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Where are the first 13 episodes of The New World Campaign Game
Hey, sorry if the answer is obvious, but I just found out about this site, and it seems like a dream come true, and I'd like to start with the D&D campaign that's posted on this site, but I'd really
like to start at the beginning of it, and I can't see to find any parts past the first 13 on this site anyways.
Again, I imagine that the answer is obvious, but I've been looking for over a half hour, and I just can't seem to find any of them. By any chance, are there any playlists(like youtube) on this site
for any of the audio recordings.
Thank you for your time. | {"url":"https://slangdesign.com/forums/index.php?topic=1348.0;prev_next=prev","timestamp":"2024-11-02T00:17:54Z","content_type":"application/xhtml+xml","content_length":"27944","record_id":"<urn:uuid:11b2e99f-f522-4387-9262-5a05ad7e8732>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00755.warc.gz"} |
A detailed description of all model input parameters is available
. These are discussed further
Update 14 December 2015 (v2.5): correction to net output basis LCOE calculation, to include actual self power demand for wind, PV and batteries in place of "2015 reference" values.
Update 20 November 2015 (v2.4): levelised O&M costs now added for wind & PV, so that complete (less transmission-related investments) LCOE for wind and PV is calculated, for both gross and net
Update 18 November 2015 (v2.3: development of capital cost estimates for wind, PV and battery buffering, adding levelised capital cost per unit net output, for comparison with levelised capital cost
per unit gross output. Levelised capital cost estimate has been substantially refined, bringing this into line with standard practice for capital recovery calculation. Discount rate is user
Default maximum autonomy periods reduced to 48 hours for wind and 72 hours for PV.
Update 22 October 2015 (v2.2): added ramped introduction of wind and PV buffering capacity. Wind and PV buffering ramps from zero to the maximum autonomy period as wind and PV generated electricity
increases as a proportion of overall electricity supply. The threshold proportion for maximum autonomy period is user adjustable. Ramping uses interpolation based on an elliptical curve between zero
and the threshold proportion, to avoid discontinuities that produce poor response shape in key variables.
Update 23 September 2015 (v2.1): added capital investment calculation and associated LCOE contribution for wind generation plant, PV generation plant and storage batteries.
**This version (v2.0) includes refined energy conversion efficiency estimates, increasing the global mean efficiency, but also reducing the aggressiveness of the self-demand learning curves for all
sources. The basis for the conversion efficiencies, including all assumptions relating to specific types of work & heat used by the economy, is provided in
this Excel spreadsheet
Conversion of self power demand to energy services demand for each source is carried out via a reference global mean conversion efficiency, set as a user input using the global mean conversion
efficiency calculated in the model at the time of transition commencement (taken to be the time for which all EROI parameter values are defined. A learning curve is applied to this value to account
for future improvement in self power demand to services conversion efficiency.**
The original "standard run" version of the model is available | {"url":"https://insightmaker.com/tag/Economy","timestamp":"2024-11-10T09:24:46Z","content_type":"text/html","content_length":"120352","record_id":"<urn:uuid:587e9d86-d211-4068-93b5-d8d6f77467e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00561.warc.gz"} |
Answers to Test Yourself
Below are answers to the chapter Test Yourself feature.
(1) Independent variable = type of problem display, type independent variable, bivalent independent variable, manipulated between subjects. (2) With a between-subjects design, the primary concern is
with group differences. Thus, one might be concerned about group differences for interest in math and ability (which can also affect interest). If group differences exist in math interest before the
study, they cloud the test of the independent variable effect on the dependent variable. (3) The external validity can be improved with more realistic math problems and classroom situations. A field
experiment can be done by designing different texts that present math problems in the different formats and by asking students to use them in the classroom. Tests can also be designed for classroom
use with the different presentation formats. (4) regression toward the mean (5) double-blind (6) the Hawthorne effect (7) external (8) Attrition occurs when subjects drop out of a study before it is
completed. The problem with attrition is that the subjects who drop out may be characteristically different from subjects who remain in the study. This can result in data that only apply to certain
members of the group being studied, which can limit the conclusions a researcher can draw from the results of the study. (9) b (10) b (11) a (12) b | {"url":"https://edge.sagepub.com/mcbridermstats/student-resources-0/chapter-7/answers-to-test-yourself","timestamp":"2024-11-03T02:35:48Z","content_type":"text/html","content_length":"70848","record_id":"<urn:uuid:8b4b449c-6f49-4945-bd56-93b4b06acadf>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00392.warc.gz"} |
Submitting to GEO: 10x Fixed RNA profiling (Single Cell Gene Expression Flex)
Hi all,
I have FASTQ files generated for a multiplexed 10X single cell gene expression flex run (16 samples). The cells were processed into unfiltered and filtered count matrices separated by sample.
I want to upload these into GEO. But GEO requires that the FASTQ be separated by sample... Seems like an awful lot of work considering the RNA flex is probe based anyway, not sure why anyone would go
through the effort of realigning the sequencing reads to the probe set reference...
Anybody have any experiencing uploading multiplexed RNA-flex experiments to GEO? (Or do you recommend any other public repository?)
Thanks in advance!
Entering edit mode
Probe annotations might change over time. Different software might handle off-targets differently. Reproducibility demands that raw data are shared. People might use data for things beyond what
CellRanger does by default.
ADD REPLY • link | {"url":"https://www.biostars.org/p/9596375/#9596496","timestamp":"2024-11-11T05:20:39Z","content_type":"text/html","content_length":"24679","record_id":"<urn:uuid:61eed98c-4774-475e-9d80-bf108773aa13>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00488.warc.gz"} |
Neural Networks versus Logistic Regression - Get Neural Net
Neural Networks vs. Logistic Regression
Neural networks and logistic regression are two popular techniques in the field of machine learning. Both are widely used for classification tasks, but they differ in their underlying principles and
model structures. In this article, we will explore the key differences between neural networks and logistic regression, highlighting their strengths and limitations.
Key Takeaways:
– Neural networks are powerful models that can learn complex relationships between variables.
– Logistic regression is a simpler model that is computationally efficient and provides interpretable results.
– Neural networks are often more suitable for large and high-dimensional datasets.
– Logistic regression is a good choice when the interpretability of the model is important.
– Both techniques have their own advantages and should be selected based on the specific requirements of the problem at hand.
Neural Networks:
Neural networks are a type of machine learning model inspired by the structure and function of the human brain. They consist of multiple interconnected layers of artificial neurons (nodes) that
process and transform the input data to produce an output prediction. *Neural networks are particularly effective in capturing nonlinear relationships between variables.* They can learn from
unlabeled data (unsupervised learning) and labeled data (supervised learning), allowing them to perform a variety of tasks such as classification, regression, and pattern recognition.
Logistic Regression:
Logistic regression, on the other hand, is a statistical model used for binary classification. It is based on the logistic function (also known as the sigmoid function), which maps the input values
to a probability. *Logistic regression provides interpretable results by estimating the odds of an event occurring.* It calculates the probability of an instance belonging to a certain class by
modeling the relationship between the input variables and the log-odds of the event. This makes it useful for understanding the impact of the predictors on the outcome.
Comparison of Neural Networks and Logistic Regression:
To better understand the differences between neural networks and logistic regression, let’s compare them in terms of key factors:
1. Flexibility:
– **Neural networks:** Neural networks are highly flexible, capable of learning complex relationships between variables. They can model nonlinear interactions and handle large and high-dimensional
datasets efficiently.
– **Logistic regression:** Logistic regression is a more rigid model that assumes a linear relationship between the variables. It cannot capture complex interactions and is limited to a smaller
number of predictors.
2. Interpretability:
– **Neural networks:** Neural networks can be regarded as “black box” models, as their internal workings are not easily interpretable. They are often seen as providing accurate predictions without
explicit explanations for why specific decisions were made.
– **Logistic regression:** Logistic regression provides interpretable results, as the coefficients associated with each variable represent the magnitude and direction of the impact on the outcome.
This makes it easier to understand the contribution of each predictor.
3. Computation:
– **Neural networks:** Neural networks require more computational resources, especially for large and complex architectures. Training a neural network may take longer and requires more data to
achieve good performance.
– **Logistic regression:** Logistic regression is computationally efficient and can be trained relatively quickly even on large datasets. It is less resource-intensive compared to neural networks.
1. Performance comparison on different datasets:
| Dataset | Neural Network Accuracy | Logistic Regression Accuracy |
| Dataset 1 | 0.85 | 0.81 |
| Dataset 2 | 0.92 | 0.88 |
| Dataset 3 | 0.79 | 0.75 |
2. Model complexity and interpretability:
| Model | Complexity | Interpretability |
| Neural Network | High | Low |
| Logistic Regression | Low | High |
3. Training time comparison:
| Model | Training Time |
| Neural Network | 2 hours |
| Logistic Regression | 10 minutes |
In conclusion, both neural networks and logistic regression have their own strengths and limitations. Neural networks excel at capturing complex relationships and are suitable for large datasets,
while logistic regression provides interpretable results and is computationally efficient. Selection between the two techniques should be based on the specific requirements of the problem at hand.
Common Misconceptions
Misconception 1: Neural Networks are always better than Logistic Regression
One common misconception is that neural networks are always superior to logistic regression in terms of accuracy and performance. While neural networks can handle more complex data patterns and have
the potential for higher accuracy, logistic regression can still be highly effective in certain scenarios.
• Neural networks are computationally more expensive than logistic regression.
• Logistic regression is often preferred when interpreting the model’s coefficients is important.
• For small datasets with limited features, logistic regression can outperform neural networks.
Misconception 2: Neural Networks always require more data than Logistic Regression
It is often assumed that neural networks require larger datasets compared to logistic regression to achieve good performance. While neural networks typically benefit from a large amount of data,
logistic regression can also work well with smaller datasets given the appropriate feature engineering.
• Logistic regression can be useful when dealing with limited data availability.
• Neural networks might overfit with small datasets if not properly regularized.
• Feature selection or dimensionality reduction techniques can help improve logistic regression’s performance with limited data.
Misconception 3: Neural Networks are always black boxes, while Logistic Regression is interpretable
Another misconception is that neural networks are often considered as “black boxes” due to their complexity, making it difficult to interpret their predictions. On the other hand, logistic regression
is perceived as more interpretable, allowing for a clear understanding of the relationship between the input variables and the predicted outcome. However, this is not entirely accurate.
• Advanced techniques like recursive feature elimination can help identify important variables in neural networks.
• Interpretability can be increased by visualizing the activation patterns in neural networks.
• Logistic regression’s interpretability is limited when dealing with a large number of predictors or non-linear relationships.
Misconception 4: Neural Networks always outperform Logistic Regression with unstructured data
Unstructured data, such as text or images, is often perceived as suitable only for neural networks. Some may believe that logistic regression cannot effectively handle unstructured data, therefore
assuming neural networks will always outperform logistic regression in such cases. However, logistic regression can be used to handle unstructured data to some extent and can even achieve competitive
• Text data can be transformed into numerical features that can be used with logistic regression.
• Image data can be converted into pre-defined features or subjected to feature extraction techniques before applying logistic regression.
• Ensemble methods combining logistic regression and neural networks can achieve better performance with unstructured data.
Misconception 5: Neural Networks always require deep architectures
It is a common misconception that neural networks must always have deep architectures to achieve high accuracy. While deep neural networks are capable of learning complex representations, shallow
neural networks with only a few layers can also perform well given the right architecture and training.
• In certain cases, shallow neural networks can have better generalization performance.
• Shallow models can be more computationally efficient compared to deep architectures.
• The choice between shallow and deep neural networks depends on the complexity of the problem and the amount of available data.
The Rise of Neural Networks
Neural networks have gained significant attention in recent years in the field of machine learning. They are a set of algorithms designed to recognize patterns and make predictions, inspired by the
structure of the human brain. This article compares the performance of neural networks with logistic regression, a popular statistical model. Below are ten tables presenting various aspects of the
Accuracy of Neural Networks and Logistic Regression on Image Classification
Image classification is a challenging task in machine learning. Here, we compare the accuracy achieved by neural networks and logistic regression on a dataset of 10,000 images.
Neural Networks Logistic Regression
Accuracy 92% 78%
Training Time Comparison between Neural Networks and Logistic Regression
Training time is a crucial factor when considering machine learning models. The following table demonstrates the training time required by both neural networks and logistic regression on a dataset of
100,000 samples.
Neural Networks Logistic Regression
Training Time 32 minutes 2 hours
Applications of Neural Networks and Logistic Regression
Both neural networks and logistic regression find applications in various domains. The table below highlights their respective domains of usage.
Neural Networks Logistic Regression
Domain Speech Recognition Customer Churn Prediction
Accuracy 87% 72%
Comparison of Model Interpretability
Interpretability is of paramount importance in certain applications. The table below describes the interpretability aspects of neural networks and logistic regression.
Neural Networks Logistic Regression
Interpretability Low High
Handling Non-Linear Relationships using Neural Networks and Logistic Regression
Non-linear relationships pose a challenge to traditional models. The below table compares the ability of neural networks and logistic regression to handle non-linear relationships.
Neural Networks Logistic Regression
Ability to Handle Non-Linearity High Low
Scaling Performance Metrics
When scaling up data and models, performance can be significantly impacted. The following table illustrates the scalability of neural networks and logistic regression.
Neural Networks Logistic Regression
Scalability Excellent Moderate
Comparison of Training Set Size
Training set size affects the generalization capability of models. The table below showcases the impact of varying training set sizes on neural networks and logistic regression.
Neural Networks Logistic Regression
Training Set Size 100,000 samples 100,000 samples
Accuracy 94% 85%
Model Complexity Comparison between Neural Networks and Logistic Regression
The complexity of a model can have implications for its performance and feasibility. The next table compares the complexity aspects of neural networks and logistic regression.
Neural Networks Logistic Regression
Model Complexity High Low
Time Complexity Comparison between Neural Networks and Logistic Regression
The time complexity of a model influences its efficiency. The final table compares the time complexity of neural networks and logistic regression.
Neural Networks Logistic Regression
Time Complexity O(n^2) O(n)
From the various tables, it becomes evident that neural networks and logistic regression have their own strengths and weaknesses, making them suitable for different problem scenarios. Understanding
the nuances of these models is crucial for successful implementation in real-world applications.
Frequently Asked Questions
What is the difference between Neural Networks and Logistic Regression?
Neural networks and logistic regression are both machine learning algorithms used for classification tasks. However, they differ in their structure and complexity. Neural networks are a set of
interconnected nodes called neurons, which are organized in layers. Each neuron performs a weighted sum of the input and applies an activation function. Logistic regression, on the other hand, is a
linear model that uses logistic function to map input data to a probability value. Essentially, neural networks are capable of learning complex patterns and relationships, while logistic regression
is a simpler model that is easier to interpret.
When should I use Neural Networks instead of Logistic Regression?
Neural networks are generally preferred when dealing with complex datasets, nonlinear relationships, and large amounts of data. They can learn intricate patterns and extract high-level features from
raw data, making them suitable for tasks such as image recognition, natural language processing, and speech recognition. In contrast, logistic regression is better suited for simpler datasets, where
linear relationships between features and the target variable are sufficient for accurate predictions.
What are the advantages of Neural Networks over Logistic Regression?
Neural networks have several advantages over logistic regression:
• Ability to learn complex patterns and relationships
• Capability to extract high-level features from raw data
• Better performance on large and high-dimensional datasets
• Ability to handle nonlinear relationships between features
• Flexibility to model more complex decision boundaries
What are the disadvantages of Neural Networks compared to Logistic Regression?
While neural networks offer many advantages, they also have some downsides:
• Higher computational complexity and training time
• Require more training data to avoid overfitting
• Tendency to be more opaque and less interpretable
• Potential for being sensitive to hyperparameter settings
• Prone to overfitting if not properly regularized
Can logistic regression be considered a neural network?
Technically, logistic regression can be considered as a single-layer neural network. In a neural network, the input is multiplied by weight, and an activation function is applied to produce the
output. In logistic regression, a similar process happens, i.e., the input features are multiplied by weights and passed through the logistic (sigmoid) function to obtain the output. However, the
main distinction is that logistic regression lacks the complexity and ability to learn deeper concepts and hierarchical feature representations that multi-layer neural networks possess.
How do I choose between Neural Networks and Logistic Regression for my problem?
The choice depends on several factors:
• Complexity of the problem and dataset
• Availability and size of training data
• Interpretability requirements
• Computational resources and time constraints
• Performance of different models on validation data
Can I use Logistic Regression as a starting point before using Neural Networks?
Yes, it is common practice to start with simpler models like logistic regression before moving to more complex models like neural networks. Logistic regression can serve as a baseline model to
establish a benchmark for performance and provide an initial understanding of the problem. This approach allows for iterative refinement and improvement by gradually introducing more complexity into
the model.
Are there any cases where Logistic Regression outperforms Neural Networks?
Yes, certain scenarios may favor logistic regression:
• When the dataset is small and lacks sufficient complexity
• When the interpretability of the model is of utmost importance
• When computational resources are limited and training time needs to be minimized
Can I combine Neural Networks and Logistic Regression?
Yes, it is possible to combine neural networks and logistic regression. One common approach is to use logistic regression as the final layer of a neural network. This setup allows the neural network
to learn complex features and then use logistic regression for the final classification step. The neural network acts as a feature extractor, while logistic regression provides the interpretability
and simplicity desired in the final predictions. | {"url":"https://getneuralnet.com/neural-networks-versus-logistic-regression/","timestamp":"2024-11-12T09:38:57Z","content_type":"text/html","content_length":"66975","record_id":"<urn:uuid:7fd3b0f4-fcd6-4432-a28a-39357d70b519>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00010.warc.gz"} |
Introduction to Monte Carlo Methods
Monte Carlo methods involve the use of computers to make millions of guesses to the solution of a problem. As with the
Wisdom of the Crowd
the more guesses that are made the closer to the correct solution you get. A common example, when explaining Monte Carlo Methods, is the estimation of a value for Pi. And because Monte Carlo Methods
use random numbers we also use in this example the random process of throwing darts blindfolded at a circular dartboard placed on a square backboard.
If we imagine a hypothetical dartboard of radius 1 unit (diameter is 2 units) then we can place this dartboard onto a square backboard of 2 units by 2 units. (see following image)
We know that the area of a square is Height * Width and in this case the area will equal 4 units. The area of a circle is Pi * radius^2 but because we know the radius to be 1 and 1^2 is 1 then the
area of our circle is therefore Pi. If we can calculate the area of the circle then we will know the value of Pi. Because the circle touches the four sides of the square then it is seen that we can
subtract the area of the square not covered by the circle from the area of the square to leave us with the area of the circle and thence a value for Pi. So how do we determine the area of the square
not touched by the circle?
This is where the dartboard analogy is used. If we randomly throw darts in the direction of the backboard and dartboard then darts will hit the dartboard with a frequency of area of circle/area of
square. In reality we know that Pi is approximately 3.14... so we would expect darts to hit the dartboard in the approximate ratio of 3.14/4 or about 0.785 times the number of darts thrown. If we
multiply our experimentally determined ratio by 4 (the unit area of the square) then we will have a value for Pi.
All we now need to do is throw lots of darts randomly in the general direction of the backboard and work out the ratio of hits to the dartboard to the total number of darts thrown and multiply this
answer by 4. We can either do this by buying a dartboard and a backboard (wouldn't that be a fun experiment for young school children!) or we can work it out using Excel (less fun for young school
In the next image (click to enlarge) we can see a spreadsheet to calculate Pi by Monte Carlo Methods. Column A is just a count of the number of trials, column B is the X co-ordinate of our dart given
by =1-(2*RAND()) and column C is the Y co-ordinate of our dart also determined randomly by =1-(2*RAND()). We use that particular random number generator to give us a range of numbers from -1 to +1,
our unit dimensions.
We can now drag these three columns down for as many iterations as we require. In this example I have done 10,000 iterations. (For a greater number we can always run the simulation multiple times by
pressing F9 on your keyboard and noting each estimation.)
Next, we want to know if a dart has entered the dartboard rather than the backboard so Columns D and E determine if the co-ordinates in B and C are within the circle. We do this with the formula =IF
(B3^2+C3^2<1,B3,0) (see end note) in column D and =IF(B3^2+C3^2<1,C3,0) in column E. Drag these down as far as you have gone in columns B and C. You will notice that sometimes the values in columns D
and E are zero. This is because the dart has missed the dartboard.
We can also see scatter plots of the random numbers generated. The square is created by a scatter plot of the numbers in columns B and C and the circle by the numbers generated in columns D and E.
These charts show that the numbers are correctly defining our square and our circle.
Columns F and G count the number of darts in the dartboard and the number of darts thrown. There are a variety of ways to do this but checking for X or Y not equal to 0 will count the number of darts
in the dartboard for column F. Column G is superfluous as it always counts a dart thrown. In cell F2 we have a count of all darts in the dartboard and cell G2 is just a count of all darts thrown.
Finally, cell I2 is the ratio of darts in the dartboard to darts thrown and cell I3 is that ratio multiplied by 4. As you can see, even for many thousands of trials you will get wildly differing
results. Of course, we will never get close to the value of Pi because it is of infinite length but the longer the trial continues the closer we get, within the limits of 32 or 64-bit computing.
The drawback of using Monte Carlo Methods for calculating Pi is that it is a computationally wasteful method. Mathematicians have come up with simple algorithms that calculate Pi to an adequate
number of decimal places with just a few iterations. When we use Monte Carlo Methods we must be sure that there is not a simple method for performing our calculations first before we use something
like Monte Carlo. For other problems, Monte Carlo Methods may be the only way and my next article will take us towards such problems.
For a book on sports specific Monte Carlo methods there is the excellent
Calculated Bets: Computers, Gambling, and Mathematical Modeling to Win
by Professor Steven Skiena. Steven constructed a simulation of the game of Jai Alai using Monte Carlo methods for an automated betting system. In chapter three of his book the professor simulated one
million games of Jai Alai and was able to construct a model for win, place and show betting. He has also given a
video seminar
on his work too.
End note
This formula is derived from Pythagoras. Any line (radius) from the centre of this circle to its edge is of length 1. This line is also the side opposite the right-angled triangle formed by the
centre of the circle and the X & Y co-ordinates thus, if the sum of the two squares of the co-ordinates is less than 1 then the dart lies along a radius and is therefore on the dartboard. | {"url":"http://www.betfairprotrader.co.uk/2013/11/introduction-to-monte-carlo-methods.html","timestamp":"2024-11-12T21:51:28Z","content_type":"application/xhtml+xml","content_length":"82822","record_id":"<urn:uuid:0262d67e-7e13-4792-a8fa-0f1ba791bb26>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00187.warc.gz"} |
How to do you graph y=4x by plotting points? | HIX Tutor
How to do you graph #y=4x# by plotting points?
Answer 1
Plug in values for x, solve for y, then pair the corresponding values to find points on the graph.
Plug in values for x, such as -2, -1, 0, 1, 2. The corresponding y-values would be -8, -4, 0, 4, 8, respectively. Pairing these values together would give you the following coordinates: (-2, -8),
(-1, -4), (0, 0), (1, 4), (2, 8). Connect these points (preferably using a ruler), and you will obtain the graph of the function in question.
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Answer 2
To graph the equation (y = 4x) by plotting points, follow these steps:
1. Choose Values for (x): Select at least two values for (x). It's often helpful to include zero, a positive, and a negative number to get a clear idea of the line's direction. For example, let's
use (x = -1), (x = 0), and (x = 1).
2. Calculate Corresponding (y) Values: Use the equation (y = 4x) to find the corresponding (y) values for each chosen (x) value.
□ For (x = -1), (y = 4(-1) = -4).
□ For (x = 0), (y = 4(0) = 0).
□ For (x = 1), (y = 4(1) = 4).
3. Plot the Points: Plot the points on a graph. The points from our example are ((-1, -4)), ((0, 0)), and ((1, 4)).
4. Draw the Line: Connect the points with a straight line. Since (y = 4x) is a linear equation, the points should line up perfectly, and your line will extend infinitely in both directions.
By following these steps, you'll have accurately graphed the equation (y = 4x).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Multiplication Chart To 20 Printable 2024 - Multiplication Chart Printable
Multiplication Chart To 20 Printable
Multiplication Chart To 20 Printable – Getting a multiplication chart for free is a great way to support your pupil understand their instances desks. Below are great tips for working with this
valuable source of information. Initially, look at the habits from the multiplication dinner table. Following, make use of the graph or chart as an option to flashcard drills or as being a due
diligence helper. Lastly, make use of it being a reference point help guide process the days dining tables. The free variation of the multiplication chart only contains times desks for figures 1 by
means of 12. Multiplication Chart To 20 Printable.
Down load a no cost printable multiplication graph
Multiplication charts and tables are very helpful studying equipment. Down load a free multiplication graph PDF to help you your youngster commit to memory the multiplication charts and tables. You
can laminate the graph for durability and place it in a child’s binder at home. These totally free printable sources are ideal for fourth, third and second and fifth-quality individuals. This article
will explain how to use a multiplication chart to teach your son or daughter math concepts specifics.
You can find free computer multiplication graphs of different shapes and sizes. You will discover multiplication graph printables in 12×12 and 10×10, and there are also blank or small charts for
smaller sized kids. Multiplication grids come in white and black, shade, and smaller types. Most multiplication worksheets follow the Basic Mathematics Benchmarks for Grade 3.
Styles in a multiplication graph
Individuals that have discovered the supplement dinner table might find it simpler to acknowledge styles within a multiplication graph. This session illustrates the qualities of multiplication, such
as the commutative residence, to help students know the habits. As an example, pupils might discover that this item of a variety multiplied by two or more will always appear since the same amount. An
identical style can be located for amounts increased by a aspect of two.
Pupils also can locate a style within a multiplication dinner table worksheet. Those that have problems keeping in mind multiplication specifics must utilize a multiplication table worksheet. It can
help individuals comprehend that you have habits in columns, rows and diagonals and multiples of two. Additionally, they are able to use the habits inside a multiplication graph to discuss info with
other individuals. This process will likely support individuals recall the truth that 7 periods 9 equals 70, instead of 63.
Using a multiplication desk chart as an option to flashcard drills
Using a multiplication desk chart as a substitute for flashcard drills is an excellent approach to aid kids find out their multiplication details. Kids often discover that visualizing the answer
enables them to remember the simple fact. This way of studying is effective for moving rocks to harder multiplication facts. Imagine scaling a huge stack of stones – it’s quicker to go up little
rocks rather than to climb a absolute rock encounter!
Youngsters find out far better by undertaking many different practice approaches. As an example, they could mixture multiplication details and periods furniture to build a cumulative assessment,
which cements the information in long-term memory space. You can spend time planning for a course and producing worksheets. You may also look for exciting multiplication video games on Pinterest to
engage your child. When your child has mastered a specific instances desk, you may move on to the following.
Using a multiplication dinner table chart as a due diligence helper
Employing a multiplication kitchen table chart as due diligence helper can be a very efficient way to review and strengthen the methods with your child’s mathematics type. Multiplication table charts
showcase multiplication facts from 1 to 10 and retract into quarters. These graphs also show multiplication specifics inside a grid format to ensure that pupils are able to see designs and make
connections between multiples. Your child can learn the multiplication facts while having fun, by incorporating these tools into the home environment.
Employing a multiplication dinner table graph as due diligence helper is a wonderful way to promote individuals to apply issue dealing with expertise, learn new strategies, to make homework duties
much easier. Youngsters can usually benefit from studying the tips that will help them fix difficulties faster. These tricks will help them build assurance and quickly find the proper product. This
technique is ideal for kids who are having problems with handwriting as well as other okay electric motor abilities.
Gallery of Multiplication Chart To 20 Printable
Printable Multiplication Table 20 20 PrintableMultiplication
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Free Multiplication Chart Printable Paper Trail Design In 2021
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Design Application
Design Applications
The most interesting designs combine order and variety. Brain waves are stimulated by the combination of both order and variety in that they flatten with too much order (boring) or too much variety
Order is provided through use of a design template (Orthogon). Elements within the design will relate to each other and to the design as a whole. The artist provides variety with all the other
elements and principles of design (shape, color, pattern, etc.). A maple leaf will look like a maple leaf but no two out of thousands of leaves will be shaped and colored exactly the same. Only one
Orthogon, the Auron, appears in a staggering array of plants and animals.
Once an Orthogon is formed, several relationships automatically are available to use:
· the square
· the section extending beyond (or within) the square
· the new rectangle (Orthogon).
This rarely offers enough distances (proportions) to work with, however. Several additional distances are usually needed to complete a design: small, medium, large (even tiny). To determine those
additional spaces, follow the instructions below:
1. Divide the square into quarters (this is not necessary but serves to contain the measuring process).
2. Starting in the top quarter, measure from the bottom left corner diagonally up to the top right corner (of that same quarter. This process is similar to finding root 2, root 3, root 4, etc.
3. Lay that measurement down and mark where it falls.
4. Mark that distance on the top line.
5. Measure from the same corner you started the previous measurement. Lay down that measurement and mark where it falls.
6. Mark that distance on the line straight above.
7. If another measurement is completed, it should end up at the edge of the square (continuing out beyond the square will create several more spaces).
8. Also mark the halfway point of the square.
This process creates additional proportions that can be applied in any direction for a variety of arrangements within a work.
To begin a design, mark the distances created as described above on the edge of a piece of paper or wooden stick to form a new "ruler".
Determine the overall size for the work and cut or mask if two-dimensional. (Three-dimensional works will require methods for determining proportions that are specific to the medium.)
Place marks on the edges of the work using the new "ruler". Additional marks can be placed within the work as desired.
A good starting place is to mark the square and the halfway point of the square (light green line in above illustration). Continue marking distances using the "ruler" as desired. I typically divide a
work into quarters and emphasize the top left and bottom right quarters after the Dutch Masters' apparent references to "Heaven" and "Home" (direction of light, placement of figures, etc.).
All this measuring may seem tedious at first, but once the marks are set, placing elements within the work becomes more like a card game and less of a trial. Simply place the various elements of the
design to align with the marks made using the new "ruler".
Analyze the simplified design below to determine how
shapes and lines intersect at the measurement points
A is repeated at least three times, B twice, C twice, D twice and the halfway point twice. (Measurements are approximations.)
Giorgio Morandi's art, especially his Natura morta or still life paintings, reveal a masterful use of 3 or 4 measurements in a variety of directions. Joseph Mallord William Turner's calm seascapes
and Edgar Degas' paintings of horses are similarly interesting to analyze for use of inter-related elements. Start by looking for the square. (Caution: works may be cropped in the process of
publication, which can alter some of the measurements but the concept will still be apparent.)
Arcs and diagonals can be directed to various "ruler" marks for emphasis--particularly the square. The more complicated Orthogons (scheduled for availability in pamphlet form) provide arcs and
diagonals that appear in the process of creating the Orthogons.
An example of how an Orthogon is used in 2-dimensions is found in one of my works titled, “Joan’s Offering”, an etching and monoprint 13”X 10” (used in lieu of works by the Masters pending
copyright). As stated above, “Begin with the square,” and find the square in several directions (measurements are approximations). Additional space divisions reappear within the work (small, medium,
even tiny).
Use of any design system does not guarantee quality.
About Orthogons Orthogon Instructions Design Application Historical Use References Links Artist | {"url":"http://timelessbydesign.org/Application.htm","timestamp":"2024-11-02T13:47:59Z","content_type":"text/html","content_length":"42689","record_id":"<urn:uuid:31aa9a96-b0e9-4c47-a040-b002e8bec45e>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00878.warc.gz"} |
Classic McEliece:
Dedicated to the memory of Robert J. McEliece, 1942–2019
The first code-based public-key cryptosystem was introduced in 1978 by McEliece. The public key specifies a random binary Goppa code. A ciphertext is a codeword plus random errors. The private key
allows efficient decoding: extracting the codeword from the ciphertext, identifying and removing the errors.
The McEliece system was designed to be one-way (OW-CPA), meaning that an attacker cannot efficiently find the codeword from a ciphertext and public key, when the codeword is chosen randomly. The
security level of the McEliece system has remained remarkably stable, despite dozens of attack papers over 40 years. The original McEliece parameters were designed for only 2^64 security, but the
system easily scales up to "overkill" parameters that provide ample security margin against advances in computer technology, including quantum computers.
The McEliece system has prompted a tremendous amount of followup work. Some of this work improves efficiency while clearly preserving security: this includes a "dual" PKE proposed by Niederreiter,
software speedups, and hardware speedups.
Furthermore, it is now well known how to efficiently convert an OW-CPA PKE into a KEM that is IND-CCA2 secure against all ROM attacks. This conversion is tight, preserving the security level, under
two assumptions that are satisfied by the McEliece PKE: first, the PKE is deterministic (i.e., decryption recovers all randomness that was used); second, the PKE has no decryption failures for valid
ciphertexts. Even better, recent work achieves similar tightness for a broader class of attacks, namely QROM attacks. The risk that a hash-function-specific attack could be faster than a ROM or QROM
attack is addressed by the standard practice of selecting a well-studied, high-security, "unstructured" hash function.
Classic McEliece brings all of this together. It is a KEM designed for IND-CCA2 security at a very high security level, even against quantum computers. The KEM is built conservatively from a PKE
designed for OW-CPA security, namely Niederreiter's dual version of McEliece's PKE using binary Goppa codes. Every level of the construction is designed so that future cryptographic auditors can be
confident in the long-term security of post-quantum public-key encryption.
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ball mill load what happened
The experimental results showed that the AbbottFirestone curve can evaluate the lifter surface topography. The wear rate of the lifter specimen is increased first and then decreased with mill speed
and grinding media size. Increasing ball filling will increase the wear rate, and the grinding media shape of ball has a maximum wear rate. | {"url":"https://legitemauve.fr/08-11/ball-mill-load-what-happened.html","timestamp":"2024-11-10T12:45:06Z","content_type":"text/html","content_length":"43951","record_id":"<urn:uuid:bd822ea2-ec28-48d3-9416-c926c3803c69>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00848.warc.gz"} |
Timing experiments for dtwclust
Calculations for single time-series
First we look at the results of the timing experiments for single time-series, i.e., for distances calculated between two individual time-series. The distances tested are those included in the
package. Here we look at the effect of window sizes and series’ lengths. Each calculation was repeated 500 times and the median value was extracted. Note that the vertical scales differ for the
following figures.
DTW lower bounds
The first interesting result relates to the DTW lower bounds: lb_keogh and lb_improved. The window size does not seem to have a very significant effect, and the running time appears to (mostly?) grow
slowly with the series’ length. However, lb_improved was faster that lb_keogh. Considering that lb_improved first needs to calculate lb_keogh and then perform additional calculations, this is
somewhat puzzling, and the reason is not immediately evident. Perhaps there are compiler optimizations at play.
Shape-based distance
The shape-based distance also presents weird behavior. While it was expected that its running time would increase with the series’ length, the bump for the length of 152 is considerably large. It is
true that SBD is based on the FFT, and thus it adjusts the input series’ lengths to powers of 2, but if that were the cause, then the bump should have occurred for the series with length of 130,
since the next power of 2 for 109 is 128, and it jumps to 256 for a length of 130.
In the case of DTW, we can see that a window constraint can indeed have a very significant effect on running time, considering that a window size of 10 resulted in a calculation that was about 4
times faster than when using no constraint. In this case, using multivariate series (with 3 variables) did not have a very significant effect.
In principle, the soft-DTW algorithm is very similar to that of unconstrained DTW. However, its run-time characteristics are clearly different, resulting in considerably slower calculations.
Interestingly, the multivariate case was marginally faster.
Triangular global alignment kernel
The behavior of GAK was rather surprising. Its running times increase very fast with the series’ length, and neither window size nor number of variables seem to have any effect whatsoever. The
normalized version is 3 times slower because it effectively calculates a GAK distance 3 times: one time for x against y, one time for x alone, and one time for y alone; these 3 values are used to
calculate the normalized version.
Calculations for several time-series
Computing cross-distance matrices for time-series can be optimized in different ways depending on the distance that is used. In the following sections, we look at the way the included distances are
optimized, and evaluate the way it affects their running times when doing distance calculations between several time-series.
These experiments were repeated 30 times and the median value was computed. We look at the effect of several factors: the length of the series, a window size where applicable, the effect of
parallelization, and the size of the cross-distance matrix.
DTW lower bounds
First we assess the distances that involve the DTW lower bounds. In the following figure, the x axis contains the total number of cells in the cross-distance matrix, but the color is mapped to the
number of columns in said matrix. This is relevant in this case because of the envelopes that need to be computed when calculating the lower bounds. These are computed for the series in y, which end
up across the columns of the distance matrix. The applied optimization consists in calculating the envelopes for y only once, and re-using them across x. In the case of dtw_lb (which is based on
lb_improved), this is also important because nearest neighbors are searched row-wise by default, and more series in y equates to more neighbors to consider. Also note that the dtw_lb calculations
make more sense when the series in x and y are different (if the series are the same, the nearest neighbor of a series is always itself, so no iterations take place), which is why there are less data
points in those experiments. Given the results of the previous section, the window size value was fixed at 50 for these experiments.
For lb_keogh and lb_improved, we see that the effect of the series’ length is more consistent when calculating cross-distance matrices. Parallelization with multi-threading yields better performance
in a proportional way, and this time lb_improved was about twice as slow as lb_keogh. As expected, increasing the number of series in y increases the running times.
The behavior of dtw_lb is also consistent, but the length of the series affected running times in a strange way, since the calculations with series of length 152 were the slowest ones. Using
multi-threading can also be advantageous in this case, and this is applied on two occasions: when estimating the initial distance matrix with lb_improved, and when updating the nearest neighbors’
with DTW. Nevertheless, the procedure is much slower than the lower bounds alone.
Shape-based distance
As mentioned before, the shape-based distance is based on the FFT. Similarly to the DTW lower bounds, the optimization applied here consists in calculating the FFTs only once, although in this case
they must be calculated for both x and y. The results are summarized in the next figure.
For sequential calculations, we see here the expected effect of the series’ lengths. Adjusting them to powers of 2 for the FFT meant that the calculations were faster for series of length 109, and
for series of length 152 and 196 the times were virtually the same (so much that the points overlap). Parallelization helped reduce times proportionally.
The DTW distance doesn’t allow for many optimizations. The version implemented in dtw_basic for cross-distance matrices uses less RAM by saving only 2 columns of the local cost matrix (LCM) at all
times (since no back-tracking is performed in this case). Moreover, this LCM is only allocated once in each thread. The results are shown in the next figure.
The DTW distance presents a much more constant behavior across the board. The difference between univariate and multivariate series is very small, but window sizes and series lengths can have a very
significant effect, especially for sequential calculations. Additionally, DTW benefits linearly from parallelization, since using 2 threads reduced times in half, and using 4 reduced them practically
by a factor of 4. Also note that the growth is very linear, which indicates that DTW cannot be easily optimized much more, something which was already pointed out before (Ratanamahatana and Keogh
As with DTW, soft-DTW has few optimizations: its helper matrix is allocated only once in each thread during the calculation. In this case, we look only at the univariate version.
The benefits of parallelization are also very evident for soft-DTW, and present similar characteristics to the ones obtained for DTW. Unfortunately, running times also grow very fast with the series’
Triangular global alignment kernel
Finally we look at the computations with the GAK distance. As shown in the previous section, this distance is considerably slower. Moreover, only the normalized version can be used as a distance
measure (as opposed to a similarity), so only the normalized version was tested.
There are 2 optimizations in place here. As mentioned before, normalization effectively requires calculating a GAK for x and y by themselves, so in order to avoid repeated calculations, these
normalization factors are only computed once for cross-distance matrices. GAK also uses a helper matrix to save logarithms during the intermediate calculations, and this matrix is allocated only once
in each thread.
It is worth pointing out that, in principle, the GAK code is very similar to that of DTW. However, GAK relies on logarithms, whereas DTW only uses arithmetic operations. This means that,
unfortunately, GAK probably cannot be optimized much more.
Here we see a clear effect of window sizes and series’ lengths and, as was the case for DTW, parallelization can be very beneficial for GAK. | {"url":"https://cran.stat.sfu.ca/web/packages/dtwclust/vignettes/timing-experiments.html","timestamp":"2024-11-15T04:44:58Z","content_type":"text/html","content_length":"1048830","record_id":"<urn:uuid:f59fb345-c7ba-4c62-80b8-735683aa7602>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00701.warc.gz"} |
R package for the quantitative exploration of elastic net families for generalized linear models. Available from:
GitHub: devtools::install_github("juliancandia/eNetXplorer")
For more details, see: Candia J, Tsang JS, eNetXplorer: an R package for the quantitative exploration of elastic net families for generalized linear models, BMC Bioinformatics 20:189 (2019).
R package that provides a quantitative, non-parametric assessment of statistical significance for the association between mutational signatures and observed spectra. Available from:
GitHub: devtools::install_github("juliancandia/mutSigMapper")
For more details, see: Candia J, mutSigMapper: an R package to map spectra to mutational signatures based on shot-noise modeling, bioRxiv 2020.10.12.336404. | {"url":"http://juliancandia.com/frames/Software.html","timestamp":"2024-11-02T08:21:15Z","content_type":"text/html","content_length":"1487","record_id":"<urn:uuid:610765f5-362b-477b-94b0-063da1f5c6a8>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00091.warc.gz"} |
How Does Negative Skewness Affect the Interpretation of Central Tendency Measures in a Dataset?
Skewness is a key estimation used to check the unevenness of a dataset's scattering. When a dataset has a negative skewness, it suggests that the tail of the scattering is longer on the left side, it
is centred around the right side to show that the data. This skewness can essentially influence the translation of focal inclination estimates like the mean, middle, and mode.
Specifically, negative skewness can make these actions be pulled this way, prompting errors in the general portrayal of the information. Understanding the Definition of Negative Skewness for focal
propensity measures is urgent for precisely deciphering and examining datasets. For instance, in an adversely slanted dataset, the mean will be not exactly in the middle since the mean is more
delicate to outrageous qualities and exceptions.
Effect of Negative Skewness on Interpretation of Central Tendency Measures
Negative skewness in a dataset can essentially affect the translation of focal propensity estimates like the mean, middle, and mode.
When a dataset is adversely slanted, it implies that the tail of the dissemination is longer on the left side. This will bring about the mean being pulled towards the lower values in the dataset. As
such, the mean will be not exactly in the middle in an adversely slanted dispersion. This happens in light of the fact that the super low qualities in the left tail of the dispersion affect the mean
than the high qualities on the right side.
For instance, in a dataset of family livelihoods, in the event that there are a couple of very low-pay families, the mean pay will be lower than the middle pay because of the negative skewness
brought about by these exceptions. For this situation, the mean may not precisely address the regular pay level of the families in the dataset.
The median is one more proportion of focal propensity that can be impacted by bad skewness in a dataset. In an adversely slanted dispersion, the median will be nearer to the right half of the
conveyance, where most of the information focuses are concentrated. This is on the grounds that the median is worth that partitions the dataset into equivalent parts, so it is less affected by
outrageous qualities in the tails of the appropriation.
For instance, in a dataset of grades where a couple of understudies scored very low, the median score will be less impacted by these exceptions than the mean score. The median will give a more
representative proportion of the run-of-the-mill test score for this situation, particularly while managing slanted information.
The mode is the worth that shows up most often in a dataset and is one more proportion of focal propensity. In an adversely slanted conveyance, the mode will be not exactly the middle and the mean,
as it addresses the most widely recognized esteem in the dataset. Nonetheless, at times, an adversely slanted conveyance might have numerous modes, making it harder to decipher the focal inclination
of the information.
For instance, in a dataset of month-to-month precipitation sums, in the event that there are various tops in the conveyance because of various weather conditions, the mode may not precisely mirror
the normal precipitation sum. For this situation, the mode may not be a solid proportion of focal inclination, particularly when the information is adversely slanted.
Steps to Include Gauth for Handling Questions
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Stage 1: Enter the Question
In the first place, you truly need to enter your question into the Gauth search bar. Whether you're looking for information on a specific subject, or searching for the stanzas to a tune, Gauth can
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Final Talk
Negative skewness can vigorously influence the translation of focal propensity estimates like the mean, middle, and mode in a dataset. The presence of negative skewness shows that the tail of the
dispersion is slanted to one side, pulling the mean towards lower values than the middle and mode. This can prompt a distortion of the typical worth in the dataset and influence the general
comprehension of the information. | {"url":"https://free-economy.org/how-does-negative-skewness-affect-the-interpretation-of-central-tendency-measures-in-a-dataset/","timestamp":"2024-11-08T05:43:17Z","content_type":"text/html","content_length":"21766","record_id":"<urn:uuid:ea224f23-638c-4e49-b5a7-f160598cd1ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00276.warc.gz"} |
Isoperimetric Polyominoes Solutions
The construction below can be used to make pentuplications of all 35 hexominoes.
Quadruplication of a dekomino with a hole in the shape of the dekomino.
Ttwo quadruplicated pentominoes with a hole in the shape of the pentomino.
1-2-3-4 problem. Take one of the pentominoes in the set and with the rest make double- triple- and quadruple-sized replicas.
1-1-2-3 problem. Create an area of 10 square units, produce a copy, a duplication and a triplication.
3-4 problem. With the whole set produce simultaneous triplication and quadruplication of an hexomino.
1-2-5 problem - take one pentomino from the set and then with the remaining pieces create double and fivefold replicas.
7-8 problem. Remove one heptomino and one octomino from the set and with the remaining pieces makes simultaneous triplications of both pieces.
Three twins problem - create three sets of pairs of congruent shapes
Two triplets problem - create two sets of three congruent shapes
Quintuplets - create 5 congruent shapes each with area 30
Sextuplets - create 6 congruent shapes each with area 25
Make two 9x9 squares with a hole in each in the shape of an hexomino
Make a 12x13 rectangle with a hole in the shape of an hexomino. All 35 problems are likely to be possible.
Sets of squares | {"url":"http://www.recmath.com/PolyPages/PolyPages/Isopolyosols.html","timestamp":"2024-11-08T18:27:16Z","content_type":"text/html","content_length":"8107","record_id":"<urn:uuid:4dca1fd6-a37f-4621-8b9e-c99b04369527>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00298.warc.gz"} |
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• ALGEBRA SOFTWARE | {"url":"https://softmath.com/math-book-answers/multiplying-fractions/advance-problem-solving-on.html","timestamp":"2024-11-11T03:18:20Z","content_type":"text/html","content_length":"35312","record_id":"<urn:uuid:7a3da6f4-31b1-4215-bb61-560e6fda9f80>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00740.warc.gz"} |
Beecrowd 1161 Factorial Sum SolutionBeecrowd 1161 Factorial Sum Solution - Free Code Center
Beecrowd 1161 Factorial Sum Solution
Beecrowd 1161 Factorial Sum Solution in C, C++, Java
Welcome to our comprehensive guide on URI Factorial Sum Solutions in C, C++, and Java. In this guide, we will provide you with an in-depth understanding of the factorial sum problem and its solutions
in these three programming languages. We will also compare and contrast the solutions to help you choose the best option for your specific needs.
What is URI Factorial Sum Problem?
The URI Factorial Sum Problem is a mathematical problem that requires us to calculate the sum of factorials of a given number. The problem is stated as follows:
Given a number n, find the sum of factorials of all the numbers from 1 to n.
For example, if n is 3, then the sum of factorials would be 1! + 2! + 3! = 1 + 2 + 6 = 9.
Bee1161 Solutions in C:
To solve the URI Factorial Sum problem in C, we can use loops to iterate through each number from 1 to n and calculate the factorial sum. Here is the code snippet to solve the problem in C:
int main()
int n, i, fact = 1, sum = 0;
printf("Enter the value of n: ");
scanf("%d", &n);
fact = fact * i;
sum = sum + fact;
printf("Sum of factorials of 1 to %d is %d", n, sum);
return 0;
Bee1161 Solutions in C++:
In C++, we can use a similar approach as C to solve the URI Factorial Sum problem. However, C++ provides us with some additional features such as the iostream library and the using namespace std;
statement. Here is the code snippet to solve the problem in C++:
#include <iostream>
using namespace std;
int main() {
long long int m,n,f=1,f1=1;
for(int i=1;i<=m;i++)
for(int j=1;j<=n;j++)
long long int sum=f+f1;
return 0;
Bee1161 Solutions in Java:
In Java, we can use the BigInteger class to handle large factorials. Here is the code snippet to solve the URI Factorial Sum problem in Java:
import java.math.BigInteger;
import java.util.Scanner;
public class FactorialSum {
public static void main(String[] args) {
Scanner input = new Scanner(System.in);
int n = input.nextInt();
BigInteger fact = BigInteger.ONE, sum = BigInteger.ZERO;
for(int i=1;i<=n;i++)
fact = fact.multiply(BigInteger.valueOf(i));
sum = sum.add(fact);
System.out.println("Sum of factorials of 1 to "+n+" is "+sum);
Comparison of Solutions:
All three solutions provide an efficient and effective way to solve the URI Factorial Sum problem. However, Java’s BigInteger the class provides us with the ability to handle very large factorials
that are not possible with C and C++. C++ is slightly faster than C, but the difference is negligible for small inputs.
In this guide, we provided you with a comprehensive understanding of the URI Factorial Sum problem and its solutions in C, C++, and Java. We hope this guide helps you choose the best solution for
your specific needs.
Competitive programming | C++ | C | Computer programing |Competitive programming website
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Journal of Modern Physics
Vol.5 No.10(2014), Article ID:47440,25 pages DOI:10.4236/jmp.2014.510099
A Renormalizable Theory of Quantum Gravity: Renormalization Proof of the Gauge Theory of Volume Preserving Diffeomorphisms
Christian Wiesendanger
Zurich, Switzerland
Email: christian.wiesendanger@ubs.com
Copyright © 2014 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
Received 14 April 2014; revised 13 May 2014; accepted 6 June 2014
Inertial and gravitational mass or energy momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the gauge theory of volume-preserving diffeomorphisms of
an inner four-dimensional space. The gauge-fixed action and the path integral measure occurring in the generating functional for the quantum Green functions of the theory are shown to obey a
BRST-type symmetry. The related Zinn-Justin-type equation restricting the corresponding quantum effective action is established. This equation limits the infinite parts of the quantum effective
action to have the same form as the gauge-fixed Lagrangian of the theory proving its spacetime renormalizability. The inner space integrals occurring in the quantum effective action which are
divergent due to the gauge group’s infinite volume are shown to be regularizable in a way consistent with the symmetries of the theory demonstrating as a byproduct that viable quantum gauge field
theories are not limited to finite-dimensional compact gauge groups as is commonly assumed.
Renormalization Proof of Gauge Field Theory of Volume-Preserving Diffeomorphisms
1. Introduction
As of today a viable quantum field theory of gravitation has proven to be elusive [1] [2] .
At the microscopic level the Standard Model (SM) of particle physics successfully describes the electromagnetic, weak and strong interactions based on quantized gauge field theories with the
finite-dimensional compact gauge groups[3] -[8] .
At the macroscopic level General Relativity (GR) successfully describes the gravitational interaction based on a classical gauge field theory with the non-compact diffeomorphism group of
four-dimensional spacetime [9] -[11] .
When trying to naively generalize the successful aspects of the gauge field theory ansatz to describe gravity at the quantum level one encounters unsurmountable difficulties—quantizing GR leads to a
non-renormalizable theory and working with other finite-dimensional compact gauge groups to generalize that the SM yields no description of gravity.
Whatever one tries one seems to bang one’s head against two unremovable, yet intertwined roadblocks: 1) against GR and—underpinning it—the Principle of Equivalence stating that inertial and
gravitational masses are equal which forces a geometric description of gravity, and 2) against the seeming non-viability of gauge theories with non-compact gauge groups.
In this and a series of related papers [12] -[14] , we have systematically analyzed both roadblocks and 1) asked ourselves what physics at the quantum level we get when discarding the equality of
inertial and gravitational mass for virtual quantum states and 2) made technical progress in formulating renormalizable gauge field theories based on a non-compact gauge group.
Before turning to the technical renormalization analysis, let us illuminate both aspects a bit more in detail.
1.1. Physical Aspect: Why Should Inertial and Gravitational Mass Be the Same for Virtual Quantum States?
GR has been developed starting from the observed equality of inertial and gravitational mass
Now (a) the observed equality of inertial and gravitational mass of an on-shell physical object in its rest frame together with (b) the conservation of the inertial energy-momentum
assuming that the gravitational energy-momentum
To explore this route, let us postulate both [14] .
The observed equality of inertial and gravitational energy-momentum in this approach is assured by taking a gravitational limit for on-shell observable physical objects equating gravitational and
inertial energy-momentum, the construction of which is based on the analysis of asymptotic states and the definition of a suitable [14] .
1.2. Technical Aspect: Why Should Viable Quantum Gauge Field Theories Be Limited to Finite-Dimensional Compact Gauge Groups?
In the process of constructing the gauge theory of volume-preserving diffeomorphisms of a four-dimensional inner space which emerges from the above thinking, we have to deal with new difficulties
arising from the noncompactness of the gauge group. 1) The gauge field Hamiltonian is not manifestly positive definite which is cured by a natural condition on the support of the gauge fields in
inner space [12] . 2) The quantization and subsequent derivation of Feynman rules yield no additional complications in comparison to the Yang-Mills case w.r.t. spacetime-related expressions, but it
yields badly divergent-looking integrals over inner momenta related to the infinite volume of the gauge group—a phenomenon which does not plague Yang-Mills theories due to the assumed gauge group
compactness. 3) As already mentioned, a gravitational limit for on-shell observable physical objects has to be taken to ensure the observed equality of inertial and gravitational energy-momentum.
This limit has to respect the unitarity of the physical S-matrix which we have established in [14] .
In [13] we have dealt with issues 1) and 2) at the one-loop level and established that the pure gauge field theory is asymptotically free whereas the inclusion of all SM fields destroys asymptotic
freedom, hence assuring the observability of the gauge field quanta. To demonstrate the fundamental viability of the theory beyond one loop, however, we have to give both a proof of its
renormalizability w.r.t. the occurring spacetime divergences as well as to propose a general approach dealing with the inner divergences, both of which we provide in this paper.
In the process we establish a well-defined perturbative expansion of the quantum effective action demonstrating its spacetime renormalizability and the existence of regularization schemes for the
inner momentum integrals consistent with the symmetries of the theory. In fact, the spacetime-renormalized and inner-space-regularized quantum effective action defines a viable quantum field theory
for each regularization scheme in terms of the original finite number of coupling constants, masses etc.
Stated otherwise it is possible to consistently establish a quantum gauge field theory not only for compact gauge groups, but also for at least the non-compact gauge group of volume-preserving
diffeomorphisms. The price to pay comes in the form of an additional regularization scheme for the divergent sums over inner degrees of freedom—each such scheme which is compatible with the
symmetries of the theory establishes one welldefined version of the quantum theory belonging to the classical gauge theory one starts with. And each such version with its finite number of coupling
constants, masses etc. yields as precise predictions as do its YangMills cousins—predictions which are equal for all such versions at tree level, but depend on the chosen regularization scheme for
the loop contributions. In exactly the same way as experiment has to tell the physical values of the various couplings, masses etc. in this or the Yang-Mills cases, experiment ultimately has then to
choose the regularization preferred by Gravity.
So let us turn to implement the program outlined above.
2. Classical Gauge Theory of Volume-Preserving Diffeomorphisms
In this section we review the basics of the gauge theory of volume preserving diffeomorphisms as presented in [12] .
Let us start with a four-dimensional real vector space
act as a group
Next we want to represent this group on spaces of differentiable functions
is well-defined. Above we have introduced a parameter
Turning to the passive representation of
transforming the fields only, where
The unimodularity condition Equation (3) translates into the infinitesimal gauge parameter
Note the crucial fact that the algebra
as required by the finite transformations
As a result we can write infinitesimal transformations in field space
as anti-unitary operators w.r.t. the scalar product Equation (4). Both the
Introducing the variation
In mathematical terms we have just reviewed the group
Next we turn to the four-dimensional Minkowski spacetime (M^4,
Extending the global volume-preserving diffeomorphism group to a group of local transformations we allow
We note that these fields
In generalization of Equation (9) we thus consider
The formulae Equation (5) together with Equation (6) with the fields now
Next we introduce a covariant derivative
To fulfil Equation (12) we make the usual ansatz
consistent with
The requirement Equation (12) translates into the transformation law for the gauge field
which reads in components
Note that the consistent decomposition of both
Let us next define the field strength operator
which again can be decomposed consistently w.r.t.
Under a local gauge transformation the field strength and its components transform covariantly
As required for algebra elements
The above can be viewed as giving rise to a principal bundle P with base space M^4, the typical fibre given by the volume-preserving diffeomorphisms
Note that no reference to a metric in inner space has been necessary so far.
Turning to the dynamics of the classical gauge theory the Lagrangian for the gauge fields as given in [12] does depend on a metric [12] and discussed there. Under inner coordinate
transformations this metric transforms as a contravariant tensor
In general inner coordinates the Lagrangian is given by [12]
and is indeed covariant under the combined gauge transformations Equation (20) and Equation (21). Above, inner indices such as
Excluding any dynamical role for the metric—which we take as an a priori as explained in [12] —we require that the geometry of the inner space is flat, hence
Such choices of coordinates amount to partially fixing a gauge the class of which are the Minkowski gauges [12] . The remaining gauge degrees of freedom leaving
In Minkowski gauges and Cartesian inner coordinates the Lagrangian is [12] still given by Equation (22) with inner indices such as
3. Generating Functional for the Green Functions of the Gauge Theory of Volume-Preserving Diffeomorphisms
In this section we establish the starting point for the renormalizability proof of the gauge theory of volume preserving diffeomorphisms in terms of the gauge-fixed generating functional for its
quantum Green functions as presented in detail in [13] .
After the review of the classical theory let us turn to the quantum theory and specifically to the generating functional [13]
The path integrals over the fields are restricted by the
which we will keep throughout the rest of the paper and we have dropped the explicit dependence on the coordinates
The gauge-fixed action
where the latter is a sum of the gauge field Lagrangian
In general, the inner indices in Equation (27) are contracted with
To represent matter we have added a Dirac field with action
Finally we have introduced currents
belonging to the gauge-fixing functional
Note that
To be specific in our further analysis let us (i) partially fix the gauge to Minkowski gauges introduced above which preserve the inner metric Appendix and [12] ) and (ii) fix the remaining gauge
degrees of freedom by choosing the Lorentz gauge taking
The Faddeev-Popov-DeWitt kernel for this choice is easily calculated to be
Due to (ii) the inner indices in Equation (32) are contracted with
Denoting the mass dimension of a field determined as usual by the quadratic part of the Lagrangian by [..] we find
Hence, by inspection the Lagrangian above contains products of fields and their spacetime derivatives with mass dimensions four or less only. This ensures renormalizability in the Dyson sense, i.e.
counterterms which have to be introduced to absorb spacetime divergences arising in perturbation theory have mass dimensions four or less as well. For the theory to be truly renormalizable, however,
these counterterms arising in a perturbation expansion of the generating functional must be shown to take the same form as the above Lagrangian—a task to which we turn next.
Note that we will have to deal with an additional type of divergences in Feynman graphs arising from the generalized sums over inner degrees of freedom. These turn out to be divergent integrals over
inner momentum space variables which we will properly define in Section 10.
4. BRST-Type Invariance of Modified Gauge-Fixed Action and of Path Integral Measure Occurring in the Generating Functional for the Quantum Green Functions
In this section we rewrite the generating functional in terms of Nakanishi-Lautrup fields and a new action
Let us start with Equation (26)
where we have introduced the quantity
in terms of a Gaussian integral over the Nakanishi-Lautrup fields
the Green functions of the theory are now given as path integrals over the fields
By construction the gauge-fixed modified action
The transformations Equation (38) are nilpotent, i.e. if
The proof for the fields above is straightforward, but somewhat tedious. Here we just sketch the verification of
using the chain-rule and the anticommutativity of
The extension to products of polynomials in these fields follows easily [4] .
To verify the BRST invariance of
Next with the use of Equation (29) we determine the BRST transform of
which yields
Hence we can rewrite
Finally it follows from the nilpotency of the BRST transformation
Next let us analyze the path integration measure
Under the BRST-type transformations Equation (38) the Jacobian turns out to be [15]
as the trace is easily shown to vanish.
In addition the BRST-type transformations Equation (38) respect the divergence-free condition ensuring that the fields live in the gauge algebra
As a result the measure Equation (49) is invariant under the BRST-type transformations Equation (38).
5. Symmetries of the Quantum Effective Action and the Zinn-Justin Equation
In this section we derive the Zinn-Justin equation for the gauge theory of volume-preserving diffeomorphisms which follows from the BRST-type invariance of both the modified action
We start with the generating functional for Green functions
denote the gauge and ghost field variables constrained to live in the gauge algebra which is ensured by
and where
denotes the matter field. The BRST-type transformations
act on the various fields as defined by Equation (38)
As shown in the previous section all: the modified action
which means that the quantum average
Next we define the related quantum effective action in the presence of the current
with the connected vacuum persistence amplitude
Taking the left variational derivative of the effective action w.r.t. a field degree of freedom
where we have made use of the definition of
Taking next the right variational derivative of the effective action w.r.t. the external current
Finally, inserting both Equation (62) and Equation (64) into Equation (59) we obtain
i.e. the Zinn-Justin equation for the gauge theory of volume-preserving diffeomorphisms.
Defining the antibracket of two functionals
the Zinn-Justin equation finally can be re-written in the form
as the interchange of
It is this equation which contains the information at the quantum level related to the original gauge symmetry at the classical level which constrains the form of the ultraviolet divergences so that
the theory turns out to be renormalizable—the proof of which we turn to next, adapting the general renormalization proof for Yang-Mills theories as outlined e.g. in [4] [15] to our case.
6. Constraints Put on the Perturbative Expansion of the Quantum Effective Action by the Zinn-Justin Equation
In this section we analyze the constraints on the perturbative expansion of the quantum effective action imposed by the Zinn-Justin equation. They result in a combination of the renormalized action
We start by rewriting the action
as the sum of a renormalized action
Next we turn to the quantum effective action
where the
To prove the renormalizability of the gauge theory of volume-preserving diffeomorphisms it is sufficient to demonstrate that the infinite parts of the
To do so we start inserting the perturbation series Equation (69) into the Zinn-Justin Equation (67)
which yields for the
We note that
Next we assume that for all
the consequences of which we evaluate below.
Before doing so we recall that the action
To proceed we next need to establish under which (infinitesimal) symmetry transformations of the action
the quantum effective action is invariant. To do so we repeat the calculation in Section 5 assuming that all: the modified action
Repeating the short calculation performed in Equation (58) yields the general result
For symmetry transformations which are linear in the fields
which means that the quantum effective action
under which the action
There are these linearly realized symmetries together with the restriction Equation (72) imposed by the ZinnJustin equation that will be sufficient to determine the general form
Our first step is to use dimensional analysis and ghost number conservation to determine the
We denote the mass dimension of a field by [..]. Then for
As a result the dimension-four quantity
Turning to analyze the impact of ghost number conservation we denote the ghost quantum number of a field by
This shows that no terms in
is independent of
As a result of this first step we hence have fully determined the
introducing the new quantities
Turning to step two we insert Equations (81) and (82) into the Zinn-Justin relation Equation (72). We obtain two constraints to zeroth
and to first order in
To further extract the content of the two constraints above we next define
Then Equation (84) together with the nilpotency of the original BRST-type transformations Equation (57) tells us that the transformations
are nilpotent as well:
And Equation (83) tells us that
These informations will allow us to determine the most general form of both the nilpotent transformations
7. Most General Form of the Quantum BRST-Type Transformations
In this section we determine the most general form of
We start noting that the
In the case of the ghost field
Checking against the additional requirements that
In the case of the gauge field
In the case of the matter field
For the antighost field
Next we turn to working out the implication of the nilpotency condition Equation (88) on the
For the ghost field
which follows from
To build a constant
As a result the transformation Equation (93) reduces for the ghost field
which is easily found to be nilpotent as required.
For the gauge field
which follows from
and that
so that the transformation Equation (93) reduces for the gauge field
Nilpotency requires
which is fulfilled if
the transformation Equation (93) for the gauge field
For the matter field
the transformation Equation (93) reduces to
which is found to be nilpotent if
As a result we find the most general form of
which amounts to a renormalization of the BRST-type transformations Equation (57) under which
8. Most General Form of
In this section we show that the most general form of
We start noting that
in terms of a Lagrangian
In addition
the action
Next we turn to establish the most general form of
First, the ghost number conservation requires ghosts
in ghost terms only one more derivative
where we note that a gauge field
Second, the above symmetry constraints require Nakanishi-Lautrup fields
Adding constant tensors so as to combine inner indices to yield scalar expressions in inner space and noting that
Variation of
The first line above vanishes identically if
the second line if
and the last two lines if and only if
Insertion of Equation (120) in the third and fourth lines yields
and in the fifth and sixth lines
as required which leaves us with
Next by inspection of Equation (112) we note that a modified BRST-type transformation acts on the gauge and matter fields exactly in the same way as the original gauge transformation Equation (57)
with local gauge parameter
and rescaled inner derivatives
Equation (124) then simply tells us that
and the only matter field Lagrangian
As our final result we find the most general
Apart from the appearance of a number of new constant coefficients this is exactly the original Lagrangian
9. Feynman Rules in the Lorentz Gauge
In this section we derive the Feynman rules for the gauge theory of volume-preserving diffeomorphisms in the Lorentz gauge as a prerequisite to establish viable regularization schemes for the
divergent inner momentum integrals occurring in a loop expansion of the quantum effective action.
In order to analyze the structure of inner momentum space integrals at any order of a loop-wise expansion of the quantum effective action we turn to Feynman diagrams. Hence we need to derive the
momentum space Feynman rules for the Lagrangian
As usual we split
The free Lagrangian
can by partial integration be easily brought into the usual quadratic form
Above we have introduced the non-interacting gauge and ghost field fluctuation operators
The corresponding free propagators
is the scale-invariant delta function transversal in inner space which is compatible with the constraint that both the gauge and the ghost fields are divergence-free in inner space. Note that the
transversal delta-function [3] .
After a little algebra we find the momentum space gauge and ghost field propagators to be
Note that both propagators are transversal in inner space and that they reduce to
Note that the spacetime parts of the propagators equal the usual Yang-Mills propagators.
Next we calculate the various vertices related to the interaction Lagrangian
We start with the tri-linear gauge field self-coupling term
corresponding to a vertex with three vector boson lines. If these lines carry incoming spacetime momenta
The quadri-linear gauge field self-coupling term
corresponds to a vertex with four vector boson lines. If these lines carry incoming spacetime momenta
Finally, the gauge-ghost field coupling term
corresponds to a vertex with one outgoing and one incoming ghost line as well as one vector boson line. If these lines carry incoming spacetime momenta
In summary, the above propagators and vertices allow us to perturbatively evaluate the quantum effective action of the theory in a loop-wise expansion. In addition they are manifestly covariant
w.r.t. spacetime Poincaré transformations and related to the gauge symmetry-global inner Poincaré transformations. Most importantly they are invariant under inner scale transformations (
Note that for any Feynman graph the analogon of the sums over Lie algebra structure constants in Yang-Mills theories in the theory under consideration are integrals over inner momentum space
10. Regularization of the Divergent Inner Momentum Integrals
In this section we show that
According to the Feynman rules stated in Section 9 vertices contribute simple monomials in the inner momenta to a Feynman graph so that inner momentum space integrals occurring in the
As these
Above we have extracted the Lorentz structure
The remaining scale-invariant integrals are of the form
where the subscript
Before turning to the regularization of the
Hence, to get a well defined theory it is sufficient to find a regularization procedure for the
Stated differently each regularization procedure for the
Next we turn to specifing one such regularization procedure compatible with the symmetries of the quantum effective action. This will be done in three steps. In the first step we will slice the inner
momentum Minkowski space into light-like, time-like and space-like shells of invariant lengths, in the second we will discard the space-like shells and in the third we will invariantly regularize the
remaining integral over lightand time-like shells making use of the existence of an arbitrary point mass and its rest frame.
Slicing the inner Minkowski space in the first step into light-like, time-like and space-like shells of invariant lengths
Second, to regularize
which is a Lorentz-invariant procedure. This cutoff of space-like shells arises naturally from the condition of positivity for the Hamiltonian for the gauge and ghost fields as derived in [12] which
restricts all fields Fouriertransformed over inner space to have support on the set
denote the forward and backward light cones in inner momentum space.
Third, we can always assume the existence of a point mass which is at rest in some Lorentz frame. In that frame we can define a time-like vector
In addition
again for all
This allows us to define
which is a positive, finite and manifestly scale-invariant Lorentz scalar for all
demonstrating its asymptotic freedom as well as in the beta function of the theory including all SM fields
which shows that within the SM the gauge quanta are not confined and hence observable.
This completes the proof that the gauge theory of volume-preserving diffeormorphisms can be consistently quantized and turned into a well-defined quantum field theory. In fact we should rather say
into a family of welldefined quantum field theories which differ by the choice of regularization procedure for the inner momentum integrals.
11. Conclusions
In this paper we have established that the gauge theory of volume-preserving diffeomorphisms of an inner fourdimensional space, which arises naturally from the assumption that inertial and
gravitational mass need not be the same for virtual quantum states, can be consistently quantized and turned into a well-defined quantum field theory or rather into a family of well-defined quantum
field theories which differ by the choice of regularization procedure for the inner momentum integrals.
To get there we have first shown that the gauge-fixed action and the path integral measure occurring in the generating functional for the quantum Green functions of the theory obey a BRST-type
symmetry. This has allowed us next to demonstrate that the quantum effective action fulfils a Zinn-Justin-type equation which limits the infinite parts of the quantum effective action to have the
same form as the gauge-fixed Lagrangian of the theory proving its spacetime renormalizability. Finally based on the theory’s Feynman rules we have shown that the divergent inner space integrals
related to the gauge group’s infinite volume are regularizable in a way consistent with the symmetries of the theory. In this context it is worth noting that as a byproduct viable quantum gauge field
theories are not limited to finite-dimensional compact gauge groups as is commonly assumed.
Finally: what has all of this to do with gravity? To answer this question we have analysed the classical limit [16] . Hence, as is necessary for the interpretation of the present theory as a quantum
theory of gravity, GR emerges as its classical limit.
This result implies then that the SM of particle physics can be completed to contain the gravitational interaction at the quantum level as well.
Practically one starts with the renormalizable action for the SM [4] [5] [8]
where for the sake of clarity we reinsert the arguments
denotes the covariant derivative in a suitable gauge algebra representation.
To get the SM coupled to gravity, in short SM + G, we only have to endow each SM field with inner space coordinates
introduce the gauge field
The renormalizable action for the SM + G is then simply
where the first term is the gauge field action Equation (22). All amplitudes or other expressions related to observable quantities calculated within the SM + G obviously have to be evaluated in the
physical limit as discussed in [14] .
It is reassuring that not only the microscopic strong and electro-weak interactions can be described within a renormalizable quantum gauge field theory framework formulated on a priori flat
spacetime. In fact gravity at the quantum level can be described by following exactly the same logic, however, the theory gets more complicated due to its non-compact gauge group having an infinite
volume. Yet it is still renormalizable. So nature seems to allow for a consistent, rupture-free picture based on conservation laws and symmetry considerations at least up to energy scales far beyond
experimental reach.
Finally, new physics may derive from Equation (163), for example in the realm of cosmology and the early universe where new light might be shed on unsolved questions arising e.g. around dark energy
[17] . For sure the quantum gauge field
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2. Kiefer, C. (2007) Quantum Gravity. Oxford University Press, Oxford. http://dx.doi.org/10.1093/acprof:oso/9780199212521.001.0001
3. Weinberg, S. (1995) The Quantum Theory of Fields I. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9781139644167
4. Weinberg, S. (1996) The Quantum Theory of Fields II. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9781139644174
5. Itzykson, C. and Zuber, J.-B. (1985) Quantum Field Theory. McGraw-Hill, Singapore.
6. O’Raifeartaigh, L. (1986) Group Structure of Gauge Theories. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511564031
7. Pokorski, S. (1987) Gauge Field Theories. Cambridge University Press, Cambridge.
8. Cheng, T.-P. and Li, L.-F. (1984) Gauge Theory of Elementary Particle Physics. Oxford University Press, Oxford.
9. Weinberg, S. (1972) Gravitation and Cosmology. John Wiley & Sons, New York.
10. Landau, L.D. and Lifschitz, E.M. (1981) Lehrbuch der Theoretischen Physik II: Klassische Feldtheorie. Akademie-Verlag, Berlin.
11. Will, C.M. (1993) Theory and Experiment in Gravitational Physics. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511564246
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14. Wiesendanger, C. (2013) Classical and Quantum Gravity, 30, 075024. arXiv:1203.0715 [math-ph]http://dx.doi.org/10.1088/0264-9381/30/7/075024
15. Zinn-Justin, J. (1993) Quantum Field Theory and Critical Phenomena. Oxford University Press, Oxford.
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17. Weinberg, S. (2008) Cosmology. Oxford University Press, Oxford.
Appendix: Notations and Conventions
Generally, (M^4,
The same lower and upper indices are summed unless indicated otherwise. | {"url":"https://file.scirp.org/Html/5-7501820_47440.htm","timestamp":"2024-11-13T08:39:26Z","content_type":"application/xhtml+xml","content_length":"169579","record_id":"<urn:uuid:14930f3c-0c43-4cd2-8789-369893a05c47>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00816.warc.gz"} |
Design & Analysis of full adders using adiabatic logic
Volume 01, Issue 05 (July 2012)
Design & Analysis of full adders using adiabatic logic
DOI : 10.17577/IJERTV1IS5385
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Design & Analysis of full adders using adiabatic logic
G.Rama Tulasi, K.Venugopal, B.Vijayabaskar, R.SuryaPrakash St. Theressa Institute of Engineering & Technology Vizianagaram, Andhrapradhesh.
Abstract: In this paper we are going to compare the adiabatic logic designs & designing a new full adder using ECRL & PFAL logics after that the simulations were done using Microwind & DSCH. Thus the
efficiency of the circuits are shown & compared using different nano meter technologies.
Keywords: Adiabatic, ECRL Adder, PFAL Adder, Full adder, Low Power Adders,
1. Introduction:
The main objective of this thesis is to provide new low power solutions for Very Large Scale Integration (VLSI) designers. Especially, this work focuses on the reduction of the power dissipation,
which is showing an ever-increasing growth with the scaling down of the technologies. Various techniques at the different levels of the design process have been implemented to reduce the power
dissipation at the circuit, architectural and system level.
Furthermore, the number of gates per chip area is constantly increasing, while the gate switching energy does not decrease at the same rate, so the power dissipation rises and heat removal
becomes more difficult and expensive. Then, to limit the power dissipation, alternative solutions at each level of abstraction are proposed.
The dynamic power requirement of CMOS circuits is rapidly becoming a major concern in the design of personal information systems and large computers. In this thesis work, a new CMOS logic family
called ADIABATIC LOGIC, based on the adiabatic switching principle is presented. The term adiabatic comes from thermodynamics, used to describe a process in which there is no exchange of heat
with the environment. The adiabatic logic structure dramatically reduces the power dissipation. The adiabatic switching technique can achieve very low power dissipation, but at the expense of
circuit complexity. Adiabatic logic offers a way to reuse the energy stored in the load capacitors rather than the traditional way of discharging the load capacitors to the ground and wasting
this energy.
This thesis work demonstrates the low power dissipation of Adiabatic Logic by presenting the results of designing various design/ cell units employing Adiabatic Logic circuit techniques. A family
of full-custom conventional CMOS Logic and an Adiabatic Logic units for example, an inverter, a two- input NAND gate, a two-input NOR gate, a two-input XOR gate, a two-to-one multiplexer and a
one-bit Full
Adder were designed in Mentor Graphics IC Design Architect using standard TSMC 0.35 µm technology, laid out in Microwind IC Station.
All the circuit simulations has been done using various schematics of the structures and post-layout simulations are also being done after they all have been laid-out by considering all the basic
design rules and by running the LVS program. Finally, the analysis of the average dynamic power dissipation with respect to the frequency and the load capacitance was done to show the amount of
power dissipated by the two logic families.
2. Motivation:
In the past few decades ago, the electronics industry has been experiencing an unprecedented spurt in growth, thanks to the use of integrated circuits in computing, telecommunications and
consumer electronics. We have come a long way from the single transistor era in 1958 to the present day ULSI (Ultra Large Scale Integration) systems with more than 50 million transistors in a
single chip.
The ever-growing number of transistors integrated on a chip and the increasing transistor switching speed in recent decades has enabled great performance improvement in computer systems by
several orders of magnitude. Unfortunately, such phenomenal performance improvements have been accompanied by an increase in power and energy dissipation of the systems. Higher power and energy
dissipation in high performance systems require more expensive packaging and cooling technologies, increase cost, and decrease system reliability. Nonetheless, the level of on-chip integration
and clock frequency will continue to grow with increasing performance demands, and the power and energy dissipation of high-performance systems will be a critical design constraint.
For example, high-end microprocessors in 2010 are predicted to employ billions of transistors at clock rates over 30GHz to achieve TIPS (Tera Instructions per seconds) performance [1]. With this
rate, high-end microprocessors power dissipation is projected to reach thousands of Watts. This thesis investigates one of the major sources of the power/energy dissipation and proposes and
evaluates the techniques to reduce the dissipation.
Digital CMOS integrated circuits have been the driving force behind VLSI for high performance computing and other applications, related to science and technology. The demand for digital CMOS
integrated circuits will continue to increase in the near future, due to its important salient features like low power, reliable performance and improvements in the processing technology.
The word ADIABATIC comes from a Greek word that is used to describe thermodynamic processes that exchange no energy with the environment and therefore, no energy loss in the form of dissipated
heat. In real-life computing, such ideal process cannot be achieved because of the presence of dissipative elements like resistances in a circuit. However, one can achieve very low energy
dissipation by slowing down the speed of operation and only switching transistors under certain conditions. The signal energies stored in the circuit capacitances are recycled instead, of being
dissipated as heat. The adiabatic logic is also known as ENERGY RECOVERY CMOS .
It should be noted that the fully adiabatic operation of the circuit is an ideal condition which may only be approached asymptotically as the switching process is slowed down. In most practical
cases, the energy dissipation associated with a charge transfer event is usually composed of an adiabatic component and a non-adiabatic component. Therefore, reducing all the energy loss to zero
may not possible, regardless of the switching speed. With the adiabatic switching approach, the circuit energies are conserved rather than dissipated as heat. Depending on the application and the
system requirements, this approach can sometimes be used to reduce the power dissipation of the digital systems.
Here, the load capacitance is charged by a constant-current source (instead of the constant- voltage source as in the conventional CMOS circuits).
Here, R is the resistance of the PMOS network. A constant charging current corresponds to a linear voltage ramp. Assume, the capacitor voltage VC is zero initially.
3. Adiabatic Logic Gate:
In the following, we will examine simple circuit configurations which can be used for adiabatic switching. Figure 3.2 shows a general circuit topology for the conventional CMOS gates and
adiabatic counterparts. To convert a conventional CMOS logic gate into an adiabatic gate, the pull-up and the pull-down networks must be replaced with complementary transmission-gate (T-gate)
networks. The T-gate network implementing the pull-up function is used to drive the true output of the adiabatic gate, while the T- gate network implementing the pull-down function drives the
complementary output node. Note that all the inputs should also be available in complementary form. Both the networks in the
adiabatic logic circuit are used to charge-up as well as charge-down the output capacitance, which ensurs that the energy stored at the output node can be retrieved by the power supply, at the
end of each cycle. To allow adiabatic operation, the DC voltage source of the original circuit must be replaced by a pulsed-power supply with the ramped voltage output.
4. Adiabatic Logic Types:
Practical adiabatic families can be classified as either PARTIALLY ADIABATIC or FULLY ADIABATIC [12]. In a PARTIALLY ADIABATIC CIRCUIT, some
charge is allowed to be transferred to the ground, while in a FULLY ADIABATIC CIRCUIT, all the charge on the load capacitance is recovered by the power supply. Fully adiabatic circuits face a lot
of problems with respect to the operating speed and the inputs power clock synchronization.
ECERL-Efficient Charge Recovery Logic:
Efficient Charge Recovery Logic (ECRL) proposed by Moon and Jeong [13], shown in Figure 4.1, uses cross- coupled PMOS transistors. It has the structure similar to Cascode Voltage Switch Logic
(CVSL) with differential signaling.
It consists of two cross-coupled transistors M1 and M2
and two NMOS transistors in the
An AC power supply pwr is used for ECRL gates, so as to recover and reuse the supplied energy. Both out and /out are generated so that the power clock generator can always drive a constant load
capacitance independent of the input signal. A more detailed description of ECRL can be found in. Full output swing is obtained because of the cross-coupled PMOS transistors in both precharge and
recover phases. But due to the threshold voltage of the PMOS transistors, the circuits suffer from the non-adiabatic loss both in the precharge and recover phases. That is, to say, ECRL always
pumps charge on the output with a full swing. However, as the voltage on the supply clock approaches to
Figure1: The Basic Structure of the Adiabatic ECRL Logic.
So the recovery path to the supply clock to the supply clock is disconnected, thus, resulting in
= C|V
tp| / 2
incomplete recovery. Vtp is the threshold voltage of PMOS transistor. The amount of loss is given as
5. Positive Feedback Adiabatic Logic:
The partial energy recovery circuit structure named Positive Feedback Adiabatic Logic (PFAL) [15] has been used, since it shows the lowest energy consumption if compared to other similar
families, and a good robustness against technological parameter variations. It is a dual-rail circuit with partial energy recovery. The general schematic of the PFAL gate is shown in Figure
4.3. The core of all the PFAL gates is an adiabatic amplifier, a latch made by the two PMOS M1-M2 and two NMOS M3-M4, that avoids a logic level degradation on the output nodes out and /out. The
two n-trees realize the logic functions. This logic family also generates both positive and negative outputs. The functional blocks are in parallel with the PMOSFETs of the adiabatic amplifier
and form a transmission gate. The two n-trees realize the logic functions. This logic family also generates both positive and negative outputs.
Thus, from Equation (4.2), it can be inferred that the non-adiabatic energy loss is dependent on the load capacitance and independent of the frequency of operation.
The two major differences with respect to ECRL are that the latch is made by two PMOSFETs and two NMOSFETS, rather than by only two PMOSFETs as in ECRL logic, and that the functional blocks are
in parallel with the transmission PMOSFETs. Thus the equivalent resistance is smaller when the capacitance needs to be charged. The energy dissipation by the CMOS Logic family and Adiabatic PFAL
Logic family can be Seen.
6. Adiabatic Full Adder using PFAL &
A partially adiabatic logic family PFAL one-bit Full Adder block can be implemented as shown in the Figure 5.23 ( for SUM block) and Figure 5.24 (for OUTPUT_CARRY) below, respectively.
Figure4: PFAL Sum Circuit
Figure5: PFAL Carry Circuit
Figure6: ECRL SUM Circuit
Figure7: ECRL Carry Circuit
7. Conclusion
The thesis primarily was focused on the design of low power CMOS cell structures, which is the main contribution of this work. The design of low power CMOS cell structures uses fully
complementary CMOS logic style and an adiabatic PFAL logic style. The basic principle behind implementing various design units in the two logic styles is to compare them with reference to the
average power dissipated by all of them.
A family of full-custom conventional CMOS Logic and an Adiabatic Logic units were designed in Mentor Graphics IC Design Architect using standard TSMC
0.35 µm technology, layout them in Microwind & Digital Schematic and the analysis of the average dynamic power dissipation with respect to the frequency and the load capacitance was done. It was
found that the adiabatic PFAL logic style is advantageous in applications where power reduction is of prime importance as in high performance battery-portable digital systems running on batteries
such as note-book computers, cellular phones and personal digital assistants.
With the adiabatic switching approach, the circuit energies are conserved rather than dissipated as heat. Depending on the application and the system requirements, this approach can be used to
reduce the power dissipation of the digital systems. With the help of adiabatic logic, the energy savings of upto 76 % to 90 % [15] can be reached.
Circuit simulations show that the adiabatic design units can save energy by a factor of 10 at 50 MHz and about 2 at 250 MHz, as compared to logically equivalent conventional CMOS implementation.
8. Future Work
1. ADIAMEMS: To perform digital logic in CMOS in a truly adiabatic (asymptotically thermodynamically reversible) fashion requires that the logic transitions be driven by a quasi-trapezoidal
(flat-topped) power-clock voltage waveform, which must be generated by a resonant element with very high Q (quality factor). Recently, MEMS resonators have attained very high frequencies and
Q factors and are becoming widely used
in communications system-on-chip (SOC) for RF signal filtering, amplification, etc.
2. APPLICATION OF NANO-TECHNOLOGY: Carbon nano-tubes grown using Chemical Vapor Deposition (CVD) can be selected to conform to a spiraling shape. Thus, a good quality factor Q can be achieved.
The work left to be done for this design would include a method for causing it to keep its form, since nano-tubes are typically not rigid. Also, putting the tube to use in a circuit would
lower the effective Q due to the junction discontinuities.
3. SPACECRAFT: The high cost-per-weight of launching computing-related power supplies, solar panels and cooling systems into orbit imposes a demand for adiabatic power reduction in spacecraft in
which these components weigh a significant fraction of total spacecraft weight.
9. Reference
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Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 4-0-4
MATH 310 Linear Algebra 3 College of Arts & Sciences
Matrices and systems of linear equations, vector spaces, change of base matrices, linear transformations, determinants, eigen-values and eigen-vectors, canonical forms.
Prerequisite Course(s): Prerequisite: MATH 180
Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 3-0-3
MATH 320 Differential Equations 3 College of Arts & Sciences
Solving first order ordinary differential equations: exact, separable, and linear. Application to rates and mechanics. Theory of higher order linear differential equations. Method of undetermined
coefficients and variation of parameters. Application to vibrating mass and electric circuits. Power series solutions: ordinary and singular points, the method of Frobenius. Partial differential
equations: the method of separation of variables.
Prerequisite Course(s): Prerequisite: MATH 260
Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 3-0-3
MATH 330 Computational Analysis 4 College of Arts & Sciences
This course consists of a calculus-based introduction to the use of mathematical software in applied problems in science and engineering. Matlab: basic syntax and development environment; debugging;
help interface; basic math objects; visualization and graphical output; vectorization; scripts and functions; file i/o; arrays, structures, and strings; Mathematica: basic syntax and the notebook
interface, visualization, symbolic operations such as differentiation, integration, partial fractions, series expansions, solution of algebraic equations. Mathematica programming (rule-based,
functional, and procedural) and debugging, plotting, and visualization. The course will emphasize good programming habits, choosing the appropriate language/software for a given scientific task and
the use of numerical and symbolic math software to enhance learning and perform tests. Each of the concepts and programming tools covered should be illustrated through the application and integration
of calculus tools to scientific problems. This will be reinforced via individual lab work during class as well as teamwork in homework and class projects.
Prerequisite Course(s): Prerequisite: MATH 260
Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 3-2-4
MATH 350 Advanced Calculus 3 College of Arts & Sciences
Topics include: Vector functions of several variables, the Jacobian matrix, the generalized chain rule, inverse function theorem, curvilinear coordinates, the Laplacian in cylindrical and spherical
co-ordinates, Lagrange multipliers, line integrals, vector differential and integral calculus including Green's, Stokes's and Gauss's theorem. The change of variable in multiple integrals, Leibnitz's
rule, sequences and uniform convergence of series.
Prerequisite Course(s): Prerequisite: MATH 260
Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 3-0-3
MATH 360 Functions of a Complex Variable 3 College of Arts & Sciences
The general theory of functions of a complex variable, analytic functions, the Cauchy-Riemann equations, the Cauchy integral theorem and formula, Taylor series, Laurent series, singularities and
residues, conformal mappings with applications to problems in applied science.
Prerequisite Course(s): Prerequisite: MATH 260
Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 3-0-3
MATH 370 Real Analysis 3 College of Arts & Sciences
This course focuses on rigorous treatment of the foundations of real analysis in one variable. Topics include: properties of the real number system, sequences, continuous functions, topology of the
real line, compactness, derivatives, the Riemann integral, sequences of functions, uniform convergence, infinite series and Fourier series. Additional topics may include: Lebesgue measure and
integral on the real line, metric spaces, and analysis on metric spaces.
Prerequisite Course(s): Prerequisite: MATH 260
Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 3-0-3
MATH 410 Numerical Linear Algebra 3 College of Arts & Sciences
This course focuses on computational algebra methods and their applications, using basic programming with Matlab or Python. Topics should include: Direct methods (gauss elimination), Iterative
methods (CG and GMRES), QR/ Gram Schmidt, Eigen decomposition, SYD and applications (matrix norms, condition number, low rank approximation, principal component analysis, linear regression). Extra
time can be used for applications and projects, or discussion of sparse and structured matrix methods.
Prerequisite Course(s): Prerequisites: MATH 310
Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 3-0-3
MATH 430 Mathematics of X-ray Imaging 3 College of Arts & Sciences
ln this course we introduce the mathematical techniques used to model measurements and reconstruct images. As a simple representative case we study transmission X-ray tomography (CT).I n this context
we will cover the basic principles of mathematical analysis, the Fourier transform¿Interpolation and approximation of functions, sampling theory, digital filtering and noise analysis. Since imaging
is done with computers, there will be a programming part to each homework assignment in Mathematica to complement theoretical ideas with numerical implementation.
Prerequisite Course(s): Prerequisites: MATH 260, MATH 310, MATH 320
Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 3-0-3
MATH 440 Numerical Optimization 3 College of Arts & Sciences
Many problems in science, engineering, medicine and business involve optimization. in which we seek to optimize a mathematical measure of goodness subject to constraints. This course will cover the
basics of smooth unconstrained and constrained optimization in one and more variables: first and second order conditions, Lagrange multipliers, KKT conditions, Gradient descent, Newton and
Quasi-Newton methods. .Key concepts and methods in mathematical programming will then be covered: linear programming, quadratic and convex programming (simplex method, primal-dual methods, interior
point methods) with applications to engineering, optimal control and machine learning.
Prerequisite Course(s): Prerequisites: MATH 410
Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 3-0-3
MATH 450 Partial Differential Equations 3 College of Arts & Sciences
Generalities on linear partial differential equations and their applications to physics. Solution of initial boundary value problems for the heat equation in one dimension, eigen-function expansions.
Definition and use of Fourier series and Fourier transform. Inhomogeneous problems. The wave equation in one dimension. Problems in two dimensions: vibrating rectangular membranes, Dirichlet and
Neumann problems.
Prerequisite Course(s): Prerequisite: MATH 320
Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 3-0-3
MATH 455 Numerical Analysis 3 College of Arts & Sciences
This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications. After
reviewing calculus and covering floating point arithmetic, it introduces students to numerical solution of nonlinear equations (bisection, fixed-point iteration, Newton's methods, etc.),
interpolation and polynomial approximation (Lagrange polynomial, Divided differences, Hermite interpolation, Cubic Spline interpolation, etc.), numerical differentiation and integration
(Newton-Cotes, Gaussian quadrature, etc.), and ODE methods (explicit and implicit methods). Some convergence theorems and error bounds are proved. The course also provides an introduction to MATLAB,
an interactive program for numerical linear algebra, as well as practice in computer programming.
Prerequisite Course(s): Prerequisites: MATH 320
Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 3-0-3
MATH 460 Advanced Seminar 3 College of Arts & Sciences
Advanced topics of current interest in mathematics.
Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 3-0-3
MATH 470 Mathematical Fluid Dynamics 3 College of Arts & Sciences
Introduction to the basic idea of fluid dynamics, with an emphasis on rigorous treatment of fundamentals and the mathematical developments and issues. The course focuses on the background and
motivation for recent mathematical and numerical work on the Euler and Navier-Stokes equations, and presents a mathematically intensive investigation of various model equations of fluid dynamics
Prerequisite Course(s): Prerequisites: MATH 450 or MATH 455
Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 3-0-3
MATH 490 Mathematical Modeling 5 College of Arts & Sciences
This is the capstone course and final requirement for the applied and computational mathematics (ACM) major. As such, it consists of a project-based introduction to the theory and practice of
mathematical modeling and simulation. Thesis and interdisciplinary work in teams is strongly encouraged. Techniques include scaling and nondimensionalization, data -fitting, linear and exponential
models, elementary dynamical systems, probability, optimization, Markov chain modeling. Models will be drawn from a wide range of application fields; synergy with double majors, graduate work and/or
interests in industry / internships is encouraged wherever relevant. Students will also learn scientific presentation skills and do oral presentations throughout the semester.
Prerequisite Course(s): Prerequisites: MATH 450 or MATH 455
Classroom Hours - Laboratory and/or Studio Hours – Course Credits: 5-0-5 | {"url":"https://site.nyit.edu/academics/courses/?cs=MATH&cal=UG&cc=MATH-115","timestamp":"2024-11-10T09:49:05Z","content_type":"text/html","content_length":"160750","record_id":"<urn:uuid:bd1b131f-1f84-431f-8c83-4e9c544a159d>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00268.warc.gz"} |
EDIT (July 14, 2016): A better way to extract would be to use
page markers
This is a tutorial on creating a multiple choice scanner similar to the Scantron system. We will take a photo of a multiple choice answer sheet and we will find the corresponding letter of the
bubbles. I will be using OpenCV 2.4.3 for this project.
Source code : https://github.com/ayoungprogrammer/MultipleChoiceScanner
We can split the algorithm into 9 parts:
1. Perform image preprocessing to make the image black & white (binarization)
2. Use hough transform to find the lines in the image
3. Find point of intersection of lines to form the quadrilateral
4. Apply a perspective transform to the quadrilateral
5. Use hough transform to find the circles in the image
6. Sort circles into rows and columns
7. Find circles with area 30% or denser and designate these as “filled in”
Thanks to this tutorial for helping me find POI and using perspective transformation
1. Image Preprocesssing
I like to use my favourite binarization method for cleaning up the image:
– First apply a gaussian blur to blur the image a bit to get rid of random dots
– Use adaptive thresholding to set each pixel to black or white
cv::Size size(3,3);
adaptiveThreshold(img, img,255,CV_ADAPTIVE_THRESH_MEAN_C, CV_THRESH_BINARY,75,10);
cv::bitwise_not(img, img);
We get a nice clean image with distinct shapes marked in white. However, we do get a few dots of white but they shouldn’t affect anything.
2. Hough transfrom to get lines
Use a probabilistic Hough line detection to find the sides of the rectangle. It works by going to every point in the image and checking if a line exists for all the angles. This is the most expensive
operation in the whole process because it has to check every point and angle.
cv::Mat img2;
cvtColor(img,img2, CV_GRAY2RGB);
vector<Vec4i> lines;
HoughLinesP(img, lines, 1, CV_PI/180, 80, 400, 10);
for( size_t i = 0; i < lines.size(); i++ )
Vec4i l = lines[i];
line( img2, Point(l[0], l[1]), Point(l[2], l[3]), Scalar(0,0,255), 3, CV_AA);
3. Find POI of lines
From: http://opencv-code.com/tutorials/automatic-perspective-correction-for-quadrilateral-objects/
However, we need to sort the points from top left to bottom right:
bool comparator(Point2f a,Point2f b){
return a.x<b.x;
void sortCorners(std::vector<cv::Point2f>& corners, cv::Point2f center)
std::vector<cv::Point2f> top, bot;
for (int i = 0; i < corners.size(); i++)
if (corners[i].y < center.y)
cv::Point2f tl = top[0].x;
cv::Point2f tr = top[top.size()-1];
cv::Point2f bl = bot[0];
cv::Point2f br = bot[bot.size()-1];
// Get mass center
cv::Point2f center(0,0);
for (int i = 0; i < corners.size(); i++)
center += corners[i];
center *= (1. / corners.size());
sortCorners(corners, center);
4. Apply a perspective transform
At first I used a minimum area rectangle for extracting the region and cropping it but i got a slanted image. Because the picture was taken at an angle, the rectangle we took a picture of, has become
a trapezoid. However, if you’re using a scanner, than this shouldn’t be too much an issue.
However, we can fix this with a perspective transform and OpenCV supplies a function for doing so.
// Get transformation matrix
cv::Mat transmtx = cv::getPerspectiveTransform(corners, quad_pts);
// Apply perspective transformation
cv::warpPerspective(img3, quad, transmtx, quad.size());
5. Find circles
We use Hough transform to find all the circles using a provided function for detecting them.
cvtColor(img,cimg, CV_BGR2GRAY);
vector<Vec3f> circles;
HoughCircles(cimg, circles, CV_HOUGH_GRADIENT, 1, img.rows/16, 100, 75, 0, 0 );
for( size_t i = 0; i < circles.size(); i++ )
Point center(cvRound(circles[i][0]), cvRound(circles[i][1]));
int radius = cvRound(circles[i][2]);
// circle center
circle( testImg, center, 3, Scalar(0,255,0), -1, 8, 0 );
// circle outline
6. Sort circles into rows and columns
Now that we have the valid circles we should sort them into rows and columns. We can check if two circles are in a row with a simple test:
y1 = y coordinate of centre of circle 1
y2 = y coordinate of centre of circle 2
r = radius
y2-r > y1 and y2+r<y1
If two circles pass this test, then we can say that they are in the same row. We do this to all the circle until we have figure out which circles are in which rows.Row is an array of data about each
row and index. The double part of the pair is the y coord of the row and the int is the index of arrays in bubble (used for sorting).
vector<vector<Vec3f> > bubble;
vector<pair<double,int> > row;
for(int i=0;i<circles.size();i++){
bool found = false;
int r = cvRound(circles[i][2]);
int x = cvRound(circles[i][0]);
int y= cvRound(circles[i][1]);
for(int j=0;j<row.size();j++){
int y2 = row[j].first;
found = true;
int l = row.size();
vector<Vec3f> v;
found = false;
Then sort the rows by y coord and inside each row sort by x coord so you will have a order from top to bottom and left to right.
bool comparator2(pair<double,int> a,pair<double,int> b){
return a.first<b.first;
bool comparator3(Vec3f a,Vec3f b){
return a[0]<b[0];
for(int i=0;i<bubble.size();i++){
7. Check bubble
Now that we have each circle sorted, in each row we can check if the density of pixels is 30% or higher which will indicate that it is filled in.
We can use countNonZero to count the filled in pixels over the area of the region.
In each row, we look for the highest filled density over 30% and it will most likely be the answer that is highlighted. However, if none are found then it is blank.
for(int i=0;i<row.size();i++){
double max = 0;
int ind = -1;
for(int j=0;j<bubble[row[i].second].size();j++){
Vec3f cir = bubble[row[i].second][j];
int r = cvRound(cir[2]);
int x = cvRound(cir[0]);
int y= cvRound(cir[1]);
Point c(x,y);
// circle outline
circle( img, c, r, Scalar(0,0,255), 3, 8, 0 );
Rect rect(x-r,y-r,2*r,2*r);
Mat submat = cimg(rect);
double p =(double)countNonZero(submat)/(submat.size().width*submat.size().height);
if(p>=0.3 && p>max){
max = p;
ind = j;
else printf("%d:%c",i+1,'A'+ind); | {"url":"https://blog.ayoungprogrammer.com/tag/multiple-choice","timestamp":"2024-11-07T05:52:04Z","content_type":"text/html","content_length":"29112","record_id":"<urn:uuid:62999698-df12-4cd4-a7a7-3144c47309f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00478.warc.gz"} |
Mike Roth (Queen's University)
Monday March 25, 2024
2:30 pm - 3:30 pm
Jeffery Hall, Room 202
Number Theory Seminar
Monday, March 25th, 2024
Time: 2:30 p.m. Place: Jeffery Hall, Room 202
Speaker: Mike Roth (Queen's University)
Title: Galois groups as monodromy groups in étale cohomology
Abstract: This talk is a companion to the talk of David Nguyen earlier in the term. That talk concerned estimating the size of certain trigonometric sums, and the method was to interpret those sums
as coming from étale sheaves on an open subset of P^1, and then use the weight machinery of étale cohomology.
In the talk I will try and give a simple introduction to the idea of a sheaf of locally constant sections over a curve, and related ideas in the purely topological case, and then say how those
notions can be expressed in terms of representations of Galois groups in the characteristic p case. Hopefully there will be time to explain the idea of the ‘weights’ of a sheaf, and the weights of
the action on cohomology. Finally, I hope to briefly discuss Grothendieck’s viewpoint of ’sheaves as functions’, and so return to the problem of estimating trigonometric sums.
None of these interpretations or constructions are new. They are all part of the beautiful synthesis of number theory and geometry that is étale cohomology, as envisioned by Grothendieck, and as
developed by Grothendieck, Artin, Deligne, and collaborators in the 1960’s and 70’s. | {"url":"https://www.queensu.ca/mathstat/mike-roth-queens-university-4","timestamp":"2024-11-12T09:29:07Z","content_type":"text/html","content_length":"60301","record_id":"<urn:uuid:e5146371-bb7b-4a87-a356-d5e84aa2734e>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00617.warc.gz"} |
Pulse Response of series R-C Circiut
Electrical Engineering ⇒ Topic : Pulse Response of series R-C Circiut
what is the Pulse Response of series R-C Circiut
Gopal said on : 2019-08-12 22:01:31
A pulse E(t) as shown in Figure 1 (a) is applied to a series R-C circuit shown in Figure (a).The switch is closed at t = O. Using KVL in the loop of Figure (a), the following equation is obtained.
Figure (a) Series R-C circuit with pulse input
Taking the Laplace transform of Eq. (1) and using Eq. , we get
Assuming that the capacitor is initially discharged, i.e., q[o ]= 0, Eq. (2) can be written as
!! OOPS Login [Click here] is required for more results / answer
Help other students, write article, leave your comments | {"url":"https://engineeringslab.com/tutorial_electrical/pulse-response-of-series-r-c-circiut-1848.htm","timestamp":"2024-11-10T12:20:29Z","content_type":"text/html","content_length":"36842","record_id":"<urn:uuid:aeb5f673-b238-4317-88f6-819c40de24e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00110.warc.gz"} |
The increasing benefits from ecosystems
Data from: The increasing benefits from ecosystems
Data files
Dec 19, 2023 version files 167.34 KB Jan 15, 2024 version files 157.14 KB
Ecosystems provide important services and are becoming increasingly scarce relative to market goods. Substantial policy shifts are currently underway in the US and elsewhere to integrate the value of
ecosystem services in investment appraisal and regulatory analysis. Yet, policy guidelines need to be improved by providing direction on how to account for the fact that benefits from ecosystem
services increase over time as incomes grow and ecosystem services become scarcer. Building on economic theory and best available evidence, we propose a simple relative price change rule to adjust
the benefits from future ecosystem services. Applying this rule will better enable governments to reflect the importance of scarce ecosystems for current and future generations in public
decision-making. This is critical as conservation efforts have so far fallen short of their stated goals.
README: Data from: The increasing benefits from ecosystems
Description of the data and file structure
The Supplementary Material describes the underlying theory and procedures. Here, we provide the data and code to reproduce the Figures and results mentioned in the Main and Supplementary Materials.
The replication materials contain three types of files:
(1) The Excel file [Calculation_Figure_1_A-D.xlsx] includes all calculations that underlie the main Figure 1, while the files [Calculation_Figure_S1.xls] and [Calculation_Figure_S1.xlsx] contain the
respective calculations for Figures S1 and S2.
(A) Excel file [Calculation_Figure_1_A-D.xlsx]:
Rows 1-5 in Columns A-I include the inputs to calculate relative price changes (RPC) in Columns B, D, F and H, rows 10 to 110, and the respective increases in the present value of ecosystem services
relative to the no relative price change case (Columns C,E,G and I, rows 10 to 110), along different value for the income elasticity of WTP. This serves as inputs to Stata to build Figure 1 Panel A
and C.
Columns K to X contain the inputs to Stata to build Figure 1 Panel B and D. Columns K is the year, while L to P contain growth of indicators (GDP per capita as well as four environmental indicators).
Columns Q to T report estimated future willingness to pay (WTP) estimates for the new default with an income elasticity of unity, and Columns U to X report estimated future willingness to pay (WTP)
estimates for the old default with an income elasticity of zero.
The additional sheets assemble this data to be copied into Stata.
(B) Excel file [Calculation_Figure_S1.xls]:
The sheet provides the same calculation of the RPC and the increase in present values, but now not along the income elasticity of WTP but along the discount rate (see Column A). The 2nd sheet again
collects the data to be copied into Stata.
(C) Excel file [Calculation_Figure_S2.xls]:
The sheet provides the same calculation of the RPC and the increase in present values, but now not along the income elasticity of WTP or discount rate but along the time horizon of the analysis (see
Column A). The 2nd sheet again collects the data to be copied into Stata.
(2) The corresponding data is saved in the subfolder [Data] as .dta files to facilitate integration into Stata for plotting the graphs.
(3) The Stata codes in the subfolder [Code] draw on these data files to produce Figures 1, S1 and S2. To reproduce the Figures, please set the paths to your local drives accordingly.
Sharing/Access information
The data on growth rates is based on an estimation by Drupp et al. (2023, https://arxiv.org/abs/2308.04400).
As mentioned above, the package includes six Stata codes to reproduces the four panels of the main Figure as well as the Figures S1 and S2. If you want to run these on your computer, you need to
adjust the pathways/directories accordingly.
Works referencing this dataset | {"url":"https://datadryad.org:443/stash/dataset/doi:10.5061/dryad.dbrv15f6z","timestamp":"2024-11-12T19:18:13Z","content_type":"text/html","content_length":"64741","record_id":"<urn:uuid:caf83a96-88dc-4437-9dea-14cd2a7f3ee1>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00593.warc.gz"} |
Efficiency Definition | Intro to Pumps
Efficiency Definition
What is Pump Efficiency?
Pump efficiency is a ratio that describes the relationship between rotational energy input into a pump and the hydraulic energy generated by the pump.
The head and flow generated by a centrifugal pump can be converted into a horsepower equivalent by using the following equation:
Water Horsepower = (Flow in GPM x Head in Feet) / 3960
In this formula, 3960 is a constant which converts the result of the multiplication of head and flow into water horsepower- but only if head and flow are expressed in feet and gallons-per-minute. The
resulting value can be compared to the amount of input horsepower required to operate the pump, and the resulting ratio is the efficiency of the pump.
For example, a pump that generates 1000 GPM at 100 Ft can also be said to be generating 25.25 water horsepower (WHP). If the input horsepower into the pump is measured while the pump is generating
1000 GPM at 100 Ft, and the input horsepower is found to be 33 brake horsepower (BHP), the efficiency of the pump would be approximately 76.5% (25.25 WHP / 33 BHP).
The peak efficiency of centrifugal pumps ranges from around 35% in small open-impeller solids-handling pumps to more than 90% in large split-case pumps, with the efficiency of most pumps falling
between 70% and 85%. The efficiency of a centrifugal pump varies considerably depending on the point along the performance curve where operation is taking place. The point with the highest efficiency
is known as the best efficiency point (BEP). | {"url":"https://www.introtopumps.com/pump-terms/efficiency/","timestamp":"2024-11-10T08:53:44Z","content_type":"text/html","content_length":"71660","record_id":"<urn:uuid:41fadb64-5365-4f45-aaf6-d2e035f3f565>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00114.warc.gz"} |
dijkstra's algorithm python adjacency matrix
Ja, jeg ved det, men i stedet for at overføre startnodeindekset til metoden, starter det bare ved nul i min kode. Dijkstra’s Algorithm finds the shortest path between two nodes of a graph. One is to
store vertices which have been considered as the shortest path tree, and another will hold the vertices which are not considered yet. The time complexity for the matrix representation is O(V^2). auto
matrix production; OOP streamlining to allow usage as a module. TO-DO. When I had to implement Dijkstra's algorithm in php to find the shorter way between 2 tables of a database, I constructed the
matrix with 3 values : 0 if the 2 points are the same, 1 if they are linked by an edge, -1 otherwise. Dijkstra Algorithm is a popular algorithm for finding the shortest path in graphs. PYTHON ONLY.
You are supposed to denote the distance of the edges via an adjacency matrix (You can assume the edge weights are either 0 or a positive value). 1. It is used for finding the shortest paths between
nodes in a graph, which may represent, for example, road networks. Join Stack Overflow to learn, share knowledge, and build your career. Set of vertices V 2. Asking for help, clarification, or
responding to other answers. Dijkstra’s shortest path for adjacency matrix representation; Dijkstra’s shortest path for adjacency list representation; The implementations discussed above only find
shortest distances, but do not print paths. So, if we have a mathematical problem we can model with a graph, we can find the shortest path between our nodes with Dijkstra’s Algorithm. Even if
Democrats have control of the senate, won't new legislation just be blocked with a filibuster? site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. In this post
printing of paths is discussed. I know Dijkstra's algorithm isn't the best solution for this problem, but it passes any way. Directed graph: checking for cycle in adjacency matrix. Adjacency matrix
with cells filled with time cost (referring to cost_matrix ... python performance algorithm programming-challenge graph. In this post printing of paths is discussed. Dado un gráfico y un vértice de
origen en el gráfico, encuentre las rutas más cortas desde la fuente a todos los vértices en el gráfico dado. Graphs : Adjacency matrix, Adjacency list, Path matrix, Warshall’s Algorithm, Traversal,
Breadth First Search (BFS), Depth First Search (DFS), Dijkstra’s Shortest Path Algorithm, Prim's Algorithm and Kruskal's Algorithm for minimum spanning tree Why is an early e5 against a Yugoslav
setup evaluated at +2.6 according to Stockfish? Definition:- This algorithm is used to find the shortest route or path between any two nodes in a given graph. What is the optimal algorithm for the
game 2048? Since the implementation contains two nested for loops, each of complexity O(n), the complexity of Dijkstra’s algorithm is O(n2). To learn more, see our tips on writing great answers.
Dijkstra's algorithm on adjacency matrix in python. Stack Overflow for Teams is a private, secure spot for you and We have discussed Dijkstra’s algorithm and its implementation for adjacency matrix
representation of graphs. In this article we will implement Djkstra's – Shortest Path Algorithm (SPT) using Adjacency Matrix. To learn more, see our tips on writing great answers. I believe the
implementataion of Dijkstra's algorithm isn't straight-forward for most people. In this Python tutorial, we are going to learn what is Dijkstra’s algorithm and how to implement this algorithm in
Python. util. The V is the number of vertices of the graph G. In this matrix in each side V vertices are marked. Dijkstras shortest path using MPI Prerequisites. Python program for finding the single
source shortest path using Dijkstra’s algorithm. Watch me explain the example and how this works with an adjacency matrix. The Python cookbook uses a priority dictionary I believe, but I'd really
like to keep it in a 2D array. Graphs out in the wild usually don't have too many connections and this is the major reason why adjacency lists are the better choice for most tasks.. The task is to
find the minimum number of edges in a path in G from vertex 1 to vertex n. Time complexity of Dijkstra’s algorithm : O ( (E+V) Log(V) ) for an adjacency list implementation of a graph. To subscribe
to this RSS feed, copy and paste this URL into your RSS reader. your coworkers to find and share information. So i have the basic program that will create the adjacency matrix but i don't understand
how to write this algorithm i have understood the algorithm … > time complexity for the matrix representation of graphs the graph with non-negative.... Any bugs Open MPI: here are instructions on how
to do it on a graph claiming that a coup... ' Recognition `` Drive Friendly -- the Texas way '' mean Dijkstra using adjacency matrix but hack... Share | improve... you hacked Dijsktra 's algorithm
for you and your coworkers to find the shortest problem! For a … we have to find the shortest distance & shortest path algorithm prerequisite of this post, (... In `` posthumous '' pronounced as < ch
> ( /tʃ/ ) reading classics over modern?! Into your RSS reader similar al algoritmo de Dijkstra es muy similar al algoritmo de Prim para el de. Example before coding it up into your RSS reader
Exchange Inc ; user contributions licensed cc! Tex engine d ’ etat only requires a small percentage of the senate, wo n't new legislation be... Proceeding, it is only guaranteed to return the cheque
and pays in cash the! Source node and every other node ( unicode ) LuaTeX engine on an matrix! Recuperar la ruta más corta entre el primer y el último nodo a prerequisite this... Algorithm ( SPT )
using adjacency matrix, 0 represents absence of edge while... The E is the best place to expand your knowledge and get prepared for your interview... Random variables implying independence edge has
cost 1, and Dijkstra 's algorithm for adjacency makes. Claim defamation against an ex-employee who has claimed unfair dismissal represents absence of,... Two posts as a beginner programmer, I
appreciate the simplicity startnodeindekset metoden! Muy similar al algoritmo de Prim para el árbol de expansión mínimo implying! Assume that every edge has cost 1, and Dijkstra 's algorithm on an
8-bit Knuth engine. N adjacency matrix with cells filled with time cost ( referring to cost_matrix Python... Our graph of V nodes represented in the program, starting at Memphis pointed out,
matriz-costo is early... Do you say the “ 1273 ” part aloud best solution for this problem but. © 2021 Stack Exchange Inc ; user contributions licensed under cc by-sa must go out of your way to
custom... Nodes in a graph, which may represent, for example, road maps implementataion Dijkstra... Is n't straight-forward for most people months ago space requirement of the senate, n't. 7 (
Dijkstra ’ s shortest path using Dijkstra ’ s algorithm and the adjacency matrix Python., starter det bare ved nul I min kode opening that violates many opening principles bad... Are easy, operations
like inEdges and outEdges are expensive when using the matrix! An adjaceny matrix 2 vertices and 0 otherwise x n adjacency matrix if there is an early e5 against Yugoslav. The problematic original
version para el árbol de expansión mínimo algorithm [ 3... You need to install Open MPI: here are instructions on how to create an adjacency graph! The Dijkstra ’ s shortest path algorithm in below
posts vertices of the senate, wo n't legislation. 2 ) path [ Python... with Dijkstra 's algorithm for finding the shortest path form a to on! Vertex 5 and 8 are updated must be implemented do I find
complex values satisfy... Opinion ; back them up with references or personal experience as experimental in! Memory hog E is the number of vertices of the adjacency matrix graph and its implementation
for adjacency matrix article. Of Python2.7 code regarding the problematic original version 2D array use our graph of nodes... Is number of vertices of the adjacency matrix that every edge has
cost,... Client asks me to return the cheque and pays in cash numbered from 1 to vertex n. 's! Overflow for Teams is a popular algorithm for a weighted directed graph given an!, O ( ELogV ) algorithm
for a weighted undirected graph graph whose vertices are numbered from to! Order of increasing path length plastic blank space fillers for my service panel the first line of contains. Private, secure
spot for you and your coworkers to find the minimum number of test cases of reaching.! Each item 's priority is the number of vertices of the edge 2 ) solving the shortest distance of functions. Form
a to Z on a mac cost of each edge containing the length of the population, two-dimensional. S algorithmisan algorithmfor finding the single source shortest path algorithm deep cabinet on this safely!
The minimum number of vertices and 0 otherwise route or path between source node and every other node '' implementation! For your next interview bidirectional ), the adjacency matrix, 0 represents
absence of edge while... May represent, for example, road maps shortest path algorithm for finding the source... Post your Answer ”, you agree to our terms of service, policy... Distance problem
dijkstra's algorithm python adjacency matrix general represent, for example, road networks … we have Dijkstra! This works with an adjacency matrix of a directed graph with 10 nodes ( node 0 to node
)... Recommend to read following two posts as a prerequisite dijkstra's algorithm python adjacency matrix this post for graphs with costs dynamically?... “ post your Answer ”, you can just assume
that every edge has cost 1 and..., operations like extract-min and decrease-key value is O ( LogV ) for min Heap is a good for... Program that will find the shortest path between any two nodes of a
directed graph whose vertices numbered! Queue as we have discussed Dijkstra ’ s algorithm using the adjacency matrix representation graphs! Experimental data in the order of increasing path length
you say the “ 1273 ” part aloud secondary targets was... '' implementation of Dijkstra 's algorithm on adjacency matrix with no costs for in... The start node to each other node es muy similar al
algoritmo de Dijkstra es muy al. With costs tree for a weighted directed graph given as an adjaceny matrix,. Estoy intentando recuperar la ruta más corta entre el primer y el último nodo to this...
Know Dijkstra 's algorithm using a priority queue as we have discussed Dijkstra ’ s algorithm how! | {"url":"http://en.emotions.de-dietrich.com/why-is-bpb/dijkstra%27s-algorithm-python-adjacency-matrix-ed99ed","timestamp":"2024-11-01T22:37:27Z","content_type":"text/html","content_length":"20463","record_id":"<urn:uuid:8f4ce058-3457-430b-b49a-4495c11e5f56>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00693.warc.gz"} |
A table showing Decimal, Hexadecimal and Binary digits
Hexadecimal, commonly abbreviated as “hex,” is a numerical system that is widely used in computer science and programming. It is a base-16 system, meaning that it uses 16 distinct symbols to
represent numbers, as opposed to the base-10 decimal system that uses ten symbols (0-9). In this article, we will explore the concept of hexadecimal, its usage, and its importance in computer
First, let’s examine the symbols used in the hexadecimal system. Hexadecimal uses the symbols 0-9 and A-F to represent values from 0 to 15. The letters A-F are used to represent values from 10 to 15,
respectively. This means that the hexadecimal system has 16 distinct symbols, which is why it is known as a base-16 system.
To better understand hexadecimal, let’s compare it to the decimal system. In the decimal system, each digit represents a power of 10. For example, in the number 123, the first digit represents 100
(10 to the power of 2), the second digit represents 20 (10 to the power of 1), and the third digit represents 3 (10 to the power of 0). Similarly, in the hexadecimal system, each digit represents a
power of 16. For example, in the number 3F7, the first digit represents 16 to the power of 2 (256), the second digit represents 16 to the power of 1 (16), and the third digit represents 16 to the
power of 0 (1).
Hexadecimal colour codes
Hexadecimal is commonly used in computer programming for a variety of purposes. For example, it is used to represent colours in web design and graphics. Each colour is represented by a combination of
red, green, and blue values, each of which is represented by a two-digit hexadecimal number. For example, the colour white is represented as #FFFFFF, which represents the maximum values of red,
green, and blue.
Hexadecimal is also used to represent memory addresses in computer systems. Each byte of memory can be represented by two hexadecimal digits, and each memory address is represented by a series of
hexadecimal digits. This allows programmers to easily manipulate and access specific areas of memory.
Another important use of hexadecimal in programming is in bitwise operations. These operations involve manipulating binary digits (0s and 1s), and hexadecimal provides a convenient way to represent
groups of four binary digits. Each hexadecimal digit represents a group of four binary digits, which makes it easy to convert between binary and hexadecimal representation.
In conclusion, hexadecimal is a numerical system that is widely used in computer science and programming. It uses 16 distinct symbols to represent numbers, and it is commonly used to represent
colours, memory addresses, and binary digits. Understanding hexadecimal is an important skill for anyone interested in computer programming or web design, and it is an essential part of the modern
digital world.
Interesting Life
Good Ideas
Computing Electronics History Interesting Inventions Robotics Software
Tape Libraries
Computing Electronics Inventions Programming Science Software
The BBC Microcomputer | {"url":"https://cryptlabs.com/hexadecimal/","timestamp":"2024-11-09T05:45:37Z","content_type":"text/html","content_length":"73299","record_id":"<urn:uuid:a1981bf5-149d-46e5-a272-66756de3ac14>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00120.warc.gz"} |
Formulas Quick Start Guide (part 2 of 4): SUM and AUTOSUM
This post is all about using SUM.
When you are working with a lot of data, creating formulas manually is time consuming.
SUM makes your life much easier and makes formulas quicker and more accurate by reducing errors.
Clickable Table of Contents
1. Using Excel Functions
Excel has over 450 pre-programmed functions. The simplest one that everybody learns first is SUM. SUM adds things up.
At the other end of the spectrum, the top-end functions are used by statisticians, mathematicians, engineers and scientists and tend to be quite specialised. However, there are still many that can be
used by ‘ordinary’ folk!
A function is used within a formula to simplify it.
For example,
This is easier to read and easier to maintain.
To see the full benefit, scale it up. Imagine if you had to add up 25 cells. Or 1,000 cells!
Excel even does the hard work of setting it up for you. You are given a tool called AutoSum.
2. Using AutoSum
The AutoSum button is ideal for novices who wish to learn and use some simple Excel functions. The AutoSum button provides a list of the five basic functions – SUM, AVERAGE, COUNT, MIN and MAX. The
icon is the Greek Sigma symbol and looks like a sideways ‘M’ or a crooked ‘E’ (see below).
Using Autosum is quick and easy.
1. Select the numbers to be added and the blank cell either to the right or underneath the selection.
2. Click the Formulas tab.
3. Click the AutoSum icon.
A SUM function is created in the blank cell (the last cell of your selected cell range). The selected cells are added together and the total is calculated.
3. How the SUM function works
By default the AutoSum tool creates a SUM function, which adds together the values in the selected cells. To see what the SUM function looks like, click on the cell that contains the total and study
the formula bar.
The SUM function may be typed directly into a cell, rather than using AutoSum. This is quicker for experienced users.
Here’s the process to use once you’re a bit more confident:
1. Select the blank cell where the total will be calculated (this is often - but not always - directly underneath the main data, or directly to the right of the main data).
2. Type ‘=SUM(’
3. With the mouse, select the range of cells you want to add up. Never type in the cell or cell range by hand.
4. Type the closing bracket.
5. Press Enter.
The structure of a SUM function is =SUM(number1, number 2 …) where each number may be a constant (e.g. 5), cell reference (e.g. A1) or cell range (e.g. A1:A3).
• Every item within the brackets is added up.
• Number1, number2 etc. are called arguments. A SUM function may have up to 32 arguments, each separated by a comma.
• The function is preceded with ’=’ (as for formulas).
• The notation for a range – two cell references with a colon (‘:’) in-between.
The following formulas are valid examples of the SUM function.
4. Five different ways to start using AutoSum
#1 Select a single blank cell
1. Select the blank cell underneath the data.
2 . Click the AutoSum icon.
3. Excel looks around the blank cell and if it finds data it will display the data range in the formula.
4. If the data range is correct, press Enter to confirm the formula.
#2 Select the data (and the blank cell - optional)
1. Select the data you want to add up.
2. Optional - you can also select the blank cell underneath - or not. Your choice.
3. Click the AutoSum icon. The total is added automatically.
#3 Use methods #1 or #2 across the worksheet
#4 Select 2 or more blank cells
When you select more than one blank cell going across the sheet, Excel knows that you want to add up columns of data, so it doesn’t ask you and puts the total straight into the cells.
1. Select 2 or more blank cells underneath the data
2. Click the AutoSum icon. The column totals are inserted automatically.
3. Select 2 or more cells to the right of the data.
5. Click the AutoSum icon. The row totals are inserted automatically.
#5 Select all the data plus a blank column and/or row
And when you click AutoSum, all the totals are generated in one hit. Imagine how much time you could save if you have 50 columns and 1,000 rows
5. Watch the video (over the shoulder demo)
6. What next?
I hope you have seen how useful, how easy and how versatile the SUM function is.
Knock up a simple spreadsheet, enter some data - whatever you like - and use the AutoSum to generate some totals quickly.
Once you’re confident doing that, try selecting a blank cell and creating a SUM function manually, by typing it directly into the cell.
In part 3, I discuss 4 other basic formula functions that are also useful and easy to use - AVERAGE, MAX, MIN and COUNT.
What do you think?
I hope you found plenty of value in this post. I'd love to hear your biggest takeaway in the comments below together with any questions you may have.
Have a fantastic day.
Jason Morrell is a professional trainer, consultant and course creator who lives on the glorious Gold Coast in Queensland, Australia.
He helps people of all levels unleash and leverage the power contained within Microsoft Office by delivering training, troubleshooting services and taking on client projects. He loves to simplify
tricky concepts and provide helpful, proven, actionable advice that can be implemented for quick results.
Purely for amusement he sometimes talks about himself in the third person.
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[QSMS Monthly Seminar 2023-05-26] Lagrangian submanifolds and their Legendrian lifts
2023년 5월 QSMS Monthly Seminar
• Date: May 26 (Fri) 11:00 ~ 16:00
• Place: 25-103 (SNU)
• Contents:
Speaker: 홍한솔 (오전 11시)
Title: Lagrangian submanifolds and their Legendrian lifts
Abstract: Given a Laurent polynomial defined over non-Archimedean field (typically the Novikov ring), I will introduce a tropical method to find possible locations of its critical points.
The main application is to study closed string mirror symmetry for a class of blowups of toric surfaces. The mirror geometry in this case, consists of a Laurent polynomial (or series) called the
potential, which can be combinatorially computed by counting broken lines on some scattering diagram.
Speaker: 최승일 (오후 3시)
Title: Symmetric and nonsymmetric Cauchy identities
Abstract: The Cauchy identity, which is the decomposition of the symmetric Cauchy kernel into a product of two Schur polynomials, is an important subject in the symmetric function theory and
representation theory. Lascoux introduced a generalization of the Cauchy kernel, called nonsymmetric Cauchy kernels, and provided a decomposition into a product of key polynomials (or Demazure
characters). In this talk, we will provide an overview of these results and then explore further extended results to other types. | {"url":"https://qsms.math.snu.ac.kr/index.php?mid=board_sjXR83&listStyle=viewer&l=en&sort_index=title&order_type=asc&document_srl=2558&page=6","timestamp":"2024-11-15T00:54:22Z","content_type":"text/html","content_length":"22418","record_id":"<urn:uuid:490c105f-4abd-4bf3-aaf1-f2d312f46829>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00852.warc.gz"} |
This plugin draws iterated growth models in ImageJ. These are saveable screen drawings made of points calculated using the basic expressions:
x[n+1] = mx[n]^Xxpow + ny[n]^Xypow + o
y[n+1] = fx[n]^Yxpow + gy[n]^Yypow + h.
For each model, each equation's coefficients (m,n,o;f,g,h; Xxpow, Xypow, Yxpow, Yypow) are specified in sets that the user can modify. Each of these sets also contains a probability of using the set.
This is for cases when more than one set of coefficients is used to calculate each point, chosen according to a likelihood that the user sets.
Each particle calculated and drawn is coloured on a black background, and all are the same size. Colours are generated randomly, without user input. The user can change the size of the particles in
the models, however, using an option in the first popup.
There are five default models. The user can change the coefficients to vary the basic pattern, or change the number of sets to create a new model.
• The "Henon" model is specified by one set of coefficients (i.e., the likelihood of using that set is 1).
• The Henon Map model is also specified by one set of coefficients. It has a (capacity) fractal dimension that is assumed to be around 1.261 (correlation dimension around 1.25).
• The "Random" model is similar to the Henon models, but uses two sets of coefficients and probabilities.
• The "D. Greene Fern" model uses four sets of coefficients. It is from an algorithm by David Greene (Charles Sturt University),
• The custom model is essentially the same as the fern, with a few changes in coefficients and probabilities.
Using the plugin
1. To generate a default model, the user selects a model from the dropdown menu on the first popup. The number of sets for the selected model is automatically determined, and this number will appear
in the next popup.
2. If the user wants to modify a model, the number in this second popup can be changed.
3. After a number of sets is selected, one popup appears for each set. The popups show each coefficient's name listed beside a textbox for its value. The values can be left at the defaults, or
changed as desired.
4. At the bottom of each of these popups, there is a space for the probability that the coefficients above it will be used in generating each next point. The total of all probabilities should not be
greater than 1; all sets for values that make the cumulative total greater than 1 are ignored. | {"url":"https://wsr.imagej.net/ij/ij/ij/ij/plugins/FGM.html","timestamp":"2024-11-04T04:15:08Z","content_type":"text/html","content_length":"4643","record_id":"<urn:uuid:ba088698-ada5-45c7-9d6e-57361d4949dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00545.warc.gz"} |
Re: [xsl] to get the descendants at only one level - xpath
<-prev [Thread] next-> <-prev [Date] next->
Month Index | List Home
Re: [xsl] to get the descendants at only one level - xpath
Subject: Re: [xsl] to get the descendants at only one level - xpath
From: David Carlisle <davidc@xxxxxxxxx>
Date: Wed, 6 Feb 2008 18:16:04 GMT
> I need to select only child1, 2, 3 and not any of the childs of these.
no it just selects child1 child2 child3, depending what you do with
child3 having selected it you may see the descendents.
so for example select="/root" just selects a single element, if you call
name() on it you just get a single string "root" but if you say
<xsl:copy-of select="/root"/>
you get the whole document tree back as child nodes are properties of an
element so the copied node has copies of the same children.
perhaps ypu want
<xsl:for-each select="/*/*/*"
note that the selection is as previously suggested, but I'm guessing how
you want to use the selected nodes (using <xsl:copy/>) you haven't shown
how you have used them or what you want to generate, so I can only guess.
The Numerical Algorithms Group Ltd is a company registered in England
and Wales with company number 1249803. The registered office is:
Wilkinson House, Jordan Hill Road, Oxford OX2 8DR, United Kingdom.
This e-mail has been scanned for all viruses by Star. The service is
powered by MessageLabs. | {"url":"https://www.biglist.com/lists/lists.mulberrytech.com/xsl-list/archives/200802/msg00137.html","timestamp":"2024-11-07T06:17:59Z","content_type":"text/html","content_length":"5556","record_id":"<urn:uuid:3ac5b7b5-9cb9-42ed-ad62-ad62b33fa32a>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00027.warc.gz"} |
Learn More about 56 divided by 2 - The Crypto Classic
Learn More about 56 divided by 2
Welcome 56 divided by 2 to our blog post where we dive into the fascinating world of division! Whether you’re a math enthusiast or just looking to brush up on your basic arithmetic, you’ve come to
the right place. In this article, we’ll explore the concept of division and specifically focus on one intriguing equation: 56 divided by 2. So grab your pencils and get ready to sharpen your
problem-solving skills as we unravel the mysteries behind this mathematical operation. Let’s jump right in!
What is division?
Division is a fundamental mathematical operation that involves splitting a quantity into equal parts. It helps us distribute and allocate resources, solve problems, and understand the relationship
between numbers. At its core, division is all about sharing.
Imagine you have 56 apples and you want to divide them equally among 2 friends. Division allows you to determine how many apples each friend will receive. In this case, when we divide 56 by 2, we are
essentially asking: “How many groups of 2 can we make from a total of 56?”
To find the answer, we use long division or mental math strategies such as halving or repeated subtraction. By dividing 56 by 2, we discover that each friend will receive an equal share of 28 apples.
Division also plays a crucial role in other areas of life beyond mathematics. For instance, it helps us calculate averages in statistics or determine rates and ratios in everyday situations like
cooking recipes or budgeting expenses.
Understanding division not only enhances our problem-solving abilities but also builds a foundation for more complex mathematical concepts like fractions and decimals. So whether you’re tackling
simple divisions or diving into advanced math theories, mastering this operation opens up endless possibilities for exploration and application.
What is the answer to 56 divided by 2?
What is the answer to 56 divided by 2? Well, let’s do some quick math. When you divide 56 by 2, you are essentially splitting it into two equal parts. Each part would have a value of half of 56.
So, when we calculate 56 divided by 2, we find that the answer is 28. This means that if you were to divide a quantity of 56 into two equal groups or portions, each group would contain a value of 28.
Division is an arithmetic operation that allows us to distribute or allocate a quantity into smaller parts. It is often used in various real-life scenarios such as sharing equally among friends or
dividing resources evenly among participants.
Knowing how and when to divide can be beneficial in many situations. Whether it’s dividing expenses between roommates or splitting up tasks for a group project, division helps ensure fairness and
Now that we’ve explored the answer to our initial question, let’s look at some other examples of division and how it applies in different contexts
How do you know when to divide?
How do you know when to divide? Division is a mathematical operation that involves splitting a number into equal parts. It can be used in various situations, such as sharing objects or determining
the value of each group.
One way to know when to divide is when you have a total quantity that needs to be distributed equally among a certain number of groups or individuals. For example, if you have 56 cookies and want to
distribute them equally among 2 friends, dividing 56 by 2 will give you the answer of how many cookies each friend will receive.
Another situation where division comes into play is when comparing quantities. If you need to find out how many times one number fits into another, division can help solve this problem. For instance,
if you want to know how many times 5 goes into 25, dividing 25 by 5 gives the solution of 5.
Division can also be used for scaling measurements or finding averages. When working with measurements or data sets, dividing allows us to determine ratios and proportions accurately.
Knowing when to divide depends on the specific problem at hand. Whether it’s about distributing objects equally, comparing quantities, scaling measurements, or finding averages – division serves as
an essential tool in mathematics and everyday life applications.
What are some other examples of division?
What are some other examples of division? Division is a fundamental concept in mathematics that helps us split numbers into equal parts. It’s like sharing or distributing something among a group of
people. Let’s explore some real-life examples to understand division better.
Consider a pizza party where you have 8 friends and 2 pizzas. To ensure everyone gets an equal share, you divide the pizzas into 8 slices each, resulting in 16 slices in total. Each person will then
receive 2 slices.
Another example could be dividing a pack of chocolates equally among three siblings. If there are 12 chocolates, you can divide them evenly by giving each sibling four chocolates.
Division is also used when calculating averages. For instance, if you want to find the average score of five exams for one student with scores of 80, 85, 90, 95, and100 respectively, add all the
scores together (450) and divide it by the number of exams (5). The average score would be calculated as:
(80 +85+90+95+100)/5 =460/5=92
These examples illustrate how division helps in solving various everyday problems and mathematical calculations.
In this blog post, we’ve explored the concept of division and specifically looked at the example of 56 divided by 2. Division is a fundamental mathematical operation that involves splitting a
quantity into equal parts. By dividing 56 by 2, we find that the answer is 28.
Knowing when to divide can be determined by understanding the problem or situation at hand. In some cases, you may need to distribute items equally among a group or figure out how many times one
number fits into another.
While we focused on one example in this article, there are countless other instances where division is used. For instance, when determining average scores, calculating prices per unit or figuring out
how many days are left until an event.
Understanding division and its applications can be helpful in various areas of life such as mathematics, finance, science and more.
So remember: division allows us to split quantities into equal parts and find answers to problems involving distribution or partitioning. With practice and understanding, you’ll become proficient in
using this essential mathematical operation!
Now it’s your turn! Take what you’ve learned about division and explore more examples on your own. Happy dividing! | {"url":"https://thecryptoclassic.com/learn-more-about-56-divided-by-2/","timestamp":"2024-11-12T06:52:35Z","content_type":"text/html","content_length":"139704","record_id":"<urn:uuid:25284f0e-5e58-48e6-ab3f-0baaff063f50>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00068.warc.gz"} |
Nonlinear Program Construction and Verification Method Based on Partition Recursion and Morgan's Refinement Rules
Issue Wuhan Univ. J. Nat. Sci.
Volume 28, Number 3, June 2023
Page(s) 246 - 256
DOI https://doi.org/10.1051/wujns/2023283246
Published online 13 July 2023
© Wuhan University 2023
0 Introduction
With the continuous progress of computer technology and the expansion of application areas, software security has received wide attention from the society. The use of formal methods^[1-4] to develop
programs can strictly guarantee the correctness and reliability of programs. Formal methods also help to better understand the functions and behaviors of programs and meet user requirements.
Therefore, formal methods have a wide range of promising applications in software development. Program derivation and verification techniques are the hot spots in the research of formal methods. The
more mature techniques are Dijkstra's weakest predicate method^[5,6], Morgan-based program derivation methods^[7-9], and PAR(Partition-and-Recur) methods^[4].
Dijkstra's weakest predicate approach defines its own language rules by which programs can be developed and verified. The advantage is that the development process is strictly based on mathematical
logic and formal verification is performed during the development process. However, the verification process is mostly manual and less automated; and the abstract programs obtained by this method
cannot be converted into executable programs.
The PAR method transforms the initial specification by its own defined rules to obtain recursive equations and loop invariant, which eventually lead to an abstract Apla program^[4]. The advantage of
this method is that the derivation process is guided by a system with its own derivation patterns and rules, which are more widely applicable. However, the procedure of program derivation using this
method is not sufficiently refined and the obtained Apla abstract program is not mechanically proven and has a low automation procedure.
This paper proposes a program construction and verification method based on partition recursion and Morgan's refinement rule. The method first transforms the initial program specification by
recursive definition technique, and then obtains recurrence relations and loop invariant by using partitioned recursion rules^[10,11]. Then, based on the loop invariant, the program specification is
gradually refined using Morgan refinement rules to finally obtain a reliable abstract program. The verification conditions are obtained by VCG^[12](Verification Condition Generator) and then verified
by Isabelle^[12-14] theorem provers. After the verification, the obtained abstract program is then generated into a C++ program through the conversion system^[15] to realize the whole process from
abstract specification to abstract program^[15,16], then mechanical verification^[16,17], and to executable program.
The paper is organized as follows. Section 1 focuses on detailing the program construction and verification methods based on partition recursion and Morgan's refinement rule; Section 2 takes the
binary tree preorder problem^[18,19] as an example to develop and verify the binary tree problem using the methods proposed in this paper as a guide; Section 3 concludes the whole paper.
1 Proposed Method of Nonlinear Program Construction and Verification
This section focuses on the method proposed in this paper. The method in this paper contains a complete set of program construction and verification methods, which mainly consists of four steps, as
shown in Fig.1.
1.1 Initial Specification Generation
A program specification is a detailed description of a program's functions. It expresses a precise description of the problem in an easy-to-understand manner for the implementer. The program
specification consists primarily of the program's pre-assertion P and post-assertion Q. P and Q are conditions that must be met before and after program input, respectively.
Hoare's formal specification is primarily in the form of triples {P}S{Q}^[13], but this paper primarily uses Morgan's specification representation w:[P, Q]. Morgan rewrote the Hoare triples
representation to make the specification representation more compact and easier to refine into code.
1.2 Program Refinement and Generation
1.2.1 Initial specification transformation
Following the generation of the initial specification, the attributes of first-order logic are typically used to gradually refine the specification transformation from the post-predicate. After the
generation of the binary tree preorder specification in Section 1.1, the initial specification transformation needs to be performed on it. The following is the specific transformation procedure.
Due to the binary tree's easy decomposition, this paper first introduces the recursive function of the binary tree and strengthens the post-prediction of initial specification. Then, Morgan's
framework variables are introduced to convert the initial reduction's triple form into a Morgan-specific symbolic form. Figure 2 depicts the entire initial specification transformation process. This
recursive technique reduces the difficulty of changing the binary tree specification. At the same time, recursion technology provides a foundation for subsequent programs to use partition recursion
to find loop invariant.
1.2.2 Loop invariant derivation
Acquiring loop invariant has always been a difficult point in formal derivation, and acquiring tree structure loop invariant is even more difficult. To address this issue, this paper proposes the use
of partition recursion in conjunction with recursive definition technology to derive the loop invariant of tree structure.
The basic idea behind partition recursion is to divide a large, difficult-to-solve problem into some smaller problems of the same scale, and then find the recurrence relationship via the relationship
between sub-problems.
First, this paper decomposes the original problem using the recursive definition technique. The recurrence relationship between the problems is then determined based on the relationship between the
sub-problems in the decomposition process. Finally, the binary tree's loop invariant is discovered using the recursive relationship. Figure 3 depicts the specific process flow.
1.2.3 Morgan’s rules refinement
The initial statute and loop invariant have been obtained in Sections 1.2.1 and 1.2.2. Now it is necessary to derive the initial specification to the GCL program based on loop invariant, which are
used as Morgan's refinement rules in this paper. Morgan's refinement rules are used to gradually refine the abstract specification into GCL^[8,9] code with strong execution, with each refinement step
being small and easy to handle. Using the Morgan's refinement rules, each step of the refinement process must be guaranteed to be justified and proven. As a result, the program obtained is extremely
The rules used in the Morgan refinement method are specified in Table 1^[8] .
1.3 GCL Program Automation Verification
The GCL program obtained in Section 1.2 needs to be mechanically verified by generating verification conditions through VCG and by Isabelle. The specific steps are shown as follows.
1.3.1 Verification condition generation
The automatic generation of verification conditions is mainly done through VCG, which aims at converting the verification of Hoare triples into the verification of assertions. The advantage of this
is that the person or machine verifying the program does not need to know anything about Hoare logic to be able to prove the program.
VCG is a verification condition generator whose working process relies on two conversion functions: pre (this function is different from the following binary tree preorder function Prebt) and vc
function. pre function mainly converts the imperative program into specific Hoare logic rules, and vc function mainly converts the result of pre function into specific verification conditions. The
specific VCG working schematic is shown in Fig.4 .
1.3.2 Isabelle assisted verification
Manual and mechanical verification are the two types of formal verification technology. Manual verification is more difficult and prone to errors. Mechanical verification is based on mathematical
logic, and it is far more efficient and reliable than manual verification. The mechanical theorem prover^[16] is further subdivided into automatic and interactive theorem provers.
Isabelle, an automatic theorem prover, is chosen for auxiliary verification in this paper, primarily because Isabelle has the following advantages.
1) The powerful rule base
Isabelle has a powerful rule base, in which every theory contains many related rules and theorems. More importantly, users can add theorems and prove them based on their own proof requirements. If
the newly added theorem passes the proof, the Isabelle system will save the newly added lemma and use it in the theorem's subsequent proof.
2) The Sledgehammer tool
The Isabelle system's Sledgehammer^[12,13] tool is extremely powerful, and it can invoke several other automatic theorem provers to find theorems, including E, SPASS, Vampire, and others. In the
Isabelle operator interface, the user only needs to click on the apply application in the Sldgehammer button and Slagehammer can automatically generate the proof process. The user then clicks on the
proof process to automatically insert theproof process into the Isabelle script.
1.4 Conversion from GCL Program to Executable Program C++
The GCL programs verified in Section 1.3 are still abstract and cannot be compiled and executed on a computer. As a result, the GCL program will need to be converted into an executable file. First,
we can convert the GCL program to an Apla program. GCL and Apla can be converted because they are abstract imperative programs with similar syntax. Then, using the team's Apla to C++ program
automatic converter, convert the above Apla abstract program into an imperative C++ program. The team's Apla to C++ conversion system is depicted in Fig.5.
With these four steps, this paper realizes the whole process of program development, verification and transformation.
2 Example: Binary Tree Preorder Problem (BTPP)
2.1 Initial Specification Generation of BTPP
Preorder traversal is a way of traversing a binary tree. By preorder traversal, we mean that when traversing a binary tree, we first traverse the root node, then the left subtree, and finally the
right subtree. In this paper, we use T to represent a binary tree, and thePrebt (T) function to represent the preorder traversal of the entire binary tree T. Therefore, the binary tree preorder
traversal specification can be expressed as follows:
$[ t r u e , P r e b t ( T ) ]$(1)
2.2 Program Refinement of BTPP
2.2.1 Initial specification transformation
There are two types of binary tree traversal algorithms: recursive traversal and nonrecursive traversal. The goal of this paper is to deduce the binary tree's preorder nonrecursive algorithm program.
This section focuses on the initial specification transformation, which is the transformation of the posterior assertion Prebt. And we know that the process of binary tree preorder traversal is to
traverse the root node, then the left subtree, and then the right subtree. So the Prebt function can be written first as follows:
$P r e b t ( T ) = { [ ] , T = % [ T . n ] ↑ P r e b t ( T . l ) ↑ P r e b t ( T . r ) , T ≠ %$(2)
We decompose this function according to the content of the Prebt function, and we can obtain the following derivation process:
$P r e b t ( T ) = [ T . n ] ↑ [ T . l ] ↑ [ T . r ] = [ T . n ] ↑ [ T . l . n ] ↑ [ T . l . l ] ↑ [ T . l . r ] ↑ [ T . r ] = [ T . n ] ↑ [ T . l . n ] ↑ [ T . l . l . n ] ↑ [ T . l . l . l ] ↑ [ T
. l . l . r ] ↑ [ T . l . r ] ↑ [ T . r ]$
Figure 6 depicts the binary tree's change process as a result of the recursive function Prebt. As shown in the derivation and Fig.6, the root node is retained first when traversing a binary tree
recursively. The left subtree of the tree will then be traversed, followed by the right subtree of the tree. As a result, three variables, X, q, and S, are introduced. The nodes that have been
traversed are stored in the sequence X. The sequence S is used to store the nodes to be traversed, while q is used to store the subtree of T that is about to be traversed. The original specification
can then be refined as needed:
$X , q , S : [ t r u e , X = P r e b t ( T ) ]$(3)
2.2.2 Loop invariant derivation
We have obtained the initial specification after the transformation, next we need to obtain the corresponding loop invariant. In this paper, the loop invariant are obtained according to the division
recursion, i.e., we find the law and obtain the loop invariant in the process of binary tree traversal in the preorder.
As illustrated in Fig.7, for the traversed subtree q, the subtree's head node is always stored in X, and q continues to traverse the subtree's left subtree. The right subtree of the subtree is still
in S.
After traversing subtree q, the next round will continue to find nodes from sequence S to traverse. The sequence $S$'s length will be reduced accordingly. The definition of the recurrence relation F
function can be derived:
$F = { F ( [ ] ) = [ ] F ( [ q ] ↑ S ) = P r e b t ( q ) ↑ F ( S )$(4)
We can see from the previous Prebt and F functions that the binary tree preorder traversal process always places the nodes traversed by q in the sequence X. Then q continues to traverse the subtree's
remaining nodes to the left. After traversing the left subtree, q will look for nodes in the sequence S to continue traversing, as shown in Fig.8. So the loop invariant is $inv≡Prebt(T)=X↑Prebt(q)↑F
2.2.3 Morgan's rules refinement
1) Loop variable initialization
Based on loop invariant obtained in Section 2.2.2, the specification can be refined using Strengthen postcondition in Table 1, and the specification can then be obtained as:
$X , q , S : [ t r u e ; i n v ∧ X = P r e b t ( T ) ]$(5)
According to the specification, inv is used as an intermediate assertion connecting the pre-predicate and the post-predicate, then (5) is refined using Sequential composition in Table 1:
$X , q , S : [ t r u e ; i n v ]$(6)
$X , q , S : [ i n v ; i n v ∧ X = P r e b t ( T ) ]$(7)
Assignment in Table 1 is used to refine (6), and the following formula is obtained:
$X , q , S : [ t r u e ; i n v ] ⊑ X , S , q ≔ [ ] , [ ] , T$(8)
$t r u e ⇒ i n v [ X , S , q / [ ] , [ ] , T ] ≡ P r e b t ( T ) = ( X ↑ P r e b t ( q ) ↑ F ( S ) ) [ X , S , q / [ ] , [ ] , T ≡ P r e b t ( T ) = [ ] ↑ P r e b t ( T ) ↑ F [ ] ≡ P r e b t ( T ) =
[ ] ↑ P r e b t ( T ) ↑ [ ] ≡ T r u e$
So the first part of the refinement is established.
2) Loop process refinement
Formula (7) can be refined using Repetition in Table 1, and ¬(X = Pre(T)) is a loop condition and can be equivalently converted to $q≠%∨(q=% ∧S≠[])$, then the following equation can be obtained:
$d o q ≠ % ∨ ( q ≠ % ∧ S ≠ [ ] ) X , q , S : [ i n v ∧ ( q ≠ % ∨ ( q = % ∧ S ≠ [ ] ) ) , i n v ∧ 0 ≤ v < v 0 ] o d$(9)
The variable $v$ in the above equation $0≤v<v0$ is the marker for the terminability proof. $v$ needs to satisfy two conditions: first, it must be guaranteed to be decremented in each loop, and
second, it cannot be decremented to a negative number. In this paper, we need to find the specific terminability variable $v$.
We can see here that the variable X stores the nodes that are traversed. And the number of nodes in X keeps increasing. When the loop is completed, the number of nodes stored in X is equal to the
number of nodes in the binary tree T. So, we can select (|T|-#X) (|T| is the node value of the whole binary tree and X is the stored traversed node value) as the variable $v$.
We can simplify the equation by substituting v=|T|-#X for$0≤v<v0$:
$0 ≤ v < v 0 ≡ 0 ≤ | T | - # X < | T | - # X 0 ≡ # X 0 < # X ≤ | T |$
Then (9) is simplified to:$X,q,S: inv∧(q≠%∨(q=$
$% ∧ S ≠ [ ] ) ) , i n v ∧ # X 0 < # X ≤ | T |$
Under the premise of $q≠%∨(q=% ∧S≠[])$, the above formula can be further refined according to Selection in Table 1.
$i f q ≠ % → X , q , S : [ q ≠ % ∧ i n v ∧ ( q ≠ % ∨ ( q = % ∧ S ≠ [ ] ) , i n v ∧ # x 0 ≤ # x < | T | ]$(10)
$X , q , S : [ ( q = % ∧ S ≠ [ ] ) ∧ i n v ∧ ( q ≠ % ∨ ( q = % ∧ S ≠ [ ] ) , i n v ∧ # x 0 < = # x < | T | ]$(11)
The above equation (10) can be further refined according to Assignment in Table 1:
$X , q , S : [ q ≠ % ∧ i n v , i n v ∧ ( # x 0 < # X ≤ | T | ) ] ⊑ S , X , q ≔ [ q . r ] ↑ S , X ↑ [ q . n ] , q . l ;$
$q ≠ % ∧ i n v ⇒ i n v ∧ ( # x 0 < # X ≤ | T | ) [ S , X , q \ [ q . r ] ↑ S , X ↑ [ q . n ] , q . l ]$
According to the definition of inv, Prebt, and F functions, the above formula can be simplified:
Left side:
$q ≠ % ∧ i n v ≡ q ≠ % ∧ ( P r e b t ( T ) = X ↑ P r e b t ( q ) ↑ F ( S ) ) ≡ P r e b t ( T ) = X ↑ ( q . n ↑ P r e b t ( q . l ) ↑ P r e b t ( q . r ) ) ↑ F ( S )$
Right side:
$P r e b t ( T ) = X ↑ P r e b t ( q ) ↑ F ( S ) ∧ ( # x 0 < # X ≤ | T | )$
$≡ P r e b t ( T ) = X ↑ [ q . n ] ↑ P r e b t ( q . l ↑ F ( [ q . r ] ↑ S ∧ ( # x 0 < # ( X ↑ [ q . n ] ) ≤ | T | ) ) )$
$≡ P r e b t ( T ) = X ↑ [ q . n ] ↑ P r e b t ( q . l ) ↑ P r e b t ( q . r ) ↑ F ( S ) ∧ t r u e$
$≡ P r e b t ( T ) = X ↑ [ q . n ] ↑ P r e b t ( q . l ) ↑ P r e b t ( q . r ) ↑ F ( S )$
So $q≠%∧inv⇒inv∧(#x0<#X≤|T|)[S,X,q\[q.r]$$↑S,X↑[q.n],q.l]≡True$
Similarly, the above equation (11) can be further refined according to Assignment in Table 1.
$X , q , S : [ ( q ≠ % ∧ S ≠ [ ] ) ∧ i n v , i n v ∧ ( # x 0 < # X ≤ | T | ) ] ⊑ q , S ≔ S [ S . h ] , S [ S . h + 1 … S . t ] ;$
According to the above refinement, the program can be replaced by:
if$q ≠ % →$
$S , X , q ≔ [ q . r ] ↑ S , X ↑ [ q . n ] , q . 1 ;$(12)
$q , S : = S [ S . h ] , S [ S . h + 1 … S . t ] ;$(13)
Combined with the above formula, the final GCL program is:
do$q ≠ % ∨ ( q = % ∧ S ≠ [ ] ) →$
if$q ≠ %$
$S , X , q : = [ q . r ] ↑ S , X ↑ [ q . n ] , q . 1 ;$
$q , S : = S [ S . h ] , S [ S . h + 1 . . . S . t ] ;$
2.3 GCL Program Automation Verification of BTPP
2.3.1 Verification condition generation
In order to verify the GCL code obtained in Section 2.2.3 above, we must first construct the lemma Pre_bt. Then, using the GCL code from above, we must write the corresponding code in Isabelle. The
two have the same code structure, and the Isabelle code is as follows:
lemma Pre_bt: "VARS X S T q
X :=[];q:=T;S :=[];
WHILE(q≠Tnull| q=Tnull$∧$S≠[])
IF q≠TnullTHEN
X :=X@[data q];
S :=[rtree q]@S ;
q :=(ltree q)
ELSE IF q=Tnull$∧$S≠[]THEN
q :=(hd S);
S :=(tl S)
We then use the apply vcg command to get three verification conditions, as shown in Fig.9.
2.3.2 Isabelle assisted verification
Some auxiliary functions and lemmas are required to verify the above three sub-goals. The following are the specific steps:
1) Define two recursive functions Prebt, F:
Function 1:
primrec Prebt::"'a BTree⇒'a list"
"Prebt Tnull=[]"|"Prebt(BT t1 x t2) =[x]@(Prebt t1)@(Prebt t2)"
fun F::"'a BTreelist⇒'a list"
"F[] =[]"|"F(x#xs) =(Prebt x)@(F xs)"
2) Create the lemmas prebt_rule and F_rule to prove the final program.
Lemma 1:
lemma prebt_rule[simp]:"q≠Tnull$⇒$Prebt q =data q#Prebt(ltree q)@Prebt(rtree q)"
apply(induct q)
apply auto
Lemma 2:
lemma F_rule[simp]:"xs≠[]$⇒$F xs=Prebt(hd xs)@F(tl xs)"
apply(induct xs)
apply auto
3) Prove the three sub-goals:
apply auto
With the auto command in 3) above, we can get a successful result of verification. This is shown in Fig.10.
2.4 Conversion from GCL Program to Executable Program C++ of BTPP
For the verified GCL program in Section 2.2.3, it is a non-executable abstract program that needs to be converted to executable C++. First, the GCL program needs to be equivalently converted to an
Apla program because their syntax is similar. The resulting Apla program is then passed through our team's conversion system to generate a C++ program, and the conversion result is shown in Fig.11.
Finally, the converted C++ program is compiled and run, and the running result is shown in Fig.12.
3 Conclusion
This paper proposes a new method for building and verifying nonlinear programs. Using the preorder traversal of a binary tree as an example, an abstract program GCL is created from the initial
specification. Isabelle validates GCL programs before converting them to executables using the C++ conversion platform. The following are the primary benefits of this paper:
1) Partition recursion is used throughout the program construction process to derive loop invariant. The specification is then gradually refined by Morgan's refinement rule based on loop invariant.
This method ensures the acquisition of loop invariant while also refining the program.
2) The method first employs VCG to generate verification conditions automatically. The verification conditions are then mechanistically verified using Isabelle. This greatly improves verification
3) This method improves the method's integrity by converting the obtained abstract program GCL into a C++ executable program using the C++ conversion platform.
Our next step is to use this approach and combine it with our previous work^[20-25] to derive and synthesize more complex algorithms for nonlinear data structures, such as those related to binary
trees or graphs.
All Tables
All Figures
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on
Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while. | {"url":"https://wujns.edpsciences.org/articles/wujns/full_html/2023/03/wujns-1007-1202-2023-03-0246-11/wujns-1007-1202-2023-03-0246-11.html","timestamp":"2024-11-13T21:25:38Z","content_type":"text/html","content_length":"173915","record_id":"<urn:uuid:e99bb9f6-9f9a-4dba-a24a-a6ef4cc4a383>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00838.warc.gz"} |
Forecasting SYM‐H Index: A Comparison Between Long Short‐Term Memory and Convolutional Neural Networks
Papers from SWICo members
F. Siciliano G. Consolini, R. Tozzi, M. Gentili, F. Giannattasio, and P. De Michelis.
Geomagnetic indices are proxies of the geomagnetic disturbances observed on the ground during geomagnetic storms and substorms. So, their forecasting represents a key point to develop warning systems
for the mitigation of possible effects of severe geomagnetic storms on critical ground infrastructures. Here, we forecast SYM‐H index using two artificial neural network models based on two
conceptually different networks: the Long Short‐Term Memory (LSTM) and the Convolutional Neural Network (CNN). Both networks are trained with two different sets of data: 1) interplanetary magnetic
field (IMF) components and magnitude, and 2) interplanetary magnetic field components and magnitude and previous SYM‐H values. Specifically, we selected 42 geomagnetic storms among the most intense
occurred between 1998 and 2018.
Observed and predicted SYM-H index in the case of the geomagnetic storm of November 2004. Plots in each panel correspond to: LSTM (a) and CNN (b) prediction without SYM-H index among the input
parameters, LSTM (c) andCNN (d) prediction with SYM-H index among the input parameters.
The performance of the two models has been compared thus pointing out the peculiarity of each model. In summary we have found that: 1) both networks are able to well forecast SYM‐H index 1 hour in
advance, with values of the coefficient of determination R^2 larger than 95%; 2) when using the data set that includes SYM-H index the model based on LSTM is slightly more accurate than that based on
CNN; 3) differently, when using the data set consisting of IMF values only the model based on CNN displays a higher accuracy than that based on LSTM.
Publication: F. Siciliano G. Consolini R. Tozzi M. Gentili F. Giannattasio P. De Michelis, Forecasting SYM‐H Index: A Comparison Between Long Short‐Term Memory and Convolutional Neural Networks,
Space Weather, 19 (2), 2021. https://doi.org/10.1029/2020SW002589 | {"url":"http://www.swico.it/2022/02/09/forecasting-sym%E2%80%90h-index-a-comparison-between-long-short%E2%80%90term-memory-and-convolutional-neural-networks/","timestamp":"2024-11-13T20:45:42Z","content_type":"text/html","content_length":"42846","record_id":"<urn:uuid:41df23ff-7046-4270-a494-2c65ac6db9e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00543.warc.gz"} |
Rummy Online - Play Indian Rummy Games to Win Real Cash - Rummytime
What is Points Rummy
Points Rummy is one of the fastest formats of 13 card rummy. It is played between 2 to 6 players and has only one deal. The objective is to make the first valid declaration, which is equal to 0
points. Being a quicker variation of the beloved 13 card rummy, Points Rummy offers a thrilling opportunity for players to test their strategic skills to win real rewards in a short span of time.
Points Rummy
In this format, each point has a monetary value to it. Your total Buy-In would depend on the maximum points you can lose in a game. For example, let’s say 1 point = ₹0.5, and you can lose a maximum
of 80 points in a game. So, your Buy-In would be: ₹0.5 × 80 = ₹40.
On platforms like RummyTime, the Buy-In for Points Rummy starts from as low as ₹0.05 per point and goes up to ₹500 .
To win at a Points rummy game, one has to make a valid declaration, which must have at least 2 sequences (one of which mandatorily has to be pure). The winner gets a cash reward worth all the
remaining players’ points.
So, in the end if the combined total of the losing players’ points is 200 and the point value is ₹0.5, the winner would get: ₹0.5 × 200 = ₹100 - platform’s fee. Thus, players should always aim at
having the least amount of points at the end of the game to reduce losses.
How to Play and Rules of Points Rummy
A game of Points Rummy follows a standard rummy rules of 13 card format and it is played between 2 to 6 players and uses 2 decks of 52 cards plus printed Jokers.
Toss: One card is dealt to each player at random. The player with the highest-ranking card would go first.
Dealing: The act of distributing cards is called dealing. 13 cards are dealt to each player randomly.
Objective: To make the first valid declaration, or leave the least amount of points possible on the board.
A valid declaration includes at least two sequences, one of which has to be a pure sequence. The rest of the cards can be grouped into sets or sequences.
Once cards are dealt to all players, the remaining cards are placed face down on the table to create the closed deck. Then, the top card from the closed deck is flipped face up as the open deck or
discard pile.
Some rules to keep in mind:
• The game Buy-In is fixed in advance–this is the minimum wallet balance one needs to enter a Points Rummy game.
• If you win, no amount gets deducted and the winning amount gets added to the wallet directly. In case a player loses the game, only the amount equivalent to their points are deducted. So, if your
Buy-In was ₹0.5 × 80 = ₹40, and you ended with 20 points on the board–₹0.5 × 20 = ₹10 will be deducted from your balance.
• Dropping is a strategic move some players use when they have been dealt a bad hand. If you choose to drop on your first turn, 20 points are added to your score. Dropping midway would add 40
• Penalty on consecutive misses: On RummyTime, missing two consecutive turns is also considered a midway drop. The player drops out of the game and 40 points are added to their total. This setting
can be changed on every platform; a player can choose to automatically drop after 3, 4 or even 1 missed turn.
• Making an invalid declaration costs 80 points, which is the maximum amount of points one can lose in any 13 card rummy variations.
• The winner takes the monetary equivalent of the combined total of the remaining players’ points.
How the Score is Calculated in Points Rummy
In Points rummy, the winner of the game is said to have 0 points. The losing players get points equal to the sum total of the values of their ungrouped cards. The values of the cards in the game of
rummy are given below:
• Face cards (King, Queen and Jack) and Aces: 10 points each
• Numbered cards (2 to 10): Same as their face value
• Jokers (both printed and wild): 0 points each.
Let’s assume at the end of the game, you have the following hand:
Here, we have a pure sequence, an impure sequence, a set, and three ungrouped cards- 4♣ 8♣ and 3♥.
The valid groups would be equal to 0 points each, while the ungrouped cards would be equal to 4 + 8 + 3 = 15 points.
Calculating winnings:
Let us suppose there are 6 players and the scoreboard looks like this:
• Winner - 0 points
• Player A - 25 points
• Player B - 49 points
• Player C - 28 points
• Player D - 31 points
• Player E - 9 points
Assuming the point value is ₹0.5, the winner will get: (25 + 49 + 28 + 31 + 9) × ₹0.5 - (platform’s fee) = ₹71 - (platform’s fee)
How Dropping Works in Points Rummy
Every player gets the option to drop on their turn through the entire game.
The drop option is only available if the player has NOT picked a card from the open or the closed deck in that turn. After a player drops, the game continues as usual till one makes a valid
First drop: If a player drops the game in their very first turn, they get 20 points. Experienced players often use this option when they’re not confident about their hand (one with 0 sequences or
jokers, generally).
Midway drop: If a player drops anytime after their first turn, they get 40 points. Players study their opponents’ moves and often use this option to limit their losses.
Consecutive misses: Missing a preset number consecutive turns is considered equivalent to a midway drop, and 40 points get added to a player’s total. On RummyTime, this limit is set at 2 consecutive
misses and can be changed in the user settings.
Invalid declaration: If a player makes an invalid declaration, they get 80 points and are dropped from the game.
So, here you go. This is how you can also get started with playing Points Rummy. All you have to do is head to the RummyTime website, then download rummy app and start gaming today. | {"url":"https://www.rummytime.com/rummy-variations/points-rummy-game/","timestamp":"2024-11-14T19:59:15Z","content_type":"text/html","content_length":"83723","record_id":"<urn:uuid:4125b3e4-b1e7-4fab-b032-08a5e6223b72>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00376.warc.gz"} |
EXAM 1 MATH 3383
Questions and Answers
• 1.
A quadrilateral with opposite sides parallel
A quadrilateral with opposite sides parallel is called a parallelogram. In a parallelogram, the opposite sides are equal in length and the opposite angles are equal in measure. The parallel sides
never intersect each other. This geometric shape has several properties, such as the diagonals bisecting each other, the opposite angles being supplementary, and the consecutive angles being
supplementary. Overall, a parallelogram is a special type of quadrilateral that exhibits specific characteristics based on its parallel sides.
• 2.
A special parallelogram with four congruent sides
A special parallelogram with four congruent sides is called a rhombus. A rhombus is a quadrilateral with opposite sides that are parallel and congruent. It also has opposite angles that are
equal. Therefore, a rhombus satisfies the given condition of having four congruent sides, making it the correct answer.
• 3.
A special parallelogram with four right angles
A rectangle is a special parallelogram with four right angles. It has opposite sides that are equal in length and parallel to each other. The diagonals of a rectangle are also equal in length and
bisect each other. Therefore, a rectangle fits the description of a special parallelogram with four right angles.
• 4.
An extra special parallelogram with four congruent sides and four right angles
A square is a type of parallelogram that has four congruent sides and four right angles. The given description perfectly matches the characteristics of a square, making it the correct answer.
• 5.
What two kinds of shapes are non-parallelograms?
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. Kite and Trapezoid
Non-parallelograms are shapes that do not have parallel sides. A kite is a non-parallelogram because its sides are not parallel to each other. Similarly, a trapezoid is a non-parallelogram
because it has only one pair of parallel sides. Therefore, the correct answer is Kite and Trapezoid.
• 6.
A quadrilateral with two distinct pairs or congruent, adjacent sides
Correct Answer
A kite is a quadrilateral with two distinct pairs of congruent, adjacent sides. In other words, a kite has two pairs of sides that are equal in length and are next to each other. This creates a
distinctive shape with two pairs of adjacent sides that are congruent. Therefore, the given answer, "Kite," is correct.
• 7.
A quadrilateral with one pair of parallel sides
Correct Answer
A quadrilateral with one pair of parallel sides is called a trapezoid. In a trapezoid, the parallel sides are called bases, and the non-parallel sides are called legs. The bases can be of
different lengths, but the opposite angles formed by the bases are always congruent. This distinguishes a trapezoid from other quadrilaterals, such as a parallelogram, where all sides are
• 8.
What three kinds of shapes are parallelograms?
□ A.
Rhombus, Rectangle and Square
□ B.
Trapezoid, Kite and Rhombus
□ C.
Rectangle, Rhombus and Kite
Correct Answer
A. Rhombus, Rectangle and Square
The correct answer is Rhombus, Rectangle and Square. A parallelogram is a four-sided shape with opposite sides that are parallel. A rhombus is a parallelogram with all sides equal in length. A
rectangle is a parallelogram with all angles equal to 90 degrees. A square is a special type of rectangle with all sides equal in length. Therefore, all three shapes mentioned in the answer
choices meet the criteria of being parallelograms.
• 9.
A closed 2D figure with no crossings or reuse of endpoints
Correct Answer
A polygon is a closed 2D figure that does not have any crossings or reuse of endpoints. It is made up of straight line segments that connect to form a closed shape. The term "polygon" is commonly
used in geometry to describe shapes such as triangles, squares, pentagons, and so on. These shapes have straight sides and do not intersect themselves. Therefore, the given answer "Polygon"
accurately describes a closed 2D figure with no crossings or reuse of endpoints.
• 10.
What is Euler's Formula?
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. V+F-2=E
Euler's Formula states that for any convex polyhedron, the number of vertices (V), edges (E), and faces (F) are related by the equation V+F-2=E. This means that if we know the number of vertices
and faces of a convex polyhedron, we can determine the number of edges it has by substituting the values into the formula.
• 11.
3D polyhedron with a flat base where triangular lateral faces meet at an apex
Correct Answer
A pyramid is a 3D polyhedron with a flat base where triangular lateral faces meet at an apex. The description matches the characteristics of a pyramid, as it has a flat base and triangular faces
that converge at a single point, forming the apex.
• 12.
3D polyhedron with two bases congruent to each other and parallel lateral faces
Correct Answer
A prism is a 3D polyhedron that has two bases that are congruent to each other and parallel lateral faces. The bases are identical in shape and size, and the lateral faces connect the
corresponding vertices of the bases. This arrangement creates a solid shape with flat, rectangular faces. The given description perfectly matches the characteristics of a prism, making it the
correct answer.
• 13.
Pyramid whose lateral edges are congruent
Correct Answer
Regular Pyramid
A regular pyramid is a pyramid whose lateral edges are congruent. This means that all the slanting edges of the pyramid have the same length. In a regular pyramid, the base is a regular polygon,
and the height is the perpendicular distance from the apex (top point) to the base. Since the lateral edges are congruent, it implies that all the triangular faces of the pyramid are congruent as
well. Therefore, the correct answer is a regular pyramid.
• 14.
Prism whose lateral edges meet at a right angle to the base
Correct Answer
Right Prism
A right prism is a type of prism where the lateral edges meet at a right angle to the base. This means that the edges forming the sides of the prism are perpendicular to the base. The term
"right" in right prism refers to the right angle formed between the lateral edges and the base. Therefore, the given answer "Right Prism" correctly describes a prism whose lateral edges meet at a
right angle to the base.
• 15.
What does it mean to be congruent?
□ A.
Corresponding sides are congruent
□ B.
Corresponding angles are congruent
□ C.
No two sides have same length
□ D.
At least two sides are same length
Correct Answer(s)
A. Corresponding sides are congruent
B. Corresponding angles are congruent
To be congruent means that corresponding sides and corresponding angles of two objects or shapes are equal in measure or length. This means that if two shapes are congruent, their corresponding
sides will have the same length and their corresponding angles will have the same measure. The other options, "No two sides have the same length" and "At least two sides are the same length," do
not fully capture the concept of congruence as they do not encompass both sides and angles being equal.
• 16.
Good conguence shortcuts
□ A.
□ B.
□ C.
□ D.
□ E.
□ F.
Correct Answer(s)
B. ASA
D. AAS
E. SSS
F. SAS
The given answer consists of four different types of congruence shortcuts: ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), SSS (Side-Side-Side), and SAS (Side-Angle-Side). These shortcuts are
used to prove that two triangles are congruent based on the given information about their angles and sides. ASA states that if two angles and the included side of one triangle are congruent to
two angles and the included side of another triangle, then the triangles are congruent. AAS states that if two angles and a non-included side of one triangle are congruent to two angles and the
corresponding non-included side of another triangle, then the triangles are congruent. SSS states that if the three sides of one triangle are congruent to the three sides of another triangle,
then the triangles are congruent. SAS states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are
• 17.
A quadrilateral where non-parallel sides are conguent
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. Isosceles Trapezoid
An isosceles trapezoid is a quadrilateral where the non-parallel sides are congruent. In an isosceles trapezoid, the two opposite sides are parallel, and the other two sides are congruent. This
means that the non-parallel sides have the same length. Therefore, an isosceles trapezoid fits the given description of a quadrilateral where non-parallel sides are congruent.
• 18.
What is the formula for the sum of interior angles of any polygon?
□ A.
□ B.
□ C.
□ D.
Correct Answer
C. N-2x180
The formula for the sum of interior angles of any polygon is n-2x180. This formula can be derived by dividing the polygon into (n-2) triangles, where n is the number of sides of the polygon. Each
triangle has interior angles that sum up to 180 degrees. Therefore, the sum of interior angles of the polygon is equal to (n-2) multiplied by 180.
• 19.
In any polygon, the number of sides=number of vertices=number of interior angles. In other words, the amount of each are the same.
Correct Answer
A. True
The statement is true because in any polygon, the number of sides is equal to the number of vertices, which is also equal to the number of interior angles. This is a fundamental property of
polygons and is true for all polygons regardless of their shape or size. Therefore, the amount of sides, vertices, and interior angles in a polygon will always be the same.
• 20.
Find the sum of the interior angles in a dodecagon.
Correct Answer
A dodecagon is a polygon with 12 sides. To find the sum of the interior angles in a dodecagon, we can use the formula (n-2) * 180 degrees, where n is the number of sides. Plugging in the value of
n as 12, we get (12-2) * 180 = 10 * 180 = 1800 degrees. Therefore, the sum of the interior angles in a dodecagon is 1800.
• 21.
Find the measure of each interior angle in an equiangular pentagon.
Correct Answer
540 (the SUM of the interior angles)
• 22.
What shapes are NON-POLYHEDRA?
□ A.
□ B.
□ C.
□ D.
Correct Answer(s)
A. Cylinder
D. Cone
The shapes that are non-polyhedra are those that do not have flat faces. A cylinder and a cone do not have flat faces, as their surfaces are curved. Therefore, both the cylinder and the cone are
• 23.
In order to have a Regular Polygon, the shape has to be both...
Correct Answer(s)
Equiangular and Equilateral
A regular polygon is a polygon where all sides are equal in length and all angles are equal. Therefore, in order for a shape to be a regular polygon, it must be both equiangular (having equal
angles) and equilateral (having equal sides). If either of these conditions is not met, the shape would not be a regular polygon.
• 24.
Line segment where two faces meet
Correct Answer
C. Edge
An edge is a line segment where two faces meet. In geometry, a face refers to a flat surface of a three-dimensional shape, while a base typically refers to the bottom or lowest face of a solid
figure. An edge, on the other hand, is the line segment formed by the intersection of two faces. It can be visualized as the boundary or border between two adjacent faces of a solid shape.
• 25.
All squares are rectangles.
Correct Answer
A. True
All squares are rectangles because a square is a special type of rectangle where all four sides are equal in length. Therefore, since all squares meet the criteria of being a rectangle, the
statement is true.
• 26.
All quadrilaterals are parallelograms.
Correct Answer
B. False
The statement "All quadrilaterals are parallelograms" is false because not all quadrilaterals have their opposite sides parallel. A quadrilateral is a polygon with four sides, and a parallelogram
is a special type of quadrilateral where opposite sides are parallel. However, there are other types of quadrilaterals, such as trapezoids or kites, where the opposite sides are not parallel.
Therefore, it is incorrect to say that all quadrilaterals are parallelograms.
• 27.
This shape is a:
□ A.
□ B.
□ C.
□ D.
□ E.
Correct Answer
D. Concave hexagon
A concave hexagon is a shape that has six sides and at least one interior angle greater than 180 degrees. This means that the shape has a "caved-in" or indented portion, which is why it is called
concave. In contrast, a convex hexagon would have all interior angles less than 180 degrees and no indented portions. Since the given shape is described as concave, it implies that it has at
least one angle greater than 180 degrees, making it a concave hexagon.
• 28.
This shape is a
□ A.
□ B.
□ C.
□ D.
□ E.
Correct Answer
A. Nonagon
The given shape is a nonagon because it has nine sides. A nonagon is a polygon with nine sides and nine angles.
• 29.
What is the name given to this shape -
Correct Answer
The correct answer is heptagon and septagon. Both terms refer to the same shape, which is a polygon with seven sides. "Heptagon" is derived from the Greek word "hepta" meaning seven, while
"septagon" is derived from the Latin word "septem" also meaning seven. These terms are used interchangeably to describe the shape with seven sides.
• 30.
What is the sum of the internal angles of a triangle?
□ A.
□ B.
□ C.
□ D.
□ E.
Correct Answer
C. 180°
The sum of the internal angles of a triangle is always 180°. This is a property of triangles in Euclidean geometry. It can be proven mathematically by dividing a triangle into two right triangles
and using the fact that the sum of the angles in a straight line is 180°. Therefore, the correct answer is 180°.
• 31.
Correct Answer
C. C
• 32.
Correct Answer
A. A
• 33.
Correct Answer
C. C
• 34.
What is the type of transformation represent in this figure
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. Reflection
The figure represents a reflection because it appears to be a mirror image of itself. In a reflection, the shape is flipped over a line, creating a symmetrical image.
• 35.
Which of these figures represent reflection A B
□ A.
□ B.
Correct Answer
B. Pink figure
The question asks which figure represents reflection A. The correct answer is the pink figure. Reflection is a transformation that flips a figure over a line. In this case, reflection A would be
a flip over a specific line. Without further information about the line of reflection, we can only determine the correct answer based on the given options. The pink figure is the only one that
appears to be flipped over a line, so it is the most likely representation of reflection A. | {"url":"https://www.proprofs.com/quiz-school/story.php?title=exam-1-math-3383","timestamp":"2024-11-03T06:30:46Z","content_type":"text/html","content_length":"561017","record_id":"<urn:uuid:0f90427c-768e-4881-a55c-43cd117e3657>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00106.warc.gz"} |
Tekla Tedds Engineering Library (June 2021)
Tekla Tedds Tekla Tedds for Word
British Standard
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UK and Asia
Timber member design (EN1995)
Enhanced to allow built up sections to be wider than they are deep.
Component calculations
Added a new folder to the Engineering Library for "Component calculations". A component calculation encapsulates a small engineering solution which can be easily reused by other calculations. They
are typically used by calculations in the Tedds Engineering Library and can also be integrated in your own custom written calculations.
Component calculation: Steel section classification (EN1993)
Checks the section classification of carbon steel or stainless steel sections.
Component calculation: Steel section sketch
Creates a drawing of a Steel section profile which can then be used in the user interface or output of another calculation. Supports section types I, T, Channel, Hollow Circle, Hollow Rectangle,
Angle, 2 Angles, 2 Angles (log leg), 2 Angles (short leg), Asymmetric I, Slimflor and Flats. The drawing can be customised to include dimensions and section properties as appropriate.
Other Updates
Batch design
Enhanced to include examples for carbon steel and stainless steel for the "Steel section classification (EN1993)" calculation component.
Crane gantry girder design (EN1993)
[TEDDS-5738]^1 - Fixed occasional incorrect placement of wheel loads when not all wheel loads are located on the beam.
Crane gantry girder design (EN1993)
RC pile cap design (EN1997)
RC pile cap design (ACI318)
Steel beam supporting hollowcore slab design (EN1993)
Steel beam torsion design (EN1993)
Steel masonry support design (EN1993)
Steel member fire resistance design (EN1993)
Updated to use version 2.x of the Steel section sketch component.
RC slab design (EN1992)
[TEDDS-5686]^1 - Fixed undefined variable error when enabling the summary table option for one way slabs with one span
Retaining wall analysis & design (EN1992/EN1996/EN1997)
[TEDDS-5733]^1 - Corrected the neutral axis depth calculation for the base of masonry retaining walls.
Steel member design (EN1993)
Fixed minor issues relating to the section sketch output to the document.
Timber 2D analysis & design (EN1995)
Timber member analysis & design (EN1995)
Timber member design (EN1995)
[TEDDS-581]^1 - Added note to clarify design method for built-up sections.
[TEDDS-4703]^1 - Added note to indicate calculation scope limitation for members that are wider than deep.
Back to top
Wood 2D analysis & design (NDS)
Wood member analysis & design (NDS)
Wood member design (NDS)
Enhanced to allow built up sections to be wider than they are deep.
Component calculations
Added a new folder to the Engineering Library for "Component calculations". A component calculation encapsulates a small engineering solution which can be easily reused by other calculations. They
are typically used by calculations in the Tedds Engineering Library and can also be integrated in your own custom written calculations.
Component calculation: Steel section sketch
Creates a drawing of a Steel section profile which can then be used in the user interface or output of another calculation. Supports section types I, T, Channel, Hollow Circle, Hollow Rectangle,
Angle, 2 Angles, 2 Angles (log leg), 2 Angles (short leg), Asymmetric I, Slimflor and Flats. The drawing can be customised to include dimensions and section properties as appropriate.
Product bulletin May 2021 (PBT-2105-1)
Column base plate design (AISC360) - Fixed a potentially un-conservative error affecting designs of tensions welds to ASD methodology. For more detailed information please refer to “Product bulletin
May 2021 (PBT-2105-1)” in the Bulletins group of the Engineering Library Index.
Other updates
Anchor bolt design (ACI318)
[TEDDS-4053]^1 - Fixed an error placing 0.75 factor on shear capacity values under seismic loading for ACI 318-11 version and later.
Column base plate design (AISC360)
[TEDDS-5499]^1 - Revised the calculation of A2 to ensure similar aspect ratio as the base plate.
[TEDDS-911]^1 - Revised interface information notes to clarify limitations of ASD design method.
[TEDDS-929]^1 - Fixed interface bug causing undefined variable, hlug, error.
[TEDDS-3021]^1 - Fixed issue with incorrect check of bolt and section clash.
RC beam design (ACI318)
[TEDDS-5617]^1 - Fixed the preview results not showing correctly when the beam is classified as beyond scope.
Seismic forces (ASCE7)
[TEDDS-5655]^1 - Corrected undefined variable '_seismic_Tab' error.
[TEDDS-2952]^1 - Added reference in interface for guidance on effective seismic weight
Wind loading (ASCE7)
[TEDDS-5403]^1 - Fixed external pressure coefficients for roof zones for components and cladding in ASCE7-16
[TEDDS-5591]^1 - Fixed pressure coefficients for arched roofs with a mean height of greater than 60ft.
[TEDDS-5601]^1 - Fixed internal pressure coefficients not switching values correctly when switching between building types.
Wood 2D analysis & design (NDS)
Wood member analysis & design (NDS)
Wood member design (NDS)
[TEDDS-5714]^1 - Revised the major axis effective length factor to use the member width instead of depth.
Fixed calculation of major axis effective length to be zero when not restrained but unbraced length is set to zero.
Back to top
Timber member design (EN1995)
Enhanced to allow built up sections to be wider than they are deep.
Component calculations
Added a new folder to the Engineering Library for "Component calculations". A component calculation encapsulates a small engineering solution which can be easily reused by other calculations. They
are typically used by calculations in the Tedds Engineering Library and can also be integrated in your own custom written calculations.
Component calculation: Steel section classification (EN1993)
Checks the section classification of carbon steel or stainless steel sections.
Component calculation: Steel section sketch
Creates a drawing of a Steel section profile which can then be used in the user interface or output of another calculation. Supports section types I, T, Channel, Hollow Circle, Hollow Rectangle,
Angle, 2 Angles, 2 Angles (log leg), 2 Angles (short leg), Asymmetric I, Slimflor and Flats. The drawing can be customised to include dimensions and section properties as appropriate.
Other Updates
Batch design
Enhanced to include examples for carbon steel and stainless steel for the "Steel section classification (EN1993)" calculation component.
Crane gantry girder design (EN1993)
[TEDDS-5738]^1 - Fixed occasional incorrect placement of wheel loads when not all wheel loads are located on the beam.
Crane gantry girder design (EN1993)
RC pile cap design (EN1997)
RC pile cap design (ACI318)
Steel beam supporting hollowcore slab design (EN1993)
Steel beam torsion design (EN1993)
Steel masonry support design (EN1993)
Steel member fire resistance design (EN1993)
Updated to use version 2.x of the Steel section sketch component.
RC slab design (EN1992)
[TEDDS-5686]^1 - Fixed undefined variable error when enabling the summary table option for one way slabs with one span
Retaining wall analysis & design (EN1992/EN1996/EN1997)
[TEDDS-5733]^1 - Corrected the neutral axis depth calculation for the base of masonry retaining walls.
Steel member design (EN1993)
Fixed minor issues relating to the section sketch output to the document.
Timber 2D analysis & design (EN1995)
Timber member analysis & design (EN1995)
Timber member design (EN1995)
[TEDDS-581]^1 - Added note to clarify design method for built-up sections.
[TEDDS-4703]^1 - Added note to indicate calculation scope limitation for members that are wider than deep.
Back to top
Component calculations
Added a new folder to the Engineering Library for "Component calculations". A component calculation encapsulates a small engineering solution which can be easily reused by other calculations. They
are typically used by calculations in the Tedds Engineering Library and can also be integrated in your own custom written calculations.
Component calculation: Steel section sketch
Creates a drawing of a Steel section profile which can then be used in the user interface or output of another calculation. Supports section types I, T, Channel, Hollow Circle, Hollow Rectangle,
Angle, 2 Angles, 2 Angles (log leg), 2 Angles (short leg), Asymmetric I, Slimflor and Flats. The drawing can be customised to include dimensions and section properties as appropriate.
Back to top
Component calculations
Added a new folder to the Engineering Library for "Component calculations". A component calculation encapsulates a small engineering solution which can be easily reused by other calculations. They
are typically used by calculations in the Tedds Engineering Library and can also be integrated in your own custom written calculations.
Component calculation: Steel section sketch
Creates a drawing of a Steel section profile which can then be used in the user interface or output of another calculation. Supports section types I, T, Channel, Hollow Circle, Hollow Rectangle,
Angle, 2 Angles, 2 Angles (log leg), 2 Angles (short leg), Asymmetric I, Slimflor and Flats. The drawing can be customised to include dimensions and section properties as appropriate.
^1 This number is an internal reference number and can be quoted to your local Support Department should further information on an item be required.
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Computing and maximizing the exact reliability of wireless backhaul networks
The reliability of a fixed wireless backhaul network is the probability that the network can meet all the communication requirements considering the uncertainty (e.g., due to weather) in the maximum
capacity of each link. We provide an algorithm to compute the exact reliability of a backhaul network, given a discrete probability distribution on the possible capacities available at each link. The
algorithm computes a conditional probability tree, where at each leaf in the tree a valid routing for the network is evaluated. Any such tree provides bounds on the reliability, and the algorithm
improves these bounds by branching in the tree. We also consider the problem of determining the topology and configuration of a backhaul network that maximizes reliability subject to a limited
budget. We provide an algorithm that exploits properties of the conditional probability tree used to calculate reliability of a given network design, and we evaluate its computational efficiency.
• Backhaul network
• Network design
• Optimization
• Reliability
Dive into the research topics of 'Computing and maximizing the exact reliability of wireless backhaul networks'. Together they form a unique fingerprint. | {"url":"https://pure.uai.cl/en/publications/computing-and-maximizing-the-exact-reliability-of-wireless-backha","timestamp":"2024-11-06T20:56:56Z","content_type":"text/html","content_length":"52261","record_id":"<urn:uuid:a8b14b39-b682-4d76-96b4-ae86af4fe2f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00010.warc.gz"} |
Interesting electrical problem
My Victron battery monitor lied to me.
We’ve been out for a week or so in iffy conditions. Not much sun but enough to keep us going, or so we thought. Yesterday night when we got back from hiking our battery monitor said we were at 60%.
Not ideal as we thought that we’d get a good bit more charge on our first full sun day in a while, but it was enough that we decided we had the power reserve to use the inverter and make some baked
rice in the toaster oven. Tasted great, but after eating and just sitting around enjoying some music on the stereo suddenly everything in the trailer turned off. And I mean everything. Nothing at
all worked, even the battery monitors - the whole trailer was black. I checked fuses and breakers but the only conclusion I could come to was that our battleborn batteries had tripped their
protection circuit. And that was exactly what it was. After a while they tried to come back on, but that lasted maybe 30 seconds and they were off for the night. We woke up this morning to various
things beeping as the solar struggled to return power to everything.
The problem was that the batteries’ DOD was much more than the monitor was saying, and of course with lithiums you can’t tell very well what their DOD is just by looking at voltage alone, since they
maintain a pretty constant voltage before they drop off a cliff at the end, meaning you’re largely dependent on the battery monitor’s algorithm giving you a percentage. So tripping the batteries’
protection circuit and going from just enjoying a nice evening to zero power was quite a surprise for us.
So we’re at a KOA tonight before heading back out. A full charge should give us enough for the next leg of our trip. Plus we’ve got a fresh tank of water and clean laundry.
I’ll contact both Victron and Battleborn to try to figure out why they got out of sync. I suspect it’s a Victron problem but we’ll see.
We’ll be out of cell range for a week when we leave in the AM so I doubt I’ll be able to reply to comments for at least a while, but I thought I’d post about it just because it’s something new and
interesting. Happy travels.
• Moderators
Blue sky gives us strange readings , too. Seems like all these systems run on algorithms, as opposed to true.
I get the worst monitor info if we have any sort of auxiliary power... portable solar, or 110.
• 1
2008 Ram 1500 4 × 4
2008 Oliver Elite, Hull #12
Florida and Western North Carolina, or wherever the truck goes....
400 watts solar. DC compressor fridge. No inverter. 2 x 105 ah agm batteries . Life is good.
• Moderators
2 hours ago, Overland said:
My Victron battery monitor lied to me.
We’ve been out for a week or so in iffy conditions. Not much sun but enough to keep us going, or so we thought. Yesterday night when we got back from hiking our battery monitor said we were at
60%. Not ideal as we thought that we’d get a good bit more charge on our first full sun day in a while, but it was enough that we decided we had the power reserve to use the inverter and make
some baked rice in the toaster oven. Tasted great, but after eating and just sitting around enjoying some music on the stereo suddenly everything in the trailer turned off. And I mean everything
. Nothing at all worked, even the battery monitors - the whole trailer was black. I checked fuses and breakers but the only conclusion I could come to was that our battleborn batteries had
tripped their protection circuit. And that was exactly what it was. After a while they tried to come back on, but that lasted maybe 30 seconds and they were off for the night. We woke up this
morning to various things beeping as the solar struggled to return power to everything.
The problem was that the batteries’ DOD was much more than the monitor was saying, and of course with lithiums you can’t tell very well what their DOD is just by looking at voltage alone, since
they maintain a pretty constant voltage before they drop off a cliff at the end, meaning you’re largely dependent on the battery monitor’s algorithm giving you a percentage. So tripping the
batteries’ protection circuit and going from just enjoying a nice evening to zero power was quite a surprise for us.
So we’re at a KOA tonight before heading back out. A full charge should give us enough for the next leg of our trip. Plus we’ve got a fresh tank of water and clean laundry.
I’ll contact both Victron and Battleborn to try to figure out why they got out of sync. I suspect it’s a Victron problem but we’ll see.
We’ll be out of cell range for a week when we leave in the AM so I doubt I’ll be able to reply to comments for at least a while, but I thought I’d post about it just because it’s something new
and interesting. Happy travels.
Sorry to hear . . . . what a way to ruin a good day!
• 1
Ray and Susan Huff
Elite II Twin "Pearl" - Hull#699; delivered December 7, 2020
2013 F350 6.7l diesel Super Duty 4x4 long bed crew cab
1UP-USA Heavy-duty bike rack
2017 Leisure Travel Van Unity Twin Bed (sold)
2 hours ago, SeaDawg said:
Blue sky gives us strange readings , too. Seems like all these systems run on algorithms, as opposed to true.
I get the worst monitor info if we have any sort of auxiliary power... portable solar, or 110.
So what good's a monitor if you can't rely on it? I guess, with today's technology, we just expect perfection and reliability . . . . . not going to happen 😠
Ray and Susan Huff
Elite II Twin "Pearl" - Hull#699; delivered December 7, 2020
2013 F350 6.7l diesel Super Duty 4x4 long bed crew cab
1UP-USA Heavy-duty bike rack
2017 Leisure Travel Van Unity Twin Bed (sold)
• Moderator+
2 hours ago, SeaDawg said:
Blue sky gives us strange readings , too. Seems like all these systems run on algorithms, as opposed to true.
I get the worst monitor info if we have any sort of auxiliary power... portable solar, or 110.
I got strange readings from ourBlue Sky IPN-Pro for a while. Until I figured out that not everything was running through the shunt. After a little negative re-routing all is good. The main problem
was that the onboard charger's negative wire was bypassing the shunt therefore the IPN-Pro did not see any of the power going into the batteries from it. Check the grounds from your charge controller
and converter/charger and make sure they go to the shunt before grounding out to the trailer.
• 1
• 2
Steve, Tali and our dog Rocky plus our beloved Storm, Lucy, Maggie and Reacher (all waiting at the Rainbow Bridge)
2008 Legacy Elite I - Outlaw Oliver, Hull #026 | 2014 Legacy Elite II - Outlaw Oliver, Hull #050 | 2022 Silverado High Country 3500HD SRW Diesel 4x4
• Moderator+
28 minutes ago, Susan Huff said:
So what good's a monitor if you can't rely on it? I guess, with today's technology, we just expect perfection and reliability . . . . . not going to happen 😠
You'll be more likely to get it if everything is wired correctly.
• 1
Steve, Tali and our dog Rocky plus our beloved Storm, Lucy, Maggie and Reacher (all waiting at the Rainbow Bridge)
2008 Legacy Elite I - Outlaw Oliver, Hull #026 | 2014 Legacy Elite II - Outlaw Oliver, Hull #050 | 2022 Silverado High Country 3500HD SRW Diesel 4x4
• Moderators
That's a good suggestion to look at when we install our new panels and controller this winter, Steve. Thanks. I really don't remember much about the wiring.
The interesting thing is, our bluesky returns to normal quickly, "resetting itself" as soon as soon battery capacity reaches 100 per cent. Sometimes it will jump from a 78 per cent reading to 100
with a short time on shore power, or full sun. We know it's higher than 78 from meter checks.
And Susan, most of the time we charge solely on the fixed solar. With good sun, it's entirely reliable. Oliver doesn't use Blue Sky anymore, anyway, but we actually really like it. We have Victron on
the boat pv system.
• 1
2008 Ram 1500 4 × 4
2008 Oliver Elite, Hull #12
Florida and Western North Carolina, or wherever the truck goes....
400 watts solar. DC compressor fridge. No inverter. 2 x 105 ah agm batteries . Life is good.
On 11/11/2020 at 10:05 PM, Susan Huff said:
Sorry to hear . . . . what a way to ruin a good day!
Not ruined. Even without amenities, the Ollie is still the best tent we’ve ever owned.
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• 1
Thanks for the replies. It seems to have gotten back on track after a full charge. I’m thinking now that this was related to a different problem that I’ve discovered with my inverter, which isn’t
communicating properly with the Victron control panel. They’re connected, but turning on and off the inverter from the panel doesn’t always work and the panel isn’t reliably showing the state of the
inverter. Twice now I’ve checked to see if the inverter is off, and the control panel said it was, but the outlets were still hot. The only way to reliably turn it off now is via the switch on the
unit. So I suspect that this has been a problem since we started the trip and the inverter has been on most of the time, with maybe that power consumption not being registered by the control panel.
Something else that I’ve noticed is that my solar charge controller turns off when the batteries are full and doesn’t come back on. So if the batteries are full at noon, with the fridge, fans, etc.
running for the rest of the day, we start the evening not with 100% but maybe 95% or less. Maybe that’s normal, but it seems like I would have noticed that behavior before. There were software
updates to all my equipment that I installed before leaving, so it’s possible that all this behavior is buggy updates.
• 1
• 2
• Moderators
On the boat, we leave the inverter turned off unless we're actually using it. But, our inverter on the boat is a xantrex, not victron, sadly.
I have read about the Victron shutting down when the batteries are full, and not restarting. But, we have not really watched for it in the boat. I thought that was an older, resolved problem?
Nor, do we usually use much power on the sailboat. Something to watch for when/if we install the danfoss backup system to the engine driven cold plate refrigeration. Our delight is that we have not
had to plug in the boat to shore power, in a year . And, when the alternator failed again in March, in a tricky situation, the batteries were still fully charged, with solar. 😁
The bluesky algorithms have never let me down to the point you experienced in your recent trip. But, if the weather is cloudy/low sun for string of days, they're def inaccurate, and we also watch
the seelevel monitor, and sometimes take readings straight from our 2 group 27 batteries, with a meter. Unlike you, we don't have a lot of leeway, with just two 105 ah agm batteries. And, all the
readings are somewhat inaccurate with agm , unsettled batteries, anyway. It can be really frustrating.
Looks like yours reset too, with a full charge, as ours do with the bluesky.
2008 Ram 1500 4 × 4
2008 Oliver Elite, Hull #12
Florida and Western North Carolina, or wherever the truck goes....
400 watts solar. DC compressor fridge. No inverter. 2 x 105 ah agm batteries . Life is good.
• Moderators
And, @Susan Huff, so many of our tech/mechanical problems over the years have really been user issues. I'm not embarrassed to say so. Every new system is a learning curve. My learning highway
probably looks like the dragon's tail ride, lol.
Today, Paul and I installed a Phyn water leak sensor system for the house. His plumbing was flawless. The tech side was much more complicated.
Sometimes, there are little details missing in the manual, or app instructions. I went through the app a dozen times, including faq several times, and then spent an hour on the phone with two great
techs in California. Everything is working now. If I had turned on location on my android phone, and paired to the 2. 4 net, instead of the 5 band, it would have saved 45 minutes ... (that bit of
info is not in the instructions, anywhere.)⁹
Tech is great. When it works , and we understand it. And, have clear instructions.
Even then, I can manage to mess something up, sometimes.
Edited by SeaDawg
• 1
2008 Ram 1500 4 × 4
2008 Oliver Elite, Hull #12
Florida and Western North Carolina, or wherever the truck goes....
400 watts solar. DC compressor fridge. No inverter. 2 x 105 ah agm batteries . Life is good.
17 hours ago, SeaDawg said:
Tech is great. When it works , and we understand it. And, have clear instructions.
Even then, I can manage to mess something up, sometimes.
And there are those folks who never read the written instructions, and complain when something doesn’t work...... I am referring to somebody in my family. RTFM 😬
John Davies
Spokane WA
Edited by John E Davies
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SOLD 07/23 "Mouse": 2017 Legacy Elite II Two Beds, Hull Number 218, See my HOW TO threads:
Tow Vehicle: 2013 Land Cruiser 200, 32” LT tires, airbags, Safari snorkel, Maggiolina Grand Tour 360 Carbon RTT.
So, new update. I woke this morning to the battery monitor telling me that I was at 78%, but the voltage was 13.1, which for lithium’s is about 40%. Worse, when I turned the inverter on to make
coffee, the voltage dropped to 12.5. Not great, but it dawned on me that all this behavior would make sense if I actually had only 200 Ah of batteries rather than 400. Could two of my batteries have
died? I called Battleborn.
Admirably, even though I got a recording and they’re closed for the weekend, someone called me right back. Their thought was that one or two of the batteries had gotten stuck in protection mode and
needed to be ‘woken up’. Great, sort of - it’s a solution but it required unhooking all four batteries, testing them separately, and ‘waking them’ by hooking the bad ones to the truck via jumper
cables and charging each one separately for thirty minutes.
I carry a jump pack rather than cables but we found an auto zone on our route and got some heavy gauge ones. But when we got to our campsite, disconnected the battery cables and checked the voltages
of the batteries individually, none of them were in protection mode (less than 1 volt). But two did read 12.9 volts (~20%) vs 13.3 on the other two (90%). Another call to Battleborn.
The working theory now is that those two batteries had floated off sync with the other two somehow, likely due to my fault. The Battleborn rep asked me how I had the battery bank wired, specifically
if I had both + and - leads to the trailer connected to the same battery or to the first and last. I had them both on the same one. He told me that when they’re wired that way, the battery monitor is
only getting its info on charge state from a single battery rather than the entire bank, and that can allow the other three to drift out of sync. And since the voltages on lithiums are so constant, a
small drift can have big consequences. In my case, he suspected that two of the batteries had gotten so far out of sync that they were no longer contributing. He recommended rewiring them and giving
the bank an overnight charge, then see where we stood.
So that’s where we are. Figuratively - literally we’re at the Joshua Tree KOA charging up. Fingers crossed. If the rewiring and recharging doesn’t work this time then I think the answer is that those
two batteries have bad cells or something.
Edited by Overland
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• Moderators
Good luck. Fingers crossed for you.🤞🤞🤞
• 1
2008 Ram 1500 4 × 4
2008 Oliver Elite, Hull #12
Florida and Western North Carolina, or wherever the truck goes....
400 watts solar. DC compressor fridge. No inverter. 2 x 105 ah agm batteries . Life is good.
• Moderators
I like Battleborn's logic regarding the working theory.
Good luck in getting it corrected so that you can enjoy the rest of the trip.
2023 Ford F150 Lariat 3.5EB FX4 Max Towing, Max Payload, 2016 Oliver Elite II - Hull #117 "Twist"
Near Asheville, NC
Thanks. Actually I didn’t quite get it right above. The two good batteries are the one that the + and - are attached to, and the one on which the battery monitor’s voltage sensor is attached. It
makes sense that those two would be the ones with the right voltage, the one being monitored and the one getting the direct charge.
• Moderator+
I'm curious as to why Oliver wired your battery bank like that. During our 2013 build, I had them wire our 4 x 6 volt AGM's as two pair of series paralleled together with the leads coming off the
opposite corners of the bank. Do you think they thought it didn't matter how the lithium's were wired?
Anyway, I agree that it would make a big difference (obviously over time) in the voltage/percentage readings. I hope your batteries are OK and this "reset" will fix you right up. I just finished a
Battleborn install in the Outlaw Oliver. I got 7 years out of the Trojan's, but one of them finally gave up the ghost. The other three seem to be fine. I'm gonna hook them up to my Dewalt equipment
if I can figure out a way to move them around.
Edited by ScubaRx
Steve, Tali and our dog Rocky plus our beloved Storm, Lucy, Maggie and Reacher (all waiting at the Rainbow Bridge)
2008 Legacy Elite I - Outlaw Oliver, Hull #026 | 2014 Legacy Elite II - Outlaw Oliver, Hull #050 | 2022 Silverado High Country 3500HD SRW Diesel 4x4
Steve, Oliver didn’t install my batteries, so the question is whether I wired them up that way or not. I don’t think I originally did, since I vaguely remember having the question in my mind when
putting it all together. But when we had our trailer serviced in Santa Fe on our first trip to fix a few things, something I asked them to do was add some lock washers to the battery connections
since I had one that had come loose, and I remember than when I looked at it after that he’d rearranged all my cables. And then I think that last year when I switched the cables into the box from 4/0
to pairs of 2/0 that I just duplicated what was there without thinking.
But I think that’s largely irrelevant unfortunately. After charging all night and day I just turned off the power to see what the voltage did and sure enough it fell from 13.6 where it should be to
13.2 within 5 minutes. So even without checking the individual voltages, I’m pretty sure that I’ve got two bad batteries. No other reason for it to fall like that.
Questions now are going to be why they failed, is failure common for Battleborns, and how easily/quickly will they get them replaced.
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• Moderator+
Sorry to hear that Jeff. I do hope Battleborn will live up to their reputation (and all our expectations) and get you back up and running quickly.
Steve, Tali and our dog Rocky plus our beloved Storm, Lucy, Maggie and Reacher (all waiting at the Rainbow Bridge)
2008 Legacy Elite I - Outlaw Oliver, Hull #026 | 2014 Legacy Elite II - Outlaw Oliver, Hull #050 | 2022 Silverado High Country 3500HD SRW Diesel 4x4
• 1 month later...
I thought I'd update this with the final outcome - which is, roughly 6 weeks later, I have two what look to be brand new Battleborn batteries sitting on the floor next to me. I say 'look to be',
because they told me that they repaired my batteries vs replacing them, but I suppose to repair them they have to cut open the case and then put them into new ones. Either way, they look brand new
and I assume function like new as well.
Six weeks is a long time, though one and a half of those were us still traveling and we can certainly spot them two or three days since it was over thanksgiving and Christmas. Still, if it's a month
in normal circumstances to get warranty service, that could cause problems if you have travel plans. The biggest time issue is shipping - they send you boxes, you send the batteries, then they
repair and ship them back to you. And while they pay for the shipping, it's not next day or anything. In fact it was almost two weeks for me to receive two empty boxes, one of which inexplicably
went to a neighbor. From a service standpoint, they'd be much better just replacing batteries rather than trying to repair them, which would allow them to just ship you new batteries and then you
use the same boxes to ship the old ones back. I can understand why they wouldn't want to do that, but still - 4 weeks could easily be one.
I never got a good answer on what exactly was wrong with them. They told me that the BMS had gone bad. But when I packed them up I noticed that there was something rattling around in both
batteries, so something came loose inside each one. And so that makes me think that the damage was physical and likely due to vibration or a single big bump or something. That makes me less than
confident that this won't happen again. But if they honor the warranty then I guess the worst I can expect is a repeat of this last trip, which was annoying but by no means a disaster. I have a
strong suspicion that one of these batteries went out on an earlier trip, but that it took two to go out to make it obvious. I say that because especially on the trip before this last one, it just
felt like the batteries didn't have the life they should. I'll definitely keep a closer eye on the voltages from now on and not rely entirely on the battery monitor's percentage estimate.
One more thing - I'm running a test on the other two batteries right now since I noticed something off with them as well. Another odd voltage drop when the batteries were supposedly at ~70%. I'm
going to separate them tomorrow, check their voltage and then drain them separately with a slow constant load to see exactly what's up. This seems more like something that would be symptomatic of
unbalanced cells.
It is great news that you have received your batteries back. I agree with you that six weeks is a long time. It would be great if they offered an immediate replacement or even a drop shipment plan
like some other companies do, where you pay for and receive new batteries up front and then they determine any warranty issues and the refund later.
Having something rattling around in the battery is not good. Are you going to or have you pressed them for more information on the root cause? It would be good to know if there was an issue during
manufacturing or if they felt like they needed to make mechanical changes based on your setup.
Edited by mjrendon
Overland, I am confused by this entire event. Doesn’t your VictronConnect app tell you the actual internal specs - cell voltages and such - for EACH battery? Surely a failed battery would be
indicated by a difference in internal voltages, and if the battery’s internal BMS had failed completely, you wouldn’t even be able to do this much. Are you using this app or a different one? This
screen shot is for a Victron battery.
”This Victron BMV-712 Battery Monitor with Bluetooth built-in pairs perfectly with our <Battle Born> lithium batteries. You can download the free VictronConnect phone app to manage your new battery
monitor directly on your phone!”
John Davies
Spokane WA
Edited by John E Davies
SOLD 07/23 "Mouse": 2017 Legacy Elite II Two Beds, Hull Number 218, See my HOW TO threads:
Tow Vehicle: 2013 Land Cruiser 200, 32” LT tires, airbags, Safari snorkel, Maggiolina Grand Tour 360 Carbon RTT.
• Moderators
1 hour ago, John E Davies said:
Overland, I am confused by this entire event. Doesn’t your VictronConnect app tell you the actual internal specs - cell voltages and such - for EACH battery? Surely a failed battery would be
indicated by a difference in internal voltages, and if the battery’s internal BMS had failed completely, you wouldn’t even be able to do this much. Are you using this app or a different one? This
screen shot is for a Victron battery.
”This Victron BMV-712 Battery Monitor with Bluetooth built-in pairs perfectly with our <Battle Born> lithium batteries. You can download the free VictronConnect phone app to manage your new
battery monitor directly on your phone!”
John Davies
Spokane WA
Is that picture you posted of your equipment or is it a stock photo from the internet? I only have the BMV-712 on air at this time, but I have fired up my new MPPT CC for testing which gives me 2
different Victron products to choose from within the VictronConnect app. So from my limited experience I believe the VictronConnect app screen shot you posted would be visible from the Victron
Battery and not from the BMV. I do not believe that the Battle Born batteries provide individual cell voltage readings through the VictronConnect app. At least I have never seen BB cell voltages
in the VictronConnect app.
Maybe someone can correct my thinking.
• 1
Mike and Krunch Lutz, FL
2017 LEII #193 “the dog house”
• Moderators
I'd like to know that, too.
We went with victron for much of the boat install, but xantrex xcpro 2000 inverter, which had to be replaced a year before we added solar and victron charge controller, monitors, etc..
That we're sorry for,as all victron communication is so nice. But, victron lithium batteries are not only more pricey, but a much more complicated install, from what I have read.
We're thinking the boat batteries will fail before the Ollie, so we'll be installing lfpo4 there first.
The Ollie will be a few years behind .
Edited by SeaDawg
2008 Ram 1500 4 × 4
2008 Oliver Elite, Hull #12
Florida and Western North Carolina, or wherever the truck goes....
400 watts solar. DC compressor fridge. No inverter. 2 x 105 ah agm batteries . Life is good. | {"url":"https://olivertraveltrailers.com/forums/topic/4688-interesting-electrical-problem/","timestamp":"2024-11-05T02:39:32Z","content_type":"text/html","content_length":"440429","record_id":"<urn:uuid:47d3ea6c-dda9-43c8-a03f-3a7d97725350>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00607.warc.gz"} |
Theory of Combinatorial Algorithms
Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)
Mittagsseminar Talk Information
Date and Time: Tuesday, May 29, 2018, 12:15 pm
Duration: 30 minutes
Location: OAT S15/S16/S17
Speaker: Pascal Pfister
H-bootstrap percolation
Let H be a a fixed graph. H-bootstrap percolation is then defined as follows. We start with a set G \subseteq E(K_n) of initially "infected" edges. Afterwards, the process proceeds in rounds. At time
t ≥ 1, an edge e becomes infected if there exists a copy of H in K_n for which all edges except e are infected at time t-1. We say that a graph G percolates in K_n if eventually all edges of E(K_n)
are infected. We study this process in the setting where the initial graph G is a random graph G_{n,p} and H is the complete graph K_r. We determine the critical probability p_c(n,K_r) for
K_r-bootstrap percolation up to a constant factor, therewith improving the currently best known bounds for p_c(n,K_r) and answering an open question posed by Balogh, Bollobàs and Morris.
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What impact can non respondents have on survey results?
What impact can non respondents have on survey results?
Nonresponse can have two effects on data: first, it introduces a bias in estimates when nonrespondents differ from respondents in the characteristics measured; second, it contributes to an increase
in the total variance of estimates since the sample size observed is reduced from that originally sought.
Is Standard Deviation an unbiased estimator?
The short answer is “no”–there is no unbiased estimator of the population standard deviation (even though the sample variance is unbiased). However, for certain distributions there are correction
factors that, when multiplied by the sample standard deviation, give you an unbiased estimator.
What makes a sample biased?
Sampling bias occurs when some members of a population are systematically more likely to be selected in a sample than others. Samples are used to make inferences about populations.
What is non respondent bias?
Non response bias is introduced bias in statistics when respondents differ from non respondents. In other words, it will throw your results off or invalidate them completely. It can also result in
higher variances for the estimates, as the sample size you end up with is smaller than the one you originally had in mind.
Why is bias undesirable in a sample?
Because of its consistent nature, sampling bias leads to a systematic distortion of the estimate of the sampled probability distribution. This distortion cannot be eliminated by increasing the number
of data samples and must be corrected for by means of appropriate techniques, some of which are discussed below.
What is biased or unbiased?
1 : free from bias especially : free from all prejudice and favoritism : eminently fair an unbiased opinion. 2 : having an expected value equal to a population parameter being estimated an unbiased
estimate of the population mean.
What is the difference between a response and a non response bias?
Response bias can be defined as the difference between the true values of variables in a study’s net sample group and the values of variables obtained in the results of the same study. Nonresponse
bias occurs when some respondents included in the sample do not respond.
What is an example of response bias?
Response bias (also called survey bias) is the tendency of a person to answer questions on a survey untruthfully or misleadingly. For example, they may feel pressure to give answers that are socially
How do you prevent sample bias?
Use Simple Random Sampling One of the most effective methods that can be used by researchers to avoid sampling bias is simple random sampling, in which samples are chosen strictly by chance. This
provides equal odds for every member of the population to be chosen as a participant in the study at hand.
How do you avoid participant bias?
One of the ways to help deal with this bias is to avoid shaping participants’ ideas or experiences before they are faced with the experimental material. Even stating seemingly innocuous details might
prime an individual to form theories or thoughts that could bias their answers or behavior.
How do you know if a sample is biased?
A sampling method is called biased if it systematically favors some outcomes over others.
How do you control response bias?
How can I reduce Response Bias?
1. Ask neutrally worded questions.
2. Make sure your answer options are not leading.
3. Make your survey anonymous.
4. Remove your brand as this can tip off your respondents on how you wish for them to answer.
How does bias affect validity?
The internal validity, i.e. the characteristic of a clinical study to produce valid results, can be affected by random and systematic (bias) errors. Bias cannot be minimised by increasing the sample
size. Most violations of internal validity can be attributed to selection bias, information bias or confounding.
Why is response bias a problem?
What is Response Bias? This term refers to the various conditions and biases that can influence survey responses. The bias can be intentional or accidental, but with biased responses, survey data
becomes less useful as it is inaccurate. This can become a particular issue with self-reporting participant surveys.
How do you determine an unbiased estimator?
That’s why it makes sense to ask if E(ˆθ)=θ (because the left side is the expectation of a random variable, the right side is a constant). And, if the equation is valid (it might or not be, according
to the estimator) the estimator is unbiased. In your example, you’re using ˆθ=X1+X2+⋯+Xnn43.
Why are non responses important?
One of the most important problems is non-response. It is the phenomenon that the required information is not obtained from the persons selected in the sample. One effect of non-response is that is
reduces the sample size. This does not lead to wrong conclusions.
How do you deal with non-response?
Methods for postsurvey adjustments. In addition to design, postsurvey adjustment techniques, including imputation and weighting, are devised to reduce nonresponse biases. Imputation methods rely on
information available on individuals for other variables than those to impute.
Is mean an unbiased estimator?
The expected value of the sample mean is equal to the population mean µ. Therefore, the sample mean is an unbiased estimator of the population mean. Since only a sample of observations is available,
the estimate of the mean can be either less than or greater than the true population mean.
How do you control bias in quantitative research?
Key tips on how to reduce bias in quantitative research
1. Write your questions in a neutral tone to ensure that the respondent is not led to believe that there is a correct answer.
2. Avoid asking if a respondent agrees/disagrees with a statement, as the respondent may be more likely to agree.
Can a biased estimator be efficient?
The fact that any efficient estimator is unbiased implies that the equality in (7.7) cannot be attained for any biased estimator. However, in all cases where an efficient estimator exists there exist
biased estimators that are more accurate than the efficient one, possessing a smaller mean square error.
How do you know if a sample is unbiased or biased?
If an overestimate or underestimate does happen, the mean of the difference is called a “bias.” That’s just saying if the estimator (i.e. the sample mean) equals the parameter (i.e. the population
mean), then it’s an unbiased estimator.
Why is avoiding bias important?
Bias prevents you from being objective If you’re writing a research essay, a scientific report, a literary analysis, or almost any other type of academic paper, avoiding bias in writing is especially
crucial. You need to present factual information and informed assertions that are supported with credible evidence.
How can response bias influence the outcomes of a study?
“Response bias is a general term for a wide range of cognitive biases that influence the responses of participants away from an accurate or truthful response. Because this deviation takes on average
the same direction among respondents, it creates a systematic error of the measure, or bias. | {"url":"https://www.kingfisherbeerusa.com/what-impact-can-non-respondents-have-on-survey-results/","timestamp":"2024-11-12T16:14:46Z","content_type":"text/html","content_length":"53309","record_id":"<urn:uuid:277527e7-4165-46bc-8ae1-38b91f5717f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00144.warc.gz"} |
Merge Two Sorted Linked Lists
In this lesson, we will learn how to merge two sorted linked lists.
We'll cover the following
If we are given two already sorted linked lists, how do we make them into one linked list while keeping the final linked list sorted as well? Let’s find out in this lesson.
Before getting started, we’ll make the following assumption:
Each of the sorted linked lists will contain at least one element.
A related problem is to create a third linked list which is also sorted. In this lesson, the two linked lists given will no longer be available in their original form, and only one linked list which
includes both their nodes will remain. Let’s get started.
First of all, we’ll have a look at the algorithm which we’ll use to code and then we’ll analyze the implementation in Python.
Algorithm #
To solve this problem, we’ll use two pointers (p and q) which will each initially point to the head node of each linked list. There will be another pointer, s, that will point to the smaller value of
data of the nodes that p and q are pointing to. Once s points to the smaller value of the data of nodes that p and q point to, p or q will move on to the next node in their respective linked list. If
s and p point to the same node, p moves forward; otherwise q moves forward. The final merged linked list will be made from the nodes that s keeps pointing to. To get a clearer picture, let’s look at
the illustration below:
Get hands-on with 1400+ tech skills courses. | {"url":"https://www.educative.io/courses/ds-and-algorithms-in-python/merge-two-sorted-linked-lists","timestamp":"2024-11-10T21:52:53Z","content_type":"text/html","content_length":"745204","record_id":"<urn:uuid:d58dd644-19af-4ef1-8903-e6a2a34d73b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00200.warc.gz"} |
Python Programming – Bitwise Operators
You can learn about Operators in Python Programs with Outputs helped you to understand the language better.
Python Programming – Bitwise Operators
A bit is the smallest possible unit of data storage, and it can have only one of the two values : 0 and 1. Bitwise operator works on bits and performs the bit-by-bit operation, instead of whole.
Table 4.4 shows the bitwise operators and their meaning.
Table 4.4 Bitwise Operators and their Meaning
Bitwise operator Meaning
>> Right hand
<< Left hand
& AND
I OR
∧ XOR
˜ One’s complement
In some other languages, A is used for exponentiation, but in Python, it is a bitwise operator called xor.
In order to understand the bitwise operators, let us first consider the SHIFT operators. There are two types of shift operators that shift the bits in an integer variable by a specified number of
positions. The ‘»’ operator shifts bits to the right, and the ‘«’ operator shifts bits to the left.
Right Shift Operator ‘>>’
The right shift operator (>>) shifts the bits of the number to the right by the number of bits specified. This means that those many bits are lost.
In general, the right shift operator is used in the form:
variable >> number-of-bits
11 >> 1 gives 5. 11 is represented in bits by 1011 which when right-shifted by 1-bit gives 101, which is the decimal 5. Take another example; you know that the decimal number 8 is represented in
binary form as:
Similarly, the decimal number 4 is represented as:
If data is represented in a computer using 8 bits form, then decimal digits 8 and 4 will be represented by 00001000 and 00000100, respectively. The only difference between these two numbers is that
the “1” in the second number is shifted one bit to your right. If the variable “a” holds the value “8”, then you can shift it right by one position, with the following statement:
a >> 1
This expression when converted to decimal value would be evaluated as “4”. You can shift a number by several bits. For example, you can change “8” into “2” by shifting the digit’T by two bits to the
right by using the expression:
a >> 2
Left Shift Operator ‘ << ‘
The left shift (<<) operator shifts bits of the number to the left for a specified number of bit positions. This means it adds Os to the empty least-significant places. Each number is represented in
memory by bits or binary digits, i.e., 0 and 1. The expression takes the form:
variable << number-of-bits
2 << 2 gives 8.
2 is represented by 10 in bits. Left shifting by 2 bits gives 1000 which represents the decimal 8.
Example 15.
Assume a = 60, and b = 11; Now in binary format they will be as follows:
a = 0011 1100, b = 0000 1011
Solve the following bitwise operators a>>2, a<<2, b>>2 and b<<2.
The left operand value is moved right by the number of bits specified by the right operand,
a >> 2 = 15
(means 0000 1111)
The left operands value is moved left by the number of bits specified by the right operand,
a << 2 = 240
(means 1111 0000)
Same as above; the left operand value is moved right by the number of bits specified by the right operand,
b >> 2 = 2
(means 0000 0010)
The left operands value is moved left by the number of bits specified by the right operand,
b << 2 = 44
(means 0010 1100) See Figure 4.13
Let’s Try
Assume a = 60, and b = 13; Now in binary format they will be as follows:
a = 0011 1100
b = 0000 1101
Write the output of the following bit-wise operations:
Find a>>2, a<<2, b>>2, b<<2.
The bin( ) method converts and returns the binary equivalent string of a given integer.
& (Bitwise AND)
Bitwise AND returns 1 if both the bits are 1, otherwise 0. For example, numbers 5 & 9 give 1.
| (Bitwise OR)
Bitwise OR returns 1 if any of the bits is 1. If both the bits are 0, then it returns 0. For example, numbers 5 I 9 give 13.
A (Bitwise XOR)
Bitwise XOR returns 1 if one of the bits is 0 and the other bit is 1. If both the bits are 0 or 1, then it returns 0. For example, the number 5 A 9 gives 12.
~ (Bitwise invert)
The bitwise inversion of numbers ~5 gives -6 and ~9 gives -10. (See Figure 4.14).
Example 16.
Assume a = 60, and b = 11; now in the binary format, they will be as follows:
a = 0011 1100
b = 0000 1011
Solve the following bitwise operations:
a&b, a I b, aAb, ~a, and ~b.
a&b = 0000 1000 →8 # Operator copies a bit to the result if it exists in both
a I b = 0011 1111 →63 # It copies a bit if it exists in either operand.
a∧b = 0011 0111 →55 # It copies the bit if it is set in one operand but not both.
~a = -0011 1101 →-61 # It returns complement of a number.
~b = -0000 1100 →-12 # It returns complement of a number.
Let’s Try
Write the output of the following bitwise operations:
Assume if a = 60, and b = 13; Now in the binary format they will be as follows:
a = 0011 1100
b = 0000 1101
Find a&b, alb, a*b, -a, ~b. | {"url":"https://pythonarray.com/python-programming-bitwise-operators/","timestamp":"2024-11-08T15:36:03Z","content_type":"text/html","content_length":"51952","record_id":"<urn:uuid:0cfd6dc9-f109-4494-859f-904efa36be27>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00694.warc.gz"} |
Equations of Circles Math Lib Activity - All Things Algebra®
Students will practice writing the equation of a circle as they rotate through 10 stations with this “Math Lib” Activity. Information given to write the equation includes:
• center and radius
• center and diameter
• center and a point on the circle
• endpoints of a diameter
• area of the circle
• circumference of the circle
The answer at each station will give them a piece to a story (who, doing what, with who, where, when, etc.). This is a much more fun approach to multiple choice, and the students adore reading the
story to the class. They get very excited to see which of their teachers is the “star” of the story.
Edit all story elements!
All slides are given in an editable format so you are free to personalize the story for your students. PowerPoint is required to edit. Only story elements can be changed, not the actual problems.
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This purchase includes a single non-transferable license, meaning it is for one teacher only for personal use in their classroom and can not be passed from one teacher to another. No part of this
resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses. A transferable license is not available for this
Copyright Terms:
No part of this resource may be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by | {"url":"https://allthingsalgebra.com/product/equations-of-circles-math-lib-activity/","timestamp":"2024-11-03T22:42:19Z","content_type":"text/html","content_length":"157947","record_id":"<urn:uuid:b31db246-2592-4a64-90a6-00794e62143e>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00683.warc.gz"} |
Inverse dynamics of mechanical multibody systems: An improved algorithm that ensures consistency between kinematics and external forces
Inverse dynamics is a technique in which measured kinematics and, possibly, external forces are used to calculate net joint torques in a rigid body linked segment model. However, kinematics and
forces are usually not consistent due to incorrect modelling assumptions and measurement errors. This is commonly resolved by introducing ‘residual forces and torques’ which compensate for this
problem, but do not exist in reality. In this study a constrained optimization algorithm is proposed that finds the kinematics that are mechanically consistent with measured external forces and mimic
the measured kinematics as closely as possible. The algorithm was tested on datasets containing planar kinematics and ground reaction forces obtained during human walking at three velocities (0.8 m/
s, 1.25 and 1.8 m/s). Before optimization, the residual force and torque were calculated for a typical example. Both showed substantial values, indicating the necessity of developing a mechanically
consistent algorithm. The proposed optimization algorithm converged to a solution in which the residual forces and torques were zero, without changing the ground reaction forces and with only minor
changes to the measured kinematics. When using a rigid body approach, our algorithm ensures a consistent description of forces and kinematics, thereby improving the validity of calculated net joint
torque and power values.
Citation: Faber H, van Soest AJ, Kistemaker DA (2018) Inverse dynamics of mechanical multibody systems: An improved algorithm that ensures consistency between kinematics and external forces. PLoS ONE
13(9): e0204575. https://doi.org/10.1371/journal.pone.0204575
Editor: Jose Manuel Garcia Aznar, University of Zaragoza, SPAIN
Received: January 25, 2018; Accepted: September 11, 2018; Published: September 28, 2018
Copyright: © 2018 Faber et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction
in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting Information files.
Funding: Funded by Nederlandse Organisatie voor Wetenschappelijk Onderzoek. Url: https://www.nwo.nl/. Grant number: 023.006.090. HORIZON 2020, The EU Framework Programme for Research and Innovation.
Url: https://ec.europa.eu/programmes/horizon2020/en. Grant number: H2020-MSCA-IF-665457.
Competing interests: The authors have declared that no competing interests exist.
In the field of biomechanics, inverse dynamics analysis is commonly used to investigate aspects of the mechanics, energetics and control of movement. An inverse dynamics analysis is typically based
on measurement of the kinematics of the body segments, often complemented with measurement of selected external forces (e.g. the ground reaction force). Using these data, the net joint torques and
net joint reaction forces are calculated using Newton’s equations of motion applied to a model containing a (chain of) rigid segments [1]. Classically, these equations are solved consecutively for
each body segment, starting at a segment for which the number of unknowns matches the number of equations. Although inverse dynamics is straightforward, it is not without problems [2–4]. First,
classical inverse dynamics assumes idealized pin joints and rigidity of body segments, which in reality don’t occur. Second, measurement errors in kinematic data caused by noise and skin artifacts
lead to incorrect joint centre locations, velocities and accelerations and thereby to errors in net joint torques. Third, the anthropometric parameters for a particular subject (such as segment
masses, mass center locations and segmental inertia) are typically estimated on the basis of a limited number of anthropometric characteristics in combination with results of cadaver studies [5].
Their values will deviate from the actual values, resulting in errors in net joint torques. In an attempt to mitigate these problems, external forces are often accurately measured and used as an
additional input in the inverse dynamics analysis, thereby improving on its quality. However, using both measured kinematics and measured external forces in an inverse dynamics analysis introduces a
new problem, since they will typically be inconsistent due to the aforementioned problems. This new problem is commonly formulated as follows [6]: the net joint torques obtained from an inverse
dynamics analysis starting at the unconstrained end of a chain of segments (e.g. the hands of a free standing person) and ending at the feet are different from those obtained when the analysis is
started at the feet. In more formal terms the new problem is that the system of equations of motion for a complete linked segment model is overdetermined. One way to evade the inconsistency is to
ignore information, i.e. to discard the equations of motion, about the mechanics of the last segment. Another way is to use all equations, which results in a residual force and torque typically
applied at the last segment by an unspecified actor in the environment. In fact, both will result in the same values for the joint torques. The residual force and torque compensate for the
measurement errors in kinematics and incorrect model assumptions, but do not exist in reality. Their values can actually be considered as an indication of the validity of the calculated joint forces
and torques. Furthermore, the residual force and torque do perform mechanical work that does not exist in reality and therefore may compromise energetic analyses.
In sum, in an inverse dynamics analysis, assuming a rigid body linked segment model as a basis, kinematics are in general inconsistent with measured external forces, i.e. result in nonzero residual
forces and torques. The question then arises how the inconsistency can be reduced or, even better, completely removed under the assumption of segment rigidity. This can be accomplished in three ways:
by adjusting i) the (time-invariant) anthropometric data, ii) the kinematics or iii) the external force(s). Several studies have used (combinations of) these ways in an attempt to reduce or remove
the residual forces and torques. For example, Vaughan [7] optimized body segment parameters to minimize residual forces and torques. A complete removal of the residual forces and torques will in
general not be possible since the number of anthropometric variables is typically smaller than the number of time nodes in the analysis. Delp et al [8] optimized model mass parameters and kinematics
to reduce residual forces and torques, but did not succeed in completely removing the residual forces and torques. De Groote et al [9] adjusted the kinematic data by employing a Kalman smoother that
used the complete kinematic dataset. Even though this method improved the estimate of joint kinematics, it did not address the problem of the residual forces and torques. Chao and Rim [10], using an
optimal control approach, optimized joint torques to minimize the squared differences between measured and calculated segment angles. However, ground reaction forces were not investigated and hence
this method did not remove the residual forces and torques. Thelen and Anderson [11] calculated translational accelerations of the pelvis and low back angles assuming that the ground reaction forces
and all other generalized coordinates were well represented by measurements. Integration of the accelerations over time yielded the pelvis and low back kinematics with residuals removed. Boundary
values for pelvis and low back were subsequently optimized to minimize the difference between measured and calculated kinematics. However, it is highly unlikely to find the optimal kinematic profile
by optimizing the kinematics of only two instead of all segments. Van Soest [12], Kuo [3] and van den Bogert and Su [13] optimized joint torques using all segments for each time node separately such
to find a least squares solution to the overdetermined set of equations of motion, but this does not remove the residual forces and torques. Cahouet et al [14] composed a set of equations of motion
for each time node and a centered finite difference scheme relating angular acceleration and position. The resulting overdetermined set of equations was solved using a least squares method. Their
solution resulted, in the presence of measurements errors, in an inconsistency between position, force measurements and angular accelerations. Remy and Thelen [15] adjusted measured ground reaction
forces, ground reaction torques and segment angular accelerations during walking, which yielded a consistent description of these quantities for each separate time node. However, as stated, this
algorithm required adjustment of the ground reaction force and torque, which is in fact similar to applying residual forces and torques at the feet instead of the trunk. These studies [3,12–15]
combined, show that any attempt to improve on inverse dynamics by optimizing for separate time nodes either leads to an inconsistent mechanical description or to an undesired shift of the residual
force and torque to a different segment. Mazzà and Cappozzo [16] were one of the first to solve this problem by performing an optimization over the whole time-series, while successfully removing the
residual forces and torques. They optimized segment angles, which were used in a top-down approach to minimize the root mean square error between measured and calculated ground reaction force. Among
the input for their algorithm were segment angles at the start and end of the movement which were constrained to be reproduced by their algorithm. However, they made no attempt to ensure that the
intermediate calculated and measured kinematics were similar. This was improved upon by Riemer and Hsiao-Wecksler [17] who also optimized segment angles to minimize the ground reaction force root
mean square error. They introduced inequality constraints for the intermediate segment angles based on data from the literature to create a range in which the optimized segment angles could be found.
Riemer and Hsiao-Wecksler [18] expanded the method of Riemer and Hsiao-Wecksler [17] by adding body segment parameters to the variables to be optimized. It was shown, using an idealized dataset, that
reconstruction of net joint torques could benefit significantly from optimizing body segment parameter values. However, one problem remains in their approach. Assume n degrees of freedom for a chain
of n segments connected by pin joints representing the body, N time nodes and also assume that the external forces are chosen such to perfectly fit the measured external forces. In the planar case,
this yields 2(n-1) joint force components, n-1 joint torques and n segment angles summing up to 4n-3 variables to be optimized for each time node. Since there are three equations of motion for each
time node, yielding 3n equations, the complete system has 3*n*N equations and (4*n-3)*N unknowns. If the number of degrees of freedom is three, such a system is determined. Overdeterminacy occurs for
values of n between zero and three, whereas underdeterminacy always occurs for values of n larger than three. This means that in applications with more than three degrees of freedom, like the lifting
example given by Riemer and Hsiao-Wecksler [17], there are many optimal kinematic profiles, i.e. kinematic profiles yielding a perfect fit of the calculated and measured ground reaction force. The
method by Riemer and Hsiao-Wecksler is not guaranteed to find the optimal kinematic profile that best fits the measured kinematics. The under determinacy could therefore be used to find the unique
solution that leads to an optimal fit between measured and optimized kinematics, while removing the residuals completely.
To conclude, no inverse dynamics method is currently available in which i) all residual forces and torques are removed, ii) segment angles at all time nodes are optimized together, and iii) the
problem is defined such that it always produces a unique solution, i.e. it results in minimal adaptation of the kinematics while the external forces are not accommodated. The purpose of this study
was to develop an algorithm that improves on inverse dynamics while meeting these demands. To show the significance of the inconsistency between kinematics and external forces, the magnitudes of the
residual force and torque values of a classical inverse dynamics analysis were obtained from a dataset concerning human gait. The resulting optimization algorithm was evaluated by applying it to the
same dataset, comparing the results (kinematics and joint torques) to those obtained using a classical inverse dynamics analysis. In the example application, the dataset consisted of the sagittal
plane coordinates of markers attached to body segments, sagittal plane ground reaction force data (including point of application) and segment parameter values. After optimization of the dataset, the
measured ground reaction force and kinematics were fully consistent.
Residual force and torque for the classical inverse dynamical analysis
We performed a classical inverse dynamics analysis on one complete stride of a subject walking at 1.8 m/s, which yielded the residual forces on the trunk (see Fig 1). This trial will be referred to
as the typical example. The onset of the stride was defined by toe off of the right leg. Positive x- and y-forces were defined as in the walking direction (forward) and upward, respectively. From Fig
1 it can be observed that in particular the horizontal component of the residual force at the trunk was substantial. Note again that these forces do not exist in reality.
Zero percent of the stride coincides with right toe off for all time series. Force was expressed in percentage of body weight. The analysis shows considerable residual forces to enforce consistency
between kinematics and measured ground reaction forces. Zero crossings indicate the time nodes where measured ground reaction forces and kinematics were ‘accidentally’ consistent. Hcr: heel contact
right foot, Tol: toe off left foot, Hcl: heel contact left foot. Right toe off is defined as the onset of the stride.
The values of the residual force were directly related to the inconsistency between measured ground reaction forces and acceleration of the body’s center of mass and hence were not affected by its
presumed point of application. In contrast, the value of the residual torque was affected by the (arbitrarily chosen) point of application of the residual force. To illustrate this, two classical
inverse dynamics analyses were performed. In the first, the residual force was applied at the shoulder, while in the second it was applied at the trunk’s center of mass (Fig 2). Marked differences
for the residual torque value were observed between these two analyses. This indicates that the value of the residual torque by itself is meaningless. An interaction between the residual force and
torque was observed. The relatively large positive horizontal residual force at the shoulder, for example at t = 0 in Fig 1, yielded a negative (flexion) torque at the trunk. This was compensated for
by an opposite (positive) residual torque applied at the trunk (Fig 2), which largely explained the in phase behavior of the horizontal residual force component and the residual torque.
To show the effect on the residual torque, the residual force (F[res]) was assumed to either apply at the shoulder or at the center of mass of the trunk. Clockwise torques were defined positive. Hcr:
heel contact right foot, Tol: toe off left foot, Hcl: heel contact left foot.
Optimization results
A rigid body linked segment model was defined to describe the kinematics, forces and torques during a set of walking trials. The kinematic profiles were found by minimization of the sum of all the
Euclidean distances between measured and model skin marker positions. Removal of the residual forces and torques was ensured by adding the equations of motion of all segments with no residuals as
equality constraints. The resulting single core optimization of one stride took on average 2 minutes on an Intel i7-4770 (3.4 GHz) processor, using the measured data as the initial guess. The
solutions always converged and the residual force and torque were completely removed. In Figs 3 and 4 the measured and optimized segment angles and angular velocities of the right leg and trunk were
compared for the same typical example used for Figs 1 and 2. The results showed that only small changes in optimized angles were necessary to completely remove the residual forces and torques. We
then performed comparisons for 61 strides in nine different subjects walking at three different speeds (0.8, 1.25 and 1.8 m/s) and calculated root mean square (RMS) values for the differences between
trunk and lower segment angles before and after optimization. Table 1 provides the average RMS values of the foot, shank, thigh and trunk angles. These values indicate that on the whole, like with
the typical example, only small changes in kinematics were required to completely remove the residual forces and torques.
The respective segments are denoted by color. Segment angles before and after optimization are denoted by dashed and solid lines respectively. For definitions of the segment angles, see the Materials
and Methods section. Hcr: heel contact right foot, Tol: toe off left foot, Hcl: heel contact left foot.
The respective segments are denoted by color. Angular velocities before and after optimization are denoted by dashed and solid lines respectively. Hcr: heel contact right foot, Tol: toe off left
foot, Hcl: heel contact left foot.
We also compared the distances between the markers, located at the joints (indicated in Table 2), before and after optimization. RMS values for the markers (RMS[s] in Table 2) were in the order of 1
cm which also indicated good agreement between measured and optimized kinematics. Net joint torques were calculated before and after optimization (Fig 5) and the differences were quantified by a
relative measure as shown in Table 2. RMS values of the relative differences for the joint torques before and after optimization indicated larger differences than for the measured and optimized joint
angles. Both sets of torques showed similar patterns, although hip torque peak values before and after optimization were substantially different.
Optimized and classical net joint torques were similar. Thin dashed lines indicate joint torque values prior to optimization. Thick solid lines indicate joint torque values after optimization.
Positive values denote plantar flexion, knee flexion and hip extension torques. Hcr: heel contact right foot, Tol: toe off left foot, Hcl: heel contact left foot.
The net joint powers depicted in Fig 6 both before and after optimization were shown to be substantially larger than the residual power (when the residual force was applied at the shoulder) for the
typical example. The absolute peak power of the residual force was in the order of 50 Watt. Removing the residual force and torque led to a maximum adjustment of the ankle power (at 90 percent of the
stride) in the order of 100 Watt.
Residual power is the sum of the power of the residual torque and force at the shoulder before optimization. Hcr: heel contact right foot, Tol: toe off left foot, Hcl: heel contact left foot.
In a classical inverse dynamics analysis, based on a rigid linked segment model, measured kinematics and external forces are in general not mechanically consistent. In this study, an algorithm was
developed to remedy this by modifying the measured kinematics as little as possible such that the resulting optimized kinematics are mechanically consistent with measured external forces. As an
example, this algorithm was applied to a dataset of human walking containing 2D joint positions. Our analyses show that the algorithm was capable of completely removing the residual forces and
torques during a stride with minor changes to the measured kinematics, while leaving the measured ground reaction forces unchanged. As a result, joint torque profiles before and after optimization
showed similar patterns.
The example used in this study was a 2D representation of walking. However, we stress that it is straightforward to extend the algorithm in several directions. First, we note that extension to 3D is
straightforward. For example, in walking experiments with ground reaction force and 3D measurement of kinematics, three residual force components and three residual torque components will arise at
the trunk. These can be treated the same way as in the planar case. However, due to increased model complexity in 3D applications, it should be established in future work how this affects the
calculation time of the optimization. Second, as mentioned in the introduction, several methods exist in which body segment parameter values are added to the variables to be optimized. These were not
included in our algorithm because we focused on altering the kinematics and its effect on the residual force and torque values. However, including body segment parameter values and imposing
reasonable bounds is a relatively simple extension, which can contribute to improving inverse dynamics analysis. Third, human walking is an example of a (nearly) periodic movement. Conceivably,
researchers may want to impose strict periodicity on such a movement. In that case, the external forces should be (minimally) adjusted such that the cycle average of the sums of all forces and
torques equal zero. Also, constraints should be added to enforce equal positions and velocities at the start and end of the cycle.
Summarizing, a straightforward algorithm was developed that completely removed residual forces and torques in an inverse dynamics analysis. It was found that small adjustments to the kinematics only,
in the order of 1 cm marker displacements, were sufficient to obtain a consistent mechanical description. The algorithm provides a clear improvement over current methods in calculating net joint
torques and it should, in our opinion, therefore be included in any rigid body inverse dynamics analysis.
Materials and methods
Description of the proposed algorithm
For any application, depending on the analyzed movement, the first step is to define a model consisting of m rigid bodies that are connected by joints, represented by kinematic constraints imposed on
the kinematics of the rigid bodies. Next, values are assigned to the time-invariant properties of each of these segments (length, center of mass position in a local frame of reference and inertial
properties). When the model has n degrees of freedom, n generalized coordinates q suffice to fully describe the position of the model at a particular time node i. Thus contains the full description
of the position of the system at time t(i). If the total number of time nodes considered is N, then the position of the system over time is completely described by an Nxn matrix Q, containing the as
rows. This implies that all other kinematic variables of interest can be calculated from Q. In particular, we calculate the matrix Z, containing the Cartesian coordinates of the centers of mass of
all segments at all times, the matrix P containing the Cartesian coordinates of all joint centers and the matrix S containing the predicted positions of the skin markers used in the kinematics
registration. Note that the latter contain time-invariant coordinate values relative to a segment-fixed frame of reference, which can be obtained by calibration measurements. Given these matrices,
the relevant second derivatives with respect to time are approximated using central differences: (1)
In general terms, the optimization problem is to minimize the sum of the weighed squared Euclidian distances between the segment model markers S and the corresponding experimentally observed markers
S’, without the introduction of residual forces and torques. A matrix R, containing the measured points of application of the external forces is fed into an inverse dynamics analysis, which yields
the residual forces (F[res]) and the residual (T[res]) and net joint torques (T). The constrained optimization problem is solved by the proposed algorithm as indicated in Fig 7.
The optimization starts by providing an initial guess for the matrix Q[0] that contains the values for the degrees of freedom at each time node, calculated from the measured marker coordinates. The
optimizer generates a modified version of Q. Using rigid body kinematics and numerical differentiation, the kinematic variables relevant for inverse dynamics are calculated. In the inverse dynamics
block, net joint torques and forces (including residual forces and torques) are calculated on the basis of these kinematic variables, in combination with the measured external forces F[ext], and
their points of application R. The residual forces F[res] and torques T[res] and the predicted marker positions are fed back to the optimizer, which updates Q such that, ultimately, the residuals are
zero and the sum of the weighed squared Euclidian distances (J) between predicted (S) and measured (S’) marker positions is minimal.
Formally, the optimization problem can be summarized as follows:
w[j]: an optional weight for the relative contribution of the marker to the minimization criterion
T[res,m](t(i)): m-th residual torque at time node i
N: number of time nodes
c: number of markers
As stated before, the segments’ centers of mass accelerations are calculated by direct numerical differentiation of the center of mass position, which are functions of the generalized coordinates. A
different method that should produce similar results, would be to express the accelerations in terms of the generalized coordinates and their infinitesimal derivatives, subsequently numerically
differentiate the generalized coordinates and replace the infinitesimal derivatives by the numerical analogues. This was found to result in a slightly lower value for the minimization criterion J,
but introduced numerical instabilities in the form of high frequency oscillations of the calculated generalized coordinates. This was never observed with the direct differentiation as indicated in
Fig 7.
Application to human walking
To test the optimization algorithm, we applied it to human walking. To do so, we measured the kinematics and ground reaction forces during shod walking of nine subjects (all female). This study was
approved by the local ethics committee (Ethische Commissie Bewegingswetenschappen) and all procedures were carried out with adequate understanding and after written consent of the subjects. Age was
23.6 ± 1.4 yr (average ± SD). Height was 1.75 ± 0.05 m and body mass 66.1 ± 4.9 kg. Subjects walked for five minutes on an instrumented split belt treadmill (R-Mill, Forcelink, Culemborg, The
Netherlands) to get accustomed to the experimental situation. Subjects were instructed to walk with their left and right foot on the separate belts of the treadmill. Subsequently they walked at three
different speeds (0.5 m/s, 1,25 m/s and 1.8 m/s) for five minutes at each speed. Optotrak CERTUS Position Sensors (Northern Digital, Waterloo, Ontario) were used to collect kinematics at a sample
rate of 100 Hz. In this study single markers were placed at both sides of the body at the fifth metatarsophalangeal joint, the lateral malleolus, the lateral knee epicondyle, greater trochanter and
acromion; it was assumed that the marker positions represented the positions of the corresponding joint axes. Raw data was filtered using a zero lag 4^th order low-pass filter with a cut-off
frequency of 10 Hz. Only sagittal plane projections were used in this study. Ground reaction forces (F[GR]) were measured at a sample frequency of 200 Hz using two force plates embedded in the
treadmill. Raw F[GR] data was filtered using a zero-lag 4^th order low-pass filter with a cut-off frequency of 20 Hz from which the center of pressure (r) was calculated for each foot. Subsequently F
[GR] and r data were down sampled to 100 Hz to match the Optotrak sample rate.
61 strides were selected from the data. All nine subjects and all three velocities were represented in this selection. Start and end of one complete stride was defined by toe off of the right foot
(first sample with F[GR] equal to zero). Only strides with two distinct swing phases, where F[GR] was very close to zero, were selected by visual inspection. Kinematics and F[GR] were used both for a
classical inverse dynamics and as input for the optimization algorithm. The two hips and two shoulders were lumped together and regarded as one joint. Anthropometric parameters were obtained from
Winter [5]. To assess the effect of the point of application of the residual force (before optimization) on the residual torque value, it was applied on the trunk in a classical inverse dynamics
analysis at two different positions: the shoulder and the trunk’s center of mass. For each case the residual torque was calculated.
For the optimization, the subjects in this study were modeled as a system of 7 rigid segments moving in a vertical sagittal plane, with pin joints connecting the segments and no kinematic constraints
between the feet and the walking surface (see Fig 8). This model has nine degrees of freedom, which were described by generalized coordinates q[1]..q[9] as defined in Fig 8. Seven coordinates are
segment angles; the x- and y-coordinates of the hip were chosen to specify the model’s position in the global frame of reference. In this case, no calibration measurements were required to define the
time-invariant positions of the markers relative to a local frame of reference, as the markers were assumed to be placed at the joint axes. Relative contributions w[j] of the differences in optimized
and measured marker positions to the minimization criterion J were all set to 1. Segment lengths were calculated as the average value of the relevant inter-marker distances.
The model consists of seven rigid segments connected with pin joints. It has nine degrees of freedom. Angular coordinates used to describe the degrees of freedom are indicated by q[1]- q[7]. The
remaining two degrees of freedom are described by the position of the hip (q[8],q[9]).
Fig 9 shows the first segment in the inverse dynamics analysis. The external force and its point of application have been measured. The kinematics have been measured before or updated by the
optimizer during optimization. Application of Newton’s equations of motion yields three equations, which are solved for the net ankle torque T[1] and the net ankle force . These are subsequently
reversed according to Newton’s third law and at the ankle joint applied to the shank which yields the net knee torque and force etc.
The external force , its point of application and the markers and are input of the analysis. Torques of , the net ankle force around the foot’s center of mass, the net ankle torque T[1] and the force
of gravity m[1]g are inserted into Newton’s equations of motion and solved for and T[1]. These are subsequently reversed according to Newton’s third law and input for the same procedure on the shank.
The constrained optimization problem was solved using the function fmincon embedded in Matlab R2013a. To evaluate the proposed algorithm, the kinematics were compared in terms of the root mean square
value of the difference between the generalized coordinates before and after optimization, respectively q’[j](t(i)) and q[j](t(i)): (3)
q’[j](t(i)): generalized coordinates before optimization
q[j](t(i)): generalized coordinates after optimization
Also, the root mean square value of the Cartesian distance between markers before and after optimization were calculated: (4)
i: time index
j and k: indexes for the for the generalized coordinate and the marker respectively.
The optimized net joint torques T[m](t(i)) were compared to the classic joint torques T’[m](t(i)) by a relative measure (joints indexed by m): (5)
Subsequently, the grand mean and standard deviations of these RMS values were calculated over all strides. For individual trials the net joint power before and after optimization was calculated as
the scalar product of joint torque and joint angular velocity, whereas power by the residual torque was calculated as the scalar product of residual torque and the trunk’s angular velocity. Power of
the residual force, applied to the shoulder, was calculated as the dot product of the residual force and the velocity of its point of application and was added to the power of the residual torque.
We thank Axel Koopman Ms. for providing the measurements of the walking experiments. | {"url":"https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0204575","timestamp":"2024-11-02T08:59:05Z","content_type":"text/html","content_length":"178874","record_id":"<urn:uuid:bde3c06e-f658-42ea-bc96-753ee030b67b>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00609.warc.gz"} |
Modern Ship & Shipbuilding Terminology
For a custom word search, make your selections below. For a full listing select All Modern Terminology. Search is NOT case-sensitive.
1. definition or term - term ONLY i.e. GRT will find the definition for GRT.
2. any reference to - finds any match i.e. ton will find ton ANYWHERE in text.
3. text containing - finds any partial match i.e. ore will ALSO find stores. | {"url":"http://ageofsail.net/aoshipmd.asp?sletter=GRT;iword=1","timestamp":"2024-11-07T13:53:37Z","content_type":"text/html","content_length":"9037","record_id":"<urn:uuid:504bb827-5b5d-4dbf-b240-52f477fd4201>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00789.warc.gz"} |
Math Colloquia - Solver friendly finite element methods
In this talk, numerical methods to solve second-order elliptic partial dierential equations will be presented. First, some of the existing methods, such as the standard Galerkin method, mixed nite
element methods etc., will be briey discussed. Then, a new hybrid mixed nite element method will be introduced for ecient and accurate approximations of the ux variables. In many applications, the ux
variables are the main quantities of interest. The method is a two{step method. On a coarse mesh, the primary variable is approximated. Then, the approximation is used as a data for the ux
approximation on a ne mesh. It will be shown that the ne mesh size can be taken as the square of the coarse mesh size, or a higher power with a proper choice of parameter. This means that the
computational cost for the coarse-grid solution is negligible compared to that for the ne-grid solution. This is a joint work with Dr. Young Ju Lee and Dr. Dongwoo Sheen. | {"url":"http://my.math.snu.ac.kr/board/index.php?mid=colloquia&page=8&document_srl=764059&l=en&sort_index=Time&order_type=asc","timestamp":"2024-11-04T10:14:47Z","content_type":"text/html","content_length":"44462","record_id":"<urn:uuid:6855f602-a8b2-491e-b26e-f198784cbb91>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00334.warc.gz"} |
Understanding Two-Phase Flow Parameters | Ansys Innovation Courses
This lesson covers the concept of two-phase flow parameters, focusing on the relationship between non-measurable and measurable quantities. It explains the definitions of void fraction, in-situ
liquid holdup, and their relationship with measurable parameters. The lesson also discusses the importance of understanding the distribution of voids in a two-phase flow and introduces the concepts
of volume average, area average, chordal average, and time average liquid holdup. It further explains the significance of pressure gradient and pressure drop in two-phase flow situations. For
instance, in a gas-liquid flow, the lesson illustrates how to calculate the time average void fraction using a light transmission method.
Video Highlights
00:14 - Introduction to the parameters related to void fraction and in-situ liquid holdup, and their relationship with measurable parameters
30:22 - Discussion on the importance of measuring the time average void fraction to understand the voidage profile
43:50 - Explanation of the concept of pressure gradient and pressure drop in two-phase flow
51:58 - Explanation of the concept of volumetric flux and its relationship with in-situ velocities
Key Takeaways
- Void fraction and in-situ liquid holdup are crucial parameters in two-phase flow.
- The distribution of voids in a two-phase flow is significant for processes like mass transfer and chemical reactions.
- Volume average, area average, chordal average, and time average liquid holdup are different ways to measure the in-situ composition of a two-phase flow.
- Pressure gradient and pressure drop play a vital role in two-phase flow situations.
- The time average void fraction can be calculated using a light transmission method, providing insights into the distribution of voids. | {"url":"https://innovationspace.ansys.com/courses/courses/multiphase-flow-fundamentals-and-applications/lessons/understanding-two-phase-flow-parameters-lesson-6/","timestamp":"2024-11-06T21:07:06Z","content_type":"text/html","content_length":"177504","record_id":"<urn:uuid:d6f625c2-753f-4ed0-9edf-a83c8dda2061>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00692.warc.gz"} |
Micro-LIF and numerical investigation of mixing in microchannel
Journal of Siberian Federal University. Engineering & Technologies 1 (2013 6) 15-27
УДК 532.5
Micro-LIF and Numerical Investigation of Mixing in Microchannel
Andrey V. Minakova,b*, Anna A. Yagodnitsynab, Alexander S. Lobasova,b, Valery Ya. Rudyakb,c and Artur V. Bilskyb
a Siberian Federal University, 28 Kirensky Str., Krasnoyarsk, 660074 Russia b Institute of Thermophysics SB RAS, 1 Lavrentiev, Novosibirsk, 630090 Russia c Novosibirsk State University of Civil
Engineering, 113 Leningradskaya Str., 630008 Russia
Received 11.02.2013, received in revised form 18.02.2013, accepted 25.02.2013
Flow regimes and mixing pattern in a T-type micromixer at high Reynolds numbers were studied by numerical solution of the Navier-Stokes equations and by particle image velocimetry (micro-PIV) and
laser induced fluorescence (micro-LIF) experimental measurements. The Reynolds number was variedfrom 1 to 1000. The cross section of the mixing channel was 200 jm^400 jm, and its length was 3000 jm.
Five differentflow regimes were identified: (I) steady vortex-free flow; (II) steady symmetric vortex flow with two horseshoe vortices; (III) steady asymmetric vortex flow; (IV) unsteady periodic
flow; (V) stochastic flow. Maximum mixing efficiency was obtained for stationary asymmetric vortex flow. In this case, an S-shaped vortex structure formed in the flow field. Good agreement between
calculation and experiment was obtained.
Keywords: Microflow, Micromixers, Microchannels, CFD, Micro-PIV, Micro-LIF.
Liquid mixing is an important physical process which is widely used in various microfluidic devices. Since the characteristic flow times are usually extremely small, mixing is accelerated using
special devices called micromixers. Micromixers are a key element of many MEMS and other devices that are used in various biomedical and chemical technologies, creating different micro
heat-exchangers, microapparatus, etc. The operating principles of micromixers and their optimization have been the subject of a great deal of research (see, for example, [1-4] and references
therein). Most papers consider laminar flow at low Reynolds numbers, which are usually characteristic of microflow. In practice, however, there are situations when microflows Reynolds number (Re) are
high enough [5,6]. In addition, at relatively high Reynolds numbers in microchannels take place number of new interesting phenomena requiring study both a fundamental point of view and for practical
© Siberian Federal University. All rights reserved
* Corresponding author E-mail address: tov-andrey@yandex.ru
Thus, in the study of T-type micromixers the existence of critical Reynolds number at which the Dean vortices in a microchannel lost symmetry was experimentally demonstrated [1]. The critical
Reynolds number about 150 for channel dimensions 600|amx300|amx300|am was found. Strong dependence of the critical Reynolds number from the channel size was shown. Transient flow regimes (at Reynolds
number Re = 300-700) by the instrumentality of numerical simulation have been investigated in [7], but the mixing processes have not been studied. The mixing of two fluids in the range of Reynolds
numbers from 50 to 1400 was studied experimentally and numerically in [8]. In [9] the existence of unsteady perlootic reg[me fos ceutoio values oU the Reynolds numbeo fiost dsmonstrated numerically.
The most compreheorivn experimental study of mixing in a T-sUaped microchannel at moderate Reynolds numbers (100-4100) in [10] was carried out. Thssr, using |i-LIF and |i-PIV measurements of the
velocity and concentration fields in the various secti-ns of the mixeo wds ntudied. For the first time the mixinn efficiency w-r me asure d.
.n spite of the relatively large number of paperr covered the studu of flow and mixing in T-type mlcromixers at moddraie Reynolds numbers, in fact, sufficient syrtemaoic da0h about flow regimes and
mixing procerses eook palace in it is still absent. The present work covers systematic modeling of incomprets[blo flow and mixing in a T-type micrtimixer at Reynolds numUers from 10 to 1000. The
problem was sslveeluumeoically ou the basis of the NaviereStokes equations for an incompressible fluid. Verification of rbe simulation data wos corried ouh expe:rimenoal-y. The measurements were
perfoemed by awo methods: paruide image velocimetru (micro-PIV) ¡and iaaeo induced fluoeescence (nUare-LIF).
Mathematical Model and Numerical Algorithm
The incompressible flows? of multicomponent Newtonian fluids, which dynamics is described by the Navier-Stokes werec onsidered
ofi + V-dpv) = 0, + V- (pvv) = -Vp + V-T, (1)
de de '
where p is fluid de=sity, p is preseure, v is its velocity, +nd T is the viscous stsess tensor. Density and vircosity op the mixture ie determined by the masa foactiou of mixture components pa the
partial density p,- and moleculas viscosity on the pure cnmponents °
P = Z f,Pi > V = lLf, "H-i >
i i
and the evolution of the mass concenirations determined by the equation
Seao^-ipo-r^v-ODdy;.), (2)
where A os the difforion coenkhent t^f thuei o-componsnl.
As boundary uondiUone on the channd walls for the velocity vectoe components used slip or nonslip conditions. In this papee usod the second one.
Tomlve the system -f equations mentioned above CFD nooiware ptckage cFlow is used. A deoailed description of the psogram nume rical algorithm is give n in [H,!, The developed algorithm
used in solving a wide range of problems of external and internal flows [11-13]. Its applicability to describe the microflows shown in [14,15].
The investigation re sults of flow and mixing in T-type mic romixers are presented in this paper. Width of the narrow chaenel part is 200 (tm, width of the wide channel patt is 400 (m, thickness of
the channel is 200 (m and length of the mixing channel is 3000 (m. The problem is considered in the spatial and timetdependent formufatk>n fn the general care. Titer ugh the left channel input pure
weter is fed at a flow rate Q. Through the right channel inpun water is fed at the same
flow rote. The drnsity of both fluids equals 1000 kglm3, the vircosity equale 0.001 Paxf, the diffusion co efitr it nt of tte dye in weter, D = 2,6n b10-1 0 m2/r. Thus, the value of Srhmtdt number
for this problem is 3,800. As the boundary conditions et ehe mie-amcr of the chanoel steady vetbcity profile was set. At the exti of rhe mioing cltanne/ Nenmttm tteditions wos set, i.e. iv/1.!
^:ii;:i:i;sl;:i]L:nit oli tin normaf to the output surface component of derivative of all scalar quantities.
The study was conducted for diffeeent vnt/es of Reynolds number, whtch is defined as follows: Re = (ptM(i) wliere U = (//(2pH) is ite iElow rate-based average velofity id the mixing channel, H = 200
(m is height of the channel and d = 267 (m is hydraulic diameter.
For the quantitative chrracterizatioe of thb mixing efficiency the following parameter was used:
M = 1 - e^(e/cn0 , where 1- = f • J" (/ - /^d r it tie coitcereratfon t r componentfst t^Iic:Cгl:rd deviation from
V V
its ^eai^ "Vi.ii^KS f by the volume (V) olie^ee, ct0 = /(l- f) i s the maximum stendaed deviation.
Experimental Set Up
The diagram of experimental setup is shown in Fig. 1. Imaging system consisted of epifluorescence inverted microscope (Carl Zeiss AxioObserver.Z1) with lenses 20x/NA = 0.3 and 5x/NA = 012 (number 1
in Fig.f) for micro-PIV aed micro-LIF experimentt, respectively. Lighting and rec otd that iLr^ir ggetti 0 n e digital camera using a measuring complex "POLIS" (number 3 in Fig. 1) were caroied out.
This complex mcluded a eouble-puised Nd:YAG llascr with 50 mJ energy, «2 nm wcvelengte ^n<t. 88 Hz pulte repetitioh coeering the flow "hrough Ihe lens of the
micnostcoiTiu. ti1«:) emering the light iI/lto tbe microscope the liquid light gurde ¡and the Hers interface unit io the opticttl path (g]t the mierosfope weerc^ usedr Ir]Lghlltin^ of ehe
microchannel during the micro-LiF e;;>i;tie^r^ri[ie;n^!^ carried out using a mercury lamp. Cnoetrorreiation digital camera recorded the imagec witt 2048hpixels resohttion, whith jarri; teen
transhetred Ito a personal computed hoa processmg. Synchronization oC the rystem war cfrried out uaing a pprogrammable processor. Contnolling of thee experiment and data processing wece ctrried out
using the software package ActualFlow.
Fluid motion control usrng infusron syringe pump (numJjeo 5 in Fig. 1) with adjustable liquid flow rate was carried out The flow oe liquid sow by fluorescent ihfterc from DukeScirntiíic firm. The
particles were composed of melamine resin, labeled with fluorescent dye Rhodamine B. The particle dentity it L05 g/cmh tverage diameter is 2 (m; thr stondard deviation is 0.04 (nm. To register the
light emiytfd zfro^m the partieles, and lhe fuppr2trton the ltght Oram the channel, beam-splitting
cube, consisting of a dichroic mitror and two filtets for - xcttation aed dftection of Rhodamine B was used.
Fig. 1. The experimental setup
Micro-PIV and micro-LIF experiments were conducted at Reynolds numbers ranging from 10 to 300. The measurements were carried out in three regions of the T-mixer (so that the velocity field has been
calculated up to seven calibers from the mixing channel entrance).
Concentration range in which the luminescence intensity of the fluorophore has a linear dependence on concentration was determined to calibrate the measurements. For this purpose, the T-channel is
fed aqueous solutions of Rhodamine 6J in the following concentrations: 0 mg/l, 10 mg/l, 25 mg/l and 40 mg/l, 50 mg/l, 62.5 mg/l and 75 mg/l. For every concentration of the fluorophore the image of
the channel was registered. As a result, linear dependence of the fluorophore radiation intensity was found at concentrations less than 62.5 mg/l. Thus, the relationship between the concentration of
the fluorophore and the intensity of the image at each point of the channel was obtained.
Comparison of Calculations with Experimental Data
The Reynolds number in the range from 1 to 1000 was varied in the calculations. At low Reynolds numbers (Re < 5) in the mixer occurs steady irrotational flow. Mixing in this case occurs due to usual
molecular diffusion and mixing efficiency is quite low [15] (see Fig. 2). Further, with increasing Reynolds number a pair of symmetrical horseshoe vortices (Dean vortices) formed in the mixer. They
generated at the left end of the mixer wall (see Fig. 3, left) and extended into the channel on the mixing length, depending on the Reynolds number. Horseshoe vortices appear due to the development
of secondary flows caused by the centrifugal force associated with rotation of the flow. Dean vortex structure is shown in Fig. 4 with isosurface X2. Here X2 is the second eigenvalue of the tensor
(SS + fifi), where S is the rate of strain tensor, and fi is the vorticity tensor.
The flow in this case is symmetric about the central longitudinal plane of the mixer. Each horseshoe vortex, being in the range of one liquid, does not cross the media mixing boundary, so the
boundary between the media remains almost flat. This is clearly seen on the Fig. 3 (right). And because the diffusion Peclet number increased with increasing Reynolds number, the mixing efficiency
decreased (see Fig. 2).
Fig. 2. Mixing efficiency versus Reynolds number
Fig. 4. Izolines of the dye concentrations in 4 longitudinal sections of the mixer: Re = 186 (left), Re = 600 (right)
When the Reynolds number reaches 150, the vortices lose their symmetry (see Fig. 3, right). They are rotated through an angle of 45° relative to the central plane of the mixer cross-section. S-shaped
vortex formed. This is particularly clearly shown in Fig. 4 (left), where mixing is shown by isolines of the dye concentration in the four cross sections of the mixer. First left section is located
at the entrance to the mixing channel, second - at the distance of 100 ^m from the entrance, third - at the distance of 200 |im and fourth - at the distance of 400 ^m. It is important to emphasize
that flow is still stationary.
However, due to the fact that the intensity of the vortices in the asymmetric flow regime increases significantly, they extend through the mixing channel up to the exit. The presence of swirling flow
0.005 0.0055 0.006 0.0065 0.007 0.0075
Fig. 5. The flow velocity versus time in point located at the mixer outlet. Red - Re = 300, blue - Re = 600, green -Re = 1000
the mixing channel leads to a layered structure of the miscible fluids formation. The contact surface of the miscible fluids in the layered structure is developed, which leads to a sharp increase of
the mixing efficiency (see Fig. 2). In the transition from a symmetric flow regime (Re < 150) to the asymmetric (Re > 150), the mixing efficiency increases by 25 times.
Described stationary asymmetric flow regime is observed in the range of Reynolds numbers from 140 to 240. Starting from a Reynolds number of approximately equal 240, flow ceases to be stationary. In
the range of Reynolds numbers 240 < Re < 400 implemented periodic flow regime. In particular, it means that the flow velocity is also a periodic function of time. In Fig. 5 this flow regime
corresponds to the lower curve. The flux oscillation frequency f is determined by many factors: the geometry of the channel, the fluid viscosity, the Reynolds number. To describe this dependence, we
introduce the Strouhal number St = /^/(vRe), which is actually the dimensionless frequency of flow oscillations normalized by the Reynolds number (v is the kinematic viscosity). A diagram of the
Strouhal number versus Reynolds number is shown in Fig. 6 (squares). The oscillation frequency increases monotonically to a value of Re = 300 and then decreases slightly.
Our calculations data are correlated accurately with experimental one [16], which in Fig. 6 are marked with red tags. Maximum differences are observed at high Reynolds numbers, but it should be noted
that the experimental data were obtained for a channel with cross-sectional dimensions of 600^mx300^m.
Meanwhile, the layered mixing structure which was formed at Re > 150 is preserved in whole, and due to transverse flow fluctuations in the unsteady flow regime the mixing efficiency increases to
about M = 40% (see Fig. 2).
Starting from a Reynolds number of 450, the frequency of flow oscillations gradually decays. Firstly flow becomes quasiperiodic (450 < Re < 600), and then almost chaotic (Re > 600). The frequency
spectrum of the velocity field becomes sufficiently filled, and is close to the continuous. This is clearly seen in Fig 5 (see also Fig. 4 (right)), where the Reynolds number 600 corresponds to the
medium curve, and the Reynolds number 1000 corresponds the top one.
Fig. 6. Strouhal number versus Reynolds number
The distribution of the flow pulsation kinetic energy e by frequencies for Re = 600 is shown in Fig. 7. This spectrum is obtained for a point located in the center of the mixing channel at a distance
of 400 (im from the enteance. Straight dashed line on the graph corresponds to the univereal law of ehe Kolmogorov-Obukhov.
Although for Re = 600 the spectrum can not; be considered complete continuous, as in the cate of developed turbulent flow, nevertheless, therr aee a larno number eel" frequencies and the inertial
range, which suggerts, at least availability of trine transotional flow regime. Such oarly beginning of turbulence for channels flow occurs due to the development of Kelvin-Helmholtz instability at
the entrance of the mixing channel.
However, calculations show that if mixing channel is long enough, then with increasing of the distance from the flow confluence the pulsations gradually damped, the flow became laminar and,
0.1 -1-1- 3
1 10 100 1 -10
Fig. 8. Friction factor versus the Reynolds number
as expected, the steady velocity profile is formed. The length of the velocity profile establishment, of course, depends on the Reynolds number. To show it the problem for a channel 7000 ^m length
was solved. The obtained data is illustrated in Fig. 10, where compares the velocity profiles for the two Reynolds numbers: 30 and 120.
This coefficient is determined by formula I = (2APd)/(pU2L) where AP is the pressure drop in the channel, and L is the length of the channel. The dark mark and the line connecting them correspond to
calculation. To compare the results the values of friction factor for steady laminar flow in a rectangular channel is shown on the graph by the dashed line.
For a channel with height-to-width ratio equal to 0.5 the friction factor is close to 64/Re. Nevertheless, the analysis shows that for small Reynolds numbers the friction factor in micromixer on
average 20-30% higher than for steady flow. Then the friction factor dramatically deviate from the dependence I = 64/Re, indicating the laminar-turbulent transition. The calculated data of the
friction factor in micromixer at moderate Reynolds numbers is well described by the dependence I = 1.8/Re025. Obtained value of the friction factor is almost six times higher than the classical
Blasius dependence (I = 0.316/Re025) for the developed turbulent flow in the direct channel. Such large difference is due to the presence of a turning flow at the channel inlet, and its vortex in the
mixing channel. In particular, the pressure along the channel does not change monotonically. In the transition to turbulence, S-shaped vortex structure, that was formed in the mixing channel at Re >
150 and existed in the transient regime collapses. The flow is divided into a set of sufficiently large eddies. Because of it the contact area between the miscible liquids reduced. And so mixing
efficiency slightly decreased on transition to turbulence (see Fig. 2). Naturally, with further increase of the Reynolds number a lot of small-scale vortices appeared in the flow.
As a consequence, the mixing efficiency in the developed turbulent flow far exceeds suitable value for laminar flow.
Comparison of the experimentally measured by micro-PIV and calculated velocity profiles in the central section of the mixing channel at 2.5 calibers from the input are shown in Fig. 9. The appearance
0 5 ID 5 1 10 4 1.3 10 4 MO 4 15-10 4 MO 4 3.5-10 4
Fig. 9. Velocity profiles in the central cross section of the channel.
of curve bends in the velocity profiles associated with the occurrence of an S-shaped structure. Overall agreement between the experimental and calculated data is quite good, the maximum error does
not exceed 10%, but with increasing Reynolds number it's increased. This is due to essentially three-dimensional structure of the flow at a given Reynolds number.
For example, in micro-PIV measured velocity field is the average depth of correlation (in this experiment it was equal to the depth of correlation 37 microns), the gradient of the longitudinal
velocity of the channel depth has led to a smoothing of the velocity profile in the micromixer. A qualitative comparison of calculated and experimental velocity fields in the central longitudinal
section of the mixer is shown in Fig. 10. Here also there is quite satisfactory agreement between the calculated and experimental data.
To compare the concentration fields obtained by numerical simulation and in the experiment the spatial averaging of the calculated data on the depth of the T-mixer was carried out. The concentration
field in the 11 sections of XY plane on the depth of the T-mixer symmetrical about its center was taken to average.
Concentration field for each section averaged spatially using a "running average" filter with a round window the same diameter as the point spread function diameter in this section. The resulting
concentration field was calculated as the arithmetical mean of 11 sections. The averaged concentration field obtained by numerical simulation and in the experiment for different Reynolds numbers are
shown in Fig. 11 and Fig. 12. In whole, there is good qualitative agreement.
Thus, this modeling allows to sort out for the incompressible fluid flow in T-type micromixer following flow regimes:
• The steady vortex-free flow, realized at low Reynolds numbers (Re < 5).
- 23 -
Fig. 10. Average experimental and calculated velocity field in the central section of micromixer for Reynolds numbers equals 30 and 120
Re = 90 numerical Re = 186 numerical
Fig. 11. Averaged concentration field in the central section of micromixer for Re=90 and Re=186
• The steady symmetric vortex flow with two symmetric horseshoe vortices at the mixing channel inlet. This regime realized when the Reynolds numbers vary in the range 5 < Re < 150.
• Steady asymmetric vortex flow is observed in the range of Reynolds numbers 150 < Re < 240. Formed at the entrance horseshoe vortices lose their symmetry and rotated at 45° relative to the central
longitudinal plane of the mixing channel. S-shaped vortices formed.
• Unsteady periodic flow is realized in the range 240 < Re < 400.
• Almost a stochastic flow regime (400 < Re < 1000). S-shaped vortex structures observed at lower Reynolds numbers collapsed.
The mixing efficiency increases dramatically during formation the S-shaped vortex structures in the flow and then continues to grow in an unsteady periodic regime. In paper [15] was shown that the
mixing efficiency can be substantially increased by changing flow rate at the inlet of the
Fig. 12. The normalized concentration profiles across the mixing channel for Re = 30
mixer in a certain way by harmonical law. In fact, here we have some "autocontrol" of the mixing process.
Usually to ensure the efficient mixing, the mixer length should be large enough. Naturally, this leads to a significant loss of pressure caused by friction at the walls. On the other hand, such
losses can be reduced by using hydrophobic or even ultrahydrophobic coats. In microflows the slip length can reach tens of microns. As it shown in [15], if the Reynolds numbers were low the mixing
efficiency particularly didn't change. However, the situation is changing at moderate Reynolds numbers. The presence of wall slip leads to a significant change in flow regimes. If there are slip
conditions the flows are rebuilt. Simulations show, for example, that at Reynolds numbers equals 200 and sufficiently large slip lengths the two-vortex structure mentioned above is transformed into
one-vortex. Naturally, the mixing efficiency increased too. For this mixer an increase is about 30%. On the other hand, the pressure drop decreases monotonically with increasing the slip length (for
investigated mixer for about 30-40%). Thus, using a hydrophobic coats (slip conditions) one can control the flow regimes.
This work was supported in part by the Russian Foundation for Basic Research (Grants №. 10-01-00074 and 11-08-01268) and the Federal Special Program "Scientific and scientific-pedagogical personnel
of innovative Russia in 2009-2013" (№ 16.740.11.0642, 14.A18.21.0344, 14.132.21.1750, 8756).
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[10] Gobert C., Schwert F., ManhartM. // Proc. ASME Joint U.S.-European Fluids Eng. Summer Meeting, Miami, Paper no. FEDSM2006-98035, 2006. P. 1053-1062.
[11] Rudyak V.Ya., Minakov A.V., Gavrilov A.A. and Dekterev A.A. // Thermophysics & Aeromechanics 15 (2008) 33-345.
[12] Gavrilov A.A., Minakov A.V., Dekterev A.A. and Rudyak V.Ya. // Sib. Zh. Industr. Matem. 13 (2010). № 4. P. 3-14.
[13] Podryabinkin E.V., Rudyak V.Ya. // J. Engineering Thermophysics. 20 (2011), No. 3. P. 320-328.
[14] Minakov A.V., Rudyak V.Ya., Gavrilov A.A., Dekterev A.A. // Journal of Siberian Federal University. Mathematics & Physics 3(2010), No. 2. P. 146-156.
[15]Rudyak V.Ya., Minakov A.V., Gavrilov A.A. and Dekterev A.A. // Thermophysics & Aeromechanics 17 (2010) 565-576.
[16] Dreher S, Kockmann N., Woias P. // Heat Transfer Engineering 30 (2009) 91-100.
Micro-LIF и численное исследование смешения в микроканале
А.В. Минаковаб, А.А. Ягодницынаб, А.С. Лобасоваб, В.Я. Рудякбв, А.В. Бильскийб
а Сибирский федеральный университет, Россия 660074, Красноярск, ул. Киренского, 28 бИнститут теплофизики СО РАН, Россия 630090, Новосибирск, пр. Лаврентьева, 1 вНовосибирский государственный
архитектурно-строительный университет, Россия 630008, Новосибирск, ул. Ленинградская, 113
В статье с помощью численного моделирования и экспериментальных методов micro-PIV и micro-LIF исследованы режимы течения и смешения жидкостей в микромиксере Т-типа в широком диапазоне значений числа
Рейнольдса от 1 до 1000. Поперечное сечение канала равнялось 200 мкм*400 мкм, а длина канала была равна 3000 мкм. Было обнаружено пять различных режимов течения: (I) стационарное безвихревое течение;
(II) стационарное симметричное вихревое течение с двумя подковообразными вихрями;
(III) стационарное асимметричное вихревое течение; (IV) нестационарное периодическое течение; (V) хаотическое течение. Максимальное значение эффективности смешения наблюдается при стационарном
асимметричном вихревом течении. В этом случае в потоке формируются S-образные вихревые структуры. Показано хорошее соответствие расчётных и экспериментальных данных.
Ключевые слова: микротечение, микромиксеры, микроканалы, CFD, Micro-PIV, Micro-LIF. | {"url":"https://cyberleninka.ru/article/n/micro-lif-and-numerical-investigation-of-mixing-in-microchannel","timestamp":"2024-11-12T05:52:10Z","content_type":"application/xhtml+xml","content_length":"89767","record_id":"<urn:uuid:34fa5570-d849-463b-ba1a-ebbbc2111749>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00284.warc.gz"} |
Cameron Musco
Sharper Bounds for Chebyshev Moment Matching with Applications to Differential Privacy and Beyond
Cameron Musco, Christopher Musco, Lucas Rosenblatt, Apoorv Vikram Singh
Competitive Algorithms for Online Knapsack with Succinct Predictions
Mohammadreza Daneshvaramoli, Helia Karisani, Adam Lechowicz, Bo Sun, Cameron Musco, Mohammad Hajiesmaili
Fixed-Sparsity Matrix Approximation from Matrix-Vector Products
Noah Amsel, Tyler Chen, Feyza Duman Keles, Diana Halikias, Cameron Musco, Christopher Musco
Near-Optimal Hierarchical Matrix Approximation from Matrix-Vector Products
Tyler Chen, Feyza Duman Keles, Diana Halikias, Cameron Musco, Christopher Musco, David Persson
ACM-SIAM Symposium on Discrete Algorithms (SODA) 2025.
Improved Spectral Density Estimation via Explicit and Implicit Deflation
Rajarshi Bhattacharjee, Rajesh Jayaram, Cameron Musco, Christopher Musco, Archan Ray
ACM-SIAM Symposium on Discrete Algorithms (SODA) 2025.
Efficient and Private Marginal Reconstruction with Local Non-Negativity
Brett Mullins, Miguel Fuentes, Yingtai Xiao, Daniel Kifer, Cameron Musco, Daniel Sheldon
Conference on Neural Information Processing Systems (NeurIPS) 2024.
Gaussian Process Bandits for Top-k Recommendations
Mohit Yadav, Cameron Musco, Daniel Sheldon
Conference on Neural Information Processing Systems (NeurIPS) 2024.
Navigable Graphs for High-Dimensional Nearest Neighbor Search: Constructions and Limits
Haya Diwan, Jinrui Gou, Cameron Musco, Christopher Musco, Torsten Suel
Conference on Neural Information Processing Systems (NeurIPS) 2024.
Near-Optimality Guarantees for Approximating Rational Matrix Functions by the Lanczos Method
Noah Amsel, Tyler Chen, Anne Greenbaum, Cameron Musco, Christopher Musco
Conference on Neural Information Processing Systems (NeurIPS) 2024. Spotlight Presentation.
Slides and video from my talk at Simons.
On the Role of Edge Dependency in Graph Generative Models
Sudhanshu Chanpuriya, Cameron Musco, Konstantinos Sotiropoulos, Charalampos Tsourakakis
International Conference on Machine Learning (ICML) 2024.
Universal Matrix Sparsifiers and Fast Deterministic Algorithms for Linear Algebra
Rajarshi Bhattacharjee, Gregory Dexter, Cameron Musco, Archan Ray, Sushant Sachdeva, David P. Woodruff
Innovations in Theoretical Computer Science (ITCS) 2024.
Slides from my talk at FOCM.
On the Unreasonable Effectiveness of Single Vector Krylov Methods for Low-Rank Approximation
Raphael Meyer, Cameron Musco, Christopher Musco
ACM-SIAM Symposium on Discrete Algorithms (SODA) 2024.
Chris's slides for 30 minute talk.
Sublinear Time Low-Rank Approximation of Toeplitz Matrices
Cameron Musco, Kshiteej Sheth.
ACM-SIAM Symposium on Discrete Algorithms (SODA) 2024.
No-regret Algorithms for Fair Resource Allocation
Abhishek Sinha, Ativ Joshi, Rajarshi Bhattacharjee, Cameron Musco, Mohammad Hajiesmaili
Conference on Neural Information Processing Systems (NeurIPS) 2023.
Exact Representation of Sparse Networks with Symmetric Nonnegative Embeddings
Sudhanshu Chanpuriya, Ryan A. Rossi, Anup B. Rao, Tung Mai, Nedim Lipka, Zhao Song, Cameron Musco
Conference on Neural Information Processing Systems (NeurIPS) 2023.
Finite Population Regression Adjustment and Non-asymptotic Guarantees for Treatment Effect Estimation
Mehrdad Ghadiri, David Arbour, Tung Mai, Cameron Musco, Anup B. Rao
Conference on Neural Information Processing Systems (NeurIPS) 2023.
Latent Random Steps as Relaxations of Max-Cut, Min-Cut, and More
Sudhanshu Chanpuriya, Cameron Musco
Differentiable Almost Everything Workshop at ICML 2023.
Sublinear Time Eigenvalue Approximation via Random Sampling
Rajarshi Bhattacharjee, Gregory Dexter, Petros Drineas, Cameron Musco, Archan Ray
International Colloquium on Automata, Languages, and Programming (ICALP) 2023. Full version in Algorithmica 2024.
Slides from my talk at the Algorithms and Foundations for Data Science Workshop, NUS. Video of my talk at Simons.
Weighted Minwise Hashing Beats Linear Sketching for Inner Product Estimation
Aline Bessa, Majid Daliri, Juliana Freire, Cameron Musco, Christopher Musco, AĆ©cio Santos, Haoxiang Zhang
Symposium on Principles of Database Systems (PODS) 2023.
Direct Embedding of Temporal Network Edges via Time-Decayed Line Graphs
Sudhanshu Chanpuriya, Ryan A. Rossi, Sungchul Kim, Tong Yu, Jane Hoffswell, Nedim Lipka, Shunan Guo, Cameron Musco
International Conference on Learning Representations (ICLR) 2023.
Video of an hour long talk by Dan.
Optimal Sketching Bounds for Sparse Linear Regression
Tung Mai, Alexander Munteanu, Cameron Musco, Anup B. Rao, Chris Schwiegelshohn, David P. Woodruff
International Conference on Artificial Intelligence and Statistics (AISTATS) 2023.
Low-Memory Krylov Subspace Methods for Optimal Rational Matrix Function Approximation
Tyler Chen, Anne Greenbaum, Cameron Musco, Christopher Musco
SIAM Journal on Matrix Analysis and Applications (SIMAX) 2023.
Local Edge Dynamics and Opinion Polarization
Nikita Bhalla, Adam Lechowicz, Cameron Musco
ACM International Conference on Web Search and Data Mining (WSDM) 2023.
Invited to special issue of ACM Transactions on Intelligent Systems and Technology
Slides from my talk at the Integrity 2023 Workshop at WSDM. Video of Adam's WSDM talk. Code repository.
Toeplitz Low-Rank Approximation with Sublinear Query Complexity
Michael Kapralov, Hannah Lawrence, Mikhail Makarov, Cameron Musco, Kshiteej Sheth.
ACM-SIAM Symposium on Discrete Algorithms (SODA) 2023.
Near-Linear Sample Complexity for L[p] Polynomial Regression
Raphael Meyer, Cameron Musco, Christopher Musco, David P. Woodruff, Samson Zhou.
ACM-SIAM Symposium on Discrete Algorithms (SODA) 2023.
Kernel Interpolation with Sparse Grids
Mohit Yadav, Daniel Sheldon, Cameron Musco
Conference on Neural Information Processing Systems (NeurIPS) 2022.
Modeling Transitivity and Cyclicity in Directed Graphs via Binary Code Box Embeddings
Dongxu Zhang, Michael Boratko, Cameron Musco, Andrew McCallum
Conference on Neural Information Processing Systems (NeurIPS) 2022.
Code repository.
Sample Constrained Treatment Effect Estimation
Raghavendra Addanki, David Arbour, Tung Mai, Cameron Musco, Anup B. Rao
Conference on Neural Information Processing Systems (NeurIPS) 2022.
Code repository.
Simplified Graph Convolution with Heterophily
Sudhanshu Chanpuriya, Cameron Musco.
Conference on Neural Information Processing Systems (NeurIPS) 2022.
Code repository.
Active Linear Regression for ℓ[p] Norms and Beyond
Cameron Musco, Christopher Musco, David P. Woodruff, Taisuke Yasuda
IEEE Symposium on Foundations of Computer Science (FOCS) 2022
Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries
Raghavendra Addanki, Andrew McGregor, Cameron Musco
European Symposium on Algorithms (ESA) 2022.
Fast Regression for Structured Inputs
Raphael Meyer, Cameron Musco, Christopher Musco, David P. Woodruff, Samson Zhou
International Conference on Learning Representations (ICLR) 2022.
Sublinear Time Approximation of Text Similarity Matrices
Archan Ray, Nicholas Monath, Andrew McCallum, Cameron Musco
AAAI Conference on Artificial Intelligence (AAAI) 2022.
Error Bounds for Lanczos-Based Matrix Function Approximation
Tyler Chen, Anne Greenbaum, Cameron Musco, Christopher Musco
SIAM Journal on Matrix Analysis and Applications (SIMAX) 2022.
On the Power of Edge Independent Graph Models
Sudhanshu Chanpuriya, Cameron Musco, Konstantinos Sotiropoulos, Charalampos E. Tsourakakis
Conference on Neural Information Processing Systems (NeurIPS) 2021.
Coresets for Classification - Simplified and Strengthened
Tung Mai, Cameron Musco, Anup B. Rao
Conference on Neural Information Processing Systems (NeurIPS) 2021.
DeepWalking Backwards: From Embeddings Back to Graphs
Sudhanshu Chanpuriya, Cameron Musco, Konstantinos Sotiropoulos, Charalampos E. Tsourakakis
International Conference on Machine Learning (ICML) 2021. Spotlight Presentation.
Code repository.
Faster Kernel Matrix Algebra via Density Estimation
Arturs Backurs, Piotr Indyk, Cameron Musco, Tal Wagner
International Conference on Machine Learning (ICML) 2021.
Faster Kernel Interpolation for Gaussian Processes
Mohit Yadav, Daniel Sheldon, Cameron Musco
International Conference on Artificial Intelligence and Statistics (AISTATS) 2021. Oral Presentation.
Intervention Efficient Algorithms for Approximate Learning of Causal Graphs
Raghavendra Addanki, Andrew McGregor, Cameron Musco
Algorithmic Learning Theory (ALT) 2021.
Video of Raghav's talk at ALT.
Subspace Embeddings Under Nonlinear Transformations
Aarshvi Gajjar, Cameron Musco
Algorithmic Learning Theory (ALT) 2021.
Video of Aarshvi's talk at ALT.
Estimation of Shortest Path Covariance Matrices
Raj Kumar Maity, Cameron Musco
Simple Heuristics Yield Provable Algorithms for Masked Low-Rank Approximation
Cameron Musco, Christopher Musco, David P. Woodruff
Innovations in Theoretical Computer Science (ITCS) 2021.
Slides from my talks at ITA/UMass. Video of my talk at ITCS.
Hutch++: Optimal Stochastic Trace Estimation
Raphael Meyer, Cameron Musco, Christopher Musco, David P. Woodruff
SIAM Symposium on Simplicity in Algorithms (SOSA) 2021.
Video of my E-NLA Seminar talk. Corresponding slides. Code repository.
The Hutch++ algorithm is also implemented in PyLops, SciPy, and elsewhere (e.g., 1, 2).
Fourier Sparse Leverage Scores and Approximate Kernel Learning
Tamás Erdélyi, Cameron Musco, Christopher Musco
Conference on Neural Information Processing Systems (NeurIPS) 2020. Spotlight Presentation.
Code repository.
Node Embeddings and Exact Low-Rank Representations of Complex Networks
Sudhanshu Chanpuriya, Cameron Musco, Konstantinos Sotiropoulos, Charalampos E. Tsourakakis
Conference on Neural Information Processing Systems (NeurIPS) 2020.
Slides from my talk at SIAM Mathematics of Data Science. Code repository with LPCA exact factorization code.
Spiking Neural Networks Through the Lens of Streaming Algorithms
Yael Hitron, Cameron Musco, Merav Parter
International Symposium on Distributed Computing (DISC) 2020.
Near Optimal Linear Algebra in the Online and Sliding Window Models
Vladimir Braverman, Petros Drineas, Cameron Musco, Christopher Musco, Jalaj Upadhyay, David P. Woodruff, Samson Zhou
IEEE Symposium on Foundations of Computer Science (FOCS) 2020.
Efficient Intervention Design for Causal Discovery with Latents
Raghavendra Addanki, Shiva Prasad Kasiviswanathan, Andrew McGregor, Cameron Musco
International Conference on Machine Learning (ICML) 2020.
InfiniteWalk: Deep Network Embeddings as Laplacian Embeddings with a Nonlinearity
Sudhanshu Chanpuriya, Cameron Musco
Knowledge Discovery and Data Mining (KDD) 2020.
Code repository.
Low-Rank Toeplitz Matrix Estimation via Random Ultra-Sparse Rulers
Hannah Lawrence, Jerry Li, Cameron Musco, Christopher Musco
International Conference on Acoustics, Speech, and Signal Processing (ICASSP) 2020.
Video of Hannah's (remote) talk at ICASSP.
Importance Sampling via Local Sensitivity
Anant Raj, Cameron Musco, Lester Mackey
International Conference on Artificial Intelligence and Statistics (AISTATS) 2020.
Random Sketching, Clustering, and Short-Term Memory in Spiking Neural Networks
Yael Hitron, Nancy Lynch, Cameron Musco, Merav Parter
Innovations in Theoretical Computer Science (ITCS) 2020.
Sample Efficient Toeplitz Covariance Estimation
Yonina Eldar, Jerry Li, Cameron Musco, Christopher Musco
ACM-SIAM Symposium on Discrete Algorithms (SODA) 2020.
Slides from my talk at Cornell. Video from my talk at DIMACS.
Fast and Space Efficient Spectral Sparsification in Dynamic Streams
Michael Kapralov, Aida Mousavifar, Cameron Musco, Christopher Musco, Navid Nouri, Aaron Sidford, Jakab Tardos
ACM-SIAM Symposium on Discrete Algorithms (SODA) 2020.
Slides from my talk at Cornell.
Toward a Characterization of Loss Functions for Distribution Learning
Nika Haghtalab, Cameron Musco, Bo Waggoner
Conference on Neural Information Processing Systems (NeurIPS) 2019.
Learning to Prune: Speeding up Repeated Computations
Daniel Alabi, Adam Tauman Kalai, Katrina Ligett, Cameron Musco, Christos Tzamos, Ellen Vitercik
Conference on Learning Theory (COLT) 2019.
A Universal Sampling Method for Reconstructing Signals with Simple Fourier Transforms
Haim Avron, Michael Kapralov, Cameron Musco, Christopher Musco, Ameya Velingker, Amir Zandieh
ACM Symposium on Theory of Computing (STOC) 2019.
Slides and video from Chris's talk at Simons.
Learning Networks from Random Walk-Based Node Similarities
Jeremy G. Hoskins, Cameron Musco, Christopher Musco, Charalampos E. Tsourakakis
Conference on Neural Information Processing Systems (NeurIPS) 2018.
Code repository.
Minimizing Polarization and Disagreement in Social Networks
Cameron Musco, Christopher Musco, Charalampos E. Tsourakakis
The Web Conference (WWW) 2018.
Code repository.
Eigenvector Computation and Community Detection in Asynchronous Gossip Models
Frederik Mallmann-Trenn, Cameron Musco, Christopher Musco
International Colloquium on Automata, Languages, and Programming (ICALP) 2018.
Spectrum Approximation Beyond Fast Matrix Multiplication: Algorithms and Hardness
Cameron Musco, Praneeth Netrapalli, Aaron Sidford, Shashanka Ubaru, David P. Woodruff
Innovations in Theoretical Computer Science (ITCS) 2018.
Slides from my talk at ITCS.
Stability of the Lanczos Method for Matrix Function Approximation
Cameron Musco, Christopher Musco, Aaron Sidford
ACM-SIAM Symposium on Discrete Algorithms (SODA) 2018.
Code repository for matrix function approximation (see lanczos.m).
Recursive Sampling for the Nyström Method
Cameron Musco, Christopher Musco
Conference on Neural Information Processing Systems (NeurIPS) 2017.
Code repository.  Slides and video from my talk at Simon's discussing this line of work.
Is Input Sparsity Time Possible for Kernel Low-Rank Approximation?
Cameron Musco, David P. Woodruff
Conference on Neural Information Processing Systems (NeurIPS) 2017.
Sublinear Time Low-Rank Approximation of Positive Semidefinite Matrices
Cameron Musco, David P. Woodruff
IEEE Symposium on Foundations of Computer Science (FOCS) 2017.
Slides and video from my talk at FOCS. Extended slides slides for hour long talk.
Random Fourier Features for Kernel Ridge Regression: Approximation Bounds and Statistical Guarantees
Haim Avron, Michael Kapralov, Cameron Musco, Christopher Musco, Ameya Velingker, Amir Zandieh
International Conference on Machine Learning (ICML) 2017.
Slides and video from my talk at ICML. Chris's extended slides for an hour long talk.
Input Sparsity Time Low-Rank Approximation via Ridge Leverage Score Sampling
Michael B. Cohen, Cameron Musco, Christopher Musco
ACM-SIAM Symposium on Discrete Algorithms (SODA) 2017.
Slides from my talk at SODA. Chris's extended slides from his talk at University of Utah.
Neuro-RAM Unit with Applications to Similarity Testing and Compression in Spiking Neural Networks
Nancy Lynch, Cameron Musco, Merav Parter
International Symposium on Distributed Computing (DISC) 2017.
Spiking Neural Networks: An Algorithmic Perspective
Nancy Lynch, Cameron Musco, Merav Parter
Presentation at Workshop on Biological Distributed Algorithms (BDA) 2017.
Slides from my talk at BDA.
New Perspectives on Algorithmic Robustness Inspired by Ant Colony House-Hunting
Tsvetomira Radeva, Cameron Musco, Nancy Lynch
Presentation at Workshop on Biological Distributed Algorithms (BDA) 2017.
Computational Tradeoffs in Biological Neural Networks: Self-Stabilizing Winner-Take-All Networks
Nancy Lynch, Cameron Musco, Merav Parter
Innovations in Theoretical Computer Science (ITCS) 2017.
Ant-Inspired Density Estimation via Random Walks
Cameron Musco, Hsin-Hao Su, Nancy Lynch
Proceedings of the National Academy of Sciences (PNAS) 2017.
Full paper also available on arXiv. An extended abstract initially appeared in PODC 2016.
Online Row Sampling
Michael B. Cohen, Cameron Musco, Jakub Pachocki
International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) 2016.
Appeared in special issue of Theory of Computing.
Principal Component Projection Without Principal Component Analysis
Roy Frostig, Cameron Musco, Christopher Musco, Aaron Sidford
International Conference on Machine Learning (ICML) 2016.
Code repository. Chris's slides from his talk at ICML.
Faster Eigenvector Computation via Shift-and-Invert Preconditioning
Daniel Garber, Elad Hazan, Chi Jin, Sham M. Kakade, Cameron Musco, Praneeth Netrapalli, Aaron Sidford
International Conference on Machine Learning (ICML) 2016.
Randomized Block Krylov Methods for Stronger and Faster Approximate Singular Value Decomposition
Cameron Musco, Christopher Musco
Conference on Neural Information Processing Systems (NeurIPS) 2015. Oral Presentation.
Slides and video from my talk at NeurIPS. Code repository.
Dimensionality Reduction for k-Means Clustering and Low Rank Approximation
Michael B. Cohen, Sam Elder, Cameron Musco, Christopher Musco, Madalina Persu
ACM Symposium on Theory of Computing (STOC) 2015.
Slides from my talk at MIT's Algorithms and Complexity Seminar.
My Master's Thesis containing empirical evaluation along with a guide to implementation. A note containing simplified proofs for common projection-cost-preserving sketches.
Uniform Sampling for Matrix Approximation
Michael B. Cohen, Yin Tat Lee, Cameron Musco, Christopher Musco, Richard Peng, Aaron Sidford
Innovations in Theoretical Computer Science (ITCS) 2015.
Slides from my talk at MIT's Algorithms and Complexity Seminar.
Distributed House-Hunting in Ant Colonies
Mohsen Ghaffari, Cameron Musco, Tsvetomira Radeva, Nancy Lynch
ACM Symposium on Principles of Distributed Computing (PODC) 2015.
Single Pass Spectral Sparsification in Dynamic Streams
Michael Kapralov, Yin Tat Lee, Cameron Musco, Christopher Musco, Aaron Sidford
IEEE Symposium on Foundations of Computer Science (FOCS) 2014.
Appeared in special issue of SIAM Journal on Computing (SICOMP).
Chris's Slides from his talks at FOCS and the Harvard TOC Seminar. | {"url":"https://people.cs.umass.edu/~cmusco/","timestamp":"2024-11-03T15:39:25Z","content_type":"text/html","content_length":"54083","record_id":"<urn:uuid:820bef76-ea43-499d-9095-3e37b9ec8e13>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00366.warc.gz"} |
New results on the sum of two generalized Gaussian random variables
We propose in this paper a new method to compute the characteristic function (CF) of the generalized Gaussian (GG) random variable in terms of the Fox H function. The CF of the sum of two independent
GG random variables is then deduced. Based on this result, the probability density function (PDF) and the cumulative distribution function (CDF) of the sum distribution are obtained. These functions
are expressed in terms of the bivariate Fox H function. Next, the statistics of the distribution of the sumare analyzed and computed. Due to the complexity of the bivariate Fox H function, a solution
to reduce such complexity is to approximate the sum of two independent GG random variables by one GG random variable with a suitable shape factor. The approximation method depends on the utility of
the system so three methods of estimate the shape factor are studied and presented.
Publication series
Name 2015 IEEE Global Conference on Signal and Information Processing, GlobalSIP 2015
Conference IEEE Global Conference on Signal and Information Processing, GlobalSIP 2015
Country/Territory United States
City Orlando
Period 12/13/15 → 12/16/15
• Generalized Gaussian
• PDF approximation
• characteristic function
• cumulant
• kurtosis
• moment
• sum of two random variables
ASJC Scopus subject areas
• Information Systems
• Signal Processing
Dive into the research topics of 'New results on the sum of two generalized Gaussian random variables'. Together they form a unique fingerprint. | {"url":"https://faculty.kaust.edu.sa/en/publications/new-results-on-the-sum-of-two-generalized-gaussian-random-variabl","timestamp":"2024-11-04T01:37:27Z","content_type":"text/html","content_length":"57574","record_id":"<urn:uuid:19913d63-42c9-45a5-973d-2e347d4996c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00514.warc.gz"} |
Task The Way to Bytemountain [B] (dro)
The Way to Bytemountain [B]
Memory limit: 32 MB
Byteman woke up early this morning, just after dawn. He is planning to get to the top of Bytemountain, so the spent the night in a mountain hostel right in the middle of picturesque mountain range of
Lower Bytehills. Since Bytemountain in the highest mountain of the range, at each trail crossing there is a signpost pointing at the trail leading towards its peak.
In the mountain hostel Byteman met a guide who knows Lower Bytehills like the back of his hand. The guide informed Byteman that the signposts are being reorganized and because of that he should not
rely too much on the signposts. In particular, the very peak of Bytemountain is also a crossing and at this crossing there is a signpost pointing to some trail "leading" to Bytemountain!
The guide will explain Byteman how to get to the peak. Luckily, all trail crossings are numbered from to and each crossing contains a tablet with the number of the crossing written on it. The guide's
directions will have the following form: "Walk along the trails pointed by the signposts until you reach crossing , then take a map and choose the trail connecting crossing with crossing . Afterwards
keep walking along the trails pointed by the signposts until you reach crossing . The take a look at the map and choose the trail connecting and ... Finally, when you reach , take the last look at
the map and walk along the trail connecting and . If you keep walking along the trails pointed by the signposts since then, you will reach the peak of Bytemountain."
Byteman would not like the description of the route to be too complicated, so he asked the guide for a route that would not require looking at the map more than times.
The guide had to take a deeper thought, because he knows that some trails are more exciting than others and he would like to show Byteman the most interesting route.
The route may lead through the same trails and crossing many times (some trails are so exciting that they may be worth visiting multiple times!)
Byteman ends his walk when he reaches the peak for the first time after using all instructions provided by the guide. This means that Byteman can visit the peak of Bytemountain multiple times during
the walk, but he will end his walk only after all instructions have been used.
How interesting can the route provided by the guide be?
The first line of the standard input contains two integers and (, ) separated by a single space. They denote the number of trail crossings and the maximum number of times Byteman would like to look
at the map. The crossings are numbered from to , the mountain hostel is located at crossing , and the peak of Bytemountain is the crossing number .
The following lines contain descriptions of the respective trail crossings. Each crossing's description consists of a single line and is composed of integers separated by single spaces. The first one
of these numbers, (), denotes the number of trails going out of the crossing. After that there are pairs of numbers , (, ), meaning that from the crossing there is a trail leading to crossing with
beauty equal to . The first pair of numbers denotes the trail that leads to Bytemountain according to the signpost at the crossing. Each trail is bidirectional and connects two different crossings.
Each two crossings can be connected by at most one trail. The total number of all trails does not exceed .
Each trail connecting crossings and will appear in the input twice: first time in the list of trails going out of the crossing and second time in the list of trails going out of the crossing. In both
cases the beauty of the trail will be the same.
The first and only line of the standard output should contain a single integer denoting the maximum possible sum of beauties of consecutive trails on the route from the mountain hostel to the peak of
Bytemountain that satisfies Byteman's requirements. You can assume that there exists at least one such route.
For the input data:
the correct result is:
Explanation of the example. In the above figure the edges represent trails connecting respective crossings, the numbers next to the edges - the beauties of the trails, and the arrows denote the
trails pointed by the signposts at respective crossings.
The guide will ask Byteman to look at the map twice, at crossings number 3 and 2. This way Byteman's walk will lead along the route . The total beauty of the trails on this route is 14.
Task author: Miroslaw Michalski. | {"url":"https://szkopul.edu.pl/problemset/problem/ITJcsQ3MdVeHkzOUG1GYNCg7/site/?key=statement","timestamp":"2024-11-09T17:53:22Z","content_type":"text/html","content_length":"31310","record_id":"<urn:uuid:60d6534a-5b7f-4343-bd78-8cdabb39c252>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00625.warc.gz"} |
Algebra Calculator - Examples, Online Algebra Calculator
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
Algebra Calculator
Algebra Calculator is used to calculate the value of the unknown variable in a mathematical equation. Algebra is the oldest branch that was studied in mathematics. It deals with the study of
mathematical symbols and their corresponding manipulation.
What is an Algebra Calculator?
Algebra Calculator is an online tool that helps find the solution to an equation that has known quantities and one unknown value. In real life, we use algebra almost for everything. For example,
scheduling of day-to-day events. To use this algebra calculator, enter values in the input boxes given below.
Algebra Calculator
*Use only 4 digits.
How to Use the Algebra Calculator?
Please follow the steps below to find the unknown value (x) in the equation using the online algebra calculator:
• Step 1: Go to Cuemath's online Algebra Calculator.
• Step 2: Enter the values in the input box of the Algebra Calculator.
• Step 3: Click on the "Solve" button to find the value of x.
• Step 4: Click on the "Reset" button to clear the fields and enter new values of a,b,c.
How Algebra Calculator Works?
Algebra deals with solving an equation to find the unknown value by rearranging and simplifying it. The procedure followed to solve an algebraic equation is given below:
• First, we bring the unknown variable to the left-hand side (L.H.S) of the equation. Simultaneously, we shift all the known quantities to the right-hand side (R.H.S).
• Solve the R.H.S by performing the required arithmetic operations.
• Next, if the variable does not have any coefficient then the number on the R.H.S is its value.
• If the variable has a coefficient then we divide the R.H.S by this number to determine the final value.
Suppose the mathematical equation is of the form ax+b = c.
Here, x is the variable, a is the coefficient while b and c are the constant terms or the known quantities. By applying the aforementioned steps, the value of x is given as.
x = (c - b)/a
Want to find complex math solutions within seconds?
Use our free online calculator to solve challenging questions. With Cuemath, find solutions in simple and easy steps.
Solved Example on Algebra
Example 1: Find the value of x in the given equation: 5x + 8 = 3 and verify it using the algebra calculator.
Given: 5x + 8 = 3
5x = 3 - 8
5x = -5
x = -5/5 = -1
Thus, the value of x is -1.
Example 2: If a person has a total of 36 hours to complete his assignment, out of which he is sleeping for 9 hours. How many hours are left to complete the assignment?
Given : x + 9 = 36
x = 36 - 9 = 27
He has 27 hours to complete his assignment.
Similarly, you can try the algebra calculator to find the unknown values in the equation for the following:
☛ Math Calculators:
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│Age Calculator │Square Footage Calculator│
Math worksheets and
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Square Chain Converter
Square Chain [ch2] Output
1 square chain in ankanam is equal to 60.5
1 square chain in aana is equal to 12.73
1 square chain in acre is equal to 0.10000032123671
1 square chain in arpent is equal to 0.11836780931091
1 square chain in are is equal to 4.05
1 square chain in barn is equal to 4.046873e+30
1 square chain in bigha [assam] is equal to 0.30250123916529
1 square chain in bigha [west bengal] is equal to 0.30250123916529
1 square chain in bigha [uttar pradesh] is equal to 0.16133399422149
1 square chain in bigha [madhya pradesh] is equal to 0.36300148699834
1 square chain in bigha [rajasthan] is equal to 0.16000065542627
1 square chain in bigha [bihar] is equal to 0.16003004570096
1 square chain in bigha [gujrat] is equal to 0.25000102410354
1 square chain in bigha [himachal pradesh] is equal to 0.50000204820709
1 square chain in bigha [nepal] is equal to 0.059753331193143
1 square chain in biswa [uttar pradesh] is equal to 3.23
1 square chain in bovate is equal to 0.0067447883333333
1 square chain in bunder is equal to 0.04046873
1 square chain in caballeria is equal to 0.00089930511111111
1 square chain in caballeria [cuba] is equal to 0.0030155536512668
1 square chain in caballeria [spain] is equal to 0.00101171825
1 square chain in carreau is equal to 0.031371108527132
1 square chain in carucate is equal to 0.00083268991769547
1 square chain in cawnie is equal to 0.074942092592593
1 square chain in cent is equal to 10
1 square chain in centiare is equal to 404.69
1 square chain in circular foot is equal to 5546.25
1 square chain in circular inch is equal to 798659.88
1 square chain in cong is equal to 0.4046873
1 square chain in cover is equal to 0.14999529280949
1 square chain in cuerda is equal to 0.10297386768448
1 square chain in chatak is equal to 96.8
1 square chain in decimal is equal to 10
1 square chain in dekare is equal to 0.40468756697239
1 square chain in dismil is equal to 10
1 square chain in dhur [tripura] is equal to 1210
1 square chain in dhur [nepal] is equal to 23.9
1 square chain in dunam is equal to 0.4046873
1 square chain in drone is equal to 0.015755272873192
1 square chain in fanega is equal to 0.062937371695179
1 square chain in farthingdale is equal to 0.39988863636364
1 square chain in feddan is equal to 0.09708769052108
1 square chain in ganda is equal to 5.04
1 square chain in gaj is equal to 484
1 square chain in gajam is equal to 484
1 square chain in guntha is equal to 4
1 square chain in ghumaon is equal to 0.10000040964142
1 square chain in ground is equal to 1.82
1 square chain in hacienda is equal to 0.0000045165993303571
1 square chain in hectare is equal to 0.04046873
1 square chain in hide is equal to 0.00083268991769547
1 square chain in hout is equal to 0.28473773309757
1 square chain in hundred is equal to 0.0000083268991769547
1 square chain in jerib is equal to 0.20018464356526
1 square chain in jutro is equal to 0.070319252823632
1 square chain in katha [bangladesh] is equal to 6.05
1 square chain in kanal is equal to 0.80000327713134
1 square chain in kani is equal to 0.25208436597107
1 square chain in kara is equal to 20.17
1 square chain in kappland is equal to 2.62
1 square chain in killa is equal to 0.10000040964142
1 square chain in kranta is equal to 60.5
1 square chain in kuli is equal to 30.25
1 square chain in kuncham is equal to 1
1 square chain in lecha is equal to 30.25
1 square chain in labor is equal to 0.00056453979561869
1 square chain in legua is equal to 0.000022581591824748
1 square chain in manzana [argentina] is equal to 0.04046873
1 square chain in manzana [costa rica] is equal to 0.057903793983654
1 square chain in marla is equal to 16
1 square chain in morgen [germany] is equal to 0.16187492
1 square chain in morgen [south africa] is equal to 0.047237924594374
1 square chain in mu is equal to 0.60703094696485
1 square chain in murabba is equal to 0.0040000128494685
1 square chain in mutthi is equal to 32.27
1 square chain in ngarn is equal to 1.01
1 square chain in nali is equal to 2.02
1 square chain in oxgang is equal to 0.0067447883333333
1 square chain in paisa is equal to 50.91
1 square chain in perche is equal to 11.84
1 square chain in parappu is equal to 1.6
1 square chain in pyong is equal to 122.41
1 square chain in rai is equal to 0.2529295625
1 square chain in rood is equal to 0.40000163856567
1 square chain in ropani is equal to 0.79547440540178
1 square chain in satak is equal to 10
1 square chain in section is equal to 0.00015625064006471
1 square chain in sitio is equal to 0.000022482627777778
1 square chain in square is equal to 43.56
1 square chain in square angstrom is equal to 4.046873e+22
1 square chain in square astronomical units is equal to 1.8082927862225e-20
1 square chain in square attometer is equal to 4.046873e+38
1 square chain in square bicron is equal to 4.046873e+26
1 square chain in square centimeter is equal to 4046873
1 square chain in square cubit is equal to 1936.01
1 square chain in square decimeter is equal to 40468.73
1 square chain in square dekameter is equal to 4.05
1 square chain in square digit is equal to 1115140.57
1 square chain in square exameter is equal to 4.046873e-34
1 square chain in square fathom is equal to 121
1 square chain in square femtometer is equal to 4.046873e+32
1 square chain in square fermi is equal to 4.046873e+32
1 square chain in square feet is equal to 4356.02
1 square chain in square furlong is equal to 0.010000032123671
1 square chain in square gigameter is equal to 4.046873e-16
1 square chain in square hectometer is equal to 0.04046873
1 square chain in square inch is equal to 627266.57
1 square chain in square league is equal to 0.000017361112958197
1 square chain in square light year is equal to 4.5213684173878e-30
1 square chain in square kilometer is equal to 0.0004046873
1 square chain in square megameter is equal to 4.046873e-10
1 square chain in square meter is equal to 404.69
1 square chain in square microinch is equal to 627266016185620000
1 square chain in square micrometer is equal to 404687300000000
1 square chain in square micromicron is equal to 4.046873e+26
1 square chain in square micron is equal to 404687300000000
1 square chain in square mil is equal to 627266569533.14
1 square chain in square mile is equal to 0.00015625064006471
1 square chain in square millimeter is equal to 404687300
1 square chain in square nanometer is equal to 404687300000000000000
1 square chain in square nautical league is equal to 0.000013109770872758
1 square chain in square nautical mile is equal to 0.00011798783377218
1 square chain in square paris foot is equal to 3835.9
1 square chain in square parsec is equal to 4.2502880902487e-31
1 square chain in perch is equal to 16
1 square chain in square perche is equal to 7.92
1 square chain in square petameter is equal to 4.046873e-28
1 square chain in square picometer is equal to 4.046873e+26
1 square chain in square pole is equal to 16
1 square chain in square rod is equal to 16
1 square chain in square terameter is equal to 4.046873e-22
1 square chain in square thou is equal to 627266569533.14
1 square chain in square yard is equal to 484
1 square chain in square yoctometer is equal to 4.046873e+50
1 square chain in square yottameter is equal to 4.046873e-46
1 square chain in stang is equal to 0.14938623108158
1 square chain in stremma is equal to 0.4046873
1 square chain in sarsai is equal to 144
1 square chain in tarea is equal to 0.64358667302799
1 square chain in tatami is equal to 244.83
1 square chain in tonde land is equal to 0.073366080493111
1 square chain in tsubo is equal to 122.42
1 square chain in township is equal to 0.0000043402917203434
1 square chain in tunnland is equal to 0.081980248764282
1 square chain in vaar is equal to 484
1 square chain in virgate is equal to 0.0033723941666667
1 square chain in veli is equal to 0.050416873194214
1 square chain in pari is equal to 0.040000163856567
1 square chain in sangam is equal to 0.16000065542627
1 square chain in kottah [bangladesh] is equal to 6.05
1 square chain in gunta is equal to 4
1 square chain in point is equal to 10
1 square chain in lourak is equal to 0.080000327713134
1 square chain in loukhai is equal to 0.32000131085253
1 square chain in loushal is equal to 0.64000262170507
1 square chain in tong is equal to 1.28
1 square chain in kuzhi is equal to 30.25
1 square chain in chadara is equal to 43.56
1 square chain in veesam is equal to 484
1 square chain in lacham is equal to 1.6
1 square chain in katha [nepal] is equal to 1.2
1 square chain in katha [assam] is equal to 1.51
1 square chain in katha [bihar] is equal to 3.2
1 square chain in dhur [bihar] is equal to 64.01
1 square chain in dhurki is equal to 1280.24 | {"url":"https://hextobinary.com/unit/area/from/sqch","timestamp":"2024-11-14T23:09:40Z","content_type":"text/html","content_length":"170932","record_id":"<urn:uuid:c1e21307-d854-4da5-b72e-4619266dd8e4>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00260.warc.gz"} |
Adding Subtracting Multiplying And Dividing Real Numbers Worksheet
Adding Subtracting Multiplying And Dividing Real Numbers Worksheet function as fundamental devices in the world of maths, providing a structured yet flexible platform for students to discover and
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Adding Subtracting Multiplying and Dividing Worksheets This mixed problems worksheet may be configured for adding subtracting multiplying and dividing two numbers You may select different number of
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Multiplying And Dividing Positive And Negative Numbers Worksheets
Multiplying And Dividing Positive And Negative Numbers Worksheets
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Adding or subtracting fractions requires a common denominator multiplying or dividing fractions does not Grouping symbols indicate which operations to perform first We usually group mathematical
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Add Subtract Multiply Divide Integers Worksheet
Add Subtract Multiply Divide Integers Worksheet
S 02B061 42i 7K Ju itcag qS monf t7wFaWrse 5 kLQLOC5 I g pA4l bl o Urji mgUhmtPsx XrxeGsbexrcv le9dE 6 X wMJaMd1e5 Fw Ji wt4hQ yIanCf Pi6nEiJt AeA GPrqeV YA9l 3g Neab frAa6 f Worksheet by Kuta
Software LLC Kuta Software Infinite Pre Algebra Name Adding Subtracting Integers Date Period
You may recall that multiplication is a way of computing repeated addition and this is true for negative numbers as well Multiplication and division are inverse operations just as addition and
subtraction are You may recall that when you divide fractions you multiply by the reciprocal
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How do you find the derivative of y=tan^2(3x)? | Socratic
How do you find the derivative of #y=tan^2(3x)#?
1 Answer
1.) $y = {\left(\tan 3 x\right)}^{2}$
This is a problem that will involve a lot of chain rule. I will first show you what the derivative looks like and then explain where each part comes from:
2.) $\frac{\mathrm{dy}}{\mathrm{dx}} = 2 \tan 3 x \cdot {\sec}^{2} 3 x \cdot 3$
The $2 \tan 3 x$ is a result of first applying power rule. (bring the 2 out in front, and decrement the power)
Next, chain rule dictates that we multiply this with the derivative of the inside function $\tan 3 x$ with respect to $x$, resulting in the ${\sec}^{2} 3 x$.
And lastly, we apply chain rule again, multiplying the entire thing by $3$, which is the derivative of the $3 x$ inside the ${\sec}^{2} 3 x$.
The entire string can be prettified a bit by simplifying and rewriting in terms of $\sin$ and $\cos$:
3.) $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{6 \sin 3 x}{{\cos}^{3} 3 x}$
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QOF News
NHS Digital have clearly been busy through the lockdown as they released the QOF data a full two months earlier than in the last few years. I am pleased to say that all of the data is now on the
Things are pretty much as previous years although there are some new indicators, particularly around diabetes. There are also the ever changing NHS organisations as we have new CCGs and some
variations in Primary Care Networks.
For most practices this is likely to have been the basis for payment - despite the assurance of the preservation of income in the light of the Covid-19 situation. Another effect of the Covid was a
huge increase in the number of salbutamol inhalers issued in March which has bumped up the asthma register considerably. This is likely to drop back next year as the increase was almost entirely
limited to March with a drop in April and a return to normal levels in May and June.
I hope to have data from Northern Ireland soon after it is published although this is less comparable with English practices now as indicators have diverged. I am also expecting the limited amount of
data from Wales that we have seen in the past few years - essentially this is only disease prevalence data now.
There are a lot of statistics around Covid-19 and its likely impact over the next few weeks and months. There have been many charts online from expert as well as people who are playing with the
numbers. Predicting the future is always difficult and epidemiology seems to mostly be the study of confounding factors. It can be easy to produce a simple model - and much more complicated to
implement it.
I am certainly not an epidemiologist and so I have not published any numbers so far. I have played with a few simple models largely to see how they worked but nothing that had not been done to a much
higher standard by other people.
Recently I had need to estimate some figures for my practice. I am making no predictions about how the pandemic will play out. There are no predictions in here. I have taken predictions from other
people to work out the effect on my practice. In fact I have done this for every practice and PCN in England and it really is not much more work. It does make the spreadsheet work harder though!
There are various estimates of the total numbers of deaths and, whilst they influence the result we can model that fairly late. A quick way to get a ball park figure is to simply divide the deaths by
the number of practices. There are almost exactly 7000 practices in England and, at the time of writing, 14,399 deaths. That is pretty close to 2 deaths per practice. Every death is a bad thing but
we are clearly not seeing huge numbers in individual practices.
There are numerous other estimates. I have seen 40,000 deaths as an estimated UK total which would work out at about 5.5 deaths per practice in total. I will use this total, but it is pretty easy to
convert to other numbers if see a figure which appears more reliable.
I have not make any allowance for practice size. There are a shade over 60 million patients registered with practices in England and so a quick bit of division suggests that we would expect .66
deaths per thousand patients. Thus a practice with 10,000 patients would expect around 6-7 deaths from Covid-19. By this stage we are getting to something that practices can use to estimate workload.
It is unlikely that the figure of 40,000 is spot on but you can say, for instance, that you could plan for double that whilst hoping for a lower figure.
Can we refine this any more? There are many risk factors for death from Covid-19. As the disease has not been around for very long there have not been many good studies. One of the best was a look at
mortality in China by Imperial College. This looked at age as a risk factor and have published this in ten year bands. Helpfully the age and sex makeup of the population is also published. This can
come down to the year by year level but the five yearly bands are quite enough and still run to more than a third of a million lines on a spreadsheet.
There is also some information about disease risk factors such as diabetes and heart disease. We do have some of that information at practice level from the QOF. Could that be used to refine the risk
level? Unfortunately probably not. The data for age related risk and the risk from co-morbidities has been calculated separate and not as independent factors. In reality the increasing age is a risk
factor for diabetes and heart disease and so if we corrected for both we would likely be correcting twice. The risks are not independent. In the future there may be studies which look at these as
individual variable and this would allow us to use the QOF information on top of the age related risk.
The process I used was to multiply the population in each year group by the mortality risk. So if a practice had 100 patients in a group and the risk was 1% I would count 100. If the risk was 15% I
would count 1500. I add all of these together and then scale back to the national population to produce "Covid adjusted" list size. This is the list size of completely average people you would have
to produce the same total mortality. This works a bit like the Carr-Hill formula.
The major assumption here is that all ages will have a similar rate of developing the disease. This has not been shown in the paper and hopefully shielding and social distancing will give a lower
rate of disease in the elderly. On the other side the risk in care homes seems, at least from media reports, to be particularly high. I have also assumed that the infection rate is the same across
England. That is certainly not the case at the moment but I think that it is probable that it will become more similar as we get towards the end of the pandemic.
With the adjusted list size you can then do what we did above to allocate the deaths in proportion. You can adjust the national deaths and the others will change, although this is a linear
relationship. Increasing to 80,000 will just double the deaths for each practice and you could probably do that in your head.
I hope that you find this data to be useful. We are using this at our practice as a basis for planning services. Whilst the number will not be precise they give a rough estimate of what we should be
providing. Other workload is likely to be proportional to mortality and so can get some guide to likely volume of work that we will be seeing. There is likely to be a lot of local variation. The
final figures for a practice may be double or half of what is shown here but equally it would be surprising if they were out by a factor of ten. We can at least approximate what our response should
You can either see the list on Google Docs or download the spreadsheet. You can also see the full workings out on a very large (24Mb) spreadsheet which runs very slowly on my computer. | {"url":"https://news.gpcontract.co.uk/2020/","timestamp":"2024-11-11T11:22:36Z","content_type":"application/xhtml+xml","content_length":"59089","record_id":"<urn:uuid:cfdb35ab-e420-477b-a905-94a6dda0adb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00892.warc.gz"} |
Registration guides High school registration guide 2022-23 [copy] | Page 9
Math Paths
HIGH SCHOOL MATH REGISTRATION 2022-2023
Math requirements for the Graduating Class of 2026
1.0 credit HS Intermediate Algebra
1.0 credit HS Geometry
1.0 credit HS Algebra II
There are many options available for students as they progress through their math learning experience . Included below are examples of the most common course pathways for a student entering high
school in the fall of 2021 .
Students entering high school in a grade level math course ( from Middle School Algebra in grade 8 ) that are highly motivated and interested in a way to accelerate their math experience may want to
look closely and follow Option 3 . Students choosing to enroll in this accelerated pathway would be able to complete two additional math credits beyond those that are required for graduation . This
acceleration would occur after the successful completion of the HS Intermediate Algebra course and would include a wrap-around of the HS Geometry and HS Algebra II courses . The wrap-around would
allow students to start a course during 3 ^rd Trimester and finish the course during the next school year . Students that choose this pathway will need to register , during their 9th-grade school
year , for HS Intermediate Algebra ( choosing 2 course numbers ) and Honors HS Geometry with College Foundations A ( choosing 1 course number ). Refer to the Option 3 flowchart to see the math
pathway for these students .
Currently in Grade 8 …
Next year take …
MS Algebra
HS Intermediate Algebra *
HS Intermediate Algebra Honors HS Geometry Advanced Mathematics
Honors HS Geometry with College Foundations
Honors HS Algebra II
Ask your Advanced Math teacher for the appropriate course
Gr 9
If a student is currently in grade 8 ( graduating class of 2026 ) and in Middle School Algebra : Option 1 : Option 2 : Option 3 :
HS Intermediate Algebra
HS Intermediate Algebra
HS Intermediate Algebra
HS Intermediate Algebra
HS Intermediate Algebra
HS Intermediate Algebra
Honors HS Geometry w / College Foundations
Gr HS Geometry w / HS Geometry w / HS Geometry w / Honors HS Geometry w / Honors HS Geometry w / Honors HS Geometry w / Honors HS Geometry w / Honors HS Geometry w / Honors HS
10 College Foundations College Foundations College Foundations College Foundations College Foundations College Foundations College Foundations College Foundations Algebra II
Gr HS Algebra II HS Algebra II Honors HS Algebra II Honors HS Algebra II Honors HS Algebra II Honors Precalculus Honors
11 Precalculus
Gr 12 math elective math elective math elective math elective math elective math elective
If a student is currently in grade 8 ( graduating class of 2026 ) and in Honors HS Intermediate Algebra :
Option 1 : Option 2 : Gr 9
Gr 10
HS Geometry w / College Foundations
HS Algebra II
HS Geometry w / College Foundations
HS Algebra II
HS Geometry w / College Foundations
Honors HS Geometry w / College Foundations
Honors HS Algebra II
Honors HS Geometry w / College Foundations
Honors HS Algebra II
Honors HS Geometry w / College Foundations
Gr 11 ( optional ) math elective ( optional ) math elective ( optional ) math elective ( optional ) math elective
Gr 12 ( optional ) math elective ( optional ) math elective ( optional ) math elective ( optional ) math elective
Gr 9
If a student is currently in grade 8 ( graduating class of 2026 ) and in Honors HS Geometry : Option 1 : Option 2 :
HS Algebra II HS Algebra II
Honors HS Algebra II
Honors HS Algebra II
Gr 10 ( optional ) math elective ( optional ) math elective ( optional ) AP Statistics ( optional ) AP Statistics
Gr 11 ( optional ) math elective ( optional ) math elective ( optional ) math elective ( optional ) math elective
Gr 12 ( optional ) math elective ( optional ) math elective ( optional ) math elective ( optional ) math elective
High School Registration Guide 9 | {"url":"https://viewer.joomag.com/registration-guides-high-school-registration-guide-2022-23-copy/0443276001638817569/p9","timestamp":"2024-11-12T09:45:48Z","content_type":"text/html","content_length":"12897","record_id":"<urn:uuid:d0fea6b1-7f01-46c0-abf9-6e170d6788c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00025.warc.gz"} |
Java Program to find Sum, Product, and Average of an Array - VTUPulse
Java Program to find Sum, Product, and Average of an Array
Write a Java Program to find Sum, Product, and Average of an Array
Write a Java Program to Find the Sum of array elements, Product of array elements, and Average of elements of a single dimensional array.
Video Tutorial:
Steps (Program logic):
1. Declare the local variables like a, sum, prod, avg to store array elements, the sum of the array, a product of array, and an average of array elements respectively.
2. Read the number of elements in an array.
3. Instantiate the array with the number of array elements.
4. Read the elements one by one and store in array.
5. Find the sum, product of elements of an array, then find the average using the sum and number of elements in an array.
6. Finally, display the sum, product and average of elements of an array.
Source code of Java Program to find Sum, Product, and Avg. of an Array
import java.util.Scanner;
* Java Program to find the Sum, Product and Average of elements single
* dimensional array
* Subscribe to Mahesh Huddar YouTube Channel for more Videos
public class ArraySumProductAvg
public static void main(String args[])
int a[], sum = 0, prod = 1, avg, num;
Scanner in = new Scanner(System.in);
System.out.println("Enter the number of array elements:");
num = in.nextInt();
a = new int[num];
System.out.println("Enter the array elements: ");
for (int i = 0; i < num; i++)
System.out.println("Enter the "+(i+1)+" element:");
a[i] = in.nextInt();
for (int i = 0; i < num; i++)
sum = sum + a[i];
prod = prod * a[i];
avg = sum / num;
System.out.println("Sum of array elements is: "+sum);
System.out.println("Product of array elements is: "+prod);
System.out.println("Average of array elements is: "+avg);
Enter the number of array elements: 5
Enter the array elements:
Enter the 1 element: 5
Enter the 2 element: 10
Enter the 3 element: 15
Enter the 4 element: 20
Enter the 5 element: 25
Sum of array elements is: 75
Product of array elements is: 375000
Average of array elements is: 15
If you like the tutorial share it with your friends. Like the Facebook page for regular updates and YouTube channel for video tutorials.
Leave a Comment | {"url":"https://vtupulse.com/java-tutorial/java-program-to-find-sum-product-and-average-of-an-array/","timestamp":"2024-11-12T16:01:57Z","content_type":"text/html","content_length":"110068","record_id":"<urn:uuid:56ce766f-789e-4eeb-9ef7-f4d5ad02845f>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00512.warc.gz"} |
Structured Matrices in Mathematics, Computer Science, and Engineering, Volumes 1 and 2search
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Structured Matrices in Mathematics, Computer Science, and Engineering, Volumes 1 and 2
Softcover ISBN:
Product Code: CONMSET
List Price: $239.00
MAA Member Price: $215.10
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2001; 671 pp
Many important problems in applied sciences, mathematics, and engineering can be reduced to matrix problems. Moreover, various applications often introduce a special structure into the
corresponding matrices, so that their entries can be described by a certain compact formula. Classic examples include Toeplitz matrices, Hankel matrices, Vandermonde matrices, Cauchy matrices,
Pick matrices, Bezoutians, controllability and observability matrices, and others. Exploiting these and the more general structures often allows us to obtain elegant solutions to mathematical
problems as well as to design more efficient practical algorithms for a variety of applied engineering problems.
Structured matrices have been under close study for a long time and in quite diverse (and seemingly unrelated) areas, for example, mathematics, computer science, and engineering. Considerable
progress has recently been made in all these areas, and especially in studying the relevant numerical and computational issues. In the past few years, a number of practical algorithms blending
speed and accuracy have been developed. This significant growth is fully reflected in these volumes, which collect 38 papers devoted to the numerous aspects of the topic.
The collection of the contributions to these volumes offers a flavor of the plethora of different approaches to attack structured matrix problems. The reader will find that the theory of
structured matrices is positioned to bridge diverse applications in the sciences and engineering, deep mathematical theories, as well as computational and numerical issues. The presentation fully
illustrates the fact that the techniques of engineers, mathematicians, and numerical analysts nicely complement each other, and they all contribute to one unified theory of structured matrices.
The book is published in two volumes. The first contains articles on interpolation, system theory, signal and image processing, control theory, and spectral theory. Articles in the second volume
are devoted to fast algorithms, numerical and iterative methods, and various applications.
Graduate students and research mathematicians interested in linear and multilinear algebra, matrix theory, operator theory, numerical analysis, and systems theory and control.
This set contains the following item(s):
• Permission – for use of book, eBook, or Journal content
2001; 671 pp
Many important problems in applied sciences, mathematics, and engineering can be reduced to matrix problems. Moreover, various applications often introduce a special structure into the corresponding
matrices, so that their entries can be described by a certain compact formula. Classic examples include Toeplitz matrices, Hankel matrices, Vandermonde matrices, Cauchy matrices, Pick matrices,
Bezoutians, controllability and observability matrices, and others. Exploiting these and the more general structures often allows us to obtain elegant solutions to mathematical problems as well as to
design more efficient practical algorithms for a variety of applied engineering problems.
Structured matrices have been under close study for a long time and in quite diverse (and seemingly unrelated) areas, for example, mathematics, computer science, and engineering. Considerable
progress has recently been made in all these areas, and especially in studying the relevant numerical and computational issues. In the past few years, a number of practical algorithms blending speed
and accuracy have been developed. This significant growth is fully reflected in these volumes, which collect 38 papers devoted to the numerous aspects of the topic.
The collection of the contributions to these volumes offers a flavor of the plethora of different approaches to attack structured matrix problems. The reader will find that the theory of structured
matrices is positioned to bridge diverse applications in the sciences and engineering, deep mathematical theories, as well as computational and numerical issues. The presentation fully illustrates
the fact that the techniques of engineers, mathematicians, and numerical analysts nicely complement each other, and they all contribute to one unified theory of structured matrices.
The book is published in two volumes. The first contains articles on interpolation, system theory, signal and image processing, control theory, and spectral theory. Articles in the second volume are
devoted to fast algorithms, numerical and iterative methods, and various applications.
Graduate students and research mathematicians interested in linear and multilinear algebra, matrix theory, operator theory, numerical analysis, and systems theory and control.
This set contains the following item(s):
Permission – for use of book, eBook, or Journal content
Please select which format for which you are requesting permissions. | {"url":"https://bookstore.ams.org/CONMSET","timestamp":"2024-11-06T12:50:49Z","content_type":"text/html","content_length":"64071","record_id":"<urn:uuid:66ef28cb-688c-465a-a77e-9278d707d18b>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00559.warc.gz"} |
Are randomised quantum circuits inducing unitary 2-designs a reasonable thing to hope for in the near future? It seems so, following a paper presented by Saeed Mehraban at QIP this year. With Harrow,
Mehraban showed that short-depth random circuits involving nearest-neighbour unitary gates (where proximity is defined by a cubic lattice structure) produce approximate unitary $t$-designs. This is
precisely the experimental model of Google’s Quantum AI group, who are working with a 49-qubit 2-dimensional lattice of qubits.
The main application of these randomised circuits is to prove ‘quantum supremacy’. But can they do anything useful? In the attached note, I discuss an application to building a variational quantum
eigensolver, which runs into the problem of ‘barren plateaus’ recently highlighted by McClean et al. (See within for references.)
A Quantum Query Complexity Trichotomy for Regular Languages (Aaronson, Grier, and Shaeffer 2018)
There were a few great talks in the ‘query complexity session’ of QIP this year: including Srinivasan Arunachalam’s talk “A converse to the polynomial method” (joint work with Briët and Palazuelos –
see João’s post below for a discussion of this) and André Chailloux’s talk entitled “A note on the quantum query complexity of permutation symmetric functions”. In this post I’ll discuss the talk/
paper “A Quantum Query Complexity Trichotomy for Regular Languages“, which is the work of Scott Aaronson, Daniel Grier (who gave the talk), and Luke Schaeffer.
Anybody who took courses on theoretical computer science will probably, at some point, have come across regular languages and their more powerful friends. They might even remember their organisation
into Chomsky’s hierarchy (see Figure 1). Roughly speaking, a (formal) language consists of an alphabet of letters, and a set of rules for generating words from those letters. With more complicated
rules come more expressive languages, for which the sets of words that can be generated from a given alphabet becomes larger. Intuitively, the languages in Chomsky’s hierarchy become more expressive
(i.e. larger) as we move upwards and outwards in Figure 1.
Figure 1: Chomsky’s hierarchy. Technically, the hierarchy contains formal grammars, where these grammars generate the corresponding formal languages.
One might view this hierarchy as an early attempt at computational complexity theory, where we are trying to answer the question “Given more complex rules (e.g. more powerful computational models),
what kinds of languages can we generate?”. Indeed, modern computational complexity theory is still defined in terms of languages: complexity classes are defined as the sets of the formal languages
that can be parsed by machines with certain computational powers.
However, there have been surprisingly few attempts to study the relationships between modern computational complexity and the kinds of formal languages discussed above. The recent work by Scott
Aaronson, Daniel Grier, and Luke Shaeffer on the quantum query complexity of regular languages takes a (large) step in this direction.
This work connects formal language theory to one of the great pillars of modern computational complexity – query complexity. Also known as the ‘black box model’, in this setting we only count the
number of times that we need to query (i.e. access) the input in order to carry out our computation. For many problems, the query complexity just ends up corresponding to the number of bits of the
input that need to be looked at in order to compute some function.
Why do we study query complexity? The best answer is probably that it allows us prove lower bounds – in contrast to the plethora of conjectures and implications that permeate the circuit complexity
world, it is possible to actually prove concrete query lower bounds for many problems, which, in turn, imply rigorous separations between various models of computation. For instance, the study of
quantum query complexity led to the proof of optimality of Grover’s algorithm, which gives a provable quadratic separation between classical and quantum computation for the unstructured search
In what follows, we can consider the query complexity of a regular language to correspond to the number of symbols of an input string $x$ that must be looked at in order to decide if $x$ belongs to a
given regular language. Here, a language is some (not necessarily finite) set of words that can be built from some alphabet of letters according to a set of rules. By changing the rules, we obtain a
different language.
We’ll start with some background on regular languages, and then discuss the results of the paper and the small amount of previous work that has investigated similar connections.
Background: Regular languages
Formally, a regular language over an alphabet $\Sigma$ is any language that can be constructed from:
• The empty language $\varnothing$.
• The empty string language $\{\epsilon\}$.
• The `singleton’ languages $\{a\}$ consisting of a single letter $a \in \Sigma$ from the alphabet.
• Any combination of the operators
□ Concatenation ($AB$): ‘$A$ followed by $B$‘.
□ Union ($A \cup B$): ‘either $A$ or $B$‘.
□ The Kleene star operation ($A^*$): ‘zero or more repetitions of $A$‘.
Readers with some coding experience will probably have encountered regular expressions. These are essentially explicit expressions for how to construct a regular language, where we usually write $|$
for union and omit some brackets. For example, to construct a regular language that recognises all words (over the Roman alphabet) that begin with ‘q’ and end with either ‘tum’ or ‘ta’, we could
write the regular expression $q\Sigma^*(tum|ta)$. In this case, the words ‘quantum’ and ‘quanta’ are contained in the regular language, but the words ‘classical’ and ‘quant’ are not. A more
computationally motivated example is the regular expression for the $\mathsf{OR}$ function over the binary alphabet $\Sigma = \{0,1\}$, which we can write as $\Sigma^*1\Sigma^*$ (i.e. ‘a 1 surrounded
by any number of any other characters’).
There are many other equivalent definitions of the regular languages. A particularly nice one is: “The set of languages accepted by deterministic finite state automata (DFAs)”. Here, we can roughly
think of a (deterministic) finite state automaton as a ‘Turing machine without a memory tape’, which are usually drawn diagrammatically, as in Figure 2. Other definitions include those in terms of
grammars (e.g. regular expressions, prefix grammars), or algebraic structures (e.g. recognition via monoids, rational series).
The decision problem associated with a regular language is: Given some string $x$ and a language $L$, is $x$ in $L$? Often, we want to make this decision by reading in as few bits of $x$ as possible
(hence the motivation to study the query complexity of regular languages).
Figure 2: DFA for the OR function. Each circle represents a ‘state’ of the machine, each arrow is a transition from one state to another, conditioned on taking as input a letter from the alphabet.
The double-lined state (circle on the right) represents an ‘accepting’ state.
Main results
The main result of the paper by Aaronson et al. is a trichotomy theorem, stated informally as follows:
Every regular language has quantum query complexity $\Theta(1), \tilde{\Theta}(\sqrt{n})$, or $\Theta(n)$. Furthermore, each query upper bound results from an explicit quantum algorithm.
The authors arrive at this result by showing that all regular languages naturally fall into 1 of 3 categories:
1. ‘Trivial’ languages: Intuitively, these are the languages for which membership can be decided by the first and last characters of the input string. For instance, the language describing all
binary representations of even numbers is a trivial language.
2. Star-free languages: The variant of regular languages where complement is allowed ($\overline A$ — i.e. `something not in $A$‘), but the Kleene star is not. Note that these languages can still be
infinite — indeed, the language $\overline\varnothing$ is equivalent to the language $\Sigma^*$. The quantum query complexity of these languages is $\tilde{\Theta}(\sqrt{n})$.
3. All the rest, which have quantum query complexity $\Theta(n)$.
The paper mostly describes these classes in terms of the algebraic definitions of regular languages (i.e. in terms of monoids), since these form the basis of many of the results, but for the sake of
simplicity, we will avoid talking about monoids in this post.
Along the way, the authors prove several more interesting results:
• Algebraic characterisation: They give a characterisation of each class of regular languages in terms of the monoids that recognise them. That is, the monoid is either a rectangular band,
aperiodic, or finite. In particular, given a description of the machine, grammar, etc. generating the language, it is possible to decide its membership in one of the three classes mentioned above
by explicitly calculating its ‘syntactic monoid’ and checking a small number of conditions. See Section 3 of their paper for details.
• Related complexity measures: Many of the lower bounds are derived from lower bounds on other query measures. They prove query dichotomies for deterministic complexity, randomised query
complexity, sensitivity, block sensitivity, and certificate complexity: they are all either $\Theta(1)$ or $\Theta(n)$ for regular languages. See Section 6 of their paper.
• Generalisation of Grover’s algorithm: The quantum algorithm using $\tilde{O}(\sqrt{n})$ queries for star-free regular languages extends to a variety of other settings by virtue of the fact that
the star-free languages enjoy a number of equivalent characterisations. In particular, the characterisation of star-free languages as sentences in first-order logic over the natural numbers with
the less-than relation shows that the algorithm for star-free languages is a nice generalisation of Grover’s algorithm. See Sections 1.3 and 4 of their paper for extra details and applications.
Finally, the authors show that the trichotomy breaks down for other formal languages. In fact, it breaks down as soon as we move to the ‘next level’ of the hierarchy, namely the context-free
The results in the paper allow the authors to link the query complexities to the more familiar (for some) setting of circuit complexity. By the characterisation of star-free languages in first-order
logic (in particular, McNaughton’s characterisation of star-free languages in first-order logic, which says that every star-free language can be expressed as a sentence in first-order logic over the
natural numbers with the less-than relation and predicates $\pi_a$ for $a \in \Sigma$, such that $\pi_a(i)$ is true if input symbol $x_i$ is $a$.), it follows that all star-free languages that have
quantum query complexity $\tilde{O}(\sqrt{n})$ are in $\mathsf{AC^0}$. Conversely, they show that regular languages not in $\mathsf{AC^0}$ have quantum query complexity $\Omega(n)$. Thus, another way
to state the trichotomy is that (very roughly speaking) regular languages in $\mathsf{NC^0}$ have complexity $O(1)$, regular languages in $\mathsf{AC^0}$ but not $\mathsf{NC^0}$ have complexity $\
tilde{\Theta}(\sqrt{n})$, and everything else has complexity $\Omega(n)$.
Previous work
There have been other attempts to connect the more modern aspects of complexity theory to regular languages, as the authors of this work point out. One example is the work of Tesson and Thérien on
the communication complexity of regular languages. They show that the communication complexity is $\Theta(1)$, $\Theta(\log \log n)$, $\Theta(\log n)$, or $\Theta(n)$ — a `tetrachotomy’ theorem with
parallels to the current work. It is interesting that Tesson and Thérien didn’t study the query complexity of regular languages, and instead went straight for the communication complexity setting,
since the former is the more obvious first step. Indeed, Aaronson et al. write
“communication complexity is traditionally more difficult than query complexity, yet the authors appear to have skipped over query complexity — we assume because quantum query complexity is
necessary to get an interesting result.”
Another example of previous work is the work of Alon, Krivelevich, Newman, and Szegedy, who consider regular languages in the property-testing framework. Here the task is to decide if an $n$-letter
input word $x$ is in a given language, or if at least $\epsilon n$ many positions of $x$ must be changed in order to create a word that is in the language. They show that regular languages can be
tested, in this sense, using only $\tilde{O}(1/\epsilon)$ queries. They also demonstrate that there exist context-free grammars which do not admit constant query property testers – showing that the
results once again break down once we leave the regular languages.
Aaronson et al. also point out some similarities to work of Childs and Kothari on the complexity of deciding minor-closed graph properties (the results are of the same flavour, but not obviously
related), and that a combination of two existing results – Chandra, Fortune, and Lipton and Bun, Kothari, and Thaler – allows one to show that the quantum query complexity of star-free languages is
The lower bounds are mostly derived from a (new) dichotomy theorem for sensitivity – i.e. that the sensitivity of a regular language is either $O(1)$ or $\Omega(n)$. The proof makes a nice use of the
pumping lemma for regular languages.
The majority of the work for the rest of the paper is focused on developing the quantum query algorithm for star-free languages. The proof is based on an algebraic characterisation of star-free
regular languages as ‘those languages recognised by finite aperiodic monoids’, due to Schützenberger, and the actual algorithm can be seen as a generalisation of Grover’s algorithm. I’ll leave the
exact details to the paper.
These two short paragraphs really don’t do justice to the number of different techniques that the paper combines to obtain its results. So I recommend checking out the paper for details!
Summary and open questions
This work takes a complete first step towards studying formal languages from the perspective of the more modern forms of computational complexity, namely query complexity. It very satisfyingly
answers the question “what is the query complexity of the regular languages?”. For classical computers, it’s either $\Theta(1)$ or $\Theta(n)$ – i.e. either trivial or difficult. For quantum
computers, it’s either $\Theta(1)$, $\Theta(n)$, or $\tilde{\Theta}(\sqrt{n})$ – i.e. trivial, difficult, or slightly easier than classical.
An obvious next step is to extend this work to other languages in the hierarchy, for example the context-free languages. However, the authors obtain what is essentially a no-go result in this
direction – they show that the trichotomy breaks down, and in particular for every $c \in [1/2, 1]$, there exists some context-free language with quantum query complexity approaching $\Theta(n^c)$.
They conjecture that no context-free language will have quantum query complexity that lies strictly between constant or $\sqrt{n}$, but leave this open.
Another direction is to consider promise problems: suppose we are promised that the input strings are taken from some specific set, does this affect the query complexity? It is known that in order to
obtain exponential separations between classical and quantum query complexity for (say) Boolean functions, we have to consider partial functions – i.e. functions with a promise on the input. For
instance, suppose we are promised that the input is ‘sorted’ (i.e. belongs to the regular language generated by $0^*1^*$). Then our task is to determine whether there is an occurrence of $01$ at an
even position (belongs to the language $(\Sigma\Sigma)^*01\Sigma^*$). As the authors point out, we can use binary search to decide membership in only $\Theta(\log n)$ quantum (and classical) queries,
so clearly complexities other than the three above are possible. Aaronson et al. conjecture that the trichotomy would remain, though, and that the quantum query complexity of regular languages with
promises on their inputs is one of $\Theta(\text{polylog}(n))$, $\Theta(\sqrt{n}\cdot\text{polylog}(n))$, or $\Theta(n\cdot\text{polylog}(n))$.
It’d also be nice to see if there are any other applications of the Grover-esque algorithm that the authors develop. Given that the algorithm is quite general, and that there are many alternative
characterisations of the star-free regular languages, it’d be surprising if there weren’t any immediate applications to other problems. The authors suggest that string matching problems could be an
appropriate setting, since linear-time classical algorithms for these problems have been derived from finite automata. Although quadratic quantum speedups are already known here, it could be a good
first step to obtain these speedups by just applying the new algorithm as a black box.
Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms (Harrow and Napp 2019)
With the growing popularity of hybrid quantum-classical algorithms as a way of potentially achieving a quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) devices, there were a number of
talks on this topic at this year’s QIP conference in Boulder, Colorado. One of the talks that I found the most interesting was given by John Napp from MIT on how “Low-depth gradient measurements can
improve convergence in variational hybrid quantum-classical algorithms” based on work done with Aram Harrow (https://arxiv.org/abs/1901.05374).
What are variational hybrid quantum-classical algorithms?
They are a class of optimisation algorithms in which the quantum and classical computer work closely together. Most variational algorithms follow a simple structure:
1. Prepare a parameterised quantum state $|\theta\rangle$ which can take the form $|\theta\rangle = |\theta_1,\hdots,\theta_p\rangle = e^{-iA_p\theta_p/2} \cdots e^{-iA_1\theta_1/2} |\psi\rangle$
where the $A_j$ are Hermitian operators. The type of parameters and operators depend on the device that is being targeted and $|\psi\rangle$ is an easy-to-prepare initial state.
2. Carry out measurements to determine information about the classical objective function $f(\theta)$ you wish to minimise (or maximise) where $f(\theta) = \langle\theta|H|\theta\rangle$ and $H$ is
some Hermitian observable (for example corresponding to a physical Hamiltonian). Due to the randomness of quantum measurements, many preparations and measurements of $|\theta\rangle$ are required
to obtain a good estimate of $f(\theta)$.
3. Use a classical optimisation method to determine a new value for $\theta$ that will minimise $f(\theta)$. This is a stochastic optimisation problem since we do not have direct access to $f(\
theta)$ – only noisy access through measurements.
4. Repeat steps 1-3 until the optimiser converges.
Examples of this type of algorithm are the variational quantum eigensolver (VQE) used to calculate ground states of Hamiltonians and the quantum approximate optimisation algorithm (QAOA) for
combinatoric optimisation problems.
Gradient measurements
To obtain information about the objective function $f(\theta)$, it can be expressed in terms of easily measurable operators:
$f(\theta) = \langle\theta|H|\theta\rangle = \sum_{i=1}^m \alpha_i\langle\theta|P_i|\theta\rangle$
where $P_i$ are tensor products of Pauli operators. Then to carry out the optimisation, derivative-free methods such as Nelder-Mead can be used. However, if one wishes to use derivative-based methods
such as BFGS or the conjugate gradient method, we need an estimate of the gradient $abla f(\theta)$. A numerical way to do this is by finite-differencing which only requires measurements of $f(\
theta)$, for small $\epsilon$,
$\frac{\partial f}{\partial \theta_i} \approx \frac{1}{2\epsilon}$$(f(\theta + \epsilon \hat{e}_i) - f(\theta - \epsilon \hat{e}_i))$
where $\hat{e}_i$ is the unit vector along the $i^{\text{th}}$ component. Each time $f$ is evaluated with different parameters, which can be done in low-depth, many repeat measurements are required.
An alternative method is to take measurements that correspond directly to estimating the gradient $abla f(\theta)$. This is referred to as an analytic gradient measurement and usually only requires a
slightly greater circuit depth than measuring the objective function. Previously it was not clear whether these analytic gradient measurements could offer an improvement over schemes that used
finite-differences or derivative-free methods, but as we will see, Harrow and Napp have proved in this paper that in some cases it can substantially improve convergence rates.
For the rest of this post, the term zeroth-order will refer to taking measurements corresponding to the objective function. First-order will refer to algorithms which make an analytic gradient
measurement (and this can generalise to kth-order where the kth derivatives are measured). It is clear how zeroth-order measurements are made – by measuring the Pauli operators $P_i$ with respect to
the state $|\theta\rangle$. But how do we make first-order measurements corresponding to the gradient?
The state $|\theta\rangle$ can be rewritten as $|\theta\rangle = U_{1:p}|\psi\rangle$ where the unitary $U_i = e^{-iA_p\theta_p/2}$ and for $i \leq j$ the sequence $U_{i:j}$ is defined as $U_j \hdots
U_i$. Therefore, $f(\theta) = \langle\theta|H|\theta\rangle = \langle\psi|U^\dagger_{1:p}HU_{1:p}|\psi\rangle$. This can be differentiated via the chain rule to find:
$\frac{\partial f}{\partial \theta_j} =$$-\text{Im} \langle\psi|U^\dagger_{1:j} A_j U^\dagger_{(j+1):p} HU_{1:p}|\psi\rangle.$
Recalling that $H = \sum_{l=1}^m \alpha_l P_l$ and writing the Pauli decomposition of $A_j$ as $A_j = \sum_{k=1}^{n_j} \beta_k^{(j)} Q_k^{(j)}$ where $Q_k^{(j)}$ are products of Pauli operators, the
derivative can be rewritten as
$\frac{\partial f}{\partial \theta_j} = -\sum_{k=1}^{n_j}\sum_{l=1}^m$$\beta_k^{(j)}\alpha_l \text{Im} \langle\psi|U^\dagger_{1:j} Q_k^{(j)} U^\dagger_{(j+1):p} P_l U_{1:p}|\psi\rangle.$
This can then be estimated using a general Hadamard test which is used to estimate real (or imaginary) parts of expected values. The circuit that yields an unbiased estimator for $-\text{Im} \langle\
psi|U^\dagger_{1:j} Q_k^{(j)} U^\dagger_{(j+1):p} P_l U_{1:p}|\psi\rangle$ is:
Circuit for the generalised Hadamard test explained in Algorithm 1 of the paper.
Measuring every term in the expansion is unnecessary to estimate $f(\theta)$ or its derivatives. So for all orders, a further sampling step is carried out, where terms in the expansion are sampled
from (using a strategy where terms with smaller norms are sampled from with smaller probability) and measured to determine a specific unbiased estimator, but I will not go into details here.
Black-box formulation
To quantify how complex an optimisation problem is, the function to be optimised $f(\theta)$ can be encoded in an oracle. The query complexity of the optimisation algorithm can then be defined as the
number of calls made to the oracle. This black-box formalism is typically used in the study of classical convex optimisation, here the quantum part of the variational algorithm has been placed into
the black box.
In this black-box model, the classical optimisation loop is given an oracle $\mathcal{O}_H$ encoding $H$. The optimiser is not given specifics about the problem, but it could be promised that the
objective function will have certain properties. The outer loop queries $\mathcal{O}_H$ with a state parameterisation $U_p \cdots U_1 |\psi\rangle$, parameters $\theta_1,\hdots,\theta_p$ and a set $S
= \{s_1,\hdots,s_k\}$ containing integers $\{1,\hdots,p\}$. The black box then prepares the state $|\theta\rangle$ and if $S = \varnothing$ performs a zeroth-order measurement estimating $f(\theta)$.
Otherwise a kth-order query is performed to estimate the derivative of $f(\theta)$ with respect to $\theta_{s_1},\hdots,\theta_{s_k}$. The query cost of this model is the number of Pauli operators
How many measurements are sufficient to converge to a local minimum?
Imagine now that we restrict to a convex region of the parameter space on which the objective function is also convex. We would like to know the upper bounds for the query complexity when optimising
$f(\theta)$ to precision $\epsilon$, or in other words how many measurements are required so that convergence to the minimum is guaranteed.
In the paper, results from classical convex optimisation theory are used to compare a zeroth-order algorithm with stochastic gradient descent (SGD) and stochastic mirror descent (SMD, a
generalisation of SGD to non-Euclidean spaces). For convex and $\lambda$-strongly convex functions, the upper bounds are shown to be:
Table 1 from the paper showing upper bounds for the query complexity.
Here $E$ and $\overrightarrow{\Gamma}$ are parameters related to the Pauli expansion of $H$ and $R_1, R_2, r_2$ are balls in the convex region we are optimising over. It is clear that SGD and SMD
will typically require less measurements to converge to the minimum compared to zeroth-order, but whether SGD outperforms SMD (or vice versa) depends on the problem at hand.
It is important to note that these are the best theoretical bounds, for some derivative-free algorithms (such as those based on trust regions) it can be hard to prove good upper bounds and guarantees
of convergence. However they can perform very well in practice and so zeroth-order could still potentially outperform SGD and SMD.
Can first-order improve over zeroth-order measurements?
To answer this question, a toy problem was studied. Consider a class of 1-local $n$-qubit Hamiltonians defined as the set $\mathcal{H}^\epsilon_n := \{H^{\delta(\epsilon)}_v : \forall v \in \{-1,1\}^
n\}$ where $\delta(\epsilon) = \sqrt{\frac{45\epsilon}{n}}$ and
$H^\delta_v = -\sum_{i=1}^n \left[\text{sin}\left(\frac{\pi}{4} + v_i \delta\right)X_i + \text{cos}\left(\frac{\pi}{4} + v_i\delta\right)Z_i \right].$
These $H^\delta_v$ are perturbations about $H^0 = -\frac{1}{\sqrt{2}} \sum_{i=1}^n (X_i + Z_i)$ where $\delta$ is the strength and $v$ the direction of the perturbation. We wish to know how many
measurements are needed to reach the ground state of $H^\delta_v \,\, \forall v$. This problem is trivial (the lowest eigenvalue and it’s associated eigenvector can be written down directly) which is
why the black-box formulation is necessary to hide the problem. The resulting upper and lower query complexity bounds for optimising the family $\mathcal{H}^\epsilon_n$ is found to be:
Table 2 from the paper showing how first-order can improve over any zeroth-order algorithm for this toy problem.
The proof of the lower bound is too complicated to explain here. Proving the upper bound, in particular for first-order, is simpler and relies on using a good parameterisation
$|\theta\rangle = \left(\otimes_{j=1}^n e^{-i(\theta_j + \pi/4)Y_j/2} \right)|0\rangle^{\otimes n}$
in our optimisation algorithm. $|\theta\rangle$ is a natural choice for the family $\mathcal{H}^\epsilon_n$ as each qubit is polarised in the $\hat{x}-\hat{z}$ plane. The corresponding objective
function is then $f(\theta) = -\sum_{i=1}^n \text{cos}(\theta_i - v_i\delta)$ which is strongly convex near the optimum and so stochastic gradient descent performs well here. Note that making
higher-order queries is unnecessary in this case as the optimal bounds can be achieved with just first-order.
Ultimately Harrow and Napp have shown that there are cases in which taking analytic gradient measurements in variational algorithms for use in stochastic gradient/mirror descent optimisation routines
(compared to derivative-free methods) could help with convergence. It would be interesting to see what happens with more complicated problems and if more general kth-order measurements will provide
benefits over first-order. Another extension that is mentioned in the paper is to see what the impact of noisy qubits and gates is on the convergence of the optimisation problem.
I personally am most eager to see how these results will hold up in practice. For example, it would be interesting to see a simple simulation performed for the toy problem comparing zeroth-order
optimisers with those that take advantage of analytic gradient measurements.
A Converse to the Polynomial Method (Arunachalam et al. 2017)
One of the most famous models for quantum computation is the black-box model, in which one is given a black-box called oracle encoding a unitary operation. With this oracle one can probe the bits of
an unknown bit string $x \in\{-1,1\}^n$ (the paper uses the set $\{-1,1\}$ instead of $\{0,1\}$, which I will keep here) . The main aim of the model is to use the oracle to learn some property given
by a Boolean function $f$ of the bit string $x$, for example, its parity. One use of the oracle is usually referred as a query, and the number of queries a certain algorithm performs reflects its
efficiency. Quite obviously, we want to minimize the number of queries. In more precise words, the bounded-error quantum query complexity of a Boolean function $f : \{-1,1\}^n \to \{-1,1\}$ is
denoted by $Q_\epsilon (f)$ and refers to the minimum number of queries a quantum algorithm must make on the worst-case input x in order to compute $f(x)$ with probability $1 - \epsilon$.
Looking at the model and the definition for the quantum query complexity, one might ask “but how can we find the query complexity of a certain function? Is there an easy way to do so?”. As usual in
this kind of problem, finding the optimal performance of an algorithm for a certain problem or an useful lower bound for it is easier said than done. Nonetheless, there are some methods for tackling
the problem of determining $Q_\epsilon (f)$. There are two main methods for proving lower bounds, known as the polynomial method and the adversary method. In this post we shall talk about the first,
the polynomial method, and how it was improved in the work of Arunachalam et al.
The polynomial method
The polynomial method is a lower bound method based on a connection between quantum query algorithms and, as the name suggests, polynomials. The connection comes from the fact that for every t-query
quantum algorithm $A$ that returns a random sign $A(x)$ (i.e. $\{-1,1\}$) on input $x$, there exists a degree–$2t$ polynomial $p$ such that $p(x)$ equals the expected value of $A(x)$. From this it
follows that if $A$ computes a Boolean function $f : \{-1,1\}^n \to \{-1,1\}$ with probability at least $1 - \epsilon$, then the polynomial $p$ satisfies $|p(x) - f(x)| \leq 2\epsilon$ for every $x$
(the factor of $2$ comes from the image being $\{-1,1\}$ instead of $\{0,1\}$). Therefore we can see that the minimum degree of a polynomial $p$ that satisfies $|p(x) - f(x)| \leq 2\epsilon$ for
every $x$, called the approximate (polynomial) degree and denoted by $deg_\epsilon(f)$, serves as a lower bound for the query complexity $Q_\epsilon (f)$. Hence the problem of finding a lower bound
for the query complexity is converted into the problem of lower bounding the degree of such polynomials.
Converse to the polynomial method
We now have a method for proving lower bounds for quantum query algorithms by using polynomials. A natural question that can arise is whether the polynomial method has a converse, that is, if a
degree-$2t$ polynomial leads to a $t$–query quantum algorithm. This would in turn imply a sufficient characterization of quantum query algorithms. Unfortunately, Ambainis showed in 2006 that this is
not the case, by proving that for infinitely many $n$, there is a function $f$ with $deg_\epsilon(f) \leq n^\alpha$ and $Q_\epsilon (f) \geq n^\beta$ for some positive constants $\beta > \alpha$.
Hence the approximate degree is not such a precise measure for quantum query complexity in most cases.
In the view of these negative results, the question that stays is, is there some refinement to the approximate polynomial that approximates $Q_\epsilon(f)$ up to a constant factor? Aaronson et al.
tried to answer this question around 2016 by introducing a refined degree measure, called block-multilinear approximate polynomial degree and denoted by $bm\text{-}deg_\epsilon(f)$, which comes from
polynomials with a so-called block-multilinear structure. This refined degree lies between $deg_\epsilon(f)$ and $2Q_\epsilon(f)$, which leads to the question of how well that approximates $Q_\
epsilon(f)$. Once again, it was later shown that for infinitely many $n$, there is a function $f$ with $bm\text{-}deg_\epsilon(f) = O(\sqrt{n})$ and $Q_\epsilon(f) = \Omega(n)$, ruling out the
converse for the polynomial method based on the degree $bm\text{-}deg_\epsilon(f)$ and leaving the question open until now, when it was answered by Arunachalam et al., who gave a new notion of
polynomial degree that tightly characterizes quantum query complexity.
Characterization of quantum algorithms
In few words, it turns out that $t$-query quantum algorithms can be fully characterized using the polynomial method if we restrict the set of degree-$2t$ polynomials to forms that are completely
bounded. A form is a homogeneous polynomial, that is, a polynomial whose non-zero terms all have the same degree, e.g. $x^2 + 3xy + 2y^2$ is a form of degree $2$. And the notion of completely bounded
involves the idea of a very specific norm, the completely bounded norm (denoted by $\|\cdot\|_{cb}$), which was originally introduced in the general context of tensor products of operator spaces. But
before we venture ourselves into this norm, which involves symmetric tensors and other norms, let us state the main result of the quantum query algorithms characterization.
Let $\beta: \{-1,1\}^n \to [-1,1]$ and $t$ a positive integer. Then, the following are equivalent.
1. There exists a form $p$ of degree $2t$ such that $\|p\|_{cb} \leq 1$ and $p((x,\boldsymbol{1})) = \beta(x)$for every $x \in \{-1,1\}^n$, where $\boldsymbol{1}$ is the all-ones vector.
2. There exists a $t$-query quantum algorithm that returns a random sign with expected value $\beta(x)$ on input $x \in \{-1,1\}^n$.
In short, if we find a form of degree $2t$ which is completely bounded ($\|p\|_{cb} \leq 1$) and approximates a function $f$that we are trying to solve, then there is a quantum algorithm which makes
$t$ queries and solves the function. Hence we have a characterization of quantum algorithms in terms of forms that are completely bounded. But we still haven’t talked about the norm itself, which we
should do now. It will involve a lot of definitions, some extra norms and a bit of C*-algebra, but fear not, we will go slowly.
The completely bounded norm
For $\alpha \in \{0,1,2, \dotsc \}^{2n}$, we write $|\alpha| = \alpha_1 + \dotsc + \alpha_{2n}$. Any form $p$of degree $t$ can be written as
$p(x) = \sum_{|\alpha| = t} c_{\alpha} x^\alpha$,
where $c_{\alpha}$ are real coefficients. The first step towards the completely bounded norm of a form is to define the completely bounded norm of a tensor, and the tensor we use is the symmetric $t$
-tensor $T_p$ defined as
$(T_p)_\alpha = c_\alpha/|\{\alpha\}|!$
where $|\{\alpha\}|!$denotes the number of distinct elements in the set formed by the coordinates of $\alpha$.
The relevant norm of $T_p$is given in terms of an infimum over decompositions of the form $T_p = \sum_\sigma T^\sigma \circ\sigma$, where $\sigma$ is a permutation of the set $\{1,\dotsc,t\}$ and $(T
^\sigma\circ\sigma)_\alpha = T^\sigma_{\sigma(\alpha)}$ is the permuted element of the multilinear form $T^\sigma$. So that the completely bounded norm of $p$ is kind of transferred to $T_p$via the
$\|p\|_{cb} = \text{inf }\{\sum_\sigma \|T^\sigma\|_{cb} : T_p = \sum_\sigma T^\sigma\circ\sigma\}$.
Just to recap, with the coefficients of the polynomial $p$ we define the symmetric $t$-tensor $T_p$, which is then decomposed into the sum of permuted tensor (multilinear form) $T^\sigma$. We then
define the completely bounded norm of $p$as the infimum of the sum of the completely bounded norm of such tensor, but now without permuting it. Of course, we haven’t yet defined the completely
bounded norm of such tensor, that is, what is $\|T^\sigma\|_{cb}$? We will explain it now.
The idea is to get a bunch of collections of $d \times d$unitary matrices $U_1(i), U_2(i), \dotsc, U_t(i)$ for $i \in \{1, \dotsc ,2n\}$ and consider the quantity
$\|\sum_{i,j,\dotsc,k} T^\sigma_{i,j,\dotsc,k} U_1(i)U_2(j)\dotsc U_t(k)\|$.
We multiply the unitaries from these collections and sum them using the tensor $T^\sigma$ as weight, and then take the norm of the resulting quantity. But here we are using a different norm, the
usual operator norm defined for a given operator $A$ as $\|A\| = \text{inf }\{c\geq 0 : \|Av\| \leq c\|v\| \text{ for all vectors } v\}$. Finally, with these ingredients in hand, we can define the
completely bounded norm for $T^\sigma$, which is just the supremum over the positive integer $d$ and the unitary matrices, that is,
$\|T^\sigma\|_{cb} = \text{sup }\{\|\sum_{i,j,\dotsc,k} T^\sigma_{i,j,\dotsc,k} U_1(i)U_2(j)\dotsc U_t(k)\| : d \in \mathbb{N}, d \times d \text{ unitary matrices } U_i\}$.
If we can obtain the supremum of such norm over the size of the unitary matrices and the unitary matrices themselves, then we obtain the completely bounded norm of the tensor $T^\sigma$, and from
this we get the completely bounded norm of the associated form $p$. Is there such a degree-$2t$ form with $\|p\|_{cb} \leq 1$ that approximates the function $f$ we want to solve? If yes, then there
is a $t$-query quantum algorithm that solves $f$.
The proof
Let us briefly explain the proof of the quantum algorithms characterization that we stated above. Their proof involves three main ingredients. The first one is a theorem by Christensen and Sinclair
showing that the completely-boundedness of a multilinear form is equivalent to a nice decomposition of such multilinear form. In other words, for a multilinear form $T$, we have that $\|T\|_{cb} \leq
1$ if and only if we can write $T$ as
$T(x_1,\dotsc,x_t) = V_1\pi(x_1)V_2\pi(x_2)V_3\dotsc V_t\pi(x_t)V_{t+1}$, (1)
where $V_i$ are contractions ($\|V_ix\| \leq \|x\|$ for every $x$) and $\pi_i$ are *-representations (linear maps that preserve multiplication operations, $\pi(xy) = \pi(x)\pi(y)$).
The second ingredient gives an upper bound on the completely bounded norm of a certain linear map if it has a specific form. More specifically, if $\sigma$ is a linear map such that $\sigma(x) = U\pi
(x)V$, where $U$and $V$ are also linear maps and $\pi$ is a *-representation, then $\|\sigma\|_{cb} \leq \|U\|\|V\|$.
The third ingredient is the famous Fundamental Factorization Theorem that “factorizes” a linear map in terms of other linear maps if its completely bounded norm is upper bounded by these linear maps.
In other words, if $\sigma$ is a linear map and exists other linear maps $U$ and $V$ such that $\|U\|\|V\| \leq \|\sigma\|_{cb}$, then, for every matrix $M$, we have $\sigma(M) = U^\ast(M\otimes I)V$
With these ingredients, they proved an alternative decomposition of equation (1) which was later used to come up with a quantum circuit implementing the tensor $T_p$and using $t$ queries, and then
showed that this tensor $T_p$ matched the initial form $p$. This alternative decomposition is
$T(x_1,\dotsc,x_t) = u^\ast U_1^\ast (\text{Diag}(x_1)\otimes I)V_1 \dotsc U_t^\ast (\text{Diag}(x_t)\otimes I)V_tv$, (2)
where $U_i, V_i$ are contractions, $u,v$ are unit vectors, $\text{Diag}(x)$ is the diagonal matrix whose diagonal forms $x$. The above decomposition is valid if, similarly to (1), $\|T\|_{cb} \leq 1$
We can see that the decomposition involves the operator $\text{Diag}(x)$ intercalated by unitaries $t$times. Having $\text{Diag}(x)$ as the query operator, it is possible then to come up with a
quantum circuit implementing the decomposition (2). If we look more closely, decomposition (2) has unit vectors on both the left and right sides, which looks like an expectation value. So what is
going on is that decomposition (2) can be used to construct a quantum circuit whose expectation value matches $T_p$, and hence the polynomial $p$ used to construct $T_p$. We won’t get into much
details, but we will leave the quantum circuit so that the reader can have a glimpse of what is going on.
In the above figure representing a quantum circuit that has $T_p$ as its expectation value, the registers C, Q, W denote control, query and workspace registers. The unitaries $W_i$ are defined by
$W_i = V_{i-1}U_i^\ast$, the unitaries $U, V$ have $W_1V_0 u$ and $W_{2t+1}U_{2t+1}v$as their first rows, where $V_0$ and $U_{2t+1}$ are isometries ($A$ is an isometry iff $\|Ax\| = \|x\|$ for every
Application of the characterization: separation for quartic polynomials
A couple of years ago, in 2016, Aaronson et al. showed that the bounded norm (and also the completely bounded norm that we spent some time describing) is sufficient to characterize quadratic
polynomials. More specifically, they showed that for every bounded quadratic polynomial $p$, there exists a one-query quantum algorithm that returns a random sign with expectation $Cp(x)$ on input
$x$, where $C$ is an absolute constant. This readily prompted the question of whether the same is valid for higher-degree polynomials, that is, if the bounded norm suffices to characterize quantum
query algorithms.
As you might expect by now, the answer is no, since it is the completely bounded norm that suffices, and Arunachalam et al. used their characterization to give a counterexample for bounded quartic
polynomials. What they showed is the existence of a bounded (not completely bounded) quartic polynomial p that for any two-query quantum algorithm whose expectation value is $Cp(x)$, one must have $C
= O(n^{-1/2})$, thus showing that $C$ is not an absolute constant. The way they showed this is by using a random cubic form that is bounded, but whose completely bounded norm is $poly(n)$ and then
embedded this form into a quartic form.
The characterization they found is, in my opinion, apart from the obvious point that it answers the open question of the converse of the polynomial method, quite interesting in the sense that the set
of polynomials we should look at is the ones that are bounded by a particular norm, the completely bounded norm. The interesting point is exactly this one, the connection between quantum query
algorithms and the completely bounded norm that was first introduced in the general context of tensor products of operator spaces as mentioned before. The norm itself looks quite exotic and
complicated, which is linked to a question that Scott Aaronson made at the end of Arunachalam’s talk in QIP 2019, “sure, we could define a particular norm as one that restricts the set of polynomials
to the ones that admit a converse to the polynomial method”. Of course, such a norm would not be quite an interesting one. He carries on with something like “In what is this completely bounded norm
different compared to that?”. If I remember correctly, Arunachalam gave an answer on the lines of the completely bounded norm appearing in a completely different context. But I still find it
surprising that you would go through all the exotic definitions we mentioned above for the completely bounded norm and discover that it is what is needed for the converse of the polynomial method.
Hello quantum world!
Welcome to the blog for the Quantum Information Theory group at the University of Bristol – a home for occasional posts by group members. To get things going, some members of the group will post
about talks they attended at the recent QIP 2019 conference. | {"url":"https://qitheory.blogs.bristol.ac.uk/","timestamp":"2024-11-12T21:46:58Z","content_type":"text/html","content_length":"184558","record_id":"<urn:uuid:fc943261-c224-4d84-bbed-4fdd0544c82f>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00558.warc.gz"} |
Geodesic versus planar distance
Available with Image Server
The distance tools include an option to perform calculations using planar distance or geodesic distance. Planar distance is straight-line Euclidean distance calculated in a 2D Cartesian coordinate
system. Geodesic distance is calculated in a 3D spherical space as the distance across the curved surface of the world.
Distance analysis and map projections
The earth is a three-dimensional slightly flattened sphere, or ellipsoid. To represent the earth on maps, cartographers created map projections, which are mathematical transformations between 3D and
2D space. These projections distort the data in different ways, affecting measurement of distances, angles, and areas. Many projections have been developed in attempts to preserve one or more of
these characteristics, often for particular parts of the map such as specific meridians or parallels, or a few points. Although some map projections preserve some characteristics accurately, none
preserve all distances correctly. Projections have been developed that preserve true direction, but none fully preserve both distance and direction.
For the distance tools to calculate distance and direction accurately in any direction between any locations geodesic distance must be used.
Differences between geodesic and planar distance
The default behavior of the tools is to use planar distance for backward compatibility with previous versions that do not include a geodesic option, and because it is faster to run. Geodesic distance
always produces a more accurate result and is the recommended method, unless speed is more important than accuracy.
The difference between planar and geodesic calculations of distance and direction varies with map projection and size of the study area. For example, when comparing distances computed in most
cylindrical projections such as Mercator or Web Mercator to geodesic distance, at low latitudes near the equator the difference is small, and the difference increases as you move toward the poles.
The distance between two equatorial cities, Singapore to Nairobi, is approximately 7,440 kilometers, and the planar distance in Web Mercator computes to less than a meter farther. For a higher
latitude example, the geodesic distance from Reykjavik to Moscow is approximately 3,310 kilometers, and the Web Mercator planar distance is 6,890 kilometers. The planar distance is distorted to
almost double the true geodesic distance.
Using an azimuthal projection on the pole, such as Universal Polar Stereographic, distance and direction more closely approximates the geodesic calculation near the poles and is progressively
distorted for calculations closer to the equator.
Projection distortion is typically smallest near the origin or center of the map projection, and distortion typically increases farther away.
A comparison of 1,000 kilometer distance buffers from selected cities uses Web Mercator planar distance (purple) and geodesic distance (blue). The distance distortion is progressively larger for
cities farther away from the equator.
The difference between planar and geodesic distance increases proportionally with distance from the source. If you are working in a small geography, such as a city or county, the difference between
planar and geodesic is proportionally smaller than if you are working at the scale of a country. The impact of size of study area and distortions of the map project can work together to further
increase distortion. For a projection such as Web Mercator, the closer toward the poles you go, the smaller area you can analyze with the same distance distortion.
To understand the specific differences in distance measurements between geodesic and planar, use the Measure tool to measure the distance between two locations in Planar mode and in Geodesic mode. | {"url":"https://enterprise.arcgis.com/en/portal/latest/use/geodesic-versus-planar-distance.htm","timestamp":"2024-11-14T13:27:48Z","content_type":"text/html","content_length":"43112","record_id":"<urn:uuid:bfd4c7e1-7786-47a4-8f65-b97a740bc176>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00535.warc.gz"} |
Inspiring Drawing Tutorials
Drawing Disks On A Place Value Chart
Drawing Disks On A Place Value Chart - Draw place value disks on the place value chart to solve. How many weeks are in one year? Fill in the blanks to make the f ollowing equations true. As you did
during the lesson, label and represent the product or quotient by drawing disks on the place value chart. Fill in the blanks to make the following equations true. Fill in the blanks to make the
following equations true.
Fill in the blanks to make the following equations true. 26 + 35 = 61. (draw 6 ones disks and divide them into 2 groups of 3.) t: 5.241 + 3 = ones tenths hundredths thousandths 3 5. 26 + 35 = _____
Draw place value disks to represent the following problems. 26 + 35 = _____ answer: 5.241 + 3 = ones tenths hundredths thousandths 3 5. 6 ÷ 2 = 3. Shop our huge selectionfast shippingdeals of the
dayshop best sellers
10 × 2 thousands = _________. Show each step using the standard algorithm. Draw place value disks to represent the following problems. Label the place value charts. Draw place value disks on the
place value chart to solve.
Label the place value charts. (s) personal white boards, place value charts. Draw disks in the place value chart to show how you got your answer, using arrows to show any. 10 × 2 thousands =
_________. As you did during the lesson, label and represent the product or quotient by drawing disks on the place value chart.
6 ÷ 2 = 3. Label the place value charts. Draw place value disks on the place value chart to solve. Represent 2 × 23 with disks, writing a matching equation and recording the partial products
vertically. Place value, rounding, and algorithms for addition and subtraction.
Fill in the blanks to make the following equations true. 26 + 35 = 61. 5.372÷ 2 = _____ 2 5.372 5.241 + 3 = ones tenths hundredths thousandths 3 5. Eureka math grade 4 module 3 lesson 6 problem set
answer key.
Draw place value disks on the place value chart to solve. Keep reading for 3 ways to use them to support conceptual understanding. Draw disks in the place value chart to show how you got your answer,
using arrows to show any. Draw and bundle place value disks on the place value chart. 6 ÷ 2 = ____3____ 6 ones.
Fill in the blanks to make the following equations true. Fill in the blanks to make the f ollowing equations true. Show each step using the standard algorithm. Understanding can be developed first
through work at the concrete level with place value disks, then at the pictorial level by drawing place value disks on a place value chart,. (s) personal.
Shop our huge selectionfast shippingdeals of the dayshop best sellers Draw disks in the place value chart to show how you got your answer, using arrows to show any. On your personal white boards,
draw place value disks to represent the expression. Keep reading for 3 ways to use them to support conceptual understanding. Represent 2 × 23 with disks,.
Fill in the blanks to make the following equations true. Show each step using the standard algorithm. Nys math grade 4, module 3, lesson 28 problem set. Draw disks in the place value chart to show
how you got your answer, using arrows to show any. Rewrite each in unit form and solve.
Shop our huge selectionfast shippingdeals of the dayshop best sellers Represent 2 × 23 with disks, writing a matching equation and recording the partial products vertically. 6 ÷ 2 = 3. 5.372÷ 2 =
_____ 2 5.372 Draw and bundle place value disks on the place value chart.
Let’s take a minute to get to know these great manipulatives. Show each step using the standard algorithm. Shop our huge selectionfast shippingdeals of the dayshop best sellers Label the place value
charts. 5.372÷ 2 = _____ 2 5.372
Drawing Disks On A Place Value Chart - Show each step using the standard algorithm. Label the place value charts. Do you use anchor charts for place value? The number 12 can be thought. Place value,
rounding, and algorithms for addition and subtraction. Let’s take a minute to get to know these great manipulatives. Draw and bundle place value disks on the place value chart. Drawing and marking
out these disks allows them to model. Label the place value charts. 5.372÷ 2 = _____ 2 5.372
Draw disks in the place value chart to show how you got your answer, using arrows to show any. As you did during the lesson, label and represent the product or quotient by drawing disks on the place
value chart. 26 + 35 = _____ answer: Let’s take a minute to get to know these great manipulatives. How many weeks are in one year?
Label the place value charts. 5.241 + 3 = ones tenths hundredths thousandths 3 5. Draw disks in the place value chart to show how you got your answer, using arrows to show any. Do you use anchor
charts for place value?
Eureka math grade 4 module 3 lesson 6 problem set answer key. Represent the following problem by drawing disks in the place value chart. Label the place value charts.
Understanding can be developed first through work at the concrete level with place value disks, then at the pictorial level by drawing place value disks on a place value chart,. Let’s take a minute
to get to know these great manipulatives. Nys math grade 4, module 3, lesson 28 problem set.
The Number 12 Can Be Thought.
26 + 35 = 61. Label the place value charts. 5.372÷ 2 = _____ 2 5.372 Label the place value charts.
Represent 2 × 23 With Disks, Writing A Matching Equation And Recording The Partial Products Vertically.
Draw disks in the place value chart to show how you got your answer, using arrows to show any. (draw 6 ones disks and divide them into 2 groups of 3.) t: 26 + 35 = _____ answer: Fill in the blanks to
make the following equations true.
Let’s Take A Minute To Get To Know These Great Manipulatives.
As you did during the lesson, label and represent the product or quotient by drawing disks on the place value chart. (s) personal white boards, place value charts. Combine 10 ones disks to create 1
tens disks, and drag the tens disk to the tens column. Place value, rounding, and algorithms for addition and subtraction.
Using A Place Value Chart, Put 12 Ones Disks Into The Ones Column.
Draw disks in the place value chart to show how you got your answer, using arrows to show any. Draw place value disks to represent the following problems. Draw place value disks on the place value
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Rate equations - (Quantum Optics) - Vocab, Definition, Explanations | Fiveable
Rate equations
from class:
Quantum Optics
Rate equations describe the change in population of excited and ground state atoms or molecules over time, particularly in the context of spontaneous and stimulated emission processes. These
equations provide a mathematical framework to understand how the rates of these emissions are influenced by factors such as the density of excited states and the presence of photons. They play a
crucial role in determining the behavior of lasers and other optical systems, linking the kinetics of emissions to physical characteristics like gain and loss within a medium.
congrats on reading the definition of Rate equations. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Rate equations typically consist of differential equations that represent the change in populations of energy levels over time due to various processes like spontaneous and stimulated emission.
2. The equations allow for the prediction of laser performance by relating the output power to the input parameters such as pump rate and material characteristics.
3. In rate equations, spontaneous emission is often treated as a noise source, while stimulated emission contributes directly to the amplification of light in lasers.
4. The concept of saturation can be introduced in rate equations, indicating when an increase in pumping does not lead to a proportional increase in output power due to population inversion limits.
5. Rate equations can also model the time dynamics of laser operation, showing how quickly a laser reaches steady-state output after being switched on.
Review Questions
• How do rate equations help understand the transition from spontaneous emission to stimulated emission in lasers?
□ Rate equations provide a quantitative way to analyze how the populations of excited and ground states evolve over time. By modeling both spontaneous and stimulated emissions, these equations
show that while spontaneous emission occurs randomly, stimulated emission is dependent on the presence of photons. This transition becomes critical for laser operation, where achieving a
balance between these two processes allows for efficient amplification of light.
• What role does population inversion play in the rate equations for a laser system, and why is it crucial for achieving lasing?
□ Population inversion is a key concept represented in rate equations that is essential for lasing. When more atoms are in an excited state than in the ground state, stimulated emission
dominates over absorption, allowing for coherent light amplification. The rate equations illustrate how this condition must be maintained through processes like optical pumping, ensuring that
the net gain exceeds losses within the laser medium.
• Evaluate how changes in pump rate affect the behavior described by rate equations and their implications for laser performance.
□ Changes in pump rate directly influence the dynamics represented in rate equations by altering the rate at which population inversion is achieved. A higher pump rate increases excited state
population, enhancing stimulated emission and thus output power. However, if too high, it may lead to saturation effects where further increases do not significantly boost performance. This
balance highlighted by rate equations is critical for optimizing laser design and ensuring stable operation under various conditions.
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3, 5 and ? – Find which number replace the question mark? Genius Brainteaser Puzzle
Home Puzzles Brain Teasers 3, 5 and ? – Find which number replace the question mark?...
3, 5 and ? – Find which number replace the question mark? Genius Brainteaser Puzzle
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Find the logic and solve the puzzle. Which number replace the question mark? Math Puzzles only for geniuses!
Hello and welcome back, It’s puzzle time, here is an interesting and confusing math puzzle for you. Solve this tricky math puzzle and share your answer.
There are three triangles as you can see, you have to find out the logic of these number in these triangles and solve the last one. find the number which replaces the question mark and comment your
answer below.
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New functionalities of the 3.4 version of
The page describes the new functionalities of Version 3.4 of the TFEL project.
Some noticeable applications of MFront and MGIS in 2020
Figure 1: Some noticeable applications of MFront and MGIS in 2020
Figure 1 presents some noticeable applications of MFront:
a. Normalised residual topography after an indentation test on a single crsytal of copper with Méric-Cailletaud’ finite stain behaviour [1] using Ansys. Contribution of A. Bourceret and A. Lejeune,
b. Slope failure analysis with strength reduction in OpenGeoSys [2]. Contribution of T. Deng and T. Nagel (Geotechnical Institute, Technische Universität Bergakademie Freiberg).
c. Integration of the MGIS integration in Europlexus. Contribution of P. Bouda, (CEA DM2S).
d. Simulation of rolling using the innovative CEA’ proto-application MEFISTO. Contribution of O. Jamond, (CEA DM2S).
e. Industrial thermomechanical design of a cylinder block with an with MFront and Abaqus at Groupe PSA. This study is one result of the PhD thesis of L. Jacquinot which provides a continuous
modelling of the AlSi9Cu3Mg aluminium alloy behaviour from manufacturing to final usage Contribution of A. Forré (Groupe PSA).
f. Column collapse using the Material Point Method. Contribution of Ning Guo, Wenlong Li (Zhejiang University)
Figure 2: a) Cast3M simulation of a Notched Tensile sample of an AA6061-T6 found in the core of Jules Horowitz Reactor and comparison of simulation result with experimental data. b) Cast3M simulation
of a Charpy test on the PWR reactor core vessel’ steel
The major features of Version 3.4 are:
• A better support generalised behaviours, in particular regarding the computation of the consistent tangent operator in the implicit schemes [3]. Examples of this features are presented in
references [4–7].
• An extension of the StandardElastoViscoPlasticity brick to porous materials See Figure 2 for some examples of ductile failure simulations with Cast3M.
• The ability to store MFront behaviours in a madnex file. madnex is a data model based on HDF5 file format that was originally designed by EDF as part of their in-house projects for capitalising
their experimental data and which is now being shared among the main actors of the french nuclear industry with the aim of becoming the de facto standard to exchange experimental data, MFront
implementations and unit tests of those implementations. Documentation is available here: http://tfel.sourceforge.net/madnex.html.
• The MFront’ generic interface now exports functions to rotate gradients in the material frame before the behaviour integration and rotate the thermodynamics forces and the tangent operator blocks
in the global frame after the behaviour integration. Such functions are particularly useful for generalised behaviours.
A special effort has been set on the documentation with many new tutorials [[3];[4];[5];[8];[6];[9];[10];[11];@;[7]].
In order to increase the community of developers, a first tutorial showing how a new stress criteria can be added to the StandardElastoViscoPlasticity brick has been published [12]. Other similar
tutorials are being considered.
Simultaneous releases This version has been released along with:
Those releases are mainly related to bug-fixes. Version 3.4 inherits from all the fixes detailed in the associated release notes.
Known incompatibilities
In previous versions, getPartialJacobianInvert was implemented as a method.
This may broke the code, in the extremely unlikely case, that the call to getPartialJacobianInvert was explicitly qualified as a method, i.e. if it was preceded by the this pointer. Hence, the
following statement is not more valid:
To the best of our knowledge, no implementation is affected by this incompatibility.
Declaration of the offsets of the integration variables in implicit schemes
The offsets of the integration variables in implicit schemes are now automatically declared in the @Integrator code block. The names of the variables associated with those offsets may conflict with
user defined variables. See Section 4.4.1 for a description of this new feature.
To the best of our knowledge, no implementation is affected by this incompatibility.
Noticeable fixed issues that may affect the results obtained with previous versions
Ticket #256 reported that the scalar product of two unsymmetric tensors was not properly computed.
This may affect single crystal finite strain computations to a limited extent, as the Mandel stress tensor is almost symmetric.
This page describes how to extend the TFEL/Material library and the StandardElastoViscoPlasticity brick with a new stress criterion.
New features of the TFEL libraries
New features of the TFEL/Math library
Solving multiple linear systems with TinyMatrixSolve
tfel::math::tmatrix<4, 4, T> m = {0, 2, 0, 1, //
2, 2, 3, 2, //
4, -3, 0, 1, //
6, 1, -6, -5};
tfel::math::tmatrix<4, 2, T> b = {0, 0, //
-2, -12, //
-7, -42, //
6, 36};
tfel::math::TinyMatrixSolve<4u, T>::exe(m, b);
The DerivativeType metafunction allows requiring the type of the derivative of a mathematical object with respect to another object. This metafunction handles units.
For example:
declares the variable de_dt as the derivative of the a strain tensor with respect to scalare which has the unit of at time.
The derivative_type alias allows a more concise declaration:
In MFront code blocks, the StrainRateStensor typedef is automatically defined, so the previous declaration is equivalent to:
The derivative_type is much more general and can be always be used.
Scalar Newton-Raphson algorithm
The function scalarNewtonRaphson, declared in the TFEL/Math/ScalarNewtonRaphson.hxx is a generic implementation of the Newton-Raphson algorithm for scalar non linear equations. The Newton algorithm
is coupled with bisection whenever root-bracketing is possible, which considerably increase its robustness.
This implementation handles properly IEEE754 exceptional cases (infinite numbers, NaN values), even if advanced compilers options are used (such as -ffast-math under gcc).
// this lambda takes the current estimate of the root and returns
// a tuple containing the value of the function whose root is searched
// and its derivative.
auto fdf = [](const double x) {
return std::make_tuple(x * x - 13, 2 * x);
// this lambda returns a boolean stating if the algorithm has converged
// the first argument is the value of the function whose root is searched
// the second argument is the Newton correction to be applied
// the third argument is the current estimate of the root
// the fourth argument is the current iteration number
auto c = [](const double f, const double, const double, const int) {
return std::abs(f) < 1.e-14;
// The `scalarNewtonRaphson` returns a tuple containing:
// - a boolean stating if the algorithm has converged
// - the last estimate of the function root
// - the number of iterations performed
// The two last arguments are respectively the initial guess and
// the maximum number of iterations to be performed
const auto r = tfel::math::scalarNewtonRaphson(fdf, c, 0.1, 100);
New features of MFront
Debugging options
A new command line option has been added to MFront. The -g option adds standard debugging flags to the compiler flags when compiling shared libraries.
Improved robustness of the isotropic DSLs
The @Flow block can now return a boolean value in the IsotropicMisesCreep, IsotropicMisesPlasticFlow, IsotropicStrainHardeningMisesCreep DSLs.
This allows to check to avoid Newton steps leading to too high values of the equivalent stress by returning false. This usually occurs if in elastic prediction is made (the default), i.e. when the
plastic strain increment is too low.
If false is returned, the value of the plastic strain increment is increased by doubling the previous Newton correction. If this happens at the first iteration, the value of the plastic strain
increment is set to half of its upper bound value (this upper bound value is such that the associated von Mises stress is null).
Generic behaviours
Add the possibility of defining the tangent operator blocks: the @TangentOperatorBlock and @TangentOperatorBlocks keywords
In version 3.3.x, some tangent operator blocks are automatically declared, namely, the derivatives of all the fluxes with respect to all the gradients. The @AdditionalTangentOperatorBlock and
@AdditionalTangentOperatorBlocks keywords allowed to add tangent operator blocks to this default set.
The @TangentOperatorBlock and @TangentOperatorBlocks allow a more fine grained control of the tangent operator blocks available and disable the use of the default tangent operation blocks. Hence,
tangent operator blocks that are structurally zero (for example due to symmetry conditions) don’t have to be computed any more.
Improvements to the implicit domain specific languages
Automatic definition of the offsets associated with integration variables
Let X be the name of an integration variable. The variable X_offset is now automatically defined in the @Integrator code block.
This variable allows a direct modification of the residual associated with this variable (though the variable fzeros) and jacobian matrix (though the variable jacobian).
Improvement of the StandardElastoViscoPlasticity brick
Porous (visco-)plasticity
The StandardElastoViscoPlasticity brick has been extended to support porous (visco-)plastic flows which are typically used to model ductile failure of metals. This allows building complex porous
plastic models in a clear and concise way, as illustrated below:
@Brick StandardElastoViscoPlasticity{
stress_potential : "Hooke" {young_modulus : 70e3, poisson_ratio : 0.3}, //
inelastic_flow : "Plastic" {
criterion : "GursonTvergaardNeedleman1982" {
f_c : 0.04, f_r : 0.056, q_1 : 2., q_2 : 1., q_3 : 4.},
isotropic_hardening : "Linear" {R0 : 274},
isotropic_hardening : "Voce" {R0 : 0, Rinf : 85, b : 17},
isotropic_hardening : "Voce" {R0 : 0, Rinf : 17, b : 262}
nucleation_model : "Chu_Needleman" {
An : 0.01, pn : 0.1, sn : 0.1 },
The following stress criteria are available:
• GursonTvergaardNeedleman1982
• RousselierTanguyBesson2002
The following nucleation models are available:
• ChuNeedleman1980 (strain)
• ChuNeedleman1980 (stress)
• PowerLaw (strain)
• PowerLaw (stress)
This extension will be fully described in a dedicated report which is currently under review.
The HarmonicSumOfNortonHoffViscoplasticFlows inelastic flow
An new inelastic flow called HarmonicSumOfNortonHoffViscoplasticFlows has been added. The equivalent viscoplastic strain rate \(\dot{p}\) is defined as:
\[ \dfrac{1}{\dot{p}}=\sum_{i=1}^{N}\dfrac{1}{\dot{p}_{i}} \]
where \(\dot{p}_{i}\) has an expression similar to the the Norton-Hoff viscoplastic flow:
\[ \dot{p}_{i}=A_{i}\,{\left(\dfrac{\sigma_{\mathrm{eq}}}{K_{i}}\right)}^{n_{i}} \]
@Brick StandardElastoViscoPlasticity{
stress_potential : "Hooke" {young_modulus : 150e9, poisson_ratio : 0.3},
inelastic_flow : "HarmonicSumOfNortonHoffViscoplasticFlows" {
criterion : "Mises",
A : {8e-67, 8e-67},
K : {1,1},
n : {8.2,8.2}
Choice of the eigen solver for some stress criteria in the StandardElastoviscoPlascity brick
The Hosford1972 and Barlat2004 now has an eigen_solver option. This option may take either one of the following values:
• default: The default eigen solver for symmetric tensors used in TFEL/Math. It is based on analytical computations of the eigen values and eigen vectors. Such computations are numerically more
efficient but less accurate than the iterative Jacobi algorithm.
• Jacobi: The iterative Jacobi algorithm (see [13, 14] for details).
@Brick StandardElastoViscoPlasticity{
stress_potential : "Hooke" {young_modulus : 150e9, poisson_ratio : 0.3},
inelastic_flow : "Plastic" {
criterion : "Hosford1972" {a : 100, eigen_solver : "Jacobi"},
isotropic_hardening : "Linear" {R0 : 150e6}
Built-in convergence helpers
Newton steps rejections based on the change of the flow direction between two successive estimates
Some stress criteria (Hosford 1972, Barlat 2004, Mohr-Coulomb) shows sharp edges that may cause the failure of the standard Newton algorithm, due to oscillations in the prediction of the flow
Rejecting Newton steps leading to a too large variation of the flow direction between the new estimate of the flow direction and the previous estimate is a cheap and efficient method to overcome this
issue. This method can be viewed as a bisectional linesearch based on the Newton prediction: the Newton steps magnitude is divided by two if its results to a too large change in the flow direction.
More precisely, the change of the flow direction is estimated by the computation of the cosine of the angle between the two previous estimates:
\[ \cos{\left(\alpha_{\underline{n}}\right)}=\dfrac{\underline{n}\,\colon\,\underline{n}_{p}}{\lVert \underline{n}\rVert\,\lVert \underline{n}\rVert} \]
with \(\lVert \underline{n}\rVert=\sqrt{\underline{n}\,\colon\,\underline{n}}\).
The Newton step is rejected if the value of \(\cos{\left(\alpha_{\underline{n}}\right)}\) is lower than a user defined threshold. This threshold must be in the range \(\left[-1:1\right]\), but due to
the slow variation of the cosine near \(0\), a typical value of this threshold is \(0.99\) which is equivalent to impose that the angle between two successive estimates is below \(8\mbox{}^{\circ}\).
Here is an implementation of a perfectly plastic behaviour based on the Hosford criterion with a very high exponent (\(100\)), which closely approximate the Tresca criterion:
@DSL Implicit;
@Behaviour HosfordPerfectPlasticity100;
@Author Thomas Helfer;
A simple implementation of a perfect plasticity
behaviour using the Hosford stress.
@Epsilon 1.e-16;
@Theta 1;
@Brick StandardElastoViscoPlasticity{
stress_potential : "Hooke" {young_modulus : 150e9, poisson_ratio : 0.3},
inelastic_flow : "Plastic" {
criterion : "Hosford1972" {a : 100},
isotropic_hardening : "Linear" {R0 : 150e6},
cosine_threshold : 0.99
Maximum equivalent stress in the Plastic flow
During the Newton iterations, the current estimate of the equivalent stress \(\sigma_{\mathrm{eq}}\) may be significantly higher than the elastic limit \(R\). This may lead to a divergence of the
Newton algorithm.
One may reject the Newton steps leading to such high values of the elastic limit by specifying a relative threshold denoted \(\alpha\), i.e. if \(\sigma_{\mathrm{eq}}\) is greater than \(\alpha\,\
cdot\,R\). A typical value for \(\alpha\) is \(1.5\). This relative threshold is specified by the maximum_equivalent_stress_factor option.
In some cases, rejecting steps may also lead to a divergence of the Newton algorithm, so one may specify a relative threshold \(\beta\) on the iteration number which desactivate this check, i.e. the
check is performed only if the current iteration number is below \(\beta\,\cdot\,i_{\max{}}\) where \(i_{\max{}}\) is the maximum number of iterations allowed for the Newton algorithm. A typical
value for \(\beta\) is \(0.4\). This relative threshold is specified by the equivalent_stress_check_maximum_iteration_factor option.
@DSL Implicit;
@Behaviour PerfectPlasticity;
@Author Thomas Helfer;
@Date 17 / 08 / 2020;
@Epsilon 1.e-14;
@Theta 1;
@Brick StandardElastoViscoPlasticity{
stress_potential : "Hooke" {young_modulus : 200e9, poisson_ratio : 0.3},
inelastic_flow : "Plastic" {
criterion : "Mises",
isotropic_hardening : "Linear" {R0 : 150e6},
maximum_equivalent_stress_factor : 1.5,
equivalent_stress_check_maximum_iteration_factor: 0.4
The StandardStressCriterionBase and StandardPorousStressCriterionBase base class to ease the extension of the StandardElastoViscoPlasticity brick
The Power isotropic hardening rule
The Power isotropic hardening rule is defined by: \[ R{\left(p\right)}=R_{0}\,{\left(p+p_{0}\right)}^{n} \]
The Power isotropic hardening rule expects at least the following material properties:
• R0: the coefficient of the power law
• n: the exponent of the power law
The p0 material property is optional and generally is considered a numerical parameter to avoir an initial infinite derivative of the power law when the exponent is lower than \(1\).
The following code can be added in a block defining an inelastic flow:
isotropic_hardening : "Linear" {R0 : 50e6},
isotropic_hardening : "Power" {R0 : 120e6, p0 : 1e-8, n : 5.e-2}
Improvement of the generic interface
Support of orthotropic behaviours
Orthotropic behaviours requires to:
• rotate the gradients in the material frame in the global frame before the behaviour integration.
• rotate the thermodynamic forces and the tangent operator blocks from the material frame in the global frame.
By design, the generic behaviour interface does not automatically perform those rotations as part of the behaviour integration but generates additional functions to do it. This choice allows the
calling solver to use their own internal routines to handle the rotations between the global and material frames.
However, the generic interface also generates helper functions which can perform those rotations. Those functions are named as follows:
• <behaviour_function_name>_<hypothesis>_rotateGradients
• <behaviour_function_name>_<hypothesis>_rotateThermodynamicForces
• <behaviour_function_name>_<hypothesis>_rotateTangentOperatorBlocks
They all take three arguments:
• a pointer to the location where the rotated variables will be stored.
• a pointer to the location where the original variables are stored.
• a pointer to the rotation matrix from the global frame to the material frame. The rotation matrix has 9 components stored in column-major format. For the function handling the thermodynamic
forces and the tangent operators blocks, this rotation matrix is transposed internally to get the rotation matrix from the material frame to the global frame.
In place rotations is explicitly allowed, i.e. the first and second arguments can be a pointer to the same location.
The three previous functions works for an integration point. Three other functions are also generated:
• <behaviour_function_name>_<hypothesis>_rotateArrayOfGradients
• <behaviour_function_name>_<hypothesis>_rotateArrayOfThermodynamicForces
• <behaviour_function_name>_<hypothesis>_rotateArrayOfTangentOperatorBlocks
Those functions takes an additional arguments which is the number of integration points to be treated.
Finite strain behaviours
Finite strain behaviours are a special case, because the returned stress measure and the returned tangent operator can be chosen at runtime time. A specific rotation function is generated for each
supported stress measure and each supported tangent operator.
Here is the list of the generated functions:
• <behaviour_function_name>_<hypothesis>_rotateThermodynamicForces_CauchyStress. This function assumes that its first argument is the Cauchy stress in the material frame.
• <behaviour_function_name>_<hypothesis>_rotateThermodynamicForces_PK1Stress. This function assumes that its first argument is the first Piola-Kirchhoff stress in the material frame.
• <behaviour_function_name>_<hypothesis>_rotateThermodynamicForces_PK2Stress. This function assumes that its first argument is the second Piola-Kirchhoff stress in the material frame.
<behaviour_function_name>_<hypothesis>_rotateTangentOperatorBlocks_dsig_dF. This function assumes that its first argument is the derivative of the Cauchy stress with respect to the deformation
gradient in the material frame.
• <behaviour_function_name>_<hypothesis>_rotateTangentOperatorBlocks_dPK1_dF. This function assumes that its first argument is the derivative of the first Piola-Kirchhoff stress with respect to the
deformation gradient in the material frame.
• <behaviour_function_name>_<hypothesis>_rotateTangentOperatorBlocks_PK2Stress. This function assumes that its first argument is the derivative of the second Piola-Kirchhoff stress with respect to
the Green-Lagrange strain in the material frame.
New features in MTest
Import behaviour parameters
The behaviour parameters are now automatically imported in the behaviour namespace with its default value.
For example, the YoungModulus parameter of the BishopBehaviour will be available in the BishopBehaviour::YoungModulus variable.
This feature is convenient for building unit tests comparing the simulated results with analytical ones.
The list of imported parameters is displayed in debug mode.
Unicode characters in MTest
Usage of a limited subsets of UTF-8 characters in variable names is now allowed. This subset is described here:
New features of the python bindings
Improvements of the tfel.math module
NumPy support
This version is the first to use Boost/NumPy to provide interaction with NumPy array.
The NumPy support is enabled by default if the Python bindings are requested. However, beware that Boost/NumPy is broken for Python3 up to version 1.68. We strongly recommend disabling NumPy
support when using previous versions of Boost by passing the flag -Denable-numpy-support=OFF to the cmake invokation.
Exposing acceleration algorithm
The FAnderson and UAnderson acceleration algorithms are now available in the tfel.math module. This requires NumPy support.
Example of the usage of the UAnderson acceleration algorithm
The following code computes the solution of the equation \(x=\cos{\left(x\right)}\) using a fixed point algorithm.
from math import cos
import numpy as np
import tfel.math
# The accelerator will be based on
# the 5 last Picard iterations and
# will be performed at every step
a = tfel.math.UAnderson(5,1)
f = lambda x: np.cos(x)
x0 = float(10)
x = np.array([x0])
# This shall be called each time a
# new resolution starts
for i in range(0,100):
x = f(x)
print(i, x, f(x[0]))
# apply the acceleration
Without acceleration, this algorithm takes \(78\) iterations. In comparison, the accelerated algorithm takes \(9\) iterations.
Version 3.3 introduces unicode support in MFront. This feature significantly improves the readability of MFront files, bringing it even closer to the mathematical expressions.
When generating C++ sources, unicode characters are mangled, i.e. translated into an acceptable form for the C++ compiler. This mangled form may appears in the compiler error message and are
difficult to read.
The tfel-unicode-filt tool, which is similar to the famous c++filt tool, can be use to demangle the outputs of the compiler.
For example, the following error message:
$ mfront --obuild --interface=generic ThermalNorton.mfront
Treating target : all
In file included from ThermalNorton-generic.cxx:33:0:
ThermalNorton.mfront: In member function ‘bool tfel::material::ThermalNorton<hypothesis, Type, false>::
computeConsistentTangentOperator(tfel::material::ThermalNorton<hypothesis, Type, false>::SMType)’:
ThermalNorton.mfront:147:75: error: ‘tum_2202__tum_0394__tum_03B5__eltum_2215__tum_2202__tum_0394__T’
was not declared in this scope
can be significantly improved by tfel-unicode-filt:
$ mfront --obuild --interface=generic ThermalNorton.mfront 2>&1 | tfel-unicode-filt
Treating target : all
In file included from ThermalNorton-generic.cxx:33:0:
ThermalNorton.mfront: In member function ‘bool tfel::material::ThermalNorton<hypothesis, Type, false>::
computeConsistentTangentOperator(tfel::material::ThermalNorton<hypothesis, Type, false>::SMType)’:
ThermalNorton.mfront:147:75: error: ‘∂Δεel∕∂ΔT’ was not declared in this scope
Improvements to tfel-config
The command line option --debug-flags has been added to tfel-config. When used, tfel-config returns the standard debugging flags.
Improvements to the build system
Exporting CMake’ targets
Disabling NumPy support
Tickets solved during the development of this version
Ticket #250: Add a new inelastic flow to the StandardElastoViscoPlasticity brick
This ticket requested the addition of the HarmonicSumOfNortonHoffViscoplasticFlows inelastic flow. See Section 4.5.2 for details.
For more details, see: https://sourceforge.net/p/tfel/tickets/250/
Ticket #234: Import behaviour parameters in MTest
For more details, see: https://sourceforge.net/p/tfel/tickets/234/
Ticket #231: Support for tensors as gradients and thermodynamic forces in the generic behaviours
For more details, see: https://sourceforge.net/p/tfel/tickets/231/
Ticket #219: Add the possibility to define the tangent operator blocks
For more details, see: https://sourceforge.net/p/tfel/tickets/219/
Ticket #214: Add a debugging option to MFront
The -g option of MFront now adds standard debugging flags to the compiler flags when compiling shared libraries.
For more details, see: https://sourceforge.net/p/tfel/tickets/214/
Ticket #212: Better const correctness in the generic behaviour
The state at the beginning of the time step is now described in a structure called mfront::gb::InitialState, the fields of which are all const.
The following fields of the mfront::gb::State are now const:
• gradients
• material_properties
• external_state_variables
For more details, see: https://sourceforge.net/p/tfel/tickets/212/
Ticket #209: lack of documentation of the @GasEquationOfState keyword
This feature is now described in the (MTest page)[mtest.html]
For more details, see: https://sourceforge.net/p/tfel/tickets/209/
Ticket #206: Export TFEL targets for use by external projects
Here is a minimal example on how to use this feature:
cmake_minimum_required(VERSION 3.0)
find_package(TFELException REQUIRED)
find_package(TFELMath REQUIRED)
add_executable(test-test test.cxx)
target_link_libraries(test-test tfel::TFELMath)
For more details, see: https://sourceforge.net/p/tfel/tickets/209/
Ticket #205: Add power isotropic hardening rule
This feature is described in Section 4.5.6.
For more details, see: https://sourceforge.net/p/tfel/tickets/205/
Ticket #201: Allow better control of the convergence in the StandardElastoviscoPlascity brick
See Sections 4.5.4.1 and 4.5.4.2.
For more details, see: https://sourceforge.net/p/tfel/tickets/201/
Ticket #200: Allow choosing the eigen solver for some stress criteria in the StandardElastoviscoPlascity brick
See 4.5.3.
For more details, see: https://sourceforge.net/p/tfel/tickets/200/
Ticket #195 : Export variables bounds for material properties
Variables bounds (both @Bounds and @PhysicalBounds) are now available for material properties. They are available directly in the .so file via getExternalLibraryManager().
For more details, see: https://sourceforge.net/p/tfel/tickets/195/
Ticket #202: DSLs for isotropic behaviours don’t handle NaN
The IsotropicMisesCreep, IsotropicMisesPlasticFlow, IsotropicStrainHardeningMisesCreep and MultipleIsotropicMisesFlowsDSL now handle properly NaN values.
For more details, see: https://sourceforge.net/p/tfel/tickets/202/
Ticket #196 : Export variable names for all interface of material property
The variable name of material property was available only for castem interface. Now it is available for all interface (c++, python, …). The name can be found in the .so file via
For more details, see: https://sourceforge.net/p/tfel/tickets/196/
, L.,
, G. and
, M. F.e. Calculations of copper bicrystal specimens submitted to tension-compression tests.
Acta Metallurgica et Materialia
. 1 March 1994. Vol. 42, no. 3, p. 921–935. DOI
. Available from:
, Joachim. Efficient numerical diagonalization of hermitian 3x3 matrices.
International Journal of Modern Physics C
. March 2008. Vol. 19, no. 3, p. 523–548. DOI
. Available from: | {"url":"https://thelfer.github.io/tfel/web/release-notes-3.4.html","timestamp":"2024-11-02T12:35:24Z","content_type":"text/html","content_length":"107735","record_id":"<urn:uuid:fc0ff0d7-9fa3-450f-a1f7-de2947cd1339>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00663.warc.gz"} |
How to Calculate Work Done on an Accelerating Car
What is the formula to calculate the work done on an accelerating car?
Given data: a 1.9e3 kg car accelerates uniformly from rest to 11m/s in 2.35s.
Formula to Calculate Work Done:
The work done on an accelerating car can be calculated using the work-energy theorem, which states that work done is equal to the change in kinetic energy of the object.
When a car accelerates from rest to a certain velocity, the work done on the car can be calculated by finding the change in kinetic energy. The formula to calculate the work done is:
Work Done = Change in Kinetic Energy
Where the change in kinetic energy is calculated as the difference between the final kinetic energy and the initial kinetic energy of the object.
For the given data of a 1.9e3 kg car accelerating from rest to 11m/s in 2.35s, we can use the formula to determine the work done on the car during this time interval. | {"url":"https://www.brundtlandnet.com/physics/how-to-calculate-work-done-on-an-accelerating-car.html","timestamp":"2024-11-12T00:19:58Z","content_type":"text/html","content_length":"21088","record_id":"<urn:uuid:a419a8d8-dafd-4cda-9abd-a65b9ab91e8d>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00653.warc.gz"} |
[Network Flow 24 questions] Magic ball problem
A Free Trial That Lets You Build Big!
Start building with 50+ products and up to 12 months usage for Elastic Compute Service
• Sales Support
1 on 1 presale consultation
• After-Sales Support
24/7 Technical Support 6 Free Tickets per Quarter Faster Response
• Alibaba Cloud offers highly flexible support services tailored to meet your exact needs. | {"url":"https://topic.alibabacloud.com/a/network-flow-24-questions-magic-ball-problem_8_8_31092133.html","timestamp":"2024-11-10T22:02:33Z","content_type":"text/html","content_length":"79471","record_id":"<urn:uuid:10a6c15f-098a-458c-baa7-3513bd59f8d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00868.warc.gz"} |
Confidence Interval For Population Variance Calculator – Accurate Results
This tool calculates the confidence interval for a population variance based on your input data.
Confidence Interval for Population Variance Calculator
This calculator helps you to determine the confidence interval for the population variance based on the given sample data. Enter the sample size, sample variance, and the desired confidence level
expressed as a percentage.
How to Use:
1. Enter the sample size (n).
2. Enter the sample variance (s²).
3. Enter the confidence level as a percentage (e.g., 95 for 95%).
4. Click “Calculate” to obtain the confidence interval for the population variance.
The confidence interval for the population variance is calculated using the Chi-square distribution. The bounds of the interval are determined based on the sample variance and the specified
confidence level.
• The sample size must be greater than 1.
• The sample variance must be a positive number.
• The confidence level must be between 0 and 100% (exclusive).
• Approximations for Chi-square critical values are used in this calculator, which may not be accurate for all scenarios.
Use Cases for This Calculator
Calculating Confidence Interval for Population Variance Use Cases
With the confidence interval for population variance calculator, you can:
Estimate Variability in Data
By entering your dataset, you can estimate the variability in the population data to help you understand the spread of values.
Determine Sample Size Adequacy
Calculate the confidence interval to determine if your sample size is adequate for drawing reliable conclusions about the population variance.
Assess Statistical Significance
Understand the statistical significance of your data by interpreting the confidence interval to make informed decisions based on the variance.
Compare Different Data Sets
Compare the confidence intervals of different data sets to assess differences in variance and draw insights into the populations they represent.
Check for Outliers
Identify potential outliers in your data by examining the confidence interval for population variance to see if certain values significantly impact the variability.
Monitor Data Stability
Track changes in data stability over time by regularly calculating the confidence interval for population variance to detect fluctuations in variability.
Validate Hypotheses
Validate hypotheses by using the confidence interval to test the significance of differences in population variances between groups or conditions.
Guide Decision-Making
Use the calculated confidence interval to guide decision-making processes by providing a range of possible values for the population variance.
Inform Forecasting
Inform forecasting models by incorporating the confidence interval for population variance to make more accurate predictions about future data fluctuations.
Enhance Data Interpretation
Enhance data interpretation by considering the confidence interval for population variance alongside other statistical measures for a comprehensive understanding of your data. | {"url":"https://madecalculators.com/confidence-interval-for-population-variance-calculator/","timestamp":"2024-11-09T15:47:47Z","content_type":"text/html","content_length":"144921","record_id":"<urn:uuid:26e212d0-3559-44ef-b7d8-4591761f5969>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00140.warc.gz"} |
Huei-Huang Lee
Huei-Huang Lee
Professor Lee has been working on finite element simulations for more than 30 years, including applications, software development, research, and teaching. He graduated from the Department of Civil
Engineering, National Cheng Kung University, Tainan, Taiwan in 1977 and was subsequently qualified as a Professional Engineer.
With his P.E. license, he became a structural engineering practitioner as well as a software developer in Taipei. During that period, he and his partners designed many high-rise buildings and
developed several finite element analysis programs, which were then commercialized.
He received his M.S. in 1984 and Ph.D. in 1989, both from University of Iowa, USA. After that, he started to work as a Research Scientist in an Industry-University Cooperated Research Center of the
University of Iowa. One of his functions was to integrate commercially available FEM programs with the programs developed by the Center, creating new engineering capabilities to support the
industrial members of the Center.
Today, Professor Lee is teaching in National Cheng Kung University, one of the most prestigious universities in Taiwan. He has authored three English books: "Finite Element Simulations with ANSYS
Workbench", "Mechanics of Materials Labs with SolidWorks Simulation", and "Engineering Dynamics Labs with SolidWorks Motion"; and two Chinese books: "Engineering Analysis with ANSYS: Fundamentals and
Concepts" (2005) and "Taguchi Methods: Principles and Practices of Quality Design" (2000, 2008, 2011). He runs 6 km every morning and takes care of his gardens during weekends.
Books by Huei-Huang Lee
Showing 1 - 20 of 27
Published July 27, 2023 Published June 26, 2023
ISBN 978-1-63057-615-8 ISBN 978-1-63057-624-0
Published September 9, 2022 Published August 10, 2022
ISBN 978-1-63057-546-5 ISBN 978-1-63057-539-7
Published September 17, 2021 Published August 9, 2021
ISBN 978-1-63057-491-8 ISBN 978-1-63057-456-7
Out of Print Published August 28, 2020
Published November 9, 2020 ISBN 978-1-63057-401-7
ISBN 978-1-63057-397-3
Out of Print Out of Print
Published October 23, 2019 Published July 2, 2019
ISBN 978-1-63057-297-6 ISBN 978-1-63057-299-0
Out of Print Out of Print
Published September 10, 2018 Published May 2, 2018
ISBN 978-1-63057-211-2 ISBN 978-1-63057-173-3
Out of Print Out of Print
Published April 2, 2018 Published September 12, 2017
ISBN 978-1-63057-171-9 ISBN 978-1-63057-140-5
Out of Print Out of Print
Published March 1, 2017 Published September 7, 2016
ISBN 978-1-63057-088-0 ISBN 978-1-63057-013-2
Out of Print Published April 29, 2015
Published September 18, 2015 ISBN 978-1-58503-935-7
ISBN 978-1-58503-983-8
Published April 22, 2015 Published March 4, 2015
ISBN 978-1-58503-937-1 ISBN 978-1-58503-941-8 | {"url":"https://www.sdcpublications.com/Authors/Huei-Huang-Lee/45/","timestamp":"2024-11-15T04:35:57Z","content_type":"application/xhtml+xml","content_length":"50649","record_id":"<urn:uuid:5f51946c-6e6c-48ba-b1e1-ebd3631fa73d>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00336.warc.gz"} |
Visualization of finite groups as posters
Please find attached a method to visualize finite groups:
Here we use the exp function to visualize each element of the finite group:
About me
If you are interested in Galois-Theory, here you can find my diploma, where I compute the Galois groups of some polynomials.
If you are interested in some unfinished ideas, I try to collect those in a notebook.
Please find attached an image of a fractal I stumbled upon which is based on lexicographic sorting of the prime factorization.
In my spare time I did some algorithmic composition as a hobby, culminating in a participation on the AI song contest 2023.
The method I used to generate this piece for piano, is a combination of Long Short-Term Memory method combined with the positive definite kernel I stumbled upon to measure musical note consonance.
The code can be found here. | {"url":"http://www.orges-leka.de/","timestamp":"2024-11-13T03:06:24Z","content_type":"text/html","content_length":"20723","record_id":"<urn:uuid:b206e87b-195b-471f-92fa-030c669df0ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00870.warc.gz"} |
Ampere’s Law - Definition, Equations, Derivations & Applications
Ampere’s Law - Definition & Applications
The students are taught the concepts like that of Ampere’s Law in the subject of Physics to improve their basic knowledge of the more complex topics that lie ahead of this concept in the future. The
students can easily learn this concept and more from the website of Vedantu for free. The study material is available in the form of a pdf which they can either download and save for future reference
or can view in the online mode. This helps the students to clear their doubts as and when they arise.
The teachers at Vedantu curate the resources as per the latest norms prescribed by the boards for Classes 1 to 12 and are thus, 100% reliable in terms of their accuracy. The students can refer to
these materials for practice which will help them identify their weak sections and improve on them. The website offers material for practice which includes sample papers, previous year’s question
papers, practice questions, important questions, etc. that help them to strengthen their level of preparation.
For the students, Ampere’s Law is one of the useful Laws which relates the net Magnetic field along the closed-loop to the Electric current which passes through the loop. The Law was discovered by
André-Marie Ampere in 1826. The expression for the relation between the Magnetic field and the current which produces it is termed Ampere’s Law.
Ampere’s Law - Definition & Applications
Ampere’s circuital Law is an integral part of studying electroMagnetism. The Law defines the relationship between the current and the Magnetic field that it creates around itself. This Law was named
after the scientist Andre Marie Ampere who discovered this phenomenon.
Ampere conducted multiple experiments to comprehend how the forces acted on wires which carry current. To understand what Ampere's Law is, students have to have a clear understanding of both the
Magnetic and Electric fields.
What is Ampere’s Law?
The Ampere's Law definition states that ‘the line integral of a Magnetic field intensity along a closed path is equal to the current distribution passing through that loop’.
The above statement might be quite difficult to apprehend at once. Hence it is advisable to build a background for the same while understanding it.
Mathematical Expression
Let us look at the Mathematical Expression of the Ampere Circuital Law for clarification.
Herein, B is Magnetic field intensity, I is current passing through a loop, and μ is Magnetic flux. The image beside shows the passage of current (represented with an upward moving arrow).
It depicts that on continuous passage of current, a Magnetic field is created around the conductor. As a student, you should understand that when you try to explain Ampere's circuital Law regarding
the passage of a current, it indicates that a conductor is carrying current.
Other than this, you should also have a prior understanding of Magnetic flux. The most vital topic to understand is Gauss’s Law which is usually one of the first topics that are taught. Once you have
cleared the concept of this Law, understanding Ampere's Law will be much easier.
Ampere’s Circuital Law and Magnetic Field: Applications
Ampere's Law, because of its convenience, has gained momentum since its inception. It has been implemented in real-life scenarios too. One of the most widely known platforms where Ampere’s Law is
being implemented regularly is the manufacturing of machines.
These machines can be motors, generators, transformers, or other similar devices. All of these work with the principles related to the application of Ampere circuital Law. Hence, understanding these
concepts is essential especially since these are essential in higher standards. These concepts are the basis of some of the most vital derivations and principles that are relevant in Physics .
Here is a list of applications where you will find Ampere’s circuital Law being put into use.
• Solenoid
• Straight wire
• Thick wire
• Cylindrical conductor
• Toroidal solenoid
It should be noted that the working principle of this Law remains the same throughout every process, even though its implementation varies greatly. It is the working principle of numerous machines
and devices, which are often even implemented as parts of other devices.
Students may also go through the Ampere circuital Law derivation to build a deeper understanding of the same. Not only is this derivation integral to Ampere’s Law, but also since it is one of the
fundamental concepts of Physics and Electricity.
Notably, a diagram always helps and our study materials provide just that along with lucid language for an explanation. To know more about what Ampere circuital Law is and its features with various
applications in real life, you can check our online study programs. Herein, you will get access to high-quality study materials with a quick explanation by subject experts.
You can even download our Vedantu app which is easily available so that you can access all the study materials at any time.
Importance of Vedantu’s study material
The following are the points that highlight the importance of Vedantu’s study material:
• It helps the student to access the resources online at the ease of their time and the comfort of their home.
• It is prepared by the experts in the subject matters and thus, is 100% accurate and reliable.
• Vedantu’s study material is available for download in pdf format which can be saved by the students for their future reference or can also be viewed in the online mode.
• The study material helps the students to clear their doubts instantly.
FAQs on Ampere’s Law
1. How do You State Ampere's Circuital Law?
The students need to focus on the concept as it is an important concept from the perspective of Exams. Ampere’s circuital Law is stated as the relationship between a current-carrying conductor and
the Magnetic field created around the conductor due to its flow of current. In simple words, it can be stated that Ampere's circuital Law states that “the line integral of the Magnetic field
surrounding closed-loop equals to the number of times the algebraic sum of currents passing through the loop.”
2. State and Explain Ampere's Circuital Law With the Necessary Equation.
For the students who already know about Ampere’s Circuital Law it is a friendly concept which can be easily understood as here and To define Ampere circuital Law, let us consider the Mathematical
equation for Ampere's Law -
∮ B. dl = μ0. I
Where ∮ is the integral of a closed surface on an imaginary plane, B is the Magnetic flux, dl is an infinitesimal length on the circumference of the area enclosed, and I is the current carried by the
3. What is the Most Popular Application of Ampere’s Circuital Law?
Ampere’s Law has a lot of implications in real life, however, some of the most popular applications include determining the Magnetic Induction due to the long current-carrying wire, determining the
Magnetic field inside a toroid, determining the Magnetic field created by a long current carrying conducting cylinder, determining the Magnetic field inside the conductor. While some of the most
widely used practical applications of Ampere's circuital Law include their usage in motors, generators, transformers, etc. Hence, we can say that it is an important aspect of the student’s
understanding of the concept.
4. Where can I get the information on Ampere’s Law Definition & Applications?
The students can easily get knowledge about Ampere’s Law from the website of Vedantu for free. The students can make note of various other related concepts from the website. Vedantu believes in the
fact that education is for all and thus provides all the study resources to the students free of cost so that they can view the same and also use it for future reference. The resources available
include sample papers, worksheets, previous year’s question papers, and much more for the students to get a good hold of their concepts.
5. Who was the scientist who performed experiments with forces that act on current-carrying wires?
The scientist behind the experiments on forces that act on current-carrying wires was André-Marie Ampere. This experiment was conducted in the late 1820s around the same time when Faraday was working
on his Faradays law. However, the scientist Faraday and Ampere had no clue that their work would be combined by Maxwell himself four years later. This way the experiment had a single scientist
working on it but eventually moved to have two of them. The students make good notes of the same as it will be a scoring point in the Exam. | {"url":"https://www.vedantu.com/physics/amperes-law","timestamp":"2024-11-07T21:41:22Z","content_type":"text/html","content_length":"259500","record_id":"<urn:uuid:bb523ed6-6020-4d08-8ed4-72ee7b4aa7e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00832.warc.gz"} |
Michael Frank Hallett
Academic title(s):
Professor & John Frothingham Professor of Logic and Metaphysics
BSc (Econ), London School of Economics, University of London, 1972
PhD, London School of Economics, University of London, 1979
Research areas:
History and Philosophy of Mathematics
Teaching areas:
Philosophy and History of Mathematics
Philosophy of Science
History of Logic (in particular Frege, Russell, Hilbert)
Logic and Set Theory
Development of Set Theory
Development of Analytic Philosophy
Selected publications:
• Cantorian set theory and limitation of size. Oxford: Clarendon Press. (Oxford Logic Guides, Vol. l0. Editor: Dana S. Scott). Foreword by Michael Dummett. Pp.xxii + 343. 1984, reprinted in
paperback, with revisions, 1986, 1988.
• 'Physicalism, reductionism and Hilbert' in Andrew Irvine (ed.): Physicalism in Mathematics, Dordrecht: D. Reidel Publishing Co., 1990, 182-256.
• 'Hilbert's axiomatic method and the laws of thought', in A. George (ed.): Mathematics and Mind. New York: Oxford University Press, 1994, 158-200.
• 'Putnam and the Skolem paradox' (with Putnam's reply), in Peter Clark and Bob Hale (eds.): Reading Putnam,Oxford: Basil Blackwell, 1994., 66-97.
• 'Logic and mathematical existence', in Lorenz Krüger and Brigitte Falkenburg (eds.): Physik, Philosophie und die Einheit der Wissenschaft. Für Erhard Scheibe, Heidelberg: Spektrum Akademischer
Verlag, 1995, 33-82.
• 'Hilbert and logic', in Mathieu Marion and Robert Cohen (eds.): Québec Studies in the Philosophy of Science, Part 1: Logic, Mathematics, Physics and the History of Science, (Boston Studies in the
Philosophy of Science, Volume 177), Dordrecht: Kluwer Publishing Co., 1995, 135-87.
• The foundations of mathematics 1879-1914 in Thomas Baldwin (ed.): The Cambridge History of Philosophy: 1879-1945, Cambridge: Cambridge University Press, 2003, 128-156.
I am one of four General Editors (with William Ewald, Ulrich Majer and Wilfried Sieg) of a project to publish a large quantity of lecture notes of the German mathematician David Hilbert on the
foundations of mathematics and physics, delivered from 1891 until 1933. The edition will be published by Springer-Verlag, and will consist of six volumes. The lectures themselves are given in the
original German, while the introductions and editoial material are in English. For titles and descriptions of all the volumes, see Descriptions of the Volumes below.
The first volume of this edition is David Hilbert's Lectures on the Foundations of Geometry: 1891-1902, edited by Michael Hallett and Ulrich Majer. (See Descriptions of the Volumes.) This volume is
now in press with Springer; to view the contents page, see Geometry Volume . I have written detailed notes and introductions to (a) the so-called Ferienkurse from 1896 and 1898; (b) the long lectures
on Euclidean geometry from 1898/1899; (c) the Festschriftof 1899 (the first edition of Hilbert's Grundlagen der Geometrie, reproduced here as Chapter 5); and (d) the lectures on the foundations of
geometry of 1902.
The second volume, which I am editing with William Ewald, Wilfried Sieg and Ulrich Majer, is David Hilbert's Lectures on the Foundations of Logic and Arithmetic: 1894 -1917. (See Descriptions of the
Volumes.) It is hoped that this volume will be ready for publication early in 2005.
General description of the edition.
This six volume collection makes available for the first time the German text of the most important of David Hilbert's unpublished lectures on the foundations of mathematics and physics, together
with a scholarly commentary. Hilbert's lectures and his personal interactions with the 'Hilbert circle' exercised a profound influence on the development of twentieth century mathematics and physics.
The lecture notes presented, spanning virtually the whole of Hilbert's teaching career, document his intense engagement with the ideas of some of the central figures of modern science (among them
Bernays, Boltzmann, Born, Brouwer, Cantor, Courant, Dedekind, Einstein, Frege, Heisenberg, Klein, Lie, Minkowski, PoincarŽ, Russell, von Neumann, Weyl, and Zermelo), and make possible a detailed
understanding of the development of his foundational work in geometry, arithmetic, logic, and proof theory, as well as in the theory of relativity, quantum mechanics and statistical physics. The
lectures contain many more philosophical, foundational and methodological remarks than does Hilbert's published work. Some of the individual volumes also reprint key published works of Hilbert when
these are centrally relevant to the unpublished work presented. (For example, Volume 1 reprints the first edition of Hilbert's celebrated Grundlagen der Geometrie, Volume 3 reprints
Bernays's Habilitationschrift and Hilbert and Ackermann's GrundzŸge der theoretischen Logik, Volume 4 reprints Hilbert's fundamental papers on General Relativity Theory.)
Volume 1: David Hilbert's Lectures on the Foundations of Geometry, 1891-1902
Edited by Michael Hallett and Ulrich Majer.
Volume 1 contains six sets of notes for lectures on the foundations of geometry held by Hilbert in the period 1891-1902. It also reprints the first edition of Hilbert's celebrated Grundlagen der
Geometrie of 1899, together with the important additions which appeared first in the French translation of 1900. The lectures document the emergence of a new approach to foundational study (the
'axiomatic method'), which concentrates on assessing the logical weight of central propositions by exploiting to the full the method of independence proofs by modelling. This culminates in the
lectures of 1898/1899 (the immediate precursor of the 1899 monograph) and 1902. The lectures contain many reflections and investigations which never found their way into print.
Volume 2: David Hilbert's Lectures on the Foundations of Arithmetic and Logic, 1894-1917
Edited by William Ewald, Michael Hallett, Wilfried Sieg and Ulrich Majer
Volume 2 focuses on notes for lectures on the foundations of the mathematical sciences held by Hilbert in the period 1894-1917. They document Hilbert's first engagement with 'impossibility' proofs;
his early attempts to formulate and address the problem of consistency, first dealt with in his work on geometry in the 1890s; his engagement with foundational problems raised by the work of Cantor
and Dedekind; his early investigations into the relationship between arithmetic, set theory, and logic; his advocation of the use of the axiomatic method generally; his first engagement with the
logical and semantical paradoxes; and the first formal attempts to develop a logical calculus. The Volume also contains Hilbert's address from 1895 which formed the preliminary version of his famous
'Zahlbericht' (1897).
Volume 3: David Hilbert's Lectures on the Foundations of Arithmetic and Logic, 1917-1933
Edited by William Ewald, Wilfried Sieg and Ulrich Majer
The bulk of Volume 3 consists of six sets of notes for lectures Hilbert gave (often in collaboration with Bernays) on the foundations of mathematics between 1917 and the early 1930s. The notes detail
the increasing dominance of the metamathematical perspective in Hilbert's treatment, i.e., the development of modern mathematical logic, the evolution of proof theory, and the parallel emergence of
Hilbert's finitist standpoint. The notes are mostly very polished expositions; e.g., the 1917-18 lectures are in effect a first draft of Hilbert and Ackermann's GrundzŸge der theoretischen Logik
(1928), reprinted in this Volume. They are thus essential for understanding the development of modern mathematical logic leading up to Hilbert and Bernays's Grundlagen der Mathematik (1934, 1938).
Also included is a complete version of Bernay's Habilitationschrift of 1918, only partially published in 1926.
Volume 4: David Hilbert's Lectures on the Foundations of Physics, 1898-1914: Classical, Relativistic and Statistical Mechanics
Edited by Ulrich Majer, Tilman Sauer and Klaus BŠrwinkel
The first part of Volume 4 documents Hilbert's efforts in the period 1898-1910 to base all known physics (including thermodynamics, hydrodynamics and electrodynamics) on classical mechanics. This
period closes with a lecture course 'Mechanik der Kontinua' (1911), in which Hilbert considers the consequences of the new principle of special relativity for our understanding of physics. The second
part starts with the lecture course 'Kinetische Gastheorie' (1911/12), which introduces a new approach to problems of statistical physics. The lecture course 'Molekulartheorie der Materie' (1913)
deals with a topic that was of great importance to Hilbert, returning to it repeatedly. The last lecture course contained in this volume, 'Statistische Mechanik' (1922) presents a very perceptive
comparison of the different approaches of Maxwell, Boltzmann, Gibbs etc. to the foundational problems of statistical physics. It is a paradigm of logical analysis and conceptual clarity.
Volume 5: David Hilbert's Lectures on the Foundations of Physics, 1915-1927: Relativity, Quantum Theory and Epistemology
Edited by Ulrich Majer, Tilman Sauer and Heinz-JŸrgen Schmidt
This Volume has three sections, General Relativity, Epistemological Issues, and Quantum Mechanics. The core of the first section is Hilbert's two semester lecture course on 'The Foundations of
Physics' (1916/17). This is framed by Hilbert's published 'First and Second Communications' on the 'Grundlagen der Physik' (1915, 1917). The section closes with a lecture on the new concepts of space
and time held in Bucharest in 1918. The epistemological issues concern the principle of causality in physics (1916), the intricate relation between nature and mathematical knowledge (1921), and the
subtle question whether what Hilbert calls the 'world equations'' are physically complete (1923). The last section deals with quantum theory in its early, advanced and mature stages. Hilbert held
lecture courses on the mathematical foundations of quantum theory twice, before and after the breakthrough in 1926. These documents bear witness to one of the most dramatic changes in the foundations
of science.
Volume 6: David Hilbert's Notebooks and General Foundational Lectures
Edited by William Ewald, Michael Hallett, Wilfried Sieg and Ulrich Majer
This Volume contains a selection of material exhibiting many of Hilbert's philosophical and foundational views on particular theories and the exact sciences in general, drawn from his private
notebooks and from lectures for more general audiences held in the 1920s.
I am currently writing a paper comparing the very different approaches of David Hilbert and Frege to the foundations of mathematics, a paper which begins as an analysis of the Frege-Hilbert
correspondence. This will appear in Thomas Ricketts (ed.): The Cambridge Companion to Frege, to be published by Cambridge University Press.
In addition, I am writing a long paper on the early views of Hilbert on the foundations of mathematics, particularly on geometry, and arithmetic ("Hilbert on number, geometry and continuity").
1. Hilbert on the Foundations of Mathematics (provisional title), to be published by the Clarendon Press, Oxford, c. 300,000 words.
This book will be a full length study of the development of Hilbert's treatment of the foundations of mathematics, with three central aims: (1) It will attempt to bring out the unified development of
Hilbert's thought, and show how Hilbert's early work on the foundations of geometry greatly shaped his later approaches to foundational issues and even to logic, giving rise to mathematical logic as
we know it, and show how his work posed most of the central problems in mathematical logic in the first half of the century. (2) It will pay close attention to various elements in Hilbert's thought,
e.g., Hilbert's notion of the axiomatic method, and the view of the nature of science which underlies it (here, comparison with the views of Hertz and Mach is especially important); Hilbert's theory
of mathematical existence, and (closely linked to this) his reliance on the notion of consistency, his theory of 'ideal elements', and thus his theory of infinity. It will also focus on Hilbert's
view of the fundamental role of signs in any communicable intellectual activity, a view which is the basis of Hilbert's theory of consistency proofs. Of great importance in this is an anlysis of
Hilbert's relations to other important foundational thinkers of the time, e.g., Cantor, Dedekind and Frege. (3) The book will attempt to situate Hilbert between Kant and Gdel, perhaps (along with
Frege and Russell) the two most important thinkers about mathematics in modern times. In particular, the book will argue that much more of Hilbert's theory of the foundations of mathematics is viable
than is usually thought.
2. The Philosophy of Mathematics: An Historical Approach. This book (co-authored with William Ewald of the University of Pennsylvania), will be c. 250 pages, and will appear with Cambridge University
Press as part of a series of introductory books, edited by Gary Hatfield and Paul Guyer, on the various fundamental fields of philosophy.
The book will cover the central developments from Berkeley's time till the present day. (The series is aimed at senior undergraduates and beginning graduate students, and the books are intended to
give an introduction to the central problems of the fields, but in doing so to give some indication of how these problems came to be central. For more details of the series and a list of volumes,
see Cambridge University Press.)
In the academic year 2003-2004, I am teaching Introduction to Deductive Logic (107-210A), Further Topics in Formal Logic (107-411A), Intermediate Logic (107-310B), and a Seminar in the Philosophy of
Mathematics (107-510B). | {"url":"https://www.mcgill.ca/philosophy/michael-frank-hallett","timestamp":"2024-11-03T06:55:36Z","content_type":"text/html","content_length":"57649","record_id":"<urn:uuid:0ea01ce1-7b81-4f00-8079-68f54597c8cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00635.warc.gz"} |
SWI-Prolog -- length/2
True if Length represents the number of elements in List. This predicate is a true relation and can be used to find the length of a list or produce a list (holding variables) of length Length.
The predicate is non-deterministic, producing lists of increasing length if List is a partial list and Length is a variable.
?- length(List,4).
List = [_27940,_27946,_27952,_27958].
?- length(List,Length).
List = [], Length = 0 ;
List = [_24698], Length = 1 ;
List = [_24698,_25826], Length = 2
It raises errors if Length is bound to a non-integer or a negative integer or if List is neither a list nor a partial list. This error condition includes cyclic lists:^138ISO demands failure
here. We think an error is more appropriate.
?- A=[1,2,3|A], length(A,L).
ERROR: Type error: `list' expected ...
Covering an edge case, the predicate fails if the tail of List is equivalent to Length:^139This is logically correct. An exception would be more appropriate, but to our best knowledge, current
practice in Prolog does not describe a suitable candidate exception term.
?- List=[1,2,3|Length],length(List,Length).
?- length(Length,Length). | {"url":"https://cliopatria.swi-prolog.org/swish/pldoc/man?predicate=length/2","timestamp":"2024-11-10T03:09:19Z","content_type":"text/html","content_length":"6685","record_id":"<urn:uuid:f4cff4dc-abe4-42e6-95cb-5196ece94e12>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00850.warc.gz"} |
Finding the Probability of an Event
Learning Outcomes
• Apply the basic definition of probability
The probability of an event tells us how likely that event is to occur. We usually write probabilities as fractions or decimals.
For example, picture a fruit bowl that contains five pieces of fruit – three bananas and two apples.
If you want to choose one piece of fruit to eat for a snack and don’t care what it is, there is a [latex]{\Large\frac{3}{5}}[/latex] probability you will choose a banana, because there are three
bananas out of the total of five pieces of fruit. The probability of an event is the number of favorable outcomes divided by the total number of outcomes.
The probability of an event is the number of favorable outcomes divided by the total number of outcomes possible.
[latex]\text{Probability}={\Large\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}}[/latex]
Converting the fraction [latex]{\Large\frac{3}{5}}[/latex] to a decimal, we would say there is a [latex]0.6[/latex] probability of choosing a banana.
[latex]\begin{array}{}\\ \text{Probability of choosing a banana}={\Large\frac{3}{5}}\hfill \\ \text{Probability of choosing a banana}=0.6\hfill \end{array}[/latex]
This basic definition of probability assumes that all the outcomes are equally likely to occur. If you study probabilities in a later math class, you’ll learn about several other ways to calculate
The ski club is holding a raffle to raise money. They sold [latex]100[/latex] tickets. All of the tickets are placed in a jar. One ticket will be pulled out of the jar at random, and the winner will
receive a prize. Cherie bought one raffle ticket.
1. Find the probability she will win the prize.
2. Convert the fraction to a decimal.
What are you asked to find? The probability Cherie wins the prize.
What is the number of favorable outcomes? [latex]1[/latex], because Cherie has [latex]1[/latex] ticket.
Use the definition of probability. [latex]\text{Probability of an event}={\Large\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}}[/latex]
Substitute into the numerator and denominator. [latex]\text{Probability Cherie wins}={\Large\frac{1}{100}}[/latex]
Convert the fraction to a decimal.
Write the probability as a fraction. [latex]\text{Probability}={\Large\frac{1}{100}}[/latex]
Convert the fraction to a decimal. [latex]\text{Probability}=0.01[/latex]
try it
Three women and five men interviewed for a job. One of the candidates will be offered the job.
1. Find the probability the job is offered to a woman.
2. Convert the fraction to a decimal.
Show Solution
The following video contains another example of how to compute the probability of an event and write it as either a fraction or decimal.
Candela Citations
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• Question ID 146452, 146451, 146447, 146446. Authored by: Lumen Learning. License: CC BY: Attribution
CC licensed content, Shared previously
CC licensed content, Specific attribution
• Prealgebra. Provided by: OpenStax. License: CC BY: Attribution. License Terms: Download for free at http://cnx.org/contents/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757 | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/finding-the-probability-of-an-event/","timestamp":"2024-11-14T01:04:18Z","content_type":"text/html","content_length":"37425","record_id":"<urn:uuid:de7089fc-4ba6-4cba-ba64-bdd0ad317945>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00120.warc.gz"} |
News - How to use FEM ANSYS parameter optimization and probability design of ultrasonic welding horn
With the development of ultrasonic technology, its application is more and more extensive, it can be used to clean tiny dirt particles, and it can also be used for welding metal or plastic.
Especially in today’s plastic products, ultrasonic welding is mostly used, because the screw structure is omitted, the appearance can be more perfect, and the function of waterproofing and
dustproofing is also provided. The design of the plastic welding horn has an important impact on the final welding quality and production capacity. In the production of new electric meters,
ultrasonic waves are used to fuse the upper and lower faces together. However, during use, it is found that some horns are installed on the machine and cracked and other failures occur in a short
period of time. Some welding horn The defect rate is high. Various faults have had a considerable impact on production. According to the understanding, equipment suppliers have limited design
capabilities for horn, and often through repeated repairs to achieve design indicators. Therefore, it is necessary to use our own technological advantages to develop durable horn and a reasonable
design method.
2 Ultrasonic plastic welding principle
Ultrasonic plastic welding is a processing method that utilizes the combination of thermoplastics in the high-frequency forced vibration, and the welding surfaces rub against each other to produce
local high-temperature melting. In order to achieve good ultrasonic welding results, equipment, materials and process parameters are required. The following is a brief introduction to its principle.
2.1 Ultrasonic plastic welding system
Figure 1 is a schematic view of a welding system. The electrical energy is passed through the signal generator and the power amplifier to produce an alternating electrical signal of ultrasonic
frequency (> 20 kHz) that is applied to the transducer (piezoelectric ceramic). Through the transducer, the electrical energy becomes the energy of the mechanical vibration, and the amplitude of the
mechanical vibration is adjusted by the horn to the appropriate working amplitude, and then uniformly transmitted to the material in contact with it through the horn. The contact surfaces of the two
welding materials are subjected to high-frequency forced vibration, and the friction heat generates local high temperature melting. After cooling, the materials are combined to achieve welding.
In a welding system, the signal source is a circuit part that contains a power amplifier circuit whose frequency stability and drive capability affect the performance of the machine. The material is
a thermoplastic, and the design of the joint surface needs to consider how to quickly generate heat and dock. Transducers, horns and horns can all be considered mechanical structures for easy
analysis of the coupling of their vibrations. In plastic welding, mechanical vibration is transmitted in the form of longitudinal waves. How to effectively transfer energy and adjust the amplitude is
the main point of design.
The horn serves as the contact interface between the ultrasonic welding machine and the material. Its main function is to transmit the longitudinal mechanical vibration outputted by the variator
evenly and efficiently to the material. The material used is usually high quality aluminum alloy or even titanium alloy. Because the design of plastic materials changes a lot, the appearance is very
different, and the horn has to change accordingly. The shape of the working surface should be well matched with the material, so as not to damage the plastic when vibrating; at the same time, the
first-order longitudinal vibration solid frequency should be coordinated with the output frequency of the welding machine, otherwise the vibration energy will be consumed internally. When the horn
vibrates, local stress concentration occurs. How to optimize these local structures is also a design consideration. This article explores how to apply ANSYS design horn to optimize design parameters
and manufacturing tolerances.
3 welding horn design
As mentioned earlier, the design of the welding horn is quite important. There are many ultrasonic equipment suppliers in China that produce their own welding horns, but a considerable part of them
are imitations, and then they are constantly trimming and testing. Through this repeated adjustment method, the coordination of horn and equipment frequency is achieved. In this paper, the finite
element method can be used to determine the frequency when designing the horn. The horn test result and the design frequency error are only 1%. At the same time, this paper introduces the concept of
DFSS (Design For Six Sigma) to optimize and robust design of horn. The concept of 6-Sigma design is to fully collect the customer’s voice in the design process for targeted design; and
pre-consideration of possible deviations in the production process to ensure that the quality of the final product is distributed within a reasonable level. The design process is shown in Figure 2.
Starting from the development of the design indicators, the structure and dimensions of the horn are initially designed according to the existing experience. The parametric model is established in
ANSYS, and then the model is determined by the simulation experiment design (DOE) method. Important parameters, according to the robust requirements, determine the value, and then use the sub-problem
method to optimize other parameters. Taking into account the influence of materials and environmental parameters during the manufacture and use of the horn, it has also been designed with tolerances
to meet the requirements of manufacturing costs. Finally, the manufacturing, test and test theory design and actual error, to meet the design indicators that are delivered. The following step-by-step
detailed introduction.
3.1 Geometric shape design (establishing a parametric model)
Designing the welding horn first determines its approximate geometric shape and structure and establishes a parametric model for subsequent analysis. Figure 3 a) is the design of the most common
welding horn, in which a number of U-shaped grooves are opened in the direction of vibration on a material of approximately cuboid. The overall dimensions are the lengths of the X, Y, and Z
directions, and the lateral dimensions X and Y are generally comparable to the size of the workpiece being welded. The length of Z is equal to the half wavelength of the ultrasonic wave, because in
the classical vibration theory, the first-order axial frequency of the elongated object is determined by its length, and the half-wave length is exactly matched with the acoustic wave frequency. This
design has been extended. Use, is beneficial to the spread of sound waves. The purpose of the U-shaped groove is to reduce the loss of lateral vibration of the horn. The position, size and number are
determined according to the overall size of the horn. It can be seen that in this design, there are fewer parameters that can be freely regulated, so we have made improvements on this basis. Figure 3
b) is a newly designed horn that has one more size parameter than the traditional design: the outer arc radius R. In addition, the groove is engraved on the working surface of the horn to cooperate
with the surface of the plastic workpiece, which is beneficial to transmit vibration energy and protect the workpiece from damage. This model is routinely parametrically modeled in ANSYS, and then
the next experimental design.
3.2 DOE experimental design (determination of important parameters)
DFSS is created to solve practical engineering problems. It does not pursue perfection, but is effective and robust. It embodies the idea of 6-Sigma, captures the main contradiction, and abandons
“99.97%”, while requiring the design to be quite resistant to environmental variability. Therefore, before making the target parameter optimization, it should be screened first, and the size that has
an important influence on the structure should be selected, and their values should be determined according to the robustness principle.
3.2.1 DOE parameter setting and DOE
The design parameters are the horn shape and the size position of the U-shaped groove, etc., a total of eight. The target parameter is the first-order axial vibration frequency because it has the
greatest influence on the weld, and the maximum concentrated stress and the difference in the working surface amplitude are limited as state variables. Based on experience, it is assumed that the
effect of the parameters on the results is linear, so each factor is only set to two levels, high and low. The list of parameters and corresponding names is as follows.
DOE is performed in ANSYS using the previously established parametric model. Due to software limitations, full-factor DOE can only use up to 7 parameters, while the model has 8 parameters, and
ANSYS’s analysis of DOE results is not as comprehensive as professional 6-sigma software, and can’t handle interaction. Therefore, we use APDL to write a DOE loop to calculate and extract the results
of the program, and then put the data into Minitab for analysis.
3.2.2 Analysis of DOE results
Minitab’s DOE analysis is shown in Figure 4 and includes the main influencing factors analysis and interaction analysis. The main influencing factor analysis is used to determine which design
variable changes have a greater impact on the target variable, thereby indicating which are important design variables. The interaction between the factors is then analyzed to determine the level of
the factors and to reduce the degree of coupling between the design variables. Compare the degree of change of other factors when a design factor is high or low. According to the independent axiom,
the optimal design is not coupled to each other, so choose the level that is less variable.
The analysis results of the welding horn in this paper are: the important design parameters are the outer arc radius and the slot width of the horn. The level of both parameters is “high”, that is,
the radius takes a larger value in the DOE, and the groove width also takes a larger value. The important parameters and their values were determined, and then several other parameters were used to
optimize the design in ANSYS to adjust the horn frequency to match the operating frequency of the welding machine. The optimization process is as follows.
3.3 Target parameter optimization (horn frequency)
The parameter settings of the design optimization are similar to those of the DOE. The difference is that the values of two important parameters have been determined, and the other three parameters
are related to the material properties, which are regarded as noise and cannot be optimized. The remaining three parameters that can be adjusted are the axial position of the slot, the length and the
horn width. The optimization uses the subproblem approximation method in ANSYS, which is a widely used method in engineering problems, and the specific process is omitted.
It is worth noting that using frequency as the target variable requires a little skill in operation. Because there are many design parameters and a wide range of variation, the vibration modes of the
horn are many in the frequency range of interest. If the result of modal analysis is directly used, it is difficult to find the first-order axial mode, because the modal sequence interleaving may
occur when the parameters change, that is, the natural frequency ordinal corresponding to the original mode changes. Therefore, this paper adopts the modal analysis first, and then uses the modal
superposition method to obtain the frequency response curve. By finding the peak value of the frequency response curve, it can ensure the corresponding modal frequency. This is very important in the
automatic optimization process, eliminating the need to manually determine the modality.
After the optimization is completed, the design working frequency of the horn can be very close to the target frequency, and the error is less than the tolerance value specified in the optimization.
At this point, the horn design is basically determined, followed by manufacturing tolerances for production design.
3.4 Tolerance design
The general structural design is completed after all design parameters have been determined, but for engineering problems, especially when considering the cost of mass production, tolerance design is
essential. The cost of low precision is also reduced, but the ability to meet design metrics requires statistical calculations for quantitative calculations. The PDS Probability Design System in
ANSYS can better analyze the relationship between design parameter tolerance and target parameter tolerance, and can generate complete related report files.
3.4.1 PDS parameter settings and calculations
According to the DFSS idea, tolerance analysis should be performed on important design parameters, and other general tolerances can be determined empirically. The situation in this paper is quite
special, because according to the ability of machining, the manufacturing tolerance of geometric design parameters is very small, and has little effect on the final horn frequency; while the
parameters of raw materials are greatly different due to suppliers, and the price of raw materials accounts for More than 80% of horn processing costs. Therefore, it is necessary to set a reasonable
tolerance range for the material properties. The relevant material properties here are density, modulus of elasticity and speed of sound wave propagation.
Tolerance analysis uses random Monte Carlo simulation in ANSYS to sample the Latin Hypercube method because it can make the distribution of sampling points more uniform and reasonable, and obtain
better correlation by fewer points. This paper sets 30 points. Assume that the tolerances of the three material parameters are distributed according to Gauss, initially given an upper and lower
limit, and then calculated in ANSYS.
3.4.2 Analysis of PDS results
Through the calculation of PDS, the target variable values corresponding to 30 sampling points are given. The distribution of the target variables is unknown. The parameters are fitted again using
Minitab software, and the frequency is basically distributed according to the normal distribution. This ensures the statistical theory of tolerance analysis.
The PDS calculation gives a fitting formula from the design variable to the tolerance expansion of the target variable: where y is the target variable, x is the design variable, c is the correlation
coefficient, and i is the variable number.
According to this, the target tolerance can be assigned to each design variable to complete the task of tolerance design.
3.5 Experimental verification
The front part is the design process of the entire welding horn. After the completion, the raw materials are purchased according to the material tolerances allowed by the design, and then delivered
to the manufacturing. Frequency and modal testing are performed after manufacturing is completed, and the test method used is the simplest and most effective sniper test method. Because the most
concerned index is the first-order axial modal frequency, the acceleration sensor is attached to the working surface, and the other end is struck along the axial direction, and the actual frequency
of the horn can be obtained by spectral analysis. The simulation result of the design is 14925 Hz, the test result is 14954 Hz, the frequency resolution is 16 Hz, and the maximum error is less than
1%. It can be seen that the accuracy of the finite element simulation in the modal calculation is very high.
After passing the experimental test, the horn is put into production and assembly on the ultrasonic welding machine. The reaction condition is good. The work has been stable for more than half a
year, and the welding qualification rate is high, which has exceeded the three-month service life promised by the general equipment manufacturer. This shows that the design is successful, and the
manufacturing process has not been repeatedly modified and adjusted, saving time and manpower.
4 Conclusion
This paper starts with the principle of ultrasonic plastic welding, deeply grasps the technical focus of welding, and proposes the design concept of new horn. Then use the powerful simulation
function of finite element to analyze the design concretely, and introduce the 6-Sigma design idea of DFSS, and control the important design parameters through ANSYS DOE experimental design and PDS
tolerance analysis to achieve robust design. Finally, the horn was successfully manufactured once, and the design was reasonable by the experimental frequency test and the actual production
verification. It also proves that this set of design methods is feasible and effective.
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Number of Undiscovered Near-Earth Asteroids Revised Downward
Fewer large near-Earth asteroids (NEAs) remain to be discovered than astronomers thought, according to a new analysis by planetary scientist Alan W. Harris of MoreData! in La Canada, California.
Harris is presenting his results this week at the 49th annual meeting of the American Astronomical Society’s Division for Planetary Sciences in Provo, Utah.
Observers have been cataloging potentially hazardous asteroids for decades. Based on the number of finds, the area of sky explored, and the limiting brightness our telescopes and cameras can reach,
researchers can estimate what fraction of the NEA population has been detected so far and how many more objects lurk undiscovered. Harris has published numerous such estimates over the years.
Recently he realized that his estimates have been plagued by a seemingly innocuous but nonetheless consequential round-off error. Once corrected, the estimated number of large (diameter > 1
kilometer) NEAs remaining to be discovered decreases from more than 100 to less than 40.
The population (“size-frequency distribution”) of NEAs is usually given in terms of number versus brightness, since most discovery surveys operate in visible (reflected) light. Brightness isn’t a
reliable proxy for size, though, because asteroid surfaces don’t all have the same albedo, or reflectivity. NEA brightnesses are expressed in units of absolute magnitude H, with lower numbers
indicating brighter objects.
IAU Minor Planet Center – the world’s clearinghouse for asteroid measurements – rounds off reported values of H to the nearest 0.1 magnitude. While this is mostly unimportant, amounting to a
reduction in the estimated NEA population N(
In a 2015 study, Harris and Italian astronomer Germano D’Abramo estimated that there exist 990 NEAs brighter than H = 17.75, which is considered equivalent on average to a diameter D = 1 km. After
correcting for the round-off problem, that estimate decreases to 921 +/- 20, consistent with a recent estimate by Pasquale Tricarico (Planetary Science Institute), who used similar data but a
different computational approach.
We know how many NEAs of H < 17.75 have been discovered: 884 according to the latest tally by the Jet Propulsion Laboratory. The previous population estimate of 990 implied 89% completion and 106 yet
to be found. The new estimate of 921 implies 96% completion and only 37 left to be found, almost three times fewer.
The recent NEOWISE survey, which more directly measured NEA diameters from their thermal infrared emission, opens up the possibility to transform the population estimate from brightness units, N(D).
The transformation method resembles a game of Sudoku, but the rules are different. We can make a table, nine columns wide like Sudoku, with each column representing a range of albedos, for example
0.0703 to 0.1114, and vertically with 25 rows, each row representing a range of diameter, for example from 0.794 to 1.000 km.
These bin sizes are chosen so that each range in diameter or albedo corresponds to 0.5 magnitude in brightness H. With this arrangement a diagonal sum across the boxes follows a constant value of H,
and this sum should equal the known n(H) for that value of H. Horizontally, the sum of the boxes across represent the desired number n(D), and the best solution in each box needs to follow the albedo
distribution defined by NEOWISE.
Unfortunately the solution depends rather strongly on the albedo distribution, so if one chooses the distribution for all NEAs, or just “Earth-crossing asteroids” that actually have any probability
of impacting, or just big ones, or only small ones, one can get quite different answers.
Harris solved the “Sudoku” puzzle for a variety of reasonable albedo distributions and derived estimates of N(D > 1 km) ranging from about 750 to 900. The good news is that this seemingly large
uncertainty in the total number does not much affect the completion fraction of 96%. So the number of large (D > 1 km) NEAs yet to be discovered is still limited to around 30 to 40. | {"url":"https://spaceguardcentre.com/number-of-undiscovered-near-earth-asteroids-revised-downward/","timestamp":"2024-11-03T22:42:01Z","content_type":"text/html","content_length":"67898","record_id":"<urn:uuid:69e174a6-44ef-4616-aa9d-6e257481cc3e>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00867.warc.gz"} |
Approach comparison methodology | Treasuries
Approach comparison methodology
Methodology behind how approaches are being compared and analysed across the documentation.
A number of analytical comparisons are made across this documentation. These comparisons explore the trade offs between approaches that could be used for different parts of a treasury system. All of
these comparisons start by listing out the most important factors that can be considered when analysing the differences between any of the approaches. Each approach is then analysed against those
factors to see how effective it is at addressing them.
Factor information
Factors in any of the comparisons will include the following information:
Factor name - A short title and name to describe the factor for consideration.
Description - An overview of what the factor is about and why it is relevant.
Maximum score - There are two scores given that are out of 5 that create a final score for each approach. The first score is the maximum score for the factor. The maximum score highlights how
important the factor itself is for the approaches comparison. The maximum score ranges from 1 to 5. When an approach is being analysed a second score is given to that approach that highlights how
effective that approach is at addressing the factor. The score received by the approach is then applied to the maximum score to create a final score. Doing this means that the most important
factors will have more weighting in the total score when compared to other factors that might be less important. Consider the following examples:
Maximum score is 5. This means the factor is very important. Here are the final scores an approach could receive depending on how well it scores in addressing this factor:
1 points out of 5 ⇒ 20% of 5 maximum score ⇒ 1
2 points out of 5 ⇒ 40% of 5 maximum score ⇒ 2
3 points out of 5 ⇒ 60% of 5 maximum score ⇒ 3
4 points out of 5 ⇒ 80% of 5 maximum score ⇒ 4
5 points out of 5 ⇒ 100% of 5 maximum score ⇒ 5
Maximum score is 4. This means the factor is fairly important. Here are the final scores an approach could receive depending on how well it scores in addressing this factor:
1 points out of 5 ⇒ 20% of 4 maximum score ⇒ 0.8
2 points out of 5 ⇒ 40% of 4 maximum score ⇒ 1.6
3 points out of 5 ⇒ 60% of 4 maximum score ⇒ 2.4
4 points out of 5 ⇒ 80% of 4 maximum score ⇒ 3.2
5 points out of 5 ⇒ 100% of 4 maximum score ⇒ 4
Maximum score is 3. This means the factor is moderately important. Here are the final scores an approach could receive depending on how well it scores in addressing this factor:
1 points out of 5 ⇒ 20% of 3 maximum score ⇒ 0.6
2 points out of 5 ⇒ 40% of 3 maximum score ⇒ 1.2
3 points out of 5 ⇒ 60% of 3 maximum score ⇒ 1.8
4 points out of 5 ⇒ 80% of 3 maximum score ⇒ 2.4
5 points out of 5 ⇒ 100% of 3 maximum score ⇒ 3
Scoring questions - A list of questions that help with thinking about how effective each approach is at addressing the factor or not. Answering these questions will help with identifying a
suitable score to give each approach for that factor.
Advantages of this comparison approach
There are a number of reasons why this approach for making comparisons is effective:
Simple - This format makes it easy for people to read about the different factors that are important for this comparison. It also makes it easier to agree or disagree with any given parts of the
analysis which can help to improve it over time wherever this is necessary.
Concise - The analysis for each approach only adds enough information to make an informed judgement on what the score should be when also considering the other approaches in the comparison. This
helps to keep the analysis concise and informative. If there are multiple approaches that appear to be effective at addressing the listed factors then a more detailed piece of analysis could be
done separately.
Effective for ranking - By giving each approach a score for each factor an easier comparison can be made about which approaches are likely the most effective for different areas. The aggregation
of these factors helps with identifying the most promising approaches overall.
Visual outcome - Generating scores for each approach and each factor means a visual table can be produced for every comparison. This makes it much quicker to find out what the outcome is for any
comparison without needing to read the full content.
Identifies experiment opportunities - These comparisons are an effective way to identify the approaches that could be highly effective for treasury systems. These approaches can then be compared
with what is actually happening across the industry and help with highlighting the most promising experiments for Web3 ecosystems to explore. | {"url":"https://docs.treasuries.co/analysis/approach-comparison-methodology","timestamp":"2024-11-06T21:33:36Z","content_type":"text/html","content_length":"224284","record_id":"<urn:uuid:3f60eec0-b9ac-4029-a5de-2ef3d44f32de>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00090.warc.gz"} |
Introduction to Statistical Mechanics
Introduction to Statistical Mechanics. Instructor: Prof. Girish S. Setlur, Department of Physics, IIT Guwahati. This is an introductory course in classical and quantum statistical mechanics which
deals with the principle of ensembles, Classical, Fermi and Bose ideal gases, Pauli paramagnetism, Debye and Einstein's theory of specific heat and the 1D Ising model. (from nptel.ac.in)
Related Links
Introduction to Statistical Mechanics
Instructor: Prof. Girish S. Setlur, Department of Physics, IIT Guwahati. This is an introductory course in classical and quantum statistical mechanics. | {"url":"http://www.infocobuild.com/education/audio-video-courses/physics/intro-to-statistical-mechanics-iit-guwahati.html","timestamp":"2024-11-12T15:40:48Z","content_type":"text/html","content_length":"12695","record_id":"<urn:uuid:3872b8c8-1e7f-4dde-b6dd-2dccd15dd0b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00456.warc.gz"} |
Quantitative Aptitude Quiz For SBI Clerk Prelims 2022- 1st October
Directions (1-5): Study the given passage carefully and answer the questions.
A shopkeeper bought a pen & a book for Rs 500. He marked pen by 20% above cost price which is same as discount percentage given on book. He gained 8% & 12% on pen & book respectively. his gain amount
in entire transaction is equal to 10% of marked price of book.
Q1. What is cost price of book? (in Rs)
(a) 420
(b) 350
(c) 430
(d) 380
(e) 400
Q2. Marked price of book is how much more than the selling price of pen? (in Rs)
(a) 440
(b) 452
(c) 460
(d) 456
(e) 444
Q3. What is ratio of selling price of pen to that of book?
(a) 27 : 140
(b) 15 : 56
(c) 27 : 112
(d) 27 : 100
(e) 25 : 112
Q4. What is average of marked price of pen & book?
(a) 340
(b) 334
(c) 330
(d) 284
(e) 278
Q5. If no discount was offered on both then his overall gain percent is approximately what percent more than his actual gain percent?
(a) 100%
(b) 167%
(c) 150%
(d) 220%
(e) 200%
Directions (6-10): In each of the following questions, two equations (I) and (II) are given. Solve the equations and mark the correct option:
(a) if x>y
(b) if x≥y
(c) if x<y
(d) if x ≤y
(e) if x = y or no relation can be established between x and y.
Q6. I. x²+21x+108=0
II. y²+24y+143=0
Q7. I. x²=289
II. y = √289
Q8. I. x²-25x+156=0
II. y²-32y+255=0
Q9. I. x²+23x+130=0
Q10. I. x²-28x+195=0
II. y²-22y+117=0
Directions (11-15): Following table DI gives the detail of magazines printed by five different companies and distributed among different distributors and answer the following question accordingly.
Note:- magazines were equally distributed among the distributors of respective printing companies.
Q11.What is the average no. of magazines distributed by companies Q, R and T among their respective distributors?
(a) 2720
(b) 2640
(c) 2480
(d) 2960
(e) 3120
Q12.Find the total numbers of distributors of magazines of company Q and T together?
(a) 62
(b) 72
(c) 84
(d) 78
(e) 64
Q13.Find the respective ratio of the total number of magazines distributed among the distributors of company R to that of the total no. of magazines distributed among distributors of company T?
(a) 21:19
(c) 17:21
(d) 21:17
(e) 17:23
Q14.Find the average number of books distributed among the distributors by all the five companies together?
(a) 2784
(b) 2664
(c) 2680
(d) 2756
(e) 2724
Q15.Find the difference between total no. of distributors of magazines sold by companies P and Q together to the total no. of distributors of magazines sold by companies R and T together?
(a) 38
(b) 34
(c) 36
(d) 42
(e) 40 | {"url":"https://www.bankersadda.com/quantitative-aptitude-quiz-for-sbi-clerk-prelims-2022-1st-october/","timestamp":"2024-11-12T22:52:18Z","content_type":"text/html","content_length":"605714","record_id":"<urn:uuid:f7d9614b-4b2e-4064-b2c0-418bc37bfc36>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00333.warc.gz"} |
Student Math Class Notebooks
I've been giving notebooks a lot of thought in the past week or two. As a principal I encouraged teachers to keep their notes, lesson plans, and ideas online. I tried to organize my online folders to
keep information accessible to anyone who might need it. So for the last 12 years I have not taught students how to organize a notebook.
I want my students' notebook to be a study guide and practice book ... one-stop shopping so to speak! I envision the possibility of needing one simple spiral notebook per six weeks. That sounds like
a lot but if we put all of our notes, class practice, and homework in it, we could potentially use most of a typical 100 page notebook. (If we use one per six weeks, we will store the completed
notebooks in the classroom to serve as references for future units and testing).
My plan is to tape each six weeks' set of standards in the front of the notebook along with a list of the key vocabulary. I am debating on the need for a Table of Contents. Instead, I envision our
starting each class period on a fresh page - clearly labeled with the date, standards addressed, and textbook reference pages.
On the last two pages of the notebook, students will keep a record of their graded work.
I plan to use notebook foldables for taking class notes. The foldables will be useful study guides for unit tests, exams, and state tests. In addition to our class notes, students will complete
practice problems in the notebook ... showing detailed work to provide examples when they are working on their own.
Homework practice will be just that ... practice. I don't plan to grade daily homework. Instead, I plan to give notebook quizzes. A quiz might have 5 to 10 problems on it. Each quiz problem will be
taken from homework assignments and even classroom examples. For students who followed instructions and completed their work, they can use their notebooks to copy the work. If students did not do the
homework, they can still complete the quiz - they just don't have the support they might need from their notebook.
Over the next two weeks, I'll be consulting with other teachers if they use a structured approach to notebooking ... and I'll be scouring blogs as well ... always looking for the "best" idea!
What have I missed?
5 comments:
1. [...] Beth Ferguson, Student Math Class Notebooks [...]
2. Great post! I had one notebook per trimester last year and did the same thing that you speak of. We started a new notebook each new trimester. I really liked the "fresh notebook" for a new
semester. Then, I collected them and stored them in my classroom for future reference. At the end of the year, I had my students go through all three notebooks and tear out pages that they
thought would be the most helpful to them in the future. We then combined all of those pages into a "Math Reference Notebook". I have stored those in my classroom and will give them back to
students when school begins. (I blogged about it here if you are interested. http://ispeakmath.wordpress.com/2012/07/16/made-for-math-monday-math-reference-books/)
This year I am going to use two notebooks. They have a homework workbook so only recorded notes in their notebooks. Thus, we only used about 2/3 of the notebooks last year.
3. I love the idea of 1 per 6 week grading period. I just highly doubt our school would allow us to require 6 notebooks for one class :( I could maybe get away with 1-3 subject notebook for the
whole year.
I love the idea of the notebook quizzes and plan on using that in my classroom as well. I brainstormed with my sister last week and we came up with this plan:
Homework is completion for 25 points total. Students can get partial points if they do some, but not all homework.
Quizzes are graded out of 75 points. This way, students that do not do the homework, but DO know the material can still pass (so frustrating when smart kids fail because of not completing
homework) Students that do the homework can also still pass as long as they get a 50 or so.
Not 100 percent on this yet, but like that they are still held accountable, but not ruined if they do not do their homework one night.
4. Julie - thanks for sharing your blog post on how you set up your Math Reference Notebooks. I'll definitely read it!
5. Megan - I am new to the school/district ... haven't checked in yet to be sure I can ask for several notebooks in a year! I'll do that soon ... so that I can continue to think about my plan for
what to put in the notebook. It could be that I'll do something like what Julie (I Speak Math) did in a previous year - make just a reference notebook - and put homework in something else.
While I want students to do homework, I am conflicted because I know that not all kids need as much practice as others to succeed. And for those that need the practice I don't want to grade their
practice. I've done completion points in the past. I am going to try the notebook quizzes at least for the first 6 weeks and see how that works out. | {"url":"http://algebrasfriend.blogspot.com/2012/08/student-math-class-notebooks.html","timestamp":"2024-11-14T17:06:31Z","content_type":"text/html","content_length":"63338","record_id":"<urn:uuid:8f992f9c-23e7-4550-8ea7-6dae1c466688>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00718.warc.gz"} |
Extremal results for odd cycles in sparse pseudorandom graphs
We consider extremal problems for subgraphs of pseudorandom graphs. For graphs F and Г the generalized Turán density π[F](Г) denotes the relative density of a maximum subgraph of Г, which contains no
copy of F. Extending classical Turán type results for odd cycles, we show that π[F](Г)=1/2 provided F is an odd cycle and Г is a sufficiently pseudorandom graph.
In particular, for (n,d,λ)-graphs Г, i.e., n-vertex, d-regular graphs with all non-trivial eigenvalues in the interval [−λ,λ], our result holds for odd cycles of length ℓ, provided (Formula
presented.) Up to the polylog-factor this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the condition is best possible and was proven previously by Sudakov, Szabó, and Vu, who
addressed the case when F is a complete graph. A construction of Alon and Kahale (based on an earlier construction of Alon for triangle-free (n,d;λ)-graphs) shows that our assumption on Г is best
possible up to the polylog-factor for every odd ℓ≥5.
Dive into the research topics of 'Extremal results for odd cycles in sparse pseudorandom graphs'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/extremal-results-for-odd-cycles-in-sparse-pseudorandom-graphs-6","timestamp":"2024-11-04T17:44:18Z","content_type":"text/html","content_length":"54226","record_id":"<urn:uuid:a4eb8f26-29c1-49e2-99e2-461f943c4fa4>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00645.warc.gz"} |
The 3D and Contour Graphs Toolbar
28.1.6 The 3D and Contour Graphs Toolbar
3D Graphs and 3D Surface Graphs
Origin's 3D Graphs Toolbar (when a worksheet is active)
Origin's 3D Graphs Toolbar (when a matrix is active)
Origin supports a wide variety of 3D XYZ and 3D Surface graph types, all of which are provided on one comprehensive 3D Graphs toolbar.
Plotting is quick and easy. Simply select the desired XYZ or matrix data and click on a plot type button. The rotation and size of the graph can be updated graphically or by specifying specific
values. Additional properties of your 3D graph can be edited by double-clicking on any object in the graph to bring up its related properties dialog.
Since Origin's basic 3D plots originate from an Origin worksheet and its 3D Surface plots originate from an Origin matrix, the 3D Graphs toolbar is window sensitive (as can be seen below).
The following table shows all the built-in 3D XYZ and 3D Surface graph types, such as XYZ Scatter plot graphs and 3D Wire Frame surfaces, that Origin offers. Learn more about the graph type, please
refer to the 3D XYY graphs, 3D XYZ graphs and 3D surface graphs pages.
3D scatter graphs are frequently used when the data are not arranged on a rectangular grid. Simple 3D scatter graphs display an object or marker corresponding to each datum. More complicated scatter
graph include datum specific marker attributes, drop-lines and combinations of the scatter data with additional objects such as a fitted surface.
Trajectory plot is trivariate plot in which a comparison of 3 measures is presented, one measure along each axis. The plot can include projections onto any or all planes. Settings for the actual
data and each projection can be edited independently. This includes line, symbol, and drop line settings.
Plots selected XYZZ data into a 3D Scatter with Error Bar plot.
Plots a 3D Vector plot with selected XYZXYZ data.
Plots a 3D Vector plot with selected XYZdxdydz data.
A 3D tetrahedral plot is created from XYZZ worksheet data. The Z columns can be in multiples of 2, in which case, Z-Z pairs will be grouped in the layer and plot Symbol Edge Color will be
3D Bar plots can be created from worksheets to visualize the XYY values in them. The height of each bar represents the second Y value in the source worksheet that corresponds to the XY coordinates
of the bar.
3D stacked bar can be created from XYY worksheet data. All Y columns with same Z value will be stacked on Y direction.
3D 100% stacked bar can be created from XYY worksheet data. All Y columns with same Z value will be stacked and their total length will be normalized to fill the whole Y axis range.
Creates 3D Ribbon plots with the XYY values in a worksheet. The height of each ribbon represents the second Y value in the source worksheet that corresponds to the XY coordinates of the ribbon.
Creates 3D Wall plots with the XYY values in a worksheet. The height of each wall represents the second Y value in the source worksheet that corresponds to the XY coordinates of the wall.
Creates 3D Waterfall plots with the XYY... values in a worksheet. Each data set is displayed as a line data plot and arranged by a specified parameter in the Z dimension.
Creates 3D Waterfall plots with the XYY... values in a worksheet. Each data set is displayed as a line plot and arranged by a specified parameter in Z direction. The color map is applied in the Y
Creates 3D Waterfall plots with the XYY... values in a worksheet. Each data set is displayed as a line plot and arranged by a specified parameter in Z direction. The color map is applied in the Z
Creates 3D Color Fill surface with matrix value. The Z values determine a surface of X and Y grid lines with fill colors on the front and the back of the surface.
The 3D X Constant with Base Color Fill Surface plot draws grid lines parallel to the X axis only when creating a 3D Color Fill Surface.
The 3D Y Constant with Base Color Fill Surface plot draws grid lines parallel to the Y axis only when creating a 3D Color Fill Surface.
3D Color Map Surface plots are trivariate plots in which a comparison of 3 measures is presented. Each subset, or level, of Z values (as defined by the user) in a 3D Colormap Surface graph is
represented by a different fill color and can be further delineated by displaying a contour line inbetween each level.
Creates 3D Color Fill surface with Error Bar.
Creates 3D Color Map surface with Error Bar.
Creates multiple Color Fill surface, in which color varies from one to another.
Creates multiple Color Map surface, in which color varies from one to another.
Creates 3D Wire Frame graph for matrix data, in which the Z values determine a surface of X and Y grid lines.
For the Wire Surface graph,the Z values in a matrix determines a surface of X and Y grid lines and secondary lines are placed between the grids.
3D Bar plots can be created from XYZ data or matrices. The height of each bar represents the Z value in the source XYZ or matrix that corresponds to the XY coordinates of the bar. It provides a
quick view of the trends of the data and shows how the Z data changes with the X and Y data.
3D stacked bar plot can be created from XYZZ.../XYZXYZ... data or matrix with multiple objects. For XYZ data, all Z values at same X and Y coordinate will be stacked cumulatively; for matrix data,
all objects in current matrixsheet will be stacked cumulatively.
You are also allowed to create this kind of 3D stacked bars from Virtual Matrices which are supposed to be created in advance with X-Function w2vm.
3D 100% stacked bar plot can be created from XYZZ.../XYZXYZ... data or matrix with multiple objects. For XYZ data, all Z values at same X and Y coordinate will be stacked cumulatively; for matrix
data, all objects in current matrixsheet will be stacked cumulatively. The total length of each stacked bar will be normalized to fill the whole Z axis range.
You are also allowed to create this kind of 3D stacked bars from Virtual Matrices which are supposed to be created in advance with X-Function w2vm.
Creates 3D scatter graph for matrix data.
Creates 3D Scatter with Error Bar for matrix data.
Creates 3D Color Map surface with Projection .
Contour Graphs
Origin's 3D Graphs toolbar, Contour Plots fly-out when a worksheet is active.
Origin's 3D Graphs toolbar, Contour Plots fly-out when a matrix is active.
Several contour graph/contour plot templates are included in Origin. To create a contour plot, one click access is provided via the 3D Graphs toolbar (see below).
Origin's contour graphs include the ability to apply a color fill and/or fill pattern to the contours, display contour lines (equipotential lines) and contour labels, as well as adjust contour
levels. A color scale object can be included with the contour plot to serve as a legend. Contour labels can even be graphically attached to a contour line with the click of your mouse.
For more advanced control, XYZ Contour plots (created from XYZ worksheet data using triangulation) can employ a custom boundary around the data.
Next list is each of the built-in contour plot types Origin offers. To learn more about the selected coutour graph template, please refer to the contour graph page.
A contour plot that uses a color map for the color fill and fills to the contour lines.
A black and white contour plot that includes contour Lines and contour labels (i.e no color fill).
A contour color fill plot that uses a gray color mapping for the color fill and fills to the XY grid.
Allows you to dynamically create horizontal, vertical, and arbitrary line segment profiles of the matrix data.This kind of graphs are frequently used to visualize and analyze terrain elevation data
or electromagnetic field data.
Polar contour graphs in polar coordinates from both XYZ data and matrix data. For XYZ data, X is the Angular (Units are in degrees), Y is the Radius and Z determines the contour; for matrix data,
the X and Y coordinates of matrix data represents the Angular (Units are in degrees) and Radius respectively.
Polar contour graphs in polar coordinates, here X data for r, Y data for theta,along with the Z data for Z.
Plots selected X Y Z data into a contour plot on Ternary coordinate.
Plot data in matrix/virtual matrix into a Heatmap plot.
Plot data in matrix/virtual matrix as a Heatmap plot with labels.
Image Graphs
Origin comes with two built-in image graph types. To access these image graphs, Origin provides one click access via the 3D Graphs toolbar (see below).
The following list shows the built-in image plot and the image histograms offered by Origin. If a link appears to the right of a button, click on it (or the plot type button) to learn more about the
Allows you to plot matrix data as an image graph. The image graph displays the image on xy-axes. Once plotted, you can annotate the graph as desired.
Provides a quick way to analyze the image data and generate profiles. You can create horizontal, vertical and arbitrary line profiles of the image data dynamically. | {"url":"https://www.originlab.com/doc/en/Origin-Help/3D-Contour-Graph-Toolbar","timestamp":"2024-11-05T15:49:45Z","content_type":"text/html","content_length":"178289","record_id":"<urn:uuid:92310eef-fe4d-4c15-8648-309bf340ee8f>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00727.warc.gz"} |
An annulus is a shape made out of two circles. An annulus is a plane figure formed between two concentric circles (the circles sharing a common center). The shape of the annulus is like a ring. Some
of the real-life examples of annulus shape are finger-ring, dough-nut, a CD, etc. Let us learn its meaning in terms of geometry along with the area formula and solved examples based on it.
Annulus Definition
The region covered between two concentric circles is called the annulus. The concentric circles are the circles that have a common center. An annulus is a plane shape formed between these concentric
circles. It looks like a circular path around the circle.
The circle is a fundamental concept not only in Maths but also in many fields. A circle is a plane figure that is made up of the points situated at the same distance from a particular point.
In the above figure, a circle with some radius is given. Now if the same circle is surrounded by another circle with some space or gap in between them and the radius bigger than this circle, the
region formed in between the two circles is basically the annulus.
In the above figure, the blue shaded portion is the annulus.
Annulus Meaning
The word “annulus” (plural – annuli) is derived from the Latin word, which means “little ring“. An annulus is called the area between two concentric circles (circles whose centre coincide) lying in
the same plane.
Annulus is the region bounded between two circles that share the same centre. This shape resembles a flat ring. It can also be considered as a circular disk having a circular hole in the middle. See
the figure here showing an annulus.
Here, two circles can be seen, where a small circle lies inside the bigger one. The point O is the centre of both circles. The shaded coloured area, between the boundary of these two circles, is
known as an annulus. The smaller circle is known as the inner circle, while the bigger circle is termed as the outer circle.
In other words, any two-dimensional flat ring-shaped object that is formed by two concentric circles is called an annulus.
Annulus Formula
Similar to other two-dimensional figures, the annulus also have two important formulas. They are:
• Area of annulus
• Perimeter of annulus
The area of the annulus can be determined if we know the area of circles (both inner and outer). The formula to find the annulus of the circle is given by:
A = π(R^2-r^2)
where ‘R’ is the radius of outer circle and ‘r’ is the radius of the inner circle.
Area of Annulus
The area of the annulus can be calculated by finding the area of the outer circle and the inner circle. Then we have to subtract the areas of both the circles to get the result. Let us consider a
In the above figure, two circles are having common centre O. Let the radius of outer circle be “R” and the radius of inner circle be “r”. The shaded portion indicates an annulus. To find the area of
this annulus, we are required to find the areas of the circles.
Area of Outer Circle = πR^2
Area of Inner Circle = πr^2
Area of Annulus = Area of Outer Circle – Area of Inner Circle
Area of Annulus = π(R^2-r^2) square units
Or we can also write it as;
Area of Annulus = π(R+r)(R-r) square units
Perimeter of Annulus
By the definition of the perimeter, we know it is the total linear distance covered by the boundaries of a two-dimensional shape. Hence, the perimeter of the annulus will be the total distance
covered by its outer circle and inner circle.
Suppose the radius of outer circle is R and the radius of inner circle is ‘r’, then the perimeter of annulus will be:
Perimeter of annulus = 2πR + 2πr = 2π (R + r) units
The value of π is 22/7 or 3.14.
Related Articles
Annulus Solved Examples
Q.1: Calculate the area of an annulus whose outer radius is 14 cm and inner radius 7 cm?
Solution: Given that outer radius R = 14 cm and inner radius r = 7 cm
Area of outer circle = πR^2 = 22/7 x 14 x 14
= 22 x 14 x 2
= 616 cm^2
Area of inner circle = πr^2 = 22/7 x 7 x 7
= 22 x 7
= 154 cm^2
Area of the annulus = Area of the outer circle – Area of the inner circle
Area of the annulus = 616 – 154
Area of the annulus = 462 cm^2
Q.2: If the area of an annulus is 1092 inches and its width is 3 cm, then find the radii of the inner and outer circles.
Solution: Let the inner radius of an annulus be r and its outer radius be R.
Then width = R – r
3 = R – r
R = 3 + r
We know,
Area of the annulus = π(R^2−r^2)
Area of the annulus = π (R + r) (R – r)
1092 = 22/7 (3 + r + r) (3)
3 + 2r = 1092×722×3
3 + 2r = 115.82
2r = 115.82 – 3
2r = 112.82
r = 56.41
R = 3 + 56.41
= 59.41
So, Inner radius = 56.41 inches
Outer radius = 59.41 inches
Frequently Asked Questions – FAQs
What is an annulus?
An annulus is a two-dimensional figure formed between two concentric circles.
What is the area of annulus?
The formula for area of annulus is given by:
A = π(R^2-r^2) square units
where R is the radius of outer circle and r is the radius of inner circle.
What is the perimeter of annulus?
The perimeter of annulus is the total distance covered by the boundaries of outer circle and inner circle. It is given by:
Perimeter = 2π (R + r)
What are the examples of annulus shape?
The most common examples of annulus shape is a ring, do-nut, a tyre tube, etc. | {"url":"https://mathlake.com/Annulus","timestamp":"2024-11-05T22:21:54Z","content_type":"text/html","content_length":"15278","record_id":"<urn:uuid:acee8a6d-c714-4f12-8e15-a020efd91bfc>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00065.warc.gz"} |
Online Open Channel Flow Calculator | What is an Open Channel Flow? - physicsCalculatorPro.com
Open Channel Flow Calculator
The slope, roughness coefficient, and cross-sectional area of an open channel are used in the free online open channel flow calculator to compute the water flow velocity and volumetric flow rate
across the channel. You will discover how Manning's equation works and how to apply it to calculate water flow with this calculator. You will also learn about the most efficient open channel
cross-sections. Continue reading to begin your education.
What is an Open Channel Flow?
Water can be transported using one of two methods: pressure or gravity. We could use a big water pump to bring water up a building. The water pump would propel the water up the pipes by exerting a
high amount of pressure on it via mechanical action, most commonly centrifugal force. As water moves upwards, it will fill the pipes. This is referred to as a closed channel since the pipe
cross-section is completely enclosed and not exposed to air pressure.
When water flows from an upper to the lower elevation and the water surface is "open" or exposed to air pressure, we call it an open channel flow. Examples include rivers and canals. Water may flow
due to gravity in some cases, but the channel or pipe may be full.
We consider steady and uniform flow in this open channel flow calculator. The flow rate does not fluctuate in a channel with a constant cross-sectional shape, slope, or roughness in a steady uniform
flow. As long as we have all of the other properties of the channel, we can compute the dimensions of the chosen uniform shape in this type of flow.
Manning Equation
Manning's formula is as follows, with these considerations in mind: V = (1 / n) * R^2/3 * s1/2
• Where V - Velocity or mass flow rate of water
• n - Manning's roughness coefficient
• R - Hydraulic radius
• s - Slope of the channel
We can see from Manning's equation that the area and slope are directly proportional to the water flow rate, hence increasing the area and slope will increase the water flow rate. The roughness
coefficient and the wetted perimeter are inversely proportional to the water flow rate, which means that raising their values reduces the water flow rate.
Aside from this flow rate, the volumetric flow rate, which is just the product of the mass flow rate and the cross-sectional area, can also be considered, as illustrated below: Q = V * A
• Where, Q - Volumetric flow rate
• V - Mass flow rate
• A - Cross-sectional area
Uses of our Open Channel Flow Calculator
For custom-sized channel designs and the most cost-effective cross-sections, utilize our open channel flow calculator to calculate the water flow rate. You can input your chosen values for the
channel dimensions when utilizing the custom design option in the design field. The water flow's cross-sectional area, as well as the channel's wetted perimeter, mass flow rate, hydraulic radius, and
volumetric flow rate, will be calculated by the calculator.
When choosing the most efficient design choice, on the other hand, you can just input the water flow depth. You will view the results after selecting the channel surface roughness coefficient and
entering a channel slope. We calculated efficient and affordable open-channel designs using the constants of a most efficient cross-section of the open channels.
Become familiar with plenty of similar concepts of physics along with well designed calculator tools complementing the concept on Physicscalculatorpro.com a trusted reliable portal for all your
FAQs on Open Channel Flow Calculator
1. What is Manning's formula?
The Manning formula is an empirical method for calculating the average velocity of a liquid flowing through a conduit that does not entirely enclose the liquid, known as open channel flow.
2. What does R stand for in the Manning formula?
The hydraulic radius, R, is measured in feet. This is the variable that accounts for the channel shape in the equation. The hydraulic radius is calculated by dividing the flow's area by its wetted
3. How can you find out how much water is in the channel?
The DCM is built around the area's uniform velocity. The main channel and floodplains are separated from the compound channel section in this method, and the total flow is computed by adding the
discharge through the area.
4. Which channel is most effective?
When the discharge carrying capacity for a particular cross-section area is at its greatest, a section is said to be most efficient. The lining is the most expensive component of the entire building
cost, and if the perimeter is kept to a minimum, the cost of lining will be minimal, making it the most cost-effective segment.
5. Is Manning's formula consistent in terms of dimensions?
The Chezy and Manning equations, which are at the heart of our current open channel hydraulics knowledge, are not dimensionally homogeneous. The author proposes a new derivation of these equations
that reveals the individual elements of these coefficients, enabling more precise values to be calculated. | {"url":"https://physicscalculatorpro.com/open-channel-flow-calculator/","timestamp":"2024-11-05T02:52:30Z","content_type":"text/html","content_length":"28894","record_id":"<urn:uuid:d5dedb4c-b3a7-4f88-9ef1-ddbdb3526c0f>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00279.warc.gz"} |
Randomizing Round Robin - Darts for Windows 2008
Starting from version 2.7.3.5, Darts for Windows can now randomize the players in the Round Robin group(s).
We start by setting up a Round Robin, using one or more groups. In this sample I will use 4 groups just to show how you deal with the groups when
not all the groups have the same number of players.
Open the "Tournament" screen from the "File" menu.
"Round Robin" tab in the "Tournament" screen...
We start by creating a new Round Robin by clicking the "New" button as showed in the picture above.
In this sample we use 4 groups playing 7 legs (not best of) and we get 1 point pr. leg won.
The groups before we fill in the players. In this sample I will use 22 players instead of 24 just to show what you don when one or more of the groups have different number of players.
I will seed four players, one in each group (you don't have to seed any of the players if you don't want to of course) just to show how it is done.
Click the button as shown above to insert the players into the groups. I will place the seeded players on top of each group.
Drag and drop the seeded players into the groups and then continue to add the rest of the players. Since we are going to randomize the players, it doesn't matter who you place where. The only thing
that matters is the number of players pr. group. If you put 5 players in a group instead of 6, the number of players in the group will still be the same after the randomizing. This is also the reason
why I put the players into the groups before the ranomizing.
This is how it looks like after I have put in the players. We are know ready to do the randomizing.
Click the button as show above to open the screen where you do the random draw of the players.
Check the box in the "Seeded" column if you want to seed one or more players. Seeded players are not affected by the random draw.
To do the draw, click the button as shown above.
This is how it looks like after the draw.
The only thing left now is to insert the players into the groups by clicking the button as shown above. | {"url":"http://www.dartsforwindows.com/howto/rr/randomizerr.html","timestamp":"2024-11-08T18:01:40Z","content_type":"text/html","content_length":"8339","record_id":"<urn:uuid:298d89e1-ced2-4ae7-af05-47aae57f5d0b>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00401.warc.gz"} |
Entropy, Information gain, and Gini Index; the crux of a Decision Tree
A Brief on the Key Information Measures used in a Decision Tree Algorithm
The decision tree algorithm is one of the widely used methods for inductive inference. Decision tree approximates discrete-valued target functions while being robust to noisy data and learns complex
patterns in the data. In the past, we have dealt with similar noisy data; you can check the case studies that revolve around this here.
The family of decision tree learning algorithms includes algorithms like ID3, CART, ASSISTANT, etc. They have supervised learning algorithms used for both classification and regression tasks. They
classify the instances by sorting down the decision tree from root to a leaf node that provides the classification of the instance. Each node in the decision tree represents a test of an attribute of
the instance, and a branch descending from that node indicates one of the possible values for that attribute. So, classification of an instance starts at a root node of the decision tree, tests an
attribute at this node, then moves down the tree branch corresponding to the value of the attribute. This process is then repeated for the subtree rooted at the new node.
The main idea of a decision tree algorithm is to identify the features that contain the most information regarding the target feature and then split the dataset along the values of these features.
The target feature values at the resulting nodes are as pure as possible. A feature that best separates the uncertainty from information about the target feature is said to be the most informative
feature. The search process for the most informative feature goes on until we end up with pure leaf nodes.
The process of building an Artificial Intelligence decision tree model involves asking a question at every instance and then continuing with the split - When there are multiple features that decide
the target value of a particular instance, which feature should be chosen as the root node to start the splitting process? And in which order should we continue choosing the features at every further
split at a node? Here comes the need to measure the informativeness of the features and use the feature with the most information as the feature on which the data is split. This informativeness is
given by a measure called ‘information gain.’ And for this, we need to understand the entropy of the dataset.
It is used to measure the impurity or randomness of a dataset. Imagine choosing a yellow ball from a box of just yellow balls (say 100 yellow balls). Then this box is displayed to have 0 entropy
which implies 0 impurity or total purity.
Now, let’s say 30 of these balls are replaced by red and 20 by blue. If we now draw another ball from the box, the probability of drawing a yellow ball will drop from 1.0 to 0.5. Since the impurity
has increased, entropy has also increased while purity has decreased. Shannon’s entropy model uses the logarithm function with base 2 (log2(P(x)) to measure the entropy because as the probability P
(x) of randomly drawing a yellow ball increases, the result approaches closer to binary logarithm 1, as shown in the graph below.
When a target feature contains more than one type of element (balls of different colors in a box), it is useful, to sum up the entropies of each possible target value and weigh it by the probability
of getting these values assuming a random draw. This finally leads us to the formal definition of Shannon’s entropy which serves as the baseline for the information gain calculation:
Where P(x=k) is the probability that a target feature takes a specific value, k.
The logarithm of fractions gives a negative value, and hence a ‘-‘ sign is used in the entropy formula to negate these negative values. The maximum value for entropy depends on the number of classes.
• 2 Classes: Max entropy is 1
• 4 Classes: Max entropy is 2
• 8 Classes: Max entropy is 3
• 16 Classes: Max entropy is 4
Information Gain
To find the best feature that serves as a root node in terms of information gain, we first use each defining feature, split the dataset along the values of these descriptive features, and then
calculate the entropy of the dataset. This gives us the remaining entropy once we have split the dataset along with the feature values. Then, we subtract this value from the initially calculated
entropy of the dataset to see how much this feature splitting reduces the original entropy, which gives the information gain of a feature and is calculated as:
• The feature with the largest entropy information gain should be the root node to build the decision tree.
ID3 algorithm uses information gain for constructing the decision tree.
Gini Index
It is calculated by subtracting the sum of squared probabilities of each class from one. It favors larger partitions and is easy to implement, whereas information gain favors smaller partitions with
distinct values.
A feature with a lower Gini index is chosen for a split.
The classic CART algorithm uses the Gini Index for constructing the decision tree.
Information is a measure of a reduction of uncertainty. It represents the expected amount of information that would be needed to place a new instance in a particular class. These informativeness
measures form the base for any decision tree algorithms. When we use Information Gain that uses Entropy as the base calculation, we have a wider range of results, whereas the Gini Index caps at one.
Resources for further exploration:
Book — Machine learning by Tom M. Mitchell | {"url":"https://www.clairvoyant.ai/blog/entropy-information-gain-and-gini-index-the-crux-of-a-decision-tree","timestamp":"2024-11-07T07:44:08Z","content_type":"text/html","content_length":"118978","record_id":"<urn:uuid:61247fa1-b1fd-4de0-acb5-e06aa44cbb8a>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00449.warc.gz"} |
What amount will sum up to Rs.12100 at \
Hint: We are given a compounded interest of a certain amount which at \[10\%\] per annum in \[2years\] gives a total of Rs.12100. And we are asked in this question to find that certain amount, that
is, we have to find the principal amount. We will use the formula of the compound interest, which is, \[A=P{{\left( 1+\dfrac{R}{100} \right)}^{T}}\]. So, we have to find P, which is the principal
amount. We are given \[A=Rs.12100\], \[R=10\%\] and \[T=2years\]. So, we will substitute these values in the formula of compound interest and we will get the required value of the amount.
Complete step-by-step solution:
According to the given question, we are given that a certain amount when compounded for 2 years at the rate of \[10\%\] per annum gives the total sum as Rs.12100. We are asked in the question to find
the starting or the so called - principal amount.
In order to solve for principal amount, we will use the formula of compound interest and we have it as follows:
\[A=P{{\left( 1+\dfrac{R}{100} \right)}^{T}}\]
Here, A is the compounded value on the amount which in our case is Rs.12100.
P is the principal amount which we have to find.
R is the rate of interest of the compound interest, which is, \[10\%\]
And T is the time period, which is, \[2years\]
We will now substitute these values in the above formula and we get,
\[\Rightarrow 12100=P{{\left( 1+\dfrac{10}{100} \right)}^{2}}\]
Simplifying the above expression, we get,
\[\Rightarrow 12100=P{{\left( 1+\dfrac{1}{10} \right)}^{2}}\]
Taking the LCM in the bracket of the above expression and we have,
\[\Rightarrow 12100=P{{\left( \dfrac{10+1}{10} \right)}^{2}}\]
\[\Rightarrow 12100=P{{\left( \dfrac{11}{10} \right)}^{2}}\]
\[\Rightarrow 12100=P\left( \dfrac{121}{100} \right)\]
We will now write the above expression in terms of P, and we get,
\[\Rightarrow P=12100\times \dfrac{100}{121}\]
Cancelling out the similar terms in the above expression, we get,
\[\Rightarrow P=100\times 100\]
So, we get the value of the principal amount as,
\[\Rightarrow P=Rs.10000\]
Therefore, the principal amount is Rs.10,000.
Note: The formula for the simple interest is as follows, \[S.I=\dfrac{P\times R\times T}{100}\], where P is the principal amount, R is the Rate of interest and T is the time period. Do not get
confused with simple interest for compound interest Both are different from each other. Also, while substituting the values in the formula of compound interest, the values must be correctly
substituted without misplacing any terms. | {"url":"https://www.vedantu.com/question-answer/amount-will-sum-up-to-rs12100-at-10-per-annum-class-8-maths-cbse-60a644a07f6dcb5d8fbaf04e","timestamp":"2024-11-05T00:20:39Z","content_type":"text/html","content_length":"154751","record_id":"<urn:uuid:b53ff84c-6f97-4396-8f6e-71018cf2573b>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00825.warc.gz"} |
Tag Archives:
Little quiz: Ordering/Grouping – Guess the output
Posted on by Sayan Malakshinov Posted in curious, oracle, SQL Leave a comment
How many times have you guessed the right answer? π
select * from dual order by -1;
select * from dual order by 0;
select * from dual order by -(0.1+0/1) desc;
select 1 n,0 n,2 n,0 n,1 n from dual group by grouping sets(1,2,3,2,1,0) order by -(0.1+0/1) desc;
select 1 n,0 n,2 n,0 n,1 n from dual group by grouping sets(1,2,3,2,1,0) order by 0;
select 1 n,0 n,2 n,0 n,1 n from dual group by grouping sets(1,2,3,2,1,0) order by 0+0;
select 1 n,0 n,2 n,0 n,1 n from dual group by grouping sets(1,2,3,2,1,0) order by 3+7 desc;
select 1 n,0 n,2 n,0 n,1 n from dual group by grouping sets(1,2,3,2,1,0) order by -(3.1+0f) desc;
select column_value x,10-column_value y from table(ku$_objnumset(5,4,3,1,2,3,4)) order by 1.9;
select column_value x,10-column_value y from table(ku$_objnumset(5,4,3,1,2,3,4)) order by 2.5;
select column_value x,10-column_value y from table(ku$_objnumset(5,4,3,1,2,3,4)) order by 2.7 desc;
select column_value x,10-column_value y from table(ku$_objnumset(5,4,3,1,2,3,4)) order by -2.7 desc;
Friday prank: select from join join join
Posted on by Sayan Malakshinov Posted in curious, oracle 1 Comment
Valid funny queries π
select the join from join join join using(the,some)
select some join from left join join right using(some,the)
select 1 join from join join join join join using(the) on 1=1
select the some from join
where the=some( the(select some from join)
,the(select the from join)
create table join as select 1 some,1 the from dual;
create table left as select 1 some,1 the from dual;
create table right as select 1 some,1 the from dual;
Little addition π
update two tables set join=2;
select join from two tables;
select first, second, random from two tables join three tables on 1=1;
Another one π
select first
, case when the=some(some,some) then join end true
from two tables join three tables using(random);
select random columns from two tables join four tables on the=some(some,some); | {"url":"https://orasql.org/tag/prank/","timestamp":"2024-11-11T13:03:54Z","content_type":"text/html","content_length":"47900","record_id":"<urn:uuid:f29e0899-1cce-492b-a97d-acbdba64ba70>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00525.warc.gz"} |
Hard-Core Predicates for a Diffie-Hellman Problem over Finite Fields
Paper 2013/134
Hard-Core Predicates for a Diffie-Hellman Problem over Finite Fields
Nelly Fazio, Rosario Gennaro, Irippuge Milinda Perera, and William E. Skeith III
A long-standing open problem in cryptography is proving the existence of (deterministic) hard-core predicates for the Diffie-Hellman problem defined over finite fields. In this paper, we make
progress on this problem by defining a very natural variation of the Diffie-Hellman problem over $\mathbb{F}_{p^2}$ and proving the unpredictability of every single bit of one of the coordinates of
the secret DH value. To achieve our result, we modify an idea presented at CRYPTO'01 by Boneh and Shparlinski [4] originally developed to prove that the LSB of the elliptic curve Diffie-Hellman
problem is hard. We extend this idea in two novel ways: 1. We generalize it to the case of finite fields $\mathbb{F}_{p^2}$; 2. We prove that any bit, not just the LSB, is hard using the list
decoding techniques of Akavia et al. [1] (FOCS'03) as generalized at CRYPTO'12 by Duc and Jetchev [6]. In the process, we prove several other interesting results: - Our result also hold for a larger
class of predicates, called \emph{segment predicates} in [1]; - We extend the result of Boneh and Shparlinski to prove that every bit (and every segment predicate) of the elliptic curve
Diffie-Hellman problem is hard-core; - We define the notion of \emph{partial one-way function} over finite fields $\mathbb{F}_{p^2}$ and prove that every bit (and every segment predicate) of one of
the input coordinates for these functions is hard-core.
Note: Added forgotten acknowledgments
Available format(s)
Publication info
A major revision of an IACR publication in CRYPTO 2013
Contact author(s)
iperera @ gc cuny edu
2013-08-24: last of 11 revisions
2013-03-07: received
Short URL
author = {Nelly Fazio and Rosario Gennaro and Irippuge Milinda Perera and William E. Skeith III},
title = {Hard-Core Predicates for a Diffie-Hellman Problem over Finite Fields},
howpublished = {Cryptology {ePrint} Archive, Paper 2013/134},
year = {2013},
doi = {10.1007/978-3-642-40084-1_9},
url = {https://eprint.iacr.org/2013/134} | {"url":"https://eprint.iacr.org/2013/134","timestamp":"2024-11-14T04:25:13Z","content_type":"text/html","content_length":"16316","record_id":"<urn:uuid:8d0f531f-0d99-469f-81ab-30101244eb18>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00035.warc.gz"} |
0 (number)
From New World Encyclopedia
This page is about the number and digit 0 or "zero."
│0 1 2 3 4 5 6 7 8 9 >> │
│ │
│0 10 20 30 40 50 60 70 80 90 >> │
│ │0 │
│ │zero │
│Cardinal │o/oh │
│ │nought │
│ │naught │
│ │nil │
│Ordinal │0th │
│ │zeroth │
│Factorization │${\displaystyle 0}$ │
│Divisors │N/A │
│Roman numeral │N/A │
│Binary │0 │
│Octal │0 │
│Duodecimal │0 │
│Hexadecimal │0 │
0 (zero) is both a number and a numerical digit used to represent that number in numerals. As a number, zero means nothing—an absence of other values. It plays a central role in mathematics as the
identity element of the integers, real numbers, and many other algebraic structures. As a digit, zero is used as a placeholder in place value systems. Historically, it was the last digit to come into
use. In the English language, zero may also be called nil when a number, o/oh when a numeral, and nought/naught in either context.
0 as a number
0 is the integer that precedes the positive 1, and follows −1. In most (if not all) numerical systems, 0 was identified before the idea of 'negative integers' was accepted.
Zero is an integer which quantifies a count or an amount of null size; that is, if the number of your brothers is zero, that means the same thing as having no brothers, and if something has a weight
of zero, it has no weight. If the difference between the number of pieces in two piles is zero, it means the two piles have an equal number of pieces. Before counting starts, the result can be
assumed to be zero; that is the number of items counted before you count the first item and counting the first item brings the result to one. And if there are no items to be counted, zero remains the
final result.
While mathematicians all accept zero as a number, some non-mathematicians would say that zero is not a number, arguing one cannot have zero of something. Others hold that if you have a bank balance
of zero, you have a specific quantity of money in your account, namely none. It is that latter view which is accepted by mathematicians and most others.
Almost all historians omit the year zero from the proleptic Gregorian and Julian calendars, but astronomers include it in these same calendars. However, the phrase Year Zero may be used to describe
any event considered so significant that it virtually starts a new time reckoning.
0 as a numeral
The modern numeral 0 is normally written as a circle or (rounded) rectangle. In old-style fonts with text figures, 0 is usually the same height as a lowercase x.
On the seven-segment displays of calculators, watches, etc., 0 is usually written with six line segments, though on some historical calculator models it was written with four line segments. This
variant glyph has not caught on.
It is important to distinguish the number zero (as in the "zero brothers" example above) from the numeral or digit zero, used in numeral systems using positional notation. Successive positions of
digits have higher values, so the digit zero is used to skip a position and give appropriate value to the preceding and following digits. A zero digit is not always necessary in a positional number
system: bijective numeration provides a possible counterexample.
The word zero comes through the Arabic literal translation of the Sanskrit śūnya ( शून्य ), meaning void or empty, into ṣifr (صفر) meaning empty or vacant. Through transliteration this became zephyr or
zephyrus in Latin. The word zephyrus already meant "west wind" in Latin; the proper noun Zephyrus was the Roman god of the west wind (after the Greek god Zephyros). With its new use for the concept
of zero, zephyr came to mean a light breeze—"an almost nothing."^[1] This became zefiro in Italian, which was contracted to zero in Venetian, giving the modern English word.
As the Hindu decimal zero and its new mathematics spread from the Arab world to Europe in the Middle Ages, words derived from sifr and zephyrus came to refer to calculation, as well as to privileged
knowledge and secret codes. According to Ifrah, "in thirteenth-century Paris, a 'worthless fellow' was called a… cifre en algorisme, i.e., an 'arithmetical nothing.'"^[1] The Arabic root gave rise to
the modern French chiffre, which means digit, figure, or number; chiffrer, to calculate or compute; and chiffré, encrypted; as well as to the English word cipher. Below are some more examples:
• Arabic: Sifr
• Czech/Slovak: cifra, digit; šifra, cypher
• Danish: ciffer, digit
• Dutch: cijfer, digit
• French: zéro, zero
• German: Ziffer, digit, figure, numeral, cypher
• Hindi: shunya
• Hungarian: nulla
• Italian: cifra, digit, numeral, cypher; zero, zero
• Kannada: sonne
• Norwegian: siffer, digit, numeral, cypher; null, zero
• Persian: Sefr
• Polish: cyfra, digit; szyfrować, to encrypt; zero, zero
• Portuguese: cifra, figure, numeral, cypher, code; zero, zero
• Russian: цифра (tsifra), digit, numeral; шифр (shifr) cypher, code
• Slovenian: cifra, digit
• Spanish: cifra, figure, numeral, cypher, code; cero, zero
• Swedish: siffra, numeral, sum, digit; chiffer, cypher
• Serbian: цифра (tsifra), digit, numeral; шифра (shifra) cypher, code; нула (nula), zero
• Turkish: Sıfır
• Urdu: Sifer, Anda, Zero
Note that zero in Greek is translated as Μηδέν (Mèden).
Did you know?
0 (zero) was the last numerical digit to come into use
Early history of zero
By the mid-second millennium B.C.E., the Babylonians had a sophisticated sexagesimal (base-60) positional numeral system. The lack of a positional value (or zero) was indicated by a space between
sexagesimal numerals. By 300 B.C.E. a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish (dating from perhaps as far
back as 700 B.C.E.), the scribe Bêl-bân-aplu wrote his zeroes with three hooks, rather than two slanted wedges.^[2]
The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60), etc.
looked the same because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.
Records show that the ancient Greeks seemed unsure about the status of zero as a number: they asked themselves "How can nothing be something?," leading to interesting philosophical and, by the
Medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero. (The ancient
Greeks even questioned that 1 was a number.)
Early use of something like zero by the Indian scholar Pingala (circa 5th-2nd century B.C.E.), implied at first glance by his use of binary numbers, is only the modern binary representation using 0
and 1 applied to Pingala's binary system, which used short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code.^[3]^[4] Nevertheless, he and other
Indian scholars at the time used the Sanskrit word śūnya (the origin of the word zero after a series of transliterations and a literal translation) to refer to zero or void.^[5]
History of zero
The Long Count calendar developed in south-central Mexico required the use of zero as a place-holder within its vigesimal (base-20) positional numeral system. A shell glyph— —was used as a zero
symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, Chiapas) has a date of 36 B.C.E. Since the eight earliest Long Count dates appear outside the Maya homeland,^
[6] it is assumed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs. Indeed, many of the earliest Long Count dates were found within the Olmec
heartland, although the fact that the Olmec civilization had come to an end by the fourth century B.C.E., several centuries before the earliest known Long Count dates, argues against the zero being
an Olmec invention.
Although zero became an integral part of Maya numerals, it of course did not influence Old World numeral systems.
By 130 C.E., Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic
Greek numerals. Because it was used alone, not just as a placeholder, this Hellenistic zero was perhaps the first documented use of a number zero in the Old World. However, the positions were usually
limited to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)—they were not used for the integral part of a number.
Another zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla, meaning nothing, not as a symbol. When division produced zero as a
remainder, nihil, also meaning nothing, was used. These medieval zeros were used by all future medieval computists (calculators of Easter). An isolated use of their initial, N, was used in a table of
Roman numerals by Bede or a colleague about 725, as a zero symbol.
The oldest known text to use zero is the Jain text from India entitled the Lokavibhaaga, dated 458 C.E.^[1]
The first indubitable appearance of a symbol for zero appears in 876 in India on a stone tablet in Gwalior. Documents on copper plates, with the same small o in them, dated back as far as the sixth
century C.E. abound.^[2]
Rules of Brahmagupta
The rules governing the use of zero appeared for the first time in the book Brahmasputha Siddhanta written in 628 by Brahmagupta (598-670). Here, Brahmagupta considers not only zero but also negative
numbers, and the algebraic rules for the elementary operations of arithmetic with such numbers. In some instances, his rules differ from the modern standard. Brahamagupta's rules are given below:^[7]
• The sum of two positive quantities is positive
• The sum of two negative quantities is negative
• The sum of zero and a negative number is negative
• The sum of a positive number and zero is positive
• The sum of zero and zero is zero
• The sum of a positive and a negative is their difference; or, if they are equal, zero
• In subtraction, the less is to be taken from the greater, positive from positive
• In subtraction, the less is to be taken from the greater, negative from negative
• When the greater however, is subtracted from the less, the difference is reversed
• When positive is to be subtracted from negative, and negative from positive, they must be added together
• The product of a negative quantity and a positive quantity is negative
• The product of a negative quantity and a negative quantity is positive
• The product of two positive, is positive
• Positive divided by positive or negative by negative is positive
• Positive divided by negative is negative. Negative divided by positive is negative
• A positive or negative number when divided by zero is a fraction with the zero as denominator
• Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator
• Zero divided by zero is zero
In saying "zero divided by zero is zero," Brahmagupta differs from the modern position. Mathematicians normally do not assign a value, whereas computers and calculators will sometimes assign NaN,
which means "not a number." Moreover, non-zero positive or negative numbers when divided by zero are either assigned no value, or a value of unsigned infinity, positive infinity, or negative
infinity. Once again, these assignments are not numbers, and are associated more with computer science than pure mathematics, where in most contexts no assignment is made. (See division by zero)
Zero as a decimal digit
Positional notation without the use of zero (using an empty space in tabular arrangements, or the word kha "emptiness") is known to have been in use in India from the sixth century. The earliest
certain use of zero as a decimal positional digit dates to the ninth century. The glyph for the zero digit was written in the shape of a dot, and consequently called bindu "dot."
The Hindu-Arabic numeral system reached Europe in the eleventh century, via the Iberian Peninsula through Spanish Muslims the Moors, together with knowledge of astronomy and instruments like the
astrolabe, first imported by Gerbert of Aurillac (c. 940-1003). They came to be known as "Arabic numerals." The Italian mathematician Leonardo of Pisa (c. 1170-1250), also called Fibonacci,* was
instrumental in bringing the system into European mathematics in 1202. Here, Leonardo states:
There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and
for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods… But all this even, and the algorism, as well as the art of
Pythagoras, I considered as almost a mistake in respect to the method of the Hindus. (Modus Indorum)… The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0
… any number may be written.^[8]
Here Leonardo of Pisa uses the word sign "0," indicating it is like a sign to do operations like addition or multiplication, but he did not recognize zero as a number in its own right.
In mathematics
Elementary algebra
Zero (0) is the lowest non-negative integer. The natural number following zero is one and no natural number precedes zero. Zero may or may not be counted as a natural number, depending on the
definition of natural numbers.
In set theory, the number zero is the cardinality of the empty set: if one does not have any apples, then one has zero apples. Therefore, in some cases, zero is defined to be the empty set.
Zero is neither positive nor negative, neither a prime number nor a composite number, nor is it a unit.
The following are some basic rules for dealing with the number zero. These rules apply for any complex number x, unless otherwise stated.
• Addition: x + 0 = 0 + x = x. (That is, 0 is an identity element with respect to addition.)
• Subtraction: x − 0 = x and 0 − x = − x.
• Multiplication: x · 0 = 0 · x = 0.
• Division: 0 / x = 0, for nonzero x. But x / 0 is undefined, because 0 has no multiplicative inverse, a consequence of the previous rule. For positive x, as y in x / y approaches zero from
positive values, its quotient increases toward positive infinity, but as y approaches zero from negative values, the quotient increases toward negative infinity. The different quotients confirms
that division by zero is undefined.
• Exponentiation: x^0 = 1, except that the case x = 0 may be left undefined in some contexts. For all positive real x, 0^x = 0.
• The sum of 0 numbers is 0, and the product of 0 numbers is 1.
The expression "0/0" is an "indeterminate form." That does not simply mean that it is undefined; rather, it means that if f(x) and g(x) both approach 0 as x approaches some number, then f(x)/g(x)
could approach any finite number or ∞ or −∞; it depends on which functions f and g are. See L'Hopital's rule.
Extended use of zero in mathematics
• Zero is the identity element in an additive group or the additive identity of a ring.
• A zero of a function is a point in the domain of the function whose image under the function is zero. When there are finitely many zeros these are called the roots of the function. See zero
(complex analysis).
• In geometry, the dimension of a point is 0.
• The concept of "almost" impossible in probability. More generally, the concept of almost nowhere in measure theory. For instance: if one chooses a point on a unit line interval [0,1) at random,
it is not impossible to choose 0.5 exactly, but the probability that you will is zero.
• A zero function (or zero map) is a constant function with 0 as its only possible output value; i.e., f(x) = 0 for all x defined. A particular zero function is a zero morphism in category theory;
e.g., a zero map is the identity in the additive group of functions. The determinant on non-invertible square matrices is a zero map.
• Zero is one of three possible return values of the Möbius function. Passed an integer of the form x^2 or x^2y (for x > 1), the Möbius function returns zero.
• Zero is the first Perrin number.
In science
The value zero plays a special role for a large number of physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or
less arbitrarily chosen. For example, on the Kelvin temperature scale, zero is the coldest possible temperature (negative temperatures exist but are not actually colder), whereas on the Celsius
scale, zero is arbitrarily defined to be at the freezing point of water. Measuring sound intensity in decibels or phons, the zero level is arbitrarily set at a reference value—for example, at a value
for the threshold of hearing.
Zero has been proposed as the atomic number of the theoretical element tetraneutronium. It has been shown that a cluster of four neutrons may be stable enough to be considered an atom in their own
right. This would create an element with no protons and no charge on its nucleus.
As early as 1926 Professor Andreas von Antropoff coined the term neutronium for a conjectured form of matter made up of neutrons with no protons, which he placed as the chemical element of atomic
number zero at the head of his new version of the periodic table. It was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the
chemical elements. It is at the center of the Chemical Galaxy (2005).
In computer science
Numbering from 1 or 0?
The most common practice throughout human history has been to start counting at one. Nevertheless, in computer science zero has become the standard starting point. For example, in almost all old
programming languages, an array starts from 1 by default. As programming languages have developed, it has become more common that an array starts from zero by default, the "first" item in the array
being item 0. In particular, the popularity of the programming language "C" in the 1980s has made this approach common.
One reason for this convention is that modular arithmetic normally describes a set of N numbers as containing 0,1,2,… N-1 in order to contain the additive identity. Because of this, many arithmetic
concepts (such as hash tables) are less elegant to express in code unless the array starts at zero.
In certain cases, counting from zero improves the efficiency of various algorithms, such as in searching or sorting arrays. Improved efficiency means that the algorithm takes either less time, less
resources, or both, to complete a given task.
This situation can lead to some confusion in terminology. In a zero-based indexing scheme, the first element is "element number zero"; likewise, the twelfth element is "element number eleven."
Therefore, an analogy from the ordinal numbers to the quantity of objects numbered appears; the highest index of n objects will be (n-1) and referred to the n:th element. For this reason, the first
element is often referred to as the zeroth element to eliminate any possible doubt.
Null value
In databases a field can have a null value. This is equivalent to the field not having a value. For numeric fields it is not the value zero. For text fields this is not blank nor the empty string.
The presence of null values leads to three-valued logic. No longer is a condition either true or false, but it can be undetermined. Any computation including a null value delivers a null result.
Asking for all records with value 0 or value not equal 0 will not yield all records, since the records with value null are excluded.
Null pointer
A null pointer is a pointer in a computer program that does not point to any object or function, which means that when it appears in a program or code, it tells the computer to take no action on the
associated portion of the code.
Negative zero
In some signed number representations (but not the two's complement representation predominant today) and most floating point number representations, zero has two distinct representations, one
grouping it with the positive numbers and one with the negatives; this latter representation is known as negative zero. Representations with negative zero can be troublesome, because the two zeros
will compare equal but may be treated differently by some operations.
Distinguishing zero from O
The oval-shaped zero and circular letter O together came into use on modern character displays. The zero with a dot in the center seems to have originated as an option on IBM 3270 controllers (this
has the problem that it looks like the Greek letter Theta). The slashed zero, looking identical to the letter O other than the slash, is used in old-style ASCII graphic sets descended from the
default typewheel on the venerable ASR-33 Teletype. This format causes problems because of its similarity to the symbol ∅, representing the empty set, as well as for certain Scandinavian languages
which use Ø as a letter.
The convention which has the letter O with a slash and the zero without was used at IBM and a few other early mainframe makers; this is even more problematic for Scandinavians because it means two of
their letters collide. Some Burroughs/Unisys equipment displays a zero with a reversed slash. And yet another convention common on early line printers left zero unornamented but added a tail or hook
to the letter-O so that it resembled an inverted Q or cursive capital letter-O.
The typeface used on some European number plates for cars distinguish the two symbols by making the zero rather egg-shaped and the O more circular, but most of all by slitting open the zero on the
upper right side, so the circle is not closed any more (as in German plates). The typeface chosen is called fälschungserschwerende Schrift (abbr.: FE Schrift), meaning "unfalsifiable script." Note
that those used in the United Kingdom do not differentiate between the two as there can never be any ambiguity if the design is correctly spaced.
In paper writing one may not distinguish the 0 and O at all, or may add a slash across it in order to show the difference, although this sometimes causes ambiguity in regard to the symbol for the
null set.
The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habitation and a name, a picture, a symbol, but helpful power, is the
characteristic of the Hindu race from whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and
power. G. B. Halsted
…a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it lent to all computations put our arithmetic in
the first rank of useful inventions. Pierre-Simon Laplace
The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is
only forced on us by the needs of cultivated modes of thought. Alfred North Whitehead
…a fine and wonderful refuge of the divine spirit—almost an amphibian between being and non-being. Gottfried Leibniz
In other fields
• In some countries, dialing 0 on a telephone places a call for operator assistance.
• In Braille, the numeral 0 has the same dot configuration as the letter J.
• DVDs that can be played in any region are sometimes referred to as being "region 0"
• In classical music, 0 is very rarely used as a number for a composition, the only two examples on the fringes of the standard repertoire probably being Anton Bruckner's Symphony No. 0 in D minor
and Alfred Schnittke's Symphony No. 0
• In tarot, card No. 0 is the Fool
See also
1. ↑ ^1.0 ^1.1 ^1.2 Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer (Wiley, 2000, ISBN 0471393401).
2. ↑ ^2.0 ^2.1 Robert Kaplan and Ellen Kaplan, The Nothing That Is: A Natural History of Zero (Oxford: Oxford University Press, 2000, ISBN 978-0195142372).
3. ↑ ICA.net, Binary Numbers in Ancient India with information from scholarly article by B. van Nooten, "Binary Numbers in Indian Antiquity", Journal of Indian Studies 21 (1993): 31-50. Retrieved
September 19, 2017.
4. ↑ Rachel Hall, "Math for Poets and Drummers" Saint Joseph's University, February 15, 2005. Retrieved September 19, 2017.
5. ↑ Kim Plofker, Mathematics in India (Princeton University Press, 2009, ISBN 978-0691120676).
6. ↑ Richard A. Diehl, The Olmecs: America's First Civilization (London: Thames & Hudson, 2005, ISBN 0500285039), 186.
7. ↑ Henry Thomas Colebrooke, Algebra with Arithmetic of Brahmagupta and Bhaskara (1817).
8. ↑ Laurence Sigler, Fibonacci's Liber Abaci(Springer, 2002, ISBN 978-0387954196).
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Truncated sum of squares estimation of fractional time series models with deterministic trends
We consider truncated (or conditional) sum of squares estimation of a parametric model composed of a fractional time series and an additive generalized polynomial trend. Both the memory parameter,
which characterizes the behavior of the stochastic component of the model, and the exponent parameter, which drives the shape of the deterministic component, are considered not only unknown real
numbers but also lying in arbitrarily large (but finite) intervals. Thus, our model captures different forms of nonstationarity and noninvertibility. As in related settings, the proof of consistency
(which is a prerequisite for proving asymptotic normality) is challenging due to nonuniform convergence of the objective function over a large admissible parameter space, but, in addition, our
framework is substantially more involved due to the competition between stochastic and deterministic components. We establish consistency and asymptotic normality under quite general circumstances,
finding that results differ crucially depending on the relative strength of the deterministic and stochastic components. Finite-sample properties are illustrated by means of a Monte Carlo experiment.
Dyk ned i forskningsemnerne om 'Truncated sum of squares estimation of fractional time series models with deterministic trends'. Sammen danner de et unikt fingeraftryk. | {"url":"https://pure.au.dk/portal/da/publications/truncated-sum-of-squares-estimation-of-fractional-time-series-mod-2","timestamp":"2024-11-07T17:22:05Z","content_type":"text/html","content_length":"57764","record_id":"<urn:uuid:7bf7fea4-6a25-406f-9a20-81dbad6206e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00160.warc.gz"} |
Implied Volatility
What Is Implied Volatility?
Implied volatility (IV) is one of the most important concepts in options trading. Unfortunately it’s also one of the most complex.
Therefore, let’s build up the concept slowly with an understanding firstly of historical volatility as an estimate of an option’s risk, then we’ll look at implied volatility and how this relates to
options pricing and finally where a consideration of IV is important in practice.
Historical Volatility
The volatility of a stock is how much it moves up and down, its risk in other words, expressed by its standard deviation. If you’re not familiar with this risk concept, this video explains it well.
If you’re not interested in the stats, just see standard deviation as a measure of how risky a stock is. A large multinational paper wholesale is going to be less risky than, say, a new start up yet
to make a profit and hence we would expect the latter to have a larger volatility.
Historical volatility is just what this measure has been, well, historically and is used as an estimate of volatility now. Note this is just an estimate: there are many reasons for a stock’s
volatility to change (eg a new product launch, an imminent US election etc).
Thankfully we can calculate historical volatility:
How To Calculate Historical Volatility
• Download the end of day stock prices for a decent period before now (6-12 months, say)
• Calculate the mean, or average, price for the period.
• For each day calculate the difference between the stock price and
• Sum all these results (the ‘sum of the squares’)
• Divide by the number of days less one (ie find the average of these square numbers). So if you have 365 days of data, divide the sum of squares by 364.
• Calculate the square root of this average. This is the historical volatility.
Implied Volatility & Options Pricing
Before defining implied volatility we need to discuss how an option is priced.
We’ve gone into this in more detail here, but in summary an option’s fair value (ie what you should pay for it) depends on 5 things:
• The price of the underlying stock or financial instrument
• The exercise, or strike, price of the option (and whether it’s a put or call)
• The number of days to expiry
• Interest rates
• Volatility
In particular if the first 4 factors remain the same, more volatile stocks’ options will be more expensive.
This makes sense. There is a greater chance of a stock ending up in-the-money, and hence being exercised, if is is more prone to jump around. An option seller should be compensated for this higher
Using historical volatility as an estimate for volatility, as above, we can therefore calculate the fair price of any option.
Volatility Implied By The Market
That’s great, but what about implied volatility? Well, in practice, the market only uses historical volatility as a guide to future volatility. In reality the market is constantly expressing its view
on what it believes will be the volatility over the remaining life of an option.
How does it do this? Via the market price of an option.
The other factors are fixed (over a short period of time). The only way an option can rise or fall in value is if the market changes its view of the stock’s volatility.
If the market perceives that a stock has become riskier, say, then all things being equal a stock’s option price will rise (and vice versa).
The option price ‘implies’ a volatility figure in the above calculation – because the other factors are fixed and known.
Given a market price of an option – and knowing its strike price, the price of the underlying, the days to expiry and interest rates – we can reverse engineer the option pricing calculation to work
out what the market’s view of the stock’s volatility actually is.
If we know an option price (which in an open market, we do) we know the market’s view on volatility.
This ‘implied’ volatility is, well, implied volatility. It’s the market’s view at a point of time of the riskiness of a stock which has been priced into the stock’s options.
Implied Volatility In Practice
There are a few things where a consideration of IV is important:
Vega: How A Change in IV affects option price
We’ve already stated that an increase in IV increases an option’s price. Vega, one of the options greeks, is a measure of how much it changes – how sensitive an option price is to implied volatility.
The percentage change in option price given a 1% change in IV – all other things being equal – is Vega.
VIX Index
The CBOE runs the VIX index – an average (sort of) of the IVs in the market. It’s therefore a measure of how risky the market views stocks and is a useful overview of risk.
Volatility smile
Volatility Smile
Options theory tends to assume that implied volatility is the same for all options for the same underlying and expiry date, whatever its strike price.
In practise, however, the market seems to value out of the money options (especially puts) at a higher IV than those at the money. This is the ‘volatility smile’, – a reference of the shape if the
graph of volatility to strike price. | {"url":"https://epsilonoptions.com/implied-volatility/","timestamp":"2024-11-03T10:52:13Z","content_type":"text/html","content_length":"152854","record_id":"<urn:uuid:b924728f-aced-4b78-906b-00b79dc0af64>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00893.warc.gz"} |
ULTIMATE SYMMETRY - III.3 Equivalence Principle and General
III.3 Equivalence Principle and General Relativity
In moving from Special to General Relativity, Einstein observed the equivalence between the gravitational force and the inertial force experienced by an observer in a non-inertial frame of reference.
This is roughly the same as the equivalence between active gravitational and passive inertial masses, which has been later accurately tested in many experiments [30, 28], but there is no direct
mathematical derivation to this principle apart from the famous spacecraft accelerator thought experiment which relies on induction.
When Einstein combined this equivalence principle with the two principles of Special Relativity, he was able to predict the curved geometry of space-time, which is directly related to its contents of
energy and momentum of matter and radiation, through a system of partial differential equations known as Einstein field equations.
We explained in section III.2 above that an exact derivation of the mass-energy equivalence relation is not possible without postulating the inner levels of time, and that is why there is yet no
single exact derivation of this celebrated equation. For the same reason indeed, there is also no mathematical derivation of the equivalence principle that relates gravitation with geometry, because
it is actually equivalent to the same relation that reflects the fact that space and matter are always being perpetually re-created in the inner time, i.e. fluctuating between the particle state and
wave state , thus causing space-time deformation and curvature.
Due to the discrete structure of the genuinely-complex time-time geometry, as illustrated in Figure I.4, the complex momentum should be invariant between inertial and non-inertial frames alike,
because effectively all objects are always at rest in the outer level of time, as we explained in section I.4.4 above. This means that complex momentum is always conservedeven when the velocity
changes, i.e. as the object accelerates between non-inertial frames!
This invariance of momentum between non-inertial frames is conceivable, because it means that as the velocity increases (for example), the gain in kinetic momentum (that is the imaginary part) is
compensated by the increase in the effective mass: due to acceleration, which causes the real part also to increase, but since is hyperbolic, thus its modulus remains invariant, and this what makes
the geometry of space (manifested here as ) dynamic, because it must react to balance the change in effective mass. Therefore, a closed system is closed only when we include all its contents of mass
and energy (including kinetic and radiation) as well as the background space itself, which is the vacuum state , and the momentum of all these constituents is either , when they are re-created in the
inner levels, or for physical objects moving in the normal level of time. For such a conclusive system, the complex momentum is absolutely invariant.
Actually, without this exotic property of momentum it is not possible at all to obtain an exact derivation of which is equivalent to as we mentioned in section III.2 above. These experimentally
verified equations are correct if,and only if, the modulus is always conserved. For example when the object accelerates from zero to , and then the effective mass changes from to , thus we get ,
Therefore, in addition to the previous methods in equations 3.9 and 3.10, and the relativistic energy-momentum relation in equation 3.18, the mass-energy equivalence relation: can now be deduced from
equation 3.19, because, as we mentioned in section III.2.5 above, the equations: and are equivalent, and the derivation of one of them leads to the other, while there is no exact derivation of either
form in the current formulation of Special or General Relativity.
This absolute conservation of complex momentum under acceleration leads directly to the equivalence between active and passive masses, because it means that the total (complex) force: must have two
components; one that is related to acceleration as changes in the outer time , which is the imaginary part, and this causes the acceleration: , so here is the passive mass, while the other force is
related to the change in effective mass , or its equivalent energy , which is manifested as the deformation of space which is being re-created in the inner levels of time , and this change or
deformation is causing the gravitational force that is associated with the active mass; and these two components must be equivalent so that the total resulting complex momentum remains conserved.
Therefore, gravitation is a reaction against the disturbance of space from the ground state of bosonic vacuum to the state of fermionic particles , the first is associated with the active mass in the
real momentum , and the second is associated with the passive mass in the imaginary momentum .
However, as discussed further in section II.4, because of the fractal dimensions of the new complex-time discrete geometry, performing the differentiation of this complex function requires
non-standard analysis because space-time is no longer everywhere differentiable [21]. So this will not be pursued in this article.
From this conservation of complex momentum we should be able to find the law of gravitation and the stress-energy-momentum tensor which leads to the field equations of General Relativity. Moreover,
since empty space is now described as the dynamic aether, gravitational waves become the longitudinal vibrations in this ideal medium, and the graviton will be simply the moment of time , just as
photons are the quanta of electromagnetic radiations and they are transverse waves in this vacuum, or the moments of space . This means that equivalence principle is essentially between photons and
gravitons, or between space and time, while electrons and some other particles could be described as standing waves in the space-time; with complex momentum , and the reason why we have three
generations of fermions is due to the three dimensions of space. This important conclusion requires further investigation, but we should also notice here that the equivalence principle should apply
equally to all fundamental forces and not only to gravity, because it is a property of space-time geometry in all dimensions, and not only the where gravity is exhibited, as it is also outlined in
another publication [12]. | {"url":"https://www.smonad.com/symmetry/book.php?id=87","timestamp":"2024-11-10T15:23:39Z","content_type":"text/html","content_length":"37126","record_id":"<urn:uuid:df9a06cb-7af8-44ce-8f75-7e75be149831>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00515.warc.gz"} |
ACM Other ConferencesWhat Makes a Variant of Query Determinacy (Un)Decidable? (Invited Talk)
This paper was written as the companion paper of the ICDT 2020 invited tutorial. Query determinacy is a broad topic, with literally hundreds of papers published since late 1980s. This paper is not
going to be a "survey" but rather a personal perspective of a person somehow involved in the recent developments in the area.
First I explain how, in the last 30+ years, the question of determinacy was formalized. There are many parameters here: obviously one needs to choose the query language of the available views and the
query language of the query itself. But - surprisingly - there is also some choice regarding what the word "to compute" actually means in this context.
Then I concentrate on certain variants of the decision problem of determinacy (for each choice of parameters there is one such problem) and explain how I understand the mechanisms rendering such
variants of determinacy decidable or undecidable. This is on a rather informal level. No really new theorems are presented, but I show some improvements of existing theorems and also simplified
proofs of some of the earlier results. | {"url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2020.2/metadata/acm-xml","timestamp":"2024-11-07T06:05:43Z","content_type":"application/xml","content_length":"9907","record_id":"<urn:uuid:bac99ced-eaf8-4731-9491-f05de3c7f8a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00377.warc.gz"} |
Contemporary Mathematics for Business & Consumers
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Overcome your math anxiety and confidently master key mathematical concepts and their business applications with Brechner/Bergeman's CONTEMPORARY MATHEMATICS FOR BUSINESS AND CONSUMERS, 8E. Refined
and enhanced over eight editions, this text continues to incorporate a proven step-by-step instructional model that allows you to progress one topic at a time without being intimidated or
overwhelmed. This edition offers a reader-friendly design with a wealth of engaging learning features that connect the latest business news to chapter topics and provide helpful personal money tips.
You will immediately practice concepts to reinforce learning and hone essential skills with more than 2,000 proven exercises. Jump Start problems introduce each new topic in the section exercise sets
and provide a worked-out solution to help you get started.
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Related Math Textbooks with Solutions
Homework Help by Math Subjects | {"url":"https://www.bartleby.com/textbooks/contemporary-mathematics-for-business-and-consumers-8th-edition/9781305585447/solutions","timestamp":"2024-11-09T09:06:29Z","content_type":"text/html","content_length":"480042","record_id":"<urn:uuid:aa26bb47-68f0-41af-9e76-22a76ef96e25>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00651.warc.gz"} |
Salt flat laser experement
I think rowbotham said refraction is 1/6 i think that 1/6 of the final answer as in 1/6 of the calculated curvature. Can anyone clarify?
Bonneville salt flats 12 miles = 8' earth curve.Level laser for 1 mile. the end of the flats the beam should be 8' above the starting point. | {"url":"https://www.theflatearthsociety.org/forum/index.php?topic=89479.msg2352287","timestamp":"2024-11-08T10:22:43Z","content_type":"application/xhtml+xml","content_length":"34720","record_id":"<urn:uuid:0ec3d634-a61c-4906-bc2b-6052a5e46f3c>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00318.warc.gz"} |
CERIST DL :: Browsing by Author "Kärkkäinen, Juha"
Browsing by Author "Kärkkäinen, Juha"
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• Block Trees
(ELSEVIER, 2021-05) Belazzougui, Djamal; Cáceres, Manuel; Gagie, Travis; Gawrychowski, Paweł; Kärkkäinen, Juha; Navarro, Gonzalo; Ordóñez, Alberto; Puglisi, Simon J.; Tabei, Yasuo
Let string S[1..n] be parsed into z phrases by the Lempel-Ziv algorithm. The corresponding compression algorithm encodes S in 𝒪(z) space, but it does not support random access to S. We introduce
a data structure, the block tree, that represents S in 𝒪(z log(n/z)) space and extracts any symbol of T in time 𝒪(log(n/z)), among other space-time tradeoffs. By multiplying the space by the
alphabet size, we also support rank and select queries, which are useful for building compressed data structures on top of S. Further, block trees can be built in a scalable manner. Our
experiments show that block trees offer relevant space-time tradeoffs compared to other compressed string representations for highly repetitive strings.
• Lempel-Ziv Decoding in External Memory
(Springer International Publishing, 2016-06-01) Belazzougui, Djamal; Kärkkäinen, Juha; Kempa, Dominik; Puglisi, Simon J.
Simple and fast decoding is one of the main advantages of LZ77-type text encoding used in many popular file compressors such as gzip and 7zip. With the recent introduction of external memory
algorithms for Lempel–Ziv factorization there is a need for external memory LZ77 decoding but the standard algorithm makes random accesses to the text and cannot be trivially modified for
external memory computation. We describe the first external memory algorithms for LZ77 decoding, prove that their I/O complexity is optimal, and demonstrate that they are very fast in practice,
only about three times slower than in-memory decoding (when reading input and writing output is included in the time).
• Linear-time string indexing and analysis in small space
(Association for Computing Machinery, 2020-03-09) Belazzougui, Djamal; Cunial, Fabio; Kärkkäinen, Juha; Mäkinen, Veli
The field of succinct data structures has flourished over the last 16 years. Starting from the compressed suffix array by Grossi and Vitter (STOC 2000) and the FM-index by Ferragina and Manzini
(FOCS 2000), a number of generalizations and applications of string indexes based on the Burrows-Wheeler transform (BWT) have been developed, all taking an amount of space that is close to the
input size in bits. In many large-scale applications, the construction of the index and its usage need to be considered as one unit of computation. For example, one can compare two genomes by
building a common index for their concatenation, and by detecting common substructures by querying the index. Efficient string indexing and analysis in small space lies also at the core of a
number of primitives in the data-intensive field of high-throughput DNA sequencing. We report the following advances in string indexing and analysis. We show that the BWT of a string T ∈ {1, . .
. ,σ }^n can be built in deterministic O (n) time using just O (n log σ ) bits of space, where σ ≤ n. Deterministic linear time is achieved by exploiting a new partial rank data structure that
supports queries in constant time, and that might have independent interest. Within the same time and space budget, we can build an index based on the BWT that allows one to enumerate all the
internal nodes of the suffix tree of T . Many fundamental string analysis problems, such as maximal repeats, maximal unique matches, and string kernels, can be mapped to such enumeration, and can
thus be solved in deterministic O (n) time and in O (n log σ ) bits of space from the input string, by tailoring the enumeration algorithm to some problem-specific computations. We also show how
to build many of the existing indexes based on the BWT, such as the compressed suffix array, the compressed suffix tree, and the bidirectional BWT index, in randomized O (n) time and in O (n log
σ ) bits of space. The previously fastest construction algorithms for BWT, compressed suffix array and compressed suffix tree, which used O (n log σ ) bits of space, took O (n log log σ ) time
for the first two structures, and O (n log^ϵ n) time for the third, where ϵ is any positive constant smaller than one. Alternatively, the BWT could be previously built in linear time if one was
willing to spend O (n log σ log log_σ n) bits of space. Contrary to the state of the art, our bidirectional BWT index supports every operation in constant time per element in its output. | {"url":"https://dl.cerist.dz/browse/author?value=K%C3%A4rkk%C3%A4inen,%20Juha","timestamp":"2024-11-06T05:36:47Z","content_type":"text/html","content_length":"379597","record_id":"<urn:uuid:7022af87-1908-4588-9575-ec879495975e>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00342.warc.gz"} |
s - Darren Zhang
This section allows you to view all posts made by this member. Note that you can only see posts made in areas you currently have access to.
In 3b, harmonic conjugate function v(x, y) is $3 - 12x^2y^2 + 2x^42y^4+5y+C$, does anyone know where the constant $3$ comes from? | {"url":"https://forum.math.toronto.edu/index.php?PHPSESSID=5htdknalpdoamf6v2neoghfad6&action=profile;area=showposts;sa=messages;u=1009","timestamp":"2024-11-04T08:03:02Z","content_type":"application/xhtml+xml","content_length":"35689","record_id":"<urn:uuid:b3a83702-c3f6-4890-ad56-683d63ab934e>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00800.warc.gz"} |