eight weight categories. Responses for subjects within each category are averaged, and then average response is regressed on average weight, so that the regression
line is estimated with eight pairs of means. This type of averaging often can reduce estimation efficiency seriously, because the quantity of data used for regression analysis is smaller, sometimes
much smaller (tenfold in our example) than the true total sample size, and an unnecessarily large standard error for estimates of regression coefficients results. In some instances, however, the loss
in effective sample size can be compensated for by the variance reduction that comes with averaging, especially if variances in response are much larger within than among averaged groups. Because
efficiency is an advanced statistical topic, the researcher is advised to consult with a statistician on these issues in his/her work.
Research results are never certain. Hypothesis testing recognizes statistical uncertainty by making probability statements about observed findings. In this section and the next, we focus on bias and
imprecision pertaining to these probability statements, beginning in this section with a discussion of bias in Type I error. Recall that Type I error is the probability of incorrectly rejecting a
null hypothesis. Useful pneumonics are to refer to Type I error as the false-positive or false-alarm rate. Typically studies set Type I error rates at, say, 5% to serve as the “nominal” level. In
this section, we define bias as a difference of consistent tendency in direction between actual and nominal Type I error rates.
In the course of reviewing data from a study, a pattern may appear to the researcher which he/she thinks may be worthy of testing for “statistical significance” with those data. The difficulty with
this “data snooping” is that it can result in a Type I error rate that is larger than the nominal level (see Ref. 9 for some discussion of this topic) and therefore represents a form of bias. One can
specifically warn the reader that a test was conducted as a result of snooping. However, within a study, perhaps the best rule to follow is that hypothesis formulation should dictate what data are
collected and used for testing rather than allowing collected data to direct which hypotheses to test.
A subtle, but at times decisive, form of bias in Type I error can arise when one chooses between one-tailed and two-tailed testing. When, say, one wishes to compare means between two groups, a
one-tailed test corresponds to an alternative hypothesis, in which one population's mean is strictly larger than the other's. In contrast, a two-tailed test applies to an alternative hypothesis,
which states that the two means differ, without specifying which is larger than the other. The advantage of a one-tailed test is that it offers greater statistical power (i.e., probability of
correctly rejecting the null) in the specified direction. One-tailed tests are scientifically justifiable in two special circumstances: 1) a difference between two means in one of two directions is
known to be impossible or 2) a difference in one direction is of no interest whatsoever in any circumstance. For example, suppose a researcher anticipates that a compound will elevate an average
glucose-cycling rate but acknowledges that the compound may, for reasons not yet understood or foreseen, depress the average rate. This possibility must be allowed by employing a two-tailed test,
unless the researcher would never be interested in recognizing such a counter-theoretical result. Because these two instances are rare in this literature, one-tailed testing should be rare, and if
employed, given very strong justification, which is not always the practice in the E and M literature. When a one-tailed test is employed outside these constraints, the nominal Type I error is less
than the actual.
Another arena in which actual Type I error rate may exceed the nominal rate is with multiple hypothesis testing. Multiple hypothesis testing is common in the E and M literature, especially where more
than one type of measurement is collected on each subject (e.g., as may be reported in a table of data on free fatty acids, triglycerides, glycerol, glucose, and so on) and a separate test performed
on each measurement. Each test conducted carries a particular Type I error rate, so that across multiple tests, the total Type I error rate compounds. For a provocative, accessible, and playful
discussion of this topic see Ref. 1. When one does wish to control for compounded Type I error, a number of powerful methods are available, including those described in Refs. 6 and 11, with the
latter suggested for the nonstatistician.
The researcher should also be aware that the results of a hypothesis test are only as good as the probability model on which they are based. Take, for example, the two-sample t-test, which is
employed very widely in the E and M literature. Contrary to many practitioners' beliefs, the two-sample t-test does not require that each group's population be approximately normally distributed
(although this helps). Rather, it assumes that the distribution of differences in sample means is approximately normally distributed. To illustrate, suppose that we sample n subjects from each group
and calculate the difference in their sample means. Then we draw another sample of n subjects from each group and calculate the difference in these two new sample means. Repeating this process
infinitely many times generates the full distribution of differences in sample means. It is this sampling distribution that we are assuming is normal in the two-sample t-test. Because of a
statistical property described by the Central Limit Theorem, this distribution of sample means tends toward the normal as n increases in size. However, when n is small and the populations
corresponding to the two groups are strongly skewed or possess multiple modes, then the sampling distribution of the differences in means may be nonnormal, and use of the t-test may lead to actual
Type I error rates that are either smaller or larger than the nominal rate.
The E and M researcher is advised to know what assumptions underlie a specific method of hypothesis testing and to assess if his/her data meet these, at least approximately. Small sample sizes,
common in the E and M literature, limit the analyst's ability to check whether data meet a test's assumptions. As a very rough rule, it is desirable to have ≥30 subjects per population sampled to
permit adequate examination of the assumptions underlying many statistical tests. Even so-called nonparametric tests (which is a misnomer, for many of these tests are often used to test hypotheses
regarding parameters) can be based on specific assumptions about the populations that have been sampled (see, for example, Ref. 2). The E and M literature also contains some examples of more
sophisticated statistical modeling methods (e.g., repeated-measures ANOVA, which is common in the E and M literature and makes very strong assumptions about correlations among repeated measurements).
With this added sophistication comes a complexity of assumptions that should be carefully examined. Statements such as “appropriate regression procedures were applied” are inadequate alone. When a
statistical method's assumptions are examined, these results should be reported. Choosing an appropriate method for fitting data and testing hypotheses requires striking a balance. Methods that rely
on many strong assumptions can be quite powerful when data clearly meet those assumptions but can introduce appreciable bias in Type I error when those assumptions are violated.
Imprecision in hypothesis testing is measured by Type II error. Type II error is the probability of failing to accept the alternative hypothesis when true. To illustrate, one might pose a null
hypothesis that two populations' means do not differ and an alternative hypothesis that they do. If, in reality, the two populations' means differ and we fail to detect this, we have committed a Type
II error. The complementary probability to Type II error is statistical power. That is, if we denote Type II error by β, power is given by 1 - β. Power is the probability of rejecting the null
hypothesis when the alternative is true.
Type II error can grow from any source of imprecision that arises in the process that leads up to hypothesis testing. Thus Type II error increases with 1) smaller sample sizes (common in the E and M
literature), 2) larger technical error (category IV), or 3) less efficient estimators (category VI).
Beyond these sources, which have already been discussed, the E and M literature contains other examples of lost opportunities to enhance statistical power. An example is the use of unequal sample
sizes. In the E and M literature, one can find statements such as “5-25 specimens were measured per group” or “each group consisted of a minimum of 4 subjects.” Broadly speaking, when comparing
groups' means for a fixed total sample size, power is greatest when sample sizes are equal for those groups (e.g., 15).
As indicated in the previous section, multiple testing is common in the E and M literature, in part because studies typically take measurement on more than one parameter for each subject. As a
result, most characterizations of subjects are multivariate, which makes sense scientifically given that most metabolic and endocrine processes involve multiple parameters. Despite this, instances of
multivariate hypothesis testing, in which one conducts tests on multiple parameters simultaneously (e.g., Hotelling's T^2 test), are rare to nonexistent in the E and M literature. Instead, several
univariate (single-parameter) tests are conducted. The shortcoming of this wholly univariate approach is that univariate hypothesis testing of multivariate processes can sometimes result in a failure
to detect patterns of scientific interest. Rencher (10) provides an introduction to multivariate hypothesis testing that is reasonably accessible to nonstatisticians.
A form of reporting bias that may be undetectable to even the most astute reader is the failure to report. Sometimes referred to as the “file drawer problem,” results go unpublished because an author
or editor chooses not to publish “statistically insignificant” findings. This generates bias in the literature. Although it is true that underpowered studies are poor candidates for funding on
ethical (4) and statistical grounds, one may not wish to strictly equate statistical insignificance with biological insignificance. For instance, an otherwise fully powered study may be worthy even
if an underpowered portion lacks a statistically significant result, because that result may nevertheless provide an empirical suggestion, admittedly weak, for hypothesis generation. Such a
hypothesis could be more rigorously tested in subsequent, fully powered studies. This is not to say, however, that the importance of findings from underpowered research should ever be overstated. On
the flipside are studies designed with ample power that generate negative findings. These too can serve an important role in the literature by directing subsequent research away from apparently
fruitless avenues.
Statistically insignificant results can also give rise to another form of reporting bias. Suppose in testing a null hypothesis of no difference between two populations against an alternative claiming
that their means differ, we obtain an attained significance level of P = 0.13. Authors are advised to avoid stating in conclusive terms that no difference exists between the populations, which is an
overstatement (and thus directional error). As an example, one may come across statements such as “incubation with the inhibitor failed to alter the rate of metabolic transport.” A more accurate
conclusion would be that “no change in rate was detected statistically.” As mentioned above, if one is seeking statistical support for the assertion that two groups do not differ, then bioequivalency
testing should be employed (see category III).
Like the file drawer problem discussed above, reporting imprecision arises when useful information is withheld from the reader, but here any resulting misunderstanding is not directional. An example
from the E and M literature is when sample means are reported “±” a second value, but it is not indicated if the second value is a sample standard deviation (estimate of the population standard
deviation), standard error (estimate of the standard deviation of the sample means), or some other measure of dispersion. Identification of which specific dispersion parameter has been estimated is
necessary if other researchers are to use these estimates to plan sample sizes in future research. Another example found in this literature is when a result is presented along with an attained
significance level (P value), but no clear information is provided on what statistical method was employed to obtain that result. This leaves the reader with no means of assessing the value of the
result of a hypothesis test.
This essay has illustrated ten categories of statistical errors. It is intended to highlight potential statistical pitfalls for those who are designing, implementing, and reviewing E and M research.
Some accessible references to the statistical literature have been given throughout to provide an entry for those who are interested in learning more about the topics discussed.
A review of the recent E and M literature finds that the most common potential cause of statistical error is small sample size (n ≤ 10). Very small sample sizes 1) result in parameter estimates that
are unnecessarily imprecise, 2) enhance potential for failed randomizations, and 3) yield hypothesis testing that is underpowered, or 4) yield hypothesis testing that is biased because assumptions
underlying the applied statistical methods could not be examined adequately. Missing data exacerbate this problem by further reducing and unbalancing sample size and, when “informative,” missing data
can introduce bias. E and M research could also see major improvements through more widespread use of stratified sampling designs, careful selection and implementation of experimental controls, use
of adaptive and stratified randomization procedures, routine reporting of technical error, application of bioequivalency testing to studies designed to demonstrate equivalency, identification of
statistically efficient means of data reduction through consultation with a statistician, more restrained use of one-tailed tests, greater controls on Type I and Type II errors for hypothesis tests
across multiple parameters, fuller descriptions of statistical methods and methods of examining methodological assumptions, and more frequent reporting on amply powered but statistically
nonsignificant results.
Despite these limitations, the E and M literature appears to contain fewer statistical errors, in kind and quantity, than other medical literatures examined by the author. In part, this may be for
lack of opportunity, as the range of statistical methods employed in the E and M literature is comparatively narrow. This is a lost opportunity, as processes studied in E and M are typically high
dimensional and time varying (e.g., electrolyte concentration vs. time since initiation of dialysis), which makes this discipline ripe for greater application of multivariate (5, 10) and more
flexible longitudinal (3) statistical designs and analyses. For example, generalized estimating equations (3) offer a highly flexible and robust method of analyzing repeated-measures data, especially
when no data are missing; dimensionality-reduction techniques, such as principal-components analysis and cluster analysis (5), are useful for forming a few strata from several baseline
characteristics; and these are just a few possibilities. The capacity for expanding the utility and sophistication of multivariate statistics in application to E and M research is tremendous. | {"url":"https://journals.physiology.org/doi/full/10.1152/ajpendo.00484.2003","timestamp":"2024-11-13T11:11:26Z","content_type":"text/html","content_length":"209574","record_id":"<urn:uuid:0e4e5eed-d7ba-4ad0-9153-308207020486>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00550.warc.gz"} |
Division of Decimals: Quotient and Remainder Calculations and Long Division ｜ Hatsudy
Decimals are taught in elementary school mathematics. After learning multiplication with decimals, you need to be able to do division.
When calculating decimals, a division is a bit more complicated. This is because it is not always divisible. You have to calculate until you can divide or use the remainder to get the answer. It is
also common to move the decimal point.
When it comes to calculating decimals, addition, subtraction, and multiplication are easy. Division, on the other hand, requires a lot of understanding. In this section, we will explain how to do
decimal division.
How to Divide Using Decimals and Integers: The Case of Divisible Numbers
Let’s start with the simple division of decimals. As long as the numbers are divisible, the calculations are not complicated. For example, how do you calculate the following?
When doing this calculation, create the following long division.
The way to do division is the same as a long division with whole numbers. Ignore the decimal point and perform the division calculation. Then, add the decimal point to the answer (quotient) in the
same position. The result is as follows.
When doing division using decimals and integers, it is not difficult to calculate as long as the numbers are divisible. After dividing, add the decimal point in the same place.
In some calculation problems, the ones place of the answer may be zero. In this case, you should always write a zero in the ones place. The following is an example.
In calculating decimals, the number can be smaller than one. Understand that in the division of decimals, the ones place can be zero.
Division of Decimals Until Divisible
On the other hand, there are some problems that need to be calculated until a number is divisible. In this case, the calculation is done by adding zeros to the right of the decimal.
There are hidden zeros in decimals. For example, 1.4 can be written as 1.400. However, since there is an infinite number of zeros, we omit the zeros in decimals. In any case, it is important to
understand that in decimals, there are many zeros to the right of the number.
Therefore, when calculating divisible decimals, add zeros to get the answer. For example, how do we calculate the following?
When doing this calculation, let’s change the value to 1.40 instead of 1.4. The result is as follows.
However, the calculation should not end here. From this point, we can do more division. So, let’s change 1.40 to 1.400 and do the next calculation.
By adding zeros to the right of the decimal, we were able to get the answer. When doing division, if it is divisible, try to get the answer by this method.
When the Divisor Is a Decimal, Move the Decimal Point to Perform the Division
So far, we have discussed division when the divisor is an integer. On the other hand, when a divisor is a decimal number, how do we calculate it?
When the divisor is a decimal, make sure to move the decimal point. By moving the decimal point to the right, you make the divisor an integer. If you don’t do this, you will not be able to divide
decimals. For example, how would you do the following calculation?
The long division is as follows.
The divisor is 1.4. If we don’t change it to an integer, we will not be able to do the division. So, instead of 1.4, let’s make it 14. Move the decimal point one place to the right.
At the same time, move the decimal point of 3.22 one place to the right as well. Since we have moved the decimal point of the divisor, we need to do the same for the dividend. Therefore, the long
division looks like this.
Understand that when the divisor is a decimal, you cannot divide the number unless you change the decimal point. Therefore, we need to move the decimal point like this.
We need to change the divisor to an integer, and how many times we need to move the decimal point to the right depends on the divisor. For example, in the following calculation, we need to shift the
decimal point two places to the right.
Let’s do the following long division calculation.
Understand how to move the decimal point and change a divisor to an integer.
Why Moving the Decimal Point Doesn’t Matter
There is a question that is asked by many people. Why is it okay to move the decimal point? The reason for this is that even if you change the position of the decimal point, the answer will be the
For example, the following will all give the same answer.
• $100÷20=5$
• $10÷2=5$
• $1÷0.2=5$
• $0.1÷0.02=5$
The answer will be the same if you multiply the number by 10 or 100 for both the divisor and the dividend. For example, compare $10÷2=5$ and $100÷20=5$.
In the same way, we can see that for $1÷0.2$ and $0.1÷0.02$, we can get the same answer by multiplying both the divisor and the dividend by 10 or 100.
For $1÷0.2$, multiply both numbers by 10 to get $10÷2$. For $0.1÷0.02$, multiply both numbers by 100 to get $10÷2$. In this way, decimals can be converted to integers. This method allows us to divide
Division of Decimals and Integers with Remainders
So far, we have discussed division when the number is divisible. On the other hand, what should we do with numbers that are not divisible? When an integer is not divisible, we use the remainder. This
is also true for decimal division.
For example, how do we calculate the following?
If you make a long division and calculate it, you will get the following.
The method of division is the same as the one explained so far. The only difference is that the remainder is given. If you calculate $22.3÷4$, the answer will be 5.5 R 0.3.
When calculating the remainder, make sure to put the decimal point directly below. When calculating decimals, it is easy to make mistakes with the position of the decimal point. If the position of
the decimal point is different, the remainder will be different. If this happens, the answer will not be correct, so be sure to check the position of the decimal point.
How to Calculate Division of Two Decimals with Remainder
On the other hand, how do we calculate if the divisor is a decimal? When there is a remainder, it is easy to make a miscalculation if the divisor is a decimal. Therefore, we need to understand how to
calculate them.
For example, how do we calculate the following?
In order to do the math, we need to make the divisor an integer. So instead of 3.3, let’s change it to 33. We can do the math as follows.
Finally, we need to find the remainder. We must pay attention to the position of the decimal point in the remainder; it must be put directly below the previous decimal point. The result is as
The answer to $10.86÷3.3$ is 3.2 R 0.3.
By multiplying the divisor and dividend by 10, the quotient is 3.2. On the other hand, for the remainder, instead of using 108.6 multiplied by 10 as the reference number, we use the original number,
10.86, as the reference number. We put the decimal point of 10.86 directly below to get the remainder.
Why the Decimal Point Is Different in the Quotient and the Remainder
The reason why it is easy to make miscalculations when dividing decimals is that the position of the decimal point is different for the quotient and remainder. Why does the decimal point change
between the quotient and remainder? Let’s understand the reason for this.
As explained before, the quotient is the same even if you multiply the divisor and the dividend by 10 (or 100). The answer is the same, as shown below.
Next, let’s try multiplying by 10 for the non-divisible numbers. What will be the answers to the following questions?
• $100÷30=3$ … $10$
• $10÷3=3$ … $1$
As you can see, the quotient is the same even if we multiply the divisor and the dividend by 10. On the other hand, we can see that the remainder is 10 times larger. Also, if we multiply the divisor
and the dividend by 100, the remainder is 100 times greater. In division, if we multiply numbers by 10 or 100, the remainder also increases accordingly.
If the remainder is 10 or 100 times larger than the original number, the answer will be different. For example, in the case of $10÷3$, if we multiply the divisor and the dividend by 10, as explained
earlier, the remainder will be 10 times larger than the original number.
Even though the divisor is 3, the remainder is 10. Therefore, it is obvious that the answer is wrong.
So when dividing, use the number before multiplying by 10 (or 100) for the remainder. By using the number before the decimal point is moved as the reference, we can add the decimal point in the
correct place for the remainder.
Calculating Decimals to Find the Quotient and the Remainder
When we do division, the expression may contain decimals. So, let’s understand how to divide them.
When dividing decimals, it is easy if the number is divisible. All you have to do is to divide in the usual way, paying attention to the position of the decimal point. If the divisor is a decimal,
you can change the divisor to an integer by moving the decimal point.
On the other hand, if there is a remainder, it is easy to make a miscalculation. If the divisor is a decimal, the quotient is based on the number after the decimal point is moved. Also, the remainder
is based on the number before the decimal point is moved, and then the decimal point needs to be added.
Understand that there are rules for these calculations and perform division calculations involving decimals. | {"url":"https://hatsudy.com/decimal-d.html","timestamp":"2024-11-01T20:58:47Z","content_type":"text/html","content_length":"43686","record_id":"<urn:uuid:c86c5864-3219-45f5-8b39-c0082d31215b>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00391.warc.gz"} |
Iron Condor: The Monthly Income Option Strategy (94% Success)
by Anchit Shethia · February 18, 2020 ·
The Iron Condor is an options trading strategy used by many option traders for generating monthly income. This strategy gives profit when the underlying stock or index stays within a certain range
over the life of the trade.
The Iron Condor is profitable when the underlying stock or index goes
• Up a little
• Sideways
• Down a little
and gives loss when it goes up or down too much.
Let’s say Nifty is currently at 10,000 and you expect it to remain in the 500 points up-down range over the next month. You can write 10500 call options and 9500 put options. If Nifty remains in this
thousand points range of 9500-10500 you make the profit.
The range you expect will give profit if things go as per expectation which doesn’t always happen in the real world. If Nifty crosses 9500-10500 range there is a possibility of bigger loss. To
prevent this loss some protection is required. This protection can be obtained by buying the 10700 call options and 9300 put options. In the case of violent movement on any side, this long options
will reduce some loss.
Which Expiry month to trade
This strategy is designed to take maximum advantage of time decay, therefore it is good to take a position in a month ahead expiry. Generally, the sweet spot for iron condors is anywhere between 40
and 60 days to expiry.
When to book profit or loss
This is a complete range-bound movement-based strategy. We are going to make a profit only when the price remains within the band. If the range is breaking losses can be very high, therefore it is
good to exit when the range is broken by 100 points, in the case of Nifty.
Like in our example we had a sell-side position in put and call options of 9500-10500 if the price touches the level of 9400 or 10600 we should exit and book the losses.
For booking profit, you can either wait for the options to go zero or you can book the profit when it is giving 50% of the calculated profit. I prefer to book, when 50% profit is there and take the
next month the same strategy trade.
Say you have
• sold the 10,500 Call option at 60
• sold the 9500 Put option at 50
• Bought the 10700 call option at 20
• Bought the 9300 put option at 15
At the expiry, all will go zero in case of range-bound movement in the band of 9500-10500
The maximum profit will be 60+50-20-15 = 75 points
and if you are getting 50% of that, i.e 35-40 points profit, you can start booking the profits.
When to take the trade
Option premium depends on volatility. Since we expect to make a profit by selling the option. It is necessary that the option premium is moderately high. In the case of Nifty, if the implied
volatility is 15+ I prefer to take the trade when it is less than 12 I prefer to avoid the trade.
Success Rate: This strategy will be profitable 90% of the time. Discipline is very important. Loses can be very high if you continue holding the position when the price is breaking the range. So
always respect the stop losses.
Hence we always suggest our readers not to trade emotionally and just book the losses if the range is broken. You can always recover in the next trade.
So, this was all about the Monthly income Iron Condor Options strategy. Let me know what you think about it in the comments.
You can also join our InvestorJi Academy to learn Fundamental and Technical Analysis and learn more such strategies to make your monthly income.
2 Comments
Hi Abhisek,
A great article! I want to clarify regarding stoploss…here it is mentioned stoploss would be 9400 or 10600, but i think it should be 9600 from put side and 10400 from call side! correct me if i
am wrong? Also, please let me know how to know when profit becomes 50% of credit received? how much time does it usually take for the profits to become 50%? If i want to book profits before
expiration nifty should gain some heavy points either up or down…..if it does not moves up or down and remains range bound then in that case i have no option but to wait till expiry???
I would really appreciate if you could throw some light!
Stop loss is 100 points beyond the strike price of sold options. Since in the example we are selling 9500 and 10500 strike options, the stop loss will be 9400/10600
In the example there is net premium of 75 points , when the premium reduces to half of that 75/2 , will be the profit booking stage
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DarkRadiant 3.4.0 released
DarkRadiant 3.4.0 is ready for download.
What's new:
• Feature: Allow Layers to be arranged into a Tree
• Fixed: Readable Editor displays "shader not found" in view
• Fixed: Undoing snap to grid with prefabs causes crash
• Fixed: Include doc in building instructions
• Fixed: Decal textures causes DR to crash - (textures/darkmod/decals/dirt/long_drip_pattern01)
• Fixed: Skin chooser: double click on materials list closes window
• Fixed: Selecting and deselecting a filtered child brush through layers leaves the brush selected
• Fixed: Material editor re-sorts stages on pasting image map resulting in wrong material stages list and wrong selected stage
• Fixed: Crash on start if engine path is choosen (Doom 3)
Feature: Layers can now be arranged to form a hierarchy
Windows and Mac Downloads are available on Github: https://github.com/codereader/DarkRadiant/releases/tag/3.4.0 and of course linked from the website https://www.darkradiant.net
Thanks to all the awesome people who keep using DarkRadiant to create Fan Missions - they are the main reason for me to keep going.
Please report any bugs or feature requests here in these forums, following these guidelines:
• Bugs (including steps for reproduction) can go directly on the tracker. When unsure about a bug/issue, feel free to ask.
• If you run into a crash, please record a crashdump: Crashdump Instructions
• Feature requests should be suggested (and possibly discussed) here in these forums before they may be added to the tracker.
The list of changes can be found on the our bugtracker changelog.
Have fun mapping!
The layer hierarchy is very welcome!
edit: it doesn't seem to work when compiling from code on Linux...
Edited by thebigh
My missions: Stand-alone Duncan Lynch series
Down and Out on Newford Road the Factory Heist
The Wizard's Treasure A House Call
I'm getting this while trying to package for Debian:
CMake Error at cmake_install.cmake:554 (file):
file INSTALL cannot find
No such file or directory.
[S:I'm currently preparing a patch, will make a PR soon.:S] PR: https://github.com/codereader/DarkRadiant/pull/31
Edited by coldtobi
On 10/9/2022 at 4:08 PM, thebigh said:
The layer hierarchy is very welcome!
edit: it doesn't seem to work when compiling from code on Linux...
Did you file a bug report yet? I noticed that I can drag the layers, but dropping them doesn't do anything.
Thanks for all the awesome work!!!!
I still have the issue with the brushes not being highlighted when selected in ortho views, clearly an AMD thing though.
I always assumed I'd taste like boot leather.
On 10/9/2022 at 10:08 PM, thebigh said:
edit: it doesn't seem to work when compiling from code on Linux...
I just pushed a fix to issue #6129 to github, I think it'll be working now
42 minutes ago, AluminumHaste said:
I still have the issue with the brushes not being highlighted when selected in ortho views, clearly an AMD thing though.
I wish I could do something about it... I have no machine with an AMD card around, neither at home nor at work, or anybody I know.
3 hours ago, greebo said:
I just pushed a fix to issue #6129 to github, I think it'll be working now
Yep, it's working now
My missions: Stand-alone Duncan Lynch series
Down and Out on Newford Road the Factory Heist
The Wizard's Treasure A House Call
6 hours ago, greebo said:
I just pushed a fix to issue #6129 to github, I think it'll be working now
I wish I could do something about it... I have no machine with an AMD card around, neither at home nor at work, or anybody I know.
You can remote into my desktop and run whatever tests you need to.
I always assumed I'd taste like boot leather.
I'm running linux; darkradiant compiled from source (most recent commit of master).
In the 'Regular' and 'RegularLeft' window layouts keyboard input doesn't work at all. Some windows (console or media browser) don't open.
The layer dialog does not work as supposed. I can't assign a parent layer by drag and drop. Hiding through the checkboxes does nothing. Everything else seems to work.
Maybe not a bug:
When the media browser is shown (or some other window with a search function) the search field captures keypresses. Which is really, really annoying and disruptive.
Edited by Baal
10 hours ago, Baal said:
In the 'Regular' and 'RegularLeft' window layouts keyboard input doesn't work at all. Some windows (console or media browser) don't open.
I can reproduce this. It doesn't surprise me much since these legacy GtkRadiant layouts are hardly maintained or tested these days. I recommend using the Dockable layout instead and dragging the dock
windows to the positions you want.
10 hours ago, Baal said:
The layer dialog does not work as supposed. I can't assign a parent layer by drag and drop. Hiding through the checkboxes does nothing. Everything else seems to work.
I also see the checkboxes not doing anything. Drag and drop partially works for me, although I can only create a parent relationship. I cannot find a way to unparent a layer and bring it back to the
top level.
10 hours ago, Baal said:
When the media browser is shown (or some other window with a search function) the search field captures keypresses. Which is really, really annoying and disruptive.
I think the Dockable layout would avoid that problem too (because the media browser would no longer be a top-level window).
these legacy GtkRadiant layouts are hardly maintained or tested these days. I recommend using the Dockable layout instead and dragging the dock windows to the positions you want.
I much prefer the old behaviour of opening the media browser or entity view in a temporary window. This way you can make them comfortably large and they dont't waste screen space when you don't need
them (which is most of the time).
I think the Dockable layout would avoid that problem too (because the media browser would no longer be a top-level window).
Mostly yes. But it is still a problem. You cannot quickly switch to the media browser and then back to another view. I think this behaviour of capturing keys and stop shortcuts to work as expected is
bad. If it's not to difficult to implement, I would say ditch the automatic search on keypress.
22 hours ago, OrbWeaver said:
I also see the checkboxes not doing anything. Drag and drop partially works for me, although I can only create a parent relationship. I cannot find a way to unparent a layer and bring it back to
the top level.
Gotta love these little platform-specific differences. The wxDataViewEvent sent in wxGTK is not delivering the correct column, so the event handler is not reacting. I rewrote the event handler in #
6130, the checkbox should work now.
About the drag-and-drop: this is another problem that seems to affect wxGTK only... in Windows, I can drag the layer to the top border of the view and it will deliver an empty target wxDataViewItem,
which means make it toplevel - in wxGTK it doesn't seem to be possible to drag it to the top.(*)
I can think of two possibilities right away
• either restrict the system such that the Default layer can never have child layers. Dragging a layer onto the Default layer will make the dragged layer a top-level layer
• Add a context menu item called "Make Top-Level"
I don't like either of the two very much. Suggestions?
(*) Yes, there's a guard in the wxGTK event handler:
wxDataViewItem item(GetOwner()->GTKPathToItem(path));
if ( !item )
return FALSE;
this prevents any empty item from being forwarded to the event.
2 minutes ago, Baal said:
Maybe drag the child onto the parent to unparent (make a sibling)?
Hm, I might try that to see if it feels intuitive.
Ok, hiding via the checkbox works now.
Drag and drop seems to work to, but I can't tell where I am dropping a node (above, below or on it). The highlight needs to colored more clearly.
Yep, hiding layers via the checkbox works for me now as well. Rearranging the layers is still a bit unintuitive but that's no big deal.
My missions: Stand-alone Duncan Lynch series
Down and Out on Newford Road the Factory Heist
The Wizard's Treasure A House Call
I think the drop indicators are out of my control, they are subject to the platform wxWidgets is running on.
But I'm open to suggestions, if anyone has ideas on how to improve the hierarchy arrangement.
Is it possible to reorder layers with drag and drop?
Parenting and unparenting layers works; but it is trial and error.
39 minutes ago, Baal said:
Is it possible to reorder layers with drag and drop?
It's sorted alphabetically right now. The layers don't persist any sort order in the map files.
40 minutes ago, Baal said:
Parenting and unparenting layers works; but it is trial and error.
Yes, drag and drop works much better in Windows, the wxGTK port is not giving good visual feedback about where you're going to drop the layer.
What is the file / where are the files where this is configured in the source code? I would like to take a look.
After working with the layer hierarchy for a bit, I've come to the conclusion that only the leaf nodes should contain geometry. You can't easily select a parent node alone; you have to move its
contents to a child if you want to do that.
So instead of making one node the child of another node, they should be selected, then grouped. That would create a new, empty parent with the grouped layers as children.
Unless there is some error in my thinking, this makes more sense to me. | {"url":"https://forums.thedarkmod.com/index.php?/topic/21615-darkradiant-340-released/#comment-479304","timestamp":"2024-11-04T15:37:17Z","content_type":"text/html","content_length":"395860","record_id":"<urn:uuid:76c16611-7f5e-4a32-9731-7023ea6e8ea9>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00155.warc.gz"} |
Understanding Mathematical Functions: What Makes A Function Constant
Introduction to Mathematical Functions
Mathematical functions are essential tools in mathematics that help us understand the relationship between two sets of numbers. By representing a specific rule or relation, functions provide a
systematic way of mapping input values to output values. Understanding functions is fundamental for various fields of mathematics and other disciplines.
A Definition of a mathematical function
A mathematical function is a rule that assigns each input value from a set (domain) to exactly one output value from another set (codomain). This relationship is often denoted by f(x) in algebraic
notation, where x represents the input value. Functions can take many forms, including algebraic, trigonometric, exponential, and logarithmic functions.
Overview of different types of functions in mathematics
In mathematics, functions can be classified into various types based on their properties. Some common types of functions include:
• Linear functions: Functions that produce a straight line when graphed.
• Quadratic functions: Functions that produce a parabolic curve when graphed.
• Exponential functions: Functions with a variable in the exponent.
• Trigonometric functions: Functions involving trigonometric ratios like sine, cosine, and tangent.
Importance of understanding the concept of constant functions
Constant functions are a special type of function where the output value is always the same regardless of the input value. In other words, a constant function produces a horizontal line when graphed.
Understanding constant functions is crucial as they provide a basic building block for more complex functions and help in understanding concepts like slope, intercepts, and transformations in
Key Takeaways
• Constant functions have the same output for all inputs.
• Graph of a constant function is a horizontal line.
• Constant functions have a constant rate of change.
• Examples of constant functions include f(x) = 5.
• Constant functions have a slope of zero.
Understanding Constant Functions
Constant functions are a fundamental concept in mathematics that play a crucial role in various mathematical applications. In this chapter, we will delve into what makes a function constant, how they
are mathematically represented, and how they differ from non-constant functions.
Defining what makes a function constant
A constant function is a function that always produces the same output regardless of the input. In simpler terms, no matter what value you input into a constant function, the output will remain
constant. For example, the function f(x) = 5 is a constant function because it always outputs 5, no matter what x is.
Mathematically, a function f(x) is considered constant if and only if f(x) = c for all x in the domain of the function, where c is a constant value. This means that the function does not depend on
the input variable x and always returns the same value.
Mathematical representation of constant functions
Constant functions can be represented in various ways in mathematics. One common way to represent a constant function is through a horizontal line on a graph. Since the output of a constant function
does not change, the graph of a constant function is a straight horizontal line at the constant value.
For example, if we have the constant function f(x) = 3, the graph of this function would be a horizontal line at y = 3, indicating that the output is always 3 regardless of the input.
Comparison with non-constant functions
It is essential to differentiate constant functions from non-constant functions to understand their unique properties. Non-constant functions, unlike constant functions, produce different outputs for
different inputs. In other words, the output of a non-constant function varies based on the input.
For example, the function g(x) = x is a non-constant function because the output changes based on the input value of x. If we input x = 2, the output would be 2, but if we input x = 5, the output
would be 5.
Constant functions serve as a foundational concept in mathematics and are essential for understanding more complex mathematical functions and relationships. By grasping the defining characteristics
and representations of constant functions, we can better comprehend the broader scope of mathematical functions.
Characteristics of Constant Functions
Constant functions are a fundamental concept in mathematics that play a significant role in various mathematical applications. Understanding the characteristics of constant functions is essential for
grasping their behavior and significance in mathematical analysis.
A Horizontal line representation in graphical form
One of the defining characteristics of a constant function is its graphical representation as a horizontal line on a Cartesian plane. This means that for every value of the independent variable, the
function produces the same constant output. Visually, this results in a straight line that does not slope up or down.
This graphical representation is a clear indicator that the function is constant, as it does not change regardless of the input value. The horizontal line serves as a visual cue for identifying
constant functions and distinguishing them from other types of functions.
B The role of the slope in defining a constant function
In the context of constant functions, the slope of the function is crucial in understanding its behavior. A constant function has a slope of zero, which means that there is no change in the output
value for any change in the input value. This is in contrast to linear functions, which have a non-zero slope and exhibit a change in output corresponding to a change in input.
The concept of slope helps to differentiate constant functions from other types of functions and provides a mathematical basis for their definition. By recognizing the role of slope in defining
constant functions, we can better understand their behavior and properties.
C Constant functions in the context of domain and range
When considering constant functions in the context of domain and range, it is important to note that the domain of a constant function is all real numbers, as there are no restrictions on the input
values that can be used. This is because the output of a constant function remains the same regardless of the input value.
Similarly, the range of a constant function consists of a single value, which is the constant output of the function. This means that the function produces the same output for every input value,
resulting in a range that is limited to a single constant value.
Understanding the relationship between constant functions, domain, and range provides insight into the behavior and properties of these functions, highlighting their unique characteristics and
significance in mathematical analysis.
Application of Constant Functions in Real-world Scenarios
Constant functions play a crucial role in various real-world scenarios, from simplifying mathematical models to programming and software development, as well as in physics and engineering.
Use in simplifying mathematical models
Constant functions are often used in simplifying mathematical models by providing a fixed value that remains unchanged throughout the model. This can help in reducing the complexity of the model and
making it easier to analyze and understand. For example, in finance, a constant function may represent a fixed interest rate or a constant growth rate in an investment portfolio.
Role in programming and software development
In programming and software development, constant functions are used to define values that do not change during the execution of a program. These constants can be used to represent fixed values such
as mathematical constants (e.g., pi) or configuration settings that remain constant throughout the program's execution. By using constant functions, developers can ensure that these values are not
accidentally modified, leading to more robust and reliable software.
Examples in physics and engineering
Constant functions are also prevalent in physics and engineering applications. For instance, in physics, a constant function may represent a physical constant such as the speed of light or the
gravitational constant. These constants play a fundamental role in various equations and models in physics. In engineering, constant functions can be used to represent fixed parameters in a system,
such as the resistance of a material or the voltage of a power source.
Calculating and Graphing Constant Functions
Constant functions are a fundamental concept in mathematics that represent a function with a fixed output value regardless of the input. Understanding how to calculate and graph constant functions is
essential for mastering mathematical functions. Let's delve into the step-by-step process of plotting a constant function on a graph, explore tools and software that can facilitate visualization, and
discuss common mistakes to avoid when dealing with constant functions.
A. Step-by-step process of plotting a constant function on a graph
• Step 1: Identify the constant value of the function. This value will remain the same for all inputs.
• Step 2: Choose a range of input values to plot on the x-axis. These values will help you visualize the function's behavior.
• Step 3: Substitute the input values into the constant function to determine the corresponding output values.
• Step 4: Plot the points (input, output) on the graph. Since the function is constant, all points will lie on a horizontal line at the constant value.
• Step 5: Connect the points with a straight line to represent the constant function on the graph.
B. Tools and software that can facilitate the visualization of constant functions
Graphing constant functions can be made easier with the help of various tools and software designed for mathematical visualization. Some popular tools include:
• Graphing calculators: Devices like TI-84 and Casio calculators have built-in functions for plotting graphs, including constant functions.
• Math software: Programs like Mathematica, MATLAB, and Desmos offer advanced graphing capabilities for visualizing mathematical functions.
• Online graphing tools: Websites like GeoGebra and Wolfram Alpha provide free platforms for graphing functions, including constant functions.
C. Common mistakes to avoid when dealing with constant functions
When working with constant functions, it's important to be aware of common errors that can arise. Here are some mistakes to avoid:
• Confusing constant functions with linear functions: Constant functions have a fixed output value, while linear functions have a constant rate of change. Be sure to differentiate between the two.
• Incorrectly plotting points: Make sure to substitute the correct input values into the constant function to determine the corresponding output values for accurate plotting.
• Ignoring the horizontal line: Since constant functions result in a horizontal line on the graph, ensure that all points lie on this line to represent the function correctly.
Troubleshooting and Overcoming Challenges
When dealing with constant functions in mathematics, it is important to be able to identify errors in calculations, interpret graphs correctly, and distinguish constant functions from other similar
functions. Let's explore some common challenges and how to overcome them.
Identifying errors in calculations involving constant functions
One common mistake when working with constant functions is incorrectly identifying a function as constant. Remember, a constant function is a function that returns the same output value regardless of
the input value. If you are getting different output values for different input values, then the function is not constant.
Another error to watch out for is mistaking a linear function for a constant function. While both types of functions have a constant rate of change, a linear function will have a non-zero slope,
whereas a constant function will have a slope of zero.
Tips for correctly interpreting the graphs of constant functions
When looking at the graph of a constant function, remember that it will be a horizontal line since the output value does not change with different input values. The line will be parallel to the
x-axis, indicating that the function is constant.
Pay attention to the y-intercept of the graph, as this will be the constant value that the function returns for all input values. Understanding the behavior of constant functions graphically can help
you interpret them correctly.
How to distinguish constant functions from other similar functions
One way to distinguish a constant function from other similar functions is to look at the rate of change. Constant functions have a rate of change of zero, meaning that the output value does not vary
with the input value.
Compare the behavior of the function with different input values. If the output value remains the same regardless of the input value, then you are likely dealing with a constant function. Be cautious
not to mistake linear functions or other types of functions with constant functions.
Conclusion & Best Practices
A Recap of the key points discussed about constant functions
Throughout this blog post, we have delved into the concept of constant functions and what makes them unique in the realm of mathematical functions. We have learned that a constant function is a
function that always produces the same output, regardless of the input. This means that the graph of a constant function is a horizontal line.
Additionally, we have explored how constant functions are represented algebraically, with the general form being f(x) = c, where c is a constant value. We have also discussed how constant functions
can be useful in various mathematical applications, providing a stable and predictable output.
Best practices in applying constant functions in various mathematical problems and projects
• When working with constant functions, it is important to understand the nature of these functions and how they differ from other types of functions.
• Constant functions can be used to represent quantities that do not change over time or in response to other variables.
• When applying constant functions in mathematical problems, it is essential to clearly define the constant value and its significance in the context of the problem.
• Constant functions can be particularly useful in modeling scenarios where a fixed value is involved, such as in budgeting, pricing, or other financial calculations.
Encouragement to further explore and understand the depth of mathematical functions beyond the constant functions
While constant functions provide a solid foundation in understanding mathematical functions, it is important to remember that they are just one piece of the larger puzzle. There are many other types
of functions, each with its own unique properties and applications.
I encourage you to continue exploring the world of mathematical functions, delving into more complex functions such as linear, quadratic, exponential, and trigonometric functions. By expanding your
knowledge and understanding of different types of functions, you will be better equipped to tackle a wide range of mathematical problems and projects. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-what-makes-a-function-constant","timestamp":"2024-11-13T02:10:31Z","content_type":"text/html","content_length":"225163","record_id":"<urn:uuid:9a87dd6c-ac79-4c5c-8e59-04495932b2b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00055.warc.gz"} |
Satyanarayana Reddy | School of Natural Sciences | Shiv Nadar University
Faculty at School of Natural Sciences
Satyanarayana Reddy
Associate Professor
School of Natural Sciences
Contact Information
Algebraic Graph Theory, combinatorial matrix theory and Algebraic Number Theory.
Ph.D, Indian Institute of Technology Kanpur M.Sc Mathematics, Andhra University M.A Education, Andhra University B.Sc, Ideal Degree College, Kakinada
06/01 - 09/06 Assistant Professor, C.V.R College of Engineering, Hyderabad, Hyderabad, Telengana 11/11 - 04/12 Assistant Professor, Sir Padampat Singania University, Rajasthan 05/12 - 04/19 Assistant
Professor, Shiv Nadar University, Greater Noida 05/19 ? Present Associate Professor, Department of Mathematics, School of Natural Sciences, Shiv Nadar University, Delhi-NCR
First position for oral presentation on Respectable Graphs, presented in a national seminar held at Brahmanand college, Kanpur on February 12, 2011.
(a) Shivani Chauhan and A. Satyanarayana Reddy, Algebraic connectivity of Kronecker products of line graphs, acceptd for publication in Discrete Mathematics, Algorithms and Applications.
(b) Shivani Chauhan and A. Satyanarayana Reddy, Spectral properties of edge Laplacian matrix, accepted for publication in Proceedings of the Jangjeon Mathematical Society.
(c) A. Satyanarayana Reddy and Kavita Samant, Generating Graphs of Finite Dihedral Groups, accepted for publication in Results in Mathematics.
(d) Shivani Chauhan and A. Satyanarayana Reddy, On the double covers of a line graph, accepted for publication in Contributions to Discrete Mathematics.
(e) A. Satyanarayana Reddy, Pattern polynomial graphs, accepted for publication in Indian journal of pure and applied mathematics.
(f) A. Satyanarayana Reddy and Chandrashekara BM, Elements of Zn, accepted for conference proceedings of International Conference on Graphs and Combinatorics-2022, with ISBN: 978-93-95283-23-6, Dec
1-3, Mangalore University.
(a) Biswajit Koley, A.Satyanarayana Reddy, An irreducible class of polynomials over integers Journal of the Ramanujan Mathematical Society, Soc. 37, No.4 (2022) 319– 330.
(b) Ayush Bhora, A. Satyanarayana Reddy, Permanent of 3×3 invertible matrices modulo n, #A67, INTEGERS 22 (2022).
(c) Veer Singh Panwar, A. Satyanarayana Reddy, Positivity of Hadamard powers of few band matrices, Electronic Journal of Linear Algebra, (2022), Vol.38, 85–90.
(d) Mallika Muralidharan and A. Satyanarayana Reddy, Alternating Sign Matrices, Black board, Bulletin of the Mathematics Teachers’ Association (India), (2022), Issue 4, 37–44.
(a) Ayush Bohra, A. Satyanarayana Reddy, Permanents of 2x2 Matrices Modulo n, The PUMP Journal of Undergraduate Research, (2021) Vol.4, 141–145.
(b) Monimala Nej and A. Satyanarayana Reddy, Exponents of Primitive Symmetric Companion Matrices, Indian J. Discrete Math., Vol. 7, No.1 (2021) pp. 43–66. 3
(a) K. Siddharth Choudary, A. Satyanarayana Reddy, Calkin-Wilf tree, Resonance, Vo.25, No.7, 2020,1001–1013. DOI: https://doi.org/10.1007/s12045-020-1015-x
(b) Siddharth Choudary, K and Satyanarayana Reddy, A. (2020) Number of tuples with a given least common multiple, Notes on Number Theory and Discrete Mathematics, 26
(2), 53–60.
(c) Biswajit Koley, A.Satyanarayana Reddy, An irreducibility criterion for polynomials over integers, Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie, Volume 63 (111)/2020,
Issue no. 1, pages 83-89.
(d) Priyanka Grover, Veer Singh Panwar and A. Satyanarayana Reddy, Positivity properties of some special matrices Linear algebra and applications, Vol 596,(2020), 203–215.
(e) B. S. Shaik, V. K. Chakka and A. Satyanarayana Reddy, Orthogonal and Non-Orthogonal Signal Representations Using New Transformation Matrices Having NPM Structure, IEEE Transactions on Signal
Processing, vol.68,(2020), 1229–1242.
(f) Biswajit Koley and A. Satyanarayana Reddy, Irreducibility of x n −a, The Mathematics Student, Vol. 89, Nos. 1-2, January-June (2020), 169–174.
(a) Monimala Nej, A. Satyanarayana Reddy, A Note on the Exponents of Primitive Companion Matrices, Rocky Mountain journal of Mathematics, Volume 49, No.5, (2019), 1633–1645.
(b) Devendra Prasad, Krishnan Rajkumar, A. Satyanarayana Reddy A survey on Fixed divisors Confluentes Mathematici, 11 no. 1 (2019), 29–52.
(c) B. S. Shaik, V. K. Chakka and A. Satyanarayana Reddy, On Complex Conjugate Pair Sums and Complex Conjugate Subspaces, IEEE signal processing letters, vol 26, issue 9, (2019), 1403–1407.
(d) Biswajit Koley, A.Satyanarayana Reddy, Irreducibility criterion for certain trinomials, Malaya Journal of Matematik, Vol. S, No. 1, (2019), 116–119.
(e) Biswajit Koley, A.Satyanarayana Reddy, Cyclotomic factors of Borwein polynomials,The Bulletin of the Australian Mathematical Society, Vol. 100, Issue- 1 (2019), 41–47.
(f) Abhijit Santha Jayanthan, A. Satyanarayana Reddy Number of non-primes in the set of units modulo n, The Mathematics Student, Vol. 88, Nos. 1-2, January-June (2019), 147–152.
(g) B. S. Shaik, V. K. Chakka and A. Satyanarayana Reddy, A new signal representation using complex conjugate pair sums, IEEE Signal Processing Letters, vol. 26, issue 2 (2019) 252–256.4
(h) Monimala Nej, A. Satyanarayana Reddy, Binary strings of length n with x zeros and longest k-runs of zeros, Indian journal of Mathematics, Vol. 61, No. 1, (2019), 111–139.
(a) Krishnan Rajkumar, A. Satyanarayana Reddy,Devendra Prasad, Fixed divisor of a multivariate polynomial over an arbitrary subset of the Cartesian product of a Dedekind domain,J. Korean Math. Soc.
55 (2018), No. 6, pp. 1305-1320. (https://arxiv.org/abs/1803.05780).
(b) A.K.Lal and A.Satyanarayana Reddy, Non-singular circulant graphs and digraphs, Electronic Journal of Linear Algebra, Volume 26,(2013), 248–257.
(c) A.Satyanarayana Reddy, Shashank K Mehta and A.K.Lal, Representation of Cyclotomic Fields and their Subfields, Indian J. Pure Appl. Math., 44(2)(2013),203–230.
(d) A.Satyanarayana Reddy, Adjacency Algebra of Unitary Cayley Graph, Journal of Global Research in Mathematical Archives, Vol 1, Issue 1(2013), 77–84.
(e) A.Satyanarayana Reddy, Few Non-derogatory Directed Graphs from Directed Cycles, International Journal of Graph Theory, Vol 1,Issue 2(2013), 41–53.
(f) A.Satyanarayana Reddy Respectable Graphs, International Journal of Mathematical Combinatorics, Vol.2 (2011),104–110.
2012 and 2013
(a) Teacher observer in MTTS program May-June 2012 held in IIT Kanpur, sponsored by National Board for Higher Mathematics, Govt of India.
(b) Teacher observer in MTTS program from 20th May 2013 to 1st June 2013, held inNIT,Surat, Gujarat, sponsored by National Board for Higher Mathematics, Govt of India.
(a) As a Resource person in mini-MTTS program from 9th June 2014 to 21st June 2014 conducted at IIT Patna, sponsored by National Board for Higher Mathematics (NBHM).
(b) Resource person in Summer School- 29th June to 20th July 2014, organized by Department of Mathematics, SNU.
(c) As a Resource person in mini-MTTS program from 2nd August 2014 to 12th August 2014 conducted at Jammu Kashmir Institute of Mathematical Sciences, Srinagar, sponsoredby NBHM.
(d) Visiting Professor, Harish Chandra Research Institute, Allahabad, from 17/10/2014 to 24/10/2014.
(a) As a tutor in AFS-II, held at SNU from 4th May to 30th, May, 2015.
(b) As a resource person in MTTS-O-level, from 1st June to 27th June 2015, to be held atSNU, sponsored by NBHM.
(c) As a resource person in AFS-III to be held at HRI, Allahabad from 29th June to 4th July 2015.
(d) As a Resource person in mini-MTTS program from 26th July 2015 to 31st July 2015 conducted at Jammu Kashmir Institute of Mathematical Sciences, Srinagar, sponsored by NBHM.
(e) As an Associate Teacher (Tutor) in IST (Instructional school for Teachers)-Number Theory held at Kerala School of Mathematics, Kozhikode, Kerala, from 5th October to 17th October, 2015.
(a) As an Associate Teacher (Tutor) in AIS(Advanced instructional School)-Matrix analysis held at Shiv Nadar university, Delhi from 2nd May to 22nd May, 2016.
(b) As a resource person in MTTS-O-level, from 30th May to 25th June 2016, to be held at SNU, sponsored by NBHM.
(c) As an Associate Teacher in IST (Instructional school for Teachers)-Algebraic Number Theory, held at S.P. Pune University, Pune, from 3rd October to 15th October 2016.
(d) External examiner for M.Sc and M.Phil Mathematics student’s comprehensive viva, Devi Ahilya university, Indore from 14th December to 16th December, 2016.
(a) As a Resource person for mini-MTTS program from 19th January, 2017 to 24th January 2017 conducted at Tripura University Agartala, sponsored by NBHM.
(b) As a Resource person for MTTS program from 29th May, 2017 to 24th June 2017 held at IIT Indore.
(c) Invited lecture on Applications of 0,1 matrices at Sri H.D Devegowda Government first grade college, Paduvalahippe, Hasan, Karnataka on 22nd September, 2017.
(d) Invited Lecture on Nth roots of unity at Smt Rukmini Shedthi Memorial National Government First Grade College, Barkur, Karnataka on 26th September, 2017
(e) Invited Lecture on Introduction to Algebraic Graph theory at Poornaprajna College, Udipi, Karnataka on 28th September, 2017.
(f) Invited Lecture Series on Linear algebra and Number Theory at Dr. G. Shankar Government Women’s first grade college and PG study center, Udipi, Karnataka. from 23rd September to 30th September,
(g) As a Resource person for mini-MTTS program from 4th December, 2017 to 9th December 2017 conducted at IIT Mandi, sponsored by NBHM.
(h) Invited speaker for Annual Foundation school-1, held at IIT Delhi from 19th Dec to 23rd Dec, 2017.
(a) As a resource person for MTTS program from 21st , May 2018 to 16th June 2018 conducted at SSN college of Engineering, Chennai, sponsored by NBHM.
(b) As a resource person for Mini-MTTS program from 24th , Sep 2018 to 29th Sep 2018 conducted at RIE, Ajmer, Rajasthan, sponsored by NBHM.
(c) Invited to give a seminar on introduction to algebraic graph theory at St Stephen’s College on 23rd October, 2018.
(d) As a resource person for Mini-MTTS program from 3rd December to 8th December,2018 conducted at IIT Mandi, sponsored by NBHM.
(e) As a resource person for Ganitha Poorna-2018, from 27th December to 31st December held at Poornaprajna College, Udipi.
(f) Invited talk on Rings and fields at Dr. G. Shankar Government Women’s first grade college and PG study center, Udipi, Karnataka on 29/12/2018.
(a) As a resource person for Mini-MTTS program from 7th January to 12th January,2019 conducted at SGTB Khalsa College, Sri Anandpur Sahib, sponsored by NBHM.
(b) Associate teacher for AFS-1 (Annual Foundation School-1) from 6th May to 18th May 2019 held at HRI, Prayagraj.
(c) As a resource person for MTTS program from 20th May to 15th June,2019 conducted at IISER Thiruvananthapuram, sponsored by NBHM.
(d) As a resource person for Mini-MTTS program from 26th August to 1st September,2019 held at Silver Jubilee Government College, Kurnool, AP, sponsored by NBHM.
(e) As a resource person for MTTS program from 2nd December to 7th December, 2019 going to be held at IIT, Mandi, sponsored by NBHM.
(f) As a resource person for Ganitha Poorna-2019, from 27th December to 31st December held at Poornaprajna College, Udipi.
(a) As a resource person for mini-MTTS program from 27th January, 2020 to 1st February,2020 held at VNIT, Nagpur, sponsored by NBHM.
(b) As a resource person for mini-MTTS program from 17th February, 2020 to 22nd February,2020 held at DAV College for Women,Firozpur Punjab, sponsored by NBHM.
(c) As an associate teacher for Virtual AFS-1, organized by SGGS, Nanded from 28th April to 30th May.
(d) As a resource person for online MTTS program from 1st June to 20th June 2020.
(e) Invited speaker for five day faculty development program from 13th July to 17th July 2020, organized by Commissionerate of collegiate edducation Govt of Andhra Pradesh.
(f) Invited talk on Division Algorithm and its applications on 8th August,2020, organized by Poornaprajna College and post graduate center, Udipi, Karnataka.
(g) Resource person for Online Foundation Course in Mathematics (OFCM-2020) organized by the MTTS Trust during 04-24 October, 2020.
(h) Resource person for Online SDP on Linear algebra and its applications organized by Department of Mathematics SGTB Khalsa College, Sri Anandpur Sahib, Punjab from 26th October to 7th November
(i) Resource person for 80th Refresher Course in Mathematics and Statistics Sciences for the college and university teachers organized by UGC-Human Development Center, Punjabi University, Patiala
from 16th November to 28th November, 2020.
(j) Invited talk on Pigeonhole Principle and Its Applications organized on 8th November, 2020 by Math club of MTTS Alumni.
(a) Invited talk, Pi and Primes, on 14th March, 2021 organized by Adamas University, Kolkata.
(b) Participated in many activities related to MTTS, some times as an instructor, some times as a coordinator.
(c) Organizer and associate teacher for the Annual foundation school-II, held at Shiv Nadar University, 29 Nov 2021 to 8 Jan 2022.
(a) Academic Coordinator for online OFCM-Followup-Real analysis, 9th Jan to 21st Jan 2022.
(b) Academic Coordinator for online OFCM-Followup-Linear algebra, 31st Jan to 13 Feb 2022.
(c) Resident Faculty for InIt Mathematics program held at IIT Bhilai from 7th March to 12th March, 2022 sponsored by NBHM.
(d) Invited talk on “Applications of Coding theory and Cryptography” in an annual symposium “Real-life Applications of Mathematical Concepts” organized by The Mathematics Society and the Department
of Mathematics of St. Stephen’s College, Delhi on 22nd April, 2022.
(e) Invited as a resource person for Ganitha Poorna 2021-22, A Mathematics Nurture Programme from 9th May to 13th May, 2022 at the Poornaprajna College and Postgraduate Centre, Udupi.
(f) Resident faculty for 30th MTTS Level-O program held at IIT Ropar from 23/05/2022 to 18/06/2022.
(g) Resource faculty for Online Student Development Programme on Foundations of Real Analysis for the students of undergraduate classes August 06, 2022 to October 22, 2022 organized jointly by
Department of Mathematics, Sidharth Government “Utkrisht” College Nadaun, Distt. Hamirpur, Himachal Pradesh Indian Science Congress Association (ISCA), Shimla Chapter.
(h) Resource faculty for Online foundation Course in Mathematics (OFCM-2022), from 16/10/2022 to 05/11/2022, organized by MTTS trust, funded by NBHM.
(i) Invited to give a short course on foundations to M.Sc students of Dr. G. Shankar Government Women’s first grade college and PG study center, Udipi, Karnataka, Nov 28-30, 2022.
(a) Invited as a resource person for Ganitha Poorna 2022-23, A Mathematics Nurture Programme from 9th March to 13th March, 2023 at the Poornaprajna College and Postgraduate Centre, Udupi.
(b) Invited to give a talk on Group actions to M.Sc students of Dr. G. Shankar Government Women’s first grade college and PG study center, Udipi, Karnataka on 11th March, 2023.
(c) Invited to give a talk on Pi and Primes in a Webinar conducted on 14/03/2023 by the Department of Mathematics, Government Science College, Hassan.
(d) Resident faculty for the program Initiation into Mathematics program held at Mangalore University from 13/03/2023 to 18/03/2023.
(e) Resident faculty for 31st MTTS Level-1 program held at IIT Madras from 22/05/2023 to 17/06/2023.
(f) Resident faculty for the program Initiation into Mathematics program held at Jammu & Kashmir Institute of Mathematical Sciences, Srinagar during July 10–18, 2023.
(g) Resource person for Online Foundation Course in Mathematics (OFCM-2023) organized by the MTTS Trust, 20/08/2023 to 02/09/2023, 2023.
Conference Presentations
1. A.Satyanarayana Reddy, Bose-Mesner algebra, open house-08, Department of mathematics and statistics, IITK, India.
2. A.Satyanarayana Reddy, Non-singular η(x)-circulant digraphs, Presented in an International conference IMST-2010-FIM -IXX at Patna University, Patna, India.
3. A.Satyanarayana Reddy, Association schemes from generously transitive graphs, Presented in a national workshop on Block designs and its applications, CMS, Banasthali University, Rajasthan on
February 8, 2011.
4. A.Satyanarayana Reddy, Respectable Graphs, Presented in a national seminar held at Brahmanand college, Kanpur on February 12, 2011.
5. A.Satyanarayana Reddy, Pattern Polynomial graphs, presented in Spring school in discrete probability, ergodic theory and combinatorics held in Graz university of Technology, Graz, Austria from
April 4 to April 15, 2011.
6. As a participant, 2011 Summer School Random Motions and Random Graphs, TU, Berlin, Germany.
7. A.Satyanarayana Reddy, Adjacency Algebra of Unitary Cayley Graph, 27th Annual Conference of the Ramanujan Mathematical Society (RMS), Organized by SNU.
8. A. Satyanarayana Reddy, Shashank K Mehta and A.K.Lal,Pattern Polynomial graphs, IndiaTaiwan Conference on Discrete Mathematics, NCTU Taiwan, November 18-22, 2013.
9. Remya Krishnan, A.Satyanarayana Reddy, and Pawan K. Dhar, Computational Analysis of miRNA “hubs” towards Therapeutic Applications at the conference and publication in Proceedings of the Kuala
Lumpur conference Sept. 2014.
10. Invited talk on Euler’s Product formula at “One-day national workshop on analysis and topology” on 27th September, 2017 Dr. G. Shankar Government Women’s first grade college and PG study center,
Udipi, Karnataka.
11. Invited talk on Graph Algebras, Two day conference on Graph theory and Discrete Mathematics, 5−6th February 2018, held at Mangalore University.
12. Invited talk on Exponents of primitive graphs, Two day national conference on recent advancements in graph theory, November 9-10, 2019 held at Gujarat University, Ahmadabad.
13. Invited talk on Edge Adjacency Matrix of a graph, Two day National Conference on emerging trends in Graph Theory (NCETGT-2020), February 27-28, 2020 held at Christ University, Bangalore.
14. Prof M. Rajesh Kannan started weekly seminar series ( http://www.facweb.iitkgp.ac.in/˜rkannan/gma.html) on Graphs, Matrices and Applications, presented a talk on Adjacency algebra of a Graph on 9
15. Presented a talk entitled “Primitive Companion Matrices” at the 24 th Conference of the International Linear Algebra Society at National University of Ireland, Galway, June 20-24, 2022.
16. invited lecture on Polynomials with integer coefficients divisible by cyclotomic polynomials during the Conference on Algebra, Analysis and Applications on 05 August 2022 at Dr. B. R. Ambedkar
University Delhi (AUD).
17. Invited talk on Numbers with a group theoretic property, National Conference on Scope of Research in Discrete mathematics as per NEP” (NCSRDMN-2022), Nov 28-30, Udipi.
18. Invited talk on Elements of Zn, International Conference on Graphs and Combinatorics-2022, Dec 1-3, Mangalore University.
19. Invited talk on On the number of cyclic subgroups of a finite group, 7th Dec, 2022 SRM-AP.
20. Contributed talk on Generating graphs of finite Dihedral group 2023 Ural Workshop on Group Theory and Combinatorics Yekaterinburg-Online, Russia, August 21–27, 2023.
Poster presentations
1. Presented a poster on Fibonacci numbers in the open house, as a part of Golden Jubilee celebrations of IITK, Kanpur.
2. Presented a poster on Circulant Matrices , Departmental day of Mathematics and Statistics, as a part of Golden Jubilee celebrations of IITK, Kanpur.
Serb Matrix project File NO. MTR/2019/001206 for three years 2020 to 2022 titled Polynomials with integer coefficients divisible by cyclotomic polynomials.
1. Faculty advisor of Math club of MTTS alumni
2. Invited member of MTTS trust | {"url":"https://snu.edu.in/schools/school-of-natural-sciences/faculty/satyanarayana-reddy/","timestamp":"2024-11-12T22:24:02Z","content_type":"text/html","content_length":"82154","record_id":"<urn:uuid:277bcba7-f2cc-4f16-b1d8-dd25ff1a1d32>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00415.warc.gz"} |
Response of Spring–Mass System to a Step Function
Transient Vibrations: Response of Spring–Mass System to a Step Function
As we have discussed so far, in many situations the long term (steady state) response of a vibrating system is of interest. For example, for a motor that will operate more than 99% of the time at a
fixed operating speed we would typically want to investigate the response at that speed so an appropriate vibration isolation system could be designed.
On the other hand, there are times when it is the transient response of the system that is of interest.
• For the motor example above, we may be concerned about the response at startup, particularly if the system passes through resonance.
• The response of an automobile suspension system when a pothole or other obstacle in the road is encountered.
• The response of various mechanical systems to shock loading.
In these situations, the loading isn’t periodic or is applied for a short period of time. We will look at the responses for some specific examples. We will limit ourselves generally to undamped
systems for simplicity so that the general form of the equation of motion we are concerned with here is
Response of a Spring-Mass System to a Step Function
(a) Schematic
(b) FBD/MAD (x, x¨ > 0)
Consider a simple spring–mass system subjected to a rectangular step load as shown in Figure 7.1(a). We would like to find the response of the mass to this transient loading condition. Figure 7.1(b)
shows the associated FBD/MAD. The equation of motion obtained using Newton’s laws is
As usual, the solution is composed of the homogeneous and particular solutions
For the RHS in (7.2), the particular solution is
so the total solution becomes
If we start with the initial conditions
the constants* become
so that the total solution is
*Note that the total solution (including the particular solution) must be used to determine the arbitrary constants which arise from the homogeneous solution.
Figure 7.2: Response of simple spring–mass system to applied step load
This response is illustrated in Figure 7.2. We can see that the maximum displacement of the mass is
where 7.3). Similarly, the velocity of the mass during this period is, by differentiation,
Therefore, at the time
which can be used to find
Consider two specific examples
As a result,
so the response in this case is
As a result,
so the response in this case is
These responses are shown in Figure 7.3. As these responses clearly show, the transient nature of the applied loading can have a significant effect on the resulting response.
Figure 7.3: Response of simple spring–mass system to applied step load which is subsequently removed
(a) τ =2π/p
Figure 7.3: Response of simple spring–mass system to applied step load which is subsequently removed
(b) τ = π/p
The previous responses can be understood on a physical basis. In the first case,
Figure 7.4: Shift of equilibrium position
At time free vibration problem with the initial conditions
so the free vibration response is
However, since
as before.
The two cases shown in Figure 7.3 can be understood on similar principles. When the force
Response of Spring–Mass System to an Impulse
Figure 7.5: Impulse Loading
We now consider the response of a spring–mass system subjected to an impulsive loading shown in Figure 7.5. (Note that the quantity impulse applied to the system.) Here the time
In such a case,
The “initial” conditions are approximately
so the initial conditions (after impulse has been applied) become
The constants
so the response becomes
This response will play an important role later when we consider the method of convolution.
We can obtain this result another way by applying the basic principle of impulse and momentum for our system.
Figure 7.6: Impulse–momentum diagram for mass m
Figure 7.6 shows an impulse–momentum diagram for our simple spring mass system when the impulsive load is applied. The system is initially at rest and after the impulse has been applied has a
resulting velocity non-impulsive.)
Applying the principle of impulse and momentum shows that
so we get
After the impulse is over, we have a free vibration problem with the initial conditions
The two constants are
so the response becomes
as in equation (7.4).
We can also consider the response to some additional useful forcing functions as in the next sections.
Video Lecture: Transient Vibrations Section 7.1 | {"url":"https://engcourses-uofa.ca/books/vibrations-and-sound/transient-vibrations/response-of-spring-mass-system-to-a-step-function/","timestamp":"2024-11-13T02:40:00Z","content_type":"text/html","content_length":"88051","record_id":"<urn:uuid:0ac965b2-02dc-4e58-843a-c653f19c75d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00706.warc.gz"} |
What is Breadth-First Search
What is Breadth-First Search?
What is Breadth-First Search
When working with Binary Search Tree as a programmer you will be asked or required to traverse through the nodes. One concept you want to have in your arsenal is the “Breadth-First Search” technique.
The basic method to this is to start from the root node and add all the additional nodes from the right or left to the queue. Once added to the queue it can be pushed to the final array. This
technique is used for finding the shortest path in a graph.
How Does it Work
In order to start this method, we need to work off a Binary Tree. Create a variable and set it equal to the root node. Here’s what it looks like below.
let currentNode = this.root}
In addition, we will create two more variables and set both to empty arrays. The reason for this as explained earlier we will add the nodes to the queue, which will then be added to the result array.
Furthermore, we will start off by pushing the root node to the queue. Here is a clear example below.
let currentNode = this.root let queue = []
let result = []
In this particular step, we initiate the traversing by looping through the queue. Create a while loop that takes an argument of queue.length. If the queue is empty the loop will not start.
let currentNode = this.root let queue = []
let result = []
queue.push(currentNode) while(queue.length){ }}
Once the loop is initiated will set the variable we had earlier for the current node to queue.shift, which will remove the node in the queue.
let currentNode = this.root let queue = []
let result = []
queue.push(currentNode) while(queue.length){
currentNode = queue.shift() }}
We will add that current node value to the result. Here’s a better example below.
let currentNode = this.root let queue = []
let result = []
queue.push(currentNode) while(queue.length){
currentNode = queue.shift()
result.push(currentNode.value) }}
The last steps are to add the conditional for both left and right nodes. If the condition passes it will then be added to the queue. In addition, the last step is to return the final/result array.
let currentNode = this.root let queue = []
let result = []
queue.push(currentNode) while(queue.length){
currentNode = queue.shift()
if(currentNode.left) queue.push(currentNode.left)
if(currentNode.right) queue.push(currentNode.right)
} return result
Thanks a lot for reading stay tuned for Depth First Search. | {"url":"https://woodelinflorveus.medium.com/what-is-breadth-first-search-f9896d975a68","timestamp":"2024-11-05T12:43:05Z","content_type":"text/html","content_length":"103591","record_id":"<urn:uuid:f7d239ea-8e88-4897-a9f0-db49f8ba8437>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00355.warc.gz"} |
If Tomorrow
How do we solve it ?
If tomorrow is thursday, today is wednesday.
A week has 7 days " Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday ".
78 / 7 = 11 ( Remainder = 1)
So you can say that after multiples of 7 days, the day will repeat itself as wednesday.
Hence after 77 days wednesday will occur again.
So we calculate after the remainder (1) , the answer is Thursday.
Simple Logic = And whatever the remainder obtained is, add that to the day. | {"url":"https://timedate.org/if-tomorrow-is-thursday-after-78-days","timestamp":"2024-11-09T03:10:38Z","content_type":"text/html","content_length":"4993","record_id":"<urn:uuid:d6683685-c02d-4a17-9956-d9f78d86c28f>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00804.warc.gz"} |
by Cockcroft-Gault Equation Creatinine Clearance Estimate by Jelliffe Formula Formula (for older subjects with low muscle mass) Creatinine Clearance
ambula- tory care center; amylase creatinine clearance; automated cell count black single female/male BSG brain stem gliomas BSI bloodstream infection; body distal forearm DFD defined formula diets;
degenerative facet disease DFE
No additional details found for the CrCl Cockroft-Gault calculator. References. Cockcroft DW, Gault MH. Prediction of creatinine clearance from serum creatinine. Nephron. 1976;16(1):31-41. Related
In-Calculator sökning och växlingsfunktion. Cockcroft-Gault CrCl = [ (140-age) x (Wt in kg) x (0.85 if female)] / (72 x Cr) Note: The original Cockroft-Gault Equation listed here provides an estimate
of creatinine clearance, but this original formula is not adjusted for body surface area and may be less accurate in obese patients. The Cockroft-Gault Equation typically overestimates creatinine
clearance by approximately 10 to 30%. The resulting CrCl is multiplied by 0.85 if the patient is female to correct for the lower CrCl in females. The C-G formula is dependent on age as its main
predictor for CrCl.
Created by. 0/4 completed. Start.
Practice Calculation. Calculate the creatinine clearance of a 41-year old female patient with 168 lb of weight. The serum creatinine level of the patient is 0.91 mg/dL. Cockroft-Gault formula: CrCl m
= 140 - Age Serum Cr × Weight in Kg 72. C r C l m = 140 − A g e S e r u m C r × W e i g h t i n K g 72. CrCl f = CrCl m × 0.85.
The serum creatinine level of the patient is 0.91 mg/dL. Cockroft-Gault formula: CrCl m = 140 - Age Serum Cr × Weight in Kg 72. C r C l m = 140 − A g e S e r u m C r × W e i g h t i n K g 72.
Patient's sex? Male, Female. Patient's age ? years. Patient's actual weight ? kg The dose calculator tool is intended to help in calculating initial Gentamicin doses when The Cockcroft and Gault
Creatinine Clearance ha
years. Patient's actual weight ? kg The dose calculator tool is intended to help in calculating initial Gentamicin doses when The Cockcroft and Gault Creatinine Clearance ha 6 Aug 2020 Retooling the
creatinine clearance equation to estimate kinetic GFR when the plasma creatinine is changing acutely. J Am Soc Nephrol 2013; 24: 7 Jul 2019 Female: 88 to 128 mL/min (1.496 to 2.18 mL/s).
Screening and Diagnosis: Self-Study CNE/CME; Lessons. HCV Epidemiology in the United States CRCL : Estimated GFR Using Serum Creatinine Alone: Estimated glomerular filtration rate (eGFR) is
calculated using the 2009 CKD Epidemiology Collaboration (CKD-EPI) equation: eGFR(CKD-EPI) =141 x min(Scr/k, 1)alpha x max(Scr/k,1)-1.209 x 0.993 age x 1.018 (if patient is female) x 1.159 (if
patient is black) -where age is in years -k is 0.7 for females and 0.9 for males -alpha is -0.329 for 2016-11-10 · Renal impairment is a major risk factor for mortality in various populations. Three
formulas are frequently used to assess both glomerular filtration rate (eGFR) or creatinine clearance (CrCl) and mortality prediction: body surface area adjusted-Cockcroft–Gault (CG-BSA),
Modification of Diet in Renal Disease Study (MDRD4), and the Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI The CrCl Cockroft-Gault calculator is created by QxMD. Created by on . 01/09/
2015 Female. Download the app for offline access.
Sälja bostadsrätt tips
Note that renal impairment (creatinine clearance <60 ml/min) increases the plasma concentration of oxycodone by 50%.
Why is the Cockcroft-Gault formula provided on this website? The Cockcroft-Gault (CG) formula is provided on this website for research purposes only.
Nobel 1912
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30 Apr 2008 Height (m) Male Female 1.52 50 45.5 1.55 52.3 47.8 1.57 54.6 50.1 Calculated Creatinine Clearance (CrCl): Cockcroft-Gault equation
0.85. In cases of persons of extreme 30 Apr 2008 Height (m) Male Female 1.52 50 45.5 1.55 52.3 47.8 1.57 54.6 50.1 Calculated Creatinine Clearance (CrCl): Cockcroft-Gault equation CRCL : Estimated
GFR Using Serum Creatinine Alone: Estimated glomerular 1 )alpha x max(Scr/k,1)-1.209 x 0.993 age x 1.018 (if patient is female) x 1.159 (if An alternative creatinine-based GFR estimating equation is
acceptable if i Renal functions include maintaining an acid-base balance; regulating fluid balance; regulating Creatinine clearance rate (CCr or CrCl) is the volume of blood plasma that is cleared of
The C-G equation assumes that a woman will ha Schwartz classical formula in patients with GFR24 ≥ 90ml/min/1.73m2 estimated The concordance was better in patients with obesity and lower in women,
Figure 1a shows UCr to be lower in women than in men and lower with older age , Table 2 compares measured urinary CrCl with eCrCl by each equation.
Systemair save vtrliu anslagstavlan litteratur
Cockroft & Gault formula {(140 - Age) x 1.2 wt (kg) x F} / (Serum Creatinine in µmol/L) Where F = 1 if male, and 0.85 if female. Salazar-Corcoran formula*
2020 — In addition, more females, about 15% were failed to detect by SCr method in contrast to males of 9%.
For healthy males, the discharge of creatinine is 95mL/minute while for females, this figure increases to 120mL/minute. CRCL facts and figures. CRCL is calculated on the basis of a properly defined
equation called the Cockcroft-Gault Equation. To calculate CRCL, you need to use a dependable calculator designed for this purpose.
Related Calculators.
§Cardiodascular and predicted creatinine clearance. Ann Clin. There was also a poor correlation between iohexol clearance and calculated creatinine clearance using the Cockcroft-Gault (R2=0.046) or
MDRD formula Results 790 participants (432 females), mean age (SD) 77.6 +/- 5.7years. | {"url":"https://valutaoxdv.web.app/94493/38507.html","timestamp":"2024-11-06T15:22:51Z","content_type":"text/html","content_length":"11607","record_id":"<urn:uuid:8c810eb2-2caa-4b03-b872-0eb3de64a8be>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00512.warc.gz"} |
McGraw Hill My Math Grade 4 Chapter 6 Lesson 6 Answer Key Interpret Remainders
All the solutions provided in McGraw Hill Math Grade 4 Answer Key PDF Chapter 6 Lesson 6 Interpret Remainders will give you a clear idea of the concepts.
McGraw-Hill My Math Grade 4 Answer Key Chapter 6 Lesson 6 Interpret Remainders
Math in My World
Example 1
Mandy wants to buy 4 books that each cost the same amount. If the total cost is $74, how much does each book cost?
Divide 74 by 4.
The remainder shows that each book will cost a little more than _____.
The remainder shows that each book will cost a little more than 18.
Example 2
Gracie has 64 prizes. She will put 3 prizes in each bag. How many bags will she have? Interpret the remainder.
So, Gracie will have ____ bags. 64 ÷ 3 = ___ R ____
The remainder, ____, shows the number of prizes that Gracie will have left.
So, Gracie will have ____ prize left.
____ × 3 = ____ Multiply.
____ + 1 = ____ Add the remainder.
Gracie will have 21 bags. 64 ÷ 3 = 21 R 1
The remainder, 1, shows the number of prizes that Gracie will have left.
So, Gracie will have 1 prize left.
21 × 3 = 63 Multiply.
63 + 1 = 64 Add the remainder.
Talk Math
What kind of information can you get from a remainder?
Guided Practice
Question 1.
There are 45 people waiting for a bus: Each seat holds 2 people. How many seats will be needed? Divide. Interpret the remainder.
45 ÷ 2 = ___ R ____
So, ____ seats will be needed.
45 ÷ 2 = 22 R1
22 seats will be needed.
Independent Practice
Divide. Interpret the remainder.
Question 2.
Gianna is at the school carnival. She has 58 tickets. It costs 3 tickets to play the basketball game. If she plays the basketball game as many times as she can, how many tickets will she have left?
58 ÷ 3 = ____
So, there is ____ ticket left.
There are 19 tickets left.
Question 3.
There are 75 people waiting in line to ride a roller coaster. Each car of the roller coaster holds 6 people. How many cars will be needed?
75 ÷ 6 = ____
The answer is the next whole number, ____
So, they will need ____ cars.
The answer is the next whole number is 13. So, they will need 13 cars.
Question 4.
There are 4 cartons of orange juice in each package. If there are 79 cartons of orange juice, how many packages can be filled?
79 ÷ 4 = ____
So, ____ packages can be filled.
A total of 20 packages can be filled.
Question 5.
The fourth grade classes are going on a field trip. There are 90 students in all. Each van can seat 8 students. How many vans will be needed?
90 ÷ 8 = ____
The answer is the next whole number, _____
So, they will need ___ vans
The answer is the next whole number 11. So, they will need 11 vans.
Problem Solving
For Exercises 6 and 7, use the following information.
Parents are driving groups of children to the science center. Each van holds 5 children. There are 32 children in all.
Question 6.
Mathematical PRACTICE 2 Reason How many vans are needed?
The total number of vans needed is 7.
Total number of children = 32
Each van has 5 children
To find: How many vans are needed?
32 ÷ 5 = 6.5
By rounding off to the nearest one the answer is 7.
Question 7.
Circle the true statement about the remainder.
• You do not need to know anything about the remainder to solve this problem.
• The remainder tells you that the answer is the next greatest whole number.
• The remainder is the answer to the question.
The remainder tells you that the answer is the next greatest whole number.
HOT Problems
Question 8.
Mathematical PRACTICE 2 Use Number Sense Brody is organizing his action figures on a shelf. He wants to divide them equally among 4 shelves. There are 37 action figures. Brody says he will have 2
left over. Find and correct his mistake.
Brody will have 1 action figure left over with each of the 4 shelves.
There are 37 action figures.
He wants to divide them equally among 4 shelves.
37 ÷ 4 = 9.25
So now multiply to check
9 × 4 = 36
Now subtract 37 – 36 = 1
So, Brody will have 1 action figure left over with each of the 4 shelves.
Question 9.
? Building on the Essential Question Why is it important to know how to interpret a remainder?
Interpreting remainders will help us to develop problem-solving skills. We will logically reason if a remainder should be dropped, rounded up or shared that is it is turned into a fraction or
decimal) depending on the given question.
McGraw Hill My Math Grade 4 Chapter 6 Lesson 6 My Homework Answer Key
Divide. Interpret the remainder.
Question 1.
Tyler is planting 60 trees at the apple orchard. He will plant 8 trees in each row. How many full rows of trees will Tyler plant?
____ rows
7 rows.
Question 2.
Ms. Ling bought party hats for the 86 fourth graders at her school. The hats come in packages of 6. How many packages did Ms. Ling buy?
____ packages
15 packages.
Problem Solving
Question 3.
Wezi has sixty-eight $1-bills. He takes the $1 -bills to the bank to exchange them for $5-bills. How many $5-bills does Wezi get?
Yes, he has 13 $5 bills.
No of $1-bills = 68
He takes the $1 -bills to the bank to exchange them for $5-bills.
68 ÷ 5 = 13
Hence Wezi gets 13 $5 bills.
Question 4.
Mathematical PRACTICE 2 Reason Henry decorates the top of each cupcake with 3 walnuts. If he has 56 walnuts, is that enough to decorate 2 dozen cupcakes? Explain.
19 cupcakes.
There are 56 walnuts in total and we put 3 walnuts on each cupcake.
So we have to divide 56 by 3.
56 ÷ 3 = 19 approx.
since 1 dozen = 12 cupcakes
2 dozen = 24 cupcakes.
Therefore he only is able to decorate approximately
Test Practice
Question 5.
Janice bought juice packets for the 15 players on the soccer team. The juice packets come in boxes of 6. How many boxes did Janice buy?
A. 5 boxes
B. 4 boxes
C. 3 boxes
D. 2 boxes
3 boxes.
that Janice bought juice packets for the 15 players on the soccer team. The juice packets come in boxes of 6.
To find the number of boxes that Janice need to buy for 25 players, we need to divide 15 by 6.
Since the juice packets come in boxes of 6, so we will round to the nearest whole number.
Therefore, Janice will buy 3 boxes of juice packets for 15 players.
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Drawing A Slope Field
Drawing A Slope Field - Equation from slope field worked example: Web a slope field is a visual representation of a differential equation in two dimensions. Web given a differential equation in x and
y, we can draw a segment with dy/dx as slope at any point (x,y). The slope field is utilized when you want to see the tendencies of solutions to a de, given that the solutions pass through a certain
localized area or set of points. Web how to sketch the slope field brian mclogan 1.27m subscribers join subscribe like share save 3.7k views 4 years ago differential equations learn how to create
slope fields and sketch the.
Learn how to draw them and use them to find particular solutions. Web a slope field is a visual representation of a differential equation in two dimensions. And this is the slope a solution \(y(x)\)
would have at \(x\) if its value was \(y\). That's the slope field of the equation. Discover any solutions of the form y= constant. Clearly, t t t is the independent variable, and y y y is a function
of t. Wolframalpha.com type your deqs into wolfram (for example if y′ f x,y ) using syntax like:
How to sketch direction fields — Krista King Math Online math help
A slope field doesn't define a single function, rather it describes a class of functions which are all solutions to a particular differential equation. Using a visualization of a slope field, it is
easy to. Sketching slope fields slope fields introduction worked example: Web this calculus video tutorial provides a basic introduction into slope fields..
Slope field plotter GeoGebra
Web draws the slope (direction) field for the given differential equation y' = f(x,y).the movable black point sets the initial condition of an approximated particular solution drawn with euler's
method. Web this calculus video tutorial provides a basic introduction into slope fields. Draw conclusions about the solution curves by looking at the slope field. Therefore.
Slope Fields
Web brian mclogan 1.29m subscribers 3.7k views 5 years ago differential equations learn how to create slope fields and sketch the particular solution to a differential equation. For instance, suppose
you had the differential equation: Therefore by drawing a curve through consecutive slope lines, you can find a solution to the differential equation. Sketching slope.
Slope Fields Calculus YouTube
Slope field from equation worked example: A direction field (or slope field / vector field) is a picture of the general solution to a first order differential equation with the form. Wolframalpha.com
type your deqs into wolfram (for example if y′ f x,y ) using syntax like: Web practice this lesson yourself on khanacademy.org right.
Sketch the slope field and sketch the particular equation YouTube
Web this calculus video tutorial provides a basic introduction into slope fields. Web explore math with our beautiful, free online graphing calculator. We'll illustrate this with a simple example:
We'll learn in a few sections how to solve this kind of equation, but for now we can't get an explicit solution. Web 5 years ago.
Slope Fields_Example 2 on how to sketch a slope field YouTube
Match a slope field to a differential equation. Slope field from equation worked example: Web practice this lesson yourself on khanacademy.org right now: Match a slope field to a solution of a
differential equation. Y ′ = t + y y' = t + y y ′ = t + y. Graph functions, plot points,.
Worked example slope field from equation AP Calculus AB Khan
That's the slope field of the equation. A slope field doesn't define a single function, rather it describes a class of functions which are all solutions to a particular differential equation. Draw
conclusions about the solution curves by looking at the slope field. Clearly, t t t is the independent variable, and y y y.
PPT Slope Field & Particular Solutions PowerPoint Presentation ID
That's the slope field of the equation. Web 5 years ago observe that you can draw infinitely many possible graphs for a given slope field. Equation from slope field worked example: Learn how to draw
them and use them to find particular solutions. Forming a slope field slope fields & equations slope fields & equations.
How do you draw the slope field of the differential equation \\[{{y
That's the slope field of the equation. So each individual point of a slope field (or vector field) tells us the slope of a function. At a point \((x,y)\), we plot a short line with the slope \(f.
Match a slope field to a solution of a differential equation. Equation from slope field worked example:.
Calculus AB/BC 7.3 Sketching Slope Fields YouTube
Web slope fields allow us to analyze differential equations graphically. Web given a differential equation in x and y, we can draw a segment with dy/dx as slope at any point (x,y). The beauty of
slope field diagrams is that. Y ′ = t + y y' = t + y y ′ = t.
Drawing A Slope Field Slope field from equation worked example: Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. The pattern produced by the slope
field aids in visualizing the. For dy dx x2 −2, this would be slope field x2 −2. This shows us the rate of change at every point and we can also determine the curve that is formed at every single
Draw Conclusions About The Solution Curves By Looking At The Slope Field.
We'll illustrate this with a simple example: The slope field is utilized when you want to see the tendencies of solutions to a de, given that the solutions pass through a certain localized area or
set of points. Y ′ = t + y y' = t + y y ′ = t + y. The beauty of slope field diagrams is that.
Web Given A Slope Field, Sketch A Solution Curve Through A Given Point.
Sketching slope fields slope fields introduction worked example: The vectors in a slope field are usually drawn without arrowheads, indicating that they can be followed in either direction. So each
individual point of a slope field (or vector field) tells us the slope of a function. Discover any solutions of the form y= constant.
See How We Determine The Slopes Of A Few Segments In The Slope Field Of An Equation.
Web the slope field is a cartesian grid where you draw lines in various directions to represent the slopes of the tangents to the solution. And this is the slope a solution \(y(x)\) would have at \(x
\) if its value was \(y\). Slope field from equation worked example: Web drawing paths in the plane that are parallel to the nearby slope marks (as in the graph above) gives you asolution curve, a
curve representing a solution to your deq.
Web This Calculus Video Tutorial Provides A Basic Introduction Into Slope Fields.
Web you are essentially correct. Web a slope field is a visual representation of a differential equation in two dimensions. Web explore math with our beautiful, free online graphing calculator. This
shows us the rate of change at every point and we can also determine the curve that is formed at every single point.
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Let f(x)={min(x,x2),max(2x,x−1),x≥0x<0, then which of the fol... | Filo
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(b) ,
Clearly it is non-differentiable at and 1 .
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Question Text Let , then which of the following is not true ?
Topic Continuity and Differentiability
Subject Mathematics
Class Class 12
Answer Type Text solution:1
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Subtract Trinomials
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The Story of the quest for a Unified Theory
Posted inPhysics Science
The Story of the quest for a Unified Theory
Originally asked: Is there a unified theory that explains all fundamental forces of nature, including gravity?
There isn’t a currently accepted theory that explains all fundamental forces of nature, including gravity. This all-encompassing theory is often called a Theory of Everything (TOE) or a Grand Unified
Theory (GUT).
Here’s the current situation:
• The Standard Model: This incredibly successful theory describes three of the four fundamental forces: electromagnetism, the strong nuclear force, and the weak nuclear force. It explains how these
forces interact with various subatomic particles.
• Gravity: General Relativity, developed by Einstein, is the reigning theory of gravity. It describes gravity as a curvature of spacetime caused by mass and energy.
The challenge lies in bridging the gap between these two successful theories. Here’s why:
• Quantum Mechanics vs. General Relativity: The Standard Model and Quantum Mechanics work well in the microscopic world, but they don’t seamlessly integrate with General Relativity, which thrives
in the macroscopic realm. Quantum mechanics relies on probability and wave functions, while General Relativity depicts spacetime as smooth. Unifying these contrasting viewpoints is a hurdle.
• Gravity’s Odd One Out: Gravity is fundamentally different from the other three forces. It’s much weaker, and current theories haven’t been able to successfully describe it within the quantum
mechanics framework.
Also read: Clock behavior at High Velocities — Temporal Flux?
Unified Theories in Work:
Physicists are actively searching for a GUT or TOE that incorporates all the forces, including gravity. Here are some promising areas of exploration:
• String Theory: This theory proposes that fundamental particles aren’t point-like but instead tiny vibrating strings. The specific vibrations of these strings determine the different types of
particles and forces, potentially including gravity.
• Loop Quantum Gravity: This theory suggests spacetime itself is quantized, meaning it has a granular structure at the incredibly small Planck scale. It attempts to reconcile gravity with the
quantum world.
• Modified Gravity Theories: These approaches tweak Einstein’s equations of General Relativity to account for quantum effects and potentially unify gravity with the other forces.
The Road Ahead:
While there’s no single, accepted theory of everything yet, physicists are constantly developing and testing new ideas. String theory, loop quantum gravity, and modified gravity theories are actively
being researched. Future experiments and discoveries might provide the key to unifying all the fundamental forces, including gravity, under a single theoretical framework.
Also Read: Apart from the moon landings, who has been the highest in orbit/space?
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MATH 560 - Introduction to Mathematical Biology
Course overview
MATH 560 provides a broad overview of Mathematical Biology at an introductory level. The scope is obviously subject to the limitations of time and instructor knowledge and interests - this is a HUGE
area of research.
It is intended for early stage math bio grad students, general applied math grad students interested in finding out more about biology applications, and grad students in other related departments
interested in getting some mathematical and computational modelling experience.
The course is organized around a sample of topics in biology that have seen a significant amount of mathematical modelling over the years. Currently, I'm including content from ecology, evolution and
evolutionary game theory, epidemiology, biochemistry and gene regulation, cell biology, electrophysiology, developmental biology. However, this list changes gradually from year to year, to reflect
students' and my own interests. The mathematical modelling methods and techniques covered are those that typically arise in the biological applications listed above. For example, I will cover models
using ordinary and partial differential equations, stochastic processes, agent-based models and introduce techniques from bifurcation theory, asymptotics, dimensional analysis, numerical solution
methods, and parameter estimation. An emphasis will be placed on reading and discussing classic and current papers.
The course does not have any official prerequisites listed on the UBC calendar but it is expected that students will have some experience with differential equations (acceptable: MATH 215, better:
MATH 361 and/or MATH 316) and some familiarity with the ideas of probability and/or statistics (e.g. MATH 302 and/or STAT 200). If you aren't sure if you have the right background, come talk to me.
• Three hours per week in class. Roughly half the time will be me lecturing, a quarter students working on problems in class, and a quarter discussing papers.
• Reading papers. One paper will be assigned every week. With a partner, you will read and discuss each paper (outside of class) and write a short report on it. Every two weeks, we will discuss the
most recent two papers in class. Everyone will be expected to contribute to each discussion, at the minimum by saying a few words about what they got out of the paper.
• Written homework. I may give a small collection of practice exercises as a warm up / background to some of the papers (not weekly, no more than 4-5 throughout the term when the papers require
it). If your background for the course is suitable, this will be light work.
• Project. See details below.
Learning goals
• Become comfortable reading papers on mathematical modelling in biology.
• Develop the ability to go from a question about a biological phenomenon observed or read about to building a model that can answer (part of) that question.
• Discern the best modelling framework (ODE, PDE, stochastic process, agent-based model etc.) for answering a particular biological question.
• Have an awareness of the analytical tools that might be required in formulating such answers (e.g. bifurcation theory, parameter estimation, matched asymptotics).
Weekly readings
Each week, there will be assigned readings. The "discussion" papers in the table below are to be read in preparation for our biweekly discussions. Together with a partner in the class, you are to
hand in a report on the paper after reading and discussing it with your partner. You should focus on the structure of the paper as well as the content. Items to consider in your report:
• Structure of the paper - How is the content organized? For example, TAIMRD is a common scientific format. Does any information seem out of place? What content is omitted or buried? For example,
is all the mathematical analysis presented or in an appendix/ supplemental material section? Think about how different disciplines make these choices as we look at different papers.
• What is the scientific focus of the paper?
• What modeling formalism(s) is(are) used? (ODE numerics, PDE bifurcation theory, agent-based modeling...)
• How would you classify the model(s) in the paper with respect to the MAW classification (see Mogilner, Allard, and Wollman, Science 2012)? Plot the MAW axes with the paper marked as a point.
Think about how you would tweak this classification scheme as we read different papers throughout the term.
• What are the main results? (mathematical and/or scientific)
• How are the main results dependent on the choice of modeling formalism? Could they have been achieved (better/worse) using other tools?
• How well are the main results highlighted and framed? That is, can you easily identify what the authors consider their most important contribution? And do they make a good case for the importance
of those results?
• Who is the intended audience? Consider elements of your answer to "structure of the paper" and "highlighted and framed".
Sample paper report (von Dassow et al. 2000) - pdf, tex
Paper-discussion schedule:
Report due Discussion date Paper
Jan 16 Jan 16 Variations and fluctuations of the number of individuals in animal species living together.
Volterra. ICES Journal of Marine Science 3(1):3-51, 1928. DOI. Focus on pages 1-15.
Jan 16 Jan 16 Optimal approaches for balancing invasive species eradication and endangered species management.
Lampert, Hastings, Grosholz, Jardine, Sanchirico. Science 344(6187):1028-1030, 2014.
Jan 23 Jan 30 A Simple Model for Complex Dynamical Transitions in Epidemics.
Earn, Rohani, Bolker, Grenfell. Science 287:667-670, 2000.
Jan 30 Jan 30 Models predict that culling is not a feasible strategy to prevent extinction of Tasmanian devils from facial tumour disease.
Beeton, McCallum. Journal of Applied Ecology 48:1315–1323, 2011.
Feb 6 Feb 13 The logic of animal conflict.
Smith, Price. Nature 246:15-18, 1973.
Feb 13 Feb 13 Social evolution in structured populations.
Debarre, Hauert, Doebeli. Nature Communications 5:3409, 2014 DOI.
Feb 27 Mar 5 Potential for Control of Signaling Pathways via Cell Size and Shape.
Meyers, Craig, Odde. Current Biology 16(17):1685-1693, 2006.
Mar 5 Mar 5 Thresholds in development.
Lewis, Slack, Wolpert. J Theo Bio 65:579-590, 1977.
Mar 12 Mar 18 The segment polarity network is a robust developmental module. von Dassow, Meir, Munro, Odell. Nature 406:188-192.
Mar 18 Mar 18 Auxin transport is sufficient to generate a maximum and gradient guiding root growth. Grieneisen, Xu, Maree, Hogeweg, Scheres. Nature 449(7165):1008-1013.
Mar 26 Apr 2 An agent-based model contrasts opposite effects of dynamic and stable microtubules on cleavage furrow positioning. Odell, Foe. J Cell Bio 183: 471–483, 2008.
Apr 2 Apr 2 Resetting and annihilation of reentrant abnormally rapid heartbeat. Glass, Josephson. PRL 75(10):2059-2062, 1995.
For this project, you will pick a paper (or a couple closely related papers) and consider the suitability of the modeling formalism used. Using an alternate formalism (or several), you will explore
the impact this alternate choice has on (a subset of) the results in the paper(s). What can and what cannot be accomplished and why? For example, if the original paper carried out stability analysis
and found a Turing instability, can you rediscover this using a numerical simulation approach or a stochastic treatment? The goal is to learn about the strengths and weaknesses of various formalisms.
The project can be carried out in groups but groups are expected to address a few related papers and/or explore multiple alternate formalisms.
• Jan 20 - Choose three papers to consider for the project.
• Jan 31 - Submit a summary of your chosen paper(s) and a plan for the project.
• Feb 3-6 - Meet with me to discuss your chosen paper and plan.
• Mar 6 - Submit a report on your results. It should be in the TAIMRD format and roughly between 5-10 pages including any figures.
• Mar 13 - Submit a plan for revising your work based on feedback on the report.
• Apr 3 - Submit the final report.
• Apr (TBD) - Presentations
Given the typical wide range of backgrounds of students in this course, the marking is, to a large extent, on a relative scale. Along with your final report, you should submit a document a few
paragraphs in length outlining your background for the course (your previous degree(s), relevant coursework, relevant research experience) and what aspects of the project you consider to represent
new learning for you.
This is a list of papers that you might want to consider for you project. Any of the "Discussion" papers above would also be acceptable.
TO BE UPDATED SOON FOR 2019
Title Author(s) Journal info
A synthetic oscillatory network of transcriptional regulators. Elowitz, Leibler. Nature 403:335-338, 2000.
Thresholds in development. Lewis, Slack, Wolpert. J Theo Bio 65:579-590, 1977.
Dynamic instability of microtubules as an efficient way to search space. Holy, Leibler. PNAS 91:5682-5685, 1994.
Computer simulations reveal motor properties generating stable antiparallel microtubule interactions. Nedelec. J Cell Bio 158(6):1005–1015, 2002.
Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Tyson, Chen, Novak. Current Opinion in Cell Biology 15:221–231, 2003..
The Chemical Basis of Morphogenesis. Turing. Bull Math Bio 52(1):153-197, 1952.
The reference textbook listed below by de Vries et al. has a collection of project ideas in "Part III" that would be appropriate for the project in this course. If you can't find a copy of that book,
ask me about borrowing mine.
Here are some ideas that have some interesting interplay between different mathematical formalisms. You can look for your own paper or come talk to me for tips.
• Spiral waves using PDEs and cellular automata.
• Stochastic resonance - noise near a Hopf bifurcation
• Turing instabilities or other patterning in a noisy environment - PDEs, PDEs+noise, stochastic (e.g. using SMOLDYN)
TO BE UPDATED SOON FOR 2019
Relevant Paper
All term Cell polarity: quantitative modeling as a tool in cell biology. Mogilner, Allard, Wollman. Science 336:175-179, 2012.
Jan 19 A contribution to the mathematical theory of epidemiology. Kermack, McKendrick. Proceedings of the Royal Society A 115(772):700-721, 1927.
Jan 24 A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. Gillespie. J Comp Phys 22(4):403-434, 1976.
Jan 24 Exact stochastic simulation of coupled chemical reactions. Gillespie. J Phys Chem 81(25):2340-2361, 1977.
Jan 24 Efficient formulation of the stochastic simulation algorithm for chemically reacting systems. Cao, Li, Petzold. J Chem Phys 121(9):4059-4067, 2004.
Mar 26-28 A quantitative description of membrane current and its application to conduction and excitation in nerve. Hodgkin, Huxley. J Physiology 117: 500-544, 1952. Reprinted in Bull Math Bio 52
(1):25-71, 1990.
Sniffers, buzzers, toggles, and blinkers: dynamics of regulatory and signaling pathways in the cell. Tyson, Chen, Novak. Curr Op in Cell Bio 15:221-231, 2003.
Relevant dates Textbook
Jan 8-26 Mathematical models in population biology and epidemiology. Brauer, Castillo-Chavez.
Jan 8-26 A course in mathematical biology - quantitative modeling with mathematical and computational methods. de Vries, Hillen, Lewis, Müller, Schönfisch.
Jan 22 - Feb 9 Evolutionary dynamics. Nowak.
Lots Mathematical models in biology. Edelstein-Keshet.
Feb 26 - Mar 2 Random walks in biology. Berg.
Other (e.g. code):
Gillespie simulation code
My matlab code for simulating stochastic realizations, solution to the Kolmogorov equation and the logistic equation for the SIS model.
Lecture notes up to Jan 24
The marks in this course will be determined by three factors: (i) Participation in the weekly paper-discussions and the submitted summary (30%), (ii) the written homework (20%), (iii) written and
oral project report (50%). The written report mark will have a self-evaluation component and the oral report will have a peer-evaluation component. | {"url":"https://personal.math.ubc.ca/~cytryn/teaching/math560/","timestamp":"2024-11-07T10:05:16Z","content_type":"application/xhtml+xml","content_length":"18655","record_id":"<urn:uuid:7a5aef88-4ecd-499b-81c1-f4fb404e9c5d>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00209.warc.gz"} |
A note about sigmas
A note about sigmas#
We are regularly asked about the “sigma” levels in the 2D histograms. These are not the 68%, etc. values that we’re used to for 1D distributions. In two dimensions, a Gaussian density is given by:
pdf(r) = exp(-(r/s)^2/2) / (2*pi*s^2)
The integral under this density (using polar coordinates and implicitly integrating out the angle) is:
cdf(x) = Integral(r * exp(-(r/s)^2/2) / s^2, {r, 0, x})
= 1 - exp(-(x/s)^2/2)
This means that within “1-sigma”, the Gaussian contains 1-exp(-0.5) ~ 0.393 or 39.3% of the volume. Therefore the relevant 1-sigma levels for a 2D histogram of samples is 39% not 68%. If you must use
68% of the mass, use the levels keyword argument when you call corner.corner.
We can visualize the difference between sigma definitions:
import corner
import numpy as np
# Generate some fake data from a Gaussian
x = np.random.randn(50000, 2)
First, plot this using the correct (default) 1-sigma level:
fig = corner.corner(x, quantiles=(0.16, 0.84), levels=(1 - np.exp(-0.5),))
_ = fig.suptitle("default 'one-sigma' level")
Compare this to the 68% mass level and specifically compare to how the contour compares to the marginalized 68% quantile:
fig = corner.corner(x, quantiles=(0.16, 0.84), levels=(0.68,))
_ = fig.suptitle("alternative 'one-sigma' level") | {"url":"https://corner.readthedocs.io/en/latest/pages/sigmas/","timestamp":"2024-11-10T17:30:51Z","content_type":"text/html","content_length":"20665","record_id":"<urn:uuid:32da286c-ec05-4d3d-ba40-8d10cecb958e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00432.warc.gz"} |
Sleazy P. Martini Goes On Trial
Sleazy P. Martini is up to his old tricks again, throwing dice on the wharf and winning lots of bets. You (an observer with statistical knowledge) suspect that Sleazy P. is pulling some shenanigans.
That is, you suspect the die he is using is loaded.
How do you prove this?
Sleazy P. Martini Goes On Trial Partial Answer
: You collect data.
You start writing down the result of each roll of Sleazy P.'s die, and record the following data:
The sample mean of the above data is $\overline{x}=4.02.$
Is this evidence that Sleazy P. has loaded the die?
The Sampling Distribution
If Sleazy P. is innocent, the sampling distributions should be very close to $$N(3.5,0.171).$$ Assuming that Sleazy P. is innocent, what is the probability of seeing the sample mean we recorded?
Sleazy P. Goes to Jail
Assuming that Sleazy P. is innocent, the chances of observing a sample mean as far from $3.5$ as $\bar{x}=4.02$ over $100$ rolls is $0.0024.$
This is very strong evidence that he is not innocent, but that he actually loaded the die.
Hypothesis Testing
We just performed a hypothesis test.
The null hypothesis is that Sleazy P. is that innocent and that the true mean of his die is $3.5.$ This is denoted as $$H_0: \mu=3.5.$$ The alternative hypothesis is that Sleazy P. is guilty, and
really did load the die, altering the fair-die mean of $3.5.$ We denote this as $$H_a: \mu \neq 3.5$$
Hypothesis Testing
We saw that the probability that Sleazy P. is innocent given the sample mean we recorded was $0.0024.$
We computed a
test statistic
: $$z=\frac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\frac{4.02-3.5}{1.71/\sqrt{100}}=3.04$$ The probability of observing a test statistic
than $3.04$ is $0.0024.$
The value $0.0024$ is called the $p\mbox{-value}$ of the test.
We say that the result of hypothesis test is
statistically significant
if the $p\mbox{-value}$ falls below a certain threshold.
Common thresholds are $\alpha=0.05$ and $\alpha=0.01.$
In this case, we say that a test is significant at the level of $\alpha$ when $p\mbox{-value} \lt \alpha.$
In the case of statistical significance ($p\mbox{-value} \lt \alpha$), we
reject the null hypothesis. Another way to say it:
When the $p\mbox{-value}$'s low, $H_0$ has got to go!
If the $p\mbox{-value}$'s low, reject the
The Way Mr. Holt Likes to Think About It
The $p\mbox{-value}$ is a measure of our belief in the null hypothesis $H_0.$
In Sleazy P's case, at the level of significance $\alpha=0.01$, there is
statistically significant
evidence that he loaded the die since $$p\mbox{-value}=0.0024<0.01.$$
The result is significant at the $0.05$ level too.
Consumer Advocacy
: Sleazy P. Gets out of Jail
After serving his sentence, Sleazy P. Martini is now rehabilitated and ready to enter society as a soft drink manufacturer.
Sleazy P. has created a new brand of cola called
Sleazy P.'s Easy Peazy
Now, a $12 \mbox{ fl oz}$ can of soda should contain $355 \mbox{ mL}$ of product, but you (the statistically savvy citizen) notice that in general, there seems on average to be less.
Hmmmmm.... Is Sleazy P. up to his old tricks AGAIN!?
Big Question
: How do we find out?
Big Answer
: Gather data and perform a hypothesis test.
Because $355 \mbox{ mL}$ is printed on the can, we know the true mean $\mu$ should be a little bigger than $355$ to prevent underfilling.
After a little research, we learn from various soft drink manufacturers that, in fact, the amount should vary according to a normal distribution with mean $\mu = 355.2 \mbox{ mL}$ and standard
deviation $\sigma = 0.5 \mbox{ mL}.$
Next, we begin collecting data...
The Data
Taking a simple random sample from around the country, we procured $40$ cans of
Sleazy P.'s Easy Peazy
Here is the data.
355.1 354.7 354.4 355.1 355.3 354.5 354.1 354.4 355.2 355.1 353.9 354.3 355.3 355.1 354.9 354.9 355.5 356.0 354.8 356.2 354.5 354.9 355.8 354.8 354.1 354.8 355.1 354.4 355.2 354.0 354.0 355.4 354.7
354.9 355.1 355.6 355.7 355.1 354.9 355.6
The Hypothesis Test
$$H_0: \mu=355.2$$ $$H_a: \mu<355.2$$
From our data we get $\overline{x}=354.935.$
Using $\sigma=0.5$ we compute the $z$-statistic: $$z=\frac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\frac{354.935-355.2}{0.5/\sqrt{40}}=-3.35$$ For this test statistic, $p\mbox{-value}=0.0004.$
Given that the null hypothesis ($H_0: \mu=355.2$) is true, the chances of seeing a test statistic of $z=-3.35$
is $0.0004.$ That is, $p\mbox{-value}=0.0004.$
At the $\alpha=0.01$ level of significance, we reject the null hypothesis.
: Reject $H_0.$ That is, Sleazy P. is underfilling. Thus, we can still count on Sleazy P. to be a shady wheeler-dealer.
The General Hypothesis Test
When you perform a hypothesis test, you need to:
Step 1
: State your hypotheses: $H_0$ and $H_a.$
Step 2
: Compute the test statistic (in this case, the $z$-statistic).
Step 3
: Determine your $p\mbox{-value}.$
Step 4
: State your conclusion (keep or reject $H_0$). If your $p\mbox{-value}$ falls below $\alpha,$ then we reject $H_0.$ Otherwise, we keep $H_0$. Also, summarize the conclusion
using the language of the problem situation
The following are a random sample of $n=30$ IQ scores of seventh-grade students from a school district in Portland:
128, 96, 114, 100, 105, 114, 111, 132, 112, 91, 119, 98, 86, 74, 103, 103, 72, 107, 118, 93, 104, 114, 111, 130, 89, 112, 102, 112, 120, 108
Assume that the IQ scores in this population has a normal distribution with standard deviation $\sigma=15.$ A previous estimate of the mean IQ $\mu$ of seventh graders from this school district is
$105.$ We suspect that the true value may actually be higher. To test our suspicion, we carry out a test of significance on the data we collected above.
At the $\alpha=0.05$ level of significance, what is the conclusion?
Step #1: We suspect that the true mean may actually be higher than $105.$ The null hypothesis says that there is "no difference: the population mean is $105,$" whereas the alternative hypotheses says
that "there is a difference: the population mean is higher than $105.$" These hypotheses are expressed as $$ \begin{array}{c} H_0: \mu=105\\ H_a: \mu \gt 105 \end{array} $$ Step #2: From our data
set, we have that $\bar{x}=105.933$ and $n=30.$ Our null hypothesis assumes that $\mu=105$ and $\sigma=15.$ From these we may now compute our test statistic, in this case, the $z$-statistic: $$ z=\
frac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\frac{105.933-105}{15/\sqrt{30}}=0.34 $$ to two decimal places.
Step #3: The probability of observing a $z$ test statistic of $0.34$ or higher is $$ P(z \gt 0.34)=1-P(z \lt 0.34)=1-0.6331=0.3669 $$ from Table A. Therefore, $p\mbox{-value}=0.3669.$
Step #4: Since our $p\mbox{-value}= 0.3669 \gt 0.05=\alpha,$ we keep the null hypothesis $H_0$. In plain language, the data set we have does not give significant evidence that that the mean IQ score
of seventh graders from this school district is higher than $105.$
Sleazy P.'s Die (Revisited):
We are now going to roll Sleazy P's crooked die and perform the following hypothesis test after each roll at significance level $\alpha=0.01.$ $$\begin{array}{c} H_0: \mu=3.5 \\ H_a: \mu \neq 3.5 \ | {"url":"https://holt.blue/MTH_243/Lecture_Notes/section_9_1.html","timestamp":"2024-11-02T20:42:15Z","content_type":"text/html","content_length":"22016","record_id":"<urn:uuid:950f29b3-5422-4c0a-aa9d-4ab04b8cad95>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00001.warc.gz"} |
What's Fuzzy Logic
What's Fuzzy Logic
Fuzzy logic is a "nuanced" logic with infinite intermediate states between zero and one.
Example. In the fuzzy logic a variable can take the value 0 and 1 but also 0.2, 0.55, 0.8, etc. The fuzzy variable is a continuous variable and can assume infinite values between zero and one.
The fuzzy logic is the evolution of Boolean logic and it is particularly useful in the study of artificial intelligence.
How fuzzy logic works
In computer science we usually use the Boolean logic which admits only two values: zero and one.
All programming languages are based on the binary logic of George Boole. A condition can be true or false, with no middle ground.
No programmer would ever develop a user interface with the yes, no, and maybe buttons.
It is completely illogical to insert a third option ( Maybe ) between Yes and No to delete a file.
However, in artificial intelligence it becomes very useful.
Why is it useful?
Fuzzy logic allows computers to better represent the complexity of reality and the natural language of men.
A practical example
In this photo the sky is clear or cloudy?
In the Boolean logic I can choose between 0 (clear sky) and 1 (cloudy sky).
I could say that the sky is clear ... but it is not completely true because the sky is also a little cloudy.
Note. Boolean logic has a limited scope because reality is much more complex. In the Boolean logic the previous information ("a little cloudy") can not be transformed into data.
In the fuzzy logic, however, zero and one are only the extremes of a continuous variable. The variable can also take intermediate values such as 0.9, 0.5, 0.1, etc ...
Thanks to the fuzzy logic I can assign to the variable the value 0.3 in a scale between 0 (clear sky) and 1 (cloudy sky).
In practice, I say that the sky is basically serene but also a little cloudy.
In doing so, I better represent the real situation.
Another practical example
In natural language, many human expressions are subjective and relative.
A Boolean machine could not understand them without falling into logical contradictions.
For example, if I say "I'm tall", what am I saying? There is no objective and absolute measure to define the height of something, it is a subjective and relative concept.
In fuzzy logic I can transform subjective information into an objective fact.
A basketball team has 7 players with different heights.
I can not say whether each of them is tall or short. So I use fuzzy logic.
• I order the players from highest to lowest.
• Then I associate the value one (1) to the highest and the value zero (0) to the lowest.
This scale allows me to assign percentages at intermediate heights.
For example, the fourth person is exactly in the middle of the scale, so the fuzzy variable has a value of 0.5 (or 50%). And so on.
Note. Thanks to the fuzzy logic I can affirm that the degree of truth of the statement "the boy is tall" is true to 0.5 for the fourth person. In the same way, I can affirm that even the statement
"the boy is short" has a degree of truth equal to 0.5. In his group the boy is at the same time a little tall and a little short.
Fuzzy logic and artificial intelligence
Fuzzy logic is very useful in artificial intelligence because it allows the rational agent to choose under conditions of uncertainty, using a nuanced and probabilistic representation of reality.
Apparently the system may seem weaker, because it is without certainty. In reality, it is considerably more powerful because it manages to deal with logical reasoning, avoiding the logical
inconsistencies typical of human reasoning.
In many cases it is impossible to solve a problem in its entirety with the classical Boolean logic, due to computational reasons (processing time, amount of memory, etc.) or lack of information
(imperfect information).
Instead, through the fuzzy logic it is possible.
The application of fuzzy logic to artificial neural networks is one of the most interesting fields of study, one that I think can make a significant contribution in the development of A.I. | {"url":"https://www.andreaminini.net/computer-science/artificial-intelligence/fuzzy-logic","timestamp":"2024-11-01T22:24:42Z","content_type":"text/html","content_length":"16526","record_id":"<urn:uuid:597e4887-fcba-4fb7-b9fd-0b0334e3ca96>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00541.warc.gz"} |
Vizly: AI-powered data analysisVizly: Optimizing Distillation Column Yield Using Historical Data
Executive Summary: Yield Prediction Analysis for Distillation Tower
This analysis aims to predict the yield of a distillation tower using various process variables. We utilized an XGBoost regression model to understand the key factors influencing the yield and to
make accurate predictions.
Data Summary
• Dataset: The dataset contains various process variables recorded over time, including temperatures, pressures, and flow rates.
• Total Features: 24 (excluding the date and yield columns)
• Target Variable: Yield
Key Findings
Correlation Analysis
• A correlation matrix was generated to understand the relationships between different process variables.
• Key observations include strong correlations between certain temperature and flow variables, indicating potential areas for process optimization.
Model Performance
• Model Used: XGBoost Regressor
• Mean Squared Error (MSE): 0.881
• R-squared (R²): 0.190
The model's performance indicates that while it captures some variability in the yield, there is room for improvement. The R² value suggests that approximately 19% of the variance in yield is
explained by the model.
Feature Importance
The most important features affecting the yield according to the XGBoost model are:
1. FlowC1 (Importance: 0.275)
2. Temp1 (Importance: 0.181)
3. TempC2 (Importance: 0.078)
4. FlowC9 (Importance: 0.072)
5. TempC3 (Importance: 0.070)
SHAP Analysis
• SHAP (SHapley Additive exPlanations) values were used to interpret the model's predictions.
• The SHAP summary plot provides a detailed view of the impact of each feature on the model's output.
• Key insights include the significant influence of FlowC1 and Temp1 on yield predictions, with FlowC1 having the highest impact.
1. Focus on Key Variables: Given the importance of FlowC1 and Temp1, process engineers should prioritize monitoring and optimizing these variables to improve yield.
2. Further Model Optimization: Consider additional feature engineering and model tuning to improve the predictive performance.
3. Regular Monitoring: Implement real-time monitoring of key variables to ensure they remain within optimal ranges.
• Correlation Matrix: Provides an overview of relationships between process variables.
• Feature Importances: Highlights the most influential features for yield prediction.
• SHAP Summary Plot: Offers detailed interpretability of the model's predictions.
This analysis provides a foundational understanding of the factors influencing distillation tower yield. By focusing on key variables and further optimizing the model, process engineers can make
data-driven decisions to enhance yield and overall process efficiency.
Would you like to explore any specific aspect further or need additional visualizations? | {"url":"https://vizly.fyi/share/8f508f51-0569-4cfe-99b1-2a43ff1b3fd2/b33c467f-5b62-4733-a339-d67fe7e5d336","timestamp":"2024-11-14T13:14:54Z","content_type":"text/html","content_length":"1049462","record_id":"<urn:uuid:760cbf74-7bdd-4a81-9f1d-a26a62babe3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00731.warc.gz"} |
The Model of Computation
Understand the theoretical running times of operations on data structures.
We can analyze the theoretical running times of operations on the data structures. To do this precisely, we need a mathematical model of computation. For this, we use the w-bit word-RAM model.
Create a free account to access the full course.
By signing up, you agree to Educative's Terms of Service and Privacy Policy | {"url":"https://www.educative.io/courses/data-structures-with-generic-types-in-cpp/the-model-of-computation","timestamp":"2024-11-08T04:59:57Z","content_type":"text/html","content_length":"755268","record_id":"<urn:uuid:806d663f-9ebf-4137-bbb1-46ad3e8596b9>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00217.warc.gz"} |
Tutorial 3: Transformation
In this tutorial, we will explain what a transformation is and how it can be performed in Model Lab.
A transformation is a function or operation that takes a point as input and outputs a new point. When talking about 3D models, the points to be transformed are called vertices.
The 3 basic transformations are:
Basic Concepts and User Interface Explanation
The Pivot is simply a point in space, also referred to as a pivot point. You can think of the pivot point as the local origin of the model in relation to the space origin.
It is basically the same thing as the position of the model with an important difference. If you change the position of the pivot point, it does not move the geometry in the model; you only move the
local origin. The geometry in the model is only moved when a transformation operation (scale, rotate or translate/position) is executed.
You should also note that the pivot point remains fixed (does not move) during scaling and rotation. You can input your own values to define the pivot point, but usually, you are better off using the
quick buttons, Lower left and Center.
The scale defines the dimensions of the model. You may adjust the scale of the loaded model by changing the slider or input field to the right of the slider. The value to the right can be thought of
as a magnifier.
For instance, by setting it to 2, you will double the model's size. Setting it to 0.1 will reduce the model's size to a tenth of the original size. You can also change the scale by inputting a
desired width, depth or height. The values in parenthesis display the original size.
If you know you have a lot of models that need to be downsized using the same scale, you can use the remember scale option.
It will automatically change the scale of the model during import. All the scaling operations are uniform, meaning if you double the width value, the depth and height values will also be doubled.
The squares seen on the floor are actually measurement points in meters.
The smaller square is a length of 0.25 meters.
A larger square, made of 4 smaller squares, is 1 meter.
Here you can input the model's position, also known as the translation.
There are some quick ways that can come in handy when you want to position your model quickly, the Position Picker and Place on floor.
The Position Picker shows 2 arrows within 9 squares and an XY axes button.
The axes are marked red, green and blue, corresponding to X, Y and Z. So, as long as you remember RGB, you will know if you are changing X, Y or Z.
The 2 arrows symbolize the 2 axes of the drawing, which are now X and Y.
The 9 squares can be clicked to place the model in relation to the origin point of the space (not the pivot point).
Clicking on the XY axes button will change the 2 axes and will show you either XY, XZ, or YZ. This determines which 2 axes the region picker buttons are working on:
Place on floor will move the model so that its pivot point is on the floor:
Used to change the rotation of the model. Keep in mind that the position of the pivot point affects the outcome of a rotate operation.
Fixing the Transformation Issues
1. Download and load the Fika chair (transformation issues) .cmsym file found at the top of this article.
2. We'll begin by adjusting the scale.
Currently, the width is 22m. A chair is more likely to be around 0.5m in width.
Input 0.5m as the width and press Enter.
Notice how the depth and height are uniformly scaled.
3. Now, let's make the chair stand upright. We need to rotate the model -90 degrees around the X-axis.
Input -90 as the rotation for X. Model Lab will automatically interpret this as 270 degrees, resulting in the same rotation as rotating -90 degrees.
4. The chair is still floating in the air. Don't worry, this can be easily fixed.
Set the pivot point to the lower left corner of the model by clicking Lower left under the Pivot.
Under Position, click Place on Floor to lower the chair to the floor level.
5. Finally, in the Region Picker, ensure the XY-axes are selected, then click on the top right square.
The fixed model should now look like this:
That's it! Now you should know:
• What a transformation is.
• What a pivot point is.
• How the scale, position, and rotate operations can be performed in Model Lab.
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Emergence of currents as a transient quantum effect in nonequilibrium systems
Most current calculations are based on equilibrium or semi-equilibrium models. However, except for very special scenarios (like ring configuration), the current cannot exist in equilibrium. Moreover,
unlike with equilibrium scenarios, there is no generic approach to confront out-of-equilibrium currents. In this paper we used recent studies on transient quantum mechanics to solve the current,
which appears in the presence of very high density gradients and fast transients. It shows that the emerging current appears instantaneously, and although the density beyond the discontinuity is
initially negligible the currents there have a finite value, and remain constant for a finite period. It is shown that this nonequilibrium effect can be measured in real experiments (such as cooled
rubidium atoms), where the discontinuity is replaced with a finite width (hundreds of nanometers) gradient.
Dive into the research topics of 'Emergence of currents as a transient quantum effect in nonequilibrium systems'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/emergence-of-currents-as-a-transient-quantum-effect-in-nonequilib-3","timestamp":"2024-11-03T19:28:05Z","content_type":"text/html","content_length":"53753","record_id":"<urn:uuid:ac45395b-0228-4d02-928f-0364176b7b23>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00523.warc.gz"} |
New Member
wen long
Join Date: May 2012
Posts: 29
Rep Power:
Hi, Peterputer
The third variable 'T' is treated as z internally and it is given values as zero when z is not explicitly defined. This seems to be a bug for the code. To overcome this problem, you can use:
%test Example 2
tdata.lines(1).zonename='myline zone';
and use x,y,T instead.
Or you can also try instead of getting it in the v, put the T value in z as below:
tdata.lines(1).zonename='myline zone';
In essence, only use v if you have more than 3 variable, v(i, | {"url":"https://www.cfd-online.com/Forums/tecplot/103860-mat2tecplot-3.html","timestamp":"2024-11-11T16:39:49Z","content_type":"application/xhtml+xml","content_length":"157740","record_id":"<urn:uuid:3f24825d-bcfc-4bbd-97cc-3aa547d805d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00014.warc.gz"} |
In March, a bank short-term investment manager has $1 million in
90 day Tbills on its...
In March, a bank short-term investment manager has $1 million in 90 day Tbills on its...
In March, a bank short-term investment manager has $1 million in 90 day Tbills on its balance sheet that it plans to sell in June for liquidity purposes, and is worried about interest rates rising
(i.e. prices falling) in the next few months, which would cause the value of the T-bills to fall. The current (spot) discount yield is 1.10% (i.e. a Discount % price of 98.90%) for a 90-day T-bill.
a. What is the price for the $ 1 million of T-bills in dollars?
T-bill Price in Dollars ________
b. On the CME Group website, a June Eurodollar Futures contract gives a price of 98.10% (i.e., a discount yield of 1.90%) for a $1 million, 90 day Eurodollar Futures contract. b. What is the contract
price for the Eurodollar Futures Contract in dollars?
Eurodollar Futures Price in Dollars ______________
What type of Eurodollar futures contract should be purchased (long or short)? Explain why.
Long or Short _______ Why?_______________________
c. Suppose in June the T-bill discount yield goes up by 20 basis points to 1.30%, and the Eurodollar Futures yield goes up by 25 basis points to 2.15%, what is the new dollar price for the 1 mil.
T-bills, and what is the new contract dollar price for the Eurodollar Futures Contract?
New T-bill Price in Dollars _______________
New Eurodollar Futures Price in Dollars ____________ | {"url":"https://justaaa.com/finance/208918-in-march-a-bank-short-term-investment-manager-has","timestamp":"2024-11-13T04:47:16Z","content_type":"text/html","content_length":"43259","record_id":"<urn:uuid:dd95fcaa-7687-4321-8c30-12693e774c6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00342.warc.gz"} |
An early fault feature extraction method for rolling bearings based on variational mode decomposition and random decrement technique
The early fault characteristics of rolling bearing are weak, and the background noise is so strong that it is difficult to diagnose. In order to solve the above problems, an early fault feature
extraction method for rolling bearings based on variational mode decomposition and random decrement technique was proposed. The variational mode decomposition was used to decompose the collected
vibration signals, and the component with the larger correlation coefficient was selected as the fault component. Then the fault component was processed by random decrement technology, and the
Hilbert envelope spectrum of the fault component was made. According to the proposed method, the early fault characteristic of outer ring of rolling bearing was extracted. Compared with the method
based on EMD, the proposed method is more effective in extracting the early fault characteristics of rolling bearings.
1. Introduction
Rolling bearing is an important component in the mechanical system and the damage of rolling bearing will cause the failure of the mechanical system. It is of great significance to extract fault
information and eliminate hidden dangers in the early stage of rolling bearing failure [1]. Therefore, it is very important to diagnose the early faults of rolling bearings. In engineering
applications, the early failure characteristics of bearings are rather weak. The vibration transmission path is very complex and the background noise is strong, making the feature extraction of early
failure of rolling bearing becomes a difficult point [2].
Huang et al proposed the Empirical Mode Decomposition (EMD) [3]. EMD has been widely used in early fault diagnosis of rolling bearings since it was put forward [4]. EMD is a powerful tool for
analyzing non-stationary signals and nonlinear signals. However, there is a lack of strict mathematical foundation, low algorithm efficiency and modal aliasing.
In 2014, Dragomiretskiy et al. [5] proposed a new adaptive signal processing method, Variational Mode Decomposition (VMD). In the process of obtaining the decomposed components, the method
iteratively searches the optimal solution of the variational model to determine the frequency center and bandwidth of each component. Thus, the frequency division of signals and the effective
separation of components can be realized adaptively. Compared with EMD, VMD has a solid theoretical foundation, and its essence is a number of adaptive Wiener filtering groups, showing better noise
robustness [6]. However, there is nothing VMD can do about the random noise in the signal.
In order to solve the problem, the random decrement technique (RDT) is introduced in this paper. Random decrement technique is a method of identification of modal parameters, which is first proposed
by Cole [7, 8] in 70s. The basic concept of RDT is to assume a system with stationary random excitation, and its response is the superposition of both deterministic and random responses. The
deterministic response is separated from the random response, and the random response is eliminated by using the statistical mean method. Finally, a deterministic free attenuation signal is obtained
by filtering. RDT has been widely used in many fields such as vibration modal analysis [9], and structural damage detection [10].
Based on VMD and random decrement technique, a new early fault diagnosis method for rolling bearings is proposed in this paper. The method proposed in this paper is applied to early fault diagnosis
of rolling bearings. The fault characteristics of rolling bearing are successfully extracted, and the practicability and effectiveness of the method are verified.
2. Feature extraction based VMD and RDT
2.1. Variational Mode Decomposition
According to the Reference [11], we get the complete algorithm for VMD, summarized in following algorithm.
Initialize$\left\{{\stackrel{^}{u}}_{k}^{1}\right\}$, $\left\{{\stackrel{^}{\omega }}_{k}^{1}\right\}$, $\left\{{\stackrel{^}{\lambda }}^{1}\right\}$, $n←0$
Repeat $n←n+1$
Update${\stackrel{^}{u}}_{k}^{n+1}$for all$w\ge 0$:
${\stackrel{^}{u}}_{k}^{n+1}\left(\omega \right)←\frac{\stackrel{^}{f}\left(\omega \right)-{\sum }_{i}{\stackrel{^}{u}}_{i}\left(\omega \right)+\frac{\stackrel{^}{\lambda }\left(\omega \right)}{2}}
{1+2\alpha \left(\omega -{\omega }_{k}{\right)}^{2}},$
Update${\omega }_{k}$:
${\omega }_{k}^{n+1}\left(\omega \right)=\frac{{{\int }_{0}^{\infty }\omega \left|{\stackrel{^}{u}}_{k}\left(\omega \right)\right|}^{2}d\omega }{{{\int }_{0}^{\infty }\left|{\stackrel{^}{u}}_{k}\left
(\omega \right)\right|}^{2}d\omega },$
end for
Dual ascent for all$\omega \ge 0$:
${\stackrel{^}{\lambda }}^{n+1}\left(\omega \right)←{\stackrel{^}{\lambda }}^{n}\left(\omega \right)+\beta \left[\stackrel{^}{f}\left(\omega \right){\sum }_{k=1}^{K}{\stackrel{^}{u}}_{k}^{n+1}\left(\
omega \right)\right],$
until convergence:
$\frac{{{\sum }_{k=1}^{K}‖{{\stackrel{^}{u}}_{k}}^{n+1}-{{\stackrel{^}{u}}_{k}}^{n}‖}_{2}^{2}}{{‖{{\stackrel{^}{u}}_{k}}^{n}‖}_{2}^{2}}<e.$
The reconfiguration signal can be expressed as:
$\stackrel{^}{f}\left(t\right)={\sum }_{k=1}^{K}{\stackrel{^}{u}}_{k},$
2.2. Random decrement technique
RDT can be used to describe the impulse response of the system, and its advantage is that it can extract free impact response from the stationary random response in the system. The core is to assume
that a system is subjected to stationary random excitation, and the response is the superposition of deterministic response determined by the initial condition and random response determined by the
initial external load. Under the same initial conditions, stationary random response is divided into several sections, and the general mean of the intercept segments is calculated, so as to extract
the free attenuation response.
The response signal is divided into $L$ segments, and each segment of the signal is expressed as ${x}_{i}\left(t\right)$, whose length is $\tau$. Each response signal has the same trigger value, and
the trigger value can be expressed as:
${x}_{i}\left({t}_{i}\right)={x}_{s}=const,i=1,2,\cdots ,L.$
The overall mean of the $L$ segment signal is calculated, and the random decrement function can be expressed as:
$x\left(\tau \right)=\frac{1}{L},\sum _{i=1}^{L}{x}_{i}\left({t}_{i}+\tau \right),$
where ${x}_{i}\left({t}_{i}\right)={x}_{s},i=1,2,\cdots ,L$.
2.3. Proposed method
The variational mode decomposition is used to decompose the collected vibration signals, and the component with the larger correlation coefficient is selected as the fault component. Then the fault
component is processed by random decrement technology, and the Hilbert envelope spectrum of the fault component is made. According to the proposed method, the early fault characteristic of rolling
bearing outer ring is extracted.
3. Experimental results
3.1. Experiment condition
The experimental system is shown in Figure 1, including vibration sensor, coupling, driving motor, torque decoder/encoder and dynamometer. The type of the test bearing is SKF 6205-2RS, and the
technical parameters of the bearing are shown in Table 1.
Fig. 1The experimental system
Electric spark is used to process faults on bearing outer ring. The fault of diameter 0.18 mm and depth 0.28 mm is used to simulate the slight faults of outer ring. The speed of the driving motor is
1750 r/min, the sampling frequency is 12000 Hz, and the number of sampling points is 32768. Through theoretical calculation, the rotation frequency of the bearing can be obtained ${f}_{r}=$ 29.16 Hz;
the fault characteristic frequency of the bearing outer ring is ${f}_{Oi}=$ 104.3Hz.
Table 1The technical parameters of the bearing
Contact angle
Internal diameter (${d}_{i}$) / mm Outside diameter (${d}_{o}$) / mm Roller diameter ($d$) / mm Pitch diameter ($D$) / mm Number of rolling element ($Y$)
$\alpha$ / (°)
3.2. Experimental data processing
The time-domain and frequency domain waveform of the early fault vibration signal of the rolling bearing outer ring is shown in Fig. 2. It can be seen from Fig. 2 that the vibration signal has
obvious impact components and there is a lot of background noise. The spectrum components are complex and can not extract the rotation frequency (29.16 Hz) and outer ring fault characteristic
frequency (104.3 Hz).
Fig. 2The time-domain and frequency domain waveform of the early fault vibration signal of the rolling bearing outer ring
b) Frequency domain waveform
In order to highlight the fault features, the vibration signal is processed by the method proposed in this paper. The number of decomposed modes $K$ is 4. The vibration signal is decomposed by VMD,
and the component U2 and U3 with the largest correlation coefficient of vibration signal are selected as fault components. Then the fault component is processed by random decrement method and its
Hilbert envelope spectrum is calculated, as shown in Fig. 3. It can be seen from Fig. 3 that the fault component of rolling bearing outer ring contains many characteristic frequencies, including 2
times the frequency 58.6 Hz, the outer ring fault characteristic frequency 100.1 Hz (the reason which is not exactly consistent with the theoretical value 104.3 Hz is the effect of the random sliding
of the rolling element) and its 2 frequency doubling 200.2 Hz. In addition, there are also modulation sidebands, such as 158.7 Hz ($\text{2}{f}_{r}+{f}_{Oi}=$ 158.7 Hz), 258.8 Hz ($\text{2}{f}_{r}+\
text{2}{f}_{Oi}=$ 258.8 Hz), 358.9 Hz ($\text{2}{f}_{r}+\text{3}{f}_{Oi}=$ 358.8 Hz), and 459 Hz ($\text{2}{f}_{r}+\text{4}{f}_{Oi}=$ 459 Hz). Therefore, the proposed method successfully extracted
the characteristic frequency of the early fault of the outer ring of the rolling bearing.
Fig. 3The Hilbert envelope spectrum of the fault component
4. Conclusions
In engineering applications, the early failure characteristics of bearings are rather weak. The vibration transmission path is very complex, and the background noise is strong, making the feature
extraction of early failure of rolling bearing becomes a difficult point. A new method of rolling bearing fault feature extraction based on RDT and VMD is presented in this paper. At the same time,
the research on early fault feature extraction of rolling bearing is carried out. The proposed method successfully extracted the characteristic frequency of the early fault of the outer ring of the
rolling bearing.
• Zhang Y., Randall R. B. Rolling element bearing fault diagnosis based on the combination of genetic algorithms and fast kurtogram. Mechanical Systems and Signal Processing, Vol. 23, Issue 5,
2009, p. 1509-1517.
• Wang X., Zi Y., He Z. Multiwavelet denoising with improved neighboring coefficients for application on rolling bearing fault diagnosis. Mechanical Systems and Signal Processing, Vol. 25, Issue 1,
2011, p. 285-304.
• Huang N. E., Shen Z., Long S. R. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proceedings Mathematical Physical and Engineering
Sciences, Vol. 454, Issue 1971, 1998, p. 903-995.
• Yu D., Cheng J., Yang Y. Application of EMD method and Hilbert spectrum to the fault diagnosis of roller bearings. Mechanical Systems and Signal Processing, Vol. 19, Issue 2, 2005, p. 259-270.
• Dragomiretskiy K., Zosso D. Variational mode decomposition. IEEE Transactions on Signal Processing, Vol. 62, Issue 3, 2013, p. 531-544.
• Zhao C., Feng Z. P. Application of multi-domain sparse features for fault identification of planetary gearbox. Measurement, Vol. 104, 2017, p. 169-179.
• Cole H. A. On-the-line analysis of random vibration. AIAA/ASME 9th Structure, Structural Dynamics and Materials Conference, AIAA Paper, 1968, p. 268-288.
• Cole H. A. Method and Apparatus for Measuring the Damping Characteristic of a Structure. United State, Patent No. 3620069, 1971.
• Mikael A., Gueguen P., Bard P. Y., et al. The analysis of long-term frequency and damping wandering in buildings using the random decrement technique. Bulletin of the Seismological Society of
America, Vol. 103, Issue 1, 2013, p. 236-246.
• Yang J. C. S., Chen J., Dagalakis N. G. Damage detection in offshore structures by the random decrement technique. Journal of Energy Resources Technology, Vol. 55, Issue 106, 1984, p. 637-642.
• Li Z. P., Chen J. L., Zi Y. Y. Independence-oriented VMD to identify fault feature for wheel set bearing fault diagnosis of high speed locomotive. Mechanical Systems and Signal Processing, Vol.
85, 2017, p. 512-529.
About this article
Fault diagnosis based on vibration signal analysis
variational mode decomposition
random decrement technique
rolling bearing
early fault feature extraction
Copyright © 2018 Chengcheng Zhu, et al.
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Ricci curvature
Short description: 2-tensor obtained as a contraction of the Rieman curvature 4-tensor on a Riemannian manifold
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a
manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.
The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space. In general relativity, which involves the pseudo-Riemannian setting, this is
reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric,
with a strikingly simple relationship between the Ricci tensor and the matter content of the universe.
Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric bilinear form (Besse 1987).^[1] Broadly, one could analogize the role of the Ricci curvature in
Riemannian geometry to that of the Laplacian in the analysis of functions; in this analogy, the Riemann curvature tensor, of which the Ricci curvature is a natural by-product, would correspond to the
full matrix of second derivatives of a function. However, there are other ways to draw the same analogy.
In three-dimensional topology, the Ricci tensor contains all of the information which in higher dimensions is encoded by the more complicated Riemann curvature tensor. In part, this simplicity allows
for the application of many geometric and analytic tools, which led to the solution of the Poincaré conjecture through the work of Richard S. Hamilton and Grigory Perelman.
In differential geometry, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf. comparison theorem) with the
geometry of a constant curvature space form. This is since lower bounds on the Ricci tensor can be successfully used in studying the length functional in Riemannian geometry, as first shown in 1941
via Myers's theorem.
One common source of the Ricci tensor is that it arises whenever one commutes the covariant derivative with the tensor Laplacian. This, for instance, explains its presence in the Bochner formula,
which is used ubiquitously in Riemannian geometry. For example, this formula explains why the gradient estimates due to Shing-Tung Yau (and their developments such as the Cheng-Yau and Li-Yau
inequalities) nearly always depend on a lower bound for the Ricci curvature.
In 2007, John Lott, Karl-Theodor Sturm, and Cedric Villani demonstrated decisively that lower bounds on Ricci curvature can be understood entirely in terms of the metric space structure of a
Riemannian manifold, together with its volume form.^[2] This established a deep link between Ricci curvature and Wasserstein geometry and optimal transport, which is presently the subject of much
Suppose that [math]\displaystyle{ \left( M, g \right) }[/math] is an [math]\displaystyle{ n }[/math]-dimensional Riemannian or pseudo-Riemannian manifold, equipped with its Levi-Civita connection
[math]\displaystyle{ \nabla }[/math]. The Riemann curvature of [math]\displaystyle{ M }[/math] is a map which takes smooth vector fields [math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/
math], and [math]\displaystyle{ Z }[/math], and returns the vector field[math]\displaystyle{ R(X,Y)Z := \nabla_X\nabla_Y Z - \nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z }[/math]on vector fields [math]\
displaystyle{ X, Y, Z }[/math]. Since [math]\displaystyle{ R }[/math] is a tensor field, for each point [math]\displaystyle{ p \in M }[/math], it gives rise to a (multilinear) map:[math]\displaystyle
{ \operatorname{R}_p:T_pM\times T_pM\times T_pM\to T_pM. }[/math]Define for each point [math]\displaystyle{ p \in M }[/math] the map [math]\displaystyle{ \operatorname{Ric}_p:T_pM\times T_pM\to\
mathbb{R} }[/math] by
[math]\displaystyle{ \operatorname{Ric}_p(Y,Z) := \operatorname{tr}\big(X\mapsto \operatorname{R}_p(X,Y)Z\big). }[/math]
That is, having fixed [math]\displaystyle{ Y }[/math] and [math]\displaystyle{ Z }[/math], then for any orthonormal basis [math]\displaystyle{ v_1, \ldots, v_n }[/math] of the vector space [math]\
displaystyle{ T_p M }[/math], one has
[math]\displaystyle{ \operatorname{Ric}_p(Y,Z) = \sum_{i=1} \langle\operatorname{R}_p(v_i, Y) Z, v_i \rangle. }[/math]
It is a standard exercise of (multi)linear algebra to verify that this definition does not depend on the choice of the basis [math]\displaystyle{ v_1, \ldots, v_n }[/math].
[math]\displaystyle{ \mathrm{Ric}_{ab} = \mathrm{R}^{c}{}_{bca} = \mathrm{R}^{c}{}_{acb}. }[/math]
Sign conventions. Note that some sources define [math]\displaystyle{ R(X,Y)Z }[/math] to be what would here be called [math]\displaystyle{ -R(X,Y)Z; }[/math] they would then define [math]\
displaystyle{ \operatorname{Ric}_p }[/math] as [math]\displaystyle{ -\operatorname{tr}(X\mapsto \operatorname{R}_p(X,Y)Z). }[/math] Although sign conventions differ about the Riemann tensor, they do
not differ about the Ricci tensor.
Definition via local coordinates on a smooth manifold
Let [math]\displaystyle{ \left( M, g \right) }[/math] be a smooth Riemannian or pseudo-Riemannian [math]\displaystyle{ n }[/math]-manifold. Given a smooth chart [math]\displaystyle{ \left( U, \varphi
\right) }[/math] one then has functions [math]\displaystyle{ g_{ij}: \varphi(U) \rightarrow \mathbb{R} }[/math] and [math]\displaystyle{ g^{ij}: \varphi(U) \rightarrow \mathbb{R} }[/math] for each
[math]\displaystyle{ i, j = 1, \ldots, n }[/math] which satisfy
[math]\displaystyle{ \sum_{k=1}^n g^{ik}(x)g_{kj}(x) = \delta^{i}_j = \begin{cases} 1 & i=j \\ 0 & i \neq j \end{cases} }[/math]
for all [math]\displaystyle{ x \in \varphi(U) }[/math]. The latter shows that, expressed as matrices, [math]\displaystyle{ g^{ij}(x) = (g^{-1})_{ij}(x) }[/math]. The functions [math]\displaystyle{ g_
{ij} }[/math] are defined by evaluating [math]\displaystyle{ g }[/math] on coordinate vector fields, while the functions [math]\displaystyle{ g^{ij} }[/math] are defined so that, as a matrix-valued
function, they provide an inverse to the matrix-valued function [math]\displaystyle{ x \mapsto g_{ij}(x) }[/math].
Now define, for each [math]\displaystyle{ a }[/math], [math]\displaystyle{ b }[/math], [math]\displaystyle{ c }[/math], [math]\displaystyle{ i }[/math], and [math]\displaystyle{ j }[/math] between 1
and [math]\displaystyle{ n }[/math], the functions
[math]\displaystyle{ \Gamma_{ab}^c &:= \frac{1}{2} \sum_{d=1}^n \left(\frac{\partial g_{bd}}{\partial x^a} + \frac{\partial g_{ad}}{\partial x^b} - \frac{\partial g_{ab}}{\partial x^d}\right)g^{cd}\\
R_{ij} &:= \sum_{a=1}^n\frac{\partial\Gamma_{ij}^a}{\partial x^a} - \sum_{a=1}^n\frac{\partial\Gamma_{ai}^a}{\partial x^j} + \sum_{a=1}^n\sum_{b=1}^n\left(\Gamma_{ab}^a\Gamma_{ij}^b - \Gamma_{ib}^a\
Gamma_{aj}^b\right) }[/math]
as maps [math]\displaystyle{ \varphi: U \rightarrow \mathbb{R} }[/math].
Now let [math]\displaystyle{ \left( U, \varphi \right) }[/math] and [math]\displaystyle{ \left( V, \psi \right) }[/math] be two smooth charts with [math]\displaystyle{ U \cap V \neq \emptyset }[/
math]. Let [math]\displaystyle{ R_{ij}: \varphi(U) \rightarrow \mathbb{R} }[/math] be the functions computed as above via the chart [math]\displaystyle{ \left( U, \varphi \right) }[/math] and let
[math]\displaystyle{ r_{ij}: \psi(V) \rightarrow \mathbb{R} }[/math] be the functions computed as above via the chart [math]\displaystyle{ \left( V, \psi \right) }[/math]. Then one can check by a
calculation with the chain rule and the product rule that
[math]\displaystyle{ R_{ij}(x) = \sum_{k,l=1}^n r_{kl}\left(\psi\circ\varphi^{-1}(x)\right)D_i\Big|_x \left(\psi\circ\varphi^{-1}\right)^kD_j\Big|_x \left(\psi\circ\varphi^{-1}\right)^l. }[/math]
where [math]\displaystyle{ D_{i} }[/math] is the first derivative along [math]\displaystyle{ i }[/math]th direction of [math]\displaystyle{ \mathbb{R}^n }[/math]. This shows that the following
definition does not depend on the choice of [math]\displaystyle{ \left( U, \varphi \right) }[/math]. For any [math]\displaystyle{ p \in U }[/math], define a bilinear map [math]\displaystyle{ \
operatorname{Ric}_p : T_p M \times T_p M \rightarrow \mathbb{R} }[/math] by
[math]\displaystyle{ (X, Y) \in T_p M \times T_p M \mapsto \operatorname{Ric}_p(X,Y) = \sum_{i,j=1}^n R_{ij}(\varphi(x))X^i(p)Y^j(p), }[/math]
where [math]\displaystyle{ X^1, \ldots, X^n }[/math] and [math]\displaystyle{ Y^1, \ldots, Y^n }[/math] are the components of the tangent vectors at [math]\displaystyle{ p }[/math] in [math]\
displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] relative to the coordinate vector fields of [math]\displaystyle{ \left( U, \varphi \right) }[/math].
It is common to abbreviate the above formal presentation in the following style:
The final line includes the demonstration that the bilinear map Ric is well-defined, which is much easier to write out with the informal notation.
Comparison of the definitions
The two above definitions are identical. The formulas defining [math]\displaystyle{ \Gamma_{ij}^k }[/math] and [math]\displaystyle{ R_{ij} }[/math] in the coordinate approach have an exact parallel
in the formulas defining the Levi-Civita connection, and the Riemann curvature via the Levi-Civita connection. Arguably, the definitions directly using local coordinates are preferable, since the
"crucial property" of the Riemann tensor mentioned above requires [math]\displaystyle{ M }[/math] to be Hausdorff in order to hold. By contrast, the local coordinate approach only requires a smooth
atlas. It is also somewhat easier to connect the "invariance" philosophy underlying the local approach with the methods of constructing more exotic geometric objects, such as spinor fields.
The complicated formula defining [math]\displaystyle{ R_{ij} }[/math] in the introductory section is the same as that in the following section. The only difference is that terms have been grouped so
that it is easy to see that [math]\displaystyle{ R_{ij}=R_{ji}. }[/math]
As can be seen from the symmetries of the Riemann curvature tensor, the Ricci tensor of a Riemannian manifold is symmetric, in the sense that
[math]\displaystyle{ \operatorname{Ric}(X ,Y) = \operatorname{Ric}(Y,X) }[/math]
for all [math]\displaystyle{ X,Y\in T_pM. }[/math]
It thus follows linear-algebraically that the Ricci tensor is completely determined by knowing the quantity [math]\displaystyle{ \operatorname{Ric}(X, X) }[/math] for all vectors [math]\displaystyle{
X }[/math] of unit length. This function on the set of unit tangent vectors is often also called the Ricci curvature, since knowing it is equivalent to knowing the Ricci curvature tensor.
The Ricci curvature is determined by the sectional curvatures of a Riemannian manifold, but generally contains less information. Indeed, if [math]\displaystyle{ \xi }[/math] is a vector of unit
length on a Riemannian [math]\displaystyle{ n }[/math]-manifold, then [math]\displaystyle{ \operatorname{Ric}(\xi, \xi) }[/math] is precisely [math]\displaystyle{ (n - 1) }[/math] times the average
value of the sectional curvature, taken over all the 2-planes containing [math]\displaystyle{ \xi }[/math]. There is an [math]\displaystyle{ (n - 2) }[/math]-dimensional family of such 2-planes, and
so only in dimensions 2 and 3 does the Ricci tensor determine the full curvature tensor. A notable exception is when the manifold is given a priori as a hypersurface of Euclidean space. The second
fundamental form, which determines the full curvature via the Gauss–Codazzi equation, is itself determined by the Ricci tensor and the principal directions of the hypersurface are also the
eigendirections of the Ricci tensor. The tensor was introduced by Ricci for this reason.
As can be seen from the second Bianchi identity, one has
[math]\displaystyle{ \operatorname{div}\operatorname{Ric} = \frac{1}{2}dR, }[/math]
where [math]\displaystyle{ R }[/math] is the scalar curvature, defined in local coordinates as [math]\displaystyle{ g^{ij}R_{ij}. }[/math] This is often called the contracted second Bianchi identity.
Informal properties
The Ricci curvature is sometimes thought of as (a negative multiple of) the Laplacian of the metric tensor (Chow Knopf).^[3] Specifically, in harmonic local coordinates the components satisfy
[math]\displaystyle{ R_{ij} = -\frac{1}{2}\Delta \left(g_{ij}\right) + \text{lower-order terms}, }[/math]
where [math]\displaystyle{ \Delta = \nabla \cdot \nabla }[/math] is the Laplace–Beltrami operator, here regarded as acting on the locally-defined functions [math]\displaystyle{ g_{ij} }[/math]. This
fact motivates, for instance, the introduction of the Ricci flow equation as a natural extension of the heat equation for the metric. Alternatively, in a normal coordinate system based at [math]\
displaystyle{ p }[/math],
[math]\displaystyle{ R_{ij} = -\frac{2}{3}\Delta \left(g_{ij}\right). }[/math]
Direct geometric meaning
Near any point [math]\displaystyle{ p }[/math] in a Riemannian manifold [math]\displaystyle{ \left( M, g \right) }[/math], one can define preferred local coordinates, called geodesic normal
coordinates. These are adapted to the metric so that geodesics through [math]\displaystyle{ p }[/math] correspond to straight lines through the origin, in such a manner that the geodesic distance
from [math]\displaystyle{ p }[/math] corresponds to the Euclidean distance from the origin. In these coordinates, the metric tensor is well-approximated by the Euclidean metric, in the precise sense
[math]\displaystyle{ g_{ij} = \delta_{ij} + O \left(|x|^2\right) . }[/math]
In fact, by taking the Taylor expansion of the metric applied to a Jacobi field along a radial geodesic in the normal coordinate system, one has [math]\displaystyle{ g_{ij} = \delta_{ij} - \frac{1}
{3} R_{ikjl}x^kx^l + O\left(|x|^3\right) . }[/math]
In these coordinates, the metric volume element then has the following expansion at p:
[math]\displaystyle{ d\mu_g = \left[ 1 - \frac{1}{6} R_{jk}x^j x^k+ O\left(|x|^3\right) \right] d\mu_\text{Euclidean} , }[/math]
which follows by expanding the square root of the determinant of the metric.
Thus, if the Ricci curvature [math]\displaystyle{ \operatorname{Ric}(\xi, \xi) }[/math] is positive in the direction of a vector [math]\displaystyle{ \xi }[/math], the conical region in [math]\
displaystyle{ M }[/math] swept out by a tightly focused family of geodesic segments of length [math]\displaystyle{ \varepsilon }[/math] emanating from [math]\displaystyle{ p }[/math], with initial
velocity inside a small cone about [math]\displaystyle{ \xi }[/math], will have smaller volume than the corresponding conical region in Euclidean space, at least provided that [math]\displaystyle{ \
varepsilon }[/math] is sufficiently small. Similarly, if the Ricci curvature is negative in the direction of a given vector [math]\displaystyle{ \xi }[/math], such a conical region in the manifold
will instead have larger volume than it would in Euclidean space.
The Ricci curvature is essentially an average of curvatures in the planes including [math]\displaystyle{ \xi }[/math]. Thus if a cone emitted with an initially circular (or spherical) cross-section
becomes distorted into an ellipse (ellipsoid), it is possible for the volume distortion to vanish if the distortions along the principal axes counteract one another. The Ricci curvature would then
vanish along [math]\displaystyle{ \xi }[/math]. In physical applications, the presence of a nonvanishing sectional curvature does not necessarily indicate the presence of any mass locally; if an
initially circular cross-section of a cone of worldlines later becomes elliptical, without changing its volume, then this is due to tidal effects from a mass at some other location.
Ricci curvature plays an important role in general relativity, where it is the key term in the Einstein field equations.
Ricci curvature also appears in the Ricci flow equation, where certain one-parameter families of Riemannian metrics are singled out as solutions of a geometrically-defined partial differential
equation. This system of equations can be thought of as a geometric analog of the heat equation, and was first introduced by Richard S. Hamilton in 1982. Since heat tends to spread through a solid
until the body reaches an equilibrium state of constant temperature, if one is given a manifold, the Ricci flow may be hoped to produce an 'equilibrium' Riemannian metric which is Einstein or of
constant curvature. However, such a clean "convergence" picture cannot be achieved since many manifolds cannot support such metrics. A detailed study of the nature of solutions of the Ricci flow, due
principally to Hamilton and Grigori Perelman, shows that the types of "singularities" that occur along a Ricci flow, corresponding to the failure of convergence, encodes deep information about
3-dimensional topology. The culmination of this work was a proof of the geometrization conjecture first proposed by William Thurston in the 1970s, which can be thought of as a classification of
compact 3-manifolds.
On a Kähler manifold, the Ricci curvature determines the first Chern class of the manifold (mod torsion). However, the Ricci curvature has no analogous topological interpretation on a generic
Riemannian manifold.
Global geometry and topology
Here is a short list of global results concerning manifolds with positive Ricci curvature; see also classical theorems of Riemannian geometry. Briefly, positive Ricci curvature of a Riemannian
manifold has strong topological consequences, while (for dimension at least 3), negative Ricci curvature has no topological implications. (The Ricci curvature is said to be positive if the Ricci
curvature function [math]\displaystyle{ \operatorname{Ric}(\xi, \xi) }[/math] is positive on the set of non-zero tangent vectors [math]\displaystyle{ \xi }[/math].) Some results are also known for
pseudo-Riemannian manifolds.
1. Myers' theorem (1941) states that if the Ricci curvature is bounded from below on a complete Riemannian n-manifold by [math]\displaystyle{ (n - 1)k \gt 0 }[/math], then the manifold has diameter
[math]\displaystyle{ \leq \pi / \sqrt{k} }[/math]. By a covering-space argument, it follows that any compact manifold of positive Ricci curvature must have finite fundamental group. Cheng (1975)
showed that, in this setting, equality in the diameter inequality occurs if only if the manifold is isometric to a sphere of a constant curvature [math]\displaystyle{ k }[/math].
2. The Bishop–Gromov inequality states that if a complete [math]\displaystyle{ n }[/math]-dimensional Riemannian manifold has non-negative Ricci curvature, then the volume of a geodesic ball is less
than or equal to the volume of a geodesic ball of the same radius in Euclidean [math]\displaystyle{ n }[/math]-space. Moreover, if [math]\displaystyle{ v_p(R) }[/math] denotes the volume of the
ball with center [math]\displaystyle{ p }[/math] and radius [math]\displaystyle{ R }[/math] in the manifold and [math]\displaystyle{ V(R) = c_n R^n }[/math] denotes the volume of the ball of
radius [math]\displaystyle{ R }[/math] in Euclidean [math]\displaystyle{ n }[/math]-space then the function [math]\displaystyle{ v_p(R) / V(R) }[/math] is nonincreasing. This can be generalized
to any lower bound on the Ricci curvature (not just nonnegativity), and is the key point in the proof of Gromov's compactness theorem.)
3. The Cheeger–Gromoll splitting theorem states that if a complete Riemannian manifold [math]\displaystyle{ \left( M, g \right) }[/math] with [math]\displaystyle{ \operatorname{Ric} \geq 0 }[/math]
contains a line, meaning a geodesic [math]\displaystyle{ \gamma : \mathbb{R} \to M }[/math] such that [math]\displaystyle{ d(\gamma(u), \gamma(v)) = \left| u - v \right| }[/math] for all [math]\
displaystyle{ u, v \in \mathbb{R} }[/math], then it is isometric to a product space [math]\displaystyle{ \mathbb{R} \times L }[/math]. Consequently, a complete manifold of positive Ricci
curvature can have at most one topological end. The theorem is also true under some additional hypotheses for complete Lorentzian manifolds (of metric signature [math]\displaystyle{ \left( + - -
\ldots \right) }[/math]) with non-negative Ricci tensor (Galloway 2000).
4. Hamilton's first convergence theorem for Ricci flow has, as a corollary, that the only compact 3-manifolds which have Riemannian metrics of positive Ricci curvature are the quotients of the
3-sphere by discrete subgroups of SO(4) which act properly discontinuously. He later extended this to allow for nonnegative Ricci curvature. In particular, the only simply-connected possibility
is the 3-sphere itself.
These results, particularly Myers' and Hamilton's, show that positive Ricci curvature has strong topological consequences. By contrast, excluding the case of surfaces, negative Ricci curvature is now
known to have no topological implications; (Lohkamp 1994) has shown that any manifold of dimension greater than two admits a complete Riemannian metric of negative Ricci curvature. In the case of
two-dimensional manifolds, negativity of the Ricci curvature is synonymous with negativity of the Gaussian curvature, which has very clear topological implications. There are very few two-dimensional
manifolds which fail to admit Riemannian metrics of negative Gaussian curvature.
Behavior under conformal rescaling
If the metric [math]\displaystyle{ g }[/math] is changed by multiplying it by a conformal factor [math]\displaystyle{ e^{2f} }[/math], the Ricci tensor of the new, conformally-related metric [math]\
displaystyle{ \tilde{g} = e^{2f} g }[/math] is given (Besse 1987) by
[math]\displaystyle{ \widetilde{\operatorname{Ric}}=\operatorname{Ric}+(2-n)\left[ \nabla df-df\otimes df\right]+\left[\Delta f -(n-2)\|df\|^2\right]g , }[/math]
where [math]\displaystyle{ \Delta = *d*d }[/math] is the (positive spectrum) Hodge Laplacian, i.e., the opposite of the usual trace of the Hessian.
In particular, given a point [math]\displaystyle{ p }[/math] in a Riemannian manifold, it is always possible to find metrics conformal to the given metric [math]\displaystyle{ g }[/math] for which
the Ricci tensor vanishes at [math]\displaystyle{ p }[/math]. Note, however, that this is only pointwise assertion; it is usually impossible to make the Ricci curvature vanish identically on the
entire manifold by a conformal rescaling.
For two dimensional manifolds, the above formula shows that if [math]\displaystyle{ f }[/math] is a harmonic function, then the conformal scaling [math]\displaystyle{ g \mapsto e^{2f}g }[/math] does
not change the Ricci tensor (although it still changes its trace with respect to the metric unless [math]\displaystyle{ f = 0 }[/math].
Trace-free Ricci tensor
In Riemannian geometry and pseudo-Riemannian geometry, the trace-free Ricci tensor (also called traceless Ricci tensor) of a Riemannian or pseudo-Riemannian [math]\displaystyle{ n }[/math]-manifold
[math]\displaystyle{ \left( M, g \right) }[/math] is the tensor defined by
[math]\displaystyle{ Z = \operatorname{Ric} - \frac{1}{n}Rg , }[/math]
where [math]\displaystyle{ \operatorname{Ric} }[/math] and [math]\displaystyle{ R }[/math] denote the Ricci curvature and scalar curvature of [math]\displaystyle{ g }[/math]. The name of this object
reflects the fact that its trace automatically vanishes: [math]\displaystyle{ \operatorname{tr}_gZ\equiv g^{ab}Z_{ab} = 0. }[/math] However, it is quite an important tensor since it reflects an
"orthogonal decomposition" of the Ricci tensor.
The orthogonal decomposition of the Ricci tensor
The following, not so trivial, property is
[math]\displaystyle{ \operatorname{Ric} = Z + \frac{1}{n}Rg. }[/math]
It is less immediately obvious that the two terms on the right hand side are orthogonal to each other:
[math]\displaystyle{ \left\langle Z, \frac{1}{n}Rg\right\rangle_g \equiv g^{ab}\left(R_{ab} - \frac{1}{n}Rg_{ab}\right) = 0. }[/math]
An identity which is intimately connected with this (but which could be proved directly) is that
[math]\displaystyle{ \left|\operatorname{Ric}\right|_g^2 = |Z|_g^2 + \frac{1}{n}R^2. }[/math]
The trace-free Ricci tensor and Einstein metrics
By taking a divergence, and using the contracted Bianchi identity, one sees that [math]\displaystyle{ Z = 0 }[/math] implies [math]\displaystyle{ \frac{1}{2}dR - \frac{1}{n}dR = 0 }[/math]. So,
provided that n ≥ 3 and [math]\displaystyle{ M }[/math] is connected, the vanishing of [math]\displaystyle{ Z }[/math] implies that the scalar curvature is constant. One can then see that the
following are equivalent:
In the Riemannian setting, the above orthogonal decomposition shows that [math]\displaystyle{ R^2 = n|\operatorname{Ric}|^2 }[/math] is also equivalent to these conditions. In the pseudo-Riemmannian
setting, by contrast, the condition [math]\displaystyle{ |Z|_g^2 = 0 }[/math] does not necessarily imply [math]\displaystyle{ Z = 0, }[/math] so the most that one can say is that these conditions
imply [math]\displaystyle{ R^2 = n \left|\operatorname{Ric}\right|_g^2. }[/math]
In particular, the vanishing of trace-free Ricci tensor characterizes Einstein manifolds, as defined by the condition [math]\displaystyle{ \operatorname{Ric} = \lambda g }[/math] for a number [math]\
displaystyle{ \lambda. }[/math] In general relativity, this equation states that [math]\displaystyle{ \left( M, g \right) }[/math] is a solution of Einstein's vacuum field equations with cosmological
Kähler manifolds
On a Kähler manifold [math]\displaystyle{ X }[/math], the Ricci curvature determines the curvature form of the canonical line bundle (Moroianu 2007). The canonical line bundle is the top exterior
power of the bundle of holomorphic Kähler differentials:
[math]\displaystyle{ \kappa = {\textstyle\bigwedge}^n ~ \Omega_X. }[/math]
The Levi-Civita connection corresponding to the metric on [math]\displaystyle{ X }[/math] gives rise to a connection on [math]\displaystyle{ \kappa }[/math]. The curvature of this connection is the
2-form defined by
[math]\displaystyle{ \rho(X,Y) \;\stackrel{\text{def}}{=}\; \operatorname{Ric}(JX,Y) }[/math]
where [math]\displaystyle{ J }[/math] is the complex structure map on the tangent bundle determined by the structure of the Kähler manifold. The Ricci form is a closed 2-form. Its cohomology class
is, up to a real constant factor, the first Chern class of the canonical bundle, and is therefore a topological invariant of [math]\displaystyle{ X }[/math] (for compact [math]\displaystyle{ X }[/
math]) in the sense that it depends only on the topology of [math]\displaystyle{ X }[/math] and the homotopy class of the complex structure.
Conversely, the Ricci form determines the Ricci tensor by
[math]\displaystyle{ \operatorname{Ric}(X, Y) = \rho(X, JY). }[/math]
In local holomorphic coordinates [math]\displaystyle{ z^\alpha }[/math], the Ricci form is given by
[math]\displaystyle{ \rho = -i\partial\overline{\partial}\log\det\left(g_{\alpha\overline{\beta}}\right) }[/math]
where ∂ is the Dolbeault operator and
[math]\displaystyle{ g_{\alpha\overline{\beta}} = g\left(\frac{\partial}{\partial z^\alpha}, \frac{\partial}{\partial\overline{z}^\beta}\right). }[/math]
If the Ricci tensor vanishes, then the canonical bundle is flat, so the structure group can be locally reduced to a subgroup of the special linear group [math]\displaystyle{ SL(n; \mathbb{C}) }[/
math]. However, Kähler manifolds already possess holonomy in [math]\displaystyle{ U(n) }[/math], and so the (restricted) holonomy of a Ricci-flat Kähler manifold is contained in [math]\displaystyle{
SU(n) }[/math]. Conversely, if the (restricted) holonomy of a 2[math]\displaystyle{ n }[/math]-dimensional Riemannian manifold is contained in [math]\displaystyle{ SU(n) }[/math], then the manifold
is a Ricci-flat Kähler manifold (Kobayashi Nomizu).
Generalization to affine connections
The Ricci tensor can also be generalized to arbitrary affine connections, where it is an invariant that plays an especially important role in the study of projective geometry (geometry associated to
unparameterized geodesics) (Nomizu Sasaki). If [math]\displaystyle{ \nabla }[/math] denotes an affine connection, then the curvature tensor [math]\displaystyle{ R }[/math] is the (1,3)-tensor defined
[math]\displaystyle{ R(X,Y)Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z }[/math]
for any vector fields [math]\displaystyle{ X, Y, Z }[/math]. The Ricci tensor is defined to be the trace:
[math]\displaystyle{ \operatorname{ric}(X,Y) = \operatorname{tr}\big(Z\mapsto R(Z,X)Y\big). }[/math]
In this more general situation, the Ricci tensor is symmetric if and only if there exists locally a parallel volume form for the connection.
Discrete Ricci curvature
Notions of Ricci curvature on discrete manifolds have been defined on graphs and networks, where they quantify local divergence properties of edges. Ollivier's Ricci curvature is defined using
optimal transport theory.^[4] A different (and earlier) notion, Forman's Ricci curvature, is based on topological arguments.^[5]
See also
External links
Original source: https://en.wikipedia.org/wiki/Ricci curvature. Read more | {"url":"https://handwiki.org/wiki/Ricci_curvature","timestamp":"2024-11-08T22:24:05Z","content_type":"text/html","content_length":"122412","record_id":"<urn:uuid:b029631e-47c6-4161-ac51-c117bda5f842>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00561.warc.gz"} |
Capacitance - GAMSAT Notes
in Physics
• Capacitors are used to store charge.
• The capacitance of a capacitor is the value that tells you how good a capacitor is at storing charge
Capacitance = Charge stored on a capacitor / Voltage across the capacitor
• Capacitance (C) = Q / V
• Q = Coulombs (c)
• V = Volt
• Coulomb / Volt = Farad (F)
Capacitors store Electrical Potential Energy:
But, E[of a capacitor] = 1/2 QV
• Not all of the charges drop through the voltage V
• As charge decreases, the voltage across the capacitor decreases
• Only the first charge drops through voltage V
Here we have two plates with charge of -6, and +6; denoted with the plus and minus symbols.
• The electrons are going to move to the positive plate
• Now, on the first go, the positive plate has 6/6 = 100%
• On the third one, the positive plate only has 4/6 at the ‘time of departure’, which is 66%
• The fifth one has 2/6 = 33%, and so on:
Recommended Reading for Physics
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What Is Liquidation Price? A Definition for Beginners
• Published
• Updated
• InLeverage
To get the complete picture of what liquidation price means in relation to what liquidation is and to understand how losses work in leveraged trading I would recommend that you read up on how
liquidation works with leverage.
In this article, I will break down the concept of the liquidation price which is misunderstood among most beginner traders since the nature of how it works might seem complicated at first glance.
Key takeaways
• Liquidation price is the distance from your entry price to the price where your leveraged position gets liquidated due to a loss.
• The greater the amount of credit, the shorter the distance becomes from your entry price to your liquidation price, increasing the risk of liquidation.
• The liquidation price can be derived by using a liquidation price formula that is derived from the chosen ratio.
Related: Use our liquidation price calculator to manage your risk.
Liquidation price explained
The liquidation price is the distance from the entry price of your leveraged position to where it gets liquidated.
Remember, a liquidation can only happen in a market with at least a 1:2 multiplier ratio.
At a ratio of 1:2, the distance to your liquidation price will be 50% of your entry price.
This is easy to remember as 1:2 cuts the liquidation price in half, hence the 50% distance.
When trading without margin, liquidation is not a factor.
Your position can fall indefinitely, but it can never get liquidated.
Now, below is a chart of BNB/USD to illustrate how liquidation price works at a 1:2 ratio.
The entry price of the trade is $277 and with a 1:2 ratio, the distance to your liquidation price is 50%.
This means that once you reach a loss of -50% with a ratio of 1:2 your position will get liquidated.
As you increase the multiplier, the distance from your entry price to the liquidation price shrinks.
• At a leverage ratio of 1:3, your liquidation price will be 33% from your entry price.
• At a leverage ratio of 1:4, your liquidation price will be 25% from your entry price.
• At a leverage ratio of 1:5, your liquidation price will be 20% from your entry price.
You can see how the distance shrinks with the increase of your multiplier.
This can further be illustrated with a chart/table to show you how the relationship between margin and liquidation price works.
The table below shows ratios from 1:1 leverage to 1:100 leverage with the corresponding price of liquidation.
For each position, are assume that the trader is trading BTC/USD and the entry price will always be $20.000. From here we get the liquidation price.
Leverage Ratio 1:1 1:2 1:5 1:10 1:15 1:20 1:35 1:50 1:75 1:100
Liquidation price – $10.000 $16.000 $18.000 $18.668 $19.000 $19.428 $19.600 $19.734 $19.800
Liquidation distance (%) – -50% -20% -10% -6.66% -5.00% -2.86% -2.00% -1.33% -1.00%
To find out the liquidation price for a ratio of more than 1:100 see the formula below.
Why does it exist in the first place?
The meaning of the liquidation price is:
• To indicate at which price level your losses would mount up to the total value of your account balance.
• To give a representation of the maximum risk for each position.
• To help traders adjust their risk profile.
Two factors play a big role in why the liquidation price exists in the first place.
This concept would not exist without credit and how margin accounts function.
When you open a position with a multiplier you are opening a position that is larger than your total account balance.
Liquidation price is a factor since all your losses are deducted from your margin balance and not the total position size including borrowed capital.
Suppose you have $800 in your forex account and you enter a trade with a 1:2 multiplier.
This would give you a total position value of $1600 which is twice the size of your initial deposit.
Since your account balance can only cover losses up to $800, your position will get closed out when this loss has been reached, which is at -50% of the total trade value of $1600.
On the contrary, when you trade without borrowed funds, your account balance will be able to cover all the losses until the underlying asset falls to literally $0.
How it works in simple terms
Liquidation is an automatic process that is controlled and executed by the broker.
Once your margin requirement hits 0% and your asset price has reached the liquidation price all your open positions are closed ut and all your funds in your account are lost.
Once the liquidation is in process, your open positions will be sold back to the market with a market order.
Before a liquidation, the trader will receive a warning signal called a margin call.
This warning is meant to warn the trader that the open losses are close to reaching the full value of the account.
If nothing is done to prevent further losses the account will get liquidated once the losses reach the threshold.
A great way to avoid liquidations is to predetermine at what price you are going to receive a margin call.
This can be done by using a margin call calculator.
Examples that describe it better
I will give you two different examples that describe this concept.
In the first example, Trader A will use a ratio of 1:25 and in the second example, Trader B will use a ratio of 1:175.
Example 1
Trader A is trading Bitcoin with a ratio of 1:25.
The current price of Bitcoin is $19,419.
If we use the same formula as above the liquidation price, in this case, would be:
100 / 25 = 4%
$19,419 x 0.04 = $776.50
$19,419 – $776.50 = $18,642.50
Example 2
Trade B is trading the Nasdaq index with a ratio of 1:175.
The current price of Nasdaq is $10,652.
We use the same formula to figure out the liquidation price, here is the result:
100 / 175 = 0.57%
$10,652 x 0.0057 = $60,70
$10,652 – $60,70 = $10,591.30
How to prevent blowing up your account
There are several ways to prevent liquidation depending on what market and what kind of broker you choose.
Here are my best tips:
1. Use a stop loss – A stop loss is a protective risk management tool that prevents further losses from happening. With a stop loss you can choose the maximum loss per position either with a dollar
value or with a percentage value.
2. Trade with a lower leverage ratio – If you have read this article from the top you will begin to understand that the risk of getting liquidated increases with increased margin. Thus, using a
lower leverage ratio will significantly reduce the chances of getting liquidated.
3. Use isolated margin when possible – Isolated margin is a way of isolating all losses to one position only. When isolated margin is used, only that one position can get liquidated. This prevents
the whole account from getting liquidated from one bad trade. The difference between cross margin and isolated margin is how the margin requirement is shared.
4. Trade fewer markets – When trading fewer markets it is easier to maintain control over your positions and your losses. Beginner traders who trade several markets at the same time with high ratios
usually suffer more frequent liquidations.
5. Learn how to calculate leverage – By calculating your margin before you enter the market you can set yourself up for success. This is one of the best ways to control your positions and risk in
How to figure out the loss
There are two ways of finding out the liquidation loss.
It can either be an account-wide liquidation or a position liquidation.
1. Full account liquidation – If your total account suffers a liquidation then the total loss will be the amount that you have deposited into your account.
2. Position liquidation – If your position gets liquidated then the total amount is the margin requirement that went into opening that position.
If your total account balance is $800 and your whole account gets liquidated then your total loss is $800.
However, if your total account balance is $800 but you only use $200 as margin requirement and your position gets liquidated then your total loss would be $200.
Can you get liquidated in spot trading?
No, it is not possible to get liquidated in a spot market since there is no leverage attached to the positions.
Only a leveraged position can get liquidated.
Positions in spot trading can fall in price and lose as much as 99.99% until the value of the asset hits zero in value.
To understand this concept further read our guide:
Anton Palovaara
Anton Palovaara is an expert leverage trader with decades of experience trading stocks and forex through proprietary software. After shifting over to leveraged crypto trading in derivatives and
futures contracts he has become an influential figure in the cryptocurrency industry. Anton's trading strategies have helped numerous investors achieve significant returns on their crypto
investments. With a keen eye for market trends and a deep understanding of technical analysis, Anton has developed a reputation as a shrewd trader who is not afraid to take calculated risks. He has a
track record of predicting market movements accurately, and his insights are highly sought after by crypto traders and investors alike.
Articles: 71 | {"url":"https://leverage.trading/what-is-liquidation-price/","timestamp":"2024-11-09T07:12:42Z","content_type":"text/html","content_length":"117788","record_id":"<urn:uuid:7c504400-2c6f-462d-a499-237ab2a8570a>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00283.warc.gz"} |
Sonmez (2006-08). Compactness of the dbar-Neumann problem and Stein neighborhood bases. Doctoral Dissertation.
• This dissertation consists of two parts. In the first part we show that for 1 k 1, a complex manifold M of dimension at least k in the boundary of a smooth bounded pseudoconvex domain in Cn is an
obstruction to compactness of the @- Neumann operator on (p, q)-forms for 0 p k n, provided that at some point of M, the Levi form of b has the maximal possible rank n − 1 − dim(M) (i.e. the
boundary is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction to compactness of the @-Neumann operator on (p, 1)-forms, provided that at
some point of the disc, the Levi form has only one vanishing eigenvalue (i.e. the eigenvalue zero has multiplicity one). We also show that a boundary point where the Levi form has only one
vanishing eigenvalue can be picked up by the plurisubharmonic hull of a set only via an analytic disc in the boundary. In the second part we obtain a weaker and quantified version of McNealA?s
Property ( eP) which still implies the existence of a Stein neighborhood basis. Then we give some applications on domains in C2 with a defining function that is plurisubharmonic on the boundary. | {"url":"https://vivo.library.tamu.edu/vivo/display/n725d6178","timestamp":"2024-11-11T00:00:42Z","content_type":"text/html","content_length":"14143","record_id":"<urn:uuid:3292707a-ecb7-4ad7-aade-23f858662c9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00825.warc.gz"} |
How to find experts for time-series forecasting and prediction in Python programming assignments? | Pay Someone To Do My Python Assignment
How to find experts for time-series forecasting and prediction in Python programming assignments? Last night, the day before I posted a blog news about computing in Python programming assignments, I
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assignments management. Let’s start with the most popular modules in Python and work on your topic structure and class lists. Python site Python Module is such a name for a top-level module that is
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python homework Given a parameter set is a tuple of variables of type T, the parameter set of GetParam function using the first argument is for the construction and the second is the firing of
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and look up code of your class in the File Browser. You can also check the view and put your web-based code along with it to help you find and search working code. We will be doing some analyzing
here. So first, we will let you to define a binary predicate function thatHow to find experts for time-series forecasting and prediction in Python programming assignments? In this post, I am going to
bring you knowledge based on time-series estimations of the following four factors in multiple occasions (2015, 2016, 2017, and 2018). The reason of doing this in Python projects is simply because I
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Begin with the four factors in the table below: Below is the line of Python code which makes it possible to find the experts in Python 5 and Python Server 2019. # Source of all arguments if(type
(input)==’float’){ global n, total, id, var id, for i in 1:n!=3:var id = str(id) if var id in n { id=0; var temp=id;var temp2=temp;var temp3=temp3;var temp4=temp4;var u1=isUnext(temp3,”U1″) temp1=
temp3; for j in 1:2:var u1 = print(name(u1,u2,u3,u4,u5,u6,u7,u8,u9)) if u1 = isUnext(temp2,”U2″){ if u1 == getUnext(“U2”) { print(“U1: “) } else { print(“U1 : “) } } fraction=1;numVar; for j in
1:numVar + varuj=isCompleting(temp3,”U1”) for m in sum(var_m+varuj):m = (row(sum(),total) / numVar+varuj)*int((number(m))-1)/sum ; if m > numVar : m navigate to this site m – m – int(result), mw=-1;
varuj[sum()]=(m-varuj)*int((number(m))-1)/sum try:sum=sum+numVar; fraction=2*result; fraction=fraction+1/numVar; if(fraction-1!=0.77)fraction*3/numVar=1; else:fraction=fraction+1/fraction; return
(sum, numVar); else: sum=getUnext(“U1”)*sum+numVar; returnHow to find experts for time-series forecasting and prediction in Python programming assignments? I`m having a tough time finding What
variables should I make in my 2nd hour prediction format? What time-series operators should I use for time-series predictions? And are there any links to them in the TPU? Edit No-no-no-no-no-no-no-no
= I think I am just oversanguint finding a good reference resource in the format explained by Guy. But I don`t have time series in Python. I have problems with time series that would be difficult to
keep up with in general programming practice. I found a nice guide online about time-series prediction and did all sorts of planning to become a Pythonic programmer. I find that the best way to use
tools in programming is to use “time-series python” is probably to think about the “time series” (with the intention to learn how to use it in a way that is clear to the programmer). You have to
understand that a series of these kind of things can be studied independently, but you must also understand the structure of your series. Here are some examples I found: As you see now, time-series
computing uses the most advanced tools that can extract the shape of a window. In this case, I`d say time-series Python and time-series ML methods. They are the most accurate for the most part, so
let`s try to find a few examples. import numpy as np import time import pandas as pd import timeformat ## Interval ## Index (optional) ## Time (optional) ## Baseline method ## Topping method ##
Function ## Set and Zero Decimal (optional) ## Topping method # Add 1(1) to the number of times the power line is plugged in import pandas as | {"url":"https://pythonhomework.com/how-to-find-experts-for-time-series-forecasting-and-prediction-in-python-programming-assignments","timestamp":"2024-11-14T08:17:20Z","content_type":"text/html","content_length":"94757","record_id":"<urn:uuid:bbaf6698-bca7-4d5a-91a8-b55847de6c85>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00832.warc.gz"} |
Shear Wave Velocity Calculator
Last updated:
Shear Wave Velocity Calculator
The shear wave velocity calculator determines the velocity of a shear wave as it moves through a body. This velocity is calculated based on the two main parameters or properties of a material, which
are density and shear modulus. An example of a shear wave is a seismic wave. That's why . The faster the velocity, the lesser the ground is shaking. Read on to understand what is shear wave velocity
and how to calculate shear wave velocity.
🙋 If you want to learn more about shear modulus, head to Omni's Poisson's ratio calculator.
What is a shear wave?
A shear wave is a type of wave occurring in bulk mediums when the particles oscillate perpendicular to the direction of wave propagation. A shear type of loading causes the shear wave. In case of an
axial strain, you may check our stress calculator.
The shear forces cause a change in the shape of the bulk medium and do not cause any change in the volume of the body. These forces occur in pairs, with either acting along the face of the medium.
The shear wave is also known as an S-wave. We define the shear wave velocity as the square root of the ratio of shear modulus (modulus of rigidity) to the density of material. The velocity of the
shear wave V[s] in the medium having shear modulus G and density ρ can be written using the shear wave velocity formula:
V[s] = √(G / ρ)
The shear wave velocity is about half of the longitudinal wave. The speed of longitudinal waves V[L] depends on the modulus of elasticity E and density.
V[L] = √(E / ρ)
🔎 You can evaluate the exact speed of longitudinal waves with our speed of sound in solids calculator.
How to use the shear wave velocity calculator?
Follow the steps below to find shear wave velocity:
• Step 1: Enter the shear modulus of the bulk medium.
• Step 2: Input the density of the material.
• Step 3: The shear wave velocity calculator will return the answer.
Example: What is shear wave velocity in a medium?
Now, let's try an example. Find the shear wave velocity for wave propagation in a copper rod.
You can select the material from the list, or you can enter the shear modulus and density for copper, which are 45 GPa and 8,940 kg/m³, respectively.
• Step 1: Enter the shear modulus, G = 45 GPa.
• Step 2: Input the density of material, ρ = 8,940 kg/m³.
• Step 3: The calculator will now use the shear wave velocity formula:
V = √(G / ρ) = √(45,000,000,000 / 8,940) = 2,243.6 m/s
The velocity of the shear wave in copper is 2.244 km/s.
What is shear wave?
The waves generated by a pair of shear forces acting along the opposite faces of a body is known as a shear wave. The particles in this wave oscillate perpendicular to the direction of wave
What is shear wave velocity?
The velocity at which shear waves propagate through bulk matter is called the shear wave velocity. It depends on the shear modulus and density of the material. Mathematically, that's: V[s] = √(G / ρ)
How do I find shear wave velocity?
To find shear wave velocity:
1. Divide the shear modulus by the density of the material.
2. Find the square root of this ratio. Mathematically, that's: V[s] = √(G / ρ).
What is velocity of shear wave in a titanium body?
The velocity of the shear wave in titanium is 3,018.5 m/s, which you can find by dividing the shear modulus, G = 41 GPa by density, ρ = 4,500 kg/m³ and then taking the square root. | {"url":"https://www.omnicalculator.com/physics/shear-wave-velocity","timestamp":"2024-11-13T09:23:48Z","content_type":"text/html","content_length":"391608","record_id":"<urn:uuid:4f6d0fb8-bc0c-4378-99ca-74bc397aa39a>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00499.warc.gz"} |
AI SL: Paper 1 (TZ2 May 2021)
Quiroz Math
14 Nov 2022120:13
TLDRThis video script offers a comprehensive educational session, covering a variety of mathematical problems and concepts. It begins with a detailed explanation of drug dosage calculation and the
percentage of drug elimination from the body over time. The script then delves into geometric problems involving the distance between points, coordinates of midpoints, and the height of 3D figures.
It continues with surface area calculations for a storage container and a box with a half-cylinder lid, followed by trigonometric problems related to angles of depression and elevation. The session
also includes a statistical analysis of sick days using a box and whisker plot, a t-test for comparing the weights of chinchilla and sable rabbits, and a compound interest problem. The script
concludes with a goodness of fit test for a newspaper vendor's sales model, a quadratic function exploration, and a calculus problem on profit maximization in electric car production.
• π The initial dose of a drug can be found by evaluating the drug concentration function at the earliest time point, which is when T equals zero.
• π § To determine the percentage of a drug that leaves the body each hour, it's important to understand the function's mechanics and not just take the number adjacent to the initial dose at face
• π For calculating the amount of drug remaining in the body after a certain number of hours, plug in the specific time (T) into the drug concentration function and evaluate.
• π The distance between two points in 3D space can be calculated using the distance formula, ensuring to correctly identify corresponding X, Y, and Z coordinates.
• π Finding the coordinates of a midpoint, such as a station between two others, involves averaging the respective coordinates of the two points.
• π To find the height of a point above the ground in a 3D scenario, identify the vertical coordinate (often Z) and apply the correct units.
• π ¨ When calculating the area to be painted for a storage container with a half-cylindrical lid, consider the exterior surface area, including the curved surface of the cylinder and the circles
at the ends.
• π The angle of depression from a point above to a point on the ground is equal to the angle of elevation from the ground point to the point above, crucial for understanding relative positions.
• π To find the distance between two points when one has moved from an initial to a final position, use the sine rule or trigonometric principles to relate the angles and sides of the triangles
• π ’ In a box and whisker diagram, the minimum, lower quartile (Q1), median (Q2), upper quartile (Q3), and maximum are represented by specific points on the graph, providing a quick summary of
data distribution.
• π The percentage of employees who took fewer or more sick days can be inferred from the spread of the box and whisker diagram, but it's important to correctly interpret the percentages
represented by each section.
Q & A
• What is the initial dose of the drug mentioned in the script, and how is it calculated?
-The initial dose of the drug is 23 milligrams. It is calculated by plugging in T equals 0 into the given function, which simplifies to 23 times 1, hence the initial dose is 23 milligrams.
• What is the percentage of the drug that leaves the body each hour, and why is the common interpretation incorrect?
-The common interpretation that 85 percent of the drug leaves the body each hour is incorrect. The correct interpretation is that 15 percent of the drug remains in the body each hour, which means
85 percent is incorrectly associated with the loss, rather than the remaining amount.
• How is the amount of drug remaining in the body after 10 hours calculated, and what is the result?
-The amount of drug remaining after 10 hours is calculated by plugging in T equals 10 into the function. The result is 4.52 milligrams, which is obtained by multiplying 23 by 0.85 to the power of
• What is the formula used to calculate the distance between two points in a 3D space, and how is it applied in the script?
-The formula used to calculate the distance between two points in a 3D space is the 3D distance formula: β ((x2 - x1)Β² + (y2 - y1)Β² + (z2 - z1)Β²). In the script, it is applied by identifying
the coordinates of points A and B and plugging the values into the formula to find the distance between them.
• How is the height of station M above the ground determined, and what is its value?
-The height of station M above the ground is determined by identifying the z-coordinate of the midpoint between stations A and B. Since the z-axis represents the vertical direction, the height of
station M is the z-coordinate of point M, which is 125 meters.
• What is the surface area of a half-cylinder, and how is it calculated in the script?
-The surface area of a half-cylinder includes the curved surface area and the area of the two half-circles at the ends. In the script, the curved surface area is calculated as Ο rh, and the area
of the two half-circles is calculated as Ο rΒ², with both results being halved because it's a half-cylinder.
• What is the angle of depression from point H to point C in the helicopter scenario, and how is it found?
-The angle of depression from point H to point C is 25 degrees. It is found by recognizing that the angle of depression is the same as the angle of elevation when observing point H from point C,
which is given as 25 degrees in the script.
• How is the distance from point A to point C calculated in the helicopter scenario?
-The distance from point A to point C is calculated using the sine rule, which relates the lengths of sides of a triangle to the sines of its opposite angles. The calculation involves finding the
unknown side (AC) using the known angle (25 degrees) and the opposite side (380 meters).
• What is the process to determine the location of the new bus stop on the road between two schools, and where should it be located?
-The process involves finding the perpendicular bisector of the line segment between the two schools. The bus stop should be located at the point where this perpendicular bisector intersects the
given road, ensuring equal distances from both schools.
• How is the equation of the perpendicular bisector found, and what is its final form?
-The equation of the perpendicular bisector is found by first determining the slope of the line segment between the two schools and then finding the negative reciprocal of this slope. Using the
midpoint of the segment and the new slope, the point-slope form of the line equation is used to derive the final form of the perpendicular bisector, which is y = -3x + 46.
π € Mathematical Drug Dosage Analysis
The paragraph discusses the mathematical modeling of a drug's concentration in the body over time after injection. It explains the concept of initial dose calculation, the percentage of drug
remaining after a certain period, and the importance of understanding the function's behavior. The speaker clarifies misconceptions about the percentage of drug elimination per hour and demonstrates
how to calculate the amount of drug left in the body after 10 hours using the formula provided.
π Calculating Distance and Coordinates in 3D Space
This section of the script focuses on using mathematical formulas to find distances between points and calculate coordinates in three-dimensional space. It introduces the distance formula and
midpoint formula, applying them to find the distance between two railway stations and the coordinates of a midpoint for a new station. The explanation emphasizes the importance of correctly pairing
coordinates and understanding the context of 3D space.
π ¨ Surface Area Calculation for a Storage Container
The script explains how to calculate the total exterior surface area of a storage container that consists of a box with a half-cylinder lid. It details the process of identifying which surfaces need
to be calculated and which do not, such as the bottom of the box. The explanation includes the formulas for the surface area of a rectangle and a cylinder, and it demonstrates the calculation of the
curved surface area of the half-cylinder and the area of the circular ends.
π Angles and Distances Involving a Helicopter and Swimmers
This paragraph explores geometric relationships involving a helicopter hovering above a lake and a swimmer observing the helicopter from different points. It discusses the concept of angle of
depression and elevation, using them to determine the angle of the helicopter from the swimmer's perspective. The script then delves into the use of sine rule to find the distance from the lake's
surface to the swimmer's point and the distance between two points on the shore, employing trigonometric principles to solve for unknown distances.
π Interpreting a Box and Whisker Diagram for Sick Days
The script provides an analysis of a box and whisker diagram representing the number of sick days taken by employees in a company over a year. It explains how to read the diagram to find the minimum,
lower quartile, median, and upper quartile values. The explanation also addresses a claim about the percentage of employees who took fewer or more sick days than certain values, using the diagram's
proportions to refute the claim.
π Locating a Bus Stop on a Road Using Perpendicular Bisector
The paragraph describes a problem involving finding the optimal location for a bus stop equidistant from two schools represented by points on a graph. It explains the concept of a perpendicular
bisector and how to find it using the slope of the line connecting the two schools and the midpoint formula. The explanation includes the process of deriving the equation of the perpendicular
bisector and finding its intersection with the given road to determine the bus stop's location.
π Determining the Range of a Function and Evaluating Its Inverse
This section discusses the process of finding the range of a given function by analyzing its graph and identifying the possible y-values it can take. It also covers the concept of an inverse function
and demonstrates how to derive it by swapping and solving for the variable. The explanation includes an example of evaluating the inverse function at a specific point to find its value.
π ° Statistical Analysis of Rabbit Weights Using a T-test
The script outlines a statistical test to determine if there is a significant difference in the weights of two types of rabbits, using a t-test. It explains how to formulate the null and alternative
hypotheses, perform the test using a calculator, and interpret the p-value obtained from the test. The explanation includes the steps to conclude the test based on the comparison of the p-value with
the significance level.
π ³ Calculating the Area and Length of a Sector in a Garden
This paragraph explains how to calculate the area of a lawn shaped like a sector of a circle and the length of the arc that forms its boundary. It introduces the formulas for the area of a sector and
the length of an arc, and demonstrates their application using the given angle and radius. The explanation includes the process of plugging the values into the formulas and performing the
calculations to find the area of the lawn and the length of the footpath around it.
π Ά Compound Interest Calculation for Savings Accounts
The script discusses the concept of compound interest, specifically when applied to savings accounts. It explains how to calculate the future value of an investment using the compound interest
formula, taking into account the principal amount, interest rate, compounding frequency, and the number of years. The explanation includes an example of calculating the amount in a college savings
account after five years with a given interest rate compounded half-yearly.
π Goodness of Fit Test for Newspaper Sales Prediction
This paragraph describes a goodness of fit test used to evaluate a newspaper vendor's model for predicting daily sales. It explains the process of estimating expected sales per day, calculating the
chi-squared value, and comparing it to a critical value to determine if the model is suitable. The explanation includes the steps for setting up the test, interpreting the results, and concluding
whether the model fits the actual sales data.
π Solving a Quadratic Function for a Parabola's Properties
The script explains how to solve for the coefficients of a quadratic function representing a parabola, given its vertex and the points where it intersects the x-axis. It demonstrates the process of
setting up a system of equations based on these points, solving the system to find the values of the coefficients, and using the formula for the axis of symmetry to find the line that runs through
the vertex of the parabola.
π Calculating Profit Function for Electric Car Production
This paragraph discusses the process of deriving a profit function for a company that produces electric cars, based on the rate of change of profit with respect to the number of cars produced. It
explains the concept of integration to find the original profit function from its derivative, using given values to solve for the constant of integration. The explanation includes an example of how
to determine the profit function and use it to analyze the company's profit at different production levels.
π ‘Medicinal drug
A medicinal drug refers to any substance or compound used for medical treatment. In the context of the video, it discusses the amount of a drug present in the body at different times after
administration, highlighting the importance of understanding drug dosage and its effects over time.
π ‘Initial dose
The initial dose is the starting amount of a drug given at the beginning of a treatment. The script explains calculating the initial dose of a drug in the body, emphasizing its significance in
determining the drug's concentration over time.
π ‘Percentage of drug leaving the body
This concept refers to the proportion of a drug that is eliminated from the body each hour. The script uses this to illustrate the decay of drug concentration, showing how understanding this rate is
crucial for calculating the drug's remaining amount in the body.
π ‘Distance formula
The distance formula is a mathematical formula used to calculate the distance between two points in space. The video script applies this formula to find the distance between two stations on a
railway, demonstrating its practical use in calculating real-world distances.
π ‘Midpoint
The midpoint is the central point between two endpoints of a line segment. In the script, the midpoint formula is used to find the coordinates of a new station located exactly halfway between two
existing stations, underscoring the concept's utility in locating positions in a 3D space.
π ‘Surface area
Surface area is the total area occupied by the surface of a three-dimensional object. The video explains calculating the surface area of a storage container, including its lid, to determine the area
that needs to be painted, illustrating the concept's relevance in real-life applications.
π ‘Angle of depression
The angle of depression is the angle formed between a horizontal line and the line of sight to an object below the horizontal plane. The script uses this concept to describe the angle at which a
helicopter is observed from a lower point, demonstrating its use in visual observations and measurements.
π ‘Box and whisker diagram
A box and whisker diagram is a graphical representation of statistical data that shows the distribution of data points. The video script uses this diagram to analyze the number of sick days taken by
employees, illustrating how it can be used to understand data distribution and identify statistical measures such as median and quartiles.
π ‘Perpendicular bisector
A perpendicular bisector is a line that is both perpendicular to a given line segment and bisects it into two equal parts. The script discusses finding the equation of the perpendicular bisector of a
line segment, showing its importance in locating a point equidistant from two given points.
π ‘Compound interest
Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. The video explains calculating the future
value of an investment using compound interest, highlighting the effect of interest compounding on investment growth.
π ‘Goodness of fit test
A goodness of fit test is a statistical test used to determine how well a set of data fits a certain distribution. The script describes using a goodness of fit test to evaluate a newspaper vendor's
model for predicting daily sales, illustrating the test's application in validating predictive models.
π ‘Quadratic function
A quadratic function is a polynomial function of degree two, often used to model phenomena with a parabolic relationship. The video script involves finding the equation of a quadratic function that
represents a parabola, demonstrating the process of determining the function's coefficients based on given points and vertex.
π ‘Integral
An integral is a fundamental concept in calculus, representing the area under a curve defined by a function. The script uses integration to find the original profit function from its rate of change,
showing how integration can be used to reverse the process of differentiation and recover the underlying function.
The initial dose of a medicinal drug is calculated using the formula DT at T equals zero.
The percentage of drug leaving the body each hour is intuitively determined by understanding the function's impact on the initial dose.
Calculating the amount of drug remaining after a certain time involves careful attention to the function's variables and units.
Finding the distance between two points involves applying the distance formula correctly with given coordinates.
The coordinates of a midpoint between two stations are calculated using the midpoint formula in a 3D space.
Determining the height of a point in a 3D coordinate system requires understanding the orientation of axes.
The total exterior surface area of a storage container is calculated by considering all relevant sides and shapes.
The curved surface area of a half-cylinder is computed using the formula for a full cylinder and adjusting for the half shape.
The angle of depression from a point is equal to the angle of elevation to the same point, a key concept in trigonometry.
Finding the distance from one point to another involves using trigonometric principles and the sine rule effectively.
Calculating the speed of an object involves converting time units and applying the correct formula for rate.
Interpreting a box and whisker diagram requires understanding the representation of minimum, lower quartile, median, and maximum values.
The percentage of employees who took fewer or more sick days is inferred from the box and whisker diagram's distribution.
Finding the equation of a perpendicular bisector involves understanding the relationship between slopes of perpendicular lines.
Determining the location of a bus stop equidistant from two schools involves finding the intersection of a perpendicular bisector and a given line.
The range of a function is determined by analyzing the graph and identifying the minimum and maximum y-values.
Calculating the value of an inverse function involves manipulating the original function's equation to solve for the inverse.
Conducting a t-test involves setting up null and alternative hypotheses, calculating the test statistic, and comparing it to a critical value or p-value.
The length of an arc in a circle is calculated using the proportion of the circle's angle and the circle's radius.
The area of a sector and a triangle within a circle are used to determine the area of a shaded lawn in a garden.
Compound interest is calculated using the formula that includes principal amount, interest rate, compounding frequency, and time.
Investment growth is modeled by setting up an equation based on the expected future value and solving for the unknown interest rate.
Goodness of fit tests are used to evaluate a model's accuracy by comparing observed and expected values.
The symmetry of a parabola is utilized to find unknown points on the graph by recognizing equal distances from the vertex.
The values of a, b, and c in a quadratic function are determined by creating and solving a system of equations derived from given points.
The axis of symmetry for a parabola is calculated using the formula that relates the coefficients a and b.
Finding the profit function involves integrating the rate of change of profit with respect to the number of cars produced.
The change in profit with varying production levels is analyzed by examining the derivative of the profit function. | {"url":"https://math.bot/blog-AI-SL-Paper-1-TZ2-May-2021-38367","timestamp":"2024-11-07T00:58:35Z","content_type":"text/html","content_length":"176738","record_id":"<urn:uuid:153c9b49-5960-43d4-a3ff-ac831b53b7ec>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00557.warc.gz"} |
Division Facts
Division is a fundamental operation in mathematics that allows us to divide numbers into equal parts and understand the concept of sharing and distribution. It plays a crucial role in various
real-life scenarios, from splitting a pizza among friends to calculating the average speed of a moving object. In this comprehensive article, we delve into the fascinating world of division,
uncovering 15 intriguing facts that will broaden your understanding of this mathematical operation. Join us as we embark on a journey of exploration and discover the beauty of division.
Division Basics: Dividend, Divisor, and Quotient
Division involves three key elements: the dividend, the divisor, and the quotient. The dividend is the number being divided, the divisor is the number by which we divide, and the quotient is the
result of the division.
The Division Symbol: ÷ and /
In mathematical notation, division is represented by the division symbol, which can be either “÷” or “/”. The dividend is written on top of the division symbol, while the divisor is written below it.
Division Terminology: Quotient and Remainder
In division, the quotient is the whole number part of the result. If there is a remainder, it represents the amount left over after dividing as completely as possible.
Image from Adobe Stock
Divisibility Rules: Dividing with Ease
Divisibility rules help determine if a number is divisible by another number without performing the actual division. For example, a number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.
Long Division: A Methodical Approach
Long division is a common method used to divide large numbers or numbers with multiple digits. It involves a step-by-step process of dividing, multiplying, subtracting, and bringing down digits until
the division is complete.
Division with Decimals: Extending the Concept
Division can also be performed with decimal numbers. In decimal division, the divisor and dividend are written with decimal points aligned, and the division is carried out as usual.
Division as the Inverse of Multiplication
Division and multiplication are inverse operations. Dividing a number by another is the same as multiplying the first number by the reciprocal of the second number.
Fraction Division: Sharing in Parts
Division can be represented as a fraction. Dividing one fraction by another is equivalent to multiplying the first fraction by the reciprocal of the second fraction.
Image from Adobe Stock
Division by Zero: Undefined
Division by zero is undefined in mathematics. It is impossible to divide a number by zero, as it leads to mathematical inconsistencies.
The Quotient Rule in Calculus
In calculus, the quotient rule is a formula used to find the derivative of a function that is the quotient of two other functions. It is an essential tool in differential calculus.
The Joy of Division Puzzles
Division puzzles are a fun way to sharpen your division skills and problem-solving abilities. From number grids to mystery quotient challenges, these puzzles provide a stimulating mental workout.
The Division Algorithm
The division algorithm is a fundamental theorem in number theory. It states that given any two integers, a dividend, and a nonzero divisor, there exist unique integers called the quotient and
remainder that satisfy the division equation.
Division in Advanced Mathematics
Division extends beyond elementary arithmetic and finds applications in advanced mathematical fields such as algebra, number theory, and calculus. It forms the basis for more complex concepts and
Image from Adobe Stock
Division in Computer Science
Division is an essential operation in computer science and programming. It is used in algorithms, data structures, and various mathematical calculations performed by computers.
Division in Everyday Life
Division plays a significant role in our daily lives. From dividing expenses among friends to calculating recipe measurements, understanding division helps us navigate practical situations.
Final Thoughts
Division is a fundamental mathematical operation that empowers us to split and distribute quantities, solve problems, and explore the intricate relationships between numbers. Through this article, we
have unveiled 15 fascinating facts about division, from its basic principles to its applications in various domains. Whether you’re a student, a math enthusiast, or someone curious about the world of
numbers, we hope this exploration has deepened your appreciation for the power and versatility of division. Division is not just a mathematical operation; it is a key that unlocks a world of
Frequently Asked Questions (FAQs)
What happens when you divide a number by zero?
Division by zero is undefined in mathematics. It leads to mathematical inconsistencies and cannot be performed.
Can you divide by a fraction?
Yes, you can divide by a fraction. Dividing by a fraction is equivalent to multiplying by its reciprocal.
What is the purpose of long division?
Long division is used to divide large numbers or numbers with multiple digits. It provides a systematic approach to division.
Are there any shortcuts for division?
Yes, divisibility rules provide shortcuts to determine if a number is divisible by another number without performing the division.
How is division used in real life?
Division is used in various real-life scenarios, such as dividing expenses among friends, calculating recipe measurements, and determining average values.
Our commitment to delivering trustworthy and engaging content is at the heart of what we do. Each fact on our site is contributed by real users like you, bringing a wealth of diverse insights and
information. To ensure the highest standards of accuracy and reliability, our dedicated editors meticulously review each submission. This process guarantees that the facts we share are not only
fascinating but also credible. Trust in our commitment to quality and authenticity as you explore and learn with us. | {"url":"https://facts.net/division-facts/","timestamp":"2024-11-14T10:57:19Z","content_type":"text/html","content_length":"230687","record_id":"<urn:uuid:b7458e7f-2cb5-4c4d-96b9-bd86da0d82bf>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00375.warc.gz"} |
How to Install Prob Package In R?
To install the "prob" package in R, you can use the install.packages() function. Simply type the following code in your R console:
1 install.packages("prob")
This will download and install the "prob" package from the Comprehensive R Archive Network (CRAN). You can then load the package using the library() function to start using its functions in your R
How to share data or results generated with the prob package in R?
Once you have generated data or results using the prob package in R, there are several ways you can share this information with others:
1. Save the data or results as an R object: You can save the data or results as an R object using the save() function. This will create a .RData file that can be shared with others. They can then
load the data or results into their own R session using the load() function.
1 # Save data or results as an R object
2 save(data_or_results, file = "data_or_results.RData")
1. Export the data or results as a CSV file: You can export the data or results as a CSV file using the write.csv() function. This will create a CSV file that can be easily shared with others and
opened in spreadsheet software.
1 # Export data or results as a CSV file
2 write.csv(data_or_results, file = "data_or_results.csv")
1. Create a report or presentation: You can create a report or presentation summarizing the data or results using R Markdown or other reporting packages in R. This report can then be shared with
others as a PDF, HTML, or Word document.
2. Share the code: If others are comfortable working with R, you can share the code you used to generate the data or results. This way, they can replicate your analysis and results in their own R
By using these methods, you can easily share the data or results generated with the prob package in R with others.
How to remove all dependencies of the prob package in R?
To remove all dependencies of the prob package in R, you can use the following steps:
1. First, load the prob package using the library() function:
1. Use the remove.packages() function to remove the prob package and all its dependencies:
1 remove.packages("prob", dependencies = TRUE)
This will remove the prob package and all its dependencies from your R environment.
How to install the prob package from CRAN in R?
To install the prob package from CRAN in R, you can use the following code:
1 install.packages("prob")
Simply run this code in your R console and the prob package will be downloaded and installed from CRAN.
How to check the version of the prob package installed in R?
You can check the version of the prob package installed in R by using the following command in the R console:
This will display the version number of the prob package that is currently installed in your R environment. | {"url":"https://tech-blog.us.to/blog/how-to-install-prob-package-in-r","timestamp":"2024-11-11T23:24:36Z","content_type":"text/html","content_length":"145491","record_id":"<urn:uuid:37400b69-39c7-4ce6-9450-6a5f3d5b2caa>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00786.warc.gz"} |
Computational Data Science
ST444 Half Unit
Computational Data Science
This information is for the 2024/25 session.
Teacher responsible
This course is available on the MSc in Data Science, MSc in Econometrics and Mathematical Economics, MSc in Statistics, MSc in Statistics (Financial Statistics), MSc in Statistics (Financial
Statistics) (Research), MSc in Statistics (Research), MSc in Statistics (Social Statistics) and MSc in Statistics (Social Statistics) (Research). This course is available with permission as an
outside option to students on other programmes where regulations permit.
Basic knowledge in calculus and linear algebra, as well as a first course in probability and statistics.
Course content
An introduction to the use of popular algorithms in statistics and data science, including (but not limit to) numerical linear algebra, optimisation, graph data and massive data processing, as well
as their applications. Examples include least squares, maximum likelihood, principle component analysis, LASSO and graphical LASSO, PageRank, etc. Throughout the course, students will gain practical
experience of implementing these computational methods in a programming language. Learning support will be provided for at least one programming language, such as R, Python or C++, but the choice of
language supported may vary between years, depending on judged benefits to students, whether in terms of pedagogy or resulting skills. This year, the default choice is Python.
This course will be delivered through a combination of classes/computer workshops/lectures/Q&A sessions totalling a minimum of 30 hours across Autumn Term. This course includes a reading week in Week
Lectures will cover:
(1) Introduction: overview of the topics to be discussed, how numbers are presented in memory, floating point arithmetic, stability of numerical algorithms
(2) Basic algorithms: overview of different types of algorithms, Big-O notation, elementary complexity analysis, and their applications in data science
(3) Tools in optimisation: convexity, bi-section, steepest descent, Newton’s method, Quasi-Newton methods, stochastic gradient, coordinate descent, other related topics (e.g. stochastic search, ADMM)
(4) Tools in numerical linear algebra: Gaussian elimination, Cholesky decomposition, LU decomposition, matrix inversion and condition, computing eigenvalues and eigenvectors, and their applications
(5) Other topics (if time permits): graph data processing, massive data processing, Monte-Carlo methods, etc
Formative coursework
Students will be expected to produce 4 problem sets in the AT.
Bi-weekly exercises, involving computer programming and theory.
Indicative reading
Computational Statistics by Givens and Hoeting
Statistical computing in C++ and R by Eubank and Kupresanin
Foundations of Data Science by Blum, Hopcoft and Kannan
Introduction to Algorithms by Cormen, Leiserson, Rivest and Stein
The Art of R Programming: A Tour of Statistical Software Design by Matloff
Think Python: How to Think Like a Computer Scientist by Downey
Exam (70%, duration: 2 hours) in the spring exam period.
Coursework (30%).
Student performance results
(2020/21 - 2022/23 combined)
Classification % of students
Distinction 33.3
Merit 33.3
Pass 25
Fail 8.3
Key facts
Department: Statistics
Total students 2023/24: 7
Average class size 2023/24: 8
Controlled access 2023/24: Yes
Value: Half Unit
Guidelines for interpreting course guide information
Course selection videos
Some departments have produced short videos to introduce their courses. Please refer to the course selection videos index page for further information.
Personal development skills
• Self-management
• Team working
• Problem solving
• Application of information skills
• Communication
• Application of numeracy skills | {"url":"https://www.lse.ac.uk/resources/calendar2024-2025/courseGuides/ST/2024_ST444.htm","timestamp":"2024-11-12T20:21:06Z","content_type":"text/html","content_length":"12827","record_id":"<urn:uuid:50c1712f-b908-4f32-952a-8be8b96d85be>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00181.warc.gz"} |
Mechanical Characteristics and Acoustic Emission Characteristics of Mortar-Rock Binary Medium
School of Resources and Safety Engineering, Central South University, Changsha 410083, China
Hunan Provincial Communications Planning, Survey and Design Institute, Changsha 410200, China
Authors to whom correspondence should be addressed.
Submission received: 19 April 2022 / Revised: 10 May 2022 / Accepted: 13 May 2022 / Published: 17 May 2022
The stability of the interface between mortar and rock is very important in engineering construction. In this paper, the all-digital acoustic emission (AE) system is used to detect the direct shear
test of the mortar-rock binary medium interface with different sawtooth angles under different normal stress states. The stress-displacement information and AE signal during the whole shearing
process are extracted. The coupling relationship between stress and AE characteristic parameters is discussed. The quantitative relationship between sawtooth angle and shear strength of binary medium
is established, and three AE characteristic parameters that can be used to predict structural instability are proposed. The research shows that: With the increase of the normal stress and the
sawtooth angle, the shear strength of the mortar-rock binary medium increases. The relationship of that is obtained by least squares fitting. The shear stress-displacement curve is divided into five
stages according to the change of deformation law. Through the analysis of AE characteristic parameters, it is found that increasing the sawtooth angle makes the AE count and AE cumulative count
increase. Based on the analysis of the characteristic parameters of RA-AF, the changes of shear cracks and tensile cracks within the whole shearing process were obtained, respectively. In the process
of binary medium shearing, the AE peak frequency is in the range of 120–340 kHz. Three acoustic emission parameters that can predict the macroscopic damage of binary media are obtained: the AE b
value, the ratio of shear crack signals, and the number of signals with a peak frequency of 220 Hz to 320 Hz.
1. Introduction
The bond interface between mortar and rock is widely distributed in geotechnical engineering and plays a crucial role in the structure, such as the bond between aggregate and mortar in concrete and
the bond between mortar and surrounding rock in the an-choring system are the key factors to system carrying capacity [
]. Once the bonding of the interface is destroyed, it will lead to concrete cracking, instability, and failure of the bolt support. Therefore, it is of great importance to study the action mechanism
of the mor-tar-rock interface. In recent years, the research on the rock-mortar interface has mainly focused on the crack propagation and mechanical properties of the interface between concrete
aggregate and mortar [
]. LYBIMOVE and Pinns [
] proposed the concept of interfacial transition zone (ITZ), and the structural properties of ITZ are the main factors determining the mechanical properties of concrete. Qiu [
] proposed a rock-mortar dual-material Brazilian disk configuration and simulated the ITZ failure process of the dual-material specimen. Dynamic direct tensile properties of concrete are related to
mortar strength and ITZ strength, and an empirical formula denotes the dynamic increase factor of tensile fracture energy of concrete [
]. The effect of aggregates on the fracture parameters of high-performance concrete was studied through experiments and acoustic emission techniques by Chen and Liu [
]. The dynamic fracture characteristics of the mortar-rock specimens were studied, and the crack propagation time and velocity were measured by Qiu [
]. A crack propagation criterion for layered mortar-rock beams is presented, and a numerical method for predicting the crack propagation process is presented by Wang [
]. Based on the bending strength and fracture energy, the relationship between the fracture parameters and the mechanical properties of the mortar-rock interface is revealed by Satoh [
]. In addition, most of the studies on the bonding effect of the anchoring interface are based on static or dynamic mortar push-out tests [
]. The operation of the mortar push-out test is simple and convenient, but sometimes the matrix cracks, which affects the test results. In contrast, the direct shear test under constant normal stress
can effectively avoid the occurrence of matrix cracking [
]. Although the above study deepens the understanding of interface fracture and failure of binary media to a certain extent, they have not paid much attention to the roughness of the mortar-rock
interface. In fact, the roughness of the interface also has a great influence on the bonding effect of the two-material interface and the shear characteristics of the structure [
]. Therefore, it is necessary to explore the effect of the roughness of the binary medium interface. In addition, the acoustic emission monitoring technology can analyze the changes of the acoustic
emission characteristic parameters when the internal cracks develop in the specimen, judge the nature of the cracks, and give an early warning of the macroscopic damage which is a powerful tool to
conduct failure characteristic [
]. A more comprehensive analysis of the shear behavior of rock-mortar binary medium using the acoustic emission monitoring technology and considering the roughness of the interface will be carried
out in this paper.
In this paper, the direct shear test of the mortar-rock binary medium is carried out. The interface of specimens is serrated with four different sawtooth angles, and the tests are carried out under
the constant normal stress of five levels. The shear process is monitored by a fully digital acoustic emission system, and the characteristic parameters and shear stress of shear acoustic emission
are coupling analyzed. The influence of normal stress and interface roughness on the interface shear characteristics is revealed, and the interface instability failure is reasonably predicted, which
provides a basis for various geotechnical engineering construction and design.
2. Materials and Methods
In order to study the effect of roughness on the shear characteristics of the binary medium of mortar and rock, the engraving machine was used to engrave 10 sawtooths on the rock surface. As shown in
Figure 1
, the sawtooth angle is set to 8°, 30°, 45°, and 55°, and the size of the rock is 70 mm, 70 mm, and 35 mm in length, width, and height, respectively. After the rock is processed, it is poured on the
serrated surface of the rock with the prepared cement mortar, the cement (Hunan Ningxiang Nanfang Cement Co., LTD, Hunan Ningxiang, China) grade is 42.5, and the particle size of the river sand is
less than 0.5 mm. The ratio of cement, sand, and water by weight are 2:1:0.6, and the size of the specimen after pouring is 70 mm × 70 mm × 70 mm (as shown in
Figure 2
). The rock selected for this test is greywacke from Hunan, China, which is a relatively uniform sedimentary rock with relatively stable mechanical properties. The uniaxial compressive strength of
the rock is 72 MPa, the bond force is 9 MPa, and the internal friction Angle is 61.8°. The uniaxial compressive strength of mortar is 15.6 MPa, the bond force is 3.12 MPa, and the internal friction
Angle is 60.4°.
As shown in
Figure 3
, the direct shear test under constant normal stress was carried out by the YZW100 multifunctional rock direct shear instrument developed by Jinan Mining and Rock (Jinan, Shandong, China). According
to the rock physical property test procedures [
], the normal stress should not be lower than the design stress, the normal stress in this test is set to 1 MPa, 2 MPa, 3 MPa, 4 MPa, and 5 MPa respectively. In order to make the shear loading
element contact the specimen quickly, the shear loading adopts the combined loading mode of pre-control and displacement control, that is, when the pre-control shear force reaches 1 KN, it is
switched to the displacement control loading mode, and the loading speed is 0.01 mm/s.
AE monitoring is carried out during the shearing process. The position of the AE sensors and the signal acquisition process is shown in
Figure 4
. The AE monitoring system used in this paper is a 16-channel fully digital acoustic emission system. AE signals were detected by two AE sensors and the sampling frequency for recording wave-forms
was 10 MHz. AE waves were amplified with 40 dB gain by a pre-amplifier. AE sensors were attached to the surface of the specimen with vaseline, so that the sensor face and the specimen surface have a
good contact for the signal detection. The thresh-old level is 45 dB.
3. Test Results and Analysis
3.1. Shear Mechanics Characteristics of Mortar-Rock Binary Medium
In order to compare the difference in shear mechanical properties of rock and mortar rock binary media, direct shear tests were carried out on intact rock and binary media samples, respectively. The
intact rock sample size, normal stress, and shear rate are all the same as those of the binary medium sample. The shear stress-displacement relationship of rock and binary medium is shown in
Figure 5
Figure 6
Figure 5
shows that the intact rock exhibits elastic-brittle failure in shear, and the shear stress before failure is linearly related to the shear displacement. In order to easily reflect the shear stiffness
of rocks and binary media, the value of k is defined as the ratio of the change of shear stress to the change of shear displacement in the elastic section, that is, k = Δτ/Δδ, in MPa/mm, and k’ is
the ratio of shear stress change to shear displacement change before the elastic stage. According to the change in shear stiffness, the shear curve can be divided into five stages: the shear
compression stage, the elastic stage, the plastic failure stage, the strain softening stage, and the residual stress stage, as shown in
Figure 7
In order to study the effect of roughness on the shear characteristics of the binary medium under different normal stress, direct shear tests were carried out on the mortar-rock binary medium with
sawtooth angles of 8°, 30°, 45°, and 55°, the relationship between shear displacement and shear stress is obtained, as shown in
Figure 6
. It can be seen from
Figure 6
that with the increase of shear displacement, the shear stress first increases, then decreases, and finally tends to remain unchanged.
During the shear compression stage, the slope of the curve increases gradually and does not change after entering the elastic phase. Due to the different shear stiffness of mortar and rock, the
deformation of the two materials is inconsistent when the binary medium is under the same external force. As the shear displacement increases, the deformation of the two materials is coordinated, and
then the specimen enters the elastic stage, and the two materials jointly provide shear resistance, so the slope of the curve does not change.
In the plastic failure stage, both the sawtooth angle and the normal stress affect the failure mode of the binary medium. In the case of a small sawtooth angle, as shown in
Figure 6
a, two stress peaks appear in the curve, the stress drops rapidly after the first peak and decreases gently at the second peak. The failure mode of the specimen is brittle failure followed by plastic
failure. Brittle failure occurs when the mortar and rock cementation surface are separated, and the subsequent ductility fluctuation is the plastic failure caused by slip friction. In the case of a
large sawtooth angle, as shown in
Figure 6
c,d, the curve quickly drops after reaching the peak value and enters the residual stress stage. The specimen showed brittle failure, and the greater the normal stress, the more obvious the brittle
failure phenomenon. When the sawtooth angle and normal stress are large, the sawtooth on the mortar side is prone to shearing damage.
Table 1
summarizes the k values of the binary medium samples and intact rocks. The relationship between the k value and normal stress is plotted in
Figure 8
. It can be seen from
Figure 8
. that with the increase of normal stress, the k values of rock and binary medium show an upward trend, and the k value of rock is greater than that of the binary medium. The sawtooth angle has
little effect on the k value of the binary medium, but the increase of the normal stress has a stronger compacting effect on the micro-cracks in the specimen, so it has an increasing effect on the
shear stiffness of the specimen. The mortar-rock binary medium is worse than the intact rock in terms of density and integrity, so its shear stiffness is much smaller than that of the intact rock.
In order to analyze the influence of the serration angle before the elastic stage on the shear stiffness, the k’ value was calculated for the shear results of the binary medium with different
serration angles under the normal stress of 2 MPa. The ratio of k’ to k reflects the shear stiffness before the elastic stage.
Figure 9
shows the relationship between shear displacement and the ratio. It can be seen from
Figure 9
. that with the increase of shear displacement, the ratio first increases slowly, then suddenly rise, and lastly remains the same. The change in the ratio indicates that the shear stiffness has
changed, and the shear transitions from the shear compression stage to the elastic stage. We use this as the basis for dividing the shear compression stage and the elastic stage in the shear process.
With the increase of the sawtooth angle of the binary medium, the displacement corresponding to the sudden change of the ratio is larger, which indicates that the greater the roughness of the
interface of the binary medium, the later the specimen transitions to the elastic stage.
In order to study the effect of sawtooth angle and normal stress on the shear strength of binary media, the shear strength of specimens with different sawtooth angles under different normal stresses
is shown in
Figure 10
. It can be seen from the Figure that with the increase of normal stress and sawtooth angle, the shear strength increases. According to the Mohr-Coulomb criterion, the relationship between the normal
stress and the shear stress at each sawtooth angle in
Figure 10
can be linearly fitted to obtain the cohesive force and internal friction angle of the rock and mortar-rock binary medium. The cohesive force of the rock is 9 MPa, and the internal friction angle is
61.8°, as shown in
Figure 11
With the increase of the sawtooth angle, the cohesive force, and the internal friction angle of the binary medium both increase. The cohesive force of the binary medium is much lower than that of the
complete rock, which is caused by the weakness of the cemented surface of the binary medium, and the cemented interface is easily damaged such as dislocation and separation. This is also the reason
why the binary medium is very important for structural stability in rock mass engineering. It can be seen from Figure that the internal friction angle of the sample with the sawtooth angle of 30–55°
is close to that of the rock, but the internal friction angle is very small at 8°. This is because the sawtooth angle is very small, the binary medium is mainly in the interface separation and slip
failure mode, and the friction between the mortar and the rock provides the main shearing effect. With the increase of the sawtooth angle, the structural surface is mostly damaged by partial or
complete shearing of the sawtooth, and the mortar and rock bear more shear resistance. Therefore, a piecewise function is used to fit the relationship between the sawtooth angle and the internal
friction angle of the binary medium, and the relationship between the internal friction angle and the sawtooth angle is obtained, as shown in formula (1). Equation (2) is obtained by linear fitting
between the cohesive force of the binary medium and the tangent of the sawtooth angle. According to the Mohr-Coulomb criterion, the relationship between the shear strength of the mortar-rock binary
medium, the normal stress, and the sawtooth angle of the structural plane can be obtained, as shown in Equation (3).
$θ ( α ) = { a 1 + k 1 α , α < α i θ i + k 2 ( α − α i ) , α ≥ α i$
is the internal friction angle of the binary medium,
$θ i = a 1 + k 1 α i$
$α i = 31.5 , a 1 = 39.1 , k 1 = 0.609 , k 2 = 0.2234$
$C ( α ) = a + b tan ( α )$
is the cohesive force of the binary medium.
= 0.78,
= 0.4.
$τ = C ( α ) + σ n tan ( θ ( α ) )$
In the residual stage, the shear stress-displacement curve of the binary medium basically remains unchanged at the residual stress.
Figure 12
shows the relationship between the residual strength of the specimens and the normal stress. It can be seen from the Figure that with the increase of the normal stress, the residual strength of the
binary medium tends to increase. According to the Mohr-Coulomb criterion, the residual internal friction angle of binary media with different roughness is obtained.
Figure 13
shows the relationship between the sawtooth angle and the residual internal friction angle. It can be seen from the Figure that the residual internal friction angle of the binary medium is in the
range of 30–48°, and increases with the increase of the sawtooth angle. The larger the sawtooth angle, the greater the roughness of the contact interface after shearing is, which provides a greater
bearing capacity in the residual stage. Interestingly, the residual internal friction angle of the specimen with the sawtooth angle of 55° is larger than the internal friction angle of the specimen
with the sawtooth angle of 8°. It shows that when the sawtooth angle is large, even if the structure is damaged, the residual bearing capacity of the binary medium is larger than that of the small
sawtooth angle binary medium without damage.
3.2. Analysis of Acoustic Emission Characteristic Parameters
3.2.1. AE Counts
AE counts is the number of times the AE signal excursions over the AE threshold.
Figure 14
shows the relationship between AE counts, AE cumulative counts, and time during the shearing process of the binary medium with different sawtooth angles under the normal stress of 3 MPa. It can be
seen from the Figure that in the shear compaction stage and the elastic stage, the AE count is low, and the AE cumulative counts increase slowly, which indicates that the internal cracks of the
specimen develop slowly before failure. In the failure stage, super large AE counts appeared, and the AE cumulative counts increased suddenly, which indicated that the micro-cracks in the specimen
rapidly expanded and penetrated each other during the failure stage, forming macroscopic failure. In the residual stage, the AE counts dropped, but were generally higher than those before shear
failure, and the cumulative AE counts showed a rapid growth pattern. The main failure mode of the specimen in the residual stage is sliding friction failure, which is more severe than the acoustic
emission in the elastic stage and the shear compaction stage, but there is almost no development and penetration of micro-cracks. Comparing
Figure 14
a–d, it can be found that the greater the interface roughness of the binary medium, the greater value of the AE counts and the AE cumulative counts. The increase of the sawtooth angle makes the shear
damage more severe, and the AE signal is easier to monitor.
3.2.2. b Value
The b value is an important parameter derived from seismology, and its application in the AE technology of geotechnical mechanics can reflect the degree of crack propagation inside the material. When
the crack propagation scale is large, the acoustic emission b value is small, and when the crack propagation scale is small, the acoustic emission b value is large. The b value is calculated by
Equation (4) [
In which, M is the AE amplitude, in dB; N is the number of amplitudes within the range of the amplitude M and M + dM; a, b are constants.
In this paper, the self-compiled MATLAB program is used to calculate the b value in the shearing process of binary media with different sawtooth angles, and the relationship between time and b value
is plotted in
Figure 15
. It can be seen from the Figure that in the shear compaction stage and the elastic stage, the b value increases slowly or does not change, indicating that in these two stages, the formation and
expansion of tiny cracks mainly occur inside the specimen but have not yet penetrated. In the failure stage, the b value dropped suddenly, indicating that a large number of micro-cracks penetrated
through the specimen to form macro-cracks, and caused the instability of the structure. In the residual stage, the b value of acoustic emission varies greatly, but the overall trend is gradually
increasing, indicating that the proportion of large-scale cracks in the specimen decreases. Observing the relationship between the change of the b value and the shear stress in the figure, before the
shear stress reaches the peak value, the AE b value will drop suddenly, indicating that the sudden drop of the b value can provide an early warning for the macroscopic damage of the structure.
3.2.3. Crack Identification Analysis
According to the Japanese standard JCMS-III B5706 [
], as shown in
Figure 16
, tensile cracks and shear cracks can be distinguished by two parameters, AF and RA. The slope of the dividing line is defined as r, and the signal is defined as shear failure signal, the signal with
AF/RA > r is defined as tension failure signal, and the r value is between 0–200. AF is the average frequency, which is the ratio of ring count to duration in kHz. RA is the ratio of AE rise time to
AE amplitude, Equation (5) [
], in ms/V. The maximum voltage value of the AE signal can be converted from the AE amplitude, as shown in Equation (5).
$RA = T V m a x = T 10 A / 20$
The T is the rise time, V[max] is the maximum voltage value in μV, and A is the amplitude in dB.
In order to analyze the development of tensile and shear cracks in the shearing process of the mortar-rock binary medium with different sawtooth angles, the relationship between RA and AF in the
shearing process of four kinds of roughness binary mediums is shown in
Figure 17
, respectively. In this paper, based on previous experience [
], the value of r is set to 50, it can be seen from the Figure that most of the signals are distributed below the critical line, that is, the shear crack acoustic emission signal is dominant in the
entire shearing process. As the sawtooth angle increases, more signal points of obvious shear cracks are shown in the figure. (High RA value, low AF value).
Figure 18
shows the change in the RA value during the shearing process. It can be seen from the Figure that the RA value is evenly distributed in the low area during the shear compaction and elastic stages. In
the plastic failure stage, the RA value suddenly increased to a peak value, and the signal was denser, indicating that the shear cracks in the specimen developed and penetrated rapidly during this
stage. In the residual stage, the RA value of acoustic emission is mostly reduced, and the distribution range is larger than the RA value of the shear stage and the elastic stage. This indicates that
in the residual stage, frictional slip between the two mediums occupies the dominant failure mode.
In order to analyze the proportion of shear cracks and tensile cracks in the whole shearing process,
Figure 19
shows the proportion of shearing and tensile cracks in all cracks during the shearing process. The red curve in the Figure is the proportion of shear cracks, and the blue curve is the proportion of
tensile cracks. It can be seen from the Figure that in most cases, shear cracks are more than tensile cracks, indicating that the tangential stress is the dominant factor leading to the failure of
the binary medium in the test, especially in the failure stage and the residual stage. In the shear compaction stage and the elastic stage, the ratio of shear cracks and tensile cracks basically
remained stable, but after entering the plastic failure stage, the shear cracks increased suddenly and fell back in the residual stage, but still maintained a high ratio, while the changes in these
stages of tensile crack are not obvious. This shows that shear cracks develop violently in the plastic failure stage and residual stage from the aspect of crack ratio, and the main factor leading to
structural instability is the development of shear cracks. Corresponding the change of shear crack ratio with the change of shear stress, the sudden increase of shear crack ratio is earlier than the
peak of stress, so the sudden increase of shear crack ratio can play a good role in predicting the failure of specimen or structure.
3.2.4. Peak Frequency Analysis
An acoustic emission signal is a kind of non-stationary signal, and fast Fourier transform (FFT) is a classical spectrum analysis method to analyze a non-stationary signal. The spectral
characteristics of the AE signal generated by the rock can characterize the stress state, structure, and mechanical properties of the rock. Many problems that are difficult to visualize in the time
domain can be easily identified through acoustic emission spectrum analysis. The peak frequency [
] (in kHz) is defined as the point in the Power Spectrum at which the peak magnitude occurs. A real time FFT is performed on the waveform associated with the AE hit. The peak frequency is the
frequency of maximum amplitude [
]. In this paper, the peak frequency information of each sample at different times during the whole loading process is collected, and the variation and distribution of the peak frequency during the
shearing process are analyzed. In this paper, the experimental results of the binary medium with the sawtooth angle of 30° and 45° under the normal stress of 3 MPa are selected as the representative
for specific analysis.
Figure 20
shows the peak frequency changes of the sheared acoustic emission signals of the two samples during the direct shearing process. According to the distribution range and concentrated area of the peak
frequency value, the peak frequency distribution map of the whole shearing process is obtained, as shown in
Figure 21
Figure 20
Figure 21
, it can be seen that the peak frequency of the AE signal of the binary medium in the direct shear test is related to the shear stress level. In the shear compaction stage and the elastic stage, the
peak frequencies of the AE signal are mainly concentrated in the range of 20–40 kHz and 200–320 kHz. During the plastic failure stage and the residual stage, the peak frequency of the AE signal
changed abruptly, with a very large peak frequency, accounting for about 0.01%, and a large number of mid-peak frequencies in the 120–200 kHz frequency band. This indicates that the peak frequency
band broadens with the increase of shear stress during the shearing process. M. Cai et al. [
] pointed out that the peak frequency of the AE signal corresponds to small-scale cracks, and the low frequency corresponds to large-scale cracks. Combined with
Figure 21
, the peak frequency range of the two specimens in the direct shear test is 0–600 kHz, but the peak frequency of 0–340 kHz accounts for more than 99.99%. Therefore, the signal with a peak frequency
of around 300 kHz is a high-frequency signal, and the signal with a peak frequency of about 100 kHz is a low-frequency signal. The high-frequency signal and the low-frequency signal correspond to the
micro-crack and through-crack generated in the specimen, respectively.
In order to further analyze the evolution law of the internal fracture of the structural plane at each stage of shearing, the percentage of each frequency interval in each time period was calculated.
Figure 22
shows the relation between the percentage of each frequency interval and time. It can be seen from the Figure that the percentage of the signal with a peak frequency of 20–40 kHz is almost unchanged
as time increases, indicating that the peak frequency signal of 20–40 kHz is a noise signal due to the equipment or experimental conditions. The peak frequency of the AE signal in the binary medium
direct shear test is the high-frequency signal at 220–340 kHz, and the low-frequency signal at 120–220 kHz. In the shear compaction stage, the high-frequency signals tend to increase, and the number
of low-frequency signals is much lower than that of the high-frequency signals, and there is almost no change, indicating that a large number of micro-cracks represented by the high-frequency signals
are formed. In the elastic stage, the percentage of high-frequency signals decreased, and the number of low-frequency signals changed little, indicating that under the combined action of normal
stress and shear stress, the micro-cracks temporarily closed and suspended development, and no large through cracks were formed. In the plastic failure stage, both high-frequency and low-frequency
signals showed a sudden increase, indicating that the micro-cracks rapidly developed and penetrated to form large-scale cracks. And the peak value of the low-frequency signal and the peak value of
the shear stress of the specimen occurred synchronously, but the peak value of the high-frequency signal occurred earlier than the two, indicating that the sudden increase of the high-frequency
signal is helpful in the prediction of structural instability. In the residual stage, the number of high-frequency and low-frequency signals decreased, but was still much larger than that in the
shear compaction stage and the elastic stage, indicating that the slip friction failure caused a large number of micro-cracks and large cracks in the specimen.
4. Conclusions
The direct shear test and acoustic emission monitoring of the mortar-rock binary medium with different sawtooth angles were carried out, and the following conclusions were obtained:
• The direct shear process of the mortar-rock binary medium is divided into five stages. There is a linear relationship between the shear strength of the binary medium and the normal stress, and
the relationship between the sawtooth angle and the shear strength of the binary medium, cohesive force, internal friction angle, and residual in-ternal friction angle is established.
• The AE count and cumulative count of the binary medium direct shear test are both affected by the interface roughness. The greater the sawtooth angle, the greater the AE count and cumulative
• In the plastic failure stage, the AE b value will decrease suddenly, and the proportion of shear cracks in the specimen will increase suddenly. The sudden drop of the AE b value and the sudden
increase of the shear crack signal ratio can be used as reference indicators for predicting the macroscopic damage of mor-tar-rock binary medium.
• The shear and tensile cracks are distinguished and their proportion is statistically analyzed. With the increase of shear stress, the number of shear cracks increases suddenly, but the change of
tensile cracks is little. The development of shear cracks plays a decisive role in the failure of specimens in the shear failure stage. The variation of AE shear crack number can play a role in
the early warning and prediction of structural failure. In the practice of monitoring practical engineering, this parameter can be used as one of the parameters to predict structural failure.
• The peak frequency of shear acoustic emission in the mortar-rock binary medium is distributed in high and low-frequency bands. The effective peak frequency of acoustic emission signal in the
direct shear process of binary medium structure is mainly concentrated in 120–340 kHz, and the sudden increase of high-frequency signal (220–320 kHz) has an efficient prediction effect on
structural surface damage.
Author Contributions
Conceptualization, H.L., W.T.; methodology, W.T.; software, W.T.; validation, Y.C.; formal analysis, W.T; investigation, J.F; resources, H.L..; data curation, H.L.; writing—original draft
preparation, W.T.; writing—review and editing, Y.C.; visualization, J.F.; supervision, H.H.; project administration, H.H.; funding acquisition, H.H. All authors have read and agreed to the published
version of the manuscript.
This paper gets its funding from Hunan provincial key research and development Program (2022SK2082); Science and Technology Project of Hunan Natural Resources Department (2021-52); Science and
Technology Progress and Innovation Plan of Hunan Provincial Department of Transportation (201003); Science and Technology Progress and Innovation Plan of Hunan Pro-vincial Department of
Transportation (202120); Hunan Civil Air Defense Research Project (HNRFKJ-2021-07). The authors wish to acknowledge these supports.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
The data used to support the findings of this study are available from the corresponding author upon request.
This paper gets its funding from Hunan provincial key research and development Program (2022SK2082); Science and Technology Project of Hunan Natural Resources Department (2021-52); Science and
Technology Progress and Innovation Plan of Hunan Provincial Department of Transportation (201003); Science and Technology Progress and Innovation Plan of Hunan Provincial Department of Transportation
(202120); Hunan Civil Air Defense Research Project (HNRFKJ-2021-07). The authors wish to acknowledge these supports.
Conflicts of Interest
The authors declare no conflict of interest.
1. Han, L.; Lin, H.; Chen, Y.; Lei, D. Effects of strength property difference on shear strength of joint of binary media. Environ. Earth Sci. 2021, 80, 712. [Google Scholar] [CrossRef]
2. Shen, Y.; Wang, Y.; Yang, Y.; Sun, Q.; Luo, T.; Zhang, H. Influence of surface roughness and hydrophilicity on bonding strength of concrete-rock interface. Constr. Build. Materials. 2019, 213,
156–166. [Google Scholar] [CrossRef]
3. Suits, L.D.; Sheahan, T.C.; Seidel, J.P.; Haberfield, C.M. Laboratory Testing of Concrete-rock Joints in Constant Normal Stiffness Direct Shear. Geotech. Test. J. 2002, 25, 391–404. [Google
4. Zhao, Y.; Chang, L.; Wang, Y.; Lin, H.; Liao, J.; Liu, Q. Dynamic response of cylindrical thick-walled granite specimen with clay infilling subjected to dynamic loading. Arch. Appl. Mech. 2022,
92, 643–648. [Google Scholar] [CrossRef]
5. Fan, X.; Yu, H.; Deng, Z.; He, Z.; Zhao, Y. Cracking and deformation of cuboidal sandstone with a single nonpenetrating flaw under uniaxial compression. Theor. Appl. Fract. Mech. 2022, 119,
103284. [Google Scholar] [CrossRef]
6. Fan, X.; Yang, Z.; Li, K. Effects of the lining structure on mechanical and fracturing behaviors of four-arc shaped tunnels in a jointed rock mass under uniaxial compression. Theor. Appl. Fract.
Mech. 2021, 112, 102887. [Google Scholar] [CrossRef]
7. LYBIMOVE, C.; Pinns, E. Crystallization Structure in Concrete Contact Zone between aggregate and cement in concrete. Colloid J. 1962, 24, 491–498. [Google Scholar]
8. Qiu, H.; Wang, F.; Zhu, Z.; Wang, M.; Yu, D.; Luo, C.; Wan, D. Study on dynamic fracture behaviour and fracture toughness in rock-mortar interface under impact load. Compos. Struct. 2021, 271,
114174. [Google Scholar] [CrossRef]
9. Chen, L.; Yue, C.; Zhou, Y.; Zhang, J.; Jiang, X.; Fang, Q. Experimental and mesoscopic study of dynamic tensile properties of concrete using direct-tension technique. Int. J. Impact Eng. 2021,
155, 103895. [Google Scholar] [CrossRef]
10. Feng, X.T.; Li, S.J.; Chen, S.L. Effect of water chemical corrosion on strength and cracking characteristics of rocks—A review. In Advances in Fracture and Failure Prevention, Pts 1 and 2;
Kishimoto, K., Kikuchi, M., Shoji, T., Saka, M., Eds.; Trans Tech Publications Ltd.: Bäch SZ, Switzerland, 2004; pp. 1355–1360. [Google Scholar]
11. Qiu, H.; Zhu, Z.; Wang, F.; Wang, M.; Mao, H. Dynamic behavior of a running crack crossing mortar-rock interface under impacting load. Eng. Fract. Mech. 2020, 240, 107202. [Google Scholar] [
12. Wang, H.W.; Wu, Z.M.; Wang, Y.J.; Yu, R.C. Investigation on crack propagation perpendicular to mortar–rock interface: Experimental and numerical. Int. J. Fract. 2020, 226, 45–69. [Google Scholar]
13. Satoh, A.; Yamada, K.; Shinohara, Y. Simulation of Adhesion Performance of Mortar—Mortar Interface with Varied Fractographic Features. Key Eng. Mater. 2013, 577, 357–360. [Google Scholar] [
14. Buzzi, O.; Hans, J.; Boulon, M.; Deleruyelle, F.; Besnus, F. Hydromechanical study of rock-mortar interfaces. Phys. Chem. Earth. 2007, 32, 820–831. [Google Scholar] [CrossRef]
15. Wang, Y.; Chen, S.J.; Zhao, H.T.; Chen, Y.Z. Acoustic emission characteristics of interface between aggregate and mortar under shear loading. Russ. J. Nondestruct. Test. 2015, 51, 497–508. [
Google Scholar] [CrossRef]
16. Lin, H.; Zhang, X.; Cao, R.; Wen, Z. Improved nonlinear Burgers shear creep model based on the time-dependent shear strength for rock. Environ. Earth Sci. 2020, 79, 149. [Google Scholar] [
17. Tang, Z.C.; Zhang, Q.Z.; Peng, J.; Jiao, Y.Y. Experimental study on the water-weakening shear behaviors of sandstone joints collected from the middle region of Yunnan province, P.R. China. Eng.
Geol. 2019, 258, 105161. [Google Scholar] [CrossRef]
18. Fan, X.; Li, K.; Lai, H.; Zhao, Q.; Sun, Z. Experimental and numerical study of the failure behavior of intermittent rock joints subjected to direct shear load. Adv. Civ. Eng. 2018, 2018, 19. [
Google Scholar] [CrossRef] [Green Version]
19. Cheng, Y.; Yang, W.; He, D. Influence of structural plane microscopic parameters on direct shear strength. Adv. Civ. Eng. 2018, 2018, 7. [Google Scholar] [CrossRef] [Green Version]
20. Xie, S.; Lin, H.; Cheng, C.; Chen, Y.; Wang, Y.; Zhao, Y.; Yong, W. Shear strength model of joints based on Gaussian smoothing method and macro-micro roughness. Comput. Geotech. 2022, 143,
104605. [Google Scholar] [CrossRef]
21. Du, S.-G.; Lin, H.; Yong, R.; Liu, G.-J. Characterization of Joint Roughness Heterogeneity and Its Application in Representative Sample Investigations. Rock Mech. Rock Eng. 2022, 1–25. [Google
Scholar] [CrossRef]
22. Zhao, Y.; Zhang, C.; Wang, Y.; Lin, H. Shear-related roughness classification and strength model of natural rock joint based on fuzzy comprehensive evaluation. Int. J. Rock Mech. Min. Sci. 2021,
137, 104550. [Google Scholar] [CrossRef]
23. Wang, C.; Wang, L.; Karakus, M. A new spectral analysis method for determining the joint roughness coefficient of rock joints. Int. J. Rock Mech. Min. Sci. 2019, 113, 72–82. [Google Scholar] [
24. Lin, Q.; Cao, P.; Cao, R.; Lin, H.; Meng, J. Mechanical behavior around double circular openings in a jointed rock mass under uniaxial compression. Arch. Civ. Mech. Eng. 2020, 20, 19. [Google
Scholar] [CrossRef] [Green Version]
25. Naderloo, M.; Moosavi, M.; Ahmadi, M. Using acoustic emission technique to monitor damage progress around joints in brittle materials. Theor. Appl. Fract. Mech. 2019, 104, 102368. [Google Scholar
] [CrossRef]
26. 0276.25-2015 DT. Part 25: Test for determining the shear strength of rock. In Regulation for Testing the Physical and Mechanical Properties of Rock; Ministry of Land and Resources: Beijing,
China, 2015.
27. Ge, Z.; Sun, Q. Acoustic emission characteristics of gabbro after microwave heating. Int. J. Rock Mech. Min. Sci. 2021, 138, 104616. [Google Scholar] [CrossRef]
28. JC MS-III B5706. Monitoring Method for Active Cracks in Concrete by Acoustic Emission. Federation of Construction Materials Industries Japan: Tokyo, Japan, 2003.
29. Physical Acoustics Corporation. SAMOS AE System User’s Manual; Physical Acoustics Corporation: Princeton Junction, NJ, USA, 2005. [Google Scholar]
30. Ohno, K.; Ohtsu, M. Crack classification in concrete based on acoustic emission. Constr. Build. Mater. 2010, 24, 2339–2346. [Google Scholar] [CrossRef]
31. Aggelis, D.G. Classification of cracking mode in concrete by acoustic emission parameters. Mech. Res. Commun. 2011, 38, 3–157. [Google Scholar] [CrossRef]
32. Shahidan, S.; Pulin, R.; Bunnori, N.M.; Holford, K.M. Damage classification in reinforced concrete beam by acoustic emission signal analysis. Constr. Build. Mater. 2013, 45, 78–86. [Google
Scholar] [CrossRef] [Green Version]
33. Degala, S.; Rizzo, P.; Ramanathan, K.; Harries, K.A. Acoustic emission monitoring of CFRP reinforced concrete slabs. Constr. Build. Mater. 2009, 23, 2016–2026. [Google Scholar] [CrossRef]
Figure 2. Specimen preparation before direct shear test. (a) Schematic diagram of specimens with different sawtooth angles. (b) Specimens after pouring.
Figure 6. Relationship between shear stress and shear displacement of mortar-rock binary medium. (a) 8° (b) 30° (c) 45° (d) 55°.
Figure 14. Relationship between AE counts, AE cumulative counts and time. (a) 8° (b) 30° (c) 45° (d) 55°.
Figure 15. Relationship between AE b value and time in the whole shearing process. (a) 8° (b) 30° (c) 45° (d) 55°.
Figure 17. Relation between the RA value and the average frequency in direct shear test of binary medium. (a) 8° (b) 30° (c) 45° (d) 55°.
Figure 18. Relation between the RA value and the time in direct shear test of binary medium. (a) 8° (b) 30° (c) 45° (d) 55°.
Figure 19. Relation between the crack ratio and the time in direct shear test of binary medium. (a) 8° (b) 30° (c) 45° (d) 55°.
The Value of k
Normal Stress Sawtooth Angle of 8° Sawtooth Sawtooth Sawtooth Intact Rock
Angle of 30° Angle of 45° Angle of 55°
1 MPa 1.74 3.46 5.72 5.61 12.01
2 MPa 1.75 4.83 6.07 7.21 13.23
3 MPa 5.85 8.06 5.79 8.03 15.58
4 MPa 6.45 5.67 6.26 8.82 16.21
5 MPa 6.92 9.29 8.02 11.59 18.56
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What is Completing the Square ⭐ Definition with Examples
Completing the Square – Definition with Examples
Updated on January 15, 2024
In the colorful and expansive world of mathematics, many concepts are like the pieces of a jigsaw puzzle, fitting together to form a coherent picture. At Brighterly, our mission is to guide young
learners on their journey through this vast mathematical landscape, one concept at a time. Today, we explore the fascinating concept of “Completing the Square”.
Completing the square might seem like a cryptic term at first, but don’t be fooled by its complex appearance. As we unpack its meaning and explore its applications, you’ll discover that it’s a
crucial mathematical tool that will open doors to advanced problem-solving techniques. Whether you’re grappling with quadratic equations or trying to make sense of geometric shapes, understanding the
process of completing the square will give you an edge. So buckle up as we embark on an enlightening mathematical adventure with Brighterly!
What is Completing the Square?
In the realm of mathematics, some terms may sound complicated, but once you explore them, you will realize they are intriguing and enlightening. One such term is “Completing the Square”. Simply put,
completing the square is a method used to solve quadratic equations. The name derives from the visual representation of the algebraic process where you arrange literal and numerical terms to form a
perfect square on one side of the equation.
Completing the square is a fundamental mathematical concept that serves as a stepping stone for more advanced mathematical operations. The technique is immensely beneficial, especially in calculus
and geometric equations.
Definition of Completing the Square
Completing the square refers to a technique employed in algebra to convert a quadratic equation of the form ax^2 + bx + c = 0 into the form (x-h)^2 = k, where x represents a variable, and a, b, c, h,
and k are constants. This transformation simplifies the equation, making it easier to solve.
This method is so named because it involves arranging terms in the equation to resemble a squared binomial.
Explanation of the Completing the Square Process
The process of completing the square involves a series of steps that require rearranging and simplifying the quadratic equation. We begin by ensuring that the coefficient of the x^2 term is 1. If it
isn’t, we divide the entire equation by the coefficient to make it so.
Next, we take the coefficient of the x term, halve it, square the result, and then add and subtract that value in the equation. This creates a perfect square trinomial in the equation, which we can
rewrite as the square of a binomial.
The process of completing the square uncovers the roots of the equation, thus making it an essential algebraic skill.
Purpose of Completing the Square
The primary purpose of completing the square is to simplify quadratic equations, making them easier to solve. This method is fundamental in the development of the quadratic formula, a universally
applicable formula for solving any quadratic equation.
Beyond solving equations, completing the square is also vital for graphing quadratic functions, as it allows us to convert quadratic equations to vertex form. This makes it easier to identify key
features of the graph, such as the vertex and the axis of symmetry.
Uses of Completing the Square in Mathematics
The uses of completing the square in mathematics are vast and varied. It’s a fundamental technique used in algebra, calculus, and geometry.
In algebra, completing the square is used to derive the quadratic formula, which can solve any quadratic equation.
In calculus, the method is used to integrate certain types of functions and also when finding the area enclosed by a curve.
Finally, in geometry, completing the square plays a key role in deriving equations for circles, ellipses, and hyperbolas.
Detailed Steps in Completing the Square
Now, let’s delve deeper into the detailed steps involved in completing the square:
1. Step 1: Start with a quadratic equation in the standard form ax^2 + bx + c = 0.
2. Step 2: If a ≠ 1, divide the entire equation by a to make the coefficient of x^2 equal to 1.
3. Step 3: Rewrite the equation as (x^2 + bx) + c = 0.
4. Step 4: Add and subtract the square of half of the coefficient of x within the parentheses.
5. Step 5: Simplify the equation, which should now be in the form (x-h)^2 = k.
By following these steps, you can complete the square of any quadratic equation.
Different Forms of Quadratic Equations and Their Relation to Completing the Square
Quadratic equations can be written in three forms: standard form, vertex form, and factored form. Completing the square serves as the bridge between these different forms, making it possible to
convert from standard form to vertex form or factored form.
Standard form ax^2 + bx + c = 0 is the common representation of a quadratic equation. However, by completing the square, we can convert this standard form into vertex form (x-h)^2 = k, which provides
a clear picture of the vertex and the axis of symmetry of the function’s graph.
Completing The Square Practice Worksheet PDF
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Completing The Square Practice Worksheet
Completing The Square Worksheet PDF
View pdf
Completing The Square Worksheet
At Brighterly, we believe that practice is the key to mastery. That’s why we invite you to explore our completing the square worksheets, where you can find an array of additional practice questions,
complete with answers.
Difference Between Factoring, Quadratic Formula, and Completing the Square
Factoring, the quadratic formula, and completing the square are all methods of solving quadratic equations. However, they differ in terms of complexity, applicability, and ease of use.
Factoring involves expressing the equation as a product of its factors. It’s the simplest method but only works when the equation can be easily factored.
The quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / 2a is a universally applicable method, derived from the process of completing the square. It can solve any quadratic equation but may be seen as
complex due to the involvement of the square root and fraction.
Completing the square stands as the middle ground between the two. It’s more versatile than factoring and simpler than the quadratic formula, but it requires understanding of how to rearrange and
simplify terms to create a perfect square trinomial.
Equations involving Completing the Square
Let’s look at an example of an equation where completing the square is used: x^2 + 6x - 7 = 0.
We first rewrite the equation as (x^2 + 6x) - 7 = 0. Then, we add and subtract (6/2)^2 = 9 within the parentheses, yielding (x^2 + 6x + 9 - 9) - 7 = 0.
Simplifying further, we get (x + 3)^2 - 16 = 0, which is easier to solve.
Writing Quadratic Equations Using Completing the Square
Writing a quadratic equation using completing the square can provide a clearer understanding of the equation’s properties.
Let’s take x^2 - 4x + 4. First, identify the value of b in the equation, which is -4. Divide this by 2 to get -2, then square it to get 4.
The equation is already in the form (x-2)^2, indicating the vertex of the graph and the axis of symmetry.
Solving Quadratic Equations by Completing the Square
Solving quadratic equations using completing the square involves a few simple steps.
For instance, given the equation (x + 3)^2 - 16 = 0, we start by isolating the squared term by adding 16 to both sides of the equation, yielding (x + 3)^2 = 16.
We then take the square root of both sides, remembering to include both the positive and negative square root, which gives us x + 3 = ±4.
Finally, we subtract 3 from both sides to get x = -3 ± 4, which gives us the solutions x = 1 and x = -7.
Practice Problems on Completing the Square
1. Complete the square for x^2 + 8x - 9 = 0.
2. Solve x^2 - 6x + 1 = 0 by completing the square.
3. Write x^2 - 10x + 25 in vertex form.
These problems will help reinforce the concepts learned and develop proficiency in completing the square.
Frequently Asked Questions on Completing the Square
What is completing the square used for?
Completing the square is a versatile tool in mathematics. It’s primarily used to simplify quadratic equations, transforming them into a format that’s easier to solve. By arranging the terms of the
quadratic equation to create a perfect square trinomial, we can easily find the roots of the equation. Furthermore, this technique plays a critical role in the derivation of the quadratic formula, an
essential tool for solving any quadratic equation. Lastly, completing the square is also employed in the graphing of quadratic functions, as it aids in converting the equation into its vertex form,
allowing us to easily identify key features of the graph, such as the vertex and axis of symmetry.
How does completing the square work?
Completing the square works by systematically manipulating the terms of a quadratic equation to form a perfect square trinomial, which can then be expressed as the square of a binomial. The process
begins by making the coefficient of the x^2 term 1. Next, we look at the coefficient of the x term, divide it by 2, and then square the result. This value is added and subtracted in the equation,
creating a perfect square trinomial that can be written as a squared binomial. The manipulation simplifies the quadratic equation and uncovers its roots, providing the solution.
Is completing the square always possible?
Yes, completing the square is a universally applicable technique and can be performed for any quadratic equation. Regardless of the coefficients of the equation, the process of completing the square
can always be used to transform the equation into a format that reveals the roots of the equation. So, no matter how complex a quadratic equation might seem, rest assured, completing the square has
got you covered!
What is the relationship between completing the square and the quadratic formula?
Completing the square and the quadratic formula are closely linked. In fact, the quadratic formula, which is used to solve any quadratic equation, is derived using the process of completing the
square. During the derivation, the quadratic equation in standard form is manipulated via the process of completing the square, and this manipulation results in the quadratic formula. Hence,
completing the square is not only a method to solve quadratic equations but also a fundamental step in the derivation of one of the most important formulas in algebra.
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double MAfTpQuad (const float R[], const float x[], int N)
Calculate a quadratic form for a symmetric Toeplitz matrix
This routine calculates a quadratic form S = x'R x, where x is a vector and R is a symmetric Toeplitz matrix,
N-1 N-1
S = SUM SUM x(i) R(i,j) x(j) .
i=0 j=0
The elements of the matrix R are constant down diagonals, R(i,j) = R(|i-h|). In this routine R is specified by its first column or row. The result is accumulated as a double value and returned as a
double value.
<- double MAfTpQuad
Resultant value
-> const float R[]
First column (or row) of the symmetric Toeplitz matrix (N values)
-> const float x[]
Vector of N values
-> int N
Number of elements in x and R
Author / revision
P. Kabal / Revision 1.3 2003/05/09
See Also
MAfSyBilin, MAfSyQuad
Main Index libtsp | {"url":"https://mmsp.ece.mcgill.ca/Documents/Software/Packages/libtsp/MA/MAfTpQuad.html","timestamp":"2024-11-05T12:36:57Z","content_type":"text/html","content_length":"1518","record_id":"<urn:uuid:b8b56c60-5d5a-4fb5-842e-b514d90c082a>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00376.warc.gz"} |
cond_false: Number of individuals for which the condition is false. in riskyr: Rendering Risk Literacy more Transparent
cond_false is a frequency that describes the number of individuals in the current population N for which the condition is FALSE (i.e., actually false cases).
to probabilities: The frequency of cond_false individuals depends on the population size N and the complement of the condition's prevalence 1 - prev and is split further into two subsets of fa by the
false alarm rate fart and cr by the specificity spec.
The frequency cond_false is determined by the population size N times the complement of the prevalence (1 - prev):
a. The frequency fa is determined by cond_false times the false alarm rate fart = (1 - spec) (aka. FPR):
fa = cond_false x fart = cond_false x (1 - spec)
b. The frequency cr is determined by cond_false times the specificity spec = (1 - fart):
cr = cond_false x spec = cond_false x (1 - fart)
to other frequencies: In a population of size N the following relationships hold:
N = dec_cor + dec_err (by correspondence of decision to condition)
N = hi + mi + fa + cr (by condition x decision)
Current frequency information is computed by comp_freq and contained in a list freq.
is_freq verifies frequencies; num contains basic numeric parameters; init_num initializes basic numeric parameters; freq contains current frequency information; comp_freq computes current frequency
information; prob contains current probability information; comp_prob computes current probability information.
Other frequencies: N, cond_true, cr, dec_cor, dec_err, dec_neg, dec_pos, fa, hi, mi
cond_false <- 1000 * .90 # => sets cond_false to 90% of 1000 = 900 cases. is_freq(cond_false) # => TRUE is_prob(cond_false) # => FALSE, as cond_false is no probability [but (1 - prev) and spec are]
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/riskyr/man/cond_false.html","timestamp":"2024-11-13T22:10:26Z","content_type":"text/html","content_length":"35607","record_id":"<urn:uuid:b9b7c99d-dc8a-4c83-a0c8-fb2e8087caf9>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00717.warc.gz"} |
LM 16.2 Microscopic description of an ideal gas Collection
16.2 Microscopic description of an ideal gas by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.
16.2 Microscopic description of an ideal gas
Evidence for the kinetic theory
Why does matter have the thermal properties it does? The basic answer must come from the fact that matter is made of atoms. How, then, do the atoms give rise to the bulk properties we observe? Gases,
whose thermal properties are so simple, offer the best chance for us to construct a simple connection between the microscopic and macroscopic worlds.
A crucial observation is that although solids and liquids are nearly incompressible, gases can be compressed, as when we increase the amount of air in a car's tire while hardly increasing its volume
at all. This makes us suspect that the atoms in a solid are packed shoulder to shoulder, while a gas is mostly vacuum, with large spaces between molecules. Most liquids and solids have densities
about 1000 times greater than most gases, so evidently each molecule in a gas is separated from its nearest neighbors by a space something like 10 times the size of the molecules themselves.
If gas molecules have nothing but empty space between them, why don't the molecules in the room around you just fall to the floor? The only possible answer is that they are in rapid motion,
continually rebounding from the walls, floor and ceiling. In chapter 12, we have already seen some of the evidence for the kinetic theory of heat, which states that heat is the kinetic energy of
randomly moving molecules. This theory was proposed by Daniel Bernoulli in 1738, and met with considerable opposition, because there was no precedent for this kind of perpetual motion. No rubber
ball, however elastic, rebounds from a wall with exactly as much energy as it originally had, nor do we ever observe a collision between balls in which none of the kinetic energy at all is converted
to heat and sound. The analogy is a false one, however. A rubber ball consists of atoms, and when it is heated in a collision, the heat is a form of motion of those atoms. An individual molecule,
however, cannot possess heat. Likewise sound is a form of bulk motion of molecules, so colliding molecules in a gas cannot convert their kinetic energy to sound. Molecules can indeed induce
vibrations such as sound waves when they strike the walls of a container, but the vibrations of the walls are just as likely to impart energy to a gas molecule as to take energy from it. Indeed, this
kind of exchange of energy is the mechanism by which the temperatures of the gas and its container become equilibrated.
Pressure, volume, and temperature
A gas exerts pressure on the walls of its container, and in the kinetic theory we interpret this apparently constant pressure as the averaged-out result of vast numbers of collisions occurring every
second between the gas molecules and the walls. The empirical facts about gases can be summarized by the relation
`PVpropnT`, [ideal gas]
which really only holds exactly for an ideal gas. Here `n` is the number of molecules in the sample of gas.
Example 7: Volume related to temperature
The proportionality of volume to temperature at fixed pressure was the basis for our definition of temperature.
Example 8: Pressure related to temperature
Pressure is proportional to temperature when volume is held constant. An example is the increase in pressure in a car's tires when the car has been driven on the freeway for a while and the tires and
air have become hot.
We now connect these empirical facts to the kinetic theory of a classical ideal gas. For simplicity, we assume that the gas is monoatomic (i.e., each molecule has only one atom), and that it is
confined to a cubical box of volume `V`, with `L` being the length of each edge and `A` the area of any wall. An atom whose velocity has an `x` component `v_x` will collide regularly with the
left-hand wall, traveling a distance `2L` parallel to the `x` axis between collisions with that wall. The time between collisions is `Deltat=2L"/"v_x`, and in each collision the `x` component of the
atom's momentum is reversed from `-mv_x` to `mv_x`. The total force on the wall is
`F=(Deltap_(x,1))/(Deltat_1)+(Deltap_(x,2))/(Deltat_2)+...["monoatomic ideal gas"]`,
where the indices 1, 2, ... refer to the individual atoms. Substituting `Deltap_(x,i)=2mv_(x,i)` and `Deltat_i=2L"/"v_(x,i)`, we have
`F=(mv_(x,1)^2)/L+(mv_(x,2)^2)/L+...["monoatomic ideal gas"]`.
The quantity `mv_(x,1)^2` is twice the contribution to the kinetic energy from the part of the atom's center of mass motion that is parallel to the `x` axis. Since we're assuming a monoatomic gas,
center of mass motion is the only type of motion that gives rise to kinetic energy. (A more complex molecule could rotate and vibrate as well.) If the quantity inside the sum included the `y` and `z`
components, it would be twice the total kinetic energy of all the molecules. By symmetry, it must therefore equal 2/3 of the total kinetic energy, so
`F=(2KE_"total")/(3L)["monoatomic ideal gas"]`.
Dividing by `A` and using `AL=V`, we have
`P=(2KE_"total")/(3V)["monoatomic ideal gas"]`..
This can be connected to the empirical relation `PVpropnT` if we multiply by `V` on both sides and rewrite `KE_"total"` as `nKE_av` where `KE_av` is the average kinetic energy per molecule:
`PV=2/3nKE_"av"` [monoatomic ideal gas].
For the first time we have an interpretation for the temperature based on a microscopic description of matter: in a monoatomic ideal gas, the temperature is a measure of the average kinetic energy
per molecule. The proportionality between the two is `KE_(av)=(3"/"2)kT`, where the constant of proportionality `k`, known as Boltzmann's constant, has a numerical value of `1.38×10^(-23)` J/K. In
terms of Boltzmann's constant, the relationship among the bulk quantities for an ideal gas becomes
`PV=nkT`, [ideal gas]
which is known as the ideal gas law. Although I won't prove it here, this equation applies to all ideal gases, even though the derivation assumed a monoatomic ideal gas in a cubical box. (You may
have seen it written elsewhere as `PV=NRT`, where `N=n"/"N_A` is the number of moles of atoms, `R=kN_A` and `N_A=6.0×10^(23)`, called Avogadro's number, is essentially the number of hydrogen atoms in
1 g of hydrogen.)
Example 9: Pressure in a car tire
`=>` After driving on the freeway for a while, the air in your car's tires heats up from `10°C` to `35°C`. How much does the pressure increase?
`=>` The tires may expand a little, but we assume this effect is small, so the volume is nearly constant. From the ideal gas law, the ratio of the pressures is the same as the ratio of the absolute
`=(308 K)"/"(283 K)`
or a 9% increase.
Example 10: Earth's senescence
Microbes were the only life on Earth up until the relatively recent advent of multicellular life, and are arguably still the dominant form of life on our planet. Furthermore, the sun has been
gradually heating up ever since it first formed, and this continuing process will soon (“soon” in the sense of geological time) eliminate multicellular life again. Heat-induced decreases in the
atmosphere's `CO_2` content will kill off all complex plants within about 500 million years, and although some animals may be able to live by eating algae, it will only be another few hundred million
years at most until the planet is completely heat-sterilized.
Why is the sun getting brighter? The only thing that keeps a star like our sun from collapsing due to its own gravity is the pressure of its gases. The sun's energy comes from nuclear reactions at
its core, and the net result of these pressure, which makes the core contract. As the core contracts, collisions between hydrogen atoms become more frequent, and the rate of fusion reactions
Example 11: A piston, a refrigerator, and a space suit
Both sides of the equation `PV=nkT` have units of energy. Suppose the pressure in a cylinder of gas pushes a piston out, as in the power stroke of an automobile engine. Let the cross-sectional area
of the piston and cylinder be `A`, and let the piston travel a small distance `Deltax`. Then the gas's force on the piston [Private Equation] does an amount of mechanical work `W=FDeltax=PADeltax=
PDeltaV`, where `DeltaV` is the change in volume. This energy has to come from somewhere; it comes from cooling the gas. In a car, what this means is that we're harvesting the energy released by
burning the gasoline.
In a refrigerator, we use the same process to cool the gas, which then cools the food.
In a space suit, the quantity `PDeltaV` represents the work the astronaut has to do because bending her limbs changes the volume of the suit. The suit inflates under pressure like a balloon, and
doesn't want to bend. This makes it very tiring to work for any significant period of time.
16.2 Microscopic description of an ideal gas by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license. | {"url":"https://www.vcalc.com/collection/?uuid=1e628f2c-f145-11e9-8682-bc764e2038f2","timestamp":"2024-11-13T08:19:05Z","content_type":"text/html","content_length":"60979","record_id":"<urn:uuid:00026ec8-26a4-49d6-9599-0f9f39252b34>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00435.warc.gz"} |
Math counts problem
I did a hard math counts problem recently, and I don’t know why the answer is 72, can someone please explain it for me? Thanks a lot!
How many perfect squares are divisors of the product 1! ∙ 2! ∙ 3! ∙ 4! ∙ 5! ∙ 6! ∙ 7! ?
Hey @modestwallaby !
Whenever tackling a divisors problem, oftentimes it's a great strategy to first consider the prime factorization.
The prime factorization of the product is \(2^{16}\cdot3^{7}\cdot5^{3}\cdot7\). Since we're dealing with perfect squares, let's think about what the prime factorization of a perfect square looks
like-- in a perfect square, all of the exponents are even! For example, \(2^4\cdot3^2\) is a perfect square, so is \(2^2 \cdot 5^2\), so is \(3^6\), but not \(2^3\cdot3^2\) or \(3^4\cdot 5\).
So, using \(2^{16}\cdot3^{7}\cdot5^{3}\cdot7\), we have to find all of the possibilities to make a perfect square using each of these factors. Let's tackle each of the factors individually!
Every perfect square that is a divisor of that product is either going to have \(2^0,2^2,2^4,2^6,2^8,2^{10},2^{12},2^{14},\) or \(2^{16}\) in its prime factorization. This gives us \(9\)
possibilities for the power of \(2\).
Then, using the same logic, every perfect square that is a divisor of that product has a power of \(3\) that's either \(3^0,3^2,3^4\) or \(3^{6}\) in its prime factorization. This gives us \(4\)
possibilities for the power of \(2\).
Finally, there are only \(2\) possibilities for a perfect square power of \(5\): \(5^0\) and \(5^2\). We can stop here since we have no other options for any factors greater than 5.
So, multiplying our possibilities together, we have \(9\cdot4\cdot2=\boxed{72}\) in total. Let me know if this made sense and if you still have any questions!
@quacker88 Thanks for your help! | {"url":"https://forum.poshenloh.com/topic/947/math-counts-problem","timestamp":"2024-11-12T11:48:06Z","content_type":"text/html","content_length":"58634","record_id":"<urn:uuid:d6214ee5-1fed-4783-8db5-b74303a2dd63>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00063.warc.gz"} |
How to Calculate Inflation with GDP Deflators: Economics 101
In a previous article here, we covered the formulas for nominal and real GDP. In that article, we also touched on GDP deflators. I’ll repeat our prior definition and a short excerpt here:
So, the real GDP in 2026 is $120, while the nominal GDP is $144. This means that roughly half of the “increase in the value” of output came from inflation, while the actual value (think of this
like purchasing power) increase stripped of inflation was only about $20 from 2024 to 2026.
Here’s one more useful formula to know:
GDP Deflator = nominal GDP/real GDP x 100
Like we said, the GDP deflator gives us a metric detailing how much of GDP increases are due to inflation versus output increases. Since we’ve now calculated both real and nominal GDP, we can run
these numbers:
Real GDP in 2026: $120
Nominal GDP in 2026: $144
GDP Deflator = 144/120 x 100 = 1.2 x 100 = 120
A GDP deflator of 120 infers that the current price level is 20% higher than the price level in the base year, or that inflation has been 20% over that three year period. If we look at the
inflation numbers described earlier — 6%, 10%, and 3% — 1.06 × 1.10 × 1.03 happens to equal 1.20098, equivalent to exactly 20%.
Now, how can we calculate inflation with GDP deflators?
Well, here’s the idea: since GDP deflators measure how much of GDP’s growth can be attributed to changes in price levels (e.g., inflation or deflation), they essentially strip GDP of changes in
actual (real) economic output and provide the raw price inflation. Thus, comparing GDP deflators from two different years—say, a base year and an end year—reflects the normalized (both on a scale of
100+) rate of inflation over that period of time.
To make this comparison work, both GDP deflators have to be based on the same base year when calculating nominal and real GDP.
This may sound complex, but it’s really simple subtraction and division to calculate the different, in percent, between two numbers. Here’s the generalized formula to derive inflation from GDP
• GDP Deflator for Year 1 (base year): D1
• GDP Deflator for Year 2 (end year): D2
*let’s say you want to calculate the inflation rate from 2005 to 2010 using GDP deflators: the deflator from 2005 would be D1, and 2010 deflator would be D2.
Inflation Rate = ((D2 —D1)/D1)*100
Let’s use a concrete example. Say we want to find the inflation rate from 2021 to 2024. The following data is presented:
• Nominal GDP in 2021: $1.1 trillion
• Real GDP in 2021 (base year): $1.0 trillion
• Nominal GDP in 2024: $1.4 trillion
• Real GDP in 2024: $1.2 trillion
GDP deflators follow this formula: nominal GDP/real GDP x 100, so the GDP deflators for 2021 and 2024 go as follows:
• 2021 deflator: (1.1t/1t)*100 = 110
• 2024 deflator: (1.4t/1.2t)*100 = 116.6
Returning to our inflation rate formula, we can plug the numbers in:
((116.6 – 110)/110)*100 = (6.6/110)*100 = 0.06*100 = 6
Thus, the price inflation rate from 2021 to 2024 is 6%.
Hopefully that helped you! If it did, let me know what else you’d like to learn about, or else check out some other econ articles of mine below: | {"url":"https://jonwlaw.medium.com/how-to-calculate-inflation-with-gdp-deflators-economics-101-12538495c928","timestamp":"2024-11-02T08:16:44Z","content_type":"text/html","content_length":"117267","record_id":"<urn:uuid:168d7add-b0fa-4305-9529-69f89fb4e560>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00167.warc.gz"} |
“Mathematical and Musical Notation as Models” by Mark Colyvan (14 Mar)
Since the demise of formalism in the philosophy of mathematics, notation has ceased to be a topic of philosophical interest. But within mathematics there are lively debates about notation, it’s just
that philosophers typically don’t weigh in. I hope to take a small step towards correcting this neglect on the part of philosophers of mathematics. I will look at the roles musical notation plays in
composition, performance, and arranging musical pieces and I will argue that there is a great deal of similarity in the functions of mathematical and musical notations. I will argue that both
notational systems serve as models of the target system in question (mathematical structures or musical pieces, respectively).
Philosophy Seminar Series.
Date: Friday, 14 Mar 2014
Time: 2 pm – 4 pm
Venue: Philosophy Resource Room (AS3 #05-23)
Speaker: Mark Colyvan, University of Sydney
Moderator: Dr. Ben Blumson
About the Speaker:
Mark Colyvan was awarded a BSc(Hons) in mathematics at the University of New England in 1994 before taking a PhD in philosophy from the Australian National University in 1998. He is currently an
Australian Research Council Future Fellow and Professor of Philosophy at the University of Sydney, Australia. His main research interests are in the philosophy of mathematics, philosophy of science,
philosophy of logic, decision theory, and environmental philosophy. He is the author of The Indispensability of Mathematics (Oxford University Press, 2001), An Introduction to the Philosophy of
Mathematics (Cambridge University Press, 2012) and, with Lev Ginzburg, Ecological Orbits: How Planets Move and Populations Grow (Oxford University Press, 2004). Two of his papers, “Applying
Inconsistent Mathematics” and “Mating, Dating, and Mathematics: It’s All in the Game” were selected by Princeton University Press as being among the best writing on mathematics for 2010 and 2012
respectively (in M. Pitici (ed.), The Best Writing on Mathematics, Princeton University Press, 2011/2013). Further information is available from his website: http://www.colyvan.com. | {"url":"https://blog.nus.edu.sg/philo/2014/03/03/mathematical-and-musical-notation-as-models-by-mary-colyvan-14-mar/","timestamp":"2024-11-02T04:41:06Z","content_type":"text/html","content_length":"46572","record_id":"<urn:uuid:8c56c593-86e9-4660-bfea-66fc9e23ec55>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00875.warc.gz"} |
The Stacks project
Lemma 42.67.2. In Situation 42.67.1 let $X \to S$ be locally of finite type. Denote $X' \to S'$ the base change by $S' \to S$. If $X$ is integral with $\dim _\delta (X) = k$, then every irreducible
component $Z'$ of $X'$ has $\dim _{\delta '}(Z') = k + c$,
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Indicator Function
An Indicator Function is a binary function that indicates the presence of some predetermined pattern within a set or system.
• AKA: Characteristic Function.
• Context:
□ It can (typically) represent the membership of an element in a subset of a larger set, returning 1 if the element is in the subset and 0 otherwise.
□ ...
□ It can be used in probability theory to define events in a sample space, where the indicator function returns 1 if the event occurs and 0 if it does not.
□ It can be applied in machine learning as a feature function that captures whether a particular condition holds for an input, such as the presence of a word in a text or a class label in
classification tasks.
□ It can be utilized in optimization problems, where the indicator function helps define constraints by indicating when certain variables or conditions are met.
□ It can be used in integration to simplify calculations by selecting parts of the domain for which the integrand is non-zero, particularly in the case of integrals over subsets.
□ It can appear in expressions for piecewise-defined functions, helping to determine which part of the function applies under given conditions.
□ It can help define loss functions in algorithms by indicating whether certain criteria or thresholds are met during training.
□ It can aid in combinatorial optimization, where it indicates the feasibility of different combinations based on a set of constraints.
□ It can be used to define characteristics in decision trees or rule-based systems by representing binary conditions.
□ It can simplify notation and make expressions more readable when defining complex systems with conditional behaviors.
□ ...
• Example(s):
□ One defined over a set of numbers to check whether each element is greater than a threshold, returning 1 for elements that meet the condition and 0 otherwise.
□ One in a machine learning feature set, where it indicates whether a particular word appears in a document or not.
□ One in a probabilistic model that defines whether a random variable falls within a specified event, contributing to the calculation of event probabilities.
□ An Impulse Response Function that describes the output of a system when presented with a brief input signal.
□ ...
• Counter-Example(s):
□ A Heaviside Step Function, which is a continuous approximation to the indicator function but can take values other than 0 and 1.
□ A Gaussian Function, which is continuous and does not have binary output but instead varies smoothly over its domain.
□ A Piecewise Function that is not necessarily binary, as it may have different non-binary values depending on the input conditions.
• See: Characteristic Function, Set Member, Subset, Probability Theory, Machine Learning.
• http://en.wikipedia.org/wiki/Indicator_function
□ In mathematics, an indicator function or a characteristic function is a function defined on a set [math]\displaystyle{ X }[/math] that indicates membership of an element in a subset A of X,
having the value 1 for all elements of [math]\displaystyle{ A }[/math] and the value 0 for all elements of [math]\displaystyle{ X }[/math] not in A. | {"url":"https://www.gabormelli.com/RKB/indicator_function","timestamp":"2024-11-13T07:44:47Z","content_type":"text/html","content_length":"42885","record_id":"<urn:uuid:9dac8c77-3eb7-48bf-a2fe-b0c44198957c>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00289.warc.gz"} |
Two Sum II - Input Array Is Sorted
Go to my solution
Go to the question on LeetCode
My Thoughts
What Went Well
I knew to use the two-pointer method because the list was sorted.
What I Learned
I further solidified the two-pointer method in my brain.
Algorithm Description
Two Pointer Algorithm - If a list is sorted and we place a pointer at the extreme values (max and min), we can iterate one pointer at a time to approach the solution.
Binary Search - A search algorithm that repeatedly halves the search interval until the target variable is found at the middle index.
Visual Examples
The two-pointer algorithm being performed on a list, click to view
Binary search being performed on an array that contains the target, click to view
Binary search being performed on an array that does not contain the target, click to view
Solution Statistics
Time Spent Coding
5 minutes
Time Complexity
O(n) - In the worst case, the target indices are near the middle, resulting in the O(n) time complexity because almost all indices will be visited.
Space Complexity
O(1) - We only need to declare two new variables, and the number of variables does not depend on the number of elements in the list (n), resulting in the O(1) space complexity.
Runtime Beats
79.62% of other submissions
Memory Beats
80.90% of other sumbissions
1 class Solution:
2 def twoSum(self, numbers: List[int], target: int) -> List[int]:
3 p0 = 0
4 p1 = len(numbers) - 1
6 while p1 > p0:
7 # If the elements at the indices sum to the target
8 if numbers[p0] + numbers[p1] == target:
9 # Return them, but add one
10 # (This is a special constraint that only applies to this problem)
11 return [p0+1,p1+1]
13 # If the number is greater than the target, then we know if we decrement
14 # p1 we will get a smaller value, there for inching closer to the target
15 if numbers[p0] + numbers[p1] > target:
16 p1 -=1
18 # The same principle applies here, except the sum is smaller than the target
19 # and incrementing p0 gets us a larger value
20 else:
21 p0 +=1 | {"url":"https://douglastitze.com/posts/Two-Sum-II-Input-Array-Is-Sorted/","timestamp":"2024-11-13T15:43:06Z","content_type":"text/html","content_length":"25946","record_id":"<urn:uuid:ed62af5b-1e98-4df2-a330-b429b6204cbd>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00657.warc.gz"} |
Universal Bayes Consistency in Metric Spaces
We show that a recently proposed 1-nearest-neighbor-based multiclass learning algorithm is universally strongly Bayes consistent in all metric spaces where such Bayes consistency is possible, making
it an "optimistically universal"Bayes-consistent learner. This is the first learning algorithm known to enjoy this property; by comparison, k-NN and its variants are not generally universally Bayes
consistent, except under additional structural assumptions, such as an inner product, a norm, finite doubling dimension, or a Besicovitch-type property.The metric spaces in which universal Bayes
consistency is possible are the "essentially separable"ones-a new notion that we define, which is more general than standard separability. The existence of metric spaces that are not essentially
separable is independent of the ZFC axioms of set theory. We prove that essential separability exactly characterizes the existence of a universal Bayes-consistent learner for the given metric space.
In particular, this yields the first impossibility result for universal Bayes consistency.Taken together, these positive and negative results resolve the open problems posed in Kontorovich, Sabato,
Weiss (2017).
Publication series
Name 2020 Information Theory and Applications Workshop, ITA 2020
Conference 2020 Information Theory and Applications Workshop, ITA 2020
Country/Territory United States
City San Diego
Period 2/02/20 → 7/02/20
• Bayes consistency
• classification
• metric space
• nearest neighbor
Dive into the research topics of 'Universal Bayes Consistency in Metric Spaces'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/universal-bayes-consistency-in-metric-spaces-5","timestamp":"2024-11-07T12:12:06Z","content_type":"text/html","content_length":"56651","record_id":"<urn:uuid:62eee2dc-aad2-427b-8f00-19db4fa36642>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00136.warc.gz"} |
A factoring success story
I covered a couple of my colleague’s classes yesterday so he could attend a math conference. The afternoon class was a somewhat boisterous grade 10 group. I was asked to teach students how to find
the greatest common factor, and if I had time, introduce them to more general factoring techniques.
I decided that the greatest common factor is a topic students find relatively easy, and so I just showed some examples of how to do it (actually, I drew the "how to" out of the class by asking them
questions, but this is my standard technique) after verifying that they understood the distributive principle. I then assigned some practice problems, which then each student wrote their solution up
on the board, and we discussed. I then showed students a couple of different techniques for multiplying binomials (like (x+2)(x+3) for example).
Next, I put up the following 4 questions.
1. x^2 + 7x + 12
2. 2x^2 + 7x + 3
3. x^2 – 25
4. x^3 + 8
I asked students to try and figure out how to write these expressions as one set of brackets times another, just like with the example from before, but I suggested to them that what we are trying to
do is undo the distributive rule.
I went around the room and encouraged students, gave them hints when they needed them, asked them questions to prod their thinking, and observed their problem solving strategies. Students were
engaged in the problem solving activity for a good 30 minutes. Once some of the students’ attentions started to wane a bit, I gave them a sheet with a description of how to do factoring by grouping
and some problems to work on the back.
A group of students though really dove into question 4, which, as you may notice is actually quite a bit more difficult than the other three problems. I ended up having to give students two hints: I
told them that the expression broke into two factors, one of which was (x+2) and the other of which was three terms long. The group of students worked feverishly on solving the 4th problem for a good
twenty minutes, and then all of a sudden, one of the girls in the group leapt out of her seat and screamed, "I GOT IT!! YES!!" I circled around to see if she had the right answer, asked her how she
was so sure it was right (she had multiplied everything back through using the distributive rule), and then gave her group x^3+27 to solve (which she did quickly) and then x^3 + a^3 to solve.
At 5:30pm that night, I received an email from the girl, excitedly telling me how she had an inspiration while she was on the bus home on how to solve the general question, and had then figured out
the general formula for how to factor a sum of cubes.
I emailed her back and congratulated her on becoming a mathematician. | {"url":"https://davidwees.com/content/factoring-success-story/","timestamp":"2024-11-14T22:12:34Z","content_type":"text/html","content_length":"69455","record_id":"<urn:uuid:753a860a-38f1-4287-8d56-cea8abec190b>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00440.warc.gz"} |
Integer Rules Practice Quiz
Note: this page contains legacy resources that are no longer supported. You are free to continue using these materials but we can only support our current worksheets, available as part of our
membership offering.
Integer Rules Practice Quiz
Related Resources
The Integers resource above is aligned (wholly or partially) with standard 7NS01 taken from the Common Core Standards For Mathematics (see the extract below). The various resources listed below are
aligned to the same standard.
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
Similar to the above listing, the resources below are aligned to related standards in the Common Core For Mathematics that together support the following learning outcome:
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
This resource is available for re-use under a Creative Commons Attribution 4.0 International License. More details and source files here | {"url":"https://helpingwithmath.com/generators/7ns1-integer-rules01/","timestamp":"2024-11-09T04:27:42Z","content_type":"text/html","content_length":"111215","record_id":"<urn:uuid:d8286057-e5b8-42c2-a2c5-724bc9e32b5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00112.warc.gz"} |
Alternating Current And Voltage Archives - Page 4 of 4 -
Question: One sine wave has a positive-going zero crossing at 15° and another sine wave has a positive-going zero crossing at 55°. The phase angle between the two waveforms is
none of the above
Answer: Option C
No answer description available for this question.
One sine wave has a positive-going zero crossing at 15° and another sine wave has a positive-going zero crossing at 55°. The phase angle between the two waveforms is Read More »
Alternating Current And Voltage, Electrical Engineering
Calculate the positive-going slope of the waveform in the given circuit.
Question: Calculate the positive-going slope of the waveform in the given circuit.
[A]. 5 V/ms
2.5 V/ms
2.5 V/s
5 V/s
Answer: Option B
No answer description available for this question.
Calculate the positive-going slope of the waveform in the given circuit. Read More »
Alternating Current And Voltage, Electronics | {"url":"https://answeringexams.com/category/alternating-current-and-voltage/page/4/","timestamp":"2024-11-12T20:46:14Z","content_type":"text/html","content_length":"129193","record_id":"<urn:uuid:1dee5370-cf3f-42f2-a617-903e9cef069a>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00237.warc.gz"} |
The figure above is the graph of the function f(x)−px2−2, where... | Filo
The figure above is the graph of the function , where is a constant. If , which of the following describes the graph of in the relation to the graph of
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From the graph, we know that because the parabola opens upwards. The value of witl also be positive and the graph of will open upwards. Alco, for , for all values of except . This implies that the
graph of will be wider (or flatter) than the graph of
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Question Text The figure above is the graph of the function , where is a constant. If , which of the following describes the graph of in the relation to the graph of
Topic Functions
Subject Mathematics
Class Grade 12
Answer Type Text solution:1
Upvotes 99 | {"url":"https://askfilo.com/mathematics-question-answers/the-figure-above-is-the-graph-of-the-function-fx-p-x2-2-where-p-is-a-constant-if","timestamp":"2024-11-09T12:53:24Z","content_type":"text/html","content_length":"366502","record_id":"<urn:uuid:ca1cadde-0361-4561-ae5f-9b78f627002b>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00293.warc.gz"} |
Business Problems Can't (Only) Be Solved With Algorithms
We discuss artificial intelligence, machine learning, what an algorithm is and the role of algorithms in solving business problems.
Machine learning and deep learning solutions have already established themselves as business tools that organisations use to improve customer experience, increase return on investment and gain a
competitive advantage in business operations, among others. Both artificial intelligence models are based on complex algorithms that, on too many occasions, are granted more power than they actually
have. We discuss the role of algorithms in business problems resolution.
The euphoria for machine learning and deep learning in the business world continues to grow. A research conducted by MarketsandMarkets predicts that, by 2022, the machine learning market will have
grown by 44.1% in 6 years; from $1.03 billion in 2016 to $8.81 billion in 2022. The same study points out that data generation and technological advances are already among the main factors driving
the market. In addition, machine learning technologies such as Azure Machine Learning are becoming increasingly prevalent in enterprises.
We explore the difference between machine learning and deep learning in this article: 'What Is the Difference Between Machine Learning and Deep Learning?'
Both are artificial intelligence technologies based on complex mathematical algorithms that enable machines to learn from data in a similar way to how humans do. Algorithms are used for a wide range
of business operations and activities.
Nowadays, algorithms are practically everywhere and the quest to develop or apply a better algorithm than the competition is as widespread as the desire to decipher all the secrets of Instagram's new
What is an algorithm?
According to Google, which has quite a bit of knowledge about algorithms, an algorithm is an "ordered set of systematic operations that allows us to make a calculation and find the solution to a
problem". In practice, an algorithm is nothing more than a mathematical equation or a set of several mathematical equations applied to technological tools so that they do exactly what we want them to
Eduardo Peña, professor at the Faculty of Computer Science at the Complutense University of Madrid, explains it this way: "Ultimately, the job of computer programmers is to translate the world's
problems into a language that machines can understand".
The algorithm is not the solution: The story of Vincent Warmerdam
In the business environment, algorithms are constantly used for the optimisation of operations and functionalities. They are data-driven and rely, more than it may seem, on the human mind. Data
scientists and engineers develop algorithms with the intention of solving problems and perfecting operations carried out by machines, technological tools, platforms, etc. However, on many occasions,
no matter how prodigious the algorithm, business problems are not solved or the expected results are not achieved.
But why is this?
Let's start at the beginning. Despite being represented as such by society, an algorithm is not some kind of magic wand with superpowers or an evil entity programmed to sneak into our brains and
reveal all our secrets. Algorithms are certainly capable of solving complex problems and doing things that in other times would have seemed extraordinary, but they do not do it on their own.
Sticking with mathematics, an algorithm is only a formula. To solve a mathematical problem, the first step is to understand the problem and then find out which formula should be applied. Applying the
wrong formula will obviously not solve the problem. This does not mean that the formula is wrong —the formula in itself is correct— it is just not well applied.
The same goes for algorithms. Vincent Warmerdam, co-founder of PyData and specialist in algorithms and machine learning, addresses this problem and talks about his experience applying algorithms to
solve business problems in his lecture 'The profession of solving (the wrong problem)'. In it, Warmerdam expresses the problem of applying algorithms through several personal stories that helped him
to realise that, indeed, the algorithm is not the solution. What really solves business problems is everything that surrounds the algorithm: databases, data quality, data analysis, A/B testing,
problem statement and, most importantly, "natural intelligence", as he calls it.
Warmerdam's story is based on a memory from his high school days. The teacher asked the students to apply statistics to a real database. At the time, Warmerdam was working in a theatre where a
possible extension was being evaluated. Warmerdam asked his superior for the theatre's annual attendance growth figures and he quickly discovered that attendance growth had been steadily decreasing
year after year. Vincent came to a very clear conclusion: the theatre should not go ahead with the expansion project, as attendance was gradually falling with each passing year. Warmerdam's discovery
astonished his professor, who rewarded him with an A. His superior at work was also impressed and congratulated him on his discovery. Problem solved, right?
The following weeks Warmerdam continued to work at the theatre and during his shit he noticed how hot it was in the hall because it was packed. All the seats were taken and there were even people
standing. The next day, the scene was the same and Warmerdam observed that every day that week, the hall was full to capacity.
Then he understood. He had not solved the problem. He had applied the wrong formula. The theatre's attendance growth was steadily decreasing year after year because there was no more space and
therefore no more people could fit. During the first years of activity, the growth was booming until the hall was filled almost every day and the space started to become too small. From then on,
theatre attendance stopped growing. Not because people stopped coming, but because there was no more space. Warmerdam had not approached the problem correctly, had applied the wrong formula and
therefore had not solved the problem, despite being congratulated by his superiors.
After that first encounter with the algorithm, Warmerdam continued to insist and managed to build a successful career working as an expert in machine learning and algorithms. His long experience in
the field has served to confirm that what happened to him in high school with statistics, happens all the time in the business world with algorithms.
The problem with the algorithm
Warmerdam is convinced that algorithms do not solve problems and that, in fact, no matter how good an algorithm is, if poorly applied, it can make the problem worse. What is more concerning is that
he himself, on more occasions than he would like to admit, has celebrated with his colleagues the resolution of a business problem after the development of an algorithm, only to realise days or weeks
later that the problem had not been solved at all and that they were celebrating a fake victory proclaimed by themselves.
This is the problem with algorithms. The world is hell-bent on believing that an algorithm can solve anything. Vincent himself paraphrases several of his colleagues who, when faced with any problem,
the first thing they say is something like: "Do you need to solve this? We're going to create a super-algorithm!" Immediately after learning that a client needs to solve a problem. Without even
understanding the problem, analysing the data or making sure of its veracity.
Another example of failure caused by blind faith in the algorithm is the famous 'Flash Crack' incident. On 6 May 2010, stock market algorithms caused the stock market to plummet by 1,000 points —p
ractically 9% of the shares— for no apparent reason. After a few minutes everything returned to normal and the points returned to their natural state. However, to this day, no one can explain why
this happened or what happened. The creators of the algorithm themselves were unable to determine why the algorithm had acted as it did, which showed that none of them really understood the whole
process or what was behind the algorithm, confirming Warmerdam's suspicions that artificial intelligence is incapable of being intelligent without natural intelligence or, in other words, human
In this respect, while machine learning, deep learning and algorithms have been a huge breakthrough in the business world, it is essential that entrepreneurs and data scientists realise that
algorithms alone do not solve problems. Applying the right formulas to the wrong problem can lead to a false sense of victory that, in the long run, always ends in defeat.
Posted by Núria Emilio | {"url":"https://blog.bismart.com/en/what-is-an-algorithm-solving-business-problems","timestamp":"2024-11-04T11:15:44Z","content_type":"text/html","content_length":"116412","record_id":"<urn:uuid:89b61a36-57a4-4a08-a496-3b4e7dac2488>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00680.warc.gz"} |
Alpha Built-in Functions - Using the GNU Compiler Collection (GCC)
6.58.2 Alpha Built-in Functions
These built-in functions are available for the Alpha family of processors, depending on the command-line switches used.
The following built-in functions are always available. They all generate the machine instruction that is part of the name.
long __builtin_alpha_implver (void)
long __builtin_alpha_rpcc (void)
long __builtin_alpha_amask (long)
long __builtin_alpha_cmpbge (long, long)
long __builtin_alpha_extbl (long, long)
long __builtin_alpha_extwl (long, long)
long __builtin_alpha_extll (long, long)
long __builtin_alpha_extql (long, long)
long __builtin_alpha_extwh (long, long)
long __builtin_alpha_extlh (long, long)
long __builtin_alpha_extqh (long, long)
long __builtin_alpha_insbl (long, long)
long __builtin_alpha_inswl (long, long)
long __builtin_alpha_insll (long, long)
long __builtin_alpha_insql (long, long)
long __builtin_alpha_inswh (long, long)
long __builtin_alpha_inslh (long, long)
long __builtin_alpha_insqh (long, long)
long __builtin_alpha_mskbl (long, long)
long __builtin_alpha_mskwl (long, long)
long __builtin_alpha_mskll (long, long)
long __builtin_alpha_mskql (long, long)
long __builtin_alpha_mskwh (long, long)
long __builtin_alpha_msklh (long, long)
long __builtin_alpha_mskqh (long, long)
long __builtin_alpha_umulh (long, long)
long __builtin_alpha_zap (long, long)
long __builtin_alpha_zapnot (long, long)
The following built-in functions are always with -mmax or -mcpu=cpu where cpu is pca56 or later. They all generate the machine instruction that is part of the name.
long __builtin_alpha_pklb (long)
long __builtin_alpha_pkwb (long)
long __builtin_alpha_unpkbl (long)
long __builtin_alpha_unpkbw (long)
long __builtin_alpha_minub8 (long, long)
long __builtin_alpha_minsb8 (long, long)
long __builtin_alpha_minuw4 (long, long)
long __builtin_alpha_minsw4 (long, long)
long __builtin_alpha_maxub8 (long, long)
long __builtin_alpha_maxsb8 (long, long)
long __builtin_alpha_maxuw4 (long, long)
long __builtin_alpha_maxsw4 (long, long)
long __builtin_alpha_perr (long, long)
The following built-in functions are always with -mcix or -mcpu=cpu where cpu is ev67 or later. They all generate the machine instruction that is part of the name.
long __builtin_alpha_cttz (long)
long __builtin_alpha_ctlz (long)
long __builtin_alpha_ctpop (long)
The following built-in functions are available on systems that use the OSF/1 PALcode. Normally they invoke the rduniq and wruniq PAL calls, but when invoked with -mtls-kernel, they invoke rdval and
void *__builtin_thread_pointer (void)
void __builtin_set_thread_pointer (void *) | {"url":"https://gcc.gnu.org/onlinedocs/gcc-5.4.0/gcc/Alpha-Built-in-Functions.html","timestamp":"2024-11-05T10:50:43Z","content_type":"text/html","content_length":"6529","record_id":"<urn:uuid:b323530e-e5be-4810-b759-956d0c5aa61e>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00741.warc.gz"} |
class tslearn.shapelets.LearningShapelets(n_shapelets_per_size=None, max_iter=10000, batch_size=256, verbose=0, optimizer='sgd', weight_regularizer=0.0, shapelet_length=0.15, total_lengths=3,
max_size=None, scale=False, random_state=None)[source]¶
Learning Time-Series Shapelets model.
Learning Time-Series Shapelets was originally presented in [1].
From an input (possibly multidimensional) time series \(x\) and a set of shapelets \(\{s_i\}_i\), the \(i\)-th coordinate of the Shapelet transform is computed as:
\[ST(x, s_i) = \min_t \sum_{\delta_t} \left\|x(t+\delta_t) - s_i(\delta_t)\right\|_2^2\]
The Shapelet model consists in a logistic regression layer on top of this transform. Shapelet coefficients as well as logistic regression weights are optimized by gradient descent on a
L2-penalized cross-entropy loss.
n_shapelets_per_size: dict (default: None)
Dictionary giving, for each shapelet size (key), the number of such shapelets to be trained (value). If None, grabocka_params_to_shapelet_size_dict is used and the size used to compute is
that of the shortest time series passed at fit time.
max_iter: int (default: 10,000)
Number of training epochs.
Changed in version 0.3: default value for max_iter is set to 10,000 instead of 100
batch_size: int (default: 256)
Batch size to be used.
verbose: {0, 1, 2} (default: 0)
keras verbose level.
optimizer: str or keras.optimizers.Optimizer (default: “sgd”)
keras optimizer to use for training.
weight_regularizer: float or None (default: 0.)
Strength of the L2 regularizer to use for training the classification (softmax) layer. If 0, no regularization is performed.
shapelet_length: float (default: 0.15)
The length of the shapelets, expressed as a fraction of the time series length. Used only if n_shapelets_per_size is None.
total_lengths: int (default: 3)
The number of different shapelet lengths. Will extract shapelets of length i * shapelet_length for i in [1, total_lengths] Used only if n_shapelets_per_size is None.
max_size: int or None (default: None)
Maximum size for time series to be fed to the model. If None, it is set to the size (number of timestamps) of the training time series.
scale: bool (default: False)
Whether input data should be scaled for each feature of each time series to lie in the [0-1] interval. Default for this parameter is set to False in version 0.4 to ensure backward
compatibility, but is likely to change in a future version.
random_stateint or None, optional (default: None)
The seed of the pseudo random number generator to use when shuffling the data. If int, random_state is the seed used by the random number generator; If None, the random number generator
is the RandomState instance used by np.random.
shapelets_numpy.ndarray of objects, each object being a time series
Set of time-series shapelets.
shapelets_as_time_series_numpy.ndarray of shape (n_shapelets, sz_shp, d) where sz_shp is the maximum of all shapelet sizes
Set of time-series shapelets formatted as a tslearn time series dataset.
Transforms an input dataset of timeseries into distances to the learned shapelets.
Returns the indices where each of the shapelets can be found (minimal distance) within each of the timeseries of the input dataset.
Directly predicts the class probabilities for the input timeseries.
Dictionary of losses and metrics recorded during fit.
10. Grabocka et al. Learning Time-Series Shapelets. SIGKDD 2014.
>>> from tslearn.generators import random_walk_blobs
>>> X, y = random_walk_blobs(n_ts_per_blob=10, sz=16, d=2, n_blobs=3)
>>> clf = LearningShapelets(n_shapelets_per_size={4: 5},
... max_iter=1, verbose=0)
>>> clf.fit(X, y).shapelets_.shape
>>> clf.shapelets_[0].shape
(4, 2)
>>> clf.predict(X).shape
>>> clf.predict_proba(X).shape
(30, 3)
>>> clf.transform(X).shape
(30, 5)
fit(X, y) Learn time-series shapelets.
fit_transform(X[, y]) Fit to data, then transform it.
from_hdf5(path) Load model from a HDF5 file.
from_json(path) Load model from a JSON file.
from_pickle(path) Load model from a pickle file.
get_metadata_routing() Get metadata routing of this object.
get_params([deep]) Get parameters for this estimator.
get_weights([layer_name]) Return model weights (or weights for a given layer if layer_name is provided).
locate(X) Compute shapelet match location for a set of time series.
predict(X) Predict class for a given set of time series.
predict_proba(X) Predict class probability for a given set of time series.
score(X, y[, sample_weight]) Return the mean accuracy on the given test data and labels.
set_output(*[, transform]) Set output container.
set_params(**params) Set the parameters of this estimator.
set_score_request(*[, sample_weight]) Request metadata passed to the score method.
set_weights(weights[, layer_name]) Set model weights (or weights for a given layer if layer_name is provided).
to_hdf5(path) Save model to a HDF5 file.
to_json(path) Save model to a JSON file.
to_pickle(path) Save model to a pickle file.
transform(X) Generate shapelet transform for a set of time series.
fit(X, y)[source]¶
Learn time-series shapelets.
Xarray-like of shape=(n_ts, sz, d)
Time series dataset.
yarray-like of shape=(n_ts, )
Time series labels.
fit_transform(X, y=None, **fit_params)[source]¶
Fit to data, then transform it.
Fits transformer to X and y with optional parameters fit_params and returns a transformed version of X.
Xarray-like of shape (n_samples, n_features)
Input samples.
yarray-like of shape (n_samples,) or (n_samples, n_outputs), default=None
Target values (None for unsupervised transformations).
Additional fit parameters.
X_newndarray array of shape (n_samples, n_features_new)
Transformed array.
classmethod from_hdf5(path)[source]¶
Load model from a HDF5 file. Requires h5py http://docs.h5py.org/
Full path to file.
Model instance
classmethod from_json(path)[source]¶
Load model from a JSON file.
Full path to file.
Model instance
classmethod from_pickle(path)[source]¶
Load model from a pickle file.
Full path to file.
Model instance
Get metadata routing of this object.
Please check User Guide on how the routing mechanism works.
A MetadataRequest encapsulating routing information.
Get parameters for this estimator.
deepbool, default=True
If True, will return the parameters for this estimator and contained subobjects that are estimators.
Parameter names mapped to their values.
Return model weights (or weights for a given layer if layer_name is provided).
layer_name: str or None (default: None)
Name of the layer for which weights should be returned. If None, all model weights are returned. Available layer names with weights are:
■ “shapelets_i_j” with i an integer for the shapelet id and j an integer for the dimension
■ “classification” for the final classification layer
list of model (or layer) weights
>>> from tslearn.generators import random_walk_blobs
>>> X, y = random_walk_blobs(n_ts_per_blob=100, sz=256, d=1, n_blobs=3)
>>> clf = LearningShapelets(n_shapelets_per_size={10: 5}, max_iter=0,
... verbose=0)
>>> clf.fit(X, y).get_weights("classification")[0].shape
(5, 3)
>>> clf.get_weights("shapelets_0_0")[0].shape
(5, 10)
>>> len(clf.get_weights("shapelets_0_0"))
Compute shapelet match location for a set of time series.
Xarray-like of shape=(n_ts, sz, d)
Time series dataset.
array of shape=(n_ts, n_shapelets)
Location of the shapelet matches for the provided time series.
>>> from tslearn.generators import random_walk_blobs
>>> X = numpy.zeros((3, 10, 1))
>>> X[0, 4:7, 0] = numpy.array([1, 2, 3])
>>> y = [1, 0, 0]
>>> # Data is all zeros except a motif 1-2-3 in the first time series
>>> clf = LearningShapelets(n_shapelets_per_size={3: 1}, max_iter=0,
... verbose=0)
>>> _ = clf.fit(X, y)
>>> weights_shapelet = [
... numpy.array([[1, 2, 3]])
... ]
>>> clf.set_weights(weights_shapelet, layer_name="shapelets_0_0")
>>> clf.locate(X)
Predict class for a given set of time series.
Xarray-like of shape=(n_ts, sz, d)
Time series dataset.
array of shape=(n_ts, ) or (n_ts, n_classes), depending on the shape
of the label vector provided at training time.
Index of the cluster each sample belongs to or class probability matrix, depending on what was provided at training time.
Predict class probability for a given set of time series.
Xarray-like of shape=(n_ts, sz, d)
Time series dataset.
array of shape=(n_ts, n_classes),
Class probability matrix.
score(X, y, sample_weight=None)[source]¶
Return the mean accuracy on the given test data and labels.
In multi-label classification, this is the subset accuracy which is a harsh metric since you require for each sample that each label set be correctly predicted.
Xarray-like of shape (n_samples, n_features)
Test samples.
yarray-like of shape (n_samples,) or (n_samples, n_outputs)
True labels for X.
sample_weightarray-like of shape (n_samples,), default=None
Sample weights.
Mean accuracy of self.predict(X) w.r.t. y.
set_output(*, transform=None)[source]¶
Set output container.
See Introducing the set_output API for an example on how to use the API.
transform{“default”, “pandas”}, default=None
Configure output of transform and fit_transform.
■ “default”: Default output format of a transformer
■ “pandas”: DataFrame output
■ None: Transform configuration is unchanged
selfestimator instance
Estimator instance.
Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each
component of a nested object.
Estimator parameters.
selfestimator instance
Estimator instance.
set_score_request(*, sample_weight: bool | None | str = '$UNCHANGED$') LearningShapelets[source]¶
Request metadata passed to the score method.
Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config()). Please see User Guide on how the routing mechanism works.
The options for each parameter are:
☆ True: metadata is requested, and passed to score if provided. The request is ignored if metadata is not provided.
☆ False: metadata is not requested and the meta-estimator will not pass it to score.
☆ None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
☆ str: metadata should be passed to the meta-estimator with this given alias instead of the original name.
The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.
This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.
sample_weightstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED
Metadata routing for sample_weight parameter in score.
The updated object.
set_weights(weights, layer_name=None)[source]¶
Set model weights (or weights for a given layer if layer_name is provided).
weights: list of ndarrays
Weights to set for the model / target layer
layer_name: str or None (default: None)
Name of the layer for which weights should be set. If None, all model weights are set. Available layer names with weights are:
■ “shapelets_i_j” with i an integer for the shapelet id and j an integer for the dimension
■ “classification” for the final classification layer
>>> from tslearn.generators import random_walk_blobs
>>> X, y = random_walk_blobs(n_ts_per_blob=10, sz=16, d=1, n_blobs=3)
>>> clf = LearningShapelets(n_shapelets_per_size={3: 1}, max_iter=0,
... verbose=0)
>>> _ = clf.fit(X, y)
>>> weights_shapelet = [
... numpy.array([[1, 2, 3]])
... ]
>>> clf.set_weights(weights_shapelet, layer_name="shapelets_0_0")
>>> clf.shapelets_as_time_series_
property shapelets_as_time_series_[source]¶
Set of time-series shapelets formatted as a tslearn time series dataset.
>>> from tslearn.generators import random_walk_blobs
>>> X, y = random_walk_blobs(n_ts_per_blob=10, sz=256, d=1, n_blobs=3)
>>> model = LearningShapelets(n_shapelets_per_size={3: 2, 4: 1},
... max_iter=1)
>>> _ = model.fit(X, y)
>>> model.shapelets_as_time_series_.shape
(3, 4, 1)
Save model to a HDF5 file. Requires h5py http://docs.h5py.org/
Full file path. File must not already exist.
If a file with the same path already exists.
Save model to a JSON file.
Full file path.
Save model to a pickle file.
Full file path.
Generate shapelet transform for a set of time series.
Xarray-like of shape=(n_ts, sz, d)
Time series dataset.
array of shape=(n_ts, n_shapelets)
Shapelet-Transform of the provided time series.
Examples using tslearn.shapelets.LearningShapelets¶ | {"url":"https://tslearn.readthedocs.io/en/latest/gen_modules/shapelets/tslearn.shapelets.LearningShapelets.html","timestamp":"2024-11-06T16:48:22Z","content_type":"text/html","content_length":"74511","record_id":"<urn:uuid:3957c497-4c7e-496f-9ffd-ef977dd1968d>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00675.warc.gz"} |
C Program - Structure - coderz.py C Keep Coding Keep Cheering!
C Program – Structure
Before we begin with our basic concepts of the C programming language let’s take a look at the C Program -Structure of a simple code for better understanding.
A C program consists of the following components−
• standard input-output library functions
• variables
• comments
• statements
• operators
• operands
• functions
Hello World Program
#include <stdio.h>
1. int main()
2. {
3. /* first program in C */
4. printf("Hell, World!");
5. return 0;
6. }
Description of the above code:
• here the first line of the code #include <stdio.h> is the standard input-output library. printf() function is defined in the stdio.h library file.
• int main() is the main function where the execution of the code begins.
• The opening curly brace { symbolizes the beginning of the block.
• The statement inside /*…*/ is the comment which is ignored by the compiler.
• The print() function prints the message “Hello World” on the output window/screen.
• The return 0 statement is the ending of the code, as well as returns the execution state of the OS.
• The closing curly brace } symbolizes the end of the block.
Compiling and Executing a C Program
Let us take a look at how to compile and run a program:
• If you are using a C code editor, you may directly click on the compile and run button resulting in the compilation of the program and an output will be generated on the screen.
• Otherwise, if you are using the command prompt then follow the below-given methods.
• open an editor and type your code in it
• save the file with the “.c” extension.
• open the command prompt go to the directory where you saved your c file
• type gcc filename.c -o filename and press enter.
$ gcc filename.c -o filename
$ ./filename
Hello, World!
Make sure the gcc compiler is in your path and that you are running it in the directory containing the source file filename.c.
Now you might have a good understanding of the C Program -Structure.
Note: Also read C Programming Language Overview C Programming Language
Staying up to the mark is what defines me. Hi all! I’m Rabecca Fatima a keen learner, great enthusiast, ready to take new challenges as stepping stones towards flying colors.
Leave a Comment | {"url":"https://coderzpy.com/c-program-structure/","timestamp":"2024-11-02T15:47:51Z","content_type":"text/html","content_length":"47697","record_id":"<urn:uuid:6aa3fab9-6f5a-4b20-aab9-ddabd6674b70>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00869.warc.gz"} |
Akelius Foundation
Are Greek letters important for calculations?
Is a special sign for square root needed?
What kind of a root is squared?
The point is not to teach Greek letters, strange names, special signs, but to find the length of a square side.
Change the word root to the side,
and visualize.
Put sixteen wooden blocks in a pile.
Ask a student to build a square with all sides of the same length.
A square has an area of sixteen wooden blocks.
How many wooden blocks are on one side?
Repeat with nine, four, and 25 blocks.
Try twenty blocks.
The student finds that the square root is more than four, but less than five.
mean value, median, mode value, range
Can a small child learn statistics?
Most adults will be disturbed when asked to calculate the mean value of the set
1, 2, 2, 2, 3, 5, 6.
Place wooden cubes in seven piles in seven heaps,
or chocolate pralines, or something else.
1, 2, 2, 2, 3, 5, 6
Ask the child
Which pile is in the middle?
Three piles to the left side, three piles to the right side. The middle pile, a two, is called the median.
How to share equally?
How many wooden blocks will there be if you move the wooden blocks so that all have the same height?
The average value is three. | {"url":"https://www.akelius-foundation.org/en/education/green-mathematics","timestamp":"2024-11-03T07:41:01Z","content_type":"text/html","content_length":"174492","record_id":"<urn:uuid:63c21cdb-3655-42d5-b5d3-b25a5ec6aa42>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00415.warc.gz"} |
A248605 - OEIS
The expression "a set of frequencies which has no binary carry," means the following: For a given partition take the set of frequencies of the summands expressed as binary numbers and add them
together. If there is a carry in the addition, then this is not an allowed set of frequencies. See the example for more explanation.
Elements of this sequence have the same parity (
) as the corresponding elements of the sequence of unrestricted partitions (
). See lemma 2.2.ii of the paper by Cooper, Eichorn and O'Bryant.
Also the number of partitions of n into parts which are powers of 2 used with a frequency which is k(3k plus or minus 1)/2.
Every set of partitions defined with the "no binary carry" condition has a dual of this sort. (End)
For n=5, there are 4 partitions which have summands coming from {1,2,5,7,...} namely: 5; 2+2+1; 2+1+1+1; and 1+1+1+1. The third of these has frequencies 1 and 3. These frequencies when written in
binary are 1 and 11. If we add these two binary numbers there will be a carry from the units column; therefore this set of frequencies is not allowed and the partition 2+1+1+1 is not counted.
For[n=1, n<=nend, n++,
summands={1, 2, 5, 7, 12, 15, 22, 26, 35, 40};
p=Partitions[n]; preduced=p;
For[i=Length[p], i>=1, i--,
For[j=1, j<=Length[p[[i]]], j++,
If[MemberQ[summands, p[[i]][[j]]]= =False, preduced=Delete[preduced, i];
For[i=Length[preduced], i>=1, i--,
For[j=1, j<=nend, j++, sum[j]=0];
For[j=1, j<=Length[t], j++,
IntDig=IntegerDigits[t[[j, 2]], 2, 7];
For[k=1, k<=7, k++, sum[k]=sum[k]+IntDig[[k]]]];
table=Table[sum[k], {k, 1, 7}];
If[Max[table]>1, preduced=Delete[preduced, i]]];
Print[Table[a[i], {i, 1, nend}]] | {"url":"https://oeis.org/A248605","timestamp":"2024-11-11T03:31:55Z","content_type":"text/html","content_length":"17323","record_id":"<urn:uuid:3c3ace52-8a7c-4c38-bba5-8ee8760e0cb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00697.warc.gz"} |
Chainstore paradox
The chainstore paradox (or "chain-Store paradox") is a concept that purports to refute standard game theory reasoning.
The chain store game
A monopolist (Player A) has branches in 20 towns. He faces 20 potential competitors, one in each town, who will be able to choose in or out. They do so in sequential order and one at a time. If a
potential competitor chooses out, he receives a payoff of 1, while A receives a payoff of 5. If he chooses in, he will receive a payoff of either 2 or 0, depending on the response of Player A to his
action. Player A, in response to a choice of in, must choose one of two pricing strategies, cooperative or aggressive. If he chooses cooperative, both player A and the competitor receive a payoff of
2, and if A chooses aggressive, each player receives a payoff of 0.
These outcomes lead to two theories for the game, the induction (game theoretically correct version) and the deterrence theory (weakly dominated theory):
Induction theory
Consider the decision to be made by the 20th and final competitor, of whether to choose in or out. He knows that if he chooses in, Player A receives a higher payoff from choosing cooperate than
aggressive, and being the last period of the game, there are no longer any future competitors whom Player A needs to intimidate from the market. Knowing this, the 20th competitor enters the market,
and Player A will cooperate (receiving a payoff of 2 instead of 0).
The outcome in the final period is set in stone, so to speak. Now consider period 19, and the potential competitor's decision. He knows that A will cooperate in the next period, regardless of what
happens in period 19. Thus, if player 19 enters, an aggressive strategy will be unable to deter player 20 from entering. Player 19 knows this and chooses in. Player A chooses cooperate.
Of course, this process of backward induction holds all the way back to the first competitor. Each potential competitor chooses in, and Player A always cooperates. A receives a payoff of 40 (2×20)
and each competitor receives 2.
Deterrence theory
This theory states that Player A will be able to get payoff of higher than 40. Suppose Player A finds the induction argument convincing. He will decide how many periods at the end to play such a
strategy, say 3. In periods 1–17, he will decide to always be aggressive against the choice of IN. If all of the potential competitors know this, it is unlikely potential competitors 1–17 will bother
the chain store, thus risking the safe payout of 1 ("A" will not retaliate if they choose "out"). If a few do test the chain store early in the game, and see that they are greeted with the aggressive
strategy, the rest of the competitors are likely not to test any further. Assuming all 17 are deterred, Player A receives 91 (17×5 + 2×3). Even if as many as 10 competitors enter and test Player A's
will, Player A will still receive a payoff of 41 (10×0+ 7×5 + 3×2), which is better than the induction (game theoretically correct) payoff.
The chain store paradox
If Player A follows the game theory payoff matrix to achieve the optimal payoff, he or she will have a lower payoff than with the "deterrence" strategy. This creates an apparent game theory paradox:
game theory states that induction strategy should be optimal, but it looks like "deterrence strategy" is optimal instead.
The "deterrence strategy" is not a Nash equilibrium: It relies on the non-credible threat of responding to in with aggressive. A rational player will not carry out a non-credible threat, but the
paradox is that it nevertheless seems to benefit Player A to carry out the threat.
Selten's response
Reinhard Selten's response to this apparent paradox is to argue that the idea of "deterrence", while irrational by the standards of Game Theory, is in fact an acceptable idea by the rationality that
individuals actually employ. Selten argues that individuals can make decisions of three levels: Routine, Imagination, and Reasoning.
Complete information?
Game theory is based on the idea that each matrix is modeled with the assumption of complete information: that "every player knows the payoffs and strategies available to other players," where the
word "payoff" is descriptive of behavior—what the player is trying to maximize. If, in the first town, the competitor enters and the monopolist is aggressive, the second competitor has observed that
the monopolist is not, from the standpoint of common knowledge of payoffs and strategies, maximizing the assumed payoffs; expecting the monopolist to do so in this town seems dubious.
If competitors place even a very small probability on the possibility that the monopolist is spiteful, and places intrinsic value on being (or appearing) aggressive, and the monopolist knows this,
then even if the monopolist has payoffs as described above, responding to entry in an early town with aggression will may be optimal if it increases the probability that later competitors place on
the monopolist's being spiteful.
Selten's levels of decision making
The routine level
The individuals use their past experience of the results of decisions to guide their response to choices in the present. "The underlying criteria of similarity between decision situations are crude
and sometimes inadequate". (Selten)
The imagination level
The individual tries to visualize how the selection of different alternatives may influence the probable course of future events. This level employs the routine level within the procedural decisions.
This method is similar to a computer simulation.
The reasoning level
The individual makes a conscious effort to analyze the situation in a rational way, using both past experience and logical thinking. This mode of decision uses simplified models whose assumptions are
products of imagination, and is the only method of reasoning permitted and expected by game theory.
Decision-making process
The predecision
One chooses which method (routine, imagination or reasoning) to use for the problem, and this decision itself is made on the routine level.
The final decision
Depending on which level is selected, the individual begins the decision procedure. The individual then arrives at a (possibly different) decision for each level available (if we have chosen
imagination, we would arrive at a routine decision and possible and imagination decision). Selten argues that individuals can always reach a routine decision, but perhaps not the higher levels. Once
the individuals have all their levels of decision, they can decide which answer to use...the Final Decision. The final decision is made on the routine level and governs actual behavior.
The economy of decision effort
Decision effort is a scarce commodity, being both time consuming and mentally taxing. Reasoning is more costly than Imagination, which, in turn is more costly than Routine. The highest level
activated is not always the most accurate since the individual may be able to reach a good decision on the routine level, but makes serious computational mistakes on higher levels, especially
Selten finally argues that strategic decisions, like those made by the monopolist in the chainstore paradox, are generally made on the level of Imagination, where deterrence is a reality, due to the
complexity of Reasoning, and the great inferiority of Routine (it does not allow the individual to see herself in the other player's position). Since Imagination cannot be used to visualize more than
a few stages of an extensive form game (like the Chain-store game) individuals break down games into "the beginning" and "towards the end". Here, deterrence is a reality, since it is reasonable "in
the beginning", yet is not convincing "towards the end".
See also
Further reading
This article is issued from
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Creative Commons Attribution/Share Alike
but additional terms may apply for the media files. | {"url":"https://ipfs.io/ipfs/QmXoypizjW3WknFiJnKLwHCnL72vedxjQkDDP1mXWo6uco/wiki/Chainstore_paradox.html","timestamp":"2024-11-09T05:59:01Z","content_type":"text/html","content_length":"18973","record_id":"<urn:uuid:cc5ace9d-18d0-4fe6-a0aa-d43be568d6fb>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00243.warc.gz"} |
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Dyadic martingales and harmonic functions, II
The interaction between probability and analysis, in particular harmonic analysis, can be traced back to the formative days of both fields. In fact, one can say that it predates the mathematical
"codification" of probability realized by Kolmogorov's axioms.
Early on this connection was rather implicit, however in the second half of the last century it was studied and developed by many famous researchers (Burkholder, Gundy, Fefferman, Stein, McKean,
Makarov, Banuelos, Peres, just to name a few) resulting in many groundbreaking advances. In addition to a new language to describe analytic phenomena they provided an abundance of deep techniques and
ideas that were instrumental in the solution of many problems of "classical" analysis.
The goal of this course is to elucidate several instances of this relationship and provide a demonstration of the symbiosis enjoyed by probability and (harmonic) analysis. This particular field is
vast and extensive, and it continues to grow in many different directions. Therefore the aim is to concentrate on the most simple (and in a way classical) examples of this kind, thus, essentially,
restricting to the discrete approaches. More specifically, the topics discussed will cover the representation of functions by dyadic martingales, the interplay between the behaviour of various
maximal functions, laws of iterated logarithm, the boundary behaviour of harmonic functions, in particular the properties of the harmonic measure.
The course is divided into two parts. In this one we use the "harmonic function-dyadic martingale" relation to study various boundary properties of harmonic functions and the boundary behaviour of
harmonic measure.
i. Introduction: recalling harmonic functions, dyadic martingales and approximations of the former by the latter.
ii. Harmonic measure on the plane and in higher dimensions.
iii. Hausdorff dimension of the harmonic measure I, history up to Makarov.
iv. Hausdorff dimension of the harmonic measure II, Makarov's theorem.
v. Hausdorff dimension of the harmonic measure III, Bourgain's upper estimate.
vi. Hausdorff dimension of the harmonic measure IV, Wolff's snowflakes.
vii. Banuelos-Moore LIL for harmonic functions: upper estimate.
viii. Banuelos-Moore LIL for harmonic functions: lower estimate and some open problems.
ix. Bourgain's theorem on the radial variation and variations thereof.
[1] N. Arcozzi, N. Chalmoukis, M. Levi, P. Mozolyako. Two-weight dyadic Hardy's inequalities. https://arxiv.org/abs/2110.05450
[2] R. Banuelos, C. N. Moore. Probabilistic behavior of harmonic functions. Birkh auser, Basel-Boston-Berlin (1999)
[3] 3C.J. Bishop. Harmonic Measure: Algortihms and Applications. Proc. Int. Cong. Math. -- 2018, Rio de Janeiro, Vol.2 (2018).
[4] J. Bourgain. On the Hausdorff dimension of harmonic measure in higher dimensions. Invent. Math., 87, 477-483.
[5] J. Bourgain. On the radial variation of bounded analytic functions on the disk, Duke Math. J. 69 (1993), no. 3, 671 682.
[6] A. Canton, J.L. Fernandez, D. Pestana, J.M. Rodri guez. On harmonic functions on trees, Potential Analysis, 15 (2001), 199-244.
[7] I. Daubechies, Ten lectures on wavelets, SIAM, Philadelphia, PA, 1992.
[8] K.S. Eikrem, E. Malinnikova, P. Mozolyako. Wavelet characterization of growth spaces of harmonic functions, Journal d'Analyze Math ematique 122 (2014), no. 1, 87 -111.
[9] J. L. Fernandez, J. Heinonen, J. G. Llorente, Asymptotic values of subharmonic functions, Proc. London Math. Soc. 73.2 (1996), no. 3, 404 430.
[10] J.B. Garnett, D.E. Marshall. Harmonic measure. Cambridge University Press, (2005), 571 pp.
[11] V.P. Havin, P.A. Mozolyako, Boundedness of variation of a positive harmonic function along the normals to the boundary. Algebra and Analysis 28 (2016), no. 3, 67–110
[12] J.G. Llorente, Boundary values of harmonic Bloch functions in Lipschitz domains: a martingale approach, Potential Analysis, 9, 229-260 (1998)
[13] J.G. Llorente. Discrete martingales and applications to analysis. Univ. of Jyv\"{a}skyl\"{a} report, (2002).
[14] N.G. Makarov, Probability methods in the theory of conformal mapping, Algebra i Analiz, 3-59 (1989), (Russian) [Engl. transl. Leningrad Math. J., 1, (1990)]
[15] Y. Meyer, Wavelets and Operators, 225 pp. Cambridge University Press, Cambridge, (1992)
[16] T. H. Wolff, Counterexamples with harmonic gradients in R3, pp. 321–384 in Essays on Fourier analysis in honor of Elias M. Stein (Princeton, 1991), Princeton Math. Ser. 42, Princeton Univ.
Press, Princeton, NJ, 1995
Lecturer Intro
Pavel Mozolyako is an associate professor at St. Petersburg State University. He leads PhD program in mathematics at the department of Mathematics and Computer Science. He got his PhD degree in 2009,
at St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences. He was a postdoc at Norwegian University of Science and Technology, University of Bologna, and a
visiting professor at Michigan State University. His research considers mostly boundary behaviour of harmonic functions and discrete models in potential theory. | {"url":"https://qzc.tsinghua.edu.cn/en/info/1124/3618.htm","timestamp":"2024-11-13T05:24:15Z","content_type":"text/html","content_length":"31112","record_id":"<urn:uuid:6ae16592-487e-47b4-9428-0c8241f8ba4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00412.warc.gz"} |
Cats Per Square Mile Calculator - Calculator Doc
Cats Per Square Mile Calculator
The density of a population within a given area is an important metric in understanding the distribution and impact of that population. This is true not only for human populations but also for
animals, such as cats. Whether you’re managing a feral cat colony or simply curious about the cat population in your neighborhood, calculating the number of cats per square mile can provide valuable
insights. This article will guide you through using a cats per square mile calculator, the underlying formula, and practical examples.
The formula to calculate cats per square mile (CPM) is:
Cats Per Square Mile (CPM) = Number of Cats (C) / Area in Square Miles (SM)
How to Use
1. Enter the Number of Cats (C): Input the total number of cats within the area you’re measuring.
2. Enter the Area in Square Miles (SM): Input the total area in square miles.
3. Click Calculate: The calculator will compute the density of cats per square mile.
Suppose you are tracking a population of 200 cats within a neighborhood that covers 5 square miles. Using the formula:
Cats Per Square Mile (CPM) = 200 cats / 5 square miles = 40 cats per square mile
This means there are 40 cats per square mile in the area you are studying.
1. What is Cats Per Square Mile (CPM)?
□ CPM is a measure of the density of the cat population within a specific area, calculated as the number of cats per square mile.
2. Why is it important to calculate cats per square mile?
□ Understanding cat density can help in managing and controlling cat populations, planning for shelters, and assessing the impact on local ecosystems.
3. Can this calculator be used for both domestic and feral cats?
□ Yes, the calculator can be used for any population of cats, whether domestic or feral.
4. What if the area is not exactly a square mile?
□ You can still use the calculator by converting the area into square miles and then inputting the value.
5. How accurate is the cats per square mile calculation?
□ The accuracy depends on the accuracy of the inputs. If you have an accurate count of cats and area, the calculation will be accurate.
6. Can this calculator be used for other animals?
□ Yes, the same formula can be applied to calculate the density of any animal population per square mile.
7. Is there a difference between urban and rural CPM?
□ Yes, urban areas typically have a higher CPM due to the availability of food and shelter, while rural areas may have a lower CPM.
8. How can I reduce the number of cats per square mile?
□ Strategies such as trap-neuter-return (TNR) programs, adoption initiatives, and public education can help manage and reduce cat populations.
9. What if the area is measured in different units?
□ Convert the area into square miles before using the calculator to ensure accuracy.
10. Does CPM account for the movement of cats?
□ CPM provides a static snapshot based on the number of cats in a given area at a specific time; it does not account for movement.
11. Can CPM be used to estimate the need for veterinary services?
□ Yes, understanding the density of cats can help in planning the availability of veterinary services and resources.
12. How do I count the number of cats in an area?
□ Counting methods may include surveys, visual counts, or using tracking technology in areas with larger or feral populations.
13. What is a high CPM, and what does it indicate?
□ A high CPM indicates a dense cat population, which may lead to issues such as overpopulation, competition for resources, and potential ecological impact.
14. How do environmental factors influence CPM?
□ Factors like food availability, shelter, climate, and human interaction can significantly influence the density of cat populations.
15. Can CPM help in planning for animal shelters?
□ Yes, CPM can assist in determining the need for shelters and resources in specific areas.
16. Is there a way to track changes in CPM over time?
□ By regularly updating the number of cats and area size, you can track changes in CPM over time to assess population trends.
17. What are the challenges in calculating CPM for large areas?
□ Challenges include accurately counting cats over large or inaccessible areas and ensuring that the area measurement is precise.
18. Can CPM be used in wildlife conservation efforts?
□ Yes, CPM can be a valuable tool in assessing the impact of domestic and feral cats on local wildlife and planning conservation strategies.
19. How does CPM vary by region?
□ CPM can vary widely depending on factors like urbanization, local policies, and climate, with some regions having much higher or lower densities.
20. Can I use CPM data to advocate for better animal control policies?
□ Yes, CPM data can be used to support advocacy for policies and programs aimed at managing cat populations more effectively.
Calculating the number of cats per square mile is a useful tool for understanding and managing cat populations in various settings. Whether you’re involved in animal welfare, urban planning, or
simply curious about your local cat population, the cats per square mile calculator provides a straightforward way to measure and analyze population density. Regular use of this tool can help in
making informed decisions that benefit both the cats and the communities they inhabit. | {"url":"https://calculatordoc.com/cats-per-square-mile-calculator/","timestamp":"2024-11-13T22:38:20Z","content_type":"text/html","content_length":"87965","record_id":"<urn:uuid:e5041aad-f435-46b9-9b90-60d5711e52ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00595.warc.gz"} |
K Nearest Neighbour For Supervised Learning
K-Nearest Neighbour (KNN) Algorithms is an easy-to-implement & advanced level supervised machine learning algorithm used for both - classification as well as regression problems. However, you can see
a wide of its applications in classification problems across various industries.
If you’ve been shopping a lot in e-commerce sites like Amazon, Flipkart, Myntra, or love watching web series over Netflix and Amazon Prime, one common thing you’ve always noticed, and that is
Are you wondering how they recommend you following your choice? They use KNN Supervised Learning to find out what you may need the next when you’re buying and recommend you with a few more products.
Imagine you’re looking for an iPhone to purchase. When you scroll down a little, you see some iPhone cases, tempered glasses - saying, “People who purchased an iPhone have also purchased these items.
The same applies to Netflix and Amazon Prime. When you finished a show or a series, they give you recommendations of the same genre. And do it all using KNN supervised learning and classify the items
for the best user experience.
Advantages Of KNN
● Quickest Calculation Time
● Simple Algorithms
● High Accuracy
● Versatile - best use for Regression and Classification.
● Doesn’t make any assumptions about data.
Where KNN Are Mostly Used
● Simple Recommendation Models
● Image Recognition Technology
● Decision-Making Models
● Calculating Credit Rating
Choosing The Right Value For K
To choose the right value of K, you have to run KNN algorithms several times with different values of K and select the value of K, which reduces the number of errors you’ve come across and come out
as the most stable value for K.
Your Step-By-Step Guide For Choosing The Value Of K
● As you decrease the value of K to 1 (K = 1), you’ll reach a query point, where you get to see many elements from class A (-) and class B (+) where (-) is the only nearest neighbor. Reasonably, you
would think about the query point to be most likely the red one. As K =1, which has a blue color, KNN incorrectly predicts the wrong color blue.
● As you increase the value of K to 2 (K=2), you get to see two elements, (-) and (+) are the only nearest neighbor. As you have two values, which are of Class A and Class B, KNN incorrectly predicts
the wrong values (Blue and Red).
● As you increase the value of K to 3 (K=3), you get to see three elements (-) and (+), (+) are the only nearest neighbor. And this time, you got three values, one from blue and two from red. As your
assumption is red, KNN correctly predicts the right value (Blue and Red, Red). Your answer is more stable this time compared to previous ones.
KNN works by finding the nearest distance between a query and all the elements in the database. By choosing the value for K, we get the closest to the query. And then, KNN algorithms look for the
most frequent labels in classification and averages of labels in regression.
Report Story | {"url":"https://www.52cs.com/archives/story/k-nearest-neighbour-for-supervised-learning","timestamp":"2024-11-01T23:56:04Z","content_type":"application/xhtml+xml","content_length":"54404","record_id":"<urn:uuid:15e0abbd-477a-4e2c-880e-f7e8e1537e95>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00054.warc.gz"} |
What's a Coefficient?
Polynomials are those expressions that have variables raised to all sorts of powers and multiplied by all types of numbers. When you work with polynomials you need to know a bit of vocabulary, and
one of the words you need to feel comfortable with is 'term'. So check out this tutorial, where you'll learn exactly what a 'term' in a polynomial is all about. | {"url":"https://virtualnerd.com/common-core/hsa-algebra/HSA-SSE-expressions-seeing-structure/A/1/1a/coefficient-definition","timestamp":"2024-11-10T01:59:09Z","content_type":"text/html","content_length":"33328","record_id":"<urn:uuid:8c53f1fa-0f6e-44e8-a194-6ef5d1e18a90>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00145.warc.gz"} |
Spherical Worlds
Recently I saw a description of spherical fractals in a blog post by Samuel Monnier.
These Julia-sets are constructed like ordinary Mandelbrots and Julias: first the argument is squared, but instead of adding a constant afterwards, a Möbius transformation is applied:
\(z = \frac{a z^2 + b}{c z^2 + d}\)
For the right choices of (complex) constants, plane-filling patterns appear.
There is an intimate connection between Möbius transformations and spherical geometry: if the plane is stereographically projected onto a sphere, a Möbius transformations corresponds to rotating and
moving the sphere, and then project stereographically back to the plane (this is nicely visualized in this video).
This connection can be visualized graphically: if the plane-filling patterns are stereographically projected onto a sphere, they fit naturally on it. There are no discontinuities or voids, and no
singularities near the poles.
Here I’ve used Fragmentarium to create some images of these plane-filling patterns, together with their stereographical projection onto a sphere. It was done by distance estimated ray marching, but
in this case we could have used ordinary ray tracing, and calculated the exact intersections.
The Fragmentarium script can be found here.
2 thoughts on “Spherical Worlds”
1. Wow! These are really beautiful. In particular the second one. It should be possible to convert the colors into a height field on the sphere and explore those planets ;o).
And thank you for the shader.
2. Thanks Knighty! Actually the second one is a heightmap (although the effect is subtle). I experimented with it, and found it somewhat difficult to get results (and it is much slower!). But it can
most likely be improved! | {"url":"http://blog.hvidtfeldts.net/index.php/2012/03/spherical-worlds/","timestamp":"2024-11-05T19:51:33Z","content_type":"text/html","content_length":"27306","record_id":"<urn:uuid:199ac8c3-0900-484f-ad0d-3e3ac530b713>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00097.warc.gz"} |
Energy Balances with Reaction: ConcepTest and Example Problem
Try to answer this ConcepTest and solve the example problem before using this module. Studies show that trying to answer the questions before studying material improves learning and retention. We
suggest that you write down the reasons for your answers. By the end of this module, you should be able to answer these on your own. Answers will be given at the end of this module.
100 mol/h of N[2] and 300 mol/h of H[2] are fed to an isothermal reactor at 350°C to carry out the following gas-phase reaction:
N[2] + 3H[2] → 2NH[3]
Assume the heat capacities for the components are the following:
C[p](N[2]) = C[p](H[2]) = 29 J/mol-K
C[p](NH[3]) = 36 J/mol-K
How much heat must be removed from the reactor if the conversion is 75%? The heat of formation of ammonia at 25°C is -46 kJ/mol. | {"url":"https://learncheme.com/quiz-yourself/interactive-self-study-modules/energy-balances-with-reaction/energy-balances-with-reaction-conceptest-and-example-problem/","timestamp":"2024-11-05T19:31:07Z","content_type":"text/html","content_length":"77016","record_id":"<urn:uuid:65b31c94-0aa6-4e18-b980-288c67a67f87>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00168.warc.gz"} |
Faster characteristic three polynomial multiplication and its application to NTRU Prime decapsulation
Efficient computation of polynomial multiplication over characteristic three fields is required for post-quantum cryptographic applications which gain importance upon the recent advances in quantum
computers. In this paper, we propose three new polynomial multiplication algorithms over F-3 and show that they are more efficient than the current state-of-the-art algorithms. We first examine
through the well-known multiplication algorithms in F-3[x] including the Karatsuba 2-way and 3-way split formulas along with the latest enhancements. Then, we propose a new 4-way split polynomial
multiplication algorithm and an improved version of it which are both derived by using interpolation in F-9, the finite field with nine elements. Moreover, we propose a 5-way split multiplication
algorithm and then compare the efficiencies of these algorithms altogether. Even though there exist 4-way or 5-way split multiplication algorithms in characteristic two (binary) fields, there has not
been any such algorithms developed for characteristic three fields before this paper. We apply the proposed algorithms to the NTRU Prime protocol, a post-quantum key encapsulation mechanism,
submitted to the MST PQC Competition by Bernstein et al., performing polynomial multiplication over characteristic three fields in its decapsulation phase. We observe that the new hybrid algorithms
provide a 12.9% reduction in the arithmetic complexity. Furthermore, we implement these new hybrid methods on Intel (R) Core (TM) i7-9750H architecture using C and obtain a 37.3% reduction in the
implementation cycle count.
Characteristic three fields
Key encapsulation
Lattice-based cryptography
NTRU Prime
Polynomial multiplication
Post-quantum cryptography
E. Yeniaras and M. Cenk, “Faster characteristic three polynomial multiplication and its application to NTRU Prime decapsulation,” JOURNAL OF CRYPTOGRAPHIC ENGINEERING, pp. 0–0, 2022, Accessed: 00,
2022. [Online]. Available: https://hdl.handle.net/11511/95275. | {"url":"https://open.metu.edu.tr/handle/11511/95275","timestamp":"2024-11-09T02:58:23Z","content_type":"application/xhtml+xml","content_length":"58843","record_id":"<urn:uuid:3a65a1fd-4d85-415f-b2e9-344ec1a580b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00235.warc.gz"} |
RFC 5990
<- RFC Index (5901..6000)
RFC 5990
Internet Engineering Task Force (IETF) J. Randall
Request for Comments: 5990 Randall Consulting
Category: Standards Track B. Kaliski
ISSN: 2070-1721 EMC
J. Brainard
S. Turner
September 2010
Use of the RSA-KEM Key Transport Algorithm
in the Cryptographic Message Syntax (CMS)
The RSA-KEM Key Transport Algorithm is a one-pass (store-and-forward)
mechanism for transporting keying data to a recipient using the
recipient's RSA public key. ("KEM" stands for "key encapsulation
mechanism".) This document specifies the conventions for using the
RSA-KEM Key Transport Algorithm with the Cryptographic Message Syntax
(CMS). The ASN.1 syntax is aligned with an expected forthcoming
change to American National Standard (ANS) X9.44.
Status of This Memo
This is an Internet Standards Track document.
This document is a product of the Internet Engineering Task Force
(IETF). It represents the consensus of the IETF community. It has
received public review and has been approved for publication by the
Internet Engineering Steering Group (IESG). Further information on
Internet Standards is available in Section 2 of RFC 5741.
Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
Randall, et al. Standards Track [Page 1]
RFC 5990 Use of RSA-KEM in CMS September 2010
Copyright Notice
Copyright (c) 2010 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
(http://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents
carefully, as they describe your rights and restrictions with respect
to this document. Code Components extracted from this document must
include Simplified BSD License text as described in Section 4.e of
the Trust Legal Provisions and are provided without warranty as
described in the Simplified BSD License.
Table of Contents
1. Introduction ....................................................3
1.1. Conventions Used in This Document ..........................4
2. Use in CMS ......................................................4
2.1. Underlying Components ......................................4
2.2. RecipientInfo Conventions ..................................5
2.3. Certificate Conventions ....................................5
2.4. SMIMECapabilities Attribute Conventions ....................6
3. Security Considerations .........................................7
4. IANA Considerations .............................................9
5. Acknowledgements ................................................9
6. References .....................................................10
6.1. Normative References ......................................10
6.2. Informative References ....................................11
Appendix A. RSA-KEM Key Transport Algorithm ......................12
A.1. Underlying Components ....................................12
A.2. Sender's Operations ......................................12
A.3. Recipient's Operations ...................................13
Appendix B. ASN.1 Syntax .........................................15
B.1. RSA-KEM Key Transport Algorithm ..........................16
B.2. Selected Underlying Components ...........................18
B.2.1. Key Derivation Functions ............................18
B.2.2. Symmetric Key-Wrapping Schemes ......................19
B.3. ASN.1 Module .............................................20
B.4. Examples .................................................25
Randall, et al. Standards Track [Page 2]
RFC 5990 Use of RSA-KEM in CMS September 2010
1. Introduction
The RSA-KEM Key Transport Algorithm is a one-pass (store-and-forward)
mechanism for transporting keying data to a recipient using the
recipient's RSA public key.
Most previous key transport algorithms based on the RSA public-key
cryptosystem (e.g., the popular PKCS #1 v1.5 algorithm [PKCS1]) have
the following general form:
1. Format or "pad" the keying data to obtain an integer m.
2. Encrypt the integer m with the recipient's RSA public key:
c = m^e mod n
3. Output c as the encrypted keying data.
The RSA-KEM Key Transport Algorithm takes a different approach that
provides higher security assurance, by encrypting a _random_ integer
with the recipient's public key, and using a symmetric key-wrapping
scheme to encrypt the keying data. It has the following form:
1. Generate a random integer z between 0 and n-1.
2. Encrypt the integer z with the recipient's RSA public key:
c = z^e mod n
3. Derive a key-encrypting key KEK from the integer z.
4. Wrap the keying data using KEK to obtain wrapped keying data WK.
5. Output c and WK as the encrypted keying data.
This different approach provides higher security assurance because
(a) the input to the underlying RSA operation is effectively a random
integer between 0 and n-1, where n is the RSA modulus, so it does not
have any structure that could be exploited by an adversary, and
(b) the input is independent of the keying data so the result of the
RSA decryption operation is not directly available to an adversary.
As a result, the algorithm enjoys a "tight" security proof in the
random oracle model. (In other padding schemes, such as PKCS #1
v1.5, the input has structure and/or depends on the keying data, and
the provable security assurances are not as strong.) The approach is
also architecturally convenient because the public-key operations are
Randall, et al. Standards Track [Page 3]
RFC 5990 Use of RSA-KEM in CMS September 2010
separate from the symmetric operations on the keying data. Another
benefit is that the length of the keying data is bounded only by the
symmetric key-wrapping scheme, not the size of the RSA modulus.
The RSA-KEM Key Transport Algorithm in various forms is being adopted
in several draft standards as well as in American National Standard
(ANS) X9.44 [ANS-X9.44]. It has also been recommended by the New
European Schemes for Signatures, Integrity, and Encryption (NESSIE)
project [NESSIE]. Originally, [ANS-X9.44] specified a different
object identifier to identify the RSA-KEM Key Transport Algorithm.
[ANS-X9.44] used id-ac-generic-hybrid, while this document uses
id-rsa-kem. These OIDs are used in the KeyTransportInfo field to
indicate the key encryption algorithm, in certificates to allow
recipients to restrict their public keys for use with RSA-KEM only,
and in SMIME Capability attributes to allow recipients to advertise
their support for RSA-KEM. Legacy implementations that wish to
interoperate with [ANS-X9.44] should consult that specification for
more information on id-ac-generic-hybrid.
For completeness, a specification of the algorithm is given in
Appendix A of this document; ASN.1 syntax is given in Appendix B.
NOTE: The term "KEM" stands for "key encapsulation mechanism" and
refers to the first three steps of the process above. The
formalization of key transport algorithms (or more generally,
asymmetric encryption schemes) in terms of key encapsulation
mechanisms is described further in research by Victor Shoup
leading to the development of the ISO/IEC 18033-2 standard
1.1. Conventions Used in This Document
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in RFC 2119 [STDWORDS].
2. Use in CMS
The RSA-KEM Key Transport Algorithm MAY be employed for one or more
recipients in the CMS enveloped-data content type (Section 6 of
[CMS]), where the keying data processed by the algorithm is the CMS
content-encryption key.
2.1. Underlying Components
A CMS implementation that supports the RSA-KEM Key Transport
Algorithm MUST support at least the following underlying components:
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o For the key derivation function, KDF3 (see [ANS-X9.44]) based on
SHA-256 (see [FIPS-180-3]). KDF3 is an instantiation of the
Concatenation Key Derivation Function defined in [NIST-SP800-56A].
o For the key-wrapping scheme, AES-Wrap-128, i.e., the AES Key Wrap
with a 128-bit key-encrypting key (see [AES-WRAP]).
An implementation SHOULD also support KDF2 (see [ANS-X9.44]) based on
SHA-1 (this function is also specified as the key derivation function
in [ANS-X9.63]). The Camellia key wrap algorithm (see [CAMELLIA])
SHOULD be supported if Camellia is supported as a content-encryption
cipher. The Triple-DES Key Wrap (see [3DES-WRAP]) SHOULD also be
supported if Triple-DES is supported as a content-encryption cipher.
It MAY support other underlying components. When AES or Camellia is
used, the data block size is 128 bits and the key size can be 128,
192, or 256 bits, while Triple-DES requires a data block size of
64 bits and a key size of 112 or 168 bits.
2.2. RecipientInfo Conventions
When the RSA-KEM Key Transport Algorithm is employed for a recipient,
the RecipientInfo alternative for that recipient MUST be
KeyTransRecipientInfo. The algorithm-specific fields of the
KeyTransRecipientInfo value MUST have the following values:
o keyEncryptionAlgorithm.algorithm MUST be id-rsa-kem (see
Appendix B);
o keyEncryptionAlgorithm.parameters MUST be a value of type
GenericHybridParameters, identifying the RSA-KEM key encapsulation
mechanism (see Appendix B);
o encryptedKey MUST be the encrypted keying data output by the
algorithm, where the keying data is the content-encryption key
(see Appendix A).
2.3. Certificate Conventions
The conventions specified in this section augment RFC 5280 [PROFILE].
A recipient who employs the RSA-KEM Key Transport Algorithm MAY
identify the public key in a certificate by the same
AlgorithmIdentifier as for the PKCS #1 v1.5 algorithm, i.e., using
the rsaEncryption object identifier [PKCS1]. The fact that the user
will accept RSA-KEM with this public key is not indicated by the use
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of this identifier. This MAY be signaled by the use of the
appropriate SMIME Capabilities either in a message or in the
If the recipient wishes only to employ the RSA-KEM Key Transport
Algorithm with a given public key, the recipient MUST identify the
public key in the certificate using the id-rsa-kem object identifier
(see Appendix B). When the id-rsa-kem algorithm identifier appears
in the SubjectPublicKeyInfo algorithm field, the encoding SHALL omit
the parameters field from AlgorithmIdentifier. That is, the
AlgorithmIdentifier SHALL be a SEQUENCE of one component, the object
identifier id-rsa-kem.
Regardless of the AlgorithmIdentifier used, the RSA public key is
encoded in the same manner in the subject public key information.
The RSA public key MUST be encoded using the type RSAPublicKey type:
RSAPublicKey ::= SEQUENCE {
modulus INTEGER, -- n
publicExponent INTEGER -- e
Here, the modulus is the modulus n, and publicExponent is the public
exponent e. The Distinguished Encoding Rules (DER)-encoded
RSAPublicKey is carried in the subjectPublicKey BIT STRING within the
subject public key information.
The intended application for the key MAY be indicated in the key
usage certificate extension (see [PROFILE], Section 4.2.1.3). If the
keyUsage extension is present in a certificate that conveys an RSA
public key with the id-rsa-kem object identifier as discussed above,
then the key usage extension MUST contain the following value:
dataEncipherment SHOULD NOT be present. That is, a key intended to
be employed only with the RSA-KEM Key Transport Algorithm SHOULD NOT
also be employed for data encryption or for authentication such as in
signatures. Good cryptographic practice employs a given RSA key pair
in only one scheme. This practice avoids the risk that vulnerability
in one scheme may compromise the security of the other, and may be
essential to maintain provable security.
2.4. SMIMECapabilities Attribute Conventions
RFC 3851 [MSG], Section 2.5.2 defines the SMIMECapabilities signed
attribute (defined as a SEQUENCE of SMIMECapability SEQUENCEs) to be
used to specify a partial list of algorithms that the software
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announcing the SMIMECapabilities can support. When constructing a
signedData object, compliant software MAY include the
SMIMECapabilities signed attribute announcing that it supports the
RSA-KEM Key Transport Algorithm.
The SMIMECapability SEQUENCE representing the RSA-KEM Key Transport
Algorithm MUST include the id-rsa-kem object identifier (see
Appendix B) in the capabilityID field and MUST include a
GenericHybridParameters value in the parameters field identifying the
components with which the algorithm is to be employed.
The DER encoding of a SMIMECapability SEQUENCE is the same as the DER
encoding of an AlgorithmIdentifier. Example DER encodings for
typical sets of components are given in Appendix B.4.
3. Security Considerations
The RSA-KEM Key Transport Algorithm should be considered for new CMS-
based applications as a replacement for the widely implemented RSA
encryption algorithm specified originally in PKCS #1 v1.5 (see
[PKCS1] and Section 4.2.1 of [CMSALGS]), which is vulnerable to
chosen-ciphertext attacks. The RSA Encryption Scheme - Optimal
Asymmetric Encryption Padding (RSAES-OAEP) Key Transport Algorithm
has also been proposed as a replacement (see [PKCS1] and [CMS-OAEP]).
RSA-KEM has the advantage over RSAES-OAEP of a tighter security
proof, but the disadvantage of slightly longer encrypted keying data.
The security of the RSA-KEM Key Transport Algorithm described in this
document can be shown to be tightly related to the difficulty of
either solving the RSA problem or breaking the underlying symmetric
key-wrapping scheme, if the underlying key derivation function is
modeled as a random oracle, and assuming that the symmetric key-
wrapping scheme satisfies the properties of a data encapsulation
mechanism [SHOUP]. While in practice a random-oracle result does not
provide an actual security proof for any particular key derivation
function, the result does provide assurance that the general
construction is reasonable; a key derivation function would need to
be particularly weak to lead to an attack that is not possible in the
random oracle model.
The RSA key size and the underlying components should be selected
consistent with the desired symmetric security level for an
application. Several security levels have been identified in the
NIST FIPS PUB 800-57 [NIST-GUIDELINE]. For brevity, the first three
levels are mentioned here:
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o 80-bit security. The RSA key size SHOULD be at least 1024 bits,
the hash function underlying the KDF SHOULD be SHA-1 or above, and
the symmetric key-wrapping scheme SHOULD be AES Key Wrap, Triple-
DES Key Wrap, or Camellia Key Wrap.
o 112-bit security. The RSA key size SHOULD be at least 2048 bits,
the hash function underlying the KDF SHOULD be SHA-224 or above,
and the symmetric key-wrapping scheme SHOULD be AES Key Wrap,
Triple-DES Key Wrap, or Camellia Key Wrap.
o 128-bit security. The RSA key size SHOULD be at least 3072 bits,
the hash function underlying the KDF SHOULD be SHA-256 or above,
and the symmetric key-wrapping scheme SHOULD be AES Key Wrap or
Camellia Key Wrap.
Note that the AES Key Wrap or Camellia Key Wrap MAY be used at all
three of these levels; the use of AES or Camellia does not require a
128-bit security level for other components.
Implementations MUST protect the RSA private key and the content-
encryption key. Compromise of the RSA private key may result in the
disclosure of all messages protected with that key. Compromise of
the content-encryption key may result in disclosure of the associated
encrypted content.
Additional considerations related to key management may be found in
The security of the algorithm also depends on the strength of the
random number generator, which SHOULD have a comparable security
level. For further discussion on random number generation, please
see [RANDOM].
Implementations SHOULD NOT reveal information about intermediate
values or calculations, whether by timing or other "side channels",
or otherwise an opponent may be able to determine information about
the keying data and/or the recipient's private key. Although not all
intermediate information may be useful to an opponent, it is
preferable to conceal as much information as is practical, unless
analysis specifically indicates that the information would not be
Generally, good cryptographic practice employs a given RSA key pair
in only one scheme. This practice avoids the risk that vulnerability
in one scheme may compromise the security of the other, and may be
essential to maintain provable security. While RSA public keys have
often been employed for multiple purposes such as key transport and
digital signature without any known bad interactions, for increased
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security assurance, such combined use of an RSA key pair is NOT
RECOMMENDED in the future (unless the different schemes are
specifically designed to be used together).
Accordingly, an RSA key pair used for the RSA-KEM Key Transport
Algorithm SHOULD NOT also be used for digital signatures. (Indeed,
the Accredited Standards Committee X9 (ASC X9) requires such a
separation between key establishment key pairs and digital signature
key pairs.) Continuing this principle of key separation, a key pair
used for the RSA-KEM Key Transport Algorithm SHOULD NOT be used with
other key establishment schemes, or for data encryption, or with more
than one set of underlying algorithm components.
Parties MAY formalize the assurance that one another's
implementations are correct through implementation validation, e.g.,
NIST's Cryptographic Module Validation Program (CMVP).
4. IANA Considerations
Within the CMS, algorithms are identified by object identifiers
(OIDs). With one exception, all of the OIDs used in this document
were assigned in other IETF documents, in ISO/IEC standards
documents, by the National Institute of Standards and Technology
(NIST), and in Public-Key Cryptography Standards (PKCS) documents.
The two exceptions are the ASN.1 module's identifier (see Appendix
B.3) and id-rsa-kem that are both assigned in this document. The
module object identifiers are defined in an arc delegated by the
former company RSA Data Security Inc. to the S/MIME Working Group.
When the S/MIME Working Group closes, this arc and its registration
procedures will be transferred to IANA.
5. Acknowledgements
This document is one part of a strategy to align algorithm standards
produced by ASC X9, ISO/IEC JTC1 SC27, NIST, and the IETF. We would
like to thank the members of the ASC X9F1 working group for their
contributions to drafts of ANS X9.44, which led to this
Our thanks to Russ Housley as well for his guidance and
encouragement. We also appreciate the helpful direction we've
received from Blake Ramsdell and Jim Schaad in bringing this document
to fruition. A special thanks to Magnus Nystrom for his assistance
on Appendix B. Thanks also to Bob Griffin and John Linn for both
editorial direction and procedural guidance.
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6. References
6.1. Normative References
[3DES-WRAP] Housley, R., "Triple-DES and RC2 Key Wrapping",
RFC 3217, December 2001.
[AES-WRAP] Schaad, J. and R. Housley, "Advanced Encryption
Standard (AES) Key Wrap Algorithm", RFC 3394,
September 2002.
[ANS-X9.44] ASC X9F1 Working Group. American National Standard
X9.44: Public Key Cryptography for the Financial
Services Industry -- Key Establishment Using
Integer Factorization Cryptography. 2007.
[ANS-X9.63] American National Standard X9.63-2002: Public Key
Cryptography for the Financial Services Industry:
Key Agreement and Key Transport Using Elliptic
Curve Cryptography.
[CAMELLIA] Moriai, S. and A. Kato, "Use of the Camellia
Encryption Algorithm in Cryptographic Message
Syntax (CMS)", RFC 3657, January 2004.
[CMS] Housley, R., "Cryptographic Message Syntax (CMS)",
RFC 5652, September 2009.
[CMSALGS] Housley, R., "Cryptographic Message Syntax (CMS)
Algorithms", RFC 3370, August 2002.
[FIPS-180-3] National Institute of Standards and Technology
(NIST). FIPS 180-3: Secure Hash Standard. October
[MSG] Ramsdell, B. and S. Turner, "Secure/Multipurpose
Internet Mail Extensions (S/MIME) Version 3.2
Message Specification", RFC 5751, January 2010.
[PROFILE] Cooper, D., Santesson, S., Farrell, S., Boeyen, S.,
Housley, R., and W. Polk, "Internet X.509 Public
Key Infrastructure Certificate and Certificate
Revocation List (CRL) Profile", RFC 5280, May 2008.
[STDWORDS] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
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6.2. Informative References
[AES-WRAP-PAD] Housley, R. and M. Dworkin, "Advanced Encryption
Standard (AES) Key Wrap with Padding Algorithm",
RFC 5649, September 2009.
[CMS-OAEP] Housley, R., "Use of the RSAES-OAEP Key Transport
Algorithm in Cryptographic Message Syntax (CMS)",
RFC 3560, July 2003.
[NESSIE] NESSIE Consortium. Portfolio of Recommended
Cryptographic Primitives. February 2003.
[NIST-GUIDELINE] National Institute of Standards and Technology.
Special Publication 800-57: Recommendation for Key
Management - Part 1: General (Revised). March
[NIST-SP800-56A] National Institute of Standards and Technology.
Special Publication 800-56A: Recommendation for
Pair-Wise Key Establishment Schemes Using Discrete
Logarithm Cryptography (Revised). March 2007.
[PKCS1] Jonsson, J. and B. Kaliski, "Public-Key
Cryptography Standards (PKCS) #1: RSA Cryptography
Specifications Version 2.1", RFC 3447, February
[RANDOM] Eastlake 3rd, D., Schiller, J., and S. Crocker,
"Randomness Requirements for Security", BCP 106,
RFC 4086, June 2005.
[SHOUP] Shoup, V. A Proposal for an ISO Standard for
Public Key Encryption. Version 2.1, December 20,
2001. http://eprint.iacr.org/2001/112.
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Appendix A. RSA-KEM Key Transport Algorithm
The RSA-KEM Key Transport Algorithm is a one-pass (store-and-forward)
mechanism for transporting keying data to a recipient using the
recipient's RSA public key.
With this type of algorithm, a sender encrypts the keying data using
the recipient's public key to obtain encrypted keying data. The
recipient decrypts the encrypted keying data using the recipient's
private key to recover the keying data.
A.1. Underlying Components
The algorithm has the following underlying components:
o KDF, a key derivation function, which derives keying data of a
specified length from a shared secret value;
o Wrap, a symmetric key-wrapping scheme, which encrypts keying Data
using a key-encrypting key.
In the following, kekLen denotes the length in bytes of the key-
encrypting key for the underlying symmetric key-wrapping scheme.
In this scheme, the length of the keying data to be transported MUST
be among the lengths supported by the underlying symmetric key-
wrapping scheme. (Both the AES and Camellia Key Wraps, for instance,
require the length of the keying data to be a multiple of 8 bytes,
and at least 16 bytes.) Usage and formatting of the keying data
(e.g., parity adjustment for Triple-DES keys) is outside the scope of
this algorithm. With some key derivation functions, it is possible
to include other information besides the shared secret value in the
input to the function. Also, with some symmetric key-wrapping
schemes, it is possible to associate a label with the keying data.
Such uses are outside the scope of this document, as they are not
directly supported by CMS.
A.2. Sender's Operations
Let (n,e) be the recipient's RSA public key (see [PKCS1] for
details), and let K be the keying data to be transported.
Let nLen denote the length in bytes of the modulus n, i.e., the least
integer such that 2^{8*nLen} > n.
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The sender performs the following operations:
1. Generate a random integer z between 0 and n-1 (see note), and
convert z to a byte string Z of length nLen, most significant byte
z = RandomInteger (0, n-1)
Z = IntegerToString (z, nLen)
2. Encrypt the random integer z using the recipient's public key
(n,e), and convert the resulting integer c to a ciphertext C, a
byte string of length nLen:
c = z^e mod n
C = IntegerToString (c, nLen)
3. Derive a key-encrypting key KEK of length kekLen bytes from the
byte string Z using the underlying key derivation function:
KEK = KDF (Z, kekLen)
4. Wrap the keying data K with the key-encrypting key KEK using the
underlying key-wrapping scheme to obtain wrapped keying data WK:
WK = Wrap (KEK, K)
5. Concatenate the ciphertext C and the wrapped keying data WK to
obtain the encrypted keying data EK:
EK = C || WK
6. Output the encrypted keying data EK.
NOTE: The random integer z MUST be generated independently at random
for different encryption operations, whether for the same or
different recipients.
A.3. Recipient's Operations
Let (n,d) be the recipient's RSA private key (see [PKCS1]; other
private key formats are allowed), and let EK be the encrypted keying
Let nLen denote the length in bytes of the modulus n.
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The recipient performs the following operations:
1. Separate the encrypted keying data EK into a ciphertext C of
length nLen bytes and wrapped keying data WK:
C || WK = EK
If the length of the encrypted keying data is less than nLen
bytes, output "decryption error", and stop.
2. Convert the ciphertext C to an integer c, most significant byte
first. Decrypt the integer c using the recipient's private key
(n,d) to recover an integer z (see note):
c = StringToInteger (C)
z = c^d mod n
If the integer c is not between 0 and n-1, output "decryption
error", and stop.
3. Convert the integer z to a byte string Z of length nLen, most
significant byte first (see note):
Z = IntegerToString (z, nLen)
4. Derive a key-encrypting key KEK of length kekLen bytes from the
byte string Z using the underlying key derivation function (see
KEK = KDF (Z, kekLen)
5. Unwrap the wrapped keying data WK with the key-encrypting key KEK
using the underlying key-wrapping scheme to recover the keying
data K:
K = Unwrap (KEK, WK)
If the unwrapping operation outputs an error, output "decryption
error", and stop.
6. Output the keying data K.
NOTE: Implementations SHOULD NOT reveal information about the
integer z and the string Z, nor about the calculation of the
exponentiation in Step 2, the conversion in Step 3, or the key
derivation in Step 4, whether by timing or other "side channels".
The observable behavior of the implementation SHOULD be the same at
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these steps for all ciphertexts C that are in range. (For example,
IntegerToString conversion should take the same amount of time
regardless of the actual value of the integer z.) The integer z, the
string Z, and other intermediate results MUST be securely deleted
when they are no longer needed.
Appendix B. ASN.1 Syntax
The ASN.1 syntax for identifying the RSA-KEM Key Transport Algorithm
is an extension of the syntax for the "generic hybrid cipher" in
ANS X9.44 [ANS-X9.44]. The syntax for the scheme is given in
Appendix B.1. The syntax for selected underlying components
including those mentioned above is given in Appendix B.2.
The following object identifier prefixes are used in the definitions
is18033-2 OID ::= { iso(1) standard(0) is18033(18033) part2(2) }
nistAlgorithm OID ::= {
joint-iso-itu-t(2) country(16) us(840) organization(1)
gov(101) csor(3) nistAlgorithm(4)
pkcs-1 OID ::= {
iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) pkcs-1(1)
x9-44 OID ::= { iso(1) identified-organization(3) tc68(133)
country(16) x9(840) x9Standards(9) x9-44(44) }
x9-44-components OID ::= { x9-44 components(1) }
NullParms is a more descriptive synonym for NULL when an algorithm
identifier has null parameters:
NullParms ::= NULL
The material in this Appendix is based on ANS X9.44.
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B.1. RSA-KEM Key Transport Algorithm
The object identifier for the RSA-KEM Key Transport Algorithm is
id-rsa-kem, which is defined in this document as:
id-rsa-kem OID ::= {
iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1)
pkcs-9(9) smime(16) alg(3) 14
When id-rsa-kem is used in an AlgorithmIdentifier, the parameters
MUST employ the GenericHybridParameters syntax. The parameters MUST
be absent when used in the SubjectPublicKeyInfo field. The syntax
for GenericHybridParameters is as follows:
GenericHybridParameters ::= {
kem KeyEncapsulationMechanism,
dem DataEncapsulationMechanism
The fields of type GenericHybridParameters have the following
o kem identifies the underlying key encapsulation mechanism,
which in this case is also denoted as RSA-KEM.
The object identifier for RSA-KEM (as a key encapsulation
mechanism) is id-kem-rsa as:
id-kem-rsa OID ::= {
is18033-2 key-encapsulation-mechanism(2) rsa(4)
The associated parameters for id-kem-rsa have type
RsaKemParameters ::= {
keyDerivationFunction KeyDerivationFunction,
keyLength KeyLength
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The fields of type RsaKemParameters have the following
* keyDerivationFunction identifies the underlying key
derivation function. For alignment with ANS X9.44, it MUST
be KDF2 or KDF3. However, other key derivation functions
MAY be used with CMS. Please see Appendix B.2.1 for the
syntax for KDF2 and KDF3.
KeyDerivationFunction ::=
AlgorithmIdentifier {{KDFAlgorithms}}
KDFAlgorithms ALGORITHM ::= {
kdf2 | kdf3,
... -- implementations may define other methods
* keyLength is the length in bytes of the key-encrypting key,
which depends on the underlying symmetric key-wrapping
KeyLength ::= INTEGER (1..MAX)
o dem identifies the underlying data encapsulation mechanism.
For alignment with ANS X9.44, it MUST be an X9-approved
symmetric key-wrapping scheme. However, other symmetric key-
wrapping schemes MAY be used with CMS. Please see Appendix
B.2.2 for the syntax for the AES, Triple-DES, and Camellia Key
DataEncapsulationMechanism ::=
AlgorithmIdentifier {{DEMAlgorithms}}
DEMAlgorithms ALGORITHM ::= {
... -- implementations may define other methods
X9-SymmetricKeyWrappingSchemes ALGORITHM ::= {
aes128-Wrap | aes192-Wrap | aes256-Wrap | tdes-Wrap,
... -- allows for future expansion
Camellia-KeyWrappingSchemes ALGORITHM ::= {
Camellia128-Wrap | Camellia192-Wrap | Camellia256-Wrap
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B.2. Selected Underlying Components
B.2.1. Key Derivation Functions
The object identifier for KDF2 (see [ANS-X9.44]) is:
id-kdf-kdf2 OID ::= { x9-44-components kdf2(1) }
The associated parameters identify the underlying hash function. For
alignment with ANS X9.44, the hash function MUST be an ASC
X9-approved hash function. However, other hash functions MAY be used
with CMS.
kdf2 ALGORITHM ::= { OID id-kdf-kdf2 PARMS KDF2-HashFunction }
KDF2-HashFunction ::= AlgorithmIdentifier {{KDF2-HashFunctions}}
KDF2-HashFunctions ALGORITHM ::= {
... -- implementations may define other methods
X9-HashFunctions ALGORITHM ::= {
sha1 | sha224 | sha256 | sha384 | sha512,
... -- allows for future expansion
The object identifier for SHA-1 is:
id-sha1 OID ::= {
iso(1) identified-organization(3) oiw(14) secsig(3)
algorithms(2) sha1(26)
The object identifiers for SHA-224, SHA-256, SHA-384, and SHA-512 are
id-sha224 OID ::= { nistAlgorithm hashAlgs(2) sha224(4) }
id-sha256 OID ::= { nistAlgorithm hashAlgs(2) sha256(1) }
id-sha384 OID ::= { nistAlgorithm hashAlgs(2) sha384(2) }
id-sha512 OID ::= { nistAlgorithm hashAlgs(2) sha512(3) }
There has been some confusion over whether the various SHA object
identifiers have a NULL parameter, or no associated parameters. As
also discussed in [PKCS1], implementations SHOULD generate algorithm
identifiers without parameters and MUST accept algorithm identifiers
either without parameters, or with NULL parameters.
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sha1 ALGORITHM ::= { OID id-sha1 } -- NULLParms MUST be
sha224 ALGORITHM ::= { OID id-sha224 } -- accepted for these
sha256 ALGORITHM ::= { OID id-sha256 } -- OIDs
sha384 ALGORITHM ::= { OID id-sha384 } -- ""
sha512 ALGORITHM ::= { OID id-sha512 } -- ""
The object identifier for KDF3 (see [ANS-X9.44]) is:
id-kdf-kdf3 OID ::= { x9-44-components kdf3(2) }
The associated parameters identify the underlying hash function. For
alignment with the draft ANS X9.44, the hash function MUST be an ASC
X9-approved hash function. However, other hash functions MAY be used
with CMS.
kdf3 ALGORITHM ::= { OID id-kdf-kdf3 PARMS KDF3-HashFunction }
KDF3-HashFunction ::= AlgorithmIdentifier { KDF3-HashFunctions }
KDF3-HashFunctions ALGORITHM ::= {
... -- implementations may define other methods
B.2.2. Symmetric Key-Wrapping Schemes
The object identifiers for the AES Key Wrap depend on the size of the
key-encrypting key. There are three object identifiers (see
id-aes128-Wrap OID ::= { nistAlgorithm aes(1) aes128-Wrap(5) }
id-aes192-Wrap OID ::= { nistAlgorithm aes(1) aes192-Wrap(25) }
id-aes256-Wrap OID ::= { nistAlgorithm aes(1) aes256-Wrap(45) }
These object identifiers have no associated parameters.
aes128-Wrap ALGORITHM ::= { OID id-aes128-Wrap }
aes192-Wrap ALGORITHM ::= { OID id-aes192-Wrap }
aes256-Wrap ALGORITHM ::= { OID id-aes256-Wrap }
The object identifier for the Triple-DES Key Wrap (see
[3DES-WRAP]) is:
id-alg-CMS3DESwrap OBJECT IDENTIFIER ::= {
iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) pkcs-9(9)
smime(16) alg(3) 6
Randall, et al. Standards Track [Page 19]
RFC 5990 Use of RSA-KEM in CMS September 2010
This object identifier has a NULL parameter.
tdes-Wrap ALGORITHM ::=
{ OID id-alg-CMS3DESwrap PARMS NullParms }
NOTE: ASC X9 has not yet incorporated AES Key Wrap with Padding
[AES-WRAP-PAD] into ANS X9.44. When ASC X9.44 adds AES Key Wrap with
Padding, this document will also be updated.
The object identifiers for the Camellia Key Wrap depend on the size
of the key-encrypting key. There are three object identifiers:
id-camellia128-Wrap OBJECT IDENTIFIER ::=
{ iso(1) member-body(2) 392 200011 61 security(1)
algorithm(1) key-wrap-algorithm(3)
camellia128-wrap(2) }
id-camellia192-Wrap OBJECT IDENTIFIER ::=
{ iso(1) member-body(2) 392 200011 61 security(1)
algorithm(1) key-wrap-algorithm(3)
camellia192-wrap(3) }
id-camellia256-Wrap OBJECT IDENTIFIER ::=
{ iso(1) member-body(2) 392 200011 61 security(1)
algorithm(1) key-wrap-algorithm(3)
camellia256-wrap(4) }
These object identifiers have no associated parameters.
camellia128-Wrap ALGORITHM ::= { OID id-camellia128-Wrap }
camellia192-Wrap ALGORITHM ::= { OID id-camellia192-Wrap }
camellia256-Wrap ALGORITHM ::= { OID id-camellia256-Wrap }
B.3. ASN.1 Module
{ iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1)
pkcs-9(9) smime(16) modules(0) cms-rsa-kem(21) }
DEFINITIONS ::=
-- EXPORTS ALL
-- IMPORTS None
-- Useful types and definitions
Randall, et al. Standards Track [Page 20]
RFC 5990 Use of RSA-KEM in CMS September 2010
OID ::= OBJECT IDENTIFIER -- alias
-- Unless otherwise stated, if an object identifier has associated
-- parameters (i.e., the PARMS element is specified), the
-- parameters field shall be included in algorithm identifier
-- values. The parameters field shall be omitted if and only if
-- the object identifier does not have associated parameters
-- (i.e., the PARMS element is omitted), unless otherwise stated.
ALGORITHM ::= CLASS {
&id OBJECT IDENTIFIER UNIQUE,
&Type OPTIONAL
WITH SYNTAX { OID &id [PARMS &Type] }
AlgorithmIdentifier { ALGORITHM:IOSet } ::= SEQUENCE {
algorithm ALGORITHM.&id( {IOSet} ),
parameters ALGORITHM.&Type( {IOSet}{@algorithm} ) OPTIONAL
NullParms ::= NULL
-- ISO/IEC 18033-2 arc
is18033-2 OID ::= { iso(1) standard(0) is18033(18033) part2(2) }
-- NIST algorithm arc
nistAlgorithm OID ::= {
joint-iso-itu-t(2) country(16) us(840) organization(1)
gov(101) csor(3) nistAlgorithm(4)
-- PKCS #1 arc
pkcs-1 OID ::= {
iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) pkcs-1(1)
-- RSA-KEM Key Transport Algorithm
id-rsa-kem OID ::= {
iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1)
pkcs-9(9) smime(16) alg(3) 14
Randall, et al. Standards Track [Page 21]
RFC 5990 Use of RSA-KEM in CMS September 2010
GenericHybridParameters ::= SEQUENCE {
kem KeyEncapsulationMechanism,
dem DataEncapsulationMechanism
KeyEncapsulationMechanism ::= AlgorithmIdentifier {{KEMAlgorithms}}
KEMAlgorithms ALGORITHM ::= { kem-rsa, ... }
kem-rsa ALGORITHM ::= { OID id-kem-rsa PARMS RsaKemParameters }
id-kem-rsa OID ::= {
is18033-2 key-encapsulation-mechanism(2) rsa(4)
RsaKemParameters ::= SEQUENCE {
keyDerivationFunction KeyDerivationFunction,
keyLength KeyLength
KeyDerivationFunction ::= AlgorithmIdentifier {{KDFAlgorithms}}
KDFAlgorithms ALGORITHM ::= {
kdf2 | kdf3,
... -- implementations may define other methods
KeyLength ::= INTEGER (1..MAX)
DataEncapsulationMechanism ::= AlgorithmIdentifier {{DEMAlgorithms}}
DEMAlgorithms ALGORITHM ::= {
X9-SymmetricKeyWrappingSchemes |
... -- implementations may define other methods
X9-SymmetricKeyWrappingSchemes ALGORITHM ::= {
aes128-Wrap | aes192-Wrap | aes256-Wrap | tdes-Wrap,
... -- allows for future expansion
X9-SymmetricKeyWrappingScheme ::=
AlgorithmIdentifier {{ X9-SymmetricKeyWrappingSchemes }}
Randall, et al. Standards Track [Page 22]
RFC 5990 Use of RSA-KEM in CMS September 2010
Camellia-KeyWrappingSchemes ALGORITHM ::= {
camellia128-Wrap | camellia192-Wrap | camellia256-Wrap,
... -- allows for future expansion
Camellia-KeyWrappingScheme ::=
AlgorithmIdentifier {{ Camellia-KeyWrappingSchemes }}
-- Key Derivation Functions
id-kdf-kdf2 OID ::= { x9-44-components kdf2(1) }
-- Base arc
x9-44 OID ::= {
iso(1) identified-organization(3) tc68(133) country(16) x9(840)
x9Standards(9) x9-44(44)
x9-44-components OID ::= { x9-44 components(1) }
kdf2 ALGORITHM ::= { OID id-kdf-kdf2 PARMS KDF2-HashFunction }
KDF2-HashFunction ::= AlgorithmIdentifier {{ KDF2-HashFunctions }}
KDF2-HashFunctions ALGORITHM ::= {
... -- implementations may define other methods
id-kdf-kdf3 OID ::= { x9-44-components kdf3(2) }
kdf3 ALGORITHM ::= { OID id-kdf-kdf3 PARMS KDF3-HashFunction }
KDF3-HashFunction ::= AlgorithmIdentifier {{ KDF3-HashFunctions }}
KDF3-HashFunctions ALGORITHM ::= {
... -- implementations may define other methods
-- Hash Functions
X9-HashFunctions ALGORITHM ::= {
sha1 | sha224 | sha256 | sha384 | sha512,
... -- allows for future expansion
Randall, et al. Standards Track [Page 23]
RFC 5990 Use of RSA-KEM in CMS September 2010
id-sha1 OID ::= {
iso(1) identified-organization(3) oiw(14) secsig(3)
algorithms(2) sha1(26)
id-sha224 OID ::= { nistAlgorithm hashAlgs(2) sha224(4) }
id-sha256 OID ::= { nistAlgorithm hashAlgs(2) sha256(1) }
id-sha384 OID ::= { nistAlgorithm hashAlgs(2) sha384(2) }
id-sha512 OID ::= { nistAlgorithm hashAlgs(2) sha512(3) }
sha1 ALGORITHM ::= { OID id-sha1 } -- NullParms MUST be
sha224 ALGORITHM ::= { OID id-sha224 } -- accepted for these
sha256 ALGORITHM ::= { OID id-sha256 } -- OIDs
sha384 ALGORITHM ::= { OID id-sha384 } -- ""
sha512 ALGORITHM ::= { OID id-sha512 } -- ""
-- Symmetric Key-Wrapping Schemes
id-aes128-Wrap OID ::= { nistAlgorithm aes(1) aes128-Wrap(5) }
id-aes192-Wrap OID ::= { nistAlgorithm aes(1) aes192-Wrap(25) }
id-aes256-Wrap OID ::= { nistAlgorithm aes(1) aes256-Wrap(45) }
aes128-Wrap ALGORITHM ::= { OID id-aes128-Wrap }
aes192-Wrap ALGORITHM ::= { OID id-aes192-Wrap }
aes256-Wrap ALGORITHM ::= { OID id-aes256-Wrap }
id-alg-CMS3DESwrap OBJECT IDENTIFIER ::= {
iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) pkcs-9(9)
smime(16) alg(3) 6
tdes-Wrap ALGORITHM ::= { OID id-alg-CMS3DESwrap PARMS NullParms }
id-camellia128-Wrap OBJECT IDENTIFIER ::=
{ iso(1) member-body(2) 392 200011 61 security(1)
algorithm(1) key-wrap-algorithm(3)
camellia128-wrap(2) }
Randall, et al. Standards Track [Page 24]
RFC 5990 Use of RSA-KEM in CMS September 2010
id-camellia192-Wrap OBJECT IDENTIFIER ::=
{ iso(1) member-body(2) 392 200011 61 security(1)
algorithm(1) key-wrap-algorithm(3)
camellia192-wrap(3) }
id-camellia256-Wrap OBJECT IDENTIFIER ::=
{ iso(1) member-body(2) 392 200011 61 security(1)
algorithm(1) key-wrap-algorithm(3)
camellia256-wrap(4) }
camellia128-Wrap ALGORITHM ::= { OID id-camellia128-Wrap }
camellia192-Wrap ALGORITHM ::= { OID id-camellia192-Wrap }
camellia256-Wrap ALGORITHM ::= { OID id-camellia256-Wrap }
B.4. Examples
As an example, if the key derivation function is KDF3 based on
SHA-256 and the symmetric key-wrapping scheme is the AES Key Wrap
with a 128-bit KEK, the AlgorithmIdentifier for the RSA-KEM Key
Transport Algorithm will have the following value:
SEQUENCE {
id-rsa-kem, -- RSA-KEM cipher
SEQUENCE { -- GenericHybridParameters
SEQUENCE { -- key encapsulation mechanism
id-kem-rsa, -- RSA-KEM
SEQUENCE { -- RsaKemParameters
SEQUENCE { -- key derivation function
id-kdf-kdf3, -- KDF3
SEQUENCE { -- KDF3-HashFunction
id-sha256 -- SHA-256; no parameters (preferred)
16 -- KEK length in bytes
SEQUENCE { -- data encapsulation mechanism
id-aes128-Wrap -- AES-128 Wrap; no parameters
Randall, et al. Standards Track [Page 25]
RFC 5990 Use of RSA-KEM in CMS September 2010
This AlgorithmIdentifier value has the following DER encoding:
06 0b 2a 86 48 86 f7 0d 01 09 10 03 0e -- id-rsa-kem
06 07 28 81 8c 71 02 02 04 -- id-kem-rsa
30 1e
06 0a 2b 81 05 10 86 48 09 2c 01 02 -- id-kdf-kdf3
30 0b
06 09 60 86 48 01 65 03 04 02 01 -- id-sha256
02 01 10 -- 16 bytes
30 0b
06 09 60 86 48 01 65 03 04 01 05 -- id-aes128-Wrap
The DER encodings for other typical sets of underlying components are
as follows:
o KDF3 based on SHA-384, AES Key Wrap with a 192-bit KEK
30 47 06 0b 2a 86 48 86 f7 0d 01 09 10 03 0e 30
38 30 29 06 07 28 81 8c 71 02 02 04 30 1e 30 19
06 0a 2b 81 05 10 86 48 09 2c 01 02 30 0b 06 09
60 86 48 01 65 03 04 02 02 02 01 18 30 0b 06 09
o KDF3 based on SHA-512, AES Key Wrap with a 256-bit KEK
30 47 06 0b 2a 86 48 86 f7 0d 01 09 10 03 0e 30
38 30 29 06 07 28 81 8c 71 02 02 04 30 1e 30 19
06 0a 2b 81 05 10 86 48 09 2c 01 02 30 0b 06 09
60 86 48 01 65 03 04 02 03 02 01 20 30 0b 06 09
60 86 48 01 65 03 04 01 2d
o KDF2 based on SHA-1, Triple-DES Key Wrap with a 128-bit KEK (two-
key Triple-DES)
30 45 06 0b 2a 86 48 86 f7 0d 01 09 10 03 0e 30
36 30 25 06 07 28 81 8c 71 02 02 04 30 1a 30 15
06 0a 2b 81 05 10 86 48 09 2c 01 01 30 07 06 05
2b 0e 03 02 1a 02 01 10 30 0d 06 0b 2a 86 48 86
f7 0d 01 09 10 03 06
Randall, et al. Standards Track [Page 26]
RFC 5990 Use of RSA-KEM in CMS September 2010
Authors' Addresses
James Randall
Randall Consulting
55 Sandpiper Drive
Dover, NH 03820
EMail: jdrandall@comcast.net
Burt Kaliski
176 South Street
Hopkinton, MA 01748
EMail: burt.kaliski@emc.com
John Brainard
RSA, The Security Division of EMC
174 Middlesex Turnpike
Bedford, MA 01730
EMail: jbrainard@rsa.com
Sean Turner
IECA, Inc.
3057 Nutley Street, Suite 106
Fairfax, VA 22031
EMail: turners@ieca.com
Randall, et al. Standards Track [Page 27] | {"url":"https://www.armware.dk/RFC/rfc/rfc5990.html","timestamp":"2024-11-13T18:11:34Z","content_type":"text/html","content_length":"55580","record_id":"<urn:uuid:a5007449-8c06-463c-b45c-07c0dcea62d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00159.warc.gz"} |
Solving logarithmic equations step-by-step
Algebra Tutorials!
solving logarithmic equations step-by-step
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Graphs of Rational
Solve Two-Step Equations Author Message
Multiply, Dividing;
Exponents; Square Roots; shattolh Posted: Saturday 30th of Dec 21:38
and Solving Equations I am in a real situation . Somebody save me please. I face a lot of problems with decimals, x-intercept and evaluating formulas and
LinearEquations especially with solving logarithmic equations step-by-step. I need to show some fast change in my math. I read there are many Software
Solving a Quadratic Tools available online which helps you in algebra. I can shell out some money too for an effective and inexpensive tool which helps me with
Equation my studies. Any reference is greatly appreciated. Thanks.
Systems of Linear Registered:
Equations Introduction 18.03.2005
Equations and From:
Solving 2nd Degree
Review Solving Quadratic kfir Posted: Monday 01st of Jan 09:18
Equations Hello , Algebrator available at the site https://rational-equations.com/test-description-for-rational-ex.html can be of great aid to you. I
System of Equations am a math coach who give private math classes to students and I recommend Algebrator to my students since that aids them a lot when they
Solving Equations & sit to work on their math problems by themselves at home.
Linear Equations Registered:
Functions Zeros, and 07.05.2006
Applications From: egypt
Rational Expressions and
Linear equations in two
variables Vnode Posted: Wednesday 03rd of Jan 07:16
Lesson Plan for I allow my son to use that program Algebrator because I believe it can effectively help him in his math problems. It’s been a long time
Comparing and Ordering since they first used that program and it did not only help him short-term but I noticed it helped in improving his solving capabilities.
Rational Numbers The program helped him how to solve rather than helped them just to answer. It’s fantastic!
Solving Equations Registered:
Radicals and Rational 27.09.2001
Exponents From: Germany
Solving Linear Equations
Systems of Linear
Solving Exponential and Hemto Posted: Wednesday 03rd of Jan 15:40
Logarithmic Equations Great! I think that’s what I need . Can you tell me where to get it?
Solving Systems of
Linear Equations
QUADRATIC EQUATIONS Registered:
Solving Quadratic 21.05.2003
Equations From:
Quadratic and Rational
Applications of Systems
of Linear Equations in Hiinidam Posted: Friday 05th of Jan 12:41
Two Variables Algebrator is the program that I have used through several math classes - Intermediate algebra, Algebra 2 and Intermediate algebra. It is a
Systems of Linear truly a great piece of algebra software. I remember of going through difficulties with inequalities, least common denominator and
Equations logarithms. I would simply type in a problem homework, click on Solve – and step by step solution to my algebra homework. I highly
Test Description for recommend the program.
RATIONAL EX Registered:
Exponential and 06.07.2001
Logarithmic Equations From: Greeley, CO, US
Systems of Linear
Equations: Cramer's Rule
Introduction to Systems
of Linear Equations Flash Fnavfy Liom Posted: Sunday 07th of Jan 12:01
Literal Equations & You can get it at https://rational-equations.com/solving-exponential-and-logarithmic-equations.html. Please do post your experience here.
Formula It may help a lot of other beginners as well.
Equations and
Inequalities with
Absolute Value Registered:
Rational Expressions 15.12.2001
SOLVING LINEAR AND From:
Steepest Descent for
Solving Linear Equations
The Quadratic Equation
Linear equations in two
solving logarithmic equations step-by-step
Related topics:
Home algebra definitions | mathmatics formulas in pdf | adding mixed fractions on ti-83 plus | eight grade algebra worksheets | tricks for solving aptitude questions
Rational Expressions | how to solve difference quotients | ratio problem solver | change log base ti 83 | solving algebra homework step by step | free graphing solver for equations
Graphs of Rational
Solve Two-Step Equations Author Message
Multiply, Dividing;
Exponents; Square Roots; shattolh Posted: Saturday 30th of Dec 21:38
and Solving Equations I am in a real situation . Somebody save me please. I face a lot of problems with decimals, x-intercept and evaluating formulas and
LinearEquations especially with solving logarithmic equations step-by-step. I need to show some fast change in my math. I read there are many Software
Solving a Quadratic Tools available online which helps you in algebra. I can shell out some money too for an effective and inexpensive tool which helps me with
Equation my studies. Any reference is greatly appreciated. Thanks.
Systems of Linear Registered:
Equations Introduction 18.03.2005
Equations and From:
Solving 2nd Degree
Review Solving Quadratic kfir Posted: Monday 01st of Jan 09:18
Equations Hello , Algebrator available at the site https://rational-equations.com/test-description-for-rational-ex.html can be of great aid to you. I
System of Equations am a math coach who give private math classes to students and I recommend Algebrator to my students since that aids them a lot when they
Solving Equations & sit to work on their math problems by themselves at home.
Linear Equations Registered:
Functions Zeros, and 07.05.2006
Applications From: egypt
Rational Expressions and
Linear equations in two
variables Vnode Posted: Wednesday 03rd of Jan 07:16
Lesson Plan for I allow my son to use that program Algebrator because I believe it can effectively help him in his math problems. It’s been a long time
Comparing and Ordering since they first used that program and it did not only help him short-term but I noticed it helped in improving his solving capabilities.
Rational Numbers The program helped him how to solve rather than helped them just to answer. It’s fantastic!
Solving Equations Registered:
Radicals and Rational 27.09.2001
Exponents From: Germany
Solving Linear Equations
Systems of Linear
Solving Exponential and Hemto Posted: Wednesday 03rd of Jan 15:40
Logarithmic Equations Great! I think that’s what I need . Can you tell me where to get it?
Solving Systems of
Linear Equations
QUADRATIC EQUATIONS Registered:
Solving Quadratic 21.05.2003
Equations From:
Quadratic and Rational
Applications of Systems
of Linear Equations in Hiinidam Posted: Friday 05th of Jan 12:41
Two Variables Algebrator is the program that I have used through several math classes - Intermediate algebra, Algebra 2 and Intermediate algebra. It is a
Systems of Linear truly a great piece of algebra software. I remember of going through difficulties with inequalities, least common denominator and
Equations logarithms. I would simply type in a problem homework, click on Solve – and step by step solution to my algebra homework. I highly
Test Description for recommend the program.
RATIONAL EX Registered:
Exponential and 06.07.2001
Logarithmic Equations From: Greeley, CO, US
Systems of Linear
Equations: Cramer's Rule
Introduction to Systems
of Linear Equations Flash Fnavfy Liom Posted: Sunday 07th of Jan 12:01
Literal Equations & You can get it at https://rational-equations.com/solving-exponential-and-logarithmic-equations.html. Please do post your experience here.
Formula It may help a lot of other beginners as well.
Equations and
Inequalities with
Absolute Value Registered:
Rational Expressions 15.12.2001
SOLVING LINEAR AND From:
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
solving logarithmic equations step-by-step
Related topics:
algebra definitions | mathmatics formulas in pdf | adding mixed fractions on ti-83 plus | eight grade algebra worksheets | tricks for solving aptitude questions | how to solve difference
quotients | ratio problem solver | change log base ti 83 | solving algebra homework step by step | free graphing solver for equations
Author Message
shattolh Posted: Saturday 30th of Dec 21:38
I am in a real situation . Somebody save me please. I face a lot of problems with decimals, x-intercept and evaluating formulas and especially with solving logarithmic
equations step-by-step. I need to show some fast change in my math. I read there are many Software Tools available online which helps you in algebra. I can shell out
some money too for an effective and inexpensive tool which helps me with my studies. Any reference is greatly appreciated. Thanks.
kfir Posted: Monday 01st of Jan 09:18
Hello , Algebrator available at the site https://rational-equations.com/test-description-for-rational-ex.html can be of great aid to you. I am a math coach who give
private math classes to students and I recommend Algebrator to my students since that aids them a lot when they sit to work on their math problems by themselves at
From: egypt
Vnode Posted: Wednesday 03rd of Jan 07:16
I allow my son to use that program Algebrator because I believe it can effectively help him in his math problems. It’s been a long time since they first used that
program and it did not only help him short-term but I noticed it helped in improving his solving capabilities. The program helped him how to solve rather than helped
them just to answer. It’s fantastic!
From: Germany
Hemto Posted: Wednesday 03rd of Jan 15:40
Great! I think that’s what I need . Can you tell me where to get it?
Hiinidam Posted: Friday 05th of Jan 12:41
Algebrator is the program that I have used through several math classes - Intermediate algebra, Algebra 2 and Intermediate algebra. It is a truly a great piece of
algebra software. I remember of going through difficulties with inequalities, least common denominator and logarithms. I would simply type in a problem homework, click
on Solve – and step by step solution to my algebra homework. I highly recommend the program.
From: Greeley, CO, US
Flash Fnavfy Liom Posted: Sunday 07th of Jan 12:01
You can get it at https://rational-equations.com/solving-exponential-and-logarithmic-equations.html. Please do post your experience here. It may help a lot of other
beginners as well.
Author Message
shattolh Posted: Saturday 30th of Dec 21:38
I am in a real situation . Somebody save me please. I face a lot of problems with decimals, x-intercept and evaluating formulas and especially with solving logarithmic equations
step-by-step. I need to show some fast change in my math. I read there are many Software Tools available online which helps you in algebra. I can shell out some money too for an
effective and inexpensive tool which helps me with my studies. Any reference is greatly appreciated. Thanks.
kfir Posted: Monday 01st of Jan 09:18
Hello , Algebrator available at the site https://rational-equations.com/test-description-for-rational-ex.html can be of great aid to you. I am a math coach who give private math
classes to students and I recommend Algebrator to my students since that aids them a lot when they sit to work on their math problems by themselves at home.
From: egypt
Vnode Posted: Wednesday 03rd of Jan 07:16
I allow my son to use that program Algebrator because I believe it can effectively help him in his math problems. It’s been a long time since they first used that program and it
did not only help him short-term but I noticed it helped in improving his solving capabilities. The program helped him how to solve rather than helped them just to answer. It’s
From: Germany
Hemto Posted: Wednesday 03rd of Jan 15:40
Great! I think that’s what I need . Can you tell me where to get it?
Hiinidam Posted: Friday 05th of Jan 12:41
Algebrator is the program that I have used through several math classes - Intermediate algebra, Algebra 2 and Intermediate algebra. It is a truly a great piece of algebra
software. I remember of going through difficulties with inequalities, least common denominator and logarithms. I would simply type in a problem homework, click on Solve – and
step by step solution to my algebra homework. I highly recommend the program.
From: Greeley, CO, US
Flash Fnavfy Liom Posted: Sunday 07th of Jan 12:01
You can get it at https://rational-equations.com/solving-exponential-and-logarithmic-equations.html. Please do post your experience here. It may help a lot of other beginners as
Posted: Saturday 30th of Dec 21:38
I am in a real situation . Somebody save me please. I face a lot of problems with decimals, x-intercept and evaluating formulas and especially with solving logarithmic equations step-by-step. I need
to show some fast change in my math. I read there are many Software Tools available online which helps you in algebra. I can shell out some money too for an effective and inexpensive tool which helps
me with my studies. Any reference is greatly appreciated. Thanks.
Posted: Monday 01st of Jan 09:18
Hello , Algebrator available at the site https://rational-equations.com/test-description-for-rational-ex.html can be of great aid to you. I am a math coach who give private math classes to students
and I recommend Algebrator to my students since that aids them a lot when they sit to work on their math problems by themselves at home.
Posted: Wednesday 03rd of Jan 07:16
I allow my son to use that program Algebrator because I believe it can effectively help him in his math problems. It’s been a long time since they first used that program and it did not only help him
short-term but I noticed it helped in improving his solving capabilities. The program helped him how to solve rather than helped them just to answer. It’s fantastic!
Posted: Wednesday 03rd of Jan 15:40
Great! I think that’s what I need . Can you tell me where to get it?
Posted: Friday 05th of Jan 12:41
Algebrator is the program that I have used through several math classes - Intermediate algebra, Algebra 2 and Intermediate algebra. It is a truly a great piece of algebra software. I remember of
going through difficulties with inequalities, least common denominator and logarithms. I would simply type in a problem homework, click on Solve – and step by step solution to my algebra homework. I
highly recommend the program.
Posted: Sunday 07th of Jan 12:01
You can get it at https://rational-equations.com/solving-exponential-and-logarithmic-equations.html. Please do post your experience here. It may help a lot of other beginners as well. | {"url":"https://rational-equations.com/in-rational-equations/graphing-equations/solving-logarithmic-equations.html","timestamp":"2024-11-06T05:07:46Z","content_type":"text/html","content_length":"100946","record_id":"<urn:uuid:19c1011b-a547-4aa1-8c5b-81a4f2aaca9e>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00803.warc.gz"} |
ROBERT BOSCH Placement Question Paper
Bosch Placement Paper
ROBERT BOSCH Placement question paper.
1. There was a figure of JK flip flop in which ~q is connected to J input and K=1. If clock signal is successively applied 6 times what is output sequence (q=?)
d) 010101
2. Frequency response of a filter is
a) Range of frequencies at which amplification of signal is employed.
b) Output voltage versus frequency (plot)
c) Filter which suppresses particular frequency
3. Gain and bandwidth of an op amp is
a) Independent of each other
b) Gain decreases as bandwidth decreases
c) Gain increases as bandwidth increases till some extent after which stability decreases
4. There was a figure of 4:1 MUX in which A and B are select lines. Inputs S0 and S1 are connected together and labeled as C where as S2 and S3 are connected together and labeled as D. Then which of
the following is true?
a) Y= B+C
b) Y= A+C
c) Y= A+B
d) Y= C+D (Where Y is the output)
5. In step up transformer (or Step down… not sure) transformation ratio is 1:5. If the impedance of secondary winding is 16 ohm then what is the impedance of primary winding?
a) 80 b) 3.2
6. There was a circuit consisting of AC voltage source and one inductance. Inductance value=0.2mH (or 0.2uH or 0.2H not sure).AC voltage =150 sin (1000t).what is the current flowing in the circuit?
a) i= 7.5 sin (1000t)
b) i= -7.5 sin (1000t)
c) i= 7.5 cos (1000t)
d) i= -7.5 cos (1000t)
7. Power gain of an amplifier having i/p gain of 20W and output gain of 20mW is
a) 60 b) 25 c) 10 d) 0
8. There was a RC circuit given with AC voltage source. Expression for capacitance was asked for charging condition. Choices were somewhat like this: a) some value multiplied by exp (-t/T)
ans –c i= (Vs/R)exp(-t/ T)
9. 2’s complement of -17
ans — 01111
10. Instrumentation amplifier is used for——— –?
a). effective shielding
b). high respective filters
c). high common mode
d). all the above.
11. In “ON CHIP” decoding memory can be decoded to
a) 2^n b)2^n +1 c)2^n -1 d) some other choice
12. Half of address 0Xffffffff is
a) 77777777 b) 80000000 c) 7FFFFFFF d) some other choice
13. Which one of the following is used for high speed power application?
a) BJT b) MOSFET c) IGBT d) TRIAC
14. One question related with SCR rotation angle given ifring angle is 30degree
ans – 150degree
15. SCR is used for
a) To achieve optimum (or maximum …not sure) dv/dt
b) For high current ratings
c) To achieve high voltage
d) Some other choice
16. State in which o/p collector current of transistor remains constant in spite of increase in base current is
a) Q point b) Saturation c) Cut off
17. A 16 bit monosample is used for digitization of voice. If 8 kHz is the sampling rate then the rate at
which bit is transferred is
a) 128 b) 48 c) d)
18. To use variable as recursive, variable should be used as
a) Static b) Global c) Global static d) Automatic
19. what is the resonant frequency of parrel RLC circuit of R= 4.7 komh L= 2 micro Henry and c=30pf.
a). 20.5 MHz
b). 2.65 KHz
c). 20.5 KHz
d). none
20.for the parallel circuit (one figure is given) Is= 10mA. R1= 2R, R2=3R, R3= 4R. R is artritary
a). 3.076mA
b). 3.76mA
21. main ()
int a=0x1234;
printf (“%x”, a);
What is the output?
a)1000 b) 2000 c) d) None of these
22. What does (*fun () []) (int) indicate?
ans…b). an array of pointer to a functions that an int as parameter and return int.
23. #define A 10+10
main ()
int a;
printf (“%d”, a);
a) 100 b) 200 c) 120 d) 400
24. One more question related with ADC like voltage is 8 volts. frequency 2 Mega hz. what is the conversion rate
25. Question related with serial in parallel out shift register…What is output sequence?
Ans….. 1010
26. Given one RLC circuit in which values of R, L and C were given. What is the value of frequency f?
27. if (fun ())
X gets incremented if and only if
a) fun () returns 0
b) fun () return 1
c) fun () return -1
d) return a value other than 0
28. In dynamic memory
a) Power dissipation is less than that of static memory
b) Clock is needed
c) Refreshing is required
d) All the above
29. Short, int and long integers have how many bytes?
a)2,2,4 b) Machine dependent c)2,4,8 d) Some other choice
30. A (n) is ———–filter combination of
a) Passive b) Active c) AMPLIFIER d) BOOSTER
31.Mobility of electron is
a) Increases as temperature increases
b) Decreases as temp decreases
c) Independent of conductivity
d) Some other choice
32. Structure comparison is done
a). yes
b) no
c) compiler dependent
33. The system in which communication occurs in both ways but not simultaneously in both ways is
a) Half simplex b) Simplex c) Half duplex d) duplex
34. main ()
int a=5, b=6;
int i=0;
i=a>b? a:b;
printf (“%d”, i);
a) 0 b) 1 c) 6 d)
35. int fun (char c)
int i;
static int y ;}
a) c, i are stored in stack and y stored in data segment
b)c stored in stack and i,y are stored in data segment
c) c is stored in text segment, y in data and i in stack
36. main ()
int *p;
short int i;
p= (char *) malloc (i*10); (code was showing error here)
printf (“%d”, p);
Value of p?
37. main ()
int *p,i[2]={1,2, 3};
printf (“%d %d %d”, i [0],*p,*p+1) ;
38. F = A’B’ + C’ + D’ + E’ then
A) F = A+B+C+D+E
B) F= (A+B)CDE
C) F = AB(C+D+E)
D) F= AB+C+D+E
39. how would you insert pre-written code into a current program?
a) #read<>
b) #get<>
c) #include<>
d) #pre<>
40.structure may contain
a) any other structure
b) any other structure expect themselves
c) any other structure except themselves and pointed to themselves
d) none of the above
41. three boys x,y,z and three girls x,y,z… sit around a round table . but x does not want any girl sitting to him and girl y does not want any boy sitting next. how many ways can they be seated.
a) 2
b) 4
c) 6
d) 8
42. k is brother of n and x. y is the mother of n and z is is the father of k . which of the following statement is not definitely true.
a) k is the son of z.
b) y is the wife of z.
c) k is the son of y.
d) n is the brother of x.
43. find the lateral surface of a prism with a triangular base if the perimeter of the base is 34 cm. and the height is 45 cm.
a) 765 square cm.
b) 3060 square cm.
c) 1530 square cm.
d) none
44.Given a< b< c < d . what is the max ratio of given equation
a) (a+b)/(c+d)
b) (b+c)/ (a+d)
c) (c+d)/ (a+b)
d) (a+c)/ (b+d)
45. A and B starts moving from points X and Y simultaneously at a speed of 5kmph and 7kmph to a destination point which is of 27 km from points X and Y. B reaches Y earlier than A and immediately
turns back and met Z. Find the distance XZ.
ans…. 22.5 km
46. Ann is shorter than Jill and Jill is taller than Tom. Which of the following inferences are true?
a) Ann is taller than Tom b) c) d) Data insufficient
47. A and B starts from same point at opposite direction. They will move 6km and take 8km left. How Far is A and B from each other?
ans …20m
48.6440 soldiers are to be arranged in the shape of square. If 40 soldiers were kept out then the number of soldiers making each straight line is?
Ans –80
49. Sum of squares of two numbers is 404 and sum of two numbers is 22.Then product of two Numbers?
a) 20 b) 40 c) d) (Answer is 40. Two numbers are 20 and 2)
50. In an examination 4 marks are assigned for correct answer and 1 mark is deducted for wrong answer. However one student attempted all 60 questions and scored 130.Number of questions he attempted
correct is?
a) 35 b) 38 c) 42 d) 35
51. Each ruby is of 0.3 kg and diamond is of 0.4 kg.Ruby costs 400 crores and diamond costs 500 crores. Ruby and diamonds have to be put into a bag. Bag cannot contain more than 12 kg.Which of the
following gives maximum profit (or in terms of wealth) (In crores)
ans …only ruby 40p
52. the cpu stack is placed in ….
a). cpu resister
b). RAM
c). ROM
d). hard disk
53). (10 | 7) would produce
a). 17
b). 3
c). 11
d). 15
ENGLISH section
fill in the blanks…… .. the answers is
61) a). impounded b). protected c). hounded d). relegated.
62).a). oblinion b). authertiory c). dejection d). deso…….
63).a). subdued b). bountiful c). tentative d). ardem.
64).a). b). esulcant c). emblerratie d). innate
65).manor — d). n. the landed estate of a land or nobleman.
66).neologism — c). n. giving a new meaning to an add word.
67).batten — b). n. a narrow strip of wood.
68).tepid — d). adj. lacking interest enhusi…., luewarm
69. discerning — d). adj. distinguishin one thing from another, having good judgment | {"url":"https://placement.freshershome.com/robert-bosche/bosch-placement-paper_1038.html","timestamp":"2024-11-14T20:22:33Z","content_type":"text/html","content_length":"91313","record_id":"<urn:uuid:5c5e7173-0f28-4118-98c0-74c55a9ed37d>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00086.warc.gz"} |
Deep learning linear algebra, probability and information theory
(3) Probability and information theory)
2, Linear algebra
Purpose: it is mainly to reduce the dimension of data and restore or build the target with the least data
2.1 eigenvalue decomposition
Only symmetric positive definite matrix can be decomposed into eigenvalues.
Eigenvalue decomposition: A = P * B * PT, where B is the diagonal matrix whose diagonal element is the eigenvalue of a, and P is the orthogonal matrix composed of the eigenvector of A.
Then let's look at whether the row vector or column vector in vec decomposed by eig in np is an eigenvector.
It can be seen that the latter is equal, indicating that the column vector of vecs decomposed by eig in np is an eigenvector.
2.2 singular value decomposition
Every real matrix has a singular value decomposition, but not necessarily an eigendecomposition. For example, the matrix of non square matrix has no eigendecomposition, so we can only use singular
value decomposition.
Examples and functions of singular value decomposition: https://www.cnblogs.com/pinard/p/6251584.html
#2.1 eigenvalue decomposition
import numpy as np
from numpy.linalg import eig
A = np.random.randint(-10,10,(4,4))
C = np.dot(A.T, A)# Generating symmetric positive definite matrix
vals,vecs = eig(C)
Lambda = np.diag(vals)#First, the eigenvalue is transformed into a matrix
np.dot(np.dot(vecs, Lambda), vecs.T) # Equal to C=A.T*A
#Then let's see whether the row vector or column vector in vec decomposed by eig in np is an eigenvector. Just verify: A*vecs[0] = vals[0]*vecs[0]
np.dot(C, vecs[0])
np.dot(C, vecs[:,0])
vals[0]*vecs[:, 0]
#2.2 singular value decomposition
from numpy.linalg import svd
a = np.random.randint(-10,10,(4, 3)).astype(float)
u, s, vh = np.linalg.svd(a)# Here vh is the transpose of V
u.shape, s.shape, vh.shape
#Transform s into singular value matrix
smat[:3, :3] = np.diag(s)
np.allclose(a, np.dot(u, np.dot(smat, vh)))
https://www.cnblogs.com/cymwill/p/9937850.html The above example follows this link. Finally, the singular value is transformed into a matrix error, which can only be re assigned into a 4 * 3 matrix.
2.3 Moore Penrose pseudoinverse
UDVT can be obtained by singular value decomposition, and the diagonal matrix D + is obtained by taking the reciprocal of the non-zero elements in D and then transposing. Examples and code references
https://www.section.io/engineering-education/moore-penrose-pseudoinverse/ . Seems to be transposed directly in the link?
Note that in the previous SVD decomposition, a new blank row is created when converting Array D into matrix. In this link, a new blank column is created. The number of rows and columns of singular
value matrix D is the same as that of the original matrix, and the number of rows and columns of matrix D + in violation is opposite to that of the original matrix.
If you calculate directly, call numpy linalg. Pinv() function.
u, s, vh = np.linalg.svd(a)# Here vh is the transpose of V, U,D,VT
dinv=np.linalg.inv(d)#Reciprocal of singular value diagonal matrix?
dmat[:3, :3]=dinv#Unified into the number of rows and columns opposite to the original matrix
aplus= np.dot(vh.T, np.dot(dmat,u.T))
2.4 PCA principal component analysis
While reducing the indicators to be analyzed, try to reduce the loss of information contained in the original indicators, so as to achieve the purpose of comprehensive analysis of the collected data.
The main idea of PCA is to map n-dimensional features to k-dimension, which is a new orthogonal feature, also known as the main component. It is a k-dimensional feature reconstructed from the
original n-dimensional features.
The job of PCA is to find a set of mutually orthogonal coordinate axes from the original space. The selection of new coordinate axes is closely related to the data itself. Among them, the first new
coordinate axis selection is the direction with the largest variance in the original data, the second new coordinate axis selection is the one with the largest variance in the plane orthogonal to the
first coordinate axis, and the third axis is the one with the largest variance in the plane orthogonal to the first and second axes. By analogy, n such coordinate axes can be obtained. Through the
new coordinate axes obtained in this way, we find that most of the variance is contained in the first k coordinate axes, and the variance of the latter coordinate axes is almost 0. Therefore, we can
ignore the remaining coordinate axes and only keep the first k coordinate axes containing most of the variance. In fact, this is equivalent to retaining only the dimension features that contain most
of the variance, while ignoring the feature dimensions that contain almost zero variance, so as to reduce the dimension of data features.
Think: how do we get these principal component directions that contain the greatest differences?
Answer: in fact, by calculating the covariance matrix of the data matrix, the eigenvalue eigenvector of the covariance matrix is obtained, and the matrix composed of the eigenvectors corresponding to
the k features with the largest eigenvalue (i.e. the largest variance) is selected. In this way, the data matrix can be transformed into a new space to reduce the dimension of data features. There
are two solutions: eigenvalue decomposition covariance matrix and singular value decomposition covariance matrix.
One problem of PCA algorithm is that it is not interpretable
2.4.1 eigenvalue decomposition covariance matrix
2.4.2 singular value decomposition
3, Probability and information theory
3.1 expected variance covariance
The covariance of the two variables is 0, indicating that they are not related, there must be no linear relationship, and they may not be independent; If the two variables are independent, the
covariance must be 0 and there is no correlation.
Common probability distribution
3.2.1 common probability distribution in probability theory
3.2.2 common probability distribution in deep learning
3.2.3 Bayesian rules
Finally have their own original code.
import pandas as pd
import numpy as np
#Find the individual a priori probability p(x|y)
def naivebayes(data,n,x,y):#n is the nth column, x is the value classified as x in the nth column, and y is the label item
numEntires = len(data) #Returns the number of rows in the dataset
labelCounts = {} #y
for featVec in data:
currentLabel = featVec[-1]
if currentLabel not in labelCounts.keys():
labelCounts[currentLabel] = 0
labelCounts[currentLabel] += 1
for i in range(len(data)):
if data[i][-1] == y and data[i][n] == x:
num = num + 1
#print (num/labelCounts[y])
print ('p(',y,')The probability is:',labelCounts[y]/numEntires)#p(y)
return (num/labelCounts[y])#p(x|y)
#Find p(y|x) = multiply individual p and * p(y)
3.2.4 technical details of continuous variables
3.3 information theory
3.3.1 information quantity: self information and Shannon entropy
3.3.2 DKL divergence and cross entropy H(P,Q)
3.5 structured probability model | {"url":"https://programmer.ink/think/deep-learning-linear-algebra-probability-and-information-theory.html","timestamp":"2024-11-03T12:39:23Z","content_type":"text/html","content_length":"17006","record_id":"<urn:uuid:ba02baa4-3311-4008-be9d-895544fae68b>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00488.warc.gz"} |
• A multiplication operator ${\displaystyle T_{f}}$ on ${\displaystyle L^{2}(X)}$ , where X is ${\displaystyle \sigma }$ -finite, is bounded if and only if f is in ${\displaystyle L^{\infty }(X)}$
. In this case, its operator norm is equal to ${\displaystyle \|f\|_{\infty }}$ .^[1]
• The adjoint of a multiplication operator ${\displaystyle T_{f}}$ is ${\displaystyle T_{\overline {f}}}$ , where ${\displaystyle {\overline {f}}}$ is the complex conjugate of f. As a consequence,
${\displaystyle T_{f}}$ is self-adjoint if and only if f is real-valued.^[4]
• The spectrum of a bounded multiplication operator ${\displaystyle T_{f}}$ is the essential range of f; outside of this spectrum, the inverse of ${\displaystyle (T_{f}-\lambda )}$ is the
multiplication operator ${\displaystyle T_{\frac {1}{f-\lambda }}.}$ ^[1]
• Two bounded multiplication operators ${\displaystyle T_{f}}$ and ${\displaystyle T_{g}}$ on ${\displaystyle L^{2}}$ are equal if f and g are equal almost everywhere.^[4]
Consider the Hilbert space X = L^2[−1, 3] of complex-valued square integrable functions on the interval [−1, 3]. With f(x) = x^2, define the operator ${\displaystyle T_{f}\varphi (x)=x^{2}\varphi
(x)}$ for any function φ in X. This will be a self-adjoint bounded linear operator, with domain all of X = L^2[−1, 3] and with norm 9. Its spectrum will be the interval [0, 9] (the range of the
function x↦ x^2 defined on [−1, 3]). Indeed, for any complex number λ, the operator T[f] − λ is given by ${\displaystyle (T_{f}-\lambda )(\varphi )(x)=(x^{2}-\lambda )\varphi (x).}$
It is invertible if and only if λ is not in [0, 9], and then its inverse is ${\displaystyle (T_{f}-\lambda )^{-1}(\varphi )(x)={\frac {1}{x^{2}-\lambda }}\varphi (x),}$ which is another
multiplication operator.
This example can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any L^p space.
See also | {"url":"https://www.knowpia.com/knowpedia/Multiplication_operator","timestamp":"2024-11-08T14:47:29Z","content_type":"text/html","content_length":"104576","record_id":"<urn:uuid:69d3ee22-7edb-4477-b25b-bc6ca2b0181a>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00696.warc.gz"} |
Re: Factor
Tuesday, December 2, 2014
Yesterday, I committed a performance improvement to the heap implementation in Factor.
There’s an interesting comment on the pypy implementation of the “heapq” module that discusses a performance optimization that takes advantage of the fact that sub-trees of the heap satisfy the heap
invariant. The strategy is to reduce the number of comparisons that take place when sifting items into their proper place in the heap.
Below, I demonstrate the time it takes to run our heaps benchmark and to sort 1 million random numbers using heapsort, before and after making the change.
IN: scratchpad gc [ heaps-benchmark ] time
Running time: 0.224253523 seconds
IN: scratchpad 1,000,000 random-units gc [ heapsort drop ] time
Running time: 2.210408992 seconds
IN: scratchpad gc [ heaps-benchmark ] time
Running time: 0.172660576 seconds
IN: scratchpad 1,000,000 random-units gc [ heapsort drop ] time
Running time: 1.688299185 seconds
Not a bad improvement! | {"url":"https://re.factorcode.org/2014/12/heaps.html","timestamp":"2024-11-11T20:17:50Z","content_type":"text/html","content_length":"7188","record_id":"<urn:uuid:d425c052-b7fb-4162-8301-776cd0a61b6a>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00616.warc.gz"} |
How to Convert Binary to Decimal
Introduction to Binary Numbers
We’ve all seen it before. You put in that new fast 1 TB SSD only to find it has 931GB of space. What happened to that other 69 GB of storage? That is another 20,000+ songs from your itunes library
you won’t be able to fit. This has to do with how computers calculate capacity vs how we as humans market products. This is where the binary system vs decimal system comes into play.
If you ever heard the phrase, “computers work in 1’s and 0’s” that is because they do. Each number or letter is some series of 1’s and 0’s. 0 is 0, 1 is 1, but after that it gets complicated. But
then 2 is (10), 3 is (11) and 4 is (100). This system is referred to as base 2, so the calculations will primarily be built on that. Everything is multiplied by 2. So remember this: 1, 2, 4, 8, 16,
32, 64. So how this works is, let’s take the binary number of 1011001 and figure out what it equals. If there is a 1 there, you add the base 2 number, if there is a zero then it is excluded.
There is a lot to unpack here …
How to convert binary numbers to decimal
A bit is a binary digit that can either have a value of 0 or 1. To convert a binary number to a decimal you can either start from the top left or the top right bit and, for each bit, apply the
following formula:
2ˆn * bit_value
(two raised at the position of the bit, multiplied by the bit value)
• n is the position of the bit, where n starts from 0 in the top right position
• and bit_value is either 0 or 1
The resulting decimal value will be the sum all the decimal numbers at each position:
If you want to simplify this equation there are 2 things to understand. What the 0’s and 1’s mean and how the second row of numbers come to be. The 0’s and 1’s simply mean either there is a digit for
calculation or not. If there is a 0 then there is nothing to add. If there is a 1 there, there is a digital to add to find what the binary number equals. The second part is, starting from right and
going left, each digit increases by a multiple of 2, starting with 1. So if this was 1111111 it would have been 64+32+16+8+4+2+1 (127), if it was 1000000 it would have been 64+0+0+0+0+0+0 (64). And
if it was 1000001 then it would be 64+0+0+0+0+1 (65).
Weirdly enough, we calculate memory in multiples of 2s but more on that later.
So decimal is based on 10 which is more or less how we do our mathematics for most of humanity. These values go from 0-9 and it is how we as humans essentially have done math for a very long time.
This is also the format where we calculate even the most complex equations by hand.
So why is your 1 TB SSD only 931GB?
That actually is not as confusing as you may think. Remember the base 2 system we discussed before? This is where base 2 and base 10 start to have some separation. We’ve all heard of Megabytes,
Gigabyte and Terabytes, they are designed in base 10. 1000MB=1GB and 1000GB=1TB. However that is for reporting purposes (marketing if you will), not how storage is calculated.
A Byte is the smallest measurement we typically deal with. Although there are bits, that will muddy the waters right now. 1,024 bytes = 1 Kilobyte (KB). Then 1024 KB = 1 Megabyte (MB), etc. So to
figure this out, if 1TB is 10^12 Bytes or 1 Trillion Bytes, we have to convert that to GB. So you take 1,000,000,000 / 1,024^3), the 3 comes from how many times you have to divide 1024 to get from
Bytes to Gigabytes. B>KB, KB>MB, MB>GB. So that number becomes 931GB.
Drawn out it looks like this (rounded decimals):
1,000,000,000,000 Bytes / 1024 = 976,562,500 KB
976,562,500KB KB / 1024 = 953,674 MB
953,674 MB / 1024 = 931 GB
What about Memory (RAM)
Funny enough, ram is the complete opposite. 2GB of ram is 2,048MB. Marketing does the opposite with memory where we take the actual amount and round down. So while most systems list 8 GB or 16 GB of
memory, it is really 8,192 MB or 16,384 MB. To avoid confusion, Windows usually hides this from the users, but either in the BIOS screen or if you run DX Diagnostic tool, you can see the full amount.
What about Letters?
Well this one is tricky because the binary system has some overlap. For example, a capital “A” is in 8-bit encoding equal 01000001 which also happens to be equal to 65 in decimal. Different software
and operating systems more or less have a binary subset table to differentiate how these are handled to make sure it calculates everything properly. And this is done over a billion times per second
without error so do not be worried about this.
So now we get to how this is applied a bit more in networking. If you recall in a previous blog where we talked about MAC Addresses, we use hexadecimals in those strings. Oftentimes you’ll see a “0x”
in front of a series of digits, numbers or a combination of both. This signifies this is a hex or hexadecimal. In the case of MAC Addresses, it is usually 2 characters. DO NOTE, this table is vastly
different than the binary table as anything past 9 is different, below is how to translate hex vs decimal vs binary:
Hexadecimal Binary Decimal
A 1010 10
B 1011 11
C 1100 12
D 1101 13
E 1110 14
F 1111 15
So you’ll notice to encode one hex you need 4-bit words or 2-digit words and like MAC Addresses, we will convert 8 characters down to 2. So if we take 10011110 and convert it, it will be 9 (1001) and
E (1110) or 9E, but often written as 0x9E to distinguish that this is a hexadecimal. This is how your computer translates things like MAC Addresses into its core language.
Binary, decimal and even hexadecimal are very important concepts to understand. In practice, most of it is used by computers, communication systems, and some software languages (e.g. assembly). So it
is good to understand the basics. But in reality, most people even in the tech space don’t actively use it on a daily basis. But at the very least, you know where that missing 69 GB of data went to! | {"url":"https://netbeez.net/blog/how-to-convert-binary-to-decimal/","timestamp":"2024-11-14T00:32:04Z","content_type":"text/html","content_length":"70699","record_id":"<urn:uuid:cd1c3b07-9be1-4eb1-b2a8-b58c73a81c75>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00357.warc.gz"} |
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Math 125 Survey of special topics in mathematicsTopic: research an ancient or contemporary mathematician or scientist. Format Directions: 1. A single page, single space, or | {"url":"https://uniquepaperhelp.com/math-homework-help/","timestamp":"2024-11-03T12:32:00Z","content_type":"text/html","content_length":"146650","record_id":"<urn:uuid:e3137aab-0352-4bc6-8805-152114568deb>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00406.warc.gz"} |