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MIS 462 Summer2022 Formative Assignment
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145 points
Review Problems, Chapters 1 to 6
Do all of these problems in a single Excel Workbook.
(25) When done, upload your file, named Formative01.xlsx, to the D2L Assignment Folder 'Formative01'.
Solve each problem on a separate Worksheet.
Your Excel File must look like this-notice the Worksheet tab names:
(50) When done, upload your file, named Formative01.xlsx, to the D2L Dropbox Folder Formative01.
Named Ranges-Chapter 1, Problem 1
1. (10) The file Stock.xlsx contains monthly stock returns for General Motors and Microsoft.
Name the ranges containing the monthly returns for each stock ("GM" and "MSFT").
Use these named ranges in a forumla that computes the average monthly return for each stock.
Lookup-Chapter 2, Problem 4
2. (10) You are thinking of advertising Microsoft products on a popular TV music program.
You pay one price for the first group of ads, but as you buy more ads, the price per ad drops as described in the following table:
Number of Ads Price Per Ad
1-5 $12,000
6-10 $11,000
11-20 $10,000
More than 20 $9,000
Write a formula that yields the total cost of purchasing any number of ads.
Do the calculation for the following ad quantities: 22, 3, 7 and 13
Index-Chapter 3, Problem 4
3. (10) Use the file Product.xlsx which contains monthly sales for six products.
Use the INDEX function to compute the sales of product 2 in March.
Use the INDEX function in a formula that computes total sales during April.
Match-Chapter 4, Problem 3
The file MatchTheMax.xlsx gives the product ID code and unit sales for 265 different products.
Use the MATCH function in a formula that yields the product ID of the product with the largest unit sales.
4. (10) Write the Product ID of that product below:
Text-Chapter 5, Problem 3
5. (10) The workbook QuarterlyGnpData.xlsx contains quarterly GNP data for the United States (in billions of 1996 dollars) in the format shown here.
Extract this data to three separate columns, where the first column contains the year, the second column contains the quarter number, and the third column contains the GNP value.
Dates-Chapter 6, Problem 2
6. (10) What is the serial format for February 14, 1950?
Dates-Chapter 6, Problem 8
7. (10) How many workdays, excluding Christmas, New Year's Day and the 4th of July, are there between July 10, 2005 and August 15, 2006? | {"url":"https://eprofessor.azurewebsites.net/MIS462/Formative/Formative01/Formative01.asp","timestamp":"2024-11-13T19:19:04Z","content_type":"application/xhtml+xml","content_length":"7870","record_id":"<urn:uuid:ee69fedf-64af-4aeb-9609-f8b4e888d42f>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00870.warc.gz"} |
How do I calculate depreciation using the sum of the years’ digits?
[restrict paid=true]
It enhances how one views the utility of fixed assets whilst resulting in tax shields for tech company ABC. An accelerated depreciation method, the double-declining method calculates depreciation
twice as fast as that in the declining balance method. It records a larger depreciation in the earlier years of the asset’s useful life. Companies typically use accelerated depreciation to minimize
their taxable income because it allows for greater depreciation expense deductions in the earlier years of the equipment or asset’s life. Accelerated depreciation methods could also be seen as more
accurate, as they assume that an asset loses a majority of its value in the first few years of its use. The formula to calculate the sum of the years’ digits depreciation divides the remaining useful
life by the sum of the years’ digits of the fixed asset (PP&E), which is then multiplied by the depreciable basis.
1. To find the SYD function on Excel, one must navigate to the formulas tab and click on the Financials drop-down menu where it can be seen.
2. Two depreciation schedules are created using the formula approach and function approach to compare the SYD formula and its corresponding Excel function.
3. Therefore, a decreasing depreciation charge will help balance the cost of maintenance of the asset.
The companies need to measure this deterioration and calculate the values of the assets as it affects their business. To illustrate SYD depreciation, assume that a service business purchases
equipment at a cost of $160,000. This asset is expected to have a useful life of 5 years at which time it will be sold for $10,000. This means that the total amount of depreciation will be $150,000
spread over the equipment’s useful life of 5 years. When looking at the function’s syntax, it can be seen how the Per component changes for each year, leading to different depreciation expenses.
Repair and Maintenance Costs
The method facilitates the calculation when the asset performance is at its highest. The sum of years’ method matches the cost of utilizing an asset and the overall utility of the asset across the
economic or useful life of the asset. A major benefit of using this method is that it considers the fact that the asset performance will decline over the years; i.e. the asset is more productive in
the early years. Therefore, it is only apt to charge a higher depreciation in the early years and decrease it in later years. All these assets have an estimated useful life across which they
This is a method that allocates higher depreciation expense in the initial years of asset use. Many companies calculate their depreciation expense using an accounting method called accelerated
depreciation. In this depreciation scenario, an asset, such as a piece of equipment, has its book value reduced on the balance sheet at a faster rate than a traditional straight-line depreciation
Changing would require a revision of all previously submitted financial statements. The above depreciation schedule can be confirmed by building a new one using the same inputs but replacing the
formula method with the SYD function in Excel. This depreciation schedule should have the same depreciation expense values as in the table above.
At the end of its useful life, the components are expected to have a residual value of $5 million (i.e. scrap value), which reflects the sale proceeds the manufacturer could hypothetically earn from
selling those used components. The SYD Function Depreciation Schedule confirms the previous findings as the total depreciation equates to the depreciation https://www.bookkeeping-reviews.com/
website-builder-for-bookkeepers-and-virtual-pa-s/ amount at year 1 for both versions of the SYD method. Another key component to notice is that the depreciation amount for year 1 is always multiplied
by the depreciation factor for each year. To find the SYD function on Excel, one must navigate to the formulas tab and click on the Financials drop-down menu where it can be seen.
Therefore, charging higher depreciation costs early on and decreasing depreciation charges in later years reflects the reality of an asset’s changing economic usefulness over time. Under this method,
the percentage of depreciation rate for each year is calculated by the years remaining in the useful life divided by the sum of remaining life every year throughout the asset’s life. The same asset,
using straight-line depreciation and zero salvage value, would be depreciated at $5,000 per year for five years ($25,000 ÷ 5) until the asset depreciates to zero value. The same company, with the
exact same assets, would appear to be earning different amounts of profit and have assets carried at different values on the balance sheet, depending upon which depreciation method was utilized.
Still, it differs from straight-line depreciation, where the amount deducted is the same for each year of an asset’s useful life.
What is the SYD Function (Sum of Years Digits)?
Deskera can help you generate payroll and payslips in minutes with Deskera People. Your employees can view their payslips, apply for time off, and file their claims and expenses online. With these
values, we move on to applying free cash flow fcf formula and calculation the sum of the years’ formula in a step-wise manner. Our example assumes ABC technologies that purchased computers for
$4,000,000. Considering the useful life of the computers to be 5 years and a salvage value of $100,000.
Their values will automatically flow to respective financial reports.You can have access to Deskera’s ready-made Profit and Loss Statement, Balance Sheet, and other financial reports in an instant.
Therefore, the company deducts its balance from the balance of the equipment account in the balance sheet. In simple terms, the company reports the net asset value in the balance sheet. The sum of
years method uses the expected life and adds the digits for every year to give the final depreciation expense amount. In the second full year of the asset’s life, the amount of depreciation will be
$40,000 (4/15 of $150,000). In the third full year of the asset’s life, the depreciation will be $30,000 (3/15 of $150,000).
The (Cost – Salvage) component refers to the depreciation amount for the specific period used; the Life and Per sections comprise the depreciation factor discussed previously. For instance, a company
might use SYD for technology-related assets as advancements are regularly introduced in the field, causing previous versions of a technological asset to become obsolete quickly. The best examples or
scenarios where applying this method is fruitful can be automobiles, computers, mobile phones. A newer model of a car or the latest technological advent can lead to quick obsolescence of these
assets. It must be noted that the final depreciation expense equals the salvage value of the asset.
Sum of Years Digits (SYD) Formula
The implicit assumption of the sum of the years’ digits depreciation method is that the fixed asset (PP&E) is more productive and provides more near-term value in the periods immediately
post-purchase. The sum of the years’ digits method of depreciation, or “SYD”, reduces the book value of a fixed asset (PP&E) at a front-loaded, accelerated depreciation rate. The SYD depreciation
schedules using the formula and Excel function showcased how the depreciation expense is distributed over the equipment’s useful life.
The result is excessively low profits in the near term, followed by excessively high profits in later reporting periods. It is also more complex to calculate than straight-line depreciation, which
can lead to errors in the calculation. Using the information from the example above, you would calculate the applicable depreciation percentage for each depreciable year. | {"url":"https://www.emobilitydirectory.com/how-do-i-calculate-depreciation-using-the-sum-of/","timestamp":"2024-11-05T16:16:29Z","content_type":"text/html","content_length":"353464","record_id":"<urn:uuid:4e7999cd-69a1-40d8-9891-8bc32ea7c260>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00185.warc.gz"} |
Symbolic Regression: The Forgotten Machine Learning Method by Rafael Ruggiero | First Glass Fencing
Symbolic Regression: The Forgotten Machine Learning Method by Rafael Ruggiero
Symbolic AI vs machine learning in natural language processing
Machine learning models, on the other hand, excels in handling such complexities. Its ability to model intricate patterns and interrelationships in high-dimensional space allows for a more nuanced
understanding and prediction of non-linear human behavior, making it a powerful tool in art research. Precise sample size justification (power analysis) for complex machine learning-based data
analysis methods is still an open matter, and to the best of our knowledge, no standards have been established. Therefore, we followed a series of available suggestions regarding a reasonable sample
size. First, suggestion is that 50 samples are required to start any meaningful machine learning-based data analysis (scikit-learn ). Second, a controversial suggestion is that 10 to 20 samples per
degree of freedom (independent variable, art-attribute) is reasonable, particularly for logistic regression, which would result in a total of 170 to 340 samples needed for our study52.
A high correlation was found between imaginativeness and symbolism with a coefficient of 0.78, but no correlations above that. These finding still align with our initial hypothesis as the predictors
were deliberately chosen to encapsulate the multifaceted and interrelated attributes of artistic creativity3,4,64. The influence of independent variables on the prediction can vary across the range
of these variables due to the capacity of RF models to capture non-linear associations between independent and dependent variables.
Implementing ensemble learning methods to predict the shear strength of RC deep beams with/without web reinforcements
However, the aforementioned models are hard to be used in practical engineering design, because the prediction process of pure data-driven approaches cannot be transformed into a useable mathematical
equation for structural engineers. Therefore, data-driven approaches are often regarded as black-box models [30]. Fiber reinforced polymer (FRP)-reinforced concrete slabs, an extension of reinforced
concrete (RC) slabs leveraged for resisting environment corrosion, are susceptible to punching shear failure due to the lower elasticity modulus of FRP reinforcement.
Bayesian approaches enable a modeller to evaluate different representational forms and parameter settings for capturing human behaviour, as specified through the model’s prior45. These priors can
also be tuned with behavioural data through hierarchical Bayesian modelling46, although the resulting set-up can be restrictive. MLC shows how meta-learning can be used like hierarchical Bayesian
models for reverse-engineering inductive biases (see ref. 47 for a formal connection), although with the aid of neural networks for greater expressive power. Our research adds to a growing
literature, reviewed previously48, on using meta-learning for understanding human49,50,51 or human-like behaviour52,53,54.
Machine learning based data analysis approach
We introduce the Deep Symbolic Network (DSN) model, which aims at becoming the white-box version of Deep Neural Networks (DNN). The DSN model provides a simple, universal yet powerful structure,
similar to DNN, to represent any knowledge of the world, which is transparent to humans. The conjecture behind the DSN model is that any type of real world objects sharing enough common features are
mapped into human brains as a symbol. Those symbols are connected by links, representing the composition, correlation, causality, or other relationships between them, forming a deep, hierarchical
symbolic network structure. Powered by such a structure, the DSN model is expected to learn like humans, because of its unique characteristics.
The four most frequent responses are shown, marked in parentheses with response rates (counts for people and the percentage of samples for MLC). The superscript notes indicate the algebraic answer
(asterisks), a one-to-one error (1-to-1) or an iconic concatenation error (IC). The words and colours were randomized for each participant and a canonical assignment is therefore shown here.
Optimisation of code generators so that they produce code satisfying various quality criteria is another important area of future work. CGBE strategies would need to be designed to favour the
production of code generation rules which result in generated code satisfying the criteria. We used the proportion p of correct translations of an independent validation set to assess the accuracy of
synthesised code generators.
Associated content
There have been several efforts to create complicated symbolic AI systems that encompass the multitudes of rules of certain domains. Called expert systems, these symbolic AI models use hardcoded
knowledge and rules to tackle complicated tasks such as medical diagnosis. But they require a huge amount of effort by domain experts and software engineers and only work in very narrow use cases. As
soon as you generalize the problem, there will be an explosion of new rules to add (remember the cat detection problem?), which will require more human labor. This is particularly true for problems
in a small number of dimensions — symbolic regression is unlikely to be useful for problems like image classification, which would require enormous formulas with millions of input parameters. A shift
to explicit symbolic models could bring to light many hidden patterns in the sea of datasets that we have at our disposal today.
How hybrid AI can help LLMs become more trustworthy … – Data Science Central
How hybrid AI can help LLMs become more trustworthy ….
Posted: Tue, 31 Oct 2023 17:35:21 GMT [source]
A T2T approach to code generation specifies the translation from source to target languages in terms of the source and target language concrete syntax or grammars, and does not depend upon metamodels
(abstract syntax) of the languages. A T2T author needs to know only the source language grammar and target language syntax, and the T2T language. To summarise our contribution, we have provided a new
technique (CGBE) for automating the construction of code generators, via a novel application of symbolic machine learning.
In our experiments, we found that the most common human responses were algebraic and systematic in exactly the ways that Fodor and Pylyshyn1 discuss. However, people also relied on inductive biases
that sometimes support the algebraic solution and sometimes deviate from it; indeed, people are not purely algebraic machines3,6,7. We showed how MLC enables a standard neural network optimized for
its compositional skills to mimic or exceed human systematic generalization in a side-by-side comparison. MLC shows much stronger systematicity than neural networks trained in standard ways, and
shows more nuanced behaviour than pristine symbolic models. MLC also allows neural networks to tackle other existing challenges, including making systematic use of isolated primitives11,16 and using
mutual exclusivity to infer meanings44.
It is about optimizing models that are capable of learning from huge amounts of data. Examples are computer vision algorithms for image recognition and general-purpose models like support vector
machines and neural networks. Symbolic regression is an alternative to these methods that works by finding explicit formulas that connect the variables, allowing hidden nonlinear patterns to be
Explore the first generative pre-trained forecasting model and apply it in a project with Python
Read more about https://www.metadialog.com/ here. | {"url":"https://www.firstglassfencing.com.au/2023/09/27/symbolic-regression-the-forgotten-machine-learning/","timestamp":"2024-11-10T18:50:57Z","content_type":"text/html","content_length":"133031","record_id":"<urn:uuid:997e40ff-0542-4a31-9a63-e0219e226bc8>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00670.warc.gz"} |
Adding and subtracting decimals worksheets
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Author Message Author Message
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IlbendF Posted: Wednesday 03rd of Jan 19:43 https://softmath.com/faqs-regarding-algebra.html.
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TensorFlow Constrained Optimization Example Using CelebA Dataset | Responsible AI Toolkit
This notebook demonstrates an easy way to create and optimize constrained problems using the TFCO library. This method can be useful in improving models when we find that they’re not performing
equally well across different slices of our data, which we can identify using Fairness Indicators. The second of Google’s AI principles states that our technology should avoid creating or reinforcing
unfair bias, and we believe this technique can help improve model fairness in some situations. In particular, this notebook will:
• Train a simple, unconstrained neural network model to detect a person's smile in images using tf.keras and the large-scale CelebFaces Attributes (CelebA) dataset.
• Evaluate model performance against a commonly used fairness metric across age groups, using Fairness Indicators.
• Set up a simple constrained optimization problem to achieve fairer performance across age groups.
• Retrain the now constrained model and evaluate performance again, ensuring that our chosen fairness metric has improved.
Last updated: 3/11 Feb 2020
This notebook was created in Colaboratory, connected to the Python 3 Google Compute Engine backend. If you wish to host this notebook in a different environment, then you should not experience any
major issues provided you include all the required packages in the cells below.
Note that the very first time you run the pip installs, you may be asked to restart the runtime because of preinstalled out of date packages. Once you do so, the correct packages will be used.
Pip installs
!pip install -q -U pip==20.2
!pip install git+https://github.com/google-research/tensorflow_constrained_optimization
!pip install -q tensorflow-datasets tensorflow
!pip install fairness-indicators \
"absl-py==0.12.0" \
"apache-beam<3,>=2.40" \
"avro-python3==1.9.1" \
Note that depending on when you run the cell below, you may receive a warning about the default version of TensorFlow in Colab switching to TensorFlow 2.X soon. You can safely ignore that warning as
this notebook was designed to be compatible with TensorFlow 1.X and 2.X.
Import Modules
import os
import sys
import tempfile
import urllib
import tensorflow as tf
from tensorflow import keras
import tensorflow_datasets as tfds
import numpy as np
import tensorflow_constrained_optimization as tfco
from tensorflow_metadata.proto.v0 import schema_pb2
from tfx_bsl.tfxio import tensor_adapter
from tfx_bsl.tfxio import tf_example_record
Additionally, we add a few imports that are specific to Fairness Indicators which we will use to evaluate and visualize the model's performance.
Fairness Indicators related imports
import tensorflow_model_analysis as tfma
import fairness_indicators as fi
from google.protobuf import text_format
import apache_beam as beam
Although TFCO is compatible with eager and graph execution, this notebook assumes that eager execution is enabled by default as it is in TensorFlow 2.x. To ensure that nothing breaks, eager execution
will be enabled in the cell below.
Enable Eager Execution and Print Versions
if tf.__version__ < "2.0.0":
print("Eager execution enabled.")
print("Eager execution enabled by default.")
print("TensorFlow " + tf.__version__)
print("TFMA " + tfma.VERSION_STRING)
print("TFDS " + tfds.version.__version__)
print("FI " + fi.version.__version__)
CelebA Dataset
CelebA is a large-scale face attributes dataset with more than 200,000 celebrity images, each with 40 attribute annotations (such as hair type, fashion accessories, facial features, etc.) and 5
landmark locations (eyes, mouth and nose positions). For more details take a look at the paper. With the permission of the owners, we have stored this dataset on Google Cloud Storage and mostly
access it via TensorFlow Datasets(tfds).
In this notebook:
• Our model will attempt to classify whether the subject of the image is smiling, as represented by the "Smiling" attribute^*.
• Images will be resized from 218x178 to 28x28 to reduce the execution time and memory when training.
• Our model's performance will be evaluated across age groups, using the binary "Young" attribute. We will call this "age group" in this notebook.
^* While there is little information available about the labeling methodology for this dataset, we will assume that the "Smiling" attribute was determined by a pleased, kind, or amused expression on
the subject's face. For the purpose of this case study, we will take these labels as ground truth.
gcs_base_dir = "gs://celeb_a_dataset/"
celeb_a_builder = tfds.builder("celeb_a", data_dir=gcs_base_dir, version='2.0.0')
num_test_shards_dict = {'0.3.0': 4, '2.0.0': 2} # Used because we download the test dataset separately
version = str(celeb_a_builder.info.version)
print('Celeb_A dataset version: %s' % version)
Test dataset helper functions
local_root = tempfile.mkdtemp(prefix='test-data')
def local_test_filename_base():
return local_root
def local_test_file_full_prefix():
return os.path.join(local_test_filename_base(), "celeb_a-test.tfrecord")
def copy_test_files_to_local():
filename_base = local_test_file_full_prefix()
num_test_shards = num_test_shards_dict[version]
for shard in range(num_test_shards):
url = "https://storage.googleapis.com/celeb_a_dataset/celeb_a/%s/celeb_a-test.tfrecord-0000%s-of-0000%s" % (version, shard, num_test_shards)
filename = "%s-0000%s-of-0000%s" % (filename_base, shard, num_test_shards)
res = urllib.request.urlretrieve(url, filename)
Before moving forward, there are several considerations to keep in mind in using CelebA:
• Although in principle this notebook could use any dataset of face images, CelebA was chosen because it contains public domain images of public figures.
• All of the attribute annotations in CelebA are operationalized as binary categories. For example, the "Young" attribute (as determined by the dataset labelers) is denoted as either present or
absent in the image.
• CelebA's categorizations do not reflect real human diversity of attributes.
• For the purposes of this notebook, the feature containing the "Young" attribute is referred to as "age group", where the presence of the "Young" attribute in an image is labeled as a member of
the "Young" age group and the absence of the "Young" attribute is labeled as a member of the "Not Young" age group. These are assumptions made as this information is not mentioned in the original
• As such, performance in the models trained in this notebook is tied to the ways the attributes have been operationalized and annotated by the authors of CelebA.
• This model should not be used for commercial purposes as that would violate CelebA's non-commercial research agreement.
Setting Up Input Functions
The subsequent cells will help streamline the input pipeline as well as visualize performance.
First we define some data-related variables and define a requisite preprocessing function.
Define Variables
ATTR_KEY = "attributes"
IMAGE_KEY = "image"
LABEL_KEY = "Smiling"
GROUP_KEY = "Young"
IMAGE_SIZE = 28
Define Preprocessing Functions
def preprocess_input_dict(feat_dict):
# Separate out the image and target variable from the feature dictionary.
image = feat_dict[IMAGE_KEY]
label = feat_dict[ATTR_KEY][LABEL_KEY]
group = feat_dict[ATTR_KEY][GROUP_KEY]
# Resize and normalize image.
image = tf.cast(image, tf.float32)
image = tf.image.resize(image, [IMAGE_SIZE, IMAGE_SIZE])
image /= 255.0
# Cast label and group to float32.
label = tf.cast(label, tf.float32)
group = tf.cast(group, tf.float32)
feat_dict[IMAGE_KEY] = image
feat_dict[ATTR_KEY][LABEL_KEY] = label
feat_dict[ATTR_KEY][GROUP_KEY] = group
return feat_dict
get_image_and_label = lambda feat_dict: (feat_dict[IMAGE_KEY], feat_dict[ATTR_KEY][LABEL_KEY])
get_image_label_and_group = lambda feat_dict: (feat_dict[IMAGE_KEY], feat_dict[ATTR_KEY][LABEL_KEY], feat_dict[ATTR_KEY][GROUP_KEY])
Then, we build out the data functions we need in the rest of the colab.
# Train data returning either 2 or 3 elements (the third element being the group)
def celeb_a_train_data_wo_group(batch_size):
celeb_a_train_data = celeb_a_builder.as_dataset(split='train').shuffle(1024).repeat().batch(batch_size).map(preprocess_input_dict)
return celeb_a_train_data.map(get_image_and_label)
def celeb_a_train_data_w_group(batch_size):
celeb_a_train_data = celeb_a_builder.as_dataset(split='train').shuffle(1024).repeat().batch(batch_size).map(preprocess_input_dict)
return celeb_a_train_data.map(get_image_label_and_group)
# Test data for the overall evaluation
celeb_a_test_data = celeb_a_builder.as_dataset(split='test').batch(1).map(preprocess_input_dict).map(get_image_label_and_group)
# Copy test data locally to be able to read it into tfma
Build a simple DNN Model
Because this notebook focuses on TFCO, we will assemble a simple, unconstrained tf.keras.Sequential model.
We may be able to greatly improve model performance by adding some complexity (e.g., more densely-connected layers, exploring different activation functions, increasing image size), but that may
distract from the goal of demonstrating how easy it is to apply the TFCO library when working with Keras. For that reason, the model will be kept simple — but feel encouraged to explore this space.
def create_model():
# For this notebook, accuracy will be used to evaluate performance.
METRICS = [
# The model consists of:
# 1. An input layer that represents the 28x28x3 image flatten.
# 2. A fully connected layer with 64 units activated by a ReLU function.
# 3. A single-unit readout layer to output real-scores instead of probabilities.
model = keras.Sequential([
keras.layers.Flatten(input_shape=(IMAGE_SIZE, IMAGE_SIZE, 3), name='image'),
keras.layers.Dense(64, activation='relu'),
keras.layers.Dense(1, activation=None)
# TFCO by default uses hinge loss — and that will also be used in the model.
return model
We also define a function to set seeds to ensure reproducible results. Note that this colab is meant as an educational tool and does not have the stability of a finely tuned production pipeline.
Running without setting a seed may lead to varied results.
def set_seeds():
Fairness Indicators Helper Functions
Before training our model, we define a number of helper functions that will allow us to evaluate the model's performance via Fairness Indicators.
First, we create a helper function to save our model once we train it.
def save_model(model, subdir):
base_dir = tempfile.mkdtemp(prefix='saved_models')
model_location = os.path.join(base_dir, subdir)
model.save(model_location, save_format='tf')
return model_location
Next, we define functions used to preprocess the data in order to correctly pass it through to TFMA.
Data Preprocessing functions for
def tfds_filepattern_for_split(dataset_name, split):
return f"{local_test_file_full_prefix()}*"
class PreprocessCelebA(object):
"""Class that deserializes, decodes and applies additional preprocessing for CelebA input."""
def __init__(self, dataset_name):
builder = tfds.builder(dataset_name)
self.features = builder.info.features
example_specs = self.features.get_serialized_info()
self.parser = tfds.core.example_parser.ExampleParser(example_specs)
def __call__(self, serialized_example):
# Deserialize
deserialized_example = self.parser.parse_example(serialized_example)
# Decode
decoded_example = self.features.decode_example(deserialized_example)
# Additional preprocessing
image = decoded_example[IMAGE_KEY]
label = decoded_example[ATTR_KEY][LABEL_KEY]
# Resize and scale image.
image = tf.cast(image, tf.float32)
image = tf.image.resize(image, [IMAGE_SIZE, IMAGE_SIZE])
image /= 255.0
image = tf.reshape(image, [-1])
# Cast label and group to float32.
label = tf.cast(label, tf.float32)
group = decoded_example[ATTR_KEY][GROUP_KEY]
output = tf.train.Example()
output.features.feature[GROUP_KEY].bytes_list.value.append(b"Young" if group.numpy() else b'Not Young')
return output.SerializeToString()
def tfds_as_pcollection(beam_pipeline, dataset_name, split):
return (
| 'Read records' >> beam.io.ReadFromTFRecord(tfds_filepattern_for_split(dataset_name, split))
| 'Preprocess' >> beam.Map(PreprocessCelebA(dataset_name))
Finally, we define a function that evaluates the results in TFMA.
def get_eval_results(model_location, eval_subdir):
base_dir = tempfile.mkdtemp(prefix='saved_eval_results')
tfma_eval_result_path = os.path.join(base_dir, eval_subdir)
eval_config_pbtxt = """
model_specs {
label_key: "%s"
metrics_specs {
metrics {
class_name: "FairnessIndicators"
config: '{ "thresholds": [0.22, 0.5, 0.75] }'
metrics {
class_name: "ExampleCount"
slicing_specs {}
slicing_specs { feature_keys: "%s" }
options {
compute_confidence_intervals { value: False }
disabled_outputs{values: "analysis"}
""" % (LABEL_KEY, GROUP_KEY)
eval_config = text_format.Parse(eval_config_pbtxt, tfma.EvalConfig())
eval_shared_model = tfma.default_eval_shared_model(
eval_saved_model_path=model_location, tags=[tf.saved_model.SERVING])
schema_pbtxt = """
tensor_representation_group {
key: ""
value {
tensor_representation {
key: "%s"
value {
dense_tensor {
column_name: "%s"
shape {
dim { size: 28 }
dim { size: 28 }
dim { size: 3 }
feature {
name: "%s"
type: FLOAT
feature {
name: "%s"
type: FLOAT
feature {
name: "%s"
type: BYTES
""" % (IMAGE_KEY, IMAGE_KEY, IMAGE_KEY, LABEL_KEY, GROUP_KEY)
schema = text_format.Parse(schema_pbtxt, schema_pb2.Schema())
coder = tf_example_record.TFExampleBeamRecord(
physical_format='inmem', schema=schema,
tensor_adapter_config = tensor_adapter.TensorAdapterConfig(
# Run the fairness evaluation.
with beam.Pipeline() as pipeline:
_ = (
tfds_as_pcollection(pipeline, 'celeb_a', 'test')
| 'ExamplesToRecordBatch' >> coder.BeamSource()
| 'ExtractEvaluateAndWriteResults' >>
return tfma.load_eval_result(output_path=tfma_eval_result_path)
Train & Evaluate Unconstrained Model
With the model now defined and the input pipeline in place, we’re now ready to train our model. To cut back on the amount of execution time and memory, we will train the model by slicing the data
into small batches with only a few repeated iterations.
Note that running this notebook in TensorFlow < 2.0.0 may result in a deprecation warning for np.where. Safely ignore this warning as TensorFlow addresses this in 2.X by using tf.where in place of
BATCH_SIZE = 32
# Set seeds to get reproducible results
model_unconstrained = create_model()
model_unconstrained.fit(celeb_a_train_data_wo_group(BATCH_SIZE), epochs=5, steps_per_epoch=1000)
Evaluating the model on the test data should result in a final accuracy score of just over 85%. Not bad for a simple model with no fine tuning.
print('Overall Results, Unconstrained')
celeb_a_test_data = celeb_a_builder.as_dataset(split='test').batch(1).map(preprocess_input_dict).map(get_image_label_and_group)
results = model_unconstrained.evaluate(celeb_a_test_data)
However, performance evaluated across age groups may reveal some shortcomings.
To explore this further, we evaluate the model with Fairness Indicators (via TFMA). In particular, we are interested in seeing whether there is a significant gap in performance between "Young" and
"Not Young" categories when evaluated on false positive rate.
A false positive error occurs when the model incorrectly predicts the positive class. In this context, a false positive outcome occurs when the ground truth is an image of a celebrity 'Not Smiling'
and the model predicts 'Smiling'. By extension, the false positive rate, which is used in the visualization above, is a measure of accuracy for a test. While this is a relatively mundane error to
make in this context, false positive errors can sometimes cause more problematic behaviors. For instance, a false positive error in a spam classifier could cause a user to miss an important email.
model_location = save_model(model_unconstrained, 'model_export_unconstrained')
eval_results_unconstrained = get_eval_results(model_location, 'eval_results_unconstrained')
As mentioned above, we are concentrating on the false positive rate. The current version of Fairness Indicators (0.1.2) selects false negative rate by default. After running the line below, deselect
false_negative_rate and select false_positive_rate to look at the metric we are interested in.
As the results show above, we do see a disproportionate gap between "Young" and "Not Young" categories.
This is where TFCO can help by constraining the false positive rate to be within a more acceptable criterion.
Constrained Model Set Up
As documented in TFCO's library, there are several helpers that will make it easier to constrain the problem:
1. tfco.rate_context() – This is what will be used in constructing a constraint for each age group category.
2. tfco.RateMinimizationProblem()– The rate expression to be minimized here will be the false positive rate subject to age group. In other words, performance now will be evaluated based on the
difference between the false positive rates of the age group and that of the overall dataset. For this demonstration, a false positive rate of less than or equal to 5% will be set as the
3. tfco.ProxyLagrangianOptimizerV2() – This is the helper that will actually solve the rate constraint problem.
The cell below will call on these helpers to set up model training with the fairness constraint.
# The batch size is needed to create the input, labels and group tensors.
# These tensors are initialized with all 0's. They will eventually be assigned
# the batch content to them. A large batch size is chosen so that there are
# enough number of "Young" and "Not Young" examples in each batch.
model_constrained = create_model()
BATCH_SIZE = 32
# Create input tensor.
input_tensor = tf.Variable(
np.zeros((BATCH_SIZE, IMAGE_SIZE, IMAGE_SIZE, 3), dtype="float32"),
# Create labels and group tensors (assuming both labels and groups are binary).
labels_tensor = tf.Variable(
np.zeros(BATCH_SIZE, dtype="float32"), name="labels")
groups_tensor = tf.Variable(
np.zeros(BATCH_SIZE, dtype="float32"), name="groups")
# Create a function that returns the applied 'model' to the input tensor
# and generates constrained predictions.
def predictions():
return model_constrained(input_tensor)
# Create overall context and subsetted context.
# The subsetted context contains subset of examples where group attribute < 1
# (i.e. the subset of "Not Young" celebrity images).
# "groups_tensor < 1" is used instead of "groups_tensor == 0" as the former
# would be a comparison on the tensor value, while the latter would be a
# comparison on the Tensor object.
context = tfco.rate_context(predictions, labels=lambda:labels_tensor)
context_subset = context.subset(lambda:groups_tensor < 1)
# Setup list of constraints.
# In this notebook, the constraint will just be: FPR to less or equal to 5%.
constraints = [tfco.false_positive_rate(context_subset) <= 0.05]
# Setup rate minimization problem: minimize overall error rate s.t. constraints.
problem = tfco.RateMinimizationProblem(tfco.error_rate(context), constraints)
# Create constrained optimizer and obtain train_op.
# Separate optimizers are specified for the objective and constraints
optimizer = tfco.ProxyLagrangianOptimizerV2(
# A list of all trainable variables is also needed to use TFCO.
var_list = (model_constrained.trainable_weights + list(problem.trainable_variables) +
The model is now set up and ready to be trained with the false positive rate constraint across age group.
Now, because the last iteration of the constrained model may not necessarily be the best performing model in terms of the defined constraint, the TFCO library comes equipped with
tfco.find_best_candidate_index() that can help choose the best iterate out of the ones found after each epoch. Think of tfco.find_best_candidate_index() as an added heuristic that ranks each of the
outcomes based on accuracy and fairness constraint (in this case, false positive rate across age group) separately with respect to the training data. That way, it can search for a better trade-off
between overall accuracy and the fairness constraint.
The following cells will start the training with constraints while also finding the best performing model per iteration.
# Obtain train set batches.
NUM_ITERATIONS = 100 # Number of training iterations.
SKIP_ITERATIONS = 10 # Print training stats once in this many iterations.
# Create temp directory for saving snapshots of models.
temp_directory = tempfile.mktemp()
# List of objective and constraints across iterations.
objective_list = []
violations_list = []
# Training iterations.
iteration_count = 0
for (image, label, group) in celeb_a_train_data_w_group(BATCH_SIZE):
# Assign current batch to input, labels and groups tensors.
# Run gradient update.
optimizer.minimize(problem, var_list=var_list)
# Record objective and violations.
objective = problem.objective()
violations = problem.constraints()
"\r Iteration %d: Hinge Loss = %.3f, Max. Constraint Violation = %.3f"
% (iteration_count + 1, objective, max(violations)))
# Snapshot model once in SKIP_ITERATIONS iterations.
if iteration_count % SKIP_ITERATIONS == 0:
# Save snapshot of model weights.
temp_directory + "/celeb_a_constrained_" +
str(iteration_count / SKIP_ITERATIONS) + ".h5")
iteration_count += 1
if iteration_count >= NUM_ITERATIONS:
# Choose best model from recorded iterates and load that model.
best_index = tfco.find_best_candidate_index(
np.array(objective_list), np.array(violations_list))
temp_directory + "/celeb_a_constrained_" + str(best_index) + ".0.h5")
# Remove temp directory.
os.system("rm -r " + temp_directory)
After having applied the constraint, we evaluate the results once again using Fairness Indicators.
model_location = save_model(model_constrained, 'model_export_constrained')
eval_result_constrained = get_eval_results(model_location, 'eval_results_constrained')
As with the previous time we used Fairness Indicators, deselect false_negative_rate and select false_positive_rate to look at the metric we are interested in.
Note that to fairly compare the two versions of our model, it is important to use thresholds that set the overall false positive rate to be roughly equal. This ensures that we are looking at actual
change as opposed to just a shift in the model equivalent to simply moving the threshold boundary. In our case, comparing the unconstrained model at 0.5 and the constrained model at 0.22 provides a
fair comparison for the models.
eval_results_dict = {
'constrained': eval_result_constrained,
'unconstrained': eval_results_unconstrained,
With TFCO's ability to express a more complex requirement as a rate constraint, we helped this model achieve a more desirable outcome with little impact to the overall performance. There is, of
course, still room for improvement, but at least TFCO was able to find a model that gets close to satisfying the constraint and reduces the disparity between the groups as much as possible. | {"url":"https://tensorflow.google.cn/responsible_ai/fairness_indicators/tutorials/Fairness_Indicators_TFCO_CelebA_Case_Study?authuser=7","timestamp":"2024-11-02T03:10:52Z","content_type":"text/html","content_length":"258316","record_id":"<urn:uuid:112bbf05-6bf1-4c50-856c-7668f917dd99>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00362.warc.gz"} |
wu :: forums - Political slugfest
wu :: forums
medium (Moderators: ThudnBlunder, Eigenray, Grimbal, Icarus, SMQ, william wu, towr) « Previous topic | Next topic »
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Author Topic: Political slugfest (Read 5507 times)
ecoist Political slugfest
Senior « on: Nov 1^st, 2008, 9:22pm » Quote Modify
4 libertarians, 13 republicans, and 17 democrats gather to argue their political philosophy. They wander about and debate each other in pairs. When two of them of different political
persuasions debate each other, they become so disallusioned that they both change to the third political persuasion. Show that it cannot happen that, after awhile, all of them acquire the
same political philosophy.
« Last Edit: Nov 1^st, 2008, 9:39pm by ecoist »
towr Re: Political slugfest
wu::riddles Moderator « Reply #1 on: Nov 2^nd, 2008, 8:29am » Quote Modify
Reminds me of a type of lizard.. Let's see if I can find a link
Ah, here we go, and more to the point, here.
Some people are average, some are just mean.
Posts: 13730 Wikipedia, Google, Mathworld, Integer sequence DB
Hippo Re: Political slugfest
Uberpuzzler « Reply #2 on: Nov 2^nd, 2008, 2:09pm » Quote Modify
let ^2 be primitive 3rd root of unity.
Look at ^2+d)
Operations are additions of (-1,-1,2),(-1,2,-1) or (2,-1,-1) to (l,r,d).
As addition of (-1,-1,-1) does not change
Starting position (4,13,17) is mod 3 equal (1,1,-1) having
Posts: 919 ^2
I am not able to finish the proof.
So try to go to 34d.
1) (4,13,17)+=(6,-3,-3)=(10,10,14)
2) (10,10,14)+=(-10,-10,20)=(0,0,34).
At least I have prooved noone else will be able
If two of l,r,d are equal (mod 3), we can equalize them as in 1). Than convert to the remaining kind as in 2).
If l,r,d are distinct mod 3 we get
« Last Edit: Nov 2^nd, 2008, 3:02pm by Hippo »
towr Re: Political slugfest
wu::riddles Moderator « Reply #3 on: Nov 2^nd, 2008, 3:04pm » Quote Modify
on Nov 2^nd, 2008, 2:09pm, Hippo wrote:
So try to go to 34d.
1) (4,13,17)+=(6,-3,-3)=(10,10,14)
2) (10,10,14)+=(-10,-10,20)=(0,0,34).
Some people are average, some are just mean.
That bodes well for Obama
Gender: « Last Edit: Nov 2^nd, 2008, 3:05pm by towr »
Posts: 13730
Wikipedia, Google, Mathworld, Integer sequence DB
Eigenray Re: Political slugfest
wu::riddles « Reply #4 on: Nov 2^nd, 2008, 3:53pm » Quote Modify
Uberpuzzler For the general case, the analysis is simpler if they're allowed to borrow people. For example, 3 libertarians left to themselves will stay libertarian. But after a couple rounds with
a republican in the room (who then leaves, still a republican), they might just find themselves all democrats.
Find a simple criterion to determine whether one state can turn into another without borrowing.
Hippo: I also thought roots of unity were the right way to go. But they don't really work for a composite number of parties. That is, with n parties, and a given number of people,
there are n^n-1 distinct states (allowing negative people), by looking at all the differences mod n. But ^, where ^2pi i/n is an n-th root of unity.
Posts: 1948 « Last Edit: Nov 2^nd, 2008, 4:08pm by Eigenray »
ecoist Re: Political slugfest
Senior « Reply #5 on: Nov 2^nd, 2008, 5:27pm » Quote Modify
I screwed up, guys! There is an elementary solution if the total number of people involved is a multiple of 3. I should have said there are only 3 libertarians.
Thinking about what Eigenray wrote, I came up with the following variation. Let there be 15 parties with the i-th party having i members, for i=1,...,15. Whenever 14 of these guys meet, no
two belonging to the same party, they all switch to the party none belong to. Show that it cannot happen that after awhile everyone belongs to the same party.
Hippo Re: Political slugfest
Uberpuzzler « Reply #6 on: Nov 3^rd, 2008, 12:41am » Quote Modify
on Nov 2^nd, 2008, 3:53pm, Eigenray wrote:
For the general case, the analysis is simpler if they're allowed to borrow people. For example, 3 libertarians left to themselves will stay libertarian. But after a couple rounds with
a republican in the room (who then leaves, still a republican), they might just find themselves all democrats.
Find a simple criterion to determine whether one state can turn into another without borrowing.
Hippo: I also thought roots of unity were the right way to go. But they don't really work for a composite number of parties. That is, with n parties, and a given number of people,
there are n^n-1 distinct states (allowing negative people), by looking at all the differences mod n. But ^, where ^2pi i/n is an n-th root of unity.
Posts: 919
Actually I didn't thing about general case and the roots of unity was used in confusing way ... I actualy thought on triangular grid mod 3. I have used 1,^2 as a coordinates. You can as
well use x,y,z space coordinates and project it to the plane perpendicular to main diagonal (t(1,1,1) line). ... There is 9 positions in the projection mod 3, 7 of them are on
projections of axis. Mod 3 just projections of (1,-1,0) and (-1,1,0) are not.
I don't think it can be easily generalised to higher number of parties. At least not from my point of view
Oh yes, it looks well for prime p number of parties, but I have had problems to imagine the mod p operation
BTW: The greasmonkey eats spaces after escape sequences as
« Last Edit: Nov 3^rd, 2008, 1:18am by Hippo »
ecoist Re: Political slugfest
Senior Riddler « Reply #7 on: Nov 11^th, 2008, 4:27pm » Quote Modify
Ok, how about this less ambitious generalization of the chameleon problem?
Suppose that there are n>2 political parties whose members satisfy two conditions.
a) For any two parties, the numbers of members of each party are incongruent modulo n.
b) If any n-1 people meet, no two of them belonging to the same party, then all n-1 switch to the party none belong to.
Show that it can never happen that everyone belongs to the same party.
Posts: 405
ecoist Re: Political slugfest
Senior « Reply #8 on: Nov 16^th, 2008, 11:44am » Quote Modify
I'm a complete idiot (thx for letting me find out for myself)! This problem, and its generalization, belongs in easy. Wowbagger suggested the easy solution, and Hippo noted what was needed
to be assumed to make his solution work.
The numbers of members of each party forms a multiset of residues modulo n. That multiset never changes when party switching occurs (see Hippo's example with (4,13,17). There are always
exactly two parties with the same number of members modulo 3). The reason for this is that, when switching occurs, it amounts to subtracting 1 from each membership modulo n (adding n-1
members to one party's membership is the same as subtracting 1 modulo n). Hence, since initially all memberships are incongruent modulo n, they must remain incongruent modulo n; and so the
guys can never all belong to the same party. The only way such a thing could happen is if, initially, n-1 memberships are congruent modulo n.
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I Hope This Old Train Breaks Down...
I am starting the Measurement Unit! yay. :) (Starting this week, to be finished in January, after the midterms.)
I did this unit with my kids last year, and it was where my whole outlook on the Geometry curriculum sort of started to turn around. (I had really dreaded my first few months of teaching Geometry
last year...)
I'll write more about the revised lessons as they unfold, but the order goes more or less like this (modelled after last year's sequence):
* Day 1: Intro to measurement. What quantities can we measure? How do we convert between units? (Including move-around activity for measuring quantities around the classroom.)
* Day 2: Indirect measurement of height. How can we measure an object's height using proportions and a mirror, or proportions and shadows? (Including outdoors activity portion)
* Day 3: Direct measurement of height. How can we accurately measure the height of a balcony using a string, a water bottle, and a meter stick? (Including group competition of results!)
* Day 4: Measurement of volumes. How can we measure/calculate the volume of a container shaped like a cylinder or prism? How can we predict how high water will rise once transfered to another
container? (Including hands-on measurements and water transfer activity)
* Day 5: Conversions between liters and cm^3. How big is a liter? A cubic meter? How many liters of water will it take to fill up our classroom? (Including demo for transfering water from a one-liter
bottle to a cube of 1000 cm^3 volume, plus move-around measurement of the classroom)
* Day 6: Measurement of irregular volumes, density, liquid density. (Stations activity - kids rotate around to measure: volumes of irregular containers via transfer of water; mass and volume of rocks
via triple-beam balance and displacement of water; weight / volume / density of liquids. Also included discussion of net weight and "Gee whiz!" demo of how mixed liquids will separate based on liquid
* Day 7: Reading about how the emperor measured the weight of the elephant, plus a bunch of conversion practice.
The big difference is that last year, we ended up doing a bunch of practice at the end of the unit, and also had little focus on estimation. This year, I will be pushing more estimation throughout
the measurement processes, and also sprinkling / spacing out more of the conversion practice throughout the unit. :)
When I was a kid, I moved with my parents to a country (ie. the States) whose language I didn't speak. In the few years that followed, I experienced being grouped with other language-learning kids in
remedial classes, where the teacher taught less material every week simply because the teacher didn't have faith in our collective ability to learn. Even as a kid, I had decided that no one else was
going to determine for me what my limits were. I went home and studied, on my own, so that I wouldn't lag behind other kids in other classes. In the end, I think I turned out doing OK, even though I
definitely took a bit of time to get there.
As a teacher, I occasionally come across kids who really, really struggle with basic instructions/material on a daily basis. (As in, 20 to 30 minutes into the problem set, and they're barely starting
#2.) Sometimes I find myself feeling very frustrated, wondering if such a kid is really placed in the right class. And then, a part of me always asks quietly, "Who has the right to say what any one
kid can achieve? If I hold them back, if I ask them to change into a more remedial class*, then I am fundamentally doubting the kid's ability to achieve. Maybe all this kid needs is a little more
time -- and a little more patience/a different approach from me."
The truth is, in the end I don't know if I have the ability to serve every child in my classroom the way they need to be helped. But I can do my best without giving up on that kid. (And, very
occasionally, that mentality translates to seeing a 60 going up to an 80 by the end of the year.)
*I find this type of mentality to be surprisingly common amongst honors-level teachers. Often it's very easy to say that a kid doesn't belong in our honors class, because they aren't quite as "quick"
as the others. To me, my favorite kids -- honors or not -- are those who give me 110% daily. If a kid is willing to REALLY try their best in an honors class, who are we to say that they don't belong?
For that, I really like my school's open-enrollment policy for honors (and AP) classes.
Geoff and I spent the last four days playing host to his parents. It was AWESOME - I actually never imagined that it would be possible to jam pack so many things into four days, in El Salvador. We
saw a beautiful beach (and did a whole all-inclusive thing) that had a stunning salt-water pool, a jacuzzi, and many very large and luxurious pools. We had an oiled massage (here they are $15 per
hour... very affordable!). We took the Coxes up to a beautiful (and very delicious/intimate) fusion restaurant at the top of the mountain, overlooking many mountains and the city while the sun was
setting. (The owners came out and talked to us, and one of them told us the story about how the restaurant came to be, and also played and sang some tunes on his guitar. Geoff's parents LOVED that!)
We visited an old Spanish colonial town, saw its church, had a drink by the lake, and even hiked down a bit to check out the awesome hexagonal-prism shaped rock columns that are completely natural.
And on the last night, we went to a beautiful restaurant that's already all decked out in Christmas spirit, on top of Torre de la Futura. (Geoff's mom LOVES Christmas, so it was a special treat for
her to see the whole place decked out already.)
All in all, it was an absolutely lovely weekend. Happy Thanksgiving, and may we all be thankful for family and for love.
I loved doing the rational functions activities with my Precalc kids!
1. The laser lab was FANTASTIC. I have to say that in El Salvador, it's not easy finding laser pointers. Even though my school has awesome resources and even a full-time driver to whom I can make
requests to run school-related errands for me during the day, he was only able to find two $16 punteros laser (muy costosos!!). In the end, I had to run around and borrow make-shift laser pointers
that were originally part of some fancy schmancy USB powerpoint clicker device. The science teachers were awesome and let me borrow 4 of their fancy clickers, plus our awesome head librarian had two
in her presentation technology collection, plus I had my two very-expensive $16 pointers. Made just enough for a class set! Yay.
But, all of the hassle was super worth it, because in the end, the kids collected beautiful data that fell neatly into a rational function pattern, and we discussed the conceptual linking between
where the domain breaks, the amount of vertical shift, and our laser setup. (The domain, which represents the laser source's horizontal distance away from the wall, is x > 30 or so, because the
activity instructions indicate that the mirror needs to be placed horizontally 25cm away from the wall, and realistically it's hard to stand/place the laser source right on top of the mirror when
you're doing the laser measurements, since the mirror rests on top of a platform made of textbooks, that juts out another few centimeters horizontally. The range of the function, which represents the
height of the reflected laser beam on the wall, is y > 15 or so, because the activity instructions indicate that the mirror platform be about 10cm high. Our kids estimated that the lowest point the
reflected laser beam could reach on the wall is just above that, or ~15cm. This gave them the partial equation y = a/(x-30) + 15, and all they had to still do was to plug in a point to solve for a.)
It was super!
2. And then today, I ran another lab with my Precalc kiddies on resistors. (See below.) Previously, I had used Megan Golding's resistors questions as intro to solving for equivalent resistances. We
did that intro on Friday. Today, for the actual lab portion, my kids first took color-coded resistors and used ohmmeters to find out the resistances across each individual resistor. (Since not all
resistors that are color-coded the same actually have the exact same resistances AND since the class needed to share a small stack of resistors, I made the kids grab two of each color to find their
average resistance to use in the calculations/predictions. That way, it didn't matter much later on if they grabbed another resistor of that color; its resistance value would be roughly the same as
the average resistance they had found earlier.)
I then gave them some series and parallel situations, and they had to make a prediction for the equivalent resistance, and then use the ohmmeter to verify their prediction. It was super cool; the
math on paper really came alive for them! They got to see their calculation results match what was popping up in their ohmmeters.
Another cool (but tricky) part of the lab was teaching kids to read ohmmeters. I'm not sure if all ohmmeters do this, but the ones I had borrowed from the physics teachers have a dial of different
settings. Depending on the resistance value, you have to change the position of the dial to measure a different maximum resistance, and the result actually takes on different decimal places /
different units. After about 10 to 15 minutes, kids were starting to get the hang of reading the different settings for the correct units, but in the beginning it was quite a bit tricky! (The physics
teacher was really pleased when I told him this afterwards; he said it's good practice for the kids to read/interpret the outputs from a machine such as an ohmmeter.)
So, first I should preface this story by saying that we have some really great kids at my school. Their parents are some of the richest and most influential people in a third-world country (some,
possibly in all of Central America), and many of them are expected to take over the family business regardless of how they do in school. Some of their parents are away from home all the time because
of work, and they are raised by maids and drivers. Yet, despite all of this, about 90% of the kids are really kind. In various circumstances, I've seen the way they treat the kids who are less
fortunate (ranging from orphans to poor kids to disabled kids), and their kindness is always genuine.
So, yesterday Geoff and I went to chaperon a volunteer trip to build houses in San Vincente for the victims of Hurricane Ida last November, who are still displaced from their homes. We took a group
of 6 juniors who voluntarily met us at school at 7am and who helped to carry rocks and to paint houses in a remote village/work site from 8am to 3pm on a Saturday (getting back to school ~4:30pm).
Geoff and I worked on the metal foundation of a house for that time, to improve its earthquake-preparedness. All in all, the kids were fabulous. They really enjoyed the experience, especially because
they got to meet the families whose houses they were helping to re-build. The families said some really powerful things, like they've had the strength to go on (after losing everything in Hurricane
Ida) only because they have seen the help that God had sent them via all of the international and local volunteers. (I'm not religious, but my kids certainly are, having grown up in a conservative
Catholic country. So, I'm sure hearing this is even more moving for them.)
But, there were strange things I observed that were characteristic of even our best kids that I wish were not. For instance, during lunch, our kids went into the school van, turned on the air
conditioning, and slept in the AC while every other Habitat for Humanity volunteer sat in the dirt and hung out. Or, at 3pm, they came to me and asked if we can stop at a gas station on the way back,
so that they could use the bathroom, because they couldn't stand using the outhouse. I was pretty embarrassed for them; I told the kids (because it was what I was feeling and I've taught more than
half of those kids) that they were re-affirming the impression that the American School kids were too good to use the same bathroom as everyone else. I also told them that in some countries/places,
those kinds of bathrooms are all people have, all the time. After I said that, a couple of the kids chuckled in embarrassment and half of the group went to use the outhouse. I took the rest up to a
"nicer" bathroom up the hill, because I figured it was better for them to use another bathroom than to hold their need in the car ride. But, I couldn't help being awed by the irony of it all.
There they were, volunteering their entire Saturday to sweat under a ridiculously warm sun in order to help out people who had lost everything a year ago in a flood. Yet they couldn't bring
themselves to use an outhouse. Amazing.
It's official: I am looking for a new job! Geoff and I had decided a few months ago that we wanted to move to Europe after this school year, the reason being that in a few years, we might be married
with kids and will not have the same freedom we have now to travel and look around.
It took me a while to tell all three of my supervisors (mostly because they're each insanely busy, and it's not one of those things you want to say during the passing period), but now the deed is
done. Next up: Looking for a job!! Scary. I don't have IB experience, which is a biggie when looking for European jobs, so Geoff and I will have to be extra flexible. But, we're hopeful that since
I'm starting relatively early (now), that I'll find a job by June 2011. :) (Geoff's working on getting his British passport in the meanwhile.) The exciting part is that we get to go to somewhere
different, that hopefully will also allow me to teach something different (ie. AP Calculus or IB)!
So, keep your fingers crossed for me that I won't be jobless (and homeless) by June.
This is the part of the year when I start to emphasize to students that every day, they're making choices towards their learning. During Quarter 1, I pretty much hand-held the freshmen through all
quiz corrections. Every time a kid did poorly on a quiz, I emailed home and convinced their parents to talk them into staying after school for some remediation. Last year, there was a change sometime
during the latter part of Q3 where kids started to be proactive on their own about their learning. I want that to happen sooner this year. Like this time, I told kids specifically if I thought they
needed extra review time with me after school before the test. Most came, although a couple of the kids didn't come because of sports commitments or other things. I told those kids sternly that
they're making a choice, and they have to understand that consequences follow their every choice. That way, if they don't end up doing too hot on the exam, it'll be a learning experience for them
about making positive choices.
We're more or less through with a few tedious, very algebraic weeks in Geometry! yay. Next big unit will be super hands-on (Methods of Measurement), so it'll be a nice break from all of the
heavy-duty algebra. In the interim, I've taken some projects from the wonderful Nancy Powell and modified them a bit. Check them out!
For the mini-golf project, I took Nancy's project and added a couple of scaffolding questions. I also added a section where the kids would design their own golf course (which I think she does make
the kids do on the computer, in GSP, but it wasn't in this version of her project).
For the string art project, since I don't actually want to spend a lot of class time making the artistic portion of the project, I made the whole sewing-with-strings thing to be optional (extra
credit). Instead, the focus of the project is on identifying symmetries and constructing regular shapes using a compass. (The kids will need to be able to construct these same shapes later, when we
begin to build nets of 3-D solids.)
That should take us to almost Thanksgiving. After Thanksgiving, we will have only a short week or so of instruction before we have to start reviewing for midterms (given twice a year)! Wildness.
I noticed on a recent quiz that my kids have trouble identifying angle relationships once there is a network of more than 3 lines. I have an idea for making kids construct parallel lines using the
angle concepts, that will hopefully help to further their ability to visualize angle relationships.
In my mind, the exercise looks like this:
1. I'll first let the kids draw a scalene triangle and label it ABC.
2. I'll ask the kids to use a protractor to construct "a line parallel to AC at point B, using the concept of alternate interior angles." (The kids should be able to do this quickly, since that's the
same angle relationship we used during our tessellations project a while back to create parallel lines. But, in my experience, kids need some help interpreting things like "a line parallel to AC at
B." It's surprisingly difficult for them to decode what that means!)
3. Then, I'll ask the kids to construct "a line parallel to BC at point A, using the concept of same-side interior angles."
4. Finally, the kids will construct "a line parallel to AB at point C, using the concept of corresponding angles."
5. Depending on if the kids feel like this has been a difficult exercise or not, at this point I might optionally insert requirements for written explanations next to each newly constructed line,
explaining which angle pairs are which type of angles. (Good practice with naming angles with 3 letters.)
Just thinking aloud.
I found a GREAT rational functions activity over at NCTM, that really focuses on helping kids understand the meaning of basic rational function equations. I'm in the process of doing some review /
test with my Precalc kids, but I have already given them the packet and we're going to be doing a good chunk of it (including the reflection activity!). yay! So excited. That, along with Megan
Golding's resistors lab and Kate Nowak's intro to rational expressions (which my kids have already seen), is going to make a neat mini-unit on rational functions! :)
I had my honors kids do a series of constructions in class with compass and a straight edge. To help them overcome the temptation to "cheat", I gave them popsicle sticks as straight edges. Most kids
figured out right away how to do an isosceles triangle (all on their own), and about half of the class figured out on their own how to make a kite. (I figured it was a small hop from being able to do
isosceles triangles.) Some kids fumbled their way to an equilateral triangle while trying to get the isosceles one. Then, everyone struggled with constructing a pair of parallel lines, so after they
struggled for a while, I let them open up to a part of the textbook that describes the "rhombus method" for constructing parallel lines. Except, the way that the book describes does not allow them to
construct 5 equally spaced parallel lines as I had requested. So, either the kids had to fiddle and figure out on their own a modified "rhombus method" (which a handful of kids did manage to do), or
they had to get a hint from me.
All in all, the kids liked the activity so much that I decided to turn it into a project. So, the next day I gave them a list of specs, and they brought me clean final drafts with explanations for
justifying why the sides are indeed congruent.
I tried to scan in the best piece of student work, but the scanner doesn't pick up on the arc marks all too well. For a kite, he started with two intersecting circles of different radii, and
connected the circles' centers to the points of intersection in their arcs. He constructed equilateral triangle using two intersecting circles of the same radius, and his parallel lines are formed
from a network of congruent circles.
Neat, eh? Lots of math with no numbers.
We'll be seeing construction again, very soon...
Admittedly, parallel lines and transversal problems are very contrived in most cases and are not very real-world relevant. But, I still like those problems because they give the kids some basic
algebra practice, while forcing them to think about the meanings of their equations.
In the last two years, I have consistently introduced the series of parallel-lines-and-transversal theorems using the same worksheets, and they have worked very well for my kids. My theory behind
these worksheets is that: A.) In order to learn the angles vocabulary, kids need to be actively engaged in visualizing the relationships. So, why not start with making them visualize the
relationships before introducing the terms? B.) Once they learn the terms, kids can discover all angle relationships via a protractor. C.) At the end, you give them some memory tools or some
color-coding shortcuts to quickly figure out angle relationships for setting up angle equations. That way, even if a kid can't remember the name of an angle relationship, they can still have enough
geometric knowledge to move through the algebra part. D.) In the end, as a quick check-in, kids should be able to quickly pick out the correct equations corresponding to different diagrams.
So, here we go.
1. Getting kids to visualize angle location relationships before introducing the terms.
2. Getting kids to conjecture about angle measurement relationships on their own.
3. Using colors to re-inforce angle relationships. (And I give them the memory tool that only "Same-side interior" and "Same-side exterior" angles are "Supplementary." Every other pair of
recognizable relationship is congruent.)
4. Final check-in. Kids should all be able to pick out the right equations.
These are worksheets and not activities, but they are very effective in teaching the basics of angle relationships. Follow it up with a day of textbook algebra problems practice, and your kids are
golden on this often-tested concept! (My Holt Geometry textbook also has an interesting angles word problems activity that I have adopted the last couple of years, which works well as an extension to
ask kids to look at the application of angles in a slightly more realistic situation.)
And of course, you can always defer to Erastothenes to show kids some gee-whiz angles math.
By the way, stay tuned for my kids' straight-edge and compass construction projects. Neatest pure-geometry thing we've done in a while.
We have spent the last two weekends away, first in Austin for a wedding, then in Panama over the four-day weekend (para Dia de los Muertos). Some observations:
* Downtown Austin is just as fun as rumor has it! But, you have to be ready for bars to smell like 18-year-olds (ie. throwup). ...And for the bars to close at 2am (demaciado temprano...).
* Taxis in Austin cost something like $2 a minute! It's pretty insane. Geoff almost asked one of our drivers whether the meter was broken. I think every single time we left our hotel to go anywhere,
it was about a $15 ride -- even if the ride takes only 5 minutes! Thank goodness we were kindly given a ride by some of Allen's non-drinking friends to and from the wedding, so that we could party it
up without having to be the DD.
* Best restaurant we found in Austin (recommended by the locals), hands down: "Moonshine." It has various Southern food, but with a unique twist. It also feels like you're sitting in someone's back
yard, having a nice brunch.
* San Antonio, TX, is also a nice town. Downtown San Antonio has a man-made river-front that's really nice, and with very friendly bartenders.
* Panama Canal is just a bunch of locks. It has got a cool history bit, obviously, but isn't actually much to look at. If you go, you should definitely pay the extra $3 to watch the introductory
movie, because you can get a sense of the past, present, and future of the canal and its continuing importance to the country and the world.
* Panama is diverse! I love that. Panama City is probably the most metropolitan city I've seen in Central America. They also have ethnic foods (including Indian, even though our taxi driver messed up
and took us to a Lebanese restaurant instead), which is really exciting. The area around the casinos is verrrry "working girl" friendly, which we discovered by accident.
* By law, only the native tribes can own the land on the islands scattered around the 365 beautiful San Blas islas. We stayed with a native family overnight, and went on some island-hopping during
the day. The village was very rustic! (For example, various households, if not the entire village, share two "toilets", which are merely two holes that hover above the ocean. They don't have a sewage
system. The villagers live in grass huts and throw their trash directly into the ocean.) It was a really neat/unique experience!
That's it for now. Progress reports are due this Friday, so things are busy with work, obviously. Soon, Geoff's parents will be here visiting us (over Thanksgiving weekend), and we'll be busy on that
end with the preparations, as well. :) I can't wait. The year is just flying by! | {"url":"https://untilnextstop.blogspot.com/2010/11/","timestamp":"2024-11-14T15:37:39Z","content_type":"application/xhtml+xml","content_length":"153877","record_id":"<urn:uuid:66e7c869-2fe6-46c2-b102-823e6efe6b63>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00242.warc.gz"} |
We shall discuss some Selection Principles and related games on multicovered spaces. A multicovered space is a pair (X,m) consisting of a set X and a family m of covers of X. The category of
multicovered space is a natural place where Theory of Selection Principles develops naturally and deeply. A typical selection principle asserts that for each sequence (u[n])[n Î w] Î m^w of
covers of a multicovered space (X,m) it is possible to select a cover v = {B[n] : n Î w} of X by u[n]-bounded subsets B[n] Ì X so that for each point x Î X the index set {n Î w: x Î B[n]} is
"large" in a suitable sense. If "large" means "non-empty" (resp. "coinfinite") then we obtain the classical Menger (resp. Hurewicz) property. The (non-trivial and highly fruitful) interplay
between Selection Principles and recently created Theory of Semifilters will be discussed as well.
The notion of uniform embedding of metric spaces plays an important role in study of large scale properties of finitely generated groups. A map f: X® Y of metric spaces (X,d[X]) and (Y,d[Y]) is
called a uniform embedding if there are two real functions r[-] and r[+] with lim[r®¥] r[-] (r) = +¥ such that r[-] (d[X](x,z)) £ d[Y] (f(x),f(z)) £ r[+](d[X](x,z)) for all x,z Î X. For example,
a bi-Lipschitz map is a uniform embedding with linear functions r[-] and r[+]. If one tries to embed a given space X uniformly into Hilbert space, how close to bi-Lipschitz could the embedding
be? We answer this question for finite dimensional CAT(0) cube complexes and for hyperbolic groups with word metric.
By the Kuratowski-Ulam theorem, if A Í R^n+1 = R^n ×R is a Borel set which has second category intersection with every ball (i.e., is "everywhere second category"), then there is a y Î R such
that the section AÇ(R^n ×{y}) is everywhere second category in R^n ×{y}. If A is not Borel, then there may not exist a large cross-section through A, even if the section does not have to be flat.
For example, a variation on a result of T. Bartoszynski and L. Halbeisen shows that there is an everywhere second category set A Í R^n+1 such that for any polynomial p in n variables, AÇgraph(p)
is finite. It is a classical result that under the Continuum Hypothesis, there is an everywhere second category set L in R^n+1 which has only countably many points in any first category set. In
particular, LÇgraph(f) is countable for any continuous function f :R^n ® R. We prove that it is relatively consistent with ZFC that for any everywhere second category set A in R^n+1, there is a
function f : R^n ®R which is the restriction to R^n of an entire function on C^n and is such that, relative to graph(f), the set AÇgraph(f) is everywhere second category. Moreover, given a
non-negative integer k, a function g : R^n® R of class C^k and a positive continuous function e: R^n ® R, we may choose f so that for all multiindices a of order at most k and for all x Î R^n, |D
^a f(x)-D^a g(x)| < e(x). The method builds on fundamental work of K. Ciesielski and S. Shelah which provides, for everywhere second category sets in 2^w×2^w, large sections which are the graphs
of homeomorphism of 2^w.
We use approximations of multivalued maps by chain mappings into singular chain complexes to prove fixed point theorems.
Let S^1 = R/Z denote the complex unit circle and define s: S^1 ® S^1 by s(t) = 2t mod 1. Thurston describes a collection of s-invariant laminations on the complex unit disk [`(D)], which gives a
combinatoric parametrization of the Mandelbrot set. Each one of these laminations defines an equivalence relation ~ on S^1 such that (s, ~ ) induces a map F : S^1/ ~ ® S^1/ ~ . Often, there
exists a quadratic polynomial P with Julia set J such that P|[J] is semi-conjugate to F. However there are obstructions to this being true in general. One of these obstructions is that S^1/ ~
could reduce to a point. In this case we call the lamination degenerate.
Bullett and Sentenac introduced the notion of a closed set having a sequence of rotation numbers for s. This notion is related to "ouady tuning". We use this concept to give a necessary and
sufficient condition for when the lamination is degenerate.
The Hahn-Mazurkiewicz Problem asks for conditions under which a Hausdorff space is the continuous image of a generalized arc. The first characterizations of continuous images of non-metric arcs
were given by Bula and Turzanski and by Nikiel. Additional results include those of Mardesi\'c, Treybig, and many others.
In a related study, we herein consider applications of selections (carriers) to the study of images of ordered compacta. In particular, let X be a compact ordered space, Y a Hausdorff space, and
let F(Y) denote the family of all nonempty closed subsets of Y with the Vietoris topology. Assuming G : X® F(Y) is continuous, we consider conditions under which G can be "lifted" to a continuous
map of X onto Y.
This work relies heavily on work of R. S. Countryman as well as the theory of selections and that of continuous images of ordered compacta.
Let E denote the class of compact groups that admit no continuous endomorphism with infinite topological entropy.
(1) There exist connected infinite-dimensional groups in E, but all abelian groups in E are finite-dimensional.
(2) An abelian compact connected K group belongs to E iff K is finite-dimensional.
The non-connected groups in E are hard to describe even in the abelian case. In particular, we do not know the answer to the following question: does there exist an abelian totally disconnected
group in E that has endomorphisms with positive entropy?
Let (G[i])[i Î I] be a family of topological Abelian groups and let Å[i Î I] G[i] denote their algebraic direct sum, that is, the subgroup of the product Õ[i Î I] G[i] formed by those families x
= (x(i))[i Î I] for which x(i)=0 for all but finitely many i Î I. If we let H^Ù denote the character group of the Abelian group H, there are canonical algebraic isomorphisms
æ ö Ù æ ö Ù
è Õ G[i] ø » Å G[i]^Ù è Å G[i] ø » Õ G[i]^Ù.
i Î I i Î I i Î I i Î I
Two significant group topologies arise on Å[i Î I] G[i] in a very natural way: the box topology T[b], given by "rectangular" neighborhoods of zero, and the coproduct topology T[f], which is the
finest group topology on Å[i Î I] G[i] which makes all canonical inclusions continuous.
Nevertheless, in general none of them turns the above-mentioned isomorphisms into topological ones, when considering Tychonoff topology on the products, and compact-open topology on all dual
groups. The analysis of such duality properties leads to the definition of an intermediate topology T[*], the so-called asterisk topology, firstly introduced by Kaplan in 1948. Actually the lack
of a natural generalization of Minkowski functional to groups gives rise to a number of variants of Kaplan's original definition. We shall survey what is known about the conditions under which
such topologies are in fact the same, their behaviour with respect to duality, reflexivity and local quasi-convexity, and their relation with each other and with the box and coproduct topologies.
It is proved that for a 3-dimensional compact metrizable space X the infinite real projective space RP^¥ is an absolute extensor of X if and only if the real projective plane RP^2 is an absolute
extensor of X.
The method of resolutions was introduced in [1] (see also [2]). This method allows us to construct new spaces using given collections of spaces. Many examples of applications of this method are
given in [3]. By applying iterated resolutions and fully closed mappings one can obtain more sophisticated examples [4]. Here we present several new applications of resolutions.
Theorem 1 For any prime p there exists a 2-dimensional homogeneous separable first countable compact space T[p] such that dim(T[p]×T[q]) = 3 for p ¹ q.
Question 1 Are there homogeneous metrizable compacta X and Y such that dim(X×Y) < dimX + dimY?
Recent results by J. L. Bryant [5] imply that if X and Y are homogeneous metrizable ANR-compacta, then
Question 2 Does the equality (1) hold if X is a homogeneous ANR-compactum and Y is an arbitrary (homogeneous) metrizable compactum?
Remark 1 As for Question 1, we cannot omit homogeneity of Y, since Pontryagin's surface P[2] is homogeneous.
Another two results are joint with A. V. Ivanov and J. van Mill.
Theorem 2 [CH; [6]] For every n Î N, there exists a family of separable compacta X[i], i Î N, such that for every non-empty finite subset M of N and every non-empty closed subset F of P[i Î M] X
[i] we have dimF = k(F)n, where k(F) is integer such that k(F) ³ 1 for infinite F. Moreover, |F|=2^c for infinite closed F.
Theorem 3 [CH; [6]] There exists an infinite separable compactum X such that for any positive integer m, if F is an infinite closed subset of X^m, then |F|=2^c and F is strongly
Question 3 Does there exist in ZFC an n-dimensional compactum Y[n], n ³ 2, such that for every m ³ 2, every non-empty closed subset F of Y^m[n] has dimension kn, where k is some integer between 0
and m?
Question 4 Does there exist in ZFC an infinite-dimensional compactum Z such that for every non-empty closed subset F of Z^2 we have either dimF = 0 or F is infinite-dimensional?
V. V. Fedorchuk, Bicompacta with non-coinciding dimensionalities. Soviet Math. Dokl. 9(1968), 1148-1150.
V. V. Fedorchuk and K. P. Hart, d-23 Special Constructions. In: Encyclopedia of General Topology (K. P. Hart, J. Nagata and J. E. Vaughan, eds.), Elsevier Science Ltd., 2004, 229-232.
S. Watson, The construction of topological spaces: planks and resolutions. In: Recent Progress in General Topology (M. Husek and J. van Mill, eds.), North-Holland Publishing Co., Amsterdam,
1992, 673-757.
V. V. Fedorchuk, Fully closed mappings and their applications. (Russian) Fundament. i Prikl. Matem. (4) 9(2003), 105-235; J. Math. Sci. (New York), to appear.
J. L. Bryant, Reflections on the Bing-Borsuk conjecture. Preprint, 2003, 1-4.
V. V. Fedorchuk, A. V. Ivanov and J. van Mill, Intermediate dimensions of products. Topology Appl., submitted.
We discuss some recent results on the topic of the title.
As a rule, most of the classical Michael-type selection theorems for the existence of single-valued continuous selections are analogues and, in certain respects, generalizations of ordinary
extension theorems. In contrast to this, most of the selection theorems for the existence of semi-continuous set-valued selections seem to have no proper analogues in the extension theory. In
this talk, we will discuss the role of the "selection" condition in such theorem, and how it is related to the metrizability of the range of the corresponding set-valued mappings.
For a compact Hausdorff space X, C(X) denotes the ring of all continuous complex-valued functions on X. The ring C(X) is said to be algebraically closed if each monic polynomial with C(X)
-coefficients has a root in C(X). Starting with a classical theorem due to Countryman, Jr., we discuss a problem on topological characterizations of X with C(X) being algebraically closed. Also
the existence of "approximate roots" and related topics will be discussed.
The present talk is based on joint works with A. Chigogidze, A. Karasev, T. Miura and V. Valov.
A trajectory of a flow on a 3-manifold is wild if the closure of at least one of the semi-trajectories is a wild arc. A trajectory is 2-wild if the closure of each semi-trajectory is a wild arc.
We describe a method of embedding wild trajectories in flows on 3-manifolds. This method yields interesting examples of dynamical systems. In particular, every boundary-less 3-manifold admits a
flow with a discrete set of fixed points and such that the closure of every non-trivial trajectory is 2-wild, which answers a question posed at the 2004 Spring Topology and Dynamics Conference.
In a widely circulated preprint (1984) William Thurston introduced the notion of a (geodesic) lamination of the unit disk. Laminations are combinatorial/geometric/topological objects used to
study Julia sets of polynomials in analytic complex dynamics. A lamination of the unit disk is a closed collection of chords of the disk that do not cross each other (they may touch at
endpoints). Consider the power map f(z)=z^d, d > 1, on the unit circle; extend f linearly to the lamination (the chords). A chord is critical if its endpoints map to one point. A lamination is
invariant if the collection maps to itself forward and backward, with d-many disjoint pre-images of each chord backward, and f extends linearly to a positively-oriented confluent map of the disk
to itself. The plan is that
(1) a lamination is determined by `pulling back' a set of critical chords,
(2) the lamination naturally induces an equivalence relation on the unit circle,
(3) the quotient space of the circle under this equivalence relation is a topological Julia set, and
(4) the topological Julia set is dynamically (and topologically) equivalent to an analytic Julia set for some degree d polynomial.
But there are obstructions to the fulfillment of the plan. Thurston completed most of the plan for d=2, but left some questions unanswered. Moreover, fundamental questions remain unanswered for d
> 2, but recent progress has been made. In particular, one obstruction is that the lamination determined by a collection of critical chords may naturally induce a degenerate equivalence relation,
collapsing the circle to a point in the quotient. In this talk, we show how the obstruction arises in degree d=2, and give some insight into degree d=3 and greater. In a subsequent talk at this
meeting, D. Childers provides a complete solution to when degeneracy occurs, for degree d=2, in terms of the dynamics of the critical chord, answering an implicit question of Thurston.
This talk is mostly joint work with members of the UAB Laminations Seminar: A. Blokh, L. Oversteegen, D. Childers, G. Brouwer, C. Curry, and P. Eslami.
A continuum X is simple triod-like if for every e > 0 there exists a continuous function g[e] : X® T such that T is a simple triod and for every t Î T, diam((g[e](t))^-1 ) < e. I will discuss the
techniques used in showing when a map f : X® X has a periodic point where X is a simple triod-like continuum.
A connected open subset U of the sphere is called pseudo convex if for all points z in U there exist at most two closest points in the boundary of U. Answering a question by David Herron and
David Minda, we show that such a set has at most two boundary components. We also provide a detailed analysis of sets with this property.
We discuss economic models for which the space of predicted future states is an inverse limit space.
The class of spaces which have the property that every cover by clopen sets has a finite subcover was introduced by A. Sostaks. These spaces are now known as CLP-compact spaces and it has emerged
that much of the interesting behaviour of this class derives from the possibility that the product of two topological spaces contains clopen sets which do not belong to the algebra generated by
the product of the algebras of clopen sets in each factor. Hence the productive nature of CLP-compactness poses certain problems not occurring in the classical case. Indeed, the problem of
finding weak hypotheses under which the product of CLP-compact spaces is CLP-compact should still be considered to be open even though some progress has been recorded. It will be shown that the
product of finitely many sequential, CLP-compact spaces is CLP-compact.
For a family of sets A, a set X and a cardinal k (usually £ w), X is said to be a k-transversal of A if X Í ÈA and 0 ¹ |aÇX| < k for each a Î A. If k=2 we will say that X is a transversal of A. X
is said to be a Bernstein set for A if Æ ¹ aÇX ¹ a for each a Î A. When an almost disjoint family admits a k-transversal or a Bernstein set was first studied in [1] motivated mainly by
applications in topology.
We consider here a weaker property:
Definition Given a family of sets A, A is said to admit a s-transversal if A can be written as A = È{A[n] :n Î w} such that each A[n] admits a transversal.
The restriction that an almost disjoint family admits a transversal is quite strong and not of much interest. However, quite a wide class of almost disjoint families admit s-transversals. We
consider the question when an almost disjoint family admits a s-transversal and present some examples and applications.
P. Erdös and A. Hajnal, On a property of families of sets. Acta Math. Acad. Sci. Hungar. 12(1961), 87-124. | {"url":"https://www2.cms.math.ca/Events/summer05/abs/gta.html","timestamp":"2024-11-07T22:30:48Z","content_type":"text/html","content_length":"40598","record_id":"<urn:uuid:98588554-8f1c-45ac-9684-2f4693640650>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00658.warc.gz"} |
An Introduction to Matplotlib – Python’s Data Visualization Library - Python Unbound
An Introduction to Matplotlib – Python’s Data Visualization Library
What is matplotlib?
Matplotlib is a popular open-source library for data visualization in Python. It provides a variety of functions and tools for creating a wide range of plots and charts, including line plots, scatter
plots, bar plots, histograms, pie charts, and more using the PyPlot Method.
One of the main features of matplotlib is its ability to create high-quality plots and charts with a simple and intuitive interface. Users can easily define the data to be plotted, the type of plot
to be created, and various formatting options such as colors, line styles, and plot titles. Matplotlib also provides a number of customization options, allowing users to fine-tune the appearance of
their plots and charts.
In addition to creating static plots and charts, matplotlib also provides tools for creating interactive plots and visualizations. Users can use matplotlib’s event handling and animation functions to
create interactive plots that respond to user input or change over time.
Matplotlib is widely used in a variety of fields, including data analysis, scientific computing, and machine learning. It is often used in conjunction with other libraries such as NumPy and Pandas
for data manipulation and analysis, and with libraries such as SciPy and scikit-learn for statistical analysis and machine learning.
PyPlot: Customization
PyPlot is the main plotting tool in the matplotlib library. It has many features that allow you to create visualizations directly in programs like jupyter notebook or Google Collab or whatever coding
environment you work in. One of the basic features that is important to master is how to customize a chart. PyPlot allows you to customize many of the chart elements. Below is an example of the basic
import matplotlib.pyplot as plt
# Sample data
x = [1, 2, 3, 4, 5]
y = [2, 4, 6, 8, 10]
# Create the figure and axes objects
fig, ax = plt.subplots()
# Plot the data
ax.plot(x, y, 'o-', color='blue', linewidth=2, markersize=10)
# Customize the x and y axis labels
ax.set_xlabel('Stuff along the X')
ax.set_ylabel('Things along the Y')
# Customize the title
ax.set_title('Customized Chart')
# Customize the grid
ax.grid(True, linestyle='--', color='gray', alpha=0.7)
# Show the plot
This code creates a simple line chart with sample data. The chart is customized by setting the labels for the x and y axes, the title of the chart, the color and style of the line and markers, and
the appearance of the grid. You can adjust the properties like color, linewidth, markersize, label and title etc to customize the chart. For more information on chart customization in PyPlot, check
out this article I found with code examples over on Python Graph Gallery
PyPlot: Plot Types
PyPlot can produce a variety of different chart types as well. This can be useful for looking at your dataset in different ways or when needing to make different types of comparisons. Its quite
likely we all remember from grade school that a line graph shows trends over time or relationships between two variables and that Bar or Column graphs are meant to show comparisons. In the Data world
we use a variety of less common charts including histograms, scatter charts, and various metric graphs to demonstrate model effectiveness. Here is an example of code that demonstrates different plot
types that the PyPlot library can produce:
import matplotlib.pyplot as plt
import numpy as np
# Sample data
x = np.linspace(0, 10, 100)
y1 = np.sin(x)
y2 = np.cos(x)
# Create the figure and axes objects
fig, axs = plt.subplots(2, 2)
#Line plot
axs[0, 0].plot(x, y1, '-', color='blue', label='sin(x)')
axs[0, 0].plot(x, y2, '-', color='red', label='cos(x)')
axs[0, 0].set_title('Line Plot')
axs[0, 0].legend(loc="best")
#Scatter plot
axs[0, 1].scatter(x, y1, color='blue', label='sin(x)')
axs[0, 1].scatter(x, y2, color='red', label='cos(x)')
axs[0, 1].set_title('Scatter Plot')
axs[0, 1].legend(loc="best")
#Bar plot
axs[1, 0].bar(x, y1, color='blue', label='sin(x)')
axs[1, 0].bar(x, y2, color='red', label='cos(x)')
axs[1, 0].set_title('Bar Plot')
axs[1, 0].legend(loc="best")
#Histogram plot
axs[1, 1].hist(y1, bins=20, color='blue', histtype='bar', label='sin(x)')
axs[1, 1].hist(y2, bins=20, color='red', histtype='bar', label='cos(x)')
axs[1, 1].set_title('Histogram Plot')
axs[1, 1].legend(loc="best")
This code creates a 2×2 grid of subplots. The top left subplot shows a line plot of the sin(x) and cos(x) functions, the top right subplot shows a scatter plot of the same data, the bottom left
subplot shows a bar plot and the bottom right subplot shows a histogram of the same data. It also shows the title, legend and x and y axis labels of each plot. You can experiment with different plot
types and customize them as demonstrated above.
PyPlot: Annotation
Pyplot can also be used to annotate a chart, which can be helpful when you are trying to accent certain information when presenting it to stakeholders. The chart below uses this function to callout
the Maximum and Minimum values presented on the graph.
import matplotlib.pyplot as plt
import numpy as np
# Sample data
x = np.linspace(0, 2*np.pi, 100)
y = np.sin(x)
# Create the figure and axes objects
fig, ax = plt.subplots()
# Plot the data
ax.plot(x, y, '-', color='blue')
# Annotate the maximum value
max_val = max(y)
max_ind = np.argmax(y)
ax.annotate(f'Max: {max_val:.2f}', xy=(x[max_ind], max_val), xytext=(x[max_ind]+0.1, max_val+0.2),
arrowprops=dict(facecolor='red', shrink=0.05))
# Annotate the minimum value
min_val = min(y)
min_ind = np.argmin(y)
ax.annotate(f'Min: {min_val:.2f}', xy=(x[min_ind], min_val), xytext=(x[min_ind]-0.3, min_val-0.2),
arrowprops=dict(facecolor='green', shrink=0.05))
# Customize the x and y axis labels
ax.set_xlabel('X Label')
ax.set_ylabel('Y Label')
# Show the plot
Visualization with pandas
matplotlib can also be used alongside other libraries like pandas. I covered pandas in a previous article here. Integrating the use of matplotlib with pandas allows you to activate the power of the
pandas DataFrame structure that is extremely versitile and user friendly. Here’s an example of code on how to get started with matplotlib and pandas together
import matplotlib.pyplot as plt
import pandas as pd
# Create a sample dataframe
data = {'name': ['John', 'Jane', 'Mike', 'Emily', 'Adam'],
'age': [35, 28, 32, 42, 25],
'income': [50000, 60000, 55000, 70000, 35000]}
df = pd.DataFrame(data)
# Use the 'plot' function of the dataframe to create a bar chart
df.plot(kind='bar', x='name', y='income', color='blue')
# Add labels and title
plt.title('Income by Name')
# Show the plot
Takeaway: Matplotlib is a powerful library for data visualization.
Overall, matplotlib is a powerful and widely-used library for data visualization in Python. Its simple and intuitive interface, along with a range of customization options, make it a popular choice
for creating high-quality plots and charts.
Get more information and code examples on matplotlib in the official documentation here.
The notebook with all the code examples I wrote for this article can be found here.
All the code examples from my articles are also available via my GitHub. | {"url":"https://pythonunbound.com/best-libraries-in-python-matplotlib","timestamp":"2024-11-06T04:43:41Z","content_type":"text/html","content_length":"144568","record_id":"<urn:uuid:27264b44-227c-4496-aa73-c7395315aaf2>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00135.warc.gz"} |
Bean plots in SPSS
It seems like I have come across alot of posts recently about visualizing univariate distributions. Besides my own recent blog post about comparing distributions of unequal size in SPSS, here are a
few other blog posts I have recently come across;
Such a variety of references is not surprising though. Examining univariate distributions is a regular task for data analysis and can tell you alot about the nature of data (including potential
errors in the data). Here are some posts on the Cross Validated Q/A site of related interest I have compiled;
In particular the recent post on bean plots and Luca Fenu's post motivated my playing around with SPSS to produce the bean plots here. Note Jon Peck has published a graphboard template to generate
violin plots for SPSS, but here I will show how to generate them in the usual GGRAPH commands. It is actually pretty easy, and here I extend the violin plots to include the beans suggested in bean
A brief bit about the motivation for bean plots. Besides consulting the article by Peter Kampstra, one is interested in viewing a univariate continuous distribution among a set of different
categories. To do this one uses a smoothed kernel density estimate of the distribution for each of the subgroups. When viewing the smoothed distribution though one loses the ability to identify
patterns in the individual data points. Patterns can mean many things, such as outliers, or patterns such as striation within the main body of observations. The bean plot article gives an example
where striation in measurements at specific inches can be seen. Another example might be examining the time of reported crime incidents (they will have bunches at the beginning of the hour, as well
as 15, 30, & 45 minute marks).
Below I will go through a brief series of examples demonstrating how to make bean plots in SPSS.
SPSS code to make bean plots
First I will make some fake data for us to work with.
set seed = 10.
input program.
loop #i = 1 to 1000.
compute V1 = RV.NORM(0,1).
compute groups = TRUNC(RV.UNIFORM(0,5)).
end case.
end loop.
end file.
end input program.
dataset name sim.
value labels groups
0 'cat 0'
1 'cat 1'
2 'cat 2'
3 'cat 3'
4 'cat 4'.
Next, I will show some code to make the two plots below. These are typical kernel density estimates of the V1 variable I made for the entire distribution, and these are to show the elements of the
base bean plots. Note the use of the TRANS statement in the GPL to make a constant value to plot the rug of the distribution. Also note although such rugs are typically shown as bars, you could
pretty much always use point markers as well in any situation where you use bars. Below the image is the GGRAPH code used to produce them.
*Regular density estimate with rug plot.
/GRAPHDATASET NAME="graphdataset" VARIABLES=V1 MISSING=LISTWISE REPORTMISSING=NO
/GRAPHSPEC SOURCE=INLINE.
SOURCE: s=userSource(id("graphdataset"))
DATA: V1=col(source(s), name("V1"))
TRANS: rug = eval(-26)
GUIDE: axis(dim(1), label("V1"))
GUIDE: axis(dim(2), label("Density"))
SCALE: linear(dim(2), min(-30))
ELEMENT: interval(position(V1*rug), transparency.exterior(transparency."0.8"))
ELEMENT: line(position(density.kernel.epanechnikov(V1*1)))
END GPL.
*Density estimate with points instead of bars for rug.
/GRAPHDATASET NAME="graphdataset" VARIABLES=V1 MISSING=LISTWISE REPORTMISSING=NO
/GRAPHSPEC SOURCE=INLINE.
SOURCE: s=userSource(id("graphdataset"))
DATA: V1=col(source(s), name("V1"))
TRANS: rug = eval(-15)
GUIDE: axis(dim(1), label("V1"))
GUIDE: axis(dim(2), label("Density"))
SCALE: linear(dim(2), min(-30))
ELEMENT: point(position(V1*rug), transparency.exterior(transparency."0.8"))
ELEMENT: line(position(density.kernel.epanechnikov(V1*1)))
END GPL.
Now bean plots are just the above plots rotatated 90 degrees, adding a reflection of the distribution (so the area of the density is represented in two dimensions), and then further paneled by
another categorical variable. To do the reflection, one has to create a fake variable equal to the first variable used for the density estimate. But after that, it is just knowing alittle GGRAPH
magic to make the plots.
compute V2 = V1.
/make V from V1 V2
/index panel_dum.
/GRAPHDATASET NAME="graphdataset" VARIABLES=V panel_dum groups MISSING=LISTWISE REPORTMISSING=NO
/GRAPHSPEC SOURCE=INLINE.
SOURCE: s=userSource(id("graphdataset"))
COORD: transpose(mirror(rect(dim(1,2))))
DATA: V=col(source(s), name("V"))
DATA: panel_dum=col(source(s), name("panel_dum"), unit.category())
DATA: groups=col(source(s), name("groups"), unit.category())
TRANS: zero = eval(10)
GUIDE: axis(dim(1), label("V1"))
GUIDE: axis(dim(2), null())
GUIDE: axis(dim(3), null())
SCALE: linear(dim(2), min(0))
ELEMENT: area(position(density.kernel.epanechnikov(V*1*panel_dum*1*groups)), transparency.exterior(transparency."1.0"), transparency.interior(transparency."0.4"),
color.interior(color.grey), color.exterior(color.grey)))
ELEMENT: interval(position(V*zero*panel_dum*1*groups), transparency.exterior(transparency."0.8"))
END GPL.
Note I did not label the density estimate anymore. I could have, but I would have had to essentially divide the density estimate by two, since I am showing it twice (which is possible, and if you
wanted to show it you would omit the GUIDE: axis(dim(2), null()) command). But even without the axis they are still reasonable for relative comparisons. Also note the COORD statement for how I get
the panels to mirror each other (the transpose statement just switches the X and Y axis in the charts).
I just post hoc edited the chart to get it to look nice (in particular settign the spacing between the panel_dum panel to zero and making the panel outlines transparent), but most of those things can
likley be more steamlined by making an appropriate chart template. Two things I do not like, which I may need to edit the chart template to be able to accomplish anyway; 1) There is an artifact of a
white line running down the density estimates, (it is hard to see with the rug, but closer inspection will show it), 2) I would prefer to have a box around all of the estimates and categories, but to
prevent a streak running down the middle of the density estimates one needs to draw the panel boxes without borders. To see if I can accomplish these things will take further investigation.
This framework is easily extended to the case where you don't want a reflection of the same variable, but want to plot the continuous distribution estimate of a second variable. Below is an example,
and here I have posted the syntax in entirety used in making this post. In there I also have an example of weighting groups inversely proportional to the total items in each group, which should make
the area of each group equal.
In this example of comparing groups, I utilize dots instead of the bar rug, as I believe it provides more contrast between the two distributions. Also note in general I have not superimposed other
summary statistics (some of the bean plots have quartile lines super-imposed). You could do this, but it gets a bit busy.
Tue January 27, 2015 01:56 AM
Thanks, Andrew, and also thanks for the link. Interesting insights on within-the-bar bias and overestimating differences with barchart error margins. Personally I liked the idea to show errors as
gradually disappearing bars around statistic.
Fri January 23, 2015 07:44 AM
It is a good question Anton. I have not used plots like these, but more typical histograms in publications. I would personally be ok with them, but I can't speak to everyone. (It would not surprise
me to see blow back from using them in a publication -- but I rather get a critique of that than many other things.)
If they do a good job showing differences that would be masked with summary statistics or other graphs I think they have a good argument for there use. Also see
Error Bars Considered Harmful: Exploring Alternate Encodings for Mean and Error (Correll & Gleicher, 2014)
, for some experimental evidence that they are more effective for showing statistical tests (and not just kde's of the original data).
Fri January 23, 2015 05:52 AM
Hi, Andrew! Thanks for excellent examples of violin plots in SPSS. A bit off-topic, do you happen to know, how favourable are editors of scientific journals for such illustrations? I mean I didn't
see much papers using violins in particular in the domain of sociology or medicine. To some they might look not so "strict" as, say, boxplots: the width of violin changes, but there is no explicit
axis underlying. Just interesting in your opinion/experience.
Thu March 27, 2014 10:04 AM
[…] histograms are not the most appropriate tool for identifying outliers (e.g. a rug plot showing individual values below the axis would help), but this is a fairly simple change to make […]
Mon June 25, 2012 12:48 PM
Hi Louise,
In theory, something like ELEMENT: point(position(summary.median(V*1*panel_dum*1*groups)), color.exterior(panel_dum), shape("Median"), size(size.large)) should work, although my quick attempts to get
it to act as desired were unsucessful. I also tried to make an actual summary variable via the TRANS command in inline GPL and that did not work either. So what I ended up doing was making a new
variable and plotting that in its own element statement. Note when you have multiple elements like this the legend gets a bit un-wieldy, and what I have been doing is editing post-hoc in a different
vector editor to get the legend to how I want it. Part of the problem I think is that SPSS does not like mapping the same aesthetic to different types of elements, so sometimes it gives errors when
trying to construct the legend.
Below is an example using the same data that is in the last plot in my blog post.
/OUTFILE=* MODE=ADDVARIABLES
/BREAK=groups panel_dum
/GRAPHDATASET NAME="graphdataset" VARIABLES=V V_median panel_dum groups MISSING=LISTWISE REPORTMISSING=NO
/GRAPHSPEC SOURCE=INLINE.
SOURCE: s=userSource(id("graphdataset"))
COORD: transpose(mirror(rect(dim(1,2))))
DATA: V=col(source(s), name("V"))
DATA: V_median=col(source(s), name("V_median"))
DATA: panel_dum=col(source(s), name("panel_dum"), unit.category())
DATA: groups=col(source(s), name("groups"), unit.category())
TRANS: zero = eval(20)
TRANS: med_point = eval(40)
GUIDE: axis(dim(1), label("V1"))
GUIDE: axis(dim(2), label("Frequency"))
GUIDE: axis(dim(3), null())
GUIDE: legend(aesthetic(aesthetic.color.exterior))
GUIDE: legend(aesthetic(aesthetic.color.interior))
GUIDE: legend(aesthetic(aesthetic.shape.interior), null())
GUIDE: legend(aesthetic(aesthetic.transparency.exterior), null())
GUIDE: legend(aesthetic(aesthetic.transparency.interior), null())
GUIDE: legend(aesthetic(aesthetic.size), null())
SCALE: linear(dim(2), min(0))
SCALE: cat(aesthetic(aesthetic.shape.interior), map(("Median", shape.square), ("Rug", shape.circle)))
SCALE: cat(aesthetic(aesthetic.transparency), map(("Median", transparency."0.0"), ("Rug", transparency."0.9")))
SCALE: cat(aesthetic(aesthetic.size), map(("Median", size."8"), ("Rug", size."4")))
ELEMENT: point(position(V*zero*panel_dum*1*groups), color.exterior(panel_dum), color.interior(panel_dum), transparency.interior("Rug"), transparency.exterior("Rug"),
shape.interior("Rug"), size("Rug"))
ELEMENT: point(position(V_median*zero*panel_dum*1*groups), color.exterior(panel_dum), color.interior(panel_dum), transparency.interior("Median"), transparency.exterior("Median"),
shape.interior("Median"), size("Median"))
ELEMENT: area(position(density.kernel.epanechnikov(V*1*panel_dum*1*groups)), transparency.exterior(transparency."1.0")), color(panel_dum), transparency.interior(transparency."0.5")))
END GPL.
Thanks for the comment and if you have any other questions let me know here in the comments or feel free to shoot me an email.
Fri June 22, 2012 11:33 AM
Hi Andrew,
Thanks for posting the syntax - it was really useful to follow the steps you took to make these. I'm just wondering how you could add a coloured point to distinguish the median in each group- would
it be something like this:
ELEMENT: point(position(summary.mean(V*1*panel_dum*1*groups))?
thanks in advance for your help! | {"url":"https://community.ibm.com/community/user/ai-datascience/blogs/archive-user/2012/05/20/bean-plots-in-spss","timestamp":"2024-11-06T15:37:22Z","content_type":"text/html","content_length":"636732","record_id":"<urn:uuid:833eba47-e03f-4e17-8029-ace9679b834c>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00202.warc.gz"} |
Understanding Mathematical Functions: Which Graph Represents A One To
Understanding Mathematical Functions and Their Importance
Mathematical functions play a critical role in various fields such as science, engineering, and economics. They are fundamental tools for analyzing and understanding relationships between different
variables. In this blog post, we will delve into the concept of one-to-one functions, their significance, and how to identify their graphs.
Define what a mathematical function is and its role in various fields such as science, engineering, and economics
A mathematical function is a relationship between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. In other words, it assigns exactly
one output to each input. Functions are used to model and describe various phenomena in the natural and physical sciences, engineering, and economics. They are used to analyze data, make predictions,
and optimize systems.
Highlight the significance of recognizing different types of functions, specifically one-to-one functions, for mathematical analysis and real-world applications
Recognizing different types of functions is crucial for mathematical analysis and real-world applications. For instance, one-to-one functions have special properties that make them valuable for
solving equations, modeling inverse relationships, and ensuring the uniqueness of solutions. Understanding one-to-one functions allows us to make accurate predictions and optimize systems in various
Outline the objectives of the blog post: to explain what a one-to-one function is, how to identify its graph, and its importance
The main objectives of this blog post are to explain what a one-to-one function is, how to identify its graph, and its importance in mathematical analysis and real-world applications. By the end of
this post, readers will have a clear understanding of the concept of one-to-one functions and their significance in various fields.
Key Takeaways
• One to one function: each input has a unique output
• Graphs of one to one functions do not intersect themselves
• Graphs of one to one functions pass the horizontal line test
• Example of a one to one function: y = x
• One to one functions have an inverse function
The Concept of One-to-One Functions
Understanding mathematical functions is essential in various fields, and one type of function that plays a critical role is the one-to-one function, also known as an injective function. Let's delve
into the concept of one-to-one functions and explore their unique characteristics and significance in mathematical concepts.
A. Define a one-to-one function (injective function)
A one-to-one function is a type of function in which each element of the domain pairs with a distinct element of the codomain. In other words, no two different elements in the domain can map to the
same element in the codomain. This unique characteristic ensures that every input has a unique output, making it a one-to-one correspondence.
B. Explain why one-to-one functions are critical in mathematical concepts
One-to-one functions are crucial in various mathematical concepts, such as inverse functions and bijective mappings. Inverse functions are functions that 'reverse' the action of another function. For
a function to have an inverse, it must be a one-to-one function, as this ensures that each output has a unique input. Additionally, bijective mappings, which are both injective and surjective (onto),
rely on the one-to-one characteristic to establish a one-to-one correspondence between the domain and codomain.
C. Provide simple algebraic examples to illustrate the concept of one-to-one functions
Let's consider a simple algebraic example to illustrate the concept of a one-to-one function. Suppose we have the function f(x) = 2x + 3. To determine if this function is one-to-one, we can use the
horizontal line test. If any horizontal line intersects the graph of the function at more than one point, the function is not one-to-one. In this case, the graph of f(x) = 2x + 3 is a straight line,
and any horizontal line intersects it at most once, indicating that it is indeed a one-to-one function.
Another example is the function g(x) = x^2. This function is not one-to-one, as for every positive value of x, there are two corresponding values of g(x) (x and -x) that map to the same output.
However, if we restrict the domain to only positive values of x or only negative values of x, the function becomes one-to-one within that restricted domain.
Characteristics of Graphs Representing One-to-One Functions
Understanding mathematical functions is essential in the field of mathematics and its applications. One important type of function is the one-to-one function, which has distinct characteristics that
set it apart from other types of functions. In this chapter, we will explore the characteristics of graphs representing one-to-one functions.
A. Introduce the Horizontal Line Test as a method to visually determine if a function is one-to-one
The Horizontal Line Test is a visual method used to determine if a function is one-to-one. When applying the Horizontal Line Test to a graph, if any horizontal line intersects the graph at more than
one point, then the function is not one-to-one. On the other hand, if every horizontal line intersects the graph at most once, then the function is one-to-one. This test provides a quick and easy way
to visually identify one-to-one functions.
B. Describe how the absence of repeated y-values for different x-values indicates a one-to-one function
In a one-to-one function, each input value (x) corresponds to a unique output value (y). This means that for different x-values, there are no repeated y-values. In other words, no two different
x-values can have the same y-value. This distinct mapping of x-values to y-values is a key characteristic of one-to-one functions and is reflected in their graphs.
C. Discuss the distinct behavior and appearance of one-to-one function graphs compared to non-one-to-one functions
The graphs of one-to-one functions exhibit specific behavior and appearance that differentiate them from non-one-to-one functions. One notable feature is that one-to-one function graphs do not have
any vertical lines that intersect the graph at more than one point. This aligns with the concept that each x-value maps to a unique y-value. Additionally, the graphs of one-to-one functions often
show a consistent increase or decrease without any sudden jumps or breaks, reflecting the continuous and distinct nature of these functions.
Real-World Examples of One-to-One Functions
One-to-one functions are prevalent in various real-world scenarios, playing a crucial role in fields such as technology, security, and data management. Let's explore some examples of how one-to-one
functions manifest in everyday life.
A. Serial numbers to products
In the retail industry, each product is assigned a unique serial number to differentiate it from others of the same type. This one-to-one relationship ensures that each product can be identified
individually, allowing for efficient inventory management and tracking of sales. For example, a barcode scanner in a supermarket uses a one-to-one function to match each product's barcode to its
corresponding information in the database.
B. Biometric data to individuals
Biometric authentication systems, such as fingerprint scanners and facial recognition technology, rely on one-to-one functions to match an individual's unique biometric data to their identity. This
ensures that only authorized individuals can access secure areas or sensitive information, making it an essential component of security in various industries, including finance and law enforcement.
C. Cryptography for secure communication
One-to-one functions are fundamental in cryptography, where they are used to encrypt and decrypt data for secure communication. In encryption, a one-to-one function is applied to transform plaintext
into ciphertext, ensuring that each input has a unique output. This prevents unauthorized parties from deciphering the original message, making it a critical aspect of secure communication over
networks and digital platforms.
D. Computer science and data structures
In computer science, one-to-one functions play a vital role in hashing algorithms and data structures. Hash functions, which map data of arbitrary size to fixed-size values, are designed to be
one-to-one to ensure that each input produces a unique hash value. This property is essential for efficient data retrieval and storage in databases, file systems, and distributed computing systems.
Overall, one-to-one functions are integral to various aspects of modern society, from retail operations and security systems to digital communication and data management. Understanding their
significance helps us appreciate their widespread impact on our daily lives.
Troubleshooting: Common Pitfalls in Identifying One-to-One Functions
When working with mathematical functions, it is important to be able to identify whether a function is one-to-one or not. However, there are common misconceptions and pitfalls that can lead to errors
in this process. In this section, we will address some of these common pitfalls and offer strategies to avoid them.
A. Address misconceptions like mistaking any increasing function as one-to-one without proper verification
One common misconception is the belief that any increasing function is automatically a one-to-one function. While it is true that one-to-one functions are often increasing or decreasing, it is not
always the case. It is important to verify the function's behavior over its entire domain to ensure that it is indeed one-to-one.
Strategy: When encountering an increasing function, it is essential to verify its one-to-one nature by checking for any repeated y-values for different x-values. This can be done by using the
horizontal line test, where a horizontal line intersects the graph of the function at most once. If there are any points where the horizontal line intersects the graph more than once, the function is
not one-to-one.
B. Offer strategies to avoid errors when working with piecewise functions which may be one-to-one on individual intervals but not on their entire domain
Piecewise functions can be particularly tricky when it comes to identifying whether they are one-to-one. While a piecewise function may be one-to-one on individual intervals, it may not be one-to-one
over its entire domain. This can lead to errors if not approached carefully.
Strategy: When dealing with piecewise functions, it is important to consider the behavior of the function on each individual interval. Verify whether the function is one-to-one on each interval
separately, and then determine whether it is one-to-one over its entire domain. This approach helps to avoid mistakenly identifying a piecewise function as one-to-one when it is not.
C. Highlight the importance of domain restrictions in defining one-to-one functions, especially within trigonometric functions
Trigonometric functions, such as sine and cosine, often require careful consideration of domain restrictions when determining whether they are one-to-one. Without proper domain restrictions, these
functions may not be one-to-one, leading to misconceptions and errors.
Strategy: When working with trigonometric functions, it is crucial to define appropriate domain restrictions to ensure that the function is one-to-one. For example, restricting the domain of the sine
function to the interval [-π/2, π/2] makes it one-to-one. Emphasizing the importance of domain restrictions helps to avoid misidentifying trigonometric functions as one-to-one when they are not.
Tools and Techniques for Analyzing Functions
When it comes to understanding mathematical functions, it is essential to have the right tools and techniques at your disposal. Whether it's using software and online graphing calculators or
employing analytical methods, there are various ways to analyze functions and determine if they are one-to-one. Let's explore some of these tools and techniques in more detail.
A. Explore software and online graphing calculators that can assist in visualizing and confirming if a function is one-to-one
One of the most effective ways to understand the nature of a function is by visualizing it. There are several software programs and online graphing calculators available that can help in this regard.
These tools allow you to input a function and generate its graph, making it easier to visualize its behavior and determine if it is one-to-one.
By plotting the graph of a function, you can observe its patterns and identify whether it passes the horizontal line test, a key characteristic of one-to-one functions. This visual confirmation can
provide valuable insight into the nature of the function and its one-to-one behavior.
B. Discuss analytical methods, such as derivative tests, to ascertain the one-to-one nature of functions algebraically
While visualizing functions can be helpful, it's also important to employ analytical methods to ascertain their one-to-one nature algebraically. One such method is using derivative tests, which can
provide valuable information about the behavior of a function.
For example, the first derivative test can be used to determine the increasing or decreasing nature of a function, which is a key characteristic of one-to-one functions. By analyzing the derivative
of a function, you can gain insights into its behavior and confirm whether it is one-to-one.
C. Encourage the use of graph sketching to understand the behavior of functions and identify one-to-one functions more effectively
Graph sketching is another valuable technique for understanding the behavior of functions and identifying one-to-one functions more effectively. By manually sketching the graph of a function, you can
gain a deeper understanding of its patterns and characteristics.
Through graph sketching, you can observe the turning points, slopes, and overall shape of the function, which can provide valuable clues about its one-to-one nature. This hands-on approach to
visualizing functions can be a powerful tool in identifying one-to-one functions and understanding their behavior.
Conclusion & Best Practices
A Recap the main points covered in the post, emphasizing the definition and identification of one-to-one functions
In this blog post, we have discussed the concept of one-to-one functions and how they are represented graphically. A one-to-one function is a type of function where each element in the domain maps to
exactly one element in the range, and no two different elements in the domain map to the same element in the range. This property makes one-to-one functions unique and valuable in various
mathematical and real-world applications.
Share best practices, such as consistently applying the Horizontal Line Test and verifying results with different methods
One of the best practices for identifying whether a function is one-to-one is to consistently apply the Horizontal Line Test. By drawing a horizontal line across the graph of a function, if the line
intersects the graph at more than one point, then the function is not one-to-one. On the other hand, if the horizontal line intersects the graph at only one point for every possible value of y, then
the function is one-to-one.
Another best practice is to verify results with different methods. This can include algebraic methods such as solving for x or y in terms of the other variable, and then checking for uniqueness of
solutions. By using multiple methods to verify whether a function is one-to-one, you can increase the confidence in your results.
Encourage readers to apply the knowledge from the post in practical situations and to delve deeper into the subject for a fuller understanding
It is important for readers to apply the knowledge gained from this post in practical situations. Understanding one-to-one functions can be beneficial in fields such as economics, engineering, and
computer science, where unique relationships between variables are essential.
Furthermore, I encourage readers to delve deeper into the subject of one-to-one functions for a fuller understanding. This can involve exploring advanced topics such as inverse functions and their
properties, as well as real-world examples where one-to-one functions play a crucial role. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-graph-one-to-one","timestamp":"2024-11-14T17:50:53Z","content_type":"text/html","content_length":"226574","record_id":"<urn:uuid:79bd33da-9965-4dad-ab91-09e380e92e3d>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00033.warc.gz"} |
The Effect of Magnetic Field Strength and Geometry on the Deposition Rate and Ionized Flux Fraction in the HiPIMS Discharge
Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland
Laboratoire de Physique des Gaz et Plasmas—LPGP, UMR 8578 CNRS, Université Paris-Sud, Université Paris Saclay, 91405 Orsay CEDEX, France
Institute of Physics v. v. i., Academy of Sciences of the Czech Republic, Na Slovance 2, 182 21 Prague 8, Czech Republic
Department of Space and Plasma Physics, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
Plasma and Coatings Physics Division, IFM-Materials Physics, Linköping University, SE-581 83 Linköping, Sweden
Author to whom correspondence should be addressed.
Submission received: 2 April 2019 / Revised: 25 April 2019 / Accepted: 6 May 2019 / Published: 13 May 2019
We explored the effect of magnetic field strength $| B |$ and geometry (degree of balancing) on the deposition rate and ionized flux fraction $F flux$ in dc magnetron sputtering (dcMS) and high power
impulse magnetron sputtering (HiPIMS) when depositing titanium. The HiPIMS discharge was run in two different operating modes. The first one we refer to as “fixed voltage mode” where the cathode
voltage was kept fixed at 625 V while the pulse repetition frequency was varied to achieve the desired time average power (300 W). The second mode we refer to as “fixed peak current mode” and was
carried out by adjusting the cathode voltage to maintain a fixed peak discharge current and by varying the frequency to achieve the same average power. Our results show that the dcMS deposition rate
was weakly sensitive to variations in the magnetic field while the deposition rate during HiPIMS operated in fixed voltage mode changed from 30% to 90% of the dcMS deposition rate as $| B |$
decreased. In contrast, when operating the HiPIMS discharge in fixed peak current mode, the deposition rate increased only slightly with decreasing $| B |$. In fixed voltage mode, for weaker $| B |$,
the higher was the deposition rate, the lower was the $F flux$. In the fixed peak current mode, both deposition rate and $F flux$ increased with decreasing $| B |$. Deposition rate uniformity
measurements illustrated that the dcMS deposition uniformity was rather insensitive to changes in $| B |$ while both HiPIMS operating modes were highly sensitive. The HiPIMS deposition rate
uniformity could be 10% lower or up to 10% higher than the dcMS deposition rate uniformity depending on $| B |$ and in particular the magnetic field topology. We related the measured quantities, the
deposition rate and ionized flux fraction, to the ionization probability $α t$ and the back attraction probability of the sputtered species $β t$. We showed that the fraction of the ions of the
sputtered material that escape back attraction increased by 30% when $| B |$ was reduced during operation in fixed peak current mode while the ionization probability of the sputtered species
increased with increasing $| B |$, due to increased discharge current, when operating in fixed voltage mode.
1. Introduction
Conventional dc magnetron sputtering (dcMS) suffers from a low degree of ionization of the sputtered material. High power impulse magnetron sputtering (HiPIMS) has emerged as a promising alternative,
providing a highly ionized material flux, while being compatible with conventional magnetron sputtering deposition systems [
]. HiPIMS operation is characterized by a pulsed high peak power in the range of several kW/cm
and consequently a high plasma density of up to
$10 19$
$− 3$
in the cathode target vicinity, which is up to three orders of magnitude higher than in dcMS [
]. Such discharge conditions result in a significant increase of ionization of the sputtered neutrals, where ionized flux fractions
$F flux$
well above 50% have been reported [
]. However, a high ionized flux fraction commonly comes at a cost of lower deposition rate, which has thus far limited the use of HiPIMS in industry [
Several reports demonstrate the lower deposition rate in (mainly non-reactive) HiPIMS when compared to dcMS operated at the same average power [
]. The seminal work of Kouznetsov et al. [
] reports up to 80% lower deposition rate for HiPIMS than for dcMS. Samuelsson et al. [
] compared the deposition rates from eight metal targets (Ti, Cr, Zr, Al, Cu, Ta, Pt, and Ag) in pure Ar for both dcMS and HiPIMS discharges applying the same average power. They observed HiPIMS
deposition rates in the range of 30–85% of the dcMS rates depending on target material.
There are several suggestions on the cause of the lower deposition rate observed in HiPIMS deposition [
]. It is generally agreed on by the scientific community that back attraction of ionized sputtered material to the target, quantified as back attraction probability
$β t$
, plays a major role in the reduction of the amount of sputtered particles reaching the substrate [
]. The reason is that atoms ionized in the cathode region are likely to be back-attracted to the target due to strong electric fields in the presheath and extended presheath [
]. Spatial measurements of the plasma potential in HiPIMS discharges [
] have shown that there commonly is a potential uphill, from the cathode sheath edge and reaching far outside the ionization region (several cm), that can vary in the range 7–100 V.
Several attempts have been made to increase the deposition rate in HiPIMS. This includes varying the pulse length [
], varying the magnetic field strength
$| B |$
], modifying the magnetic field geometry [
], adding an external magnetic field in the target vicinity [
], chopping the pulse into a train of shorter pulses [
], and increasing the target temperature [
]. Several of these reports propose that modifying the magnetic field, using either permanent magnets or electromagnets [
], is one of the most promising approaches. For example, Čapek et al. [
] showed that lowering
$| B |$
in HiPIMS can have a profound effect on increasing the deposition rate. Using spacers of different thicknesses behind the cathode to reduce
$| B |$
at the target (and also increasing the average discharge voltages to achieve nominally similar power levels), the deposition rate of Nb was increased by roughly a factor of 5. Similarly, Mishra et
al. [
] found a six-fold increase in the deposition rate of Ti by weakening
$| B |$
by 33%. Bradley et al. [
] reported on a deposition rate increase by a factor of 2 for a Ti target when the magnetic field strength at the target was reduced by 45%. In addition, while weakening
$| B |$
by 82% a factor of 2.6 higher deposition rate was observed while depositing vanadium films by HiPIMS, although for the weaker magnetic field the films exhibited significantly higher surface roughness
and were not as dense [
There have also been a few attempts to modify the magnetic field geometry in order to improve the deposition rate. This includes the work of Yu et al. [
], who used a 36 cm diameter magnetron with a spiral-shaped magnet pack assembly to increase the plasma uniformity in the substrate vicinity and to improve target utilization. More recently, Raman et
al. [
] modified the magnetic field topology of a HiPIMS discharge, which increased the deposition rate by up to a factor of 2 [
]. In the cited studies, the modified magnet pack had a strong magnetic field region over three concentric race track regions (referred to as a TriPack magnetron assembly), but the magnetic field
strength fell off more steeply than for a conventional magnet pack when moving away from the target surface. However, those designs encounter some difficulties when scaled down to a smaller cathode
The combined effect of weakening
$| B |$
, the correlation between the deposition rate increase and the lower ionized flux fraction to the substrate, is still poorly understood. One reason is that most HiPIMS studies on ionization indeed
$F flux$
, but have so far not focused on changing the magnetic field strength/topology. For example, Lundin et al. [
] explored the ionized flux fraction for Al, C and Ti targets using a gridless ion meter. For a Ti target, they found an increase in the ionized flux fraction from roughly 20% to 68% with increased
peak discharge current density in the range of 0.7–2.5 A/cm
. These values are in line with the work of Poolcharuansin et al. [
] (30–50%) and Kubart et al. [
] (20–60%) for current densities in the range 1–2.5 A/cm
. Another reason is that the studies on
$F flux$
did not in parallel systematically investigate the deposition rate (or the change thereof). The exception is the study of Raman et al. [
], who, in addition to the previously discussed deposition rate study, also estimated the ionized flux fraction during HiPIMS operation using conventional and TriPack magnetrons. They recorded an
ionized flux fraction of Cu of approximately 5% for the conventional magnetron and 16% utilizing the TriPack magnetron assembly, which indicates that optimization of the magnetic field can in fact
result in increased deposition rate as well as increased ionized flux fraction.
In the present study, we therefore systematically investigated the relationships among
$| B |$
, the magnetic field geometry (level of balancing), the deposition rate, and the ionized flux fraction during HiPIMS and dcMS operation. Such an approach enabled us to study the combined effects of
HiPIMS pulse parameters and magnetic configurations. In the analysis, we used the well known materials pathway model [
] to assess both the ionization probability
$α t$
and the back attraction probability
$β t$
from the experimental data. Finally, we attempted to explain our observations based on the physics behind the transport of charged particles in these devices.
2. Materials and Methods
All experiments were carried out in a custom-built cylindrical vacuum chamber (height 50 cm and diameter 45 cm) made of stainless steel. A base pressure of
$4 × 10 − 6$
Pa was achieved using a turbo molecular pump backed by a roughing pump. The working gas pressure was adjusted to 1 Pa by injecting 50 sccm Ar into the chamber and adjusting a butterfly valve located
between chamber and the turbo pump. The deposition system was equipped with a circular 4 inch diameter VTec Magnetron assembly (Gencoa, Liverpool, UK). The magnetron assembly, as well as a probe
holder used during measurements, was mounted on movable bellows controlled with millimeter precision, as shown in
Figure 1
. This made it possible to perform radial as well as axial scans with high precision. The absolute magnetic field strength
$| B |$
as well as the geometry of the magnetic field (degree of balancing) above the magnetron target was varied by displacing the center magnet (C) and the outer ring magnet at the target edge (E) using
two micrometer screws located on the outer side of the magnetron assembly. We refer to each configuration using the displaced distance (in mm) of each magnet from the target backing plate. Thus, the
notation C0E0 refers to a magnetron configuration where the center and outer magnets touch the backing plate (zero displacement, i.e., the strongest magnetic field above the target).
In this work, we investigated seven different magnet configurations: C0E0, C5E5 and C10E10, C0E5, C0E10, C5E0, and C10E0. For all of these configurations, the magnetic field above the target was
mapped using a Lake Shore 425 Gauss meter (Lake Shore Cryotronics, Westerville, OH, USA) equipped with a Hall probe. The magnetic field distribution above the target for each configuration is shown
Figure 2
. Axial symmetry was assumed. For the configurations investigated, it was found that a magnetic null point was always present, which means that all configurations ware categorized as unbalanced type
II [
]. The magnetic null was used as a measure of the degree of balancing. The magnetic null point for the different cases was located at 43–74 mm from the target surface above the target center and is
given in
Table 1
for each configuration. Note, however, that the case C0E10 was only weakly unbalanced, i.e., close to being balanced (
$z null = 74$
mm), whereas C10E0 was the most strongly unbalanced (
$z null = 43$
Table 1
also lists the radial component of the magnetic field strength next to the target surface over the race track
$| B r , rt |$
. These values were recorded at
$z = 11$
mm, which was the closest distance that could be probed for the
$B r$
A dc power supply (SR1.5-N-1500, Technix, Créteil, France) and a HiPIMS power supply (HiPSTER 1, Ionautics, Linköping, Sweden) were used to ignite the discharge in dc and HiPIMS modes, respectively.
For both cases, an average discharge power was maintained at 300 W. The HiPIMS pulse was always kept at constant length of 100
s and the discharge was regulated in two different ways. The first mode is referred to as fixed voltage mode, and was realized by keeping the cathode voltage fixed at 625 V and varying the pulse
frequency to achieve the desired average power. The second mode is referred to as fixed peak current mode and was realized by changing the cathode voltage to maintain the peak discharge current at
$I D , peak$
= 40 A, corresponding to current density
$J D , peak$
= 0.5 A/cm
for the ionized flux fraction measurements, and
$I D , peak$
= 80 A and
$J D , peak$
= 1.0 A/cm
for the measurements of deposition rate. Again, the pulse frequency was varied to achieve the desired average power. The discharge parameters are summarized in
Table 1
for dcMS operation and both operating modes of HiPIMS for all the seven magnet configurations investigated.
We captured the discharge current–voltage (
$I D$
$V D$
) waveforms when operating the HiPIMS discharges at different magnet configurations.
Figure 3
a depicts the cathode voltage and
Figure 3
b the discharge current for all seven magnetic field configurations explored when operating in fixed voltage mode. When moving both the central and outer magnets together,
$I D , peak$
changed from 80 A to 36 A to finally 12 A for the C0E0, C5E5 and C10E10 magnet configurations, respectively.
Figure 3
b shows that
$I D , peak$
occurred before the pulse end using the C0E0 configuration while for two other magnet configurations the discharge current waveforms had an ascending trend over the entire pulse length. The value of
$I D , peak$
was more sensitive to the absolute strength of the magnetic field than to the degree of balancing. The C5E0 and C0E5 configurations gave
$I D , peak$
= 53–54 A and the C10E0 and C0E10 configurations 31–35 A.
Figure 3
c depicts the discharge current waveforms captured at fixed peak current mode with various magnet configurations. Although
$I D , peak$
was very similar in all cases, the current rise rate was different and as a result the discharge current peaked at different times. Note that different cathode voltages were applied to achieve the
$I D , peak$
Table 1
), but the voltage was not correlated to the time of peak current. For example, the C5E5 magnet configuration exhibited sharper current rise than C0E0 while the corresponding cathode voltage was 150
V higher than for the C0E0 magnet. In contrast, looking at discharge current waveforms for the C5E0 and C10E0 magnets showed that using the C5E0 magnet resulted in sharper current rise than the C10E0
magnet, although the corresponding cathode voltage was approximately 100 V lower.
A quartz crystal micro-balance (QCM) with native frequency of 5 MHz and gold coated surface was used to measure the deposition rate. It was mounted on the probe holder shown in
Figure 1
. By moving the probe holder and/or the magnetron assembly, it was possible to investigate a region defined by
$0 ≤ r ≤ 50$
$20 ≤ z ≤ 70$
mm, where
is the radial coordinate parallel to the target surface and
is the axial coordinate perpendicular to the target surface, and
$( r , z ) = ( 0 , 0 )$
marks the center of the target surface. The center of the target race track was located at approximately
$( r , z ) = ( 30 , 0 )$
mm. In this work, only axial material fluxes were investigated, i.e., mimicking a conventional sputtering setup with a substrate facing the target surface.
The QCM sensor was also used as a main component in the ion meter (or gridless QCM/m-QCM) used for measuring the ionized flux fraction
$F flux$
. The device is described in detail in a previous work [
] and is here only summarized. The ion meter can measure either the deposition rate from ions and neutrals or from neutrals only by varying a voltage applied to the biased top QCM electrode, allowing
for fast (roughly 1 min) determination of the ionized fraction of material flux to the sensor head. The gridless sensor uses a magnetic field configuration consisting of a ferromagnetic yoke and
magnetic pole pieces (cylindrical SmCo magnets with a diameter of 8 mm and a length of 5 mm) placed in front of the sensor. This configuration produces a localized homogeneous magnetic field of about
4000 Gauss, which does not significantly affect the magnetic field of the magnetron assembly [
]. The QCM control unit with the oscillator was connected directly to the crystal electrode. The electrode was either grounded for measurements of both ions and neutrals, or biased to +40 V to
collect only the neutrals without positive ions. The dc bias voltage was connected to the QCM collecting electrode through a 1 k
resistor, to protect the crystal in case of arcing, and the ground of the oscillator and the readout unit were connected to the crystal collecting electrode through a 150 nF capacitor such that dc
current was blocked while rf current could flow from the crystal through this capacitor back to the ground of the oscillator and give a readout (see
Figure 1
). In this configuration, the top crystal electrode could be readily biased without any influence on the QCM operation. The ionized fraction of the metal flux
$F flux = R t − R n R t ,$
was determined from the total mass deposition rate
$R t$
and the mass deposition rate of neutral metal atoms
$R n$
, as discussed by Wu et al. [
]. The deposition rates were recorded by manually recording the film thickness at a chosen time on a readout unit connected to the QCM. In addition, we tried to minimize errors due to the QCM crystal
heating up during the process by making short measurements (typically less than 120 s). The total error of
$F flux$
was estimated to be up to 15% for a single result mainly based on the accuracy of the mass deposition rate determination. Since the QCM electrode was grounded during the measurement of the total
deposition rate, no significant collimation of the ions [
] was expected at this stage due to the low plasma potential, which potentially could introduce additional errors in the measurements. The ion meter was mounted on the probe holder shown in
Figure 1
and could thereby map out the same region of interest as the standard QCM. However, due to interference with the plasma discharge, it was not possible to move it closer than
$z ≥ 30$
mm. In addition, high peak currents in the HiPIMS mode sometimes resulted in strong fluctuations of
$( R t − R n )$
, which meant that the HiPIMS series with fixed peak current had to be limited to
$I D , peak$
= 40 A (
$J D , peak$
= 0.5 A/cm
$2 )$
when measuring
$F flux$
3. Results
The deposition rates as well as the ionized flux fractions for each of the magnetron configurations shown in
Figure 2
and listed in
Table 1
are presented here. For the deposition rate results, we chose to focus on the data recorded at a typical target-to-substrate distance of
$z = 70$
mm, which also includes three radial points (
$r = 0 , 25 , 50$
mm) to determine the expected film thickness profile at that axial distance. However, the deposition rate was also recorded closer to the target and comparisons were made where appropriate.
Concerning the ionized flux fraction at
$z = 70$
mm, we only show data recorded above the target center, i.e.,
$( r , z ) = ( 0 , 70 )$
mm, although all radial positions were used in the analysis. We also show the flux fractions at
$( r , z ) = ( 25 , 30 )$
mm due to the interest in comparing with other reports of
$F flux$
, which are typically recorded at the outer edge of the ionization region (the dense plasma region) above the target race track. We refer to the region where the substrate is typically located as the
diffusion region.
3.1. Deposition Rate
The deposition rates measured above the center of the target (
$r = 0$
mm) at an axial distance of 70 mm (substrate position) are plotted as a bar chart in
Figure 4
for the different discharge types as well as all magnetic configurations investigated. The magnet configurations on the
-axis are ordered from highest
$| B |$
at the left to the lowest
$| B |$
on the right. We have here used the recorded
$| B r , rt |$
value above the race track as a measure of
$| B |$
. Overall, the dcMS discharges exhibited the highest deposition rates independent of magnetron configuration, with deposition rates in a rather narrow range (92–116 Å/min). Much larger differences
were observed for the HiPIMS discharge operated in the fixed voltage mode, where the deposition rate varied between 45 Å/min and 96 Å/min. However, for the fixed peak discharge current mode, the
deposition rate varied between 34 Å/min and 47 Å/min with an increasing trend of 38% larger deposition rate at the weakest
$| B |$
compared to the strongest
$| B |$
Let us start by comparing the three cases C0E0, C5E5, and C10E10, exhibiting the same magnetic topology but approximately a reduction of 63% of the absolute magnetic field strength at the center of
the target surface and a reduction of
$| B r , rt |$
by 53% (configurations C0E0 and C10E10). For the dcMS discharges, only small differences were found. The strongest magnetic field (C0E0) showed the lowest deposition rate (92 Å/min) and the weakest
magnetic field (C10E10) showed the highest deposition rate (103 Å/min), i.e., a deposition rate increase of 11%. The HiPIMS discharges operated in fixed voltage mode showed a much more pronounced
deposition rate dependence on changes in
$| B |$
, where a weaker
$| B |$
resulted in a considerably higher deposition rate. For example, C0E0 exhibited the lowest deposition rate (45 Å/min) and C10E10 the highest deposition rate (96 Å/min), i.e., a rate increase of 113%.
It was also observed that this latter HiPIMS case resulted in a deposition rate, which was around 90% of the dcMS rate, i.e., a significantly higher value than what is commonly reported for HiPIMS,
as discussed in the Introduction. In contrast, the HiPIMS discharges operated in fixed peak current mode exhibited smaller changes in the measured deposition rate when
$| B |$
was varied, as observed when comparing cases C0E0 and C5E5 (no data from C10E10), about 38% increase of the deposition rate when weakening
$| B |$
. In this discharge mode, the HiPIMS deposition rate was around 40% of the dcMS rate for the equivalent magnetron configurations, which was closer to the value of 30% reported by Samuelsson et al. [
For completeness, it is also noted that a significant deposition rate increase could be achieved at closer target-to-substrate distances, as expected. The highest deposition rate values, independent
of discharge type and magnet configuration, were recorded at the closest axial distance investigated, $z = 20$ mm, with on the average, 2.3, 2 and 1.9 times higher values for dcMS, fixed voltage and
fixed peak current HiPIMS discharges, respectively, compared to the values measured at $z = 70$ mm (results not shown here). In general, similar trends in the deposition rate for the different
configurations investigated were observed at all distances from the target. However, the closer was the distance to the target, the larger was the radial variation in the recorded deposition rates,
which is generally not desired in thin film deposition.
To address the issue of the expected radial film thickness profile at the substrate position, the relative standard deviation (RSD) of the deposition rate was calculated from recorded deposition
rates at three radial points,
$r = 0 , 25 , 50$
mm at
$z = 70$
mm from the target surface. RSD is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed as a percentage, and is defined as
is the standard deviation and
is the mean of the dataset. The standard deviation of the deposition rates was calculated as the square root of its variance. Overall, we found a weak trend of decreasing RSD with increasing degree
of magnetic balancing. This is illustrated in
Figure 5
, where the magnet configurations on the
-axis are ordered with increasing
$z null$
(increasing degree of magnetic balancing) from left to right (see
Table 1
In addition, the dcMS discharges exhibited the lowest sensitivity to $| B |$, as can be seen when comparing the three cases C0E0, C5E5, and C10E10. Note that this does not imply that the coating
uniformity was the best since RSD was still rather high. Changing the magnet configuration from weakly to strongly unbalanced configurations (C0E5 to C5E0 and C0E10 to C10E0) barely affected the dcMS
deposition rate uniformity, which remained in the range of 16% to 19%.
The deposition rate of a HiPIMS discharge operated in fixed voltage mode exhibited the most uniform deposition rate profile of all cases investigated when using the C5E0 magnetic field geometry with
RSD of $12 %$. The C0E5 and C0E10 configurations led to similar RSDs (15%). The lowest uniformity (highest RSD) achieved was observed for C10E0, just below 20%, i.e., similar to the corresponding
dcMS value. In the fixed peak current mode, the maximum RSD recorded was 22% for C10E0 and C5E5, while using C0E5 and C5E0 resulted in RSD values of 14% and 16%, respectively. A similar analysis of
the fixed voltage HiPIMS mode showed that the highest RSD was 23% when using the C10E10 configuration and the lowest RSD was 12% with the C5E0 configuration. For the strongest $| B |$ case C0E0, the
deposition rate profile was similar to the dcMS case. However, the RSD values found for the fixed peak current HiPIMS mode were generally higher with RSD of $19 %$ for C0E0 and RSD of $22 %$ for
C5E5. Overall, the deposition uniformity was more dependent on the magnetic configuration than the discharge type. Moving closer to the target ($z = 20$ mm), the deposition rate became significantly
less uniform (about two times higher RSD values) compared to a typical substrate position ($z = 70$ mm).
3.2. Ionized Flux Fraction
The ionized flux fractions
$F flux$
measured above the center of the target (
$r = 0$
mm) at an axial distance of
$z = 70$
mm are plotted as a bar chart in
Figure 6
for the two HiPIMS operating modes (fixed voltage and fixed peak current modes) as well as for all magnet configurations investigated. Note that the magnet configurations on the
-axis are now ordered from highest
$| B |$
at the left to the lowest
$| B |$
on the right where, again,
$| B r , rt |$
Table 1
was used as a suitable indicator of
$| B |$
. No dcMS values are presented here, since
$F flux$
was always very close to 0%, i.e., within the margin of error, and thus in line with the results reported by Kubart et al. [
] using the same technique.
Figure 6
shows that the ionized flux fraction decreased with decreasing
$| B |$
when the HiPIMS discharge was operated in fixed voltage mode. For the HiPIMS discharges operated in fixed voltage mode, significant differences were found when comparing the three cases C0E0, C5E5,
and C10E10 (reduced absolute magnetic field strength
$| B |$
, while maintaining the same magnet topology). The strongest magnetic field (C0E0) showed the highest
$F flux$
(18%) and the weakest magnetic field (C10E10) showed the lowest
$F flux$
(4.7%). In the fixed voltage mode,
$F flux$
seemed to decrease with the decreased absolute magnetic field strength
$| B |$
which is correlated with the peak discharge current presented above in
Figure 3
. This is analyzed in more detail in the next section. However, the corresponding HiPIMS discharges operated in fixed peak current mode clearly did not exhibit such a behavior. Instead, the ionized
flux fraction
$F flux$
increased slightly with decreasing
$| B |$
. The ionized flux fraction increased from 11% to 16.8% when comparing cases C0E0 and C5E5 (no data from C10E10), i.e., by a factor 1.5 when decreasing
$| B |$
When focusing on changes in the degree of balancing, i.e., comparing the configuration pairs C0E5/C5E0 and C0E10/C10E0, the following observations could be made. For the HiPIMS discharges operated in
fixed voltage mode, it was somewhat surprising that the highest
$F flux$
was recorded for the weakly unbalanced C0E5 configuration (16%), whereas the most strongly unbalanced configuration C10E0 exhibited a much lower value (8.5%), although there was only a small
difference compared to C5E0 and C0E10. Any influence on
$F flux$
from the unbalance was masked by the strong influence of the peak discharge current values on
$F flux$
. For the HiPIMS discharges operated in fixed peak current mode, the trend observed for
$F flux$
Figure 6
) was somewhat more expected. The more strongly unbalanced cases exhibited higher
$F flux$
with a maximum of 17.2% for C10E0.
In addition, by making radial scans of
$F flux$
, we attained radial profiles at
$z = 70$
mm in the fixed voltage mode (not shown here). In general, only minor differences compared to the results at
$r = 0$
mm were observed. The maximum
$F flux$
was commonly reached above the target race track position, and it was approximately 2–5 precentage points higher compared to the values reported in
Figure 6
. However, a few exceptions are worth noting. For the strong
$| B |$
configuration C0E0, there was a sudden jump in
$F flux$
towards the region above the target edge (
$r = 50$
mm). Here,
$F flux$
increased to 27% compared to just below 20% above the target center and race track. In addition, the configuration C5E5 exhibits a striking increase in
$F flux$
compared to the result presented in
Figure 6
, and
$F flux$
peaks at 11.5% above the target race track compared to 8.5% above the target center.
We now turn to investigate
$F flux$
in the ionization region, since this would provide a better basis for comparison with other reports of the ionized flux fraction, as discussed in the Introduction. Furthermore, these values were
indispensable for our ongoing modeling efforts of the internal parameters in HiPIMS using the ionization region model [
]. Measurements were therefore taken above the target race track at
$( r , z ) = ( 25 , 30 )$
mm and a summary is shown in
Figure 7
For HiPIMS discharges operated in fixed voltage mode, we observed the same general trend as shown in
Figure 6
but with approximately a 72% increase in
$F flux$
for C0E10 and C5E0, 66% increase for C0E0, 55% increase for C5E5 and C10E0, 12% increase for C0E5, and almost no change for C10E10 compared to
$F flux$
measured at
$( r , z ) = ( 0 , 70 )$
mm. However, HiPIMS discharges operated in fixed peak current mode clearly did not exhibit such a behavior. Instead
$F flux$
using C0E0 showed 55% increase and reached 17% while
$F flux$
of C5E5 remained at 17% with no change compared to our measurements at 70 mm (
Figure 6
). By focusing on changes in the degree of balancing, 17%, 34% and 54% increases in
$F flux$
were observed using C0E5, C0E10, C5E0, respectively, while C10E0 showed negligible change compared to what is shown in
Figure 6
. As a consequence, the C5E0 configuration led to the highest
$F flux$
(20.5%) over the race track and
$z = 30$
mm. In the fixed peak discharge current mode with peak current density of
$J D , peak = 0.5$
, the measured values were in the range 14.2–20.5%. For comparison, Lundin et al. [
] reported ionized flux fraction in the range 22–31%, over the race track 40 mm from the target surface, increasing with increased working gas pressure in the range 0.5–2 Pa for a Ti target when
operating at peak current density of
$J D , peak = 0.5$
, pulse length 100
s, and time averaged power of 200 W. Similarly, Kubart et al. [
] reported ionized flux fraction of 24% 43 mm above the target race track for a Ti target with argon as the working gas at 1 Pa and operating at peak current density of
$J D , peak = 0.5$
for 100
s long pulses and time averaged power of 200 W. The values reported here were thus somewhat lower than the values reported in these earlier studies.
4. Discussion
4.1. Discharge Physics
As a background to how the magnetic field influences the ionized flux fraction and the deposition rate, let us discuss how it influences the discharge physics. The magnetic field in sputtering
magnetrons makes the discharge more efficient through two mechanisms, Ohmic heating and electron confinement. Ohmic heating [
] allows for energizing the large majority of the electrons, those that are created by ionization within the plasma discharge, in addition to the energization through acceleration across the cathode
sheath of the small minority of electrons that are created by secondary emission at the target [
]. Electron confinement adds further to the efficiency by reducing the loss of the energetic electrons out of the discharge volume (the ionization region). The magnetic field therefore enables more
ionization for a given input energy. Since most of the discharge current at the target surface is carried by ions [
], this results in a higher discharge current for a given voltage. This effect was clearly observed in our experiments.
Figure 8
shows two sets of data, both plotted as functions of the magnetic field strength at the race track center, i.e.
$| B r , rt |$
Table 1
: the peak discharge current when operating at fixed voltage, and the discharge voltage when operating at fixed peak current. Let us first look at the fixed voltage case. The peak discharge current
varied with
$| B |$
as expected, from 12 A for the weakest magnetic field, configuration C10E10, to 80 A for the strongest
$| B |$
configuration, C0E0 (
Figure 3
b). Extrapolation to weaker
$| B |$
indicated that, below about 50 Gauss, it would not be possible to ignite a discharge at the set pressure. The fixed peak discharge current case confirmed this picture. A higher voltage was needed to
drive the discharge for weaker
$| B |$
, and, for the weakest
$| B |$
, a 40 A discharge could not be reached due to the voltage limitation of the power supply.
A consequence of these effects was that the peak power in the individual HiPIMS pulses varied between the different magnetic field configurations. This variation was around 50% in the fixed peak
current studies, and almost an order of magnitude in the fixed voltage studies. For a normalization of the deposition rates to dcMS, it was most practical to operate at constant average power. This
was achieved by varying the pulse repetition frequency
$f pulse$
, as given in
Table 1
. This variation of the discharge impedance between the magnetic field configurations and our compensation by adjusting
$f pulse$
to have constant power are important to keep in mind in the analysis presented below. The most important consequence is that, even if both the cathode voltage and the average power were kept
constant, the peak discharge current could vary by almost an order of magnitude between the different magnetic field configurations. This implies a variation of the plasma density of the same order,
which in turn implies a large variation in the probability of ionization of the sputtered material as it passes through the plasma [
4.2. Deposition Rate and Ionized Flux Fraction
Figure 4
shows that, for HiPIMS operated in the fixed voltage mode, the deposition rate increased with decreasing
$| B |$
. For dcMS operation, there was only a small change in the deposition rate when
$| B |$
was varied. However, when operating the HiPIMS discharge in fixed peak current mode, there was a slight increase in the deposition rate as the
$| B |$
was decreased, as shown in
Figure 4
. Bradley et al. [
] recently explored the difference in the discharge behavior between dcMS and HiPIMS operation with changing
$| B |$
. For dcMS and pulsed-dc operation they found that the deposition rate decreases by 25–40% when decreasing
$| B |$
. They found the opposite for HiPIMS operation and the deposition rate increases significantly with decreasing
$| B |$
. They used a simple phenomenological model (pathway model) to relate the sputtered particle fluxes and the measured deposition rates to find the combined probabilities of ionization
$α t$
and subsequent back attraction
$β t$
of the ions of the sputtered species
$α t β t$
$| B |$
is varied. They found a drop in
$α t β t$
with decreasing
$| B |$
and proposed it being due to the weakening of the electrostatic ion back attraction, due to a potential hill seen by the ions of the sputtered material. A fall in
$α t β t$
gives higher deposition rates.
Here, we expanded on the approach of Bradley et al. [
] and explore how the measured parameters, the deposition rate and the ionized flux fraction
$F flux$
, depend separately on the probability of ionization
$α t$
and back attraction of the sputtered species
$β t$
. We derived a few general equations that relate the measured quantities to the parameters
$α t$
$β t$
. Let us call the total flux (atoms/s) of atoms sputtered from the target
$Γ 0$
and the flux of sputtered species (ions and neutrals) that leave the ionization region (IR) towards the diffusion region (DR)
$Γ DR$
. The useful fraction of the sputtered species becomes
$F DR = Γ DR Γ 0 = 1 − α t β t .$
Note that this equation does not need to take into account ion focusing (or spreading) en route towards the substrate [
]. This equation indicates a reduced fraction of the sputtered species reaching the substrate when the ionization of the sputtered material increases. Recall that the main drawback using HiPIMS is
the low deposition rate. As can be seen in Equation (
), the fraction of the sputtered species leaving the ionization region
$F DR$
and thus the deposition rate can be increased by decreasing the product
$α t β t$
. Two different mechanisms can achieve this: decrease the probability of ionization of the sputtered atoms
$α t$
, and/or decrease the ion back attraction probability
$β t$
. There is experimental support for both approaches. Mishra et al. [
] showed that the back attracting electric field
$E z$
in front of the target decreases with a decreasing
$| B |$
and thus reduces
$β t$
. In addition, a lower
$| B |$
with a fixed discharge voltage generally leads to lower peak discharge currents and thus lower
$α t$
. We also show in
Figure 3
b that, when operating in the fixed voltage mode, the peak discharge current
$I D , peak$
decreased as
$| B |$
decreased. This was a consequence of lower magnetic confinement, which led to lower plasma density. For our three magnetic field configurations, where the magnetic pack was moved as a whole, the peak
discharge currents were 80 A for the strongest
$| B |$
(C0E0 configuration), 36 A for the intermediate (C5E5 configuration), and 12 A for the weakest
$| B |$
(C10E10 configuration). The lower discharge currents at weaker
$| B |$
corresponded to lower plasma densities in front of the target, which should reduce the probability of ionizing sputtered atoms that pass through the ionization region, i.e., reduce
$α t$
. As pointed out by Bradley et al. [
], poorer magnetic confinement, lower plasma densities, and lower discharge currents give rise to increased deposition rates. However, this increased deposition rate is at the cost of decreased
ionized flux fraction, as discussed in
Section 3.2
. Thus, decreased discharge current and lower plasma density lead to decreased ionization probability of the sputtered material
$α t$
. The fraction of the sputtered species reaching the substrate, which is proportional to
$( 1 − α t β t )$
, then increases if
$β t$
remains roughly fixed, which is explored in more detail below in the fixed voltage mode. In the fixed peak current mode, we could assume that the plasma density remained fixed, thus
$α t$
and decreasing
$β t$
with decreasing
$| B |$
gave increased deposition rate, which was also examined.
A relationship between the ionization flux fraction
$F flux$
and the parameters
$α t$
$β t$
has been derived from the pathway model [
$F flux = Γ DR , ions Γ DR = Γ 0 α t ( 1 − β t ) Γ 0 ( 1 − α t β t ) = α t ( 1 − β t ) ( 1 − α t β t )$
where no additional ionization of the sputtered material in the diffusion region is assumed. Note that that there is a slight difference from the equation derived by Butler et al. [
] as here we neglected ion focusing. Our goal was to assess how much
$| B |$
and the magnetic field structure influence
$α t$
$β t$
, respectively. To this purpose, we plot a graph that shows
$F DR$
on the horizontal axis, and
$F flux$
on the vertical axis in
Figure 9
. In this graph, we have used Equations (
) and (
) to plot two sets of lines: (i) lines of constant
$β t$
$α t$
varied from 0 to 1 (green dashed lines in
Figure 9
); and (ii) lines of constant
$α t$
, with
$β t$
varied from 0 to 1 (blue solid lines in
Figure 9
). This gives us a coordinate system (
$α t$
$β t$
) transformed into the (
$F DR$
$F flux$
) plane. Plotting the experimentally determined combinations of
$F DR$
$F flux$
in this plane gives us estimates of the corresponding values of
$α t$
$β t$
. The ionized flux fraction
$F flux$
generally increases with increasing ionization probability
$α t$
, as shown in
Figure 9
(blue solid lines). Thus, for a fixed
$β t$
, we found that, for decreased ionization probability
$α t$
, the ionized flux fraction decreased. This is indeed what we observed for the fixed voltage mode operation. In the HiPIMS discharge,
$F flux$
was lower than
$α t$
because only a small fraction of the ions left in the direction of the substrate as
$β t$
was high [
]. At high
$α t$
, the flux of neutrals was reduced due to high ionization and this flux was only partially replaced by ions since most of the ions were drawn back to the target.
For an exact calculation of
$F DR$
from Equation (
), we needed the total flux of sputtered atoms that (before ionization) were headed towards the position
$( r , z )$
where the deposition flux
$Γ DR$
is measured. This is not a measured quantity, but it can be estimated from the measured deposition rates in a dcMS discharge operated at the same average power
$Γ dcMS$
as follows. First, we note that all discharges studied here were run at the same average power. This means that the average discharge current obeyed the relation
$I D , dcMS V D , dcMS = 〈 I D , HiPIMS 〉 V D , HiPIMS$
which gives
$〈 I D , HiPIMS 〉 I D , dcMS = V D , dcMS V D , HiPIMS$
$〈 I D , HiPIMS 〉$
is the time averaged discharge current of the HiPIMS discharge. We neglect the small contribution of secondary electron emission to the current at the target surface, and also assume only singly
charged ions. In the dcMS discharge, all the sputtering was due to ions of the working gas, the primary ions. The flux of the sputtered material in the dcMS case was then
$Γ sput , dcMS = I D , dcMS e Y tg ( V D , dcMS )$
$Y tg ( V D , dcMS )$
is the sputter yield for Ar
ions at the ion energy
$E Ar + = e V D , dcMS$
. The situation in the HiPIMS discharge was more complex and both ions of the working gas and ions of the target material participated in the sputter process [
]. In the HiPIMS case, a fraction
$ζ = I D , Ar + / I D , i$
of the total ion current to the target was due to ions of the working gas and sputtered the target with sputter yield
$Y tg ( V D , HiPIMS )$
, and the remaining discharge current fraction
$( 1 − ζ )$
was due to ions of the target material (self sputtering) with sputter yield
$Y SS ( V D , HiPIMS )$
. This gives the total flux of sputtered species from the target
$Γ 0 = 〈 I D , HiPIMS 〉 e ζ Y tg ( V D , HiPIMS ) + ( 1 − ζ ) Y SS ( V D , HiPIMS ) .$
Using Equation (
) to replace currents with voltages in Equations (
) and (
) then gives
$Γ 0 = Γ sput , dcMS V D , dcMS V D , HiPIMS ζ Y tg ( V D , HiPIMS ) + ( 1 − ζ ) Y SS ( V D , HiPIMS ) Y tg ( V D , dcMS ) ≡ Γ sput , dcMS Ψ$
All the parameters in this expression were known and easily accessible except the fraction
of the ion current to the target that was carried by Ar
ions. We used the concept of a critical discharge current introduced by Huo et al. [
] along with the generalized recycling model [
] to estimate this fraction. With argon as the working gas, a gas temperature of 300 K, and the approximation that the race track area
$S RT$
was half the full target area
$S T$
, the critical discharge current could be approximated as [
$p g$
is the working gas pressure in Pa and
$S T$
is the target area in cm
. In our case,
$p g = 1$
Pa and
$S T ≈ 80$
, giving a critical current of 16 A. At the critical current, about half the ion current was carried by the working gas ions, and the other half by self-sputter recycling [
]. The discharge current waveforms and peak discharge currents, for the different cases studied here, are given in
Figure 3
. With only one exception, they were above 30 A, far above
$I crit$
. In this current range, the ion current was carried mainly by recycled ions, of both the working gas and of the sputtered material. The relative fraction of these depends mainly on the self-sputter
yield of the target material [
]. For a Ti target, with argon as working gas, the fraction was typically
$ζ ≈ 50$
% when
$I D ≥ I crit$
]. We assumed here that the titanium was only singly charged, neglecting the fact that, for HIPIMS discharges with Ti target, significant amounts of multiply charged titanium ions are known to exist
For Ar
ions sputtering titanium, the sputter yield is
$Y tg = 0.0425 × E Ar + 0.443$
and for Ti
ions sputtering titanium (self-sputtering) the sputter yield is
$Y SS = 0.0285 × E Ti + 0.484$
]. The ratio
$Ψ = Γ 0 / Γ sput , dcMS$
for the fixed voltage case can be calculated using Equation (
) and the discharge voltages during dcMS and HiPIMS operation given in
Table 1
. For the case of 50/50 Ar
ions sputtering the target, this ratio is
$Ψ = 0.66$
. For solely Ar
ions, the ratio is 0.61 and, for solely Ti
ions, it is 0.71. The experimental data
$F flux$
$F DR = Γ DR / Γ 0 = Γ DR / ( Γ sput , dcMS Ψ )$
from the fixed voltage operation and taken 70 mm from the target are plotted in
Figure 9
for all three locations, center
$( r = 0 )$
mm, over the race track
$( r = 25$
mm), and edge
$( r = 50$
mm). We assumed here that 50% of the ions were Ar
and the other 50% were Ti
$Ψ = 0.66$
. We note that all the experimental data fall in a narrow range for the back attraction probability
$β t$
$0.90 − 0.95$
while they span a wide range in ionization probability
$α t$
or 0.38–0.8. Thus, in the fixed voltage mode,
$β t$
was almost constant while
$α t$
was varied by varying the magnetic field strength. For the fixed current case, the ratio
was in the range 0.64–0.74 assuming 50/50 Ar
ions sputtering the target and the variation was due to variation in the discharge voltage.
Finally, we can derive an equation that gives the back attraction probability
$β t$
as a function of the measured quantities
$F flux$
$F DR$
. An expression in which
$α t$
is eliminated from Equations (
) and (
) allows estimating
$β t$
directly from the measured quantities:
$β t = 1 − F DR 1 − F DR ( 1 − F flux )$
and similarly we can derive an equation that gives
$α t$
as a function of the measured quantities
$α t = 1 − F DR ( 1 − F flux ) .$
The ionization probability and back attraction probability for the ions of the sputtered species calculated using the measured quantities
$F flux$
$F DR$
are shown in
Figure 10
a,b, respectively, versus the magnetic field strength above the race track for various combination of operating modes, magnetic field configurations and locations over the target surface.
Figure 10
a shows the ionization probability
$α t$
above the race track
$( r = 25$
mm) and in the target center
$( r = 0$
mm) versus the magnetic field strength over the race track. When operating in the fixed voltage mode, the ionization probability increased with increased magnetic field strength. The back attraction
probability was always high, in the range 0.89–0.96, over the entire range of
$B r , rt$
shown in
Figure 10
b. In the fixed current mode,
$β t$
increased slightly with increased
$| B |$
in the range 0.93–0.96 while
$α t$
was almost constant in a narrow range 0.75–0.79. If we make linear fit of the increase in
$β t$
$| B |$
, the fraction
$( 1 − β t )$
was roughly 30% higher at the highest
$| B |$
than at the lowest
$| B |$
. This was important since the total flux of ions of the sputtered material away from the target toward the substrate was
$Γ DR , ions = α t ( 1 − β t ) Γ 0$
, as a fraction
$β t$
of the ions of the sputtered material went back to the target. Recall that, as shown in
Figure 4
, there was a 38% increase in the deposition rate when
$| B |$
decreased from 238 to 111 Gauss when operating at fixed peak discharge current. For the fixed peak current mode the ionization probability
$α t$
was roughly constant independent of the location of the magnetic null (not shown). In the fixed voltage mode, there was some spread in the ionization probability values independent of the location of
the magnetic null and no clear trend observed (not shown).
Figure 11
a shows the ionization probability
$α t$
above the race track and in the target center versus the peak discharge current. We observed that the ionization probability increased roughly linearly with the peak discharge current. Similarly, we
observed an increase in the ionized flux fraction with increased peak discharge current in
Figure 11
Figure 6
shows that, for operation in the fixed voltage mode, the stronger was the magnetic field, the higher was the
$F flux$
. We can explain why:
Figure 3
b shows that higher magnetic field strength led to higher peak discharge current, and
Figure 11
b that higher discharge current gave higher
$F flux$
5. Conclusions
The effect of the magnetic confinement on the deposition rate and the ionized flux fraction was explored for both dcMS and HiPIMS deposition from a Ti target. The experimental findings at $z = 70$ mm
indicate that, for the dcMS case, there was a small, about 10%, decrease in deposition rate as $| B |$ was increased from its weakest value to its strongest value. In the dcMS case, the ionized flux
fraction was too small to be of interest. For HiPIMS operated in the fixed voltage mode, we found opposing trends with increasing $| B |$ in the studied range: a trade-off between the deposition rate
(decreased by more than a factor of two) and the ionized flux fraction (increased by a factor of 4–5). The back attraction probability of the ions of the sputtered material in a HiPIMS discharge was
found to be high and roughly constant independent of $| B |$ and the ionization probability of the sputtered species increased with increasing $| B |$ due to a increased discharge current when
operating in the fixed voltage mode. For HiPIMS operated in the fixed peak current mode, we found concurring, but smaller trends in the two parameters: Decreasing $| B |$ improved both the deposition
rate (by 38%) and the ionized flux fraction (by 53%). When operating in the fixed peak current mode, the ionization probability of the sputtered species was roughly constant while the parameter $( 1
− β t )$ increased roughly 30% with decreasing $| B |$. In short, when operating a HiPIMS discharge in fixed voltage mode, the ionization probability $α t$ varied with $| B |$ and $β t$ remained
roughly constant, while, in the fixed peak current mode, $β t$ varied with $| B |$ and $α t$ remained roughly constant.
Author Contributions
Conceptualization, H.H., J.T.G., M.Č., Z.H., M.A.R., N.B. and D.L.; experiment, H.H., S.Ü., M.Č., Z.H. and D.L.; writing—original draft preparation, H.H., J.T.G. and D.L.; writing—review and editing,
H.H., J.T.G., N.B., M.A.R., S.Ü., M.Č., Z.H., and D.L.; and funding acquisition, M.Č., Z.H., D.L. and J.T.G.
This work was partially supported by the University of Iceland Research Fund for Doctoral students, the Icelandic Research Fund Grant Nos. 130029 and 196141, the Czech Science Foundation through
project 19-00579S and by Operational Programme Research, Development and Education financed by European Structural and Investment Funds and the Czech Ministry of Education, Youth and Sports (Project
No. SOLID21 CZ.02.1.01/0.0/0.0/16_019/0000760).
The authors acknowledge the support of Benjamin Seznec at Université Paris-Sud, Orsay, for his work on interpolating the recorded magnetic field data to draw
Figure 2
. The authors also acknowledge stimulating discussion with Tiberiu M. Minea at Université Paris-Sud, Orsay.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 1. A schematic of the magnetron sputtering chamber. The magnetron assembly and the probe holder with the m-QCM are mounted on movable bellows that can be controlled with millimeter precision.
The red arrows indicate linear motion.
Figure 2. The measured magnetic field (flux density $B$) and field line directions for the various magnetic field configurations. Normalized arrows indicate the magnetic field direction, the color
scale indicates the magnitude of the magnetic field $| B | = B r 2 + B z 2$. The value of $B r$ above the race track at $z = 11$ mm is given in the inset for each case.
Figure 3. The HiPIMS discharge current and voltage waveforms recorded for various magnetic field configurations: (a) the discharge voltage in fixed voltage mode; (b) the discharge current in fixed
voltage mode; and (c) discharge current in fixed peak current mode. The Ar pressure was set to 1 Pa. The pulse width was 100 $μ$s at an average power of 300 W.
Figure 4. The Ti deposition rate from both dcMS and HiPIMS discharges operated in fixed voltage mode and fixed peak current mode using various magnetic field configurations, measured at 70 mm axial
distance over center of cathode. The magnet configurations on the x-axis are ordered from high $| B |$ at the left to low $| B |$ on the right. The recorded $| B r , rt |$ value above the race track
was used as a measure of $| B |$.
Figure 5. The RSD of Ti deposition rates from both dcMS and HiPIMS discharges operated in fixed voltage mode and fixed peak current mode using various magnetic field configurations. The rates
measured at 70 mm axial distance over center, race track and edge of cathode. The magnet configurations on the x-axis are ordered with increasing $z null$ from left to right.
Figure 6. The Ti ionized flux fraction in a HiPIMS discharge using various magnet configurations measured at 70 mm axial distance over the center of the cathode. The discharge is operated in the
HiPIMS fixed voltage and fixed peak current modes. The magnet configurations on the x-axis are ordered from high $| B |$ at the left to low $| B |$ on the right. The recorded $| B r , rt |$ value
above the race track was used as a measure of $| B |$.
Figure 7. The Ti ionized flux faction in a HiPIMS discharge using various magnet configurations measured at 30 mm axial distance over the center of the cathode. The discharge was operated in the
HiPIMS fixed voltage and fixed peak current modes. The magnet configurations on the x-axis are ordered from high $| B |$ at the left to low $| B |$ on the right. The recorded $| B r , rt |$ value
above the race track was used as a measure of $| B |$.
Figure 8.
The peak discharge current (left
-axis) when operating in fixed voltage mode (
$V D = 625$
V) and the discharge voltage (right
$y –$
axis) when operating in fixed peak discharge current mode (
$I D , max = 40$
A) as a function of the magnetic field strength over the race track (
$B r , rt$
Table 1
all magnets moved together (C0E0, C5E5, and C10E10) and fixed voltage operation,
magnets mixed (C0E5, C5E0, C10E0 and C0E10) and fixed voltage operation,
all magnets moved together (C0E0, C5E5, and C10E10) and fixed peak current operation, and
magnets mixed (C0E5, C5E0, C10E0 and C0E10) and fixed peak current operation.
Figure 9.
Experimentally determined combinations of
$F DR$
$F flux$
$z = 70$
mm, for all three radial positions, and for all magnetic field configurations. The configurations C0E0, C5E5, and C10E10 are denoted by
corresponding to variable
$| B |$
when all the magnets were moved together. The configurations C0E5, C5E0, C10E0 and C0E10 where the two magnets were moved relative to each other are denoted by
. The discharges were operated at constant voltage and constant average power. Lines of constant
$α t$
(solid blue lines) and constant
$β t$
(dashed green lines), calculated using Equations (
) and (
), respectively, give approximate estimate of these parameters for the studied discharges.
Figure 10. (a) The ionization probability $α t$ and (b) the back attraction probability $β t$ for the ions of the sputtered species versus the magnetic field strength above the race track $( r = 25$
mm). o both magnets moved together (C0E0, C5E5, and C10E10) over race track in fixed voltage operation, x both magnets moved together (C0E0, C5E5, and C10E10) over center in fixed voltage operation,
+ magnets mixed (C0E5, C5E0, C10E0 and C0E10) over race track in fixed voltage operation, △ magnets mixed (C0E5, C5E0, C10E0 and C0E10) over center in fixed voltage operation, ◇ both magnets moved
together (C0E0, C5E5, and C10E10) over center in fixed peak current operation, and □ magnets mixed (C0E5, C5E0, C10E0 and C0E10) over center in fixed peak current operation.
Figure 11. (a) The ionization probability of the sputtered species; and (b) the ionized flux fraction above the race track versus the peak discharge current. o both magnets moved together (C0E0,
C5E5, and C10E10) over the race track in fixed voltage operation, x both magnets moved together (C0E0, C5E5, and C10E10) over center in fixed voltage operation, + magnets mixed (C0E5, C5E0, C10E0 and
C0E10) over race track in fixed voltage operation, △ magnets mixed (C0E5, C5E0, C10E0 and C0E10) over center in fixed voltage operation, ◇ both magnets moved together (C0E0, C5E5, and C10E10) over
center in fixed peak current operation, and □ magnets mixed (C0E5, C5E0, C10E0 and C0E10) over center in fixed peak current operation.
Table 1. Discharge operating parameters for the investigated dcMS and HiPIMS discharges in fixed voltage and in fixed peak current modes. The average discharge power was kept at 300 W for all the
discharges. For HiPIMS discharges, the pulse length was 100 $μ$s while the pulse frequency was varied to maintain a constant averaged power. The absolute magnetic field strength and the degree of
balancing was varied by displacing the center magnet (C) and the outer ring magnet at the target edge (E). Each configuration is referred to using the displaced distance (in mm) of each magnet from
the target backing plate. In this notation, C0E0 refers to a magnetron configuration where the center and outer magnets touch the backing plate.
Magnet dcMS HiPIMS HiPIMS HiPIMS
Fixed Voltage Fixed Peak Current Fixed Peak Current
$B r , rt$ $z null$ $V D$ $I D$ $V D$ $I D , peak$ $f pulse$ $V D$ $I D , peak$ $f pulse$ $V D$ $I D , peak$ $f pulse$
[Gauss] [mm] [V] [A] [V] [A] [Hz] [V] [A] [Hz] [V] [A] [Hz]
C0E0 238 66 339 0.885 625 80 54 510 40 143 555 80 60
C0E5 217 70 308 0.974 625 54 76 565 40 123 580 80 56
C0E10 213 74 311 0.964 625 35 115 650 40 111
C5E0 181 53 317 0.946 625 53 80 557 40 129 582 80 58
C5E5 161 59 334 0.926 625 36 97 655 40 97 649 80 295
C10E0 137 43 312 0.961 625 31 134 660 40 99 636 80 295
C10E10 111 52 330 0.909 625 12 450
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/
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MDPI and ACS Style
Hajihoseini, H.; Čada, M.; Hubička, Z.; Ünaldi, S.; Raadu, M.A.; Brenning, N.; Gudmundsson, J.T.; Lundin, D. The Effect of Magnetic Field Strength and Geometry on the Deposition Rate and Ionized Flux
Fraction in the HiPIMS Discharge. Plasma 2019, 2, 201-221. https://doi.org/10.3390/plasma2020015
AMA Style
Hajihoseini H, Čada M, Hubička Z, Ünaldi S, Raadu MA, Brenning N, Gudmundsson JT, Lundin D. The Effect of Magnetic Field Strength and Geometry on the Deposition Rate and Ionized Flux Fraction in the
HiPIMS Discharge. Plasma. 2019; 2(2):201-221. https://doi.org/10.3390/plasma2020015
Chicago/Turabian Style
Hajihoseini, Hamidreza, Martin Čada, Zdenek Hubička, Selen Ünaldi, Michael A. Raadu, Nils Brenning, Jon Tomas Gudmundsson, and Daniel Lundin. 2019. "The Effect of Magnetic Field Strength and Geometry
on the Deposition Rate and Ionized Flux Fraction in the HiPIMS Discharge" Plasma 2, no. 2: 201-221. https://doi.org/10.3390/plasma2020015
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Jaynes Notes 1
This is an opinionated and incomplete book. These are sloppy notes.
I nearly cut my finger off a week ago and it’s hard to type with 9 fingers. Jaynes died midway through writing his book, and it’s hard to write with 0 lives. So we both have excuses.
Dedicated to the memory of Sir Harold Jeffreys, who saw the truth and preserved it.
“Saw the truth and preserved it”. I like this quote.
This section is perhaps the most interesting in the book, and that’s not an insult. It concisely describes a fascinating idea:
Probability theory is the extension of logic to uncertainty.
As a slogan: logic is probability theory where all probabilities are 0 or 1.
Jaynes does not seem to like infinity. This is a recurring theme. Don’t mention the Axiom of Choice or nonmeasurable sets around him.
As a mathematician, I counter with: No one shall expel us from the Paradise that Cantor has created.
Chapter 1
The takeaway is that probability gives you a way to quantify how likely the converse of a proposition is to be true.
A can imply B, but that does not mean B implies A. On the other hand, if A implies B, then B being true can’t make A less likely to be true than if B was false.
So \(A \implies B\) gives a fuzzy sort of reverse implication. This fuzziness is where probability comes into the picture. It lacks certainty, but often certainty is too much to ask for, especially
in real life. We’ll see later that Bayes rule gives us this sort of reverse fuzzy modus ponens.
Jaynes lists out a bunch of desired principles of logical thinking that are each relatively reasonable to accept. He will later use this to set up the axioms of probability theory as consequences of
this alternative foundation.
This is nice because the Kolmogorov axioms are simple, but less intuitive. These principles are more complicated but more intuitive.
Boolean Algebra
This section introduces notation used in the rest of the book, so don’t skip it completely.
Chapter 2
This is full of sloppy writing.
I think math books can either be monographs that cover everything in technical correctness to a sometimes painful degree, or they can teach you the core ideas, but they should pick one and stick with
it. It’s too easy to get lost in the details otherwise.
This chapter walks through a convoluted functional equation to show that the product rule of probability is true.
The insistence on writing everything as a conditional probability for philosophical reasons is nice in theory, but is a pain to read.
You can safely skip the derivation. I promise it will grant no insight, and it’ll make you want to die while reading it.
It’s also half-assed. It says it wants to do everything from first principles to show how general these principles of plausible reasoning are, then goes and tacks on assumptions like
differentiability because otherwise the proof is apparently 11 pages long. Then why waste my time with 6 pages that didn’t teach me anything about plausible reasoning?
Takeaway: Bayes’ rule is fuzzy reverse modus ponens. The product rule is the fundamental rule of how to chain implications.
Principle of Indifference
This is derived as a special case of maximum entropy from symmetry equations.
Jaynes “Comments” (Rant)
In 2.6, Jaynes starts ranting against the use of infinite sets that are not derived from finite ones via some limiting process. You do you, man.
This part bugged me:
In this example, the undecidability is not an inherent property of the proposition or the event; it signifies only the incompleteness of our own information. But this is equally true of abstract
mathematical systems; when a proposition is undecidable in such a system, that means only that its axioms do not provide enough information to decide it. But new axioms, external to the original
set, might supply the missing information and make the proposition decidable after all.
In the future, as science becomes more and more oriented to thinking in terms of information content, Godel’s result will be seen as more of a platitude than a paradox. …
These considerations seem to open up the possibility that, by going into a wider field by invoking principles external to probability theory, one might be able to prove the consistency of our
rules. At the moment, this appears to us to be an open question.
The Continuum Hypothesis’s truth is independent of ZFC. There’s all sorts of reasonable-sounding systems that make it true or false. So this is hardly a platitude.
And if you extend a system to prove your original system is consistent, your new system has no proof that it itself is consistent. Further extensions give the same issue.
Venn Diagrams
Venn diagrams are a useful tool, and Jaynes gives a convoluted reason to avoid them that doesn’t convince me.
Kolmogorov axioms
Jaynes argues against interpreting probabilities in terms of sets, which is essentially the measure-theoretic foundation. He tells the reader to look at Appendix A if they’ve studied probability
Appendix A
Jaynes gives an excellent reason for why probabilities should sum to 1, and not just any finite number.
He also makes a good case that Kolmogorov handles conditional probability awkwardly, something I’ve thought too.
The requirement that degrees of plausibility be represented by real numbers can be replaced by any totally ordered set (at least for finitely many propositions).
The totality of a total order may feel unnecessary to some, so Jaynes looks at weakening it to partial orders and lattices.
The cool idea to me was that as your knowledge grows, your lattice collapses to a line. | {"url":"https://alok.github.io/2017/12/09/jaynes-notes-1/","timestamp":"2024-11-05T18:16:59Z","content_type":"text/html","content_length":"15485","record_id":"<urn:uuid:41d3bd4e-1f65-4805-8c12-03e89b58a412>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00059.warc.gz"} |
Alexandre Thiéry - Deriving Langevin MCMC
Julian Besag (1945 – 2010)
Consider a target density \(\pi(x)\) in \(\mathbb{R}^D\). Since the Langevin diffusion
\[ dX_t = \nabla \log \pi(X_t) \, dt + \sqrt{2} \, dW \tag{1}\]
is reversible with respect to \(\pi\), it is natural to use a Euler-Maruyama discretization of Equation 1 to build MCMC proposals: in a MCMC simulation and for a time discretization parameter \(\
varepsilon> 0\), if the current position is \(x \in \mathbb{R}^D\), a proposal \(y \in \mathbb{R}^D\) can be generated as
\[ y = x + \varepsilon\, \nabla \log \pi(X_t) + \sqrt{2 \varepsilon} \, \xi \]
with \(\xi \sim \mathcal{N}(0,I)\) before being accepted-or-reject according to the usual Metropolis-Hastings ratio. This MCMC method, first proposed by Julian Besag in 1994, is commonly referred to
as the Metropolis-Adjusted-Langevin-Algorithm (MALA). But how can one come-up with this proposal mechanism without knowing before hand the existence of this reversible Langevin diffusion Equation 1?
While it is intuitively clear that following the direction of \(\nabla \log \pi\) is not such a bad idea, i.e. one would like to move towards areas of “high probability mass”, where does this \(\sqrt
{2}\) comes from? Naturally, one could look at proposals of the type \(y = x + \nabla \log \pi(X_t) \, \varepsilon+ \lambda \, \xi\) for some free parameter \(\lambda > 0\) and study the behavior of
the Metropolis-Hastings ratio in the regime \(\varepsilon\to 0\): as simple as it sounds, it is not entirely straightforward and requires quite a bit of algebra (do it!). Instead, I very much like
the type of approaches described in (Titsias and Papaspiliopoulos 2018). To summarize, we would like to generate a MCMC proposal \(y \in \mathbb{R}^D\) that stays in the vicinity of the current
position \(x \in \mathbb{R}^D\) while exploiting the knowledge of \(\nabla \log \pi(x)\). One cannot simply approximate the target distribution as \(\pi(x) \approx \pi(x_k) e^{\left< \nabla \log \pi
(x_k), x-x_k \right>}\) and sample from this approximation since it is typically does not define a probability distribution. Instead, consider the following extended target distribution
\[ \overline{\pi}(x,z) \, \propto \pi(x) \, \exp {\left\{ -\frac{1}{2\varepsilon}\|z-x\|^2 \right\}} . \]
In other words, the Gaussian auxiliary variable \(z \in \mathbb{R}^D\) is centred at \(x\) and at distance about \(\sqrt{\varepsilon}\) of it. Now, given the current position \(x_k\), to generate a
proposal \(y_\star\) that stays in the vicinity of \(x_k\), one can proceed in two steps, in the spirit of a Gibbs-sampling approach:
1. First, generate \(z_\star \sim \overline{\pi}(dz | x_k) \sim \mathcal{N}(x_k, \sqrt{\varepsilon}I)\)
2. Second, sample from \(y_\star \sim \overline{\pi}(dx | z_\star)\).
Unfortunately, the second step is typically not tractable. Nevertheless, the conditional density \(\overline{\pi}(dx | z_\star)\) is concentrated in a \(\sqrt{\varepsilon}\)-neighborhood of \(z_\star
\) and a simple Gaussian approximation around \((x_k, z_\star)\) should be enough for our purpose. We have:
\[ \log \overline{\pi}(dx | z_\star) &= \log \pi(x) - \frac{1}{2 \varepsilon} \|x - z_\star\|^2 + \textrm{(Cst)}\\ &\approx \left< \nabla \log \pi(x_k), x-x_k \right> - \frac{1}{2 \varepsilon} \|x -
z_\star\|^2 + \textrm{(Cst)}\\ &= - \frac{1}{2 \varepsilon} \|x - [z_\star + \varepsilon\, \nabla \log \pi(x_k)]\|^2 + \textrm{(Cst)}. \]
This shows that the conditional \(\overline{\pi}(dx | z_\star)\) can be approximated by a Gaussian distribution centred at \([z_\star + \nabla \log \pi(x_k)]\) and variance \(\varepsilon\, I\). This
means that the final proposal \(y \in \mathbb{R}^D\) can be generated as \(y \sim z_\star + \nabla \log \pi(x_k) + \xi\) where \(\xi \sim \mathcal{N}(0,\varepsilon)\). But that is equivalent to
\[ y \sim x + \varepsilon\, \nabla \log \pi(x_k) + \sqrt{2 \varepsilon} \, \xi \]
with \(\xi \sim \mathcal{N}(0,I)\) since \(z_\star \sim \mathcal{N}(x, \sqrt{\varepsilon} I)\). It is exactly the MALA proposal. Naturally, one can also try to be slightly more clever and use an
extended distribution
\[ \overline{\pi}(x,z) \, \propto \, \pi(x) \, \exp {\left\{ -\frac{1}{2\varepsilon} \left< (z-x), M^{-1} \, (z-x) \right> \right\}} \]
for some appropriate positive-definite “mass” matrix \(M \in \mathbb{R}^{D,D}\). Indeed, this immediately leads to preconditioned MALA methods. I really like this approach since it can be adapted and
generalized to quite a few other situations!
Titsias, Michalis K, and Omiros Papaspiliopoulos. 2018. “Auxiliary Gradient-Based Sampling Algorithms.” Journal of the Royal Statistical Society Series B: Statistical Methodology 80 (4). Oxford
University Press: 749–67. | {"url":"https://alexxthiery.github.io/notes/on_Langevin_MCMC/on_Langevin_MCMC.html","timestamp":"2024-11-12T10:22:12Z","content_type":"application/xhtml+xml","content_length":"24117","record_id":"<urn:uuid:13f5bb35-b564-4093-92ac-1b0099d77ff8>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00459.warc.gz"} |
Division Worksheets Grade 3 Pdf - Divisonworksheets.com
Division Worksheets Grade 3 Pdf
Division Worksheets Grade 3 Pdf – With the help of worksheets for division to help your youngster review and practice their division skills. You can design your own worksheets. There are a variety of
choices for worksheets. They can be downloaded at no cost, and you can modify them in any way you want. They’re perfect for kindergarteners as well as first-graders.
Two can do huge quantities of work
While dividing huge numbers, children should be practicing with worksheets. These worksheets can be limited to two, three, and sometimes four different divisors. This method won’t cause the child
anxiety about not remembering how to divide the large number or making errors on their times tables. To help your child develop this mathematical ability you can either download worksheets, or search
them online.
Multi-digit division worksheets are used by children to practice and increase their knowledge. It’s an important mathematical ability which is needed to carry out complicated calculations, as well as
other tasks in everyday life. These worksheets build on the idea by providing interactive questions and activities which are based on divisions of multi-digit numbers.
It’s not easy for students to split huge numbers. These worksheets typically are constructed using a similar algorithm that provides step-by-step instructions. They may not offer the needed
understanding required by students. Teaching long division can be taught using base 10 blocks. After learning the steps, long division should be a natural process for students.
Students can learn division of big numbers using various worksheets and practice questions. The worksheets also provide fractional results in decimals. Worksheets for hundredths are even available,
which are especially useful for learning to divide large amounts of money.
Sort the numbers into small groups.
It can be difficult for smaller groups to make use of numbers. While it might sound appealing on paper, the majority of small group facilitators do not like this procedure. It is a reflection of the
way the human body evolves. This process is beneficial for the Kingdom’s endless expansion. In addition, it encourages others to reach out to the lost and fresh leadership to assume the reigns.
It can also be useful for brainstorming. You can create groups of people who share similar qualities and experiences. This allows you to generate creative ideas. After you have created your groups,
present everyone to you. It’s a great way to encourage creativity and innovative thinking.
To break large numbers down into smaller pieces of information, the basic arithmetic operation division is used. It is beneficial in situations where you have to create equal amounts of items for
different groups. For instance, a big class might be divided into five classes. The original 30 pupils are made by adding the groups.
Keep in mind that when you divide numbers, there’s a divisor and a quote. Divide one by five yields the same result, while two times two gives the similar result.
It is a good idea to use the power of 10 for big numbers.
You can divide massive numbers into powers of 10 in order to make it easier for us to evaluate them. Decimals are an extremely frequent part of shopping. They are usually seen on receipts as well as
food labels, price tags as well as receipts. They are also used by petrol stations to display the price per gallon, as well as the amount of gas that comes via a sprayer.
There are two methods to split a large number into its powers of ten. The first is by shifting the decimal line to the left and using a multiplier of 10-1. The second method uses the power of ten’s
association feature. Once you’ve mastered the associative feature of powers of ten, you can break large numbers down into smaller powers of 10.
The first method uses mental computation. There is a pattern when you divide 2.5 by the power of ten. The decimal position will shift to one side for every 10th power. Once you have mastered this
concept, it’s feasible to apply it to solve any issue.
The second method is to mentally dividing extremely large numbers into powers of ten. The other method involves rapidly writing large numbers in scientific notation. Large numbers should be expressed
with positive exponents if written in the scientific notation. You can convert 450,000 numbers to 4.5 by shifting the decimal mark five spaces to the left. To divide a huge amount into smaller powers
10, you can make use of exponent 5, or divide it in smaller powers 10 until it becomes 4.5.
Gallery of Division Worksheets Grade 3 Pdf
Grade 3 Multiplication Worksheets Multiplying Whole Tens K5 Learning
Division Grade 3 Math Worksheets Pdf Complete With Quizzes Homework
Printable Division Worksheets 3rd Grade
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Top 20 Best Math Apps for Android to Solve Math Homework Easily
Mathematics is an important subject. To be more decisive in math, we need to practice more and more. So we feel the requirement of the best math apps for Android phones. Those math tutor apps will
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In the series of Apps review today, we will review the Top 20 Best Math Apps for Android. Those apps are selected based on popularity, user feedback, and quality of content. So have fun with the
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1. Photomath
Photomath is one of the most popular and best math apps to solve math on an Android phone. It can read and solve math problems instantly by using mobile camera apps. It shows how to approach to solve
any mathematical problem, even if it is handwritten or printed.
Photomath is the math superpower for every student. It is best for solving home works. The parents are also dependent on this lucrative math tutor app.
• Beautiful graphs
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Math in Medicine: The Actual Requirement
In the healthcare sector, math plays a vital role. It assists the doctors in prescribing the correct doses and helps the surgeon apply the perfect medicine. There is a big difference between mg/pound
and mg/kg. Moreover, complexity arises in individual blood pressure, body temperature, and repository rate. Math apps help professionals to circulate in a better way. Brighterly can help to solve all
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Final Thought
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Trapped Inside | On Beyond Darwin - Chapter 12
CHAPTER 12
Trapped Inside
In this century, two very startling and revolutionary changes took place in our thinking about the universe. One of these changes is the idea that at the microscopic level there is inherent
unpredictability in the behavior of particles. This unpredictability is negligible with macroscopic objects so that for a long time it was not evident that the uncertainties were of any importance.
Newton's deterministic mechanics was perfectly adequate to describe the behavior of baseballs and planets. The second revolutionary change was called the theory of relativity and was devised by
Albert Einstein in 1905. All Einstein did in his theory of relativity was to accept, as true, Newton's principle of relativity for the interaction between charged particles. I have said that Newton's
laws of motion, his mechanics, do not properly describe the interaction between the fundamental charged particles except in the special case where the interacting particles are released from rest and
then interact. Newton's laws can be used for the electrostatic interaction. But, the fact that the interaction is not instantaneous means that any mechanics that says that at every instant the forces
of the interacting particles on each other are equal and oppositely directed, as Newton's does, cannot possibly be right.
Einstein did not begin directly by trying to develop new laws of motion but instead started to examine the situation by using the principle of relativity as devised by Newton. This principle comes
from Newton's first law of motion:
Law I: Every body perseveres in its state of rest or uniform motion in a straight line unless change in that state is compelled by impressed forces.
The important piece of information in the law is that the natural state of a body is either rest or uniform motion. These two states must be equally natural. It was discovered that only certain
frames of reference provided an environment in which an uninfluenced body could stay at rest. These were called inertial frames of reference. A frame moving at uniform speed relative to an inertial
frame must also be an inertial frame, or Newton's first law would not be true. Newton stated the principle of relativity this way:
The motions of bodies included in a given space are the same among themselves whether that space is at rest or moves uniformly forward in a straight line.
In this statement Newton claimed that it is a fact that the interaction between two bodies is the same in all inertial frames. Of course, Newton thought of the interactions as being the kind he
described in his other two laws of motion: instantaneous, and either attractions or repulsions along the line joining the interacting bodies. Einstein stated his principle of relativity by saying
that the laws of Physics are invariant from one inertial frame of reference to another. But, to work out the theory, he basically used only one set of laws--Maxwell's laws of electromagnetism. And
from these he used only the piece of information that the interaction between charged particles is not instantaneous but retarded. The interaction is straight-line with speed c and independent of the
motion of the source particle. Einstein said that, in all inertial frames of reference, the behavior of interacting charges is the same.
The next step in the development is to try to relate observations of a single interaction as made in two different inertial frames. It is easy to appreciate that the position coordinates of objects
relative to two different frames will be different. What was not as obvious is that the times of the events as measured in the different frames will be different. Suppose that we are observing two
particles; I will speak first of the description of the interaction in frame 1 . To describe the position of an object in a frame, we specify three numbers called coordinates. The frame of reference
can be thought of as three rods stuck together so that each is at right angles to the other two, the way the edges of a box come together at a corner. The three rods are called coordinate axes and
labelled by the names: x -axis, y -axis, and z -axis. To specify the position of an object relative to this frame, we give the three distances of a trip you take from the point where the rods meet,
which we call the origin of coordinates, up to the object. We are describing a trip with three legs, all at right angles to each other. You travel first along the x -axis, then in the x-y plane
parallel to the y -axis, then out of the plane in a direction parallel to the z -axis. The three distances are written as (x, y, z) and they describe the position of the object. We must also write
the time at which the particle has this position because the times for the two interacting particles will be different; remember the interaction is retarded. So, in frame 1 , particle 1 is at (x[1],
y[1], z[1]) at time t[1] and is being influenced by particle 2 at (x[2], y[2], z[2]) at time t[2] . The time t[2] is earlier than t[1] by the amount of time the interaction takes to travel from (x
[2], y[2], z[2]) to (x[1], y[1], z[1]) at speed c . In terms of the coordinates--which by the way, are called Cartesian coordinates after Descartes--the distance d between the two particles is
You may know this fact about Cartesian geometry; it is nothing magic. It all comes from Pythagoras' theorem--Remember, the square on the hypotenuse is equal to the sum of the squares on the other two
sides of a right-angled triangle. The relationship between the two instants of time is thus
The difference in time is the time required for the interaction to travel the distance d at speed c . Sometimes the coordinates of the particles are written to include the time as a coordinate. We
say that particle 1 being at (x[1], y[1], z[1]) at time t[1] is an event described by the coordinates (x[1], y[1], z[1], t[1]) . We can call it event 1 . The event 2 is described by (x[2], y[2], z
[2], t[2]) ; this means that particle 2 is at position (x[2], y[2], z[2]) at time t[2] . In frame 1 , event 2 is influencing event 1 ; we say that there is a causal connection between event 1 and
event 2 . (The talk gets quite fancy.)
Now suppose we look at exactly the same two events from frame 2 . We will call their coordinates:
The two events must still be causally connected. This means that
The two basic premises of Einstein's relativity are: first, that a particle moving at constant velocity in one inertial frame of reference, and thus judged to be uninfluenced, will be observed as
moving at constant velocity in another; and second: that two events judged as causally connected in one frame will be so judged in another. Nothing whatsoever about either of these two premises is
weird. Where do we get the idea that Einstein's relativity is weird? We get it when we try to find the mathematical relationship between the coordinates of an event in one frame, and the coordinates
of the same event in another. We find that this relationship, which is called the Lorentz transformation, says that the time t' of an event in frame 2 depends not only on the time t in frame 1 but
also the position (x, y, z) of the event in frame 1 . Events that happen at the same time in frame 1 are not assigned the same time in frame 2 , unless they also happen at the same place. This means
that we may judge events as simultaneous in frame 1 and find that they are not simultaneous in frame 2 . This leads us to say that time is not absolute; it depends on your frame of reference. This is
in conflict with the Newtonian point of view. Here is Newton's statement in the Principia :
I. Absolute, true, and mathematical time, of itself and from its own nature, flows equably without relation to anything external, and by another name is called duration; relative, apparent, and
common time, is some sensible and external (whether accurate or unequable) measure of duration.
He went on to say this about space:
II. Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute space.
You can imagine, if time in frame 2 depends on position as well as time in frame 1 , that position in frame 2 depends on time in frame 1 as well as position. This means that the length of an object
depends on the frame. If it is at rest in one frame and has a certain length, in the second frame it will have a different length; it will be measured to be contracted. If two events in frame 1 have
a time interval between them and happen at the same place--think of a pendulum bob swinging out and back--the time interval, as measured in frame 2 , will not be the same. It will be longer--we say
that the time is dilated. If I call the pendulum swinging in frame 1 a clock, its tick, as judged in frame 2 , will be longer. It appears to be running slow. You have often heard that clocks in
moving frames run slow compared to clocks at rest. And what is more, all these effects are relative. If in frame 1 you made measurements on a clock or a distance (a ruler) held fixed in frame 2 , you
would say that the clock was running slow and that the ruler was contracted.
Newton's mechanics was based on the idea that the measurement of time and space (distance) was independent of the frame as long as it was inertial. Maxwell used the same notion:
We shall find it more conducive to scientific progress to recognise, with Newton, the ideas of time and space as distinct, at least in thought, from that of the material system whose relations these
ideas serve to coordinate. [1]
How did we get to the point of having to mix time and space in a sort of four-dimensional world where we must give (x, y, z, t) instead of (x, y, z) for every event? We got to this point, I believe,
because the things we are using to make measurements involve what we are trying to measure. We are trapped inside the universe and cannot stand outside with an independent clock and a ruler and make
measurements. Schlegel says this in his book Completeness in Science :
Physically, we come to strong and hitherto unknown limitations on our knowledge of nature when the object of investigation and physical entities by which we study the object become the same. [2]
But, what are we using for a clock or a ruler? Larmor notes this in the appendix to Maxwell's book Matter and Motion :
It is impossible to ignore the rays of light as messengers of direction and duration from all parts of the visible universe. [3]
Rulers must be straight. We use light--or electric interaction--to define straight lines. How do we judge whether or not light--or electric interaction--travels in a straight line? That is the
definition of a straight line. That is how we tell if something is straight; by sighting along it or shining a laser along it; light is the messenger of direction. So it is not surprising that
electric interaction travels in straight lines. What about the speed of electric interaction--light? Why is it the same in two frames of reference?
Imagine an experiment in one frame to measure the speed of light by timing light as it goes along a ruler from one end to a mirror at the other and back again. I will call this the "speed" ruler. To
time it we would need a clock. As a clock let us use another ruler with a mirror at each end. Start a beam of light at one end and say that one tick of the clock is the time it takes the light to go
down to the far mirror and back. (Remember a tick must be a motion that comes back to the same place.) Perhaps you can see that if the two rulers are the same length that it takes exactly one tick of
the clock for the light to go down and back on the speed ruler. How could it be otherwise, since they are really identical instruments: one for the speed of light experiment, and the second a clock
for the measurement of time? With this experimental set up you can easily see that no matter what inertial frame you are in, you get the same value for the speed of light. But is it really the same
in all frames? Again, we are trapped. We could never tell whether it was or not.
But, you may say, why not use a different kind of clock? There are no different kinds of clocks. All clocks are basically electromagnetic, with the possible exception of radioactive decay clocks.
So the peculiar results of Einstein's relativity stem from the fact that we assume that, in all inertial frames of reference, light--or electric interaction--travels in straight lines at speed c . We
come to recognize that there is no independent clock--or ruler--or else we could tell whether light really travelled in straight lines at constant speed or not. We admit that we are trapped inside
with instruments that depend on the thing that we are measuring. We assume light travels at a uniform speed but how do we know? This is Mach on the subject, some time before Einstein formulated his
special relativity:
... time is an abstraction, at which we arrive by means of the changes of things... A motion may, with respect to another motion, be uniform. But, the question whether a motion is in itself uniform,
is senseless. [4]
As he points out, we judge that light travels at a uniform speed by comparing it with something that we believe travels at a uniform speed--with light of course. Sometimes we say light can never
overtake light; it all travels at the same speed. The electric interaction is simple as far as these two facets of it are concerned because they define our universe. They are the basis of all our
knowledge, and the basis cannot be independently checked.
It is generally accepted that there is no such thing as time apart from the ticking of clocks and clocks are only judged as good time keepers relative to each other. Feynman says:
... "best" clocks are those which we have reason to believe are accurate because they agree with each other. [5]
I have tried to explain why the speed of electromagnetic interaction is the same in all inertial frames, because that is the really peculiar thing about Einstein's relativity. If you accept the idea
of laws of nature you say--How reasonable it is that a law which says the speed of light--electric interaction--is constant, is the same--invariant--in all inertial frames! That is the nature of
nature. But if you are, like me, unable to accept the idea of general laws you must ask for an explanation of this fact. Otherwise, it seems like a design feature of the universe. I have not tried to
explain in detail about time dilation because it is not central. The Berkeley text on relativity says this:
There is nothing mysterious about clocks. If there is anything mysterious about special relativity, it is the constancy of the speed of light. Granted that, everything else follows directly and
fairly simply. [6]
I cannot just grant "the constancy of the speed of light." I must explain it to myself. Then I can see why the speed is invariant from one inertial frame to another. Here is Feuer on the subject:
The logical content of the principle of relativity was indeed an absolutist one, a statement of a principle of invariance. Given the requirement, however, of the conformity of laws of nature to the
Lorentz transformation, and the principle of the constancy of the velocity of light, there followed the remarkable consequences of the relativized status of time and spatial distances... The
startling relativist consequence rather than the absolutist postulate was what affiliated Einstein's theory emotionally with the relativist school. [7]
It is the "absolutist postulate" that to me must be explained, not all the relativistic consequences. They are easily explained once the invariance of the speed of light--electric interaction--is
So we end up with two facts--first, that as far as fundamental particles alone are concerned one inertial frame is equivalent to another. Second, when two particles interact we just assume that the
interaction travels in straight lines at constant--or uniform--speed and take the speed to be, by definition, the same in all inertial frames. In this way, we get distance and time measurements all
interwoven with rather bizarre consequences like time dilation and length contraction.
But, Einstein's principle of invariance of the laws of physics from one inertial frame to another is, in itself, a general law. He says all laws are invariant. In order for me to show that it is not
a general law but really a specific fact, either about electromagnetic interaction or about the universe, I must be sure that there are no other general laws to which it has been found to apply. It
does apply to the wave-particle duality of matter. De Broglie used relativity to derive his wavelength of a particle of matter. I have argued that the uncertainty which gives rise to the appearance
of a dual nature of matter can be attributed to a fluctuating influence of the rest of the universe on the particle. I indicated that this produced a Brownian type of motion, in which the product of
the uncertainties in position and velocity were characterized by a constant h , Planck's constant.
Einstein's principle of invariance would say that this law is the same from one inertial frame to another and that the constant h would also be the same. This to me must be a fact about the
universe--that the inertial effect on a particle is the same in all inertial frames, and that the fluctuating effect is also the same. If it were not a fact, we would be able to distinguish one
inertial frame from another on the basis of a quantum mechanical effect. If we assume that the fluctuations are due to the continuous radiation of atoms in their ground states, the Planck blackbody
radiation graph can be derived from the fact that the radiation environment is the same from one inertial frame to another. It is, as they say, "Lorentz invariant." This means that the fluctuations
are the same in all inertial frames. So the real information content of the blackbody radiation distribution curve is that atoms radiate in the ground state--at absolute zero--and this radiation is
the same from one inertial frame to another. This explains why radioactive clocks agree with electromagnetic clocks.
Einstein believed that the laws of physics were invariant from one inertial reference frame to another. Newton's mechanics was not invariant so Einstein discarded it and substituted his own
mechanics--relativistic mechanics. To quote myself:
Underlying the principle of relativity is the idea that what happens in a physical situation [say two particles interacting] should not depend on the frame of reference in which the happening is
described. This means that the same laws that analyze the behavior of bodies interacting among themselves relative to one inertial frame can be used to analyze their behavior relative to any other
inertial frame. For the purpose of analyzing motions one inertial frame is as suitable as another; that is, there is no preferred frame. [8]
What Einstein was doing in producing relativistic mechanics was designing a set of generalities that could be used to analyze interactions in different inertial frames and which would themselves
remain the same. One set of laws would be used for all frames. So he set about to find this set of laws. Again quoting myself:
Newton's laws of mechanics are not invariant under the Lorentz transformations. This is most obvious from the fact that the acceleration of an object is not invariant under these transformations.
Acceleration [remember F=m×a ] must certainly lose the position on center stage that it had in Newtonian mechanics. Force, mass, and momentum too are quantities that are intimately linked with
Newton's laws and, as understood by Newton, cannot be of service. But, Newton's laws had certain features which would be good to perpetuate if possible. For one thing, the ideas of force, mass,
momentum and later energy have been built up as strong intuitive notions over the centuries of thinking in Newtonian terms. Another and absolutely invaluable feature is the fact that Newton's laws
enabled us to treat a system of interacting bodies as a single entity whose interactions with other such systems could be calculated. In this way we could ignore, if we wanted to, any internal
complexities of a system. [9]
This analysis is too involved mathematically to present here, but two of the results are simple enough. We redefine mass so that it is no longer a constant independent of the velocity of a body but
is a quantity which increases as velocity increases, approaching infinity as the particle's speed approaches the speed of light. This explains for some people why things cannot travel faster than
light. I prefer to think that a charged particle accelerates in response to the presence of another charged particle and that it could not be accelerated to a greater speed than the speed at which
the electric action between them travels. How could it be induced to go any faster?
In trying, to obtain conservation laws for relativistic mechanics Einstein found that the laws of conservation of momentum and energy must go as a pair. In Newton's mechanics they were separate.
Also, the quantity which would be called energy by Einstein is given by the famous formula
E = mc^2
This is sometimes construed as saying that mass can be converted into energy. But, it is no different from the expression for kinetic energy in Newton's mechanics
E = ½mv^2
In fact, at low speeds Einstein's formula becomes Newton's formula but with an additional term which is called the rest energy of the particle.
The rest energy is always there since we cannot destroy electrons or protons; so it is not much use. In Einstein's mechanics, the total mass of two interacting particles can be different when they
are close from what it is when they are far apart. This sometimes is construed as changing energy to mass. It takes energy to bring them close if they repel, and the mass will be greater together
than apart. If they were held together as protons are in the nucleus by an attraction at short range then energy would be available if the attraction were broken. This is what happens in nuclear
fission. A neutron entering a uranium nucleus breaks it into pieces which repel each other electromagnetically, and energy is released. There is, in fact, a mass difference between the nucleus and
the final pieces that agrees with Einstein's formula ( E=mc^2 ) but the energy released does not require relativity to understand. Einstein's mechanics applies equally well to any chemical reaction.
In a chemical reaction the amount of energy released in each interaction represents a very small mass equivalent. So we do not notice the difference between the mass before and after the reaction. In
chemistry, we use the law of conservation of mass, but it is not precisely correct. The need to use relativistic mechanics is more evident in nuclear reactions. Perhaps that is why we associate
Einstein more with nuclear energy than with chemical energy.
There is a law in the list I gave from Constant's book on the Fundamental Laws of Physics that I have been ignoring--besides the law of gravitation--that is the Pauli exclusion principle. The reason
I have left it until now is that I needed to have looked at both Schrödinger wave mechanics--quantum mechanics--and the theory of special relativity. Here is a summary of the situation as described
in a book Fundamentals of Quantum Mechanics by Persico:
The necessity for this refinement [to use relativistic mechanics rather than classical mechanics for the electron in a hydrogen atom] becomes evident when we consider that the results of Schrödinger
wave mechanics are not invariant under a Lorentz transformation.
Another fact which was partly neglected... is the existence of an intrinsic angular momentum (spin) and a magnetic moment both in the electron and in the proton... At first, an attempt was made to
deal separately with these two causes of inexactitude of quantum mechanics... Pauli succeeded in introducing the spin hypothesis into (nonrelativistic) quantum mechanics, constructing a remarkable
theory... But the most satisfactory solution of both these questions was found by Dirac who showed that the two modifications--the one concerning relativity and the one concerning the spinning
electron--are conceptually reduced to one and the same modification... When wave mechanics is given a suitable relativistic form, there follows the existence of the spin and of the magnetic moment,
with their correct values and rules... without the necessity of introducing them by an ad hoc hypothesis. From the Dirac theory we may then obtain the Pauli theory as a first nonrelativistic
approximation. [10]
From my point of view, the Schrödinger equation is a good representation for the stochastic atom in which the electron and proton are charged particles subject to their mutual electromagnetic
interaction and a Brownian motion due to the influence of the rest of the universe. Dirac was able to use relativity, which remember incorporates the finite speed of the interaction between the
electron and proton, and produce from Schrödinger's equation the idea of a spinning electron--and proton--with a magnetic moment that he could calculate. Where did it come from? It needs explanation
as far as I am concerned. A magnetic moment comes when a charge spins or moves around in a circle. Certainly in the atom the electron is moving about. Its motion is complex, but it can be resolved
into a number of basic motions. That is what I believe happens. There is one component that corresponds to a spinning motion, and one corresponding to an orbiting motion as well as the random motion.
If the interaction between the electron and proton were instantaneous, there would be no velocity dependent part to the interaction, but it is not instantaneous, so that the velocity dependent part
is apparent and is identified as a magnetic interaction. We get spin-orbit interaction and so on. The proton also is not standing still; it is jiggling about and creates a magnetic effect. It has a
The Pauli exclusion principle is used most often to try to understand atoms that are more complex than hydrogen. In a many-electron atom, the electrons repel each other in addition to being attracted
to the positive nucleus. We try to understand their behavior in terms of the solutions of the Schrödinger equation for the hydrogen atom. As I said, the Schrödinger equation has stable--non
time-dependent--solutions for the ground state and for a series of excited states. Each of these solutions has a set of integers associated with it. These integers are called quantum numbers. When
there are a number of electrons in an atom, we say that each one is associated with a hydrogen stable state. We assign quantum numbers to the electrons and Pauli's exclusion principle says that no
two electrons can have the same set of quantum numbers. This means that each electron is described by one of the stable excited state wave functions for the hydrogen atom. Here is Feynman:
... in a situation where there are many electrons, it turns out that they try to keep away from each other. If one electron is occupying a certain space, then another does not occupy the same space.
More precisely, there are two spin cases, so that two can sit on top of each other, one spinning one way and one the other way. But after that we cannot put any more there. We have to put others in
another place, and this is the real reason that matter has strength. [11]
For complex atoms, the probability functions for electrons are built up from hydrogen-like probability functions, and the reason for insisting that only one of a kind must be used is said to be that
electrons are all identical. Since, in fact, you cannot keep track of any one electron in order to say it has such and such a probability distribution--with a given set of quantum numbers--it is
really just a way of describing the behavior of all electrons present. From my point of view, there cannot be a precise one-to-one correspondence between excited state wave functions and electrons
because of the nodes. It is just that, as a group, the set of wave functions represents the set of electrons. Certainly, the Pauli principle has to do with electrons being identical. This fact--that
all electrons are identical should be explained. Hanson says this:
It might be objected: No two things are ever perfectly identical. Identical twins can be remarkably similar, but they can always be distinguished ultimately. Two postage stamps, fresh from the same
block, will be quite different in detail under a microscope. The finer the scale of observation, the more discrepancies will be found. What is the physicist claiming? That two particles of the same
kind are completely alike, with no possible difference between them whatever? Even were they created perfectly identical, could they remain thus? They 'collide' with their neighbours millions of
times a second. Would they not become deformed with all this pounding? [12]
I have already suggested that a fundamental particle may constantly be being renewed. That is why they do not become "deformed with all this pounding."
Dirac made the Schrödinger theory relativistic but in the 1950's work was done to show how the quantization of radiation--which I reject--causes certain deviations from Dirac's theory. The theory is
called quantum electrodynamics (Q.E.D.). In an article on The Concept of the Photon in 1972 Scully and Sargent indicate that Q.E.D. is necessary to explain certain things:
The quantized field is fundamentally required for accurate description of certain processes involving fluctuations in the electromagnetic field: for example, spontaneous emission, the Lamb shift, the
anomalous magnetic moment of the electron, and certain aspects of electromagnetic radiation... Perhaps the greatest triumph of the photon concept to the explanation of the Lamb shift between, for
example, the 2S[1/2] and 2P[1/2] levels in a hydrogenic atom. [13]
According to the relativistic Dirac theory these hydrogen levels have the same energy, in contradiction of the experimentally observed frequency splitting of 1057.8 MHz. We can understand the shift
intuitively by picturing the electron forced to fluctuate about its "Dirac" position because of the fluctuating vacuum field. The situation is clearly complicated and I do not pretend to be able to
disentangle it. But, there does seem to be room for ambiguity. Quantum electrodynamics is very difficult to understand and is certainly not part of a low-level course in Physics. Some Physicists
would swear by it; most do not understand it.
Must our explanation be quite so esoteric?
1. In an inertial frame, light--electromagnetic interaction--travels in straight lines by definition; light travels at a constant speed by definition. These define distance and time in any inertial
2. The fluctuating effect of the rest of the universe on a particle of matter is the same in all inertial frames. | {"url":"https://onbeyonddarwin.com/the-book/trapped-inside.html","timestamp":"2024-11-10T17:34:06Z","content_type":"application/xhtml+xml","content_length":"55872","record_id":"<urn:uuid:8ff0a3e3-14cb-4f1d-8a51-2d31a8ee6250>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00519.warc.gz"} |
B.Sc Part-II Mathematics Practical All Experiments - NKG Academy
B.Sc Part-II Mathematics Practical All Experiments
Syllabus Overview of Rajasthan University B.Sc PART-II Maths Practical
List of All Experiments
• EXP: A1 – Write a C program to display n terms of Fibonacci sequence/series. 0 1 2 3 5 8 …. N terms.
• EXP: A3 – Write a C program to Defining a function and finding sum of n terms of a series / sequence whose general term is given (E.g. a[n] = (n^2 + 3)/(n+1)).
• EXP: A5 – Write a C program to find the GCD and LCM of two number by Euclid’s algorithm. (GCD-Greatest Common Divisor & LCM = Lowest Common Multiple).
• EXP: A7 – Write a C program to finding numbers of prime less than n, n ∈ Z.
• EXP: A8.1 – Write a C program to finding mean and standard deviation.
• EXP: A8.2 – Write a C program to finding ^nP[r] ^nC[r] for different n and r.
• EXP: B1 – Write a C program to evaluate Numerical integration by trapezoidal rule.
• EXP: B2 – Write a C program to evaluate Numerical integration by Simpson’s one third (1/3) rule.
• EXP: B3 – Write a C program to evaluate Numerical integration by Simpson’s three eight (3/8) rule.
• EXP: B4 – Write a C program to evaluate Numerical integration by Waddle’s rules. | {"url":"https://nkgacademy.com/b-sc-part-ii-mathematics-practical-all-experiments/","timestamp":"2024-11-08T22:02:01Z","content_type":"text/html","content_length":"62814","record_id":"<urn:uuid:5036454e-f433-4c7f-b951-272b7618bc8a>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00643.warc.gz"} |
Negative probabilities: what are they for?
Theory Seminar
Negative probabilities: what are they for?
Yuri GurevichUniversity of Michigan
(Zoom seminar; please follow the event website link.)
Abstract: The title may sound nonsensical, but negative probabilities are profitably used in quantum physics and elsewhere. So what are negative probabilities? Formally, signed probabilities can be
defined as special signed measures.
But what is the intrinsic meaning of negative probabilities? We don’t know. The standard frequency-based interpretation of probabilities makes no sense for negative probabilities. There are attempts
in the literature to provide meaning for negative probabilities but, in our judgement, the problem is wide open.
Instead, we address a more pragmatic question: What are negative probabilities good for? It is not rare in science to use a concept without understanding its intrinsic meaning.
Consider early uses of complex numbers. The standard quantity-based interpretation of numbers makes no sense for imaginary numbers. And the intrinsic meaning of imaginary numbers wasn’t clear (and is
debatable even today). Yet complex numbers were profitably used to solve algebraic equations. “What are numbers and what are they for?” asked Richard Dedekind in 1888.
It turns out that the disparate quantum applications of negative probabilities can be seen as examples of a certain application template. Our first achievement is to make this template explicit. To
this end, we introduce observation spaces. An observation space S is a family of (usual) probability distributions P1, P2, … on a common sample space. A question arises whether there is a single
probability distribution P (called a grounding for S) which yields all P1, P2, … as marginal distributions. That P may be necessarily signed. Our second achievement is solving the grounding problem
for a number of observation spaces of note.
Greg Bodwin
Euiwoong Lee | {"url":"https://eecs.engin.umich.edu/event/negative-probabilities-what-are-they-for/","timestamp":"2024-11-08T06:20:10Z","content_type":"text/html","content_length":"61095","record_id":"<urn:uuid:2147948c-1389-4564-a9c2-75033a8c0111>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00668.warc.gz"} |
Waves, Sound and Light: Sound Waves
Sound Waves: Equation Overview
There are 23 ready-to-use problem sets on the topic of Sound Waves. The problems target your ability to determine wave quantities such as frequency, period, wavelength, intensity and speed from
verbal descriptions and diagrams of physical situations pertaining to sound waves and resonance in strings and air columns. Problems range in difficulty from the very easy and straight-forward to the
very difficult and complex.
The Speed of Sound
The speed of any wave (v) is defined as the distance traveled (d) per time of travel (t) and is described by the following equation:
v = d / t
Thus, the distance traveled by a wave is related to the time required for it to travel that distance.
The speed of sound, like the speed of any wave, is dependent upon the properties of the medium through which it is moving. For a sound wave moving through air, the primary property of air that
effects its speed is the temperature of the air. There are a couple of equations that describe the speed (v) - temperature (T) relationship; the following might be the easiest to remember:
v = 331 m/s + (0.6 m/s/°C) •T
where T is the Celsius temperature of the air through which the sound wave is moving. At 0°C, the speed of sound is 331 m/s. For every degree Celsius above 0°C, the speed of sound increases by
approximately 0.6 m/s. This equation provides a rather accurate estimate of the speed of sound for temperatures upwards towards 50°C.
Sound Intensity
Sound waves are produced by a vibrating object - vocal chords, guitar string, or diaphragm on a speaker. The sound waves begin at a point (or approximately a point) and then propagate through space
in three dimensions. As the sound wave propagates, it creates a wave front that fills the surface area of an ever-expanding sphere. A sound wave (like any wave) is often referred to as an
energy-transport phenomenon. The rate at which energy is put into the wave is referred to as the power of the source. Power (P) is expressed in units of Watts. As these vibrations propagate through
space, they become less intense as the size of the spherical wave front expands. Whatever energy is created by the wave at the source fills the surface area of a sphere some distance R away. Because
the sphere is constantly expanding, the energy is becoming diluted with increasing distance from the source. The sound intensity (I) at any given location is defined as the rate at which energy
arrives at that location. Because the wave energy is spreading over a surface area, intensity is often expressed in units of Watts/meter^2 and given by the equation
I = P / (4 • π • R^2)
You might recognize the 4•π•R^2 as being the surface area of a sphere having a radius of R.
The deciBel Scale
The human ear is sensitive enough to detect the smallest of vibrations. The lowest amplitude vibration which most humans can hear is defined as the threshold of hearing (TOH). While such a sound will
be different for different people, the intensity associated with this sound is defined as 1.0 x 10^-12 W/m^2. The range of sound intensities that a typical human can detect is enormous. Sound which
100 billion times more intense than the threshold of hearing (1 x 10^-1 W/m^2) is typically detectable without pain. Intensities beyond this level begin to cause pain and possibly the risk of hearing
loss. Because there is an enormous range of intensities from the threshold of hearing to the threshold of pain, a logarithmic scale - known as the decibel scale - is often used to express sound
intensity. This deciBel scale simply expresses the intensity level of any sound in terms of how many factors of 10 greater its intensity is compared to the threshold of hearing (1 x 10^-12 W/m^2). A
sound which is 10 times more intense than the TOH is 1 bel. A sound which is 100 (10^2) times more intense than the TOH is 2 bels. And a sound which is 1000 (10^3) times more intense than the TOH is
3 bels. More commonly, the sound level is expressed in the smaller unit decibel (1/10-th of a Bel), abbreviated dB. A sound rated at 1 bel is a 10 deciBel sound; a sound rated at 2 bel is a 20
deciBel sound; and so forth.
The deciBel level can be properly determined from the intensity (I) of a sound using the following equation:
dB = 10 • log ( I / 1.0 x 10^-12 W/m^2 )
Often times, the deciBel level is known and the intensity in W/m^2 is desired. In such instances the equation below can be used.
I = 1.0 x 10^-12 W/m^2 • 10^x where x = dB/10
Decibel levels decrease as the distance from the source of sound increases. This is due to the intensity-distance relationship discussed in the previous section. As the distance from the source
doubles, the intensity decreases by a factor of 4 (2^2). As distance from the source triples, the intensity decreases by a factor of 9 (3^2). And as distance from the source quadruples, the intensity
decreases by a factor of 16 (4^2). This is known as the inverse square relationship between intensity and distance. But since the deciBel level is not linearly related to the intensity, the same
inverse square relationship does not exist for deciBel level and distance. So if given the deciBel level at one location and asked to determine the deciBel level at another location, it is important
to approach the problem in three steps.
1. Calculate the intensity at the first location.
2. Apply the inverse square law to determine the intensity at the second location.
3. Calculate the deciBel level at the second location using the intensity value at that location.
(It is worth noting that there are short-cuts to this three step method that is dependent upon an undestanding of logarathmic relationships. For those confident with logarithms, we invite you to give
the short-cut a try ... but do make an effort to understand the physics of the inverse square law.)
The Doppler Effect
As a source of sound moves toward or away from an observer, the pitch of the sound is different than the actual source frequency. This phenomenon is known as the Doppler effect. As the source of
sound approaches you, there is an upward shift in pitch or frequency relative to the source frequency. And as the source of sound moves away from you, there is a downward shift in frequency relative
to the source frequency. This phenomenon is often noticed as an ambulance with its siren on passes by you as you are parked along the roadside. The pitch is higher as the ambulance approaches and
noticeably lower after it has passed and is moving away from you. The frequency (f') that is heard can be calculated by using the Doppler Shift equations.
f' = f / (1 ± v[s]/v)
where v = the speed of sound, v[s] = the speed of the source of sound, f = the frequency of the source, and f' = observed frequency. In the denominator of this equation, the minus sign is used for
situations in which the source is approaching the observer and the plus sign is used in situations in which the source is moving away from the observer.
The same phenomenon would be observed when the observer is approaching a stationary source or moving away from the stationary source. Once more, the frequency that is heard can be calculated by using
the Doppler Shift equations.
f' = (1 ± vo/v)•f
where v = the speed of sound, v[o] = the speed of the observer, f = the frequency of the source, and f' = observed frequency. In this equation, the plus sign is used for situations in which the
observer is approaching the source and the minus sign is used in situations in which the observer is moving away from the source.
Speed of Waves in Strings, Ropes, Wires and Cables
The speed of a wave depends upon the properties of the medium through which it is transmitted, and NOT upon the properties of the wave itself. For sound waves being transmitted through strings,
wires, ropes and cables, the primary properties effecting wave speed are the tension of the medium and the mass density of the medium. Tension pertains to the force with which the two ends of the
medium are pulled tight. Being a force, it is expressed with the unit Newton. Mass density pertains to the mass per unit length of the string, wire, rope or cable and is expressed in standard units
of kilogram/meter (kg/m). The equation expressing the relationship between these variables is
v = √(T/μ)
where v represents the wave speed, T represents the tension, and μ represents the mass density. When using these equations, it is important to pay attention to the units with which the given
quantities are expressed and to make appropriate conversions where necessary. It is recommended that substitutions be made into the equation using standard metric units. For speed, use m/s; for
tension, use Newtons (abbreviated N); and for mass density, use kg/m.
Frequency-Wavelength-Speed Relationship
Many of the questions in this unittarget your ability to analyze physical situations involving the wavelength-frequency-speed relationship for standing wave patterns in strings, wires, ropes, cables
and air columns. Any wave, whether a standing wave or a traveling wave, will have a wavelength-frequency-speed relationship which follows the equation:
v = f • λ
where v represents the speed (or velocity) of the wave, f represents the frequency of the wave, and λ represents the wavelength of the wave. As mentioned above, the speed of a wave is dependent upon
the properties of the medium. For strings, wires, ropes and cables, the properties of importance are the tension and mass density. For air columns, the property of importance is the temperature of
the air. As such, the properties of the medium are also related to the frequency and the wavelength of the wave.
Resonance in Vibrating Strings, Ropes, Wires and Cables
Waves introduced into a string, wire, rope or cable will typically travel the length of the medium and reflect back upon reaching its end. At certain frequencies, the reflected portion of the wave
meets up with the original wave to create a pattern known as a standing wave pattern. In a standing wave pattern there are points along the medium that appear as if they are always standing still.
These points are known as nodes and are easily remembered as the points of no desplacement (properly spelled as displacement). Separating the nodes are anti-nodes: points of maximum positive and
negative displacement. In such standing wave patterns, there is a unique half-number relationship between the length of the medium and the wavelength of the waves that have established the pattern
seen. These relationships are shown below for the standing wave patterns having one anti-node (first harmonic), two anti- nodes (second harmonic) and three anti- nodes (third harmonic).
It is clear from the above graphic that the length of the string, wire, rope or cable is related to the wavelength of the standing wave that is established within it. As such, the length of the
medium is related mathematically to the frequency of the wave and the speed of the waver (or the properties of the medium upon which wave speed depends).
A standing wave pattern such as those shown in the graphic above is established within a medium only when it is being disturbed at specific frequencies. Not any frequency will result in a standing
wave pattern; only the discrete frequencies that lead to wavelength values that are mathematically related to the length of the medium as illustrated by these equations. Such frequencies are known as
the harmonic frequencies of the string, wire, rope or cable. As observed in the graphic, the pattern associated with the second harmonic has a wavelength that is one-half the wavelength of the
pattern associated with the first harmonic. And the pattern associated with the third harmonic has a wavelength that is one-third the wavelength of the pattern associated with the first harmonic.
Continuing this same logic, one would reason, that the pattern associated with the fifth harmonic has a wavelength which is one-fifth the wavelength of the pattern associated with the first harmonic.
And in general, the pattern associated with the n^th harmonic has a wavelength that is 1 / n the wavelength of the pattern associated with the first harmonic. Each harmonic pattern - whether the
second, third, fifth or n^th - is characterized by a wavelength which is smaller than the wavelength of the first by a factor of n. In this case, n is known as the harmonic number.
λ[n] = λ[1] / n
The speeds of the waves for each of these harmonics is the same. Thus, the decrease in wavelength which results from a progression from the first to the third (and higher) harmonic must correspond to
an increase in the frequency by the same factor n. That is, the frequency of the second harmonic is two times the frequency of the first harmonic; the frequency of the third harmonic is three times
the frequency of the first harmonic; and the frequency of the n^th harmonic is n times the frequency of the first harmonic. Put in equation form, one could state that
f[n] = n • f[1]
where f[n] is the frequency of any harmonic pattern, n is the harmonic number associated with that pattern and f[1] is the frequency of the first harmonic. The frequency of the first harmonic (f[1])
is the fundamental frequency; it is the lowest possible frequency at which a standing wave could be established within the medium.
An Effective Problem-Solving Strategy for Resonating Strings
Solving problems relating to resonating strings targets a student's ability to relate the frequency, wavelength and speed of waves to properties of the string and to the length of the string. The
graphic below depicts the relationships between the various quantities in such problems. As is the usual case, when approaching a problem, first identify what you know and what you are trying to
find. Locate the knowns and unknowns on the graphic below and plot out a strategy that allows you to determine the unknown quantity. The strategy for solving for the unknown will be centered around
the relationships depicted in the graphic. The stated equations provide the mathematical expression of those relationships.
Resonance in Air Columns
As just mentioned, a standing wave can be established in a string when vibrating at one of its resonance frequencies. Similarly, a column of air can resonate as well, provided that another object
vibrating at one of the resonance frequencies forces the air into vibration. The resonance vibrations of an air column is the basis of many of the later problems in this unit. The air column is
either open at both ends (open-end air column) or open at one end and closed at the other (closed-end air column). A resonance situation in an open-end air column is characterized by the presence of
a vibrational anti-nodes at each of the open ends, creating the standing wave patterns shown below. Each pattern is referred to as a harmonic and has its own unique wavelength and frequency. As shown
in the graphic, there is a distinct relationship between the wavelength of the standing wave and the length of the air column. Knowing the pattern allows one to relate the length to the wavelength
and ultimately to the frequency and the speed.
A closed-end air column is open to the surrounding air at one end and closed at the other end. A resonance situation in a closed-end air column is characterized by the presence of a vibrational
anti-node at the open end and a vibrational node at the closed end, creating the standing wave patterns shown below. Again, there is a distinct relationship between the length of the air columns and
the wavelength for each of the harmonics. Knowing this length-wavelength relationship allows one to relate the length of an air column to the speed and the frequency at which the air inside naturally
An Effective Problem-Solving Strategy for Resonating Air Columns
A strategy for solving problems related to resonating strings was discussed above. A similar strategy can be applied in the approach to problems involving resonating air columns. Such problems target
a student's ability to relate the frequency, wavelength and speed of waves to properties of the air column (temperature) and to the length of the air column. The graphic below depicts the
relationships between the various quantities in such problems. As is the usual case, when approaching a problem, first identify what you know and what you are trying to find. Locate the knowns and
unknowns on the graphic below and plot out a strategy which allows you to determine the unknown quantity. The strategy for solving for the unknown will be centered around the relationships depicted
in the graphic. The stated equations provide the mathematical expression of those relationships.
Interference and Beats
When two sound sources of very similar yet different frequencies meet at an observer's ears the phenomenon of beats is observed. This phenomenon is perceived by an observer as a sound that fluctuates
in amplitude very rapidly over the course of time. For instance, if two tuning forks - one with a frequency of 256 Hz and the other with a frequency of 254 Hz - produce sound waves, then an observer
would hear a fluctuation in amplitude at a frequency of 2 Hz. This 2 Hz fluctuation in frequency is known as the beat frequency and is equivalent to the absolute value of the difference in frequency
between the two sources.
Beat Frequency = | f[2] - f[1] |
The fluctuations in amplitude which are observed is the result of the interference of the two waves. A low amplitude sound is observed when a compression from one source meets up with a rarefaction
from the other source. A high amplitude sound is observed when a compression from one source meets up with a compression from the other source (or a rarefaction with a rarefaction).
Habits of an Effective Problem-Solver
An effective problem solver by habit approaches a physics problem in a manner that reflects a collection of disciplined habits. While not every effective problem solver employs the same approach,
they all have habits which they share in common. These habits are described briefly here. An effective problem-solver...
• ...reads the problem carefully and develops a mental picture of the physical situation. If needed, they sketch a simple diagram of the physical situation to help visualize it.
• ...identifies the known and unknown quantities, often times recording them on the diagram itself. They equate given values to the symbols used to represent the corresponding quantity (e.g., v =
345 m/s, λ = 1.28 m, f = ???).
• ...plots a strategy for solving for the unknown quantity; the strategy will typically center around the use of physics equations and be heavily dependent upon an understanding of physics
• ...identifies the appropriate formula(s) to use, often times writing them down. Where needed, they perform the needed conversion of quantities into the proper unit.
• ...performs substitutions and algebraic manipulations in order to solve for the unknown quantity.
Additional Readings/Study Aids:
The following pages from The Physics Classroom Tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems. | {"url":"https://direct.physicsclassroom.com/calcpad/sound/Equation-Overview","timestamp":"2024-11-12T16:20:18Z","content_type":"application/xhtml+xml","content_length":"230933","record_id":"<urn:uuid:51861934-a801-40a0-90a0-2228da5cbab9>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00142.warc.gz"} |
Adaptive Internal Model Control of a DC Motor Drive System Using Dynamic Neural Network
Adaptive Internal Model Control of a DC Motor Drive System Using Dynamic Neural Network ()
1. Introduction
During recent decades, adaptive internal model control has been studied in several research works of which we mention [1-11]. It has been exploited in several industrial fields. It is usually
interesting for its performances in servo and control where systems to be controlled are dynamic, complex, finite dimensional, open-loop stable and in addition if they have numerous delays and
Another advantage of the structure of this control lies in its simple construction and easy interpretation of the roles of its building blocks. It includes an internal model which is an explicit
process model to be controlled, a controller which can be chosen the inverse of this model and, if necessary, robustness filters.
The modeling process is to find a model whose dynamic behavior of the process approach based either on theoretical analysis, either on an experimental analysis, or on theoretical and experimental
analysis. This model will be used to make predictions of the output of the process for learning the controller, and even to simulate the processes within the control system [12-16].
The inversion of model is one of the main problems of the approach of adaptive internal model control, since the direct inversion of the model for most physical systems provides an unrealizable
structure (systems characterized by a function of transfer with the order of numerator less than the order of the denominator, the systems of nonminimum phase systems with delay, etc.). In this
contextour work presents solutions that enable the design of a controller who comes to the best of the inverse model [17-23].
In the case of a model which is not perfect, a robust filter is useful to avoid destabilization of the command structure in the presence of modeling errors and/or major disturbances [24]. The
robustness filter is usually synthesized based on the Nyquist criterion [25].The method of synthesis of this filter is based on small gain theorem [26, 27]. By cons, this filter does not affect
system stability but slows down the system in the case of perfect model [28].
Thus neural network based control systems which have the desirable proprieties of nonlinear mapping, generalization and learning can offered as candidates for solution to high performance electrical
Our objective is to apply a neural network adaptive control scheme to a Drive Motor system. The proposed scheme can control the speed of the considered Motor Drive system to tack the reference speed
with fast and damped response.
The rest of the paper is organized as follows: In Section 2, we present a description discrete time models of the considered DC Motor Drive. The adaptive conventional internal model control scheme
application of the considered system is developed in Section 3. The Section 4, provide the neural adaptive internal model control scheme to our system. The stability analysis of these two adaptive
internal model control systems is developed in Section 5.A comparative study between these two control schemes is illustrated in Section 6 and a conclusion is drawn in Section 7.
2. Description of DC Motor
A DC motor can be used in a variety of industrial applications [29]. The type DC motor is characterized by the following equations [30]:
The electrical equation:
The mechanical equation:
The electromechanical coupling equations:
U(t) is the armature voltage (V), [m](t) the motor torque[r](t) the load torque^2), f the viscous friction coefficient (N·m·s·rad^–^1), ^–^1), ^–^1), K[m] is the back torque (V·s·rad^–^1).
From the above equations, the DC motor is schematically as follows (Figure 1):
with: Laplace transforms of
The transfer function between the input and output of the DC motor can be written as follows:
The study of the stability of the system can be made by the Routh criterion (or Routh-Hurwitz) which is to form the following table (Table 1).
The terms
From the Equation (5), the load torque can be consi- dered as a perturbation.
The discrete time model system can be calculated by
replacing in Equation (5)s by the approximate expression
Hence the expression of discrete system:
3. Adaptive Conventional Internal Model Control System of DC
The basic structure of the adaptive conventional internal model control with robustness filter of DC motor is shown in figure 2 [31].
The Figure 2 can be redrawn as follows (Figure 3).
For having
The robustness filter is often taken as first order:
Figure 2. Basic structure of the adaptive conventional internal model control of DC motor with robustness filter.
Figure 3. Structure of adaptive conventional internal model control with robustness filter in the feedback loop.
In our work, we fix
The disturbance affects both the output and system status. The model of DC motor which is of type ARMAX (Autoregressive moving average model with exogenous inputs model), is therefore given by the
following equation:
From Equation (12), we obtain:
The model of the motor is as follows:
The relations
This inverse model is used in adaptive conventional internal model control system replacing the
From the above equation, we can deduce:
Such as:
So we multiply the relation (20) by
For direct and inverse models are physically feasible, the parameters
It is possible to rewrite the Equations (14) and (19) under the following matrix forms:
In which the vectors of parameters and observations are defined by:
The vectors of observations written yet in the forms:
The number of parameters to adjust in this command structure
The adjustment procedure of the vector of model parameters
This algorithm which is part of the simple gradient methods can be expressed in terms of criterion (32) as follows:
According to [33-35], this procedure is given by the system of equations:
Initialization for the matrix
The model output can be written as follows:
The difference between the system output and the model output can be expressed by the Equation (45):
The input of the equalizer is given by the following equation:
From Equation (27), the vector of the controller parameters
The command applied to the DC motor is given by:
The difference between the reference signal and the system output is determined by:
4. Adaptive Neural Internal Model Control System of DC
The structures of internal model control using nonlinear neural networks have been proposed in [36-38]. The Figure 4 illustrates the structure of an adaptive neural internal model control of DC
motor, where the model and the controller are replaced by recurrent neural networks with multilayer internal and external closure.
The choice of the structures of neural networks takes into account the knowledge and assumptions about the behavior of the system [39-43]. The Figures 5 and 6 present respectively the architecture of
the model and the neuronal controller.
Figure 4. Basic structure of adaptive neural internal model control with filter robustness of DC motor.
Figure 6. Neural controller of the DC motor.
The two vectors of model parameters and the neural controller are respectively the following forms:
These two vectors are of size
In this section, the number of fitting parameters
The output of neural model of DC motor
Note by
The neural model is updated to minimize the error function
The adjustment of the online parameter vector
such as:
The update of parameters depends on the desired performance indices (accuracy, stability). She is stopped if the number of iterations is reached or
In our work, we fix
The neural model parameters
The matrix elements
I the identity matrix of size
If all conditions are met a good identification, the model output
The learning of neural controller is implemented so as to minimize the following quadratic criterion:
Using the Levenberg-Marquardt method, the vector of parameters of the controller
The online adjustment of the parameters must be repeated until such time as the number of iterations is reached or
The neural controller parameters
The elements of matrices
5. Stability Analysis
If the system to be controller is stable, and if its model is perfect, then the system controlled by the adaptive internal model control structure is stable if and only if the controller is stable
[1,28,52]. The parametric adaptation algorithm and the Levenberg-Marquardt algorithm are algorithms stable. The neural model, neural controller and the conventional model are stable. We can therefore
conclude that the control systems mentioned previously of the DC motor are stable.
6. Comparative Study
In our work, we will assume thatFigure 7).
The considered DC motor having the following characteristics:
The choice of the sampling period can have a dramatic effect on the results of the identification and control. If the sampling period is chosen large enough, this causes a bad description of the
dynamics of the system and can lead to failure. By cons, if the sampling period is low, then the values of some parameters may become very small, so it is difficult to estimate with good accuracy.
According to [53], the sampling period
Figure 7. DC motor parameters: (a) Load torque; (b) Back EMF; (c) Back torque.
The Figure 8 shows that:
Figure 8. Step response of DC motor: determined of the sampling period T[s].
We fix
The Figure 9 shows the sequence input and output used to calculate the model parameters. These sequences
Figure 9. Sequences Data: (a) Input sequence; (b) Output sequence.
represent the system response to a random signal of zero mean and variance 1.
From Table 2, the best model to correctly approach the system dynamics model is
We fix
To validate the model chosenFigure 10) and cross-correlation between input and residuals (Figure 11). We note that these two functions are almost within the confidence intervals, thus validating the
use of the resulting network as a model system studied.
The curves representing the evolution of the model parameters are plotted in Figure 12.
The Figures 13 and 14 represent the evolutions of zeros and poles of the model. The modules of the poles and zeros of the models are less than 1, thus validating the stability of this control system.
To build the neural model of the engine, we used the
Table 2. Nash criterion values of the different candidate models.
Figure 10. Auto-correlation function of residuals.
Figure 11. Intercorrelation function between input and residuals.
data sequences that are represented in Figure 9. The evolution of the Nash criterion of different candidate models of the DC motor (Table 3) leads to the conclusion that 5 neurons in the hidden layer
are necessary and sufficient for a neuronal model with satisfactory accuracy. It therefore sets
The autocorrelation functions of residuals and cross-
Figure 12. Evolution of model parameters.
Figure 13. Evolution of the zeros of the model.
Figure 14. Evolution of the poles of the model.
correlation between input and residuals (Figure 15) are within the confidence intervals, thus validating the use of the network chosen as the model system studied.
Table 3. Evolution of the Nash criterion of different neural models.
The performance of the adaptive conventional internal model control law applied to the DC motor is illustrated in Figures 16, 17 and 18. On the other side, the Figures 19, 20 and 21 show the results
of adaptive neural internal model control of the DC motor. By comparing the results of two control systems of the DC motor mentioned previously, we see clearly:
-A sudden change of system parameters implies a sudden change in the amplitude of the model output and the controlled system.
-The system output and the neural model output are almost identical, which confirms that the neural network is synthesized with more high precision than the output
Figure 15. Tests of model validation: (a) Autocorrelation function of residuals; (b) Intercorrelation function between input and output residuals.
Figure 16. Results of the adaptive conventional internal model control of DC motor in the case of a reference signal amplitude random uniform distribution: (a) Output system and output model; (b)
Evolution of the criterion ε; (c) Control signal applied to DC motor; (d) Response system; (e) Evolution of the criterion ε[1].
Figure 17. Results of adaptive conventional internal model control of DC motor in the case of a sinusoidal reference signal: (a) Output system and output model; (b) Evolution of the criterion ε; (c)
Control signal applied to DC motor; (d) Response system; (e) Evolution of the criterion ε[1].
Figure 18. Results of the adaptive conventional internal model control of DC motor in the case of a triangular: (a) Output system and output model; (b) Evolution of the criterion ε; (c) Control
signal applied to DC motor; (d) Response system; (e) Evolution of the criterion ε[1].
Figure 19. Performance of adaptive neural internal model control of the DC motor in the case of a reference signal amplitude random uniform distribution: (a) Output system and output model; (b)
Evolution of the criterion ε; (c) Control signal applied to DC motor; (d) Response system; (e) Evolution of the criterion ε[1].
Figure 20. Performance of adaptive neural internal model control of the DC motor in the case of a sinusoidal reference signal: (a) Output system and output model; (b) Evolution of the criterion ε;
(c) Control signal applied to DC motor; (d) Response system; (e) Evolution of the criterion ε[1].
Figure 21. Performance of adaptive neural internal model control of the DC motor in the case of a triangular reference signal reference signal: (a) Output system and output model; (b) Evolution of
the criterion ε; (c) Control signal applied to DC motor; (d) Response system; (e) Evolution of the criterion ε[1].
model obtained by the conventional identification method.
-The control signal shows fluctuations in the case of adaptive conventional internal model control. By cons in the case of adaptive neural internal model control, fluctuations are greatly diminished.
-The output of the adaptive neural internal model control system adequately follows the reference signal compared to the output of conventional adaptive internal model control system.
-The evolutions of the criterion
7. Conclusion
In this paper, two adaptive internal model control structures are used so that the speed of DC motor follows the given trajectories.
The comparative study showed the effectiveness of the adaptive neural internal model control system compared to conventional adaptive internal model control system. Indeed, it was found that the
adaptive control system by neural internal model meets the desired objectives: the powerful regulation and robustness of the speed, the disturbance rejection and system stability.
The model obtained from the estimation of its parameters is valid strictly used for the experiment. So check it is compatible with other forms of input in order to properly represent the system
operation to identify. Most static tests of model validation are based on the criterion of Nash, on the auto-correlation of residuals, based on cross-correlation between residues and other inputs to
the system. According to [54], the Nash criterion is given by the following equation:
In our work, the number of samples N is equal to 2251.
In [55,56], the correlation functions are:
-autocorrelation function of residuals:
-crosscorrelation function between the residuals and the previous entries:
Ideally, if the model is validated, the results of correlation tests and the Nash criterion following results:
Typically, we verify that
The calculation of partial derivatives is carried by the following equations:
-for the neuron in the output layer:
-for a neuron in the hidden layer:
The calculation of the coefficient matrix of the network D is conducted via the two following relations [39]:
If we denote:
We can write the equations (107) and (108) as follows:
The terms
We have then the following approximate relations:
The Jacobiens | {"url":"https://scirp.org/journal/paperinformation?paperid=18298","timestamp":"2024-11-15T04:51:35Z","content_type":"application/xhtml+xml","content_length":"226123","record_id":"<urn:uuid:c16f6b93-a4e0-4791-908f-6f1a4b437142>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00415.warc.gz"} |
The universal accidental (updated 8/8/16)
Just a friendly word of warning: labeling your own work a “breakthrough” is a good way to get yourself labeled as a crackpot. (-:
I just submitted the following to “The Journal of Mathematics and Music”.
“Using the traditional accidentals of western music theory, a musical space dubbed accidental space is introduced in three contexts. The first is as an algebra reminiscent of that used in quantum
mechanics; this version places special significance on palindromic modes such as Dorian. The second is as a network reminiscent of that used in graph theory; this version shows clear patterns
regarding chord quality clustering. The final is as a category with modes as objects and accidentals as morphisms; this version provides a singular context which encompasses both algebra and network.
I’m fairly certain this is a mathematical something, but I’m not a professional mathematician; I’m looking forward to being proven right or wrong by others outside of me.
Thanks Mike.
I am fully aware of my choice of language and the effect that it has on a science community that is used hearing about FTL and perpetual energy “breakthroughs”.
I submit to you however that public confidence in the scope of one’s work does not a crackpot make.
In the case of this paper:
I have found Feynman-esque diagrams in music…
I have created a self-consistent algebra using Dirac’s Bra-Ket notation and able to link for the first time in history AFAIK all 7-note musical scales into one self-consistent mathematical system…
I have found a way to categorize these observations in a way that give insight not only into my theory and music but also just how endemic the CT-POV really is…
I have made pretty pictures and harmonious sounds. :)
Further, a score of 5-8 on the Crackpot Index, though admittedly for physics, allows me to allude to a breakthrough without being labeled a crackpot by those that read my paper. Those that don’t read
my paper can and will say anything about it but those opinions don’t carry any weight of course.
A -5 point starting credit.
1 point for every statement that is widely agreed on to be false. NO
2 points for every statement that is clearly vacuous. NO
3 points for every statement that is logically inconsistent. TBD
5 points for each such statement that is adhered to despite careful correction. NO
5 points for using a thought experiment that contradicts the results of a widely accepted real experiment. NO
5 points for each word in all capital letters (except for those with defective keyboards). NO
5 points for each mention of “Einstien”, “Hawkins” or “Feynmann”. NONE
10 points for each claim that quantum mechanics is fundamentally misguided (without good evidence). NONE
10 points for pointing out that you have gone to school, as if this were evidence of sanity. NO
10 points for beginning the description of your theory by saying how long you have been working on it. (10 more for emphasizing that you worked on your own.) NOPE
10 points for mailing your theory to someone you don’t know personally and asking them not to tell anyone else about it, for fear that your ideas will be stolen. NEVER
10 points for offering prize money to anyone who proves and/or finds any flaws in your theory. NOPE
10 points for each new term you invent and use without properly defining it. ALL NEW TERMS DEFINED
10 points for each statement along the lines of “I’m not good at math, but my theory is conceptually right, so all I need is for someone to express it in terms of equations”. NOPE
10 points for arguing that a current well-established theory is “only a theory”, as if this were somehow a point against it. NOPE
10 points for arguing that while a current well-established theory predicts phenomena correctly, it doesn’t explain “why” they occur, or fails to provide a “mechanism”. NOT HERE
10 points for each favorable comparison of yourself to Einstein, or claim that special or general relativity are fundamentally misguided (without good evidence). NOPE
10 points for claiming that your work is on the cutting edge of a “paradigm shift”. YES
20 points for emailing me and complaining about the crackpot index. (E.g., saying that it “suppresses original thinkers” or saying that I misspelled “Einstein” in item 8.) NOPE
20 points for suggesting that you deserve a Nobel prize. NOPE
20 points for each favorable comparison of yourself to Newton or claim that classical mechanics is fundamentally misguided (without good evidence). NOPE
20 points for every use of science fiction works or myths as if they were fact. NOPE
20 points for defending yourself by bringing up (real or imagined) ridicule accorded to your past theories. NOPE
20 points for naming something after yourself. (E.g., talking about the “The Evans Field Equation” when your name happens to be Evans.) ANONYMOUS PAPER
20 points for talking about how great your theory is, but never actually explaining it. NOPE
20 points for each use of the phrase “hidebound reactionary”. LOL
20 points for each use of the phrase “self-appointed defender of the orthodoxy”. ROFLMAO
30 points for suggesting that a famous figure secretly disbelieved in a theory which he or she publicly supported. (E.g., that Feynman was a closet opponent of special relativity, as deduced by
reading between the lines in his freshman physics textbooks.) NOPE
30 points for suggesting that Einstein, in his later years, was groping his way towards the ideas you now advocate. NOPE
30 points for claiming that your theories were developed by an extraterrestrial civilization (without good evidence). NOPE
30 points for allusions to a delay in your work while you spent time in an asylum, or references to the psychiatrist who tried to talk you out of your theory. NOPE
40 points for comparing those who argue against your ideas to Nazis, stormtroopers, or brownshirts. NO
40 points for claiming that the “scientific establishment” is engaged in a “conspiracy” to prevent your work from gaining its well-deserved fame, or suchlike. NOPE
40 points for comparing yourself to Galileo, suggesting that a modern-day Inquisition is hard at work on your case, and so on. NOPE
40 points for claiming that when your theory is finally appreciated, present-day science will be seen for the sham it truly is. (30 more points for fantasizing about show trials in which scientists
who mocked your theories will be forced to recant.) NOPE
50 points for claiming you have a revolutionary theory but giving no concrete testable predictions. EVERYTHING IS TESTABLE
If you have the time and interest, Mike, I would of course welcome your observations on the paper and its validity.
After all, the true sign of a crackpot is not in the presentation of a work but in their defense of it, in their un-willingness to hear reason, to modify their POV in light of conflicting evidence.
let me prove to you my willingness to hear reason and change my POV; let me prove to you that my “crackpot ideas” have merit… ;)
Okay, I suggest that we stop here. We’ve had other people who want to use the nLab and nForum to advertise their work, and it’s usually turned out not too well for the advertiser, and sometimes badly
for everyone.
fastlane, you’ve now put your work out there so that if people are interested, they can correspond with you. Any further advertisement (such as bragging about a “breakthrough”) is very much unwanted
and discouraged.
I suggest that if someone else (whose credibility is known to the more established members of this enterprise – the Steering Committee would certainly be sufficient) feels that the material in the
paper warrants incorporation into the nLab, and the author is also agreeable to this, then we could proceed along such lines. But that someone needs to be someone other than the author.
Finally, let me give my opinion that a number of discussions recently have become a bit too prolix about matters not really germane to nLab and nForum business and enquiries, and that nForum
discussions work much, much better when all participants exercise self-discipline and make themselves maximally helpful in asking good questions and addressing the questions of others.
So, does anyone want to say anything about the paper instead of the choice of language in the OP?
The topic is irrefutably in line with nLab; do you judge me a “braggart” and “crackpot” before even reading the paper? ;)
“would certainly be sufficient) feels that the material in the paper warrants incorporation into the nLab, and the author is also agreeable to this, then we could proceed along such lines. But that
someone needs to be someone other than the author.”
I perhaps misunderstood the purpose of this “preprints and publications” forum.
I thought it was a place to share preprints and publications, not a staging area for possible inclusion into nLab which, while I would welcome, was not the intent, tone, or request in the OP.
If I have posted in error, I apologize.
All I want is to have professional mathematicians, of which I am not, look at my amateur work and see if what I think “breaks through” actually does…. that’s all I asked for and I’m sorry if I didn’t
make that clearer or if you read my “proven right or wrong” as a challenge and not the request for help and external mathematical validation that it actually is.
So, does anyone want to say anything about the paper …?
Yes, that should be the question of the thread. If anyone has anything to say on it, then I suppose they will say it.
Whether the paper is a breakthrough and in line with the nLab may now be left for others to judge, but repeated declarations from you that this is the case are really not helping your cause.
I thought it was a place to share preprints and publications
I think usually people have used this category to bring to attention the preprints and publications of others here that seem relevant to nLab interests. One can mention one’s own submissions and
publications here, certainly, but if you want to say anything more about it than the brief abstract, then it should not take the form of “I think this is a mathematical breakthrough”. Please don’t
get defensive about this; we are just trying to say that this is frowned upon here.
Problem solved: Breakthrough —> Something.
Now it is my hope that everyone on this thread will spend as much time on my paper as they have on my OP and maybe help me find out what that “something” is.
I’ve updated my categorical music research paper.
I would be interested in particular in your assessment if I used QUIVER correctly as it is a reference that I make directly from the n-lab.
Also, In section 3.2 I make ample use of Leinster’s 2014 “Basic Category Theory” to establish my objects and relationships as a category; critique in that area would be most useful and welcome.
A quiver is nothing more and nothing less than a directed graph, with loops allowed and allowing multiple edges all with the same source and target. Usually though when people use the word “quiver”,
they have in mind certain resonances having to do with representation theory. We used it in the nLab I think mostly because “directed graph” has different denotations for different groups of
mathematicians, whereas “quiver” at least has just one denotation, even if the connotations differ.
Figure 1 certainly has an underlying quiver. But if for your purposes there is an important distinction between solid arrows and dashed arrows (or if colors are important), then it looks like you’re
dealing not with a mere quiver, but a quiver with some extra structure (namely, the data that tells you which arrows are dotted or dashed, etc.).
There are multiple issues in how mathematical language is being used in section 3.2. For example, “category” is defined incorrectly. For one thing, it is simply not true that there must be one or
more morphisms going from and object $A$ to an object $B$. The word “relationship” is question-raising (what do you mean by a “relationship”?), and should be avoided. Saying that the morphisms
“commute” as part of the definition is “not even wrong” – as presented it’s close to meaningless (I’m sorry if that sounds harsh, but it’s true): what you want instead is to introduce a composition
function as part of the data.
As a general rule of thumb, when you define a mathematical concept, you introduce (1) sets of the sorts you will be discussing (e.g., a set $O$ of objects, a set $M$ of morphisms), (2) some structure
that connects these sets in some way, in the form of specified functions and relations (e.g., a function $id: O \to M$, functions $dom: M \to O$, $cod: M \to O$, etc.), and (3) some conditions which
the data introduced in (1) and (2) must satisfy (e.g., identity axiom, associativity axiom). If any of these steps is skipped or is unclear, then the definition will be incomplete and confusing. For
example, saying “morphisms commute” is, grammatically, of type (3), whereas you haven’t even introduced the composition data of type (2) for the reader to know what that phrase might be talking
about. (Try going back to Leinster’s book, and see if you can view his definition of category as fitting into this general scheme.)
There may be some interesting ideas in your paper, but I believe a lot of clean-up will be required to bring this up to mathematical standards.
Thanks a million for taking the time to comment. Thanks for your insight on quivers and the mis-use of “commute” in the definitions. I hope I can trouble you for a little extra time though….
I tried to follow Leinster p. 10, Definition 1.1.1 step by step in defining my category but you point out three concerns I’d like to address:
1) I’m at a lost as to how to better define my objects. I’ve listed them in table 1, I’ve given a bra-ket construction of these objects in section 1.1, a matrix definition in 1.5, and then I define
them again in section 3.3 when I discuss identities. Can you give me any advice on how I might more clearly define my objects?
2) Also, you say
it is simply not true that there must be one or more morphisms going from and object A A to an object B B.
but Leinster’s defintion say
for each A,B in ob( A A ), a collection A A (A,B) of maps or arrows or morphisms from A to B
How am I misreading this definition and/or how can I bring my definition and usage more in line with convention?
3) Finally, the statement that “all triangles and squares commute” is the topic of section 3.4. Was your original concern exclusively the mis-use of the word “commute” as part of the definition or
that the procedure outlining how “triangles and squares commute” in 3.4 is inadequate as well?
Thanks again!
The collection might be empty…
Okay, I hadn’t commented yet on “accidental categories”; so far I only commented on how you presented the general notion of category. Let me turn to how you present accidental categories.
One thing that sticks out is notation. You write ${|atural|}_A$ for (apparently) the identity morphism $A \to A$ on a mode $A$. Why the absolute value symbolism? (I’ll come back to this in a moment.)
And it seems you have for any pair of modes $A, B$ a flat “operator” (not morphism?) ${|\flat|}: A \to B$. Should that be ${|\flat|}_{A, B}: A \to B$, to avoid ambiguity? (You don’t want the same
symbol ${|\flat|}$ for two different morphisms $A \to B$, $C \to D$.)
Ordinarily, in category theory, the composite of $f: A \to B$ and $g: B \to C$ is written by a simple juxtaposition $g f: A \to C$ (or $g \circ f$). So ordinarily you would write the composite of ${|
atural|}_A: A \to A$ and ${|\flat|}_{A, B}: A \to B$ by the juxtaposition ${|\flat|}_{A, B}{|atural|}_A: A \to B$. You have instead ${|\flat atural_A|}$ which strictly speaking makes little sense, as
the bare notations $\flat$, $atural$ were never introduced to apply an absolute value to.
I think it would be simpler and notationally less confusing just to drop the absolute value notation, unless there is really a compelling reason for it. So you could write simply $\flat_{A, B}
atural_A$ for the composite. Similarly, if I’m parsing you write, you have a morphism $\sharp_{A, B}: B \to A$ for any pair of modes $A, B$.
“The accidental operators between modes A, B, and C commute; thus accidental operators compose naturally and universally.” What does that mean?? “Composing naturally and universally” is just not a
phrase in my lexicon. Does this mean (I’m dropping the absolute value notation) for example $\flat_{B, C} \flat_{A, B} = \flat_{A, C}: A \to C$? Do we have $\sharp_{A, B}\flat_{A, B} = atural_A$?
Relying on handwaving (and idiosyncratic) phrases like the above should be avoided. If you mean to convey equational axioms, then write those out precisely.
Can there be, in an accidental category, other morphisms besides the accidental ones? If so, would the natural morphisms $atural_A$ behave like identities with respect to other morphisms?
(By the way: I have just a rudimentary grasp of music theory. I am aware of things like the canonical modes and relationships in the circle of fifths and I know basic musical notation, e.g., for the
piano. Musically, I am a piano student, for about seven years now. So I don’t think your article would be totally beyond my ken, although I haven’t tried hard to grasp what you’re really driving at;
right now I’m commenting just at the formal level of mathematical presentation.)
As always, let me start off by thanking you for taking the time to review my work. I don’t have anybody with any categorical knowledge to discuss this work with and therefore your perspective is
overwhelmingly useful.
Why the absolute value symbolism?
This is a throwback to my physics days where operators are annotated with the absolute value sign to distinguish them from other objects in the theory. Thus |b3| is an operator but |b3> is a mode.
The other reason is that it does have an absolute value (a vacuum state more accurately) and thus it can actually be mapped to a real number. But your point is well met and I think it best if I just
drop the notation in text and keep the absolute value sign to reflect a mapping from an accidental to a real number and thus bring it more in line with math convention as you state. Let me know how
that reads to you or if you have another suggestion for how to conventionally annotate these symbols
And it seems you have for any pair of modes A,B A, B a flat “operator” (not morphism?)
I see no difference between operator, morphism, map, path, or arrow; they all take one object and turn it into the same or another The sense of the word is that Ionian is in one “shape” to which when
we apply the b7 operator (for example) and it changes “shape” to Myxolydian; hence the operator b7 is the morphism. Again, this is a throwback to my physics days and how we view and talk about
Should that be |♭|A,B:A→B {|\flat|}_{A, B}: A \to B, to avoid ambiguity? (You don’t want the same symbol |♭| {|\flat|} for two different morphisms A→B A \to B, C→D C \to D.)
This point is a critical part of my paper and personal understanding so I’m glad you brought this up. Let’s use figure one, the Major Scale, as a reference.
In terms of notation, the “juxtaposition” of operators you mention comes about because I think of them as paths. Thus the composite b3b7 reflects a path whereby you take a step in the b7 direction
first and another step in the b3 direction; hence a statement like b3b7 stands for a composite path. As for my use of subscripts, I’ll need to give that more thought. The same operator can connect
two different modes… so the b7 operation connecting Ionian and Myxolydian in figure one is the same operator connection Lydian and Lydian Dominant. Figure three shows this and other mapping
redundancies. As such, the b7 operation doesn’t carry any of these modes as a subscript. I’ll have to give this more thought though…
AFAIK, we have only one object, the mode, so we are dealing with a monoidal category… so really A -> A is the only morphism (what I call the free natural operator). In figure one we see this insofar
that every mode leads directly to another mode with a single arrow. Take away all color and identifying marks and then any transition between two modes looks the same. But the morphisms, the arrows,
are invertable and thus we are dealing with a group category. To me, this means that now there is a proper A-> B transition where A is in the “sharp” category and B is in the opposite “flat”
category. This create an identity based on this adjunction, what I call the identity natural operator.
The extra structure on the quiver that you mentioned in #12 comes, as best I an tell, from the Z12 parent group. We have six generators in the form of the six arrows in figure 1 (what I may be
mistakenly referring to as my quiver). These are the six generators (as I see it) of a free Z7 group. In effect, my belief is that there is a map from Z12 to Z7 that reflects a chromatic scale
(12-tone equal temperament) “symmetry breaking down” to a heptatonic (7-tone unequal temperament) scale. This map picks two intervals in Z12 to create a 2-generator free group; hence the major scale
uses two intervals from Z12 ($e^1$, a half-step H, and $e^2$, a whole-step W) to create the major scale as a two generator free group WWHWWWH. Thus my thesis that these six arrows comprise a
“universal map” from which any heptatonic scale can be built (which again encouraged the “quiver” designation as in a finite set of arrows).
Bear in mind that that the nature of the paper is I’m just discovering the nature of this structure and I see no evidence of this discovery or this structure in the literature. Thus I’m stumbling
about, not quite blindly, but certainly out of my ken as a physicist trying to mathematically explain these structures I see in musical notation.
“The accidental operators between modes A, B, and C commute; thus accidental operators compose naturally and universally.”
Yes, as pointed out in your last, this will be confusing and I’m re-writing this up to be more in line with Leinster’s definition and less “paraphrasing” on my part.
Can there be, in an accidental category, other morphisms besides the accidental ones?
As near as I can tell, the answer is “no”. Referring to a musical mode or scale by accidentals is necessary and sufficient (or so I claim) to map out all heptatonic scales and modes. There is no need
to introduce any other object or morphism to accomplish this. In effect, the Accidental Category reflects the fact that accidentals are the only morphism in this space.
If so, would the natural morphisms ♮A \natural_A behave like identities with respect to other morphisms?
HOWEVER, if we consider the mapping of an accidental to a real number, to a musical interval like a perfect fifth, then we can see the identity still behaves like an identity since it will map to
unison, the “do nothing” music interval consisting of zero half-steps. I “think” this is the correct answer though I get the sense that this mapping takes us out of the Accidental Category and into
the category of real numbers and thus is a functor; not sure how that affects my model to be honest.
although I haven’t tried hard to grasp what you’re really driving at; right now I’m commenting just at the formal level of mathematical presentation
Absolutely! What I’m presenting is a mathematical model derived from musical notation. In fact, you don’t have to know any music theory to appreciate my model since it relys entirely on structure,
not form. As such, to reference my work all you need is my table 1 that list the scales I use or an outside reference by way of a book of scales and modes (Here’s the one I used) and then see how my
structure put them together. As such, I would LOVE for you to understand the musical ramifications of what I propose but I am equally THANKFUL for your formal mathematical presentation… which makes
me more formal in turn!
Random thought which just entered my brain: is there a rough analogy between the operators $\flat$, $\sharp$ and ladder operators in physics?
Anyway, I’m glad you mentioned why that “absolute value” notation. If I were reading more carefully I might have picked up on that. So modes are like states and accidentals are like operators or
Random thought which just entered my brain: is there a rough analogy between the operators ♭ \flat, ♯ \sharp and ladder operators in physics?
From section 1.1:
“Accidental operators thus raise and lower scale degrees in our modal bra-kets much like ladder operators raise and lower quantum numbers in a physics bra-ket (Shankar, 1988, eqn. 12.5.3). “
So modes are like states and accidentals are like operators or observables?
From the introduction
“Chapter one casts Accidental Space as an algebra acting on the scale degrees of a heptatonic mode; here we get accustomed to thinking of modes as states and accidentals as operators”
The observables are given by the operator vacuum state as outlined in section 1.2 and reflect musical intervals we can hear (i.e., observe) like a perfect fifth or major second.
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Transactions Online
Xu ZHANG, Masatake AKUTAGAWA, Qinyu ZHANG, Hirofumi NAGASHINO, Rensheng CHE, Yohsuke KINOUCHI, "Measurement System of Jaw Movements by Using BP Neural Networks Method and a Nonlinear Least-Squares
Method" in IEICE TRANSACTIONS on Information, vol. E85-D, no. 12, pp. 1946-1954, December 2002, doi: .
Abstract: The jaw movements can be measured by estimating the position and orientation of two small permanent magnets attached on the upper and lower jaws. It is a difficult problem to estimate the
positions and orientations of the magnets from magnetic field because it is a typical inverse problem. The back propagation neural networks (BPNN) are applicable to solve this problem in short
processing time. But its precision is not enough to apply to practical measurement. In the other hand, precise estimation is possible by using the nonlinear least-square (NLS) method. However, it
takes long processing time for iterative calculation, and the solutions may be trapped in the local minima. In this paper, we propose a precise and fast measurement system which makes use of the
estimation algorithm combining BPNN with NLS method. In this method, the BPNN performs an approximate estimation of magnet parameters in short processing time, and its result is used as the initial
value of iterative calculation of NLS method. The cost function is solved by Gauss-Newton iteration algorithm. Precision, processing time and noise immunity were examined by computer simulations.
These results shows the proposed system has satisfactory ability to be applied to practical measurement.
URL: https://global.ieice.org/en_transactions/information/10.1587/e85-d_12_1946/_p
author={Xu ZHANG, Masatake AKUTAGAWA, Qinyu ZHANG, Hirofumi NAGASHINO, Rensheng CHE, Yohsuke KINOUCHI, },
journal={IEICE TRANSACTIONS on Information},
title={Measurement System of Jaw Movements by Using BP Neural Networks Method and a Nonlinear Least-Squares Method},
abstract={The jaw movements can be measured by estimating the position and orientation of two small permanent magnets attached on the upper and lower jaws. It is a difficult problem to estimate the
positions and orientations of the magnets from magnetic field because it is a typical inverse problem. The back propagation neural networks (BPNN) are applicable to solve this problem in short
processing time. But its precision is not enough to apply to practical measurement. In the other hand, precise estimation is possible by using the nonlinear least-square (NLS) method. However, it
takes long processing time for iterative calculation, and the solutions may be trapped in the local minima. In this paper, we propose a precise and fast measurement system which makes use of the
estimation algorithm combining BPNN with NLS method. In this method, the BPNN performs an approximate estimation of magnet parameters in short processing time, and its result is used as the initial
value of iterative calculation of NLS method. The cost function is solved by Gauss-Newton iteration algorithm. Precision, processing time and noise immunity were examined by computer simulations.
These results shows the proposed system has satisfactory ability to be applied to practical measurement.},
TY - JOUR
TI - Measurement System of Jaw Movements by Using BP Neural Networks Method and a Nonlinear Least-Squares Method
T2 - IEICE TRANSACTIONS on Information
SP - 1946
EP - 1954
AU - Xu ZHANG
AU - Masatake AKUTAGAWA
AU - Qinyu ZHANG
AU - Hirofumi NAGASHINO
AU - Rensheng CHE
AU - Yohsuke KINOUCHI
PY - 2002
DO -
JO - IEICE TRANSACTIONS on Information
SN -
VL - E85-D
IS - 12
JA - IEICE TRANSACTIONS on Information
Y1 - December 2002
AB - The jaw movements can be measured by estimating the position and orientation of two small permanent magnets attached on the upper and lower jaws. It is a difficult problem to estimate the
positions and orientations of the magnets from magnetic field because it is a typical inverse problem. The back propagation neural networks (BPNN) are applicable to solve this problem in short
processing time. But its precision is not enough to apply to practical measurement. In the other hand, precise estimation is possible by using the nonlinear least-square (NLS) method. However, it
takes long processing time for iterative calculation, and the solutions may be trapped in the local minima. In this paper, we propose a precise and fast measurement system which makes use of the
estimation algorithm combining BPNN with NLS method. In this method, the BPNN performs an approximate estimation of magnet parameters in short processing time, and its result is used as the initial
value of iterative calculation of NLS method. The cost function is solved by Gauss-Newton iteration algorithm. Precision, processing time and noise immunity were examined by computer simulations.
These results shows the proposed system has satisfactory ability to be applied to practical measurement.
ER - | {"url":"https://global.ieice.org/en_transactions/information/10.1587/e85-d_12_1946/_p","timestamp":"2024-11-03T02:38:25Z","content_type":"text/html","content_length":"64493","record_id":"<urn:uuid:7fee13e3-189b-4a2b-91db-3255f89b2b55>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00159.warc.gz"} |
Population Parameters Common Core Algebra 2 Homework Answers
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Department of Statistics
B.S. in Statistics
Major requirements
The major requires at least 31 credit hours (46 with the Addenda requirements), including the requirements listed below.
1. Core Statistics Courses.
1. Statistical Inference. One (1) course:
☆ STAT-S 320 Introduction to Statistics
☆ STAT-S 350 Introduction to Statistical Inference
2. Data Modeling and Inference. One (1) course:
☆ STAT-S 352 Data Modeling and Inference
3. Introduction to Statistical Theory. One (1) course:
☆ STAT-S 420 Introduction to Statistical Theory
4. Applied Linear Models I. One (1) course:
☆ STAT-S 431 Applied Linear Models I
5. Applied Linear Models II. One (1) course:
☆ STAT-S 432 Applied Linear Models II
6. Statistical Consulting. One (1) course:
☆ STAT-X 498 Statistical Consulting
2. Probability. One (1) course :
□ MATH-M 463 Introduction to Probability Theory I
□ MATH-S 463 Honors Course in Probability Theory I
3. Statistics Electives. Two (2) courses:
□ STAT-S 425 Nonparametric Theory and Data Analysis
□ STAT-S 426 Bayesian Theory and Data Analysis
□ STAT-S 437 Categorical Data Analysis
□ STAT-S 439 Multilevel Models
□ STAT-S 440 Multivariate Data Analysis
□ STAT-S 445 Covariance Structure Analysis
□ STAT-S 450 Time Series Analysis
□ STAT-S 455 Longitudinal Data Analysis
□ STAT-S 460 Sampling
□ STAT-S 470 Exploratory Data Analysis
□ STAT-S 475 Statistical Learning and High-Dimensional Data Analysis
□ STAT-S 481 Topics in Applied Statistics
□ STAT-S 482 Topics in Mathematical Statistics
4. Computer Proficiency. One (1) course:
□ CSCI-C 200 Introduction to Computers and Programming
5. Addenda Requirements*.
□ Calculus I. One (1) course:
☆ MATH-M 211 Calculus I
☆ MATH-S 211 Honors Calculus I
□ Calculus II. One (1) course:
☆ MATH-M 212 Calculus II
☆ MATH-S 212 Honors Calculus II
□ Calculus III. One (1) course:
☆ MATH-M 311 Calculus III
☆ MATH-S 311 Honors Course in Calculus III
□ Linear Algebra. One (1) course:
☆ MATH-M 301 Linear Algebra and Applications
☆ MATH-M 303 Linear Algebra for Undergraduates
☆ MATH-S 303 Honors Course in Linear Algebra
6. GPA, Minimum Grade, and Other Requirements. Each of the following:
1. Major Residency. At least 18 credit hours in the major must be completed in courses taken through the Indiana University Bloomington campus or an IU-administered or IU co-sponsored Overseas
Study program.
2. Major Upper Division Courses. At least 18 credit hours in the major must be completed at the 300–499 level.
3. Minimum Grade. Except for the GPA requirement, a grade of C- or higher is required for a course to count toward a requirement in the major.
4. Major GPA. A GPA of at least 2.000 for all courses taken in the major—including those where a grade lower than C- is earned—is required.
* Courses used to fulfill addenda requirements require a grade of C- or higher and do not count toward the Major GPA or Major Hours.
Bachelor of Science requirements
The Bachelor of Science degree requires at least 120 credit hours, to include the following:
1. College of Arts and Sciences Credit Hours. At least 100 credit hours must come from College of Arts and Sciences disciplines.
2. Upper Division Courses. At least 36 credit hours (of the 120) must be at the 300–499 level.
3. College Residency. Following completion of the 60^th credit hour toward degree, at least 36 credit hours of College of Arts and Sciences coursework must be completed through the Indiana
University Bloomington campus or an IU-administered or IU co-sponsored Overseas Study program.
4. College GPA. A cumulative grade point average (GPA) of at least 2.000 is required for all courses taken at Indiana University.
5. CASE Requirements. The following College of Arts and Sciences Education (CASE) requirements must be completed:
1. CASE Foundations
1. English Composition: 1 course
2. Mathematical Modeling: 1 course
2. CASE Breadth of Inquiry
1. Arts and Humanities: 3 courses
2. Natural and Mathematical Sciences: 4 courses
3. Social and Historical Studies: 3 courses
3. CASE Culture Studies
1. Diversity in the United States: 1 course
2. Global Civilizations and Cultures: Not required
4. CASE Critical Approaches: 1 course
5. CASE Foreign Language: Proficiency in a single foreign language through the first semester of the second year of college-level coursework
6. CASE Intensive Writing: 1 course
7. CASE Public Oral Communication: 1 course
6. Major. Completion of the major as outlined in the Major Requirements section above.
Most students must also successfully complete the Indiana University Bloomington General Education program. | {"url":"https://stat.indiana.edu/undergraduates/degrees/bs-degree.html","timestamp":"2024-11-02T23:50:36Z","content_type":"text/html","content_length":"40700","record_id":"<urn:uuid:0bcd8dae-183f-4bb4-b5e3-81e2a753a422>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00673.warc.gz"} |
Characteristic polynomial of symbolic matrix of size 7
Characteristic polynomial of symbolic matrix of size 7
There is a problem when computing the characteristic polynomial of a matrix of size greater than 7 containing a large number of symbolic variables.
a = SR.var('a', 100)
M = identity_matrix(SR, 7)
for i in range(7):
for j in range(7):
M[i,j] = a[i*7+j]
print(M.charpoly().degree()) # prints 5
The value it should print is 7. Over $\mathbb{Z}[a_0,a_1,\dots]$, the result is correct.
I use Sage 8.3 (Release Date: 2018-08-03), installed from the official repository of Archlinux. The bug is present both in command line and with sage file.sage (if I copy and paste the code above).
2 Answers
Sort by ยป oldest newest most voted
Hi, This should be fixed now with pynac 0.7.22-5
edit flag offensive delete link more
Thanks for the clarification !
tmonteil ( 2018-10-08 00:24:06 +0100 )edit
It works for me (Sage 8.4.beta7 complied on Debian stretch 64bit, run from the command line).
Could you please give us some informations so that someone can try to reproduce your problem:
• which version of Sage did you use ?
• which OS ?
• did you install Sage from the binaries, and which ones ?
• did you compile Sage yourself ?
• which notebook did you use (Sage notebook or jupyter notebook) ?
• did you use the command line ?
• which commands did you type precisely to get the error ?
• which error message did you get ?
• ... ?
EDIT thanks for reporting, it seems to be an issue with Archlinux port, since it works well on Sage Cell which also ships 8.3:
Second EDIT: this is now trac ticket 26427, i will contact Archlinux devs about that issue.
edit flag offensive delete link more
I added information in the original post. I tried to install the 8.4 beta7 version but it failed.
ScreenName ( 2018-10-07 07:03:05 +0100 )edit | {"url":"https://ask.sagemath.org/question/43839/characteristic-polynomial-of-symbolic-matrix-of-size-7/?sort=oldest","timestamp":"2024-11-05T23:29:17Z","content_type":"application/xhtml+xml","content_length":"64094","record_id":"<urn:uuid:b57c07d4-3de7-4d7b-8cd8-fa440c6d8640>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00204.warc.gz"} |
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Statistical Background and Definitions (2019)
Basal metabolic rate (BMR) and basal energy expenditure (BEE) are the same and resting metabolic rate (RMR) and resting energy expenditure (REE) are the same. However, there is a small difference
between basal and resting. The terms BMR and BEE are interchangeable, and the terms RMR and REE are interchangeable, but the terms basal and resting are not. BMR is the metabolic rate of the body at
absolute rest defined as fasted overnight and awake but measured within 30 minutes of awakening in the morning without movement prior to measurement. RMR is the metabolic rate of the body at relative
rest defined as fasted for at least 4 hours and awake, preferably but not necessarily in the morning and without the requirement for no movement in the hours prior to measurement. A rest recovery
period is required immediately before the test. A measurement of RMR allows for dressing and traveling to the test site whereas a BMR measurement usually involves an overnight stay at the test site.
BMR is slightly lower than RMR. In critically ill patients, the fasting requirement is waived because feedings are usually continuous rather than intermittent.
Total metabolic rate (TMR), also called total energy expenditure (TEE) is the sum total of all energy expended by the body in 24 hours (basal metabolic rate, thermic effect of feeding and movement/
Statistical Background
Bias and accuracy are qualitative terms whose definitions can vary slightly depending on their application. Most applications of these concepts relate to epidemiology or to the comparison of
measurement techniques against a gold standard. The application in the current review is to compare a mathematical calculation (not a measurement) to a measurement considered to be a gold standard.
As such, we need to define bias and accuracy as used in the current review.
Bias will be defined as a significant tendency for the predicted value to underestimate or overestimate the measured value. An unbiased predictive equation is one in which the differences between the
estimated and measured value does not trend in either direction. The preferred statistical definition of bias is either a 95% confidence interval of the difference between the two values to exclude
zero, or the mean difference between the two values to be significantly different from zero. A less preferred statistical definition is a simple statistically significant difference between the mean
predicted value and the mean measured value. When this is the only bias indicator available in the source papers, the prediction method will be labeled as probably biased.
Accuracy will be defined as the percentage of predicted values falling within ±10% of the measured value. Although this may be the most clinically relevant measure of validity, it is limited by the
fact that the threshold for declaring a prediction method valid is up to the individual user of the information. Another small limitation is that the selection of a 10% threshold is somewhat
arbitrary. However, given that day-to-day biological and instrument variation in measurements of RMR can be 3-5%, a 10% threshold for accuracy of a predictive equation seems reasonable, and in fact
this threshold has been used at least since the 1920’s in this type of validation work.
Of special note, Bland Altman statistics have become among the most common validity tests in this field of work and therefore should be explained. A Bland Altman plot compares the difference between
two quantities (on the y-axis) against the mean of the two quantities on the x-axis. The reason for comparing to the mean of the two quantities rather than the one quantity considered the gold
standard is that sometimes the gold standard is in error and the alternate quantity is true (this seems highly unlikely in the scenario of the current work in which the alternate quantity is a
calculation and not a measurement). Three pieces of information can be produced from a Bland Altman plot:
• Mean difference between the two quantities, which can be tested for being significantly different from zero and if so, be labeled fixed bias (underestimation or over estimation). Not all authors
report the mean difference and even fewer test the difference inferentially.
• Correlation of the difference between the two quantities against the mean of the two quantities. If there is a linear relationship present, proportional bias is said to exist. Again, not all
authors report if this correlation exists, but it is sometimes visually obvious in a Bland Altman plot.
• Calculation of the value 1.96 standard deviations above the mean difference and 1.96 standard deviations below the mean difference. This is an indicator of the degree of variation around the
means of the two quantities and is labeled Limit of Agreement (LOA). The LOA should capture 95% of all the differences observed between the two quantities being compared, and the lower the LOA
the better. A limitation of the LOA is that an inferential test statistic does not exist to determine if the LOA is within an acceptable range. This is completely a judgement on the part of the
reader. The standard applied for this project for an acceptable LOA is 25% of mean RMR. This range is based on two studies of predictive equations in healthy people.
Ludbrook J. Comparing methods of measurement. Clin Exp Pharm Physiol 1997; 24: 193-203.
Walther BA, Moore JL. The concepts of bias, precision, and accuracy, and their use in testing the performance of species richness estimators, with a literature review of estimator performance.
Ecography. 2005; 28: 815-829.
Boothby WM, Sandiford I. Summary of the basal metabolism data on 8,614 subjects with especial reference to the normal standards for the estimation of the basal metabolic rate. J Biol Chem 1922; 54:
Krouwer JS. Why Bland Altman plots should use X, not (Y+X)/2 when X is a reference method. Stat Med. 2008; 27: 778-780.
Weijs PJM. Validity of predictive equations for resting energy expenditure in US and Dutch overweight and obese class I and II adults aged 18-65 y. Am J Clin Nutr. 2008; 88: 959-970.
Frankenfield DC. Bias and accuracy of resting metabolic rate equations in non-obese and obese adults. Clin Nutr. 2013; 32: 976-982. | {"url":"https://www.andeal.org/topic.cfm?menu=5658&pcat=5659&cat=5981","timestamp":"2024-11-12T18:32:04Z","content_type":"text/html","content_length":"37629","record_id":"<urn:uuid:ed2f8ce8-c808-441d-9ed5-6c24413474e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00715.warc.gz"} |
Is 7 a factor of 49?
Here we will show you two different methods you can use to determine if 7 is a factor of 49.
The first method entails simply listing all factors of 49 and then seeing if 7 is one of them. The factors of 49 are 1, 7, and 49. Looking at the list, you see that 7 is on the list, and the answer
to "Is 7 a factor of 49?" is therefore yes.
For the second and perhaps easier method, we divide 49 by 7 to see if the quotient is a whole number or fractional number. If the quotient is a whole number, then 7 is a factor of 49. If the quotient
is a fractional number, then 7 is not a factor of 49.
49 divided by 7 is 7 which is a whole number. Thus, once again, the answer to "Is 7 a factor of 49?" is yes.
Factor Checker
Need to do another factor check? No problem! You are welcome to submit a similar problem below.
Is 7 a factor of 50?
Now you know that 7 is a factor of 49. Here is the next problem on our list that our Factor Checker explained and solved.
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Probability for Pragmatists
Probability for Pragmatists
What does it mean that a probability is “correct” and how could you possibly know?
Photo by Nick Fewings on Unsplash
Mathematical modelling of uncertainty stands or falls on our ability to assess probabilities, but how do you know if your assessments are any good and what does that even mean? This article takes a
pragmatic approach to answering these questions. We’ll start in the shadows of the valley of frequentism, step out on to the Bayesian foothills of subjectivism and stride on to the summits of
pragmatism and an objective form of the Bayesian view.
From this standpoint, we will see what can go wrong with probabilistic assessments and discuss systematic biases. Finally, the pragmatist interpretation will bring us to a methodology that not only
reveals bias, but also makes it clear how well supported the inference of bias is in the data, by showing an uncertainty range that reflects both consistency of bias and the number of data used to
reveal it.
To give a sense of the problem, say we assessed at 20% the probability that Stochastic Steel will win the contract they’re negotiating with Bigger Ball Better Ball Ball-Bearings. They win the
contract. Was that assessment correct? What does that even mean?
Frequentists believe in objectively correct probabilities, but their only access to these objectively correct probabilities is as frequencies in large (infinite) numbers of repeated identical
experiments. They say things like if Stochastic Steel negotiate that contract an infinite number of times then the average number of times they win the contract is the correct probability. But we all
know that how ever many times we negotiate the same contract under the same conditions, we either win every time or never at all. If we ever repeat negotiations, it certainly isn’t under identical
circumstances. There are workarounds (involving ensembles of possible worlds fixed with respect to the information we have, but variable with respect to everything else), but there’s something very
unsatisfactory about an utterly inaccessible notion of probability.
Subjective Bayesianism
Bayesianism is the natural foil to frequentism. Bayesians start with the notion that probability is simply quantified belief and then work out coherent ways to modify beliefs in the light of data.
This is Bayes’ theorem, preached in its purest form by the high priest of Bayesianism, E.T. Jaynes. The problem for many Bayesians (though not Jaynes) is that while we know how to update beliefs,
it’s not clear where our belief should start before we bring any data to bear. One way out is to give up on objectively correct probabilities and claim it is legitimate to assess that “prior” belief
While it’s certainly true that we can elicit subjective degrees of belief, as a normative theory, I find a subjective notion of probability almost as unsettling as an inaccessible one. It’s a
terrible licence for all sorts of silliness. Jaynes seeks (and finds) answers in a coherent, objective description of pure ignorance. I will here come at it from the other end and argue that a
pragmatic approach to our final assessments of probability brings us, like Jaynes, to a working notion of objective Bayesianism.
Pragmatism is a philosophical tradition that started in the late 19th century and continues to this day. It is summed up beautifully in this quote from Charles Peirce, one of its founding members
Consider the practical effects of the objects of your conception.
Then your conception of those effects is the whole of your conception of the object
What Peirce is saying here is don’t get too hung up on what objective probability is, think about how you might use it. Think about its practical effects.
The problem with my 20% chance of success is that the outcome is too uncertain: 20%, 4:1 against. Not great great odds, but still more likely than getting out of jail in one throw in monopoly. So
even if the 20% is correct, it doesn’t really have any practical effect. To be able to say something a bit more practical, I need to reduce this uncertainty.
Aggregation to reduce uncertainty
If I have a list of probabilities and outcomes — all the contracts Stochastic Steel negotiated in the last quarter, for example — I can look at the total number of wins, which will reduce the
uncertainty a bit.
Assuming a notion of correct probability, I can ask myself if my probabilities were all “correct” what would the probability distribution for the total number of wins look like?
If the probabilities are all the same, this is just a binomial distribution, if not then I explain how to do this (and provide an Excel spreadsheet with the methods implemented) in my article on
auditing probability sequences. The result looks something like the figure here, where I have assessed twenty contract negotiations.
On the horizontal axis here, we have the number of successful wins and vertically we have the probability of getting that number, calculated directly and with an approximation.
Let’s say we we came in on the low side of our prediction with three wins in the quarter.
A frequentist would say, “the hypothesis that the probabilities are correct is accepted at 90% confidence, but not at 95%”. This is bonkers. No one ever imagined for a second that every single
assessment in that list was perfectly accurate, so the null hypothesis whose truth we are trying to establish at some level of confidence starts with a probability that is both practically and
theoretically zero. But it’s nonetheless “not refuted” with a particular level of confidence. Happy days. As you were. All is well.
Lucky that there were three and not just two wins, because then we’d have been refuted at 90%. Sad feelings. Refuted. Switch the lights out on the way out
Still, at least frequentists are trying. The subjective Bayesians just quit on objectively correct. They’ve gone fishing.
Can pragmatism do better? Can a pragmatic interpretation of probability help us understand whether that low result was bad judgement or just bad luck?
Assuming, as we are, that there is such a thing as an objectively “correct” probability, but for whatever reason — congenital optimism, human heuristics, mathematical ineptitude — I consistently and
systematically get it wrong.
Interestingly, such systematic biases are the only things we can hope to capture in this kind of analysis. Random errors — going a bit high here, a bit low there, will tend to cancel out in the
aggregation we do to reduce the uncertainty. This is the price we pay for aggregation — we reduce uncertainty, but we give up being able to say anything about individual assessments.
A taxonomy of systematic bias
It turns out that because of the ways random variables behave when you add them up, there are only four kinds of systematic bias we can catch.
• Optimism: Consistently pitching probabilities too high
• Pessimism: Consistently pitching probabilities too low
• Polarization: setting probabilities that are a little higher than average much higher than average and probabilities that are a bit lower than average way lower than average— if it’s good it’s
very very good, if it’s bad it’s horrid.
• Vagueness: Moving probabilities that are removed from average back towards the average — fence-sitting.
Polarization is a reflection of over-confidence in your ability to assess, or reading too much in to the data. Geologists are terrible at this. Vagueness is rarer, but you do see it, though often as
an overcorrection to polarization or because people are gaming their probabilities to get good results on average.
Modelling bias
The pragmatic notion of at least the theoretical existence of a correct probability allows us to build a model for what goes wrong with assessments of probabilities. The basic idea is to use Bayes’
theorem, which as I mentioned tells you how to change probabilities in the light of data or evidence, to alter the true probability on the basis of spurious data. In the case of optimism and
pessimism, these spurious data are data supporting or undermining the positive outcome. In the case of polarization and vagueness, they are additional data magnifying the available data that have
moved the probability away from its starting point.
It turns out that optimism and pessimism can be captured in a single number, which is positive for optimism and negative for pessimism; and polarization and vagueness can be captured in a single
number, which is positive for polarization and negative for vagueness. For the mathematically initiated, bias is modelled simply as a linear transformation of evidence, as I discussed in my article
last week. See the endnote below.
The figures above show how true probabilities (red line) are lifted by optimism (left figure, solid lines) and depressed by pessimism (left figure, dashed lines). Polarization (right figure, solid
lines) pushes low probabilities lower and high probabilities higher. Vagueness (right figure, dashed lines) brings probabilities back towards an uninformative 50:50.
So now we have a concept of objective, correct probabilities and we have an elegant little two-parameter model for the effects of bias. How do we put these things together to get something useful
that also explains what we might mean by objective probability?
The pragmatists “practical effects” of precise probability.
The crucial step is to realize that the concept of objectively correct probabilities allows us to treat our outcomes as samples of the probability distributions described by those correct
probabilities. That, in turn, allows us to set up what is essentially a regression problem to work out the bias parameters — those two numbers that describe all the possible forms of bias.
Here’s how it works. We start with our list of biased assessments and outcomes and we wash the biassed assessments backwards through our bias model for some choice of bias parameters to get our
objective, putatively correct probabilities. We then compare the outcome predicted by these with the actual outcomes and measure how well that choice of bias parameters fit the data.
There are two ways of closing the loop. The first just says, OK, choose the bias parameters that give the best fit between prediction and outcome. So here I’ve plotted the mean squared deviation
between predictions and outcomes (sometime known as a Brier score). The point where it’s lowest is my regression-fitted bias parameter.
The horizontal axis is the optimism / pessimism parameter and the vertical axis is the vagueness / polarization parameter. So we can see our data our best explained by a modest optimism, flavoured
with a squeeze of polarization.
Another, I think, better way of looking at this is to ask, what is the probability of seeing the data we see for a range of choices of bias parameter. This is a likelihood function — it gives the
maximum likelihood estimator, which isn’t massively different from the least squares estimate above, but which comes equipped with an uncertainty distribution, which gives a very real sense of how
well we have nailed that parameter.
Defining probability in terms of what you do with it and leaping lightly over the ontological bog has proven remarkably fertile. It led us first to the recognition that there are only four systematic
biases we can hope to extract from an analysis of probabilistic prediction, and we were able to develop a forward model for these. This in turn allowed us to invert back to the parameters in the
model and answer the question we asked ourselves in the first place: are these probabilities any good?
Of course the answer to that is never better than “probably”, at least for any reasonably sized set of outcomes, but at least now we have a coherent sense of the uncertainty that circumscribes that
Mathematical endnote
We first transform the true probability to an evidence value, as discussed in my earlier article
Then the bias transform just utilizes that the impact of data (spurious or otherwise) in evidence space is linear. So it’s simply a linear transform
Systematic optimism is a constant addition of spurious supporting data (apositive), pessimism a constant addition of spurious detracting data (a negative). When b is positive, events that are more
likely than not are increased in their relative likelihood and the converse is true for b negative. Thus b captures, respectively for positive and negative values, polarization and vagueness.
Finally, we transform back to get back to probability | {"url":"https://www.cantorsparadise.org/probability-for-pragmatists-c781d4e68c53/","timestamp":"2024-11-05T09:19:51Z","content_type":"text/html","content_length":"44977","record_id":"<urn:uuid:c1389bb8-7d6d-4dd9-ab3e-b57373ff2b9b>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00035.warc.gz"} |
Maharashtra Board Class 7 Science Solutions Chapter 7 Motion, Force and Work
Balbharti Maharashtra State Board Class 7 Science Solutions Chapter 7 Motion, Force and Work Notes, Textbook Exercise Important Questions and Answers.
Maharashtra State Board Class 7 Science Solutions Chapter 7 Motion, Force and Work
Class 7 Science Chapter 7 Motion, Force and Work Textbook Questions and Answers
1. Fill ¡n the blanks with the proper words from the brackets.
(stationary, zero, changing, constant, displacement, velocity, speed. acceleration, stationary but not zero. inc reuses)
Question a.
If a body traverses a distance in direct proportion to the time, the speed of the body is ……………… .
Question b.
If a body is moving with a constant velocity, its acceleration is ……………… .
Question c.
……………. is a scalar quantity.
Question d.
…………….. is the distance traversed by a body in a particular direction in unit time.
2. Observe the figure and answer the questions.
Sachin and Sarneer started on a motorbike from place A, took the turn at 13, did a task at C, travelled by the route CD to D and then went on to E. Altogether, they took one hour for this journey.
Find out the actual distance traversed by them and the displacement from A to E. From this, deduce their speed. What was their velocity from A to E in the direction AE’? Can this velocity be called
average velocity?
Question a.
Observe the figure and answer the questions
Sachin and Sarneer started on a motorbike from place A, took the turn at 13, did a task at C, travelled by the route CD to D and then went on to E. Altogether, they took one hour for this journey.
Find out the actual distance traversed by them and the displacement from A to E. From this, deduce their speed. What was their velocity from A to E in the direction AE’? Can this velocity be called
average velocity?
1. Actual distance = \(\overline{\mathrm{AB}}\) + \(\overline{\mathrm{BC}}\) + \(\overline{\mathrm{CD}}\) + \(\overline{\mathrm{DE}}\) = 3 + 4 + 5 + 3
Actual distance = 15 km
2. Displacement = \(\overline{\mathrm{AB}}\) + \(\overline{\mathrm{BD}}\) + \(\overline{\mathrm{DE}}\)
= 3 + 3 + 3
Displacement = 9 km
3. Speed = \(\frac{\text { Distance travelled }}{\text { Total time }}\)
Distance = 15 km = 15 × 1000 = 15000 m
Time = 1 hr = 1 × 60 × 60 = 3600 sec.
s = \(\frac{15000}{3600}\) or s = \(\frac{15 \mathrm{~km}}{1 \text { hour }}\) = 15km/hour
= 4.16 m/sec. or 15 km/hour
4. Velocity = \(\frac{\text { Distance travelled }}{\text { Total time }}\)
Displacement = 9 km = 9 × 1000 = 9000 m
Time = 1 hr = 1 × 60 × 60 = 3600 sec
V = \(\frac{9000}{3600}\) or V = \(\frac{9 \mathrm{~km}}{1 \text { hour }}\) = 9 km/hour
= 2.5 m/sec. or 9 km/hour
5. Yes, this velocity can be called as average velocity.
3. From the groups B and C, choose the proper words, for each of the words in group A.
Question a.
From the groups B and C, choose the proper words, for each of the words in group A.
┃Group ‘A’ │Group B’│Group ‘C’ ┃
┃Work │Joule │erg ┃
┃Force │Newton │dyne ┃
┃Displacement │Metre │cm ┃
4. A bird sitting on a wire, flies, circles around and comes back to its perch. Explain the total distance it traversed during its flight and its eventual displacement.
Question a.
The total distance the bird has traversed is the length of the distance covered by circling, but the eventual displacement are the bird is zero as its initial and final position are one and the same.
5. Explain the following concepts in your own words with everyday examples: force, work, displacement, velocity, acceleration, distance.
Question a.
Explain the following concepts in your own words with everyday examples: force, work, displacement, velocity, acceleration, distance.
1. Force: The interaction that brings about the acceleration is called force.
e.g: An ox is pulling a cart, applying brakes to a bicycle, lifting heavy iron object with a crane.
2. Work: When an object is displaced by applying a force on it, work is said to be done.
e.g: A bucketful of water is to be drawn from a well and taken to the home by walking from well to home.
3. Displacement: The minimum distance
traversed by a moving body in one direction from the original point to reach the final point is called displacement.
e.g: A rolling of a ball from point A to point B in the same direction.
4. Velocity: Velocity is the distance traversed by a body in a specific direction in unit time.
e.g: A truck is covering a distance of 40km from A to D in a straight line in 1 hour.
5. Acceleration: It is change in velocity per second. It can be deduced.
Acceleration = \(\frac{\text { Change in velocity }}{\text { Time taken for change }}\)
(i) In the above example a truck covered the distance AB at velocity of 60 km/hr, BC at 30 km/hr and CD at 40 km/hr. (ii) It means that the velocity for the distance CD is greater than the velocity
for the distance BC. (iii) From the number of seconds required for this change in velocity to take place, the change in velocity per second can be deduced. This is called acceleration (iv) Distance:
The length of the route actually traversed by a moving body irrespective of the direction is called distance.
e.g: Ranjit travelled 1km. from his home to school.
6. A ball is rolling from A to D on a flat and smooth surface. Its speed is 2 cm/s. On reaching B, it was pushed continuously up to C. On reaching D from C, its speed had become 4 cm/s. It took 2
seconds for it to go from B to C. What is the acceleration of the ball as it goes from B to C.
Question a.
A ball is rolling from A to D on a flat and smooth surface. Its speed is 2 cm/s. On reaching B, it was pushed continuously up to C. On reaching D from C, its speed had become 4 cm/s. It took 2
seconds for it to go from B to C. What is the acceleration of the ball as it goes from B to C.
As its initial and final positions are one and the same.
Initial Velocity = 2 cm/s.
Final Velocity = 4 cm/s
Time taken for the change in velocity from B to
D = 4 cm/s – 2 cm/s = 2 cm/s
7. Solve the following problems.
Question a.
A force of 1000 N was applied to stop a car that was moving with a constant velocity. The car stopped after moving through 10m. How much is the work done?
Force (F) = 1000 N
displacement (s) = 10m
work done (W) = ?
W = Fs
= 1000 × 10
W = 10,000 Joule
Question b.
A cart with mass 20 kg went 50 m in a straight line on a plain and smooth road when a force of 2 N was applied to it. How much work was done by the force?
Force (F) = 2 N
Displacement (s) = 50 m
Work done (W) = ?
W = Fs
= 2 × 50
W = 100 Joule
Question a.
Collect information about the study made by Sir Isaac Newton regarding force and acceleration and discuss it with your teacher.
Class 7 Science Chapter 7 Motion, Force and Work Important Questions and Answers
Fill in blanks:
Question 1.
Displacement is a …………. quantity.
Question 2.
The …………. of an object can change even while it is moving along a straight line.
Question 3.
The …………. velocity can be different at different times.
Question 4.
Change in velocity per second is called …………. .
Question 5.
The interaction that brings about the acceleration is called …………. .
Question 6.
The scientist …………. was the first to study force and the resulting acceleration.
Sir Isaac Nezvton
Question 7.
Ability to do work is called …………. .
Question 8.
W = …………. × S.
Question 9.
Unit of work is …………. and …………. .
Joule, erg
Question 10.
Unit of force is …………. and …………. .
Newton, dyne
Question 11.
Force is a …………. quantity.
Question 12.
The velocity at a particular time is called …………. velocity.
Question 13.
The …………. of a body is the distance traversed per unit time.
Question 14.
Unit of acceleration is …………. and …………. .
m/s^2 and cm/s^2
Question 15.
Force is measured by the …………. that it produces.
Question 16.
Work done by a body with no displacement will be …………. .
Say whether True or False, correct the false 1 statements:
Question 1.
Velocity is distance travelled per unit of time.
False. Speed is distance travelled per unit of time
Question 2.
In displacement, both distance and direction are taken into account.
Question 3.
Speed = Distance/time.
Question 4.
Change in speed per second is acceleration.
False. Change in velocity per second is acceleration
Question 5.
Work done depends on the force and the displacement.
Question 6.
C.G.S. unit of acceleration is m/s^2.
False. C.G.S. unit of acceleration is cm/s^2.
Question 7.
M.K.S. unit of force is dyne.
False. M.K.S. unit of force is Newton
Question 8.
Force is measured by the acceleration that it produces.
Write the difference between the following:
Question 1.
Speed and Velocity
┃Speed │Velocity ┃
┃1. Speed is distance travelled per unit of time. │1. Velocity is the distance traversed by a body in a specific direction in unit time.┃
┃2. It is a scalar quantity. │2. It is a vector quantity. ┃
┃3. Formula: │3. Formula: ┃
┃Speed = \(\frac{\text { Distance traversed }}{\text { Total time }}\)│Velocity = \(\frac{\text { Displacement }}{\text { Total time }}\) ┃
Question 2.
Distance and Displacement
┃Distance │Displacement ┃
┃1. The length of the route actually traversed by a moving body, irrespective of the │1. The minimum distance traversed by a moving body in one direction from the original point to reach the ┃
┃direction is called distance. │final point is called displacement. ┃
┃2. It is a scalar quantity. │2. It is a vector quantity. ┃
Solve the following problems!
Question 1.
A bus travelled 200 km in the first 3 hours and then 100 kms for the next one and a half hours and then 120 kms for the next one and a half hours. What is the average velocity of the bus if it has
moved in a straight line for the whole journey.
Question 2.
See the diagram and calculate the Distance and Displacement travelled by the body from A to I.
Distance travelled =
A → B → C → D → E → F → G → H + I
= 5 + 7 + 6 + 3 + 5 + 4 + 6 + 5
= 41 m
Displacement = A → I in a straight line shortest distance
= 1m
Use your brainpower:
Question 1.
The unit of acceleration is m/s^2, verify this.
Question 2.
Acceleration is a vector quantity. Is force a vector quantity too?
Yes, acceleration and force both are vector quantities, because both can be expressed completely only when magnitude and direction are given and the quantity which needs direction and magnitude both
is called a vector quantity. | {"url":"https://maharashtraboardsolutions.com/maharashtra-board-class-7-science-solutions-chapter-7/","timestamp":"2024-11-10T17:57:28Z","content_type":"text/html","content_length":"97857","record_id":"<urn:uuid:1f930543-8853-4dd7-990d-1e4bd5c3b4f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00283.warc.gz"} |
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Joules to G Force Calculator, Formula, J to GF Calculation | Electrical4u
Joules to G Force Calculator, Formula, J to GF Calculation
Joules to G Force Calculator
Enter the values of energy in joules, J[(J)], mass of the object, m[(kg)] and distance over which the energy is applied, d[(m)] to determine the value of G-force, GF.
Joules to G Force Formula
The conversion from joules (a measure of energy) to G-force (a measure of acceleration relative to Earth’s gravity) involves calculating how much force in terms of gravitational force units (Gs) is
associated with the release or absorption of energy over a certain distance, taking into account the mass of the object involved.
G-force, GF in g-forces is calculated by dividing the potential energy in joules, J[(J)] in joules by the mass of the object, m[(kg)] in kilograms, the distance over which the energy is applied, d
[(m)] in metres and then normalizing by the acceleration due to gravity, 9.81m/s^2.
G-force, GF = J[(J)] * m[(kg)] / d[(m)] * 9.81
GF = G force in g-forces.
J[(J)] = energy in joules, J.
m[(kg)] = mass in kilograms, kg.
d[(m)] = distance in metres, m.
Joules to G Force Calculation:
1. Calculate the G-force exerted when a device with 150 joules of kinetic energy and a mass of 30 kg comes to a stop over a distance of 0.25 metres.
Given: J[(J)] = 150J, m[(kg)] = 30kg, d[(m)] = 0.25m.
G-force, GF = J[(J)] * m[(kg)] / d[(m)] * 9.81
GF = 150 * 30 / 0.25 * 9.81
GF = 1835.49Gs.
2. Suppose a device uses 500 joules of energy to achieve a G-force (GF) of 100 Gs over a distance of 2 metres. Calculate the mass of the device.
Given: J[(J)] = 500J, GF = 100Gs, d[(m)] = 2m.
G-force, GF = J[(J)] * m[(kg)] / d[(m)] * 9.81
m[(kg)] = GF * d[(m)] * 9.81 / J[(J)]
m[(kg)] = 100 * 2 / 9.81 * 500
m[(kg)] = 3.924kg.
How to use the Joules Formula
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About the course
This state course is a variation of the state’s Advanced Mathematics and Decision Making (AMDM) course. This particular version focuses on deterministic and probabilistic mathematics used in solving
problems that arise in industry and government. Such situations include maximizing the output of productions facilities, minimizing distribution costs, determining the length of a queue, and finding
the best shipping routes. The mathematics involved includes programming (linear, integer, and binary), graphs and networks, probability, and decision trees. The course makes heavy use of Microsoft
Excel to analyze data.
This course can be viewed as an introduction to operations research. The course was created through the efforts of the Mathematics Instruction using Decision Science and Engineering Tools (MINDSET)
Project. MINDSET is a collaboration between educators, engineers, and mathematicians at three universities (Wayne State, NC State, UNC Charlotte) to create, implement, and evaluate a new mathematics
curriculum and textbook. You can find out more about MINDSET
This is currently being taught as a one semester course. I first taught this course, and the first time it was offered at the Magnet School, was the 2012-13 school year.
The course was on hiatus for the 19-20 school year, but is back for the 20-21 school year!
The textbook
The text is produced by the MINDSET Project and is only available through training provided by the MINDSET group. It is called When Will We Ever Use This: Making Decisions Using Advanced Mathematics.
There are two volumes: one for the deterministic and one for the probabilistic. (Shown above is the deterministic volume.)
The coursework and book is a successor to the activities found in the book Does This Line Ever Move: Everyday Applications of Operations Research. This book has a wealth of activities for classroom
More such activities can be found at High School Operations Research. | {"url":"http://drchuckgarner.webmate.me/math-of-industry-government","timestamp":"2024-11-01T22:46:40Z","content_type":"text/html","content_length":"25177","record_id":"<urn:uuid:52913e5a-b000-435c-8df0-001271656314>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00451.warc.gz"} |
npx(1) -- execute npm package binaries
npx [options] <command>[@version] [command-arg]...
npx [options] [-p|--package <pkg>]... <command> [command-arg]...
npx [options] -c '<command-string>'
npx --shell-auto-fallback [shell]
npm install -g npx
Executes <command> either from a local node_modules/.bin, or from a central cache, installing any packages needed in order for <command> to run.
By default, npx will check whether <command> exists in $PATH, or in the local project binaries, and execute that. If <command> is not found, it will be installed prior to execution.
Unless a --package option is specified, npx will try to guess the name of the binary to invoke depending on the specifier provided. All package specifiers understood by npm may be used with npx,
including git specifiers, remote tarballs, local directories, or scoped packages.
If a full specifier is included, or if --package is used, npx will always use a freshly-installed, temporary version of the package. This can also be forced with the --ignore-existing flag.
• -p, --package <package> - define the package to be installed. This defaults to the value of <command>. This is only needed for packages with multiple binaries if you want to call one of the other
executables, or where the binary name does not match the package name. If this option is provided <command> will be executed as-is, without interpreting @version if it's there. Multiple --package
options may be provided, and all the packages specified will be installed.
• --no-install - If passed to npx, it will only try to run <command> if it already exists in the current path or in $prefix/node_modules/.bin. It won't try to install missing commands.
• --cache <path> - set the location of the npm cache. Defaults to npm's own cache settings.
• --userconfig <path> - path to the user configuration file to pass to npm. Defaults to whatever npm's current default is.
• -c <string> - Execute <string> inside an npm run-script-like shell environment, with all the usual environment variables available. Only the first item in <string> will be automatically used as
<command>. Any others must use -p.
• --shell <string> - The shell to invoke the command with, if any.
• --shell-auto-fallback [<shell>] - Generates shell code to override your shell's "command not found" handler with one that calls npx. Tries to figure out your shell, or you can pass its name
(either bash, fish, or zsh) as an option. See below for how to install.
• --ignore-existing - If this flag is set, npx will not look in $PATH, or in the current package's node_modules/.bin for an existing version before deciding whether to install. Binaries in those
paths will still be available for execution, but will be shadowed by any packages requested by this install.
• -q, --quiet - Suppressed any output from npx itself (progress bars, error messages, install reports). Subcommand output itself will not be silenced.
• -n, --node-arg - Extra node argument to supply to node when binary is a node script. You can supply this option multiple times to add more arguments.
• -v, --version - Show the current npx version.
Running a project-local bin
$ npm i -D webpack
$ npx webpack ...
One-off invocation without local installation
$ npm rm webpack
$ npx webpack -- ...
$ cat package.json
...webpack not in "devDependencies"...
Invoking a command from a github repository
$ npx github:piuccio/cowsay
$ npx git+ssh://my.hosted.git:cowsay.git#semver:^1
Execute a full shell command using one npx call w/ multiple packages
$ npx -p lolcatjs -p cowsay -c \
'echo "$npm_package_name@$npm_package_version" | cowsay | lolcatjs'
< your-cool-package@1.2.3 >
\ ^__^
\ (oo)\_______
(__)\ )\/\
||----w |
|| ||
Run node binary with --inspect
$ npx --node-arg=--inspect cowsay
Debugger listening on ws://127.0.0.1:9229/....
Specify a node version to run npm scripts (or anything else!)
npx -p node@8 npm run build
You can configure npx to run as your default fallback command when you type something in the command line with an @ but the command is not found. This includes installing packages that were not found
in the local prefix either.
For example:
$ npm@4 --version
(stderr) npm@4 not found. Trying with npx...
$ asdfasdfasf
zsh: command not found: asfdasdfasdf
Currently, zsh, bash (>= 4), and fish are supported. You can access these completion scripts using npx --shell-auto-fallback <shell>.
To install permanently, add the relevant line below to your ~/.bashrc, ~/.zshrc, ~/.config/fish/config.fish, or as needed. To install just for the shell session, simply run the line.
You can optionally pass through --no-install when generating the fallback to prevent it from installing packages if the command is missing.
For bash@>=4:
$ source <(npx --shell-auto-fallback bash)
For zsh:
$ source <(npx --shell-auto-fallback zsh)
For fish:
$ source (npx --shell-auto-fallback fish | psub)
Huge thanks to Kwyn Meagher for generously donating the package name in the main npm registry. Previously npx was used for a Tessel board Neopixels library, which can now be found under npx-tessel.
Written by Kat Marchan.
Please file any relevant issues on Github.
This work is released by its authors into the public domain under CC0-1.0. See LICENSE.md for details.
• npm(1)
• npm-run-script(1)
• npm-config(7) | {"url":"https://www.npmjs.com/package/npx?utm_source=bhdouglass.com","timestamp":"2024-11-02T19:30:41Z","content_type":"text/html","content_length":"125365","record_id":"<urn:uuid:8a713ce2-d6e8-4fad-8ccc-9a74ce909a2a>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00111.warc.gz"} |
Which is One Heavier, Cement or Sand?
By weight a ton of cement and a ton of sand weigh the same amount, but by volume in an identical bags/containers of same capacity, we will get more cement (heavier) than sand i.e. 1 cubic meter of
cement will be heavier to 1 cum of sand, this is because of cement mineralogical composition (composition of its raw materials) differ than that of natural sand.
To clarify further, let us understand specific gravity.
The specific gravity of a material gives us how much more or less dense the material is than water (taken as standard material).
Water has a specific gravity of 1.000 (near 4°C). If a material is denser than water, then its specific gravity is greater than 1. If it is less dense than water, then the specific gravity is less
than 1.
Specific gravity of sand ranges 2.6 – 2.7 and that of cement is 3.14 – 3.15, whereas diesel has specific gravity ranges 0.82 to 0.95. Since the specific gravity of diesel is less than water it floats
on top of it.
As specific gravity of sand is 2.6 – 2.7 and that of cement is 3.14 – 3.15, i.e. for the same volume occupied by cement and sand, cement is “3.15/2.7 = 1.16 times “heavier than sand.
People often get confused between bulk density and specific gravity , dry bulk density of cement is 1440 kg/cum and that of natural sand is 1600 kg/cum. This does not imply that sand is heavier to
cement. The right comparison would be specific gravity of each of them or any other construction material to understand which one is “Heavier” | {"url":"https://happho.com/which-is-one-heavier-cement-or-sand/","timestamp":"2024-11-02T15:11:15Z","content_type":"text/html","content_length":"211836","record_id":"<urn:uuid:b1527aa9-1c1b-4265-9f9f-ae0ef4b10743>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00183.warc.gz"} |
How Much does Massive MIMO Improve the Spectral Efficiency?
It is often claimed in the academic literature that Massive MIMO can greatly improve the spectral efficiency. What does it mean, qualitatively and quantitatively? This is what I will try to explain.
With spectral efficiency, we usually mean the sum spectral efficiency of the transmissions in a cell of a cellular network. It is measured in bit/s/Hz. If you multiply it with the bandwidth, you will
get the cell throughput measured in bit/s. Since the bandwidth is a scarce resource, particularly at the frequencies below 5 GHz that are suitable for network coverage, it is highly desirable
to improve the cell throughput by increasing the spectral efficiency rather than increasing the bandwidth.
A great way to improve the spectral efficiency is to simultaneously serve many user terminals in the cell, over the same bandwidth, by means of space division multiple access. This is where Massive
MIMO is king. There is no doubt that this technology can improve the spectral efficiency. The question is rather “how much?”
Earlier this year, the joint experimental effort by the universities in Bristol and Lund demonstrated an impressive spectral efficiency of 145.6 bit/s/Hz, over a 20 MHz bandwidth in the 3.5 GHz band.
The experiment was carried out in a single-cell indoor environment. Their huge spectral efficiency can be compared with 3 bit/s/Hz, which is the IMT Advanced requirement for 4G. The remarkable
Massive MIMO gain was achieved by spatial multiplexing of data signals to 22 users using 256-QAM. The raw spectral efficiency is 176 bit/s/Hz, but 17% was lost for practical reasons. You can
read more about this measurement campaign here:
256-QAM is generally not an option in cellular networks, due to the inter-cell interference and unfavorable cell edge conditions. Numerical simulations can, however, predict the practically
achievable spectral efficiency. The figure below shows the uplink spectral efficiency for a base station with 200 antennas that serves a varying number of users. Interference from many tiers of
neighboring cells is considered. Zero-forcing detection, pilot-based channel estimation, and power control that gives every user 0 dB SNR are assumed. Different curves are shown for different values
of τ[c], which is the number of symbols per channel coherence interval. The curves have several peaks, since the results are optimized over different pilot reuse factors.
Uplink spectral efficiency in a cellular network with 200 base station antennas.
From this simulation figure we observe that the spectral efficiency grows linearly with the number of users, for the first 30-40 users. For larger user numbers, the spectral efficiency saturates due
to interference and limited channel coherence. The top value of each curve is in the range from 60 to 110 bit/s/Hz, which are remarkable improvements over the 3 bit/s/Hz of IMT Advanced.
In conclusion, 20x-40x improvements in spectral efficiency over IMT Advanced are what to expect from Massive MIMO.
37 thoughts on “How Much does Massive MIMO Improve the Spectral Efficiency?”
1. I note that optimum, sometimes, is to serve up to K = 70 users with M = 200 base station antennas. It is thus not necessarily true that M ≫ K, i.e. there is not always an order of magnitude more
antennas than users in massive MIMO.
1. You are absolutely right! To maximize the sum spectral efficiency it is often preferable to serve relatively many users, with M/K<10. Even if the spectral efficiency per user is not
extraordinary, the sum spectral efficiency can be huge.
However, if we want to achieve high spectral efficiency per user, at the cost of lower sum spectral efficiency, we might want to have M/K>10.
2. Correct: Myth 6…
2. I guess, all mentioned numbers correspond to quasi-static channel conditions (or instant ideal CSI), right? Because channel dynamics brings more and more degradation of SE if many layers are
Actually, it could be interesting to discuss the same issues for moving users with Doppler spread 10-20 Hz at least.
1. In the numerical analysis, we consider Rayleigh fading channels that are static within a coherence block, and independent between blocks. The channel variability is thus captured by the size
of the coherence blocks. As you say, more channel dynamics lead to smaller blocks, which in turn give smaller SE.
For any given scenario (carrier frequency, Doppler spread, etc.), you can compute an approximate coherence time and coherence bandwidth, multiply them together and then you have the number of
channel uses per coherence block.
2. Importantly: The Massive MIMO analysis does *not* rely on “ideal CSI” assumptions. Channels are estimated, once per coherence interval, from uplink pilots – and the resulting channel
estimation errors (and the pilot overhead) are accounted for in the performance bounds.
3. Most of the research works consider linear techniques such as ZF, P-ZF, MRC and MMSE to improve the spectral efficiency, where SE is mostly related to SINR and rate. Is there any other techniques
to increase SE? Next question is whether DL achievable SE is higher or UL achievable SE?
1. I would say that there is little need to develop new uplink receive combining or downlink precoding schemes. The schemes that you mention are the ones of main interest. Another way to improve
the SE is power control. It can have a substantial impact on the SE:
Regarding uplink versus downlink, there is not absolute answer to that question, because there can be substantial transmit power difference between the uplink and downlink. Traditionally, the
downlink uses higher power and thereby achieves higher SE. If the total transmit power is the same, then the uplink SE can be larger since the base station has direct access to the channel
estimates and can thus decode the signals more accurately.
1. Hi Emil,
Can you please explain further how BS has direct access to the channel estimation which the MS does have?
1. The user device transmit a pilot signal in the uplink, which enables channel estimation at the base station.
4. Thank you Emil Björnson for your sparing your valuable time. I also wonder how to optimize SE and EE in massive MIMO, what is the trade-off between them? Most papers considered the impact of
interference when evaluating the SE.
1. Interference plays an important role, both when evaluating the SE and the EE (energy efficiency). There are several papers that analyzes this tradeoff:
The good news is that Massive MIMO can achieve both high SE and EE, since both of these goals are achieved by multiplexing of many UEs, which share the energy costs and achieve a high sum SE.
I believe that this is a topic we will return to on the blog.
7. Sir, can we have the expression for spectral efficiency of massive MIMO?
1. There are many different expressions for the computing spectral efficiency in Massive MIMO, depending on if it is uplink or downlink, or which processing scheme that is used.
You can find many of these things in Section 4 of my book Massive MIMO Networks, which you can download here: https://massivemimobook.com/wp/
8. Hi – great website BTW, very helpful.
Sorry for being two months late to this thread, hopefully still open for questions… I’m debating this subject with colleagues, some of whom expect the overall spectral efficiency of a c-band
mMIMO sector to be in the 3-6 bps/Hz range, which seems a very long way below what I’m hearing from vendors (and of course the Bristol results). They claim this is due to “real world” adjustments
to expected performance, but that seems like a big adjustment. Can such conflicting levels of SE performance be explained? Perhaps currently available mMIMO equipment doesn’t implement digital
beamforming, and hence no effective MU capability? What is the expectation of equipment evolution with respect to support for more effective spatial multiplexing – over the next 5 yeats for
Many thanks
1. 3-6 bps/Hz is what you can achieve for single-user transmission under good conditions. Uncoded 64-QAM transmission gives 6 bps/Hz.
Hence, I think your colleagues are having single-user transmission with massive beamforming in mind.
By spatially multiplexing, say, 8 users the overall spectral efficiency will rather be in the range of 24-48 bps/Hz.
1. Hi Emil,
By uncoding/coding I guess here you indicated to channel coding. How channel coding impacts the spectral efficiency?
1. When the channel supports a particular spectral efficiency, you need to find a modulation+coding scheme that delivers that spectral efficiency. For example, if you support 5 bit/s/Hz,
then you can pick 64-QAM (6 bit/s/Hz) and then apply a channel code with rate 5/6.
9. sir, what are the objectives of spectral efficiency
1. The data rate is computed as “spectral efficiency” multiplied with “bandwidth”. Since the bandwidth is often determined by external factors such as licenses, the spectral efficiency is
determining how efficiently the technology itself operates.
10. Dear Prof.
Thank you so much for you this website and your book on massive MIMO networks. I am learning a lot of things from this book, as I am new in this field.
Matlab codes of all figures are given except Figure 7.1 (Example of an SE region (shaded) with different combinations of SEs).
I would like to request you please can you provide the Matlab code of Figure 7.1 also, because as a beginner, I think this will help the students who start to learn massive MIMO.
Thanks in advance
1. This figure is mainly drawn in Adobe Illustrator as an illustration, so that is why we don’t have any simulation code for it. But if you want to learn how to generate rate regions like this,
I recommend my previous book “Optimal Resource Allocation in Coordinated Multi-Cell Systems”. Figure 3.1 generates rate regions and there is simulation code that reproduces these figures, or
at least figures of the same kind (the exact region depends on the random seed).
1. Dear professor
Thanks a lot for creating such a wonderful platform and for the book “Massive MIMO Networks”. This book is very helpful for me as I am new in this field. I have seen most of your books
and journals related to massive MIMO. Currently, I am doing my thesis on spectral and energy efficiency in massive mimo. But I also want to show a relative study between Massive MIMO and
Massive MIMO-OFDM in my work. So I want to write all the equations (like SE, EE, and circuit power consumption equations) for Massive MIMO-OFDM for “N” subcarriers too. Can you please
suggest me some good papers related to this? I will be always grateful to you.
1. Thank you for reading the book!
The block-fading model that we are considering is an abstraction of OFDM that captures its essential properties (e.g., the channels vary over frequency, and all the subcarriers within
the coherence bandwidth are placed in the same coherence block) but without the complicated math. Writing all the same things using an exact MIMO-OFDM notation will be much, much more
The introduction chapters to the thesis “High-end performance with low-end hardware: Analysis of massive MIMO base station transceivers” explains some of the things rather nicely.
One of the relevant papers in the thesis is “Waveforms for the Massive MIMO Downlink:Amplifier Efficiency, Distortion and Performance“.
11. Thank you so much for this article. I was wondering where I can get the Matlab code for the simulation in this article sir?
12. Dear Professor Bjornson
I have a question. As we know massive MIMO works based on SDMA and here we have the adjacent cells that serve users on the same frequency. Is not it in conflict with the classic definition of
Can I ask what is the define of a cell in a massive MIMO network?
1. A cell is the geographical region where the users will connect to the base station that controls that cell. It is the same definition in Massive MIMO as other types of networks.
Maybe what you are thinking about is that traditional cellular networks divided the radio resources using frequency division. For example, each cell might only use 1/4 of the frequencies, to
avoid that neighboring cells use the same frequency and interfere too much. Nowadays, we use beamforming (a key part of SDMA) to limit the inter-cell interference and therefore we can a full
reuse of frequencies between all cells.
13. Dear Professor Bjornson
I have seen most of your books and journals I appreciate you mostly. Now I am doing my thesis on spectral efficiency analysis. In my work, I want to show the effect of code block length on SE.
Can you help me? I want some simulation code like the effect of code block length on SE.
1. What we call spectral efficiency is normally an achievable lower bound on the channel capacity, which means that it is achieved as the block length goes to infinity. Hence, I don’t know how
connect SE with the block length as you are suggesting.
Perhaps a better option would be to relate to BER with the code block length. You can simulate such thing using Sionna: https://nvlabs.github.io/sionna/
14. Hi! I might be rusty in comm theory but from my understanding with 20 MHz channel, 256-QAM and with as many rx and tx antennas I don’t understand where the massive increase in spectral efficiency
comes from as from definition spectral efficiency is bit/s/Hz assuming log2 of 256 bits =8 so spectral eff is 8 bits/s/Hz can some one please correct me or explain where the over 100 bits/s/Hz is
coming thanks
1. Hi! If you can send 256-QAM signals to 22 users simulateneously, the spectral efficiency per user is 8 bit/s/Hz and the total spectral efficency is 8*22 = 176 bit/s/Hz.
Massive MIMO enables you to serve the users at the same time and frequency, and protects their signals from interference using beamforming/precoding/combining (many names for roughtly the
same thing).
15. I understand with spatial multiplexing it is possible but with beamforming MIMO how do they get such high spectral efficiency as beamforming is just advanced constructive interference but its not
spatial multiplexing or does the increased SNR from beamforming allow for spatial multiplexing thanks again?
1. With multi-user beamforming/precoding/combining, you can achieve constructive interference at the desired locations and destructive interference at the undesired location. In this way, you
can communicate with many users at the same time and frequency, almost as if they were alone. This is the spatial multiplexing.
16. Hi Sir!!! Very intersting topic. Just a question
Is it really possible to achieve those capacities in real environments?
For average throughput gain, what kind of improvement can we expect comparing it to 2×2 MIMO? I have read somewhere that it is like 3x.
1. The values were obtained by measurements so they can be achieved in real environments, but it will not be the average performance.
With 2×2 MIMO, you can ideally transmit 2 layers of data without interference. This leads to 2x the capacity compared to 1×1 SISO. | {"url":"https://ma-mimo.ellintech.se/2016/10/18/how-much-does-massive-mimo-improve-spectral-efficiency/","timestamp":"2024-11-05T20:29:16Z","content_type":"text/html","content_length":"138354","record_id":"<urn:uuid:ec95219b-4662-4a88-ad4a-d772eb3a1917>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00811.warc.gz"} |
Properties of matter Homework Help, Questions with Solutions - Kunduz
Properties of matter Questions and Answers
Properties of matter
Example 2 6 Two clocks are being tested against a standard clock located in a national laboratory At 12 00 00 noon by the standard clock the readings of the two clocks are Monday Tuesday Wednesday
Thursday Friday Saturday Sunday Clock 1 12 00 05 12 01 15 11 59 08 12 01 50 11 59 15 12 01 30 12 01 19 Clock 2 10 15 06 10 14 59 10 15 18 10 15 07 10 14 53 10 15 24 10 15 11 If you are doing an
experiment that requires precision time interval measurements which of the two clocks will you prefer
Properties of matter
The mean pressure is 6K 5 which rain renders to vertical windshield of automobile moving with constant velocity of magnitude v 12 m s Consider that raindrops fall vertically with speed u 5 m s The
intensity of rainfall deposits h 2 cm of sediments in time t 1 minute p 10 kg m is the density of liquid Assume collisions are inelastic
Properties of matter
8 A thick rope of rubber of length 8 m and density 1 5 10 kg m has Young s modulus 3 5x 106 Nm 2 When hung from ceiling of a room the increase in length due to its own weight is a 96 10 3 m b 19 2x
10 5 m c 9 4 cm d 9 6 mm
Properties of matter
22 Aballoon of volumie V contains a gas of mass mat a pressure P and temperature 15 C Gas is pumped into the balloon so that its volume is doubled and the pressure is trebled If the temperature
increases 6 C in the process finc the ratio of the increase in mass to the origi nal mass 1 34 5 2 239 49
Properties of matter
Two equal and opposite forces each of magnit F is applied along a rod of transverse sectional area A The normal stress to a section PQ inclined to transverse section is ar 0 F 1 3 F sine A sin 20 2 4
P F A A Q AS cose Ecos 0 F
Properties of matter
In a Quincke s tube there are two positions where sound becomes minimum the sliding distance between the is 16 6 cm Find the freq of sound source Sound velocity in air 332 m sec A tuning fork having
frequency 300 Hz produce four beats ner ser with x If we file arm of unknown and ag
Properties of matter
13 Volume of a substance at normal atmospheric pressure is 3500 10 6 m What will be its change in volume at the pressure of 25 standard atmosphere Given that bulk modulus of the substance is 1011 N m
2 85 x 10 9 m
Properties of matter
stration 5 2 Assume that if the shear stress in steel exceeds about 4 00 x 108 N m the steel reptures Determine the shearing force necessary to a shear a steel bolt 1 00 cm in diameter and b punch a
1 00 cm diameter hole in a steel plate 0 500 cm thick
Properties of matter
A pipe of circular cross section of inner radius r and outer radius 2r is bent into a semi circle as shown in the diagram A fluid of density p 10 kg m is flowing through it The breaking stress of the
material of the pipe is 3 x 10 N m The maximum velocity with which the fluid can flow in the pipe is kx 100 m s Find value of k 9
Properties of matter
34 Density of ice is p and that of water is o What will be decrease in volume when a mass M of ice melts 1 M o p 1 3 M 2 4 o p M M 10
Properties of matter
11 The increase in pressure required to decrease the 200 L volume of a liquid by 0 008 in kPa is Bulk modulus of the liquid 2100 MPa is 1 8 4 2 84 3 92 4 4 168
Properties of matter
30 Two steel rods and an aluminium rod of equal length l and equal cross section are joined rigidly at their ends as shown in the figure below All the rods are in a state of zero tension at 0 C Find
the length of the system when the temperature is raised to 0 Coefficient of linear expansion of aluminium and steel area and a respectively Young s modulus of aluminium is Y and of steel is Y Steel
1390 Aluminium Steel O
Properties of matter
13 A steel rod is 4 cm in diameter at 30 C A brass ring has an interior diameter of 3 992 cm at 30 C In order that the ring just slides on to the steel rod the common temperature of the two rods
should be a 11x10 6 1 C a 19x10 6 1 C 1 200 C 2 350 C 3 280 C 4 300 C
Properties of matter
25 If S is stress and Y is young s modulus of material of a wire the energy stored in the wire per unit volume is a S 2Y b 2S Y c S 2Y d 2Y S
Properties of matter
Two objects A B of equal density and radius TA 1 mm and r 2 mm are moving in same medium then find the ratio of their terminal velocity VB B in the medium 2 1 2 VA 1 1 4 43 4 4 2
Properties of matter
The diagram shows a simple mercury barometer The mercury level is at a heighth when the atmospherich P pressure is 100000 Pa What is the pressure at P A 40000 Pa B 60000 Pa C 100000 Pa D 140000 Pa 0
4 h
Properties of matter
A wire elongates by mm when a load W is hanged from it If the wire goes over a pulley and two weights W each are hung at the two ends the elongation of the wire will be in mm 4 zero 1 l 2 2 l 3 2 Y
in mode of a material of density o It is falling through a liquid of density p p AIEEE 2006
Properties of matter
4 length 50 cm diameter 0 5 mm 3 length 300 cm diameter 3 mm Copper of fixed volume V is drawn into wire of length e When this wire is subjected to a constant force F extension produced in the wire
is Al Which of the following graph is a straight line 4 Al versus l AIPMT 201 1 Al versus 1 2 Al versus l 3 Al versus 1 1
Properties of matter
4 Length 300 cm Diameter 3 mm 3 Length 200 cm Diameter 2mm A catapult s string made of rubber having cross section area 25 mm and length 10 cm To throw a 5 gm pabble it is stret up to 5 cm and
released Velocity of projected pabble is Young coefficient of elasticity of rubber is 5 108 N m 1 20 m s 2 100 m s 3 250 m s RPMT 25 4 200 m s 9 x 101 N m Force required to increa
Properties of matter
A 5m aluminium wire Y 7 x 1010 N m of diameter 3 mm supports a 40 kg mass In order to have the sam elongation in a copper wire Y 12 x 10 0 N m of the same length under the same weight the diameter
should be RPMT 2007 mm 1 1 75 2 2 0 3 2 3 4 5 0
Properties of matter
4 One end of a uniform wire of length L and of Wo is attached rigidly to a point in the roof and a weight W is suspended from its lower end If S is the area of cross section of the wire the stress in
the wire at a height L 4 from its lower end is 1 W S 2 W 2 W W 4 S 3 W 3W 4 S 4 W W S
Properties of matter
Heat is being supplied at a constant rate to a sphere of ice which is melting at the rate of 0 1 gm sec It melts completely in 100 sec The rate of rise of temperature thereafter will be Assume no
loss of heat 0 8 C sec 2 5 4 C sec 3 3 6 C sec 4 will change with time 1
Properties of matter
3 450 MJ m 25 A uniform wire of length 3m and mass 10 kg is suspended vertically from one end and loaded at other end by a block of mass 10 kg The radius of cross section of wire is 0 1 m The stress
in the middle of wire is g 10 m s 1 1 4 x 104 N m 2 4 8 x 103 N m 3 96 104 N m 4 35 10 N m
Properties of matter
A steel wire of length 4 2 m and cross sectional area 3 x 10 5 m stretches by the same amount as a copper wire of length 3 5 m and cross sectional area 4 105 m under given load What is ratio of Young
s modulus of steel to that of copper 05 8
Properties of matter
4 viscosity A certain number of spherical drops of a liquid of radius r coalesce to form a single drop of radius R and volume V If T is the surface tension of the liquid then AIPMT 2014 1 Energy 4VT
1 3 Energy 3VT R is released is released 2 Energy 3VT is released 4 Energy is neither released nor absorbed of capillary tube above the surface of water is made less than
Properties of matter
A soap bubble of diameter 8 mm is formed in air The surface tension of liquid is 30 dyne cm The excess pressu inside the soap bubble is 1 150 dyne cm 2 300 dyne cm 3 3 x 10 3 dyne cm 4 12 dyne cm
Properties of matter
17 Water is filled up to a height h in a beaker of radius R as shown in the figure The density of water is p tension of water is T and the atmospheric pressure is P Consider a vertical section ABCD
of the water column through a diameter of the beaker The force on water on one side of this section by water on the other side of this section has magnitude 1 2P Rh nR pgh 2RT 3 PR R pgh 2RT 2R A B 2
2P Rh Rpgh 2RT 4 PR R pgh 2RT
Properties of matter
A capillary tube of radius R is immeresed in water and water rises in it to a height H Mass of water in capillary tube is M if the radius of the tube is doubled mass of water that will rise in
capillary tube will be 1 2M 2 M 3 M 4 4M 2 bubble of radius r r The radius R of the soapy film
Properties of matter
The lengths and radii of two rods made of same material are in the ratios 1 2 and 2 3 respectively If the temperatu difference between the ends for the two rods be the same then in the steady state
the amount of heat flowing per seco through them will be in the ratio 1 1 3 2 4 3 3 8 9 4 3 2
Properties of matter
A partition divides a container having insulated walls into two compartments I and II The same gas fills the two compartments see Fig The ratio of the number of molecules in compartments I and II is
ya fue dart ist m area fearger ergen at CI P V T ima a A 1 6 B 6 1 C 4 1 D 1 4 2P 2V T
Properties of matter
Water rises in a capillary tube of radius r upto a height h The mass of water in a capillary r of radius will be 4 m The mass of water that will rise in a A m B 4m C m enc capilar m aque
Properties of matter
9 What kind of elastic materials are derived from a strain energy density function a Cauchy elastic materials b Hypo elastic materials c Hyper elastic materials d None of the mentioned I
Properties of matter
8 As the elastic limit reaches tensile strain a Increases more rapidly b Decreases more rapidly c Increases in proportion to the stress d Decreases in proportion to the stress
Properties of matter
A gas undergoes a process in which its pressure P and volume V are related as VP constant The bulk modulus the gas in the process is 1 nP 2 p n 3 P n 4 Pn
Properties of matter
Q1 The pressure at a depth of 700 m in the ocean is around 7 kPa By how much does the volume of 0 2 m aluminum cube contract when it is dropped this depth of the ocean O A 0 2 m O 8 Aluminum cube
will not contract O c 20 mm O D 0 2 mm
Properties of matter
When a spring is subjected to 4N force its length is a metre and if 5N is applied length is b metre If 9N is applied its length is A 4b3a C 5b 4a B 5b a D 5b 2a 135
Properties of matter
72 Two soap bubbles of the same soap solution have radii 3 cm and 1 5 cm If the excess pressure inside the bigger bubble is 40 dyn cm what is the excess pressure inside the smaller bubble 1 mark
Properties of matter
The temperature gradient in a rod of 0 5 m length is 80 C m If the temperature of hotter end of the rod is 30 C then the temperature of the cooler end is Question Type Single Correct Type 1 40 C 2 10
C 3 10 C
Properties of matter
Ques 4 A pipe 10 cm in diameter contains steam at 100 C It is covered with an insulating material that is 5 cm thick This insulating material has a constant of thermal conductivity K 0 0006 The
outside surface of the system is 30 C Find the heat loss per hour from a meter length of the pipe
Properties of matter
9 Two similar rods of length 1 and I each of iron and b respectivel expansion of iron and brass are a and raised then determine the relation 1 1 A Oa a a B Oa a a C Oa a ap D O aa
Properties of matter
A glass capillary tube is of the shape of truncated cone with an apex angle a so that its two ends have cross sections of different radii When dipped in water vertically water rises in it to a height
h where the radius of its cross section is b If the surface tension of water is S its density isp and its contact angle with glass is the value of h will be g is the acceleration due to gravity 2014
Properties of matter
1 Rises 2 Falls 3 A large ship can float but a steel needle sinks because of 1 Viscostiy 2 Surface tension 3 Density The Working of an atomizer depends upon 4 None of these
Properties of matter
3 A steel ring of radius r and cross sectional area A is fitted on to a wooden disc of radius R R r It Young s modulus of steel is Y then stress in the steel ring is 1 3 RY r 2 4 R ry r F y
Properties of matter
A steel wire of length 2 m is stretched through 2 mm What is elastic potential energy stored in a wire of cross sectional area 4 mm in stretched condition Y 2 x 10 1 Nm 0 8 J O 1 2 J 2 4 J 1 6 J
Properties of matter
4 atoms contains negatively charged particles 32 Graph of total number of a particles scattered at different angles is Number of a particles Number of 14 20 160 0 20 160 0 a particles 4 32 fafan 19
Lin 40 70 Ita F 14 10 42 70 2 F341 20 160 0 20 160 0 E
Properties of matter
why is the Reynolds number taken as 2000 while as per ERT the laminar or steady flow is less than 1000 plit increases Discuss qualitatively 10 26 a What is the largest average velocity of blood flow
in an artery of radius 2x10 m if the flow must remain lanimar b What is the corresponding flow rate Take viscosity of blood to be 2 084 x 10 3 Pa s
Properties of matter
4 0 12 15 Two exactly similar wires of steel and copper are stretched by equal forces If the total elongation is 2 cm then how much is the elongation in steel and copper wire respectively Given Y 20
1011 dyne cm2 Ycopper 12 x 10 1 dyne cm 1 1 25 cm 0 75 cm 2 0 75 cm 1 25 cm 3 1 15 cm 0 85 cm 4 085 cm 115 cm
Properties of matter
The excess pressure inside an air bubble of radius r just below the surface of the water is P The pressure inside a drop of the same radius just outside the surface is P2 If T is the surface tension
then AP 2P Your Answer B P P Correct Answer P 2P DP 0 P 0
Properties of matter
Problem 249 A vertical cylinder of radius R is rotating with a constant angular velocity o along with holders A and B about the axis as shown The vertical movement of cylinder is restricted by the
holders Oil is only in between the bottom of cylinder and surface as shown The thickness of the oil layer is t Assume that coefficient of viscosity is n and that cylinder can only rotate if there is
no friction elsewhere Find the power required to overcome the viscous resistance A R
Properties of matter
38 A wire fixed at the upper end stretches by length by applying a force F The work done in stretching is AIEEE 2004 a F 21 b Fl FI | {"url":"https://kunduz.com/questions/physics/properties-of-matter/","timestamp":"2024-11-02T07:37:30Z","content_type":"text/html","content_length":"325993","record_id":"<urn:uuid:fc6e5cb8-8e17-4200-afc7-1ab03d162706>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00761.warc.gz"} |
The consensus algorithm is the dynamic method through which nodes in a blockchain system reach an agreement and make decisions.
Bitxor utilizes an innovative mechanism called the Proof-of-Stake Plus (PoS+), a modified version of the popular Proof-of-Stake (PoS) consensus.
In a basic PoS consensus algorithm, the formation of a new block in the blockchain is stochastically assigned to a node based on a combination of factors related exclusively to the node owner’s
The PoS+ mechanism considers the account’s stakes too, but it also promotes the ecosystem’s health by rewarding participants based on their activity.
The algorithm considers the following factors when calculating an account’s importance, the measure that will ultimately be used to choose the next harvesting node:
• Stake: The total amount of harvesting token held, since owners with larger balances have the incentive to see the ecosystem flourish. Only accounts holding more than 10’000 harvesting tokens (
high-value accounts) are eligible for harvesting.
• Transactions: The total amount of fees paid by an account. This encourages being an active account in the network.
• Nodes: The number of times an account has been the beneficiary of the fees collected by a node. Thus the network incentivizes accounts which run nodes.
Periodically, an importance score based on these three factors is calculated for all high-value accounts. The importance score determines an account’s probability to harvest the next block.
Partial scores
The network calculates first the following partial scores for all high-value accounts at the end of each importance period (720 blocks, roughly 6 hours. See importanceGrouping in Network
• Stake Score (\(S\)): Account’s balance divided by the balance of all high value accounts, at the end of the period.
• Transaction Score (\(T\)): Total amount of fees paid by the account divided by the total amount of fees paid by all high value accounts during the period.
• Node Score (\(N\)): Number of times the account has been the beneficiary of a node fee divided by the number of times all high value accounts have been the beneficiary of a node fee, during the
• Activity Score (\(A\)): Average of the \(T\) and \(N\) scores weighted 80% and 20% respectively, divided by the account’s balance. Dividing by the account’s balance gives some boost to small
accounts, because their importance score will depend more on their activity and less on their stake.
An absolute activity score (\(A'\)) is calculated first:
\[A' = \frac{10000}{Balance}(0.8T+0.2N)\]
And the actual activity score (\(A\)) is calculated by dividing \(A'\) by the sum of the absolute activity scores of all high value accounts.
The importance score is then calculated based on the above partial scores.
Importance score
The importance score \(I\) is calculated as the average of the \(S\) and \(A\) scores, weighted by an activity factor \(\gamma\):
\[I = \gamma A + (1-\gamma)S\]
In the Bitxor network \(\gamma\) is 0.05 (5%)
Finally, among all accounts eligible for harvesting, the probability that a particular one is chosen is proportional to its effective importance score, which is defined as the smaller of the previous
two importance scores \(I\).
Since scores are calculated every 720 blocks (roughly 6 hours) and the smaller of the previous two scores is used when calculating harvesting probabilities, when you first fund an account it will
require 12 hours to have a probability greater than zero.
Private networks can customize the consensus algorithm by changing the following configuration properties. See Network configuration.
Property Default Description
importanceGrouping 720 blocks How often importance is calculated.
minHarvesterBalance 10000 Minimum balance required to be eligible for harvesting.
importanceActivityPercentage 0.05 Contribution of the activity score (\(\gamma\)). When it is 0, PoS+ consensus behaves like conventional PoS. | {"url":"https://docs.bitxor.org/en/concepts/consensus-algorithm.html","timestamp":"2024-11-11T15:24:51Z","content_type":"text/html","content_length":"25171","record_id":"<urn:uuid:90edf412-1147-4ebf-bac8-f932ccaa76f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00478.warc.gz"} |
Molecular Vision: Wang, Mol Vis 2019; 25:583-592. Figure 5
Figure 5. Correlation of IL-33, IL-13, and IL-5 levels with ocular surface parameters. Only the eight graphs that indicated statistically significant correlation are presented. A–D: In the DE group,
the level of IL-33 positively correlated with the OSDI score (R=0.4483, P=0.0415, n=21) and CFS (R=0.5657, P=0.0075, n=21) while negatively correlated with the Schirmer I test (R=-0.4448, P=0.0433, n
=21) and BUT (R=-0.4459, P=0.0427, n=21). E–F: The level of IL-13 positively correlated with the CFS (R=0.5005, P=0.0208, n=21) while negatively correlated with the Schirmer I test (R=-0.4348, P=
0.0433, n=21). G–H: The level of IL-5 positively correlated with both OSDI score (R=0.4551, P=0.0382, n=21) and CFS (R=0.5261, P=0.0143, n=21). The levels of IL-33, IL-13, IL-5 in tears showed no
correlation with ocular surface parameters in controls. Pearson correlation coefficients. DE, dry eye. | {"url":"http://www.molvis.org/molvis/v25/583/mv-v25-583-f5.html","timestamp":"2024-11-10T21:54:25Z","content_type":"text/html","content_length":"2816","record_id":"<urn:uuid:86a54415-a977-4cb9-9b58-1e534d7532b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00820.warc.gz"} |
Java Daily Coding Problem #004
Daily Coding Problem is a website which will send you a programming challenge to your inbox every day. I want to show beginners how to solve some of these problems using Java, so this will be an
ongoing series of my solutions. Feel free to pick them apart in the comments!
Given an array of integers, find the first missing positive integer in linear time and constant space. In other words, find the lowest positive integer that does not exist in the array. The array
can contain duplicates and negative numbers as well.
For example, the input [3, 4, -1, 1] should give 2. The input [1, 2, 0] should give 3.
You can modify the input array in-place.
Spoilers! Don't look below unless you want to see my solution!
The way I interpret this problem is that we're looking for the smallest integer value, greater than 0, which does not appear in the array. So if an array contains 1, 2, and 3, but not 4, the code
should return 4. Duplicates and negative numbers can be ignored. The maximum value which should be returned is the length of the array plus one, as shown in the second example, so the solution will
always be between 1 and N+1, inclusive, where N is the length of the given array.
My first thought is to create a new array of length N, fill it with falses, and flip those falses to true if the index exists in the given array. But this solution is not a constant-space solution,
as it requires an array of length N. (Though it is linear in time, since we only need to pass through the original array once.) What can we do instead?
The example arrays aren't sorted, so we can't make any assumption that they will be in general. In researching a constant-space, linear-time sorting algorithm, I found this SO answer, which
prescribes the following algorithm that fits those requirements:
1. Start at the first array element.
2. If its array index matches its value, go to the next one.
3. If not, swap it with the value at the array index corresponding to its value.
4. Repeat step 3, until no more swaps are necessary.
5. If not at the end of the array, go to the next array element, otherwise go to step 7
6. Go to step 2.
7. Done
Note that there is a problem with this for our purposes -- our array can contain duplicates and negative numbers. Suppose the integer at index 0 is 7, and the integer at index 7 is also 7 -- we would
be stuck in an infinite loop! So this solution is also out.
This is a tough problem!
The prompt says that we can modify the input array in place, which -- to me -- seems like the only way we can keep this algorithm as constant space complexity. (Apart from making a true/false array,
as described above, with a length equal to Integer.MAX_VALUE.) So I think we need to do something similar to the above, but not exactly the same.
After a bit more research, I found this solution to this problem, which suggests simply ignoring the negative values. I wonder if that will work? So the (slightly clarified) algorithm would be
1. Move to the next element of the array (or start at the beginning of the array if this is the first time that this step has been encountered):
2. If this is the last element of the array, move to step 5.
3. If this element's array index matches its value, or if its value is zero or negative, go back to step 1. Otherwise, continue to step 4.
4. Swap this element's value with the value contained at the index represented by this element's value and go back to step 3.
5. Step through the array again, from the beginning, and compare the array indices (1-based) to the values they contain. If the index and value do not match, that index is the smallest positive
integer value which is not contained within the array.
For the example array given in the prompt, this process looks like (all 1-based indices):
[ 3, 4, -1, 1] -- original array
[-1, 4, 3, 1] -- after swapping 1st and 3rd elements
[-1, 1, 3, 4] -- after swapping 2nd and 4th elements
[ 1, -1, 3, 4] -- after swapping 2nd and 1st elements
Then, we can just walk the array a second time to find the first element whose value doesn't match its index. Walking the array twice requires N + N time, or 2N (aka. linear) time. Since we're
modifying the array in place, this is also a constant space solution. Let's consider another example:
The algorithm should return 3 for this array. But when we get to the third element (8), we have a problem: the array has fewer than 8 elements. We need to add another condition to step 3 of our
If this element's array index matches its value, or if its value is zero, negative, or larger than the length of the array, go back to step 1. Otherwise, continue to step 4.
Stepping through this second example:
[ -1, -1, 8, 1, 2, 2] -- original array
[ 1, -1, 8, -1, 2, 2] -- after swapping 4th and 1st elements
[ 1, 2, 8, -1, -1, 2] -- after swapping 5th and 2nd elements
[ 1, 2, 8, -1, -1, 2] -- ...
And here we run into another problem -- duplicates. We will have an infinite loop if we keep trying to swap the 6th and 2nd elements in the above example. So we should check if the "target" and
"source" value are the same, and if they are, move to the next element of the array:
Swap this element's value with the value contained at the index represented by this element's value, unless the two values are the same, in which case go back to step 1.
So the final algorithm looks like:
1. Move to the next element of the array (or start at the beginning of the array if this is the first time that this step has been encountered):
2. If this is the last element of the array, move to step 5.
3. If this element's array index matches its value, or if its value is zero, negative, or larger than the length of the array, go back to step 1. Otherwise, continue to step 4.
4. Swap this element's value with the value contained at the index represented by this element's value, unless the two values are the same, in which case go back to step 1.
5. Step through the array again, from the beginning, and compare the array indices (1-based) to the values they contain. If the index and value do not match, that index is the smallest positive
integer value which is not contained within the array.
...I think this should work. Let's try coding it!
So the first thing I do is set up a shell class. We'll want a method which finds the first missing integer and maybe a main which runs the two examples in the prompt:
public class DCP004 {
public static void main (String[] args) {
findSmallestMissing(new int[]{ 3, 4, -1, 1 });
findSmallestMissing(new int[]{ 1, 2, 0 });
public static int findSmallestMissing (int[] array) {
// to be implemented
return -1;
Now, within findSmallestMissing(), we'll want to implement our algorithm outlined above. First, let's set up a loop to move through the array:
public static int findSmallestMissing (int[] array) {
// if array is null or empty, 1 is the smallest missing number
if (array == null || array.length < 1) return 1;
// get the length of the array
int len = array.length;
// assume 1 is smallest missing number until proven otherwise
int smallestMissing = 1;
// loop over input array (steps 1 and 5)
for (int ii = 0; ii < len; ++ii) {
// implement steps 2-4 in here
// to be implemented
return -1;
Within that loop is where most of the work of this algorithm happens:
// loop over input array (steps 1 and 5)
for (int ii = 0; ii < len; ++ii) {
// step 3 (make sure to use 1-based indexing)
if (array[ii] == (ii+1) || array[ii] < 1 || array[ii] > len)
// step 4
while (array[ii] != ii) {
// index of the element to swap with
int swap = array[ii]-1;
// value of the element to swap with
int temp = array[swap];
// swap the values
array[swap] = array[ii];
array[ii] = temp;
// if the new value is < 1 or > len, move to next one
if (temp < 1 || temp > len) break;
Since we've modified array in place, the last thing we need to do is loop through it and find the first element whose value doesn't match its (base-1) index:
// loop over modified array
for (int ii = 0; ii < len; ++ii) {
if (array[ii] != ii+1) return ii+1;
return len+1;
I've written this code in my markdown editor without testing it, so let's paste it all together and see if it works!
jshell> /open DCP004.java
jshell> DCP004.main(new String[]{})
jshell> DCP004.findSmallestMissing(new int[]{1, 2, 0})
$3 ==> 3
jshell> DCP004.findSmallestMissing(new int[]{3, 4, -1, 1})
$4 ==> 1
...alright, so we've got a few small issues. First, in main, nothing is printed. Let's add System.out.println statements around those function calls in main:
public static void main (String[] args) {
System.out.println(findSmallestMissing(new int[]{ 3, 4, -1, 1 }));
System.out.println(findSmallestMissing(new int[]{ 1, 2, 0 }));
Next, it looks like the first array returns the correct result (3), but we have a problem with the second array. It's returning 1 when it should be returning 2. Is this just an off-by-one error?
Let's try another array:
jshell> DCP004.findSmallestMissing(new int[]{2, 4, -1, 1})
$7 ==>
...this has no return value because I had to kill it -- I think there was an infinite loop! I've found one problem at least. The following line:
while (array[ii] != ii) {
...should instead be
while (array[ii] != (ii+1)) {
That seems to have fixed the problem!
jshell> DCP004.findSmallestMissing(new int[]{2, 4, -1, 1})
$9 ==> 3
jshell> DCP004.findSmallestMissing(new int[]{3, 4, -1, 1})
$10 ==> 2
There is another issue, though. This array also gives an infinite loop:
jshell> DCP004.findSmallestMissing(new int[]{1, 1, 2, 2, 3, 5})
$11 ==>
...what did we miss this time? To debug, let's print out the array before each swap in the while loop:
// step 4
while (array[ii] != (ii+1)) {
// DEBUG: print array
// index of the element to swap with
int swap = array[ii]-1;
The output for the last example looks like:
[1, 1, 2, 2, 3, 5]
[1, 1, 2, 2, 3, 5]
[1, 1, 2, 2, 3, 5]
[1, 1, 2, 2, 3, 5]
[1, 1, 2, 2, 3, 5]
[1, 1, 2, 2, 3, 5]
It looks like nothing was ever swapped! (Or, the two 1s at the beginning are swapped over and over again.) It looks like we forgot to break out of the while when the two swapped values are identical.
Let's add a catch for that:
// if the new value is < 1 or > len, move to next one
if (temp < 1 || temp > len) break;
// if new value equals old value, move to next one
if (temp == array[ii]) break;
...has that fixed the problem?
jshell> DCP004.findSmallestMissing(new int[]{1, 1, 2, 2, 3, 5})
[1, 1, 2, 2, 3, 5]
[1, 1, 2, 2, 3, 5]
[1, 2, 1, 2, 3, 5]
[1, 2, 1, 2, 3, 5]
[1, 2, 3, 2, 1, 5]
$15 ==> 4
jshell> DCP004.findSmallestMissing(new int[]{-1, -1, 8, 1, 2, 4})
[-1, -1, 8, 1, 2, 4]
[1, -1, 8, -1, 2, 4]
[1, 2, 8, -1, -1, 4]
$16 ==> 3
...it looks so!
There we go! A solution in constant time and linear space which also prints the modified array before each swap for diagnostic purposes. It works fine on the example arrays and the few additional
arrays I've thrown at it.
This problem took a bit of thinking to stick to the constraints, but it was fun! Can anyone find any arrays that break my code?
All the code for my Daily Coding Problems solutions is available at github.com/awwsmm/daily.
Suggestions? Let me know in the comments.
Top comments (3)
l-Suraj-Patil-l • • Edited on • Edited
I tried using minimum and maximum value to get the solution.
public static int minValue(int[] i)
int minValue=i[0];
for(int x : i)
if(x < minValue)
minValue = x;
return minValue;
public static int maxValue(int[] i)
int maxValue=i[0];
for(int x : i)
if(x > maxValue)
maxValue = x;
return maxValue;
public static int findSmallestMissingValue(int[] x)
int missingValue = 0;
for(int i = minValue(x); i <= maxValue(x); i ++)
boolean flag = false;
for(int j = 0; j < x.length ; j ++)
flag = true;
missingValue = i;
return missingValue;
Andrew (he/him) •
What are the time and space complexities of your solution?
Anthony Breeganzo •
We can solve this problem in a simple way but time is O(N log N)
We just need to sort the array and start checking from 1 if there is any number missing.
public static int fmp(int[] a)
int n=a.length;
int ans=1;
for(int i=0;i<n;i++)
return ans;
For further actions, you may consider blocking this person and/or reporting abuse | {"url":"https://dev.to/awwsmm/java-daily-coding-problem-004-2ogb","timestamp":"2024-11-07T16:33:06Z","content_type":"text/html","content_length":"151803","record_id":"<urn:uuid:8bc00c4b-9ef8-45ec-ad77-018fe37c40f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00313.warc.gz"} |
Weyl tensor
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In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature
tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on
how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. The Ricci curvature, or trace component of the Riemann tensor contains precisely the
information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor. It is a tensor that has the same symmetries as the Riemann
tensor with the extra condition that it be trace-free: metric contraction on any pair of indices yields zero.
In general relativity, the Weyl curvature is the only part of the curvature that exists in free space—a solution of the vacuum Einstein equation—and it governs the propagation of gravitational waves
through regions of space devoid of matter.^[1] More generally, the Weyl curvature is the only component of curvature for Ricci-flat manifolds and always governs the characteristics of the field
equations of an Einstein manifold.^[1]
In dimensions 2 and 3 the Weyl curvature tensor vanishes identically. In dimensions ≥ 4, the Weyl curvature is generally nonzero. If the Weyl tensor vanishes in dimension ≥ 4, then the metric is
locally conformally flat: there exists a local coordinate system in which the metric tensor is proportional to a constant tensor. This fact was a key component of Nordström's theory of gravitation,
which was a precursor of general relativity.
The Weyl tensor can be obtained from the full curvature tensor by subtracting out various traces. This is most easily done by writing the Riemann tensor as a (0,4) valence tensor (by contracting with
the metric). The (0,4) valence Weyl tensor is then (Petersen 2006, p. 92) ${\displaystyle C=R-{\frac {1}{n-2}}\left(\mathrm {Ric} -{\frac {s}{n}}g\right)\wedge \!\!\!\!\!\!\bigcirc g-{\frac {s}{2n
(n-1)}}g\wedge \!\!\!\!\!\!\bigcirc g}$ where n is the dimension of the manifold, g is the metric, R is the Riemann tensor, Ric is the Ricci tensor, s is the scalar curvature, and ${\displaystyle h\
wedge \!\!\!\!\!\!\bigcirc k}$ denotes the Kulkarni–Nomizu product of two symmetric (0,2) tensors:
${\displaystyle (h\wedge \!\!\!\!\!\!\bigcirc k)(v_{1},v_{2},v_{3},v_{4})=}$ ${\displaystyle h(v_{1},v_{3})k(v_{2},v_{4})+h(v_{2},v_{4})k(v_{1},v_{3})\,}$
${\displaystyle {}-h(v_{1},v_{4})k(v_{2},v_{3})-h(v_{2},v_{3})k(v_{1},v_{4})\,}$
In full tensor notation, this can be written as ${\displaystyle C_{ik\ell m}=R_{ik\ell m}+{\frac {1}{n-2}}\left(R_{im}g_{k\ell }-R_{i\ell }g_{km}+R_{k\ell }g_{im}-R_{km}g_{i\ell }\right)+{\frac {1}
{(n-1)(n-2)}}R\left(g_{i\ell }g_{km}-g_{im}g_{k\ell }\right).\ }$
The ordinary (1,3) valent Weyl tensor is then given by contracting the above with the inverse of the metric.
The decomposition (1) expresses the Riemann tensor as an orthogonal direct sum, in the sense that
${\displaystyle |R|^{2}=|C|^{2}+\left|{\frac {1}{n-2}}\left(\mathrm {Ric} -{\frac {s}{n}}g\right)\wedge \!\!\!\!\!\!\bigcirc g\right|^{2}+\left|{\frac {s}{2n(n-1)}}g\wedge \!\!\!\!\!\!\bigcirc g\
This decomposition, known as the Ricci decomposition, expresses the Riemann curvature tensor into its irreducible components under the action of the orthogonal group (Singer & Thorpe 1968). In
dimension 4, the Weyl tensor further decomposes into invariant factors for the action of the special orthogonal group, the self-dual and antiself-dual parts C^+ and C^−.
The Weyl tensor can also be expressed using the Schouten tensor, which is a trace-adjusted multiple of the Ricci tensor,
${\displaystyle P={\frac {1}{n-2}}\left(\mathrm {Ric} -{\frac {s}{2(n-1)}}g\right).}$
${\displaystyle C=R-P\wedge \!\!\!\!\!\!\bigcirc g.}$
In indices,^[2]
${\displaystyle C_{abcd}=R_{abcd}-{\frac {2}{n-2}}(g_{a[c}R_{d]b}-g_{b[c}R_{d]a})+{\frac {2}{(n-1)(n-2)}}R~g_{a[c}g_{d]b}}$
where ${\displaystyle R_{abcd}}$ is the Riemann tensor, ${\displaystyle R_{ab}}$ is the Ricci tensor, ${\displaystyle R}$ is the Ricci scalar (the scalar curvature) and brackets around indices refers
to the antisymmetric part. Equivalently,
${\displaystyle {C_{ab}}^{cd}={R_{ab}}^{cd}-4S_{[a}^{[c}\delta _{b]}^{d]}}$
where S denotes the Schouten tensor.
Conformal rescaling[edit]
The Weyl tensor has the special property that it is invariant under conformal changes to the metric. That is, if ${\displaystyle g_{\mu u }\mapsto g'_{\mu u }=fg_{\mu u }}$ for some positive scalar
function ${\displaystyle f}$ then the (1,3) valent Weyl tensor satisfies ${\displaystyle {C'}_{\ \ bcd}^{a}=C_{\ \ bcd}^{a}}$. For this reason the Weyl tensor is also called the conformal tensor. It
follows that a necessary condition for a Riemannian manifold to be conformally flat is that the Weyl tensor vanish. In dimensions ≥ 4 this condition is sufficient as well. In dimension 3 the
vanishing of the Cotton tensor is a necessary and sufficient condition for the Riemannian manifold being conformally flat. Any 2-dimensional (smooth) Riemannian manifold is conformally flat, a
consequence of the existence of isothermal coordinates.
Indeed, the existence of a conformally flat scale amounts to solving the overdetermined partial differential equation
${\displaystyle Ddf-df\otimes df+\left(|df|^{2}+{\frac {\Delta f}{n-2}}\right)g=\operatorname {Ric} .}$
In dimension ≥ 4, the vanishing of the Weyl tensor is the only integrability condition for this equation; in dimension 3, it is the Cotton tensor instead.
The Weyl tensor has the same symmetries as the Riemann tensor. This includes:
${\displaystyle C(u,v)=-C(v,u)_{}^{}}$
${\displaystyle \langle C(u,v)w,z\rangle =-\langle C(u,v)z,w\rangle _{}^{}}$
${\displaystyle C(u,v)w+C(v,w)u+C(w,u)v=0_{}^{}.}$
In addition, of course, the Weyl tensor is trace free:
${\displaystyle \operatorname {tr} C(u,\cdot )v=0}$
for all u, v. In indices these four conditions are
${\displaystyle C_{abcd}^{}=-C_{bacd}=-C_{abdc}}$
${\displaystyle C_{abcd}+C_{acdb}+C_{adbc}^{}=0}$
${\displaystyle {C^{a}}_{bac}=0.}$
Bianchi identity[edit]
Taking traces of the usual second Bianchi identity of the Riemann tensor eventually shows that
${\displaystyle abla _{a}{C^{a}}_{bcd}=2(n-3)abla _{[c}S_{d]b}}$
where S is the Schouten tensor. The valence (0,3) tensor on the right-hand side is the Cotton tensor, apart from the initial factor.
See also[edit] | {"url":"https://static.hlt.bme.hu/semantics/external/pages/tenzorszorzatok/en.wikipedia.org/wiki/Weyl_tensor.html","timestamp":"2024-11-03T23:20:10Z","content_type":"text/html","content_length":"114925","record_id":"<urn:uuid:1c4305b9-c235-46c9-8a2c-f9d6055376b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00727.warc.gz"} |
23. The taxi fare in a city is as follows: For the first kilome... | Filo
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23. The taxi fare in a city is as follows: For the first kilometre, the fare is Rs.8 and for the subsequent distance it is Rs.5 per km. Taking the distance covered as and total fare as Rs. , write a
linear equation for this information. OR If the point lies on the graph of the equation , find the value of a?
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Question 23. The taxi fare in a city is as follows: For the first kilometre, the fare is Rs.8 and for the subsequent distance it is Rs.5 per km. Taking the distance covered as and total fare as
Text Rs. , write a linear equation for this information. OR If the point lies on the graph of the equation , find the value of a?
Updated On Mar 16, 2023
Topic All topics
Subject Mathematics
Class Class 9
Answer Type Video solution: 1
Upvotes 85
Avg. Video 11 min | {"url":"https://askfilo.com/user-question-answers-mathematics/23-the-taxi-fare-in-a-city-is-as-follows-for-the-first-34363231353338","timestamp":"2024-11-08T02:50:12Z","content_type":"text/html","content_length":"175611","record_id":"<urn:uuid:6a5ccd7d-1a82-4e2e-83ef-a2496eb4ad39>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00483.warc.gz"} |
Math Games for Middle School
top of page
Make 100 Percent
Engaging students in learning about percentages can be challenging, but our Make 100 Percent game makes it fun and interactive. This free print and play game is designed for Year 4, Year 5, and Year
6 students and is perfect for reinforcing percentage concepts in a playful setting. You can download the game by clicking the button at the bottom of the page.
What is the Make 100 Percent Game?
Make 100 Percent is a hands-on game that
helps students understand and practice percentages. It is played with 2 or 3 players, and each player takes turns spinning a wheel to determine the percentage they need to color in on a 10 by 10
How to Play
1. Set Up:
□ Print the game sheets provided in the download.
□ Each player needs a 10 by 10 grid.
□ A spinner (teachers can attach a spinner to the game board or use a makeshift spinner by spinning a paperclip around the tip of a pencil).
2. Gameplay:
□ Each player takes a turn to spin the wheel.
□ The spinner will land on a percentage (e.g., 10%).
□ Players color in the corresponding number of boxes on their grid (each box represents 1%, so for 10%, they color in 10 boxes).
3. Winning the Game:
□ Continue taking turns until all the boxes on the grid are filled.
□ Players add up their total percentage colored in.
□ The player with the highest overall percentage is the winner.
Benefits of the Make 100 Percent Game
• Interactive Learning: The game engages students in a hands-on activity, making learning about percentages enjoyable and memorable.
• Visual Representation: Coloring in the grid helps students visualize percentages and understand their value.
• Collaborative Play: Encourages teamwork and friendly competition among students.
• Skill Reinforcement: Reinforces percentage calculation and helps students practice addition of percentages in a fun way.
Why Use the Make 100 Percent Game?
• Effective Learning Tool: Helps students grasp the concept of percentages through practical application.
• Easy to Implement: Simple setup and instructions make it easy for teachers to integrate into lessons.
• Flexible Use: Suitable for classroom activities, math centers, or homework.
How to Get Started
Ready to make learning about percentages fun and interactive? Download the free Make 100 Percent game by clicking the button below. Print the game sheets, prepare the spinners, and let the fun begin!
bottom of page | {"url":"https://www.smartboardingschool.com/make-100-percent","timestamp":"2024-11-10T01:49:53Z","content_type":"text/html","content_length":"1040087","record_id":"<urn:uuid:43248199-a502-42a4-aa0a-72aed19dc9ff>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00604.warc.gz"} |
Data Set
A collection of data. Most commonly a data set corresponds to the contents of a single database table, or a single statistical data matrix, where every column of the table represents a particular
variable, and each row corresponds to a given member of the data set in question. The data set lists values for each of the variables, such as height and weight of an object, for each member of the
data set. Each value is known as a datum. The data set may comprise data for one or more members, corresponding to the number of rows. | {"url":"https://www.openmv.org/glossary/data-set/","timestamp":"2024-11-02T02:17:36Z","content_type":"text/html","content_length":"278680","record_id":"<urn:uuid:6383c359-7c41-44a7-8635-20a8addc2f64>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00342.warc.gz"} |
A slightly question about EvalDivide function
The first question I’d like to ask is that in cryptocontext.h I see the EvalMult function described as Evaluate approximate division function 1/x where x >= 1 on a ciphertext using the Chebyshev
approximation. But in MWE I used x= -5 to do the calculation without any error and the result was calculated (but incorrectly). Is this allowed behavior or a bug?
The second question is whether I have set this parameter incorrectly and when I change x to 5 the calculation is still incorrect. I don’t think the upper and lower bounds for MWE are set incorrectly.
Is the result error because of the degree issue?
Here is a MWE that reproduces the issue:
int main() {
uint32_t multDepth = 5;
uint32_t scaleModSize = 58;
uint32_t batchSize = 8;
CCParams<CryptoContextCKKSRNS> parameters;
CryptoContext<DCRTPoly> cc = GenCryptoContext(parameters);
std::cout << "CKKS scheme is using ring dimension " << cc->GetRingDimension() << std::endl << std::endl;
auto keys = cc->KeyGen();
cc->EvalRotateKeyGen(keys.secretKey, {1, -2});
std::vector<double> x1 = {-5};
Plaintext ptxt1 = cc->MakeCKKSPackedPlaintext(x1);
// Plaintext ptxt2 = cc->MakeCKKSPackedPlaintext(x2);
std::cout << "Input x1: " << ptxt1 << std::endl;
// std::cout << "Input x2: " << ptxt2 << std::endl;
auto c1 = cc->Encrypt(keys.publicKey, ptxt1);
// auto c2 = cc->Encrypt(keys.publicKey, ptxt2);
double lowerBound = -10;
double upperBound = 10;
// std::complex<double> invalid_coefficient(std::nan(""), 0.0);
std::vector<double> coefficients = {1e-10, -1.0e-10};
auto result = cc->EvalDivide(c1, lowerBound, upperBound, 1);
Plaintext res;
std::cout << std::endl << "Results of homomorphic computations: " << std::endl;
cc->Decrypt(keys.secretKey, result, &res);
std::cout << res << std::endl;
return 0;
I would really appreciate any insights or explanations regarding this behavior.
Thank you for your help!
Best regards,
Just my 2 cents here: perhaps it is not the right choice to use [-5,5] as the approximation interval since it contains x=0, where 1/x is not defined.
Plus, the fourth argument of EvalDivide is the polynomial degree, which should be set to a larger value, otherwise you are just creating a line.
I refer you to: https://openfhe.discourse.group/t/how-to-evaluate-approximate-1-x-function-with-high-precision/
Hi @wowblk8,
EvalDivide will only work correctly for x\ge1, as written in the documentation. As @narger noted, there is a discontinuity at 0, which complicates the division for the range you are trying to use.
The last argument should be much higher than 1 to achieve reasonable precision (when the right range is used).
I see. but I tried some experiments with x <= 0 and lower/upper bound set to the negative scale. Such as I set x = -5 and lower/upper bound to (-1, -5). Then the calculated result is right. On the
other hand, I tried the same number but with a positive one. Like x = 5 and lower/upper bound = (1, 5). they come up with the same result but positive one. So I think there is a gap between
EvalDivide. EvalDivide function actually can calculate the negative divide equation. So I think the description is wrong or when in situations evaldivide function can’t guarantee OpenFHE library
security. Anyways, I think the EvalDivide function is based on the EvalChebyshevSeries function, it should calculate -1…-∞ except 0. If the developer has any concerns about the negative side of the
EvalDivide function. I think it should throw an error when the user setting to a negative one（value and lower/upper bound).
It is indeed normal for the Chebyshev approximation of division to work on intervals [-5, -1] and [1, 5]. The problem is when the interval contains 0, as division is not supported there. I think
Yuriy meant |x| \geq 1, not x \geq 1. In theory, it should be |x| \geq \epsilon, but the smaller the \epsilon, the larger the required degree of the approximation.
@andreea.alexandru Thank you for correcting. Indeed, I meant |x| \ge 1. | {"url":"https://openfhe.discourse.group/t/a-slightly-question-about-evaldivide-function/1624","timestamp":"2024-11-12T15:33:32Z","content_type":"text/html","content_length":"34803","record_id":"<urn:uuid:d4369033-24ac-4d3e-9e59-e67c173e0de8>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00557.warc.gz"} |
QC—How to build a Quantum Computer with Superconducting Circuit?
QC — How to build a Quantum Computer with Superconducting Circuit?
In quantum computers, many university research groups bet on trapped ions. But the industrial giants do not necessarily agree with that. Indeed, the superconducting circuit seems to be their top
choice. Ironically, some of those decisions are not completely based on technical merit. Many universities have strong expertise in Atomic Physics and have the know-how on manipulate quantum
information at the atomic level. But scaling up the solution is not necessarily their strength. On the other hand, many industrial corporations acquire semiconductor experts with years of experience
in scaling systems. Instead of using atoms to store quantum information, engineers print artificial “quantum system” in a circuit for the qubits (where quantum computers store information- please
spend some time on qubit if you are not familiar with it). Therefore, different organizations may adopt different approaches depending on their expertise. In this article, we will focus on the
superconducting circuit and reserve another article for the trapped ion quantum computer. But we will have a high-level overview for some of the most promising approaches at the end of the article.
Nobel prizes were given (including Superconducting, Josephson Junction) for theories mentioned in this article. So feel free to skip some details if it is not explained thoroughly.
Quantum Processors
We have many ways to implement qubits. We can explore the quantum property of atoms, or we can build an artificial quantum system. A trapped ions quantum computer uses lasers to change the energy
level of laser-cooled ions trapped in an electric field. On the other hand, IBM Q quantum computer holds superconducting circuits to create a quantum system.
In a superconducting circuit computer, the quantum processor is the soul of the computer. The square block below holds four qubits.
Modified from IBM source
This processor is host at the bottom of a cylinder below.
Left (Inside IBM 50 qubits quantum computer) Right (enclosing the quantum computer)
As often, to operate at the precision of the quantum scale, we need to go big. The IBM Q contains cables to send microwave pulses at different frequencies and durations to control and to measure the
qubits. The qubit information is easily destroyed by thermal heats and other disturbance from the environment. To isolate the qubits, the computer contains a dilution refrigerator to cool down the
quantum processor to 15 miliKelvin. In addition, the circuitry contains superconductor requiring an extremely low temperature to operate.
IBM Q (IBM Quantum computer)
We don’t want our circuitry implementing the artificial quantum system to have resistance. Otherwise, it will dissipate energy and destroy the quantum information. So it is constructed with
superconducting material which has zero resistance when cooled below a certain temperature (about 1K° for superconducting aluminum).
When the temperature drops below a critical value, two electrons form a weak bound and becomes a Cooper pair that experience no resistance when traveling through the metal. The pairing opens a gap in
the energy state, which any excitation requires some minimum energy. This gap leads to superconductivity since not any random increase in energy is allowed. Many excitations such as scattering of
electrons (resistance) that lead to an illegal energy state are not allowed (yet another non-intuitive behavior from quantum mechanics).
Superconducting circuitry
We build semiconductor circuits for the qubits. Each qubit is actually an LC circuit, an inductor, and a capacitor. We manipulate its energy state to represent a superposition of |0⟩ and|1⟩.
The energy level can be modeled as a quantum harmonic oscillator with quantized energy level.
Our first challenge is the energy difference between levels is evenly spaced. As recalled in a trapped ions computer, its energy levels are uneven. The control signal that promoting quantum state |0⟩
to |1⟩ will not accidentally promote the quantum state to a higher level, the superposition will confine between |0⟩ and |1⟩.
Josephson Junction
To overcome that, the superconducting circuit includes a Josephson Junction.
The junction behaves as a non-linear non-dissipating inductor. It contains two Aluminum superconducting electrodes which are weakly couple and is separated by a thin insulator about a thousandth of a
hair thick.
It is non-linear such that the energy level is unevenly separated so we can use two lower states as the bases for our superpositions. In short, the junction provides the non-linearity such that we
can control the states unambiguously.
This inductor is combined with a linear capacitor using Niobium superconductor to create an LC resonator.
Modified from IBM source
With correct tuning, the circuit behaves like an atom with two quantum energy level, i.e. our qubit.
The qubit transition frequency depends on the capacitances and inductances in the circuit. Quantum operations are performed by sending electromagnetic impulses at microwave frequencies (around 4–6
KHz) to the resonator coupled to the qubit. This frequency resonates with the energy separation between the energy levels for |0⟩ and |1⟩.
Modified from IBM source
And the duration of the pulse controls the angle of rotation of the qubit state around a particular axis of the Bloch sphere.
Therefore, different pulses form different quantum gates.
Source: IBM
In Trapped ions computer, we use laser beams to control ions. To address two arbitrary ions in the string of trapped ions, it can be done with two laser beams (the red ones). This method can entangle
two arbitrary qubits on the string of ions.
In superconducting qubits, the coupling of two qubits is done by coupling them to a quantum bus. We print a superconducting circuit in a 2-D space (relatively speaking). To enable entanglement, a
separate bus is used to couple two neighboring qubits. Connecting all two qubits together with a resonator is hard. Therefore, not all qubits are connected with each other.
For example, for the 5 qubits IBM Q, there are 20 possible combinations in pairing qubits, but only 6 of them are implemented. Here is the connection supported by the IBM Q20 (20-qubits).
IBM Q 20 Tokyo (20-qubits) — Modified from source
To make a measurement, it sends a microwave tone to the resonator and analyzes the signal it reflects back. The amplitude and phase of the reflected signal depend on the qubit state. Once it is
amplified, we know the energy level and therefore we can determine the state of the qubit.
Type of qubit
We need to map |0⟩ and |1⟩ to two different energy states of the physical system. There are three major ways to do it in a superconducting quantum computer: Charge Qubit, Flux Qubit & Phase Qubit. In
the charge qubit, different energy levels correspond to an integer number of Cooper pairs on a superconducting island.
This creates an artificial “quantized” system.
The IBM Q computer is a Transmon qubit: an improved charge qubit in addressing the charging noise.
Dilution refrigerator
Maintain an extremely low temperature is important in a superconducting quantum computer. This dilution refrigerator has a system of pipes that contain a mixture of two helium isotopes (isotope ③ and
④). As the lighter isotope ③ is diluted into the heavier isotopes ④ inside the mixing chamber below, it absorbs heat as the entropy increases — a mixed solution is more disorder and has higher
entropy. So the temperature surrounding the mixing chamber will drop. The mixing chamber is connected to the upper distilling chamber. Since isotopes ③ has a higher boiling point, it will vaporize.
This reduces the concentration of isotopes ③ and draws more isotopes ③ to be diluted into the mixture in the other end, i.e. inside the mixing chamber. So the cooling process creates a cycling
Modified from source
The diagram below shows how the IBM Q’s dilution refrigerator cools temperature from 4° Kelvin to 15 Millilevins at the Cryoperm shield (lower right). This shield hosts the quantum processor. This
diagram also shows the cables used to send microwave pulses down to the processor and how it readouts measurements using the amplifier on the left.
Source: IBM
The quantum processor sits inside the Cryoperm shield, a light-tight and a magnetic-field shield, used to isolate the qubits from environmental disturbances. The following is an end-to-end flow on
how microwave pulses are sent down to control the qubits and how the qubits are readout.
Source: IBM
Here is another view of the IBM Q machine and the quantum chip.
Source: IBM
The quality of a quantum computer is often measured by its decoherence. T1 and T2 measures how fast the quantum information stored in the qubit may lose. As shown below, we make very good progress in
the superconducting quantum computers over the years.
Source: IBM
Tapped ions & superconducting circuits
Trapped ions quantum computer is another popular realization of quantum computers. We trap ions (for example, positively charged Calcium ions) with an oscillating electrical field inside a
high-vacuum chamber. We laser-cooled the ions so it is close to stationary. A string of ions is formed and float between electrodes. Scientists have studied Atomic Physics for a century. We know its
different energy states and how to manipulate between them. i.e., we know how to use these ions to create qubits. To manipulate and to measure the qubits, we shine lasers of different frequency and
duration to the ions.
Source: University of Innsbruck
The following is a recap on the superconducting and trapped ions quantum computer:
Trapped ions have longer coherence time compared to a superconducting circuit but the gate operation time is faster in the superconducting circuit. Google has built a 72-qubit superconducting quantum
computer in March 2018. As we write this article (Dec. 2018), IonQ has just announced a 79-qubit quantum computer. The competition is fierce and in an early phase. It is hard to determine who is the
winner for now. Trapped ions need a vacuum chamber while the superconducting needs a diluted refrigerator. Trapped ions are all “natural” and identical while the gate performance for each qubit in
the superconducting computer is slightly different. Not all qubits in a superconducting computer are connected to form 2-qubit operations. But, as the number of trapped ions increases, trapped ions
are suspectable to noises and the error rates become intolerable. Superconducting circuits work with microwaves while trapped ions system often involves lasers which are harder to integrate into the
Other technologies
There are over a dozen other technologies. Let’s have a quick overview.
Neutral Atoms
Neutral atoms quantum computers use neutral atom instead of charged ions in the trapped ions computer. Neutral atoms can be held in close confinement to create complex 2-D or 3-D qubit array. This
helps to scale the number of qubits.
In fact, we can construct complex 3-D structures which cannot be done with trapped ions due to strong interactions.
D. Barredo Et al/Nature 2018
Atoms do not interact with others so it can be a bad candidate for the purpose of building 2-qubit operations. But with timed laser pulses, the outermost electron can be excited. It inflates the atom
to billion times and reaches the Rydberg state — which the electrons are excited to a high principal quantum number close to 100 and become sensitive to external influences including microwave
radiation. i.e. it behaves more like an ion that can interact with neighboring atoms. This behavior will be exploited to create entanglement. Currently, researchers are still working on the gate
fidelity (error rate).
Quantum dots quantum computer
A single electron trapped in a semiconductor structure can form the basic building block for a qubit. We use the microwave to control the spin of the elections to construct the silicon quantum dots.
Source: Matthieu Delbecq and Shinichi Amaha, RIKEN Center for Emergent Matter Science
Here is another view of the quantum dots:
Nitrogen-vacancy (NV) centers in diamond
Diamond is made up of carbon atoms. It has a type of impurity (defect) called a Nitrogen-vacancy (NV) center which a nitrogen atom replaces a carbon atom and a vacancy takes the place of a
neighboring carbon.
Isolated spins can be extremely stable and a good choice for the qubit. The NV center harbors a spin triplet electronic ground state that can be polarized, manipulated and optically detected.
In quantum computing, we often look for energy states that have long coherence time with easy ways to initialize them and manipulate them.
So similar to the trapped ion, once these transition rules are known and quantify, we can design a quantum computer from them. However, how to create entanglement (2-qubit operator) can be tricky for
NV center quantum computers.
Phosphorus atoms in silicon
For those that want to know the phosphorous implants. Here is a 7 minutes video for your reference. Basically, we use the spin of Phosphorus to create qubits. We put Phosphorus atoms into a silicon
chip and reuse our expertise in transistors to perform the measurements.
Quick summary
Now we have a nice picture of how quantum computers are built. Next, we will look into another popular approach, the trapped ions. | {"url":"https://jonathan-hui.medium.com/qc-how-to-build-a-quantum-computer-with-superconducting-circuit-4c30b1b296cd","timestamp":"2024-11-02T23:57:07Z","content_type":"text/html","content_length":"273831","record_id":"<urn:uuid:1e2d270b-a808-4c13-a13a-549ebb0f4f75>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00478.warc.gz"} |
The Stacks project
Lemma 59.50.1 (Mayer-Vietoris for étale cohomology). Let $X$ be a scheme. Suppose that $X = U \cup V$ is a union of two opens. For any abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ there exists
a long exact cohomology sequence
\[ \begin{matrix} 0 \to H^0_{\acute{e}tale}(X, \mathcal{F}) \to H^0_{\acute{e}tale}(U, \mathcal{F}) \oplus H^0_{\acute{e}tale}(V, \mathcal{F}) \to H^0_{\acute{e}tale}(U \cap V, \mathcal{F}) \phantom
{\to \ldots } \\ \phantom{0} \to H^1_{\acute{e}tale}(X, \mathcal{F}) \to H^1_{\acute{e}tale}(U, \mathcal{F}) \oplus H^1_{\acute{e}tale}(V, \mathcal{F}) \to H^1_{\acute{e}tale}(U \cap V, \mathcal{F})
\to \ldots \end{matrix} \]
This long exact sequence is functorial in $\mathcal{F}$.
Comments (2)
Comment #1257 by Emmanuel Kowalski on
The trailing dots "\ldots" are missing at the end of the long exact sequence in the statement.
Comment #1268 by Johan on
OK, thanks. Fixed here.
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Basic Calendar Verbal Reasoning Online MCQ Test, & Quiz
Basic Calendar Problems Practice Questions Answers Test With Solutions & More Shortcuts
Question : 1 C.G. PSC 2013
Which of the following is a leap year?
a) 2800
b) 1800
c) 2600
d) 3000
e) All of these
Answer »
Answer: (a)
The century year which is completely divisible by 400, is a leap year.
Thus, the year 2800 is a leap year.
Question : 2 [U.P. PSC 2013]
The day on 18.09.1977 was Sunday A couple was married on this date. How many marriage anniversaries would fall on Sunday in the next 15 yr?
a) 1
b) 2
c) 5
d) 9
Answer »
Answer: (b)
1977 is an ordinary year.
We know that the calendar of an ordinary year repeats after 6 yrs or 11 yrs.
Let us check for the number of odd days in 6th and 11th years.
┃Year│Number of odd days ┃
┃1978│1 ┃
┃1979│1 ┃
┃1980│2 ┃
┃1981│1 ┃
┃1982│1 ┃
┃1983│1 ┃
┃1984│2 ┃
┃1985│1 ┃
┃1986│1 ┃
┃1987│1 ┃
┃1988│2 ┃
From the above table, Number of odd days from 18.09.1977 to 18.09.1983 = 7,
i.e., 0 odd days
It means that in 1983, 18th September would fall on Sunday.
From the above table, a number of odd days from 18.09.1977 to 18.09.1988 = 14
i.e., 0 odd days.
Now, it is clear that 2 marriage anniversaries would fall on Sunday in the next 15 yr.
Question : 3 S.N.A.P. 2012
How many Monday's are there in a particular month of a particular year if the month ends on Wednesday?
a) 4
b) 5
c) 3
d) Cannot be specified
Answer »
Answer: (d)
There are months of 30, 31 and 28 days and last day of the month are Wednesday.
So, using 28 and 30 days, there are 4 Mondays.
Using 31 days, there are 5 Mondays
So, it cannot be specified.
Question : 4 C.G. PSC 2013
In a month of 31 days, third Thursday falls on 16th, then what will be the last day of the month?
a) 5^th Friday
b) 4^th Saturday
c) 5^th Wednesday
d) 5^th Thursday
e) None of these
Answer »
Answer: (a)
Number of days left in the month after 16th = 31 - 16 = 15
Number of odd days = 15/7 = 2 weeks + 1 odd day
Therefore, Required day = Thursday + 1 odd day = Friday
As, 16th of the month is third Thursday, the day which is two weeks after this day is fifth Thursday.
So, one day after 5th Thursday is 5th Friday.
Question : 5
The calendar for the year 2007 will be the same for the year:
a) 2014
b) 2016
c) 2017
d) 2018
Answer »
Answer: (d)
Count the number of odd days from the year 2007 onwards to get the sum equal to 0 odd days.
Sum = 14 odd days = 0 odd day.
Therefore, Calendar for the year 2018 will be the same as for the year 2007.
Basic Calendar Problems Shortcuts »
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The hypotenuse is 2 times the length of either leg, so the. So you wanna add these three together, but you want. The ratios of the sides of a right triangle are called trigonometric ratios. This one
requires a special trick. It states that the sum of the squares of the legs of a. Answer by jim_thompson5910 (35256) ( show. We will shortly describe how to simplify the radical expressions given by
each of them. Express your answer in simplified radical form. Web write all answers in simplest radical form. For example \(\sqrt{72}\) can be reduced to \(\sqrt{4 \times 18} = 2. Web write answers
in simplest radical form. | {"url":"https://form.uame.edu.mx/simplest-radical-form-of-a-triangle.html","timestamp":"2024-11-15T03:01:35Z","content_type":"text/html","content_length":"20139","record_id":"<urn:uuid:7587191a-d99f-46c2-9221-5f64cabb0900>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00115.warc.gz"} |
IF Statement with Four Outcomes | Basic Excel Tutorial
IF Statement with Four Outcomes
IF statement Returns one value if a condition you specify evaluates to TRUE and another value if it evaluates to FALSE. An if statement with four outcomes requires you to test three conditions
The excel If () function is used to test a single condition that returns a value if the condition is met otherwise it will return the other value.
Structure of a single condition if statement
IF(logical_test, value_if_true, value_if_false)
1. Value_if_true is the value that is returned if the logical test is TRUE.
2. Value_if_false is the value that is returned if the logical test is FALSE.
Structure of If statement with three conditions and four outcomes.
=IF (CONDITION X, OUTPUT B, IF (CONDITION Y, OUTPUT C, IF (CONDITION Z, OUTPUT D, OUTPUT E))))
In this structure, we have four outcomes with three conditions
We test IF condition X is true, we return output B
But if condition X is false, then we test condition Y
If condition Y is true, you have an output C and if it is false, we test condition Z
If condition z is true we return output D, otherwise, we return output E.
Example: Three conditions and four outcomes
Find the Grade of the student given the condition that;-
Average Grade
>=70 DISTINCTION
>=60 CREDIT
>50 PASS
<50 FAIL
Type the first if statement put a comma, and type the next condition until the last condition is tested.
Use the Autofill option to copy the formula to the rest of the cells | {"url":"https://basicexceltutorial.com/if-statement-with-four-outcomes/","timestamp":"2024-11-07T19:32:31Z","content_type":"text/html","content_length":"62088","record_id":"<urn:uuid:2ff3b58f-b819-43b9-9c13-bf80089908e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00001.warc.gz"} |
Magical Word | Hackerearth practice problem solution
Dhananjay has recently learned about
values.He is very fond of experimenting. With his knowledge of
values and character he has developed a special word and named it Dhananjay's Magical word.
A word which consist of alphabets whose
values is a prime number is an Dhananjay's Magical word. An alphabet is Dhananjay's Magical alphabet if its
value is prime.
Dhananjay's nature is to boast about the things he know or have learnt about. So just to defame his friends he gives few string to his friends and ask them to convert it to Dhananjay's Magical word.
None of his friends would like to get insulted. Help them to convert the given strings to Dhananjay's Magical Word.
Rules for converting:
1.Each character should be replaced by the nearest Dhananjay's Magical alphabet.
2.If the character is equidistant with 2 Magical alphabets. The one with lower
value will be considered as its replacement.
Input format:
First line of input contains an integer
number of test cases. Each test case contains an integer
(denoting the length of the string) and a string
Output Format:
For each test case, print Dhananjay's Magical Word in a new line.
1 <= T <= 100
1 <= |S| <= 500
ASCII values of alphabets in AFREEN are 65, 70, 82, 69 ,69 and 78 respectively which are converted to CGSCCO with ASCII values 67, 71, 83, 67, 67, 79 respectively. All such ASCII values are prime
Time Limit:0.5 sec(s) for each input file.
Memory Limit:256 MB
Source Limit:1024 KB
Solution by using ( C language):-
int main()
int t,n,i,x,j,k=0,l,k1=0,r,r1=0;
char s[500];
return 0;
Recommended Post:-
• Hackerearth Problems:-
Data structure:-
Key points:-
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Financial Accounting - Online Tutor, Practice Problems & Exam Prep
So it's important to be able to analyze financial statements as well as create them in this class. Well, another way we can analyze them is through a vertical analysis. Let's check it out. So you may
have previously learned about a horizontal analysis where we do a percentage change from 1 year to the next. If not, you'll learn about it in a different video. Here we're going to learn about the
vertical analysis. A vertical analysis is still dealing with percentage changes, but we're not dealing with year over year changes anymore. We're going to be dealing with a percentage of the base
amount on the financial statement. So we're going to do this for the income statement and the balance sheet. So let's see what the base amount is going to be for each of those statements.
On the income statement, our base amount is going to be our net sales. Our net sales or just our sales revenue, right? Or our sales revenue if we don't have a net sales amount. We're going to use
that top line, sales revenue. Whatever the top line is on your income statement, that's what we're using as our base. And then we're going to have on the balance sheet, our base is going to be our
total assets and we're also going to use so for the assets part, we're going to use total assets And for the liabilities and equity, we're going to use total liabilities and equity. But remember,
these numbers are the same, aren't they, right? Total assets equal total liabilities and equity. So they're going to be the same number. So either way, we're going to have that number, as our balance
sheet base.
What do I mean when I say base? That's what we're going to use as the denominator in our percentage formula. So remember that when we find the percentage of the base, what we're going to do is we're
going to take the line item amount, say, selling expense like we have down here and we're going to divide it by the base amount. So if that was on the income statement, it would be the selling
expense, whatever that amount is for the year divided by the net sales, because we're dealing with the income statement. And remember that we're getting a percentage, so we are going to multiply this
by 100 to move the decimal place 2 places and get a percentage.
Let me go ahead and show you how we do a vertical analysis here on an income statement and then you guys can get some practice on a balance sheet. So let's go ahead and do it for 2 years here. We've
got 2018 and we've got our net sales. So remember that net sales, this is our base amount. This is always going to be the denominator. In every single one of our calculations, the base amount is net
sales for that year. So 2018, the base amount is always going to be 65455 and when we do 2017, it'll always be 58081.
Let's go ahead and do a couple of them and then I'm going to speed it up. I've done a lot of these calculations ahead of time. It's just going to be a lot of number crunching, that's why we became
accountants because we love using our calculator. Hit all those buttons. Let's go ahead and do it. So net sales, this one's always going to be 100%, right? It's going to be 100% of itself. This
calculation, we would do 65455, the line item amount divided by the base amount which is net sales. So that's going to be obviously 1, we multiply it by 100 to get the percentage and that one is
100%. So net sales is 100% of itself. Wow, that's very revealing information. Let's go on down and we'll start getting some better information.
How about cost of goods sold? So remember the numerator is going to be the line item. So 54912 is our numerator and we divide it by net sales, right? Net sales is always going to be the denominator.
65455. Alright. So let's go ahead and you just do that division and we'll get it as a percentage and we see that it's 83.9%. I'm going to be rounding just to one decimal here.
We can do our vertical analysis to subtotals as well. Remember, gross profit is a subtotal. It's not its own expense or revenue. It's a subtotal. It's just net sales minus cost of goods sold. Well,
we can find out what gross profit is out of net sales. So you do the same thing, 10543 divided by net sales, the same number 65455 and we get it as a percentage. We multiply by 100 to get a
percentage and we'll get 16.1%.
Already we can see that out of net sales, 83%. So for every dollar of net sales, almost 84% of it is going to pay for the cost of goods sold and we're left with 16% at this point. And then we still
got to pay for other stuff. We're going to have other expenses and we're going to end up with our net income. So this lets us know how much of a percent of net sales. So we get a dollar of sales, how
much of that is going to different places.
Let's keep going here with our operating expenses. So I'm going to write this one out, but after this, I think you should get the point of how we do this calculation and I'm going to start putting in
our percentages. So 2411, right? Our line item is our numerator divided by 65455 still using that same, net sales amount as our numerator, and let's go ahead do that division and we get 3.7%. We keep
going 982 divided by 65455 and we get 1.5%. So we know for every dollar of sales, what 1.5% of that is going to selling expenses. Maybe some commission we pay to our salesperson, they get 1.5% off of
every sale. Depreciation expense, 1400 divided by 65455. Same thing. 2.1% here and we can do it with total operating expenses. How much are our total operating expenses as a percentage of our sales?
And we get 7.3% here.
Another subtotal, income from operations, right? This is where we take our gross profit minus those operating expenses. This is all of our core business. How much are we making? Well, 5750 out of
that 65455 that comes out to be 8.8%. That means for every dollar of sales, well, we're keeping 8.8% from operations, and then we've got a couple more things we got to pay for, and then we're left
with our net income.
So let's do those. Next, we've got interest expense, 480 divided by 65455 and we're going to get 0.7%. How about the next one? Other expenses, look how small this is $70. That doesn't seem like a
very big part of net sales. We do our calculation 70 divided by 65455, 0.1%. Vertical analysis is pretty easy, right? We're just doing this simple division. All you g | {"url":"https://www.pearson.com/channels/financial-accounting/learn/brian/ch-14-financial-statement-analysis/vertical-analysis?chapterId=3c880bdc","timestamp":"2024-11-09T13:16:08Z","content_type":"text/html","content_length":"381065","record_id":"<urn:uuid:37811bb9-6cdb-4bc5-ba67-5a2ea066ec57>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00463.warc.gz"} |
Getting poly normal and CreatePhongNormals()
Just FYI: To calculate the normal of a single CPolygon, there is CalcFaceNormal().
See also NormalTag Manual.
What version of Cinema 4D are you using? GeFree() is pretty outdated.
i keep forgetting this CalcFaceNormal...
Thanks Manuel and PStudent,
I'm going around in circles. When I try blending between the 3 Phong normals, it comes out in variants like this:
The following is the code I use to calc the Phong normal:
//... hit test function here...
// f = 1 / area
Vector uvw;
uvw.x = f * s.Dot(h);
uvw.y = f * dir.Dot(q);
uvw.z = 1 - uvw.x - uvw.y;
//... end of hit test, return true if hit, uvw passed over...
// if hit function returns true, use uvw to calc normal at hit spot
// norms = CreatePhongNormals()
// let's test first polygon in poly 4(5)
Vector normal = uvw.x * norms->operator[](4*4) + uvw.y * norms->operator[](4*4+1) + uvw.z * norms->operator[](4*4+2);
// repeat for second tri in quad, if exists
// etc
I can get the plain polygon face normal fine. But blended ones I can't get my head around. Where am I going wrong here?
at least to me it is not quite clear what you are trying to do. One reason is that there are quite a few symbols floating around in your code which are not self-explanatory (what are s, q, h and
dir?). I also do not quite understand why you are computing something into the vector uvw what seems to be a linear combination.
If you are interested just in the polygonal normal, you can compute that by evaluating the cross product of the vertices of a corner of your choice in the polygon in a clockwise fashion (e.g. d-a
% b-a for the vertex a in a four point polygon). In non-planar polygons you will have to evaluate two vertices, or more realistically all four, since that is faster than finding "the right ones",
and compute the arithmetic mean of them.
If you want to to interpolate between the already interpolated point normals (a.k.a. Phong normals), a simple bi-linear interpolation between these four normals should be enough. Where the u and
the v coordinates of your texture data (or wherever they are coming from) are the parameters for the respective interpolation on that axis. You won't need barycentric coordinates or something
like that.
same here, not sure what you are trying to achieve at the end.
Thanks Zipit / Manuel,
@zipit said in Getting poly normal and CreatePhongNormals():
If you want to interpolate between the already interpolated point normals (a.k.a. Phong normals), a simple bi-linear interpolation between these four normals should be enough
Yep - that's what I'm trying to do!!
but can't figure out the bi-linear interpolation. This is where I'm going around in circles. Maybe we take a step back for a moment and do this in two steps.
Step 1: I have a function that tests each polygon triangle using the Moller-Trumbore algorithm. Assuming the hit is true, how do I calculate the UV? I'm asking this step to check if I'm setting
UV right (which, is where some of the code above comes from).
Step 2: how do I use UV to interpolate between Phong normals a,b and c (triangle)?
The end goal is to draw a normals map.
a bilinear interpolation is quite straight forward. If you have the quadrilateral Q with the the points,
| |
then the bilinear interpolation is just,
ab = lerp(a, b, t0)
cd = lerp(c, d, t0)
res = lerp(ab, cd, t1)
where t0, t1 are the interpolation offset(s), i.e. the texture coordinates in your case (the ordering/orientation of the quad is obviously not set in stone). I am not quite sure what you do when
rendering normals, but when you render a color gradient, in a value noise for example, you actually want to avoid linear interpolation, because it will give you these ugly star-patterns. So you
might need something like a bi-quadratic, bi-cubic or bi-cosine interpolation, i.e. pre-interpolate your interpolation offsets.
If I am not overlooking something, this should also work for triangles when you treat them as quasi-quadrilaterals like Cinema does in its polygon type.
@WickedP said in Getting poly normal and CreatePhongNormals():
what I forgot to mention, but I already hinted at in my previous posting, is that you did not state in which space your interpolation coordinates are formulated. When you talk about
uv-coordinates I am (and probably also everyone else is) assuming that you have cartesian coordinates, just like Cinemas texture coordinate system is formulated in.
If you have another coordinate system, some kind of linear/affine combination like your code snippet suggests, then the interpolated normal is just the linear combination of the neighbouring
normals (I think that was what you were trying to do in that code).
If this fails, you should either check your MΓΆller-Trumbore code for errors (Scratch-Pixel has a nice article on it and how to calculate correct affine space/barycentric coordinates for it), or
more pragmatically use Cinema's GeRayCollider instead, which will also give you texture coordinates.
I am not so well versed in the C++ SDK, maybe there is even a more low level (i.e. triangle level) version of GeRayCollider there, i.e. something where you do not have the overhead of casting
against a whole mesh.
Oooh..... I feel like such a dope. I had almost everything right to begin with, except I was accessing the returned CreatePhongNormals() SVector incorrectly. I was using operator()[] when I
should have been using ToRV(). Here's a smoothed result:
Thanks guys, we got there
One last thing, how do I 'globalise' the normals? They look local to me (see floating cube, it's slightly rotated but has the same shading as walls in the background)?
all polygonal data is in object space. So if you want your normals to be in global/world space, you will have to multiply them with the frame of the object they are attached to. To get the frame,
you will only have to zero out the offset of the global matrix of the object and then normalise the axis components.
Thanks zipit,
makes sense being local. Zeroing out and multiplying by the frame matrix does it:
Thanks for your time contributors, been a big help.
can we considered this thread as resolved ?
Thanks Manuel,
I didn't realise I had to mark the thread as solved. Done.
@WickedP Uhh, Ohh what are you coding there is that a brdf map shader ? I am searching for something like this !
Hi @mogh,
apologies I didn't see your reply.
That display is for a normals engine. I had a 2D draw engine but needed to expand on it because the native renderers wouldn't let me get what I wanted in code. I didn't want to make a render
engine, it's just kind of how it's turning out...!
It's only meant to be a basic one that serves the needs though. There's a spiel about it here on my website: [https://wickedp.com/hyperion/](link url). You'll have to excuse the site layout, I'm
stuck in the middle of revamping it until I deliver some other big projects.
Thanks for the reply, Wicked.
interessting, but not what i hoped for ...
kepp on chrunching | {"url":"https://developers.maxon.net/forum/topic/12791/getting-poly-normal-and-createphongnormals/12","timestamp":"2024-11-08T06:05:10Z","content_type":"text/html","content_length":"185383","record_id":"<urn:uuid:f31f4688-dbed-4a43-9056-a6c81c9a6887>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00695.warc.gz"} |
Where in the World Are World-Class Standards?
A current television ad portrays a neatly dressed young girl walking into a middle school classroom, a pile of books cradled in her arms. With a pleasant voice, the teacher calls roll. The scene is
genial and familiar, until we notice that there is only one long row of chairs, and that the names being called are the names of countries: “Taiwan, Korea, Switzerland....” As the girl takes her
place in the very last seat, the teacher intones ominously, “the United States of America.”
The ad expresses our national concern over how well U.S. students are doing in school. Readers of an article on international standards typically expect a similarly gloomy perspective, buttressed
with handy charts comparing students from around the world. Because the international standing of U.S. children is well-known, we will spare you another such chart. Instead, we want to look at
countries known for producing high-performing students to discover why these school systems look so good on the honor roll of nations.
Documents Don't Tell the Story
At a recent forum on 21st century education, a co-panelist asked us to send him a copy of the world-class standards in education. We had to chuckle; would that it were so easy! If world-class
standards were defined and available in the local library's reference section, researchers and policymakers alike would make frequent use of it. Unfortunately, no such volume exists.
In 1993, when the New Standards Project at the University of Pittsburgh began its international benchmarking efforts, we hoped to collect and analyze the standards documents of other countries. We
began by concentrating on mathematics, thinking it might suffer less from cultural differences than do other areas.
• those whose students perform well on international tests (for example, France),
• those whose education systems enjoy international esteem (for example, the Netherlands),
• those with a federal structure much like our own (for example, Germany), and
• those representing major economic competitors of the United States (for example, Japan).
We quickly discovered that standards are not in neat volumes on the shelves of education ministries, but instead arise out of a complex interaction of curriculums, textbooks, exams, classroom
practice, and student work.
Moreover, when we sought to compile a library of materials, teachers, both at home and abroad, warned us that documents alone cannot tell the whole standards story. After all, teachers do not teach
all and only what is in a textbook. They advised us to (1) find out what happens to kids who do not meet the standards, and (2) look at student work, which is where one really finds out what is
expected of students.
• What is the structure of the education systems in other countries?
• What are students expected to know and be able to do at key junctures in their schooling careers?
• What kinds of performances are used to demonstrate competence?
• What counts as “good enough” in these performances?
• What percentage of the students is meeting the standards?
• What reform efforts are under way or on the horizon?
• There is more than one way to help students achieve excellence.
• To successfully serve a large number and variety of students, schools must work as systems whose parts are focused on coherent, consistent, publicly articulated goals.
Let's look at mathematics education in Japan, France, and the Netherlands.
Tracking: Results are Mixed
Historically, U.S. schools were among the first to provide secondary schooling for all students. Many argue that this is why the United States fares poorly in international comparisons: while we are
committed to the education of all children, other countries practice strict ability tracking that creams off the best students. Hence, our average students are compared with their best students.
In fact, our research shows that things are no longer that simple. Other developed nations have caught up with and even surpassed us in terms of retaining students throughout the years of secondary
schooling (Centre for Educational Research and Innovation 1993). Further, tracking practices do not correlate in any straightforward way with high performance internationally. In such comparisons,
tracking and achievement appear to be independent of one another.
In practice, U.S. students are often tracked into classes for the gifted and talented, vocational education, and the like. On the other hand, those systems in which students outperform ours include
both highly tracked and highly untracked systems.
Tracking is common in the Netherlands, where secondary school students elect one of four levels of study based on both their career goals and their past experience in school. French students are
untracked through age 13; thereafter, about 85 percent of students are in a single track. In Japan, there is no tracking throughout the years of compulsory schooling.
In other words, tracking and education's availability to the whole populace cannot by themselves explain away the poor performance of U.S. students on international tests.
We have, however, learned some important lessons while examining tracking.
Japan. Japanese schools prefer heterogeneous grouping because it seems to produce higher performance all around: High performing students actually learn more, it is argued, by serving as tutors to
their classmates. Further, a central focus of Japanese schools is to help form the moral and cultural character of students. High performance is valued because it contributes to the well-being of the
group. In school, this means that students see their own excellence as compromised by another student's failure! One of the standards for all Japanese students, then, is to be a contributing member
of an effective work group.
The Netherlands. Tracking in the Netherlands is not determined by achievement tests, which are a predominant means of sorting students in the United States. In fact, achievement tests are not used in
the Netherlands. Instead, secondary students, in consultation with parents, teachers, and—sometimes—school administrators, choose the track that is most appropriate for them.
The major factor in the student's choice of track is his or her career goals, because each of the four tracks in Dutch secondary schools leads to a broad set of careers and levels of specialization.
Further, throughout the first two years of secondary school, and in some cases beyond that, students may switch tracks if their goals change. Early on, then, Dutch students have a clear sense that
their studies are directly connected with life after school.
A defining characteristic of tracking in Dutch schools is that all students are expected to perform well. Mathematics exams at the conclusion of high school are a case in point: Although students who
intend to go on to a university are asked to perform at a more sophisticated level than those who wish to enter the work force, the latter group faces very difficult exams. As Figure 1 shows, these
exams involve complex applications of algebra and geometry. Students are also expected to show how they arrived at their answers; there is more than one right way. Dutch educators have been
developing a mathematics program geared to helping all students perform well.
(Dutch National Institute for Educational Measurement)
Exercise 4. A swimming pool of 16 x 50 meters has a shallow part A (depth 1 meter) and a deep part C (depth 4 meters). In between, the depth increases regularly from 1 meter to 4 meters. (See the
drawing.) All measurements are given in meters. The swimming pool is filled up to the edge.
• Calculate how many m3 of water are in the swimming pool.
On the border between part A and part B is a sign that indicates the depth (see the drawing). The lifeguard wants to put another sign on the edge saynig "depth 1.80 meters."
• Calculate the distance between the two signs.
The Dutch approach contrasts sharply with that used in the United States today, where educators are hotly debating the relationship between tracking and achievement. Some argue that tracking results
in a weak curriculum for students whose work has been weak in the past; others argue that a failure to track means holding back highly motivated students, forcing them into heterogeneous groups with
a dumbed-down curriculum.
Curriculums: Common Goals Are Crucial
Many countries whose schools have achieved academic excellence have a national curriculum. Many educators maintain that a single curriculum naturally leads to high performance, but the fact that the
United States values local control of schools precludes such a national curriculum. This argument would have us throw in the towel regarding raising achievement.
Our research has shown that national curriculums are a diverse group of documents. Some express the educational philosophy or traditions of the country, while others concentrate on prevailing
cultural needs. Some describe teaching strategies or content considered important. Most are very sketchy. They do not detail lesson plans that mandate uniform classroom practice throughout the
In Japan, for example, the curriculum includes brief objectives for each grade and content level, and a few specific items that should be mastered. Teachers must and do go far beyond the guidelines.
The same is true in France and in the Netherlands.
Still, a centrally articulated set of goals, even if vaguely stated, plays important roles: It organizes the development of exams and curriculums, informs textbook writing, and determines the
direction of teacher training. As a result, high-stakes exams, texts, curriculums, and lesson plans do not work at cross-purposes. When all parties involved in these diverse activities have their
eyes on the same set of goals, students get a consistent message about what they should know to be well educated.
France. France offers the clearest example of this convergence of goals. In texts and exams, the influence of the national curriculum is obvious. For example, a French math text for 16-year-olds
begins by spelling out the national curriculum for the year so that all 16-year-olds know what they are expected to study. The book's similar table of contents shows that the text developers referred
to the curriculum. Moreover, the text makes frequent references to math exams the regional school districts have given in the past. Students practice on these exams to help them prepare for the exam
they will face; they know where to concentrate to meet the standard.
One could draw a tempting but fallacious inference from these examples. Can simply having a coherent system of curriculums, texts, and exams produce excellent student performance? In fact, coherence
is not enough; Sweden offers the counter-example.
Sweden. As in France, the Swedish national curriculum strongly influences texts and exams, giving students a clear message about what is expected. Still, the mathematics exam for Swedish 16-year-olds
shows that a clear message, too, can set a low standard (see fig. 2). Unlike its Dutch counterpart, the Swedish exam does not ask for complex mathematical reasoning, but focuses instead on relatively
low-skill computation. The lesson here: Unless coherent schooling elements set high academic standards, we can't expect student achievement to rise.
Figure 2. Portion of a Swedish Mathematics Exam for 16-year-olds
(PRIM Group of the School Administration for the Stockholm Teacher Training Institute)
Part B. For exercises 12–15, complete solutions must be given. NOTE! If you only give the answer you will get 0 points.
12. What is the price of a piece of ham weighing 6 hg if the price is 150 krona per kilogram?
13. How much is the telephone bill for a quarter of the year if it shows 300 units of use for that time? Each unit costs 23 Öre. In addition, there is a charge of 187 krona per quarter in
subscription fees.
Exams: Upholding Standards
To understand how certain systems produce excellence, we also must find out how students demonstrate what they know and can do. Many countries give an exam at the end of compulsory schooling, at
about age 16, and that exam often is the last measure we have of how all students are performing. After this point, not all students are expected to work to high standards.
France. In France, virtually all students attempt to qualify for the Brevet certificate at the end of middle school, and more than 75 percent succeed. This qualification is awarded on two bases:
final exams in several subjects, and classroom teachers' continuous assessment during the last two years. The exams differ among the country's 27 regional school districts, but multiple choice is
virtually unknown. Students must write essays, argue for positions, and solve problems while giving evidence of their reasoning. Texts and curriculums support these practices. This means that
students can prepare for the Brevet, because it reflects the very same skills and knowledge they have been honing in school.
The Netherlands. In the Netherlands, all students take high school leaving exams. The final grade is the average of the exam's two parts: one generated nationally, the other compiled by the school.
Dutch schools have four tracks and give four corresponding national exams in most subjects. As in France, multiple-choice and short-answer questions are rare.
U.S. teachers often marvel at these exams that require a lot of writing. “How are they graded?” they want to know. Obviously, with few exceptions, machine grading is impossible. By and large,
teachers do it. If selected as graders, they are either freed from other duties for a time, paid a stipend, or both. To be sure grades are given fairly across regions, all scorers receive scoring
guides, and auditors check a random sample of scores.
Germany. In Hessen, Germany, however, teachers both compile and grade the exam for their own students. When questioned about the possibility of teachers artificially inflating grades or helping their
students cheat, one university professor seemed puzzled. “Why would they cheat,” she asked us, “when they are professionals who care about their work?” This trust in and respect for teachers as
professionals is common in countries whose students are noted for excellence.
What We've Learned
• Setting clear, consistent, demanding, public standards helps students perform well.
• Tracking and grouping practices must make sense in the culture of the school and for both the student's and community's future goals.
• Exams should test what students have been asked to learn, preferably in the same ways they must perform in class.
• Exams that call for complex, demanding tasks can be given to a wide range of students, perhaps to all students.
• As the front-line professionals in the education process, teachers should have much to say about what goes into exams and how they are graded.
None of these results is surprising. They represent what good teachers in good programs with hard-working students have always done. For the New Standards Project, the good news from international
comparisons is that it is possible to set high standards and expect all students to work to achieve them.
One caveat is in order: The route to high performance is not necessarily to simply implement the good practices of other countries. When we aim for world-class standards, we are not aiming at a
target that is standing still and waiting for us. Far from it.
Concerns about preparing students for the challenges of work and community in the 21st century are not unique to the United States. The Netherlands continues to stress the development of improved
mathematics curriculums as a national priority. Around the world, schools are seeking to improve the technological abilities of all students.
Sweden and France are piloting creative means for teaching children of immigrants. All over the world, in fact, educators are working to improve school services to traditionally —alized groups,
including children from low-income families and girls of all economic classes. Issues of equity, or the performance of language, racial, and ethnic minorities are not unique to the United States.
The challenge for the United States is to create a national agenda of excellence that can raise the performance of all students without creating a national exam or curriculum. Each community must
adapt the agenda in unique ways that nonetheless work in unison.
The image of a symphony comes to mind: each instrument has its own score, its own qualities, its own goals, but the scores must harmonize if a satisfying performance is to result. Just so with state
and local reforms: they must and will vary in ways that make sense to local schools and communities. But they must also share a common vision of the high performance we must expect from all students. | {"url":"https://ascd.org/el/articles/where-in-the-world-are-world-class-standards","timestamp":"2024-11-09T19:28:19Z","content_type":"application/xhtml+xml","content_length":"244050","record_id":"<urn:uuid:c5c95340-5829-444a-941a-dd81d9e717cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00488.warc.gz"} |
What are different real world problems that are modeled by Linear equations? | Socratic
What are different real world problems that are modeled by Linear equations?
1 Answer
For people paid the same amount per hour for every hour worked, total pay is a linear function of hours worked.
(Overtime pay would make this a piecewise linear relationship.)
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If a radioactive isotope has a half life of 1000 years how long does it take for 3/4 of the original sample to decompose? | Socratic
If a radioactive isotope has a half life of 1000 years how long does it take for 3/4 of the original sample to decompose?
1 Answer
If you begin with a 1 gram sample of the isotope, at the end of 1000 years you would have
1 gram x $\left(\frac{1}{2}\right)$ = $\frac{1}{2}$ gram would remain
At the end of the 2nd 1000 years you would
$\left(\frac{1}{2}\right) g r a m$ x $\left(\frac{1}{2}\right)$ = $\frac{1}{4}$ gram would remain
Since you began with 1 gram and after 2000 year you would have
$\frac{1}{4}$ of a gram remaining. This means $\frac{3}{4}$ of the original amount would have decayed.
1 gram - $\frac{1}{4}$ gram = $\frac{3}{4}$ gram
The answer therefore is 2000 years.
I hope this was helpful.
Also, if you are more of a mathematical person, you can use the equation: m=ca^(t/h) where;
"m" is the final mass of the sample
"c" is the starting mass of the sample
"a" is how the substance is decaying (half life = multiplying by 1/2)
"t" is the time it is alive
"h" is the half life time.
For example, if I said: "If you begin with a radioactive isotope weighing 100g and it has a half life of 1000 years, how long does it take to decay to 3/4 (25g) of it's original mass?"
In this case, you know that the final mass, "m", is 25g, you know that the starting mass, "c" is 100g, "a" is 1/2 (it is halving), "t" is the time it is alive (you don't know this), and you know the
half life time, "h", is 1000 years. Knowing these, you sub them into the equation as such:
25 = (100)(1/2)^(t/1000) --> now you divide by 100 on both sides to simplify the expression. YOU CANNOT DIVIDE OUT THE 1/2 BECAUSE IT HAS AN EXPONENT ATTACHED TO IT!
(25/100) = (1/2)^(t/1000) --> next, if you are firmilliar with solving for exponents using logs, you write log on both sides in front of each side, then bring the exponent down in front of the log ON
log(25/100) = (t/1000)log(1/2) --> now you can divide by log(1/2) on both sides to simplify the equation:
((log(25/100))/(log(1/2)) = (t/1000)
2 = (t/1000) --> now you can multiply by 1000 on both sides to isolate "t".
2000 = t --> That's it! I hope this was of help! :)
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Sliding window technique
Sliding windows can be useful in searching through arrays. A window is a selected sub array of a main array. While searching for something, a specific word or duplicate characters in a string you can
start by picking a window and then sliding that window using two pointers to demark the two boundaries. Move the right pointer to slide the window to the right while using left pointer to narrow down
the window.
Try the contains duplicates problems on leetcode to practice the sliding windows.
The problem input is a linear data structure, linked list, array or string. Task is usually to find substrings, subarrays with some condition.
From LinkedIn
🚀 Day 32/90: Solved 3 DSA problems in Java and learnt about sliding window Pattern 📚💻 I solved an easy #geekforgeeks problem and an easy and medium #leetcode problem. The problems and code are in the
image below, as well as in my leetcode profile (https://lnkd.in/gcnZ89PX) and GFG profile(https://lnkd.in/esBsNhGb).
🔺 Max Sum Subarray of size K: Time complexity - O(N), Space complexity - O(1) 🔺 121. Best Time to Buy and Sell Stock: Time complexity - O(N), Space complexity - O(1) 🔺 424. Longest Repeating
Character Replacement: Time complexity - O(N), Space complexity - O(1) (two implementations with an optimized code)
🌐💡 Unleashing the Power of Sliding Window Algorithms: A Dynamic Approach to Data Processing 🌐💡 Understanding the versatility of sliding windows is a game-changer in problem-solving. 🌊🔍💻 🧐 What are
Sliding Window Algorithms? Sliding window algorithms process data streams by segmenting them into fixed-size chunks. The window then glides over the data, one element at a time, performing operations
on each chunk. It's a technique that brings efficiency and versatility to the table.
🎯 Identifying Sliding Window Problems: 🕵️♂️ Substrings/subarrays: If a problem involves finding a particular substring or subarrays in a larger string or array, sliding windows might be the key. 📊
Calculating Statistics: Problems requiring the calculation of statistics over a subset of data points, with updates as the window slides, often align with sliding window solutions. ⬆️⬇️ Finding
Extremes: Whether it's identifying the maximum or minimum value within a window of data points, sliding windows excel in such scenarios. 🔄 Types of Sliding Window Algorithms: 🚪 Fixed-size Window: The
simplest type, where the window size remains constant. 📏 Variable-size Window: Adaptable windows that change size based on data characteristics. 📈 Count-based Window: Tracks the frequency of a
specific element within the window. ➕ Sum-based Window: Keeps tabs on the sum of elements within the window.
🌐 Real-world Applications: 📉 Max Subarray Sum: Finding the maximum sum of a subarray of a given size. 🚀 Unique Substring Length: Discovering the longest substring without repeating characters. 🔍
Substring Occurrences: Counting the number of occurrences of a substring in a string.
🎬 Conclusion: Sliding window algorithms are a potent and versatile tool, simplifying the solution to a myriad of problems. They are easy to implement and exhibit remarkable efficiency. | {"url":"https://blog.nisalap.com/algo/arrays/sliding-window/","timestamp":"2024-11-02T18:15:12Z","content_type":"text/html","content_length":"49089","record_id":"<urn:uuid:88a3404a-b3ff-4468-842d-36e033ab6641>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00613.warc.gz"} |
Introduction to quantum computation
The course introduces the paradigm of quantum computation in an axiomatic way. We introduce the notion of quantum bit, gates, circuits and we treat the most important quantum algorithms. We also
touch upon error correcting codes. This course is independent of COM-309.
Introduction to quantum computation
- Classical circuit model, reversible computation.
- Quantum bits, Hilbert space of N qubits, unitary transformations, measurement postulate.
- Quantum circuit model, universal sets of gates.
- Deutsch and Josza's problem and algorithm.
Basic algorithms
- Hidden sub-group problem and Simon's algorithm
- Mathematical parenthesis: factoring integers and period of arithmetic functions. Notions on continued fraction expansions.
- Quantum Fourier transform and the period finding algorithm
- Shor's factoring algorithm.
- Grover's search algorithm.
Error correcting codes
- Models of noise and errors.
- Shor and Steane error correcting codes.
- Stabilizer codes.
- Calderbank-Shor-Steane construction.
Quantum computation, quantum circuits, universal gates, quantum Fourier transform, Deutsch-Josza's algorithm. Simon algorithm, Shor's algorithm, Grover's algorithm, entanglement, quantum error
Learning Prerequisites
Required courses
Linear algebra course, basic probability course
Important concepts to start the course
Matrices, unitary matrices, eigenvectors, eigenvalues, inner product, algebra of complex numbers
Learning Outcomes
By the end of the course, the student must be able to:
• Explain the concept of quantum algorithm on the circuit model
• Describe universal gates
• Describe basic quantum algorithms
• Compute the evolution of a state through a circuit
• Apply the measurement postulate
• Manipulate algebraic expressions involving the Dirac notation
• Carry out implementation on public NISQ devices
• Give an example of an error correcting code
Teaching methods
Ex cathedra classes. Exercices. Use of IBM Q NISQ devices
Expected student activities
Participation in class, exercise sessions, use of IBM Q NISQ devices
Assessment methods
• Mini project on IBM Q experience
• Graded homeworks
• Written final exam
Office hours No
Assistants Yes
Forum Yes
Others Assistants answer questions during exercise sessions
N. David Mermin. Quantum Computer Science. An Introduction. Cambridge University Press.
Nielsen and Chuang. Quantum Computation and Information. Cambridge University Press.
Ressources en bibliothèque
Moodle Link
In the programs
• Semester: Spring
• Exam form: Written (summer session)
• Subject examined: Introduction to quantum computation
• Lecture: 3 Hour(s) per week x 14 weeks
• Exercises: 1 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Spring
• Exam form: Written (summer session)
• Subject examined: Introduction to quantum computation
• Lecture: 3 Hour(s) per week x 14 weeks
• Exercises: 1 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Spring
• Exam form: Written (summer session)
• Subject examined: Introduction to quantum computation
• Lecture: 3 Hour(s) per week x 14 weeks
• Exercises: 1 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Spring
• Exam form: Written (summer session)
• Subject examined: Introduction to quantum computation
• Lecture: 3 Hour(s) per week x 14 weeks
• Exercises: 1 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Spring
• Exam form: Written (summer session)
• Subject examined: Introduction to quantum computation
• Lecture: 3 Hour(s) per week x 14 weeks
• Exercises: 1 Hour(s) per week x 14 weeks
• Type: optional
Reference week
Mo Tu We Th Fr | {"url":"https://edu.epfl.ch/coursebook/en/introduction-to-quantum-computation-CS-308","timestamp":"2024-11-09T16:42:54Z","content_type":"text/html","content_length":"31765","record_id":"<urn:uuid:bbacb9e2-a5b3-4083-ab06-7c1dd926314e>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00449.warc.gz"} |
Cubic Mile to Perch Converter
⇅ Switch toPerch to Cubic Mile Converter
How to use this Cubic Mile to Perch Converter 🤔
Follow these steps to convert given volume from the units of Cubic Mile to the units of Perch.
1. Enter the input Cubic Mile value in the text field.
2. The calculator converts the given Cubic Mile into Perch in realtime ⌚ using the conversion formula, and displays under the Perch label. You do not need to click any button. If the input changes,
Perch value is re-calculated, just like that.
3. You may copy the resulting Perch value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Cubic Mile to Perch?
The formula to convert given volume from Cubic Mile to Perch is:
Volume[(Perch)] = Volume[(Cubic Mile)] × 5947391999.999999
Substitute the given value of volume in cubic mile, i.e., Volume[(Cubic Mile)] in the above formula and simplify the right-hand side value. The resulting value is the volume in perch, i.e., Volume
Calculation will be done after you enter a valid input.
Consider that a lake contains 2 cubic miles of water.
Convert this volume from cubic miles to Perch.
The volume in cubic mile is:
Volume[(Cubic Mile)] = 2
The formula to convert volume from cubic mile to perch is:
Volume[(Perch)] = Volume[(Cubic Mile)] × 5947391999.999999
Substitute given weight Volume[(Cubic Mile)] = 2 in the above formula.
Volume[(Perch)] = 2 × 5947391999.999999
Volume[(Perch)] = 11894784000
Final Answer:
Therefore, 2 cu mi is equal to 11894784000 per.
The volume is 11894784000 per, in perch.
Consider that the total volume of a large glacier is 0.5 cubic miles.
Convert this volume from cubic miles to Perch.
The volume in cubic mile is:
Volume[(Cubic Mile)] = 0.5
The formula to convert volume from cubic mile to perch is:
Volume[(Perch)] = Volume[(Cubic Mile)] × 5947391999.999999
Substitute given weight Volume[(Cubic Mile)] = 0.5 in the above formula.
Volume[(Perch)] = 0.5 × 5947391999.999999
Volume[(Perch)] = 2973696000
Final Answer:
Therefore, 0.5 cu mi is equal to 2973696000 per.
The volume is 2973696000 per, in perch.
Cubic Mile to Perch Conversion Table
The following table gives some of the most used conversions from Cubic Mile to Perch.
Cubic Mile (cu mi) Perch (per)
0.01 cu mi 59473920 per
0.1 cu mi 594739200 per
1 cu mi 5947392000 per
2 cu mi 11894784000 per
3 cu mi 17842176000 per
4 cu mi 23789568000 per
5 cu mi 29736960000 per
6 cu mi 35684352000 per
7 cu mi 41631744000 per
8 cu mi 47579136000 per
9 cu mi 53526528000 per
10 cu mi 59473920000 per
20 cu mi 118947840000 per
50 cu mi 297369599999.9999 per
100 cu mi 594739199999.9999 per
1000 cu mi 5947391999999.999 per
Cubic Mile
The cubic mile is a unit of measurement used to quantify large three-dimensional volumes, particularly in geology, environmental science, and astronomy. It is defined as the volume of a cube with
sides each measuring one mile in length. Originating from the Imperial system, the cubic mile is used to measure vast quantities of space and volume, such as the volume of large bodies of water,
geological formations, or planetary features. Today, it remains relevant in fields where large-scale volume measurements are necessary, such as in studies of Earth's water resources, large-scale
environmental assessments, and space exploration.
The perch is a unit of measurement used to quantify volume, area, and length, primarily in historical and specific regional contexts. As a volume measure, it is often associated with a cubic
measurement of 1 cubic yard or approximately 0.7646 cubic meters. Historically, the perch was used in land measurement, particularly for timber and stone, and was commonly employed in construction
and trade. Today, while its use has largely declined, the perch is still referenced in some historical contexts and in certain industries where traditional units are preserved.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Cubic Mile to Perch in Volume?
The formula to convert Cubic Mile to Perch in Volume is:
Cubic Mile * 5947391999.999999
2. Is this tool free or paid?
This Volume conversion tool, which converts Cubic Mile to Perch, is completely free to use.
3. How do I convert Volume from Cubic Mile to Perch?
To convert Volume from Cubic Mile to Perch, you can use the following formula:
Cubic Mile * 5947391999.999999
For example, if you have a value in Cubic Mile, you substitute that value in place of Cubic Mile in the above formula, and solve the mathematical expression to get the equivalent value in Perch. | {"url":"https://convertonline.org/unit/?convert=cubic_mile-perch","timestamp":"2024-11-04T17:23:14Z","content_type":"text/html","content_length":"92738","record_id":"<urn:uuid:5cdb1fa9-8f46-4bd2-b780-1829b6d21acf>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00772.warc.gz"} |
Reflection Of Light [o0mzmm162mld]
Reflection of Light Ray-tracing with plane, convex, and concave mirrors Goals: To investigate the image-forming ability of various mirror systems To use the method of ray-tracing to locate images
Equipment: Cork board Plane mirror Convex mirror Concave mirror Supply of pins Metric ruler
Introduction: When light is reflected off of any surface, the angle that the incoming (incident) light ray makes with respect to the normal to the surface will be equal to the angle that the outgoing
(reflected) light ray makes with respect to the normal. We refer to this rule of nature as the law of reflection, and write it in equation form as θi = θr.
(Eq. 1)
This is true of all reflections, but in most situations we encounter in our daily lives, we will only see diffuse reflections, in which the surface in question is microscopically rough, so that the
normal to the surface varies randomly at each point along the surface. Because of this, the light incident on the surface is bounced in many directions at once, effectively scattering (diffusing) it.
In this lab, we will be interested in specular reflections, in which the surface in question is smooth, and as a result, the normal to the surface points essentially in one direction over a given
area, thus providing us with a very clear reflection of the incident light, what in everyday language we could call a “mirror-like reflection.” A mirror, then, is simply a material that has been
polished so much that its surface is smooth enough to allow for specular reflection to occur. We will look at three particular types of mirrors: a plane (flat) mirror, a convex mirror, and a concave
mirror. In all of these experiments, we will place an object in front of the mirror, and explore the different ways in which an image of that object appears in the mirror. In all situations, the
location of the image is defined to be the place from which it appears that light is coming (where the viewer perceives the reflection to be). Plane Mirrors Reflections off of plane mirrors are made
simply in accordance with the law of reflection, Eq. 1 above. The image formed is always a virtual image, i.e., the image will appear be located at a place from which the light rays could not have
actually traveled to the observer. (In the case of plane mirrors, the image always appears to be located behind the mirror, as if the mirror were a “window” into another place. Obviously, no light is
reaching our eye from behind the mirror, so we see why the image is called virtual.) We will contrast this later with a real image, in which Reflection of Light
the light rays emanating from the object do actually reach the location of the image and pass through it. If we placed a piece of paper or a screen at the location where a real image is formed, we
would see that image clearly on the paper. Plane mirrors are easier to understand than curved mirrors simply because they provide an unmodified reflection of the object, whereas, as we’ll see in this
lab, curved mirrors distort their images, making them smaller or larger, in some cases right-side-up and sometimes upside-down (inverted). We’ll explore these different possibilities, and learn how
to trace the light rays along their paths to understand how these images are formed. Concave Mirrors Curved mirrors also obey the law of reflection, but the normal to the surface no longer points
uniformly in the same direction as it did with plane mirrors. For concave mirrors in particular, incoming light rays from very distant objects will be directed, or focused, to a particular point,
which we call the focal point of the mirror. As an example of how this happens, we draw diagrams such as Figure below, in which we see an object placed in front of the mirror. We find the image
formed by the mirror by following the path of several light rays from a point on the source to the point where all those rays cross.
Figure 1 Ray-tracing for a concave mirror. Only three conveniently chosen rays are drawn from a point in the object in order to keep the figure clear, but bear in mind that, in reality, there are an
infinite number of light rays emanating from every point on the object. For the situation shown here, the image is inverted, reduced, and real.
In the diagram shown, C is the center of curvature for the mirror, F is the focal point for the mirror, s is the distance of the object from the mirror, s’ is the distance from the mirror of the
image, and h and h’ are the heights of the object and image, respectively. A line drawn from the center of the mirror that passes through the focal point and the center of curvature defines the
principal axis. An infinite number of rays can be drawn from a single point on the object to the corresponding point on the image, but there are three rays that are especially easy to draw for curved
mirrors: A ray that comes in parallel to the principal axis of the mirror will be reflected such that it passes through the focal point. Reflection of Light
A ray that passes through the focal point of the mirror will be reflected such that it comes out parallel to the principal axis. A ray that strikes the mirror at the principal axis with a particular
incident angle will reflect making the same angle with the principal axis. These three rays can be drawn for any curved mirror and are shown in Figure . In Figure 2, we focus on two of these special
rays, following one ray that reflects off of the mirror at the principal axis (middle ray) and another ray travels parallel to the principal axis and then is reflected through the focal point
(parallel ray). Looking at ray 1 in particular, we see that the triangle ABD is similar to triangle GED (both of which are shaded).
s A
h E
G s'
Figure 2 Looking at two specific rays coming from the tip of the object in order to find geometric relationships between the object and image distances, and the object and image heights.
The sides of similar triangles are all in proportion to each other, so we can say AB GE . BD ED Rewriting this result in terms of our measurements of distances and sizes of the object and image, we
h s
h s'
(Eq. 2)
where the minus sign appears on the right to indicate that the image is inverted. Looking again at Figure 2, we can see that triangle HIF must be similar to triangle FEG. This yields the relation FI
GE . IH EF Again, rewriting this in terms of the distances and sizes in which we are interested, we find
h f Reflection of Light
h s
(Eq. 3)
By solving both Eqs. 2 and 3 for the ratio −s’/s, we can set the two expressions equal to each other and find
f f
s f
s s s . s
By dividing both sides of this equation by s’, and re-arranging terms, we come to the mirror equation:
1 f
1 s
1 . s
(Eq. 4)
We can also define the lateral magnification, M, to be the ratio of the image height to the object height. Based on Equation 2, we then also have:
h h
s s
(Eq. 5)
where a negative magnification implies an inverted image. Convex Mirrors The same mirror equation (Eq. 4), and the same equations relating object height, image height, object distance, and image
distance (Eq. 5) apply to convex mirrors as well. Figure 3 demonstrates how to ray-trace an image created by a convex mirror. A few things need to kept in mind: For convex mirrors, the focal length
is negative. Convex mirrors can only form virtual, upright, reduced images.
h h'
Figure 3 Ray-tracing for a convex mirror. The image will always appear upright, reduced, and virtual.
Reflection of Light
Name: ____________________________
Sect.: _______
Name: ____________________________
Name: ____________________________
Directions: In all of the steps to follow, try to place your pins in the corkboard as vertically as you can, and attempt to align your pins with the images in the mirrors by aligning the bottoms of
the pins (i.e., where they enter the corkboard). Activity 1: Plane Mirrors 1. Pin the Plane Mirror Template onto the cork board, and set the plane mirror on top of the boxed-in area on the template,
with the mirrored (shiny) side facing the arrow on the template. The arrow will represent our object. Q1. Describe, in your own words, the image of the object arrow you see in the mirror.
2. Place a pin at the tip of the arrow. This will be our “object” pin and will show the location of the tip of the arrow. Look in the mirror and verify that you can see an image of the pin reflected
in the mirror. Move your head from side to side to see if the image changes or appears to come from a different location.
Figure 4 Determining the location of the image. After you place the object pin on the paper template, you’ll attempt to align several pairs of “alignment pins” with where the image of the object pin
appears in the mirror.
3. Place two more pins into the corkboard such that they line up with the image of our object pin seen in the mirror, such as in the example depicted in Figure 4. (It may help to elevate your view a
bit above the corkboard.) For simplicity, you may wish to place these two new pins such that they appear to line up with the image of lines 1 or 2 already drawn on the template.
Reflection of Light
4. At a different viewing position, place two more pins, again aligning these pins with the image of the object pin. You may choose to place a third set of pins if you wish. 5. Set the mirror aside
for the moment, and use your ruler to draw lines connecting each pair of aligned pins. (You can leave the pins in the corkboard and hold your ruler against them, or remove the pins and draw your
lines through the holes they left behind.) Extend these lines to find the location where they cross. The location where they cross is where the image of our object pin appeared to be located. 6. At
the point where the reflected rays cross, draw a line down to the principal axis, representing the image arrow. Measure the distance between the location of this image and the shiny surface of the
mirror. This is the image distance, s’. Also measure the object distance, s, the distance between the object and the surface of the mirror. Object distance, s: _____________ Image distance, s’:
_____________ Q2. How does your object distance compare to your image distance?
7. Measure the height of the image h’ and the height of the object h. Object height, h: _____________ Image height, h’: _____________ Q3. How does your image height compare to your object height? Is
the image upright or inverted?
Q4. What is the magnification M of this plane mirror (cf. Eq. 5.)?
Activity 2: Convex Mirrors 8. Setup the Convex Mirror Template and the convex mirror on the cork board, again with the mirrored (shiny) side facing the object arrow. Place a pin at the tip of the
object arrow to be used as our object pin. Q5. Describe, in your own words, the image of the object arrow you see in the mirror.
Reflection of Light
9. As before, place two pins into the corkboard such that they line up with the image of our object pin. You will probably want to place your two new pins such that they appear to line up with the
image of lines 1 or 2 already drawn on the template. Do this for at least two pairs of pins. 10. Remove the mirror, and draw the lines indicating the reflected light rays of the object pin, extending
them all until they cross. From the point where the reflected rays cross, draw a perpendicular line to the principal axis, representing the image arrow. Q6. Is the image upright or inverted? Enlarged
or reduced? Virtual or real?
11. Measure the image distance, s’, and the object distance, s. Knowing these two values (and keeping our sign conventions in mind!), calculate the focal distance f for this mirror. Record these
quantities in the table below. 12. From your knowledge of s and s’, calculate the lateral magnification M of this mirror. Record this magnification in the table below. 13. Recall that the lateral
magnification also relates the image height and the object height (cf. Eq. 5). Measure the object height, and using your calculated value for the magnification M, calculate a predicted value for the
image height. After you’ve recorded your object height and predicted image height in the table below, measure the actual image height and record it in the table as well. Table 1: Convex Mirror Object
distance s Image distance
Focal length
Lateral magnification
Object height
Image height (predicted) Image height (measured)
Q7. How well does your predicted value of the image height compare with your actual measured value obtained from your ray-tracing?
Activity 3: Concave Mirrors 14. Setup the Concave Mirror Template and the concave mirror on the cork board, again with the mirrored (shiny) side facing the object arrow. Place a pin at the tip of the
object arrow to be used as our object pin. It will be necessary for you to line up your pairs of pins with the images of lines 1 and 2 already drawn on the template.
Reflection of Light
15. When you are looking at the reflected image of a line drawn on the template, you should see two separate halves to it that seem to veer off to the right or the left. When you place your eye
directly along the reflection of this line, you should see two separate images of the ray converge as shown here. Place your pins so that they line up with this cross-over point (ask your teaching
assistant for help if you have difficulty here). 16. Line up a pair of pins with both of the rays on the template. Remove the mirror, and draw the lines indicating the reflected rays of light from
the object. Extend the lines to the point where they cross, and fill out the following table (repeating the procedure in steps 11–13). Table 2: Concave Mirror Object distance s Image distance
Focal length
Lateral magnification
Object height
Image height (predicted) Image height (measured)
Q8. Is the image upright or inverted? Enlarged or reduced? Virtual or real?
Q9. How well does your predicted value of the image height compare with your actual measured value obtained from your ray-tracing?
Analysis Q10. You used the mirror equation to determine the focal length of the concave and convex mirrors. Describe an independent method you could use to determine the focal length of the convex
and concave mirrors. (Hint: you may be able to use the reflected rays of lines 1 or 2 drawn on the template to help you find the focal point.)
Reflection of Light
Plane Mirror Template
Convex Mirror Template
Concave Mirror Template | {"url":"https://doku.pub/documents/reflection-of-light-o0mzmm162mld","timestamp":"2024-11-04T02:37:53Z","content_type":"text/html","content_length":"42826","record_id":"<urn:uuid:1db111c6-3d85-436b-9e96-2c4cf2e31c78>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00487.warc.gz"} |
WEEK 14-SLOPE-OF-LINEAR-FUNCTIONS
Want to make creations as awesome as this one?
WEEK 14-SLOPE-OF-LINEAR-FUNCTIONS
Created on September 10, 2024
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Slope of Linear Functions
What is Slope?
Classification of Slopes
Finding Slope
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What is Slope?
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Before getting started, let's see how to find the change of a variable from a graph or table.
Suppose that a car starts moving from 10m of the cross walk to 60m.
Someone takes time in a stopwatch. Time measured is 20 seconds.
Let's find the changes in position and time.
Change of time=|20s-0s|=20s
Change of distance=|60m-10m|=50m
In the previous example we saw that changes are given by differences between initial and final values.
That is: Change of a variable where x₂: Final value x₁: Initial Value.
= x₂-x₁,
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The Slope Calculation.
Definition: The slope is the rate of change of y (dependent variable) when x is changed.
If (x₁,y₁), (x₂,y₂) are two points on a line, the slope is computed as follows:
Change of x
Change of y
Let's see some examples:
Example 1: Find the slope of the line passing through points (1,2), (3,5)
Example 2: Find the slope of the line passing through (0,6) and (2,2)
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1. Identify in a graph: 1st point: (x1, y1) = (1,2) 2nd point: (x2, y2) = (3,4)
2. Replace values into slope's formula:
3. Subtract values.
4. Divide numerator and denominator.
Finding Slope from Graphs and Tables
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Let's see some examples:
Example 1: Find the slope of the line shown in the graph below.
Example 2: Find the slope in the table shown. What's the meaning of the slope?
Each additional apple costs $9
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Problem might be solved drawing the total (light orange) and subtracting the additional (10).
Classification of Slopes
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Interpreting Key Features of Linear Equations
Positive slope
negative slope
Zero slope
Increasing function/Line upwards. m=+1
Decreasing function/ Line downwards. m=-3/4
Horizontal Line m=0:
Meaning For each increase (run) of x, y increases (raises) m units. Example: m=+1, m=+2,...
Meaning For each increase (run) of x, y decreases m units. Example: m=-1, m=-2,...
Meaning For each increase (run) of x, y does not change. Example: m=0
Let's see some examples:
Example 1: Classificate the function according to the slope.
Example 2: For the situation discussed on , find the slope and interpret their meaning
Decreasing function
f(x) = -4x+2
Slope is the term accompanying the variable.
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4. The slope is the velocity and is +, so it moves to the right or upwards (direction of +x axis)
Try it by yourself:
( , )
( , )
( , )
( , )
( , )
( , )
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Try it by yourself:
Rate of change of a line
Given graph?
Given pointstable?
Go back to menu
Given (x1,y1), (x2,y2) use definition formula: m= y2-y1 x2-x1
Look up one point in the graph and see how y raises after raising x one unit.
Welcome 6th graders!
A journey soon begin through Social Science experiences!
Great job!
See you next time
8TH-SLOPE-OF-LINEAR-FUNCTIONS-EN © 2024 by CASURID is licensed under CC BY-NC-ND 4.0
It is highly advised to have:
• Grid paper.
• Pencils of different colors.
• Eraser.
• A rule.
• A compass.
• A Protactor.
• A calculator.
• Geogebra installed on your phone/tablet/computer (or use online version).
"MA.8.AR.3 Extend understanding of proportional relationships to two- variable linear equations." MA.8.AR.3.2 Given a table, graph or written description of a linear relationship, determine the
slope. ELA.K12.EE.1.1 Cite evidence to explain and justify reasoning. | {"url":"https://view.genially.com/66dfc45bb2c54e1ea057b72c/interactive-content-week-14-slope-of-linear-functions","timestamp":"2024-11-07T16:52:43Z","content_type":"text/html","content_length":"42950","record_id":"<urn:uuid:4af7d368-c459-4e3e-99be-37d24b258ee9>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00404.warc.gz"} |
How to Bake PI
What is math? And how exactly does it work? In How to Bake Pi, math professor Eugenia Cheng provides and accessible introduction to the logic of mathematics-sprinkled throughout with recipes for
everything from crispy duck to cornbread-that illustrates to the general reader the beauty of math. Rather than dwell on the math of our high school classes, with formulas to memorize an confusing
symbols to decipher, Cheng takes us into a world of abstract mathematics, showing us how math can be so much more than we ever thought possible.
Cheng is an expert on category theory, a cutting-edge subject that is all about figuring out how math works, a kind of mathematics of mathematics. In How to Bake Pi, Cheng starts with the basic
question "What is math?" to explain concepts like abstraction, generalization, and idealization. By going back to the logical foundation of the math we all know (and may or may not love), she shows
that math is actually designed to make difficult things easier. From there, she introduces us to category theory, explaining how it works to organize and simplify the whole discipline of mathematics,
bridging the gaps between different mathematical concepts and shedding light on some of math's most puzzling mysteries. Though the ideas are for from simple, Cheng outlines everything in
crystal-clear terms, drawing on a wide range of analogies and examples to show that doing math uses the same skills we rely on when we read a map, cook a new dish, or complete a jigsaw puzzle. The
result is a book that combines some of the most satisfying features of popular math books-the thrill of truly understanding things that may or may not have been confounding in high school, while
still look long and hard into unexplored territory.
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Factorization in coNP- in other domains?
I had on an exam in my grad complexity course to show that the following set is in coNP
FACT = { (n,m) : there is a factor y of n with 2 \le y \le m }
The answer I was looking for was to write FACTbar (the complement) as
FACTbar = { (n,m) | (\exists p_1,...,p_L) where L \le log n
for all i \le L we have m < p_i \le n and p_i is prime (the p_i are not necc distinct)
n =p_1 p_2 ... p_L
INTUITION: Find the unique factorization and note that the none of the primes are < m
To prove this work you seem to need to use the Unique Factorization theorem and you need
that PRIMES is in NP (the fact that its in P does not help).
A student who I will call Jesse (since that is his name) didn't think to complement the set so instead he wrote the following CORRECT answer
FACT = { (n,m) | n is NOT PRIME and forall p_1,p_2,...,p_L where 2\le L\le log n
for all i \le L, m< p_i \le n-1 , (p_i prime but not necc distinct).
n \ne p_1 p_2 ... p_L
(I doubt this proof that FACT is in coNP is new.)
INTUITION: show that all possible ways to multiply together numbers larger than m do not yield n,
hence n must have a factor \le m.
Here is what strikes me- Jesse's proof does not seem to use Unique Factorization. Hence it can be used in other domains(?). Even those that do not have Unique Factorization (e.g. Z[\sqrt{-5}]. Let D=
Z[\alpha_1,...,\alpha_k] where the alpha_i are algebraic. If n\in D then let N(n) be the absolute value of the sum of the coefficients (we might want to use the product of n with all of its
conjugates instead, but lets not for now).
FACT = { (n,m) : n\in D, m\in NATURALS, there is a factor y in D of n with 2 \le N(y) \le m}
Is this in NP? Not obvious (to me) --- how many such y's are there.
Is this the set we care about? That is, if we knew this set is in P would factoring be in P? Not obv (to me).
I suspect FACT is in NP, though perhaps with a diff definition of N( ). What about FACTbar?
I think Jesse's approach works there, though might need diff bound then log L.
I am (CLEARLY) not an expert here and I suspect a lot of this is known, so my real point is
that a students diff answer then you had in mind can be inspiring. And in fact I am inspired to
read Factorization: Unique and Otherwise by Weintraub which is one of many books I've been
meaning to read for a while.
9 comments:
1. The various definitions of FACT and FACT-bar seem have the form { (n,m) | }. I'm pretty good at figuring out what you mean, but I can't tell what Jesse means (Are there more typos in there? Did
you really mean it when you said "n \ne p_1 p_2 ... p_L"?)
It would also be helpful to get an inline LaTeX parser for the blog!
2. Has Jesse's proof been garbled? If I am understanding it right, it says FACT is the set of (n,m) such that every factorization of n contains a factor between 2 and m, which is not correct.
3. YES, I garbled things and also my editor garbled things. But its fixed now. I hope.
4. Maybe I'm confused, but is Jesse's proof really correct? A pair (n=64, m=6) belongs to FACT (2 is a factor). But in this definition, for L=2, p1=p2=8 we have that it does not belong.
1. It looks like you're right.
Couldn't we add in the condition that each p_i is prime? PRIMES is clearly in co-NP: for all q, r, q*r=p -> q=1 or r=1.
5. maybe someone should answer malcin's concern ?
1. Malcin is correct, and Andy is correct, so I used Andy's correction.
6. Here http://npcomplete-001-site1.myasp.net/ resolved Exact cover ploblem, on line solver aviable
7. What about adding MathJax to the site? (one line of code :-) | {"url":"https://blog.computationalcomplexity.org/2014/04/factorization-in-conp-in-other-domains.html?m=0","timestamp":"2024-11-04T14:33:22Z","content_type":"application/xhtml+xml","content_length":"189683","record_id":"<urn:uuid:ebcba1a6-4836-47b2-bcfc-fd122b12d3e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00863.warc.gz"} |
Simple Interest Worksheets
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Simple Interest Worksheet Maryam F PDF Interest Percentage
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The Mathematics Major
Steps to declare the Math Major (Declaration Periods: Fall 9/15- 12/9 & Spring 2/1 - 4/17)
1. Complete the calculus requirement (Math 1400 and Math 1410+2400 or Math 1610+2600) and one proof-based major course.
2. Go to Path @ Penn to formally request to add or remove your major to the Math Office (otherwise it won't show on your transcripts).
3. Use this link to complete the top portion of our course plan worksheet and email it to the major advisor we assign to you after we process your Path @ Penn request.
4. After consulting with your advisor, he/she will review your worksheet for processing.
5. Allow 15 business days for your major to appear on your transcript. If its not visible by day 15th email me at rtoney@math.upenn.edu
6. Contact your math advisor once a semester. Email your advisor to answer questions or make changes to your course plan.
7. Once you enter your semester of graduation, you must email your math major advisor to certify your math major worksheet for completion.
On This Page
Admission to the Major Program
Permission to major in mathematics is normally obtained by the end of the sophomore year, but planning for it should begin as early as possible. It is important that majors entering their junior year
commence satisfying the algebra and analysis requirements.
To be admitted to the major, a student must have completed successfully (i.e., with grades of C or better) the calculus requirement as well as one proof-based math class (such as Math 2020, 2030,
1610, 2600, 3140) in the freshman and sophomore years. A higher-level proof-based class may be substituted at the discretion of a math major advisor.
Students who plan to have math as their second major should have a cumulative GPA of at least 3.0, an average of at least 3.0 in their math courses, and no math grades lower than B-.
The Major is open to SEAS undergraduates (as a second major) as well as to students in the College.
The Math Major and its Goals
Mathematical training allows one to take a problem, abstract its essential features, and investigate them further. This ability can assist greatly in such diverse fields as economics, law, medicine,
engineering, and computer science -- as well as in the more traditional activities of research and teaching.
The goals of the major program are to assist students in acquiring both an understanding of mathematics and an ability to use it. We wish to inspire the discovery of new mathematics as well as the
application of mathematics to other fields.
The mathematics major provides a solid foundation for graduate study in mathematics as well as background for study in economics, the biological sciences, the physical sciences and engineering, as
well as many non-traditional areas. This flexibility is available through an appropriate choice of electives within the major. A variety of electives are offered. They are designed to serve the needs
of mathematics majors and others who want more advanced training in mathematics and its applications. Most of these courses presume our basic two year calculus sequence.
The mathematics major is also excellent training for students interested in elementary and secondary education. For information on the elementary education undergraduate major or the secondary
education submatriculation program which leads to a Master's degree, students should consult the Undergraduate Chair as well as the Director of Teacher Education in the Graduate School of Education.
Highly qualified and motivated students should note the possibility of obtaining both the B.A. and M.A. degrees in four years. This is discussed below.
Given the widening role of mathematics, students with special interests and needs may wish to consider the possibility of an individualized program of study, perhaps in conjunction with a major in
another field. The Major Coordinator should be consulted about this.
How to Plan a Mathematics Major Prospective majors should first check the information listed under Advanced Placement. We strongly encourage students to master the basic material as early as
possible, and AP credit is equal to credit for a course taken at Penn. Students are urged to read the Major Program Requirements carefully, and use it as a guideline to plot the plan. You should also
read Other Useful Experience and Further Recommendations for a complete overview.
What can I do with a math major?
See Careers in Mathematics . It is maintained by the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics.
See Math in the Media for pointers to current general articles involving mathematics and its applications.
Advanced Placement
Note: The AP policy and more details on this subject (and on "transfer credit") can be found on our web page AP/Transfer Credit Information.
We strongly encourage students to master the basic material as early as possible. It is our policy to waive prerequisite course requirements for those students who can pass an examination that
demonstrates that they know the material. These remarks apply especially to the first-year calculus courses. For these, a student may receive credit towards the degree (in addition to the waiving of
prerequisites) by either of the following methods:
1. Passing the external Advanced Placement BC Exam administered by the College Entrance Examination Board with a score of 5 gives credit for Math 1400. Lower scores on the BC Exam receive no course
credit. No credit is given for the AB Exam. Students taking first semester calculus, math 1400, are expected to have had an AB calculus course in high school.
2. Passing the internal Advanced Placement Examination administered in the first week of the fall and spring semester by the mathematics department. A student may take the examination regardless of
whether he or she took the external exam described under (1) above.
Those receiving advanced placement and planning to enroll in more advanced courses should see the Major Coordinator, who will help them plan a program of study.
The Major Program Requirements (Minimum 13 c.u.)
The Mathematics Major comprises 13 courses organized into eight basic requirements. Each of the 13 courses must be taken for a grade (i.e., not pass/fail), and must be completed with a grade of C
(2.0) or better. (A student who receives a grade lower than C in a requirement consisting of more than one course may still count that course toward the major by achieving a grade of C or above in a
more advanced course within the same requirement.) The math department also expects the completion of at least one proof-based math class in the freshman or sophomore year in order to be admitted
into the major (usually Math 2020, 2030, 1600, 2600, or 3140), or permission of a math major advisor. Courses taken on a pass/fail basis will not count toward fulfilling the following requirements.
1. Three Semester Calculus Requirement. This is satisfied by any of the two sequences 1400-1410-2400 or 1400-1610-2600. The 1400 requirement can be satisfied by AP credit for the Calculus BC exam
with a score of 5. The courses 1610-2600 are proof-based and provide the best preparation for higher mathematics, and in particular for Math 3600 and 3610. Math 1300 does not count toward the
Math major.
2. Advanced Calculus Requirement Math majors must take either a fourth semester of calculus, Math 2410, or partial differential equations Math 4250.
3. Complex Analysis Requirement All math majors must take Complex Analysis Math 4100.
4. Seminar Requirement This is satisfied by taking either Math 2020 (intro to analysis) or Math 2030 (intro to algebra). These courses carry one-unit of credit and are intended to be taken
concurrently with calculus. For students taking honors calculus (Math 1610-2400) the seminar requirement is replaced by a higher math elective course.
Students who begin with Math 1400 in their freshman year usually postpone this requirement until their second year. Students who have already taken one of Math 2410, Math 4100 or Math 4250 can
substitute a higher math elective course for the Seminar Requirement. Under exceptional circumstances, other students may also make such a substitution with the permission of the Undergraduate
Chair. In general, though, we recommend that prospective math majors take a freshman seminar to gain an overview of the subject.
5. Linear algebra requirement.
Math Majors must take Advanced Linear Algebra Math 3140. Math 3140 is a prerequisite for Math 3700 and Math 5020.
6. Algebra Requirement This is satisfied by taking the sequence Math 3700-3710 or the more theoretical Math 5020-5030. However, you can't get credits for both Math 3700 and Math 5020, or both Math
3710 and Math 5030.
These courses all overlap considerably.
7. Analysis Requirement This is satisfied by taking the sequence Math 3600-3610 or the more theoretical Math 5080-509- However, you can't get credits for both Math 3600 and Math 5080, or both Math
3610 and Math 5090.
Note: Majors who begin their mathematics studies with Math 1410-2400 plus a seminar should fulfill at least one of the linear algebra, algebra, and analysis requirements in their sophomore year.
8. Mathematics Electives The total number of approved math course units required for a math major is 13. Students should determine how many course units they still need for a math major after
completing requirements 1 through 6 above. This will depend on which options have been chosen in completing the requirements. The remaining courses may then be made up from Math 2100 and
mathematics courses numbered 3200 or above. One mathematics elective course unit may be taken from the list of approved Cognate Courses given outside the math department. Students who are double
majors may take two Cognate courses units.
Students may, for example, take Statistics 4300 (or Systems Engineering 3010 or Econ 103 or ENM 5030), and count such a course as being within the Mathematics Department. Thus by taking one of
these courses, one does not lower the number of cognate courses one can take outside the math department, as explained on the page of Cognate Courses.
Example 1: A student is double majoring in math and engineering, did not take a freshman seminar, and completed the Advanced Calculus requirement by taking math 2410. This student thus takes 4
courses related to the Calculus requirements, 4 courses to complete the algebra and analysis requirements, and Math 3140 and Math 4100 for a total of 10 courses. They must take 3 electives to
bring their course total up to 13. Because the student is double majoring in math and engineering, two of these electives can be Cognate Courses in other departments. Notice that on the above
list of cognate courses, some courses given in other departments are listed as being counted as within the math department as far as the math major and minor are concerned. For example, the
student could take Stat 4300 (which is counted as within the math department), use Physics 0150 and Physics 0151 as their cognate courses not counted as within the math department, and then
choose two more electives from within the math department to complete their math major requirements.
Example 2: A student is majoring only in math, took a freshman seminar, and completed the Advanced Calculus requirement by taking math 4250. This student thus takes the freshman seminar, 3
calculus courses, math 4100 and 4250 and four algebra and analysis courses in the course of completing the above requirements, for a total of 10 courses. They must take three further electives
for a math major. Only one of these can be a Cognate Course, because the student is not a double major.
Planning your Mathematics Major
Students who do not plan graduate study in mathematics or in a highly mathematics-related subject should, as a means of acquiring more background, consider Math 4100, 4200, 4250, and 4300. For
glimpses of several beautiful mathematical subjects beyond the basic core, students should consider Math 3500, 5420, 5480, 5490, 5800, 5000, 5300, 4800.
Students who are interested in the physical sciences should consider Physics 0150-0151 or 0170-0171 and the courses beyond. Those interested in the social or biological sciences should consider Math
4300 or Statistics 4300-4310. Those interested in computer science should consider CSE 110, 120-121 and the courses beyond as well as Math 4500, 5700 (previously 473 and 670). For computer
programming and numerical methods, students should learn a programming language such as Pascal or C and learn to use symbolic manipulation software such as Mathematica or Maple. They also should
consider Math 3200-3210. For discrete methods, in addition to Math 3400 and 3410, 450, 5700 (previously 473), and 5800 (previously 440), students should consider Math 5240-5250 (previously 470) and
5810 (previously 441).
For students who plan to do research in mathematics, or in a highly mathematical subject such as statistics, the considerations which are listed just above still apply. However, since a great deal of
further theoretical training is necessary, such students are directed to the basic graduate courses in mathematics: 6000, 6010, 6020, 6030, 6080, and 6090.
All this material must eventually be mastered. It needs to be understood clearly that what is required is a comprehensive grasp of theoretical mathematics. Thus, the student's attention is directed
to Method B for obtaining honors in mathematics, and to the joint B.A./M.A. program, pursuing a master's degree at the same time as their undergraduate degree.
Other Useful Experiences for a Math Major
The first order of business is to satisfy the first four requirements discussed above. When this has been done, the student usually has sufficient experience and direction to complete the program in
consultation with the Major Advisor . It needs to be emphasized strongly, however, that apart from the strict requirements, there are certain other things which all mathematics majors should do.
These are:
• Learn to program a computer and learn how to use mathematical symbolic manipulation packages. The latter skill is taught in our Calculus courses.
• Learn statistics. This may be done by taking Math 4300 or Stat 4300 followed by Stat 4310.
• Learn how mathematics is actually used. This can be done by learning something of an applied but highly mathematical field. Operations research, engineering and physics provide examples, but
there are many others. (See below.)
• Obtain some job experience. This should be done, if possible, in the summer following the junior year. It should involve some interface between mathematics and the real world.
The importance of the above four recommendations cannot be sufficiently emphasized. Equipped with them, a mathematics major is an attractive candidate for entrance into a great many fields. Without
them, job opportunities are limited. These remarks apply to the most theoretical, as well as to the most practical of careers.
The Honors Program
To be eligible for honors in mathematics, a student must have an average of at least 3.5 in his/her major and major-related courses. If this condition is satisfied, honors may be obtained by either
of the following methods.
Method A. By preparing, through independent study, a body of material approximately equal in amount to a one-semester course and giving a lecture on it as the Honors Committee shall direct.
The area of study chosen should be one that is not normally covered in the department and should involve reading sources outside normal course material. The selected topic may be picked from one
field of mathematics or may involve assimilation of topics from different fields. Before beginning the project, the student should ask two members of the faculty, at least one affiliated with the
Mathematics Department, to serve as the Honors Committee. The Honors Committee must approve the selected topic and serve as examiners for the lecture (which should be approximately an hour long,
seminar-style talk).
Method B. By passing the written Preliminary Examination in undergraduate mathematics. This is required of all incoming Ph.D candidates. Details concerning this examination may be found in the
Graduate Admissions Catalogue (also see below).
For further guidance, prospective honors students should consult with their Major Advisor during their junior year. The honors project must be completed by the end of February of the senior year.
The Master's Program
Undergraduates who wish to take courses beyond the math major program should consider submatriculation and pursuing a master's degree. The minimum requirements are a A- average in 3600-3610 or
5080-5090 and 3700-3710 or 5020-5030, and permission of the Graduate Chair. Students who plan on a master's degree should submatriculate as early as possible because only courses taken subsequently
to this may be counted toward the degree. The degree itself requires the successful completion of eight graduate courses and the written examinations for the Ph.D. The requirements can sometimes be
completed by the end of the fourth undergraduate year, but often a fifth year is required.
For more information see the SAS web page Submatriculation and the Math Department Submatriculation page.
1. Gifted high school students from the Philadelphia area are encouraged to take courses (usually Math 2400-2410) in the department while they are still in high school. This is done through the
Young Scholars Program which is administered by the College of Liberal and Professional Studies.
2. High school seniors who wish to major in mathematics and think that they might like to attend the University are invited to visit the mathematics department to meet the faculty and visit classes.
They should email ugrad AT math.upenn.edu or call 898-8178 for an appointment.
3. For our Math Majors and Minors there is a list of courses often approved as COGNATES for Mathematics Majors (these are courses from other departments often approved for mathematics majors or
minor credit). All cognates require the approval of the Undergraduate Chair and must be part of a well-planned selection of electives within the major. The statistics courses enjoy a special
status: since they count as being inside the Mathematics Department as far as the major or minor is concerned. Thus, a student who takes one or two of these may count additional outside courses
toward the Mathematics Elective requirement. Additional courses may also be approved as cognates upon application to the Undergraduate Chair.
4. Undergraduates who plan to teach in secondary schools should refer to the section on the Bachelor of Arts/Master of Science in Education.
5. Penn has an active Undergraduate Mathematics Society which conducts seminars, colloquia and other activities for those who wish to encounter Math outside the classroom. Information about Society
membership and schedules of its activities can be obtained in the Math Department office or by clicking on the link above.
6. Opportunities for summer research exist at many Universities. The Undergraduate Chair is a good source of information about such programs, which are usually announced in October or November.
Some External Links relevant to math majors | {"url":"https://www.math.upenn.edu/undergraduate/math-majors-and-minors/mathematics-major","timestamp":"2024-11-11T20:28:12Z","content_type":"text/html","content_length":"76298","record_id":"<urn:uuid:efcef9fc-8c84-4784-9e2d-24bc4f6818a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00033.warc.gz"} |
Haar Wavelet-Based Perceptual Similarity Index
The Haar wavelet-based perceptual similarity index (HaarPSI) is a similarity measure for images that aims to correctly assess the perceptual similarity between two images with respect to a human
In most practical situations, images and videos can neither be compressed nor transmitted without introducing distortions that will eventually be perceived by a human observer. Vice versa, most
applications of image and video restoration techniques, such as inpainting or denoising, aim to enhance the quality of experience of human viewers. Correctly predicting the similarity of an image
with an undistorted reference image, as subjectively experienced by a human viewer, can thus lead to significant improvements in any transmission, compression, or restoration system.
The HaarPSI has the following advantages over previous full reference quality metrics such as (MS-)SSIM, FSIM, PSNR, GSM, or VIF:
• It achieves higher correlations with human opinion scores on large benchmark databases in almost every case (see experimental results).
• It can be computed very efficiently and significantly faster than most other metrics (see Table II in our paper). | {"url":"http://www.math.uni-bremen.de/cda/HaarPSI/","timestamp":"2024-11-06T02:41:48Z","content_type":"text/html","content_length":"21877","record_id":"<urn:uuid:13482313-6d4d-4383-b1d0-23a3a652ed69>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00766.warc.gz"} |
9 tricky facts about computer science -
Moore's Law's Demise: Moore's Law, which states that the number of transistors on a microchip doubles approximately every two years, is slowing down. The challenges of maintaining this exponential
growth in computing power have profound implications for the future of hardware development.
Quantum Computing Uncertainty: While the promise of quantum computing is exciting, building practical and scalable quantum computers is extremely challenging. The field is still in its infancy, and
there are uncertainties about when we'll have reliable and powerful quantum computers.
P vs NP Problem: It remains unknown whether every problem that can be verified quickly (in polynomial time) can also be solved quickly (in polynomial time). This is one of the most important open
problems in computer science and mathematics.
The Two Generals' Problem: In distributed computing, there is a theoretical problem called the Two Generals' Problem, which explores the difficulty of coordinating actions between two entities that
can only communicate through an unreliable channel.
Rice's Theorem: This theorem, named after mathematical logician Henry Gordon Rice, states that all non-trivial properties about the behavior of a program are undecidable. In other words, it's
impossible to write a general algorithm that can decide all interesting properties of computer programs. | {"url":"https://inprogrammer.com/web-stories/9-tricky-facts-about-computer-science-2/","timestamp":"2024-11-04T04:23:00Z","content_type":"text/html","content_length":"57644","record_id":"<urn:uuid:3d0182ec-350c-4c13-a41a-c431c01daff6>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00445.warc.gz"} |
Send More Money
Problem H
Send More Money
You may have seen puzzles, sometimes called cryptarithmetic puzzles, or simply cryptarithms, that look like this:
The goal of this kind of puzzle is to replace each letter with a (base-$10$) digit so that the resulting arithmetic equality holds. Here are the rules:
• A given letter must always be replaced by the same digit.
• Different letters must be replaced by different digits.
• The leading (leftmost) letter of any word cannot be replaced by 0.
For example, if we try to solve the above puzzle, we find that the following replacements work:
S$\rightarrow $9, E$\rightarrow $5, N$\rightarrow $6, D$\rightarrow $7, M$\rightarrow $1, O$\rightarrow $0, R$\rightarrow $8, Y$\rightarrow $2
This gives us a correct equality:
Your task is to write a program to solve such puzzles. Note that some cryptarithmetic puzzles are impossible to solve. An example is:
Here the only way to make the equality hold is to replace A with 0, but this violates the rule that the first (and in this case only) letter of any word cannot be replaced by 0.
Also note that some puzzles may have multiple solutions. In this situation, find the solution in which the alphabetically lowest letter is replaced by lowest possible digit, then the second lowest
letter is replaced by the lowest possible unused digit, then the third lowest letter is replaced by the lowest possible unused digit, etc. An example is:
Clearly this has more than one solution, but the minimal solution, as described above, is:
The input consists of a line containing a single puzzle. A puzzle is a string of characters of the form $w_1$+$w_2$=$w_3$, where $w1$, $w_2$, $w_3$ are words, and ‘+’ and ‘=’ are the usual “plus” and
”equals” characters (ASCII values $43$ and $61$, respectively). A word is a nonempty string of uppercase letters (A–Z). The maximum length of any puzzle is $100$.
For each puzzle, output a single line containing the minimal solution if the puzzle is solvable, or “impossible” if the puzzle is not solvable. Note that a solution must contain exactly the same
number of characters as the corresponding puzzle, with ‘+’ and ‘=’ characters in the same positions as in the puzzle, and all letters replaced by digits (0–9).
Sample Input 1 Sample Output 1
SEND+MORE=MONEY 9567+1085=10652
Sample Input 2 Sample Output 2
A+A=A impossible
Sample Input 3 Sample Output 3
C+B=A 2+1=3 | {"url":"https://nus.kattis.com/courses/CS2040DE/CS2040DE_S1AY2425/assignments/qcfvo4/problems/sendmoremoney","timestamp":"2024-11-04T10:30:46Z","content_type":"text/html","content_length":"33222","record_id":"<urn:uuid:60adec79-1306-4a06-b727-b976150e0011>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00864.warc.gz"} |
Mastering Formulas In Excel: What Is Poly Cell Formula
Mastering formulas in Excel is crucial for anyone looking to efficiently analyze and manipulate data. One formula that is incredibly useful for this purpose is the poly cell formula, which allows
users to perform calculations across multiple cells simultaneously. In this blog post, we will take a closer look at the poly cell formula and its applications in Excel.
Key Takeaways
• Mastering formulas in Excel is crucial for efficient data analysis and manipulation.
• The poly cell formula is incredibly useful for performing calculations across multiple cells simultaneously.
• Understanding the syntax and working with poly cell formula is essential for efficiency and accuracy.
• Integration with other Excel functions and customization for specific data sets are advanced features of the poly cell formula.
• Practicing with poly cell formula and exploring real-world applications can improve proficiency and understanding.
Understanding Poly Cell Formula
A. Definition of poly cell formula
A poly cell formula in Excel refers to a formula that can be applied across multiple cells at once, allowing for efficient and streamlined calculations. It is a powerful feature that can save time
and effort when working with large datasets or complex calculations.
B. How poly cell formula differs from regular formulas in Excel
1. Range of cells
While regular formulas in Excel are typically applied to individual cells or a specific range of cells, poly cell formulas can be applied to a larger selection of cells, making it easier to perform
calculations on multiple data points at once.
2. Flexibility
Poly cell formulas offer greater flexibility in terms of applying the same formula to different sets of data without having to manually input the formula for each cell.
3. Efficiency
Using poly cell formulas can significantly improve efficiency, especially when working with large datasets, as it eliminates the need to manually input formulas for each individual cell.
C. Examples of situations where poly cell formula is useful
• Calculating the total sales for multiple products or regions
• Applying the same percentage increase or decrease to multiple values
• Performing complex calculations across a large dataset, such as financial modeling or data analysis
Syntax of Poly Cell Formula
When it comes to mastering formulas in Excel, understanding the syntax of the poly cell formula is essential for effectively manipulating data. The poly cell formula is a powerful tool that allows
you to perform complex calculations and analysis on a range of cells.
Explanation of the syntax of the poly cell formula
The syntax of the poly cell formula consists of several components that work together to generate the desired result. Understanding each component and its role is crucial for using the formula
Breakdown of each component of the syntax
The poly cell formula typically consists of the following components:
• Range: This is the range of cells on which the formula will be applied. It can be specified using cell references or named ranges.
• Condition: This is the condition that the cells in the range must meet in order to be included in the calculation. It can be a logical expression, such as a comparison or a function.
• Operation: This is the operation that will be performed on the cells that meet the condition. It can be a mathematical operation, a statistical function, or any other supported calculation.
Tips for remembering the syntax
Remembering the syntax of the poly cell formula can be challenging, especially for beginners. Here are some tips to help you remember the syntax:
• Practice: The more you practice using the formula, the more familiar you will become with its syntax.
• Use examples: Work through examples of the formula to understand how each component fits into the overall syntax.
• Reference guides: Keep a reference guide or cheat sheet handy for quick access to the syntax when needed.
Working with Poly Cell Formula
Excel is a powerful tool for data analysis and calculations, and mastering formulas is key to unlocking its full potential. One such formula that is commonly used in Excel is the poly cell formula,
which allows you to perform calculations across multiple cells. In this chapter, we'll explore how to work with poly cell formulas, common errors to avoid, and best practices for using them
How to input the poly cell formula into Excel
Inputting the poly cell formula into Excel is a straightforward process. To begin, select the cell where you want the formula result to appear. Then, type the formula using the appropriate syntax,
taking into account the range of cells you want to include in the calculation. For example, if you want to sum the values in cells A1 to A10, the formula would be =SUM(A1:A10). After typing the
formula, press Enter to apply it to the selected cell.
Common errors to avoid when using poly cell formula
• Incorrect cell references: One common error when using poly cell formulas is using incorrect cell references, which can result in inaccurate calculations. Always double-check the cell references
to ensure they are accurate.
• Mismatched ranges: Another common error is using mismatched ranges in the formula, such as trying to perform a calculation on cells with different numbers of rows or columns. Ensure that the
ranges you are using in the formula are consistent and match the data you want to include in the calculation.
• Failure to lock cell references: When copying a poly cell formula to other cells, failing to lock the cell references can lead to errors as the formula adjusts to the new cell locations. Use
absolute cell references (e.g., $A$1) when necessary to prevent this from happening.
Best practices for using poly cell formula efficiently
• Use named ranges: To make your formulas more readable and easier to manage, consider using named ranges for the cells you want to include in the calculation. This can help to avoid errors and
make your formulas more transparent.
• Document your formulas: When working with complex poly cell formulas, it can be helpful to document the formula logic and the ranges being used. This can make it easier to understand and
troubleshoot the formulas in the future.
• Test and verify: Before relying on poly cell formulas for important calculations, it's important to test and verify the results to ensure accuracy. Use sample data to confirm that the formula is
producing the expected output.
Advanced Features of Poly Cell Formula
In Excel, the poly cell formula is a powerful tool for analyzing and manipulating data. It goes beyond basic cell formulas to provide advanced features for handling complex data sets. Here are some
of the advanced features of the poly cell formula:
A. Integration with other Excel functions
• 1. Nested functions: The poly cell formula can be nested within other Excel functions to perform complex calculations and data manipulation.
• 2. Array functions: It can be used with array functions to process multiple data points at once, making it a versatile tool for handling large data sets.
B. Customizing poly cell formula for specific data sets
• 1. Conditional formatting: The poly cell formula can be customized with conditional formatting to highlight specific data points based on defined criteria.
• 2. Using logical operators: Logical operators can be integrated with the poly cell formula to customize the output based on specified conditions.
C. Automating poly cell formula for large datasets
• 1. Using macros: Macros can be created to automate the application of poly cell formula across large datasets, saving time and effort.
• 2. Data validation: Data validation rules can be set to automatically apply the poly cell formula to new data entries, ensuring consistency and accuracy in data analysis.
Tips for Mastering Poly Cell Formula
Mastering poly cell formulas in Excel can greatly enhance your data analysis and reporting skills. Here are some tips to help you improve your proficiency with poly cell formulas:
A. Practice exercises for improving proficiency with poly cell formula
• 1. Start with simple formulas: Begin by practicing with basic poly cell formulas to understand the syntax and functionality.
• 2. Use real data sets: Apply poly cell formulas to real data sets to gain practical experience and improve your problem-solving skills.
• 3. Experiment with complex scenarios: Challenge yourself with complex scenarios and data structures to deepen your understanding of poly cell formulas.
B. Resources for further learning and mastering poly cell formula
• 1. Online tutorials and courses: Explore online resources offering tutorials, courses, and webinars focused on advanced Excel formulas, including poly cell formulas.
• 2. Books and publications: Invest in books or publications dedicated to Excel and data analysis, which often provide in-depth coverage of poly cell formulas and their applications.
• 3. Community forums and user groups: Engage with Excel user communities and discussion forums to seek advice, share experiences, and learn from others who have mastered poly cell formulas.
C. Real-world applications of poly cell formula in different industries
• 1. Finance and accounting: Utilize poly cell formulas to analyze financial data, perform forecasting, and create complex financial models for decision-making.
• 2. Marketing and sales: Apply poly cell formulas to track sales performance, calculate marketing ROI, and analyze customer data for targeted campaigns.
• 3. Operations and supply chain management: Use poly cell formulas to optimize inventory levels, monitor production processes, and analyze supply chain efficiency.
Mastering formulas in Excel is crucial for anyone looking to streamline their data management and analysis. The ability to use poly cell formula effectively can greatly enhance the efficiency and
accuracy of your work.
Understanding and utilizing poly cell formula effectively in Excel is essential for:
• Handling complex calculations
• Automating repetitive tasks
• Creating dynamic and interactive spreadsheets
Final thoughts
By mastering formulas in Excel, including the poly cell formula, you can take your data manipulation skills to the next level and become more productive in your day-to-day tasks. Keep practicing and
exploring the various formulas Excel has to offer to become a proficient user.
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Calculating Tension: Forces and Vectors in a 1.8 kg Pendulum with 2.3 m String
• Thread starter alynnaD
• Start date
In summary, the formula for calculating tension in a pendulum is T = (m x g x l) / (l + h), the force of gravity in a pendulum is determined by multiplying the mass by the acceleration due to
gravity, the length of the string is a crucial factor in calculating tension, different units of measurement can be used for length and mass, and the tension in a pendulum changes as the mass and
length of the string are altered.
A bob of mass 1.8 kg attached to a string of length 2.3 m. The pendulum is held at an angle of 30 degrees from the vertical by a horizontal string attached to a wall.
What is the tension in the horizontal string?
Science Advisor
Homework Helper
Gold Member
2023 Award
Hi alynnaD and welcome to PF. Please follow the rules of this forum and use the template when you seek help with homework. Show us the relevant equations and tell us what you tried and what you think
about the problem. We just don't give answers away.
FAQ: Calculating Tension: Forces and Vectors in a 1.8 kg Pendulum with 2.3 m String
1. What is the formula for calculating tension in a pendulum?
The formula for calculating tension in a pendulum is T = (m x g x l) / (l + h), where T is the tension, m is the mass of the pendulum, g is the acceleration due to gravity, l is the length of the
string, and h is the height of the pendulum's center of mass.
2. How do you determine the force of gravity in a pendulum?
The force of gravity in a pendulum is determined by multiplying the mass of the pendulum by the acceleration due to gravity, which is approximately 9.8 m/s^2 on Earth.
3. What is the significance of the length of the string in calculating tension?
The length of the string is a crucial factor in calculating tension in a pendulum because it affects the angle at which the pendulum swings and the distance between the pendulum's center of mass and
the pivot point. A longer string will result in a larger angle and a greater distance, thus increasing the tension.
4. Can you use a different unit of measurement for length and mass in the tension formula?
Yes, the length can be measured in any unit of length (such as centimeters or inches) and the mass can be measured in any unit of mass (such as grams or pounds). However, it is important to ensure
that the units are consistent throughout the calculation.
5. How does the tension in a pendulum change as the mass and length of the string are altered?
If the mass of the pendulum is increased, the tension will also increase because there is more weight pulling on the string. Similarly, if the length of the string is increased, the tension will also
increase because the pendulum will have a larger swing and thus exert more force on the string. Conversely, decreasing the mass or length of the string will result in a decrease in tension. | {"url":"https://www.physicsforums.com/threads/calculating-tension-forces-and-vectors-in-a-1-8-kg-pendulum-with-2-3-m-string.430367/","timestamp":"2024-11-09T15:51:13Z","content_type":"text/html","content_length":"74866","record_id":"<urn:uuid:2f4fd7c4-5a9d-4ebc-86eb-12f1582bee65>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00347.warc.gz"} |
SPECIAL RELATIVITY AS A THEORY OF LIGHT
RATHER THAN SPACETIME
The purpose of this page is to examine an alternative SR model with the capacity to account for the light postulate, the phenomenon of relativity of simultaneity, and other SR phenomena. This
alternative was not examined, or even mentioned, by Einstein in his original 1905 paper, or by anyone since. The proposed model attributes SR phenomena to an undiscovered property associated with
light itself, rather than to a peculiarity in the nature of the underlying spacetime. The result is a version of SR consistent with the transformation equations and established experimental results.
It identifies a Galilean spacetime with Lorentzian, and not Galilean, interactions. It at least hints, in addition, at a possible basis for modern experiments with light, involving nonlocality and
which path erasure phenomena, and may also be able to support a better explanation for the existence of superluminal quasar jets, which cannot at present be adequately explained.
In his 1905 paper, On the Electrodynamics of Moving Bodies, in section 2 (On the Relativity of Lengths and Times), Einstein uses the light postulate to show that moving clocks, synchronised according
to a stationary system, do not appear to be synchronised according to an observer moving with the moving system.
The initial method which he uses to show the existence of the nonsimultaneity referred to, is based on the result of sending a light ray along a moving rod, reflecting it off the end of the rod, and
returning it again to the start of the rod, subject to the requirement of the light postulate, which declares that light travels at a speed independent of the speed of the source which emits it. The
consequence of this is that the light travels at the same speed, c, both within the moving system and also within the stationary system, despite the relative motion of the two systems. The peculiar
result of all this is best understood if one imagines two pulses of light to be sent in opposite directions along the rod, from the centre to both ends.
Within the moving system the rod is at rest, and the light travels at velocity c in both directions, so the light pulses will arrive simultaneously at the ends of the rod. Since, however, according
to the light postulate, the light must travel at velocity, c, in both directions, in the stationary system also, it will take longer for the light pulse travelling in the same direction as the
relative motion to reach the end of the rod than the light pulse travelling against the relative motion. This is because the light travelling with the relative motion is trying to catch up with the
end of the rod that is moving away from it, whereas the pulse travelling against the relative motion has its end of the rod coming to meet it. This means that, while the arrivals of the pulses at the
ends of the rod are both simultaneous in the moving system, they cannot be simultaneous in the stationary system.
Einstein says, in his paper:
"So we see that we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of co-ordinates, are simultaneous, can no longer be looked
upon as simultaneous events when envisaged from a system which is in motion relatively to that system."
(Note that, here, Einstein refers to 'events' as single objects common to different systems of coordinates. He does not refer to the possibility that an event in one system of coordinates might not
be the same event in another system of coordinates. He has, in effect, already assumed that the transformation equations, which he will
derive, are to be considered as simply coordinate transformations, without any consideration of the possibility they could be transformations of any other kind.)
The contradiction in respect of simultaneity, of course, is contrary to normal expectations, and this is because the light postulate, that declares light to travel at the same velocity in both
systems at the same time, is itself contrary to normal expectations. This situation must therefore be given some rational explanation, if the light postulate that gives rise to it is to have any
There are, therefore, three questions that arise. Einstein, however, appears to have never clearly distinguished these three possibilities, or subjected them to a full examination, but simply assumed
that an affirmative answer to only one of them (no 2 below) was the correct answer.
Since the argument about nonsimultaneity is determined by a situation involving the use of both light and spacetime, and not either alone, the following are the three questions that must arise:
1) Is the conclusion of nonsimultaneity, and the working of the light postulate, an appearance caused solely by the working of the light itself, in accordance with the light postulate, and not by any
other cause that exists independently of the light?
2) Is the conclusion of nonsimultaneity, and the working of the light postulate, an appearance caused by some underlying property of space and time themselves, in the context of the relative velocity
of inertial frames, that the working of the light only reveals, but doesn't actually cause?
3) Is the conclusion of nonsimultaneity an appearance caused by a combination of both of the above suggested possibilities?
These three questions have never been properly examined, and it has always been simply assumed that the answer to the second question is yes, to the point that the three questions appear to have
never even been formulated and distinguished, by Einstein himself, or anyone else.
It would seem to me that, if an affirmative answer to the second question is problematic, as I have argued elsewhere on this website that it is, an affirmative answer to the third question, involving
a combination of affirmative answers to both the first and second questions, is not likely to be any less problematic. Therefore, I would say that the most important course of action is to
concentrate on examining the possibility of an affirmative answer to the first question, combined with a negative answer to the second. This is not only because the matter has never yet been
considered, but also because important criticisms can be made against an affirmative answer to the second question.
The historical, affirmative answer to the second question led to a new concept of the nature of space and time, which is referred to as Minkowski spacetime.
PART 1 - THREE QUESTIONS THAT EINSTEIN DID NOT CONSIDER:
Here, I shall attempt to examine what might be the result of a supposition that the correct model that should be used to interpret the paradoxical phenomenon of relativity of simultaneity involves an
affirmative answer to the first question, which was:
Is the conclusion of nonsimultaneity an appearance caused solely by the working of the light itself, in accordance with the light postulate, and not by any other cause that exists independently of
the light?
We do not need to examine, here, the derivation, or the mathematics, of the SR equations themselves, since we have to consider only the interpretation of the working of the light postulate and the
consequent conclusions regarding nonsimultaneity effects between two relatively moving frames. Since we are to consider an alternative to the spacetime rotations concept, we must return to the
original context of the dynamical derivation of the equations, in terms of the two postulates of SR, and not make any assumption that any spatial or time coordinate axis is rotated in spacetime
relative to any other.
Let us, therefore, return to Einstein on his railway embankment, and consider how light appears to travel within a moving train from the perspective of both the observer on the embankment and the
observer on the train. Let rays of light be emitted, at the same time, from the centre of a train carriage to both ends. For both observers, the rays of light are emitted simultaneously, since they
form only a single event for either observer. As we know already, according to the light postulate, and the conclusion of nonsimultaneity, the light rays will simultaneously reach the ends of the
carriage in the view of the observer on the train, whereas they will not simultaneously reach the ends of the carriage in the view of the observer on the embankment.
Such results were interpreted to indicate that time cannot be the same for both observers, which then led to the conclusion that there is no such thing as a universal, common time for all observers
and, thereafter, to the concept of Minkowski spacetime. This was thus an assumption that the answer to the second question must be yes. It is a pity, I would say, that no one was there at the time to
say "wait a minute - is that really the only possible explanation for the above nonsimultaneity effect?" This time, therefore, we shall attempt to examine an alternative explanation, in terms of an
affirmative answer to the first of the three questions.
It might appear obvious that time cannot be the same for both observers, since both see the same light, and yet see it describing the same events nonsimultaneously. Is it not merely an assumption,
however, that both observers 'see the same light'? What does this assumption actually mean?
We might suppose it to be obvious that a particle of light, a photon, is a single entity, a single focus of propagation, that can be picked up by either observer. But, granted that a photon can be
regarded as a single entity in itself, is it not, nevertheless, merely an assumption that it is, as it were, identical with its focus of propagation? Suppose, therefore, we postulate a distinction
between a photon, as an entity, and its focus of propagation, and ask the question: could a single photon, perhaps, have more than one focus of propagation at the same time? Might the observer on the
train and the observer on the embankment really not be seeing the 'same' light after all?
If, indeed, a photon has a focus of propagation in the carriage reference frame distinct from its focus of propagation in the frame of the embankment, it is then easy to consider the possibility that
their individual trajectories in space and time might be distinct also. The criterion for identifying the nature of the difference between the trajectories of the two foci of propagation will then be
simply the light postulate. That is, a photon will have a focus of propagation travelling at velocity c in the carriage reference frame, and a different focus of propagation travelling at velocity c
in the embankment reference frame.
Such a condition can fully account for the nonsimultaneity effect in the observation of the same event by the two different observers. That is, the observer on the train encounters a photon by means
of a different focus of propagation, at a different time, than that by which the observer on the embankment would encounter the same photon. Thus, the implication that time, itself, is not the same
for both observers no longer exists. The explanation, instead, is that it is light, and not time, that is not the same for both observers and, in general, for different inertial observers in relative
The consequence of this is then that, as the foci of propagation in the two frames are distinct from one another, they also create distinct events in the two frames. That is, the focus of propagation
that strikes the right hand end of the carriage in the moving frame creates a spacetime event that is distinct from the spacetime event at which the stationary frame focus of propagation appears to
strike the right hand end of the carriage in the stationary frame.
The Lorentz transformation equations will thus be no longer simple coordinate transformations of a single event from one frame to another; instead they must become a kind of one-to-one mapping of the
coordinates of an event in one frame to corresponding coordinates of a different event in the other.
OF THE LIGHT POSTULATE (1):
A proposal that photons have different foci of propagation in moving and stationary frames creates immediate implications regarding the intrinsic nature of such photons that must be considered. There
is obviously nothing special about the two reference frames in the example above, so that, if a photon has two distinct foci of propagation in the two frames, it must be supposed to have distinct
foci of propagation in all possible inertial reference frames at all possible relative velocities less than the velocity of light. In other words, it must have an entire spectrum of foci of
propagation, together with an entire spectrum of trajectories associated with such foci of propagation. If, despite this, the photon is to remain as a single entity, that can still be regarded as a
single photon, all these foci of propagation, and their trajectories, must necessarily be connected together in such a way that, together, they can constitute a single entity.
I suggest that the easiest way in which this might be thought possible is to postulate that they all form part of a single geometry of the photon in space and time. That is, a photon will be
considered to project, into space and time, a geometry of some material kind, belonging to, or internal to, itself, within which all its foci of propagation, and their trajectories, will be defined,
determined, and connected together to form a single reality.
In such a scenario, it is important to try to clearly understand what might be meant by a 'focus of propagation'. Does it mean, for example, that a photon occupies many different locations at the
same time? I would say that the answer to that would have to be in the negative, in order to avoid possible paradoxical misconceptions of the reality involved. Instead, it would be more satisfactory
to suppose that the foci of propagation of the photon are merely features of the photon's own spacetime geometry, that define the locations at which, alone, the photon is able to be detected. If,
then, a particular focus of propagation is selected for the detection of the photon, the photon will manifest at that location with, at the same time, the withdrawal of its entire, internal spacetime
geometry, along with all the other foci of propagation, and their defined trajectories. It will be necessary, however, to distinguish between a photon reflection event, at which it is not absorbed,
and an event at which it is absorbed, or detected, by a target.
Note that, for all this to work, the photon will have to somehow 'know' which of its foci of propagation is approaching any target at velocity c. This is the means by which the photon can be enabled
to distinguish between one inertial frame and another. This may seem strange but, I think, no more strange than the fact that, in modern which-path erasure experiments, it has been found that the
photon seems to 'know' that which-path information has been erased, even after it has passed the section of the experiment which determines whether which-path information can exist or not.
It is worth remarking, here, that, in this explanation, there is no spacetime rotation of inertial frames, and space and time axes remain parallel, as they were always thought to do in the former
Galilean era. Thus, all the problems that arise in connection with such axial rotations are swept aside.
The above proposal regarding the nature of light is strengthened by an added benefit automatically associated with the concept of such a suggested internal spacetime geometry of photons (not as a
Minkowski type geometry, by the way). Such a proposal involves a projection of the
internal geometry of a photon into space and time at a speed so great that the nature of the photon becomes effectively 'nonlocal' in its conception. Such a speed of projection of the photon's
internal geometry does not, by the way, violate the principle of the propagation velocity of light, c, as the maximum possible. Within such a geometry, the photon can appear only at the location of
one of its foci of propagation, which propagate, within the photon's geometry, only at velocity c.
The purpose of projecting such a geometry into space and time at such a speed is to enable the creation and calculation of the entire spectrum of trajectories of all the foci of propagation, which
they will then follow, at their normal propagation velocity.
Such an internal geometry, attributed to the photon, can thus easily also form the basis of an explanation of the working of photon non-locality, which-path erasure, and entanglement, as seen in a
variety of recent experiments. In other words, the geometry projected into space and time by a photon will be withdrawn, or will 'collapse' when the photon reaches its destination, where it is
Entanglement, for example, in this view, would require only the proposition that two photons are sharing a common spacetime geometry, rather than having separate geometries, in the usual way. Then,
the detection of one photon, with a withdrawal of its internal geometry, would automatically withdraw the internal geometry of the other photon also, thus necessitating it be also detected virtually
simultaneously with the first, however far apart they might be.
Again, in a which-path erasure experiment, it can easily be conceived that, since the photon does not appear until one of its foci of propagation is selected, it is quite possible that it could
virtually instantaneously modify and 'recalculate' its internal geometry, past, present, and future, in response to any modification of its environment, at any time up to the point where it actually
reaches its target, even if its foci of propagation have already passed the double-slit part of an experiment, or its equivalent.
If it turns out that the above explanation is the correct interpretation of SR, then we will have arrived at the end of the Minkowski spacetime era, and Minkowski spacetime will have been revealed as
one of the great conceptual errors in the history of science.
The new reality, if established, will involve a restoration of the former Galilean spacetime with, however, a preservation of the Lorentz transformation equations. This implies that interactions in
this spacetime, as mediated by lightspeed bosons, are Lorentzian in nature, and not Galilean. The transformation equations, as mentioned already, will no longer be simple geometrical coordinate
transformations but, rather, a special one-to-one mapping of events, as observed in one frame, onto those as observed in another. Along with the restoration of Galilean spacetime, we will also have
the restoration of a common, universal time for all observers, with the difference that, due to the working of light, it will not be directly visible across all inertial frames to all observers. I
have already argued this on a page dealing with the twins paradox.
As mentioned here, it may also provide a single, unified theory that will underlie both SR and all nonlocality effects, as observed in recent experiments.
OF THE LIGHT POSTULATE (2):
One might argue that it seems peculiar that the quadratic nature of the equation that gave rise to the Minkowski spacetime interpretation exists, if no such spacetime exists. However, the source of
the quadratic nature of the equation can easily be seen in terms of the movement of a light ray transverse to the direction of relative motion, in accordance with the light postulate.
In the stationary frame, the focus of propagation moves along a hypotenuse, ct, the moving frame moves a distance, vt, in the direction of motion, and the focus of propagation of the light moves at
right angles to the direction of motion in the moving frame, (ct), thus creating a right triangle, which immediately gives the equation that forms the basis for Minkowski spacetime, as
(ct)^2 = (ct)^2 - (vt)^2 = (ct)^2 - x^2 - y^2 - z^2.
This can be seen in the right triangle made by ct and vt in the diagram on the right.
The above illustrations suggest how the concept of an internal spacetime geometry of a photon might help to explain the working of which-path erasure experiments and the like. I do not attempt to
explain how the foci of propagation tie in with the wave theory of light. That is a problem that exists in any case.
In the left illustration, a double focus of propagation, within the same frame, is used to roughly indicate a splitting of the focus of propagation by the double slit experiment. This concept is
sufficient for present purposes.
The essential value of these illustrations is in the concept of the projection of an internal spacetime geometry of the photon, indicated by the shaded areas, being projected instantaneously into
spacetime, within which the trajectory of the photon from source to target is established, independently of the actual propagation of the photon. The focus, or foci, of propagation then determine the
location along the trajectory at which the photon can be detected at any particular time.
The photon itself is identified with its spacetime
geometry, and not merely with its foci of propagation. When a focus of propagation strikes the target, the entire spactime geometry collapses onto that location, and a photon appears there.
Any relevant environmental change, affecting the photon's geometry, will create an instantaneous response by the photon, throughout its entire geometry, which will be retransmitted, instantaneously,
from source to target, and will include, if necessary, a new trajectory. This will be independent of where the focus of propagation currently happens to be, and the focus of propagation will appear
on the new trajectory, at the appropriate position.
This could potentially serve to explain how the interference pattern on the target screen can be made to disappear, even after the focus, or foci, of propagation have already passed the slits
themselves. In a similar manner, the interference pattern can be restored again by opening the slit which has been closed, before the focus of propagation strikes the target. This thus erases the
which-path information that previously existed after closing one of the slits.
The illustration shows two rods, at right angles to one another, lying along the x and y directions, and moving at velocity, v, in a stationary frame along the positive x direction. The blue rays
show a lateral pulse of light reflected off the end of the rod in the y direction, AB[t]C, and another pulse reflected off the end of the rod in the x direction, ABC.
There are two diagrams, which show separate foci of propagation of light: in the moving frame (the blue lines in the smaller diagram), and in the stationary frame (the blue lines in the larger
In the smaller diagram, the blue lines are symmetrical in the x and y directions, since they are in the moving frame, and unaffected by the relative motion. In the larger diagram, the blue lines are
not symmetrical, being affected by the relative motion, and the reflection off the end of the rod at B involves an apparent foreshortening of the length of the rod in the x direction.
The blue lines, laterally and in the direction of motion, coincide at C in both diagrams. That is, the relative motion does not cause the frequency of oscillation of light to differ along the x or y
axes, in the stationary frame diagram. This would not be the case, in the stationary frame, if there were no length contraction effect produced by the focus of propagation in the stationary frame.
Since spacetime here is Galilean, the spacetime coordinate systems of both diagrams are parallel to one another. That is, there is no relative rotation of coordinate axes, as is the case with
Minkowski spacetime. It is assumed that light moves at velocity, c, in the direction of time, as well as in the spatial direction.
In view of this, the position of event C(x,y,z,t),
in the moving frame, with respect to the stationary frame starting event A(x,y,z,t), is at (vt, ct), or ((1/g)x, ct), as is shown more clearly in the diagram below. This event is not visible in the
stationary frame, since it is visible, via the moving frame focus of propagation, only to the moving observer. In the stationary frame, this event becomes visible, via the stationary frame focus of
propagation, at event C(x,y,z,t), at (vt, ct), or (x, ct), or, more accurately (x, 0, 0, ct), and this expression of the stationary frame manifestiation of a moving frame event can be used to
construct the so-called spacetime interval, and the associated 4-vectors (see below).
Since event C(x,y,z,t), at time t, is a stationary frame manifestation of the moving frame event, C(x,y,z,t), at the earlier time t, a moving clock will appear time dilated. As I have argued
elsewhere, this is to be interpreted as the stationary frame observer viewing a moving clock via a present view of its past history.
The reason for using events C is that they represent one complete back-and-forth oscillation of a photon, and the (x, 0, 0, ct) specification can be applied to any event C that is determined by any
number of completed back-and-forth oscillations of a photon. The single back and forth oscillation of a photon shown, can thus represent the general internal oscillations of bosons within any moving
object, and hence the relationship between the object as it is in the moving frame, and as it is manifested in the stationary frame.
The first diagram, below, shows two complete oscillations, corresponding to the single oscillation shown in the second diagram, below it, to indicate the existence of multiple oscillations being
described in the same way mathematically, in which (2x, 0, 0, 2ct) would become (nx, 0, 0, nct)
One of the problems with SR as a theory of light rather than a theory of spacetime is the consequences caused for other theories in Physics by the loss of the Minkowski metric, and Minkowski
4-vectors, in which these are used extensively.
However, on the supposition that all reference frames travel at velocity c, in a fourth dimensional direction through time, it is still possible to construct these 4-vectors. They will simply have a
different interpretation, as follows:
In a spacetime view of an inertial reference frame, an object is travelling in the time direction at velocity, c. The distance travelled in this direction, in a Galilean spacetime, is ct, which is
the so-called spacetime interval in Minkowski spacetime. This distance terminates in a spacetime event, which is defined by the position 4-vector within the moving frame. Dividing by the time, t will
give the 4-velocity, c, and multiplying by the rest mass of the relevant object at this event will give the momentum 4-vector in this direction, m[0]c. These are Lorentz invariants and, although they
are not visible in a stationary frame, they can be identified via stationary frame vector components, which will be different in different stationary frames, relative to which the moving frame will
have different velocities.
The position 4-vector in the moving frame connects events (0,0,0,0) and (0,0,0,ct). To set up the stationary frame vector components
requires using the second event as it appears in the stationary frame, via the stationary frame focus of propagation, at stationary frame event (x,0,0,ct). The above diagram shows how this can be
used to set up the following equation, expressing the length of the position 4-vector in terms of stationary frame components:
Position 4-vector:
S[4]^2 = c^2t^2 = c^2t^2 - x^2
Velocity 4_vector:
V[4]^2 = (S[4]/t)^2 = c^2 = g^2c^2 - g^2v^2
Momentum 4_vector:
P[4]^2 = (m[0]V[4])^2 = m[0]^2c^2 = m[0]^2g^2c^2 - m[0]^2g^2 v^2
m[0]^2c^2 = m[0]^2g^2c^2 - p^2 = (E^2/c^2) - p^2
with E as energy, from which we get:
E^2 = c^2p^2 + m[0]^2c^4
with p as relativistic momentum. All this appears here, in the context of SR as a theory of light, without the existence of any Minkowski spacetime.
PART 5 - FURTHER CONSIDERATIONS (added March 2011):
This question requires examination because of the fact that, in Special Relativity as a Theory of Light, the measurement of a moving length, in a spacelike manner, in a stationary frame, does not
give the same result as a measurement obtained by sending a light ray along it. This difference is illustrated in the above diagram, in which the measurements s constitute the 'spacelike' method of
locating the positions of the rockets, while the distance x, to locate the position of the right hand rocket, is obtained by sending a light pulse along this distance. (See also my page on Bell's
Spaceship Paradox for an argument about this).
This means that a spacelike measurement of the
relativistic composition of velocities will also not give the same as the usual result, in accordance with the transformation equations. That is, from a spacelike perspective, a moving object,
accelerated within the moving frame, will be further ahead of where it would be according to the usual composition of velocities equation. This corresponds to the distance D, in the illustration,
being greater than the distance D'.
The question is, would this provide for a spacelike based composition of velocities to allow an object to be accelerated to faster than light velocities? The diagram below will help to answer this
We will examine an accelerating rocket in terms of a stepwise transfer from an inertial frame at a lower velocity to an inertial frame at a slightly higher velocity. In the above diagram the rocket
speed increases, in a single step, from v to v + Dv, in the stationary frame, still in the original direction.
In the stationary frame, the rocket has been travelling at velocity v, which means that the rocket is in a moving frame travelling at this velocity. The rocket instantly increases velocity to v + Dv.
It is now in a second moving frame, at velocity Dv', measured within the first moving frame. It has travelled a distance DvDt, as measured in the stationary frame, in this second moving frame, in
stationary frame time Dt.
We have to find the relationship between this distance, measured in the stationary frame, and this distance, measured in the first moving frame. This is the same problem as measuring the length of a
moving rod, except that the 'moving rod' is now a distance created at a velocity, Dv', in a moving frame. This is similar to a case where the moving rod is changing its length as it moves. That is,
we might have a moving rod, AB, in which the end B is moving at some velocity relative to A within the first moving frame (i.e., B's motion constitutes a second moving frame).
The current example would correspond to such a moving rod, in which the initial length is zero. This means that finding this 'moving rod' length will enable us to find a relationship between the
velocity Dv', within the first moving frame, and its value, Dv, as seen in the stationary frame. If there can be a simple addition of velocities, v + Dv', in the stationary frame, (i.e. if Dv' = Dv),
then faster than light travel is possible.
This, of course, is a matter of the spacelike measurement of the velocity in the stationary frame, as mentioned already (i.e. position measurements based on s, not x, in the initial illustration).
The current illustration shows two positions for the rear end of the rocket, which is used to calculate velocities. The position slightly to the left indicates the length contraction of the distance,
as specified by the transformation equations. The position to the right is the spacelike determined position of the rocket.
The distance, as given by the transformation equations, is:
x = g(x - vt)
(1/g)x = (x - vt)
(1/g)x is a length contracted distance, so that the non_length_contracted distance, in the stationary frame, representing a spacelike measurement of the distance, is simply x, which is also Dv'Dt. So
we have:
x = Dv'Dt = DvDt
and, since Dt = (1/g)Dt (see first illustration above), we have:
(1/g)Dv' = Dv
Since, however
(1/g) = (1 - (v/c)^2)^1/2
approaches zero as v approaches c, we thus have the possible result that Dv approaches zero as v approaches c, which would mean that faster than light velocities would be impossible to achieve.
One has to be careful, however, in interpreting this equation, since this result is due to a time dilation effect only, without the involvement of any corresponding spatial distance effect. That is,
it is necessary to consider the alternative interpretation that Dv' increases, rather than Dv decreasing, with increase in v, for the following reason:
Since Dv' = x/Dt, and Dt is a time dilated value in the stationary frame, while x is a true value (in the spacelike perspective), it follows that Dv' appears larger, in the stationary frame, than it
is in reality, in the moving frame. This is because Dt approaches zero, and consequently Dv' increases towards infinity, as seen from the stationary frame, as v approaches c, and the equation
therefore does not mean that Dv approaches zero.
It must be remembered that, in SR as a Theory of Light, Dt is not an invariant, if it is related to the current stationary frame time, as it would be in the spacetime version of the theory, and time
is the same in all frames, although it is not observed to be so. The interval Dt is perceived within the past history of the moving frame, although in terms of the present distance x. So the
corresponding real time interval in the moving frame is greater than Dt
The key to the correct interpretation is, therefore, the fact that x is the same value in both frames, which really means that the true relationship between the velocities in the two frames is:
Dv' = Dv
This means that faster than light travel can actually occur, because the resultant velocity in the stationary frame (if measured in a spacelike manner) is a simple addition of velocities, v + Dv'
(using the real version of Dv' within the moving frame).
It is to be noted, however, that an object cannot be accelerated to faster than light speed in a single frame only, since lightspeed bosons limit the acceleration interaction in any single frame. An
object has to be accelerated within at least one moving frame in order to reach such speeds.
Two separate arguments are necessary to answer this problem.
Firstly, let us consider a theoretical object already moving faster than light. Let a rod AB be moving faster than light in the direction AB. It is clear that a photon (i.e. its focus of propagation
in the stationary frame) leaving A will fall behind the rod, and can never reach B. Similarly, if a rod AB' is connected to the original rod, at right angles to the direction of motion, a photon
leaving A can also never reach B'. This means that, in this context, the SR transformation equations cannot be derived, and are therefore irrelevant. That is, they cannot have any application in a
faster than light context. The same argument applies at lightspeed also.
Secondly, let us more closely examine the nature of relativistic momentum. This is given as:
P[r] = gm[0]v
where m[0] is rest mass. In this equation g can, in fact, be applied either to the rest mass, creating relativistic mass, or to the velocity which, in our case, will prove more useful. We therefore
P[r] = m[0]gv = m[0]gds/dt = m[0]ds/(1/g)dt
= m[0]ds/dt
In this version of the equation, v is replaced by a time dilated velocity, which approaches infinity as v approaches c. We must consider, however, that all measurable quantities must be quantised, so
that velocity really does increase in a discrete, stepwise manner. This means that the time dilated velocity cannot actually become infinite, but must reach a finite maximum value, which therefore
provides for a theoretical possibility that could enable a moving object to cross the light speed barrier.
The above objection, therefore, is not an absolute impediment to superluminal velocities, where these are achieved by multiple stages of acceleration in multiple moving frames. In addition, at
superluminal velocities, there is no relativistic mass effect, since the subluminal relativistic mass effect can be alternatively regarded as a relativistic velocity effect.
A Note on Kinetic Energy (KE): It is possible to derive the usual approximate expression for kinetic energy from the relativistic energy form, in a way that avoids assigning g to the mass, rather
than to velocity, as follows:
KE = relativistic energy - m[0]c^2
= (p^2c^2 + m[0]^2c^4)^1/2 - m[0]c^2
we can convert p^2c^2 to a more convenient form:
p^2c^2 = m[0]^2g^2v^2c^2
= (m[0]^2v^2c^2)/(1 - v^2/c^2)
= m[0]^2c^4(1/[c^2/v^2 - 1])
Here we started with the form m[0]^2g^2v^2c^2, which allows assigning g to v rather than m[0]. Adding m[0]^2c^4 in the original relativistic energy expression, we get
[m[0]^2c^4(1 + c^2/v^2 -1)/(c^2/v^2 -1)]^1/2 = m[0]c^2g
which gives us
KE = m[0]c^2(g - 1)
from which, using the usual Taylor series expansion for g, we get, finally
KE = 1/2(m[0]v^2)
approximately, where v << c, in the usual way.
OF LIGHT?:
Quasars have been observed to emit thin jets of charged particles enormous distances in both directions along a single axis. These observations have been used to calculate apparent superluminal
velocities of the jets, up to numerous times the speed of light. Explanations have been proposed to suggest how such results might be possible without violating the light speed limitation that is
apparently intrinsic to Special Relativity. That is, the superluminal
velocities are argued to be illusory, due to optical effects. The previous sections, however, show that, in SR as a theory of light, superluminal velocities are not forbidden. This conclusion
therefore allows the possibility that such jets may indeed be travelling at superluminal velocities. There would, however, have to be multiple stages of acceleration - i.e. within, and along, the
jets themselves, and not only in the reference frame of the quasar itself.
© Alen, March 2007; update Dec 2010.
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Getting the 2nd matching value from a list using VLOOKUP formula » Chandoo.org - Learn Excel, Power BI & Charting Online
This article is part of our VLOOKUP Week.
Read more
We know that VLOOKUP formula is useful to fetch the first matching item from a list. So what would you do if you need 2nd (or 3rd etc.) matching item from a list?
For eg. If you have below data, and you want to find out how much sales John made 2nd time, then VLOOKUP formula becomes quite useless. Or is it?!?
A simple solution to this problem would be sorting our data on sales person’s name. That way all Johns would line up one beneath another. And we just have to find the first John’s position and add 1
to it to get to 2nd occurrence. Like this =MATCH("John", C5:C17, 0) + 1
But sorting is not an option all the time. So there should be a better way to do this?
Well, there is. We just add a helper column before the sales person name and fill it with sales-person’s name & occurrence. (see the below data table).
For this we can use COUNTIF() Formula, like this: =C5&COUNTIF($C$5:C5,C5). Notice the $C$5:C5?, well the mix of absolute & relative references does the trick here and gets John1, John2… etc.
Now, to lookup 2nd occurance of John, all we do is, simply write =VLOOKUP("John2",...) and we are done.
Sample File
Download Example File – Getting the 2nd matching item from a list using VLOOKUP formula
The file includes few examples on how to fetch 2nd, 3rd etc. matches using lookup formulas. It also has some interesting (and challenging) home work for you. Download & play with it.
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58 Responses to “Getting the 2nd matching value from a list using VLOOKUP formula”
1. 4. Is there way to fetch 2nd sales amount for Josh without using the helper column (B5:B17) ?
Select Cell I6
Define a name : cName = If(Person=$H6,Row(Person)-Row($C$3))
In cell I6 Type = Index(Sales,Small(cName,2))
This will return the 2nd instance.
□ Hi friends,
go for =sumif() formula..
2. Essentially the same as sam's idea, but you could combine it all into one formula if you use an array formula:
You can, of course, make these formulas dynamic by replacing the text "John" and the number 2 with cell references.
3. Stumped! I got the first one, but can figure out the other ones! 🙁 Any place to find the answers?
4. Here is what I have:
The matrix formula Luke M has is definitely simpler. I was just trying to avoid a matrix.
I did come up with a more complicated matrix formula for #4. I was thinking of how I could make a matrix form of a running total of the number of occurrences of value. Maybe it will help me some
other time.
Here it is:
□ Would you mind untangling your formulas for questions 1 and 3 please? Thanks
5. I had a strong desire to use a vlookup on problem 4. Here is the matrix formula I worked out for doing that.
A few named formulas and this can be simplified.
6. All Array Formulas
1) =INDEX(NetSales,LARGE(IF(SalesPerson="Jamie",ROW(SalesPerson)-MIN(ROW(SalesPerson))+1),1))
2) =MAX(IF(SalesPerson="John",NetSales))
3) ="John"&COUNTIF(C5:INDIRECT("C"&MAX(IF(IF(SalesPerson="John",NetSales)=MAX(SalesPerson="John",NetSales),ROW(NetSales)))),"John")
4) =INDEX(NetSales,SMALL(IF(SalesPerson="John",ROW(SalesPerson)-MIN(ROW(SalesPerson))+1),2))
7. @All.. good solutions.
@Perry: Here is a set of solutions. You can obviously improve or comeup with diff. approaches. http://chandoo.org/img/f/vw/vlookup-2nd-value-instructor-copy.xls
@Tristan: Can you explain =INDIRECT(“E”&SMALL(INDEX(((C5:C17=”Josh”)*ROW(C5:C17))+((C5:C17?Josh”)*10^8),0,0),2)) for us?
8. It's an awesome job!!!... thank you
9. I will endeavor to do so, Chandoo.
First off, I just now noticed that there is a not equal sign missing between the second C5:C17 and “Josh” (It must not render incorrectly)
So here is the idea behind how it works:
The overall idea is to return the row of the second occurrence of “Josh”
First the array is checked to see it if an item is equal to “Josh” (C5:C17=”Josh”) this generates and array of True and False
Then multiply the true and false array by the array of row numbers (Row(C5:C17)) this transforms the array into an array of zeros and row numbers where the value is “Josh”
The problem here is that there are an unknown number of zeros in the array. This causes the small function to fail. So I want to replace the zeros with values that are larger than any encountered
as a row.
This I did by adding the array of items not equal to “Josh” (C5:C17 not equal “Josh”), multiplied by 10^8
This results in an array of large numbers (where the value is not equal to “Josh”) and row numbers where the value is equal to “Josh”
The index function causes everything to be treated as an array, and then small picks the second smallest number which is the row number of the second occurrence of “Josh”
□ Can you get the this to work but on a different sheet in the same file?
=INDIRECT("H"&SMALL(INDEX((('Above Ground'!B6:B800='Pipe Stringing_Lowering'!C8)*ROW('Above Ground'!B6:B800))+(('Above Ground'!B6:B800<>'Pipe Stringing_Lowering'!C8)*10^8),0,0),2))
I get it to work in the sheet I am looking up the value but I can't find a way to get this to work if I want it on another sheet.
Help would be great!
☆ Never Mind Got It. Just one of those moments.
10. Thank you for the post, it actually helped me to make an auto-fill list, depending on an specific value on a cell. Thanks again!
11. Dear Chandoo:
Can you explain what exactly happens with the formula you have written as answer for the 2nd question in the exercise, i.e.:
= SUMPRODUCT(MAX(($C$5:$C$17="John")*($E$5:$E$17))).
12. Also in the answer of the 3rd question, how does Countif arrives at the right answer? Does Excel check the Countif first or the calculates Offset first?
13. sorry, my bad. I was little confused in the 3rd answer, i got it now. Will wait for your reply for the 2nd.
14. @Rahul... The sumproduct formula checks for all records and returns true where person=John. This gets multiplied to the actual sales values. Since true is 1 and false is 0, we get a bunch of
values with actual sales for john and rest as zeros. The max, then returns the maximum value among these. Refer to this article for a tutorial on sumproduct formula: http://chandoo.org/wp/2009/11
15. @Tristan: Very clever approach. Thanks for teaching us this technique. Here is a donut for you 🙂
16. DISCOUNT -7% & -5% = ?
EX 880
( Two discount in one MRP - FIRSR SEVEN PERCENTAGE & AFTER 5% DISCOUNT , HOW CAN I GIVE THE FORMULA FOR ROWS IN NEXT CLOUMN DIRECT VALUE?
17. Dear All
I think I need to give more time to catch up with you all... thanks to guru chandoo... i am confident that i can learn with these techniques.... thanks to all
V S Venkatraman
18. Hi Chandoo,
It was nice reading your blog. I got introduced to your blog by my brother 'Ayush Jain'. Really good good stuff. Very innovative and different approach.
Ashish Jain
19. [...] Get 2nd Match from a list using LOOKUP Formulas [...]
20. When you are posting examples, you need to post the column letters and row numbers. Otherwise, the reader has no idea to which cells a formula is referencing. What makes your example above
especially egregious is that you skipped an arbitrary number of rows and columns before the data table. Therefore, no one could possibly know which cell is C5.
Yeah, I know we can download the spreadsheet. That's not the point.
21. You could try either of these links if you want a custom function (thanks to ozgrid!)
22. Hi All,
I was preparing Birthday Reminder Sheet and I was using VLOOKUP. If the date of birth and month of a person matches with todays date and month(dd-mm), then their name should display. I used
simple VLOOKUP formula
However, if there is more than 1 Birthday's then this formula doesnt work.:(
Kindly resolve this and provide me solution.
Thanks in advance.
Raghuram Bhat
23. Thanks Heaps for this solution, allows me to look up values based on the first return so that each return after is relevant.
To the complainers and wingers, if you dont understand how the data is used in the downloaded spreadsheet, look up what you are not understanding. Stop bagging someone who is trying to help you.
24. Just ON-TIME when i needed it the most.... No Words to thank
25. Hi!
There's a more simple way to solve above problem, by only using vlookup and countif, but it need extra column
Row column A Column B column C
1 Name_C Name Score
2 mark1 Mark 3
3 Jay1 Jay 4
4 Mark2 Mark 5
6 mark 3 1st value
7 mark 5 2d value
A2 formula = b2 & countif(b$1:b2,b2)
to be copied until a4
b6 formula = vlookup(a6&1,$a$2:$c$4,3,false)
b7 formula = vlookup(a7&2,$a$2:$c$4,3,false)
□ Use a combination of the Vlookup function, Match function, and the Offset function.
What the does is:
1. Returns the first occurrece of the lookup value as an integer and adds one (Match function)
2. Takes the lookup_range and offsets it by the the total number of rows returned in step 1 (Offset function returns new range)
3. Runs a Vlookup using the new range and the vlookup value
Hope this helps.
26. Awesome approach. Solved my problem perfectly.
Thanks Much!!!
27. Good stuff! I have a problem. I'm trying to populate a field based on an occurence of a value in a range and it works fine. Shown below:
But I also want to populate the 2nd, 3rd, and 4th occurence of the same value in the given range and its not working. I need to nest it into the current formula I have but I'm unsuccessful in
doing so.
Can anyone help???
28. How do I CONCATENATE 1st, 2nd .. values
□ Excel Concatenate() formula does not take arrays or ranges. It can take separate values though.
One way is to use a small UDF to do this. See here for one:
29. What also works very well is to add the returned value in the same row. This will allow you to compile a list off the result of one vlookup. Choose the one you wish to see, and then use another
vlookup to return those values.
30. Hi all,
I am having difficulty wrapping my head around this formula:
Could someone please explain this.....
I tried to go different way: through MATCH function to determine the position of John's max sale. But all it gives to me is 13, instead of 4. If I have an following array :
{0;1088;0;0;0;1540;0;0;0;726;0;0;2682}, how can I make it look like this {1088;1540;726;2682}, in other words how can I get rid of zeros(false) and keep the order of non-zero(true) values.
My formula was like: =match(max((salesman="john")*(sales));((salesman="john")*(sales));0)
Thank you.
31. I am trying to lookup the sequence of values using this MATCH INDEX formula. I trying to find the second and subsequent values but i can"t get it. Please help me.
=IFERROR(INDEX('2013 MBR'!M3:M6000,MATCH("1"&$F$2&"Day",'2013 MBR'!$S$3:$S$6000&'2013 MBR'!$Z$3:$Z$6000&'2013 MBR'!$AA$3:$AA$6000,0)),IFERROR(INDEX('2013 MBR'!M3:M6000,MATCH("1"&$F$2&"Day",'2013
MBR'!$S$3:$S$6000&'2013 MBR'!$AC$3:$AC$6000&'2013 MBR'!$AD$3:$AD$6000,0)),IFERROR(INDEX('2013 MBR'!M3:M6000,MATCH("1"&$F$2&"Day",'2013 MBR'!$S$3:$S$6000&'2013 MBR'!$AF$3:$AF$6000&'2013 MBR'!
$AG$3:$AG$6000,0)),IFERROR(INDEX('2013 MBR'!M3:M6000,MATCH("1"&$F$2&"Day",'2013 MBR'!$S$3:$S$6000&'2013 MBR'!$AI$3:$AI$6000&'2013 MBR'!$AJ$3:$AJ$6000,0)),IFERROR(INDEX('2013 MBR'!M3:M6000,MATCH
("1"&$F$2&"Day",'2013 MBR'!$S$3:$S$6000&'2013 MBR'!$AL$3:$AL$6000&'2013 MBR'!$AM$3:$AM$6000,0)),IFERROR(INDEX('2013 MBR'!M3:M6000,MATCH("1"&$F$2&"Day",'2013 MBR'!$S$3:$S$6000&'2013 MBR'!
$AO$3:$AO$6000&'2013 MBR'!$AP$3:$AP$6000,0)),"NA"))))))
□ @Karthik
It may be more use if instead of asking how to fix a formula that you explain the problem and post a sample file
As often in Excel there are many ways to skin the proverbial cat.
You may also want to post the question in the forums
☆ @Hui..
Thanks for your reply... I will post the sample file in the above link.
32. Hi Guys
1 question
A vlookup when done with a wild card for example =VLOOKUP(A1&"*",C:C,1,0) always returns the exact match first, how can you get the 2nd match with extensions (for which we use wild card) first
without sorting the list.
□ my question is :- If one sku is sitting on multiple locations than how we can find all locations with the help of Vlookup
I used =VLookup($c18,list2,2,false)
In column 2 i have multiple location for sku but i m getting only one in Vlookup, How i can get all location of this sku.
Please advise
Thanks & regards
33. I want to lookup completion date for X employee in a sheet wherein multiple entries are there, how to Pick up the Latest completion date for the employee
34. Why can't we add another helper column so that we can lookup
a diiferent value?
35. Dear all,
I am average user of MS Excel. I just read all formulae above but I couldn't understand & I have same problem too. I wanted to arrange two sheets on single sheet to refer data. On both sheets
common column is "name of item". but in that few items are repeated. thus when I apply vlookup on it, it shows same value for repeated items. whereas it suppose to show different values.
which formula should I apply?
Please help!
Thank you!
36. Great solution, and yet so simple to replicate! Thanks a lot for such a useful tip! Cheers.
37. Thanks for this! Worked a treat!
38. Thanks for your website and truely appreciate your sharing. Help me a lots ( Both Stat and Excel Function)
39. Of course sometimes it might be better to use a pivot table.
Generally, I now prefer to use a nested match within a index as it is more flexible.
I generally find countif(s) and sumif(s) are handy and have my own user defined functions for MAXifs, MINifs, which could be used to find earliest or latest/ largest or smallest entries. I have
also combined sumifs to do a kind of averageifs for example if an item is sold at different prices in different volumes working out the average price per unit.
40. Here is the thing. It worked well with the text (John and etc,) and if the value is number. It wont work.
200 A
200 B
200 C
200 D
how to find the following with formula
200 A B C D
41. Hi, can u pls upload a video so that we can understand it more
42. Excellent. You are the best dear.
43. Chandoo please unravel this formula for me. Thanks. =INDEX($E$5:$E$17,SUMPRODUCT(SMALL(($C$5:$C$17="Josh")*(ROW($C$5:$C$17)-4),2+COUNTIF($C$5:$C$17,"Josh"))))
44. So I've used this guide (really good by the way) to allow me to return a cell from a VLOOKUP based upon selections from 2 drop downs.
I want to be able to drag the code down and return the 1, 2, 3, 4 etc cell until i have all of the cells listed, then i have a hlookup returning the data relating to the cells.
How do I make my VLOOKUP formula automatically increase the number that it's searching for as i drag it down the column?
I've tried this, it just returns the same cell over and over and I can't think how to increment the 1 up:
=VLOOKUP($C$2&$C$3&1,'Raw Data'!$M$2:$N$499,2,FALSE)
Hope someone can help!
45. I have been looking for this answer for a very long time. I use vloolup all the time but have been stumped with multi row data. All the previous solutions I have found out there where to
complicated. This solution is so simple and is amazing. Thanks
Doug Meadows
46. Hey - relating to the original question here. Is there a way of returning multiple matches for a single date rather than for a list of names using vlookup?
Date Price
1/5/16 25
1/5/16 30
3/10/16 35
instead of John and John, an equivalent for dates? eg: 1/5/16 a and 1/5/16 b or something to separate the two dates and return 25 and 30 respectfully?
many thanks!
47. I want to know that i have one invoice having 2 lots.for instance.
Invoice no Lot no
INV17038 421C22
INV17038 421C23
Suppose i put INV17038 I Should be able to get all lots in connection to one invoice. which formula need to be putted
48. Chandoo, u r simply gr8!
U make things look so simple.
49. I have something similar to below:-
A B C
101 Bill Fish
101 Bill Chips
103 Joe Chicken
104 Don Pizza
103 Joe Prawn
101 Bill Peas
107 Sam Pie
103 Joe Curry
101 B1ll Gravy
search Column (A) above and produce below from Colomn (C) in a separate sheet
A B C D E
101 Fish Chips Peas Gravy
103 Chicken Prawn Curry
104 Pizza
107 Pie
Please can you help me ? | {"url":"https://chandoo.org/wp/vlookup-second-value/","timestamp":"2024-11-14T18:48:29Z","content_type":"text/html","content_length":"512958","record_id":"<urn:uuid:1505cfd8-7eaf-4177-a15c-b2b94d6cbff6>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00463.warc.gz"} |
T Distribution - Explained
What is a T Distribution?
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If you still have questions or prefer to get help directly from an agent, please submit a request.
We’ll get back to you as soon as possible.
What is a T Distribution?
The T distribution, also known as Students t-distribution, is a probability distribution that is used to estimate population parameters in small-sized samples as well as samples with an unknown
population variance. The T distribution shares a lot of similarities with normal distributions (a.k.a. Bell Curves); however, it is discernibly shorter as well as fatter with heavier tails. This
essentially means that T distributions are much more likely to depict extreme values, compared to normal distributions. Tail heaviness in T distributions can be attributed to a parameter known as the
degrees of freedom the smaller the value, the heavier the tail. Conversely, for higher values, i.e for sample sizes greater than 30, the T distribution assumes the shape of a standard normal
distribution, with a mean of 0 and a standard deviation of 1.
How is a T Distribution Used?
The T distribution was introduced in 1908 by a chemist by the name of William Sealy Gosset. Hired by the celebrated Guinness brewery in its quest for formulating better industrial processes through
the application of biochemistry, Gosset wasted no time in devising the T test as a cost-effective method of monitoring the quality of stout. However, it was company policy at Guinness during that
time to forbid chemists from publishing their findings. This did not deter Gosset and he promptly published his statistical work under the pseudonym Student. To understand the basic principles behind
a T distribution, let us consider a sample size of n observations picked from a standard normal population distribution with a mean denoted by M and a standard deviation denoted by D. Now, it will be
observed that the sample mean m and the sample standard deviation d differ from M and D. This variance can be attributed to the randomness of the sample. From the above considerations, it will be
possible to calculate a Z-score by using the following formula. Z = (m M)/{D/sqrt(n)} The Z-score calculated using the above formula has a normal distribution value with a mean of 0 and a standard
deviation of 1. Now, let us consider the same Z-score calculated by applying the estimated standard deviation with the following formula T = (m M)/{d/sqrt(n)} It will be observed that the difference
between d and D turns the normal distribution into a T distribution that exhibits (n - 1) degrees of freedom.
The Rationale behind Using a T Distribution
The central limit theorem postulates that in large sample sizes, the sampling distribution of a statistic is inclined to follow a normal distribution. As such, it is a straightforward process to
calculate a z-score, as long as the standard deviation of the population is known.This, in turn, allows statisticians to evaluate probabilities with the sample mean by using the normal distribution.
However, in the case of smaller samples, the standard deviation of the population is often unknown, and as such statisticians use the distribution of the t statistic. In short, the t distribution
allows statisticians to perform statistical analyses on smaller data sets that otherwise cannot be analyzed using the normal distribution. According to statisticians, Frederick Mosteller and John
Tukey, the value of students work is not on great numerical change, but rather in the assumption that it is possible to make allowances for the uncertainties of small samples, even in studies that
vastly differ from the students original problem. According to Mosteller and Tukey, the value of Students work also lay in the provision of numerical assessment of how small the numerical adjustments
of confidence points were in the Students problem and how they relied on the extremeness of the probabilities that were involved. Lastly, the value of Students work also lay in presentation of tables
that could be used to assess the uncertainty associated with even minute data samples.
Limitations of the T Distribution
According to Mosteller and Tukey, the t distribution also suffered from certain drawbacks and limitations. To begin with, statisticians using the t distribution were easily prone to neglecting the
proviso that the solutions would stand to be true if and only if appropriate assumptions were being held. Secondly, a t distribution usually tends to overemphasize on the accuracy of Students
solution for his idealized problem. Lastly, the t distribution helped to divert attention of theoretical statisticians to the development of exact ways of treating other problems. | {"url":"https://thebusinessprofessor.com/research-analysis-decision-science/t-distribution-definition","timestamp":"2024-11-08T11:33:37Z","content_type":"text/html","content_length":"99466","record_id":"<urn:uuid:7c4710df-fb57-4608-b326-e66a2440a02e>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00764.warc.gz"} |
Constrained Dix Inversion
Dix inversion estimates interval velocities from picked stacking velocities, usually as a function of vertical two-way time. The stacking velocities are assumed to be explained by a root-mean-square
(RMS) averaging of the interval velocities. A conventional method [3] uses an explicit solution that inverts the RMS integral. This explicit solution easily produces wildly unrealistic interval
velocities from small variations in stacking velocities.
Constrained inversion fits stacking velocities with a smooth, bounded interval velocity function. This method is slower but almost always preferable to the fast explicit solution. Damped
least-squares minimizes errors in picked velocities and also minimizes unnecessary complexities in interval velocities.
Constrained inversion distributes errors uniformly when fitting the squared reciprocal of stacking velocity. This distribution corresponds to uniform errors in residual normal moveouts.
Interval velocities are constructed as a sum of overlapping bell curves extending in all spatial directions. Coefficients of these curves are damped to avoid unnecessary sharpness in the estimated
interval velocities. Rough changes in interval velocity are allowed only if strongly required by the input data. Finally, interval velocities are not allowed to exceed specified minimum and maximum
An explicit Dix solution inverts one vertical function at a time, whereas least-squares finds a global solution. Each estimated coefficient must explain stacking velocities over a range of spatial
positions on the map. Redundancy greatly improves, so a single bad stacking function does not easily corrupt the solution. A few bad data points are largely ignored when contradicted by many
neighboring values.
Many geophysical programmers familiar with damped least-squares have developed similar methods [5,2,1].
We assume interval velocities to be smooth in all physical directions. This assumption is most appropriate for “soft” rocks, where fluid pressure dominates seismic velocities. In “hard” rocks,
velocities tend to be homogeneous in intervals, with abrupt discontinuities at changes in lithology. Soft constraints can still accurately describe the time/depth conversion of hard media.
A smoothing operator with unit area (DC) does not introduce any bias into smoothed values. On average, values are no larger or smaller than before. If interval velocities are sampled as a function of
vertical traveltime, then depth is just the integral of velocity over time. If smoothing does not bias the interval velocity, then it also does not bias depth conversions. Away from the immediate
vicinity of a large discontinuity, smoothing has no effect on time/depth conversions.
Our convolutional smoothing operator is a bell-shaped curve described by a third-order polynomial. The curve has unit area to preserve magnitudes. The convolution is renormalized at boundaries to
preserve unit area when the convolution is truncated. A smoothing width is the “half-width” of the curve, the span over which the curve drops to half the peak value. The total width of the curve is
twice the smoothing interval. Over this interval, the third-order curve is , where is the distance from the peak divided by the smoothing distance. The curve has zero slope at the peak and endpoints.
(The half-width in the Fourier domain is approximately the reciprocal of the half-width in the untransformed domain.)
A stacking “velocity” is a parameter for the hyperbolic curve that best fits the moveout of reflection times over source-receiver offset. Stacking velocities are estimated from prestack seismic data
by scanning ranges of acceptable values and examining weighted sums of the data over offset. Resolution depends on the width of seismic wavelets at the largest recorded offsets. Regular sampling of
stacking velocities does not correspond to regular sampling of wavelets.
However, the squared reciprocal of stacking velocity, which we call squared stacking slowness (or “sloth”), does regularly sample wavelets at the farthest offset. We prefer to minimize errors in
squared slowness as the best way to minimize errors in corresponding reflection times.
Interpreters tend to pick stacking velocities at locations where moveouts change the most. The locations of picks are not necessarily more significant or reliable than others. Interpreters also
examine moveout adjustments at locations well away from the picks. If the interpolated behavior is acceptable, then no new picks are added. For this reason, we give interpolated stacking velocities
the same significance as picked values.
We treat an interpolated regular grid of squared stacking slownesses as our hard data to be inverted. Usually input stacking velocities are interpolated linearly between picked times, with constant
values off the ends. Functions are then triangulated and interpolated linearly over spatial directions. A regular grid of values needs enough resolution to represent all useful information in the
original functions.
For this inversion, we assume a stacking “velocity” to be equivalent to the root-mean-square (RMS) average of interval velocities. This equivalence holds exactly only for infinitesimal offsets in a
horizontally stratified medium.
Let a single sampled function of squared stacking slownesses be represented by the one-dimensional vector , and interval velocities by . Vector indices mark samples of vertical traveltime. Index zero
corresponds to zero time. We write the RMS average of in discrete form as
A fast, explicit inverse does exist for the RMS equation (1):
This equation (2) is typically referred to as the Dix equation, although the original reference [3] preferred more accurate variations. This explicit solution can easily fail when required to take
the square root of negative numbers. Worse, statistically meaningless variations in stacking velocities can cause interval velocities to vary wildly.
For a constrained inversion, we also find it useful to write the linearization of this equation. A small perturbation of interval velocity results in the following perturbation of squared stacking
Unperturbed variables retain their reference values.
Finally, the adjoint linearized equation gives the perturbation of interval velocity required to explain a small perturbation of squared stacking slowness:
Gradient optimization methods like conjugate-gradients usually require the adjoint.
Damped least-squares attempts to balance data errors with minimal complexity in the model.
Let be a linear smoothing operator with unit area. Define a smooth interval velocity with the convolution
where the vector contains the coefficients of smooth, shifted basis functions. Implicitly, this smoothing operator also convolves over all spatial indices, which we suppress in our equations.
The best coefficients should minimize the following objective function:
The small damping factor is the ratio of the variance of data errors to the variance of interval velocities. A large range of plausible values will give similar results. Damping ensures that small
variations in squared stacking slowness will not cause extreme variations in interval velocity. For a purely quadratic objective function, the damping is equivalent to pre-whitening, which adds a
small constant to the diagonal of the least-squares “normal” equations.
Once we have written the objective function (6), we have unambiguously specified a solution, although only implicitly. Much has been written on the optimization of objective functions, so we will not
cover the details here. See Luenberger [4] for more information on the Gauss-Newton method and conjugate-gradients.
The objective function (6) is not a perfectly quadratic function of the interval velocities but behaves similarly to a quadratic. The objective function has a clear global minimum and is convex far
away from that minimum. In the vicinity of the minimum, the objective function is indistinguishable from a quadratic.
If a suboptimum set of coefficients produce a particular set of squared stacking slownesses , then the actual picked slownesses may differ by an error . With linearization (3), we can say that the
best perturbation of coefficients should minimize the following objective function:
This approximate objective function (7) is perfectly quadratic. The optimum solution is a linear function of the data error . Quadratic objective functions are easily optimized by the
conjugate-gradient algorithm.
In our implementation, an outer Gauss-Newton loop iteratively replaces the objective function by the quadratic approximation (7). Each Gauss-Newton iteration begins with the best interval velocity
function so far. The first iteration uses a constant interval velocity function far from the correct solution. An inner conjugate-gradient loop minimizes the objective function that has been
approximated as a quadratic to find a perturbation to the reference interval velocity. A non-linear line-search finds the best factor to scale this perturbation before adding to the reference
interval velocity function. (The line-search algorithm uses a combination of a parabolic Newton method for speed and a golden-section search for robustness.) Finally, the Gauss-Newton loop begins
again with a new approximation of the objective function. Typically, some four to eight iterations are necessary for the Gauss-Newton and conjugate-gradient loops.
We apply hard constraints (minimum and maximum values) to interval velocities immediately after updating with a perturbation. These constraints are honored during the non-linear line-search, but not
during the temporary linearization for conjugate-gradients.
As a final optimization, early iterations begin with a large smoothing operator, and thus few degrees of freedom. After full optimization with an over-simplified interval velocity, the smoothing is
reduced. Finer details are allowed into the velocity model only when the background velocity is known to be near the final correct solution. Because of damping, rough details are introduced only when
justified to fit a sufficiently large error in the picked data.
Jon Claerbout.
Geophysical Estimation by Example.
http://sepwww.stanford.edu/sep/prof/toc_html/toc_html/gee/toc_html/, 1999.
R. Clapp, P. Sava, and J.F. Claerbout.
Interval velocity estimation with a null space.
Stanford Exploration Project Report, http://sepwww.stanford.edu/research/reports/, 97:147–156, 1998.
C. Hewett Dix.
Seismic Prospecting for Oil.
Harper and Brothers, 1952.
David G. Luenberger.
Introduction to Linear and Nonlinear Programming.
Addison Wesley, 1973.
J. L. Toldi.
Velocity analysis without picking.
PhD thesis, Stanford University, 1985. | {"url":"https://www.billharlan.com/papers/rmsinv/index.html","timestamp":"2024-11-08T04:48:26Z","content_type":"text/html","content_length":"19947","record_id":"<urn:uuid:9a3d8db0-dca2-43cb-9eaf-971395c660f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00323.warc.gz"} |
One And Two Step Equations Worksheets - Tessshebaylo
One And Two Step Equations Worksheets
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MFM2P - Day 57: Pyramids
Instead of starting with a counting circle today, I had them multiply two binomials. They haven't practiced this lately (and were really doing well with counting circles) so I thought it would be
more beneficial. I have 8 of these ready to go on little slips of paper (found here) so we can do one each day this week and again later in the semester.
Next up, pyramids! Alex Overwijk and I had talked about creating a new activity involving tents in the shape of pyramids, but that hasn't happened. This week is crazy for me so I went with some
not-very-exciting worksheets. We did start by building pyramids. I brought in the bins of G-O-Frames and asked them to build pyramids. That's all I said. Here is some of the collection:
There were a lot of tetrahedrons and some square-based pyramids. There is one hexagonal-based pyramid and one right-angled triangular-based pyramid. I asked what they noticed about the shapes and
they said that the sides are always triangles. The base could be a variety of shapes. I asked how they could find the surface area of a pyramid and took a squared-based one apart. I asked the same
question while holding up the one with a hexagon as its base. They seemed to understand the idea of adding up all the areas. I had them work on the first example from this handout and I circulated
helping some of them get started.
I showed them the right-angled triangular-based prism and said that if we had three of them we could make a cube. I also showed them this video. I could have done the demo myself, but I would just
make a mess and I wasn't up for that first thing on a Monday morning. I tried to really emphasize that to find the volume of a prism you take the area of the base and multiply it by the height
because the height represents the number of "layers". To find the volume of a pyramid, you perform that same process and divide the result by 3. Again, I let them work through some of the examples
before going over this one together:
Along the way we talked about the difference between vertical height and slant height. They figured out that the slant height of the pyramid is the height of the triangle. They also were able to
determine that if you were missing one of the heights in a square-based prism, you could use the sum of squares (Pythagorean theorem) to calculate it.
We will do a little more with pyramids tomorrow and work on some trig again. | {"url":"https://marybourassa.blogspot.com/2015/05/mfm2p-day-57-pyramids.html","timestamp":"2024-11-14T00:35:40Z","content_type":"text/html","content_length":"72381","record_id":"<urn:uuid:c1c19491-addd-458f-8e87-8e145f78f722>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00075.warc.gz"} |
Length and Distance Calculator - CalculatorBox
Length and Distance Calculator
Figures rounded to max decimal places.
A length and distance calculator is an essential tool for solving problems that involve conversions between different units of length and distance. This user-friendly online calculator is designed to
help you perform various length and distance calculations with ease, and it is suitable for both professionals and students. It is a practical utility that will save you time and effort in your daily
life, whether you are a civil engineer working on a construction project or a student learning about measurements in school.
In this article, we will discuss the formulas necessary to perform length and distance calculations and provide a worked example to illustrate how easy it is to use our calculator. Keep in mind that
this calculator is a valuable resource for converting between various units of measurement, such as meters, kilometers, miles, feet, inches, and more.
Formulas Necessary for Length and Distance Calculations
To perform length and distance calculations, you need to understand the formulas required for converting different units of measurement. The basic formula for converting between two units of length
converted ~length \\= original ~length ~×~ conversion ~factor
The conversion factor is a constant value that is used to convert one unit of length to another. Here are some common conversion factors for length and distance:
Unit Unit Symbol Quantity Unit 2 Unit 2 Symbol
meter m 1 centimeters m 100
kilometer km 1 meters m 1,000
mile mi 1 meters m 1,609.344
foot ft 1 meters m 0.3048
inch in 1 meters m 0.0254
These conversion factors can be used to convert any unit of length to meters. To convert between other units of length, you can use the following formula:
converted ~length \\= original ~length * \\(conversion ~factor ~of ~the ~original ~unit \\/ ~conversion ~factor ~of ~the ~desired ~unit)
This formula allows you to convert between any two units of length directly, without needing to convert to meters first.
Worked Example Calculation
Let’s say you want to convert a distance of 5.5 miles to kilometers using the length and distance calculator. Here’s how you can do it step by step:
Step 1: Identify the original length and units.
In this example, the original length is 5.5 miles.
Step 2: Identify the desired units.
We want to convert the distance to kilometers.
Step 3: Find the conversion factors for the original and desired units.
The conversion factor for miles to meters is 1,609.344, and the conversion factor for kilometers to meters is 1,000.
Step 4: Apply the conversion formula.
converted ~length \\= original ~length * \\(conversion ~factor ~of ~the ~original ~unit \\/ ~conversion ~factor ~of ~the ~desired ~unit)
converted length (in ~km) \\= 5.5 mi * (1,609.344 ~m/mi / 1,000 ~m/km)
converted length (in ~km) \\= 5.5 * 1.609344
converted length (in ~km) ≈ 8.851
So, 5.5 miles is approximately equal to 8.851 kilometers.
Using the Length and Distance Calculator
To make this calculation even easier, you can use our online length and distance calculator. Simply enter the original length and units, select the desired units, and click “Calculate” to obtain the
converted length instantly. The calculator will perform the necessary conversions using the correct conversion factors and provide you with the result in a matter of seconds.
The length and distance calculator is an invaluable tool for performing quick and accurate conversions between different units of measurement. By understanding the formulas and conversion factors
involved in these calculations, you can ensure that your work is precise and reliable. Whether you are a professional working on a project or a student learning about measurement, this calculator
will save you time and effort, allowing you to focus on more important tasks. So, go ahead and give our length and distance calculator a try, and see for yourself how easy it is to perform
conversions and calculations with this handy, user-friendly tool. | {"url":"https://calculatorbox.com/calculator/length-and-distance-calculator/","timestamp":"2024-11-10T21:57:04Z","content_type":"text/html","content_length":"150089","record_id":"<urn:uuid:099a6e35-619a-4eeb-a3c8-0edbe85074f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00536.warc.gz"} |
Subtracting from 10 with Hot Cocoa Marshmallows
Subtracting from 10 with Hot Cocoa Marshmallows
Price: 300 points or $3 USD
Subjects: math
Grades: 14,13,1,2
Description: Students read a number sentence and then count the marshmallows to see how many they need to take from 10. Students then click the number to show how many to subtract from ten to make
the new number in the equation. Digital practice on Google Slides can be assigned through Google Classroom, SeeSaw, or other online learning platforms. Kindergarten Math Skills include: subtracting
breaking up 10 number recognition number sense subitizing K.CC.B.5 Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as
10 things in a scattered configuration; given a number from 1-20, count out that many objects. SK.CC.B.4b Understand that the last number name said tells the number of objects counted. The number of
objects is the same regardless of their arrangement or the order in which they were counted. K.CC.B.4a When counting objects, say the number names in the standard order, pairing each object with one
and only one number name and each number name with one and only one object. K.CC.B.4 Understand the relationship between numbers and quantities; connect counting to cardinality. K.CC.A.3 Write
numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects). 30 cards | {"url":"https://wow.boomlearning.com/deck/SGE6puhdvYErP3Td8","timestamp":"2024-11-14T04:58:47Z","content_type":"text/html","content_length":"3162","record_id":"<urn:uuid:9b8f370e-ef6e-4e2d-a968-5241bacc830a>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00293.warc.gz"} |
Molecular adsorption on surfaces
Molecular adsorption on surfaces is a fundamentally important process in catalysis, gas storage, water purification, and many other areas. A key challenge in understanding the underlying mechanisms
in molecular adsorption and desorption processes is having accurate electronic structure theories that we can solve for realistic extended surface models.
In our group we aim to develop and apply periodic quantum chemical wavefunction-based methods in surface problems, such as water adsorption and hydrogen dissociation on periodic surfaces. We rely on
a canonical periodic coupled-cluster theory implementation interfaced to the Vienna ab-initio simulation package (VASP). The representation of virtual orbitals based on Gaussian basis-sets expanded
in plane-waves[1], a low-rank factorization of the Coulomb integrals[2] and a finite-size correction scheme[3] allows for reducing the computational cost of the employed coupled-cluster methods.
[1] G.H. Booth, T. Tsatsoulis, G.K. Chan, A. Grüneis, J. Chem. Phys. 145, 084111 (2016).
[2] F. Hummel, T. Tsatsoulis, A. Grüneis, J. Chem. Phys. 146, 124105 (2017).
[3] T. Gruber, K. Liao, T. Tsatsoulis, F. Hummel, A. Grüneis, Phys. Rev. X, 8, 021043 (2018).
Tsatsoulis, Theodoros, et al. “A comparison between quantum chemistry and quantum Monte Carlo techniques for the adsorption of water on the (001) LiH surface.” The Journal of chemical physics 146.20
Al-Hamdani, Yasmine S., et al. “Properties of the water to boron nitride interaction: From zero to two dimensions with benchmark accuracy.” The Journal of chemical physics 147.4 (2017): 044710.
Brandenburg, Jan Gerit, et al. “Physisorption of water on graphene: Subchemical accuracy from many-body electronic structure methods.” The journal of physical chemistry letters 10.3 (2019): 358-368.
Tsatsoulis, Theodoros, et al. “Reaction energetics of hydrogen on Si (100) surface: A periodic many-electron theory study.” The Journal of chemical physics 149.24 (2018): 244105. | {"url":"https://cqc.itp.tuwien.ac.at/research/surfaces/index.html","timestamp":"2024-11-07T04:28:49Z","content_type":"text/html","content_length":"6612","record_id":"<urn:uuid:bb008d86-0fa1-48f1-94eb-ca8b6b6456f3>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00120.warc.gz"} |
DSA Ford-Fulkerson Algorithm
The Ford-Fulkerson algorithm solves the maximum flow problem.
Finding the maximum flow can be helpful in many areas: for optimizing network traffic, for manufacturing, for supply chain and logistics, or for airline scheduling.
The Ford-Fulkerson Algorithm
The Ford-Fulkerson algorithm solves the maximum flow problem for a directed graph.
The flow comes from a source vertex (\(s\)) and ends up in a sink vertex (\(t\)), and each edge in the graph allows a flow, limited by a capacity.
Max flow: {{maxFlow}}
The Ford-Fulkerson algorithm works by looking for a path with available capacity from the source to the sink (called an augmented path), and then sends as much flow as possible through that path.
The Ford-Fulkerson algorithm continues to find new paths to send more flow through until the maximum flow is reached.
In the simulation above, the Ford-Fulkerson algorithm solves the maximum flow problem: It finds out how much flow can be sent from the source vertex \(s\), to the sink vertex \(t\), and that maximum
flow is 8.
The numbers in the simulation above are written in fractions, where the first number is the flow, and the second number is the capacity (maximum possible flow in that edge). So for example, 0/7 on
edge \(s \rightarrow v_2\), means there is 0 flow, with a capacity of 7 on that edge.
Note: The Ford-Fulkerson algorithm is often described as a method instead of as an algorithm, because it does not specify how to find a path where flow can be increased. This means it can be
implemented in different ways, resulting in different time complexities. But for this tutorial we will call it an algorithm, and use Depth-First-Search to find the paths.
You can see the basic step-by-step description of how the Ford-Fulkerson algorithm works below, but we need to go into more detail later to actually understand it.
How it works:
1. Start with zero flow on all edges.
2. Find an augmented path where more flow can be sent.
3. Do a bottleneck calculation to find out how much flow can be sent through that augmented path.
4. Increase the flow found from the bottleneck calculation for each edge in the augmented path.
5. Repeat steps 2-4 until max flow is found. This happens when a new augmented path can no longer be found.
Residual Network in Ford-Fulkerson
The Ford-Fulkerson algorithm actually works by creating and using something called a residual network, which is a representation of the original graph.
In the residual network, every edge has a residual capacity, which is the original capacity of the edge, minus the the flow in that edge. The residual capacity can be seen as the leftover capacity in
an edge with some flow.
For example, if there is a flow of 2 in the \( v_3 \rightarrow v_4 \) edge, and the capacity is 3, the residual flow is 1 in that edge, because there is room for sending 1 more unit of flow through
that edge.
Reversed Edges in Ford-Fulkerson
The Ford-Fulkerson algorithm also uses something called reversed edges to send flow back. This is useful to increase the total flow.
For example, the last augmented path \(s \rightarrow v_2 \rightarrow v_4 \rightarrow v_3 \rightarrow t\) in the animation above and in the manual run through below shows how the total flow is
increased by one more unit, by actually sending flow back on edge \( v_4 \rightarrow v_3 \), sending the flow in the reverse direction.
Sending flow back in the reverse direction on edge \( v_3 \rightarrow v_4 \) in our example meas that this 1 unit of flow going out of vertex \( v_3 \), now leaves \( v_3 \) on edge \( v_3 \
rightarrow t \) instead of \( v_3 \rightarrow v_4 \).
To send flow back, in the opposite direction of the edge, a reverse edge is created for each original edge in the network. The Ford-Fulkerson algorithm can then use these reverse edges to send flow
in the reverse direction.
A reversed edge has no flow or capacity, just residual capacity. The residual capacity for a reversed edge is always the same as the flow in the corresponding original edge. In our example, the edge
\( v_3 \rightarrow v_4 \) has a flow of 2, which means there is a residual capacity of 2 on the corresponding reversed edge \( v_4 \rightarrow v_3 \).
This just means that when there is a flow of 2 on the original edge \( v_3 \rightarrow v_4 \), there is a possibility of sending that same amount of flow back on that edge, but in the reversed
direction. Using a reversed edge to push back flow can also be seen as undoing a part of the flow that is already created.
The idea of a residual network with residual capacity on edges, and the idea of reversed edges, are central to how the Ford-Fulkerson algorithm works, and we will go into more detail about this when
we implement the algorithm further down on this page.
Manual Run Through
There is no flow in the graph to start with.
To find the maximum flow, the Ford-Fulkerson algorithm must increase flow, but first it needs to find out where the flow can be increased: it must find an augmented path.
The Ford-Fulkerson algorithm actually does not specify how such an augmented path is found (that is why it is often described as a method instead of an algorithm), but we will use Depth First Search
(DFS) to find the augmented paths for the Ford-Fulkerson algorithm in this tutorial.
The first augmented path Ford-Fulkerson finds using DFS is \(s \rightarrow v_1 \rightarrow v_3 \rightarrow v_4 \rightarrow t\).
And using the bottleneck calculation, Ford-Fulkerson finds that 3 is the highest flow that can be sent through the augmented path, so the flow is increased by 3 for all the edges in this path.
The next iteration of the Ford-Fulkerson algorithm is to do these steps again:
2. Find a new augmented path
3. Find how much the flow in that path can be increased
4. Increase the flow along the edges in that path accordingly
The next augmented path is found to be \(s \rightarrow v_2 \rightarrow v_1 \rightarrow v_4 \rightarrow v_3 \rightarrow t\), which includes the reversed edge \(v_4 \rightarrow v_3\), where flow is
sent back.
The Ford-Fulkerson concept of reversed edges comes in handy because it allows the path finding part of the algorithm to find an augmented path where reversed edges can also be included.
In this specific case that means that a flow of 2 can be sent back on edge \(v_3 \rightarrow v_4\), going into \(v_3 \rightarrow t\) instead.
The flow can only be increased by 2 in this path because that is the capacity in the \( v_3 \rightarrow t \) edge.
The next augmented path is found to be \(s \rightarrow v_2 \rightarrow v_1 \rightarrow v_4 \rightarrow t\).
The flow can be increased by 2 in this path. The bottleneck (limiting edge) is \( v_1 \rightarrow v_4 \) because there is only room for sending two more units of flow in that edge.
The next and last augmented path is \(s \rightarrow v_2 \rightarrow v_4 \rightarrow t\).
The flow can only be increased by 1 in this path because of edge \( v_4 \rightarrow t \) being the bottleneck in this path with only space for one more unit of flow (\(capacity-flow=1\)).
At this point, a new augmenting path cannot be found (it is not possible to find a path where more flow can be sent through from \(s\) to \(t\)), which means the max flow has been found, and the
Ford-Fulkerson algorithm is finished.
The maximum flow is 8. As you can see in the image above, the flow (8) is the same going out of the source vertex \(s\), as the flow going into the sink vertex \(t\).
Also, if you take any other vertex than \(s\) or \(t\), you can see that the amount of flow going into a vertex, is the same as the flow going out of it. This is what we call conservation of flow,
and this must hold for all such flow networks (directed graphs where each edge has a flow and a capacity).
Implementation of The Ford-Fulkerson Algorithm
To implement the Ford-Fulkerson algorithm, we create a Graph class. The Graph represents the graph with its vertices and edges:
class Graph:
def __init__(self, size):
self.adj_matrix = [[0] * size for _ in range(size)]
self.size = size
self.vertex_data = [''] * size
def add_edge(self, u, v, c):
self.adj_matrix[u][v] = c
def add_vertex_data(self, vertex, data):
if 0 <= vertex < self.size:
self.vertex_data[vertex] = data
Line 3: We create the adj_matrix to hold all the edges and edge capacities. Initial values are set to 0.
Line 4: size is the number of vertices in the graph.
Line 5: The vertex_data holds the names of all the vertices.
Line 7-8: The add_edge method is used to add an edge from vertex u to vertex v, with capacity c.
Line 10-12: The add_vertex_data method is used to add a vertex name to the graph. The index of the vertex is given with the vertex argument, and data is the name of the vertex.
The Graph class also contains the dfs method to find augmented paths, using Depth-First-Search:
def dfs(self, s, t, visited=None, path=None):
if visited is None:
visited = [False] * self.size
if path is None:
path = []
visited[s] = True
if s == t:
return path
for ind, val in enumerate(self.adj_matrix[s]):
if not visited[ind] and val > 0:
result_path = self.dfs(ind, t, visited, path.copy())
if result_path:
return result_path
return None
Line 15-18: The visited array helps to avoid revisiting the same vertices during the search for an augmented path. Vertices that belong to the augmented path are stored in the path array.
Line 20-21: The current vertex is marked as visited, and then added to the path.
Line 23-24: If the current vertex is the sink node, we have found an augmented path from the source vertex to the sink vertex, so that path can be returned.
Line 26-30: Looping through all edges in the adjacency matrix starting from the current vertex s, ind represents an adjacent node, and val is the residual capacity on the edge to that vertex. If the
adjacent vertex is not visited, and has residual capacity on the edge to it, go to that node and continue searching for a path from that vertex.
Line 32: None is returned if no path is found.
The fordFulkerson method is the last method we add to the Graph class:
def fordFulkerson(self, source, sink):
max_flow = 0
path = self.dfs(source, sink)
while path:
path_flow = float("Inf")
for i in range(len(path) - 1):
u, v = path[i], path[i + 1]
path_flow = min(path_flow, self.adj_matrix[u][v])
for i in range(len(path) - 1):
u, v = path[i], path[i + 1]
self.adj_matrix[u][v] -= path_flow
self.adj_matrix[v][u] += path_flow
max_flow += path_flow
path_names = [self.vertex_data[node] for node in path]
print("Path:", " -> ".join(path_names), ", Flow:", path_flow)
path = self.dfs(source, sink)
return max_flow
Initially, the max_flow is 0, and the while loop keeps increasing the max_flow as long as there is an augmented path to increase flow in.
Line 37: The augmented path is found.
Line 39-42: Every edge in the augmented path is checked to find out how much flow can be sent through that path.
Line 44-46: The residual capacity (capacity minus flow) for every forward edge is reduced as a result of increased flow.
Line 47: This represents the reversed edge, used by the Ford-Fulkerson algorithm so that flow can be sent back (undone) on the the original forward edges. It is important to understand that these
reversed edges are not in the original graph, they are fictitious edges introduced by Ford-Fulkerson to make the algorithm work.
Line 49: Every time flow is increased over an augmented path, max_flow is increased by the same value.
Line 51-52: This is just for printing the augmented path before the algorithm starts the next iteration.
After defining the Graph class, the vertices and edges must be defined to initialize the specific graph, and the complete code for the Ford-Fulkerson algorithm example looks like this:
class Graph:
def __init__(self, size):
self.adj_matrix = [[0] * size for _ in range(size)]
self.size = size
self.vertex_data = [''] * size
def add_edge(self, u, v, c):
self.adj_matrix[u][v] = c
def add_vertex_data(self, vertex, data):
if 0 <= vertex < self.size:
self.vertex_data[vertex] = data
def dfs(self, s, t, visited=None, path=None):
if visited is None:
visited = [False] * self.size
if path is None:
path = []
visited[s] = True
if s == t:
return path
for ind, val in enumerate(self.adj_matrix[s]):
if not visited[ind] and val > 0:
result_path = self.dfs(ind, t, visited, path.copy())
if result_path:
return result_path
return None
def fordFulkerson(self, source, sink):
max_flow = 0
path = self.dfs(source, sink)
while path:
path_flow = float("Inf")
for i in range(len(path) - 1):
u, v = path[i], path[i + 1]
path_flow = min(path_flow, self.adj_matrix[u][v])
for i in range(len(path) - 1):
u, v = path[i], path[i + 1]
self.adj_matrix[u][v] -= path_flow
self.adj_matrix[v][u] += path_flow
max_flow += path_flow
path_names = [self.vertex_data[node] for node in path]
print("Path:", " -> ".join(path_names), ", Flow:", path_flow)
path = self.dfs(source, sink)
return max_flow
g = Graph(6)
vertex_names = ['s', 'v1', 'v2', 'v3', 'v4', 't']
for i, name in enumerate(vertex_names):
g.add_vertex_data(i, name)
g.add_edge(0, 1, 3) # s -> v1, cap: 3
g.add_edge(0, 2, 7) # s -> v2, cap: 7
g.add_edge(1, 3, 3) # v1 -> v3, cap: 3
g.add_edge(1, 4, 4) # v1 -> v4, cap: 4
g.add_edge(2, 1, 5) # v2 -> v1, cap: 5
g.add_edge(2, 4, 3) # v2 -> v4, cap: 3
g.add_edge(3, 4, 3) # v3 -> v4, cap: 3
g.add_edge(3, 5, 2) # v3 -> t, cap: 2
g.add_edge(4, 5, 6) # v4 -> t, cap: 6
source = 0; sink = 5
print("The maximum possible flow is %d " % g.fordFulkerson(source, sink))
Run Example ยป
Time Complexity for The Ford-Fulkerson Algorithm
The time complexity for the Ford-Fulkerson varies with the number of vertices \(V\), the number of edges \(E\), and it actually varies with the maximum flow \(f\) in the graph as well.
The reason why the time complexity varies with the maximum flow \(f\) in the graph, is because in a graph with a high throughput, there will be more augmented paths that increase flow, and that means
the DFS method that finds these augmented paths will have to run more times.
Depth-first search (DFS) has time complexity \(O(V+E)\).
DFS runs once for every new augmented path. If we assume that each augmented graph increase flow by 1 unit, DFS must run \(f\) times, as many times as the value of maximum flow.
This means that time complexity for the Ford-Fulkerson algorithm, using DFS, is
\[ O( (V+E) \cdot f ) \]
For dense graphs, where \( E > V \), time complexity for DFS can be simplified to \(O(E)\), which means that the time complexity for the Ford-Fulkerson algorithm also can be simplified to
\[ O( E \cdot f ) \]
A dense graph does not have an accurate definition, but it is a graph with many edges.
The next algorithm we will describe that finds maximum flow is the Edmonds-Karp algorithm.
The Edmonds-Karp algorithm is very similar to Ford-Fulkerson, but it uses BFS instead of DFS to find augmented paths, which leads to fewer iterations to find maximum flow. | {"url":"https://www.w3schools.com/dsa/dsa_algo_graphs_fordfulkerson.php","timestamp":"2024-11-09T17:43:20Z","content_type":"text/html","content_length":"416061","record_id":"<urn:uuid:cdd6f331-e21a-42e0-b80e-bb2312620064>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00706.warc.gz"} |
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Survey of Calculus - Applied
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Survey of Calculus Online Course Info - Distance Calculus Course Information
Survey of Calculus course can best be described as a "single semester course introducing Differential and Integral Calculus to non-STEM majors".
This course has many names, all being equivalent:
• Survey of Calculus
• Liberal Arts Calculus
• Applied Calculus
• Business Calculus
• Brief Calculus
• Calculus for Life Science
At Distance Calculus, we call our "Survey of Calculus" course as Applied Calculus - DMAT 201 - 3 credits.
Below are some links for further information about the Survey of Calculus course via Distance Calculus @ Roger Williams University.
Distance Calculus - Student Reviews
Date Posted: May 3, 2020
Review by: Andris H.
Courses Completed: Applied Calculus
Review: I found out from my MBA program that I needed to finish calculus before starting the MBA. They told me 3 weeks before term started! I was able to finish Applied Calculus from Distance
Calculus. Definitely a great class. Thanks Distance Calculus!
Transferred Credits to: SUNY Stony Brook
Date Posted: Apr 6, 2020
Review by: Paul Simmons
Courses Completed: Multivariable Calculus, Differential Equations
Review: I took Multivariable and Diff Eq during the summer. The DiffEq course was awesome - very useful for my physics and engineering course. I was unsure about Mathematica at first, but I got the
hang of it quickly. Thank you Distance Calculus!
Transferred Credits to: University of Oregon
Date Posted: Mar 17, 2020
Review by: Rebecca M.
Courses Completed: Calculus II, Multivariable Calculus
Review: Fantastic courses! I barely made it through Cal 1, and halfway through Cal 2 I found this program. I took Cal 2 and then Multivariable and I just loved it! SOOOOOOO much better than a
classroom+textbook class. I highly recommend!
Transferred Credits to: Tulane University | {"url":"https://www.distancecalculus.com/survey-of-calculus/","timestamp":"2024-11-03T17:18:57Z","content_type":"text/html","content_length":"41189","record_id":"<urn:uuid:9eb9b57e-eb86-4c72-b823-209021f6e2da>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00100.warc.gz"} |
An extension of the Nemhauser-Trotter theorem to generalized vertex coverwith applications
The Nemhauser-Trotter theorem provides an algorithm which is frequently used as a subroutine in approximation algorithms for the classical Vertex Cover problem. In this paper we present an extension
of this theorem so it fits a more general variant of Vertex Cover, namely, the Generalized Vertex Cover problem, where edges are allowed not to be covered at a certain predetermined penalty. We show
that many applications of the original Nemhauser-Trotter theorem can be applied using our extension to Generalized Vertex Cover. These applications include a (2-2/d)-approximation algorithm for
graphs of bounded degree d, a polynomial-time approximation scheme (PTAS) for planar graphs, a (2 - lglg n/2lg n)-approximation algorithm for general graphs, and a 2k kernel for the parameterized
Generalized Vertex Cover problem.
• Approximation algorithms
• Generalized vertex cover
• Local ratio technique
• Nemhauser-trotter theorem
ASJC Scopus subject areas
Dive into the research topics of 'An extension of the Nemhauser-Trotter theorem to generalized vertex coverwith applications'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/an-extension-of-the-nemhauser-trotter-theorem-to-generalized-vert","timestamp":"2024-11-06T05:37:59Z","content_type":"text/html","content_length":"56642","record_id":"<urn:uuid:8c93cd58-a123-4875-a51e-69378d6eef4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00432.warc.gz"} |
Beast Academy Review (*NEW* BA Level 1a)
"Trust the process" is something I can say much more easily than I can do. There might not be a more relevant description for Beast Academy math.
You see, I was a Saxon mom. It was plain, straight-forward, and simple. Uncomplicated. It was how I had begun homeschooling my kids (Disclosure: We started with Saxon 1, not Saxon K, because holy
smokes I know I could not handle the redundant 180 pages of counting 1-5 that is in Saxon K). I appreciated things about Saxon. I liked the "spiral learning method" of constant review. I enjoyed the
assessment pages. It was plain and boring, like me.
My oldest (now 16) was in 6th grade taking Saxon 7/8 when I started to hear the hype about Beast Academy. He was decently gifted at math, but I was interested in challenging him more. We decided to
try Beast Academy for him. Naturally, I assumed since he was in 6th grade and doing 7/8 math at home, that Beast level 5 would be an appropriate start for him, I even anticipated it being too easy!
One thing you absolutely need to know about Beast is that the levels do NOT correspond well to grades. Beginning algebraic concepts begin to be introduced in Beast level 3. In level 5, students will
be doing prime factorization, solving complex arithmetic sequences, and much more complicated algebra. Level 5 BA is not comparable at all to Saxon 5, not that a very gifted 5th grade math student
couldn't handle it.
Another thing, these concepts are introduced in ways that are totally foreign to 30 something and 40 something mom "instructors." I would be completely lying if I told you that BA was not also
challenging to me. The way the concepts are introduced is different, sometimes not in ways that I can logically understand. And this is where I say "trust the process." No, this isn't some silly
Common Core way to do math! Its just different.
Every level of Beast has 4 parts, A through D. You will find incredibly helpful placement tests on beastacademy.com The placement tests are self-graded with answers also online.
Also online are sample pages from the book:
You can print them out and give them a try with your student. Keep in mind that the answer key is at the back of every book, no separate answer book to purchase. The answer key is also very thorough
and helpful with step-by-step instructions for solutions.
All the BA levels and books have corresponding guide books. The guide books are meant to teach the chapter/lesson in a comic book form. I have not found the guide books to be 100% necessary. When my
oldest decided he no longer wanted to read the comic, we stopped buying the guide book. The only challenge then was on me to be able to explain the lesson and the math. No problem, but if you
struggle with math, you may want to consider buying the guide books. The guide books are reusable, so if you have multiple children, you will get your value from them.
In the practice book you will find a number of practice problems for every lesson. Additionally, you will find questions marked with either one or two stars. These are challenge questions, and I will
tell you that they are indeed challenging! Regarding the challenge questions, from the BA website:
Most real problems in math, science, or life can’t be solved with a simple, cookie-cutter solution. Instead, real problems usually require applying knowledge in a novel way. We include many
challenging problems in Beast Academy because we believe they help kids become flexible, logical thinkers who persevere in the face of challenge. Students also learn math at a much deeper level
when they solve difficult problems, since they have to think for themselves rather than mimicking prescribed procedures. As an added benefit, difficult problems make math much more interesting
and fun! Most kids find unraveling one challenging puzzle far more satisfying than mechanically completing pages of simple problems.
Also available now is Beast Academy Online. My 6th grade son used Beast Online (starting at level 3) for almost an entire year. It is the same comprehensive program and the same practice problems. A
subscription includes a digital version of the guide book. Every lesson is taught by an instructor that is quite a dynamic character. He is decently weird and entertaining, and my son has enjoyed
him. You can enroll for the month ($15/month) or year ($96). For us, the value was in using the books. If it was taking him multiple months to go through a workbook online, then economically the
books were a better choice. I may have stuck it out with BA Online if I had not needed to help my son as much as I did (he required some kind of additionally explanation or assistance from me at
least 50% of the time). That really took away from the convenience of using BA Online.
Beast Academy only goes up to level 5D. After that, you move on to AoPS Pre-algebra. For my son, he ended up completing Beast Academy level 5 concurrent with Saxon pre-algebra (he does not mind math
and never balked at the additional work). This set him up incredibly well to do Saxon Algebra 1 in 8th grade, he did fantastic. We did not move forward with the AoPS program because my son returned
to a 2 day classical school that used the Saxon curriculum.
Early in 2021, BA came out with a level 1a book. It looks very much like basic first grade math, equivalent to grade 1 level (so far). It is exciting to have a younger book option, previously BA
started at level 2. I am looking forward to seeing what the level 1 books encompass.
It seems there might be a focus in 1A on very basic number concepts.
Excerpts from Beast Academy 1A | {"url":"https://www.ahomeschoolexperiment.com/post/beast-academy-review-new-ba-level-1a","timestamp":"2024-11-11T17:07:43Z","content_type":"text/html","content_length":"994315","record_id":"<urn:uuid:51db0262-46f2-458c-ae05-1f9c28335245>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00785.warc.gz"} |
After defining a statistical model, in addition to sampling from its distributions, one may be interested in finding the parameter values that maximise for instance the posterior distribution density
function or the likelihood. This is called mode estimation. Turing provides support for two mode estimation techniques, maximum likelihood estimation (MLE) and maximum a posterior (MAP) estimation.
To demonstrate mode estimation, let us load Turing and declare a model:
using Turing @model function gdemo(x) s² ~ InverseGamma(2, 3) m ~ Normal(0, sqrt(s²)) for i in eachindex(x) x[i] ~ Normal(m, sqrt(s²)) end end
Once the model is defined, we can construct a model instance as we normally would:
DynamicPPL.Model{typeof(gdemo), (:x,), (), (), Tuple{Vector{Float64}}, Tuple{}, DynamicPPL.DefaultContext}(Main.Notebook.gdemo, (x = [1.5, 2.0],), NamedTuple(), DynamicPPL.DefaultContext())
Finding the maximum aposteriori or maximum likelihood parameters is as simple as
# Generate a MLE estimate. mle_estimate = maximum_likelihood(model) # Generate a MAP estimate. map_estimate = maximum_a_posteriori(model)
The estimates are returned as instances of the ModeResult type. It has the fields values for the parameter values found and lp for the log probability at the optimum, as well as f for the objective
function and optim_result for more detailed results of the optimisation procedure.
Under the hood maximum_likelihood and maximum_a_posteriori use the Optimization.jl package, which provides a unified interface to many other optimisation packages. By default Turing typically uses
the LBFGS method from Optim.jl to find the mode estimate, but we can easily change that:
using OptimizationOptimJL: NelderMead @show maximum_likelihood(model, NelderMead()) using OptimizationNLopt: NLopt.LD_TNEWTON_PRECOND_RESTART @show maximum_likelihood(model,
maximum_likelihood(model, NelderMead()) = [0.062500666047239, 1.7499991015123628] maximum_likelihood(model, LD_TNEWTON_PRECOND_RESTART()) = [0.06249999999999884, 1.7499999999999971]
The above are just two examples, Optimization.jl supports many more.
We can also help the optimisation by giving it a starting point we know is close to the final solution, or by specifying an automatic differentiation method
using ADTypes: AutoReverseDiff import ReverseDiff maximum_likelihood( model, NelderMead(); initial_params=[0.1, 2], adtype=AutoReverseDiff() )
When providing values to arguments like initial_params the parameters are typically specified in the order in which they appear in the code of the model, so in this case first s² then m. More
precisely it’s the order returned by Turing.Inference.getparams(model, Turing.VarInfo(model)).
We can also do constrained optimisation, by providing either intervals within which the parameters must stay, or costraint functions that they need to respect. For instance, here’s how one can find
the MLE with the constraint that the variance must be less than 0.01 and the mean must be between -1 and 1.:
The arguments for lower (lb) and upper (ub) bounds follow the arguments of Optimization.OptimizationProblem, as do other parameters for providing constraints, such as cons. Any extraneous keyword
arguments given to maximum_likelihood or maximum_a_posteriori are passed to Optimization.solve. Some often useful ones are maxiters for controlling the maximum number of iterations and abstol and
reltol for the absolute and relative convergence tolerances:
We can check whether the optimisation converged using the optim_result field of the result:
badly_converged_mle.optim_result = retcode: Failure u: [-1.4628222951067338, 1.6998581911219297] Final objective value: 0.655794502892618
For more details, such as a full list of possible arguments, we encourage the reader to read the docstring of the function Turing.Optimisation.estimate_mode, which is what maximum_likelihood and
maximum_a_posteriori call, and the documentation of Optimization.jl.
Turing extends several methods from StatsBase that can be used to analyze your mode estimation results. Methods implemented include vcov, informationmatrix, coeftable, params, and coef, among others.
For example, let’s examine our ML estimate from above using coeftable:
Coef. Std. Error z Pr(> z )
s² 0.0625 0.0625 1.0 0.317311 -0.0599977 0.184998
m 1.75 0.176777 9.89949 4.18383e-23 1.40352 2.09648
Standard errors are calculated from the Fisher information matrix (inverse Hessian of the log likelihood or log joint). Note that standard errors calculated in this way may not always be appropriate
for MAP estimates, so please be cautious in interpreting them.
You can begin sampling your chain from an MLE/MAP estimate by extracting the vector of parameter values and providing it to the sample function with the keyword initial_params. For example, here is
how to sample from the full posterior using the MAP estimate as the starting point: | {"url":"https://turinglang.org/docs/tutorials/docs-17-mode-estimation/","timestamp":"2024-11-10T12:22:51Z","content_type":"application/xhtml+xml","content_length":"67667","record_id":"<urn:uuid:819a6902-6c1c-41ca-8903-aad3a818598a>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00431.warc.gz"} |
Exact explainer
This notebooks demonstrates how to use the Exact explainer on some simple datasets. The Exact explainer is model-agnostic, so it can compute Shapley values and Owen values exactly (without
approximation) for any model. However, since it completely enumerates the space of masking patterns it has \(O(2^M)\) complexity for Shapley values and \(O(M^2)\) complexity for Owen values on a
balanced clustering tree for M input features.
Because the exact explainer knows that it is fully enumerating the masking space it can use optimizations that are not possible with random sampling based approaches, such as using a grey code
ordering to minimize the number of inputs that change between successive masking patterns, and so potentially reduce the number of times the model needs to be called.
import xgboost
import shap
# get a dataset on income prediction
X, y = shap.datasets.adult()
# train an XGBoost model (but any other model type would also work)
model = xgboost.XGBClassifier()
model.fit(X, y);
Tabular data with independent (Shapley value) masking
# build an Exact explainer and explain the model predictions on the given dataset
explainer = shap.explainers.Exact(model.predict_proba, X)
shap_values = explainer(X[:100])
# get just the explanations for the positive class
shap_values = shap_values[..., 1]
Exact explainer: 101it [00:12, 8.13it/s]
Plot a global summary
Plot a single instance
Tabular data with partition (Owen value) masking
While Shapley values result from treating each feature independently of the other features, it is often useful to enforce a structure on the model inputs. Enforcing such a structure produces a
structure game (i.e. a game with rules about valid input feature coalitions), and when that structure is a nest set of feature grouping we get the Owen values as a recursive application of Shapley
values to the group. In SHAP, we take the partitioning to the limit and build a binary herarchial clustering tree to represent the structure of the data. This structure could be chosen in many ways,
but for tabular data it is often helpful to build the structure from the redundancy of information between the input features about the output label. This is what we do below:
# build a clustering of the features based on shared information about y
clustering = shap.utils.hclust(X, y)
# above we implicitly used shap.maskers.Independent by passing a raw dataframe as the masker
# now we explicitly use a Partition masker that uses the clustering we just computed
masker = shap.maskers.Partition(X, clustering=clustering)
# build an Exact explainer and explain the model predictions on the given dataset
explainer = shap.explainers.Exact(model.predict_proba, masker)
shap_values2 = explainer(X[:100])
# get just the explanations for the positive class
shap_values2 = shap_values2[..., 1]
Plot a global summary
Note that only the Relationship and Marital status features share more that 50% of their explanation power (as measured by R2) with each other, so all the other parts of the clustering tree are
removed by the the default clustering_cutoff=0.5 setting:
Plot a single instance
Note that there is a strong similarity between the explanation from the Independent masker above and the Partition masker here. In general the distinctions between these methods for tabular data are
not large, though the Partition masker allows for much faster runtime and potentially more realistic manipulations of the model inputs (since groups of clustered features are masked/unmasked | {"url":"https://shap.readthedocs.io/en/latest/example_notebooks/api_examples/explainers/Exact.html","timestamp":"2024-11-05T20:11:30Z","content_type":"text/html","content_length":"23304","record_id":"<urn:uuid:ece5f93b-015a-4051-ae8f-2fab7258f699>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00062.warc.gz"} |
Cube Root Calculator Online | Kody Tools
Welcome to our convenient online Cube Root Calculator! This handy tool is designed to effortlessly calculate the cube root of any number you input. It's incredibly user-friendly, requiring no
advanced configuration or customization options. Whether you're a student delving into the world of mathematics or a professional seeking accurate calculations, our Cube Root Calculator is here to
simplify your journey. Let's dive into the practical applications of this calculator and discover how it can be of great use in various contexts.
For students, the Cube Root Calculator is an invaluable resource in their mathematical endeavors. Say goodbye to tedious manual calculations and welcome a streamlined approach to finding cube roots.
By instantly obtaining the cube root of a number, students can focus on grasping the core concepts and principles behind cube roots. This powerful tool enables them to solve equations, tackle
geometric problems, and explore the fascinating realm of cubic functions. With the Cube Root Calculator at their disposal, students can develop a solid foundation in mathematics, gaining confidence
to tackle more complex mathematical concepts in the future.
Professionals from a wide range of fields can also reap significant benefits from the Cube Root Calculator. Architects, engineers, and scientists often encounter situations where calculating cube
roots is essential. Whether it's determining dimensions, analyzing volumes, or working with cubic equations, this tool provides a quick and reliable solution. By simplifying the calculation process,
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The Cube Root Calculator is also a valuable tool in the fields of physics and engineering. Researchers conducting scientific experiments often encounter situations where calculating cube roots is
crucial for understanding physical phenomena and determining critical parameters. Whether it involves calculating velocities, analyzing volumes, or exploring cubic relationships, this calculator
offers a precise and efficient solution. Researchers can focus on interpreting data, drawing conclusions, and advancing scientific knowledge while relying on the calculator for accurate cube root
Moreover, computer programmers and software developers can harness the power of the Cube Root Calculator in their coding endeavors. When designing algorithms, simulations, or mathematical models that
involve cube roots, this tool proves to be a time-saving asset. By incorporating the calculator's functionality into their code, programmers can simplify complex mathematical operations and enhance
the performance of their software applications. The calculator's simplicity and reliability contribute to the overall efficiency and accuracy of the developed software.
In conclusion, the Cube Root Calculator is a user-friendly and powerful tool that benefits students, professionals, researchers, and math enthusiasts alike. By simplifying the calculation process,
enhancing problem-solving capabilities, and saving valuable time and effort, it empowers users to explore the magic of cubes with confidence and ease. Embrace the convenience and accuracy of our Cube
Root Calculator to unlock the mysteries of cube roots and embark on an exciting journey of mathematical exploration. | {"url":"https://www.kodytools.com/cube-root-calculator","timestamp":"2024-11-12T23:05:20Z","content_type":"text/html","content_length":"56496","record_id":"<urn:uuid:8a1bfcba-2a12-4f94-b7c1-7de0892a088a>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00594.warc.gz"} |
How to Convert a TensorFlow Model to ONNX - reason.townHow to Convert a TensorFlow Model to ONNX
How to Convert a TensorFlow Model to ONNX
If you’re working with TensorFlow, you may want to convert your models to the ONNX format. ONNX is a open format for neural networks that allows you to exchange models between different frameworks.
In this blog post, we’ll show you how to convert a TensorFlow model to ONNX.
Checkout this video:
ONNX is an open standard format for representing machine learning models. TensorFlow models can be converted to ONNX with the help of the TensorFlow-ONNX converter. This repository contains a set of
scripts that allow you to convert a TensorFlow model to ONNX.
The conversion process involves three steps:
1. Export the TensorFlow model to a Protobuf file.
2. Convert the Protobuf file to an ONNX file.
3. Validate the ONNX file to make sure that it is convertible and that there are no errors.
What is TensorFlow?
TensorFlow is a powerful open-source software library for data analysis and machine learning. Originally developed by researchers and engineers working on the Google Brain team, TensorFlow has been
used in a variety of applications, including image classification, natural language processing, and predictive modeling.
ONNX (Open Neural Network Exchange) is an open format for representing deep learning models. ONNX provides a way for developers to more easily move models between different frameworks and tools.
There are a few ways to convert a TensorFlow model to ONNX:
1. Use the tf2onnx converter: This converter converts TensorFlow models into the ONNX format.
2. Use the onnxruntime converter: This converter lets you convert your TensorFlow model into the ONNX format and then run it using the onnxruntime inference engine.
3. Use the WinMLTools converter: This toolkit provides a set of utilities for converting machine learning models into the ONNX format.
What is ONNX?
ONNX is an open format developed by Facebook and Microsoft for representing deep learning models. It is designed to work with a number of different deep learning frameworks, including TensorFlow,
PyTorch, Caffe2, and MXNet. ONNX allows models to be transferred between these different frameworks with minimal effort.
In order to convert a TensorFlow model to ONNX, you will need to use the ONNX exporter library. This library can be installed using pip:
pip install onnx-tf
Once the library is installed, you can export your TensorFlow model using the following code:
import onnx-tf
onnx_model = onnx-tf.export_model(model)
Why Convert a TensorFlow Model to ONNX?
There are a number of reasons you might want to convert a TensorFlow model to ONNX:
– To run your model on a different platform than TensorFlow (e.g. Windows, Mac, Linux)
– To use a different version of TensorFlow than what is installed on your platform
– To take advantage of ONNX’s advanced optimization techniques
How to Convert a TensorFlow Model to ONNX?
There are a few steps involved in converting a TensorFlow model to ONNX:
1. Export the model from TensorFlow using the tf.onnx.export_model() function. This will produce an ONNX file containing the model’s graph and weights.
2. Optionally, you can then optimize the model with the onnxruntime.optimize() function. This can improve performance by reducing the size of the model and/or improving execution time on the target
3. Finally, you can use the onnxruntime.evaluate() function to run inference on your model and make predictions.
Tools for Converting a TensorFlow Model to ONNX
There are a few different ways to convert a TensorFlow model to ONNX. You can use the open-source TensorFlow-ONNX converter, or you can use the ONNX Converter for TensorFlow from Microsoft.
The open-source TensorFlow-ONNX converter is still in beta, but it should be able to handle most models. The ONNX Converter for TensorFlow from Microsoft is another option, and it’s a good choice if
you’re using Windows and Visual Studio.
Once you’ve chosen a tool, you’ll need to export your model from TensorFlow. The format of the model will depend on which tool you’re using. For the open-source converter, you’ll need to export your
model as a Protobuf (.pb) file. For the Microsoft converter, you can export your model as a frozen graph (.pb) or HDF5 (.h5) file.
Once you’ve exported your model, you can use the tool of your choice to convert it to ONNX.
Tips for Successfully Converting a TensorFlow Model to ONNX
There are a few things to keep in mind when converting a TensorFlow model to ONNX:
1. Make sure that you have the latest version of TensorFlow installed. The current release is TensorFlow 2.0.0.
2. Use the onnx-tf converter tool to convert your model. This tool can be found here: https://github.com/onnx/onnx-tf
3. Make sure that your input and output nodes have the same names as in your TensorFlow model. Otherwise, the converter tool will not be able to find them and the conversion will fail.
4. Specify the target opset when converting your model. The supported opsets for ONNX are 1-12. By specifying the target opset, you ensure that your model will be compatible with that version of
5. Finally, test your converted model to make sure it works as expected!
-Open the ONNX converter
-Choose your model and input tensor
-Hit “convert”
You now have an ONNX model that you can use with a variety of tools and frameworks.
-TensorFlow: https://www.tensorflow.org/
-ONNX: https://onnx.ai/
There are many reasons you may want to convert a TensorFlow model to ONNX format. Perhaps you want to run inference on a device that doesn’t have TensorFlow installed, or maybe you want to share your
model with someone who doesn’t have TensorFlow. Whatever your reasons, this guide will show you how to convert a TensorFlow model to ONNX format.
First, let’s take a look at the resources you’ll need:
-TensorFlow: https://www.tensorflow.org/
-ONNX: https://onnx.ai/
The conversion process is actually quite simple. We start by exporting our TensorFlow model to a protobuf file, then we use the onnx-tf converter to convert the protobuf file to ONNX format. Let’s
take a look at how to do each of these steps.
Exporting a TensorFlow model to protobuf format can be done with just a few lines of code. We start by creating an instance of the Exporter class, then we call the export_graph method passing in the
absolute path of where we want our protobuf file saved, and finally we pass in our TensorFlow graph object:
exporter = tf2onnx.Exporter(graph)
exporter.export_graph(‘/path/to/file’, graph)
About the Author
Hi, I’m Yaman Umuroglu, a software engineer on the ONNX Runtime team at Microsoft. ONNX Runtime is a performance-focused inference engine for ONNX (Open Neural Network Exchange) models. It’s
cross-platform and supports hardware acceleration with NVIDIA CUDA and TensorRT on Linux and Windows. | {"url":"https://reason.town/tensorflow-model-to-onnx/","timestamp":"2024-11-14T18:09:33Z","content_type":"text/html","content_length":"97850","record_id":"<urn:uuid:11be8a37-332d-4442-801f-30da903556a9>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00691.warc.gz"} |
Difference between SOLVE and solve
05-21-2015, 08:47 AM
Post: #1
akmon Posts: 199
Member Joined: Jun 2014
Difference between SOLVE and solve
Hello again.
I would like to understand definitly the differences between commands in upper or lowercase.
In this case, I search "solve" in the catalog, I type solve(x^2+x+1) in CAS mode and I get {1/2*(SQR(3)*i-1), 1/2*(-SRQ(3)*i-1)}. If I type solve(X^2+X+1) in HOME I get the same result (X in
SOLVE in uppercase does not exist in catalog, but, when I manually type it in HOME mode I don´t get Syntax error. It runs and outputs {Error: Extremum found, -0,4921875}.
Even if I type SOLVE in CAS mode, I get ["Error: Extremum found"].
So resuming all the combinations:
solve is a CAS command that works on CAS or HOME mode, I only have to type variables in upper or lowercase.
SOLVE does not exists in catalog, but the calculator recognizes it and it works, with differents results.
So, how is the way that solve or SOLVE works? I have used SOLVE in a little program (IRR, see thread) and it works flawlessly, but I don´t know when is more suitable using solve or SOLVE.
I suppose this example can be applied to other lowercase commands.
Thank you.
05-21-2015, 03:10 PM
Post: #2
Tim Wessman Posts: 2,293
Senior Member Joined: Dec 2013
RE: Difference between SOLVE and solve
SOLVE is an app function in the solve application (toolbox->app->solve). It is the same thing as FNROOT in home and calls the classic HP numerical solver.
Although I work for HP, the views and opinions I post here are my own.
05-21-2015, 04:21 PM
Post: #3
akmon Posts: 199
Member Joined: Jun 2014
RE: Difference between SOLVE and solve
Thank you Tim. Understood. I thought app commands were present in catalog, the same as on hp49 when you installed a library.
05-21-2015, 04:41 PM
Post: #4
Tim Wessman Posts: 2,293
Senior Member Joined: Dec 2013
RE: Difference between SOLVE and solve
No, they are not. I'd like to change that in the future though.
Although I work for HP, the views and opinions I post here are my own.
05-21-2015, 04:46 PM
Post: #5
Marcio Posts: 438
Senior Member Joined: Feb 2015
RE: Difference between SOLVE and solve
05-21-2015, 05:01 PM
Post: #6
salvomic Posts: 1,396
Senior Member Joined: Jan 2015
RE: Difference between SOLVE and solve
as you know, Akmon, I'm using these command to calculate IRR, and I get different results with the same data:
Solve.SOLVE() output only one result even if there are more results;
solve() output a list
fsolve() doesn't go in Home
Still I'm not decided what to do use, 'cause sometimes I'll would have complete results, sometimes only one...
And this for my program about IRR is a problem: the program should decide which of them use based on ...type of results, not so easy...
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05-21-2015, 07:32 PM
Post: #7
Marcio Posts: 438
Senior Member Joined: Feb 2015
RE: Difference between SOLVE and solve
(05-21-2015 05:01 PM)salvomic Wrote: as you know, Akmon, I'm using these command to calculate IRR, and I get different results with the same data:
Solve.SOLVE() output only one result even if there are more results;
solve() output a list
fsolve() doesn't go in Home
Still I'm not decided what to do use, 'cause sometimes I'll would have complete results, sometimes only one...
And this for my program about IRR is a problem: the program should decide which of them use based on ...type of results, not so easy...
SOLVE, with capital letters, never failed me.
05-21-2015, 08:03 PM
Post: #8
akmon Posts: 199
Member Joined: Jun 2014
RE: Difference between SOLVE and solve
To add more wood to the fire.
Look at this screenshot. If I type SOLVE in the emulator it outputs a syntax error. If I type Solve.SOLVE it works. But in the physical calculator, if I type SOLVE it works well. What the hell is
05-21-2015, 08:13 PM
Post: #9
Tim Wessman Posts: 2,293
Senior Member Joined: Dec 2013
RE: Difference between SOLVE and solve
Which application is running the the calculator? What does it say in the menu bar?
Although I work for HP, the views and opinions I post here are my own.
05-21-2015, 10:00 PM
Post: #10
akmon Posts: 199
Member Joined: Jun 2014
RE: Difference between SOLVE and solve
Excuse me Tim. I don´t understand exactly what you asked. I have reseted the emulator to factory settings, but the syntax error still appears. It´s strange. It works on the physical calculator, and
the firmware is exactly the same. Can you try on yours?
05-21-2015, 10:04 PM
Post: #11
salvomic Posts: 1,396
Senior Member Joined: Jan 2015
RE: Difference between SOLVE and solve
(05-21-2015 07:32 PM)Marcio Wrote: SOLVE, with capital letters, never failed me.
no, dear Marcio, here it doesn't fail, but my particular example, IRR some times gives more results (one of sign's change), and I noted SOLVE gave one (the principal) result, solve() a list with all
+ one, the first, that in case of many results it erratic or not financially relevant...)
I meant this only.
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05-21-2015, 10:26 PM
(This post was last modified: 05-21-2015 10:28 PM by Tim Wessman.)
Post: #12
Tim Wessman Posts: 2,293
Senior Member Joined: Dec 2013
RE: Difference between SOLVE and solve
(05-21-2015 10:00 PM)akmon Wrote: Excuse me Tim. I don´t understand exactly what you asked. I have reseted the emulator to factory settings, but the syntax error still appears. It´s strange. It
works on the physical calculator, and the firmware is exactly the same. Can you try on yours?
SOLVE is an app function in the solve app. If the solve app is not the active app, it will error like you report. So unless you see SOLVE at the top of your screen, you are running another
application. If you press APPS-> select solve app, then home and try, does it work?
Although I work for HP, the views and opinions I post here are my own.
05-21-2015, 10:52 PM
(This post was last modified: 05-21-2015 10:55 PM by akmon.)
Post: #13
akmon Posts: 199
Member Joined: Jun 2014
RE: Difference between SOLVE and solve
Bingo! It works in SOLVE app on the emulator. I´ve checked the calculator and it was running Solve app. That´s the reason of being working there, too. Conclusion, It´s better to use Solve.SOLVE
command instead of SOLVE. only
Thank you very much Tim, you nailed it and I´ve learnt an useful lesson.
05-22-2015, 05:30 AM
Post: #14
cyrille de brébisson Posts: 1,047
Senior Member Joined: Dec 2013
RE: Difference between SOLVE and solve
In the case of IRR, where you are doing purely numerical calculations, Solve.SOLVE is the best solution.
If you are using the SOLVE command in an app program that belong to an app that is based on Solve, there is no need to do Solve.SOLVE, but SOLVE is enough.
If you are doing it from the command line and the current app is a copy of Solve, SOLVE is also enough.
To do something generic that will work everywhere, you need to do Solve.SOLVE.
solve and fsolve are CAS functions and are designed to work more on CAS objects. Since the equation that you are trying to solve is a numerical function, using CAS functions will be very inefficient
as you will be repetitively having to convert from CAS to non CAS.
05-22-2015, 07:53 AM
Post: #15
salvomic Posts: 1,396
Senior Member Joined: Jan 2015
RE: Difference between SOLVE and solve
(05-22-2015 05:30 AM)cyrille de brébisson Wrote: fsolve are CAS functions and are designed to work more on CAS objects. Since the equation that you are trying to solve is a numerical function,
using CAS functions will be very inefficient as you will be repetitively having to convert from CAS to non CAS.
yes, I know, but in the program about IRR I needed to explore every result, not only the (only) numeric given by Solve.SOLVE...
Mostly, well suited problems for IRR give only this result, and Solve.SOLVE is the better solution, but to test the problem I used some series of flows "at limit", where or IRR gives more solutions,
or it's needed MIRR and not IRR... So, both solution "solve" the problem, depending on what happen...
However, generally we treat (about IRR) lists (or matrix) with relatively few items (10-100 max), so inefficiency is a problem that's "acceptable"...
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05-22-2015, 09:19 AM
Post: #16
toml_12953 Posts: 2,192
Senior Member Joined: Dec 2013
RE: Difference between SOLVE and solve
Well, I'm glad that's SOLVEd (solve.SOLVEd? solved?)
Tom L
Cui bono?
05-22-2015, 09:54 AM
(This post was last modified: 05-22-2015 09:59 AM by Marcio.)
Post: #17
Marcio Posts: 438
Senior Member Joined: Feb 2015
RE: Difference between SOLVE and solve
(05-22-2015 09:19 AM)toml_12953 Wrote: Well, I'm glad that's SOLVEd (solve.SOLVEd? solved?)
Tom, if you can't solve things with solve.SOLVE which is suppose to solve independently of the solve app, things are not solved!
05-22-2015, 10:30 AM
Post: #18
salvomic Posts: 1,396
Senior Member Joined: Jan 2015
RE: Difference between SOLVE and solve
(05-22-2015 09:19 AM)toml_12953 Wrote: Well, I'm glad that's SOLVEd (solve.SOLVEd? solved?)
Tom, if you can't solve things with solve.SOLVE which is suppose to solve independently of the solve app, things are not solved!
indeed, we are solving the solution, to find a true Solve or solve the solution..., better so, than without a SOLVE method
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IEC 60050 - International Electrotechnical Vocabulary
logarithm of the number of events in a finite set of n mutually exclusive events
H[0]=log n
EXAMPLE The decision content of a set of three events is:
$H 0 =( lb 3 )Sh≈1,585 Sh =( ln 3 )nat≈1,099 nat =( lg 3 )Hart≈0,477 Hart MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX
garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz 3bqee0evGueE0jxyaibaieYdNi=xH8yiVC0xbbL8F4rqqrFfpeea0x e9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKk Fr0xfr=xfr=
xb9adbaGaaiGadiWaamaaceGaaqaacaqbaaGceaqabe aajugibiaadIeakmaaBaaaleaajugWaiaabcdaaSqabaGccaqG9aqc fa4aaeWaaOqaaKqzaeGaaeiBaiaabkgacaqGGaGaae4maaGccaGLOa
GaayzkaaqcLbqacaqGtbGaaeiAaiabgIKi7kaabgdacaqGSaGaaeyn aiaabIdacaqG1aGaaGjbVlaabofacaqGObaakeaajugabiaaywW7ca aMe8UaaGjcVlaab2dajuaGdaqadaGcbaqcLbqacaqGSbGaaeOBaiaa
bccacaqGZaaakiaawIcacaGLPaaajugabiaab6gacaqGHbGaaeiDai abgIKi7kaabgdacaqGSaGaaeimaiaabMdacaqG5aGaaGjbVlaab6ga caqGHbGaaeiDaaGcbaqcLbqacaaMf8UaaGjbVlaayIW7caqG9aqcfa
4aaeWaaOqaaKqzaeGaaeiBaiaabEgacaqGGaGaae4maaGccaGLOaGa ayzkaaqcLbqacaqGibGaaeyyaiaabkhacaqG0bGaeyisISRaaeimai aabYcacaqG0aGaae4naiaabEdacaaMe8UaaeisaiaabggacaqGYbGa aeiDaaaaaa@7EED@$
Note 1 to entry: The base of the logarithm determines the unit used. Commonly used units are: shannon (symbol Sh) for logarithms of base 2, natural unit (symbol nat) for logarithms of base e, hartley
(symbol Hart) for logarithms of base 10.
Conversion table:
$1 Sh ≈ 0,693 nat ≈ 0,301 Hart 1 nat ≈ 1,433 Sh ≈ 0,434 Hart 1 Hart ≈ 3,322 Sh ≈ 2,303 nat MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX
garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz 3bqee0evGueE0jxyaibaiGc9aspC0FXdbbc9asFfpec8Eeeu0lXdbb a9frFj0xb9Lqpepeea0xd9s8qiYRWxGi6xij=hbba9q8aq0=yq=He9 q8qiLsFr0=vr0=
vr0db8meaabaGacmaadiWaaiWabaabaiaafaaake aafaqaaeWafaaaaeaaqaaaaaaaaaWdbiaaigdacaGGGcGaaGjcVlaa bofacaqGObaapaqaa8qacqGHijYUa8aabaWdbiaaicdacaGGSaGaaG
OnaiaaiMdacaaIZaGaaiiOaiaayIW7caqGUbGaaeyyaiaabshaa8aa baWdbiabgIKi7cWdaeaapeGaaGimaiaacYcacaaIZaGaaGimaiaaig dacaGGGcGaaGjcVlaabIeacaqGHbGaaeOCaiaabshaa8aabaWdbiaa
igdacaGGGcGaaGjcVlaab6gacaqGHbGaaeiDaaWdaeaapeGaeyisIS lapaqaa8qacaaIXaGaaiilaiaaisdacaaIZaGaaG4maiaacckacaaM i8Uaae4uaiaabIgaa8aabaWdbiabgIKi7cWdaeaapeGaaGimaiaacY
cacaaI0aGaaG4maiaaisdacaaMi8UaaiiOaiaabIeacaqGHbGaaeOC aiaabshaa8aabaWdbiaaigdacaGGGcGaaGjcVlaabIeacaqGHbGaae OCaiaabshaa8aabaWdbiabgIKi7cWdaeaapeGaaG4maiaacYcacaaI
ZaGaaGOmaiaaikdacaGGGcGaaGjcVlaabofacaqGObaapaqaa8qacq GHijYUa8aabaWdbiaaikdacaGGSaGaaG4maiaaicdacaaIZaGaaiiO aiaayIW7caqGUbGaaeyyaiaabshaaaaaaa@91F0@$
Note 2 to entry: The decision content is independent of the probabilities of the occurrence of the events.
Note 3 to entry: The number of decisions needed to select a specific event out of a finite set of mutually exclusive events equals the smallest integer which is greater than or equal to the decision
content when the base of the logarithm is the number of choices on each decision.
Note 4 to entry: When the same integer base is used for the same number of events, the decision content equals the maximum entropy. | {"url":"https://electropedia.org/iev/iev.nsf/IEVref_xref/en:171-07-10","timestamp":"2024-11-09T07:47:38Z","content_type":"text/html","content_length":"19131","record_id":"<urn:uuid:7c0381c7-e634-416e-907a-1cb4b55fca28>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00409.warc.gz"} |
Example of mixture distributions
It is well known that adult male heights follow a normal (Gaussian) distribution. The same is true of adult female heights. What does the distribution of adults in general look like? There are
several qualitatively different answers depending on minor changes to some basic assumptions.
First, assume adult male heights are normally distributed with mean 70 inches and standard deviation 3 inches. Assume also that adult female heights are normally distributed with mean 64 inches and
standard deviation 3 inches. These numbers are approximately correct for Americans; the averages vary by a few inches from country to country.
Under these assumptions, the probability density for a woman’s height is as follows.
The corresponding density for men is the same, shifted to the right.
If we assume an equal number of men and women, the probability density for the height of an adult without regard to sex is given below.
Note that this density is not Gaussian at all. Instead, it is very flat on top. You might reason that since the average of normal random variables is normal, adult heights should be normal. But we
don’t have an average, we have a mixture. The density for the general adult population is a mixture of the male and female distributions. If you assigned a height to married couples as an average of
the husband’s height and the wife’s height, the resulting value would be an average than a mixture and would follow a normal density.
The flat top of the density above is not typical. If you have two populations with the same standard deviation and take a 50-50 mixture, the mixture will be symmetric about the average of the two
population means. The second derivative of the density at the point of symmetry will be negative if the two population means are less than two standard deviations apart. For example, if the standard
deviation had been 3.2 rather than 3.0, the two population means, 64 inches and 70 inches, would be less than two standard deviations apart, and the density of the mixture would be rounded at the
mode of 67 inches.
The second derivative of the density will be positive in the middle if the two population means are more than two standard deviations apart. For example, if the standard deviations had been 2.8, the
population means would be more than two standard deviations apart and the middle value of 67 inches would be a local minimum. (The value of 2.8 may be fairly accurate. One website I found said the
standard deviation is 2.8, but I have no idea whether that site was reliable.)
If the two population means are close to two standard deviations apart, the mixture density is still approximately flat on top, flatter than a normal density. But only when the population means are
exactly two standard deviations apart is the mixture distribution completely flat on top, i.e. only then is the second derivative zero in the middle.
The calculations above have assumed the proportions of men and women were exactly equal. If we assume women form 51% of the population, then the density becomes slightly asymmetrical.
After this page was first posted, I found out about a couple related references.
The American Statistician had an article Is Human Height Bimodal? on this topic in 2002.
That same year Andrew Gelman and Deborah Nolan published Teaching Statistics: A Bag of Tricks which also deals with this subject. Gelman and Nolan point out that the variance of men’s heights is
slightly larger than that for women. If we assume a 50-50 split of men and women, but assume male heights have a standard deviation of 3 inches while female heights have a standard deviation of 2.8
inches, this tilts the graph to the left more than assuming equal variance but unequal proportions above.
See also Why heights are normally distributed. | {"url":"https://www.johndcook.com/blog/mixture_distribution/","timestamp":"2024-11-06T09:23:27Z","content_type":"text/html","content_length":"49431","record_id":"<urn:uuid:c6eccd7c-cef8-4275-a66b-0f7e72538b03>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00677.warc.gz"} |
Bellman Ford Algorithm | KUK CSE | LearnLoner
The Bellman-Ford Algorithm is a popular algorithm for computing the shortest paths from a single source vertex to all the other vertices in a weighted digraph. It is widely used in network routing
protocols and is also useful in other applications such as detecting negative cycles in a graph.
Here are the steps involved in the Bellman-Ford Algorithm:
• Initialize the distance to the source vertex as 0 and the distance to all other vertices as infinity.
• Repeat the following step V – 1 times, where V is the total number of vertices in the graph:
a. For each edge (u, v) in the graph, calculate the distance to vertex v as the minimum of its current distance and the distance to vertex u plus the weight of the edge (u, v).
• Check for the presence of negative cycles in the graph by repeating step 2 and checking if any distance is updated. If a distance is updated, then there is a negative cycle in the graph.
• Return the distances to all the vertices as the output of the algorithm.
Example of finding the shortest paths from vertex A to all the other vertices in the following weighted digraph using the Bellman-Ford Algorithm:
-1 4
A --------> B
| | \
| 2 | 3
v v v
C --------> D ---> E
2 -3
1. Initialize the distance to vertex A as 0 and the distance to all other vertices as infinity: dist[A] = 0, dist[B] = dist[C] = dist[D] = dist[E] = infinity.
2. Repeat the following step 4 times: a. For each edge (u, v) in the graph, calculate the distance to vertex v as the minimum of its current distance and the distance to vertex u plus the weight of
the edge (u, v).
□ For edge (A, B): dist[B] = min(dist[B], dist[A] + weight(A, B)) = min(infinity, 0 + 4) = 4.
□ For edge (A, C): dist[C] = min(dist[C], dist[A] + weight(A, C)) = min(infinity, 0 + 2) = 2.
□ For edge (B, D): dist[D] = min(dist[D], dist[B] + weight(B, D)) = min(infinity, 4 + 3) = 7.
□ For edge (B, E): dist[E] = min(dist[E], dist[B] + weight(B, E)) = min(infinity, 4 + -3) = 1.
□ For edge (C, D): dist[D] = min(dist[D], dist[C] + weight(C, D)) = min(7, 2 + 2) = 4.
□ For edge (D, E): dist[E] = min(dist[E], dist[D] + weight(D, E)) = min(1, 7 + -3) = 1.
b. The distances after each iteration are:
□ dist[A] = 0, dist[B] = 4, dist[C] = 2, dist[D] = 4, dist[E] = 1.
3. Check for the presence of negative cycles by repeating step 2 and checking if any distance is updated. Since no distance is updated in this step, there is no negative cycle in the graph.
4. Return the distances to all the vertices as the output of the algorithm: dist[A] = 0, dist[B] = 4, dist[C] =
(a) What do you understand by a recurrence ? write a detailed note on the substitution method for solving a recurrence.
(b) Discuss the concept of asymptotic notation and its properties.
What do you understand by time complexity ? What is big O notation ? Write the Quick sort algorithm and discuss its complexity.
What do you understand by dynamic programming ? discuss using the example of matrix chain multiplication.
What is Huffman code ? discuss the greedy algorithm for constructing the Huffman code.
What is minimum spanning tree ? discuss the steps for finding minimum spanning tree using Prim’s Algorithm
Discuss Bellman-Ford algorithm that compute shortest paths from a single source vertex to all of the other vertices in a weighted diagraph
What is bitonic sequence ? discuss bitonic sort algorithm and it’s time complexity.
What is sorting Network ? Discuss the structure of bubble sorting network. | {"url":"https://learnloner.com/bellman-ford-algorithm/","timestamp":"2024-11-03T20:19:43Z","content_type":"text/html","content_length":"311144","record_id":"<urn:uuid:1f00c78e-8dbb-42c5-9001-d0fb2626c9f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00708.warc.gz"} |
Formative Assessment Lessons
Interpreting Distance–Time Graphs
Mathematical goals
This lesson unit is intended to help you assess how well students are able to interpret distance–time graphs and, in particular, to help you identify students who:
• Interpret distance–time graphs as if they are pictures of situations rather than abstract representations of them.
• Have difficulty relating speeds to slopes of these graphs.
• Before the lesson, students work on a task designed to reveal their current understandings and difficulties. You review their work and create questions for students to answer in order to improve
their solutions.
• A whole-class introduction provides students with guidance on how to work through the first task. Students then work in small groups, matching verbal interpretations with graphs. As they do this,
they translate between words and graphical features, and begin to link the representations.
• This is followed by a whole-class discussion about applying realistic data to a graph.
• Students next work in small groups, matching tables of data to the existing matched pairs of cards. They then explain their reasoning to another group of students.
• In a final whole-class discussion, students draw their own graphs from verbal interpretations.
• Finally, students return to their original task and try to improve their individual responses.
Materials required
• Each student will need two copies of the assessment task Journey to the Bus Stop, a mini-whiteboard, a pen, and an eraser.
• Each small group of students will need copies of the cut-up Card Set A: Distance–Time Graphs, Card Set B: Interpretations, Card Set C: Tables of Data, a large sheet of paper, and a glue stick.
• A supply of graph paper to give to students who request it. There are some projector resources.
Time needed
15 minutes before the lesson, a 100-minute lesson (or split into two shorter lessons), and 10 minutes in a following lesson (or homework). Timings are approximate and will depend on the needs of the
Lesson Type
Mathematical Practices
This lesson involves a range of mathematical practices from the standards, with emphasis on:
Mathematical Content Standards
This lesson asks students to select and apply mathematical content from across the grades, including the content standards: | {"url":"https://www.map.mathshell.org/lessons.php?unit=8225&collection=8&redir=1","timestamp":"2024-11-08T17:35:36Z","content_type":"text/html","content_length":"20558","record_id":"<urn:uuid:69d87777-377a-494f-bef3-804ca16ca016>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00101.warc.gz"} |
How to Plot Pandas Dataframe Using Sympy?
To plot a pandas dataframe using sympy, you can start by creating a matplotlib figure and axis. Then, you can use the plot method on the dataframe to generate the plot. You may need to specify the
columns you want to plot by passing them as arguments to the plot method. Additionally, you can customize the plot by setting labels, titles, and other formatting options. Finally, you can show the
plot by calling plt.show().
What is the use of subplot in plotting pandas dataframes?
Subplots in plotting pandas dataframes are used to display multiple plots in a single figure. This can be useful for comparing different aspects of the data or for visualizing different variables in
relation to each other. By using subplots, you can create a more organized and comprehensive visualization of the data, making it easier to interpret and analyze.
What is the difference between pandas plot and sympy plot?
The main difference between pandas plot and sympy plot is the purpose and functionality of the two libraries.
Pandas plot is a part of the pandas library, which is used for data manipulation and analysis in Python. The plot method in pandas is used to quickly visualize data stored in a pandas DataFrame or
Series. It provides an easy way to create various types of plots such as line plots, bar plots, histograms, scatter plots, etc., directly from the data without needing to use external plotting
On the other hand, sympy plot is a part of the sympy library, which is used for symbolic mathematics in Python. The plot function in sympy is used to visualize mathematical expressions, equations,
and functions. It allows users to plot mathematical functions and equations using symbolic notation rather than working directly with data.
In summary, pandas plot is primarily used for visualizing data stored in pandas data structures, while sympy plot is used for plotting mathematical expressions and functions.
How to add a title and labels to a pandas dataframe plot using sympy?
To add a title and labels to a pandas dataframe plot using sympy, you can use the plot method provided by pandas dataframe along with the pyplot module from matplotlib to customize the plot. Here's
an example:
1 import pandas as pd
2 import matplotlib.pyplot as plt
4 # create a pandas dataframe
5 df = pd.DataFrame({
6 'A': [1, 2, 3, 4, 5],
7 'B': [10, 20, 30, 40, 50]
8 })
10 # plot the dataframe
11 ax = df.plot(kind='bar')
13 # add title and labels
14 ax.set_title('Title of the Plot')
15 ax.set_xlabel('X-axis Label')
16 ax.set_ylabel('Y-axis Label')
18 # show the plot
19 plt.show()
In this example, we first create a pandas dataframe df with some sample data. We then use the plot method with kind='bar' to create a bar plot of the dataframe. We then use the set_title, set_xlabel,
and set_ylabel methods to add a title, x-axis label, and y-axis label to the plot, respectively. Finally, we use plt.show() to display the plot with the added title and labels.
What is the advantage of using symbolic computation for plotting pandas dataframes?
One advantage of using symbolic computation for plotting pandas dataframes is that it allows for the creation of more complex and customizable plots. Symbolic computation libraries like matplotlib or
seaborn provide a wide range of options for customizing plot appearance, adding annotations, combining multiple plots, and creating interactive visualizations. This can be particularly useful for
data analysis and visualization tasks that require a high degree of customization or that involve large or complex datasets. | {"url":"https://japblog.chickenkiller.com/blog/how-to-plot-pandas-dataframe-using-sympy","timestamp":"2024-11-12T16:21:14Z","content_type":"text/html","content_length":"145693","record_id":"<urn:uuid:7c137492-9ef1-4170-833f-8fea413f359e>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00698.warc.gz"} |
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Pseudo-Boolean functions valued on hypershere
Fixing some ordering on the domain of real-valued functions of n Boolean variables (i. e. pseudo-Boolean functions) we can identify these functions (or rather tables of their values) with vectors in
the Euclidean space R 2 n of dimension 2 n . From a perspective of the Boolean function theory the integer-valued pseudo-Boolean functions are of special interest. It is due to the fact that the
Walsh–Hadamard transform of a Boolean function gives the integer-valued pseudo-Boolean function that identically corresponds to the Boolean function. If we represent such pseudo-Boolean functions by
points of Euclidean space then all of them appear to be placed on the (2n−1)-dimensional sphere with radius 2 n . Previously the mapping of the n-variables Boolean function set on the Euclidean
hypersphere in R 2 n was already studied. This paper represents an attempt to extend the results obtained in those settings to the subset of pseudo-Boolean functions corresponding to the points on
the hypersphere. In particular, we consider new concepts of curvature and nonlinearity of such pseudo-Boolean functions. We set relations between them and express curvature value via some metric
parameters related to the described geometric representation of the pseudo-Boolean functions. One of the aims of this investigation is to work out an approach to bounding maximum nonlinearity of
Boolean functions with odd number of variables.
Logachev O. A., Fedorov S. N., Yashchenko V. V. Boolean functions as points on the hypersphere in the Euclidean space // Discrete Mathematics and Applications. 2019. Vol. 29, no. 2. P. 89–101.
Logachev O. A., Salnikov A. A., Yashchenko V. V. Boolean functions in coding theory and cryptography. Providence (Rhode Island, USA) : American Mathematical Society, 2011.
Tokareva N. N. Nonlinear Boolean functions : bent functions and their generalizations. Saarbrücken (Germany) : LAMBERT Academic Publishing, 2011 [in Russian].
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How to build a modular arithmetic library in PythonHow to build a modular arithmetic library in Python - HedgeDoc
<center> # How to build a modular arithmetic library in Python <big> **We’ll use class operator overloading and redefining NumPy’s built-in functions to solve the Lights Out Game.** </big> *Written
by Alejandro Sánchez Yalí. Originally published 2022-10-05 on the [Monadical blog](https://monadical.com/blog.html).* </center> Years ago, I was building my version of the [Lights Out Game](https://
www.lightsout.ir/) using Python. For those who are unfamiliar with the original game, it basically consists of a grid of $5$-by-$5$ lights. When the game starts, a random number or a stored pattern
of lights is switched on. Pressing any one of the lights will toggle it and the four adjacent lights. The goal is to turn off all the lights with the fewest number of keystrokes possible.  <center> Image [source](http://cau.ac.kr/~mhhgtx/courses/LinearAlgebra/references/MadsenLightsOut.pdf): Madsen,
2010. </center> In this game, a change of state is represented by a change in colour. In the original game, the states only change between two states (two colours). When creating my version of the
game, I wanted to see if the states could change between more than two states demonstrating a cyclic sequence of states. Mathematically, the game can be modeled as a $5$-by-$5$ matrix ([Lights Out:
Solutions Using Linear Algebra](http://cau.ac.kr/~mhhgtx/courses/LinearAlgebra/references/MadsenLightsOut.pdf)) where each entry represents the state of a light bulb. Modular arithmetic is an
excellent way to find the winning strategy because the state of each light bulb occurs cyclically. Plus, in this case, the usual matrix operations are inefficient due to their unbounded nature. What
that means is that it’s possible to get results that are outside the range of the game states, such as decimal numbers or very large numbers that don’t match the ‘on’, ‘off’, or other states in the
game. This is article one of two. In this blog, I'll demonstrate how I have successfully implemented modular arithmetic in Python. In the second post, I'll demonstrate how I used modular arithmetic
to model the Lights Out Game's state change as a cyclic sequence. In doing so, we'll learn how to: * use matrix algebra in modular arithmetic * use concepts from graph theory to determine when a
Lights Out Game is solvable. On the programming side, we'll learn how to: * articulate these mathematical concepts to extend Python's capabilities * use operator overloading and built-in functions to
redefine NumPy’s universal operators. By the end of this tutorial, we’ll have a Python library with the ability to perform computation using modular arithmetic. This tool can effectively be used in
other projects that are related to cryptography, graphs, modeling of cyclic sequence processes, etc. ## Problem with NumPy Before we start our tutorial, I’d like to highlight one of the problems I
encountered with NumPy while building my game. NumPy doesn't have support for performing matrix calculations with modular arithmetic, and this was exactly the type of calculation I needed to make.
More specifically, I needed to solve the computation of inverse matrices in modular arithmetic in any modulo. In this case, using only the native Python mod operator (`%`) was not going to be enough
to perform this task. Let's see why in the simple example below. If we have, for example: ```python import numpy as np a = np.array([[1, 2], [3, 8]]) % 7 a ``` ```python >>> array([[1, 2], [ 3, 1]])
``` and calculate the inverse using NumPy, we get: ```python np.linalg.inv(a) ``` ```python >>> array([[-0.2, 0.4], [ 0.6, -0.2]]) ``` This matrix is not the inverse in modular arithmetic with modulo
$7$, mainly because the result is in decimals and not integers. Another way we could try is: ```python np.linalg.inv(a) % 7 ``` ```python >>> array([[6.8, 0.4], [0.6, 6.8]]) ``` Again the result is
incorrect because the expected result in modular arithmetic modulo $7$ is: ```python >>> array([[4 6] [2 4]]) ``` We encountered this problem because NumPy does not make use of modular arithmetic to
perform its internal calculations. We’ll later solve this problem in step 3 of this tutorial by overloading the operator and redefine NumPy’s universal operator. If you don’t understand modular
arithmetic very well, then this problem could be confusing. So let’s take a moment to learn the basics of modular arithmetic in the next section before we dive into our tutorial. ## Introduction to
modular arithmetic Before building our library, it’s important to understand what modular arithmetic (also called clock arithmetic) is. If you’re already familiar with modular arithmetic, feel free
to skip to the [next section](https://docs.monadical.com/asanchezyali-modular-arithmetic?view#Tutorial-How-to-build-a-modular-arithmetic-library-in-Python). In mathematics, modular arithmetic is a
system of arithmetic for integers that deal primarily with operations and applications regarding remainders. In this system, numbers “cycle” or repeat when reaching a certain value, called the
**modulo**. Let's see what this means by referring to a tool we use every day, a clock! Let’s look at the $12$-hour analog clock below. Suppose the clock reads $5$ o’clock. After $3$ hours the clock
would read $8$ o’clock. This is determined by simply adding $5 + 3 = 8$. <center> Image [source]
(https://www.cemc.uwaterloo.ca/events/mathcircles/2019-20/Fall/Junior78_Nov12_Soln.pdf): University Waterloo, 2019. </center> Now, what time would it be after $12$ hours? After $12$ hours, it would
be $5$ o’clock again since the clock cycles back to its original position every $12$ hours. However, if we simply add $5$ and $12$, we get $5 + 12 = 17$. But, $17$ isn’t on the clock! What happened?
<center> Image [source](https://www.cemc.uwaterloo.ca/events/mathcircles/2019-20/Fall/
Junior78_Nov12_Soln.pdf): University Waterloo, 2019. </center> On an analog clock, the numbers go from $1$ to $12$, but when we arrive at $13$ o’clock, it appears again as $1$ on the clock. So, $13$
becomes $1$, $14$ becomes $2$, $15$ becomes $3$, and so on. Every time we go past $12$ on the clock, we start counting the hours from $1$ again. Thus, we may view $17$ o’clock as the same as $5$
o’clock. We write this mathematically as: $$17\cong 5\:(mod\;12)$$ We use the modular operator mod to indicate that they mean the same thing on a clock. This means that $17$ o’clock is the same as
$5$ o’clock in a $12$ hour system. The $(mod\; 12)$ indicates that the clock cycles every $12$ hours. Similarly, we can add $12$ hours again to $17$ ($12+17= 29$) to get $29$ o’clock. We still
understand that it is the same as $5$ o’clock. We write this as: $$29\cong 5\:(mod\;12)$$
<center> Image [source](https://www.cemc.uwaterloo.ca/events/mathcircles/2019-20/Fall/Junior78_Nov12_Soln.pdf): University Waterloo, 2019. </center> In general, all those numbers that leave the same
remainder as $17$ divided by $12$, represent the same thing on the clock. In fact, the numbers $... -7, 5, 17, 29, ...$ divided by $12$ leave the remainder $5$. Therefore they all represent $5$
o’clock. This set $\{... -7, 5, 17, 29, ...\}$ is called the congruence class of $5$ modulo $12$, which is usually represented as $5\; (mod\; 12)$. Please note that the $12$ clock hours define $12$
different congruence classes as shown in the following figure: <center> Image [source]
(https://www.cemc.uwaterloo.ca/events/mathcircles/2019-20/Fall/Junior78_Nov12_Soln.pdf): University Waterloo, 2019. </center> Mathematically, we will name the hours of the clock as $\mathbb{Z}/12\
mathbb{Z}$ and list it like this: $$\mathbb{Z}/12\mathbb{Z} =\{0\;(mod\;12), 1\;(mod\;12), 2\;(mod\;12), ..., 11\;(mod\;12)\}$$ The elements of this set are all the possible remainders left by the
integers when divided by $12$. Note that instead of listing $12\; (mod\; 12), 0\; (mod\; 12)$ has been listed because $0\cong 12\; (mod\; 12)$. There are many other things in our lives that repeat or
cycle after a certain amount of time, like days of the week, months in a year, degrees in a circle, or seasons in a year. Modular arithmetic can be used to model any events like this that repeat. If
the length of the cycle is $n$ we refer to it as **modulo** $n$. For example, since a clock cycles every $12$ hours, we refer to it as modulo $12$. In a circle where one full revolution is $360$
degrees, it’s modulo $360$. Since a week has $7$ days, we refer to it as modulo $7$. In general, for a modulo $n$, all the remainders can be listed as the elements of the set: $$\mathbb{Z}/n\mathbb
{Z} =\{0\;(mod\;12), 1\;(mod\;12), 2\;(mod\;12), ..., n-1\;(mod\;n)\}$$ For the Lights Out Game, each bulb will be modeled as a light that switches between a finite list of colors in a cyclic
sequence. Therefore, we can use modular arithmetic $\mathbb{Z}/n\mathbb{Z}$ to tell the computer what state each of the bulbs is in. As we saw in the clock example (in general in $\mathbb{Z}/n\mathbb
{Z}$), it’s possible to add hours together if we take into account the cycle of the hours. For example, if we want to calculate $(13 + 15)\; (mod\; 12)$, we’ll add $13$ and $15$ together to get $28$.
Then, we find that the number $12$ divides into $28$ a total of $2$ times with $4$ left over. We can write this as: $$(13 + 15)\cong 28 \cong 4\;(mod\;12).$$ Alternatively, we can find the remainder
of $13$ and $15$ separately. This makes calculations easier because we are dealing with smaller numbers. Of course, this is even more useful when dealing with bigger numbers. For example, first we
note that: $$13\;(mod\;12)\cong 1 \text{ and } 15 \cong 4\;(mod\;12).$$ Therefore, $$(13 + 15)\;(mod\;12)\cong 1 + 3 \cong 4\;(mod\;12).$$ Similarly, we can also do subtraction and multiplication. In
general, addition, subtraction and multiplication are defined over $\mathbb{Z}/n\mathbb{Z}$ by the following formulae: 1. $a\;(mod\;n) \pm b\;(mod\;n) \cong (a\pm b)\;(mod\;n)$ 2. $a\;(mod\;n) \times
b\;(mod\;n) \cong (a\times b)\;(mod\;n)$ Moreover, if $n$ is prime, in the set $\mathbb{Z}/n\mathbb{Z}$, we can calculate the divisions using the [Euclidean algorithm](https://en.wikipedia.org/wiki/
Euclidean_algorithm). With this introduction to modular arithmetic, I hope that you have a better understanding of how it works. Now, let's look at how I used modular arithmetic to build a library
that allows me to model the lights in Lights Out as a cyclic sequence of states. Let’s get started! ## Tutorial: How to build a modular arithmetic library in Python For this tutorial we’ll need: 1.
Python 3.10 2. Familiarity with [operator overloading](https://docs.python.org/3/reference/datamodel.html?highlight=overloading#special-method-names) in Python, or read my [article](https://
monadical.com/posts/operator-overloading-in-python.html) on the topic. 3. A couple of beers, and a lot of patience. ### Step 1: Build a Python class to manipulate the elements of $\mathbb{Z}/n\mathbb
{Z}$ To start, let’s define a class named `Zmodn` (integers module $n$): ```python class Zmodn: def __init__(self, module: int = 2): self.representative = None self.module = module def __call__(self,
integer: int): congruence_class = self.__class__(self.module) congruence_class.representative = integer % self.module return congruence_class def __repr__(self): return f'{self.representative} (mod
{self.module})' ``` Note that Zmodn receives only the variable `module`, which allows defining the set of equivalence classes $\mathbb{Z}/n\mathbb{Z}$ according to the value of the module. To
generate the elements of $\mathbb{Z}/n\mathbb{Z}$, we redefine the `__call__` method, which allows us to call our class as a function to build different instances of `Zmodn` by `congruence_class =
self.__class__(self.module)`, and calculate the representative (`self.representative`) of the class as the remainder when dividing `integer` by `self.module`. Finally, we redefine the method
`__repr__` to be represented by the console `Zmodn` class as `a (mod n)`. To see how this works, let’s generate the elements of $\mathbb{Z}/n\mathbb{Z} =\{0\; (mod\; 2), 1\; (mod\; 2)\}$. To do this,
we first write: ```python mod2 = Zmodn(2) ``` At this point, `mod2` is a function that we can pass any integer to. Let’s look at some examples: ```python mod2(-3) >>> 1 (mod 2) mod2(4) >>> 0 (mod 2)
mod2(5) >>> 1 mod(2) ``` As we saw, the only results returned by the function are `0 (mod 2)` and `1 (mod 2)`. This is because any integer divided by two has a remainder of $0$ or $1$. We can try
other modular classes such as `mod3 = Zmodn(3)`, `mod4 = Zmodn(4)`, etc. As we saw in my post about [operator overloading](https://monadical.com/posts/operator-overloading-in-python.html), it’s
possible to redefine all the basic Python operations $+,-,\times,/$. The next step is to redefine the methods `__add__`($+$), `__sub__`($-$), `__mul__`($\times$), and `__truediv__` ($/$). For this,
we’ll make use of what we learned in the operator overloading post. So we have: ```python def __add__(self, other: int): if not self.module == other.module and not isinstance(other, self.__class__):
raise ValueError( f'Cannot add {self} and {other}, they are not in the same module' ) return self.__call__(self.representative + other.representative) def __sub__(self, other: int): if not
self.module == other.module and not isinstance(other, self.__class__): raise ValueError( f'Cannot subtract {self} and {other}, they are not in the same module' ) return self.__call__
(self.representative - other.representative) def __mul__(self, other: int): if not self.module == other.module and not isinstance(other, self.__class__): raise ValueError( f'Cannot multiply {self}
and {other}, they are not in the same module' ) return self.__call__(self.representative * other.representative) ``` Note that before calculating any operation, we must ensure that the elements that
we’re adding are calculated with respect to the same module. For example, it doesn’t make sense to add an element of $\mathbb{Z}/2\mathbb{Z}$ with an element of $\mathbb{Z}/3\mathbb{Z}$. We also
verify that `other` is an instance of `Zmodn` by using `isinstance(other, self.__class__)`. Then we compute the operations ($+,-,\times, /)$ with respect to the class representatives to finally
return the resulting class. As for the division between two elements $a$ and $b$ of $\mathbb{Z}/n\mathbb{Z}$, it only makes sense if $b$ is [coprime](https://en.wikipedia.org/wiki/Coprime_integers)
with $n$. In that case, we first make use of the [Euclidean algorithm](https://en.wikipedia.org/wiki/Euclidean_algorithm) for remainders and then calculate the multiplicative inverse of $b$. So we
have: ```python def multiplicative_inverse(self): if self.representative == 0: raise ZeroDivisionError('Cannot compute the multiplicative inverse of 0') aux1 = 0 aux2 = 1 y = self.representative x =
self.module while y != 0: q, r = divmod(x, y) x, y = y, r aux1, aux2 = aux2, aux1 - q * aux2 if x == 1: return self.__call__(aux1 % self.module) else: raise ValueError( f'{self.representative} is not
coprime to {self.module}' ``` And after this, we can define the division as follows: ```python def __truediv__(self, other: int): if not self.module == other.module and not isinstance(other,
self.__class__): raise ValueError( f'Cannot divide {self} and {other}, they are not in the same module' ) return self.__call__(self.representative) * other.multiplicative_inverse() ``` In the next
step, we’ll need the integer representation. So we define the following method: ```python def __int__(self) -> int: return self.representative ``` Finally, our `Zmodn` class would be: ```python class
Zmodn: def __init__(self, module: int = 2): self.representative = None self.module = module def __call__(self, integer: int): congruence_class = self.__class__(self.module)
congruence_class.representative = integer % self.module return congruence_class def __int__(self): return self.representative def __repr__(self): return f'{self.representative} (mod {self.module})'
def __add__(self, other: int): if not self.module == other.module and not isinstance(other, self.__class__): raise ValueError( f'Cannot add {self} and {other}, they are not in the same module' )
return self.__call__(self.representative + other.representative) def __sub__(self, other: int): if not self.module == other.module and not isinstance(other, self.__class__): raise ValueError(
f'Cannot subtract {self} and {other}, they are not in the same module' ) return self.__call__(self.representative - other.representative) def __mul__(self, other: int): if not self.module ==
other.module and not isinstance(other, self.__class__): raise ValueError( f'Cannot multiply {self} and {other}, they are not in the same module' ) return self.__call__(self.representative *
other.representative) def multiplicative_inverse(self): if self.representative == 0: raise ZeroDivisionError('Cannot compute the multiplicative inverse of 0') aux1 = 0 aux2 = 1 y =
self.representative x = self.module while y != 0: q, r = divmod(x, y) x, y = y, r aux1, aux2 = aux2, aux1 - q * aux2 if x == 1: return self.__call__(aux1 % self.module) else: raise ValueError( f'
{self.representative} is not coprime to {self.module}' ) def __truediv__(self, other: int): if not self.module == other.module and not isinstance(other, self.__class__): raise ValueError( f'Cannot
divide {self} and {other}, they are not in the same module' ) return self.__call__(self.representative) * other.multiplicative_inverse() ``` And therefore, we can use it to do our calculations on any
set of type $\mathbb{Z}/n\mathbb{Z}$. Let’s look at some examples: ```python mod5 = Zmodn(5) a, b = mod5(7), mod5(9) a, b >>> (2 (mod 5), 4 (mod 5)) a + b >>> 1 (mod 5) a - b >>> 3 (mod 5) a * b >>>
3 (mod 5) a / b >>> 3 (mod 5) c = a.multiplicative_inverse() c >>> 3 (mod 5) a * c >>> 1 (mod 5) ``` Other methods we could implement are:`__eq__`, `__neg__`, `__isub__`, and `__iadd__`. ### Step 2:
Create arrays using `Zmodn` and NumPy functionalities In this step, we’re going to look at how to create arrays using `Zmodn` with some of NumPy’s functionalities. For this, we’ll build the following
class: ```python import numpy as np class ZmodnArray: def __init__(self, module): self.module = module self.representatives = None self.congruence_class = Zmodn(module) def __call__(self, integers:
list): congruence_class_array = self.__class__(self.module) congruence_class = np.vectorize(self.congruence_class) congruence_class_array.representatives = np.array(congruence_class(integers)) return
congruence_class_array def __repr__(self): return f'{self.representatives.astype(int)} (mod {self.module})' ``` What’s new here is the line `congruence_class = np.vectorize(self.congruence_class)`.
In essence, we’re [vectorizing](https://numpy.org/doc/stable/reference/generated/numpy.vectorize.html) the `self.congruence_class` function using the `np.vectorize` method. This allows us to pass a
complete array to the `congruence_class` function as done in the line `congruence_class_array.representatives = congruence_class(integers)`. To complete this class, we add the `__add__`, `__sub__`
and `__mul__` methods. This has a similar structure to the methods already implemented in `Zmodn`, the only difference here is that the operations ($+,-,\times$) are executed through NumPy, and we
make use of the `astype(int)` method to obtain the integer representatives of the `Zmodn` classes. These methods would be: ```python def __add__(self, other: list): if not self.module == other.module
and not isinstance(other, self.__class__): raise ValueError( f'Cannot add {self} and {other}, they are not in the same module' ) return self.__call__((self.representatives +
other.representatives).astype(int)) def __sub__(self, other: list): if not self.module == other.module and not isinstance(other, self.__class__): raise ValueError( f'Cannot subtract {self} and
{other}, they are not in the same module' ) return self.__call__((self.representatives - other.representatives).astype(int)) def __mul__(self, other: list): if not self.module == other.module and not
isinstance(other, self.__class__): raise ValueError( f'Cannot multiply {self} and {other}, they are not in the same module' ) return self.__call__( (np.dot(self.representatives,
other.representatives)).astype(int) ) ``` So far, we can execute some basic operations between arrays of `Zmodn` elements. Let’s look at some examples: ```python mod7_array = ZmodnArray(7) a =
mod7_array([[1, 2], [3, 8]]) b = mod7_array([[10, 7], [3, 8]]) ``` The result per console would be: ```python a >>> [[1 2] [3 1]] (mod 7) b >>> [[3 0] [3 1]] (mod 7) ``` For elementary operations we
have: ```python a + b >>> [[4 2] [6 2]] (mod 7) a - b >>> [[5 2] [0 0]] (mod 7) a * b >>> [[2 2] [5 1]] (mod 7) ``` ### Step 3: Calculate the inverse matrix with modular arithmetic using `ZmodnArray`
Now let’s see how to implement more advanced operations, such as calculating the inverse matrix. A quick way to implement this is to calculate the inverse matrix by the [adjugate matrix](https://
en.wikipedia.org/wiki/Adjugate_matrix) using the formula for matrix inverse: $$A^{-1}=det(A)^{-1}Adj(A).$$ For the case of modular arithmetic with matrices in $\mathbb{Z}/n\mathbb{Z}$, the
determinant of a matrix must be coprime with $n$. We don’t need to get into the technical details of the following functions. For now, what’s important to know is that they’re used to calculate the
inverse of a square matrix with non-zero determinant and coprime of $n$. Here’s the method for calculating the adjugate matrix: ```python @staticmethod def adjoint_of_matrix(matrix): adjoint =
np.zeros(matrix.shape, dtype=np.int16) amount_of_rows, amount_of_columns = matrix.shape for i in range(amount_of_rows): for j in range(amount_of_columns): cofactor_i_j = np.delete(np.delete(matrix,
i, axis=0), j, axis=1) determinant = int(np.linalg.det(cofactor_i_j)) adjoint[i][j] = determinant * (-1) ** (i + j) return np.transpose(adjoint) ``` And the method for calculating the inverse matrix:
```python def inv(self): matrix = self.representatives.astype(int) if matrix.shape[0] != matrix.shape[1]: raise ValueError('Matrix is not square') determinant = int(np.linalg.det(matrix)) if
determinant == 0: raise ValueError('Matrix is not invertible') adjoint = ZmodnArray.adjoint_of_matrix(matrix) return self.__call__( int(self.congruence_class(1) / self.congruence_class(determinant))
* adjoint.astype(int) ) ``` Although calculating the inverse matrix by the method of the adjugate matrix is not the most efficient, it does allow us to illustrate the possibilities we have with the
`Zmodn` classes. If everything goes well, the result we should obtain by using `inv` method would be: ```python a = mod7_array([[1, 2], [3,8]]) a * a.inv() >>> [[1 0] [0 1]] (mod 7) ``` As a further
exercise, I recommend overloading the `__getitem__` method because it’ll allow us to read each entry of a `Zmodn` array as it is done in NumPy arrays. If this still isn’t clear, please take a second
look at my post about [operator overloading](https://monadical.com/posts/operator-overloading-in-python.html). ### Step 4: Redefine NumPy’s universal functions So far, we’ve seen how to build a
`Zmodn` class that allows us to work with the $\mathbb{Z}/n\mathbb{Z}$ modular classes, and we learned how to use these elements with NumPy. Next, we’ll solve the problem in a different way by
redefining NumPy's universal functions. For this, we’ll use the special method `__array_function__`, which allows us to override NumPy’s native methods. The structure of this method is: ```python def
__array_ufunc__(self, ufunc, method, *inputs, **kwargs) ... return result ``` Here: * `ufunc` parameter is NumPy’s universal function to be called. * `method` is a string indicating how the ufunc
parameter is to be called, either `__call__` to indicate that it’s called directly, or one of its methods such as: `reduce`, `accumulate`, `reduceat`, `outer` or `at`. * `*inputs` is a tuple for the
ufunc arguments. * `**kwargs` contains any optional or keyword arguments passed to the function. This includes any output arguments, which are always contained in a tuple. Therefore, the arguments
are normalized; only the necessary input arguments are passed as positional arguments. All others are passed as a `dict` of keyword arguments (`**kwargs`). In particular, if there are output
arguments (positional or not) that are not `None`, they are passed as a tuple in the `out` keyword argument. This goes even for the `reduce`, `accumulate` and `reduceat` methods where all actual
cases make sense as a single output. With this in mind, we can build a class to work with modular arithmetic as follows: ```python import numpy as np HANDLED_FUNCTIONS = dict() class ZmodnArrays: def
__init__(self, intergers, module): self.representatives = np.array(intergers) % module self.module = module def __repr__(self): return f'{self.representatives} (mod {self.module})' def
__array_function__(self, func, types, args, kwargs): if func not in HANDLED_FUNCTIONS: return NotImplemented if not all(issubclass(t, ZmodnArrays) for t in types): return NotImplemented return
HANDLED_FUNCTIONS[func](*args, **kwargs) def implements(numpy_function): def decorator(method): HANDLED_FUNCTIONS[numpy_function] = method return method return decorator ``` Note three important
elements: 1. the `HANDLED_FUNCTIONS` variable, which is used to store the new definitions for NumPy’s universal functions. 2. the implementation of the `__array_function__method`, which is in charge
of managing the execution of the definitions. 3. the `implements` decorator, which allows us to update the `HANDLED_FUNCTIONS` variable with the implementation of the new methods. NumPy has a list of
universal functions that can be overridden with this strategy. The list can be found in the [documentation](https://numpy.org/neps/nep-0013-ufunc-overrides.html#list-of-operators-and-numpy-ufuncs).
For our case, we’re only going to redefine `__add__`, `__sub__` and `__mul__`. In effect, the redefinitions would be: ```python @implements(np.add) def __add__(self, other): if not self.module ==
other.module and not isinstance(other, self.__class__): raise ValueError( f'Cannot add {self} and {other}, they are not in the same module' ) repr_sum = np.array(self.representatives) + np.array
(other.representatives) return self.__class__(repr_sum % self.module, self.module) @implements(np.subtract) def __sub__(self, other): if not self.module == other.module and not isinstance(other,
self.__class__): raise ValueError( f'Cannot subtract {self} and {other}, they are not in the same module' ) repr_sub = np.array(self.representatives) - np.array(other.representatives) return
self.__class__(repr_sub % self.module, self.module) @implements(np.multiply) def __mul__(self, other): if not self.module == other.module and not isinstance(other, self.__class__): raise ValueError(
f'Cannot multiply {self} and {other}, they are not in the same module' ) repr_dot = np.dot(np.array(self.representatives), np.array(other.representatives)) return self.__class__(repr_dot %
self.module, self.module) ``` As we can see, the structure is similar to what we already implemented in the previous sections. What’s new here is the implementation of the decorator `@implements
(numpy_function)`, which is named after the universal function we want to rewrite. Furthermore, in the redefinition, we implement addition, subtraction, and multiplication using NumPy’s native
operations, but we’re careful to extract the remainder. For example, in the case of an addition, we return the following: ```python repr_sum = np.array(self.representatives) + np.array
(other.representatives) return self.__class__(repr_sum % self.module, self.module) ``` Note that the sum is calculated through the native sum for objects of type `np.array`, and the result is that
the remainders are computed using `%`. Then, an instance of the class is made with the method `self.__class__`. The same is true for the other two basic operations. Our final class would be:
```python import numpy as np HANDLED_FUNCTIONS = dict() class ZmodnArrays: def __init__(self, intergers, module): self.representatives = np.array(intergers) % module self.module = module def __repr__
(self): return f'{self.representatives} (mod {self.module})' def __array_function__(self, func, types, args, kwargs): if func not in HANDLED_FUNCTIONS: return NotImplemented if not all(issubclass(t,
ZmodnArrays) for t in types): return NotImplemented return HANDLED_FUNCTIONS[func](*args, **kwargs) def implements(numpy_function): def decorator(method): HANDLED_FUNCTIONS[numpy_function] = method
return method return decorator @implements(np.add) def __add__(self, other): if not self.module == other.module and not isinstance(other, self.__class__): raise ValueError( f'Cannot add {self} and
{other}, they are not in the same module' ) repr_sum = np.array(self.representatives) + np.array(other.representatives) return self.__class__(repr_sum % self.module, self.module) @implements
(np.subtract) def __sub__(self, other): if not self.module == other.module and not isinstance(other, self.__class__): raise ValueError( f'Cannot subtract {self} and {other}, they are not in the same
module' ) repr_sub = np.array(self.representatives) - np.array(other.representatives) return self.__class__(repr_sub % self.module, self.module) @implements(np.multiply) def __mul__(self, other): if
not self.module == other.module and not isinstance(other, self.__class__): raise ValueError( f'Cannot multiply {self} and {other}, they are not in the same module' ) repr_dot = np.dot(np.array
(self.representatives), np.array(other.representatives)) return self.__class__(repr_dot % self.module, self.module) ``` To test this class, we can do: ```python mod7_array = ZmodnArrays(7) a =
mod7_array([[1, 2], [3, 8]]) b = mod7_array([[10, 7], [3, 8]]) ``` So for the elementary operations we would have: ```python a + b >>> [[4 2] [6 2]] (mod 7) a - b >>> [[5 2] [0 0]] (mod 7) a * b >>>
[[2 2] [5 1]] (mod 7) ``` Finally, in the documentation of the [data model](https://docs.python.org/3/reference/datamodel.html?highlight=overloading#data-model) we can explore what other
functionalities we can add to make this class more versatile. ## Conclusions Let’s wrap up what we’ve covered today in our four steps: 1. The special methods `__add__`,`__sub__`, `__mul__`, and
`__call__` allow us to redefine the basic Python operators, which allows us to build other types of arithmetic we can work with. This is particularly interesting for problems related to cryptography,
graphs, etc. 2. Using the `__array_function__` method, we can tell NumPy how to execute the universal operations according to the needs of our objects. 3. According to the NumPy [documentation]
(https://numpy.org/devdocs/reference/arrays.classes.html#numpy.class.__array_ufunc__), the standard structure for building a class to redefine NumPy’s universal functions is: ```python
HANDLED_FUNCTIONS = {} class MyArray: def __array_function__(self, func, types, args, kwargs): if func not in HANDLED_FUNCTIONS: return NotImplemented # Note: this allows subclasses that don't
override # __array_function__ to handle MyArray objects if not all(issubclass(t, MyArray) for t in types): return NotImplemented return HANDLED_FUNCTIONS[func](*args, **kwargs) def implements
(numpy_function): """Register an __array_function__ implementation for MyArray objects.""" def decorator(func): HANDLED_FUNCTIONS[numpy_function] = func return func return decorator @implements
(np.concatenate) def concatenate(arrays, axis=0, out=None): ... # implementation of concatenate for MyArray objects @implements(np.broadcast_to) def broadcast_to(array, shape): ... # implementation
of broadcast_to for MyArray objects ``` With this approach of overloading operators and redefining NumPy’s universal functions, we’ve built a modular arithmetic library that can be successfully used
in my version of the Lights Out Game. We'll have to meet again in my next post to learn how the library functions in the game. If you’re like me, you may be interested in building this library to
learn more about topics such as graph theory, cryptography, and number theory, among others. To learn more about these topics in relation to the Lights Out Game, please check out these articles: 1.
[A Survey of the Game “Lights Out!”](https://link.springer.com/chapter/10.1007/978-3-642-40273-9_13) 2. [Lights Out: Solutions Using Linear Algebra](http://cau.ac.kr/~mhhgtx/courses/LinearAlgebra/
references/MadsenLightsOut.pdf) 3. [Lights Out!: A Survey of Parity Domination in Grid Graphs](https://www.unf.edu/~wkloster/termpaper.pdf) 4. [Lights Out on graphs](https://arxiv.org/pdf/
1903.06942.pdf) 5. [Lights Out on a Random Graph](https://journals.calstate.edu/pump/article/view/2593/2983) The code used in this blog can be found in this notebook: [Modular Arithmetic with Python]
(https://colab.research.google.com/drive/1o27dB5ywv2h0jS9KOfOJV4PsdPAPIi4-?usp=sharing#scrollTo=S7AgP0ydmPrK). If you enjoyed building this modular arithmetic library through my tutorial, please
share it with your friends and colleagues! ## References 1. NumPy Documentation: [Standard array subclasses](https://numpy.org/devdocs/reference/arrays.classes.html#numpy.class.__array_ufunc__) 2.
NumPy Documentation: [NEP 13 - A mechanism for overriding Ufuncs](https://numpy.org/neps/nep-0013-ufunc-overrides.html) 3. W3schools: [Create your own ufunc](https://www.w3schools.com/python/numpy/
numpy_ufunc_create_function.asp) 4. Stackoverflow: [Python linear algebra in a finite field](https://stackoverflow.com/questions/71078626/python-linear-algebra-in-a-finite-field) --- <center> <img
src="https://monadical.com/static/logo-black.png" style="height: 80px"/><br/> Monadical.com | Full-Stack Consultancy *We build software that outlasts us* </center> | {"url":"https://monadical.com/posts/modular-arithmetic.html","timestamp":"2024-11-02T14:22:35Z","content_type":"text/html","content_length":"73078","record_id":"<urn:uuid:3bdfb11a-ea5f-449b-bb93-75d9c16b848d>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00308.warc.gz"} |
A Validated Model, Scalability, and Plant Growth Results for an Agrivoltaic Greenhouse
Department of Mechanical Engineering, Villanova University, Villanova, PA 19085, USA
Department of Biology, Center for Biodiversity and Ecosystem Stewardship, Villanova University, Villanova, PA 19085, USA
Department of Engineering Leadership and Society, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA
Author to whom correspondence should be addressed.
Submission received: 23 March 2022 / Revised: 10 May 2022 / Accepted: 12 May 2022 / Published: 19 May 2022
We developed an agrivoltaic greenhouse (a ‘test cell’) that partially trapped waste heat from two photovoltaic (PV) panels. These panels served as parts of the roof of the enclosure to extend the
growing season. Relative humidity, internal air temperature, incident solar radiation, wind speed, and wind direction were measured for one year. A locally 1-D transient heat and moisture transport
model, as well as a shadowing model, was developed and validated with experimental data. The models were used to investigate the effects of altering various parameters of the greenhouse in a
scalability study. The design kept test cell air temperatures generally above ambient throughout the year, with the test cell temperature below freezing for 36% less of the year than ambient. Plant
growth experiments showed that kale, Brassica oleraceae, a shade-tolerant plant, can be grown within the test cell throughout the winter. The simulations showed that enlarging the greenhouse will
increase cell air temperatures but that powering an electric load from the PV panels will reduce cell air temperatures.
1. Introduction
Local and national governments and private industry continue efforts to expand the deployment of renewable energy sources. A central pillar of renewable energy generation is utility-scale
photovoltaic (PV) solar farms. These farms require large amounts of land with high solar irradiance. Growing global populations also require more food, which necessitates more flat land with high
solar irradiance for growing crops. The needs of solar energy production have prompted significant debate and even legislative action on the uses of high-quality agricultural land [
]. Some local communities and governments have already adopted rules limiting the development of solar farms [
], but some farmers prefer renting their land to solar companies instead of growing crops, as the income can be more reliable [
]. Agrivoltaics may allow farmers to avoid the conflict by combining PV arrays and agriculture in a way that preserves the yield of both.
Different ideas for combining PV arrays and agriculture have been proposed, but they generally fall into the three categories shown in
Figure 1
: interspersed PV arrays, greenhouse-mounted PV arrays, and stilt-mounted PV arrays [
]. For interspersed PV arrays, the PV panels are mounted between rows of plants [
]. Greenhouse-mounted PV arrays use the framework of a traditional greenhouse to mount the panels on the roof, although the position and orientation of the solar panels may be sub-optimal to
accommodate needs of the greenhouse. The greenhouse may also take advantage of primary (electricity) or secondary (heat) products of the panels. Finally, stilt-mounted PV arrays position the panels
at a desired height above an otherwise open field, with plants growing under them. The space beneath allows access to the crops by farming machinery.
In [
], Joudi and Farhan replaced a greenhouse roof with a solar air heater panel to partially block light during warmer months and remove heat from the system. It also absorbed light during colder months
and redirected heat into the greenhouse. The greenhouse maintained a temperature 16 °C higher than ambient during the winter and 10 °C cooler than ambient during the summer, which showed that the
solar air heaters could reduce the cooling load in the summer and support the heating load during the winter.
Marucci et al. [
] placed PV panels in a checkerboard pattern across the roof of a Quonset-type greenhouse. The purpose of their study was to examine the effects of shading by the panels. Another design by Cossu et
al. [
] replaced the covering material of a traditional greenhouse with a version of semi-transparent PV panels (STPs). The STPs in this design were interconnected PV cells pressed between two sheets of
glass. It was found that the STPs could only partially offset the energy costs of the greenhouse.
There are multiple examples of large commercial greenhouses being converted into agrivoltaic greenhouses by replacing parts of the pitched roofs with PV panels [
]. The fraction of roof covered by the PV panels varies, but these designs have the same general limitations. They are quite large, and the shading caused by the PV panels can reduce crop yields
inside the structures. Finally, structures that have already been built are difficult or expensive to retrofit.
A stilt-mounted PV array, like the system shown in
Figure 1
c, was developed by Sekiyama and Nagashima [
]. They compared the effects of a low-density configuration (panel spacing of 1.67-m apart) and a high-density configuration (panel spacing of 71-cm apart) on the growth of corn, a shade-intolerant
crop. The panels were mounted at a height of 4 m above the ground, high enough for a tractor to pass under. Sekiyama and Nagashima found improved crop yield for corn grown beneath the low-density
configuration and a slightly reduced yield for corn grown beneath the high-density configuration. The configuration of
Figure 1
c does not extend the growing season due to the lack of a heat-trapping enclosure.
Vanthoor suggested that a comprehensive greenhouse model should consider internal temperature, relative humidity (RH), and CO
concentration [
]. Nearly all thermal models in [
] were locally 1-D transient models, with the discrete components considered as lumped masses. The models differed in regard to the other factors considered. Combinations of RH, CO
concentration, and heat flow due to certain heat transfer modes were neglected in many of the models. Jouid et al. [
] neglected the evaporation of water from the soil, assumed there was no water in the greenhouse soil, and assumed there were no plants in the greenhouse. Mohammadi et al. [
] assumed that there were no crops in the greenhouse, i.e., no evapotranspiration in the greenhouse, a negligible effect of CO
concentration on evapotranspiration, and no evaporation from the soil. They also assumed that any water condensed on the inside of the roof or screens was removed from the system. Cooper and Fuller [
] neglected edge losses, and Sethi [
] neglected radiation heat exchange between the walls and the roof, which we note below is not negligible.
Tiwari et al. [
] considered heat loss from the floor to the ground but handled it as steady-state heat transfer, unlike the rest of the model, which was quasi-steady state. They also considered solar radiation to
be partially absorbed by the plants, an effect we neglected by using a model that addresses the overall impact of the solar gain and evapotranspiration. Abdel-Ghani 2011 [
] took an entirely different modeling approach, treating a greenhouse as a solar collector and assuming that elements such as coverings, air, and plant canopies have radiation properties, such as
transmittance, absorbance, and reflectance.
There are several differences in our model compared with the above. We chose to neglect CO
concentration and the effects of the plant canopy inside the greenhouse (the test cell), because CO
concentrations were beyond the scope of our project, and the fraction of the base shadowed by the canopy would be highly uncertain. For our design, radiation heat exchange among elements inside the
cell was found to be larger than free convection. In fact, heat flow rates for radiation were at least double those for free convection. Marucci et al. [
] and Cossu et al. [
] only modeled the solar radiation transmitted from outside the greenhouse to the interior. Most of the effects in our model are included in other models [
], but only our model contains all the effects we included. Our model also included transient heat conduction in the ground below the greenhouse, as earth coupling effects are generally important
during fall and spring periods.
Unlike [
], we modeled transient transfer from floor to ground and included thermal contact conductance between concrete blocks and the ground, as well as contact conductance between possible multi-layers of
concrete blocks and modelling transient heat transfer from floor to ground. No individual greenhouse model we examined considered all the heat transfer modes included in this work, although several
considered other factors that we neglected, such as CO
The test cell (
Figure 2
) has a 20.3-cm (8-in) window of transparent polycarbonate at the top placed between two PV panels, which allows light into the area below and allows plants to grow at the floor level. The PV panels
transfer heat by free convection and thermal radiation to the cell inside surfaces when exposed to solar radiation on the outside. The heat is partially trapped in the enclosed space, increasing the
cell air temperature. At each instant in time, the value of the cell air temperature results from an energy balance, which includes all cell surfaces and masses, the thermal resistances of the cell
walls, and radiation and forced convection on all outer cell surfaces. Due to the traditional “greenhouse” effect (of poor thermal radiation through the polycarbonate from the cell inside surfaces),
the energy balance maintains internal temperatures at above the ambient throughout the day, extending the growing season without the use of heaters during periods of cool ambient temperature.
2. Materials and Methods
The test cell used two 0.7 m × 2 m PV panels donated by Solar States LLC [
]. The two panels were positioned facing east and west at 35.5° from the horizontal and were held in position by two wooden frames constructed from 3.81 cm × 8.9 cm (a U.S. “2 × 4”) wooden beams. As
shown in
Figure 2
, the frame has two vertical beams about 78-cm-long, spaced 15.2-cm-apart, and two horizontal beams of about 1.1 m. There is an approximately 20.3-cm gap between the upper edges of the two PV panels
covered by a transparent glazing of 1.6-mm-thick transparent polycarbonate.
The north- and south-facing sides of the enclosure are sealed by plywood walls. The north walls are supported by brackets, and the southern walls are held in place by hinges and locks that allow
interior access. The test cell is insulated by R-10 foamboard insulation mounted on the inside of the north- and south-facing side walls.
As shown in
Figure 2
b, the base of the test cell was covered with low-density concrete blocks of density 1920 kg/m
, which added thermal mass to the system to attempt to make cell air temperatures more uniform over short times. Three total solar radiation pyranometers, two temperature, and two RH sensors were
installed in the test cell, as shown in
Figure 2
b. The internal sensors were positioned about 1.3 m from the south-facing sidewall and about 65.3 cm from each other and from the east- and west-facing PV panels. A pyranometer and temperature, RH,
wind speed, and wind direction sensors were installed outside of the test cell to collect ambient data. Manufacturers, models, and specifications of sensors used are presented in
Appendix A
The temperature and RH sensors collected data every 5 min, while the pyranometers and wind speed and direction sensors collected data every minute and record five-minute averages. Approximately every
55 days, data were collected, and sensors were reset. The data sets were concatenated to produce a full year data set from 17 January 2020 to 17 January 2021. Gaps in data caused by brief
instrumentation failures and brief periods when snow covering rendered the data invalid were filled using linear interpolation.
Three plant growth experiments were carried out in the test cell. In each experiment, two Italian heirloom Toscano Lacinato kale (also known as “dinosaur” kale) seeds (Brassica oleraceae) were
planted together six-millimeters-deep in potting soil. Pairs of Toscano kale seeds were about 3.8-cm apart, with six sets of seeds in each planter. Plants within the test cell were watered by a
gravity-fed irrigator. Control plants growing adjacent to the test cell and in a conventional greenhouse were watered by hand every few days. Samples of soil were taken for gravimetric water content
analysis to assure all plants were receiving sufficient and approximately equal soil moisture.
In the October-2019 experiment, four planters were placed in the test cell, and one planter was placed outside as a control. In the October-2020 growth experiment, four planters were placed in the
test cell, four planters were placed outside as a control, and four planters were placed in a nearby conventional greenhouse. Finally, in the March-2021 growth experiment, two planters in the test
cell from the October-2020 growth experiment were replaced with new planters, and four new planters were placed outside as controls.
During the October-2020 growth experiment, the total leaf area of sample plants from the test cell and the control was estimated by measuring leaf length and width and assuming an ellipsoid shape for
the leaves. The same data were collected for the control plants in the greenhouse; however, those plants were then taken and dried at 62.8 °C (145 °F) in an oven for 24–48 h to estimate dry mass. An
allometric function (discussed in
Appendix B
) correlated total leaf area to plant dry mass. This function was used with the measurements of leaves from the test cell and the adjacent control plants to determine the average dry mass of the
plants and thus compare the dry mass of the kale produced in the test cell, greenhouse, and control test.
During the third growth experiment, the same measurements as those in the second growth experiment were taken. These new plants were started in March 2021 in the test cell and in the control area.
The allometric function found previously was used to compare the dry mass of kale produced in the test cell and in the control area.
3. Mathematical Modeling
Three models were developed to help understand the relative contributions of the heat and mass transfer processes in the greenhouse and to predict the effects of changes in the test cell design
parameters (i.e., scalability) in sensitivity studies. The first was a locally 1-D thermal model with temperature nodes throughout the system, based on an energy balance at each node. The moisture
transport model, like the thermal model, had nodes throughout the system and was based on a mass balance at each node. Thermal and moisture transport interact through the latent heat of water, so
their equations must be solved simultaneously. The thermal model had a strong effect on the moisture transport model results, but the moisture transport model only weakly affected the thermal model.
This interaction is discussed below.
The third model predicted shadowing of the beam and diffuse solar radiation and used the geometry of the test cell and the sun’s transit across the sky to predict which parts of the test cell base
were shadowed.
In the thermal model (
Figure 3
), several processes act at each node. Some nodes experienced incident solar radiation, such as the nodes on the outside of the solar panels. This was found from measured values of surface area and
solar radiation flux. Values for the angle of incidence and transmissivity (for the transparent polycarbonate, its absorptivity) were calculated from the instantaneous solar positions and published
data. The solar panel nodes also experience radiation heat exchange with the sky, which was found using calculated values of node temperature, sky temperature, and the radiation view factor. Solar
radiation transmitted through the transparent polycarbonate was calculated based on the transmissivity of the material and was assumed to be distributed uniformly over the internal surfaces. Forced
convection due to external wind occurred at all external surface nodes and was calculated using the measured wind speed and air temperature and the calculated surface temperatures. Free convection
occurred at all internal surface nodes. Radiation heat transfer also occurred at all internal nodes and was calculated using the net radiation method (
Appendix C
Air in the test cell is coupled with the ambient air by conduction through the north- and south-facing side walls, as well as through infiltration through gaps in the side walls. Heat flow due to
conduction through the side walls was determined using measured ambient and cell air temperatures and a calculated wall thermal conductance. All properties, such as thermal conductivity, were taken
from published values. Heat flow due to infiltration was found using the measured ambient air temperature and calculated cell air temperature. The infiltration air flow rate was determined using
measured values of wind speed and direction to calculate the stagnation pressure on the outside of the south-facing wall. In addition, infiltration due to the stack effect was calculated from cell
air temperature and measured ambient temperature. Conduction in the concrete and soil was modeled by numerically solving the 1D, transient heat conduction equation. The resulting linear algebraic
equations were solved implicitly using a matrix method. At the concrete soil interface, the heat flow due to thermal contact conductance was found from calculated temperatures and a thermal contact
resistance, the value of which was determined during model validation (by comparison with measured data).
Key assumptions in the models are as follows:
• Heat and mass transfer are lumped in a node, and heat flow is locally 1-D;
• Solar radiation entering the test cell is treated as diffuse over all internal surfaces, except on the polycarbonate sheet;
• No thermal nodes are located in the north- and south-facing side walls (these function only as thermal resistances, coupling the cell temperature with the outside air. The mass of the walls is
small due to the small size and the low thermal mass of the wall insulation);
• The thermal diffusivity and thermal conductivity of the soil are assumed to be constant based on the partial water infiltration theory [
• The irrigation rate and the rate of water diffusion from the soil are known and constant (
Appendix D
• The infiltration of ambient air and the accompanying moisture and heat transfer occurs by wind-driven infiltration through cracks in the walls and by the stack effect. The cracks are lumped into
a single gap area determined during model validation;
• Only beam radiation is considered in the shadowing model; diffuse radiation, generally smaller than beam, is not considered.
Governing equations for the thermal model are given below. The heat flow rate equations for Equations (1)–(10) are discussed in
Appendix C
, along with mass flow rates from Equations (11)–(18).
dT[lex]/dt = (q[sol,lex] + q[wind,lex] + q[sky,lex] + q[a,lex] − q[rad,lex])/ρ[lex]V[lex]C[plex]
dT[sp,out]/dt = (q[sol,sp] + q[wind,sp] + q[sky,sp] + q[cond,sp,inner])/ρ[sp]V[sp]C[Psp]
dT[sp,in]/dt = (q[a,sp]-q[rad,sp] + q[cond,sp,outer] + q[ligh,sp])/ρ[sp]V[sp]C[Psp]
dT[cell]/dt = (q[sp,west,a] + q[sp,east,a] + q[lex,a]+q[con,a] + q[sidp,a]-q[infi])/ρ[a]V[a]C[Pa]
dT[con,up]/dt = (α[con]/Δx^2[con])(2T[con,up+1] − 2T[con,up] + (2Δx[con]/k[con])[q[flux,ligh,con] + q[flux,a,con] + q[flux,evap] + q[flux,rad,con]])
dT[i,con]/dt = (α[con]/Δx[con]^2)(T[i,con+1] − 2T[i,con] + T[i,con-1])
dT[con,low]/dt = (q[cond,con] + q[tcc,con])/ρ[con]A[con][Δx[con]/2]C[Pcon]
dT[soil,up]/dt = (q[cond,soil] + q[tcc,soil])/ρ[soil]A[soil][Δx[soil]/2]C[Psoil]
dT[i,soil]/dt = (α[soil]/Δx[soil]^2)(T[i,soil+1] − 2T[i,soil] + T[i,soil-1])
dT[soil,low]/dt = (α[soil]/Δx^2[soil])(2T[soil,low-1] − 2T[soil,low])
In Equations (7) and (8), we assumed there is no heat storage in the interface. Equation (10) is a variant of Equation (9) and predicts temperature changes in the lower soil surface layer, which is
It is assumed that liquid water can condense and collects on the concrete base and that this liquid water can subsequently evaporate. Three water mass nodes are considered, one each for the air, the
accumulated liquid water, and the planter (
Figure 4
). In the moisture transport model, a mass balance is written for each node, just as is done for heat in an energy balance.
Condensation begins when the humidity ratio of air reaches the saturation humidity ratio. When the humidity ratio is less than the saturation humidity ratio, evaporation occurs.
The equations used in the moisture transport model are as follows:
dM[a]/dt = Mdot[infi,h2o] + ET[plant] + E[liq]
dM[a]/dt = Mdot[infi,h2o] − M[cd]
dM[liq]/dt = −E[liq] + Mdot[soil]
dM[liq]/dt = M[cd](Acon − A[plant])/A[con] + Mdot[soil]
dM[plant]/dt = Mdot[irrig] − ET[plant]
dM[plant]/dt = Mdot[irrig] + Mdot[cd](A[plant]/A[con])
Equations (11)–(15) are used when evaporation occurs, and Equations (12)–(16) are for condensation. ET
is evapotranspiration from the plants. It is an adapted form of the Hargraves equation [
] and is discussed in
Appendix C
A separate set of equationsgoverns the overflow of water from the planters when the water in that node exceeds the maximum capacity of the soil. If M
is greater than M
, Equations (17) and (18) are used. Note that Equation (18) modifies the mass change value predicted by Equations (13) or (14).
dM[over]/dt = (M[plant] − M[plant,max])
dM[liq]/dt = dM[liq]/dt + dM[over]/dt
The thermal and moisture transport models are solved using the algorithm below in Matlab (Mathworks, Natick, MA, USA) with the integration package ODE15s. See
Figure 5
Figure 6
The equations for the shadowing model are presented below [
]. The hour angle (ω = 0 is assumed to occur at midnight) and cell-based surface azimuthal angle are defined by
sin(α[s]) = cos(φ) cos(δ) cos(ω) + sin(φ) sin(δ)
The shadow projection point SP is the distance in the north–south direction, measured from the base of the south-facing side wall (see
Figure 7
). It determines the length of the shadow projected by the sidewall. The value of cos(ω) was found from [
]. A point must be at a distance greater than SP from the south-facing side wall to be sunlit.
A point on the base must also be between two different hour angles, defined as ω
and ω
, to be sunlit. These angles determine when the west-facing and east-facing solar panels no longer block sunlight from reaching the point under consideration. The angles are found by trigonometric
relations based on the horizontal distance, y, and vertical distance, h, to each panel edge from the point under consideration.
ω[east] = atan(h[1]/y[1]) − 90°
ω[west] = atan(h[2]/y[2]) + 90°
The sunrise and sunset angles are
ω[set] = acos(−tan(φ)tan(δ))
Once the values calculated in Equations (19)–(25) are known, the algorithm to determine if a point at the base of the test cell is shaded is as follows. First, a check is made to determine if the
current hour angle in the simulation is before sunrise or after sunset (if so, no points are illuminated).
For daytime, a check is performed to determine whether a point on the base is within the projected shadow of the side panel. If so, then the point is not illuminated.
Finally, if the point is not within the projected shadow, a check is made to see if the current hour angle is between the east and west blocking angles for that point. If it is, then the point is
irradiated (if not, then it is shaded). Note that the shadowing model only considers beam radiation. The model is illustrated in
Figure 7
Figure 8
. Values of constants used in the above models are presented in
Appendix D
4. Results
The models were validated by comparing the calculated minimum and maximum test cell air temperatures with measurements and, by trial-and-error, adjusting key parameters that were identified as most
uncertain. These included the area of the air infiltration gap, irrigation flow rate, ground-source moisture infiltration rate, Hargreaves Equation scaling factor, the thermal contact resistances
(between the soil, first concrete layer, and second concrete layer), the coefficients of free convection (between the cell air and the concrete, as well as the cell air and every other internal
surface), the SOF factors for both solar panels, and the transmittance-absorptance for the solar panels. Once validated, measured data were used as inputs to the models in a parametric study to
predict performance of the test cell subject to a variety of changes in dimensions, configurations, and other operational parameters.
4.1. Validation
4.1.1. Thermal Model
Figure 9
Figure 10
show that the predicted cell temperatures were in good agreement with experimental data. Predicted and measured daily maximum temperatures differed by an average of 1.9 °C (3.5 °F), and predicted and
measured daily minimum temperatures differed by an average of 1.5 °C (2.7 °F).
4.1.2. Moisture Transport Model
When comparing the moisture transport model with measurements, the predicted relative humidity (RH), seen in
Figure 11
Figure 12
, showed a larger disagreement than that in
Figure 9
Figure 10
. The predicted RH was noticeably greater during the winter than the observed RH, and opposite was the case during the summer. Cell air temperatures have a more pronounced effect on the RH values
during the winter. From psychrometrics in classical thermodynamics, RH is inversely proportional to the saturation vapor pressure (p
(T)) at the air temperature of the test cell.
Consider the case of air temperature at −10 °C (14 °F), where p
is nearly an order of magnitude smaller than that at 35 °C (95 °F) [
]. For a given change in the product of the humidity ratio, λ, and the air vapor pressure, p
, this translates into an order of magnitude greater change in RH at −10 °C (14 °F) compared to that at 35 °C (95 °F). Thus, the sensitivity of RH to λp
is much greater in the winter than in the summer.
Figure 13
compares model-predicted RH values calculated using both the model predicted and measured temperatures. The RH calculated using measured temperatures was significantly closer to the measured RH. The
high sensitivity of RH to temperature, as seen in
Figure 13
, is due, at least in part, to the high RH at low temperatures, as noted above.
Another possible reason for the differences in
Figure 13
is the use of a constant irrigation rate in Equations (15) and (16), while the actual irrigation rate probably fluctuated due to interruptions caused by winter freezing (We selected a constant value
for the irrigation rate by first finding a piecewise irrigation rate function that produced good agreement like that shown in
Figure 11
, and then finding the average value of this function over the course of a year.). We also assumed constant rate of water infiltration to the test cell from the moist soil below concrete blocks,
although this value was uncertain and likely varied with time.
4.1.3. Shadowing Model
The shadowing model was validated by comparing the solar radiation data collected on a near completely clear day with that predicted by the shadowing model at the locations of the sensors.
Figure 14
shows excellent agreement between the model and the experimental data. Late in the day, the model over-predicted the incident radiation, possibly due to light cloud cover or atmospheric haze near the
4.2. Experimental Data
4.2.1. Test-Cell Temperature and Ambient Temperature
Figure 15
shows that daytime temperatures inside the test cell were greater than the ambient temperature for most of the year. Furthermore, as shown in
Figure 16
, even near day 350 and approaching the winter solstice, the minimum temperatures observed at night were typically higher than ambient. This occurred throughout nearly a year of observations.
4.2.2. Plant Growth
The October-2019 plant growth experiment is shown in photographs taken weekly over the course of the experiment.
Figure 17
shows the progression of plant growth for kale in the test cell. The kale was planted in early October 2019 and grew slowly over the course of the next few months. In February 2020, plant growth
accelerated, presumably due to improving conditions, and continued in March 2020.
Allometric measurements were used to find the average dry mass of the plants in the test cell and control cell throughout the October-2020 growth experiment. The allometric function is discussed in
Appendix B
Figure 18
shows the same progression as
Figure 17
. Plant growth was slow over the course of the winter, and during this time, the control cell plant’s average dry mass was greater. At the end of the winter, the plants in the test cell caught up
with those of the control cell plants and resumed growth. The control cell plants, which were outside the test cell and unprotected, died on 21 January 2021 (six weeks after data collection started)
due to exposure to snow and prolonged freezing temperatures.
4.3. Numerical Results
4.3.1. Parametric Study
A parametric study using the above models was undertaken to evaluate test cell scalability, where we interpreted scalability as the performance of the cell under conditions different than those of
the base case. The base-case parameter values are defined in
Appendix D
, and variations of the parameter values from those of the base case appear in
Table 1
. The base-case parity plot appears in
Figure 10
The results of the parametric study are presented in
Figure 19
. The maximum and minimum temperatures predicted by the model for each day of the year and those predicted by the base case are compared in each parity plot.
Based on the results in
Figure 19
, the following conclusions could be reached regarding the test-cell scalability.
• Minimum temperatures in all parametric runs,
Figure 19
a–n, are only weakly affected by the structural, electrical load (the “load”), and thermal mass changes. Cases 2 (cell twice the length;
Figure 19
a) and 11 (no blocks;
Figure 19
j) showed a slightly elevated minimum air temperature, as the cell air received greater heat input from twice the number of PV panels in the absence of the load-leveling feature of energy storage
in the concrete block. This suggests that the system was saturated with thermal mass, so additional mass may not have improved heat retention. This conclusion is supported by
Figure 19
i, in which additional thermal mass was added to the system, with only a slight reduction in daytime maximum temperatures.
• Figure 19
a, for which the north–south length of the test cell was doubled (thereby doubling the number of PV panels), shows that the change led to increased heating during the day. This increase in
temperature was larger during the summer than during the winter. As expected, the greatest increases in cell temperature were during the highest temperature days.
• Figure 19
b shows an increase in daily maximum temperatures due to the widening of the gap area (i.e., increasing the insolation) between the two panels from 20.3 to 40.6 cm. Temperatures much above
ambient have an adverse effect on plant growth, so this increase is undesirable. However, the increase in solar gain from the widened glazing is an improvement. The relative benefit of the
increased glazing area is addressed in
Section 4.3.2
Section 4.3.4
• Figure 19
c–h shows that powering a load from the PV panels reduces the internal maximum temperatures during the day, especially during the summer, but as noted in comment 1 above, it does not
significantly affect the minimum temperatures at night. This is discussed further in
Section 4.3.3
• Figure 19
j shows that the removal of the concrete blocks reduces the maximum temperatures slightly during the hottest days of the year and slightly increases the minimum temperatures. The thermal contact
conductance between the soil and the concrete blocks, which is removed in this case, reduces heat flow into the soil during the day and reduces heat flow out into the system at night This
reduction in daily maximum temperatures (if only slight) can help keep the average cell temperature near the ideal of 20 °C (68 °F). (As thermal diffusivity is the ratio of the thermal
conductivity to the product of density and specific heat, the smaller thermal diffusivity material is better at heat storage. The thermal diffusivities of concrete and soil are α
= 0.45 · 10
/s and α
= 0.99 · 10
/s, indicating superior heat storage for concrete blocks on a per-mass basis. However, soil was the largest portion of the thermal mass in the system. Note that contact resistance is created by
placing the blocks over the soil, which reduces the heat flow to and from the dominant thermal mass.)
• Figure 19
k,l, shows some increase in temperatures during the spring and fall compared with the base case. This can be explained as follows:
□ Figure 19
k,l has more planters spread over the base of the test cell than the base case. Due to the increase in planters, less of the test cell base is exposed for evaporation, reducing evaporative
cooling during these periods. Reduced evaporative cooling leads to the higher temperatures during the spring and fall as observed.
□ Summertime temperatures in
Figure 19
k,l are similar to those in
Figure 19
a, because in all three cases, all of the water contained in the liquid water node evaporated. Thus, with no water left in the liquid water node, evaporative cooling was reduced in all three
• Figure 19
m,n show similar behavior to that in
Figure 19
k,l, respectively. We suspect this is because the planters have a much higher potential water storage, and thus, less water spills into the liquid water node during the test period.
4.3.2. Instantaneous Photosynthetic Rate (P[n])
Our plant growth experiments confirmed that kale can be grown in the test cell. Using the shadowing model and the equation to calculate P
], more information about where kale can grow at the base of the test cell could be determined. P
> 0 means that a leaf is fixing more CO
then needed for cellular respiration. P
= 0 means that a leaf is fixing the necessary amount of CO
, and P
< 0 means that a leaf is not fixing enough CO
. The instantaneous value of P
can be found using the Mitscherlich equation [
P[n](I) = (1 − e^−k(I−I[0])) P[max]
where P
is the instantaneous photosynthetic rate; P
is a constant (the maximum photosynthetic rate of 20.3–21.0 µmols CO
s [
]); I
is the PAR irradiance at the compensation point where the P
is equal to zero (which, for kale, is 13 µmols photons/m
s = 2.85 W/m
); I is the instantaneous PAR irradiance; and k is Mitscherlich function, reported as 0.0030 [
]. Our shadowing model produces values of instantaneous solar irradiance. These values can be converted into instantaneous PAR (PAR stands for Phototsynthetically Active Radiation, which is solar
radiation between 400 and 700 nm) irradiance using a conversion factor of 2.43 µmols photons/J, found using the methods described in [
] (The conversion factor for PAR in W/m
to µmol photons/m
s is about 4.57 µmols photons/J. However, the pyranometers used in this study measure solar radiation intensity between 300 and 1100 nm (visible light spectrum). PAR makes up only a fraction of that
energy; as such, the factor of 4.57 must be reduced to 2.43 µmols photons/J when applied to readings from the pyranometers in this work.). If P
is integrated over a 24-h period, a value for net photosynthetic gain per day can be found. If the net photosynthetic gain is positive valued, then a leaf could grow that day. If the net
photosynthetic gain is zero, the plant would maintain its current biomass. Finally, if the net photosynthetic gain is negative, the leaf is likely to lose biomass.
The net photosynthetic gain per day is shown for the irradiance predicted by our shadowing model (
Figure 20
) and the measured irradiance (
Figure 21
Figure 20
shows that the only viable location for plant growth in the test cell is the area near the centerline between the east and west side of the test cell (dashed green line). If plants are placed close
to the east (dashed blue line) or the west side (dashed red line), they would not be able to grow, because the net photosynthetic gain there is always negative. Clearly, the net photosynthetic gain
inside the test cell was found to be substantially lower than outside, as expected.
The results based on solar radiation measurements (
Figure 21
) show that throughout most of the year, leaves would be able to grow in the center of the test cell (solid green line), as well as the east (solid blue line) and western sides (solid red line) of
the test cell. The net photosynthetic gain values were higher than those observed in
Figure 20
for the interior and exterior points. The solar radiation measurements included diffuse radiation, while the model-predicted solar radiation included only beam radiation.
4.3.3. Ideal Temperature Zones
Kale grows optimally at a temperature in small band centered around 20 °C (68 °F) [
]. Temperatures above freezing are desirable for nearly all plants. An analysis was conducted to determine how much of the year our test cell spent within three temperature bands surrounding the
ideal temperature, based on parametric simulations and the ambient conditions (
Table 2
). Another analysis was conducted to determine how much time the test cell, the ambient conditions, and our parametric simulations spent below the freezing point (
Table 2
A brief examination of
Table 2
shows a few trends discussed here.
• Plants in ambient conditions are expected to spend more time near the ideal temperature. Compared with the test cell data,
Figure 15
reveals that the smaller time near the ideal temperature was mostly due to overheating during summer months. Simple modifications, such as ventilation fans, could reduce this overheating effect
substantially and allow for better control of relative humidity. Addressing high temperatures during the summer should result in the test cell performing better than the ambient.
• The test cell spent fewer hours below freezing 0 °C (32 °F) than the ambient (36% fewer hours).
• Doubling the north–south length, as in case 2, reduced the number of hours spent near 20 °C (68 °F).
• Doubling the glazing area, as in case 3, reduced the number of hours spent near 20 °C (68 °F).
• Powering a load from the solar panels, as in cases 4–6 and 7–9, showed a progressive increase in the number of hours spent near 20 °C (68 °F) with increasing load.
• Table 2
shows that changes which increased cell temperature reduced the time spent near 20 °C (68 °F) and that changes that decreased cell temperature increased the time spent near 20 °C (68 °F)
throughout the year. This suggests that the test cell overheated during the summer. This is an undesirable characteristic of the design. Keeping the test-cell air near the ideal temperature range
would promote better plant growth.
The reason that the predicted temperatures were below freezing for more of the year than the measured temperatures is likely because the model underpredicted nighttime temperatures by an average of
1.5 °C (2.7 °F). As a result, the predicted temperatures may have fallen just below freezing at times when the experimental values were higher than 0 °C (32 °F). For example, if the measured
temperature was 1.1 °C (34 °F), the model may have predicted −0.4 °C (31.3 °F).
4.3.4. Shadow Model: Transmitted Solar Radiation Compared with a Stilt-Mounted PV Array
Using the shadowing model developed in this work, the solar radiation transmitted to the cell base could be determined and compared with that in other agrivoltaic designs. A stilt-mounted PV array
was selected (refer to
Figure 1
c), and the shadowing model was used to determine the fraction of solar beam radiation transmitted to the base beneath each array as compared to an area equal to the floor of the test cell.
Two types of stilt-mounted PV arrays with geometries from [
] were considered. The low-density stilt-mounted PV array in [
] is composed of 4 PV panels placed 1.67-m apart. The high-density array had 8 PV panels, positioned 0.71-m apart. The calculation was for the summer solstice. The low-density panel configuration of
the stilt-mounted PV array allowed 82% of radiation to reach the base beneath the panels, compared to an unshaded area. The high-density configuration allowed 65% to reach the base beneath the
panels. By comparison, the test cell allowed 5.2% to reach the base beneath the solar panels (
Figure 22
5. Discussion
The present work focused primarily on evaluating the viability of our test cell, developing a computer model of the system, and on identifying potential improvements and scalability through a
parametric study. The project has produced promising results and highlighted several areas where the design or models could be improved.
Based on the results presented in
Section 3
, we were able to grow kale in the test cell. However, only plants located directly under the glazing grew well. Those located more east and west in the test cell did not grow as well. However, our
test cell also kept kale alive throughout the winter and allowed it to resume growth with the arrival of spring. Temperatures within the test cell were greater than those outside (ambient) for almost
the entire year, which confirms our hypothesis that the heat produced by the PV panels could be used to augment the normal greenhouse effect and extend the growing season of crops. Based on the net
photosynthetic gain (
Appendix B
) analysis of the interior of the test cell, it is clear that kale leaves can grow within the test cell, despite the lower light compared to other agrivoltaic designs. The test cell may be suitable
for other shade-tolerant plants, such as arugula, lettuce, carrots, or potatoes. Trials with multiple plant species should be carried out. Furthermore, if the glazing were removed and the panels
completely sealed against solar radiation, then crops such as mushrooms may grow quite well.
The proposed design would be difficult to use with traditional farming techniques. Possible modifications include increasing the height of the test cell to allow more comfortable access to the
interior and the integration of automated crop-tending systems.
One option for greater sunlight transmission is to widen the glazing. However, as demonstrated in
Section 4.3.3
, this would increase cell air temperatures. Based on results from
Section 4.3.3
, drawing electric power from the PV panels decreases internal temperatures. If the glazing area was widened, then more light would be transmitted to the interior of the test cell, and if the PV
panels had at least an efficiency of 14%, then no significant increase in test-cell air temperature would occur. An efficiency of 21% would lead to a reduction in daytime temperatures throughout the
summer. The drop in test-cell temperatures during the day would keep the test-cell air closer to the ideal temperature of 20 °C (68 °F).
Several improvements to the computer models have been identified. Including temperature-difference-dependent free convection heat transfer coefficients would improve the fidelity of the heat
transport model (constant values are currently in place based on assumed average thermal conditions). Improving the heat transport model would also improve the moisture transport model by increasing
the accuracy of the RH calculation (see
Section 4.1.2
and Equation (26)). The addition of a flow meter to the irrigation system would produce data that could further improve the moisture transport model. Adding diffuse radiation [
] would improve the accuracy of the shadowing model, especially in overcast climates. The effect of thermal earth coupling should be improved by modeling the ground as a two-dimensional (depth
direction and radial outward direction normal to this), transient thermal conductor.
6. Movement to a Comprehensive Agrivoltaic-Based Plant Growth Model and System Optimization
One of the next steps in this work is to produce a model that incorporates the thermal and humidity transport models above with a model for plant growth rate. The most comprehensive of the latter is
STICS [
]. Once completed, the agrivolatic greenhouse can be optimized by maximizing the monetary value of the two outputs, namely the value of the electricity produced as predicted by the solar input and PV
efficiency from above and the value of the wet mass of the agricultural product [
]. These two outputs are predicted by the plant growth from STICS subject to changes in greenhouse geometry, construction materials, plant types, numbers and locations in the greenhouse, and
environmental conditions. Clearly, STICS requires values for many input parameters that we have not considered, since growth-rate models were beyond the scope of the present study. One influencing
growth-rate factor is the CO
history in the greenhouse, which we have not modeled. However, a first-order approximation would be to the consider air infiltration (already included in the model) sufficient to keep the CO
concentration equal to about 400 ppm, that of the outside ambient air [
], which ignores the CO
contribution from soil respiration.
7. Comment on Scalability
Scalability, the extent to which the models may be accurately applied to agrivoltaic greenhouse designs of different sizes, dimensions, and materials enters the problem in several ways. For the
thermal and moisture transport models, constant values of the convective heat transfer coefficients (h), radiation surface properties, and radiation view factors were used based on fundamentals from
the heat transfer literature and were adjusted slightly during model calibration as described above. Good scalability may be assured if the designer refers to these fundamentals in carrying out the
design. For example, free convection (in addition to radiation) heat transfer occurs between the inside of PV panels and the concrete block on the test-cell floor. Correlations for h exist in the
literature for the geometry of the test cell and the intensity of the convection (i.e., laminar or turbulent flow). These should be sought out and used in the design. The same holds for radiation
view factors and the need for accurate radiation surface properties for test-cell materials. The equations for the shadowing model and the model for heat transfer in the concrete block and soil mass
are both general (no correlations are used) and will scale without any restrictions. Scalability of the relative humidity model is more challenging as we used the Hargreaves equation for
evapotranspiration of water from the planters with a correction factor (0.525) determined by comparing with measured humidity data. We have no data to suggest the general nature of this correction
factor. See Equation (A10).
8. Conclusions
Our test cell has shown promising results, and further work on developing its design is already underway. Low-light-tolerant plants, such as kale, can be grown in the test cell during extended time
periods outside of the normal growing season. Light levels should be increased to improve the instantaneous photosynthetic rate. A design change such as widening the glazing area between the panels
would have such an effect. The test cell reduces the number of hours the cell air spends below 0 °C (32 °F) by 36.6%, as compared to ambient. Powering an electrical load from the PV panels will
generally improve growing performance by keeping temperatures near 20 °C (68 °F) for more of the year. More importantly, the load tends to reduce daily maximum temperatures, especially during the
summer, but has a negligible effect on daily minimum temperatures. Days with lower maximum temperatures, like those during the winter, are also affected less.
Stilt-mounted PV arrays, as seen in [
], block much less sunlight than the panels in our test cell. However, stilt-mounted PV arrays do not extend the growing season, as the plants grow in their normal ambient environment. One of our
major objectives was to extend the growing season, which stilt-mounted PV arrays cannot do.
Based on the net photosynthetic gain each day per unit area shown in
Section 4.3.2
, it is possible to grow kale in the test cell, because plants inside it receive sufficient PAR irradiance to produce a net gain in energy and biomass on most days. When both beam and diffuse
radiation are considered, it is clear that most of the base area can be productively used for plant growth. The best growing area is along a line running north–south, which is centered between the
east and west sides of the test cell.
Increasing the panel area results in an increase in cell temperatures during periods of intense solar radiation over those seen in the base case. This increase in temperatures can be counteracted, in
full or in part, by powering a load from the panels. The same behavior is expected if the glazing area between the panels is doubled. If the system is scaled up for use in agricultural land, and the
glazing area is increased, it is expected that the system will function as intended and that temperatures in the test cell will remain above ambient throughout the year.
Author Contributions
Conceptualization, F.R.S. and G.F.J.; methodology, G.F.J. and M.E.E.; software, M.E.E.; validation, M.E.E., G.F.J. and F.R.S.; formal analysis, M.E.E.; investigation, M.E.E.; resources, G.F.J.; data
curation, M.E.E.; writing—original draft preparation, M.E.E.; writing—review and editing, G.F.J., F.R.S. and J.A.L.; visualization, M.E.E.; supervision, G.F.J. and F.R.S.; project administration,
G.F.J. and F.R.S.; funding acquisition, G.F.J. All authors have read and agreed to the published version of the manuscript.
This research was funded by internal supporting research at Villanova University, which supported co-author M.E.E. The authors appreciate this support through the College of Engineering.
Data Availability Statement
Two solar panels were donated by Solar States LLC.
Conflicts of Interest
The authors declare no conflict of interest. The funder for co-author M.E.E. had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the
manuscript; or in the decision to publish the results.
Variable Definition
A Surface area, m^2
B Solar radiation transmitted through a transparent surface, W/m^2
C[p] Specific heat capacity, J/kg·K
E[b] Emissive power, W/m^2
E Evaporation rate, kg/s
ET Evapotranspiration rate, kg/s
h Coefficient of free convection, W/m^2·K
J Radiosity, W/m^2
k Thermal conductivity, W/m^2·K
L Through-thickness length, m
M Mass, kg
Mdot Mass flow rate, kg/s
p Index of the five-minute period per day from 0 to 288
p[a] Air vapor pressure, Pa
p[g] Saturation air vapor pressure, Pa
H Heigh of obstruction (i.e., side panel), m
q Heat flow rate, W
RH Relative humidity
R[a] Extraterrestrial solar radiation, W/m^2
R[c] Thermal contact resistance, m^2·K/W
S Incident solar radiation, W/m^2
SOF Sky obstruction factor
SP Shadow projection point, m
T Temperature, K
U Transmitted solar radiation, W/m^2
V Volume, m^3
y Horizontal distance from panel edge, m
α Thermal diffusivity, m^2/s
β Angle from horizontal, degrees
Δx Material layer thickness, m
δ Declination angle, degrees
ε Emissivity
η Solar altitude angle, degrees
λ Humidity ratio
φ Latitude, degrees
ρ Density, kg/m^3
σ Stefan-Boltzmann constant, W/m^2·K^4
ω Hour angle, degrees
a Test cell air
con Concrete
cond Conduction
cd Condensation
east Eastern facing solar panel
evap Evaporative cooling
cv Free Convection
flux Heat Flux
i ith node
in Inner
infi Infiltration
irrig Irrigation
lex Polycarbonate (i.e., Lexan)
ligh SR transmitted through Lexan
liq Liquid water node
low Lower thermal node
max Maximum
out Outer
over Overflow
plant Planters
rad Radiation heat exchange
rise Sunrise
sidp Side Panel
sky Radiation heat exchange with sky
soil Soil
sol Incident solar radiation
sp Solar Panel
set Sunset
tcc Thermal Contact Conductance
up Upper thermal node
west Western facing solar panel
wind Free convection with wind
Appendix A
The test cell was instrumented with solar radiation, temperature, humidity, windspeed, and wind direction sensors. To collect solar radiation data, an Onset S-LIB-M003 silicon pyranometer was
selected, with a measurement range of 0–1280 W/m
and a resolution of 1.25 W/m
. The sensor has an uncertainty of ±10 W/m
or ±5% of the reading (whichever is greater), and an additional ±0.38 W/m
/°C (±0.21 W/m
/°F) for a temperature greater or less than 25 °C (77 °F) [
The temperature and humidity data measurements were made with Elitech GSP-6 temperature and humidity data loggers. This logger has two separate sensors, one for the temperature and one for the
humidity. The temperature measurement range is −40 °C (−40 °F) to 85 °C (185 °F). The temperature accuracy is ±0.5 °C (0.9 °F) when within the temperature range of −20 °C (−4 °F) to 40 °C (104 °F)
and ±1 °C (1.8 °F) when outside of that range. The temperature resolution is 0.1 °C (0.18 °F) [
For the humidity sensors, the measurement range is 10–99% RH. The accuracy for this RH measurement is ±3% at 25 °C (77 °F) between 20–90% RH and ±5% outside of this range. The resolution of the RH
measurements is 0.1%.
In total, the GSP-6 can collect 16,000 data points between uploads [
Wind speed and direction were measured by a Davis wind speed and direction smart sensor from Onset [
]. It has a measurement range from 0 to 76 m/s and wind direction from 0 to 355 degrees. The resolution of the wind speed sensor is 0.5 m/s, and the resolution of the wind direction sensor is 1
degree. The wind speed sensor has an accuracy of ±1.1 m/s or ±5%, whichever is greater. The accuracy for the wind direction sensor is ±7 degrees [
Appendix B
As a nondestructive means of estimating a plant’s dry mass, allometry is a predictive method based on the measurement of a characteristic dimension of a plant, such as its stem length, leaf area, or
average leaf dimensions. The dry mass of plants in the test cell and control cell were determined using measurements of total leaf area to determine an empirical allometric function, thus avoiding
the need for destructive leaf testing.
An allometric function is created by measuring several aspects of a selected plant, in our case, leaf area, stem length, leaf number, and stem thickness. Then, the plant is harvested and dried, and
its dry mass is obtained through measurement. Dry mass is then plotted against measured plant quantities, and the allometric function is determined by a regression analysis.
Figure A1
shows plant dry mass vs total leaf area and several trial curve fits. A second-order polynomial was found to give the best fit and was used in
Section 4.2.2
. To test the second-order polynomial, several plants were harvested, measured, and dried. Their dry mass was compared to the value predicted by the selected function. This is shown in
Table A1
. Both
Table A1
Figure A1
show an outlier. It is believed that this plant grew abnormally and in contradiction to the previously established trend.
Figure A1. Dry mass as a function of total leaf area. Trial curve fits include linear, a power function, and a second-order polynomial. The second-order polynomial showed best agreement with the
Table A1. Comparison of experimentally derived plant dry mass sampled from the test cell to the value predicted by the allometric correlation of the same plants by a second-order polynomial.
Plant # Leaf Area (cm^2) Correlation Result (kg) Experimental Measurement (kg) % Error
1 15.3 0.065 0.063 3.1
2 26.1 0.112 0.113 1.1
3 33.3 0.161 0.112 43.7
4 38 0.164 0.151 8.3
Appendix C
Several well-known equations were used to model phenomenon discussed regarding Equations (1)–(10). Equations (A1)–(A3) and (A6)–(A9) were found from [
], and Equations (A4) and (A5) were found from [
]. For convection heat transfer, Newton’s law of cooling is
where h is from appropriate correlations for the convection type (laminar or turbulent) and geometry [
For radiation heat transfer, the Stephan-Boltzmann law was used. For example, for radiation between the sky temperature and a Lexan surface at temperature T, we have
q[sky,lex] = σ ε (T[sky]^4 − T^4) A
With tilt of the solar panels and the effect of blockage from trees, buildings, and other obstructions being included, we obtain
q[sky,sp] = SOF σ ε (T[sky]^4 − T^4) A (1 + cos β)/2
where the dimensionless SOF (0 ≤ SOF ≤ 1) accounts for the above obstructions in the manner of a radiation view factor.
The heat flow rate due to incident solar radiation is
The heat flow rate due to light transmitted to the interior of the test cell incident on an interior surface is
We assume that all light transmitted to the interior of the test cell is diffuse and distributed uniformly over all internal surfaces.
The net radiation method is used to calculate the thermal radiation exchange among surfaces that can view one another. This is
q[rad] = (E[b] − J) A / ((1 − ε)/ε)
where E
is the blackbody emissive power, J is the radiosity, and ε is the emissivity of the participating surface [
For conduction heat transfer, Fourier’s law for 1D heat flow is
q[cond] = (k/L) (T[i] — T[j]) A
The equation for heat flow due to thermal contact conductance follows the form of Fourier’s law and is
For heat flow due to air infiltration, we obtain
q[infi] = Mdot[a] C[pa] (T[int] — T[amb])
where T
is the test cell air temperature and T
is the ambient temperature.
The evapotranspiration of water from the planters, Equation (A10), is modeled using the Hargreaves equation [
]. The Hargreaves equation predicts evapotranspiration based primarily on the maximum and minimum daily temperature. It is converted from mm/day into kg/s for the purposes of our equation, and a
scaling factor of 0.525 was applied based on the parametric study, as the equation was being used in an environment for which it was not developed. It is important to note that the daily evaporation
predicted by the Hargreaves equation was assumed to be constant throughout the day to convert the equation from mm/day to kg/s. The Hargreaves equation used in the present work is
ET[plant] = 0.525 × 0.0022 · R[a] · TR^0.5 (TC + 17.8)
In these equations, TR is the difference between the maximum and minimum temperatures of the day, and TC is the average temperature of the day in °C.
The equation for mass flow rate due to infiltration is
M[infih2o] = Mdot[a] (λ[amb] − λ[cell])
Appendix D
All parameters for the thermal and moisture transport model are presented in
Table A2
Table A3
Table A4
Table A5
Table A6
Table A7
Table A8
Table A9
Table A10
. All other parameters and values necessary for the model were derived from those presented here. Some of the parameters listed below are universal constants, like G
, σ, and R
which are the solar constant, the Stefan-Boltzmann constant, and the gas constant, respectively. Other constants, such as all angles and lengths, were measured values from the test cell, some of
which were adjusted for model calibration in the parametric simulations. Finally, values such as specific heat capacities, densities, emissivity values, and other material properties were set as
general values accepted for those materials. Some values in this appendix were slightly varied during parametric simulations. The values used in the base case are presented here.
$G s c = 1367 W m 2$ $β w e s t , s p = 0.63 rad$ $β e a s t , s p = 0.61 rad$
$β l e x = 0 rad$ $σ = 5.67 · 10 − 8 W m 2 K 4$ $R g a s = 8.3144 kg · m 2 K · mol · s 2$
$ρ l e x = 1210 kg m 3$ $A l e x = 0.394 m 2$ $α l e x = 0.19$
$Δ x l e x = 0.0024 m$ $C P l e x = 1250 J kg · K$ $ϵ l e x = 0.7$
$W S P = 1.4 m$ $L S P = 1.97 m$ $M S P = 32.3 kg$
$k S P , e f f = 1.05 W m · K$
Table A5. Constant parameters for soil. Also included here are values for the planters which contain soil.
$ρ s o i l = 1650 kg m 3$ $A p l a n t = 0.48 m 2$ $C P S o i l = 1000 J kg · K$
$k s o i l = 1.56 W m · K$ $α s o i l = 0.99 · 10 − 6 m 2 s$ $ϵ s o i l = 0.93$
$Δ x s o i l = 0.046 m$
$ρ a i r = 1.2 kg m 3$ $C P a i r = 1005 J kg · K$ $h e f f , s p , l e x = 2 W m 2 · K$
$h e f f , c o n = 3.75 W m 2 · K$
$ρ c o n = 1920 kg m 3$ $Δ x c o n = 0.05 m$ $C P c o n = 835 J kg · K$
$ϵ c o n = 0.94$ $k c o n = 0.72 W m · K$
Table A8. Miscellaneous constant parameters related to thermal contact conductance, liquid water properties, irrigation rates, sky view factors, and the Hargreaves equation scale factor.
$R c c o n , c o n = 0.04 K · m 2 W$ $R c c o n , s o i l = 0.06 K · m 2 W$ $M ˙ s o i l = 6.57 · 10 − 6 kg s$
$S O F e a s t = 0.6$ $M ˙ i r r i g = 1.06 · 10 − 5 kg s$
$S O F w e s t = 0.5$ $H G s c a l e , f a c t o r = 0.525$
ThicknessGlass = 4 mm ThicknessSi = 0.5 mm ThicknessCoat = 0.25 mm
CpSPglass = 700 J/K*kg (quartz glass) CpSi = 705 J/K*kg (silicon)
Cpcoat = 1900 J/K*kg (EVA) EpsSPb = 0.85 for aluminum (originally 0.77) (Emissivity) EpsSPt = 0.93 for glass (originally 0.93) (Emissivity)
Tasp = 0.915 for glass (transmittance–absorbtance product)
$k q = 0.003$ $I 0 = 13 μ mols photons m 2 · s$ $P m a x = 20.3 μ mols CO 2 fixed m 2 · s$
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Figure 1.
Categories of agrivoltaic arrays: (
) interspersed PV arrays; (
) greenhouse-mounted PV arrays (PVs shown as horizontal, but they can be appropriately inclined); (
) stilt-mounted PV arrays [
] (open access article distributed under the Creative Commons Attribution License).
Figure 2. The test cell: (a) exterior view of the south-facing side; (b) interior view of the north-facing side.
Figure 4. Mass transfer nodes and interactions in the computational model. The white bar is the transparent polycarbonate, the brown bars are wood side walls, the tan bars are foam board insulation,
the red rectangle represents planters, and the grey rectangle is the concrete block.
Figure 6. Solution flow chart. Maximum and minimum daily temperatures are model-predicted maximum and minimum daily temperatures for test cell air. All terms ‘mass’ refer to water mass.
Figure 9. Yearlong simulation results and experimental temperature data. “Exp sensor 1” refers to the temperature sensor located on the east side of the test cell base, and “Exp sensor 2” refers to
the sensor on the west side. Periods of sensor interruption or snow cover, where data were linearly interpolated, are highlighted.
Figure 10. Model-predicted daily maximum temperatures (red asterisks) and minimum temperatures (circles) vs. experimental maximum and minimum temperatures for the same day.
Figure 11. Predicted RH and from measurement. Note: over-prediction occurred mostly during the winter and the under-prediction during the summer. Start date of 17 January 2020.
Figure 12. Model-predicted daily minimum RH based on the model-predicted cell temperature (diamonds) and the model-predicted minimum RH based on measured temperature (circles) plotted against
measured minimum RH for the same day.
Figure 13. Example that shows improvement in RH using measured cell air temperature instead of the model predicted. Note: the change in temperature for the two identified points was 6.1 °C (11 °F)
from that predicted by the model. 27 February 2020 to 7 March 2020 test period.
Figure 14. Predicted (pred) solar radiation on 5 October 2019 and experimental (exp) data for the same day. Sensors 1, 2, 3, and 4 are located west in the test cell, center in the test cell, east in
the test cell, and ambient, respectively.
Figure 16. Expanded view of test-cell air temperature measurements compared with ambient. In general, wintertime test-cell air temperatures were greater than ambient.
Figure 17. Progression of the October-2019 plant-growth experiment: (a) 15 October 2019; (b) 20 November 2019; (c) 19 February 2020; (d) 26 March 2020.
Figure 18. Average aboveground plant dry mass in the test cell and control cell for each sampling since the start of the October-2020 growth test. The arrow indicates the point at which control cell
plants died.
Figure 19. Parity plots of maximum and minimum temperature test-cell air values from the model cases versus the base-case results.
Figure 20. Net photosynthetic flux of CO[2] per day per unit area as found by using solar irradiance from the shadowing model. (a) Shows the net photosynthetic flux within the test cell at the
points, and (b) shows the ambient. The zero line delineates a value of no net gain or loss.
Figure 21. Net photosynthetic gain of CO[2] per day per unit area as found using solar irradiance from measured solar radiation. This was found using an assumed clearness index of 1 for the whole
year. The zero line delineates a value of no net gain or loss.
Figure 22. Distribution of solar radiation at the base of the test cell over a day (the summer solstice). The vertical axis is the ratio of the actual beam radiation incident on each discrete node
surface to an unshaded node surface.
Table 1. Cases for the parametric study in which the calculations converged (only case 16 did not). Note: Case 14 had the same number of planters as in the base case. For this case the planter height
was doubled.
Case Change from Base Case Figure 19
1 Base Case -
2 Twice the length in the N–S dir. a
3 Twice the glazing width in the E–W dir. b
4 Case 2 with 7% load c
5 Case 2 with 14% load d
6 Case 2 with 21% load e
7 Case 3 with 3% load f
8 Case 3 with 14% load g
9 Case 3 with 21% load h
10 Two layers of concrete blocks i
11 No concrete blocks j
12 Twice the planters in the test cell k
13 Four times the number of planters in the test cell l
14 Twice the soil mass in the planters m
15 Four times the soil mass in the planters n
16 Twice the test-cell height -
Table 2. Percentage of the year each parametric run, the data from the test cell, and the ambient conditions spent within bands of ±1.1, ±2.2, and ±3.3 °C in relation to 20 °C (68 °F).
Case % of Year % of Year % of Year % of Year below Freezing
in ±1.1 Band in ±2.2 Band in ±3.3 Band
Test Cell 4.83 10.19 15.83 5.70
Ambient 7.61 15.26 21.98 9.01
Base 4.97 10.00 15.34 9.22
2 4.14 8.77 13.79 8.61
3 4.44 9.03 14.31 8.99
4 4.67 9.42 14.82 9.41
5 4.93 9.93 15.26 9.73
6 4.83 9.77 15.13 8.92
7 5.10 10.02 15.53 9.21
8 5.19 10.43 15.82 9.45
9 5.31 10.89 16.81 9.71
10 4.88 9.96 15.27 8.92
11 4.26 8.85 14.09 9.13
12 5.04 10.01 15.42 9.14
13 5.01 10.02 15.47 9.05
14 5.06 10.04 15.43 9.19
15 5.05 10.04 15.47 9.18
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Evans, M.E.; Langley, J.A.; Shapiro, F.R.; Jones, G.F. A Validated Model, Scalability, and Plant Growth Results for an Agrivoltaic Greenhouse. Sustainability 2022, 14, 6154. https://doi.org/10.3390/
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Evans, Michael E., J. Adam Langley, Finley R. Shapiro, and Gerard F. Jones. 2022. "A Validated Model, Scalability, and Plant Growth Results for an Agrivoltaic Greenhouse" Sustainability 14, no. 10:
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Can a Beamtube provide power for a whole room?
Technical Resources- Answers to Questions you have:
Can a Beamtube provide power for a whole room?
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Mathematically, this is a cube factor, so 10 times the distance is 10 x 10 x 10, or 1,000 times the area, dividing the energy of the original square by 1000- or 0.001 watts for a 1 watt source.
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