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Fuzzy networks for complex systems: a modular rule base approach
This book introduces the novel concept of a fuzzy network whose nodes are rule bases and the connections between the nodes are the interactions between the rule bases in the form of outputs fed as
inputs. The concept is presented as a systematic study for improving the feasibility and transparency of fuzzy models by means of modular rule bases whereby the model accuracy and efficiency can be
optimised in a flexible way. The study uses an effective approach for fuzzy rule based modelling of complex systems that are characterised by attributes such as nonlinearity, uncertainty,
dimensionality and structure.The approach is illustrated by formal models for fuzzy networks, basic and advanced operations on network nodes, properties of operations, feedforward and feedback fuzzy
networks as well as evaluation of fuzzy networks. The results are demonstrated by numerous examples, two case studies and software programmes within the Matlab environment that implement some of the
theoretical methods from the book. The book shows the novel concept of a fuzzy network with networked rule bases as a bridge between the existing concepts of a standard fuzzy system with a single
rule base and a hierarchical fuzzy system with multiple rule bases.
Original language English
Place of Publication Berlin
Publisher Springer
Number of pages 316
Edition 259
ISBN (Print) 9783642155994
Publication status Published - 2010
Publication series
Name Studies in Fuzziness and Soft Computing
Publisher Springer
No. 259
Dive into the research topics of 'Fuzzy networks for complex systems: a modular rule base approach'. Together they form a unique fingerprint. | {"url":"https://researchportal.port.ac.uk/en/publications/fuzzy-networks-for-complex-systems-a-modular-rule-base-approach","timestamp":"2024-11-13T05:12:51Z","content_type":"text/html","content_length":"52550","record_id":"<urn:uuid:589b9054-a469-4de7-8008-0ec682879a2e>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00644.warc.gz"} |
defining loop using weighted sum-multi objective
Dear All,
I am trying to use loop for objective function which include two different objective. I am trying to use weighted sum and trying to observed the effect of weighted sum to each objective function.
I could define the parameter and I can run the model without an error, but I observed that for each weighted sum (different value), I keep getting the same objective value.
I am not sure where I am making a mistake but some how it looks like the loop I defined is not taking all the scenarios.
Could you please take a look at the below model and let me know is there any other way to define the loop?
Thank you in advance.
set p(t) /year1*year4/;
set w’ lambda’/1*9/;
lambda(w) ‘weight on objective function’
1 0.1
2 0.2
3 0.3
4 0.4
5 0.5
6 0.6
7 0.7
8 0.9
9 0.9
objective … h =e=sum((w,t),CS(t)*lambda(w))+(-sum((i,m,t,w), l(i)*u(i,m)x(i,t,m)(1-lambda(w))));
parameter lam(w), resobj(w);
solve thesisdeterministic using MIP maximizing h;
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Hi Deniz,
If I get you right, you want to solve your model several times with different values for lambda.
In your current code you loop over p (year1 to year4) and assign the values of lambda(w) to lam(w) which is completely independent from the loop controlling set p. Hence you always get identical
results. I also think that you actually want to use a scalar lambda in your objective function instead of the sum over all lambda(w).
What you can do is to define parameter lam as a scalar and adjust the objective function accordingly. Then you loop over w, assign the value of lambda(w) to lam and solve your model.
set p(t) /year1*year4/;
set w ‘lambda’/1*9/;
lambda(w) ‘weight on objective function’ /
1 0.1
2 0.2
3 0.3
4 0.4
5 0.5
6 0.6
7 0.7
8 0.9
9 0.9
parameter lam, resobj(w);
objective … h =e=sum(t,CS(t))*lam + (-sum((i,m,t), l(i)*u(i,m)x(i,t,m)))(1-lam);
lam = lambda(w);
solve thesisdeterministic using MIP maximizing h;
Hope this helps.
On Monday, March 7, 2016 at 7:55:02 AM UTC+1, deniz wrote:
Dear All,
I am trying to use loop for objective function which include two different objective. I am trying to use weighted sum and trying to observed the effect of weighted sum to each objective function.
I could define the parameter and I can run the model without an error, but I observed that for each weighted sum (different value), I keep getting the same objective value.
I am not sure where I am making a mistake but some how it looks like the loop I defined is not taking all the scenarios.
Could you please take a look at the below model and let me know is there any other way to define the loop?
Thank you in advance.
set p(t) /year1*year4/;
set w’ lambda’/1*9/;
lambda(w) ‘weight on objective function’
1 0.1
2 0.2
3 0.3
4 0.4
5 0.5
6 0.6
7 0.7
8 0.9
9 0.9
objective … h =e=sum((w,t),CS(t)*lambda(w))+(-sum((i,m,t,w), l(i)*u(i,m)x(i,t,m)(1-lambda(w))));
parameter lam(w), resobj(w);
solve thesisdeterministic using MIP maximizing h;
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Thank you so much!
You are absolutely right! I appreciated your help very much!
On Monday, March 7, 2016 at 12:55:02 AM UTC-6, deniz wrote:
Dear All,
I am trying to use loop for objective function which include two different objective. I am trying to use weighted sum and trying to observed the effect of weighted sum to each objective function.
I could define the parameter and I can run the model without an error, but I observed that for each weighted sum (different value), I keep getting the same objective value.
I am not sure where I am making a mistake but some how it looks like the loop I defined is not taking all the scenarios.
Could you please take a look at the below model and let me know is there any other way to define the loop?
Thank you in advance.
set p(t) /year1*year4/;
set w’ lambda’/1*9/;
lambda(w) ‘weight on objective function’
1 0.1
2 0.2
3 0.3
4 0.4
5 0.5
6 0.6
7 0.7
8 0.9
9 0.9
objective … h =e=sum((w,t),CS(t)*lambda(w))+(-sum((i,m,t,w), l(i)*u(i,m)x(i,t,m)(1-lambda(w))));
parameter lam(w), resobj(w);
solve thesisdeterministic using MIP maximizing h;
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Grade 3 Fact or Fib Showdown Quarter 3
This google slide show will be used by third grade teachers with their students as a numeracy routine. Students should be given time to notice and wonder about each slide. Students then engage in
discourse to tell whether they believe the math presented is a fact (true) or a fib (false) and why. | {"url":"https://wlresources.dpi.wi.gov/authoring/1214-grade-3-fact-or-fib-showdown-quarter-3/view","timestamp":"2024-11-09T04:16:58Z","content_type":"text/html","content_length":"57931","record_id":"<urn:uuid:f6633a6f-0773-48bb-acae-529afc140b9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00638.warc.gz"} |
Piercing convex sets and the Hadwiger-Debrunner (p, q)-problem
A family of sets has the (p, q)property if among any p members of the family some q have a nonempty intersection. It is shown that for every p ≥ q ≥ d + 1 there is a c = c(p, q, d) < ∞ such that for
every family J of compact, convex sets in R^d which has the (p, q) property there is a set of at most c points in R^d that intersects each member of J. This settles an old problem of Hadwiger and
All Science Journal Classification (ASJC) codes
Dive into the research topics of 'Piercing convex sets and the Hadwiger-Debrunner (p, q)-problem'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/piercing-convex-sets-and-the-hadwiger-debrunner-p-q-problem","timestamp":"2024-11-02T02:25:43Z","content_type":"text/html","content_length":"44956","record_id":"<urn:uuid:b5f61fcb-38c2-4e3d-b738-3c98dbaf91a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00773.warc.gz"} |
Tutorial: Working with Numerical Data using Pandas - MachineLearningTutorials.org
Pandas, Python
Tutorial: Working with Numerical Data using Pandas
Pandas is a popular Python library that provides data manipulation and analysis tools, especially well-suited for working with structured data. One of its key features is its ability to handle
numerical data efficiently, making it an essential tool for data scientists, analysts, and researchers. In this tutorial, we will explore how to work with numerical data using Pandas, focusing on
various operations, data cleaning, and analysis techniques. We’ll cover the following topics:
1. Loading and Inspecting Numerical Data
2. Basic Numerical Operations
3. Dealing with Missing Values
4. Aggregation and Summary Statistics
5. Visualizing Numerical Data
Throughout this tutorial, we’ll provide explanations and examples to help you understand each concept thoroughly.
1. Loading and Inspecting Numerical Data
To get started, you’ll need to have Pandas installed. If you haven’t installed it yet, you can do so using the following command:
pip install pandas
Now, let’s begin by importing Pandas and loading a dataset containing numerical data:
import pandas as pd
# Load a CSV file into a Pandas DataFrame
data = pd.read_csv('numerical_data.csv')
# Display the first few rows of the DataFrame
Replace 'numerical_data.csv' with the path to your dataset file. The head() function displays the first few rows of the DataFrame, allowing you to inspect the structure of the data.
2. Basic Numerical Operations
Pandas provides various functions to perform basic numerical operations on your data. Let’s explore some of these operations using a hypothetical dataset of student exam scores:
# Suppose the DataFrame 'scores' contains columns 'math' and 'physics'
math_mean = data['math'].mean() # Calculate the mean of the 'math' column
physics_max = data['physics'].max() # Find the maximum value in the 'physics' column
print(f"Mean math score: {math_mean}")
print(f"Maximum physics score: {physics_max}")
Here, we used the mean() function to calculate the mean and the max() function to find the maximum value in the specified columns.
3. Dealing with Missing Values
Real-world datasets often contain missing values, which can hinder analysis. Pandas provides tools to handle missing data effectively. Let’s use the same ‘scores’ dataset to demonstrate:
# Count the number of missing values in each column
missing_values = data.isnull().sum()
# Drop rows with any missing values
cleaned_data = data.dropna()
print("Missing values per column:")
print("\nCleaned data shape:", cleaned_data.shape)
In this example, the isnull().sum() function counts the number of missing values in each column, and the dropna() function removes rows containing any missing values. This helps in cleaning the
dataset before analysis.
4. Aggregation and Summary Statistics
Pandas simplifies the process of calculating summary statistics and aggregating data. Let’s use a sales dataset to demonstrate how to calculate total sales for each product:
# Suppose the DataFrame 'sales' contains columns 'product' and 'sales_amount'
total_sales_per_product = data.groupby('product')['sales_amount'].sum()
print("Total sales per product:")
The groupby() function groups the data based on the ‘product’ column, and then the sum() function calculates the total sales amount for each product.
5. Visualizing Numerical Data
Visualizations can provide insights into your numerical data. Pandas can work well with visualization libraries like Matplotlib and Seaborn. Here’s an example of plotting a histogram of exam scores:
import matplotlib.pyplot as plt
import seaborn as sns
# Set up the visualization settings
plt.figure(figsize=(10, 6))
# Create a histogram of math scores
sns.histplot(data['math'], bins=10, kde=True)
plt.title("Distribution of Math Scores")
plt.xlabel("Math Score")
In this example, we used Seaborn to create a histogram of the ‘math’ scores, showing the distribution of scores across different ranges.
In this tutorial, we explored the fundamental concepts of working with numerical data using Pandas. We covered loading and inspecting data, performing basic numerical operations, handling missing
values, calculating summary statistics, and visualizing data. These skills are crucial for anyone involved in data analysis and manipulation tasks. By mastering these techniques, you’ll be better
equipped to handle and make sense of numerical data in various real-world scenarios.
Remember that practice is key to mastering these concepts. Feel free to experiment with different datasets and scenarios to deepen your understanding of Pandas and its capabilities. Happy data | {"url":"https://machinelearningtutorials.org/tutorial-working-with-numerical-data-using-pandas/","timestamp":"2024-11-04T02:20:20Z","content_type":"text/html","content_length":"100722","record_id":"<urn:uuid:50b921ff-234d-4956-802a-59a3f7f39481>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00292.warc.gz"} |
Cool and Luminous Transients from Mass-Losing Binary Stars
Cool and Luminous Transients from Mass-Losing Binary Stars
Pejcha et al
We study transients produced by equatorial disk-like outflows from catastrophically mass-losing binary stars with an asymptotic velocity and energy deposition rate near the inner edge which are
proportional to the binary escape velocity v_esc. As a test case, we present the first smoothed-particle radiation-hydrodynamics calculations of the mass loss from the outer Lagrange point with
realistic equation of state and opacities. The resulting spiral stream becomes unbound for binary mass ratios 0.06 less than q less than 0.8. For synchronous binaries with non-degenerate
components, the spiral-stream arms merge at a radius of ~10a, where a is the binary semi-major axis, and the accompanying shock thermalizes 10-20% of the kinetic power of the outflow. The
mass-losing binary outflows produce luminosities proportional to the mass loss rate and v_esc, reaching up to ~10^6 L_Sun. The effective temperatures depend primarily on v_esc and span 500 less
than T_eff less than 6000 K. Dust readily forms in the outflow, potentially in a catastrophic global cooling transition. The appearance of the transient is viewing angle-dependent due to vastly
different optical depths parallel and perpendicular to the binary plane. The predicted peak luminosities, timescales, and effective temperatures of mass-losing binaries are compatible with those
of many of the class of recently-discovered red transients such as V838 Mon and V1309 Sco. We predict a correlation between the peak luminosity and the outflow velocity, which is roughly obeyed
by the known red transients. Outflows from mass-losing binaries can produce luminous (10^5 L_Sun) and cool (T_eff less than 1500 K) transients lasting a year or longer, as has potentially been
detected by Spitzer surveys of nearby galaxies. | {"url":"https://thedragonsgaze.blogspot.com/2015/10/cool-and-luminous-transients-from-mass.html","timestamp":"2024-11-07T04:46:12Z","content_type":"text/html","content_length":"76911","record_id":"<urn:uuid:80138ff3-1e9f-4afb-9479-c381935b3f0f>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00134.warc.gz"} |
Reward Mechanism | Fraction AI Docs
Contributors and verifiers are paid $k\%$ of the total Reward Pool at the end of every epoch. The reward per participant is dependent on their reputation at the time of submission/verification.
Dataset creator specifies the following values:
$\lambda_c \in (0, 1]$: Reward multiple for contributors
$\lambda_v \in (0, 1]$: Reward multiple for verifiers
Contributor Rewards
Let $m_i$be the number of submissions by contributor $i$ at the end of current epoch
Now for any submission $j$,
$r_{ij}$: reputation of contributor $i$ while making submission j
$C_i$ : reward for contributor $i$
$\delta_j= \begin{cases} 1 &\text{if j is accepted} \\ 0 &\text{otherwise } \\ \end{cases}$
$C_i =\lambda_c\sum_{j=1}^{m_i} \delta_j r_{ij}$
Verifier Rewards
Reward Tokens
Suggested Hyperparameters | {"url":"https://docs.fractionai.xyz/data-quality-control/reward-mechanism","timestamp":"2024-11-06T01:12:36Z","content_type":"text/html","content_length":"370678","record_id":"<urn:uuid:66be2791-576b-46de-84f4-ce586add4d1d>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00547.warc.gz"} |
Calculating Change in Pressure/Volume
Calculating Change in Pressure/Volume
Calculating Change in Pressure/Volume
Calculating Change in Pressure/Volume
P x V = constant
If you increase the volume of a gas, what effect will this have on the number of collisions with the walls of the container?
If you increase the volume of the gas, what will be the effect on the pressure?
As you increase the volume of a gas, the pressure decreases by the same amount due to the decrease in the number of collisions. The value of this is shown as $P\times V$. We say that $P$ and $V$ are
inversely proportional. When one goes up, the other goes down by the same amount.
If you increase the volume by a set amount, do you always get the same change in pressure?
If $P$ is pressure and $V$ is volume for a given gas, what can you always say about the value of $P\times V$?
Have a look at this image. What do you notice about the value of $P\times V$, no matter how much you change $V$ and $P$?
A) Always the same B) Increasing C) Decreasing
The volume and pressure of a gas are directly related - they are inversely proportional. This means that for any given gas you will always have the same value of $P\times V$, no matter what the
volume is or the pressure - as long as the temperature is constant.
If the volume of a gas is 2, and the pressure exerted is 20, then what is the value of $P\times V$?
If a gas has a pressure of 50 when the volume is 4, then what is the value of $P\times V$?
A gas has a pressure of 25 and a volume of 4 giving it a $P\times V$ value of 100. If you reduce the volume of the gas what will happen to the value of $P \times V$?
Imagine a gas with a $P$value of 5 and a $V$ value of 20. If the value of $P$ changed to 10, what would be the new value of $V$?
A gas has a volume of 5 and a pressure of 4. If the volume of gas is increased to 10, what will the new pressure be? | {"url":"https://albertteen.com/uk/gcse/physics/the-particle-model/calculating-change-in-pressurevolume","timestamp":"2024-11-03T09:23:47Z","content_type":"text/html","content_length":"132784","record_id":"<urn:uuid:3c431e9b-04db-4587-93d3-0ff371148a09>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00767.warc.gz"} |
Mastering Natural Frequency and Forced Vibrations: A Comprehensive Guide
Natural frequency and forced vibrations are fundamental concepts in the field of physics and engineering, particularly in the study of mechanical systems. These phenomena play a crucial role in the
design, analysis, and optimization of various mechanical structures, from bridges and buildings to machinery and vehicles.
Understanding Natural Frequency
Natural frequency refers to the frequency at which a system vibrates naturally when disturbed. This frequency is determined by the system’s physical properties, such as its mass and stiffness. The
natural frequency formula, given by the equation f = √(k/m), where f is the natural frequency, k is the stiffness of the system, and m is the mass of the system, is a critical parameter for designing
and analyzing mechanical systems.
Theorem: Natural Frequency Formula
The natural frequency formula is given by:
f = √(k/m)
where f is the natural frequency, k is the stiffness of the system, and m is the mass of the system.
Physics Formula: Natural Frequency
The natural frequency formula is given by:
f = √(k/m)
where f is the natural frequency, k is the stiffness of the system, and m is the mass of the system.
Physics Example: Calculating Natural Frequency
Consider a mass-spring system with a mass of 1 kg and a spring constant of 100 N/m. The natural frequency of the system can be calculated as:
f = √(k/m) = √(100/1) = 10 Hz
Physics Numerical Problem: Determining Natural Frequency
A mass-spring system has a mass of 2 kg and a spring constant of 500 N/m. Calculate the natural frequency of the system.
f = √(k/m) = √(500/2) = 22.36 Hz
Forced Vibrations
Forced vibrations occur when an external force is applied to a system at a specific frequency. When a system is subjected to forced vibrations, it may exhibit resonance at certain frequencies,
leading to excessive vibrations and potential structural failure. Measuring and analyzing the frequency response of a system under forced vibrations is crucial for ensuring safe and reliable
Frequency Response Curve
A typical frequency response curve for a system under forced vibrations is shown in the figure below:
Data Points: Frequency Response Curve
The frequency response curve can be represented by a set of data points, such as:
Frequency (Hz) Amplitude (m/s^2)
10 0.1
20 0.2
30 0.3
40 0.4
50 0.5
Value: Calculating Natural Frequency
The natural frequency of the system can be calculated as 10 Hz, based on the given mass and spring constant.
Measurement: Frequency Response Curve
The frequency response curve can be measured using a vibration sensor or accelerometer, which can measure the amplitude and phase of vibrations at different frequencies.
Quantifying Vibrations
Vibration measurement is a common method for obtaining measurable and quantifiable data on natural frequency and forced vibrations. This involves measuring the amplitude, frequency, and phase of
vibrations using specialized instruments such as accelerometers, vibration sensors, and laser vibrometers.
Vibration Amplitude Quantification
The relationship between the peak-to-peak level, peak level, average level, and RMS level of a sinewave can be used to quantify the amplitude of vibrations. The RMS value is the most relevant measure
of amplitude as it takes into account the time history of the wave and gives an amplitude value that is directly related to the energy content and destructive abilities of the vibration.
Importance of Understanding Natural Frequency and Forced Vibrations
Understanding natural frequency and forced vibrations is crucial for designing and analyzing mechanical systems. By accurately calculating the natural frequency of a system and analyzing its
frequency response under forced vibrations, engineers can:
1. Predict the system’s response to external forces and identify potential resonance issues.
2. Optimize the design of mechanical structures to minimize the risk of excessive vibrations and potential failure.
3. Ensure safe and reliable operation of machinery and equipment by monitoring and controlling vibration levels.
4. Develop effective vibration mitigation strategies, such as the use of dampers or tuned mass dampers, to reduce the impact of harmful vibrations.
Natural frequency and forced vibrations are fundamental concepts in the field of physics and engineering, with far-reaching applications in the design, analysis, and optimization of mechanical
systems. By mastering the theoretical principles, formulas, and measurement techniques related to these phenomena, engineers and physicists can develop a deep understanding of the behavior of complex
mechanical structures and ensure their safe and reliable operation.
1. Vibration Measurement: The Complete Guide – HBK
2. Natural Frequency Formula: What Is It and Why Is It Important?
3. Introduction to Experiment 2: Vibrations
The techiescience.com Core SME Team is a group of experienced subject matter experts from diverse scientific and technical fields including Physics, Chemistry, Technology,Electronics & Electrical
Engineering, Automotive, Mechanical Engineering. Our team collaborates to create high-quality, well-researched articles on a wide range of science and technology topics for the techiescience.com
All Our Senior SME are having more than 7 Years of experience in the respective fields . They are either Working Industry Professionals or assocaited With different Universities. Refer Our Authors
Page to get to know About our Core SMEs. | {"url":"https://techiescience.com/natural-frequency-and-forced-vibrations/","timestamp":"2024-11-07T19:01:03Z","content_type":"text/html","content_length":"102613","record_id":"<urn:uuid:2f49e181-e282-4bed-a0d0-d38434f2bfdd>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00379.warc.gz"} |
Three consecutive approximation coefficients: Asymptotic frequencies in semi-regular cases
Denote by p n /q n ,n=1,2,3,…, pn/qn,n=1,2,3,…, the sequence of continued fraction convergents of a real irrational number x x . Define the sequence of approximation coefficients by θ n (x):=q n |
q n x−p n |,n=1,2,3,… θn(x):=qn|qnx−pn|,n=1,2,3,… . In the case of regular continued fractions the six possible patterns of three consecutive approximation coefficients, such as θ n−1 <θ n
<θ n+1 θn−1<θn<θn+1 , occur for almost all x x with only two different asymptotic frequencies. In this paper it is shown how these asymptotic frequencies can be determined for two other semi-regular
cases. It appears that the optimal continued fraction has a similar distribution of only two asymptotic frequencies, albeit with different values. The six different values that are found in the case
of the nearest integer continued fraction will show to be closely related to those of the optimal continued fraction.
• Continued fractions
• Metric theory
Dive into the research topics of 'Three consecutive approximation coefficients: Asymptotic frequencies in semi-regular cases'. Together they form a unique fingerprint.
Research output
• 1 Dissertation (TU Delft)
• de Jonge, J.
133 p.
Research output: Thesis › Dissertation (TU Delft) | {"url":"https://research.tudelft.nl/en/publications/three-consecutive-approximation-coefficients-asymptotic-frequenci","timestamp":"2024-11-01T23:15:42Z","content_type":"text/html","content_length":"61652","record_id":"<urn:uuid:6d0b6851-2ccc-4b65-8398-436b42813d6e>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00801.warc.gz"} |
Publications Abstracts
• Brannan, Michael; Eifler, Kari; Voigt, Christian; Weber, M.
Quantum Cuntz-Krieger algebras
arXiv:2009.09466 [math.OA, math.QA], 40 pages (2020).
Motivated by the theory of Cuntz-Krieger algebras we define and study C*-algebras associated to directed quantum graphs. For classical graphs the C*-algebras obtained this way can be viewed as
free analogues of Cuntz-Krieger algebras, and need not be nuclear. We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the
trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix
block up to KK-equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic
to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these C*-algebras are all pairwise non-isomorphic for complete quantum graphs of
different dimensions, even on the level of KK-theory. We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation. We also discuss
quantum symmetries of quantum Cuntz-Krieger algebras in general.
• Schmidt, Simon; Vogeli, Chase; Weber, M.
Uniformly vertex-transitive graphs
arXiv:1912.00060 [math.CO], 17 pages (2019).
We introduce uniformly vertex-transitive graphs as vertex-transitive graphs satisfying a stronger condition on their automorphism groups, motivated by a problem which arises from a Sinkhorn-type
algorithm. We use the derangement graph of a given graph to show that the uniform vertex-transitivity of the graph is equivalent to the existence of cliques of sufficient size in the derangement
graph. Using this method, we find examples of graphs that are vertex-transitive but not uniformly vertex-transitive, settling a previously open question. Furthermore, we develop sufficient
criteria for uniform vertex-transitivity in the situation of a graph with an imprimitive automorphism group. We classify the non-Cayley uniformly vertex-transitive graphs on less than 30
vertices outside of two complementary pairs of graphs.
• Eder, Christian; Levandovskyy, Viktor; Schanz, Julien; Schmidt, Simon; Steenpass, Andreas; Weber, M.
Existence of quantum symmetries for graphs on up to seven vertices: a computer based approach
arXiv:1906.12097 [math.QA, math.CO], 15 pages + appendix (2019).
The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica
capturing the quantum symmetries of the graph. In general, there are more quantum symmetries than symmetries and it is a non-trivial task to determine when this is the case for a given graph:
The question is whether or not the algebra associated to the quantum automorphism group is commutative. We use Gröbner base computations in order to tackle this problem; the implementation uses
GAP and the SINGULAR package LETTERPLACE. We determine the existence of quantum symmetries for all connected, undirected graphs without multiple edges and without self-edges, for up to seven
vertices. As an outcome, we infer within our regime that a classical automorphism group of order one or two is an obstruction for the existence of quantum symmetries.
• Weber, M.
Partition C*-algebras
arXiv:1710.06199 [math.OA, math.CO, math.QA], 19 pages + 11 pages of appendix and references (2017).
We give a definition of partition C*-algebras: To any partition of a finite set, we assign algebraic relations for a matrix of generators of a universal C*-algebra. We then prove how certain
relations may be deduced from others and we explain a partition calculus for simplifying such computations. This article is a small note for C*-algebraists having no background in compact
quantum groups, although our partition C*-algebras are motivated from those underlying Banica-Speicher quantum groups (also called easy quantum groups). We list many open questions about
partition C*-algebras that may be tackled by purely C*-algebraic means, ranging from ideal structures and representations on Hilbert spaces to K-theory and isomorphism questions. In a follow up
article, we deal with the quantum algebraic structure associated to partition C*-algebras.
• Weber, M.
Partition C*-algebras II - links to compact matrix quantum groups
arXiv:1710.08662 [math.OA, math.CO, math.QA], 27 pages (2017).
In a recent article, we gave a definition of partition C*-algebras. These are universal C*-algebras based on algebraic relations which are induced from partitions of sets. In this follow up
article, we show that often we can associate a Hopf algebra structure to partition C*-algebras, and also a compact matrix quantum group structure. This follows the lines of Banica and Speicher's
approach to quantum groups; however, we access them in a more algebraic way circumventing Tannaka-Krein duality. We give criteria when these quantum groups are quantum subgroups of Wang's free
orthogonal quantum group. As a consequence, we see that even if we start with (generalized) categories of partitions which do not contain the pair partitions, in many cases we do not go beyond
the class of Banica-Speicher quantum groups (aka easy quantum groups). However, we also discuss possible non-unitary Banica-Speicher quantum groups.
• Cébron, Guillaume; Weber, M.
Quantum groups based on spatial partitions
arXiv:1609.02321 [math.QA, math.OA], 32 pages (2016).
We define new compact matrix quantum groups whose intertwiner spaces are dual to tensor categories of three-dimensional set partitions - which we call spatial partitions. This extends
substantially Banica and Speicher's approach of the so called easy quantum groups: It enables us to find new examples of quantum subgroups of Wang's free orthogonal quantum group O[n]^+ which do
not contain the symmetric group S[n]; we may define new kinds of products of quantum groups coming from new products of categories of partitions; and we give a quantum group interpretation of
certain categories of partitions which do neither contain the pair partition nor the identity partition.
• Weber, M.
Basiswissen Mathematik auf Arabisch und Deutsch
Springer Spektrum, 2018
168 pages
Dieses Lehrbuch ist speziell für angehende Studierende mit arabischem Sprachhintergrund verfasst, die ein Studium im deutschen Sprachraum aufnehmen wollen. Um ihnen sowohl den sprachlichen als
auch den fachlichen Einstieg zu erleichtern, ist die Gestaltung zweisprachig. Dies ermöglicht sowohl das Anknüpfen an bekannte Inhalte in der Muttersprache als auch das Erlernen der deutschen
Begriffe. Inhaltlich frischt das Buch sehr konzentriert und konkret das nötigste mathematische Abiturwissen auf, das in Studiengängen wie Mathematik, Informatik, Natur- und
Ingenieurwissenschaften vorausgesetzt wird. Das Buch ist grob in Analysis und Algebra gegliedert und beinhaltet möglichst wenige formale Definitionen, dafür aber viele anschauliche Beispiele und
Verfahren sowie Beispielaufgaben.
• Voiculescu, Dan-Virgil; Stammeier, Nicolai; Weber, M. (eds)
Free probability and operator algebras
Münster Lecture Notes in Mathematics
European Mathematical Society (EMS)
132 pages
Zürich, 2016
Free probability is a probability theory dealing with variables having the highest degree of noncommutativity, an aspect found in many areas (quantum mechanics, free group algebras, random
matrices etc). Thirty years after its foundation, it is a well-established and very active field of mathematics. Originating from Voiculescu's attempt to solve the free group factor problem in
operator algebras, free probability has important connections with random matrix theory, combinatorics, harmonic analysis, representation theory of large groups, and wireless communication.
These lecture notes arose from a masterclass in Münster, Germany and present the state of free probability from an operator algebraic perspective. This volume includes introductory lectures on
random matrices and combinatorics of free probability (Speicher), free monotone transport (Shlyakhtenko), free group factors (Dykema), free convolution (Bercovici), easy quantum groups (Weber),
and a historical review with an outlook (Voiculescu). In order to make it more accessible, the exposition features a chapter on basics in free probability, and exercises for each part.
This book is aimed at master students to early career researchers familiar with basic notions and concepts from operator algebras.
Chapters in monographs
• Weber, M.
Basics in free probability, 6 pages
in Free probability and operator algebras, ed. by Dan-V. Voiculescu, N. Stammeier, M. Weber, EMS, 2016.
• Weber, M.
Easy quantum groups, 23 pages
in Free probability and operator algebras, ed. by Dan-V. Voiculescu, N. Stammeier, M. Weber, EMS, 2016.
In peer reviewed journals
• Gromada, Daniel; Weber, M.
Generating linear categories of partitions
to appear in Kyoto Journal of Mathematics, 2021
arXiv:1904.00166 [math.CT, math.QA], 19 pages (2019).
We present an algorithm for approximating linear categories of partitions (of sets). We report on concrete computer experiments based on this algorithm and how we found new examples of compact
matrix quantum groups (so called ``non-easy'' quantum groups) with it. This also led to further theoretical insights regarding the representation theory of such quantum groups.
• Nechita, Ion; Schmidt, Simon; Weber, M.
Sinkhorn algorithm for quantum permutation groups
to appear in Experimental Mathematics, 2021
arXiv:1911.04912 [math.QA], 16 pages (2019).
We introduce a Sinkhorn-type algorithm for producing quantum permutation matrices encoding symmetries of graphs. Our algorithm generates square matrices whose entries are orthogonal projections
onto one-dimensional subspaces satisfying a set of linear relations. We use it for experiments on the representation theory of the quantum permutation group and quantum subgroups of it. We apply
it to the question whether a given finite graph (without multiple edges) has quantum symmetries in the sense of Banica. In order to do so, we run our Sinkhorn algorithm and check whether or not
the resulting projections commute. We discuss the produced data and some questions for future research arising from it.
• Mang, Alexander; Weber, M.
Non-Hyperoctahedral Categories of Two-Colored Partitions, Part II: All Possible Parameter Values
to appear in Applied Categorial Structures, 2021
arXiv:2003.00569 [math.CO, math.QA], 34 pages (2020).
This article is part of a series with the aim of classifying all non-hyperoctahedral categories of two-colored partitions. Those constitute by some Tannaka-Krein type result the
co-representation categories of a specific class of quantum groups. However, our series of articles is purely combinatorial. In Part I we introduced a class of parameters which gave rise to many
new non-hyperoctahedral categories of partitions. In the present article we show that this class actually contains all possible parameter values of all non-hyperoctahedral categories of
partitions. This is an important step towards the classification of all non-hyperoctahedral categories.
• Mang, Alexander; Weber, M.
Non-hyperoctahedral categories of two-colored partitions, Part I: New categories
to appear in Journal of Algebraic Combinatorics
arXiv:1907.11417 [math.CO, math.QA], 30 pages (2019).
Compact quantum groups can be studied by investigating their co-representation categories in analogy to the Schur-Weyl/Tannaka-Krein approach. For the special class of (unitary) "easy" quantum
groups these categories arise from a combinatorial structure: Rows of two-colored points form the objects, partitions of two such rows the morphisms; vertical/horizontal concatenation and
reflection give composition, monoidal product and involution. Of the four possible classes O, B, S and H of such categories (inspired respectively by the classical orthogonal, bistochastic,
symmetric and hyperoctahedral groups) we treat the first three - the non-hyperoctahedral ones. We introduce many new examples of such categories. They are defined in terms of subtle combinations
of block size, coloring and non-crossing conditions. This article is part of an effort to classify all non-hyperoctahedral categories of two-colored partitions. The article is purely
combinatorial in nature; The quantum group aspects are left out.
• Mang, Alexander; Weber, M.
Categories of two-colored pair partitions, Part II: Categories indexed by semigroups
Journal of Combinatorial Theory, Series A, Vol. 180, 105509, 43 pp., 2021
arXiv:1901.03266 [math.CO, math.QA], 37 pages (2019).
Within the framework of unitary easy quantum groups, we study an analogue of Brauer's Schur-Weyl approach to the representation theory of the orthogonal group. We consider concrete combinatorial
categories whose morphisms are formed by partitions of finite sets into disjoint subsets of cardinality two; the points of these sets are colored black or white. These categories correspond to
"half-liberated easy" interpolations between the unitary group and Wang's quantum counterpart. We complete the classification of all such categories demonstrating that the subcategories of a
certain natural halfway point are equivalent to additive subsemigroups of the natural numbers; the categories above this halfway point have been classified in a preceding article. We achieve
this using combinatorial means exclusively. Our work reveals that the half-liberation procedure is quite different from what was previously known from the orthogonal case.
• Gromada, Daniel; Weber, M.
New products and Z[2]-extensions of compact matrix quantum groups
to appear in Annales de l'Institut Fourier
arXiv:1907.08462 [math.QA, math.OA], 39 pages (2019).
There are two very natural products of compact matrix quantum groups: the tensor product G x H and the free product G * H. We define a number of further products interpolating these two. We
focus more in detail to the case where G is an easy quantum group and H = Z[2], the dual of the cyclic group of order two. We study subgroups of G * Z[2] using categories of partitions with
extra singletons. Closely related are many examples of non-easy bistochastic quantum groups.
• Junk, Luca; Schmidt, Simon; Weber, M.
Almost all trees have quantum symmetry
Archiv der Mathematik, Vol. 115, 267-278, 2020.
arXiv:1911.02952 [math.CO, math.QA], 11 pages (2019).
From the work of Erdös and Renyi from 1963 it is known that almost all graphs have no symmetry. In 2017, Lupini, Mancinska and Roberson proved a quantum counterpart: Almost all graphs have no
quantum symmetry. Here, the notion of quantum symmetry is phrased in terms of Banica's definition of quantum automorphism groups of finite graphs from 2005, in the framework of Woronowicz's
compact quantum groups. Now, Erdös and Renyi also proved a complementary result in 1963: Almost all trees do have symmetry. The crucial point is the almost sure existence of a cherry in a tree.
But even more is true: We almost surely have two cherries in a tree - and we derive that almost all trees have quantum symmetry. We give an explicit proof of this quantum counterpart of Erdös
and Renyi's result on trees.
• Jung, Stefan; Weber, M.
Models of quantum permutations. With an appendix by Alexandru Chirvasitu and Pawel Joziak
Journal of Functional Analysis, Vol. 279, Issue 2, 1 August 2020, 108516
See also: arXiv:1906.10409 [math.OA, math.QA], 24 pages.
For N greater or equal to 4 we present a series of *-homomorphisms from C(S[N]^+) to B[n] where S[N]^+ is the quantum permutation group. They are not necessarily representations of the quantum
group S[N]^+ but they yield good and somewhat ``small'' operator algebraic models of quantum permutation matrices. In the inverse limit however they produce a quantum group which turns out to be
isomorphic to S[N]^+; the latter fact is thanks to an argument by Alexandru Chirvasitu and Pawel Joziak building on topological generation.
• Gromada, Daniel; Weber, M.
Intertwiner spaces of quantum group subrepresentations
Communications in Mathematical Physics, Vol. 376, 81-115, 2020.
See also: arXiv:1811.02821 [math.QA, math.OA], 38 pages.
We consider compact matrix quantum groups whose N-dimensional fundamental representation decomposes into an (N-1)-dimensional and a one-dimensional subrepresentation. Even if we know that the
compact matrix quantum group associated to this (N-1)-dimensional subrepresentation is isomorphic to the given N-dimensional one, it is a priori not clear how the intertwiner spaces transform
under this isomorphism. In the context of so-called easy and non-easy quantum groups, we are able to define a transformation of linear combinations of partitions and we explicitly describe the
transformation of intertwiner spaces. As a side effect, this enables us to produce many new examples of non-easy quantum groups being isomorphic to easy quantum groups as compact quantum groups
but not as compact matrix quantum groups.
• Speicher, Roland; Weber, M.
Quantum groups with partial commutation relations
Indiana University Mathematics Journal, Vol. 68, 1849-1883, 2019.
See also: arXiv:1603.09192 [math.QA, math.OA], 44 pages (2016).
We define new noncommutative spheres with partial commutation relations for the coordinates. We investigate the quantum groups acting maximally on them, which yields new quantum versions of the
orthogonal group: They are partially commutative in a way such that they do not interpolate between the classical and the free quantum versions of the orthogonal group. Likewise we define
non-interpolating, partially commutative quantum versions of the symmetric group recovering Bichon's quantum automorphism groups of graphs. They fit with the mixture of classical and free
independence as recently defined by Speicher and Wysoczanski (rediscovering Lambda-freeness of Mlotkowski), due to some weakened version of a de Finetti theorem.
• Mang, Alexander; Weber, M.
Categories of two-colored pair partitions, Part I: Categories indexed by cyclic groups
The Ramanujan Journal, Vol. 53, 181-208, 2020.
arXiv:1809.06948 [math.CO, math.QA], 25 pages (2018).
We classify certain categories of partitions of finite sets subject to specific rules on the colorization of points and the sizes of blocks. More precisely, we consider pair partitions such that
each block contains exactly one white and one black point when rotated to one line; however crossings are allowed. There are two families of such categories, the first of which is indexed by
cyclic groups and is covered in the present article; the second family will be the content of a follow-up article. Via a Tannaka-Krein result, the categories in the two families correspond to
easy quantum groups interpolating the classical unitary group and Wang's free unitary quantum group. In fact, they are all half-liberated in some sense and our results imply that there are many
more half-liberation procedures than previously expected. However, we focus on a purely combinatorial approach leaving quantum group aspects aside.
• Weber, M.; Zhao, Mang
Factorization of Frieze patterns
Revista de la Union Matematica Argentina, Vol. 60 (2), 407-415, 2019
See also: arXiv:1809.00274 [math.CO], 9 pages.
In 2017, Michael Cuntz gave a definition of reducibility of quiddity cycles of frieze patterns: It is reducible if it can be written as a sum of two other quiddity cycles. We discuss the
commutativity and associativity of this sum operator for quiddity cycles and its equivalence classes, respectively. We show that the sum is neither commutative nor associative, but we may
circumvent this issue by passing to equivalence classes. We also address the question whether a decomposition of quiddity cycles into irreducible factors is unique and we answer it in the
negative by giving counterexamples. We conclude that even under stronger assumptions, there is no canonical decomposition.
• Jung, Stefan; Weber, M.
Partition quantum spaces
Journal of Noncommutative Geometry, Vol. 14, Issue 3, 821-85, 2020
arXiv:1801.06376 [math.OA, math.FA], 35 pages (2018).
We propose a definition of partition quantum spaces. They are given by universal C*-algebras whose relations come from partitions of sets. We ask for the maximal compact matrix quantum group
acting on them. We show how those fit into the setting of easy quantum groups: Our approach yields spaces these groups are acting on. In a way, our partition quantum spaces arise as the first d
columns of easy quantum groups. However, we define them as universal C*-algebras rather than as C*-subalgebras of easy quantum groups. We also investigate the minimal number d needed to recover
an easy quantum group as the quantum symmetry group of a partition quantum space. In the free unitary case, d takes the values one or two.
• Schmidt, Simon; Weber, M.
Quantum symmetries of graph C*-algebras
Canadian Mathematical Bulletin 61, 848-864, 2018
See also: arXiv:1706.08833 [math.OA, math.FA], 18 pages.
The study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have never been computed so far. We close this gap by proving that the quantum
automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the
quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition
in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two
definitions) for undirected graphs on four vertices.
• Tarrago, Pierre; Weber, M.
The classification of tensor categories of two-colored noncrossing partitions
Journal of Combinatorial Theory, Series A, Vol. 154, Feb 2018, 464-506
See also: arXiv:1509.00988 [math.CO, math. QA], 40 pages.
Our basic objects are partitions of finite sets of points into disjoint subsets. We investigate sets of partitions which are closed under taking tensor products, composition and involution, and
which contain certain base partitions. These so called categories of partitions are exactly the tensor categories being used in the theory of Banica and Speicher's orthogonal easy quantum
groups. In our approach, we additionally allow a coloring of the points. This serves as the basis for the introduction of unitary easy quantum groups, which is done in a separate article. The
present article however is purely combinatorial. We find all categories of two-colored noncrossing partitions. For doing so, we extract certain parameters with values in the natural numbers
specifying the colorization of the categories on a global as well as on a local level. It turns out that there are ten series of categories, each indexed by one or two parameters from the
natural numbers, plus two additional categories. This is just the beginning of the classification of categories of two-colored partitions and we point out open problems at the end of the
• Weber, M.
Introduction to compact (matrix) quantum groups and Banica-Speicher (easy) quantum groups
Notes of a lecture series at IMSc Chennai, India, 2015
Indian Academy of Sciences. Proceedings. Mathematical Sciences, Vol. 127, Issue 5, pp 881-933, Nov 2017.
This is a transcript of a series of eight lectures, 90 minutes each, held at IMSc Chennai, India from 05--24 January 2015. We give basic definitions, properties and examples of compact quantum
groups and compact matrix quantum groups such as the existence of a Haar state, the representation theory and Woronowicz's quantum version of the Tannaka-Krein theorem. Building on this, we
define Banica-Speicher quantum groups (also called easy quantum groups), a class of compact matrix quantum groups determined by the combinatorics of set partitions. We sketch the classification
of Banica-Speicher quantum groups and we list some applications. We review the state of the art regarding Banica-Speicher quantum groups and we list some open problems.
• Mai, Tobias; Speicher, Roland; Weber, M.
Absence of algebraic relations and of zero divisors under the assumption of finite full
non-microstates free entropy dimension
Advances in Mathematics, Vol. 304, 2 January 2017, pages 1080-1107.
See also: arXiv:1502.06357 [math.OA], 25 pages.
(we extended the former version arXiv:1407.5715 substantially)
We show that in a tracial and finitely generated W*-probability space existence of conjugate variables excludes algebraic relations for the generators. Moreover, under the assumption of maximal
non-microstates free entropy dimension, we prove that there are no zero divisors in the sense that the product of any non-commutative polynomial in the generators with any element from the von
Neumann algebra is zero if and only if at least one of those factors is zero. In particular, this shows that in this case the distribution of any non-constant self-adjoint non-commutative
polynomial in the generators does not have atoms. Questions on the absence of atoms for polynomials in non-commuting random variables (or for polynomials in random matrices) have been an open
problem for quite a while. We solve this general problem by showing that maximality of free entropy dimension excludes atoms.
• Tarrago, Pierre; Weber, M.
Unitary easy quantum groups: the free case and the group case
International Mathematics Research Notes, 18, 1 Sept 2017, 5710-5750.
See also: arXiv:1512.00195 [math. QA, math.OA], 39 pages.
Easy quantum groups have been studied intensively since the time they were introduced by Banica and Speicher in 2009. They arise as a subclass of (C*-algebraic) compact matrix quantum groups in
the sense of Woronowicz. Due to some Tannaka-Krein type result, they are completely determined by the combinatorics of categories of (set theoretical) partitions. So far, only orthogonal easy
quantum groups have been considered in order to understand quantum subgroups of the free orthogonal quantum group O[n]^+. We now give a definition of unitary easy quantum groups using colored
partitions to tackle the problem of finding quantum subgroups of U[n]^+. In the free case (i.e. restricting to noncrossing partitions), the corresponding categories of partitions have recently
been classified by the authors by purely combinatorial means. There are ten series showing up each indexed by one or two discrete parameters, plus two additional quantum groups. We now present
the quantum group picture of it and investigate them in detail. We show how they can be constructed from other known examples using generalizations of Banica's free complexification. For doing
so, we introduce new kinds of products between quantum groups. We also study the notion of easy groups.
• Gabriel, Olivier; Weber, M.
Fixed point algebras for easy quantum groups
SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) 12 (2016), 097, 21 pages.
See also: arXiv:1606.00569 [math.OA, math.KT], 21 pages.
Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence
they are completely determined by their K-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the K-groups of
their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group S[n]^+, the free orthogonal quantum group O[n]^+ and the quantum reflection groups H
[n]^s+. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups - which are related to Hopf-Galois extensions.
• Freslon, Amaury; Weber, M.
On bi-free De Finetti theorems
Annales Mathématiques Blaise Pascal, 23(1), 21-51, 2016.
See also: arXiv:1501.05124 [math.PR, math.OA, math.QA], 16 pages.
We investigate possible generalizations of the de Finetti theorem to bi-free probability. We first introduce a twisted action of the quantum permutation groups corresponding to the combinatorics
of bi-freeness. We then study properties of families of pairs of variables which are invariant under this action, both in the bi-noncommutative setting and in the usual noncommutative setting.
We do not have a completely satisfying analogue of the de Finetti theorem, but we have partial results leading the way. We end with suggestions concerning the symmetries of a potential notion of
• Raum, Sven; Weber, M.
The full classification of orthogonal easy quantum groups
Communications in Mathematical Physics, 341(3), 751-779, Feb 2016.
See also: arXiv:1312.3857 [math.QA], 38 pages.
We study easy quantum groups, a combinatorial class of orthogonal quantum groups introduced by Banica-Speicher in 2009. We show that there is a countable descending chain of easy quantum groups
interpolating between Bichon's free wreath product with the permutation group S[n] and a semi-direct product of a permutation action of S[n] on a free product. This reveals a series of new
commutation relations interpolating between a free product construction and the tensor product. Furthermore, we prove a dichotomy result saying that every hyperoctahedral easy quantum group is
either part of our new interpolating series of quantum groups or belongs to a class of semi-direct product quantum groups recently studied by the authors. This completes the classification of
easy quantum groups. We also study combinatorial and operator algebraic aspects of the new interpolating series.
• Raum, Sven; Weber, M.
Easy quantum groups and quantum subgroups of a semi-direct product quantum group
Journal of Noncommutative Geometry, Vol. 9(4), 1261-1293, 2015
See also: arXiv:1311.7630 [math.QA], 26 pages.
We consider compact matrix quantum groups whose fundamental corepresentation matrix has entries which are partial isometries with central support. We show that such quantum groups have a simple
representation as semi-direct product quantum groups of a group dual quantum group by an action of a permutation group. This general result allows us to completely classify easy quantum groups
with the above property by certain reflection groups. We give four applications of our result. First, there are uncountably many easy quantum groups. Second, there are non-easy quantum groups
between the free orthogonal quantum group and the permutation group. Third, we study operator algebraic properties of the hyperoctahedral series. Finally, we prove a generalised de Finetti
theorem for easy quantum groups in the scope of this article.
• Freslon, Amaury; Weber, M.
On the representation theory of partition (easy) quantum groups
Journal für die reine und angewandte Mathematik [Crelle's Journal], Vol. 2016, Issue 720 (Nov 2016), 2016.
See also: arXiv:1308.6390 [math.QA], 42 pages.
Compact matrix quantum groups are strongly determined by their intertwiner spaces, due to a result by S. L. Woronowicz. In the case of easy quantum groups (also called partition quantum groups),
the intertwiner spaces are given by the combinatorics of partitions, see the initial work of T. Banica and R. Speicher. The philosophy is that all quantum algebraic properties of these objects
should be visible in their combinatorial data. We show that this is the case for their fusion rules (i.e. for their representation theory). As a byproduct, we obtain a unified approach to the
fusion rules of the quantum permutation group S[N]^+, the free orthogonal quantum group O[N]^+ as well as the hyperoctahedral quantum group H[N]^+. We then extend our work to unitary easy
quantum groups and link it with a ''freeness conjecture'' of T. Banica and R. Vergnioux.
• Raum, Sven; Weber, M.
The combinatorics of an algebraic class of easy quantum groups
Infinite Dimensional Analysis, Quantum Probability and related topics Vol 17, No. 3, 2014.
See also: arXiv:1312.1497 [math.QA], 16 pages.
Easy quantum groups are compact matrix quantum groups, whose intertwiner spaces are given by the combinatorics of categories of partitions. This class contains the symmetric group and the
orthogonal group as well as Wang's quantum permutation group and his free orthogonal quantum group. In this article, we study a particular class of categories of partitions to each of which we
assign a subgroup of the infinite free product of the cyclic group of order two. This is an important step in the classification of all easy quantum groups and we deduce that there are
uncountably many of them. We focus on the combinatorial aspects of this assignment, complementing the quantum algebraic point of view presented in another article.
• Weber, M.
On the classification of easy quantum groups
Advances in Mathematics, Volume 245, 1 October 2013, pages 500-533.
See also: arXiv:1201.4723 [math.OA], 39 pages.
In 2009, Banica and Speicher began to study the compact quantum groups G with S[n] c G c O[n]^+ whose intertwiner spaces are induced by some partitions. These so-called easy quantum groups have
a deep connection to combinatorics. We continue their work on classifying these objects, by introducing some new examples of easy quantum groups. In particular, we show that the six easy groups
O[n], S[n], H[n], B[n], S[n]' and B[n]' split into seven cases O[n]^+, S[n]^+, H[n]^+, B[n]^+, S[n]'^+, B[n]'^+ and B[n]^#+ on the side of free easy quantum groups. Also, we give a complete
classification in the half-liberated and in the nonhyperoctahedral case.
• Weber, M.
On C*-Algebras Generated by Isometries with Twisted Commutation Relations
Journal of Functional Analysis, Volume 264, Issue 8, pages 1975-2004, 2013.
See also: arXiv:1207.3038 [math.OA], 35 pages.
In the theory of C^*-algebras, interesting noncommutative structures arise as deformations of the tensor product, e.g. the rotation algebra A[theta] as a deformation of C(S^1)\otimes C(S^1). We
deform the tensor product of two Toeplitz algebras in the same way and study the universal C^*-algebra T \otimes[theta]T generated by two isometries u and v such that uv = e^2 pi i thetavu and u
^*v = e^-2 pi i thetavu^*, for theta in R. Since the second relation implies the first one, we also consider the universal C^*-algebra T *[theta] T generated by two isometries u and v with the
weaker relation uv = e^2 pi i thetavu. Such a "weaker case" does not exist in the case of unitaries, and it turns out to be much more interesting than the twisted "tensor product case" T \otimes
[theta]T. We show that T \otimes[theta]T is nuclear, whereas T *[theta] T is not even exact. Also, we compute the K-groups and we obtain K[0](T *[theta] T) = Z and K[1](T *[theta] T) = Z, and
the same K-groups for T \otimes[theta]T.
Further publications
• Weber, M.
Berichte über Arbeitsgruppen. SFB/TRR 195 Symbolic Tools in Mathematics and their Application (Part 4/5).
Random matrices, free probability theory and compact quantum groups
Computeralgebra-Rundbrief Nr. 65, Oktober 2019.
• Weber, M.
Quantum symmetry
Snapshots of modern mathematics from Oberwolfach, 2020.
• Weber, M.
Auffrischungskurs Mathematik für Geflüchtete - ein best practice example
in Kergel, D., Heidkamp, B. (Hrsg.) (2018). Praxishandbuch habitus- und diversitätssensible Hochschullehre,
VS Springer, 2019.
arising from the preparatory math courses for refugees for studies in MINT-subjects
Further publications by students under my supervision
Postal address
Saarland University
Department of Mathematics
Postfach 15 11 50
66041 Saarbrücken
Physical address
Saarland University
Campus building E 2 4
66123 Saarbrücken | {"url":"https://www.uni-saarland.de/lehrstuhl/weber-moritz/research/publications/publications-abstracts.html","timestamp":"2024-11-02T17:14:47Z","content_type":"text/html","content_length":"112289","record_id":"<urn:uuid:26e16c42-cd82-42b0-86d4-1832afa064f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00362.warc.gz"} |
From cppreference.com
inline namespace /* unspecified */ {
inline constexpr /* unspecified */ weak_order = /* unspecified */; (since C++20)
Call signature
template< class T, class U >
requires /* see below */
constexpr std::weak_ordering weak_order(T&& t, U&& u) noexcept(/* see below */);
Compares two values using 3-way comparison and produces a result of type std::weak_ordering
Let t and u be expressions and T and U denote decltype((t)) and decltype((u)) respectively, std::weak_order(t, u) is expression-equivalent to:
• If std::is_same_v<std::decay_t<T>, std::decay_t<U>> is true:
□ std::weak_ordering(weak_order(t, u)), if it is a well-formed expression with overload resolution performed in a context that does not include a declaration of std::weak_order,
□ otherwise, if T is a floating-point type:
☆ if std::numeric_limits<T>::is_iec559 is true, performs the weak ordering comparison of floating-point values (see below) and returns that result as a value of type std::weak_ordering,
☆ otherwise, yields a value of type std::weak_ordering that is consistent with the ordering observed by T's comparison operators,
□ otherwise, std::weak_ordering(std::compare_three_way()(t, u)), if it is well-formed,
□ otherwise, std::weak_ordering(std::strong_order(t, u)), if it is well-formed.
• In all other cases, the expression is ill-formed, which can result in substitution failure when it appears in the immediate context of a template instantiation.
Expression e is expression-equivalent to expression f, if
• e and f have the same effects, and
• either both are constant subexpressions or else neither is a constant subexpression, and
• either both are potentially-throwing or else neither is potentially-throwing (i.e. noexcept(e) == noexcept(f)).
Customization point objects
The name std::weak_order denotes a customization point object, which is a const function object of a literal semiregular class type. For exposition purposes, the cv-unqualified version of its type is
denoted as __weak_order_fn.
All instances of __weak_order_fn are equal. The effects of invoking different instances of type __weak_order_fn on the same arguments are equivalent, regardless of whether the expression denoting the
instance is an lvalue or rvalue, and is const-qualified or not (however, a volatile-qualified instance is not required to be invocable). Thus, std::weak_order can be copied freely and its copies can
be used interchangeably.
Given a set of types Args..., if std::declval<Args>()... meet the requirements for arguments to std::weak_order above, __weak_order_fn models
Otherwise, no function call operator of __weak_order_fn participates in overload resolution.
Strict weak order of IEEE floating-point types
Let x and y be values of same IEEE floating-point type, and weak_order_less(x, y) be the boolean result indicating if x precedes y in the strict weak order defined by the C++ standard.
• If neither x nor y is NaN, then weak_order_less(x, y) == true if and only if x < y, i.e. all representations of equal floating-point value are equivalent;
• If x is negative NaN and y is not negative NaN, then weak_order_less(x, y) == true;
• If x is not positive NaN and y is positive NaN, then weak_order_less(x, y) == true;
• If both x and y are NaNs with the same sign, then (weak_order_less(x, y) || weak_order_less(y, x)) == false, i.e. all NaNs with the same sign are equivalent.
This section is incomplete
Reason: no example
See also
the result type of 3-way comparison that supports all 6 operators and is not substitutable
(C++20) (class)
performs 3-way comparison and produces a result of type std::strong_ordering
(C++20) (customization point object)
performs 3-way comparison and produces a result of type std::partial_ordering
(C++20) (customization point object)
(C++20) performs 3-way comparison and produces a result of type std::weak_ordering, even if operator<=> is unavailable
(customization point object) | {"url":"https://cppreference.patrickfasano.com/en/cpp/utility/compare/weak_order.html","timestamp":"2024-11-04T14:47:28Z","content_type":"text/html","content_length":"48333","record_id":"<urn:uuid:50f12403-16b1-4f46-9198-c8d2f4db73f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00213.warc.gz"} |
External magnetic fields
SLaSi supporst the next list of in-build field types:
• constant,
• linear $H = h_1q + h_2$,
• harmonic $H = h \cos (\omega q + \phi)$,
where q means space or time coordinate.
Each the field can be defined repeatedly with different parameters for the axes X, Y or Z. For fields a smoothing hat function $A \exp(-t^2/T^2)$ can be defined for avoiding of beats. The common
syntax follows:
field<Axis> = select <fieldParams1> <fieldSpec> <fieldParams2> <hatTime>
Here <Axis> defines the axis for field, X, Y or Z. For example, fieldX, fieldY or fieldZ. <fieldParams1> means the spatial and temporal coordinates in which field is applied. <fieldSpec> can be
const, lin or harm and <fieldParams> list depends on it. <hatTime> defines $T$ from the hat function (can be 0 for absence of smoothing).
<fieldParams1> are defined as set of commands select with coordinate specification and > or < symbol with number. They can repeat for defining an interval. For example
fieldX = select t > 0.0 select y < 10 ...
defines the field along X axis which starts from the beginning of the simulations and acts from the first to ninth nodes along Y axis.
For the <fieldSpec> equals const <fieldParams2> is only one number: field strength. For example:
fieldX = select t > 10.0 const 0.3 30
Such command generates magnetic field along X axis with strength 0.3 units with smoothing during 30 time units after time equal 10 units.
For the <fieldSpec> equals lin <fieldParams2> is a construction like h1 q h2, which means $H = h_1q + h_2$ where q takes values x, y, z or t and h1 and h2 are numbers. For example:
fieldX = select t > 0.0 lin 0.1 t 0.0 0.0
Such command generates magnetic field along X axis which linearly increase in time with speed 0.1.
For the <fieldSpec> equals harm <fieldParams2> is a construction like h omega q phi, which means $H = h \cos (\omega q + \phi)$ where q takes values x, y, z or t, h is a field strength and phi is a
phase. For example:
fieldX = select t > 0.0 harm 0.1 -6.2 t 1.57 0.0
Such command generates alternating sinusoidal magnetic field along X with frequency equal 1.
You can specify till 100 different fields along each axis. | {"url":"http://ritm.knu.ua/products/slasi/slasi-documentation/external-magnetic-fields/","timestamp":"2024-11-03T10:32:26Z","content_type":"text/html","content_length":"69388","record_id":"<urn:uuid:6a9f7b63-dda6-48e1-935a-92e35e16942f>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00677.warc.gz"} |
lme4 News
CHANGES IN VERSION 1.1-35.5
• in predict, the synonyms ReForm, REForm, and REform of re.form are now hard-deprecated, giving an error (after 10 years of soft deprecation)
• minor adjustments in test tolerances
• correct Matrix dependency to >= 1.2-3
CHANGES IN VERSION 1.1-35.4 (2024-06-19)
• predict(., re.form=...) works in a wider range of cases (GH #691)
• Gamma simulation now uses correct shape parameter (GH #782)
• Avoid triggering RcppEigen UBSAN bug(?) in the case of profiling fixed effects in a merMod object with a single fixed effect parameter (GH #794: lots of help from Dirk Eddelbuettel and Mikael
• fix bug in plot methods (cbind'ing zero-length objects)
CHANGES IN VERSION 1.1-35.3 (2024-04-16)
• bug-fix for ASAN/memory access problem in CholmodDecomposition (Mikael Jagan)
CHANGES IN VERSION 1.1-35.2 (2024-03-28)
• This is primarily a 'bump' release to ensure that package repositories rebuild binaries with the latest version of the Matrix package.
• simulate works (again) with re.form=NULL when NA values are present in the data (GH #737, @frousseu)
• updated tests of upstream Matrix version; should now warn only on ABI incompatibility, not on package version mismatch alone
CHANGES IN VERSION 1.1-35.1 (2023-11-05)
• lFormula and glFormula once again do allow matrix-valued responses (for use in downstream packages like galamm)
CHANGES IN VERSION 1.1-35 (2023-11-03)
• predict.merMod now has a se.fit method, which computes the standard errors of the predictions, conditional on the estimated theta (variance-covariance) parameters
• using lmer with a matrix-valued response now throws a more informative error message, directing the user to ?refit
CHANGES IN VERSION 1.1-34 (2023-07-04)
• summary(<merMod>) now records if correlation was specified explicitly and to what; and its print() method takes it into account; notably summary(<merMod>, correlation=TRUE) will by default print
the correlation (matrix of the fixed effects) fixing GH #725
• refit gains a newweights argument
CHANGES IN VERSION 1.1-33 (2023-04-25)
• a boundary check could fail occasionally when large data produced an NA value in a computed gradient; now warns instead (GH #719, Mathias Ambuehl)
• allFit now works better when optimx and dfoptim packages are not installed (GH #724)
• refit reset internal degrees of freedom component incorrectly for REML fits (resulted in incorrect reported REML criteria, but otherwise harmless: side effect of GH #678)
• dotplot and qqmath methods gain a level argument to set the width of confidence intervals
• dotplot method is now more flexible, using ".v" options (lty.v, col.line.v, lwd.v) to set appearance of vertical lines (Iago Giné Vázquez)
• refit gains a newweights argument (GH #678)
CHANGES IN VERSION 1.1-32 (2023-03-14)
• formatVC() gets a new optional argument corr indicating if correlations or covariances should be used for vector random effects; this corresponds to print(<merMod>, ranef.corr = ...) . By
default, it is FALSE for comp = "Variance", fixing (GH #707).
• qqmath.merMod adds a (useless) data argument for S3 compatibility. Going forward, the id and idLabels arguments should always be specified by name. We have added code to try to detect/warn when
this is not done.
• nobars now retains the environment of its formula argument (GH #713, Mikael Jagan)
CHANGES IN VERSION 1.1-31 (2022-11-01)
• confint(fm, <single string>) now works (after years of being broken) again.
• simulating from binomial model with a factor response, when the simulated response contains only a single factor level, now works (Daniel Kennedy)
CHANGES IN VERSION 1.1-30 (2022-07-08)
• nl (term names) component added to output list of mkReTrms (GH #679)
• eliminate partial-matching of eta (for etastart) (GH #686: not actually "user-visible" unless getOption("warnPartialMatchDollar") is TRUE)
• summary method doesn't break for GLMMs other than binomial/Poisson when merDeriv's vcov.glmerMod method is attached (GH #688)
• better handling of simulate(., re.form = NULL) when model frame contains derived components (e.g. offset(), log(x)) (https://github.com/florianhartig/DHARMa/issues/335)
• bootMer works with glmmTMB again (broken in 1.1-29)
• maxfun argument to allFit controls max function evaluations for every optimizer type (GH#685)
CHANGES IN VERSION 1.1-29 (2022-04-07)
• prediction with new levels (when not allowed) returns a more informative error message (displays a list of unobserved levels)
• glmer.nb now works when lme4 is not loaded (GH #658, @brgew)
• tests for singularity (check.conv.singular) now run independently of derivative computation (e.g., when calc.derivs=FALSE) (GH #660, @palday)
• influence.merMod now works when data were originally specified as a tibble
• fixed bug in cooks.distance method for influence.merMod (i.e., objects created via influence(fitted_model)) (John Fox) (GH #672)
• predict works for formulas containing . when newdata is specified (GH #653)
• bootMer now correctly inherits control settings from original fit
CHANGES IN VERSION 1.1-28 (2022-02-04)
• construction of interacting factors (e.g. when f1:f2 or f1/f2 occur in random effects terms) is now more efficient for partially crossed designs (doesn't try to create all combinations of f1 and
f2) (GH #635 and #636)
• mkNewReTrms is exported
• singular-fit message now refers to help("isSingular") rather than ?isSingular
• fix all.equal(p1,p2,p3) and similar expect_equal() thinkos
• fix some tests only run when lme4:::testLevel() > 1; adapt tests for upcoming Matrix 1.4-1 which has names(diag(<sparse>))
• reOnly preserves environment (GH #654, Mikael Jagan)
• backward-compatibility hooks changed to evaluate at run-time (i.e., in .onLoad()) rather than at build time (GH #649)
• lmList no longer warns when data is a tibble (GH #645)
CHANGES IN VERSION 1.1-27.1 (2021-06-22)
• influence.merMod allows user-specified starting parameters
• cleaned up performance vignette
• cooks.distance now works with objects computed by influence method
• influence.merMod now works with glmer models using nAGQ=0
• predict (with new data) and simulate methods now work for models with >100 levels in a random effect grouping variable (GH #631)
CHANGES IN VERSION 1.1-27 (2021-05-15)
• improvements from Lionel Henry (via https://github.com/lme4/lme4/pull/587) to fix corner cases in data checking; also resolves GH #601 (allFit scoping)
• getME(., "lower") now has names (request of GH #609)
• improved detection of NaN in internal calculations (typically due to underflow/overflow or out-of-bounds linear predictors from non-constraining link functions such as identity-link Gamma models)
• influence.merMod allows parallel computation
• the statmod package is no longer required unless attempting to simulate results from a model with an inverse Gaussian response
• long formulas work better in anova headings (GH #611)
CHANGES IN VERSION 1.1-26 (2020-11-30)
• predict, model.frame(.,fixed.only=TRUE) work with variable names containing spaces (GH #605)
• simulate works when original response variable was logical
• densityplot handles partly broken profiles more robustly
• thpr method for densityplot() (for plotting profiles scaled as densities) gets new arguments
CHANGES IN VERSION 1.1-25 (2020-10-23)
• Set more tests to run only if environment variable LME4_TEST_LEVEL>1
CHANGES IN VERSION 1.1-24 (never on CRAN)
• anova() now returns a p-value of NA if the df difference between two models is 0 (implying they are equivalent models) (GH#583, @MetaEntropy)
• speedup in coef() for large models, by skipping conditional variance calculation (Alexander Bauer)
• simulate.formula machinery has changed slightly, for compatibility with the ergm package (Pavel Krivitsky)
• informational messages about (non-)convergence improved (GH #599)
• improved error messages for 0 non-NA cases in data (GH #533)
• getME(.,"devfun") now works for glmer objects. Additionally, profile/confint for GLMMs no longer depend on objects in the fitting environment remaining unchanged (GH #589). This change also
affects likelihood profiling machinery; results of glmer profiling/CIs may not match results from previous versions exactly.
• improved handling/documentation of glmer.nb controls (GH #556)
• predict works better for gamm4 objects (GH #575)
• resolved some long-standing UBSAN issues (GH #561)
CHANGES IN VERSION 1.1-23 (2020-03-06)
This is primarily for CRAN compliance (previous submission was retracted to allow time for downstream package adjustments).
• Some PROTECT/UNPROTECT fixes
CHANGES IN VERSION 1.1-22 (never on CRAN)
• prediction now works better for factors with many levels (GH#467, solution by @sihoward)
• minor changes to argument order in [g]lmerControl; default tolerance for convergence checks increased from 0.001 to 0.002 for glmerControl (now consistent with lmerControl)
• lmer(*, family="<fam>") is no longer valid; it had been deprecated since 2013-06.
• lmer(), glmer(), and nlmer() no longer have a formal ... argument. This defunctifies the use of a sparseX = . argument and will reveal some user errors, where extraneous arguments were previously
• In isSingular(x, tol), the default tolerance (tol) has been increased from 1e-5 to 1e-4, the default of check.conv.singular in g?lmerControl().
• for clarity and consistency with base R methods, some column names of anova() output are changed: "Df" becomes "npar", "Chi Df" becomes "Df" (GH #528)
• simulate() now works with inverse-Gaussian models (GH #284 revisited, @nahorp/Florian Hartig)
• single-model mode of anova() now warns about unused arguments in ... (e.g. type="III")
• default tolerances for nloptwrap/BOBYQA optimizer tightened (xtol_abs and ftol_abs were 1e-6, now 1e-8). (To revert to former tolerances, use control=lmerControl(optimizer="nloptwrap", optCtrl=
list(xtol_abs=1e-6, ftol_abs=1e-6)).)
• improved checking for missing data (@lionel-)
• internal checkZrank() should be able to deal with (Matrix package) rankMatrix() returning NA.
• allFit(fm) now works for a model that had an explicit control = lmerControl(..) call.
• internal getStart() now works when model's start was specified as a list, and when called from drop1() on a submodel, fixing GH #521.
• internal function mkdevfun now works even if there is an extraneous getCall function defined in the global environment (GH #535)
• allFit() works even if a variable with symbol i is used somewhere in the original model call (GH #538, reported by Don Cohen); generally more robust
• glmer.nb works even if an alternative version of negative.binomial (other than the one from MASS) is loaded in the workspace (e.g. by the GLMMadaptive package) (GH#516)
• level argument is now honoured by confint(..., type="boot", level=...) (GH #543)
CHANGES IN VERSION 1.1-21 (2019-03-05)
• bootMer now traps and stores messages, warnings, and errors
• bootMer returns an object of class c("bootMer","boot"); new print and confint methods for class bootMer
• small changes to wording of singular-fit messages
CHANGES IN VERSION 1.1-20 (2019-02-04)
• default value for condVar (whether to return conditional variances as part of the ranef.merMod object) is now TRUE
• changed default optimizer to "nloptwrap" (BOBYQA implementation from the nloptr package) for lmer models. To revert to the old default, use control=lmerControl(optimizer="bobyqa")
• adapted tests to work with R-devel's more consistent formula(model.frame(.)) behavior.
CHANGES IN VERSION 1.1-19 (2018-11-10)
• influence measure code from car rolled in (see ?influence.merMod)
• mkReTrm gets new arguments reorder.terms, reorder.vars to control arrangement of RE terms and individual effects with RE terms within model structures
• adding material from the RePsychLing package (on GitHub; see Bates et al 2015 arXiv:1506.04967) to show orthogonal variance components.
• new utility isSingular() function for detecting singular fits
• allFit function/methods have been moved to the main package, rather than being included in an auxiliary source file; computations can (in principle) be done in parallel
• by default a message is now printed for singular fits (i.e., fits with linear combinations of variance components that are exactly zero)
• as.data.frame.merMod finds conditional variance information stored either as attr(.,"postVar") or attr(.,"condVar") (for glmmTMB compatibility)
• change to defaults of [g]lmerControl to print a message when fits are singular
• post-fitting convergence checks based on estimated gradient and Hessian (see troubleshooting) are no longer performed for (nearly-)singular fits (see isSingular)
CHANGES IN VERSION 1.1-18-1 (2018-08-17)
• This is a minor release; the only change is to roll back (unexport) the influence.merMod method, pending resolution of conflicts with the car package
CHANGES IN VERSION 1.1-18 ((2018-08-16)
• ranef(.,condVar=TRUE) now works when there are multiple random effects terms per factor
• rstudent and influence methods are available for merMod objects
• devfun2 function (for generating a deviance function that works on the standard deviation/correlation scale) is now exported
• lmList now obeys its pool argument (instead of always using what currently is the default, GH #476)
CHANGES IN VERSION 1.1-17 (2018-04-03)
• This is a maintenance release only (fixes CRAN problems with cross-platform tests and examples)
CHANGES IN VERSION 1.1-16 (2018-03-28)
• lmList no longer ignores the subset argument (John Fox)
• fixed several minor issues with predicting when (1) grouping variables have different levels from original model (e.g. missing levels/factor levels not explicitly specified in newdata) or (2)
re.form is a subset of the original RE formula and some (unused) grouping variables are omitted from newdata (GH #452, #457)
• lmList tries harder to collect errors and pass them on as warnings
• documented as.function method (given a merMod object, returns a function that computes the deviance/REML criterion for specified parameters)
• print method for summary.merMod objects no longer collapses small values of the t-statistic to zero
CHANGES IN VERSION 1.1-15 (2017-12-21)
• model.frame(., fixed.only=TRUE) now handles models with "non-syntactic" (e.g. space-containing/backtick-delimited) variables in the formula.
• confint(<merMod>) now works again for the default method "profile".
• exported dotplot.ranef.mer
CHANGES IN VERSION 1.1-14 (2017-09-27)
• Primarily an R-devel/CRAN-compatibility release.
• added transf argument to dotplot.ranef.mer to allow back-transformation (Ferenci Tamás, GH #134)
• added as.data.frame.ranef.mer convenience method
• user can specify initial value for overdispersion parameter in glmer.nb (Timothy Lau, GH #423)
• fix bug where NAs in fitting data were carried over into predictions on new data (!) (lmwang9527, GH #420)
• fix bug with long terms in models with || notation
• nlmer now respects user-specified lower/upper bounds (GH #432)
• confint.thpr (confint method applied to an already-computed profile now respects "theta_"/"beta_" specifications to return all random-effect or all fixed-effect confidence intervals,
• document need to export packages and objects to workers when using bootMer with snow
• improved warning message when using lmerControl() with glmer (GH #415)
• avoid deparsing big data frames when checking data (GH #410)
• pass verbose options to nloptr optimizers when using nloptwrap (previously ignored, with a warning)
• the fl (factor list) component of mkReTrms objects is now returned as a list rather than a data frame
CHANGES IN VERSION 1.1-13 (2017-04-18)
• added prof.scale argument to profile.merMod, documented caveats about using varianceProf/logProf transformation methods for correlation parameters
• suppressed spurious contrast-dropping warning (GH #414)
• fixed bug in confint.lmList4 (GH #26)
• fixed bug when FUN returned an unnamed vector in confint(.,FUN=FUN,method="boot")
• fixed small bug relating to nAGQ0initStep=FALSE
• fixed time stamps on compiled versions of vignettes
CHANGES IN VERSION 1.1-12 (2016-04-15)
This release is primarily a bump for compatibility with the new Windows toolchain. Some small documentation and test changes.
• reduced default print precision of fixed-effect correlation matrix in summary.merMod (related to GH #300)
• fixed bug in de novo Gamma-response simulations
CHANGES IN VERSION 1.1-11 (2016-02-11)
• change VarCorr method signature (for compatibility with upstream nlme changes)
• several glmer.nb bugs fixed (generally not changing results, but causing warnings and errors e.g. during bootstrapping)
• fixes to some lmList bugs (Github #320)
• minor documentation, vignette updates
• minor fix to plot.merMod with id specified
• bootMer now handles separate offset term properly (Github #250)
CHANGES IN VERSION 1.1-10 (2015-10-05)
This release is primarily a version bump for the release of the paper in J. Stat. Software.
• We export set of about a dozen printing utility functions which are used in our print methods.
• bootMer now allows the use of re.form.
• fixed reordering bug in names of getME(.,"Ztlist") (terms are reordered in decreasing order of the number of levels of the grouping variable, but names were not being reordered)
• fixed issue with simulation when complex forms (such as nested random effects terms) are included in the model (Github #335)
CHANGES IN VERSION 1.1-9 (2015-08-20)
• explicit maxit arguments for various functions (refit, mkGlmerDevfun, ...)
• terms and formula methods now have random.only options
• getME gains a glmer.nb.theta option. It is now (an S3) generic with an "merMod" method in lme4 and potentially other methods in dependent packages.
• simulate now works for glmer.nb models (Github #284: idea from @aosmith16)
• prediction and simulation now work when random-effects terms have data-dependent bases (e.g., poly(.) or ns(.) terms) (Github #313, Edgar Gonzalez)
• logLik for glmer.nb models now includes the overdispersion parameter in the parameter count (df attribute)
• lmList handles offsets and weights better
• lots of fixes to glmer.nb (Github #176, #266, #287, #318). Please note that glmer.nb is still somewhat unstable/under construction.
• import functions from base packages to pass CRAN checks
• tweak to failing tests on Windows
CHANGES IN VERSION 1.1-8 (2015-06-22)
• getME gains a "Tlist" option (returns a vector of template matrices from which the blocks of Lambda are generated)
• hatvalues method returns the diagonal of the hat matrix of LMMs
• nlminbwrap convenience function allows use of nlminb without going through the optimx package
• as.data.frame.VarCorr.merMod gains an order option that allows the results to be sorted with variances first and covariances last (default) or in lower-triangle order
• allow more flexibility in scales for xyplot.thpr method (John Maindonald)
• models with only random effects of the form 1|f have better starting values for lmer optimization (Gabor Grothendieck)
• glmer now allows a logical vector as the response for binomial models
• anova will now do (sequential) likelihood ratio tests for two or more models including both merMod and glm or lm models (at present, only for GLMMs fitted with the Laplace approximation)
• deviance() now returns the deviance, rather than half the negative log-likelihood, for GLMMs fitted with Laplace (the behaviour for LMMs and GLMMs fitted with nAGQ>1 has not changed)
• convergence warning and diagnostic test issues are now reported in print and summary methods
• update now (attempts to) re-evaluate the original fit in the environment of its formula (as is done with drop1)
• refit of a nonlinear mixed model fit now throws an error, but this will hopefully change in future releases (related to bug fixes for Github #231)
• lmList now returns objects of class lmList4, to avoid overwriting lmList methods from the recommended nlme package
• names of random effects parameters in confint changed (modified for consistency across methods); oldNames=TRUE (default) gives ".sig01"-style names, oldNames=FALSE gives "sd_(Intercept)|
Subject"-style names
• confint(.,method="Wald") result now contains rows for random effects parameters (values set to NA) as well as for fixed-effect parameters
• simulate and predict now work more consistently with different-length data, differing factor levels, and NA values (Github #153, #197, #246, #275)
• refit now works correctly for glmer fits (Github #231)
• fixed bug in family.merMod; non-default links were not retrieved correctly (Alessandro Moscatelli)
• fixed bootMer bug for type=="parametric", use.u=TRUE (Mark Lai)
• gradient scaling for convergence checks now uses the Cholesky factor of the Hessian; while it is more correct, this will lead to some additional (probably false-positive) convergence warnings
• As with lm(), users now get an error for non-finite (Inf, NA, or NaN) values in the response unless na.action is set to exclude or omit them (Github #310)
CHANGES IN VERSION 1.1-7 (2014-07-19)
• the nloptr package is now imported; a wrapper function (nloptwrap) is provided so that lmerControl(optimizer="nloptwrap") is all that's necessary to use nloptr optimizers in the nonlinear
optimization stage (the default algorithm is NLopt's implementation of BOBYQA: see ?nloptwrap for examples)
• preliminary implementation of checks for scaling of model matrix columns (see check.scaleX in ?lmerControl)
• beta is now allowed as a synonym for fixef when specifying starting parameters (Github #194)
• the use of deviance to return the REML criterion is now deprecated; users should use REMLcrit() instead (Github #211)
• changed the default value of check.nobs.vs.rankZ to "ignore" (Github #214)
• change gradient testing from absolute to relative
• fix confint(.,method="boot") to allow/work properly with boot.type values other than "perc" (reported by Alan Zaslavsky)
• allow plot() to work when data are specified in a different environment (reported by Dieter Menne)
• predict and simulate work for matrix-valued predictors (Github #201)
• other simulate bugs (Github #212)
• predict no longer warns spuriously when original response was a factor (Github #205)
• fix memory access issues (Github #200)
CHANGES IN VERSION 1.1-6 (2014-04-13)
This version incorporates no changes in functionality, just modifications to testing and dependencies for CRAN/backward compatibility.
• change drop1 example to prevent use of old/incompatible pbkrtest versions, for 2.15.3 compatibility
• explicitly require(mlmRev) for tests to prevent cyclic dependency
• bump RcppEigen Imports: requirement from >0.3.1.2.3 to >=0.3.2.0; Rcpp dependency to >= 0.10.5
CHANGES IN VERSION 1.1-5 (2014-03-14)
• improved NA handling in simulate and refit
• made internal handling of weights/offset arguments slightly more robust (Github #191)
• handle non-positive-definite estimated fixed effect variance-covariance matrices slightly more generally/robustly (fall back on RX approximation, with a warning, if finite-difference Hessian is
non-PD; return NA matrix if RX approximation is also bad)
• Added output specifying when Gauss-Hermite quadrature was used to fit the model, and specifying number of GHQ points (Github #190)
CHANGES IN VERSION 1.1-4
• Models with prior weights returned an incorrect sigma and deviance (Github issue #155). The deviance bug was only a practical issue in model comparisons, not with inferences given a particular
model. Both bugs are now fixed.
• Profiling failed in some cases for models with vector random effects (Github issue #172)
• Standard errors of fixed effects are now computed from the approximate Hessian by default (see the use.hessian argument in vcov.merMod); this gives better (correct) answers when the estimates of
the random- and fixed-effect parameters are correlated (Github #47)
• The default optimizer for lmer fits has been switched from "Nelder_Mead" to "bobyqa" because we have generally found the latter to be more reliable. To switch back to the old behaviour, use
• Better handling of rank-deficient/overparameterized fixed-effect model matrices; see check.rankX option to [g]lmerControl. The default value is "message+drop.cols", which automatically drops
redundant columns and issues a message (not a warning). (Github #144)
• slight changes in convergence checking; tolerances can be specified where appropriate, and some default tolerances have changed (e.g., check.conv.grad)
• improved warning messages about rank-deficiency in X and Z etc. (warnings now try to indicate whether the unidentifiability is in the fixed- or random-effects part of the model)
• predict and simulate now prefer re.form as the argument to specify which random effects to condition on, but allow ReForm, REForm, or REform, giving a message (not a warning) that they are
deprecated (addresses Github #170)
• small fixes for printing consistency in models with no fixed effects
• we previously exported a fortify function identical to the one found in ggplot2 in order to be able to define a fortify.merMod S3 method without inducing a dependency on ggplot2. This has now
been unexported to avoid masking ggplot2's own fortify methods; if you want to add diagnostic information to the results of a model, use fortify.merMod explicitly.
• simulate.formula now checks for names associated with the theta and beta parameter vectors. If missing, it prints a message (not a warning); otherwise, it re-orders the parameter vectors to match
the internal representation.
• preliminary implementation of a check.scaleX argument in [g]lmerControl that warns about scaling if some columns of the fixed-effect model matrix have large standard deviations (relative to 1, or
to each other)
CHANGES IN VERSION 1.1-3
• The gradient and Hessian are now computed via finite differencing after the nonlinear fit is done, and the results are used for additional convergence tests. Control of the behaviour is available
through the check.conv.* options in [g]lmerControl. Singular fits (fits with estimated variances of zero or correlations of +/- 1) can also be tested for, although the current default value of
the check.conv.singular option is "ignore"; this may be changed to "warning" in the future. The results are stored in @optinfo$derivs. (Github issue #120; based on code by Rune Christensen.)
• The simulate method will now work to generate simulations "from scratch" by providing a model formula, a data frame holding the predictor variables, and a list containing the values of the model
parameters: see ?simulate.merMod. (Github issue #115)
• VarCorr.merMod objects now have an as.data.frame method, converting the list of matrices to a more convenient form for reporting and post-processing. (Github issue #129)
• results of fitted(), predict(), and residuals() now have names in all cases (previously results were unnamed, or named only when predicting from new data)
• the anova method now has a refit argument that controls whether objects of class lmerMod should be refitted with ML before producing the anova table. (Github issues #141, #165; contributed by
Henrik Singmann.)
• the print method for VarCorr objects now has a formatter argument for finer control of standard deviation and variance formats
• the optinfo slot now stores slightly more information, including the number of function evaluations ($feval).
• dotplot.ranef.mer now adds titles to sub-plots by default, like qqmath.ranef.mer
• fitted now respects na.action settings (Github issue #149)
• confint(.,method="boot") now works when there are NA values in the original data set (Github issue #158)
• previously, the code stored the results (parameter values, residuals, etc.) based on the last set of parameters evaluated, rather than the optimal parameters. These were not always the same, but
were almost always very close, but some previous results will change slightly (Github issue #166)
CHANGES IN VERSION 1.1-0
• when using the default method="profile", confint now returns appropriate upper/lower bounds (-1/1 for correlations, 0/Inf for standard deviations) rather than NA when appropriate
• in a previous development version, ranef returned incorrect conditional variances (github issue #148). this is now fixed
CHANGES IN VERSION 1.0-6 (2014-02-02)
CHANGES IN VERSION 1.0-5 (2013-10-24)
• confint.merMod and vcov.merMod are now exported, for downstream package-author convenience
• the package now depends on Matrix >=1.1-0 and RcppEigen >=0.3.1.2.3
• new rename.response option for refit (see BUG FIXES section)
• eliminated redundant messages about suppressed fixed-effect correlation matrices when p>20
• most inverse-link functions are now bounded where appropriate by .Machine$double.eps, allowing fitting of GLMMs with extreme parameter values
• merMod objects created with refit did not work with update: optional rename.response option added to refit.merMod, to allow this (but the default is still FALSE, for back-compatibility) (reported
by A. Kuznetsova)
• fixed buglet preventing on-the-fly creation of index variables, e.g. y~1+(1|rownames(data)) (reported by J. Dushoff)
• predict now works properly for glmer models with basis-creating terms (e.g. poly, ns)
• step sizes determined from fixed effect coefficient standard errors after first state of glmer fitting are now bounded, allowing some additional models to be fitted
CHANGES IN VERSION 1.0-4 (2013-09-08)
• refit() now works, again, with lists of length 1, so that e.g. refit(.,simulate(.)) works. (Reported by Gustaf Granath)
• getME(.,"ST") was returning a list containing the Cholesky factorizations that get repeated in Lambda. But this was inconsistent with what ST represents in lme4.0. This inconsistency has now been
fixed and getME(.,"ST") is now consistent with the definition of the ST matrix in lme4.0. See https://github.com/lme4/lme4/issues/111 for more detail. Thanks to Vince Dorie.
• Corrected order of unpacking of standard deviation/correlation components, which affected results from confint(.,method="boot"). (Reported by Reinhold Kliegl)
• fixed a copying bug that made refitML() modify the original model
CHANGES IN VERSION 1.0-1 (2013-08-17)
• check.numobs.* and check.numlev.* in (g)lmerControl have been changed (from recent development versions) to check.nobs.* and check.nlev.* respectively, and the default values of
check.nlev.gtreq.5 and check.nobs.vs.rankZ have been changed to "ignore" and "warningSmall" respectively
• in (g)lmerControl, arguments to the optimizer should be passed as a list called optCtrl, rather than specified as additional (ungrouped) arguments
• the postVar argument to ranef has been changed to the (more sensible) condVar ("posterior variance" was a misnomer, "conditional variance" – short for "variance of the conditional mode" – is
• the REform argument to predict has been changed to ReForm for consistency
• the tnames function, briefly exported, has been unexported
• getME(.,"cnms") added
• print method for merMod objects is now more terse, and different from summary.merMod
• the objective method for the respMod reference class now takes an optional sigma.sq parameter (defaulting to NULL) to allow calculation of the objective function with a residual variance
different from the profiled value (Vince Dorie)
CHANGES IN VERSION 1.0-0 (2013-08-01)
• Because the internal computational machinery has changed, results from the newest version of lme4 will not be numerically identical to those from previous versions. For reasonably well- defined
fits, they will be extremely close (within numerical tolerances of 1e-4 or so), but for unstable or poorly-defined fits the results may change, and very unstable fits may fail when they
(apparently) succeeded with previous versions. Similarly, some fits may be slower with the new version, although on average the new version should be faster and more stable. More numerical tuning
options are now available (see below); non-default settings may restore the speed and/or ability to fit a particular model without an error. If you notice significant or disturbing changes when
fitting a model with the new version of lme4, please notify the maintainers.
• VarCorr returns its results in the same format as before (as a list of variance-covariance matrices with correlation and stddev attributes, plus a sc attribute giving the residual standard
deviation/scale parameter when appropriate), but prints them in a different (nicer) way.
• By default residuals gives deviance (rather than Pearson) residuals when applied to glmer fits (a side effect of matching glm behaviour more closely).
• As another side effect of matching glm behaviour, reported log-likelihoods from glmer models are no longer consistent with those from pre-1.0 lme4, but are consistent with glm; see glmer
• More use is made of S3 rather than S4 classes and methods: one side effect is that the nlme and lme4 packages are now much more compatible; methods such as fixef no longer conflict.
• The internal optimizer has changed. [gn]lmer now has an optimizer argument; "Nelder_Mead" is the default for [n]lmer, while a combination of "bobyqa" (an alternative derivative-free method) and
"Nelder_Mead" is the default for glmer. To use the nlminb optimizer as in the old version of lme4, you can use optimizer="optimx" with control=list(method="nlminb") (you will need the optimx
package to be installed and loaded). See lmerControl for details.
• Families in GLMMs are no longer restricted to built-in/hard- coded families; any family described in family, or following that design, is usable (although there are some hard-coded families,
which will be faster).
• [gn]lmer now produces objects of class merMod rather than class mer as before.
• the structure of the Zt (transposed random effect design matrix) as returned by getME(.,"Zt"), and the corresponding order of the random effects vector (getME(.,"u")) have changed. To retrieve Zt
in the old format, use do.call(Matrix::rBind,getME(.,"Ztlist")).
• the package checks input more thoroughly for non-identifiable or otherwise problematic cases: see lmerControl for fine control of the test behaviour.
• A general-purpose getME accessor method allows extraction of a wide variety of components of a mixed-model fit. getME also allows a vector of objects to be returned as a list of mixed-model
components. This has been backported to be compatible with older versions of lme4 that still produce mer objects rather than merMod objects. However, backporting is incomplete; some objects are
only extractable in newer versions of lme4.
• Optimization information (convergence codes, warnings, etc.) is now stored in an @optinfo slot.
• bootMer provides a framework for obtaining parameter confidence intervals by parametric bootstrapping.
• plot.merMod provides diagnostic plotting methods similar to those from the nlme package (although missing augPred).
• A predict.merMod method gives predictions; it allows an effect-specific choice of conditional prediction or prediction at the population level (i.e., with random effects set to zero).
• Likelihood profiling for lmer and glmer results (see link{profile-methods}).
• Confidence intervals by likelihood profiling (default), parametric bootstrap, or Wald approximation (fixed effects only): see confint.merMod
• nAGQ=0, an option to do fast (but inaccurate) fitting of GLMMs.
• Using devFunOnly=TRUE allows the user to extract a deviance function for the model, allowing further diagnostics/customization of model results.
• The internal structure of [gn]lmer is now more modular, allowing finer control of the different steps of argument checking; construction of design matrices and data structures; parameter
estimation; and construction of the final merMod object (see ?modular).
• the formula, model.frame, and terms methods return full versions (including random effect terms and input variables) by default, but a fixed.only argument allows access to the fixed effect
• glmer.nb provides an embryonic negative binomial fitting capability.
• Adaptive Gaussian quadrature (AGQ) is not available for multiple and/or non-scalar random effects.
• Posterior variances of conditional models for non-scalar random effects.
• Standard errors for predict.merMod results.
• Automatic MCMC sampling based on the fit turns out to be very difficult to implement in a way that is really broadly reliable and robust; mcmcsamp will not be implemented in the near future. See
pvalues for alternatives.
• "R-side" structures (within-block correlation and heteroscedasticity) are not on the current timetable.
• In a development version, prior weights were not being used properly in the calculation of the residual standard deviation, but this has been fixed. Thanks to Simon Wood for pointing this out.
• In a development version, the step-halving component of the penalized iteratively reweighted least squares algorithm was not working, but this is now fixed.
• In a development version, square RZX matrices would lead to a pwrssUpdate did not converge in 30 iterations error. This has been fixed by adding an extra column of zeros to RZX.
• Previous versions of lme4 provided the mcmcsamp function, which efficiently generated a Markov chain Monte Carlo sample from the posterior distribution of the parameters, assuming flat (scaled
likelihood) priors. Due to difficulty in constructing a version of mcmcsamp that was reliable even in cases where the estimated random effect variances were near zero (e.g. https://stat.ethz.ch/
pipermail/r-sig-mixed-models/2009q4/003115.html), mcmcsamp has been withdrawn (or more precisely, not updated to work with lme4 versions >=1.0).
• Calling glmer with the default gaussian family redirects to lmer, but this is deprecated (in the future glmer(...,family="gaussian") may fit a LMM using the penalized iteratively reweighted least
squares algorithm). Please call lmer directly.
• Calling lmer with a family argument redirects to glmer; this is deprecated. Please call glmer directly.
CHANGES IN VERSION 0.999375-16 (2008-06-23)
• The underlying algorithms and representations for all the mixed-effects models fit by this package have changed - for the better, we hope. The class "mer" is a common mixed-effects model
representation for linear, generalized linear, nonlinear and generalized nonlinear mixed-effects models.
• ECME iterations are no longer used at all, nor are analytic gradients. Components named 'niterEM', 'EMverbose', or 'gradient' can be included in the 'control' argument to lmer(), glmer() or nlmer
() but have no effect.
• PQL iterations are no longer used in glmer() and nlmer(). Only the Laplace approximation is currently available. AGQ, for certain classes of GLMMs or NLMMs, is being added.
• The 'method' argument to lmer(), glmer() or nlmer() is deprecated. Use the 'REML = FALSE' in lmer() to obtain ML estimates. Selection of AGQ in glmer() and nlmer() will be controlled by the
argument 'nAGQ', when completed.
• The representation of mixed-effects models has been dramatically changed to allow for smooth evaluation of the objective as the variance-covariance matrices for the random effects approach
singularity. Beta testers found this representation to be more robust and usually faster than previous versions of lme4.
• The mcmcsamp function uses a new sampling method for the variance-covariance parameters that allows recovery from singularity. The update is not based on a sample from the Wishart distribution.
It uses a redundant parameter representation and a linear least squares update.
• CAUTION: Currently the results from mcmcsamp look peculiar and are probably incorrect. I hope it is just a matter of my omitting a scaling factor but I have seen patterns such as the parameter
estimate for some variance-covariance parameters being the maximum value in the chain, which is highly unlikely.
• The 'verbose' argument to lmer(), glmer() and nlmer() can be used instead of 'control = list(msVerbose = TRUE)'. | {"url":"http://rsync.jp.gentoo.org/pub/CRAN/web/packages/lme4/news.html","timestamp":"2024-11-05T16:27:08Z","content_type":"text/html","content_length":"59014","record_id":"<urn:uuid:d632c76b-9c79-4265-a88e-0c4d78d8de7f>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00728.warc.gz"} |
Will physicists be replaced by robots? - Jakob Schwichtenberg
in Miscellaneous
Will physicists be replaced by robots?
I always thought that such a suggestion is ridiculous. How could a robot ever do what physicists do? While many jobs seem to be in danger because of recent advances in automation – up to 47 %
according to recent studies – the last thing that will be automated, if ever, are jobs like that of physicists which need creativity, right?
For example, this site, which was featured in many major publications, states that there is only a 10% chance that robots will take the job of physicists:
As physicist I have only 10% Probability of Automation in Future! Robots will have to work harder to get my job 🤖 https://t.co/NSLtxuzXDa
— Freya Blekman (@freyablekman) June 18, 2017
Recently author James Gleick commented on how shocked professional “Go” players are by the tactics of Google’s software “AlphaGo”:
If humans are this blind to the truth of Go, how well are we doing with theoretical physics? https://t.co/UeeGqyxRD8
— James Gleick (@JamesGleick) October 18, 2017
Sean Caroll answered and summarized how most physicists think about this:
We’re doing great with theoretical physics! It’s the worst possible analogy to how AI does better than humans at complex games like Go. https://t.co/Cki1q0hRgS
— Sean Carroll (@seanmcarroll) October 18, 2017
Fundamental physics is analogous to "the rules of Go." Which are simple and easily mastered. Go *strategy* is more like bio or neuroscience.
— Sean Carroll (@seanmcarroll) October 18, 2017
A Counterexample
Until very recently I would have agreed. However, a few weeks ago I discovered this little paper and it got me thinking. The idea of the paper is really simple. Just feed measurement data into an
algorithm. Give him a fixed set of objects to play around with and then let the algorithm find the laws that describe the data best. The authors argue that their algorithm is able to rediscover
Maxwell’s equations. These equations are still the best equations to describe how light behaves. Their algorithm was able to find these equations “in about a second”. Moreover, they describe their
program as a “computational embodiment of the scientific method: observation, consideration of candidate theories, and validation.” That’s pretty cool. Once more I was reminded that “everything seems
impossible until it’s done.”
Couldn’t we do the same to search for new laws by feeding such an algorithm the newest collider data? Aren’t the jobs of physicists that safe after all?
What do physicists do?
First of all, the category “physicist” is much too broad to discuss the danger of automation. For example, there are experimental physicists and theoretical physicists. And even inside these
subcategories, there are further important sub-sub-categories.
On the experimental side, there are people who actually build experiments. Those are the guys who know how to use a screwdriver. In addition, there are people who analyze the data gathered by
On the theoretical side, there are theorists and phenomenologists. The distinction here is not so clear. For example, one can argue that phenomenology is a subfield of theoretical physics. Many
phenomenologists call themselves theoretical physicists. Broadly, the job of a theoretical physicist is to explain and predict how nature behaves by writing down equations. However, there are many
different approaches how to write down new equations. I find the classification outlined here helpful. There is:
1. Curiosity Driven Research; where “anything goes, that is allowed by basic principles and data. […] In general, there is no further motivation for the addition of some particle, besides that it is
not yet excluded by the data.”
2. Data-Driven Research; where new equations are written down as a response to experimental anomalies.
3. Theory-Driven Research; which is mostly about “aesthetics” and “intuition”. The prototypical example, is of course, Einstein’s invention of General Relativity.
The job of someone working in each such sub-sub-category is completely different to the jobs in another sub-sub-category. Therefore, there is certainly no universal answer to the question how likely
it is for “robots” to replace physicists. Each of sub-sub-categories mentioned above must be analyzed on its own.
What could robots do?
Let’s start with the most obvious one. Data analysis is an ideal job for robots. Unsurprisingly several groups are already working or experimenting with neural networks to analyze LHC data. In the
traditional approach to collider data analyses, people have to invent criteria for how we can distinguish different particles in the detector. If the angle of two detected photons is large than X°,
the overall energy of them smaller than Y GeV the particle is with a probability Z% some given particle. In contrast, if you use a neural network, you just have train it using Monte-Carlo data, where
you know which particle is where. Then you can let the trained network analyze the collider data. In addition, after the training, you can investigate the network to see what it has learned. This way
neural networks can be used to find new useful variables that help to distinguish different particles in a detector. I should mention that this approach is not universally favored because some feel
that a neural network is too much of a black box to be trusted.
What about theoretical physicists?
In the tweet quoted above, Sean Carroll argues that “Fundamental physics is analogous to “the rules of Go.” Which are simple and easily mastered. Go *strategy* is more like bio or neuroscience.” Well
yes, and no. Finding new fundamental equations is certainly similar to inventing new rules for a game. This is broadly the job of a theoretical physicist. However, the three approaches to “doing
theoretical physics”, mentioned above, are quite different.
In the first and second approach, the “rules of the game” are pretty much fixed. You write down a Lagrangian and afterward compare its predictions with measured data. The new Lagrangian involves new
fields, new coupling constants etc., but must be written down according to fixed rules. Usually, only terms that respect rules of special relativity are allowed. Moreover, we know that the simplest
possible terms are the most important ones, so you focus on them first. (More complicated terms are “non-renormalizable” and therefore suppressed by some large scale.) Given some new field or fields
writing down the Lagrangian and deriving the corresponding equations of motion is a straight-forward. Moreover, while deriving the experimental consequences of some given Lagrangian can be quite
complicated, the general rules of how to do it are fixed. The framework that allows us to derive predictions for colliders or other experiments starting from a Lagrangian is known as Quantum Field
This is exactly the kind of problem that was solved, although in a much simpler setting, by the Mark A. Stalzer and Chao Ju in the paper mentioned above. There are already powerful algorithms, like,
for example, SPheno or micrOMEGAs which are capable of deriving many important consequences of a given Lagrangian, almost completely automagically. So with further progress in this direction, it
seems not completely impossible that an algorithm will be able to find the best possible Lagrangian to describe given experimental data.
As an aside: A funny name for this goal of theoretical physics that deals with the search for the “ultimate Lagrangian of the world” was coined by Arthur Wightman, who called it “the hunt for the
Green Lion”. (Source: Conceptual Foundations of Quantum Field Theory: Tian Yu Cao)
What then remains on the theoretical side is “Theory-Driven Research”. I have no idea how a “robot” could do this kind of research, which is probably what Sean Caroll had in mind in his tweets. For
example, the algorithm by Mark A. Stalzer and Chao Ju only searches for laws that consist of some predefined objects, vectors, tensors and uses predefined rules of how to combine them: scalar
products, cross products etc. It is hard to imagine how paradigm-shifting discoveries could be made by an algorithm like this. General relativity is a good example. The correct theory of gravity
needed completely new mathematics that wasn’t previously used by physicists. No physicists around 1900 would have programmed crazy rules such as those of non-Euclidean geometry into the set of
allowed rules. An algorithm that was designed to guess Lagrangian will always only spit out a Lagrangian. If the fundamental theory of nature cannot be written down in Lagrangian form, the algorithm
would be doomed to fail.
To summarize, there will be physicists in 100 years. However, I don’t think that all jobs currently done by theoretical and experimental physicists will survive. This is probably a good thing. Most
physicists would love to have more time to think about fundamental problems like Einstein did.
P.S. I wrote a textbook which is in some sense the book I wished had existed when I started my journey in physics. It's called
"Physics from Symmetry"
and you can buy it, for example, at
. And I'm now on
too if you'd like to get updates about what I'm recently up to.
If you want to get an update whenever I publish something new, simply put your email address in the box below. | {"url":"http://jakobschwichtenberg.com/physicist_robots/","timestamp":"2024-11-03T21:47:33Z","content_type":"text/html","content_length":"36781","record_id":"<urn:uuid:2f849d3e-5482-482b-be8c-8ac8d0cc31e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00081.warc.gz"} |
Mass-Mole Calculations Chemistry Tutorial (2024)
Key Concepts
⚛ 1 mole of a pure substance has a mass equal to its molecular mass^(1) expressed in grams.
· This is known as the molar mass, M, and has the units g mol^-1
(g/mol, grams per mole of substance)
⚛ The relationship between molar mass, mass and moles can be expressed as a mathematical equation as shown below:
g mol^-1 = g ÷ mol
molar mass = mass ÷ moles
M = m ÷ n
M = molar mass of the pure substance (measured in g mol^-1)
m = mass of the pure substance (measured in grams, g)
n = amount of the pure substance (measured in moles, mol)
⚛ This mathematical equation can be rearranged to give the following:
(i) n = m ÷ M
moles = mass ÷ molar mass
(ii) m = n × M
mass = moles × molar mass
⚛ To calculate the moles of pure substance: n = m ÷ M
⚛ To calculate mass of pure substance: m = n × M
⚛ To calculate molar mass of pure substance: M = m ÷ n
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Calculating the Mass of a Pure Substance (m=nM)
1 mole of a pure substance is defined as having a mass in grams equal to its relative molecular mass.
This quantity is known as the molar mass (symbol M).
So, mass of 1 mole of a pure substance = relative molecular mass in grams
And, mass of 1 mole of a pure substance = molar mass of the pure substance (g mol^-1)
Or, mass of 1 mole = M (g mol^-1)
The table below gives the mass of 1 mole of a number of common pure substances:
name molecularformula relative molecular mass molar mass mass of 1 mole
(g mol^-1) (g)
helium gas He 4.003 4.003 g mol^-1 4.003 g
oxygen gas O[2] 2 × 16.00 = 32.00 32.00 g mol^-1 32.00 g
carbon dioxide gas CO[2] 12.01 + (2 × 16.00) = 44.01 44.01 g mol^-1 44.01 g
liquid water H[2]O (2 × 1.008) + 16.00 = 18.016 18.016 g mol^-1 18.016 g
From the table we see that 1 mole of water has a mass of 18.016 grams, which isn't very much (about the mass of water in a couple of small ice-cubes you'd make in your family freezer).
But what if you had 10 moles of water? What would be the mass of 10 moles of water?
If 1 mole of water has a mass of 18.016 g, then 10 moles of water must have ten times more mass:
mass of 10 moles of water = 10 × mass of 1 mole of water
mass of 10 moles of water = 10 × 18.016 = 180.16 g (about the mass of water you could put in a small glass)
So, if we only had ½ mole of water, what mass of water would we have?
If 1 mole of water has a mass of 18.016 g, then ½ mole of water must have ½ the mass:
mass of ½ mole of water = ½ × mass of 1 mole of water
mass of ½ mole of water = ½ × 18.016 = 9.008 g
In both of the examples above, we can calculate the mass of water in grams by multiplying the moles of water by the mass of 1 mole of water in grams:
mass water = moles of water × mass of 1 mole water
because the mass of 1 mole of water in grams is known as its molar mass, we can write:
mass water = moles of water × molar mass of water
The table below compares the mass of different amounts of water in moles:
mass of water moles of water mass of 1 mole of water
(g) = (mol) × (molar mass of water)
(g mol^-1)
0 = 0.00 × 18.016
9.008 = 0.50 × 18.016
18.016 = 1.00 × 18.016
27.024 = 1.50 × 18.016
180.16 = 10.00 × 18.016
270.24 = 15.00 × 18.016
From the data in the table we can generalise and say that for any pure substance the mass of substance in grams is equal to the moles of substance multiplied by the mass of 1 mole of the substance:
mass = moles × mass of 1 mole
and since mass of 1 mole of a substance (in grams) = molar mass (in grams per mole)
mass (g) = moles × molar mass (g mol^-1)
m = n × M
m = mass of pure substance in grams
n = amount of pure substance in moles
M = molar mass of pure substance in grams per mole
We could also plot the data in the table above on a graph as shown below:
This graph above shows a straight line that passes through the origin (0,0) so the equation for the line is:
y = slope × x
y is mass of water (g)
x is moles of water (mol)
slope (gradient) of the line = vertical rise ÷ horizontal run
We can determine the slope of the line using 2 points on the straight line, for example, (0,0) and (15.0, 270.24):
slope = (270.24 g - 0 g) ÷ (15 mol - 0 mol) = 18.016 g mol^-1
Since 18.016 g mol^-1 is the molar mass of water, we can say:
slope = molar mass of water (g mol^-1)
Therefore the equation for this line is:
mass (H[2]O) = molar mass (H[2]O) × moles (H[2]O)
In general:
mass (g) = molar mass (g mol^-1) × moles (mol)
From the data in the table and its graphical representation, we can generalise and say that for any pure substance the mass of substance in grams is equal to the moles of substance multiplied by the
mass of 1 mole of the substance:
mass = moles × mass of 1 mole
and since mass of 1 mole of a substance (in grams) = molar mass (in grams per mole)
mass (g) = moles × molar mass (g mol^-1)
m = n × M
Follow these steps to calculate the mass of a pure substance given the amount of substance in moles:
Step 1. Extract the data from the question:
mass = m = ? (units are grams)
moles = n = write down what you are told in the question
molar mass = M = write down what you are told in the question (units are g mol^-1)
(you may need to calculate this using the molecular formula of the pure substance and a Periodic Table)
Step 2. Check the units for consistency and convert if necessary:
The amount of substance must be in moles (mol) !
If amount is given in millimoles (mmol), divide it by 1,000 to give the amount in moles (mol).
If amount is given in micromoles (μmol), divide it by 1,000,000 to give an amount in moles (mol).
If amount is given in kilomoles (kmol), multiply it by 1,000 to give an amount in moles (mol).
Step 3. Write the mathematical equation (mathematical formula):
mass = moles × molar mass
m = n × M
Step 4. Substitute in the values and solve the equation to find the value of mass, m, in grams (g).
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Calculating the Moles of a Pure Substance (n=m/M)
In the discussion above, we discovered that we could calculate the mass of a pure substance using the moles and molar mass of the substance:
mass (g) = moles (mol) × molar mass (g mol^-1)
How would we calculate the moles of pure substance if we knew the mass of the substance?
(a) We could use some algebra: divide both sides of the equation by the molar mass:
mass = moles × [DEL:molar mass:DEL]
molar mass [DEL:molar mass:DEL]
moles = mass ÷ molar mass
n = m ÷ M
(b) We could use some logic:
we know the mass with units of grams (g)
we know the molar mass with units of grams per mole (g mol^-1)
we need to find moles with units of mole (mol)
By inspection of units we see that dividing molar mass by mass will give us a quantitiy in units of "mol^-1"
molar mass/mass = [DEL:g:DEL] mol^-1/[DEL:g:DEL] = mol^-1
If we turn this upside down (in mathematical terms, take the reciprocal) we get a quantity with units of "mol" which is what we want:
mass/molar mass = [DEL:g:DEL]/[DEL:g:DEL] mol^-1 = mol
moles = mass ÷ molar mass
n = m ÷ M
Follow these steps to calculate the amount of pure substance in moles given the mass of substance:
Step 1. Extract the data from the question:
mass = m = write down what you are told in the question
moles = n = ? (units are mol)
molar mass = M = write down what you are told in the question (units are g mol^-1)
(you may need to calculate this using the molecular formula of the pure substance and a Periodic Table)
Step 2. Check the units for consistency and convert if necessary:
Mass must be in grams !
If mass is given in milligrams (mg), divide it by 1,000 to give the mass in grams (g).
If mass is given in micrograms (μg), divide it by 1,000,000 to give a mass in grams (g).
If mass is given in kilograms (kg), multiply it by 1,000 to give a mass in grams (g).
Step 3. Write the mathematical equation (mathematical formula):
moles = mass ÷ molar mass
n = m ÷ M
Step 4. Substitute in the values and solve the equation to find moles of substance (mol).
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Calculating the Molar Mass of a Pure Substance (M=m/n)
What if you knew the amount of a pure substance in moles and its mass?
Could you calculate its molar mass?
Recall that mass = moles × molar mass or m = n × M
(a) We could use some algebra: divide both sides of the equation by the moles:
mass = [DEL:moles:DEL] × molar mass
moles [DEL:moles:DEL]
molar mass = mass ÷ moles
M = m ÷ n
or (b) We could use some logic:
By inspection of units we see that dividing the mass in grams by the amount in moles we arrive at a quantity with the units grams per mole (g mol^-1) which are the units for molar mass.
Therefore, molar mass (g mol^-1) = mass (g) ÷ moles (mol)
or you can write
M = m ÷ n
Follow these steps to calculate the molar mass of a pure substance given the amount of substance in moles and the mass of substance:
Step 1. Extract the data from the question:
mass = m = write down what you are told in the question
moles = n = write down what you are told in the question
molar mass = M = ? (units are g mol^-1)
Step 2. Check the units for consistency and convert if necessary:
Mass must be in grams (g)!
Amount, moles, must be in moles (mol)!
Step 3. Write the mathematical equation (mathematical formula):
molar mass = mass ÷ moles
M = m ÷ n
Step 4. Substitute in the values and solve the equation to find the molar mass of the substance in grams per mole.
↪ Back to top
Worked Examples of Calculating Mass, Moles, Molar Mass
In each of the worked examples below, you will be asked to calculate either the moles, mass, or molar mass of a pure substance.
To answer each question correctly you will need to:
1. Step 1. Extract the information from the question
2. Step 2. Check the data for consistency of units
3. Step 3. Choose a mathematical equation to find the unknown
4. Step 4. Substitute your values into the equation and solve
Worked Example: mass = moles × molar mass (m=n×M)
Question: Calculate the mass of 0.25 moles of water, H[2]O.
Step 1. Extract the data from the question:
moles = n = 0.25 mol
molar mass = M = (2 × 1.008) + 16.00 = 18.016 g mol^-1
(Calculated using the periodic table)
mass = m = ? g
Step 2. Check the data for consistency:
Is the amount of water in moles (mol)? Yes.
We do not need to convert this.
Step 3. Write the mathematical equation (mathematical formula):
mass = moles × molar mass
m = n × M
Step 4. Substitute the values into the equation and solve for mass (g):
mass = m = 0.25 [DEL:mol:DEL] × 18.016 g [DEL:mol^-1:DEL] = 4.5 g
(only 2 significant figures are justified)
Worked Example: moles = mass ÷ molar mass (n=m/M)
Question: Calculate the amount of oxygen gas, O[2], in moles present in 124.5 g of oxygen gas.
Step 1. Extract the data from the question:
mass = m = 124.5 g
molar mass = M = 2 × 16.00 =32.00 g mol^-1
(Calculated using the periodic table)
moles = n = ? mol
Step 2. Check the data for consistency:
Is the mass of oxygen gas in grams (g)? Yes.
We do not need to convert this.
Step 3. Write the mathematical equation (mathematical formula):
moles = mass ÷ molar mass
n = m ÷ M
Step 4. Substitute the values into the equation and solve to find moles of oxygen gas:
moles = n = 124.5 [DEL:g:DEL] ÷ 32.00 [DEL:g:DEL] mol^-1 = 3.891 mol
(4 significant figures are justified)
Worked Example: molar mass = mass ÷ moles (M=m/n)
Question: Calculate the molar mass of a pure substance if 1.75 moles of the substance has a mass of 29.79 g.
Step 1. Extract the data from the question:
mass = m = 29.79 g
moles = n = 1.75 mol
Step 2. Check the data for consistency:
Is the mass of in grams (g)? Yes. We do not need to convert this.
Is the amount of substance in moles (mol)? Yes. We do not need to convert this.
Step 3. Write the equation:
molar mass = mass ÷ moles
M = m ÷ n
Step 4. Substitute the values into the equation and solve for molar mass:
molar mass = M = 29.79 g ÷ 1.75 mol = 17.0 g mol^-1
(3 significant figures are justified)
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Problem Solving Using Moles, Mass, and Molar Mass
The Problem: Calcium carbonate, CaCO[3], is an important industrial chemical.
Chris the Chemist has an impure sample of calcium carbonate.The mass of the impure sample is 0.1250 kg and it is composed of 87.00% (by mass) calcium carbonate.Before Chris can use this calcium
carbonate in a chemical reaction, Chris needs to know the amount, in moles, of calcium carbonate present in this sample.
Calculate the amount of calcium carbonate in moles present in this impure sample of calcium carbonate.
Solving the Problem using the StoPGoPS model for problem solving:
What is the question asking you to do?
STOP! State the question. Calculate the amount of calcium carbonate in moles
n(CaCO[3]) = moles of calcium carbonate = ? mol
What chemical principle will you need to apply?
Apply stoichoimetry (n = m ÷ M)
What information (data) have you been given?
• molecular formula of calcium carbonate: CaCO[3]
• mass of sample = m(sample) = 0.1250 kg
• percentage composition of sample = 87.00% by mass CaCO[3]
What is your plan for solving this problem?
(i) Write the mathematical equation to calculate moles of calcium carbonate:
n(mol) = m(g) ÷ M(g mol^-1)
(ii) Calculate the mass of calcium carbonate in the sample in kilograms (kg).
PAUSE! Pause to Plan.
mass of calcium carbonate (kg) = 87.00% of mass of sample (kg)
m(CaCO[3]) in kilograms = (87.00/100) × m(sample)
(iii) Convert the mass of calcium carbonate in kilograms (kg) to mass in grams (g)
m(CaCO[3]) in grams = m(CaCO[3]) in kg × 1000 g/kg
(iv) Calculate the molar mass of calcium carbonate
(use Periodic Table to find molar mass of each element):
molar mass = M(CaCO[3]) = M(Ca) + M(C) + [3 × M(O)] =
(v) Substitute the values for m(CaCO[3]) in grams and M(CaCO[3]) in g mol^-1 into the mathematical equation and solve for n (mol)
n(mol) = m(g) ÷ M(g mol^-1)
(i) Write the mathematical equation to calculate moles of calcium carbonate:
n(mol) = m(g) ÷ M(g mol^-1)
n(CaCO[3]) = m(CaCO[3]) ÷ M(CaCO[3])
(ii) Calculate the mass of calcium carbonate in the sample in kilograms (kg).
mass of calcium carbonate (kg) = 87.00% of mass of sample (kg)
m(CaCO[3]) in kilograms = (87.00/100) × m(sample)
m(CaCO[3]) in kilograms = (87.00/100) × 0.1250 kg = 0.10875 kg
(iii) Convert the mass of calcium carbonate in kilograms (kg) to mass in grams (g)
m(CaCO[3]) in grams = m(CaCO[3]) in kg × 1000 g/kg
GO! Go with the Plan.
m(CaCO[3]) in grams = 0.10875 kg × 1000 g/kg = 108.75 g
(iv) Calculate the molar mass of calcium carbonate
(use Periodic Table to find molar mass of each element):
molar mass = M(CaCO[3]) = M(Ca) + M(C) + [3 × M(O)]
M(CaCO[3]) = 40.08 + 12.01 + [3 × 16.00] = 40.08 + 12.01 + 48.00 = 100.09 g mol^-1
(v) Substitute the values for m(CaCO[3]) in grams and M(CaCO[3]) in g mol^-1 into the mathematical equation and solve for n (mol)
n(CaCO[3]) = m(CaCO[3]) ÷ M(CaCO[3])
n(CaCO[3]) = m(CaCO[3]) ÷ M(CaCO[3]) = 108.75 [DEL:g:DEL] ÷ 100.09 [DEL:g:DEL] mol^-1 = 1.087 mol
(4 significant figures are justified)
Have you answered the question that was asked?
Yes, we have calculated the moles of calcium carbonate in the sample.
Is your solution to the question reasonable?
Let's work backwards to see if the moles of calcium carbonate we have calculated will give us the correct mass for the sample.
PAUSE! Ponder Plausability. Roughly calculate mass of CaCO[3] in 1.087 mol (≈ 1 mol):
m(CaCO[3]) = n × M = 1 × (40 + 12 + 3 × 16) = 100 g
Roughly calculate the mass of sample if 87% of its mass is due to CaCO[3]:
m(CaCO[3]) = 87/100 × m(sample)
m(sample) = 100/87 × m(CaCO[3]) = 100/87 × 100 = 115 g = 0.115 kg
Since this approximate value for the mass of the sample is about the same as the mass of sample given in the question, we are reasonably confident that our answer is
How many moles of calcium carbonate are present in the sample?
STOP! State the solution.
n(CaCO[3]) = 1.087 mol
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Sample Question: Moles, Mass, and Molar Mass
Determine the mass in grams of 0.372 moles of solid rhombic sulfur (S[8]).
↪ Back to top
(1) Molecular mass is also known as molecular weight, formula weight or formula mass
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Physicists To Probe Beyond The Beginning Of Time, Before Creation Itself
Previously it was believed that the beginning of the universe was a ‘singularity’ – a moment beyond physics, beyond examination and simply beyond human understanding. But an international team of
researchers have re-written the physics books allowing cosmologists to see and investigate the very moment of creation – and perhaps even further.
The implications are profound. The physics to precisely explain the moment of creation would effectively render God and world religions redundant in a way which would make the Copernican revolution
(proving the Earth was not the center of the universe) seem a relatively minor moment in scientific history.
According to current thinking of scientific giants like Stephen Hawking, Roger Penrose, Edwin Hubble and Alan Guth, post-Einstein physicists dictates that it is impossible to see or investigate the
moment of origin of the universe because physics simply breaks down.
The same collapse in understanding applies to black holes where the observer eventually hits a ‘singularity’ – the point at which all attempts to comprehend the math and physics are doomed. But now
an international team of scientists and mathematicians have turned that notion on its head in a new paper which dispels the notion of the impossible-to-comprehend ‘singularity’.
Using mind-bogglingly complex maths and the extraordinary strangeness of the quantum world they claim the physics of both the origin of the universe and black holes ARE comprehensible and can be
examined using existing, if cutting-edge, physics.
It should then be possible to not only probe the interior of a black hole but also to investigate and PROVE the origin of the universe and the creation of everything. The paper, called Quantum
no-singularity theorem from geometric flows, introduces the super-strange but provable quantum physics to disprove established notions of the singularity. The paper has been accepted for publication
in Int J Mod Phys A, and is present on arXiv as arXiv:1705.00977
Author of the paper Prof. Mir Faizal, who is affiliated to University of Lethbridge, Canada and University of British Colombia, Okanagan, Canada, said:
“It is known that general relativity predicts that the universe started with a big bang singularity and the laws are physics cannot be meaningfully applied to a singularity.
“Furthermore, the Penrose-Hawking singularity theorems demonstrate that the occurrence of these singularities is an intrinsic property built into the structure of classical general relativity. It has
been argued that such singularities will be removed by quantum effects, but such work to date has been done using different approaches to quantum gravity, and all these approaches have problems
associated with them.
“To prove that the singularities are actually removed by quantum effects, we would need a quantum version of the Penrose-Hawking singularity theorems, and this is what we have obtained in our paper.
So, our paper proves quantum effects do remove singularities from general relativity, just as Penrose-Hawking singularity theorems proves that singularities are an intrinsic property of classical
general relativity. ''
The paper, co-written by Salwa Alsaleh, Lina Alasfar and Ahmed Farag Ali, opens by explaining how Einstein’s General Relativity predicts its own downfall due to singularities.
It also explains that even though it had been proposed that singularities can be removed by quantum effects, the previous work relied on specific models. All these models had their own problems. The
absence of singularities had been studied in string theory, but string theory has its own problems.
It predicts the existence of supersymmetry, and supersymmetry has not been discovered till date. Another approach to study quantum effects in general relativity is the loop quantum gravity, and but
it has its own problems too ( for example, no one has been able to reproduce Einstein equations from loop quantum gravity by neglecting the quantum effects).
The theorem proven in this paper does not rely on a specific model of quantum gravity, but is proven by very general considerations, and is expected to hold no matter which approach to quantum
gravity is the correct one. However, when quantum effects are neglected, the origin Hawking and Penrose results are reproduced in their formalism. The paper ends by concluding the center of black
holes are not singularities and thus one day could be scientifically investigated.
The paper states: “The absence of singularity means the absence of inconsistency in the laws of nature describing our universe, that shows a particular importance in studying black holes and | {"url":"http://www.thespaceacademy.org/2018/03/physicists-to-probe-beyond-beginning-of.html","timestamp":"2024-11-06T13:54:25Z","content_type":"application/xhtml+xml","content_length":"176869","record_id":"<urn:uuid:8c6baa73-67ab-4dba-9511-8e29d357302e>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00623.warc.gz"} |
The Power of the Bayes Theorem | Hire Someone To Take My Proctored Exam
In statistics and probability, the Bayes Theorem describes how the likelihood of an event depends on previous information about possible conditions that could be associated with that event. The
statement says, “the occurrence of any event A is more likely than not B,” where A and B are the probabilities of different events, such as a coin coming up tails or a coin coming up heads. A
generalization is that the ratio of A’s and B’s probabilities can be compared to the ratio of their prior probabilities.
To make this work, a person needs to have prior data that indicates how likely it is that the event is true. Bayesian statisticians typically use the following form to express this, where p(A) is the
prior probability that A would occur, and p(B) is the prior probability that B would occur. If you use a different form, you will get different results, which depend on the data and assumptions you
have made. To get a Bayesian estimate of A’s prior probability, first determine whether the A/B ratio and its posterior probability is greater than zero. Then calculate p(A|B) from this.
This Bayesian method of calculation of Bayesian estimates involves a model that allows you to combine information obtained in different ways and allows you to estimate both A and B at once. In other
words, Bayesian statisticians have to be able to generate a model that allows you to combine information from different sources and estimates both of the probabilities of A and B.
There are many models that Bayesian statisticians use to build their estimates, but two popular models are: the conditional random variable model and the random equation model. The conditional model
is used when you have a set of A/B data and you want to know whether the Bayesian estimate of A/B is a good fit to the data; on the other hand, the random equation model is used when you have A and B
data and you want to find the posterior probability that the data give rise to a Bayesian estimate of A/B.
In the conditional model, you look at the A/B data in one way and then ask yourself what is the chance that if the data came up tails you would have made the same decision as if it came up heads. If
the odds are high, this means that the data was not very well-fitting to the data, so the Bayesian estimate of A/B is high, and if they are low then the odds are low, and this means the data was very
well-fitting to the data.
Then, you model the odds as follows: If the prior has A and B data, you use the conditional model and assume that both A and B come out tails and you use the same prior and the posterior. The
conditional probability is given by the formula p(A/B) = p(B). And, if the prior has nothing, then you assume that the data are independent, so that the data come up tails and the Bayesian estimate
is zero, p(A/B) = zero. The posterior is then given by taking the posterior of the conditional probability and dividing it by the total number of data to be estimated, or by the total number of
observations to be estimated.
The Bayesian model allows you to get a better fit to the data, especially if you want a high prior and a low posterior. A lot of Bayesian statistics calculations are based upon this basic model. A
person may also assume that if the data come out tails he/she would also have drawn the data in the opposite direction (toward tails) if the prior had tails. If the posterior is negative, that means
that the data have come up tails and you would have drawn the data in the same direction if the prior were positive. and vice versa.
Random variables are typically used in the Bayesian method, because they do not have any prior distributions, so they are able to give a much better fit. A person who is not familiar with Bayes can
still use this type of model. It is easier for them to estimate the probabilities if they know about Bayes. It is also easy for them to make correct estimations of the posterior distributions.
Bayesian statisticians often use random variable theory as an example when explaining their method to students who are not yet completely aware of Bayesian theory. | {"url":"https://crackmyproctoredexam.com/the-power-of-the-bayes-theorem/","timestamp":"2024-11-03T12:26:31Z","content_type":"text/html","content_length":"106738","record_id":"<urn:uuid:2cb85254-011e-4b33-8b88-80a141c6b93d>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00171.warc.gz"} |
Convert Decade to Attosecond
Please provide values below to convert decade to attosecond [as], or vice versa.
Decade to Attosecond Conversion Table
Decade Attosecond [as]
0.01 decade 3.15576E+24 as
0.1 decade 3.15576E+25 as
1 decade 3.15576E+26 as
2 decade 6.31152E+26 as
3 decade 9.46728E+26 as
5 decade 1.57788E+27 as
10 decade 3.15576E+27 as
20 decade 6.31152E+27 as
50 decade 1.57788E+28 as
100 decade 3.15576E+28 as
1000 decade 3.15576E+29 as
How to Convert Decade to Attosecond
1 decade = 3.15576E+26 as
1 as = 3.1688087814029E-27 decade
Example: convert 15 decade to as:
15 decade = 15 × 3.15576E+26 as = 4.73364E+27 as
Popular Time Unit Conversions
Convert Decade to Other Time Units | {"url":"https://www.unitconverters.net/time/decade-to-attosecond.htm","timestamp":"2024-11-06T14:03:26Z","content_type":"text/html","content_length":"9556","record_id":"<urn:uuid:dea8bd53-9e6a-4070-a270-db1e51108b7b>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00614.warc.gz"} |
How do you write a sequence that has three geometric means between 256 and 81? | HIX Tutor
How do you write a sequence that has three geometric means between 256 and 81?
Answer 1
$256 , 192 , 144 , 108 , 81$
#256=4^4# and #81 = 3^4#
So the geometric mean of #256# and #81# is:
#sqrt(256*81) = sqrt(4^4*3^4) = 4^2*3^2 = 144#
The geometric mean of #256# and #144# is:
#sqrt(256*144) = sqrt(4^4 4^2 3^2) = 4^3*3 = 192#
The geometric mean of #144# and #81# is:
#sqrt(144*81) = sqrt(4^2 3^2 3^4) = 4*3^3 = 108#
The sequence: #256, 192, 144, 108, 81# is a geometric sequence with common ratio #3/4#
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Spiking Neural P Systems: An improved normal form
Spiking neural P systems (in short, SN P systems) are computing devices based on the way the neurons communicate through electrical impulses (spikes). These systems involve various ingredients; among
them, we mention forgetting rules and the delay in firing rules. However, it is known that the universality can be obtained without using these two features. In this paper we improve this result in
two respects: (i) each neuron contains at most two rules (which is optimal for systems used in the generative mode), and (ii) the rules in the neurons using two rules have the same regular expression
which controls their firing. This result answers a problem left open in the literature, and, in this context, an incompleteness in some previous proofs related to the elimination of forgetting rules
is removed. Moreover, this result shows a somewhat surprising uniformity of the neurons in the SN P systems able to simulate Turing machines, which is both of a theoretical interest and it seems to
correspond to a biological reality. When a bound is imposed on the number of spikes present in a neuron at any step of a computation (such SN P systems are called finite), two surprising results are
obtained. First, a characterization of finite sets of numbers is obtained in the generative case (this contrasts the case of other classes of SN P systems, where characterizations of semilinear sets
of numbers are obtained for finite SN P systems). Second, the accepting case is strictly more powerful than the generative one: all finite sets and also certain arithmetical progressions can be
accepted. A precise characterization of the power of accepting finite SN P systems without forgetting rules and delay remains to be found. | {"url":"http://www.gcn.us.es/?q=node/648","timestamp":"2024-11-13T23:08:53Z","content_type":"application/xhtml+xml","content_length":"39836","record_id":"<urn:uuid:712a7d6d-bd2e-4e3a-b56d-b2ba86bdb008>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00687.warc.gz"} |
Math, Grade 6, Equations and Inequalities, Applying Multiplication Properties
Keep Balanced
Keep Balanced
Look at the balance scale pictured here.
• What number can you multiply the number on each side of the scale by but still keep the scale balanced?
• Based on your new scale, what number can you divide both sides by to keep the scale balanced? | {"url":"https://oercommons.org/courseware/lesson/1629/student/?section=1","timestamp":"2024-11-04T11:03:42Z","content_type":"text/html","content_length":"34412","record_id":"<urn:uuid:ad016585-8a00-42e6-801f-3b60f0c15aea>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00816.warc.gz"} |
Accuracy, Precision and Uncertainty
Counting is the only type of measurement which is free of uncertainties, as long as the number of objects or elements being counted does not change during the counting process. The result of this
type of counting measurement is an example of an exact number. When we count eggs in a box, we know the exact number of eggs present in the box. Also, the number of the defined quantities are exact.
From definition, 1 foot is exactly equal to 12 inches, 1 inch is exactly equal to 2.54 centimeters, and 1 gram is exactly equal to 0.001 kilograms. However, measurements other than counting are
uncertain to varying degrees due to practical limitations of the measurement method used.
Accuracy, Precision & Uncertainty
Accuracy is the agreement between a measurement value and true value. When a clock strikes twelve and the sun is directly overhead, then the clock is considered accurate. The measurement of clock and
the phenomena which is meant to measure (the sun is located or present at the zenith) are in agreement. Accuracy cannot meaningfully be discussed unless the correct value is not known or cannot be
Accuracy Definition
The ability of a device to measure an exact or accurate value is called accuracy. In other words, accuracy is the closeness of measured value to a standard value or true value. Accuracy is achieved
with small readings. The small reading decreases the error in the calculation.
Precision is the repeatability of the measurement. It is not necessary for us to know the correct value or true value. When a clock reads exactly 10:17 a.m. when the sun is at the zenith, this clock
is said to be very precise.
The proximity or closeness of two or more measurements to each other is referred to as the precision of a material or substance. If we weigh a given substance 5 (five) times and get 3.2 kg each time,
our measurement is very precise, but not necessary that it is accurate. Precision is independent from accuracy.
The uncertainty of a measured value is an interval around that value, so each repetition of the measurement produces a new result which falls within that interval. The experimenter allocates this
uncertainty interval according to established principles of uncertainty estimation.
Uncertainty instead of error is the important term for the working scientist. Uncertainty miraculously enables the scientist to form completely certain statements.
Uncertainty in Measurement
All scientific measurements include some degree of error or uncertainty. Precision and accuracy are two essential factors related to uncertainty. Precision means how well each measurement agrees with
each other, and the accuracy means how well the experimental measurement agrees with true or correct values.
Accuracy and Precision
Each measurement of the experiment is slightly different from the other and the errors and uncertainties found in it depend on the efficiency of the measuring device and the person who makes the
measurement. Accuracy denotes the value that comes closest to the actual (true) value, that is it defines the difference between average value of experiment and the actual value. While precision
refers to the closeness of the values obtained through the measurement.
Difference Between Accuracy and Precision
There are many differences between accuracy and precision some of which are given here.
1. The degree of agreement between the actual measurement and absolute measurement is termed as accuracy. But the degree (level) of variation found in the values of multiple measurements of the
same factor is termed as precision.
2. The accuracy indicates how close the measurement is to the actual measurement. But the precision determined how close a single measurement is to others.
3. Accuracy is based on only one factor, but precision is based on more than one factor.
4. Precision focuses on systematic errors, errors caused by the problem in the appliance. In contrast, precision refers to random errors that occur periodically with no discernible pattern.
The ability of a device to measure an exact or accurate value is called accuracy.
The proximity or closeness of two or more measurements to each other is referred to as the precision of a material or substance.
The uncertainty of a measured value is an interval around that value, so each repetition of the measurement produces a new result which falls within that interval.
The accuracy indicates how close the measurement is to the actual measurement. But the precision determined how close a single measurement is to others.
Accuracy is based only one factor, but the precision is based on more than one factors | {"url":"https://unacademy.com/content/cbse-class-11/study-material/physics/accuracy-precision-and-uncertainty/","timestamp":"2024-11-13T00:59:29Z","content_type":"text/html","content_length":"645105","record_id":"<urn:uuid:9d50d8ac-98e3-4cf9-a69b-c32ce652f9a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00883.warc.gz"} |
Effective Methods For Typesetting Matlab Code In Latex Documents - Latexum
Effective Methods For Typesetting Matlab Code In Latex Documents
Importing and Formatting MATLAB Code
When including MATLAB code in a LaTeX document, it is important to properly format the code to make it more readable. The matlab-prettifier is a useful LaTeX package for importing and syntax
highlighting MATLAB code.
To import an entire MATLAB script file into LaTeX, the \lstinputlisting command can be used. The matlab-prettifier package will automatically detect MATLAB syntax and highlight keywords, comments,
strings, etc. in different colors. Additionally, line numbers can be enabled using the numbers=left option.
The font size, line spacing, margins, and other typographical elements can be customized by passing additional options to \lstlisting. Generally, a smaller monospaced font with 1.0 or 1.1 line
spacing provides the best results for readability.
Inserting Figures and Plots
MATLAB's plotting functions can generate high-quality graphics that may need to be included alongside code listings or referenced in text. The preferred approach is to export plots as external image
To generate plots, the print or export_graphics functions can be used to save plots in PDF, PNG, JPEG, etc. formats. The graphic image files can then be included in the LaTeX document using the
standard \includegraphics macro.
\caption{MATLAB plot caption}
Make sure to set appropriate image widths and positioning captions or labels for plot images inserted. Using the figure environment with [ht] placement specifiers allows LaTeX more flexibility in
placing larger images.
Displaying Code Output and Results
When showing output generated from MATLAB code, the \verbatim environment can be highly effective for preserving all whitespace, text formatting, line breaks, etc.
By default, \verbatim uses a monospaced font and disables LaTeX special characters. This allows output text to be copied directly from the MATLAB command prompt without worrying about alignment or
font issues.
>> format long
>> myvar = 3.14159265358979
Margin sizes may need to be adjusted using \begin{verbatim*} to handle extra width from long text lines in output. Referring to output values in the main text is also common and can be formatted
nicely by escaping out of verbatim mode.
Troubleshooting Common Issues
Getting MATLAB and LaTeX integration running smoothly can involve some debugging of packages, file paths, compilers, etc. Here are solutions to some common pitfalls.
Missing packages - If unusual LaTeX errors occur when importing MATLAB code, ensure matlab-prettifier and its token library dependencies are installed on your system and referenced correctly.
Path errors - Including graphic image files relies heavily on relative file paths. Double check that image file locations are specified correctly relative to the main LaTeX .tex document.
Formatting inconsistencies - With multiple syntax elements importing code, output, and images, font sizes, margins, and text flow may not always align. Careful troubleshooting of spacing options is
Example MATLAB Code Listings
Below are examples demonstrating effective strategies for incorporating MATLAB code, plots, and output in LaTeX documents:
Importing .m File and Highlighting Syntax
• Imports lines 1-10 of surfaceplot.m
• Applies syntax highlighting for MATLAB code
• Shows line numbers on left in tiny font
• Uses smaller monospaced font for code
Inserting Plot Image File with Caption
\caption{Surface plot for 3D quadratic function}
• Places surface plot image file with 80% width
• Allows LaTeX to position image in optimal location
• Adds descriptive caption below figure
Showing Output in Verbatim Environment
>> format long
>> myvar = 3.14159265358979
>> disp(myvar)
The value of \texttt{myvar} shown in the output is the first 8 digits of $\pi$.
• Preserves whitespace and text formatting
• Handles extra width from long output line
• Escapes verbatim to reference \texttt{myvar} in text | {"url":"https://latexum.com/effective-methods-for-typesetting-matlab-code-in-latex-documents/","timestamp":"2024-11-07T07:26:56Z","content_type":"text/html","content_length":"106085","record_id":"<urn:uuid:d1333e11-a91f-4134-9b88-bab375f16b2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00584.warc.gz"} |
Printable Calendars AT A GLANCE
Graphing Using Slope Intercept Form Worksheet
Graphing Using Slope Intercept Form Worksheet - Web this lesson covers in detail three different methods of graphing the equation y=0 on a coordinate plane. It may be printed, downloaded or saved and
used in. Graphing lines using standard form; Free trial available at kutasoftware.com. Web finding slope from a graph; 2.when an equation in slope intercept form is. These linear equations worksheets
are a good resource for students in the 5th grade through the 8th grade. Follow these steps to play the activity and complete your worksheet. Finding slope from an equation; This helpful worksheet
starts with an introduction that walks students through how to graph an.
Web worksheets for slope and graphing linear equations. This helpful worksheet starts with an introduction that walks students through how to graph an. Sketch the line given the equation. 1 date_____
period____ ©j y2_0l1u6r. Graphing lines with integer slopes. These linear equations worksheets are a good resource for students in the 5th grade through the 8th grade. Web this lesson covers in
detail three different methods of graphing the equation y=0 on a coordinate plane.
Follow these steps to play the activity and complete your worksheet. 2.when an equation in slope intercept form is. 3.when you have a slope of −3 what is the rise and run that you would use to graph
the equation? It is the measure of the steepness of the line. Graphing lines with integer slopes.
Graphing Slope Intercept Form Worksheets Pinterest
Web graphing a line using the equation in slope intercept form. Finding slope from an equation; This graphing slope intercept form practice can be printed in color or black and white, whichever suits
your preferences and needs. You may select the type of solutions that the students must perform. Finding slope from two points;
Graphing Lines Using Slope Intercept Form worksheets
Highlight an activity using $ or #, and press b. Sketch the line given the equation. These linear equations worksheets are a good resource for students in the 5th grade through the 8th grade. 1
date_____ period____ ©j y2_0l1u6r. Web finding slope from a graph;
Graphing Equations In Slope Intercept Form Worksheet 133 13 —
How can you describe the graph of the equation y mx b? You can select from two activities — match it! 3.when you have a slope of −3 what is the rise and run that you would use to graph the equation?
1 date_____ period____ ©l t2m0j1f6y hksujthal ysuocfntfwwalr_ea olnldcu.f w nayldlb zrtibgvhwthsb hrhevscesrnv^exdn. You may select different configuration for.
Graphing using slope intercept form YouTube
Sketch the line given the equation. 1 date_____ period____ ©j y2_0l1u6r. State two ways that you could use a slope of 5 3 − to graph a line: 1 date_____ period____ ©l t2m0j1f6y hksujthal
ysuocfntfwwalr_ea olnldcu.f w nayldlb zrtibgvhwthsb hrhevscesrnv^exdn. Web finding slope from a graph;
16 Best Images of SlopeIntercept Graph Worksheet YIntercept and
2.when an equation in slope intercept form is. Web y = mx + b. 1 date_____ period____ ©j y2_0l1u6r. Let's graph y = 2 x + 3. Finding slope from two points;
Slope Intercept Form Worksheets with Answer Key
2.when an equation in slope intercept form is. You can select from two activities — match it! Draw a line joining those two points. 1) y = −8x − 4 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 −6 −5 −4 −3 −2 1 2 3
4 5 6 2) y = −x + 5.
Slope Intercept Form Graphing Worksheets
Graphing lines using standard form; You can select from two activities — match it! 1 date_____ period____ ©l t2m0j1f6y hksujthal ysuocfntfwwalr_ea olnldcu.f w nayldlb zrtibgvhwthsb hrhevscesrnv^exdn.
Web graphing a line using the equation in slope intercept form. Finding slope from two points;
Graphing Lines Using Slope Intercept Form worksheets
Traditional algebra 1 slope intercept form y = mx + b flippedmath. Representing horizontal and vertical lines on coordinate plane; How can you describe the graph of the equation y mx b? 1 date_____
period____ ©j y2_0l1u6r. Highlight an activity using $ or #, and press b.
Slope Intercept Form Worksheet
You may select the type of solutions that the students must perform. How can you describe the graph of the equation y mx b? You may select different configuration for the problems to test different
concepts. 3.when you have a slope of −3 what is the rise and run that you would use to graph the equation? It may be.
Graphing Using Slope Intercept Form Worksheet - Web this worksheet includes twenty equations that they will need to solve. How can you describe the graph of the equation y mx b? Web finding slope
from a graph; Web y = mx + b. 2.when an equation in slope intercept form is. Representing horizontal and vertical lines on coordinate plane; This graphing slope intercept form practice can be printed
in color or black and white, whichever suits your preferences and needs. 1 date_____ period____ ©j y2_0l1u6r. 3.when you have a slope of −3 what is the rise and run that you would use to graph the
equation? It is the measure of the steepness of the line.
Sketch the line given the equation. Finding slope from two points; 1 date_____ period____ ©j y2_0l1u6r. Sketch the line given the equation. Graphing lines using standard form;
State two ways that you could use a slope of 5 3 − to graph a line: These linear equations worksheets are a good resource for students in the 5th grade through the 8th grade. Let's graph y = 2 x + 3.
Finding slope from two points;
Sketch The Line Given The Equation.
Let's graph y = 2 x + 3. Web finding slope from a graph; Web graphing a line using the equation in slope intercept form. 2.when an equation in slope intercept form is.
It Is The Measure Of The Steepness Of The Line.
You may select the type of solutions that the students must perform. Representing horizontal and vertical lines on coordinate plane; 1) y = −8x − 4 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 −6 −5 −4 −3 −2 1 2
3 4 5 6 2) y = −x + 5 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 −6 −5 −4 −3 −2 1 2 3. Finding slope from two points;
Web This Lesson Covers In Detail Three Different Methods Of Graphing The Equation Y=0 On A Coordinate Plane.
Graphing lines using standard form; This helpful worksheet starts with an introduction that walks students through how to graph an. Web this worksheet includes twenty equations that they will need to
solve. You may select different configuration for the problems to test different concepts.
It May Be Printed, Downloaded Or Saved And Used In.
Graphing lines with integer slopes. Follow these steps to play the activity and complete your worksheet. Highlight an activity using $ or #, and press b. How can you describe the graph of the
equation y mx b?
Related Post: | {"url":"https://ataglance.randstad.com/viewer/graphing-using-slope-intercept-form-worksheet.html","timestamp":"2024-11-04T02:50:25Z","content_type":"text/html","content_length":"36474","record_id":"<urn:uuid:0482a857-7bfc-4680-952f-43a92e6e4ea0>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00487.warc.gz"} |
THE STORY OF THE DISCOVERY OF THE TRUE NATURE OF REALITY.
A MAJOR PARDDIGM SHIFT
© 2018, Edward R. Close
The history of this story goes back at least 5000 years, with concepts originating in the East, Middle East, Arabia, and Northern Africa, inspiring Diophantus of Alexandria around 250 BC, Pierre de
Fermat around 1640 AD, and Max Planck and Albert Einstein from 1900 to 1935, to look at the nature of reality in terms of multi-dimensional models. Modern mainstream science has had no major paradigm
shift with regard to expanding the dimensional domain of science since quantum physics and general relativity were proved valid around 1935. Advances have happened, but they were within the
scientific paradigm known as the Standard Model (SM). The new shift is from the SM based on materialism to a paradigm in which consciousness is primary, revealing the true nature of reality. It is
difficult to do justice to the importance of the discovery of the true nature of reality in a short essay, but I must try. I have neither the time nor the patience to wait for the media and
mainstream science to catch up.
I introduced the concept of the non-quantum receptor in a poster presentation at Tucson II (the second Toward a Science of Consciousness Convention) at the University of Arizona in 1996, and
published it with an introduction to the calculus of distinctions in my third book, Transcendental Physics in 1997, re-released in 2000. Discussing these ideas in the Science Within Consciousness
Journal and the Karl Jaspers Forum online, in 2000 to 2005, someone asked: “Putting forth such revolutionary ideas on line, aren’t you afraid someone will steal your ideas?” Someone else responded:
“If you are introducing truly new science, no one in mainstream science will understand it even if you push it down their throats!”
The person making the second comment was right. The history of western science shows us that a truth outside the boundaries of the established paradigm, however valid, is initially almost universally
ignored, and condemned as un-scientific nonsense. That was the case for paradigm breakers like Copernicus, Planck and Einstein, and it is now the case for ideas introduced by Close and Neppe. As Max
Planck said:
“A scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it
The purpose of this post is not to convince the believers in the current materialistic paradigm, it is simply to present a scientific truth as simply as possible and hope that the new generation of
scientists will be open to going beyond the box of the current mainstream paradigm.
AN UNAMBIGUOUS DESCRIPTION OF OUR UNIVERSE AND THE COSMOS
(A Comprehensive Description of Everything)
Behind it all is surely an idea so simple, so beautiful, that when we grasp it
- in a decade, a century, or a millennium - we will all say to each other,
how could it have been otherwise? How could we have been so stupid?
– John A. Wheeler, theoretical physicist
We know now that the physical universe available to our five senses and their physical extensions is only a very small part of reality. If we call all of reality the cosmos, then the physical
universe we perceive through our senses is to the cosmos as a single sentence is to a 1000-page book, or as one novel is to all the books in all the libraries in the world. It is virtually a single
unfolding thought in the infinite mind of God.
The Triadic Dimensional Vortical Paradigm (TDVP) is a major shift in the basis of scientific thought from a narrow materialistic view of reality limited to consideration of the physical universe, to
a comprehensive description of everything. But, it is not a theory of everything (TOE). To clarify this, and present an unambiguous description of everything (DOE), it is necessary to define some
basic terms.
There are some basic terms that most people interested in science, and even many professional scientists, often use ambiguously, so they need to be defined accurately and carefully for the purposes
of this discussion. The definitions given below will clarify what the words mean in the new description of reality presented here. It makes no difference whether or not these definitions agree with
dictionary definitions or with your personal understanding of what the terms mean. The definitions given below specify their meanings in the discussion that follows. Please refer to them any time the
discussion seems unclear.
SCIENCE: The formal, organized effort to understand the nature of the reality we experience.
THE SCIENTIFIC METHOD: The practice of the process of proposing reasonable hypotheses, also called theories, and testing them against the experience of direct knowledge through observation and
measurement. If a hypothesis is validated by experience, it is accepted, if not, then it should be discarded.
SCIENTIST: A person who engages in the practice of science.
TDVP: The Triadic Dimensional Vortical Paradigm.
TRIADIC: Consisting of three components, or a multiple of three components.
VORTICAL: An adjective describing a vortex spinning in three or more dimensions.
DIMENSIONAL: An adjective describing domains of extension.
PARADIGM: A comprehensive understanding of reality based on empirical data and mathematical proof.
DIMENSION: A measure of extent, like length, width or height. This term is often used incorrectly to mean a dimensional domain.
DIMENSIONAL DOMAIN: A specific well-defined extent of space-time and consciousness. For example, a line, extending infinitely is a one-dimensional domain; a plane, extending indefinitely in two
mutually orthogonal directions is a two-dimensional domain; a volume extending indefinitely in three mutually orthogonal directions is a three-dimensional domain; and, in general, a volume extending
indefinitely in n mutually orthogonal directions is an n-dimensional domain.
EXTENT: The measure of the variable distance, area or volume, of a dimensional domain.
CONTENT: That which occupies a dimensional Domain of three or more dimensions. Note that dimensional domains of less than three dimensions have no capacity for content.
ORTHOGONAL DIMENSIONS: Dimensions separated by an angle of rotation of 90 degrees, given that one complete rotation is 360 degrees.
VORTICAL SPIN: The rotation of a vortex.
INTRINSIC SPIN: The increase in vortical spin caused by simultaneous rotations around more than one axis, i.e., in more than one plane.
FIELD: A field is a dimensional domain of finite extent in three or more dimensions, with a well-defined distribution of mass-energy-consciousness content, for example, the gravitational field of a
planet, the energy field of a magnet, or the extent of an individual’s consciousness.
MASS: The resistance to motion due to vortical spin.
ENERGY: Any force capable of creating, sustaining and altering a vortex, or distorting the distribution of a field.
UNIVERSE: A finite domain of three or more dimensions along with all its contents.
COSMOS: The infinite sum of all possible universes, past, present and future.
PRIMARY CONSCIOUSNESS: The Infinite Reality within which all things are embedded. The Source of all of the logical patterns of reality.
INDIVIDUALIZED CONSCIOUSNESS: Finite manifestations of specific limited fields and image content originating in Primary Consciousness.
THEORY: A hypothesis to be proved or disproved.
THEOREM: A hypothesis that has been expressed mathematically and confirmed by mathematical logic based on known axioms expressing direct experiences of reality.
CALCULATION: The process of transforming the form of a given representation of a known feature of reality to a different, equivalent form.
CALCULUS: A system of logical operations that transform the form of a given description of a known feature of reality to a different equivalent form. For example, the fundamental operations of
arithmetic transform expressions of numerical values as in: 1+1=2, and (3 x 4)+1=13. Other examples include algebraic transformations such as: (x + y)(x – y) = x^2 – y^2, and differential and
integral calculations like: d/dx(x^n) = nx^n-1, ∫nx^n-1 = x^n + C, etc.
DISTINCTION: Any form that can be distinguished from the rest of reality in some way.
THE CALCULUS OF DISTINCTIONS: The logical system of calculations that changes the form of a distinction or combinations of distinctions into different, but equivalent forms.
THE CALCULUS OF DIMENSIONAL DISTINCTIONS (CoDD): The logical system of calculations that transform the form of a dimensional distinction occupying a volumetric domain or combinations of volumetric
distinctions into different, but equivalent forms.
DIMENSIONAL ANALYSIS: In a series of calculations involving mathematical expressions describing a stable physical relationship (most often expressions of one or more of the known laws of physics) in
terms of distinctions measurable in units of mass/energy, space and time, the number and symmetry of the basic units of the final expression must match exactly the number and symmetry of the units of
the initial expression prior to the series of mathematical transformations. Otherwise, there is an error either in physical conceptualization or mathematical logic. This a very useful analysis
usually taught to first-year university physics students.
VOLUME: The extent of a dimensional domain of three or more orthogonal dimensions.
THE TRIADIC ROTATIONAL UNIT OF EQUIVALENCE (TRUE): The basic quantum equivalence unit of the CoDD, derived from the mas/energy equivalence of the electron.
QUANTUM: The smallest possible measurable unit of reality.
THE QUANTUM EQUIVALENCE PRINCIPLE: All observable and measurable objects in the universe consist of integral multiples of the quantum equivalence unit (TRUE).
DIMENSIONAL EXTRAPOLATION: The projection from an n-dimensional domain into an (n+1)-dimensional domain. The process of dimensional extrapolation from an n-dimensional domain where the numerical
types of the dimensions are known, results in the definition of the mathematical nature of the (n+1)^th dimension. For example, extrapolation from a 3-dimensional domain into a 4-dimensional domain
reveals that the 4^th dimension is measurable in the primary type of complex numbers, i.e., integer multiples of the so-called imaginary unit, the square-root of negative one (√-1).
THE DIMENSIONAL INVARIANCE PRINCIPLE: An n-dimensional domain can only be observed and/or measured from an (n+1)-dimensional domain.
DIOPHANTINE EQUATIONS: Diophantine equations are polynomial equations, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is a solution
such that all the unknowns take integer values). They are named after Diophantus of Alexandria (210 -294 AD).
Don’t worry if you don’t understand all of the details of the definitions of the terms listed above. They will become clear and meaningful as they are used in this discussion.
DISCUSSION: THE TRUE NATURE OF REALITY
I will begin by explaining how the practice of science based on belief rather than knowledge leads to erroneous conclusions about the nature of reality. Then I will explain how replacing belief with
an analysis of experience replaces belief with knowledge and leads to a new paradigm. This will be followed by a broad-brush description of the nature of reality revealed by the new paradigm, and a
description of how the new paradigm was discovered. Finally, I will list some of the major problems with the current mainstream belief-based paradigm that are explained by the new experience-based
paradigm. I will provide references to publications containing the detailed derivations from empirical data and mathematical proofs of the basic parameters establishing the new paradigm that
describes the true nature of reality and explain some of the conundrums and paradoxes of the current belief-based paradigm of mainstream science.
The Belief-based Standard Model of Reality
The current mainstream model promoted by most modern scientists is based on the metaphysical belief system of materialism, also called physicalism. In this belief system, the entirety of reality is
believed to consist of matter and energy in the form of combinations of elementary particles and weak and strong forces evolving and interacting in the arena of a universal relativistic space-time
domain. Consciousness is believed to be an epiphenomenon of physical evolution, i.e., something secondary to physical reality, arising when a sufficiently sophisticated level of physical complexity
is attained. Mainstream scientists have not explained how this complexity could evolve from particles flying apart in a universe expanding from a big-bang explosion. They have not discovered what
consciousness is, or how it arises from matter, but they express confidence that it will all be explained when a real “theory of everything” based on physical principles is finally discovered. But a
physical theory of everything is an unachievable goal, because not everything we experience is physical.
The job of science is to explain everything we experience. Materialism is an attractive hypothesis because of its simplicity, but should not be used as the basis of scientific investigation because
it fails the test of falsifiability. The hypothesis that a physical universe can exist without consciousness cannot be tested. To discover what is wrong with the mainstream theory, and understand why
it leads to puzzles and paradoxes at both the quantum and cosmic levels of measurement, we only need to return to what we actually experience. Recall that the first definition listed above identifies
science as the effort to understand the nature of the reality we experience, not what we believe or imagine might exist. The physical theory of everything envisioned by mainstream science is not
really a theory of everything, because it does not include everything we experience in its axiomatic basis. Limited to physical reality, science cannot explain more than about 5% of everything we
experience, and produces no answers for our most important questions concerning the ultimate nature of reality, the source of consciousness, and the meaning or purpose of existence.
All of the observations and measurements of scientific experimentation are possible only because of the conscious drawing of distinctions, not because of the pre-observation existence of an
independent physical universe assumed by physicalists. The first distinction drawn is the distinction of self from other, the direct conscious experience of the separation of ‘in-here” from ‘
out-there’. The first mistake of materialism occurs when reality is assumed to be binary, leading them to focus on the distinction between an object of observation and its surroundings, ignoring the
third component, which is the conscious entity drawing the distinction. By relegating consciousness to a dimensionless point outside the domain of scientific observation and measurement, physicalism
misses the key to understanding the nature of reality.
In fact, mainstream science is not science as defined. It has, however, played a very important role in the slow development of human civilization. By limiting the goal of research and
experimentation to understanding the mechanics of physical reality, mainstream science has been very successful solving practical problems related to physical survival and the manipulation of the
physical environment. But, that is not science as defined above; it is pragmatic technological engineering. By focusing on the mechanics of physical reality, mainstream science has ignored the
ontological connection of consciousness with reality, and has therefore no effective way to study the nature of the relationship of consciousness to physical reality.
On the other hand, the limited practical approach of current mainstream science has served us well in one respect. Because of the successes of engineering technology, we no longer have to fight wild
animals and the environmental elements to survive. We have created a safety buffer called modern civilization, creating a comfortable physical existence and providing the leisure time needed for a
deeper look into the nature of reality. It is critical that we do this now, because if we squander the anxiety-free time provided by labor-saving devices produced by engineering technology in the
pursuit of short-term gratification, the lack of a deeper understanding of the nature of reality will result in the decay and self-destruction of civilization.
Consciousness is actually the only thing that we experience directly, so it must be included in any serious scientific endeavor to understand reality. Everything else is perceived indirectly through
the senses. To think of consciousness as less real than the objects it perceives indirectly is a fatal mistake, dooming mainstream science to the pseudoscience of physicalism and the dead end of
materialism. This short-sightedness has led to loss of meaning, decadence and the decay of modern civilization. With an understanding of what is at stake, it is of paramount importance that we
rectify the errors of materialistic science and physicalism as soon as possible. How do we begin to do that? By including consciousness in the equations describing reality. This is what is done in
Close & Neppe's TDSVP. Let me explain how.
The Road to a Reality Paradigm
If the bricks of the yellow brick road leading to the Land of Oz were the elementary particles of particle physics, the Emerald City would be the mainstream paradigm. When the light of scientific
inspection is expanded to the scope of the reality we experience, the hypothetical particles of mainstream science fade away and disappear like the bricks of Dorothy’s dream. When individualized
consciousness dons the ruby slippers of the calculus of dimensional distinctions and clicks its heels, it awakens from the dream of materialism and returns to the reality of the greater cosmos. The
discoveries of relativity and quantum physics reveal quantized building blocks at the bottom of physical reality, but, as Planck indicated when he said “there is no matter as such”, they are not
physical particles at all. Instead, TDVP shows us that they are energy vortices, spinning simultaneously in multiple dimensions. To understand how and why this is true, we must apply the calculus of
dimensional dimensions (CoDD) to analyze experience.
Going back to experience, then, we realized that conscious experience is triadic, not binary as assumed by the scientists who developed the standard model. We experience , the resistance of mass
(1.), the impact of energy (2.) and the mutable image of a finite volumetric expanse of space and time as our field of awareness (3.). To properly describe quantized reality, we must have a set of
unitary quantum distinctions to use, just as we use units of size, weight, and time, e.g., inches, pounds and minutes, to measure any normal size physical object. But, these conventional units are
far too large to use to measure quanta. Trying to do so is like trying to measure the diameters of dust particles in units of light years, the distance between galaxies!
Planck defined quantum units for quantum reality by naturalizing certain fundamental constants of physics: the speed of light, the Coulomb constant, Boltzmann’s constant, and the gravitational
constant. ”Naturalizing” them means setting the unit values of these constants equal to one. This is not some arcane definitional concept done for mysterious reasons. We unitize measures of physical
objects all the time: We measure distances in multiples of one inch, or one meter, mass in multiples of one pound or one gram, and time in multiples of one second. However, the unitary length of one
inch, the weight of one pound, and the duration of one second, are arbitrarily chosen for convenience of measurement and calculation. Setting fundamental constants of nature to unity at the quantum
level provides “natural” units of measurement, which physicists call Planck units in honor of Max Planck.
The table below shows the relationships between these fundamental universal constants and conventional international units of measurement.
│ Constant │Symbol│Metric units │ Types │Dimensional Analysis│
│ │ │ │ │ │
│ │ │ │ of Units │ │
│ Speed of light │ c │meters/second│ L/T │ U/U = 1 │
│ Gravity │ G │ m^3kg/s^2 │ L^3/M∙T^2 │ U^3/U∙U^2 = 1 │
│Coulomb (electric charge) │ K[e] │m^3kg/s^2q^2 │L^3M/T^2Q^2│ U^3U/U^2U^2 = 1 │
│ Boltzmann (temperature) │ K[B] │ m^2kg/s^2ϴ │L^2M/T^2 ϴ │ U^2U/U^2 U = 1 │
│ Planck’s Constant │ h │ m^2kg/s^2 │ L^2M/T^2 │ U^2U/U^2 = U │
Where L implies units of length, M→ mass, T→ time, Q → electrical charge, and ϴ → temperature in degrees, m = meters, kg = kilograms and s = seconds.
Notice that the basic unit types for measurement of the speed of light are length and time, and for the gravitational constant, they are length, mass and time. Considering the mathematical
equivalence of mass and energy (E – mc^2), these unit types, M, L and T, (mass, length and time) are all that are needed to describe physical reality. All other measurable variables can always be
expressed in mathematical combinations of these three basic units. For example, density is mass per unit volume (M/L^3), Force = mass times acceleration = Ma = ML/T^2, etc. The reader can verify this
for other physical parameters. The other two constants, K[e] and K[B], contain linear measures of electrical charge and temperature that may vary over the field of observation.
The CoDD requires naturalized units of measurement for use in calculation, just as the standard model does. So, one might ask, why not just use Planck units? To answer this question, we turn to TRUE,
the quantum equivalence unit of the CoDD, combined with dimensional analysis (see definitions above. In the CoDD, we have defined the TRUE, the quantum equivalence unit derived from the physical
characteristics of the electron, as the basic unit of the calculus. For this application of dimensional analysis, let U represent the TRUE, the quantum equivalence unit.
Looking at the fourth column of the table above, we see that the dimensional analysis of four of these universal constants shows that they are symmetric. That is, in quantum equivalence units, the
dimensional domains of 1, 2 or 3 dimensions cancel out in the dimensional analysis, making the constant dimensionless. This means that c, G, K[e], and K[B] are unitary regardless of the size of the
units used, whether inches, meters, grams, pounds, etc. when they are quantized and naturalized. Thus they are verified as being universal constants in the CoDD, just as they are in the standard
model. The fifth constant, h, Planck’s constant, however, proves to be asymmetric, because the dimensional domains do not cancel. Thus h is not a universal constant, because its value depends on the
units of measurement chosen.
This is not the only thing that makes the TRUE (quantum equivalence units) different than Planck units. The TRUE, the ultimate quantum units used in CoDD calculations, are natural quantum units based
on the mass and volume of the electron, the elementary object with the smallest mass in the stable components of the natural elements. It is thus the true building block of the physical universe, and
the Quantum Equivalence Principle (see definition above) implies that if the TRUE is the true quantum building block of the universe, then all real objects will contain integer (whole number)
multiples of the TRUE, and thus the equations describing real phenomena will be Diophantine equations with integer solutions.
The fact that the Planck constant is not an integer in any system of units, tells us that Planck units are not quantum units, while triadic rotational units of equivalence (TRUE) are. This conclusion
is verified by the many explanations of empirical observations and agreements with experimental data obtained by applying the CoDD with the TRUE as the basic distinction. These verifications with
real data and logical explanations of observed phenomena that are not explained in the standard model paradigm, establishes TDVP as a valid scientific paradigm, not just a theory.
With the mathematical proofs provided in our published papers, TDVP attains the status of a theorem. It is no longer just a theoretical hypothesis. The following examples of successful solutions of
some of the paradoxes and puzzles of the standard model, with references to the detailed presentations of derivations and proofs are offered as evidence of the validity of TDVP.
1.) Why are protons and neutrons combinations of three quarks and not some other electrically neutral combination? Applications of the CoDD with TRUE to the Diophantine combinatorial equations show
that other combinations are mathematically and dimensionally impossible because they would violate Fermat’s Last Theorem. The proof has beeen published in several of the references below.
2.) Why do fermions like protons have an intrinsic one-half integer spin? In standard model physics, intrinsic spin is considered part of quantum weirdness that cannot be explained in classical
terms. The half-integer spin of fermions and whole-integer spin of bosons are postulated as numerical features of the quantum states of elementary particles, that cannot be derived from first
principles and have nothing to do with physical rotation, even though they contribute to the total angular momentum of the particle. However, the application of 9-D spin dynamics in TDVP explains
intrinsic spins perfectly well as the direct result of simultaneous rotation in multiple dimensions. Dimensional mathematical proof has been published. See References.
3.) Why do protons and neutrons have so much more mass than the combined mass of the quarks of which they are composed? The standard model posits particles called gluons that hold the quarks together
and impart the extra mass to the combination even though in theory, they themselves have zero mass. TDVP derives the mass of the proton and neutron from CoDD applications with TRUE that agree exactly
with the results of exhaustive statistical analysis of experimental data from the LHC. See these derivations in published reference listed below.
4.) The standard model does not explain from theory or from first principles why the Cabibbo quark mixing angle is 13.04 degrees, while TDVP provides a straight-forward derivation from 9-D CoDD
dynamics calculating the angle at 13.0392 degrees. See References.
5.) The standard model does not explain why there is something rather than nothing. The standard model scientist assumes that consciousness is an epiphenomenon of matter that has no direct causative
relationship in the formation, evolution and ultimate nature of reality. Because of this assumption, consciousness has no place in the equations used to describe reality in the theories of mainstream
science. TDVP, on the other hand, by following the data of quantum physics experiments where they lead, found that there would be no physical universe if some form of primary consciousness did not
exist prior to the formation of protons, neutrons and the natural elements. With the discovery of gimmel, the third (non-mass, non-energy) form of the essential substance of reality, TDVP explains
why there is something rather than nothing. See References.
6.) Gimmel, the third form of reality discovered by applying the logic of the CoDD to the mathematical description of the combination of quarks that form stable protons in the 9-D dimensional domain
of the finite cosmos, is the link between physical reality and primary consciousness, the substrate of reality, in which the 9-D finite domains of the physical universe and the cosmos are embedded.
This is only the beginning of a long list of fifty-some phenomena, paradoxes and puzzles not explained by the standard model, that are explained by TDVP using CoDD Diophantine integer mathematics
with the TRUE quantum unit derived from data on the electron and quarks from DHC data. See References.
Max Planck discovered the quantization of energy, and Albert Einstein provided the equations expressing the equivalence of mass and energy. The Large Hadron Collider, the largest, most sophisticated
machine developed by mainstream science so far, has produced mega terabytes of physical data defining the mass and energy of the building blocks of physical reality, providing very accurate estimates
of the mass and energy equivalence of electrons and quarks for use as the basis for defining the true quantum equivalence units needed for the calculus of dimensional distinctions.
Reality is triadic, consisting of three sequentially embedded dimensional domains: space, time and consciousness, which are describable in variables of extent, and three forms of the essence of
reality: mass, energy and consciousness, describable in variables of content.
Since the new paradigm has been validated with empirical data from the Large Hadron Collider and mathematical proof, as prerviously statedit is no longer a theory. Thus, since it is not a theory,
TDVP is not a theory of everything, instead, it is a description of everything.
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2 comments:
1. The history of western science shows us that a truth outside the boundaries of the established paradigm, however valid, is initially almost universally ignored, and condemned as un-scientific
nonsense. 1z0-1027 dumps
2. Students should download the DHSE Kerala Plus Two Question Paper 2022 for practise in order to achieve good results in the test. Candidates can get the Kerala Plus two question paper 2022 from
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Aspirants can learn about the marking method and types of questions asked by looking at sample papers. The test for Kerala board Class 12th may be held between March 17 and March 30, 2022. To
learn more about the DHSE Kerala 2022 Plus Two Question Paper, read the entire article. | {"url":"https://www.erclosetphysics.com/2018/09/describing-true-nature-of-reality.html","timestamp":"2024-11-03T00:17:31Z","content_type":"text/html","content_length":"338796","record_id":"<urn:uuid:4ac3c4d3-0d8a-4e38-b7ae-d2320321ade9>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00368.warc.gz"} |
Developmental Math Emporium
Learning Outcomes
• Use the substitution method to solve systems of equations
• Express the solution of an inconsistent system of equations containing two variables
• Express the solution of a dependent system of equations containing two variables
Solve a system of equations using the substitution method
In the last couple sections, we verified that ordered pairs were solutions to systems, and we used graphs to classify how many solutions a system of two linear equations had. Solving a linear system
in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method. What if we are not given
a point of intersection, or it is not obvious from a graph? Can we still find a solution to the system? Of course you can, using algebra!
In this section we will learn the substitution method for finding a solution to a system of linear equations in two variables. We have used substitution in different ways throughout this course. For
example, when we were using the formulas for the area of a triangle and simple interest we substituted values that we knew into the formula to solve for values that we did not know. The idea is
similar when applied to solving systems, there are just a few different steps in the process. In the substitution method we solve one of the equations for one variable and then substitute the result
into the other equation to solve for the second variable. Recall that we can solve for only one variable at a time which is the reason the substitution method is both valuable and practical. Let’s
start with an example to see what this means.
Find the value of [latex]x[/latex] for this system.
Equation A: [latex]4x+3y=−14[/latex]
Equation B: [latex]y=2[/latex]
Show Solution
You can substitute a value for a variable even if it is an expression. Here’s an example.
Solve for [latex]x[/latex] and [latex]y[/latex].
Equation A: [latex]y+x=3[/latex]
Equation B: [latex]x=y+5[/latex]
Show Solution
Remember, a solution to a system of equations must be a solution to each of the equations within the system. The ordered pair [latex](4,−1)[/latex] does work for both equations, so you know that it
is a solution to the system as well.
In the examples above, one of the equations was already given to us in terms of the variable [latex]x[/latex] or [latex]y[/latex]. This allowed us to quickly substitute that value into the other
equation and solve for one of the unknowns.
Sometimes you may have to rewrite one of the equations in terms of one of the variables first before you can substitute. In the example below, you will first need to isolate one of the variables
before you can substitute it into the other equation.
Solve the following system of equations by substitution.
[latex]\begin{array}{l}-x+y=-5\hfill \\ \text{ }2x - 5y=1\hfill \end{array}[/latex]
Show Solution
Here is a summary of the steps we use to solve systems of equations using the substitution method.
How To: Given a system of two equations in two variables, solve using the substitution method
1. Solve one of the two equations for one of the variables in terms of the other.
2. Substitute the expression for this variable into the second equation, and then solve for the remaining variable.
3. Substitute that solution into either of the original equations to find the value of the other variable. If possible, write the solution as an ordered pair.
4. Check the solution in both equations.
Let’s look at some examples whose substitution involves the distributive property.
Solve for [latex]x[/latex] and [latex]y[/latex].
[latex]\begin{array}{l}y = 3x + 6\\−2x + 4y = 4\end{array}[/latex]
Show Solution
Solve for [latex]x[/latex] and [latex]y[/latex].
Show Solution
In the following video, you will be given an example of solving a systems of two equations using the substitution method.
If you had chosen the other equation to start with in the previous examples, you would still be able to find the same solution. It is really a matter of preference because sometimes solving for a
variable will result in having to work with fractions. As you become more experienced with algebra, you will be able to anticipate what choices will lead to more desirable outcomes.
Try It
Identify systems of equations that have no solution or an infinite number of solutions
Recall that an inconsistent system consists of parallel lines that have the same slope but different y-intercepts. They will never intersect. When searching for a solution to an inconsistent system,
we will come up with a false statement such as [latex]12=0[/latex].
When we learned methods for solving linear equations in one variable, we found that some equations didn’t have any solutions, and others had an infinite number of solutions. We saw this behavior
again when we started describing solutions to systems of equations in two variables.
Recall this example from Module 1 for solving linear equations in one variable:
Solve for [latex]x[/latex]. [latex]12+2x–8=7x+5–5x[/latex]
[latex] \displaystyle \begin{array}{l}12+2x-8=7x+5-5x\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,2x+4=2x+5\end{array}[/latex]
[latex]\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,2x+4=2x+5\\\,\,\,\,\,\,\,\,\underline{-2x\,\,\,\,\,\,\,\,\,\,-2x\,\,\,\,\,\,\,\,}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4= \,5\end{array}[/
This false statement implies there are no solutions to this equation. In the same way, you may see an outcome like this when you use the substitution method to find a solution to a system of linear
equations in two variables. In the next example, you will see an example of a system of two equations that does not have a solution.
Solve for [latex]x[/latex] and [latex]y[/latex].
Show Solution
You get the false statement [latex]−8=4[/latex]. What does this mean? The graph of this system sheds some light on what is happening.
The lines are parallel, they never intersect and there is no solution to this system of linear equations. Note that the result [latex]−8=4[/latex] is not a solution. It is simply a false statement
and it indicates that there is no solution.
Let’s look at another example in which there is no solution.
Solve the following system of equations.
[latex]\begin{array}{l}x=9 - 2y\hfill \\ x+2y=13\hfill \end{array}[/latex]
Show Solution
In the next video, we show another example of using substitution to solve a system that has no solution.
Try It
We have also seen linear equations in one variable and systems of equations in two variables that have an infinite number of solutions. In the next example, you will see what happens when you apply
the substitution method to a system with an infinite number of solutions.
Solve for [latex]x[/latex] and [latex]y[/latex].
Show Solution
This time you get a true statement: [latex]−4.5x=−4.5x[/latex]. But what does this type of answer mean? Again, graphing can help you make sense of this system.
This system consists of two equations that both represent the same line; the two lines are collinear. Every point along the line will be a solution to the system, and that’s why the substitution
method yields a true statement. In this case, there are an infinite number of solutions.
Try It
In the following video you will see an example of solving a system that has an infinite number of solutions.
In the following video you will see an example of solving a system of equations that has no solutions.
The substitution method is one way of solving systems of equations. To use the substitution method, use one equation to find an expression for one of the variables in terms of the other variable.
Then substitute that expression in place of that variable in the second equation. You can then solve this equation as it will now have only one variable. Solving using the substitution method will
yield one of three results: a single value for each variable within the system (indicating one solution), an untrue statement (indicating no solutions), or a true statement (indicating an infinite
number of solutions). | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/3-2-1-the-substitution-method/","timestamp":"2024-11-09T10:52:15Z","content_type":"text/html","content_length":"71258","record_id":"<urn:uuid:3d08ce62-7935-4b73-bc00-75e2ecf3845a>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00528.warc.gz"} |
A Simple Model of International Environmental ... - P.PDFKUL.COM
A Simple Model of International Environmental Agreements with Bayesian Learning Hei Sing (Ron) Chan∗ University of Maryland First Draft: April 13, 2011 This Draft: May 10, 2011
Abstract In this paper I study the economics of self-enforcing international environmental agreements where agents never know what exactly the state of the world is. Explicitly, I consider countries
using Bayesian learning to update their beliefs on the state of the world. Using a very simple framework of allowing pollution as a common bad, I study how Bayesian learning conveys message to
countries and whether a full disclosure of information can necessarily improve the aggregate welfare. Interestingly, I find that the value of information is always negative which suggests that
strategic interactions between countries significantly make the countries worse off. I also consider a dynamic setting where countries emissions can affect the learning process and surprisingly I
find that the equilibrium breaks down to a coalition that cannot have more than two countries.
How to cooperate? While research on climate change and other international environmental problems often point to a call for cooperation among nations to cut carbon emissions, we do not observe a lot
of international environmental agreements (IEA) in the world1 . Since the seminal work by Barrett (1994), scholars have attempted to explain low participation rates of international environmental
agreements. ∗ I would like to thank Rob Williams for his encouragement and valuable suggestions for the paper. I would also like to thanks Ian Page and other participants in the class for their
feedback. The remaining errors are all of my own. Contact:
[email protected]
. 1 See the discussion in Stern (2007) and Barrett (2003).
Barrett (1994) models the game as a two stage game where countries choose to become signatories or not in the first stage, then they will choose emissions in the second stage. Signatories (who agreed
to join the agreement in the first stage) will choose emissions to maximize the joint surplus. Since the agreements are assumed to be self-enforcing, we have to impose internal stability and external
stability conditions to get the equilibrium number of signatories (in a stable equilibrium). In most scenarios that Barrett (1994) showed, the maximum equilibrium number of signatories is three2 .
Potentially there are two problems of such modelling. The first problem is the lack of treatment of uncertainty. Authors often assume the cost-benefit ratio is deterministic, meaning that all
countries know exactly what the cost-benefit ratio is. There are still controversies over the quantification in atmospheric temperature changes (as well as its potential damage) due to accumulation
of greenhouse gases (Intergovernmental Panel on Climate Change, 2007a,b,c), therefore it is impossible for the government to know the exact benefits when they decide to cut carbon emissions. Na and
Shin (1998) is the first paper attempting to introduce individual uncertainty into the framework of self-enforcing IEA. Nevertheless, they only allow for three countries therefore it is difficult to
generalize their results in a broader framework. Kolstad (2007) allows for systematic uncertainty so all countries are subject to risk on the identical cost-benefit ratio of carbon emissions
abatement. The second problem is the static nature of the game. In the early models of IEA, scholars consider static models only because what matters for the interaction of agents is a flow
pollutant. More papers start looking at a stock pollutant instead and therefore we need a model that turns the IEA into a dynamic game. Ulph (2004) is one of the papers that features a stock
pollutant while also allowing countries to re-consider their position each period by consider a 2-period version of Barrett’s type of game. Rubio and Ulph (2007) consider a similar game with infinite
horizon, at the same time allowing for dynamic membership as well. In both papers the cost is a function of the stock of the pollutant instead of the emission in that period, meaning that if the
country chooses not to abate by a sufficient amount, the cost can go up the next period. It is natural to think about the problem of learning in the dynamic context of IEA, and the above papers also
feature certain kinds of learning by agents. Kolstad and Ulph (2008) look at different forms of learning on the timing that countries learn about the true cost-benefit ratio using a simple, flow
pollutant format. Regardless, there are still shortcomings in the papers above. In reality, countries never know the true state of the world, yet they have to guess what the cost-benefit ratio is
when they make abatement 2 Rubio
and Ulph (2006) critically review some of the claims made by Barrett (1994) and some of the results are overturned when some assumptions are relaxed.
decisions. In other words, countries probably have priors over the true state of the world (in our case, the cost-benefit ratio of abatement) and they will update (through Bayes’ Rule) their beliefs
by observing ”events” (catastrophes, hereafter) that suggest what the state of world is. Previous papers have a mixed result on whether learning can sustain more cooperation. While intuition suggests
that value of information should be positive as shown in many papers (Na and Shin, 1998; Ulph, 2004), certain kinds of learning that allow country to take strategic actions after information is
revealed can make the value of information negative (Ulph and Maddison, 1997; Kolstad and Ulph, 2008). Since countries never know what exactly the state of the world is, it seems that in some cases
strategic actions (that some authors are worrying that it can make a negative impact on the number of signatories) are muted. Imagine a case where countries strongly believe the pollution is very
costly, then it will create a huge incentive to free-ride and hence it results in a lower coalition (in the world where marginal benefits and costs of polluting is constant), In other words, it is
going to depend on the prior and the realization of signals. Scholars have previously considered Bayesian learning in the literature of the environmental economics. Kelly and Kolstad (1999) introduce
Bayesian learning into an optimal growth model and they study how the learning process affect the benefits of pollution control. Karp and Zhang (2006) show that information in the case of anticipated
learning has negative value because learning implies a more optimistic view of future damages. My paper will be the first paper to study Bayesian learning in the framework of self-enforcing IEA. The
result can be different from what these paper found because we are looking at the internal and external stability conditions in which countries have to balance their benefits and costs of joining and
not joining the agreement, and catastrophes are going to impact both benefits and costs. The above papers assumed there is a single regulator who is setting the control measures.3 In my model, all
countries know that the cost-benefit ratio (of pollution) is either γh or γl , but they do not know the probability what this is. Instead, they formed common priors over γ. Catastrophes happen
according to a probability distribution related to γ (which are costly) and countries update (and form posterior on) their believes on γ. I am going to consider first a simple static model that
country faces a binary choice of either to pollute or abate as in Kolstad (2007) to demonstrate how Bayesian learning is going to impact our equilibrium. It may look weird that I am talking about a
static model while also allowing for Bayesian learning, but the context of the problem is the same each period so it allows me to focus on a single period. 3 This
point is emphasized in Ulph (2004).
I start by characterizing the signatories equilibrium in a setting where both benefits and costs of polluting are linear in private and global emissions respectively. Since the game is finitely
repeated, the equilibrium number of signatories in each stage game is a function of the expected cost-benefit ratio, using the information set at each time period. Given that it depends on the
realization of the catastrophes variable, I run a Monte Carlo simulation to give an estimate of the welfare under Bayesian learning. I contrast the welfare measures with the ones where information
(in this case, the true cost-benefit ratio) is fully revealed, surprisingly I find that countries prefer Bayesian learning to full disclosure of information, due to the nature of the equilibrium that
we have fewer cooperation when the cost is high. Countries would like the expected state to converge to the true faster if the cost is low and it to converge slower if the cost is high. In
expectation, I find that the latter effect dominates the former. After setting the stage up, I am going to add more dynamics to the model by considering the probability that a catastrophe will happen
as a function of emissions one period ago. In other words, I allow country decisions to impact the learning process directly so now the pollution decision also includes the information tradeoff on
top of the cost-benefit tradeoff. I start by describing a two-period framework to focus on the pollution decision in period 1 that can potentially affect the learning process, while agents stick with
the same plan in period 2 as described in the static framework. I find that the equilibrium number of signatories can go up - which would imply that Bayesian learning did actually enforce more
cooperation in this case. Contrast to the static framework where countries cannot control for the learning process, now signatories know that if they decide to pollute, the learning process will be
faster and this harms their expected welfare as we saw in static model. There is an extra benefit of controlling emissions of signatories that drives more cooperation and hence we observe a bigger
coalition. Still, this model will not be a completely illustration of what the world looks like, but this is certainly a big step towards that. This paper only attempts to compare exogenously given
benefits and costs, instead of constructing how the impact of the greenhouse gas control diffuses to the economy (and hence results in benefits and costs). Understanding how this diffusion works is
very important in designing climate change and other kinds of environmental policies, and it will certainly be important to look at the distributional issues of the international environmental
agreements as well. The rest of the paper is organized as follows. In section 2, we are going to look at a simple “static” model of self-enforcing international environmental agreement. After
presenting the IEA equilibria under different learning path, I compare the corresponding welfare using Monte Carlo simulations. In
section 3, I present another framework that emissions can directly affect the Bayesian learning process. Analytically, I look at a simple two-period model so that the equilibrium properties in
section 2 can be carried over. After that I conclude in section 4.
“Static” Model
Consider a world where there are N identical countries. Both benefits and costs of pollution are assumed to be linear in emissions, however, benefit is linear in private emission while cost is linear
in global emission. Each country can choose either to pollute (qi = 1) or not (qi = 0). The ratio of the costs to benefits is γ which is not known to countries. Nonetheless, countries do know that γ
can take either γ H or γ L where 1 > γ H > γ L . γ not only governs the cost-benefit ratio, but also determines the frequency of catastrophes. Catastrophes zt can take a value of either Z or 0. Under
a high cost state (γ H ), catastrophes zt = Z happen more often4 with probability pH > pL . Bear in mind that I normalize the benefit of pollution to be 1, so Z and gamma can be thought as relative
cost of catastrophes and global pollution in respective order. In this static model, I am going to assume that both pH and pL are constant and this is a common knowledge to all countries. I assume
that all countries have a common prior α that the cost state is high (γ H ). The timing of the game is as follows. At the beginning of the period, catastrophes are revealed (so zt = Z or zt = 0) and
all countries update their beliefs over γ. After that, countries simultaneously decide whether to become a signatory or not (the ‘membership phase’). Signatories will then come together and decide
whether to pollute or not, and all signatories will have to follow the direction. On the other hand, non-signatories simultaneously and non-cooperatively decide whether to pollute or not (the
‘pollution phase’). The timing of the game allows us not to consider the economic cost of catastrophes because catastrophes happened with some exogenous probabilities at the time of coalition
negotiation, and their decision at time t is not going to affect realization of catastrophes at time t + 1. Effectively, country i’s payoff is, Vi = qi − γ(qi + Q−i )
where Q−i is the total emissions beside i. I am going to denote γt to be the expected value of γ using 4 This assumption is equivalent to the monotonic likelihood ratio condition requires for agents
to translate the information into learning about the state.
the information set at time t, i.e. γt = α ˆ t γ H + (1 − α ˆ t )γ L
where α ˆ t is the posterior probability that countries updated using the information at time t contingent on the value of zt .
Non-Cooperative and Cooperative Equilibria
Throughout the paper, I am going to make the following assumption: Assumption 1 1 > γ H > γ L >
1 N
This assumption guarantees an interior solution of the membership game. If the cost-benefit ratio is bigger than one, then each country would not find it optimal to pollute. On the other hand, if
costbenefit ratio is smaller than 1/N , even a coalition consisting of all countries in the world (that is, n = N , or the full coalition and co-operative equilibrium) will not find it optimal to
abate since,
˜ = 1) = 1 − N γt > 0 WtC (n = N, Q
Under assumption 1, the non-cooperative solution is straightforward. Since benefits outweigh the costs, all countries will choose to pollute. This is also the optimal strategy for non-signatories.
The aggregate payoff (in each period) is given by, WtN C = N (1 − γt N )
As derived earlier, the cooperative equilibrium is for all countries to abate. The aggregate welfare is 0.
IEA Equilibrium - Full Learning Case
After characterizing the cooperative and non-cooperative equilibria, I will now discuss the equilibrium with international environmental agreement (IEA). Before I move to the Bayesian learning case,
it is convenient for me to outline the full learning case as in Kolstad and Ulph (2008) so we can contrast the welfare of the two equilibria in the next subsection. In the full learning case,
countries learn about the true type of the cost-benefit ratio γ before both the membership phase and the pollution phase. of the first period. 6
Let V S (n) and V N (n) be the payoff to each signatory and non-signatory respectively, given that there are n signatories in the coalition. Here are some definitions that will be useful. Definition
I(x) gives the smallest integer that is strictly greater than x. Definition The IEA with n signatories is internally stable if V S (n) > V N (n − 1) Definition The IEA with n signatories is
externally stable if V N (n) > V S (n + 1) The internal and external stability conditions guarantee that we have a Nash equilibrium in the (sub)game. If the IEA is internally stable, it means any
signatory would not find it optimal to switch to become a non-signatory; if the IEA is externally stable, no non-signatory would find it optimal to join the coalition and become a signatory. Whenever
I am talking about an equilibrium number of signatories being n∗ in this paper, it means that n∗ satisfies both internal and external stability conditions. The following lemma would be useful in many
propositions that follow. Lemma 1 For a given coalition of n signatories and γ, it is NOT optimal to pollute if γ > 1/n. Proof Signatories know that non-signatories are always going to pollute. If
signatories decide to pollute, each of them has a payoff of V S (n) = 1 − γN since the overall emissions will be N . If signatories decide not to pollute, each of them has a payoff of V S (n) = −γ(N
−n) as only (N −n) non-signatories are going to pollute. Therefore, it is not optimal for signatories to pollute if −γ(N − n) > 1 − γN ⇒ γ > 1/n. The result below is a replication of the result in
Kolstad (2007). Proposition 1 Under Assumption 1 and full learning mechanism, the expected number of signatories in the equilibrium is αI( γ1H ) + (1 − α)I( γ1L ) and it is unique. Proof Starting at
the time when all countries now what the true state is. Since all countries the true γ at the membership phase, I will just use γ here to denote the true one. When n = I(1/γ), it means n > 1/γ. From
Lemma 1, it implies all signatories will agree not to pollute. More importantly, from the definition of the I(·) function, n − 1 < 1/γ. In other words, if a single signatory leaves the coalition, all
signatories will choose to pollute. The internal validity condition hence becomes V S (n) > V N (n − 1) ⇒ −γ(N − n) > 1 − γN ⇒ γ > 1/n which is satisfied. The external validity is satisfied under
Assumption 1: V N (n) > V S (n + 1) ⇒ 1 − γ(N − n) > −γ(N − (n + 1)) ⇒ 1 > γ. The uniqueness of the equilibrium 7
is coming from the fact than n is selected to represent the optimality of the signatories not to pollute at the margin. If there are more signatories than n, then at least one country would find it
optimal to ”free-ride” due to the face that γ < 1. To complete the proof, we just move the game back to the time when the uncertainty is not resolved - and the expected number of signatories in the
equilibrium is just the expectation of the signatories in two possible subgames.
This standard result implies that in a high cost-benefit ratio state, the number of signatories is lower, which implies the aggregate welfare (W = (N − n∗ )(1 − N γ) < 0) is lower. This is primarily
driven by the strategic action by non-signatories. This result will also be useful for the analysis in the next section, where we focus on Bayesian learning.
IEA Equilibrium - Bayesian Learning Case
If we look at the tradeoff facing both non-signatories and signatories carefully, we notice that countries never need to learn about the cost-benefit ratio - they are basing all their decisions on
the expected value of the cost. Since the decision in period t (which is to pollute or not) does not affect the learning process, damage or the decision in period t + 1, I can analyze this game
‘statically’ since all periods can be treated the same, except that the expected cost (γt ) would be different in each subgame. We can view the game as a finitely repeated game of each signatories
stage game, and we know that the Nash equilibrium of the repeated game is equivalent to the Nash equilibrium of individual subgames. Given that the uncertainty is “systematic” as it applies to all
countries equally, we can just repeat the exercise earlier of showing the internal and external validity by replacing the true γ (which we have in the previous case) with its expected value γt . By
Bayes’ Rule and given the prior α , if countries observe a catastrophe,
α ˆ (z1 = Z) =
pH α pL + (pH − pL )α
Similarly, if countries do not observe a catastrophe,
α ˆ (z1 = 0) =
(1 − pH )α 1 − (pL + (pH − pL )α)
The rest of the problem is almost identical to the one we saw in the full learning case. Therefore it is straight-forward to prove the following, 8
Proposition 2 Under Assumption 1 and bayesian learning mechanism, the number of signatories is I( γ1t ) in each period and it is unique. We should notice that in each period, the equilibrium numbers
of signatories are potentially different when we are using a Bayesian learning mechanism. It is intuitive to think that why the number of signatories is changing. As agents learn about the true
cost-benefit ratio, agents update their priors and form updated expectation on the cost-benefit ratio. We can interpret the Bayesian mechanism (Bayes) as a slow convergence to the true state, while
the full disclosure (Full) as an immediate convergence. Therefore, the above result motivates us to look at the value of information (in rough sense) in this context: if all countries are told the
true state, what would the improvement of the aggregate welfare be? Information in this setting can be positive or negative - information generally has positive value because it allows agents to act
early, while strategic actions could bring the value of information down to negative5 . High / Bayes
High / Full
Low / Bayes
Low / Full
Exp Bayes
Exp Full
-7293.1 -7515.2 -3880.6 -3832.7
-7793.6 -7793.6 -4135.4 -4135.4
-2590.4 -2684.0 -957.58 -956.62
-2485.2 -2485.2 -795.26 -795.26
-3530.9 -5099.6 -2419.1 -2394.7
-3546.9 -5139.4 -2465.3 -2465.3
0.2/0.5 0.2/0.5 0.1/0.3 0.1/0.3
0.05/0.2 0.05/0.2 0.05/0.2 0.05/0.1
0.2 0.5 0.5 0.5
Note: ‘Bayes’ estimates are computed using Monte-Carlo simulations of 1000 observations. ‘Full’ scenarios are computed by assuming countries are being told the true cost-benefit ratio before the
membership phase of the first period begins. Expected values (‘Exp’) are then calculated by weighting the corresponding high and low values using priors as weights. The payoffs do not include the
economic cost of catastrophes because it will enter equally to both ‘Bayes’ and ‘Full’ scenarios. The following parameters are used in all specifications: (1) Number of Countries = 30, (2) Discount
Factor = 0.95,(3) Time periods = 100.
Table 1: Aggregate Welfare under Different Scenarios and Specifications In light of that, I estimate the expected aggregate welfare using Monte-Carlo simulations to calculate the discounted sum of
aggregate welfare (because the timing of information determines crucially on the welfare). Table 1 above compares the aggregate welfare under four different scenarios using different parameters. The
very surprising result is that the expected value of aggregate welfare under full disclosure is always lower than the one under Bayesian learning. This acts as an indirect piece of evidence that the
information under IEA framework can be negative. If we investigate the numbers more closely, our main result is primarily driven by the difference between the aggregate welfare when cost-benefit
ratio is high. Our argument above goes through here - since the strategic actions between agents make the number of signatories essentially low (which harms the aggregate welfare), agents would not
want to learn the information when the state is high. On the other hand when the cost is 5 It is inconclusive in the literature of self-enforcing IEA on whether information has positive or negative
values. This is left for further theoretical research as other papers only attempt to quantify the value information in corresponding settings. See the discussion in Kolstad and Ulph (2008).
low, agents would like the equilibrium to converge faster hence we observe a higher aggregate welfare under full disclosure. We could also find interesting results as we look across different rows.
By comparing rows 1 and 2, when countries are more “optimistic” (by having a lower prior α) the difference between the values of information increases when cost is high while it shrinks when cost is
low. This is mostly driven by the change in convergence rate when α decreases: if the true state is low (high), it means the convergence is faster (slower) and the value of information will change
accordingly as in the discussion above. I get similar results when I change the γ and p pairs. The aggregate welfares increase as a result of a decrease in cost.
Dynamic Model
In this section, I attempt to introduce some dynamics to the simple model in order to study the repeated game of the static model above. There are two basic departures that we have to take into
account. First, catastrophes probability next period is a function of current global emissions. I assume the probability that a catastrophe will happen under a high cost state depends on the total
emissions in the period before while the probability is still a constant under a low cost state to keep things simple, i.e.
L p˜L t+1 = p L p˜H t+1 (Qt ) = p +
(5a) Qt H Qt p (p − pL ) ≡ pL + Λ N N
H where Λp = pH − pL > 0. Assume that the emissions before the first period is N therefore p˜H 1 (Q0 ) = p .
Under the convenient timing of the game, countries observe both the catastrophes and the number of signatories before they make the pollution decision. This implies, however, that the next period
posterior probability is going to depend on the current emissions since
α ˆ t+1 (zt+1 = Z) = α ˆ t+1 (zt+1 = 0) =
p˜L t+1
ˆt p˜H (1 − α ˆ t )pL t+1 α =1− L H L p p +Λ α ˆ t Qt /N + (˜ pt+1 − p˜t+1 )ˆ αt
(1 − p˜H αt (1 − α ˆ t )(1 − pL ) t+1 )ˆ = 1 − 1 − (pL + Λp α ˆ t Qt /N ) 1 − (˜ pL pH ˜L ˆt) t+1 )α t+1 + (˜ t+1 − p
By taking derivatives respect to Qt , we get the following corollary: Corollary 3 (a)
∂α ˆ t+1 (zt+1 =Z) ∂Qt
> 0; (b)
∂α ˆ t+1 (zt+1 =0) ∂Qt
(6a) (6b)
It will be useful in subsequent analysis of the equilibrium. The above corollary is due to the fact by polluting more, the difference between p˜H and p˜L increases. It means that now the volatility
of the process - if the catastrophe happens, countries therefore can strongly believe that the state of the world is high cost; same argument goes through if countries observe no catastrophes, in
that case. Consequently, higher pollution generally increases the convergence rate of the equilibria. Recall that I assume that countries will re-negotiate at each period on the size of the coalition
and we get an equilibrium number of signatories. As noted in last section, the size of the coalition can be different as well, as a result, this paper features a dynamic membership game. Then, a
question that one can ask is, what do countries expect to get in period 2 when they are at period 1? I am going to follow the simple approach used by Rubio and Ulph (2007) and assume a random
assignment rule so that countries will endogenize the effect of the emission on the expected payoff (in terms of both signatory status and catastrophes).6 If the equilibrium number of signatories in
period 2 is large, then the country in period 1 would also take into account that she has a higher chance of becoming a signatory in period 2. Second, we have to deal with the economic cost of
catastrophes zt now. It matters because when countries are deciding whether they should pollute, pollution now leads to a higher chance of catastrophes next period, and consequently the expected cost
goes up in the next period. To avoid confusion and build our results step-by-step, I will first assume this effect away in the first part of the results in this section. I will bring this extra cost
back and see how our results change. I start the analysis by looking at the decision of signatories, assuming that non-signatories always pollute. I will then come back and evaluate the claim that
non-signatories always pollute, and study when this assumption hold. To present my argument in the simplest way, I am going to present a simple 2-period model that global pollution in period 1
affects the posterior probabilities in period 2. I conclude this section by studying how results change when we depart from a two-period model.
Two-Period Model
The value to each signatory at period 1 can be written as follows:
V1S (ˆ α1 , n) = max {q1 − γ1 (ˆ α1 ) · (N − n(1 − q1 )) + δE1 V2 } q1 ∈{0,1}
6 Rubio
and Ulph (2007) also describe other ways that one can model this in a dynamic programming setting.
E1 V2 = [ˆ α1 p˜H ˆ 1 )˜ pL α2 (z2 = Z), n2 (z2 = Z)) − Z] 2 (Q1 ) + (1 − α 2 ] · [V2 (ˆ + [α ˆ 1 (1 − p˜H ˆ 1 )(1 − p˜L α2 (z2 = 0), n2 (z2 = 0)) 2 (Q1 )) + (1 − α 2 )] · V2 (ˆ = [pL + α ˆ 1 Λp Q1 /
N ] · [V2 (ˆ α2 (z2 = Z), n2 (z2 = Z)) − Z] + [1 − (pL + α ˆ 1 Λp Q1 /N )] · V2 (α ˆ 2 (z2 = 0), n2 (z2 = 0)) (8) and V2 (α, n) =
1−n N n S V2 (α, n) + V2 (α, n) N N
We solve the game backwards by starting at the last period. The payoffs in last period will be identical to the ones we solved in the previous section; we expect the number of signatories to be equal
to I(1/γ2 ) in the stage game. The following lemma will be found useful. Lemma 2 If there exists a significant amount of non-signatories or the change in γ2 is small, V2 (α, n) is decreasing in γ2 .
Proof I am going to treat n as a differentiable function of γ to keep notations simple. Using (9) and results in previous section, we can write V2 (α, n) as
V2 (α, n2 ) =
n2 1 − n2 1 (−γ2 (N − n2 )) + (1 − γ2 (N − n2 )) = (−n2 + 1 − γ2 (N − n2 )) N N N
By taking derivative with respect to γ2 (and taking into account the fact that n2 is a function of γ2 ), ∂n2 ∂V2 = −(1 − γ2 ) − (N − n2 − 1) ∂γ2 ∂γ2 given the fact that is large or
∂n2 ∂γ2
∂n2 ∂γ2
< 0 we cannot sign the derivative. However, under the condition that N − n2
maybe 0 for small changes in γ, the second effect is going to dominate and hence the
derivative is negative. Define V2Z ≡ V2 (ˆ α2 (z2 = Z), n2 (z2 = Z)), V20 ≡ V2 (α ˆ 2 (z2 = 0), n2 (z2 = 0)) and ΛV2 ≡ V2Z − V20 . Following from the fact that γ and α are positively correlated and
Corollary 3, it is straightforward to prove the following, Corollary 4 V2Z , V20 and ΛV2 have the following properties, (a) ΛV2 < 0; 12
(b) V2Z is decreasing in Q1 ; (c) V20 is increasing in Q1 ; (d) ΛV2 is decreasing in Q1 . The signs implied in this corollary are intuitive. When a catastrophe happened, the information is updated
such that the expected cost increases. It will result in a potential decrease in the equilibrium number of signatories hence the global pollution in period 2 is going to increase and harm the welfare
(opposite when catastrophe did not happen). Global pollution in period 1 controls for the convergence rate of γ. In the case where information is beneficial (no catastrophe), more pollution in period
1, which results in more information, creates an extra gain in welfare. Similarly, when information is harmful (catastrophe happened), more pollution in period 1 creates an extra cost in welfare. Now
we can start analyzing the signatories decisions. Signatories will NOT pollute if and only if
−γ1 (N − n) + δV20 |lowQ + δ(pL + α ˆ 1 ΛP − α ˆ 1 ΛP n/N )ΛV2 |lowQ − δ[pL + α ˆ 1 ΛP (N − n)/N ]Z > 1 − γ1 N + δV20 |highQ + δ(pL + α ˆ 1 ΛP )ΛV2 |highQ − δ[pL + α ˆ 1 ΛP ]Z
Denote ∆x = x|highQ − x|lowQ and using Corollaries 3 and 4, nγ1 > 1 + δ∆V20 + δ(pL + α α1 ΛP n/N )ΛV2 |lowQ − δ α ˆ 1 ΛP nZ/N ˆ 1 ΛP )∆ΛV2 + δ(ˆ | {z } | {z } | {z } {z } | >0
= 1 + δ(1 − (pL + α ˆ 1 ΛP ))∆V20 + δ(pL + α ˆ 1 ΛP )∆V2Z + δ(ˆ α1 ΛP n/N )ΛV2 |lowQ − δ α ˆ 1 ΛP nZ/N | {z } | {z } | {z } {z } | >0
Similarly, nonsignatories will pollute if
1+δ(1−(pL +α ˆ 1 ΛP (N −n)/N ))∆V20 +δ(pL +α ˆ 1 ΛP (N −n)/N )∆V2Z +δ(ˆ α1 ΛP /N )ΛV2 |lowQ > γ1 +δ(ˆ α1 ΛP /N )Z (14) Notice that that the high Q and low Q scenarios for nonsignatories imply Q1 = N
−n and Q1 = N −n−1 respectively. In this subsection, I am going to assume that catastrophes are not costly, i.e. Z = 0 and the fourth term in (13) will drop out from the subsequent analysis. When we
look at (13), it is not immediately obvious how this IEA equilibrium is going to differ from the one that we found in the static model. The first term is the change in value when there is no
catastrophe happened (due to a potential increase in signatories and decrease in global pollution). The difference is positive because period-1 pollution contributes the updating of information and
the update is beneficial to society. The
second term is the change when catastrophe happened, it is negative because the update is harmful to society as it results in a potential increase in global pollution. These two terms isolate the
effect in which global pollution in period 1 only affects the potential number of signatories in period 2. The third term takes the second effect of global pollution in period 1 into account, that is
to increase the chance that the catastrophe will happen in period 2, hence it is negative. If the first three terms are greater than zero, it would mean that the number of signatories is at least as
large as the IEA equilibrium in the static model. The following intermediate proposition showed that if the learning is slow and the expected γt does not change a lot in different scenarios (whether
pollution is high or low; or whether catastrophes happened or not in period 2), and this leads to the fact that the number of signatories is the same in all four cases, then the three terms as in
(13) summed up to zero. Lemma 3 Suppose ∆x = x|highQ − x|lowQ where highQ : Q1 = K and lowQ : Q1 = K − h. If the number of signatories in period 2 is the same under all scenarios, Θ ≡ δ(1 − (pL + α ˆ
1 ΛP K/N ))∆V20 + δ(pL + α ˆ 1 ΛP K/N )∆V2Z + δ(ˆ α1 ΛP h/N )ΛV2 |lowQ = 0 Proof See Appendix. We can apply the lemma to the problem of signatories by replacing K = N and h = n. The intuition is that
when countries completely consider all the possibilities (with appropriate probabilities), if the global emissions is going to be the same (since n2 is always the same) anyway, then the net effect is
zero when signatories in period 1 consider whether or not to pollute. The above striking result also implies that internal validity is satisfied. With some algebra, the external validity condition is
also satisfied because non-signatories are looking at the same difference in expected period-2 values as the signatories (using the Lemma again, by replacing the K and h accordingly). By studying
(14) carefully (and recall that we assumed Z to be 0 here), nonsignatories will always pollute. Therefore, it leads to the following corollary, which states that the result in previous section
follows. Corollary 5 Under (i) Assumptions 1, (ii) the Dynamic Bayesian learning mechanism, (iii) if the number of signatories in period 2 is the same under all scenarios and (iv) catastrophes
involve no economic loss, the equilibrium number of signatories in period 1 is I( γ11 ) and it is unique. 0 Can we use the above to generalize some results? Let nZ 2 (n2 ) be the number of
signatories when
catastrophe in period 2 is (not) revealed. I am going to make the following assumption throughout the paper: 14
Assumption 2 The pollution decision of signatories cannot change the number of signatories in any state of z2 . This assumption simplifies the problem into one dimension when signatories only
consider the case whether catastrophes happened or not in the next period, and the signatories can only affect the chance they lie on each state of z2 through pH .7 We know that when catastrophes are
revealed, the number of signatories can decrease because the expected γ is larger. The following proposition presents another result of the paper: Lemma 4 Under Assumption 2, Θ =
mδ ˆ 1 ΛP h/N )(1 N (α
− γ H ) ≥ 0, where m is the (positively-defined)
difference of the equilibrium number of signatories whether catastrophes are revealed in period 2. When m > 0, Θ > 0. Proof See Appendix. This is the main result of the paper. Equations (13) and (14)
can be boiled down to
Signatories: Non-Signatories:
where ΘS ≡
mδ α1 ΛP n/N )(1 N (ˆ
− γ H ) ≥ 0 and ΘN ≡
nγ1 > 1 + ΘS
1 + ΘN > γ1
mδ α1 ΛP /N )(1 N (ˆ
− γ H ) ≥ 0. Equation (16) always
holds and similarly external validity also holds. Internal validity implies the number of signatories in S
1 this case is n∗ = I( 1+Θ γ1 ) ≥ I( γ1 ). It is summed up in the following proposition.
Proposition 6 Under Assumptions 1 and 2, and catastrophes involve no economic loss, then the equilibrium number of signatories in period 1 under the dynamic Bayesian framework cannot be smaller than
that under the static Bayesian framework. The results from this section lead us to think more about the convergence of the state into the decisionmaking of agents. As we showed in the simulations in
last part, countries do prefer the convergence to be slower in expectation. In the static model in section 2, countries have no way to change the learning process itself because all catastrophes are
happened according to a predetermined exogenous process. In this dynamic setting however, countries know that they can now alter this learning process by choosing to pollute or not. The marginal
signatory now has an extra cost of leaving the coalition - if 7 It is very hard to quantify the assumption because the number of signatories in next period is a non-differentiable function of current
emissions. I omitted this for the sake of representation.
she decides to leave the coalition, not only the global pollution will go up (which is a cost to them), now the increase in the Bayesian learning process convergence rate creates an extra cost to the
country as this would decrease their expected welfare if the number of signatory goes down.
When Catastrophes are Costly
Now let us bring the costly catastrophes back to the big picture. Non-signatories will pollute if
1 − γ1 + δ α ˆ 1 ΛP /N
mδ (1 − γ H ) − Z N
Equivalently, using (15), signatories will not pollute if mδ (ˆ α1 ΛP n/N )(1 − γ H ) − δ α ˆ 1 ΛP nZ/N N n m = 1 + δα ˆ 1 ΛP (1 − γ H ) − Z N N
nγ1 > 1 +
Therefore whether the number of signatories increases or decreases as a result of Bayesian learning mechanism depends on the relative strength of learning and cost. As I showed earlier in Lemma 3, if
the number of signatories is not changing, m = 0 and the term in the bracket is always negative, hence the equilibrium number of signatories will decrease. If (17) holds, we can rearrange that and
get δα ˆ 1 ΛP N
mδ (1 − γ H ) − Z N
> −(1 − γ1 )
Substitute (19) into (18), we get
nγ1 > 1 + δ α ˆ 1 ΛP
n m (1 − γ H ) − Z N N
> 1 + n(−(1 − γ1 )) nγ1 > 1 − n + nγ1
(20) is satisfied for any coalition of signatories that consist of more than one member. This is a very surprising result. When non-signatories always pollute, signatories will never pollute. It also
implies that any members in the coalition now has incentive to free-ride. External validity condition for non-signatories is always satisfied under the assumption that they always pollute and Lemma
4. To 16
satisfy the internal validity condition, the maximum number of signatories can only be 2. The result is wrapped up in the following proposition: Proposition 7 Under Assumption 2 and when catastrophes
are costly, if non-signatories always pollute, the equilibrium number of signatories in the dynamic Bayesian learning framework cannot exceed 2. The intuition is that both non-signatories and
signatories are looking at the same margin: between the benefits of information term Θ and cost of catastrophes Z. Once non-signatories are willing to pollute such that the cost of catastrophes is
not too high, it also implies that signatories would always find polluting more costly.
Generalization to More Periods
If the game is finitely repeated, we can treat the analysis as the last two periods (T − 1, T ) of the model. As the results in last subsection is ‘robust’ to the prior knowledge entering into the
period (ˆ αt ) if we are willing to make an assumption that non-signatories will always pollute in all periods. When t = T − 2, given that the number of signatories is always 2, we can invoke from
Lemma 3 that the Θ = 0 and the tradeoff for signatories is essentially
nγ1 > 1 − δ α ˆ T −2 ΛP nZ/N
and the equilibrium number of signatories will be lowered than the one in our static setting. Qualitatively, we should expect for all (finite) periods (but not the last period), that the number of
signatories should be 2 by the above arguments. It is much harder to analyze the results qualitatively for more than 3 periods - one will have to rely on a computational model to solve the dynamic
programming problem. The reason is that the results will be contingent on the value and realizations of catastrophes such that α ˆ t will respond differently. As illustrated in (21), α ˆ t and Z
directly impact the number of signatories through the internal validity condition. In other words, the period-2 problem in a two-period problem also involves this whole chain of implications from the
pollution decision in period t. I left this part for interesting readers.
Concluding Remarks
Models on self-enforcing international environmental agreement (IEA) starts in a simple static and deterministic framework. Different scholars try to make the model more realistic by relaxing some
assumptions and considering other components like uncertainty on the cost (Na and Shin, 1998; Kolstad, 2007), stock pollutant instead of flow pollutant (Ulph, 2004; Rubio and Ulph, 2007),
heterogeneity and learning mechanism (Kolstad and Ulph, 2008). This paper is the first paper to consider Bayesian learning in the framework of self-enforcing IEA. I first consider a rough sense of
Bayesian learning that there is some exogenous process that determines the learning process. Using this ‘naive’ framework, I am able to compare the aggregate expected welfare under a Bayesian
learning process and a full disclosure. I find that the expected welfare is always higher in the Bayesian learning process. The next thing that I attempt to do is to consider another Bayesian
learning process that agents would have some control over it. I assume, for simplicity, that when agents pollute they can be informed ‘more’ of the state of the world (yet assuming that there is no
cost of catastrophes). As I argue in the paper, this may be implied by the simulation exercise in the static model that countries ‘prefer’ a slower learning process. As a consequence, pollution will
create an extra cost to the marginal signatory and this will increase the size of the coalition. When I allow for the cost of catastrophes, I show that the equilibrium number of signatories breaks
down to a maximum coalition of 2. One thing to note here is that this paper is just a start of this fruitful branch of research in the literature of self-enforcing IEA. My paper allows readers to
think more about how learning shapes the equilibrium. Of course, this paper does not intend to represent the reality, but it offers some important insight that other papers missed earlier. I show
that information potentially has a negative expected value. It will be interesting to see if such a negative value of information is robust to the settings of the model. As I mentioned earlier,
modelling the pollutant as a stock pollutant will be a one big step closer to representing the reality. I left this as a potentially interesting topic for enthusiastic readers. I am aware that the
last result of my paper is subject to the particular setting in my model so I look forward to work where other scholars try to bring Bayesian learning in the model of international environmental
agreements differently. In particular, information can potentially enter and affect individuals in many different ways other than a cost shock.
Appendix Proof of Lemma 3 Denote n and n2 to be the number of signatories in period 1 and 2 respectively. To simplify the analysis, Iet ZH denote high Q1 and z2 = Z, similarly for 0H, 0L and ZL. Call
P ≡ pL + α ˆ 1 ΛP K/N and Λγ ≡ γ H − γ L > 0. Using (6a) and (6b), we have (1 − α ˆ 1 )(1 − pL ) 1−P (1 − α ˆ 1 )pL =1− P (1 − α ˆ 1 )(1 − pL ) =1− 1 − P + ΛP α ˆ 1 h/N L (1 − α ˆ 1 )p =1− P − ΛP α ˆ
1 h/N
α ˆ 20H = 1 −
α ˆ 2ZH
α ˆ 20L α ˆ 2ZL
(22c) (22d)
By our assumption that n2 is the same under all scenarios and the definitions of ∆V20 and ∆V2Z , ∆V20 = V20 |highQ − V20 |lowQ 1 0L (γ − γ20H )(N − n2 ) N 2 1 α20L − α ˆ 20H ) = Λγ (N − n2 )(ˆ N 1 =
Λγ (N − n2 )(ˆ α2ZL − α ˆ 2ZH ) N
(23a) (23b)
Using (22a)-(22d), (1 − α ˆ 1 )(1 − pL ) (1 − α ˆ 1 )(1 − pL )ΛP α ˆ 1 h/N (1 − α ˆ 1 )(1 − pL ) − = >0 P P 1−P 1−P +Λ α ˆ 1 h/N (1 − P )(1 − P + Λ α ˆ 1 h/N ) (1 − α ˆ 1 )pL (1 − α ˆ 1 )pL (1 − α ˆ
1 )pL ΛP α ˆ 1 h/N = − =− <0 P P P −Λ α ˆ 1 h/N P (P − ΛP α ˆ 1 h/N )
α ˆ 20L − α ˆ 20H = α ˆ 2ZL − α ˆ 2ZH
(24a) (24b)
Combining (23a), (23b), (24a), (24b) and simplifying, we get
δ(1 − P )∆V20 + δP ∆V2Z =
δ γ p Λ Λ (N − n2 )(1 − α ˆ 1 )α ˆ1h N2
1 − pL pL − 1 − P + ΛP α ˆ 1 h/N P − ΛP α ˆ 1 h/N | {z }
Using the definition of P , 1 (1 − pL )P − ΛP α ˆ 1 h/N − pL + P pL C ΛP α ˆ 1 (K − h)/N = >0 C
where C ≡ (1 − P + ΛP α ˆ 1 h/N )(P − ΛP α ˆ 1 h/N ) (26)
Now we deal with the remaining term. By the similar derivations as above, 1 γ Λ (N − n2 )(ˆ α20L − α ˆ 2ZL ) N 1 (1 − α ˆ 1 )pL (1 − α ˆ 1 )(1 − pL ) = Λγ (N − n2 ) − N P − ΛP α ˆ 1 h/N 1 − P + ΛP α
ˆ 1 h/N (1 − α ˆ1) L 1 γ p − P + ΛP α ˆ 1 h/N = Λ (N − n2 ) N C 1 (1 − α ˆ1) P = − Λγ (N − n2 ) Λ α ˆ 1 (K − h)/N < 0 N C
ΛV2 |lowQ =
Putting all the pieces together,
δ(1 − (pL + α ˆ 1 ΛP ))∆V20 + δ(pL + α ˆ 1 ΛP )∆V2Z + δ(ˆ α1 ΛP h/N )ΛV2 |lowQ =
δ γ p ΛP α ˆ 1 (K − h)/N δ (1 − α ˆ1) P Λ Λ (N − n )(1 − α ˆ )ˆ α h − 2 (ˆ α1 ΛP h)Λγ (N − n2 ) Λ α ˆ 1 (K − h)/N 2 1 1 2 N C N C
Proof of Lemma 4 0 Assume that nZ 2 = n2 − m where m ≥ 0 by the arguments made above that the equilibrium number of
signatories is smaller if catastrophes are revealed. If m = 0 the results follow from Lemma 3, so I am just going to focus on the case that m > 0. Recall the results in (23a) and (23b),
∆V20 = ∆V2Z = = ΛV2 |lowQ = =
1 γ Λ (N − n02 )(ˆ α20L − α ˆ 20H ) N 1 γ Λ (N − nZ ˆ 2ZL − α ˆ 2ZH ) 2 )(α N 1 γ m Λ (N − n02 )(ˆ α2ZL − α ˆ 2ZH ) + Λγ (ˆ α2ZL − α ˆ 2ZH ) N N 1 1 ZL Z (−nZ (−n02 + 1 − γ20L (N − n02 )) 2 + 1 − γ2
(N − n2 )) − N N 1 γ m Λ (N − n02 )(ˆ α20L − α ˆ 2ZL ) + (1 − γ2ZL ) N N
Using (2) and (22d),
γ2ZL = γ L + α ˆ 2ZL Λγ (1 − α ˆ 1 )pL Λγ = γL + 1 − P − ΛP α ˆ 1 h/N (1 − α ˆ 1 )pL Λγ P P −Λ α ˆ 1 h/N (1 − α ˆ 1 )pL = (1 − γ H ) + Λγ P P −Λ α ˆ 1 h/N = γH −
1 − γ2ZL
By substituting (24b), (28b), (28c) and (29) into the last terms in (13), as well as using Lemma 3 to cancel out the terms with (N − n02 ), Θ = δ(1 − (pL + α ˆ 1 ΛP K/N ))∆V20 + δ(pL + α ˆ 1 ΛP K/N )
∆V2Z + δ(ˆ α1 ΛP h/N )ΛV2 |lowQ mδ (1 − α ˆ 1 )pL ΛP α ˆ 1 h/N γ mδ mδ (1 − α ˆ 1 )pL P H P Λ + (ˆ α Λ h/N )(1 − γ ) + ( α ˆ Λ h/N ) Λγ 1 1 N P − ΛP α ˆ 1 h/N N N P − ΛP α ˆ 1 h/N mδ = (ˆ α1 ΛP h/N )
(1 − γ H ) > 0 N
References Barrett, Scott (1994), “Self-enforcing international environmental agreements.” Oxford Economic Papers, 46, 878–94. Barrett, Scott (2003), Environment and Statecraft: the strategy of
environmental treaty-making. Oxford University Press, New York. Intergovernmental Panel on Climate Change (2007a), Climate Change 2007: The Physical Science Basis. Cambridge University Press.
Intergovernmental Panel on Climate Change (2007b), Climate Change 2007: Impacts, Adaptation, and Vulnerability. Cambridge University Press. Intergovernmental Panel on Climate Change (2007c), Climate
Change 2007: Mitigation of Climate Change. Cambridge University Press. Karp, Larry S. and Jiangfeng Zhang (2006), “Regulation with anticipated learning about environmental damages.” Journal of
Environmental Economics and Management, 51, 259–279. Kelly, David L. and Charles D. Kolstad (1999), “Bayesian learning, growth, and pollution.” Journal of Economic Dynamics and Control, 23, 491–518.
Kolstad, Charles D. (2007), “Systematic uncertainty in self-enforcing international environmental agreements.” Journal of Environmental Economics and Management, 53, 68–79. Kolstad, Charles D. and
Alistair Ulph (2008), “Uncertainty, learning and heterogeneity in international environmental agreements.” mimeo. Na, Seong-lin and Hyun Song Shin (1998), “International environmental agreements
under uncertainty.” Oxford Economic Papers, 50, 173–85. Rubio, Santiago Jose and Alistair Ulph (2006), “Self-enforcing international environmental agreements revisited.” Oxford Economic Papers, 58,
233–263. Rubio, Santiago Jose and Alistair Ulph (2007), “An infinite-horizon model of dynamic membership of international environmental agreements.” Journal of Environmental Economics and Management,
54, 296–310. Stern, Nicholas (2007), The Economics of Climate Change: The Stern Review. Cambridge University Press, Cambridge and New York. 22
Ulph, Alistair (2004), “Stable international environmental agreements with a stock pollutant, uncertainty and learning.” Journal of Risk and Uncertainty, 29, 53–73. Ulph, Alistair and James Maddison
(1997), “Uncertainty, learning and international environmental policy coordination.” Environmental and Resource Economics, 9, 451–466. | {"url":"https://p.pdfkul.com/a-simple-model-of-international-environmental-_5a16dad61723dd2b6215d4a0.html","timestamp":"2024-11-05T12:55:02Z","content_type":"text/html","content_length":"105512","record_id":"<urn:uuid:b8b6db48-3742-4127-b067-edbae4ddd526>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00567.warc.gz"} |
RFC 5830: GOST 28147-89: Encryption, Decryption, and Message Authentication Code (MAC) Algorithms
GOST 28147-89: Encryption, Decryption, and Message Authentication Code (MAC) Algorithms
RFC 5830
This RFC was published on the Independent Submission stream. This RFC is
not endorsed by the IETF
and has
no formal standing
in the
IETF standards process
RFC - Informational (March 2010) Errata
Document Type Updated by
RFC 8891
Author Vasily Dolmatov
Last updated 2020-01-21
RFC stream Independent Submission
IESG Responsible AD Russ Housley
Send notices to (None)
RFC 5830
Independent Submission V. Dolmatov, Ed.
Request for Comments: 5830 Cryptocom, Ltd.
Category: Informational March 2010
ISSN: 2070-1721
GOST 28147-89: Encryption, Decryption,
and Message Authentication Code (MAC) Algorithms
This document is intended to be a source of information about the
Russian Federal standard for electronic encryption, decryption, and
message authentication algorithms (GOST 28147-89), which is one of
the Russian cryptographic standard algorithms called GOST
algorithms). Recently, Russian cryptography is being used in
Internet applications, and this document has been created as
information for developers and users of GOST 28147-89 for encryption,
decryption, and message authentication.
Status of This Memo
This document is not an Internet Standards Track specification; it is
published for informational purposes.
This is a contribution to the RFC Series, independently of any other
RFC stream. The RFC Editor has chosen to publish this document at
its discretion and makes no statement about its value for
implementation or deployment. Documents approved for publication by
the RFC Editor are not a candidate for any level of Internet
Standard; see Section 2 of RFC 5741.
Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
Dolmatov Informational [Page 1]
RFC 5830 GOST 28147-89 March 2010
Copyright Notice
Copyright (c) 2010 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
(http://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents
carefully, as they describe your rights and restrictions with respect
to this document.
This document may not be modified, and derivative works of it may not
be created, except to format it for publication as an RFC or to
translate it into languages other than English.
Table of Contents
1. Introduction ....................................................3
1.1. General Information ........................................3
2. Applicability ...................................................3
3. Definitions and Notations .......................................3
3.1. Definitions ................................................3
3.2. Notation ...................................................4
4. General Statements ..............................................4
5. The Electronic Codebook Mode ....................................6
5.1. Encryption of Plain Text in the Electronic Codebook Mode ...6
5.2. Decryption of the Ciphertext in the Electronic
Codebook Mode ..............................................9
6. The Counter Encryption Mode ....................................10
6.1. Encryption of Plain Text in the Counter Encryption Mode ...10
6.2. Decryption of Ciphertext in the Counter Encryption Mode ...13
7. The Cipher Feedback Mode .......................................13
7.1. Encryption of Plain Text in the Cipher Feedback Mode ......13
7.2. Decryption of Ciphertext in the Cipher Feedback Mode ......14
8. Message Authentication Code (MAC) Generation Mode ..............15
9. Security Considerations ........................................17
10. Normative References ..........................................17
Appendix A. Values of the Constants C1 and C2 .....................18
Appendix B. Contributors ..........................................19
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1. Introduction
1.1. General Information
[GOST28147-89] is the unified cryptographic transformation algorithm
for information processing systems of different purposes, defining
the encryption/decryption rules and the message authentication code
(MAC) generation rules.
This cryptographic transformation algorithm is intended for hardware
or software implementation and corresponds to the cryptographic
requirements. It puts no limitations on the encrypted information
secrecy level.
2. Applicability
GOST 28147-89 defines the encryption/decryption model and MAC
generation for a given message (document) that is meant for
transmission via insecure public telecommunication channels between
data processing systems of different purposes.
GOST 28147-89 is obligatory to use in the Russian Federation in all
data processing systems providing public services.
3. Definitions and Notations
3.1. Definitions
The following terms are used in the standard:
Running key: a pseudo-random bit sequence generated by a given
algorithm for encrypting plain texts and decrypting encrypted texts.
Encryption: the process of transforming plain text to encrypted data
using a cipher.
MAC: an information string of fixed length that is generated from
plain text and a key according to some rule and added to the
encrypted data for protection against data falsification.
Key: a defined secret state of some parameters of a cryptographic
transformation algorithm, that provides a choice of one
transformation out of all the possible transformations.
Cryptographic protection: data protection using the data
cryptographic transformations.
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Cryptographic transformation: data transformation using encryption
and (or) MAC.
Decryption: the process of transforming encrypted data to plain text
using a cipher.
Initialisation vector: initial values of plain parameters of a
cryptographic transformation algorithm.
Encryption equation: a correlation showing the process of generating
encrypted data out of plain text as a result of transformations
defined by the cryptographic transformation algorithm.
Decryption equation: a correlation showing the process of generating
plain text out of encrypted data as a result of transformations
defined by the cryptographic transformation algorithm.
Cipher: a set of reversible transformations of the set of possible
plain texts onto the set of encrypted data, made after certain rules
and using keys.
3.2. Notation
In this document, the following notations are used:
^ is a power operator.
(+) is a bitwise addition of the words of the same length modulo 2.
[+] is an addition of 32-bit vectors modulo 2^32.
[+]' is an addition of the 32-bit vectors modulo 2^32-1.
1..N is all values from 1 to N.
4. General Statements
The structure model of the cryptographic transformation algorithm (a
cryptographic model) contains:
- a 256-bit key data store (KDS) consisting of eight 32-bit registers
(X0, X1, X2, X3, X4, X5, X6, X7);
- four 32-bit registers (N1, N2, N3, N4);
- two 32-bit registers (N5 and N6) containing constants C1 and C2;
- two 32-bit adders modulo 2^32 (CM1, CM3);
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- a 32-bit adder of bitwise sums modulo 2 (CM2);
- a 32-bit adder modulo (2^32-1) (CM4);
- an adder modulo 2 (CM5), with no limitation to its width;
- a substitution box (K);
- a register for a cyclic shift of 11 steps to the top digit (R).
A substitution box (S-box) K consists of eight substitution points
K1, K2, K3, K4, K5, K6, K7, K8, with 64-bit memory. A 32-bit vector
coming to the substitution box is divided into eight successive 4-bit
vectors, and each of them is transformed into a 4-bit vector by a
corresponding substitution point. A substitution point is a table
consisting of 16 lines, each containing four bits. The incoming
vector defines the line address in the table, and the contents of
that line is the outgoing vector. Then, these 4-bit outgoing vectors
are successively combined into a 32-bit vector.
Remark: the standard doesn't define any S-boxes. Some of them are
defined in [RFC4357].
When adding and cyclically shifting binary vectors, the registers
with larger numbers are considered the top digits.
When writing a key (W1, W2, ..., W256), Wq = 0..1, q = 1..256, in the
KDS the value:
- W1 is written into the 1st bit of the register X0;
- the value W2 is written into the 2nd bit of the register X0 (etc.);
- the value W32 is written into the 32nd bit of the register X0;
- the value W33 is written into the 1st bit of the register X1;
- the value W34 is written into the 2nd bit of the register X1
- the value W64 is written into the 32nd bit of the register X1;
- the value W65 is written into the 1st bit of the register X2
- the value W256 is written into the 32nd bit of the register X7.
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When rewriting the information, the value of the p-th bit of one
register (adder) is written into the p-th bit of another register
The values of the constants C1 and C2 in the registers N5 and N6 are
in the Appendix 1.
The keys defining fillings of KDS and the substitution box K tables
are secret elements and are provided in accordance with the
established procedure.
The filling of the substitution box K is described in GOST 28147-89
as a long-term key element common for a whole computer network.
Usually, K is used as a parameter of algorithm, some possible sets of
K are described in [RFC4357].
The cryptographic model contemplates four working modes:
- data encryption (decryption) in the electronic codebook (ECB) mode,
- data encryption (decryption) in the counter (CNT) mode,
- data encryption (decryption) in the cipher feedback (CFB) mode, and
- the MAC generation mode.
[RFC4357] also describes the CBC mode of GOST 28147-89, but this mode
is not a part of the standard.
5. The Electronic Codebook Mode
5.1. Encryption of Plain Text in the Electronic Codebook Mode
The plain text to be encrypted is split into 64-bit blocks. Input of
a binary data block Tp = (a1(0), a2(0), ... , a31(0), a32(0), b1(0),
b2(0), ..., b32(0)) into the registers N1 and N2 is done so that the
value of a1(0) is put into the first bit of N1, the value of a2(0) is
put into the second bit of N1, etc., and the value of a32(0) is put
into the 32nd bit of N1. The value of b1(0) is put into the first
bit of N2, the value of b2(0) is put into the 2nd bit of N2, etc.,
and the value of b32(0) is input into the 32nd bit of N2.
The result is the state (a32(0), a31(0), ..., a2(0), a1(0)) of the
register N1 and the state (b32(0), b31(0), ..., b1(0)) of the
register N2.
The 256 bits of the key are entered into the KDS. The contents of
eight 32-bit registers X0, X1, ..., X7 are:
Dolmatov Informational [Page 6]
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X0 = W32, W31, ..., W2, W1
X1 = W64, W63, ..., W34, W33
. . . . . . . . . . . . . . .
X7 = W256, W255, ..., W226, W225
The algorithm for enciphering 64-bit blocks of plain text in the
electronic codebook mode consists of 32 rounds.
In the first round, the initial value of register N1 is added modulo
2^32 in the adder CM1 to the contents of the register X0. Note: the
value of register N1 is unchanged.
The result of the addition is transformed in the substitution block
K, and the resulting vector is put into the register R, where it is
cyclically shifted by 11 steps towards the top digit. The result of
this shift is added bitwise modulo 2 in the adder CM2 to the 32-bit
contents of the register N2. The result produced in CM2 is then
written into N1, and the old contents of N1 are written in N2. Thus,
the first round ends.
The subsequent rounds are similar to the first one:
- in the second round, the contents of X1 are read from the KDS;
- in the third round, the contents of X2 are read from the KDS, etc.;
- in the 8th round, the contents of X7 are read from the KDS.
- in rounds 9 through 16 and 17 through 24, the contents of the KDS
are read in the same order:
X0, X1, X2, X3, X4, X5, X6, X7.
- in the last eight rounds from the 25th to the 32nd, the contents of
the KDS are read backwards:
X7, X6, X5, X4, X3, X2, X1, X0.
Thus, during the 32 rounds of encryption, the following order of
choosing the registers' contents is implemented:
X0, X1, X2, X3, X4, X5, X6, X7, X0, X1, X2, X3, X4, X5, X6, X7,
X0, X1, X2, X3, X4, X5, X6, X7, X7, X6, X5, X4, X3, X2, X1, X0
Dolmatov Informational [Page 7]
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In the 32nd round, the result in the adder CM2 is written into the
register N2, and the old contents of register N1 are unchanged.
After the 32nd round, the contents of the registers N1 and N2 are an
encrypted data block corresponding to a block of plain text.
The equations for enciphering in the electronic codebook mode are:
|a(j) = (a(j-1) [+] X(j-1)(mod 8))*K*R (+) b (j-1)
| j = 1..24;
|b(j) = a(j-1)
|a(j) = (a(j-1) [+] X(32-j))*K*R (+) b(j-1)
| j = 25..31; a32 = a31;
|b(j) = a(j-1)
b(32) = (a(31) [+] X0)*K*R (+) b(31) j=32,
a(0) = (a32(0), a31(0), ..., a1(0)) constitutes the initial contents
of N1 before the first round of encryption;
b(0) = (b32(0), b31(0), ..., b1(0)) constitutes the initial contents
of N2 before the first round of encryption;
a(j) = (a32(j), a31(j), ..., a1(j)) constitutes the contents of N1
after the j-th round of encryption;
b(j) = (b32(j), b31(j), ..., b1(j)) constitutes the contents of N2
after the j-th round of encryption, j = 1..32.
R is the operation of cyclic shift towards the top digit by 11 steps,
as follows:
R(r32, r31, r30, r29, r28, r27, r26, r25, r24, r23, r22, r21,
r20, ..., r2, r1) =
(r21, r20, ..., r2, r1, r32, r31, r30, r29, r28, r27, r26, r25,
r24, r23, r22)
The 64-bit block of ciphertext Tc is taken out of the registers N1,
N2 in the following order:
the first, second, ..., 32nd bit of the register N1, then the first,
second, ..., 32nd bit of the register N2, i.e.,
Tc = a1(32), a2(32), ..., a32(32), b1(32), b2(32), ..., b32(32)).
Dolmatov Informational [Page 8]
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The remaining blocks of the plain text in electronic codebook mode
are encrypted in the same fashion.
5.2. Decryption of the Ciphertext in the Electronic Codebook Mode
The same 256-bit key that was used for encryption is loaded into the
KDS, the encrypted data to be deciphered is divided into 64-bit
blocks. The loading of any binary information block
Tc = (a1(32), a2(32), ..., a32(32), b1(32), b2(32), ..., b32(32))
into the registers N1 and N2 is done in such a way that:
- the contents of a1(32) are written into the first bit of N1;
- the contents of a2(32) are written into the second bit of N1 (and
so on);
- the contents of a32(32) are written into the 32nd bit of N1;
- the contents of b1(32) are written into the first bit of N2 (and so
- and the contents of b32(32) are written into the 32nd bit of N2.
The decryption procedure uses the same algorithm as the encryption of
plain text, with one exception: the contents of the registers X0, X1,
..., X7 are read from the KDS in the decryption rounds in the
following order:
X0,X1,X2,X3,X4,X5,X6,X7, X7,X6,X5,X4,X3,X2,X1,X0,
X7,X6,X5,X4,X3,X2,X1,X0, X7,X6,X5,X4,X3,X2,X1,X0.
The decryption equations are:
|a(32-j) = (a(32-j+1) [+] X(j-1))*K*R (+) b(32-j+1)
| j = 1..8;
|b(32-1) = a(32-j+1)
|a(32-j) = (a(32-j+1) [+] X(j-1)(mod 8))*K*R (+) b(32-j+1)
| j = 9..31;
|b(32-1) = a(32-j+1)
|a(0) = a(1)
| j=32.
|b(0) = (a(1) [+] X0)*K*R (+) b1
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The fillings of the adders N1 and N2 after 32 working rounds are a
plain text block.
Tp = (a1(0), a2(0), ... , a32(0), b1(0), b2(0), ..., b32(0))
corresponding to the encrypted data block:
- the value of a1(0) of the block Tp corresponds to the contents of
the first bit of N1;
- the value of a2(0) corresponds to the contents of the second bit of
N1 (etc.);
- the value of b1(0) corresponds to the contents of the first bit of
- the value of b2(0) corresponds to the contents of the second bit of
N2 (etc.);
- the value of b32(0) corresponds to the contents of 32nd bit of N2;
- the remaining blocks of encrypted data are decrypted similarly.
The encryption algorithm in the electronic codebook mode of a 64-bit
block Tp is denoted by A, that is:
A(Tp) is A(a(0), b(0)) = (a(32), b(32)) = Tc.
6. The Counter Encryption Mode
6.1. Encryption of Plain Text in the Counter Encryption Mode
The plain text divided into 64-bit blocks Tp(1), Tp(2), ..., Tp(M-1),
Tp(M) is encrypted in the counter encryption mode by bitwise addition
modulo 2 in the adder CM5 with the running key Gc produced in 64-bit
blocks, that is:
Gc = (Gc(1), Gc(2), ..., Gc(M-1), Gc(M))
where M is defined by the size of the plain text being encrypted.
Gc(i) is the i-th 64-bit block where i=1..M, the number of bits in a
block Tp(M) can be less than 64. In this case, the unused part of
the running key block Gc(M) is discarded.
Dolmatov Informational [Page 10]
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256 bits of the key are put into the KDS. The registers N1 and N2
accept a 64-bit binary sequence (an initialisation vector) S = (S1,
S2, ..., S64), that is, the initial filling of these registers for
subsequent generation of M blocks of the running key. The
initialisation vector is put into the registers N1 and N2 so:
- the value of S1 is written into the first bit of N1;
- the value of S2 is written into the second bit of N1 (etc.);
- the value of S32 is written into the 32nd bit of N1;
- the value of S33 is written into the first bit of N2;
- the value of S34 is written into the 33th bit of N2 (etc.);
- the value of S64 is written into the 32nd bit of N2.
The initial filling of the registers N1 and N2 (the initialisation
vector S) is encrypted in the electronic codebook mode in accordance
with the requirements from section 5.1. The result of that
encryption A(S) = (Y0, Z0) is rewritten into the 32-bit registers N3
and N4 so as the contents of N1 are written into N3, and the contents
of N2 are written into N4.
The filling of the register N4 is added modulo (2^32-1) in the adder
CM4 to the 32-bit constant C1 from the register N6; the result is
written into N4. The filling of the register N3 is added modulo 2^32
in the adder CM3 with the 32-bit constant C2 from the register N5;
the result is written into N3.
The filling of N3 is copied into N1, and the filling of N4 is copied
into N2, while the fillings of N3 and N4 are kept.
The filling of N1 and N2 is encrypted in the electronic codebook mode
according to the requirements of section 5.1. The resulting
encrypted filling of N1 and N2 is the first 64-bit block of the
running key Gc(1), this block is bitwise added modulo 2 in the adder
CM5 with the first 64-bit block of the plain text:
Tp(1) = (t1(1), t2(1), ..., t63(1), t64(1)).
The result of this addition is a 64-bit block of the encrypted data:
Tc(1) = (tau1(1), tau2(1), ..., tau63(1), tau64(1)).
Dolmatov Informational [Page 11]
RFC 5830 GOST 28147-89 March 2010
The value of tau1(1) of the block Tc(1) is the result of the addition
of modulo 2 in the CM5 the value t1(1) of the block Tp(1) to the
value of the first bit of N1; the value of tau2(1) of the block Tc(1)
is the result of addition modulo 2 in the CM5 the value of t2(1) from
the block Tp(1) to the value of the second bit of N1, etc.; the value
of tau64(1) of the block Tc(1) is the result of addition modulo 2 in
the CM5 of the value t64(1) of the block Tp(1) to the value of the
32nd bit of N2.
To get the next 64-bit block of the running key Gc(2), the filling of
N4 is added modulo (2^32-1) in the adder CM4 with the constant C1
from N6; the filling of N3 is added modulo 2^32 in the adder CM3 with
the constant C2 from N5. The new filling of N3 is copied into N1;
the new filling of N4 is copied into N2; the fillings of N3 and N4
are kept.
The filling of N1 and N2 is encrypted in the electronic codebook mode
according to the requirements of section 5.1. The resulting
encrypted filling of N1 and N2 is the second 64-bit block of the
running key Gc(2); this block is bitwise added modulo 2 in the adder
CM5 with the first 64-bit block of the plain text Tp(2). The
remaining running key blocks Gc(3), Gc(4), ..., Gc(M) are generated
and the plain text blocks Tp(3), Tp(4), ..., Tp(M) are encrypted
similarly. If the length of the last M-th block of the plain text is
less than 64 bits, then only the corresponding number of bits from
the last M-th block of the running key is used; remaining bits are
The initialisation vector S and the blocks of encrypted data Tc(1),
Tc(2), ..., Tc(M) are transmitted to the telecommunication channel or
to the computer memory.
The encryption equation is:
Tc(i) = A(Y[i-1] [+] C2, Z[i-1]) [+]' C1) (+) Tp(i)
= Gc(i) (+) Tp(i) i=1..M
Y[i] is the contents of the register N3 after encrypting the
i-th block of the plain text Tp(i);
Z(i) is the contents of the register N4 after encrypting the
i-th block of the plain text Tp(i);
(Y[0], Z[0]) = A(S).
Dolmatov Informational [Page 12]
RFC 5830 GOST 28147-89 March 2010
6.2. Decryption of Ciphertext in the Counter Encryption Mode
256 bits of the key that was used for encrypting the data Tp(1),
Tp(2), ..., Tp(M) are put into the KDS. The initialisation vector S
is put into the registers N1 and N2 and, like in the section 6.1 M
blocks of the running key, Gc(1), Gc(2), ..., Gc(M) are generated.
The encrypted data blocks Tc(1), Tc(2), ..., Tc(M) are added bitwise
modulo 2 in the adder CM5 with the blocks of the running key, and
this results in the blocks of plain text Tp(1), Tp(2), ..., Tp(M),
and Tp(M) may contain less than 64 bit.
The decryption equation is:
Tp(i) = A (Y[i-1] [+] C2, Z[i-1] [+]' C1) (+) Tc(i)
= Gc(i) (+) Tc(i) i = 1..M
7. The Cipher Feedback Mode
7.1. Encryption of Plain Text in the Cipher Feedback Mode
The plain text is divided into 64-bit blocks Tp(1), Tp(2), ..., Tp(M)
and encrypted in the cipher feedback mode by bitwise addition modulo
2 in the adder CM5 with the running key Gc generated in 64-bit
blocks, i.e., Gc(i)=(Gc(1), Gc(2), ..., Gc(M)), where M is defined by
the length of the plain text, Gc(i) is the i-th 64-bit block, i=1,M.
The number of bits in the block Tp(M) may be less than 64.
256 bits of the key are put into the KDS. The 64-bit initialisation
vector S = (S1, S2, ..., S64) is put into N1 and N2 as described in
section 6.1.
The initial filling of N1 and N2 is encrypted in the electronic
codebook mode in accordance with the requirements in section 6.1. If
resulting encrypted filling N1 and N2 is the first 64-bit block of
the running key Gc(1)=A(S), then this block is added bitwise modulo 2
with the first 64-bit block of plain text Tp(1) = (t1(1), t2(1), ...,
The result is a 64-bit block of encrypted data
Tc(1) = (tau1(1), tau2(1), ..., tau64(1)).
The block of encrypted data Tc(1) is simultaneously the initial state
of N1 and N2 for generating the second block of the running key Gc(2)
and is written on feedback in these registers. Here:
- the value of tau1(1) is written into the first bit of N1;
Dolmatov Informational [Page 13]
RFC 5830 GOST 28147-89 March 2010
- the value of tau2(1) is written into the second bit of N1, etc.;
- the value of tau32(1) is written into the 32nd bit of N1;
- the value of tau33(1) is written into the first bit of N2;
- the value of tau34(1) is written into the second bit of N2, etc.;
- the value of tau64(1) is written into the 32nd bit of N2.
The filling of N1 and N2 is encrypted in the electronic codebook mode
in accordance with the requirements in the section 6.1. The
encrypted filling of N1 and N2 makes the second 64-bit block of the
running key Gc(2), this block is added bitwise modulo 2 in the adder
CM5 to the second block of the plain text Tp(2).
The generation of subsequent blocks of the running key Gc(i) and the
encryption of the corresponding blocks of the plain text Tp(i) (i =
3..M) are performed similarly. If the length of the last M-th block
of the plain text is less than 64 bits, only the corresponding number
of bits of the M-th block of the running key Gc(M) is used; remaining
bits are discarded.
The encryption equations in the cipher feedback mode are:
|Tc(1) = A(S) (+) Tp(1) = Gc(1) (+) Tp(1)
|Tc(i) = A(Tc(i-1)) (+) Tp(i) = Gc(i) + Tp(i), i = 2..M.
The initialisation vector S and the blocks of encrypted data Tc(1),
Tc(2), ..., Tc(M) are transmitted into the telecommunication channel
or to the computer memory.
7.2. Decryption of Ciphertext in the Cipher Feedback Mode
256 bits of the key used for the encryption of Tp(1), Tp(2), ...,
Tp(M) are put into the KDS. The initialisation vector S is put into
N1 and N2 similar to 6.1.
The initial filling of N1 and N2 (the initialisation vector S) is
encrypted in the electronic codebook mode in accordance with the
subsection 6.1. The encrypted filling of N1, N2 is the first block
of the running key Gc(1) = A(S), this block is added bitwise modulo 2
in the adder CM5 with the encrypted data block Tc(1). This results
in the first block of plain text Tp(1).
Dolmatov Informational [Page 14]
RFC 5830 GOST 28147-89 March 2010
The block of encrypted data Tc(1) makes the initial filling of N1, N2
for generating the second block of the running key Gc(2). The block
Tc(1) is written in N1 and N2 in accordance with the requirements in
the subsection 6.1, the resulted block Gc(2) is added bitwise modulo
2 in the adder CM5 to the second block of the encrypted data Tc(2).
This results in the block of plain text Tc(2).
Similarly, the blocks of encrypted data Tc(2), Tc(3), ..., Tc(M-1)
are written in N1, N2 successively, and the blocks of the running key
Gc(3), Gc(4), ..., Gc(M) are generated out of them in the electronic
codebook mode. The blocks of the running key are added bitwise
modulo 2 in the adder CM5 to the blocks of the encrypted data Tc(3),
Tc(4), ..., Tc(M), this results in the blocks of plain text Tp(3),
Tp(4), ..., Tp(M); here, the number of bits in the last block of the
plain text Tp(M) can be less than 64 bit.
The decryption equations in the cipher feedback mode are:
|Tp(1) = A(S) (+) Tc(1) = Gc(1) (+) Tc(1)
|Tp(1) = A(Tc(i-1)) (+) Tc(i) = Gc(i) (+) Tc(i), i=2..M
8. Message Authentication Code (MAC) Generation Mode
To provide the protection from falsification of plain text consisting
of M 64-bit blocks Tp(1), Tp(2), ..., Tp(M), M >= 2, an additional
l-bit block is generated (the message authentication code I(l)). The
process of MAC generation is the same for all the
encryption/decryption modes.
- The first block of plain text:
Tp(1) = (t1(1), t1(2), ..., t64(1)) = (a1(1)[0], a2(1)[0], ...,
a32(1)[0], b1(1)[0], b2(1)[0], ..., b32(1)[0])
is written to the registers N1 and N2;
- the value of t1(1) = a1(1)[0] is written into the first bit of N1;
- the value of t2(1) = a2(1)[0] is written into the second bit of N1,
- the value of t32(1) = a32(1)[0] is written into the 32nd bit of N1;
- the value of t33(1) = b1(1)[0] is written into the first bit of N2,
- the value of t64(1) = b32(1)[0] is written into the 32nd bit of N2.
Dolmatov Informational [Page 15]
RFC 5830 GOST 28147-89 March 2010
The filling of N1 and N2 is transformed in accordance with the first
16 rounds of the encryption algorithm in the electronic codebook mode
(see the subsection 6.1). In the KDS, there exists the same key that
is used for encrypting the blocks of plain text Tp(1), Tp(2), ...,
Tp(M) in the corresponding blocks of encrypted data Tc(1), Tc(2),
..., Tc(M).
The filling of N1 and N2 after the 16 working rounds, looking like
(a1(1)[16], a2(1)[16], ..., a32(1)[16], b1(1)[16], b2(1)[16], ...,
b32(1)[16]), is added in CM5 modulo 2 to the second block Tp(2) =
(t1(2), t2(2), ..., t64(2)).
The result of this addition
(a1(1)[16](+)t1(2), a2(1)[16](+)t2(2), ..., a32(1)[16](+)t32(2),
b1(1)[16](+)t33(2), b2(1)[16](+)t34(2), ..., b32(1)[16](+)t64(2))
(a1(2)[0], a2(2)[0] ..., a32(2)[0], b1(2)[0], b2(2)[0], ...,
is written into N1 and N2 and is transformed in accordance with the
first 16 rounds of the encryption algorithm in the electronic
codebook mode.
The resulting filling of N1 and N2 is added in the CM5 modulo 2 with
the third block Tp(3), etc., the last block Tp(M) = (t1(M), t2(M),
..., t64(M)), padded if necessary to a complete 64-bit block by
zeros, is added in CM5 modulo 2 with the filling N1, N2 (a1(M-1)[16],
a2(M-1)[16], ..., a32(M-1)[16], b1(M-1)[16], b2(M-1)[16], ...,
The result of the addition
(a1(M-1)[16](+)t1(M), a2(M-1)[16](+)t2(M), ..., a32(M-1)[16](+)
t32(M), b1(M-1)[16](+)t33(M), b2(M-1)[16](+)t34(M), ...,
(a1(M)[0], a2(M)[0] ..., a32(M)[0], b1(M)[0], b2(M)[0], ...,
is written into N1, N2 and encrypted in the electronic codebook mode
after the first 16 rounds of the algorithm's work. Out of the
resulting filling of the registers N1 and N2:
Dolmatov Informational [Page 16]
RFC 5830 GOST 28147-89 March 2010
(a1(M)[16], a2(M)[16] ..., a32(M)[16], b1(M)[16], b2(M)[16], ...,
an l-bit string I(l) (the MAC) is chosen:
I(l) = [a(32-l+1)(M)[16], a(32-l+2)(M)[16], ..., a32(M)[16]].
The MAC I(l) is transmitted through the telecommunication channel or
to the computer memory attached to the end of the encrypted data,
i.e., Tc(1), Tc(2), ..., Tc(M), I(l).
The encrypted data Tc(1), Tc(2), ..., Tc(M), when arriving, are
decrypted, out of the resulting plain text blocks Tp(1), Tp(2), ...,
Tp(M). The MAC I'(l) is generated as described in the subsection 5.3
and compared with the MAC I(l) received together with the encrypted
data from the telecommunication channel or from the computer memory.
If the MACs are not equal, the resulting plain text blocks Tp(1),
Tp(2), ..., Tp(M) are considered false.
The MAC I(l) (I'(l)) can be generated either before encryption (after
decryption, respectively) of the whole message or simultaneously with
the encryption (decryption) in blocks. The first plain text blocks,
used in the MAC generation, can contain service information (the
address section, a time mark, the initialisation vector, etc.) and
they may be unencrypted.
The parameter l value (the bit length of the MAC) is defined by the
actual cryptographic requirements, while considering that the
possibility of imposing false data is 2^-l.
9. Security Considerations
This entire document is about security considerations.
10. Normative References
[GOST28147-89] "Cryptographic Protection for Data Processing System",
GOST 28147-89, Gosudarstvennyi Standard of USSR,
Government Committee of the USSR for Standards, 1989.
(In Russian)
[RFC4357] Popov, V., Kurepkin, I., and S. Leontiev, "Additional
Cryptographic Algorithms for Use with GOST 28147-89,
GOST R 34.10-94, GOST R 34.10-2001, and GOST R
34.11-94 Algorithms", RFC 4357, January 2006.
Dolmatov Informational [Page 17]
RFC 5830 GOST 28147-89 March 2010
Appendix A. Values of the Constants C1 and C2
The constant C1 is:
The bit of N6 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18
The bit value 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
The bit of N6 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
The bit value 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0
The constant C2 is:
The bit of N6 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18
The bit value 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
The bit of N6 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
The bit value 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
Dolmatov Informational [Page 18]
RFC 5830 GOST 28147-89 March 2010
Appendix B. Contributors
Dmitry Kabelev
Cryptocom, Ltd.
14 Kedrova St., Bldg. 2
Moscow, 117218
Russian Federation
EMail: kdb@cryptocom.ru
Igor Ustinov
Cryptocom, Ltd.
14 Kedrova St., Bldg. 2
Moscow, 117218
Russian Federation
EMail: igus@cryptocom.ru
Irene Emelianova
Cryptocom Ltd.
14 Kedrova St., Bldg. 2
Moscow, 117218
Russian Federation
EMail: irene@cryptocom.ru
Author's Address
Vasily Dolmatov, Ed.
Cryptocom, Ltd.
14 Kedrova St., Bldg. 2
Moscow, 117218
Russian Federation
EMail: dol@cryptocom.ru
Dolmatov Informational [Page 19] | {"url":"https://datatracker.ietf.org/doc/rfc5830/","timestamp":"2024-11-07T03:43:22Z","content_type":"text/html","content_length":"68035","record_id":"<urn:uuid:144f79e0-ef76-45f9-afa5-822ffe2b9bb5>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00706.warc.gz"} |
How to Easily Expose NetSuite Unapplied Customer Deposit or Payment Amount in a Saved Search. - NetSuite InsightsHow to Easily Expose NetSuite Unapplied Customer Deposit or Payment Amount in a Saved Search.
How to Easily Expose NetSuite Unapplied Customer Deposit or Payment Amount in a Saved Search.
6 min read
In this article, I explain two ways you can create a single saved search that exposes the NetSuite unapplied Customer Deposit or Customer Payment alongside the applied amount.
By following the steps outlined in this article, you will produce saved search output similar to the one illustrated below.
While working with a client recently, he asked me if it was possible to produce a control search that captures all payments and deposits with unapplied balances. In my client's business, deposits are
created when money is received against a sales order that is pending fulfillment. Typically, these deposits are automatically generated as prepaid orders from their eCommerce frontend are synced to
NetSuite. Upon fulfilling the sales order for which the funds were captured, the customer deposit is used to pay the invoice and recognize revenue accordingly. Using customer deposits represents a
superior accounting practice as opposed to using cash sale transactions for this use case.
Thus, for this client, unapplied deposit balances were an indication of an anomaly that needed to be investigated and fixed. In some cases, taxes were off due to issues with the tax computation in
the front end, orders had changed and a refund was pending, etc. Understandably, reviewing all deposits/payments and/or performing lookups in Excel to determine which ones required attention was
undesirable. My client wanted to "manage by exceptions" i.e. only review transactions with unapplied balances. The reasoning is similar for customer (pre-)payments resulting from incoming wire
Based on prior experience and results of Google searches on the subject, I was inclined to believe that it was impossible to capture this information in a single saved search. The basic limitation of
conventional approaches is that the applied amount is available at the line level. Thus, to compute the unapplied amount on a given deposit or payment, one would somehow need to sum the applied
amount across all lines and subtract it from the header-level total amount. The latter step of subtraction is where the challenge lies.
While SuiteQL would likely produce the desired result using inner queries or similar strategies, the client required that the results be automatically emailed to individuals responsible for reviewing
and handling these exceptions. At Prolecto, we have developed a Query Renderer tool that has the requisite email capability. Thus, I had a fallback. However, before going down the SuiteQL route, I
stumbled upon an article that offered a breakthrough!
Breakthrough With Exposing the NetSuite Unapplied Customer Deposit Amount
SuiteAnswer ID: 22423^[I]NetSuite (January 23, 2023). Transaction Saved Search > Show the Amount of Customer Deposit, Applied and Unapplied Amounts Sorted by Customer Name. Available at: … Continue
reading offered the basic pattern for customer deposits, which I extended to cover customer payments as well.
Thanks to an amazing pointer from one of our regular readers, Lee Hopper, I learned about a simpler solutions. Many thanks to Lee for this tip!
Next, I'll explain the two solutions, beginning with a simpler one.
Solution 1: Easy Approach
This solution is simple, elegant and gets the job done without complicated formulas.
We create a Transaction saved search (Lists > Search > Saved Searches > New > Transaction). The search criteria are pretty basic: Add filters Type is any of Payment, Customer Deposit and Main Line is
false as illustrated below. If necessary, add other criteria e.g. to limit the results to a particular subsidiary or GL accounts. In case of a GL account filter, be sure to use the "Account (Main)"
field rather than the "Account" field as the bank account will be on the header (i.e. main) line.
In the Results subtab, enter the columns illustrated below:
The formulas should be pretty straightforward to understand.
• The key is the amountpaid search result field amountpaid which captures the sum of the paid amounts and is available on each line - exactly what we need! Interestingly, this field is always zero
at header-level.
• The unapplied amount is thus the difference between amount and amountpaid. We take the absolute value of the amount as it is negative on Payment transaction lines.
That's it!
Note that there is also a paidamount field which captures the amount paid per line. This could be interesting as well. Note though that it is only available on Payments, not Customer Deposits. Here
are the results of the search to help clarify the difference between amountpaid and paidamount.
Pro Tip: If you'd rather not have a summary search, all you need to do add search criterion "Line Sequence Number is 1". This is guaranteed to always look at the first non-header transaction
line. See related article "Understand Line ID vs. Line Sequence Number in NetSuite Transactions" if you do not understand why this works.
Solution 2: Advanced Approach
Next, I'll present the original solution that I came up with. As you will see, it is complex and hard to understand. Thus, I recommend the previous solution. However, I've left this one in as I
suspect there are other use cases where this pattern will be useful (I recently worked on an item search where I leveraged the same technique).
Creating the Saved Search
Again, we create a Transaction saved search (Lists > Search > Saved Searches > New > Transaction). The search criteria are identical to those in solution 1 above. Thus, we focus on the results. In
the Results subtab, enter the columns illustrated below. I will focus on explaining the formula fields as the others should be self-explanatory.
1. Field: Formula (Text)
□ Summary Type = Group
□ Formula: '<a href="/app/accounting/transactions/transaction.nl?id='||{internalid}||'">'||{tranid}||'</a>'
□ Summary Label = View
□ This adds a convenient link to open the transaction without having to first drill down which would be the case if we added the Document Number field directly. It is an optional but handy
trick to eliminate one extra click.
2. Field: Amount
□ Summary Type = Maximum
□ Function = Absolute Value
□ Summary Label = Total Deposit / Payment Amount
□ This captures the total deposit or payment amount. We take the absolute value as the value is negative on payment transactions. Note that we use the Maximum summary type to ensure that the
column is visible at the summary search level. Minimum or Average will also work as the (header) amount is the same on all lines. This observation also applies to the following formula fields
3. Field: Formula (Currency)
□ Summary Type = Minimum
□ Formula = abs(sum(NVL(DECODE({typecode}, 'CustDep', {applyingtransaction.amount}, {appliedtolinkamount}),0)))
□ Custom Label = Summary Label = Applied Amount
□ This formula sums the total applied amount. I will explain this formula in more detail in the next section.
4. Field: Formula (Currency)
□ Summary Type = Minimum
□ Formula = max(abs({amount}))-abs(sum(NVL(DECODE({typecode}, 'CustDep', {applyingtransaction.amount}, {appliedtolinkamount}),0)))
□ Custom Label = Summary Label = Unapplied Amount
□ This formula captures the total unapplied amount. We will explore how it works shortly.
If you've followed the above steps accurately, you should now have a search that produces the desired output. Congratulations! But before you jump to the next challenge, hang on a bit, and let's try
to understand what is happening here.
Decoding the Saved Search
Let's zoom in on the last two formulas. To help you understand, I've placed images of the line details of the four transactions in our example next to each other. Click on the image to enlarge it if
Formula for Applied Amount
Observe the following:
• Unapplied deposits and payments have a non-header line albeit with zeroes or blanks in the fields on interest (recall that our search was filtered to mainline = false so every line in the output
is a non-header line).
• We are deliberately using {appliedtotransaction.amount} instead of {appliedtolinkamount} on the one hand and {applyinglinkamount} instead of {applyingtransaction.amount} on the other hand for
reasons explained below.
□ For payments, the applied to transaction is what we care about. However, whereas Applied To Transaction : Amount ({appliedtotransaction.amount}) gives the total amount of the applied to
transaction, we need to use the Applied to Link Amount ({appliedtolinkamount}) as it captures the exact amount that was applied. For example, in line 3 of PYMTH00003429, only $50 was applied
to the JE and that is the amount we need instead of the total JE amount of $100!
□ For deposits, the applying transaction is the deposit application and the Applying Transaction : Amount ({applyingtransaction.amount}) captures what was applied to the target invoice, JE,
etc. In our example, we deliberately used a deposit that was partially applied to confirm. Notice in the screenshot of the deposit application below that the $113 corresponds to the total
amount applied to the two invoices which is less than the total original amount ($170.73) and the total amount due ($136). Thus, we confirm that we are capturing exactly what was applied
using this field. Moreover, the Applying Link Amount ({applyinglinkamount}) is blank for unclear reasons.
To summarize, the formula abs(sum(NVL(DECODE({typecode}, 'CustDep', {applyingtransaction.amount}, {appliedtolinkamount}),0))) sums the total applied amounts, taking empty values (i.e. unapplied
payments) as well as nuances between the various amount fields into account. The abs() function is necessary as the applying transaction amount is negative.
Formula for Unapplied Amount
This formula for the unapplied amount is the essence of the solution. Pay close attention. It reuses the previous formula and can be expressed conceptually as max(abs({amount})) -
This formula is challenging to understand, at least for me. It produces the desired results by somehow computing the total applied amount across all lines and subtracting it from the total deposit/
payment amount! Notice that the max function only applies to the total deposit/payment amount and not the entire expression. It is not very intuitive to have a max here because the {amount} is the
same across all lines as we saw in the screenshot above. Moreover, I observed from experimenting that the max is somewhat of a placeholder; min or avg also produce identical results. It thus appears
that max is simply used to trick NetSuite to bypass the normal formula restrictions and allow us to access powerful aggregation/analytics functionality. A similar pattern is used to access Oracle's
analytics functions as Marty Zigman explains here. This might be yet another instance of the same trick. By the way, removing the max breaks the search: "An unexpected error has occurred. Please
click here to notify support and provide your contact information." It remains a bit mysterious to me and I would be glad if a more knowledgeable reader can explain precisely what is happening here.
Someday, I am sure I will better understand the advanced solution. In the meantime, I am glad that Lee led us to a simpler approach. Never say never!
Finally, while researching, I came across SuiteAnswer ID: 65454 which explains how to create a saved search that shows the total applied and unapplied amount on all customer deposits for each
customer. It approaches the problem using a customer search rather than a transaction search. I assume that we will achieve identical results using the above transaction search by stripping all
Summary Type = group fields except the Name (i.e. customer) field. The interested reader is invited to explore this further and share their findings.
I hope you have found this article insightful. If so, let me know by dropping a comment. Keep an open mind, keep learning, and keep sharing.
NetSuite Insights is on a mission to raise the standards around NetSuite practices, one insight at a time. If that resonates with you, learn how you can become an author/collaborator here.
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12 Replies to “How to Easily Expose NetSuite Unapplied Customer Deposit or Payment Amount in a Saved Search.”
1. This is exactly what I needed, thank you for the solution
1. Great to hear that. You’re most welcome!
2. Hi Chidi, Thanks for sharing this amazing solution.
Just want to know if there is any way that we can export those unapplied customer deposits, that their status was unapplied before a specific date.
For example, find those customer deposits were Unapplied before 01/02/2024 (However, their current status is applied).
I would really appreciate it if you could help.
1. Hi Reza, perhaps you can join to the System Notes to filter on the status change/date criteria you mentioned?
3. Hi Chidi, thank you – this is very helpful, I could find unapplied Customer Deposit amount. I was wondering if you know how I could tweak this to find unapplied Credit Note amount? Cheers
1. Hi, I don’t have a definite answer but I suspect if you follow the pattern explained in this article, you can get answers. If you succeed, please share your findings!
4. Brilliant! This is very useful.
1. You’re welcome, Mike. Please review the revised article as I’ve updated it with an even more elegant solution.
5. Thanks for the blog post.
I created a solution for this a few years ago as I had a similar issue. My solution was a little simpler, so I am wondering if we have the same outcome.
My criteria is as follows:
Type is Customer Deposit
Account is [select the account used for Customer Deposits]
Formula (Numeric) is greater than 0: {amount}-{amountpaid}
I then have the relevant columns including:
Amount (Custom label: Deposit Amount)
Amount Paid (Custom label: Amount Applied)
Formula (Numeric): {amount}-{amountpaid}
I have a separate Saved Search for Open Payments, but you could probably combine the 2 with “CASE”:
Type is Payment
Account is [select the account used for payments – in our case Accounts Receivable – Debtors]
Formula (Numeric) is not 0: {amount}+{amountpaid}
I then have the relevant columns including:
Amount (Custom label: Payment Amount)
Amount Paid (Custom label: Amount Applied)
Formula (Numeric): {amount}+{amountpaid}
1. Thanks for sharing your solution, Lee. I will look into it and share my findings. If it produces the same results that would be a great simplification!
1. Lee, I just tested your solution and it works. Way more elegant. I’ve updated the article with attribution. Thanks for sharing. Consider becoming a NetSuite Insights’ contributor to share
more of your valuable insights with the community!
1. You’re welcome. Keep up the good work! | {"url":"https://netsuite.smash-ict.com/how-to-expose-netsuite-unapplied-customer-deposit/","timestamp":"2024-11-05T19:16:18Z","content_type":"text/html","content_length":"409154","record_id":"<urn:uuid:563ed192-54f4-497c-9709-d2c0c49b571c>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00307.warc.gz"} |
Chernova P.D., Goloveshkin V.A., Myagkov N.N.
Model of interaction of a rigid mesh with a deformable target
An analytical model of the high-velocity interaction of a rigid mesh with a semi-infinite deformable target, which is modeled by a rigid-plastic body, is proposed. We consider the so-called “normal”
impact of the mesh on the target: we assume that at the initial moment and subsequent moments of time the mesh is parallel to the target surface, and the mesh velocity vector is perpendicular to the
target surface. The model reproduces the most interesting case when the mesh aperture is comparable to or less than the diameter of the wire from which the mesh is woven. The dependence of the mesh
penetration depth on the impact velocity and the geometric parameters of the mesh, which are characterized by one dimensionless parameter equal to the ratio of the wire diameter to the mesh period,
is studied. Two versions of the model are considered: with and without taking into accounts the fragmentation of the ejected material of the target. The results obtained on the basis of the proposed
model are compared with the numerical solutions based on the complete system of equations of the deformable solid mechanics. Numerical simulations were performed using the LS-DYNA package. The
example of the penetration of a steel mesh into an aluminum-alloy target with impact velocities of 1-3 km/s is analyzed. It is shown that the model taking into account fragmentation agrees well with
the numerical simulations for the mesh parameter interval , which the lower boundary decreases with increasing impact velocity: for km/s, respectively.
Pages: 3-23 Elibrary
Yankovskii A.P.
Refined model of viscoelastic-plastic deformation of flexible shallow shells with spatial reinforcement structures
A model of viscoelastic-plastic deformation of spatially reinforced flexible shallow shells is developed. The instant elastoplastic behavior of the components of the composition is determined by the
theory of plastic flow with isotropic hardening. The viscoelastic deformation of these materials is described by the equations of the Maxwell – Boltzmann model. The geometric nonlinearity of the
problem is taken into account in the Karman approximation. The obtained relations make it possible to determine with varying degrees of accuracy the displacements of shell points and the
stress-strain state in the components of the composition (including residual ones). In this case, the weakened resistance of the composite structure to transverse shear is modeled. In a first
approximation, the obtained equations and boundary conditions correspond to the traditional non-classical Reddy theory. The solution of the formulated problem is constructed numerically using an
explicit «cross» type scheme. The viscoelastic-plastic dynamic behavior of composite cylindrical rectangular panels under the action of a load generated by an air blast wave is investigated. Designs
have a «flat»-cross or spatial reinforcement structure. It has been demonstrated that in some cases, even for relatively thin composite curved panels, the Reddy theory is unacceptable for adequate
calculations of their dynamic viscoelastic-plastic deformation. It is shown that the size and shape of the residual deflections of the reinforced gently shallow shells substantially depend on which
of their front surfaces (concave or convex) is subjected to an external load. It was found that in both cases of loading residual longitudinal folds are formed in a thin cylindrical shallow composite
shell. It has been demonstrated that even for a relatively thin panel, replacing a «flat»-cross reinforcement structure with a spatial reinforcement structure can significantly reduce the amount of
residual deflection and the intensity of residual strain in the binder. In the case of relatively thick shallow shells, the effect of such a replacement of the reinforcement structures is manifested
even more.
Pages: 24-42 Elibrary
Grishanina T.V., Rybkina N.M.
To the calculation of a straight high aspect-ratio wing in an incompressible flow using a nonstationary aerodynamic theory
Bending-torsional vibrations of a straight high aspect-ratio wing in an incompressible flow of an ideal gas are considered. Linear aerodynamic loads (lift and torque) are determined by non-stationary
and quasi-stationary theories of cross sections flat flow. The displacements and twisting angles of the wing console cross sections during bending-torsional vibrations are represented by the Ritz
method as an expansion for given functions with unknown coefficients, which are considered as generalized coordinates. The equations of aeroelastic wing oscillations are composed as Lagrange
equations and written in matrix form as first-order differential equations. The problem of determining eigenvalues is solved on the basis of the obtained equations. The main purpose of this work is
to compare calculations for determining the dynamic stability boundary (flutter) obtained using non-stationary and quasi-stationary aerodynamic theories. Calculations are performed for a wing model
with constant cross-section characteristics. As the set functions eigenmodes of bending and torsional vibrations of a cantilever beam of constant cross-section were used. Calculations are performed
to determine the flutter boundary for a different number of approximating functions. The results obtained allow us to conclude that when using the quasi-stationary and refined quasi-stationary
theories for determining aerodynamic loads, the values of the critical flutter velocity are obtained less than when calculating using the non-stationary theory. This makes it possible to use a
simpler (from the point of view of labor intensity) quasi-stationary theory to determine flutter boundaries. It is also found that the influence of the attached air masses, which is taken into
account in the non-stationary and refined quasi-stationary theories, is very small.
Pages: 43-57 Elibrary
Starovoitov E.I., Zakharchuk Yu.V.
The physically nonlinear deformation of circular sandwich plates with a compressible filler
Three-layer structural elements are used in aerospace and transport engineering, construction, production and transportation of hydrocarbons. The theory of deformation of three-layer plates with
incompressible fillers is currently developed under external force influences quite well. Here is the formulation of the boundary value problem on the bending of an elastoplastic circular three-layer
plate with a compressible filler. For thin bearing layers, the Kirchhoff hypothesis is accepted. In a relatively thick lightweight filler, the hypothesis of Tymoshenko is performed with a linear
approximation of radial displacements and deflection along the layer thickness. The work of shear stresses and compression stresses is assumed to be small and is not taken into account. The contour
is assumed to have a rigid diaphragm that prevents the relative shift of the layers. The physical equations of state in the bearing layers correspond to the theory of small elastic-plastic
deformations of Ilyushin. The filler is nonlinear elastic. The inhomogeneous system of ordinary nonlinear differential equations of equilibrium is obtained by the Lagrange variational method.
Boundary conditions are formulated. The solution of the boundary value problem is reduced to finding the four desired functions – the deflection of the lower layer; shear, radial displacement and
compression function in the filler. The method of successive approximations based on the method of elastic solutions is applied for the solution. The General iterative analytical solution of the
boundary value problem in Bessel functions is obtained. Its parametric analysis is carried out at uniformly distributed load and rigid sealing of the plate contour. The influence of the
compressibility of the filler on the stress-strain state of the plate is numerically investigated. The comparison of the calculated deflection values obtained by the traditional model with
incompressible filler and in the case of its compression is given.
Pages: 58-73 Elibrary
Bobok D.I.
Analytical solution of the problem of bending of a round plate made of shape memory alloy
In this paper, we consider the problem of the solids mechanics about the bending of a circular plate made of a shape-memory alloy (SMA) during a direct thermoelastic martensitic phase transformation
under the action of a constant in magnitude and uniformly distributed transverse load radius. The problem of relaxation in a similar plate during direct phase transformation has also been solved. As
the second problem, a normal load uniformly distributed over the radius is applied to the plate surface in the austenitic phase state. Next, the plate material is cooled through the temperature range
of direct thermoelastic martensitic transformation. It is required to determine how the uniformly distributed load should decrease during such a transition so that the deflection of the plate remains
unchanged. During the work, rigidly and articulated plates were investigated. The solution was obtained in the framework of the Kirchhoff-Love hypotheses. To describe the behavior of the plate
material, we used the well-known model of linear deformation of SMA during phase transformations. The solution was obtained under the assumption that the phase composition parameter at each moment of
the process under consideration is uniformly distributed over the plate material, which corresponds to non-coupled statement of the problem for the case of uniform distribution of temperature over
the material. The possibility of structural transformation in the plate material is not taken into account. It neglects the variability of the elastic moduli during the phase transition, as well as
the property of the SMA diversity resistance. To obtain an analytical solution to all the equations of the boundary value problem, the Laplace transform method with respect to martensite volume
fraction parameter was used. After transformation in the space of images, an equivalent elastic problem is obtained. As a result of solving the equivalent elastic problem, the Laplace images of the
desired quantities are obtained in the form of analytical expressions, which include operators that are Laplace images of elastic constants. These expressions are fractional rational functions of the
Laplace image of the phase composition parameter. After returning to the original space, which is carried out analytically by decomposing the expressions for the desired quantities in the image space
into simple factors, the desired analytical solutions are obtained.
Pages: 74-97 Elibrary
Firsanov Vic.V.
Computational models of beam bending taking into account shear deformation
The classical model of beam bending is based on Bernoulli’s hypotheses: there is no transverse linear strain, no shear strain in the plane, where is the longitudinal and is the transverse coordinates
of the beam, and no transverse normal stress. At the same time, both the transverse normal and tangent stresses are preserved in the equilibrium equations, since without them the problem of bending
the beam has no solution. Implementation of the corresponding physical relations is neglected. For isotropic and orthotropic linear elastic materials, the shear strain is determined by dividing the
tangent stress by the shear modulus. The larger the shear modulus, for example, compared to the elastic modulus in tension and bending, the closer we are to the hypothesis of no shear deformations,
and Vice versa, the smaller the shear modulus, the more problematic the use of this hypothesis. This is especially true for the problem of bending orthotropic plates that are not reinforced in the
transverse direction. Then the shear modules in the transverse direction are mainly determined by the properties of the weak binder and can be significantly less than the physical characteristics of
an orthotropic package with planar reinforcement. In a beam, the reinforcement is carried out in the plane, and if the beam cannot be reinforced in the transverse direction due to too small a normal
transverse stress, then a small number of layers at angles must be applied, since the bent beam also works for shear. Therefore, the shear modulus is determined not only by the binder, but also by
the reinforcing fibers, and can be commensurate with the elastic modulus, and be several times smaller, depending on the number of reinforcing fibers. The aim of the work is to assess the effect of
shear deformation on the stress-strain state of the beam.
Pages: 98-107 Elibrary
Saganov E.B., Sharunov A.V.
Solution of problem on sphere of alloy with memory of shape under action of constant pressure, taking into account divergence of material
In the work, a numerical solution of the problem on the stress-strain state (SSS) of a thick-walled sphere made of a shape memory alloy (SMA), which is under the influence of constant internal or
external pressure in the mode of martensitic inelasticity (MI) taking into account elastic deformations and the property of material tension-compression asymmetry. Under the property,
tension-compression asymmetry refers to the dependence of the material constants of these alloys on the type parameter of the state of stress. The parameter associated with the third invariant of the
stress deviator is used as a parameter of the type of stress state. The solution was obtained on the basis of the model of nonlinear deformation of SMA during phase and structural transformations.
When solving the problem without taking into account elastic deformations, the provision on active processes of proportional loading is used. In the framework of the deformation process under
consideration, the influence of the SMA diversity resistance as well as elastic deformations on the distribution of radial and ring stresses in the sphere cross section is demonstrated. It has been
established that the distribution of radial and circular stresses over the sphere cross section is nonlinear, and the stresses themselves can vary nonmonotonously during loading. In the course of
work, the module of the finite element complex Simulia Abaqus was verified, which was developed for the analysis of the SSS of structures from SMA in the MI mode. As a verification basis, the
obtained numerical solution of the spatial three-dimensional boundary-value problem of SSS of a thick-walled spherical shell made of SMA under the loading of internal or external pressure, taking
into account the different resistance of these alloys, was used. The obtained numerical solution converges to the analytical solution of the corresponding problem without taking into account elastic
deformations with increasing of Young’s modulus.
Pages: 108-121 Elibrary
Artamonova N.B., Sheshenin S.V.
Coupling consolidation problem in a nonlinear formulation. theory and method of solution
The consolidation problems are related to the study of soil deformation under load in the presence of fluid outflow. In the process of joint deformation of the porous skeleton and the fluid contained
in the pores, the solid and liquid phases of the soil interact. The filtration processes in the soil mass are described by a coupling system of differential equations with rapidly oscillating
coefficients. To solve such equations, averaging over the representative volume element (RVE) is used. In the paper, the equations of the nonlinear consolidation model are written from the general
laws of conservation of continuum mechanics (the equilibrium equation, the law of mass conservation of solid and liquid phases of the soil, and Darcy’s filtration law) using spatial averaging over
the representative volume element. The following assumptions were made: the fluid fills the pores entirely, the fluid is Newtonian and homogeneous, the deformation of the fluid with a change in pore
pressure obeys the law of barotropy, and the soil skeleton material is incompressible. To determine effective properties, an approach based on solving local problems in a representative volume
element is possible. As a result, a coupled physically and geometrically nonlinear formulation of the boundary value problem was obtained using the Lagrange approach with adaptation for the solid
phase and the ALE (Arbitrary Lagrangian-Eulerian) approach for the fluid under the assumption of quasistatic deformation of the rock skeleton. In the method of solving the coupled problem,
linearization of variational equations is carried out in combination with internal iterations according to the Uzawa method for connecting at each time step. For spatial discretization, the finite
element method is used: trilinear type elements for approximating the filtration equation and quadratic elements for approximating the equilibrium equation. An implicit time scheme can be used to
take into account the inertia forces.
Pages: 122-138 Elibrary
Russkikh S.V., Shklyarchuk F.N.
Numerical solution of nonlinear motion equations of compound elastic systems with joins
The unsteady motion of two elastic systems described by nonlinear differential equations in generalized coordinates is considered. It is believed that in the initial state or during the
transformation process, these two systems are connected to each other in a finite number of points by elastic or geometric holonomic bonds. Based on the principle of virtual displacements
(D’Alembert-Lagrange), the equations of motion of the composite system in the same generalized coordinates are obtained taking into account the constraints. In this case, elastic bonds are taken into
account by adding the potential energy of deformation of the connecting elements, which is expressed using the conditions of the connection through the generalized coordinates of the two systems.
Geometric bonds are taken into account in the variational equation by adding variations to the work of unknown reactions of bond retention at their small possible changes and are expressed through
variations of the generalized coordinates of the systems under consideration. From this extended variational equation, equations of the composite system are obtained, to which algebraic equations of
geometric relationships are added. This approach is equivalent to the approach of obtaining equations in generalized coordinates with indefinite Lagrange multipliers representing reactions in bonds.
As an example, we consider a system consisting of a bending elastic, inextensible cantilever beam that performs non-linear quadratic longitudinal-transverse vibrations, at the end of which a heavy
rigid body is pivotally connected, which rotates through a finite angle. The beam bending is represented by the Ritz method by two generalized coordinates. Two linear constraints on the displacements
of the beam and the body in the hinge are satisfied exactly, and the third nonlinear constraint, representing the condition of inextensibility of the beam, is added to the equations of motion of the
system, including an unknown reaction to hold this constraint. Numerical solutions of the initial problem of forced nonlinear vibrations of a beam with an attached body are obtained in two versions
with comparisons: 1) the nonlinear coupling is satisfied analytically accurately, and the unknown reaction is excluded from the vibration equations; 2) the connection is differentiated in time and is
satisfied by numerical integration together with the differential equations of motion of the system.
Pages: 139-150 Elibrary | {"url":"https://iampress.ru/en/n1-2020/","timestamp":"2024-11-09T04:26:46Z","content_type":"text/html","content_length":"83584","record_id":"<urn:uuid:42a30db6-ec6c-4cdd-a601-24ae616a04d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00779.warc.gz"} |
Contour plots for XYZ interpolation
SharpPlot Tutorials > Chart Samples > Contour plots for XYZ interpolation
Contour plots for XYZ interpolation
Contour plots offer a 2-dimensional alternative to CloudCharts when it is necessary to visualize ‘height’ data in addition to x and y values. The data may be geographic (rainfall totals from a number
of weather stations) or simply a model of 3 variables (house prices as a function of the age of the property and its floor area).
Depending on the type of data, SharpPlot can fit a variety of models to generate the contours for the height dimension. These four examples show how very similar numbers can create very different
maps depending on the chosen approach. The key properties are the order of fit (which creates an underlying model) and flexibility (which determines how far the computed surface can deviate from the
Note that the ContourPlot is just an extension of the simple ScatterPlot, and shares many of its style settings.
A Simple 2-variable Regression Surface
This example fits the same Quadratic surface as the third CloudChart tutorial. You can see that several of the points fall on the ‘wrong’ side of the line, which is very reasonable for a noisy
dataset where the Z-values may be subject to a large random error.
sp.Heading = "Modelled Surface using Contours";
zdata = new int[] {12,65,77,117,9,112};
xdata = new int[] {17,31,29,21,30,24};
ydata = new int[] {190,270,310,300,190,230};
sp.Flexibility = 0;
sp.EquationFormat = "z = C0 + C1x + C2x² + C3y";
sp.ContourPlotStyle = ContourPlotStyles.ValueTags|ContourPlotStyles.ExplodeAxes|
The Cloudchart is probably a better tool for an initial visualisation, but the Contourplot is much more suitable if you want to answer questions like “what is the best estimate for z, given x and y”
as you can easily read off he numbers.
An Approximate Trend Surface
The remaining examples all use the same set of data-points as the final example in the Cloudchart tutorial. The same data-set can produce very different ‘landscapes’ depending on the model chosen.
zdata = new int[] {100,15,27,117,19,112};
xdata = new int[] {17,31,29,21,30,24};
ydata = new int[] {190,270,310,300,190,230};
sp.Heading = "Trend Surface (Rough fit)";
sp.ContourPlotStyle = ContourPlotStyles.ValueTags|ContourPlotStyles.ExplodeAxes|
sp.Flexibility = 5;
sp.MeshDensity = 2;
The first surface shows the effect of setting the flexibility quite low. Each computed point on the xy grid then ‘sees’ many of the nearby points and the effect is create a quite smooth (but strongly
averaged) surface. This would be a suitable model if the data were known to be noisy, and a rough feel for the shape of the surface was all that was required.
Fitting an Accurate set of Spot-heights
If the z-values really represent accurately measured values (spot heights in a landscape) then the map should be forced to fit itself around them.
zdata = new int[] {100,15,27,117,19,112};
xdata = new int[] {17,31,29,21,30,24};
ydata = new int[] {190,270,310,300,190,230};
sp.Heading = "Trend Surface (Close Fit)";
sp.SetAltitudeColors(new Color[]{Color.Navy,Color.Green,Color.GreenYellow,
sp.ContourPlotStyle = ContourPlotStyles.ValueTags|ContourPlotStyles.ExplodeAxes|
sp.Flexibility = 8;
sp.MeshDensity = 5;
This example increases the mesh-density (to compute the contours at many more points) and sets the flexibility high enough to force the contours to behave correctly with respect to the points nearby.
Whether this map is any better than any other is (of course) arguable. It is an interesting exercise to take a set of points like this and attempt to make the map by hand. Altitude shading has been
used with ‘realistic’ colouring to give the effect of an aerial view of a landscape.
Fitting a Flexible Cubic Model
The final example generates the most ‘satisfying’ map, from a purely visual point of view. This allows SharpPlot to fit a cubic regression surface in the x-direction, then apply a little flexibility
to this to finalise the shape of the surface. No underlying model is assumed in the y-direction.
zdata = new int[] {100,15,27,117,19,112};
xdata = new int[] {17,31,29,21,30,24};
ydata = new int[] {190,270,310,300,190,230};
sp.Heading = "Cubic Model with False Colours";
sp.SetAltitudeColors(new Color[]{Color.SlateBlue,Color.Navy,Color.Green,Color.Red,
sp.ContourPlotStyle = (ContourPlotStyles.ValueTags|ContourPlotStyles.ExplodeAxes|
sp.Flexibility = 8;
sp.MeshDensity = 3;
This combination of modelfit and flexibility is a good approach when you know that the z-value is composed of several effects, some of which are expected to obey a known model, but some of which are
effectively ‘random’ values. Altitude shading can be used with an array of suitable color values to help bring out the shape of the final surface.
Contour plots can be an excellent way to display 3-dimensional data on a 2-dimensional chart, but creating the ‘best’ surface for any given dataset will require some prior knowledge of the underlying
model, and a certain amount of experimentation.
Send comments on this topic
© Dyalog Ltd 2021 | {"url":"https://www.sharpplot.com/ContourPlots.htm","timestamp":"2024-11-11T01:24:01Z","content_type":"text/html","content_length":"14037","record_id":"<urn:uuid:707f51c8-ad1d-44b8-aaba-5f974366cb5a>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00447.warc.gz"} |
Find elements that appear more than N/3 times
Open-Source Internship opportunity by OpenGenus for programmers. Apply now.
In this article, we have solved the problem of finding elements that appear more than N/3 times. This involve ideas of Hash Map and Boyer Moore algorithm.
Let's look at the problem first and understand what we have to do here.
There are $N$ numbers of elements in an array, and we have to find the element or elements that may occur more than $\frac{N}{3}$ times in the array.
For example, if we have the following list:
[1, 1, 1, 1, 2]
you can notice that obviously 1 is the element that satisfies the condition of having count more than $N/3$.
Now, let's look at this problem in a little different way, that is if we make a list that may have only three different elements, then we can see a few specific cases it may create.
• Case-I: Perfect 3 Elements
In this case, we'll get the only empty list in return because there are 3 elements and every element has the same number of occurrences.
For example, in the below list $N/3$ is 3 and every element has occurred only 3 also note this happened because they all are sharing equal space.
[1, 1, 1, 2, 2, 2, 3, 3, 3]
• Case-II: Result gives us one element
If an element occurs 4 times in the list of 9 then it has surely taken space of other elements.
[1, 1, 1, 1, 2, 2, 3, 3, 3]
• Case-III: Result gives us two elements
Or else, there could be two elements that may take space from other remaining space.
[1, 1, 1, 1, 2, 3, 3, 3, 3]
To conclude what we just saw above, we can see that we're looking for an element that takes up more than $\frac{1}{3}$^rd of the space of total space.
Therefore, no matter how much big the space is we can only have two elements which will have a greater count than other ( if available ) third or many other elements.
How to solve
One could use many different ways to solve this,
• Naive Approach
One way could be to look at every element then count it and then check if the element satisfies the check and then add it to another array and then return that "another array" as a result.
This way we will have
Time Complexity of $O(N^2)$
Space Complexity of $O(1)$.
An example, code.
example_list = [1, 2, 2, 2, 4, 5, 1, 1, 1, 1, 2]
## This is a solution one may use when they don't know of
## HashMap and may have learned bubble sort before.
def more_than_N_by_3(l):
length = len(l)
nums = []
for i in range(length):
n = l[i]
count = 0
for j in range(length):
if l[j] == n:
count += 1
if count > length//3 and l[i] not in nums:
return nums
Output: [1, 2]
• using HashMap
The second way could be to use the smart method of counting and keeping elements in a list, that is to use the HashMap to store the element and counts and then at last look at every count and
return the ones that have a higher count than N/3.
using HashMap, cause to have
Space Complexity of $O(N)$
Time Complexity of $O(N)$
An example, code.
example_list = [1, 2, 2, 2, 4, 5, 1, 1, 1, 1, 2]
def count(l):
counts = {}
for i in l:
if counts.get(i) is not None:
counts[i] += 1
counts[i] = 1
return counts
def more_than_N_by_3(counts, N):
result = []
for key in counts.keys():
if counts[key] > N//3:
return result
l = count(example_list)
print(more_than_N_by_3(l, len(example_list)))
Output: [1, 2]
• Boyer Moore's vote Algorithm
Let's first look at how we will implement this, and then we will look at an explanation to understand why this algorithm works?
So, first, we take two pairs of variables, each pair will have a variable to count and a variable to store the previous value we were counting on.
Then we perform three kinds of operations according to the element we currently while parsing the list.
First, If we find the current iteration value in the list, same as the previous element's value then we increment the counter.
Second, if we have the count is at zero then we change the element we're counting for and also set the count of a particular element to 1.
Third, if the above does not happen then just decrement count for elements.
The above is three kinds, not the right order to perform the operations, for right order look at the code.
After we are done with the above parsing, we will reset the counters and then count for the values we found to be dominant in the previous parsing of the list.
And at last, we will take the value count and if it passes the condition of being greater than $N/3$ and add it to the list, which we will return as a result.
The Code,
example_list = [1, 2, 2, 2, 4, 5, 1, 1, 1, 1, 2]
class Elem:
val = 0
count = 0
def inc(self):
self.count += 1
def dec(self):
self.count -= 1
def more_than_N_by_3(l):
elem1 = Elem()
elem2 = Elem()
for n in l:
if elem1.count == 0:
elem1.val = n
elif elem1.val == n:
elif elem2.count == 0:
elem2.val = n
elif elem2.val == n:
nums = []
N = len(l)
# Reseting the counter
elem1.count = 0
elem2.count = 0
# counting again to confirm
for n in l:
if n == elem1.val:
elif n == elem2.val:
## adding if counter statisfy our problem's condition
if elem1.count > N//3:
if elem2.count > N//3:
return nums
Output: [1, 2]
Why it works?
If you may remember at the very beginning we discussed how there can only be at most two elements that will satisfy the condition of having the count of elements greater than $N/3$.
This algorithm takes advantage of that fact and assumes that there do exist two such elements and starts to count up for every consecutive similar element we find and count down if we find different
consecutive elements.
Let's take a simple example,
[1, 1, 1, 1, 2, 3, 4, 5, 5]
We can separate this list into 3 different categories.
1. The dominant element, which in the above case occurs for 4 times
2. The second dominant element, which in this case is 5 and occurs for 2 times.
3. Other non-dominant elements.
Now, ask yourself if we ran the above algorithm then what would elem1 and elem2 will look like?
Note: we're elif, so, only one thing will occur at every iteration.
So, therefore, the elem1 will get updated until we reach 2 and its counter will become 4 after this there are three consecutive different numbers, for which elem1's counter will decrement only when
the elem2 counter is not zero.
Let's say after elem1's counter reaches 4, then after that, we have 7 continuous different element ( 7 because if we take more than that then 1 will become less than $N/3$. ) then we will have 7
iterations of which there will be 3 iterations where elem1's counter will be decremented and other times elem2's counter being 0 will make skip the decrement.
Well for time complexity, since we're parsing the list two times ( separately ) therefore it should be O(N).
Time Complexity: O(N)
And for space complexity, we're not creating anything more than four variables and a list of two elements, therefore,
Space Complexity: O(1)
With this article at OpenGenus, you must have the complete idea of finding elements that appear more than N/3 times. | {"url":"https://iq.opengenus.org/elements-that-appear-more-than-n-3-times/","timestamp":"2024-11-03T06:28:33Z","content_type":"text/html","content_length":"68834","record_id":"<urn:uuid:7e0447d6-3593-4c16-a624-7be0cf076444>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00103.warc.gz"} |
Formally Verifying Loops: Part 2
This is the second introductory article on formal verification of loops in the Solidity and EVM smart contracts. In the first article, we covered loop unrolling, an automatic reasoning technique that
enforces termination by cutting off reachable states, which can lead to missed bugs and vulnerabilities. We also saw that increasing the looping limit is insufficient for unbounded loops. This
follow-up article solves this problem by introducing loop invariants to prove properties about unbounded loops!
Be warned that proving loop invariants is generally considered one of the most (if not the most) difficult challenge of formal verification. Unlike loop unrolling, loop invariants are not fully
automated and require human input to guide the prover. From the tools presented in the first article, Kontrol is the only prover capable of proving properties about unbounded loops; consequently the
other tools won’t reappear in this article. We will cover some of the most advanced features Kontrol has to offer.
In this blog post, you will learn:
• What are loop invariants?
• What is natural induction?
• How to prove loop invariants with pen and paper
• How to spot bytecode level loops
• How to inspect EVM states
• The difference between black-box and glass-box formal verification
• How to prove loop invariants with Kontrol
Recap: Gauss’ Looping Problem
This article continues our attempt to prove the equivalence of the sumToN and triangleNumber functions introduced in the first post. Here is a quick reminder of the code and the specification (the
full source code used in this article can be found on GitHub).
function sumToN(uint256 n) public pure returns (uint256) {
uint256 result = 0;
uint256 i = 0;
while (i < n) {
i = i + 1;
result = result + i;
return result;
function triangleNumber(uint256 n) public pure returns (uint256) {
return n * (n + 1) / 2;
function check_equivalence(uint256 n) public pure returns (bool) {
vm.assume(n < 2**128); // prevent overflow
assert(sumToN(n) == triangleNumber(n));
An Inductive Approach
The problem in front of us is proving a property about an unbounded loop in finite time. So, just proving the property for 1,2,3... iterations is hopeless. Let’s take inspiration from Gauss: Gauss
summed up all numbers from 1 to 100 without adding one number at a time. Instead, Gauss only calculated the result of the expression 100 * 101 / 2. How can we know that these numbers are the same? A
mathematician's solution usually involves natural induction.
In this exercise, we aim to prove Gauss’ childhood observation to remind ourselves of the natural induction principle: a proof technique that can be used for claims of the form $\forall n.P(n)$. It
tells us that it is sufficient to prove $P(0)$ (called the base case) and $\forall n .P(n) => P(n+1)$ (called the inductive case). The principle can also be given as an inference rule:
$\frac{P(0) \quad \forall n .P(n) \implies P(n+1)}{\forall n.P(n)}$
Now, let’s apply this principle to Gauss’ formula: $\forall n. \Sigma_{i=0}^{n} i = \frac{n(n+1)}{2}$
First, we must show the base case, where we substitute $n$ for $0$.
$\Sigma_{i=0}^{0} i = 0 = \frac{0(0+1)}{2}$
Clearly, both sides simplify to 0, so the base case holds.
Next, we must show the inductive case:
$\forall n.(\Sigma_{i=0}^{n} i = \frac{n(n+1)}{2}) \implies (\Sigma_{i=0}^{n+1} i = \frac{(n+1)((n+1)+1)}{2})$
The trick here is first to extract the final summand from the series $\Sigma_{i=0}^{n+1} i$:
$\forall n.(\Sigma_{i=0}^{n} i = \frac{n(n+1)}{2}) \implies ((\Sigma_{i=0}^{n} i) + n + 1 = \frac{(n+1)((n+1)+1)}{2})$
Now, we can apply the induction hypothesis by replacing $\Sigma_{i=0}^{n} i$ with $\frac{n(n+1)}{2}$.
$\forall n.(\Sigma_{i=0}^{n} i = \frac{n(n+1)}{2}) \implies (\frac{n(n+1)}{2} + n + 1 = \frac{(n+1)((n+1)+1)}{2})$
Finally, we can show the remaining equality by some simple arithmetic manipulation:
$\forall n.(\Sigma_{i=0}^{n} i = \frac{n(n+1)}{2}) \implies (\frac{(n+1)((n+1)+1)}{2} = \frac{(n+1)((n+1)+1)}{2}) \quad \blacksquare$
This completed our natural induction proof. We now return our focus to formally verifying the equivalence of two Solidity algorithms. The idea stays the same: we develop a closed formula that
captures the computational effect of the loop, and then prove the equivalence of the closed formula and the loop inductively.
The Pen and Paper Method
Before we use a mechanized approach to explore loop invariants, let’s do it once with pen and paper.
Forgetting our running example for a minute, we will perform the manual proof on an even simpler codebase.
Using loop invariants in formal proofs is usually a three-step process:
1. Hypothesizing a closed formula
2. Proving the closed formula holds after any number of loop iterations
3. Apply the loop invariant to the original problem
Step 1: Hypothesizing a Closed Formula
This is arguably the most challenging part. Coming up with closed formulas to capture the essence of a loop often requires creative thinking, and that is why it is so hard to automate this process.
Our best chance in tackling looping programs is human creativity.
Have a look at the following program:
uint256 result = 0;
uint256 i = 0
while (i < n) {
i += 1;
result += 2;
assert(result == 2 * n);
We aim to prove that the result equals two times n after the loop. Looking carefully at the loop, we see that the variable i increases by one and the result by two in each iteration of the loop. In
other words, the result is increasing at twice the rate of i, so we come up with the following closed formula: result = 2 * i
Step 2: Prove the Closed Formula Holds After Any Number of Loop Iterations
So far, our loop invariant is just a hypothesis, not a theorem. So, let’s prove it. We do so by induction on the number of loop iterations.
In our base case, we must prove that the invariant holds after 0 loop iterations. We have result = 0 and i = 0, hence result = 2 * i is true.
In the inductive case, we must show that the loop's body preserves the loop invariant:
Assuming that result = 2 * i holds for some fixed i. After incrementing i by one, we have result = 2 * (i - 1); after incrementing the result by two, we have result = 2 * (i - 1) + 2, which
simplifies to result = 2 * i.
Concluding, the loop invariant holds before the loop, and every loop iteration preserves the loop invariant. Hence, the loop invariant holds. Notice that it temporarily breaks inside the loop body -
but that is okay.
In the literature, this argument is often associated with partial correctness. It is called partial because we have not proven that the loop terminates. After all, it could be an infinite loop. So,
let’s prove termination: i increases by 1 in every loop iteration, and n stays constant. Further, the body of the loop terminates in each iteration. Therefore, the loop eventually terminates when i =
n. Partial correctness and termination establish total correctness.
Step 3: Applying the Loop Invariant
Our final goal is to show that result = 2 * n. After the loop, we know that i = n and result = 2 * i. A simple substitution of n for i gives us the final result = 2 * n $\blacksquare$
The mechanized method using Kontrol
Let's return our attention to the original running example, and formally verify our claim using Kontrol.
Kontrol helps us to prove and apply invariants. Unfortunately, it cannot hypothesize the closed formulas for us - we must supply it to Kontrol. That doesn’t mean Kontrol trusts us - we will verify
the invariant before applying it. Moreover, Kontrol operates on the bytecode level, not the Solidity level, where loops are encoded with JUMPI instructions instead of for or while statements, but the
principle stays the same. As a final word of caution, Kontrol cannot read even your prettiest handwriting and therefore expects the loop invariants in a machine-readable format called K. K is special
purpose programming language for designing, implementing and verifyging other programing languages. Ever wondered why it's called Kontrol? Now you know.
We’re now getting to the bottom of smart contracts on the implementation and specification language levels. So far, we have only looked at Solidity. Now, we’re turning our attention to EVM bytecode,
and K specifications. We will take one step at a time, and break down our process as follows.
1. Hypothesize a closed formula at the Solidity level
2. Translate the formula into an KEVM formula
3. Formally prove the bytecode-level invariant
4. Apply the bytecode-level invariant to the original program.
Step 1: Hypothesizing a Closed Formula
As a reminder, we’re looking for an invariant for this loop:
function sumToN(uint256 n) public pure returns (uint256) {
uint256 result = 0;
uint256 i = 0;
while (i < n) {
i = i + 1;
result = result + i;
return result;
Fortunately, Gauss already figured it out for us. The loop invariant we’re looking for is result = i * (i + 1) / 2
Step 2: Translate the formula into an KEVM formula
The first step in our translation from paper to a KEVM-claim is figuring out what the loop looks like at the EVM level. This requires the use of a disassembler (we recommend the one shipped with
Simply run the following command:
forge inspect GaussSpec assembly
Here, we manually annotated the bytecode with the corresponding source code, and draw arrows to depict the control flow for our convinience.
Interestingly, the EVM loop inverted the condition of the original while loop. The reason is that the JUMPI instruction jumps to the desired JUMPDEST only if the condition differs from zero.
The next step is understanding how the EVM state changes when we jump out of the loop. Now it would be really beneficial for our purpose to set breakpoints at the entry and exit of the loop. While
breakpoints are common for concrete execution, they are rarely found in symbolic execution engines. Fortunately, Kontrol is an exception to this rule. Setting breakpoints on JUMP and JUMPI
instructions turns out to be so valuable that there is even a flag for it. Now let’s run Kontrol with breakpoints and in bounded loop unrolling mode (remember, we have not proven the loop invariant
yet, so loop unrolling is our best chance).
kontrol prove --match-test 'GaussSpec.check_equivalence' --bmc-depth 3 --break-on-jumpi
After the exploration is complete, we can inspect our (symbolic) execution trace. There are two built-in subcommands for inspection: kontrol view-kcfg launches an interactive terminal user interface
(TUI) and kontrol show prints the output non-interactively. In this example, we will use the interactive TUI. If you’re about to use the kcfg viewer for the first time, you will definitely want to
review the documentation first.
To launch the TUI, use the following command:
kontrol view-kcfg GaussSpec.check_equivalence
Take your time and play around in the TUI a little.
Let's take a step back for a minute and talk about a unique feature of Kontrol: State inspection is an invaluable asset for all kinds of proofs, useful way beyond just loop invariants. The other
tools we discussed earlier, including the Certora Prover, Halmos, and HEVM, treat the EVM as a black box. This means the internal workings are hidden, making it difficult to understand what’s
happening when the symbolic execution gets stuck or times out.
Kontrol, on the other hand, treats the EVM as a glass box. With this approach, you can observe how the EVM state evolves, seeing every internal detail and mechanism at work. Imagine watching all the
gears turning inside, gaining insight into how each part contributes to the overall process. If you've used formal verification before, you might be familiar with tools getting stuck or timing out
unexpectedly. With Kontrol’s glass box approach, you can inspect the exact state where progress halted, identify the stuck gearwheel, and make the necessary adjustments to keep the machine moving
In formal verification, black box tools are typically referred to as fully automated provers, while glass box tools are known as interactive provers. This terminology can be a bit misleading, as it
might imply that interactive provers require more manual intervention and are, therefore, less automated. However, this is not the case. Kontrol, for example, excels in automation, often
outperforming others in automated reasoning tasks, as demonstrated by its results in the eth-sc-comp benchmark suite. The key distinction lies in Kontrol's ability to combine powerful automation with
the transparency of a glass box, offering both insight and efficiency.
Now, let's see how it works in practice. With the help of the kcfg viewer, we can identify that node 31 marks the loop's entry, and node 34 marks its exit after 0 loop iterations. Not visible in the
screenshot but equally interesting are nodes 43 and 46, which mark the loop's entry and exit after one iteration. Nodes 55 and 58 mark the entry and exit after two iterations.
We can ask Kontrol to get a summary of the changes between two nodes. Let's ask for the difference between the first entry of the loop and the exit after two iterations. This gives us a good
impression of which parts of the EVM state change during the loop. We removed some pieces from the output for brevity. Watch out for the arrow => telling us exactly which pieces change:
$ kontrol show GaussSpec.check_equivalence --node-delta 31,58
State Delta 31 => 58:
( JUMPI 898 bool2Word ( VV0_n_114b9705:Int <=Int 0 ) => JUMP 898 )
~> #pc [ JUMPI ]
~> #execute
~> CONTINUATION:K
( ( 0 => 2 ) : ( ( 0 => 3 ) : ( 0 : ( VV0_n_114b9705:Int : ( 1816 : ( ( ( VV0_n_114b9705:Int *Int ( VV0_n_114b9705:Int +Int 1 ) ) /Int 2 ) : ( VV0_n_114b9705:Int : ( 402 : ( selector ( "check_equivalence(uint256)" ) : .WordStack ) ) ) ) ) ) ) ) )
The beauty of Kontrol is that the summary is actually a rewrite rule telling us exactly how to go from node 31 to 58. The rule tells us what every single bit of the internal EVM state looked like
before and after we executed two loop iterations. There is no hidden or private state. Kontrol makes every single bit inspectable: the memory, the word stack, the program counter, the bytecode,
accounts storage, etc. We will use this rewrite rule as the foundation for our loop invariant! However, the rule is overly specific. For our loop invariant, this detail can get in the way; after all,
most of the state components don’t change and do not constrain the rule in any specific way. Let’s remove all the noise from the rule and only keep the relevant parts:
( JUMPI 898 bool2Word ( N:Int <=Int 0 ) => JUMP 898)
~> #pc [ JUMPI ]
~> #execute
~> _CONTINUATION:K
<pc> 860 </pc>
( ( 0 => 2 ) : ( ( 0 => 3 ) : ( 0 : ( VV0_n_114b9705:Int : ( 1816 : ( ( ( VV0_n_114b9705:Int *Int ( VV0_n_114b9705:Int +Int 1 ) ) /Int 2 ) : ( VV0_n_114b9705:Int : ( 402 : ( selector ( "check_equivalence(uint256)" ) : .WordStack ) ) ) ) ) ) ) ) )
<useGas> false </useGas>
<program> #binRuntime </program>
<jumpDests> #computeValidJumpDests( #binRuntime ) </jumpDests>
Now we have a rule that takes us from the $0^{th}$ loop iteration to the loop exit after two iterations. What we actually need is a rule to take us from the $i^{th}$ iteration to the exit after $n$
iterations. Hence, we must generalize our rule, so it matches at every iteration. It's handy to look at some more node deltas to learn which parts of the state we must generalize. Eventually, we came
up with the following hypothesis for our loop invariant:
claim <k>
( JUMPI 898 bool2Word ( N:Int <=Int I:Int ) => JUMP 898 )
~> #pc [ JUMPI ]
~> #execute
~> _CONTINUATION:K
<program> #binRuntime </program>
<pc> 860 </pc>
<wordStack> ( I => N ) : ( RESULT => N *Int ( N +Int 1 ) /Int 2 ) : 0 : N : WS </wordStack>
<useGas> false </useGas>
<activeTracing> false </activeTracing>
<jumpDests> DESTS </jumpDests>
requires 0 <=Int N andBool N <Int 2 ^Int 128
andBool 0 <=Int I andBool I <=Int N
andBool RESULT ==Int I *Int (I +Int 1) /Int 2
andBool #sizeWordStack(WS) <Int 1013
andBool DESTS ==K #computeValidJumpDests( #binRuntime )
Let's take a closer look at the claim above.
• <k> JUMPI 898 bool2Word ( N:Int <=Int I ) => JUMP 898 … </k>
This line claims that when we reach the loop head, we will eventually jump out of the loop (to program counter 898). Hence, it claims termination of our while loop.
• The <program> cell contains the runtime bytecode of our smart contract. The value inside is just a human-readable variable name that maps to the actual bytecode. Nobody wants to type out the
entire bytecode.
• The <pc> cell tells us that this claim applies only to the program counter 860 (the offset of the JUMPI instructions).
• The <wordStack> is the most exciting part of the loop invariant. It tells us how the I and RESULT variables are supposed to change during the loop. We know that these variables are allocated on
top of the stack and we claim that I will be equal to N, and RESULT will be equal to N *Int ( N +Int 1 ) /Int 2 after the loop.
• The <useGas> cells disables all reasoning about gas. This is Kontrol’s default setting for performance. If we enable gas reasoning, we must develop a closed formula for the total amount of gas
consumed by the loop - but we want to to keep it simple for the sake of this article.
• The <jumpDests> is a bitstring marking all valid JUMPDEST locations. It is needed only for technical reasons and is always computed from the bytecode.
• After the requires keyword we list a bunch of constraints about our claim. For example, we require that N is in the range [0, 2^128] and I is in the range [0, N].
• The most noteworthy contraint is RESULT ==Int I *Int (I +Int 1) /Int 2 telling us that our claim only applies if the RESULT variable equals the closed formula.
To summarize, here is a recipe to develop loop invariants with Kontrol.
1. Unroll the loop a couple of times with kontrol prove --bmc-depth
2. Find nodes corresponding to the loop’s entry and exit using kontrol view-kcfg or kontrol show.
3. Use kontrol show --node-delta to obtain a rewrite rule from the entry to the exit node.
4. Generalize the rule to go from the entry of the $i^{th}$ iteration to the loop’s exit.
Step 3: Proving the Invariant
So far, the loop invariant is just a hypothesis; it is still our obligation to prove it before we apply it in a bigger context. This is where Kontrol really shines. Recall the manual effort of the
natural induction proof at the beginning of this article, and then again when we used the pen and paper method to prove a simple invariant at the Solidity level. Try to imagine how tedious and nasty
these proofs can get on the bytecode level! However, with Kontrol this process is mostly automated.
Let’s make an attempt with kevm prove: We first copy our invariant into a file called lemmas.k and then run the following commands to prove it (again the full source code is available on GitHub).
$ kontrol build --require src/lemmas.k --module-import GaussSpec:GAUSS-CONTRACT --regen --rekompile
$ kevm prove --definition out/kompiled --spec-module LOOP-INVARIANTS src/lemmas.k --break-on-jumpi --max-depth 100
PROOF PASSED: a5f38b84d9889082be3e485d304f57b9a559a94e32726fdd0bb55f3fa7850fa9
Success! The Kontrol prover was smart enough to prove the invariant all by itself!
Notice that we are not always this lucky. Sometimes, the prover can get stuck and demand human intervention to drive the proof forward. Overcoming stuck states is out of the scope for this blog post,
but the interested reader can continue reading our official documentation.
Step 4: Applying the Loop Invariant
Kontrol will not apply the loop invariant to our program automatically. After all, throwing random theorems at math problems is rarely a wise idea. How would Kontrol know that this theorem is
actually making progress towards the final proof obligation in a bigger context? We must introduce this theorem as a rewrite rule with a higher priority than the default rules, thereby ensuring that
when Kontrol reaches our loop it will apply the loop invariant instead of unrolling it one iteration at a time.
This is a simple mechanical process; we just copy the claim and replace the claim keyword with rule. When rebuilding the project, K will complain about non-functional symbols on the left-hand side of
the rule, which is an unfortunate technical limitation but simple to resolve by a process we call defunctionalization: We must move all function calls from the left-hand side of our rule into the
rule <k>
( JUMPI 898 CONDITION => JUMP 898 )
~> #pc [ JUMPI ]
~> #execute
~> _CONTINUATION:K
<program> BYTECODE </program>
<pc> 860 </pc>
<wordStack> ( I => N ) : ( RESULT => N *Int ( N +Int 1 ) /Int 2 ) : 0 : N : WS </wordStack>
<useGas> false </useGas>
<activeTracing> false </activeTracing>
<jumpDests> DESTS </jumpDests>
requires 0 <=Int N andBool N <Int 2 ^Int 128
andBool 0 <=Int I andBool I <=Int N
andBool RESULT ==Int I *Int (I +Int 1) /Int 2
andBool #sizeWordStack(WS) <Int 1013
andBool DESTS ==K #computeValidJumpDests( #binRuntime )
andBool CONDITION ==K bool2Word ( N:Int <=Int I )
andBool BYTECODE ==K #binRuntime
For example, we moved the function call bool2Word ( N:Int <=Int I ) into a side condition CONDITION ==K bool2Word ( N:Int <=Int I ) and replaced all occurrences of the call with CONDITION.
The alert reader might wonder if this rule is still identical to the proven claim after defunctionalizing it. Indeed, we must be very careful not to introduce unsound rules. At this stage, it is best
to modify the original claim to match the defunctionalized rule and prove it again. We must ensure that the module containing the claim does not include the module defining the rule, or we would end
up with a circular argument!
Finally, let’s prove our original claim with the loop invariant:
$ kontrol build --require src/lemmas.k --module-import GaussSpec:GAUSS-LEMMAS --regen --rekompile
$ kontrol prove --match-test GaussSpec.check_equivalence
🏃 Running Kontrol proofs 🏃
Add `--verbose` to `kontrol prove` for more details!
Selected functions: src%GaussSpec.check_equivalence(uint256)
Running setup functions in parallel:
Running test functions in parallel: src%GaussSpec.check_equivalence(uint256)
0:00:47 src%GaussSpec.check_equivalence(uint256):0 Finished PASSED: 5 nodes: 0 pending|1 passed|0 failing|0 vacuous|0 refuted|0 stuck
✨ PROOF PASSED ✨ src%GaussSpec.check_equivalence(uint256):0
⏳ Time: 48s ⏳
This brings us to the end of our journey into the depths of proving loop invariants. Let's take a moment to reflect on what we've covered. You've journeyed through some of the most complex
territories in formal verification, tackling one of the hardest challenges: proving properties about unbounded loops. Together, we've demystified loop invariants, ventured through natural induction,
and explored a mechanized approach with Kontrol. What’s more, we’ve witnessed how Kontrol’s unique glass-box approach allows for unparalleled transparency and control during symbolic execution.
Feel proud of these accomplishments. While formal verification can feel daunting, you've taken steps toward mastering techniques that most developers only touch on the surface. And with Kontrol,
you're not just using any prover — you’re utilizing a tool capable of handling even the most challenging aspects of smart contract verification.
So, take a breath and pat yourself on the back. You’re now well on your way to mastering the intricacies of formal verification, and each loop you prove adds another notch to your growing expertise. | {"url":"https://runtimeverification.com/blog/formally-verifying-loops-part-2","timestamp":"2024-11-05T15:25:48Z","content_type":"text/html","content_length":"188814","record_id":"<urn:uuid:5f037464-ffb0-4978-8faf-1ff62427d59f>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00075.warc.gz"} |
Collection of Solved Problems
Heat Given Off by Nitrogen
Task number: 3950
Mow much heat is it necessary to be given off by 56 g of nitrogen in order to isothermally compress it from the pressure of 100 kPa to the pressure of 500 kPa at the temperature of 300 K?
Note: Consider nitrogen to be an ideal gas.
• Hint 1
Realize that at constant temperature, the internal energy of ideal gas does not change and the given-off heat is therefore equal to work performed by the surroundings during the compression of
nitrogen gas, or rather the absolute value of work performed by nitrogen (during compression the performed work is negative).
• Hint 2
We need to determine the performed work using integral calculus due to the fact that the pressure is not constant.
• Hint 3
To determine pressure p as a function of volume V (and conversely), we can use Boyle's Law for isothermal process.
• Hint 4
To determine initial volume V[1] and final volume V[2], use the equation of state for ideal gas.
• Analysis
Since at constant temperature the internal energy of the gas does not change, from the First Law of Thermodynamic it follows that the given-off heat is equal to the absolute value of work
performed by the nitogen gas during its compression. To determine it we need to use integral calculus due to the fact that pressure and volume are not constant.
The pressure of nitrogen gas as a volume function is determined from Boyle's Law that applies for isothermal processes and perform an itegration of resulting function with respect to volume.
After expressing the unknown initial volume and final volume we finally use the equation of state for ideal gas.
• Given Values
m = 56 g = 0.056 kg nitrogen mass
T = 300 K nitrogen temperature
p[1] = 100 kPa = 1·10^5 Pa initial pressure of nitrogen
p[2] = 500 kPa = 5·10^5 Pa final pressure of nitrogen
Q = ? given-off heat
Table values:
M[m] = 28 g mol^−1 molar mass of nitrogen N[2]
R = 8.31 JK^−1mol^−1 molar gas constant
• Solution
During isothermal process the internal energy of the gas does not change which according to the First Law of Thermodynamics means that the given-off heat Q is equal to work W' performed by the
surroundings during gas compression, or rather the absolute value of work W performed by the gas (it is negative during compression). It is true that
\[W=\int\limits_{V_1}^{V_2}p\, \text{d}V,\]
where V[1] and V[2] are initial and final volume of the gas and p is the gas pressure that continuously changes during the compression (it is a volume function).
To determine pressure p as a function of volume V, we use Boyle's Law:
From here we can express pressure p:
\[p = \frac{p_1V_1}{V}.\]
Now we can perform the integration:
\[W = \int\limits_{V_1}^{V_2}p\, \text{d}V = \int\limits_{V_1}^{V_2}\frac{p_1V_1}{V}\, \text{d}V =\]
we factor the constants out of the integral
\[=p_1V_1 \int\limits_{V_1}^{V_2}\frac{1}{V}\, \text{d}V = \]
we perform the integration and substitute the limits
\[=p_1V_1[\ln V]_{V_1}^{V_2} = p_1V_1 \ln \frac{V_2}{V_1}.\]
Now we determine the unknown initial volume V[1] and final volume V[2] from the equation of state for ideal gas
\[p_1V_1=\frac{m}{M_m}RT \qquad \Rightarrow \qquad V_1=\frac{mRT}{p_1M_m},\] \[p_2V_2=\frac{m}{M_m}RT \qquad \Rightarrow \qquad V_2=\frac{mRT}{p_2M_m}.\]
After substitution we obtain
The given-off heat Q is then determined as
\[Q= -W =-\frac{mRT}{M_m}\,\ln{\frac{p_1}{p_2}} =\frac{mRT}{M_m}\,\ln{\frac{p_2}{p_1}}.\]
• Numerical Solution
\[Q=\frac{mRT}{M_m}\,\ln{\frac{p_2}{p_1}}=\frac{0.056\cdot{8.31}\cdot{300}}{0.028}\cdot \ln{\frac{500}{100}}\,\mathrm{J}\dot{=}8025\,\mathrm{J}\dot{=}8\,\mathrm{kJ}\]
• Answer
It is necessary to give off heat of approximately 8 kJ.
• Alternative Solution
As is stated in the solution of this task, the given-off heat Q is equal to the absolute value of work performed by the nitrogen gas during compression. For this work it holds
\[W=\int\limits_{V_1}^{V_2}p\, \text{d}V,\]
where V[1] and V[2] are initial and final volume of the gas and p is pressure of the gas.
Given the fact that we know initial and final pressure of the gas and not its volumes, we adjust this formula in such a way that the work would be determined by volume as a pressure function.
We start with the equation of state
that we differentiate:
Since it is an isothermal process, we can say that dT = 0 and we obtain
\[p\,\text{d}V=-V\text{d}p. \]
If we substitute this expression to our original integral expression for work and adjust the limits of the integral, we obtain
\[W=-\int\limits_{p_1}^{p_2}V\, \text{d}p,\]
where p[1] and p[2] are initial and final pressures.
Now we need to express volume V as a function of pressure p. We use Boyle's Law
\[p_1V_1=pV, \]
and from here we determine volume V:
Unknown initial volume V[1] is determined from the equation of state for ideal gas
\[p_1V_1=\frac{m}{M_m}RT. \]
Now we can perform the integration:
\[W=-\int\limits_{p_1}^{p_2}V\, \text{d}p = -\int\limits_{p_1}^{p_2}\frac{p_1V_1}{p}\text{d}p = -\int\limits_{p_1}^{p_2}\frac{p_1mRT}{pp_1M_m}\text{d}p = \]
we factor the constants out of the integral
\[=-\frac{mRT}{M_m}\int\limits_{p_1}^{p_2}\frac{1}{p}\text{d}p = \]
we perform the integration and substitute the limits
\[=-\frac{mRT}{M_m}\,\left[\ln p\right]_{p_1}^{p_2}= -\frac{mRT}{M_m}\,\ln{\frac{p_2}{p_1}}.\]
The given-off heat Q is then determined as follows
which is the same relationship we obtained in the Solution section. | {"url":"https://physicstasks.eu/3950/heat-given-off-by-nitrogen","timestamp":"2024-11-11T16:35:34Z","content_type":"text/html","content_length":"34000","record_id":"<urn:uuid:5b89d957-2d47-4d1b-adab-3293de3d7ee1>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00883.warc.gz"} |
Previously … on Machine Learning here
From my blog on Artificial Intelligence (AI), we know that when it comes to processing data and creating patterns for decision making, the aim is to imitate the workings of the human brain. Deep
learning (DL) is a subset of Machine Learning (ML) and AI. It has neural networks which are capable of learning unsupervised from data that is unstructured or unlabeled.
In the digital era, we are getting information from everywhere. Big Data is found from sources such as social media, internet search engines, online platforms etc. Big Data is accessible and is
shared through FinTech applications such as cloud computing. However, this data is unstructured, and so huge that it would take you and me years to understand and get the relevant information from
it. There is boundless potential in decoding Big Data, so companies are using AI techniques to do this.
A common techniques to process Big Data is ML. ML uses self-adapting algorithms to get better at analysis and find patterns with experience or additional data. More information on ML can be found in
my previous blog article.
Deep learning, a subset of machine learning, utilizes a hierarchical level of Artificial Neural Networks (ANN) to carry out the process of machine learning. To understand how Deep Learning Works, we
must first understand ANN.
Artificial Neural Networks (ANN)
In DL, the computing systems are designed to simulate how the human brain analyzes and processes information. ANN has self-learning capabilities that allow it to produce positively correlated results
with the amount of available information.
How does ANN work?
Artificial neural networks are built like the human brain, with neuron nodes interconnected like a web. The human brain has hundreds of billions of cells called neurons. Each neuron is made up of a
cell body that is responsible for processing information by carrying information towards (inputs) and away (outputs) from the brain. ANN has hundreds or thousands of artificial neurons called
processing units which are interconnected by nodes.
These processing units are made up of input and output units. The input units receive various forms and structures of information. The neural network attempts to learn about the information presented
in order to produce one output report. Just like humans need rules and guidelines to come up with a result or output, ANNs also use a set of learning rules called Backpropagation to perfect their
output results.
An ANN goes through a training phase where it learns to recognize patterns in data. During this supervised phase, the network compares its actual output produced with the desired output. The
difference between both outcomes is adjusted using backpropagation. At this stage, the network works backwards, going from the output unit, through the hidden units to the input units in order to
adjust the weight of its connections between the units. It iterates until the difference between the actual and desired outcome produces the lowest possible error.
During the training and supervisory stage, the ANN is taught what to look for and what its output should be, using Yes/No questions with binary numbers. For example, a bank that wants to detect
credit card fraud on time may have four input units fed with these questions:
1. Is the transaction in a different country from the user’s resident country?
2. Is the website the card is being used at affiliated with companies or countries on the bank’s watch list?
3. Is the transaction amount larger than $2000?
4. Is the name on the transaction bill the same as the name of the cardholder?
The bank wants the “fraud detected” responses to be Yes Yes Yes No which in binary format would be 1 1 1 0. If the network’s actual output is 1 0 1 0, it adjusts its results until it delivers an
output that coincides with 1 1 1 0. After training, the computer system can alert the bank of pending fraudulent transactions, saving the bank money.
ANN Application to Deep Learning
While traditional programs build analysis with data in a linear way, the hierarchical function of deep learning systems enables machines to process data with a non-linear approach. A traditional
approach to detecting fraud or money laundering might rely on the amount of transaction that ensues, while a deep learning non-linear technique to weeding out a fraudulent transaction would include
time, geographic location, IP address, type of retailer, and any other feature that is likely to make up a fraudulent activity.
The first layer of the neural network processes, is raw data input, such as the amount of the transaction. This is then passed on to the next layer as output. The second layer processes the previous
layer’s information by including additional information like the user’s IP address and passes on its result. The next layer takes the second layer’s information and includes raw data like geographic
location and makes the machine’s pattern even better. This continues across all levels of the neuron network until the best output is determined. This is an example of a FeedForward Neural Network.
warning Little Bit of Maths on the Horizon! The next section is here purely for higher understanding of neural networks. If Maths is something you left in your dust at school, simply skip forward to
Examples. (Click on the Equations for a GIF(t) to help you along!)
Sigmoid Neuron
In ANN we mention weights of neurons. These weighted neurons are called Sigmoid Neurons. Some neural networks use perceptron outputs that have discrete values of 0 and 1. A sigmoid neuron outputs a
smooth continuous range of values between 0 and 1. This is defined by the Sigmoid Function defined below,
In ANN, values change very slowly with each iteration and input. Observing how a small change in the bias value or the weights (associated with the artificial neurons) affects the overall value of
the output of the the neuron is very important. A Perceptron may have its outputs flipped suddenly with a small change in the input value. To observe the tiny changes in the output to arrive at the
correct value of input, we need a function to be applied on the dot product of weights and bias value so that overall output is smooth. For now, the function could have been any function f() that is
smooth in nature, such as quadratic functions or cubic functions. The reason the sigmoid function is chosen, is that, exponential functions generally are similar to handle mathematically and, since
learning algorithms involve lots of differentiation, choosing a function that is computationally cheaper to handle is wise.
The Sigmoid Neurons are organized into layers, similar to how they are in the human brain. Neurons on the bottom layer (inputs) receive signals from the inputs, where neurons in the top layers
(outputs) have their outlets connected to the “answer,” Usually there are no connections between neurons in the same layer. This is an optional restriction, more complex connectivity’s require more
involved mathematical analysis. This is a feed-forward network so there are no connections that lead from a neuron in a higher layer to a neuron in a lower layer. Alternatively, there are Recursive
Neural Networks (RNN). Again, these are much more complicated to analyze and train.
A fundamental idea of DL and ANN is BackPropagation. We have briefly touched on backpropagation in previous blog articles but we will dive into the Algorithm into more detail.
The idea behind backpropagation is that we don’t know what the hidden units (the neurons between the input and output) should be doing, but we do know how fast the error between the iteration and
actual result changes as we change the hidden activity. We want to find the steepest path from our starting results to the actual result.
Each hidden unit can affect many different output units. We start by looking at the error derivatives for one layer of hidden units. Once we have the error derivatives for one layer of hidden units,
we use them to compute the error derivatives for the activities of the layer below. Once we find the error derivatives for the activities of the hidden units, it’s relatively easy to get the error
derivatives for the weights leading into a hidden unit.
The symbol “y” will refer to the activity of a neuron. The symbol “z” will refer to the logic of a neuron. We start by looking at the base case of the dynamic programming problem, the error function
derivatives at the output layer:
For the inductive step, let’s presume we have the error derivatives for layer j. We now aim to calculate the error derivatives for the layer below it, layer i. To do so, we must accumulate
information for how the output of a neuron in layer i affects the logic of every neuron in layer j. The partial derivative of the logic, with respect to the incoming output data from the layer
beneath, is merely the weight of the connection wij:
To complete the inductive step
Combining, we express the partial derivatives of layer i in terms of the partial derivatives of layer j.
Next, we determine how the error changes with respect to the weights. This gives us how to modify the weights after each training example:
For backpropagation with training examples, we sum up the partial derivatives over all training examples. This gives us the following modification formula:
This is the Backpropagation Algorithm for Feed-Foward Neural Networks using Sigmoidal Neurons. In better news, this also the end of the Maths. You made it through, and I’m sure you’re a stronger
person for it!
You can find a worked example of BackPropagation here.
Examples Of Deep Learning:
1. Email service providers use ANN to detect and delete spam from a user’s inbox
2. Asset managers use it to forecast the direction of a company’s stock
3. Credit rating firms use it to improve their credit scoring methods
4. E-commerce platforms use it to personalize recommendations to their audience
5. Chatbots are developed with ANN for natural language processing
6. Deep learning algorithms use ANN to predict the likelihood of an event
7. Colorization of Black and White Images
8. Adding Sound to Silent Movies (Charlie Chaplain will never be the same)
9. Automation of:
□ Translation of Text
□ Handwriting Generation
□ Text Generation
□ Image Captions
The possibilities for the future of Deep Learning are endless. As algorithms and technology advances, we may see models that use less training cases to learn, maybe diving into unsupervised learning.
This could kick start a race between global companies. Apple has recently been on a hiring mission, seeking 80-plus AI experts to help make Siri smarter than Google Now or Microsoft’s Cortana. Google
paid $400 million for DeepMind who specialised in Deep Learning, with Deep Learning experts commanding seven-figure salaries. They are the Premier League Footballers of the programming world.
There’s a simple reason why, when it comes to Deep Learning and AI, Google are one of, if not the, market leaders. Data! They have it coming out of their ears. We have seen that the more training
cases you have for your model, the better it becomes at predictions. Apple, in this scenario, are hamstrung by their own privacy policies. As an iPhone encrypts and holds data on the device itself,
Apple has little user data to exploit. A former Apple employee told Reuters that Siri retains user data for six months, but Apple Maps user data can be gone in as little as 15 minutes. This pales in
comparison to the amount of data that Google aggregates from Android users around the globe. Potentially, this disparity may stifle Apple advances in AI-driven technology, and especially where big
data is essential to refine and perfect the learning process.
Eventually we may see Neural networks running on our mobile devices. Mobile device may have the ability to conduct machine learning tasks locally, opening up a wide range of opportunities for object
recognition, speech, face detection, and other innovations for mobile platforms. | {"url":"https://www.juxt.pro/blog/deep-learning/","timestamp":"2024-11-07T19:37:38Z","content_type":"text/html","content_length":"64232","record_id":"<urn:uuid:9d43cd91-7a84-4b99-bf6a-7f0f6e18804e>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00639.warc.gz"} |
Kinetic theory of the high-frequency part of the slow electromagnetic mode in weakly ionized gas-discharge argon plasma with inelastic collisions
The spectral characteristics of the high-frequency part of the slow electromagnetic mode, specific for plasmas placed in an external d.c. electric field, as well as the features of the corresponding
instability, are analysed for weakly ionized argon gas-discharge plasmas with E/n ranging from 25 to 150 Td, and with electron temperatures between 60000 and 70000 K. The analysis is based on the
linear theory of perturbation, and the dynamics of the electrons is described by appropriately modified kinetic equations for the one-particle distribution function. Attention is focused on the
collisional processes between electrons and neutrals, and both elastic and excitational collisions are taken into account. Apart from the ‘indirect’ collision effects (modifications of the form of
the electron steady-state distribution function, evaluated here analytically, with the thermal motion of the neutrals included), their ‘direct’ influence (arising from perturbations of the collision
integrals) is also significant in the electron temperature range considered. As a consequence of the ‘direct’ influence of inelastic collisions, in particular, the mode studied was found to exist in
two distinctly separate wavelength ranges. The instability was found to develop only in the one corresponding to shorter wavelengths (below some 30 cm).
Journal of Plasma Physics
Pub Date:
October 1987
□ Argon Plasma;
□ Collisional Plasmas;
□ Kinetic Theory;
□ Plasma Frequencies;
□ Electron Distribution;
□ Electron Energy;
□ Magnetohydrodynamic Stability;
□ Spectral Energy Distribution;
□ Plasma Physics | {"url":"https://ui.adsabs.harvard.edu/abs/1987JPlPh..38..223Z/abstract","timestamp":"2024-11-07T19:12:10Z","content_type":"text/html","content_length":"39787","record_id":"<urn:uuid:23c50efe-f837-425e-b15b-d7db19099cf5>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00038.warc.gz"} |
Self-similarity of the third type in ultra-relativistic blastwaves
A new type of self-similarity is found in the problem of a plane-parallel, ultra-relativistic blastwave, propagating in a power-law density profile of the form ρ ∝ z − k . Self-similar solutions of
the first kind can be found for k < 7∕4 using dimensional considerations. For steeper density gradients with k > 2, second type solutions are obtained by eliminating a singularity from the equations.
However, for intermediate power-law indices 7 / 4 < k < 2 , the flow does not obey any of the known types of self-similarity. Instead, the solutions belong to a new class in which the self-similar
dynamics are dictated by the non-self-similar part of the flow. We obtain an exact solution to the ultra-relativistic fluid equations and find that the non-self-similar flow is described by a
relativistic expansion into vacuum, composed of (1) an accelerating piston that contains most of the energy and (2) a leading edge of a fast material that coincides with the interiors of the
blastwave and terminates at the shock. The dynamics of the piston itself are self-similar and universal and do not depend on the external medium. The exact solution of the non-self-similar flow is
used to solve for the shock in the new class of solutions.
Bibliographical note
Publisher Copyright:
© 2024 Author(s).
Dive into the research topics of 'Self-similarity of the third type in ultra-relativistic blastwaves'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/self-similarity-of-the-third-type-in-ultra-relativistic-blastwave","timestamp":"2024-11-10T08:56:20Z","content_type":"text/html","content_length":"49009","record_id":"<urn:uuid:5d71f149-6f05-4abf-b610-019f4e293c61>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00233.warc.gz"} |
Scattering of electromagnetic waves by many small perfectly conducting or impedance bodies
Citation: Ramm, A. G. (2015). Scattering of electromagnetic waves by many small perfectly conducting or impedance bodies. Journal of Mathematical Physics, 56(9), 21. doi:10.1063/1.4929965
A theory of electromagnetic (EM) wave scattering by many small particles of an arbitrary shape is developed. The particles are perfectly conducting or impedance. For a small impedance particle of an
arbitrary shape, an explicit analytical formula is derived for the scattering amplitude. The formula holds as a -> 0, where a is a characteristic size of the small particle and the wavelength is
arbitrary but fixed. The scattering amplitude for a small impedance particle is shown to be proportional to a(2-k), where k epsilon [0,1) is a parameter which can be chosen by an experimenter as he/
she wants. The boundary impedance of a small particle is assumed to be of the form zeta = ha(-k), where h = const, Reh >= 0. The scattering amplitude for a small perfectly conducting particle is
proportional to a(3), and it is much smaller than that for the small impedance particle. The many-body scattering problem is solved under the physical assumptions a << d << lambda, where d is the
minimal distance between neighboring particles and lambda is the wavelength. The distribution law for the small impedance particles is N(Delta) similar to 1/a(2-k) integral N-Delta(x) dx as a -> 0.
Here, N(x) >= 0 is an arbitrary continuous function that can be chosen by the experimenter and N(.) is the number of particles in an arbitrary sub-domain Delta. It is proved that the EM field in the
medium where many small particles, impedance or perfectly conducting, are distributed, has a limit, as a -> 0 and a differential equation is derived for the limiting field. On this basis, a recipe is
given for creating materials with a desired refraction coefficient by embedding many small impedance particles into a given material. (C) 2015 AIP Publishing LLC. | {"url":"https://krex.k-state.edu/items/a4c672f5-8e69-4b23-87a7-a4ff83ea7fcb","timestamp":"2024-11-09T18:53:39Z","content_type":"text/html","content_length":"470008","record_id":"<urn:uuid:2a5a6509-ba03-4736-8cf0-ae2c56139bc5>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00517.warc.gz"} |
(1) Simple Interest.pdf [6lk9exk5o3q4]
Simple interest
When people need to secure funds for some purposes, one of the ways they usually resort to is borrowing. On the other hand, the person or institution, which lends the money would also wish to get
something in return for the use of money.
the person who borrows money for any purpose
the person or institution which loans the money
the payment for the use of borrowed money the capital or sum of money invested
the fractional part of the principal that is paid on the loan the number of units of the time for which the money is borrowed and for which interest is calculated the sum of the principal and
interest which is accumulated at a certain time the amount received by the borrower
interest in which only the original principal bears interest for the entire term of the loan
interest added to the principal at the end of a certain period of time after which the interest is computed on the new principal, and this process is repeated until the end of the term of the loan is
John F borrows P10,000 at the rate of 12% per year. If the loan is a simple interest loan, then the interest on P10,000 is P1,200. At the end of one year, John F should pay the lender a total amount
of P11, 200.
Formula: π °=π ·β π β π
π =π ·+π ° π °=π β π ·
π Ό - simple interest π - principal amount π - rate or percent of interest π ‘ - units of time (usually in years)
π Ό - simple interest π - principal amount π Ή - final amount
The term or time may be stated in any of the following ways: When the time is expressed in number of year(s), our formula will be:
π Ό= π Γ π Γ π π π π π π π π π π π π π When the time is expressed in number of month(s): π π π π π π π π π π π π π π π Ό= π Γ
π Γ 12
Express the following in decimal form:
1.) 1.25% 2.) 96% 3.) 5 1β 2% 4.) 25 3β 4% 5.) 0.17% Express the following in years: 1.) 36 months 2.) 6 months 3.) 75 months 4.) 15 months
Example 1 Find the interest and amount on P800 at 6 Β½ % simple interest for 5 years.
Example 2 Find the interest and amount on P900 at 7 ΒΌ% simple interest for 9 months.
Find the missing value: 1.) π = P 2,300 π = 4% π ‘= 3 ΒΌ years 2.)
π = P 5,500 π Ό= 610 π = 7 7β 8%
Example 1 If a principal of P 2,500 earns interest of P185 in 3 years and 3 months, what interest rate is in effect?
Example 2 A principal earns interest of P 385 in 2 years and 9 months at a simple interest rate of 9.5 %. Find the principal invested.
Example 3 How long will it take for P8,000 to earn P2, 400, if it is invested at 6Β½% simple interest? | {"url":"https://doku.pub/documents/1-simple-interestpdf-6lk9exk5o3q4","timestamp":"2024-11-07T06:18:47Z","content_type":"text/html","content_length":"29857","record_id":"<urn:uuid:36d7d3ec-7a39-4304-b9b4-88796761c644>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00254.warc.gz"} |
NCERT solution of Chapter 5 Exercise 5.5 Class 12 Maths - Infinity Learn by Sri Chaitanya
Study MaterialsNCERT SolutionsNCERT Solutions for Class 12NCERT solution of Chapter 5 Exercise 5.5 Class 12 Maths
NCERT solution of Chapter 5 Exercise 5.5 Class 12 Maths
Class 12 Maths is a comprehensive course that includes important chapters based on advanced ideas. These chapters have been introduced to help pupils expand their expertise. They will be able to lay
a solid foundation for themselves and perform well in board exams and competitive tests. The continuity and differentiability are the topics of the fifth chapter of Class 12 Maths. The Ex 5.5 Class
12 Maths NCERT Solutions can be used to solve this new chapter. For the benefit of the students, Infinity Learn’s experienced Maths teachers have described the solutions to the specific task.
You may quickly grasp new concepts and apply them to solve problems on your own by following Exercise 5.5 Class 12 Maths NCERT Solutions. Follow the road established by Exercise 5.5 Class 12 Maths to
develop your problem-solving skills and finish preparing this chapter.
Do you need help with your Homework? Are you preparing for Exams? Study without Internet (Offline)
Exercise 5.5
NCERT solution of Class 12 Chapter 5 Exercise 5.5
For scientific students, mathematics is an extremely important topic. It also lays the groundwork for other subjects. Solving mathematical problems in higher courses develop problem-solving and
analytical skills. Solving the next chapter’s conceptual challenges can be scary. Students like to use NCERT Solutions for Class 12 Maths Chapter 5.5. Exercise 5.5 in Chapter 5 Maths Class 12 helps
assess a student’s problem-solving abilities after completing learning continuity and differentiability. All courses in Class 12 Mathematics have a large syllabus. Having to deal with all of the
things at once might be overwhelming. To avoid confusion, you must be very perceptive when constructing your study curriculum utilizing the NCERT Class 12 Maths Chapter 5 Exercise 5.5 Solutions. Now
is the time to think wiser and create a well-organized study schedule. To do so, you’ll need NCERT Solutions for Class 12 Maths Ex 5.5 to finish your study materials and get started studying.
It would help if you considered dividing the chapters according to their weight in the board exams once you have a good comprehension of the subject’s syllabus. When you consider the value of
understanding continuity and differentiability, you’ll see how crucial it is to comprehend functions. Later in the advanced chapters of higher mathematics, you’ll see how these new concepts are
applied. Use the NCERT Maths Class 12 Exercise 5.5 Solutions to help you understand your concepts. It will be extremely simple to understand Chapter 5 Class 12 Maths Exercise 5.5. The concepts in
NCERT Solutions Class 12 Maths Chapter 5 Ex 5.5 have been perfectly presented in a simpler style. You may rest assured that completing Maths Class 12 Chapter 5 Exercise 5.5 would be a breeze if you
follow the qualified teachers’ lead.
Why should you Learn Continuity and Differentiability Exercise 5.5 with Infinity Learn?
Exercise 5.5 Class 12 Maths NCERT Solutions produced by expert teachers are available at Infinity Learn. They understand students' normal skepticism when learning new ideas like continuity and
differentiability. Complete Exercise 5.5 Class 12 Maths by finding the exact answers to your doubts described simply.
How is the 'Infinity Learn' study material for Class 12 Maths Chapter 5 prepared?
On Infinity Learn, the study material for Class 12 Maths is organized by chapter. You won't have any complaints about the curriculum because the mentors here are the best, with extensive knowledge in
the subject field, and can answer any questions you may have. Class 12 Chapter 5 solutions are accessible for free on the Infinity Learn website and the Infinity Learn app.
Apart from practicing for Chapter 5, how can I achieve complete marks in Class 12 Boards Maths?
Practice makes perfect, and mathematics is nothing more than practice. Brushing up on workouts regularly will lead to perfection and accuracy. It will help you improve your pace and finish your paper
on time. It will also assist you in obtaining a perfect score on this topic. You can do this by visiting the Infinity Learn website or downloading the Infinity Learn app, which will give you the best
solutions for all chapters at no cost, allowing you to achieve your goal of getting top scores in Math. | {"url":"https://infinitylearn.com/surge/study-materials/ncert-solutions/class-12/maths/chapter-5-continuity-and-differentiability-exercise-5-5/","timestamp":"2024-11-07T10:56:25Z","content_type":"text/html","content_length":"175841","record_id":"<urn:uuid:f2fef1a3-4842-4f7f-a71e-d3f86affa591>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00298.warc.gz"} |
Quantum Chaos @ Bernoulli
While the concept of chaos is well-established for classical interacting systems, a quantum mechanical formulation of chaos has yet to attain a comparable level of maturity. On one hand, classical
notions like the butterfly effect need to be reimagined to accommodate the absence of phase-space trajectories and the inherent linearity of quantum dynamics. On the other hand, the operatorial
formulation of quantum mechanics introduces fundamental objects such as spectral statistics, offering crucial insights into dynamics but without immediate classical counterparts.
The significance of quantum chaos goes beyond the foundations of quantum statistical mechanics; it plays a pivotal role in addressing the information paradox in black holes and more generally in
gauge-gravity dualities. Moreover, as the field of quantum optics makes remarkable progress in incorporating increasingly many controllable quantum degrees of freedom in its devices, the onset of
chaos and the subsequent scrambling of information have become pressing practical concerns for the development of quantum-information processing schemes.
This intricate yet incomplete landscape prompts various communities across physics and mathematics, with diverse cultures and methodologies, to tackle the challenges posed by quantum chaos. This
workshop aims to bring together experts in quantum chaos from high-energy physics, hybrid quantum systems (both theory and experiment), quantum information, condensed matter, mathematical physics,
random matrices, and free probability to foster interactions and collaborations.
the external participants are staying in the Starling Hotel Lausanne (Map), which is a 10 minutes walk away from the Bernoulli Center. | {"url":"https://indico.psi.ch/event/16539/timetable/?view=standard_numbered_inline_minutes","timestamp":"2024-11-15T00:20:28Z","content_type":"text/html","content_length":"150380","record_id":"<urn:uuid:91c115d7-2312-40c2-87a9-2b38fdf3ac28>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00093.warc.gz"} |
Understanding Liquidity Ratio And Its Importance For The
By Alexander Shishkanov
Alexander Shishkanov has several years of experience in the crypto and fintech industry and is passionate about exploring blockchain technology. Alexander writes on topics such as cryptocurrency,
fintech solutions, trading strategies, blockchain development and more. His mission is to educate individuals about how this new technology can be used to create secure, efficient and transparent
financial systems.
Understanding Liquidity Ratio And Its Importance For The Company
Imagine a famous tech giant that just bought a lot of new equipment to increase production. Suddenly, a main supplier asks for an immediate payment, which strains the company’s available cash.
Despite having a wealth of assets like new machines, the company finds it hard to pay the unexpected expense.
This situation highlights the importance of liquidity – the ability to turn assets into cash to cover immediate debts quickly. One key indication of a company’s financial health and operational
efficacy is its liquidity ratio, which measures its ability to pay its short-term obligations. But what is liquidity ratio, exactly, and how can you calculate it?
Key Takeaways
1. The liquidity ratio is a financial metric showing a company’s capacity to cover its short-term debts with its liquid assets.
2. These ratios reflect a company’s financial stability, helping investors and creditors evaluate its readiness to handle short-term obligations.
3. There are three main types: current ratio, quick ratio, and cash ratio, each providing a different lens on a company’s liquidity.
4. High liquidity ratios generally suggest better financial health. However, extremely high or low ratios might signal financial inefficiency or liquidity risks. Benchmarks for good liquidity are
typically a current ratio of 2:1, a quick ratio of 1:1, and a cash ratio of 0.2:1.
5. Companies can boost liquidity ratios by controlling overheads, selling excess assets, adjusting payment cycles, securing lines of credit, and reevaluating debt strategies.
Liquidity ratio is a financial metric that measures a company’s ability to meet its short-term obligations and manage its current liabilities. It provides valuable insights into a company’s liquidity
position and capacity to promptly fulfill financial commitments.
The liquidity ratio is derived by comparing a company’s current assets, such as cash, tradable securities, accounts receivable, and inventory, to its current liabilities, which include short-term
debts and obligations due within a year. By analyzing these ratios, investors, creditors, and financial analysts can evaluate a company’s ability to pay and handle its financial responsibilities
Why Does The Liquidity Ratio Matter?
The liquidity ratio can tell a lot about the company that can determine its reputation and success on the market. Here are some critical points:
Fulfillment of Short-term Obligations
One crucial aspect of these ratios is their ability to gauge a company’s capacity to fulfill short-term financial commitments. Ideally, a liquidity ratio of 2 or 3 suggests a company’s strong
position to manage immediate liabilities. Conversely, a ratio below one might denote a negative working capital scenario, indicating potential liquidity challenges.
Determination of Creditworthiness
Liquidity ratios play a vital role in determining a company’s creditworthiness. Creditors scrutinize these ratios to verify a company’s financial capacity to repay debts. Indications of financial
instability might deter loan provisions, potentially labeling the company as a risky borrower.
Verification of Investment Viability
Investors employ liquidity ratios to assess a company’s financial health and investment viability. A robust working capital scenario is attractive to investors as it assures operational flexibility,
enabling the company to manage unforeseen circumstances without adverse operational implications.
Balancing Liquid Assets and Profitable Investments
While a company needs to maintain a liquidity ratio that allows for safe coverage of bills, an excessively high ratio could signify mismanagement of resources. It suggests that the company might be
retaining an unnecessary cash surplus, which isn’t being utilized effectively. The company can potentially increase its returns and enhance shareholder value by redirecting such reserves to
higher-yield investments.
Ensuring Operational Sustainability
Another pivotal role of these ratios is their contribution to business continuity. A company exhibiting consistent strong liquidity ratios signifies financial resilience, suggesting an ability to
sustain operations amidst uncertain events or economic downturns. This financial strength instills confidence in all stakeholders, including employees, suppliers, and customers.
Impact on Company’s Reputation
liquidity ratios can have a significant impact on a company’s market reputation. Steady, healthy liquidity ratios may serve as indicators of financial stability, positively influencing the company’s
image. This could affect the decision-making process of stakeholders, including investors, creditors, suppliers, and customers. On the other hand, poor liquidity ratios can potentially tarnish a
company’s reputation, making it challenging to attract investments or negotiate favorable credit terms.
Vendor and Supplier Negotiations
Such ratios can also play a role in negotiations with suppliers and vendors. Suppliers may extend more favorable credit terms or discounts to companies with strong liquidity, as they present less
risk of late or defaulted payments.
Employee Satisfaction and Retention
Lastly, liquidity ratios can indirectly impact employee satisfaction and retention. Employees want assurance of the company’s ability to meet payroll obligations and other employee-related expenses.
A healthy liquidity ratio can help foster a sense of financial security, contributing to job satisfaction and employee retention.
Types of Liquidity Ratios and Their Calculations
In financial analysis, several liquidity ratios are commonly used, each offering a different perspective on a company’s liquidity status. You can choose the relevant formula depending on what you
want to analyze.
The current ratio is the most straightforward measurement of liquidity. It represents the company’s ability to pay off its short-term debts. To calculate liquidity ratio, use the following formula:
Current Ratio = Current Assets / Current Liabilities
Cash, marketable securities, accounts receivable, and inventory are all examples of current assets. Meanwhile, “current liabilities” refers to debts expected to be paid off within a year.
Unlike the current ratio, which includes inventories in current assets, the Acid-Test ratio, or quick ratio, does not, making it a more strict measure of liquidity. The quick ratio formula is as
Quick Ratio = (Cash + Marketable Securities + Accounts Receivable) / Current Liabilities
Marketable securities typically comprise quickly sellable assets like stocks or bonds.
The cash ratio, also known as the absolute liquidity ratio, provides an even more conservative view by only comparing the most liquid assets – cash and marketable securities – with current
liabilities. The formula for this ratio is:
Each of these ratios allows a unique glimpse into a company’s liquidity position and can be used to understand the business’s financial health better.
Liquidity Ratio Analysis: An Illustrative Example
Let’s consider a company with the following figures in its balance sheet:
• Cash: $60,000
• Marketable Securities: $30,000
• Accounts Receivable: $120,000
• Inventories: $40,000
• Current Liabilities (Accounts Payable): $100,000
Applying the formulas from the previous section, we can compute the company’s liquidity ratios:
• Current Ratio = ($60,000 + $30,000 + $120,000 + $40,000) / $100,000 = 2.5 or 250%
• Quick Ratio = ($60,000 + $30,000 + $120,000) / $100,000 = 2.1 or 210%
• Cash Ratio = ($60,000 + $30,000) / $100,000 = 0.9 or 90%
Analysis & Interpretation of Liquidity Ratios
These calculations shed light on the firm’s financial fluidity. For instance, a current ratio of 250% signifies that the firm’s short-term assets surpass its short-term liabilities by 2.5 times—an
encouraging sign of monetary stability.
Nevertheless, when we look at the quick ratio—a more rigorous liquidity measurement—it dwindles to 210%. This reduction suggests the firm possesses the financial capacity to satisfy all its
liabilities swiftly without resorting to the sale of inventory assets.
Lastly, the cash ratio, the strictest measure of liquidity, further descends to 90%. This figure suggests that if the firm were required to settle all its short-term obligations instantaneously, it
could cover 90% of them using its most readily available assets. To fulfill the remaining liabilities, it would either need to liquidate a part of its inventory or await the inflow from accounts
What Is a Good Liquidity Ratio?
While it may seem that the higher liquidity ratio is, the better, it’s not always the case.
For the current ratio, a benchmark of 200% is considered solid—it indicates that the company has twice the amount of current assets required to pay off its short-term liabilities.
In the case of the quick ratio, a value of 100% is ideal, indicating the firm can cover its short-term liabilities without the need to sell its inventories.
For the cash ratio, a value of 20% is considered adequate. Although it suggests that the firm could cover only a fifth of its short-term liabilities with its most liquid assets, many companies accept
this risk to fuel growth. A high cash ratio implies idle cash, which could be invested for further expansion.
To summarize, the acceptable benchmarks for liquidity ratios are:
• Current Ratio: 200%
• Quick Ratio: 100%
• Cash Ratio: 20%
Differentiating Liquidity and Solvency Ratios
Solvency is another critical metric used to evaluate a company’s financial health. However, unlike liquidity ratios, primarily concerned with short-term financial obligations, solvency ratios assess
a company’s capacity to meet all its financial commitments, including long-term debts.
In essence, while liquidity zeroes in on a company’s short-term financial position, solvency takes a broader view of its ability to sustain operations and repay debts over the long haul.
To be solvent, a company’s total assets must exceed its total liabilities. Similarly, for a company to be considered liquid, its current assets must surpass its current liabilities. Even though
solvency isn’t directly linked to liquidity, liquidity ratios can provide an initial gauge of a company’s solvency.
The solvency ratio is computed by dividing a company’s net income plus depreciation by its total (short-term and long-term) liabilities. It offers insight into whether the company’s net earnings
sufficiently cover its total liabilities. A higher solvency ratio typically signifies a more creditworthy and appealing investment.
How Can A Company Improve Its Liquidity Ratio?
Improving the liquidity ratio not only enhances a company’s financial strength but also boosts the confidence of investors and lenders. Here are a few strategies a company can employ to enhance its
liquidity ratio:
1. Supervise Overhead Expenditures
The first pivotal step a company can take is efficiently managing overhead expenses. This can include items like rent, insurance, utilities, and more. Negotiating better deals, shopping for more
affordable options, and streamlining operations such as digitizing paperwork can help trim these costs.
2. Liquidate Nonessential Assets
Selling off surplus or underutilized assets can be a smart move. This strategy provides an immediate boost to the company’s liquid assets and can also lower maintenance costs tied to these assets.
3. Modify the Payment Cycle
Adjusting the payment cycle with both suppliers and customers can enhance liquidity. Negotiating early payment discounts with vendors can result in cost savings while incentivizing customers to pay
ahead of schedule can increase cash inflows.
4. Utilize Lines of Credit
A business line of credit can serve as a valuable buffer for managing intermittent cash flow gaps, thereby improving liquidity. However, it’s crucial to meticulously review the terms and conditions
of different credit offerings before committing.
5. Reconsider Debt Arrangements
Reassessing a company’s debt structure can positively impact liquidity. Transitioning short-term debts to long-term arrangements can lessen monthly payments and alleviate immediate financial
Conversely, shifting from long-term to short-term debt may raise monthly payments but can speed up clearing debt. Debt consolidation or loan refinancing can lower monthly payments and lead to
long-term monetary benefits.
6. Strengthen Cash Flow Management
Effective cash flow management is vital. Implementing strategies such as punctual invoicing, monitoring receivables closely, and maintaining a cash reserve for unforeseen expenses can contribute to
better liquidity.
7. Diversify Income Sources
Expanding the company’s sources of revenue can boost liquidity. By venturing into new markets or introducing new products or services, a company can increase its revenue, thereby enhancing its
liquidity position.
8. Enhance Inventory Management
Finally, effective inventory management can help free up cash tied up in unsold goods. Employing strategies like just-in-time inventory management allows a company to align its inventory purchases
more closely with demand, reducing the cash held up in stock.
Overall, liquidity ratios are instrumental in dissecting a company’s financial robustness and capacity to meet short-term debts. Their careful interpretation can prevent financial hiccups,
safeguarding stakeholders and the company’s reputation.
What is a good liquidity ratio?
A good liquidity ratio varies by industry and specific circumstances, but a current ratio of 2:1 is generally considered solid, indicating a company has twice as many current assets as
liabilities. A 1:1 ratio is desirable for the quick ratio, and a cash ratio of at least 0.2:1 is considered sound.
Does high liquidity mean high risk?
No, high liquidity does not mean high risk. High liquidity often implies lower risk, indicating that a company can quickly meet its short-term financial obligations. However, excessively high
liquidity could suggest the company is not efficiently using its assets to generate profits.
What companies have the best liquidity ratios?
Pinterest, Shopify, Twilo, Beyond Meat, and Twitter take the top positions as the top companies with high current liquidity ratios.
Seeking answers or advice?
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Sébastien Loriot, Olga Sorkine-Hornung, Yin Xu and Ilker O. Yaz
This package offers surface mesh deformation algorithms which compute new vertex positions of a surface mesh under positional constraints of some of its vertices, without requiring any additional
structure other than the surface mesh itself
This package implements the algorithm described in [5] together with an alternative energy function [3]. The algorithm minimizes a nonlinear deformation energy under positional constraints to
preserve rigidity as much as possible. The minimization of the energy relies on solving sparse linear systems and finding closest rotation matrices.
A surface mesh deformation system consists of:
• a triangulated surface mesh (surface mesh in the following),
• a set of vertices defining the region to deform (referred to as the region-of-interest and abbreviated ROI),
• a subset of vertices from the ROI that the user wants to move (referred to as the control vertices),
• a target position for each control vertex (defining the deformation constraints).
A vertex from the ROI that is not a control vertex is called an unconstrained vertex. These definitions are depicted in Figure 53.1.
In this package, two algorithms are implemented:
• The As-Rigid-As-Possible (ARAP) method described in [5];
• The Spokes and Rims method [3].
Given an edge weighting scheme, both methods iteratively minimize an energy function and produce a different surface mesh at each step until convergence is reached.
Spokes and Rims is the default method proposed by the package. It provides an unconditional convergence while the ARAP method requires the edge weights to be positive. However, the results obtained
using the Spokes and Rims method are more dependent on the discretization of the deformed surface (See Figure 53.2).
More details on these algorithms are provided in section Deformation Techniques, Energies and Weighting Schemes.
User Interface Description
The deformation methods implemented rely on solving a sparse linear system. The sparse matrix definition depends on the weighting scheme and on the unconstrained and control vertices. The right term
depends only on the target positions of the control vertices.
The deformation process is handled by the class Surface_mesh_deformation and the surface mesh is represented as a halfedge data structure that must be a model of the HalfedgeGraph concept.
The class Surface_mesh_deformation provides two groups of functions for the preprocessing (sparse matrix definition) and the deformation (right hand term definition).
The preprocessing consists in computing a factorization of the aforementioned sparse matrix to speed up the linear system resolution. It requires the ROI to be defined. The following conventions are
used for the definition of the ROI:
• A vertex inserted in the set of control vertices is inserted in the ROI;
• A control vertex erased from the ROI is no longer considered as a control vertex;
• A control vertex that is erased is not erased from the ROI.
Each time the ROI is modified, the preprocessing function preprocess() must be called. Note that if it is not done, the first deformation step calls this function automatically and has a longer
runtime compared to subsequent deformation steps.
The function Surface_mesh_deformation::preprocess() returns true if the factorization is successful, and false otherwise.
Rank deficiency is the main reason for failure. Typical failure cases are:
• All the vertices are in the ROI and no control vertices are set
• The weighting scheme used to fill the sparse matrix (model of SurfaceModelingWeights) features too many zeros and breaks the connectivity information
The choice of the weighting scheme provides a mean to adjust the way the control vertices influences the unconstrained vertices. The defaults provides satisfactory results in general but other
weighting schemes may be selected or designed to experiment or improve the results in specific cases.
The ROI does not have to be a connected component of the graph of the surface mesh. However, for better performances it is better to use an individual instance of the deformation object for each
connected component.
The deformation of the surface mesh is triggered by the displacement of the control vertices. This is achieved through setting the target positions of the control vertices (directly or by using an
affine transformation to be applied to a control vertex or a range of control vertices).
Note that a rotation or a translation of a control vertex is always applied on its last target position set: they are cumulative.
The deformation of the surface mesh happens when calling the function Surface_mesh_deformation::deform(). The number of optimization iterations varies depending on whether the user chooses a fixed
number of iterations or a stopping criterion based on the energy variation.
After the call to the deformation function, the input surface mesh is updated and the control vertices are at their target positions and the unconstrained vertices are moved accordingly. The function
Surface_mesh_deformation::deform() can be called several times consecutively, in particular if the convergence has not been reached yet (otherwise it has no effect).
Vertices can be inserted into or erased from the ROI and the set of control vertices at any time. In particular, any vertex that is no longer inside the ROI will be assigned to its original
position when Surface_mesh_deformation::preprocess() is first called. The original positions can be updated by calling Surface_mesh_deformation::overwrite_initial_geometry() ( which will also
require a new preprocessing step). This behavior is illustrated in Video 1.
As-Rigid-As-Possible and Spokes-and-Rims Deformation Techniques
Two deformation techniques are provided by this package. This section summarizes from the user point of view what is explained in details in the section Deformation Techniques, Energies and Weighting
The As-Rigid-As-Possible deformation technique requires the use of a positive weighting scheme to guarantee the correct minimization of the energy. When using the default cotangent weighting scheme,
this means that the input surface mesh must be clean. That is, that for all edges in the surface mesh the sum of the angles opposite to the edge in the incident triangles is less that \( \pi \). If
this is not the case and the targeted application allows the modification of the surface mesh connectivity, a solution (amongst other) is to bissect (possibly recursively) the problematic edges. See
Figure 53.3.
If the mesh connectivity must be preserved, the Spokes and Rims deformation technique is guaranteed to always correctly minimize the energy even if the weights are negative. However, this technique
is more dependent on the discretization of the deformed surface (See Figure 53.2).
Using the Whole Surface Mesh as Region-of-Interest
In this example, the whole surface mesh is used as ROI and a few vertices are added as control vertices. Surface_mesh_deformation::set_target_position() is used for setting the target positions of
the control vertices.
File Surface_modeling/all_roi_assign_example.cpp
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Polyhedron_3.h>
#include <CGAL/Polyhedron_items_with_id_3.h>
#include <CGAL/IO/Polyhedron_iostream.h>
// HalfedgeGraph adapters for Polyhedron_3
#include <CGAL/boost/graph/graph_traits_Polyhedron_3.h>
#include <CGAL/boost/graph/properties_Polyhedron_3.h>
#include <CGAL/Surface_mesh_deformation.h>
#include <fstream>
typedef boost::graph_traits<Polyhedron>::vertex_descriptor vertex_descriptor;
typedef boost::graph_traits<Polyhedron>::vertex_iterator vertex_iterator;
int main()
Polyhedron mesh;
std::ifstream input("data/plane.off");
if ( !input || !(input >> mesh) || mesh.empty() ) {
std::cerr<< "Cannot open data/plane.off" << std::endl;
return 1;
// Init the indices of the halfedges and the vertices.
// Create a deformation object
Surface_mesh_deformation deform_mesh(mesh);
// Definition of the region of interest (use the whole mesh)
vertex_iterator vb,ve;
boost::tie(vb, ve) = vertices(mesh);
deform_mesh.insert_roi_vertices(vb, ve);
// Select two control vertices ...
// ... and insert them
// The definition of the ROI and the control vertices is done, call preprocess
bool is_matrix_factorization_OK = deform_mesh.preprocess();
std::cerr << "Error in preprocessing, check documentation of preprocess()" << std::endl;
return 1;
// Use set_target_position() to set the constained position
// of control_1. control_2 remains at the last assigned positions
deform_mesh.set_target_position(control_1, constrained_pos_1);
// Deform the mesh, the positions of vertices of 'mesh' are updated
// The function deform() can be called several times if the convergence has not been reached yet
// Set the constained position of control_2
deform_mesh.set_target_position(control_2, constrained_pos_2);
// Call the function deform() with one-time parameters:
// iterate 10 times and do not use energy based termination criterion
deform_mesh.deform(10, 0.0);
// Save the deformed mesh into a file
std::ofstream output("deform_1.off");
output << mesh;
// Add another control vertex which requires another call to preprocess
// The prepocessing step is again needed
std::cerr << "Error in preprocessing, check documentation of preprocess()" << std::endl;
return 1;
// Deform the mesh
deform_mesh.set_target_position(control_3, constrained_pos_3);
deform_mesh.deform(15, 0.0);
output << mesh;
Using an Affine Transformation on a Range of Vertices
In this example, we use the functions translate() and rotate() on a range of control vertices. Note that the translations and the rotations are defined using a 3D vector type and a quaternion type
from the Eigen library.
File Surface_modeling/k_ring_roi_translate_rotate_example.cpp
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Polyhedron_3.h>
#include <CGAL/IO/Polyhedron_iostream.h>
#include <CGAL/Polyhedron_items_with_id_3.h>
// HalfedgeGraph adaptors for Polyhedron_3
#include <CGAL/boost/graph/graph_traits_Polyhedron_3.h>
#include <CGAL/boost/graph/properties_Polyhedron_3.h>
#include <CGAL/Surface_mesh_deformation.h>
#include <fstream>
#include <map>
#include <queue>
typedef boost::graph_traits<Polyhedron>::vertex_descriptor vertex_descriptor;
typedef boost::graph_traits<Polyhedron>::vertex_iterator vertex_iterator;
typedef boost::graph_traits<Polyhedron>::halfedge_descriptor halfedge_descriptor;
typedef boost::graph_traits<Polyhedron>::out_edge_iterator out_edge_iterator;
typedef Eigen::Vector3d Vector3d;
// Collect the vertices which are at distance less or equal to k
// from the vertex v in the graph of vertices connected by the edges of P
std::vector<vertex_descriptor> extract_k_ring(const Polyhedron &P, vertex_descriptor v, int k)
std::map<vertex_descriptor, int> D;
std::vector<vertex_descriptor> Q;
Q.push_back(v); D[v] = 0;
std::size_t current_index = 0;
int dist_v;
while( current_index < Q.size() && (dist_v = D[ Q[current_index] ]) < k ) {
v = Q[current_index++];
out_edge_iterator e, e_end;
for(boost::tie(e, e_end) = out_edges(v, P); e != e_end; e++)
halfedge_descriptor he = halfedge(*e, P);
vertex_descriptor new_v = target(he, P);
if(D.insert(std::make_pair(new_v, dist_v + 1)).second) {
return Q;
int main()
Polyhedron mesh;
std::ifstream input("data/plane.off");
if ( !input || !(input >> mesh) || mesh.empty() ) {
std::cerr<< "Cannot open data/plane.off";
return 1;
// Init the indices of the halfedges and the vertices.
// Create the deformation object
Surface_mesh_deformation deform_mesh(mesh);
// Select and insert the vertices of the region of interest
vertex_iterator vb, ve;
boost::tie(vb,ve) = vertices(mesh);
std::vector<vertex_descriptor> roi = extract_k_ring(mesh, *
(vb, 47), 9);
deform_mesh.insert_roi_vertices(roi.begin(), roi.end());
// Select and insert the control vertices
std::vector<vertex_descriptor> cvertices_1 = extract_k_ring(mesh, *
(vb, 39), 1);
std::vector<vertex_descriptor> cvertices_2 = extract_k_ring(mesh, *
(vb, 97), 1);
deform_mesh.insert_control_vertices(cvertices_1.begin(), cvertices_1.end());
deform_mesh.insert_control_vertices(cvertices_2.begin(), cvertices_2.end());
// Apply a rotation to the control vertices
Eigen::Quaternion<double> quad(0.92, 0, 0, -0.38);
deform_mesh.rotate(cvertices_1.begin(), cvertices_1.end(), Vector3d(0,0,0), quad);
deform_mesh.rotate(cvertices_2.begin(), cvertices_2.end(), Vector3d(0,0,0), quad);
// Save the deformed mesh
std::ofstream output("deform_1.off");
output << mesh;
// Restore the positions of the vertices
// Apply a translation on the original positions of the vertices (reset() was called before)
deform_mesh.translate(cvertices_1.begin(), cvertices_1.end(), Vector3d(0,0.3,0));
deform_mesh.translate(cvertices_2.begin(), cvertices_2.end(), Vector3d(0,0.3,0));
// Call the function deform() with one-time parameters:
// iterate 10 times and do not use energy based termination criterion
// Save the deformed mesh
output << mesh;
Using Polyhedron without Ids
In the previous examples, we used an enriched polyhedron storing an ID in its halfedges and vertices together with the default property maps in the deformation object to access them. In the following
example, we show how we can use alternative property maps.
For practical performance however we recommend relying upon the former examples instead, as using a std::map to access indices increases the complexity from constant to logarithmic.
File Surface_modeling/deform_polyhedron_with_custom_pmap_example.cpp
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Polyhedron_3.h>
#include <CGAL/IO/Polyhedron_iostream.h>
// Halfedge adaptors for Polyhedron_3
#include <CGAL/boost/graph/graph_traits_Polyhedron_3.h>
#include <CGAL/boost/graph/properties_Polyhedron_3.h>
#include <CGAL/property_map.h>
#include <CGAL/Surface_mesh_deformation.h>
#include <fstream>
typedef boost::graph_traits<Polyhedron>::vertex_descriptor vertex_descriptor;
typedef boost::graph_traits<Polyhedron>::vertex_iterator vertex_iterator;
typedef boost::graph_traits<Polyhedron>::halfedge_descriptor halfedge_descriptor;
typedef boost::graph_traits<Polyhedron>::halfedge_iterator halfedge_iterator;
// Define the maps
typedef std::map<vertex_descriptor, std::size_t> Vertex_id_map;
typedef std::map<halfedge_descriptor, std::size_t> Hedge_id_map;
typedef boost::associative_property_map<Vertex_id_map> Vertex_id_pmap;
typedef boost::associative_property_map<Hedge_id_map> Hedge_id_pmap;
int main()
Polyhedron mesh;
std::ifstream input("data/plane.off");
if ( !input || !(input >> mesh) || mesh.empty() ) {
std::cerr<< "Cannot open data/plane.off";
return 1;
// Init the indices of the vertices from 0 to num_vertices(mesh)-1
Vertex_id_map vertex_index_map;
vertex_iterator vb, ve;
std::size_t counter = 0;
for(boost::tie(vb, ve) = vertices(mesh); vb != ve; ++vb, ++counter)
// Init the indices of the halfedges from 0 to 2*num_edges(mesh)-1
Hedge_id_map hedge_index_map;
counter = 0;
halfedge_iterator eb, ee;
for(boost::tie(eb, ee) = halfedges(mesh); eb != ee; ++eb, ++counter)
Surface_mesh_deformation deform_mesh( mesh,
Hedge_id_pmap(hedge_index_map) );
// Now deform mesh as desired
// .....
Using a Custom Edge Weighting Scheme
Using a custom weighting scheme for edges is also possible if one provides a model of SurfaceModelingWeights. In this example, the weight of each edge is pre-computed and an internal map is used for
storing and accessing them.
Another example is given in the manual page of the concept SurfaceModelingWeights.
File Surface_modeling/custom_weight_for_edges_example.cpp
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Polyhedron_3.h>
#include <CGAL/IO/Polyhedron_iostream.h>
// HalfedgeGraph adapters for Polyhedron_3
#include <CGAL/boost/graph/graph_traits_Polyhedron_3.h>
#include <CGAL/boost/graph/properties_Polyhedron_3.h>
#include <CGAL/Surface_mesh_deformation.h>
#include <fstream>
#include <map>
#include <CGAL/property_map.h>
typedef boost::graph_traits<Polyhedron>::vertex_descriptor vertex_descriptor;
typedef boost::graph_traits<Polyhedron>::vertex_iterator vertex_iterator;
typedef boost::graph_traits<Polyhedron>::halfedge_descriptor halfedge_descriptor;
typedef boost::graph_traits<Polyhedron>::halfedge_iterator halfedge_iterator;
typedef std::map<vertex_descriptor, std::size_t> Internal_vertex_map;
typedef std::map<halfedge_descriptor, std::size_t> Internal_hedge_map;
typedef boost::associative_property_map<Internal_vertex_map> Vertex_index_map;
typedef boost::associative_property_map<Internal_hedge_map> Hedge_index_map;
// A model of SurfaceModelingWeights using a map of pre-computed weights
struct Weights_from_map
typedef Polyhedron Halfedge_graph;
Weights_from_map(std::map<halfedge_descriptor, double>* weight_map) : weight_map(weight_map)
{ }
template<class VertexPointMap>
double operator()(halfedge_descriptor e, Polyhedron& /*P*/, VertexPointMap /*vpm*/) {
return (*weight_map)[e];
std::map<halfedge_descriptor, double>* weight_map;
int main()
Polyhedron mesh;
std::ifstream input("data/plane.off");
if ( !input || !(input >> mesh) || mesh.empty() ) {
std::cerr << "Cannot open data/plane.off" << std::endl;
return 1;
std::map<halfedge_descriptor, double> weight_map;
// Store all the weights
halfedge_iterator eb, ee;
for(boost::tie(eb, ee) = halfedges(mesh); eb != ee; ++eb)
weight_map[*eb] = 1.0; // store some precomputed weights
// Create and initialize the vertex index map
Internal_vertex_map internal_vertex_index_map;
Vertex_index_map vertex_index_map(internal_vertex_index_map);
vertex_iterator vb, ve;
std::size_t counter = 0;
for(boost::tie(vb, ve) = vertices(mesh); vb != ve; ++vb, ++counter) {
put(vertex_index_map, *vb, counter);
// Create and initialize the halfedge index map
Internal_hedge_map internal_hedge_index_map;
Hedge_index_map hedge_index_map(internal_hedge_index_map);
counter = 0;
for(boost::tie(eb, ee) = halfedges(mesh); eb != ee; ++eb, ++counter) {
put(hedge_index_map, *eb, counter);
Surface_mesh_deformation deform_mesh(mesh,
// Deform mesh as desired
// .....
How to Use the Demo
A plugin for the polyhedron demo is available to test the algorithm. The following video tutorials explain how to use it. When the deformation dock window is open, the picking of control vertices and
of the ROI is done by pressing Shift and clicking with the left button of the mouse. The displacement of the vertices is triggered when the Ctrl button is pressed.
Deformation Techniques, Energies and Weighting Schemes
This section gives the theoretical background to make the user manual self-contained and at the same time explains where the weights comes in. This allows advanced users of this package to tune the
weighting scheme by developing a model of the concept SurfaceModelingWeights used in the class Surface_mesh_deformation.
Laplacian Representation
The Laplacian representation (referred to as Laplace coordinates in [2]) of a vertex in a surface mesh is one way to encode the local neighborhood of a vertex in the surface mesh. In this
representation, a vertex \( \mathbf{v}_i \) is associated a 3D vector defined as:
\[ $$L(\mathbf{v}_i) = \sum_{\mathbf{v}_j \in N(\mathbf{v}_i)} w_{ij}(\mathbf{v}_i - \mathbf{v}_j), \label{eq:lap_open}$$ \]
• \(N(\mathbf{v}_i)\) denotes the set of vertices adjacent to \(\mathbf{v}_i\);
• \(w_{ij}\) denotes a weight for the directed edge \(\mathbf{v}_i\ \mathbf{v}_j\).
The simplest choice for the weights is the uniform scheme where \( w_{ij}=1/|N(\mathbf{v}_i)| \) for each adjacent vertex \(\mathbf{v}_j\). In this case, the Laplacian representation of a vertex is
the vector between this vertex and the centroid of its adjacent vertices (Figure 53.6).
In the surface mesh deformation context, a popular choice is the cotangent weight scheme that derives from the discretization of the Laplace operator [4] : Given an edge of the surface mesh, its
corresponding cotangent weight is the mean of the cotangents of the angles opposite to the edge. It was shown to produce results that are not biased by the surface mesh of the approximated surface
Considering a surface mesh with \(n\) vertices, it is possible to define its Laplacian representation \(\Delta\) as a \(n \times 3\) matrix:
\[ $$\mathbf{L}\mathbf{V} = \Delta, \label{eq:lap_system}$$ \]
• \(\mathbf{L}\) is a \(n \times n\) sparse matrix, referred to as the Laplacian matrix. Its elements \( m_{ij} \), \(i,j \in \{1 \dots n\} \) are defined as follows:
□ \( m_{ii} \) = \( \sum_{\mathbf{v}_j \in N(\mathbf{v}_i)}w_{ij} \),
□ \( m_{ij} = -w_{ij} \) if \( \mathbf{v}_j \in N(\mathbf{v_i}) \),
□ \( 0 \) otherwise.
• \(\mathbf{V}\) is a \(n \times 3\) matrix made of the Cartesian coordinates of the vertices.
Laplacian Deformation
This section is an introduction to provide the background for the next two sub-sections describing the algorithms implemented in this package. A system relying only on the approach described below
results in non-smooth transitions in the neighborhood of the control vertices. For a survey on different Laplacian-based editing techniques we refer to [1].
The main idea behind Laplacian-based deformation techniques is to preserve the Laplacian representation under deformation constraints. The Laplacian representation of a surface mesh is treated as a
representative form of the discretized surface, and the deformation process must follow the deformation constraints while preserving the Laplacian representation as much as possible.
There are different ways to incorporate deformation constraints into the deformation system [1]. This package supports hard constraints, that is, target positions of control vertices are preserved
after the deformation.
Given a surface mesh deformation system with a ROI made of \( n \) vertices and \( k \) control vertices, we consider the following linear system:
\[ $$\left[ \begin{array}{ccc} \mathbf{L}_f\\ 0 \; \mathbf{I}_c \end{array} \right] \mathbf{V} = \left[ \begin{array}{ccc} {\Delta}_f \\ \mathbf{V}_c \end{array} \right], \label{eq:lap_energy_system}
$$ \]
• \(\mathbf{V}\) is a \(n \times 3\) matrix denoting the unknowns of the system that represent the vertex coordinates after deformation. The system is built so that the \( k \) last rows correspond
to the control vertices.
• \(\mathbf{L}_f\) denotes the Laplacian matrix of the unconstrained vertices. It is a \( (n-k) \times n \) matrix as defined in Eq. \(\eqref{eq:lap_system}\) but removing the rows corresponding to
the control vertices.
• \(\mathbf{I}_c\) is the \(k \times k\) identity matrix.
• \({\Delta}_f\) denotes the Laplacian representation of the unconstrained vertices as defined in Eq. \(\eqref{eq:lap_system}\) but removing the rows corresponding to the control vertices.
• \(\mathbf{V}_c\) is a \(k \times 3\) matrix containing the Cartesian coordinates of the target positions of the control vertices.
The left-hand side matrix of the system of Eq. \(\eqref{eq:lap_energy_system}\) is a square non-symmetric sparse matrix. To solve the aforementioned system, an appropriate solver (e.g. LU solver)
needs to be used. Note that solving this system preserves the Laplacian representation of the surface mesh restricted to the unconstrained vertices while satisfying the deformation constraints.
As-Rigid-As Possible Deformation
Given a surface mesh \(M\) with \( n \) vertices \( \{\mathbf{v}_i\} i \in \{1 \dots n \} \) and some deformation constraints, we consider the following energy function:
\[ $$\sum_{\mathbf{v}_i \in M} \sum_{\mathbf{v}_j \in N(\mathbf{v}_i)} w_{ij} \left\| (\mathbf{v}'_i - \mathbf{v}'_j) - \mathbf{R}_i(\mathbf{v}_i - \mathbf{v}_j) \right\|^2, \label{eq:arap_energy}$$
• \(\mathbf{R}_i\) is a \( 3 \times 3 \) rotation matrix
• \(w_{ij}\) denotes a weight
• \(N(\mathbf{v}_i)\) denotes the set of vertices adjacent to \(\mathbf{v}_i\) in \(M\)
• \(N(\mathbf{v}'_i)\) denotes a new position of the vertex \(N(\mathbf{v}_i)\) after a given deformation
An as-rigid-as possible surface mesh deformation [5] is defined by minimizing this energy function under the deformation constraints, i.e. the assigned position \( {v}'_i\) for each vertex \( \mathbf
{v}_i\) in the set of control vertices. Defining the one-ring neighborhood of a vertex as its set of adjacent vertices, the intuitive idea behind this energy function is to allow each one-ring
neighborhood of vertices to have an individual rotation, and at the same time to prevent shearing by taking advantage of the overlapping of one-ring neighborhoods of adjacent vertices (see Figure
There are two unknowns per vertex in Eq. \(\eqref{eq:arap_energy}\): the new positions ( \(\mathbf{v}'_k\)) of the unconstrained vertices and the rotation matrices ( \(\mathbf{R}_i\)). If the energy
contribution of each vertex is positive, this boils down to minimizing the energy contribution of each vertex \(\mathbf{v}_i\).
Each such term of the energy is minimized by using a two-step optimization approach (also called local-global approach).
In the first step, the positions of the vertices are considered as fixed so that the rotation matrices are the only unknowns.
For the vertex \(\mathbf{v}_i\), we consider the covariance matrix \(\mathbf{S}_i\):
\[ $$\mathbf{S}_i = \sum_{\mathbf{v}_j \in N(\mathbf{v}_i)} w_{ij} (\mathbf{v}_i - \mathbf{v}_j)(\mathbf{v}'_i - \mathbf{v}'_j)^T, \label{eq:cov_matrix}$$ \]
It was shown [6] that minimizing the energy contribution of \(\mathbf{v}_i\) in Eq. \(\eqref{eq:arap_energy}\) is equivalent to maximizing the trace of the matrix \(\mathbf{R}_i \mathbf{S}_i\). \(\
mathbf{R}_i \) is the transpose of the unitary matrix in the polar decomposition of \(\mathbf{S}_i\).
In the second step, the rotation matrices are substituted into the partial derivative of Eq. \(\eqref{eq:arap_energy}\) with respect to \(\mathbf{v}'_i\). Assuming the weights are symmetric, setting
the derivative to zero results in the following equation:
\[ $$\sum_{\mathbf{v}_j \in N(\mathbf{v}_i)} w_{ij}(\mathbf{v}'_i - \mathbf{v}'_j) = \sum_{\mathbf{v}_j \in N(\mathbf{v}_i)} w_{ij} \frac{(\mathbf{R}_i + \mathbf{R}_j)}{2} (\mathbf{v}_i - \mathbf{v}
_j). \label{eq:lap_ber}$$ \]
The left-hand side of this equation corresponds to the one of Eq. \(\eqref{eq:lap_open}\), and we can set \(\Delta\) to be the right-hand side. Solving the linear system in Eq. \(\eqref
{eq:lap_energy_system}\) gives the new positions of the unconstrained vertices.
This two-step optimization can be applied several times iteratively to obtain a better result.
The matrix built with the Laplacian matrix of the unconstrained vertices in the left-hand side of Eq. \(\eqref{eq:lap_energy_system}\) depends only on the initial surface mesh structure and on
which vertices are control vertices. Once the control vertices are set, we can use a direct solver to factorize the sparse matrix in Eq. \(\eqref{eq:lap_energy_system}\), and reuse this
factorization during each iteration of the optimization procedure.
The original algorithm [5] we described assumes that:
A method minimizing another energy function is described next to avoid the latter issue.
Spokes and Rims Version
The elastic energy function proposed by [3] additionally takes into account all the opposite edges in the facets incident to a vertex. The energy function to minimize becomes:
\[ $$\sum_{\mathbf{v}_i \in M} \sum_{(\mathbf{v}_j, \mathbf{v}_k) \in E(\mathbf{v}_i)} w_{jk} \left\| (\mathbf{v}'_j - \mathbf{v}'_k) - \mathbf{R}_i(\mathbf{v}_j - \mathbf{v}_k) \right\|^2, \label
{eq:arap_energy_rims}$$ \]
where \(E(\mathbf{v}_i)\) consists of the set of edges incident to \(\mathbf{v}_i\) (the spokes) and the set of edges in the link (the rims) of \(\mathbf{v}_i\) in the surface mesh \(M\) (see Figure
The method to get the new positions of the unconstrained vertices is similar to the two-step optimization method explained in As-Rigid-As Possible Deformation. For the first step, the Eq. \(\eqref
{eq:cov_matrix}\) is modified to take into account the edges in \(E(\mathbf{v}_i)\):
\[ $$\mathbf{S}_i = \sum_{(\mathbf{v}_j, \mathbf{v}_k) \in E(\mathbf{v}_i)} w_{jk} (\mathbf{v}_j - \mathbf{v}_k)(\mathbf{v}'_j - \mathbf{v}'_k)^T, \label{eq:cov_matrix_sr}$$ \]
For the second step, setting partial derivative of Eq. \(\eqref{eq:arap_energy_rims}\) to zero with respect to \(\mathbf{v}_i\) gives the following equation:
\[ $$\sum_{\mathbf{v}_j \in N(\mathbf{v}_i)} (w_{ij} + w_{ji})(\mathbf{v}'_i - \mathbf{v}'_j) = \sum_{\mathbf{v}_j \in N(\mathbf{v}_i)} \frac{w_{ij}(\mathbf{R}_i + \mathbf{R}_j + \mathbf{R}_m) + w_
{ji}(\mathbf{R}_i + \mathbf{R}_j + \mathbf{R}_n)}{3} (\mathbf{v}_i - \mathbf{v}_j). \label{eq:lap_ber_rims}$$ \]
where \(\mathbf{R}_m\) and \(\mathbf{R}_n\) are the rotation matrices of the vertices \(\mathbf{v}_m\), \(\mathbf{v}_n\) which are the opposite vertices of the edge \(\mathbf{v}_i \mathbf{v}_j\) (see
Figure 53.8). Note that if the edge \( \mathbf{v}_i \mathbf{v}_j \) is on the boundary of the surface mesh, then \( w_{ij} \) must be 0 and \( \mathbf{v}_m \) does not exist.
An important property of this approach compared to As-Rigid-As Possible Deformation is that the contribution to the global energy of each vertex is guaranteed to be non-negative when using the
cotangent weights [3]. Thus even with negative weights, the minimization of the energy with the iterative method presented is always guaranteed. However, this method is more dependent on the
discretization of the deformed surface (See Figure 53.2).
The implementation in this package uses the cotangent weights by default (negative values included) as proposed in [3].
Design and Implementation History
An initial version of this package has been implemented during the 2011 Google Summer of Code by Yin Xu under the guidance of Olga Sorkine and Andreas Fabri. Ilker O. Yaz took over the finalization
of the package with the help of Sébastien Loriot for the documentation and the API. The authors are grateful to Gaël Guennebaud for his great help on using the Eigen library and for providing the
code to compute the closest rotation. | {"url":"https://doc.cgal.org/4.5/Surface_modeling/index.html","timestamp":"2024-11-12T06:24:37Z","content_type":"application/xhtml+xml","content_length":"74264","record_id":"<urn:uuid:00e1aa46-180c-444d-9044-d8290f945d0e>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00611.warc.gz"} |
Alternating Current (AC) - Page 2 of 2 - Electronics Area
The Crystal Oscillators (Piezoelectric Oscillator) A quartz crystal has a property called: piezoelectric effect. This effect causes, by applying a mechanical pressure on the surface of the crystal,
that a voltage is developed on the opposite sides. In a similar way, a voltage applied on the faces of the crystal produces a mechanical distortion on their surface. An AC voltage causes […] | {"url":"https://electronicsarea.com/category/alternating-current/page/2/","timestamp":"2024-11-09T23:57:21Z","content_type":"text/html","content_length":"55705","record_id":"<urn:uuid:e63f11b5-0264-462c-a158-23c6936227c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00824.warc.gz"} |
Section: Application Domains
Applications of optimal transport
Optimal Transportation in general has many applications. Image processing, biology, fluid mechanics, mathematical physics, game theory, traffic planning, financial mathematics, economics are among
the most popular fields of application of the general theory of optimal transport. Many developments have been made in all these fields recently. Two more specific fields:
- In image processing, since a grey-scale image may be viewed as a measure, optimal transportation has been used because it gives a distance between measures corresponding to the optimal cost of
moving densities from one to the other, see e.g. the work of J.-M. Morel and co-workers [54] .
- In representation and approximation of geometric shapes, say by point-cloud sampling, it is also interesting to associate a measure, rather than just a geometric locus, to a distribution of points
(this gives a small importance to exceptional “outlier” mistaken points); this was developed in Q. Mérigot’s PhD [56] in the GEOMETRICA project-team. The relevant distance between measures is again
the one coming from optimal transportation.
- A collaboration between Ludovic Rifford and Robert McCann from the University of Toronto aims at applications of optimal transportation to the modeling of markets in economy; it was to subject of
Alice Erlinger's PhD, unfortunately interrupted.
Applications specific to the type of costs that we consider, i.e. these coming from optimal control, are concerned with evolutions of densities under state or velocity constraints. A fluid motion or
a crowd movement can be seen as the evolution of a density in a given space. If constraints are given on the directions in which these densities can evolve, we are in the framework of non-holonomic
transport problems. | {"url":"https://radar.inria.fr/report/2014/mctao/uid28.html","timestamp":"2024-11-09T01:25:19Z","content_type":"text/html","content_length":"39138","record_id":"<urn:uuid:03df91e0-24e2-4925-b4b1-02b40693a225>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00051.warc.gz"} |
Laura Toma
Professor of Computer Science.
Department of Computer Science
Bowdoin College Office: Searles 219
8650 College Station Email: ltoma@bowdoin.edu
Brunswick, ME 04011 Phone:
Publications | Student research | Teaching | CV | About me
My research is in the theory and practice of cache-efficient algorithms for large data, and in particular applications that involve large, high-resolution data in Geographic Information Systems
(GIS). Together with great students I explore algorithms for fundamental problems on terrains such as visibility, flooding, sea level rise and least-cost-path surfaces. Our goal is to come up with
approaches that are resource-efficient (CPU, IO, cache, parallel), are backed by algorithms that we can theoretically prove efficient, and at the same time work well in practice. Ultimately, our goal
is to transfer these algorithms into free and open-source software. I am grateful for the past support of NSF award 0728780 (2007-2013) which enabled me to launch this research with students at
Bowdoin. Memory-efficient algorithms and parallel algorithms share many techniques and insights which brought me towards exploring high-performance computing using Bowdoin's HPC grid. My DBLP page |
Google Scholar page.
I finished my Ph.D in 2003 at Duke University, Department of Computer Science. My thesis advisor was Lars Arge. My dissertation focused on IO-efficient algorithms for modeling flow on very large
terrains (terraflow | terrastream), as well algorithms for basic graph problems like IO-efficient breadth-first search and depth-first search, IO-efficient topological sort, IO-efficient minimum
spanning trees and IO-efficient shortest paths.
In Fall 2023 I am teaching Algorithms for GIS (csci3225) and Algorithms (csci 2200).
Other classes I taught (see teaching for links to course websites): Introduction to Computer Science (1101), Data Structures (2101), Algorithms (2200), Computational geometry (3250), Algorithms for
GIS, Spatial data structures, Computing with massive data.
(Some) Projects
• A multi-resolution apprach for visibility [with Lily Smith and Herman Haverkort]
• Faster algorithms for viewsheds and total viewsheds in 2D [with Drew Prescott and Herman Haverkort]
• Sea-level rise for the coast of Maine [with Cory Alini and Eileen Johnson]
• Exploring self-efficacy and its impact in teaching and learning algorithms [with Jan Vahrenhold]
• IO- and cache-efficient algorithms for viewsheds on grid terrains (r.viewshed) [with Bob Wei, Jeremy Fishman and Herman Haverkort]
• Computing multi-source shortest path surfaces on terrains in external memory (r.terracost) [with Tom Hazel and Jan Vahrenhold]
• Algorithms for flow-related indices on terrains in external memory (r.terraflow)
See publications for a complete list. | {"url":"https://tildesites.bowdoin.edu/~ltoma/","timestamp":"2024-11-10T01:59:03Z","content_type":"text/html","content_length":"4878","record_id":"<urn:uuid:d2e31026-e6e8-464d-bd0e-05cf89b696ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00179.warc.gz"} |
ACT Math Prep For Dummies, 2nd Edition » FoxGreat
ACT Math Prep For Dummies, 2nd Edition
• Length: 352 pages
• Edition: 2
• Language: English
• Publisher: For Dummies
• Publication Date: 2024-08-01
• ISBN-10: 1394242263
• ISBN-13: 9781394242269
Improve your score on the math section of the ACT
A good math score on the ACT exam can set you on the path to a number of rewarding college programs and future careers, especially in the STEM fields. ACT Math Prep For Dummies walks you through this
challenging exam section, with simple explanations of math concepts and proven test-taking strategies. Now including access to an all-new online test bank―so you can hammer out even more practice
sessions―this book will help you hone your skills in pre-algebra, algebra, geometry, trigonometry and beyond. Handy problem-solving tips mean you’ll be prepared for the ever-more-advanced questions
that the ACT throws at students each year.
• Learn exactly what you’ll need to know to score well on the ACT math section
• Get tips for solving problems quicker and making good guesses when you need to
• Drill down into more complex concepts like matrices and functions
• Practice, practice, practice, with three online tests
If you’re a high school student preparing to take the ACT and you need extra math practice, ACT Math Prep For Dummies has your back.
To access the link, solve the captcha. | {"url":"https://foxgreat.com/act-math-prep-for-dummies/","timestamp":"2024-11-09T03:22:37Z","content_type":"text/html","content_length":"47926","record_id":"<urn:uuid:90b16bf8-999f-4d1b-a0b7-60b8175988f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00281.warc.gz"} |
CAT Analytical Reasoning
Expert tips on CAT Analytical Reasoning
CAT Analytical Reasoning: Introduction
Analytical Reasoning is one of the most important topics when it comes to CAT & other MBA exams. Probing the candidate on his/her reasoning skills, the topic is essentially a test of your
temperament. On the one hand, this question type hardly requires you to be in possession of any previous knowledge, and on the other, it requires you to possess an ability to think on your feet. The
best way to prepare for Analytical Reasoning is to expose yourself to multifarious question types and make sure you are able to develop an approach for a variety of contexts. Let's explore some more
tips and tricks for this topic that will help you gain a competitive edge.
CAT Analytical Reasoning Syllabus
The following is the list of important topics that form the Analytical Reasoning section:
• Sets based on games like Cricket, Football, Hockey, Tennis etc.
• Share trading
• Sitting Arrangement – Linear, Circular.
• Directions & Ranking.
• Blood Relations.
• Sets based on Playing cards.
CAT Analytical Reasoning: Tips and Tricks
• Read Carefully: Read the information given in the Analytical sets very carefully. Remember, the devil is in the detail.
• Do not use prior INFORMATION: NEVER assume or use any information that is not given in the directions. Remember, this is not an assessment of how much you know; this section tests your logical
and analytical ability and checks how well you interpret the information given and how intelligently you derive the information required for answering the questions.
• Pay special attention to Special words: Many times you will encounter some special words like "could be" or "must be" etc. Do underline or circle such words. These words can change your answer
completely. "Could be" or "may be" is different from "must be" as the former deals with one of the possible outcomes whereas the latter deals with the outcome which is essential as per the
conditions mentioned. The solved examples at the end of the article would further clear your doubts about this little tip.
• Knowledge of basic terminologies: You should acquire the basic knowledge with regards to directions, relations, sitting arrangements, rules of games (like those in cricket about runs, overs etc.,
in football/hockey about goals-for, goals- against etc.), shares, debentures, playing cards etc. for easy and quick understanding of the case-lets.
• Universal conditions and Local conditions: There are universal conditions, which all questions must satisfy at all times and there are local conditions, which are followed by particular
questions. These local conditions are specific for a particular question only and always keep this in mind. When you move to the next question, you have to abide by the universal conditions along
with any local condition given within this very question, if any.
• Practice hard: Practice makes a man perfect and Analytical Reasoning is no exception to this. As the scope is limited in this section, unlike Quantitative Aptitude, practice is the only way to
acquaint yourself with the different problems and attain perfection.
CAT Analytical Reasoning Shortcuts
Well, the elusive search for shortcuts in Analytical Reasoning brings you here. Unlike the CAT Quantitative Ability section, there are no universal short-cuts for Analytical Reasoning. Rather, what
we have are important methodologies that you can adopt in your problem solving approach for this area. Two such shortcuts/methodologies are:
CAT AR Practice Sets/ Practice Questions
AR Set 1:
Answer the following questions based on the information given below: In a sports event, six teams (A, B, C, D, E and F) are competing against each other. Matches are scheduled in two stages. Each
team plays three matches in stage – I and two matches in Stage – II. No team plays against the same team more than once in the event. No ties are permitted in any of the matches. The observations
after the completion of Stage – I and Stage – II are as given below:
• One team won all the three matches.
• Two teams lost all the matches.
• D lost to A but won against C and F.
• E lost to B but won against C and F.
• B lost at least one match.
• F did not play against the top team of Stage-I.
• The leader of Stage-I lost the next two matches.
• Of the two teams at the bottom after Stage-I, one team won both matches, while the other lost both matches.
• One more team lost both matches in Stage-II.
Q.1 The two teams that defeated the leader of Stage-I are:
(1) F & D
(2) E & F
(3) B & D
(4) E & D
(5) F & D
Q.2 The only team(s) that won both matches in Stage-II is (are):
(1) B
(2) E & F
(3) A, E & F
(4) B, E & F
(5) B & F
Q.3 The teams that won exactly two matches in the event are:
(1) A, D & F
(2) D & E
(3) E & F
(4) D, E & F
(5) D & F
Q.4 The team(s) with the most wins in the event is (are):
(1) A
(2) A & C
(3) F
(4) E
(5) B & E
Solution: As per the instructions given for stage – I, we can reach the following conclusions:
(a) As B lost at least one match, hence A won all the 3 matches.
(b) The two teams who lost all the matches cannot be A (as explained above), cannot be B (E lost to B), cannot be D (D won against C & F). Hence, the two teams must be C and F.
(c) F did not play against the top team (i.e. A). We get the following table for stage – I. (To be read from rows)
A B C D E F
A X W W W
B L X W W
C L X L L
D W X W
E L W X X
F L L L X
As per the instructions given for Stage-II, we can reach the following conclusions. (d) A lost both its matches against E and F.
(e) F won against A, hence is the bottom team (out of C & F) which won both the matches ⇒ F won against C as well. This also means that C lost both its matches against B and F.
(f) Apart from A and C, one more team lost both the matches in Stage-II. That team can neither be E (A lost to E), nor B (as C lost to B), nor F (as F won both its matches). Hence, the team must be
We get the following table for Stage-II.
A B C D E F
A X W W W
B L X W W
C L X L L
D W X W
E L W X X
F L L L X
Therefore, the answers are:
• Option 2- E & F defeated A. [Please note that in this question option (1) and (5) were the same]
• Option 4- B, E & F won both the matches in Stage-II.
• Option 5- D & F won exactly two matches in the event.
• Option 5- B & E has most wins, 4 each.
AR Set 2:
K, L, M, N, P, Q, R, S, U and W are the only ten members in a department. There is a proposal to form a team from within the members of the department, subject to the following conditions:
The size of a team is defined as the no. of members in the team.
• A team must include exactly one among P, R, and S.
• A team must include either M or Q, but not both.
• If a team includes K, then it must also include L, and vice versa.
• If a team includes one among S, U, & W, then it must also include the other two.
• L & N cannot be members of the same team.
• L & U cannot be members of the same team.
Q.1 Who cannot be a member of a team of size 3?
Q.2What would be the size of the largest possible team?
1. 8
4. 5
5. Can't Say
Q.3In how many ways a team can be constituted so that the team includes N?
Solution: In this case, it is mentioned that exactly one among P, R, S can be there in a team. This means only one of these will be there in the team. Also either M or Q must be there but they cannot
be there together in the team. K and L will always be together either inside or outside the team. Similarly S, U, W will be together either inside or outside the team. L cannot be with either N or U
in the team. These are the conditions given in the questions.
Sol 1: From the conditions it is very clear that one among P, R, S will definitely be there in the team along with either M or Q. That means two persons are fixed. Now we need one more person as team
size should be of 3. Now if L is there then K must be there and thus team can never be of 3 persons with L. Hence, the answer is the first option.
Sol 2: This is simply a hit and trial process. Go on forming largest possible team with the conditions mentioned. You will find maximum size possible is 5 like S, U, W, M, N. The reason for this to
be maximum is as S is taken, P and R are rejected. M is taken so Q is rejected. U being there L is rejected, along with that K automatically gets rejected. Hence, the answer is the fourth option.
Sol 3:In this question the condition is that 'N' must be there in the team. Now the size of team is not given that means any size would be appropriate. Thus start forming teams with 'N'.
NMP, NQP, NMR, NQR, SUWNM, SUWNQ are the required teams. Hence, the answer is the fifth option.
AR set 3:
A person can have at the most 10 books. At least one book of Maths, Quality control, Physics and Fine Arts. For every Maths book more than two Fine Arts books are required. For every Quality control
book more than two Physics books are required. Maths, Quality Control, Physics & Fine Arts books carry 4, 3, 2 and 1 points respectively. Find the maximum points that can be earned?
In this case, one can have at the most 10 books; at least one each of all the four subjects. Now it is given that for every Maths book more than two Fine Arts books are required. Here more than two
indirectly means at least three as question is talking about books and the required number has to be integral. Thus, if one maths book is there, then minimum three Fine Arts books are required and
similarly, if one Quality Control book is there, then at least three Physics books are required. Now if we take one book of Maths, one of Quality Control, three of Physics and three of Fine Arts then
we have total of eight books while we can have a maximum of ten books. Now in order to maximize the points our intention is to use all the books. Now the question arises for which subject we should
select the two remaining books. Now Maths and Quality Control books cannot be used as then in that case we need to allot certain books for Fine Arts and Physics also and that is not possible so only
option left with us to allot the two books with Physics subjects to maximize the points. So, the final configuration would be M-1, QC-1, P-3+2=5, FA–3. Calculating points: (1*4) + (1*3) + (5*2) +
(3*1) = 20. Thus, the answer is 20. | {"url":"https://mba.hitbullseye.com/cat/analytical-reasoning.php","timestamp":"2024-11-12T16:24:05Z","content_type":"text/html","content_length":"99690","record_id":"<urn:uuid:482b6870-2088-48eb-928d-d7acb3d021ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00782.warc.gz"} |
[molpro-user] help regarding MRCI calculations
rama chandran rcchelat at yahoo.co.in
Mon Dec 24 08:49:04 GMT 2007
Dear friends,
I am very new to molpro suite of programs.
I calculated the energy of a linear HeHH+ 3 electron
system reported in a paper.
Pls. see the input below.
a1= 180.0
According to that paper, the calculations have done
using CASSCF method and recovered the dynamic
correlation energy by the interanally contracted MRCI
method. The basis set used were cc-pV5Z of Dunning
(8s4p3d2f1g) contracted to [5s4p3d2f1g].
As you see in the input file, I tried the same method
and basis set for the geometry specified in
However, I am getting a slightly different energy.
For example, the energy reported is -3.51889301 where
as I got the value of -3.518185866
Although the difference is very small, I afraid
something is wrong in the calculations (Input file).
I couldn't reproduce the result for other geometries
He is approaching H2^+ in a linear fashion.
So as per molpro symmetry constraints it can have C2v
As it is a 3 electron system now, it would have filled
sigma-g and sigma-u orbitals.
So the occupied levels as per the notation I guess,
I doubt whether I made any mistake in this step or
I am a novice in these calculations and from manual I
got only these much.
(2) If He is approaching by making an angle <180
it will have Cs symmetry.
So the irreducible representation will be A' and A''
So what should be the configuration?
Can anyone help me?
Thanks in advance.
5, 50, 500, 5000 - Store N number of mails in your inbox. Go to http://help.yahoo.com/l/in/yahoo/mail/yahoomail/tools/tools-08.html
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Information theory: questions and answers
Information theory
is fundamentally about
We understand information itself in terms of questions and answers: 1 bit of information is the uncertainty in the answer to a question with a 50-50 outcome, e.g. "will this coin flip give tails?".
Just as importantly though, the measures of information theory themselves are all about questions and answers too.
For the basic measures, the questions they ask seem fairly obvious. The
Shannon Entropy
"How much uncertainty is there in the state of this variable X?".
Mutual information
asks "
how much information does the state of variable X tell me about the state of Y?
", while
conditional mutual information
asks "
how much information does the state of variable X tell me about the state of Y, given that I already know the state of Z?
But I want to make a few more subtle points about these questions and answers.
In my opinion (which is of course the only correct one), the answers that the measures give are
correct. If you think they're wrong, then you're asking the wrong question, or have malformed the question in some way. There are plenty of ways to do this, or at least to inadvertently change the
question that you're asking.
I see the sample data itself as part of the question that a measure is answering. When you estimate the probability distribution functions (PDFs) empirically from a given sample data set, your
original question about entropy really becomes:
"How much uncertainty is there in the state of this variable X, given what we're assuming to be a representative sample of realisations x of X here?"
Of course, your representative sample could simply be too short, and thereby completely misrepresent the PDF. Or you could get into trouble with stationarity (1) of the process - you might implicitly
have appended "given what we're assuming to be a representative stationary sample here" to the question, but that assumption may not be true.
In both cases, the measure will give the correct answer to your question, but it might not be the question you really intended to ask.
As another way of inadvertently changing the question, one must realise that for the same information-theoretic measure, different estimators (or indeed different parameter settings for the same
estimator) answer different questions. Take the mutual information, for example, which one could measure on continuous-valued data via (box)
kernel estimation
. Using this estimator, the measure asks: "
how much information does knowing the state of variable X within radius r tell me about the state of variable Y within radius r?
" Clearly, using different parameter values for r amount to asking different questions - potentially the questions are very different if one uses radically different scales for r. Going further, one
could measure the mutual information using the enhanced
Kraskov-Grassberger kernel estimation
technique. With this estimator, the mutual information measure asks "
how much information does knowing the state of variable X tell me about the state of variable Y, to the precision defined in their k closest neighbours of the sample data set in the joint X-Y space?
" Apart from that being something of a mouthful, it's obviously a different question to what the box kernel estimation is asking. And again, changing the parameter k changes the question being asked
as well.
So to reiterate, information theory is fundamentally about questions and answers - the better you can keep that in mind, the better you will understand information theory and its tools.
UPDATE- 13/12/12
- My colleague
Oliver Obst
provided a perfect quote about this: "Better a rough answer to the right question than an exact answer to the wrong question" - attributed to Lord Kelvin.
(1) Here's a controversial statement: I suggest that it can be valid to make information-theoretic measurements on non-
stationary processes
. This simply changes the question that is being asked to something like: "
how much uncertainty is there in the state of this non-stationary variable X, if we don't know how the joint probability distribution of the non-stationary process is operating at this specific time,
given what we're assuming to be a representative sample of the joint probability distribution weighted over all possible ways it may operate?
". Now, obviously that's quite a mouthful, but I'm trying to capture that intuition that one could validly consider how much information it takes to predict X if we don't know the specifics of the
non-stationarity at this particular point in time, but do know the overall distribution of X (covering all possible behaviours). So long as one bears in mind that a different question is being asked
(indeed a question that is quite different to the intended use of the measure), then certainly the answer can be validly interpreted. Of course, the bigger issue is in properly sampling the PDF of X
over all possible behaviours, but that's another story.
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How To Find The Area of A Parallelogram? | TIRLA ACADEMY
We multiply the base and height to find the Area of the Parallelogram. Or We can use the formula Area of a parallelogram = Base✕Height
Let's take an example:
Q- Find parallelogram area with base=8 cm and height=6 cm.
Area of Parallelogram = Base✕Height
= 8 cm ✕ 6 cm
= 48 cm² | {"url":"https://www.tirlaacademy.com/2024/04/how-to-find-area-of-parallelogram.html","timestamp":"2024-11-03T01:00:22Z","content_type":"application/xhtml+xml","content_length":"310788","record_id":"<urn:uuid:ed12fd66-ddf4-4f8c-b212-4ab7ff535d05>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00774.warc.gz"} |
Math Problem Statement
Consider the initial value problem given below. y primeequals1 minus y plus y cubed, y(0)equals0 Use the improved Euler's method with tolerance to approximate the solution to this initial value
problem at xequals0.8. For a tolerance of epsilonequals0.003, use a stopping procedure based on the absolute error.
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Math Problem Analysis
Mathematical Concepts
Numerical Methods
Differential Equations
Error Tolerance
Euler's Method
y_{n+1} = y_n + (h / 2) [f(x_n, y_n) + f(x_{n+1}, y_n + h f(x_n, y_n))]
f(x, y) = 1 - y + y^3
Improved Euler's Method (Heun's Method)
Suitable Grade Level
Undergraduate level (Calculus / Numerical Analysis) | {"url":"https://math.bot/q/initial-value-problem-euler-method-tolerance-FMOFQMRt","timestamp":"2024-11-11T04:21:18Z","content_type":"text/html","content_length":"85914","record_id":"<urn:uuid:2bb4880a-f786-4213-be28-d9fe0a486c36>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00425.warc.gz"} |
Talk Abstract: 2012a
Title: Why are the theories of controllability and stabilisability so different, and should they be?
Detail: 5th Biennial Meeting on Systems and Control Theory, 2012/05/07
The theories of controllability (from a state) and stabilisability (to a state) are among the most fundamental in control theory. Anyone seeing the definitions of these notions for the first time and
possessing no preexisting socio-academic bias would think that these subjects are very closely linked. Such a (fictitious) person would then be surprised to see that the literature on these subjects
is virutally disjoint. The theory of controllability is geometric and is about Lie algebras of vector fields and the like. The theory of stabilisability is analytic in nature and is about Lyapunov
functions and the like.
In this talk I will say a few words about these subjects, attempting to strip away the fact that they have developed along almost entirely separate lines. I will focus on a few simple questions and
some results (some obvious and some not) related to these questions that indicate that the distinctions we see in these areas of research are not real, but human-made.
No online version avaliable.
Andrew D. Lewis (andrew at mast.queensu.ca) | {"url":"https://mast.queensu.ca/~andrew/talks/abstracts/2012a.html","timestamp":"2024-11-03T13:54:13Z","content_type":"text/html","content_length":"2013","record_id":"<urn:uuid:bb7a7e9f-88cd-42e5-9723-4c8dca87fa02>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00399.warc.gz"} |
A study of the electrostatic properties of the interiors of low-mass stars: Possible implications for the observed rotational properties
Issue A&A
Volume 690, October 2024
Article Number A228
Number of page(s) 13
Section Stellar structure and evolution
DOI https://doi.org/10.1051/0004-6361/202450670
Published online 10 October 2024
A&A, 690, A228 (2024)
A study of the electrostatic properties of the interiors of low-mass stars: Possible implications for the observed rotational properties
^1 Instituto Superior de Gestão, Rua Prof. Reinaldo dos Santos 46 A, 1500-552 Lisboa, Portugal
^2 Centro de Astrofísica e Gravitação – CENTRA, Departamento de Física, Instituto Superior Técnico IST, Universidade de Lisboa – UL, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
Received: 9 May 2024
Accepted: 8 August 2024
Context. In the partially ionized material of stellar interiors, the strongest forces acting on electrons and ions are the Coulomb interactions between charges. The dynamics of the plasma as a whole
depend on the magnitudes of the average electrostatic interactions and the average kinetic energies of the particles that constitute the stellar material. An important question is how these
interactions of real gases are related to the observable stellar properties. Specifically, the relationships between rotation, magnetic activity, and the thermodynamic properties of stellar interiors
are still not well understood. These connections are crucial for understanding and interpreting the abundant observational data provided by space-based missions, such as Kepler/K2 and TESS, and the
future data from the PLATO mission.
Aims. In this study, we investigate the electrostatic effects within the interiors of low-mass main sequence (MS) stars. Specifically, we introduce a global quantity, a global plasma parameter, which
allows us to compare the importance of electrostatic interactions across a range of low-mass theoretical models (0.7−1.4M[⊙]) with varying ages and metallicities. We then correlate the
electrostatic properties of the theoretical models with the observable rotational trends on the MS.
Methods. We use the open-source 1D stellar evolution code MESA to compute a grid of main-sequence stellar models. Our models span the log g−T[eff] space of a set of 66 Kepler main-sequence stars.
Results. We identify a correlation between the prominence of electrostatic effects in stellar interiors and stellar rotation rates. The variations in the magnitude of electrostatic interactions with
age and metallicity further suggest that understanding the underlying physics of the collective effects of plasma can clarify key observational trends related to the rotation of low-mass stars on the
MS. These results may also advance our understanding of the physics behind the observed weakened magnetic braking in stars.
Key words: stars: evolution / stars: fundamental parameters / stars: general / stars: interiors / stars: low-mass / stars: rotation
© The Authors 2024
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.
1. Introduction
It is well understood that various types of plasma exist in nature, each characterized by significant variations in temperature and electron density. Within the innermost layers of low-mass stars, we
commonly encounter what is generally referred to as a hot and dense plasma. Chen (2016) describe plasma as a quasi-neutral gas of charged and neutral particles that exhibits collective behavior. In
this definition there are two key concepts. Quasi-neutrality refers to the fact that the ionic density is not exactly equal to the electronic density (n[i]≈n[e]). Small local imbalances of electric
charge are not only possible but unavoidable, according the plasma definition. Therefore, the plasma maintains an overall state of neutrality, but is not so neutral that it loses all its interesting
and important electromagnetic properties. Collective behavior implies that the motions of plasma particles depend not only on local conditions but also on the state of the plasma in distant regions.
Thus, the collective behavior of plasma is a consequence of the long-range properties of electromagnetic forces.
Two important quantities that classify the properties of all astrophysical plasmas, including those found in stellar interiors, are the Debye length and the plasma coupling parameter (e.g., Maeder
2009). The Debye length is a characteristic length scale over which the Coulomb potential of a charged particle is “screened” by the redistribution of the surrounding charged particles of opposite
sign. Hence, as a consequence of this Debye shielding, and for scales substantially larger than the Debye length, such as the stellar radii, a state of overall quasi-neutrality is achieved. The
second important quantity, the plasma coupling parameter, measures the degree of coupling within the plasma, indicating the strength of Coulomb interactions between its constituent particles (e.g.,
Potekhin et al. 2009; Stanton & Murillo 2016). More specifically, the plasma coupling parameter is a ratio between two magnitudes: the average energy of electrostatic interactions and the average
kinetic energies of the particles. This ratio can serve as an indicator of the dynamics of the plasma as a whole; that is, it is an indicator of the significance of the collective effects of the
In the initial part of this study, we investigate the properties of critical quantities from the perspective of plasma physics. These quantities enable us to describe the electrostatic effects within
the interiors of low-mass stars. Specifically, we examine the properties of the Debye length and the energy density of electrostatic interactions throughout the stellar interiors. Special attention
is given to the plasma coupling parameter, along with some related quantities such as electron number density, electron degeneracy parameter, and mean molecular weight. For the first time, we explore
how the internal Coulomb interactions between charged particles compare across a large set of theoretical models of main sequence (MS) low-mass stars. We computed the internal profiles of the plasma
coupling parameter for a group of models with masses ranging from 0.7 to 1.4M[⊙]. For each model, we define a global parameter that allows us to classify the star from the perspective of Coulomb
interactions. Additionally, we analyze how this global parameter varies with mass, age, and metallicity.
Stellar magnetic fields are crucial for understanding stellar structure and evolution, stellar oscillations, and the magnetic activity and space weather around the stars. In particular, for low-mass
MS stars, the dynamo-generated magnetic fields are thought to be tied to the formation of stellar winds, which carry away angular momentum, supporting the stellar spin-down observed in these stars
(e.g., Schatzman 1962; Weber & Davis 1967; Kawaler 1988; Vidotto et al. 2014). Magnetic activity manifestations, such as chromospheric or coronal emissions, have long since revealed strong
correlations between stellar rotation and magnetic activity (e.g., Hall et al. 1991; Hempelmann et al. 1995; Böhm-Vitense 2007; García et al. 2014; Marsden et al. 2014; do Nascimento 2014; Oláh et
al. 2016). Generally, it is found that rapid rotators exhibit higher levels of magnetic activity than slow rotators (e.g., Kraft 1967). The transition from slow to fast rotators is believed to be
related to the efficiency of magnetic braking, which occurs due to angular momentum loss through magnetized stellar winds. Low-mass stars below the Kraft break have deep convective envelopes capable
of hosting dynamos that produce efficient magnetized winds, contributing to the observed spin-down in these stars. On the other hand, stars above the Kraft break have shallower convective envelopes
that cannot host efficient dynamos, resulting in weaker magnetized winds and rapid rotation (e.g., van Saders & Pinsonneault 2013).
Collectively, the observational evidence of the activity–rotation relationship obtained over recent decades has allowed us to establish empirical relations between the stars’ rotation periods and
their ages. Skumanich (1972) was the first to show that the equatorial rotational velocities of low-mass stars are proportional to the inverse of the square root of stellar age (v[eq]∝t^−0.5).
Therefore, rotation periods can be used to estimate the ages of stars that spin down on the MS, which is the core principle of gyrochronology, a method used to estimate the ages of isolated stars
(e.g., Barnes 2003; Mamajek & Hillenbrand 2008; Barnes & Kim 2010; Epstein & Pinsonneault 2014).
It is now clear that observations related to the stellar rotation of MS stars reveal that rotation is an intricate function of mass, age, and metallicity. Hence, the study of the internal structure
and internal thermodynamics of stars is fundamental for understanding all the available observational traits. We are interested in studying the microphysics of stellar interiors, which can be linked
to the rotational patterns observed on the MS for low-mass stars. Therefore, in the second part of this work, we study the relationship between the electrostatic properties of the interiors of
low-mass MS stellar models and the observed rotational properties of stars with similar characteristics. In particular, given the plasma-like characteristics of stellar interiors and the importance
of Lorentz forces in the transport of angular momentum (e.g., Zaire et al. 2022), we relate the properties of the herein introduced global plasma parameter with the different observed rotational
behaviors of low-mass stars on the MS. We find that the electrostatic properties of stellar interiors correlate with the observed MS rotational trends for low-mass stars.
The first part of this work, consisting of sections 2 and 3, focuses on the study and description of electrostatic effects within stellar interiors. Section 2 describes the stellar models and the
modeling process. In section 3, we examine the main properties of the Debye length, the total energy density of electrostatic interactions, and we define the global plasma coupling parameter that
allows us to compare the electrostatic properties of stellar interiors across different stellar models. We also define and examine global values for the electron number density, the electron
degeneracy parameter, and the mean molecular weight.
In the second part of the work, we explore the correlations between the global plasma parameter and the observed rotational trends on the MS. Section 4 is dedicated to studying how the global plasma
parameter varies with age and metallicity, whereas section 5 compares the theoretical scaling of the global plasma parameter with the observed scaling of stellar rotation rates. Finally, we present
our conclusions in section 6.
2. Modeling the log g – T[eff] space of Kepler MS stars
We based our grid of theoretical models on a set of MS stars taken from the catalog of dwarf stars with asteroseismic rotation rates (Hall et al. 2021). This dataset contains a total of 94 stars. In
the catalog, these 94 stars are classified into three groups by “type.” Hall et al. (2021) identified three different types: “subgiants” (SG) with only 4 stars, “hot stars” (H) with 24 stars, and
“main sequence” (MS) with 66 stars. The stars we selected, represented in red in Figure 1, are the 66 stars from the catalog clearly classified as MS. These are 66 low-mass stars with oscillation
frequencies detected at high signal-to-noise ratios. Thus, these stars, which serve as a basis for our theoretical models, have T[eff]<6250 K and logg>4. Specifically, we considered all stars in
the catalog of Hall et al. (2021) except those classified as H and SG.
Fig. 1.
Kiel diagram of the sample of 66 Kepler stars that serve as a basis for the theoretical models. These stars are represented in red. The theoretical data points, representing stellar model values,
are colored according to age in the top panel and to metallicity in the bottom panel. The location of the Sun is also represented with its usual symbol for reference.
The idea is to cover the log g−T[eff] range of the subsample of 66 MS stars. We did not intend to model the stars to exactly match the log g−T[eff] values of the observed stars. Our goal is to
study the electrostatic properties of stellar interiors using a set of models for which we know there are observed stars with similar properties in terms of mass, age, and metallicity. We proceeded
as follows. First, we computed a grid from 0.7 to 1.4 M[⊙] (with a step of 0.1) as this is the stellar mass range of the subsample of observed stars. The age of the models is based on the average age
of the observed stars for each mass category. Here, “mass category” refers to all the stars in the selected subsample with mass values rounded to the closest model mass values (0.7, 0.8, 0.9, 1.0,
1.1, 1.2, 1.3, or 1.4 M[⊙]). For each stellar mass, we computed models with the following metallicities: Z = 0.005, 0.01, 0.02, 0.03, and 0.04. By doing this, we obtained a set of 40 models. We then
allowed some of the models from this initial set to vary in age to fill the gaps in the log g−T[eff] values of the observed stars. This procedure resulted in 37 models. Our final set of models
comprises 77 models.
Figure 1 shows the distribution of our models in the log g−T[eff] diagram. In this Kiel diagram, the observational values for the set of 66 Kepler stars are shown in red. The models that extend to
lower and higher effective temperatures and fall outside the log g−T[eff] values of the observed stars are, in general, models with either higher masses and low metallicities or lower masses and
high metallicities. As these models are interesting for the study of electrostatic properties, we decided to keep them in the study.
In the top panel of Figure 1, the theoretical data are colored according to the ages of the models, whereas in the bottom panel, the same model data are colored according to metallicity. Table 1
summarizes some basic statistical details to help the reader better understand the two datasets. Specifically, we compare the average values and their standard deviation (SD) for stellar parameters
such as mass, age, metallicity, effective temperature, and surface gravity.
Table 1.
Basic statistics (theoretical models vs observational sample).
Concerning the input physics, our choices are as follows. Theoretical models were computed with the stellar evolution code Modules for Experiments in Stellar Astrophysics (MESA v15140; Paxton et al.
2011, 2013, 2015, 2018, 2019; Jermyn et al. 2023). The code was compiled using https://doi.org/10.5281/zenodo.4587206. All models were evolved with the metallicity mixture from Asplund et al. (2009).
The opacities used are from OPAL tables at high temperatures (Iglesias & Rogers 1993, 1996) complemented at low temperatures with opacities from Ferguson et al. (2005). MESA relies on an equation of
state that is a blend of several equations of state (Saumon et al. 1995; Timmes & Swesty 2000; Rogers & Nayfonov 2002; Irwin 2004; Potekhin & Chabrier 2010; Jermyn et al. 2021). The nuclear reaction
rates were obtained from JINA REACLIB (Cyburt et al. 2010) with included screening effects using the prescription of Chugunov et al. (2007). Additionally, all models include atomic diffusion
according to Thoul et al. (1994), as previous studies describe it as an important element-transport process for low-mass stars (e.g., Nsamba et al. 2018; Moedas et al. 2022). Although radiative
accelerations have also been shown to be important at low metallicities and for higher masses within the range studied here (e.g., Deal et al. 2018), we did not include them in this study. This is
because our aim in this initial study of the electrostatic properties of stellar interiors is to keep our models simple and maintain consistent input physics across all models. However, in subsequent
studies, it will be very interesting to investigate the impact that radiative accelerations can have on the electrostatic properties of the theoretical models. Convection is treated according to the
mixing-length theory from Böhm-Vitense (1958), without overshoot, and using a mixing-length parameter value of α[MLT]=1.8, common to all models. Finally, the outermost layers on the models are
described by a Grey-Eddington atmospheric structure. A set of files that allows the reader to reproduce all the models in this work, and consequently all the findings in this study, is openly
available through Zenodo ^1.
3. Electrostatic properties of stellar interiors
In this section, we study electrostatic effects in the interiors of the theoretical models represented in Figure 1. We also investigate the variations of these effects with mass, age, and
3.1. The Debye length and the total energy density of the electrostatic interactions
In Section 1, we highlight quasi-neutrality as a key concept defining plasma. Quasi-neutrality is a state that the plasma actively seeks to attain by continuously readjusting the local distribution
of charged particles in response to perturbations in charge. These effects are commonly known as plasma screening effects, and were initially developed in the pioneering work of Debye & Hückel (1923)
. Thus, a crucial property of plasma is that the charged particles arrange themselves in such a way as to shield any electrostatic fields within a certain distance. This distance, usually represented
by λ[D], is called the Debye length. The Debye length is a measure of the shielding distance. Let us consider the electric charge density, that is, the net density of electric charges at a given
location, ρ[e], defined as the difference between the ionic electric charge density and the electronic charge density (e.g., Weiss et al. 2004):
$ρ e = ∑ i n i Z i e − n e e .$(1)
Here, the index i represents the different atomic species that constitute the stellar plasma and e is the elementary charge. Boltzmann’s distributions developed to first order allows us to write the
ionic concentration as
$n i ( r ) = n 0 i ( 1 − Z i e Φ ( r ) kT ) ,$(2)
as well the concentration of electrons as
$n e ( r ) = n 0 e ( 1 + e Φ ( r ) kT ) .$(3)
The values n[0i] and n[0e] represent, respectively, the concentrations of ions and electrons when the stellar matter is unperturbed. Because unperturbed matter is neutral, we have n[0i]Z[i]=n[0e]
which represents the well-known neutrality condition. Finally, Φ(r) is the electric potential, which is given by the Poisson equation
$∇ 2 Φ ( r ) = − 4 π ρ e .$(4)
The Poisson equation relates Φ(r) to the number densities n[i](r) and n[e](r) (e.g., Rose 1998; Mestel 1999). Specifically, by substituting the ionic and electronic densities, given respectively by
Equations (2) and (3), into Equation (4), it is possible to obtain an expression for the electrostatic potential of a charge Ze:
$Φ ( r ) = Ze r e − r / λ D ≃ Ze r − Ze λ D .$(5)
This is the potential that takes into account the fact that electrons tend to surround an ion of positive charge Ze. The quantity λ[D] is the Debye length and can be written as
$λ D = kT 4 π e 2 ( n e + ∑ i n i Z i 2 ) .$(6)
This length scale represents a distance over which the thermal fluctuations of the stellar material can lead to an important separation between negative and positive charged particles (e.g., Mestel
We computed the internal profiles of the Debye lengths for several of our models. In Figure 2 (left panel), we show how the Debye length varies with mass for Z=0.02 metallicity models at an
intermediate age on the main sequence (IAMS), which means that the age of the model is 50% of the entire MS lifetime. Typically, for the same age and metallicity, the Debye length increases as the
mass of the stellar model also increases. Debye lengths also vary with the age of a star. The central panel of Figure 2 shows the variation of λ[D] along the MS for the less massive and more massive
models. As the stars evolve on the MS, the Debye length increases with the mass of the stellar model by a few orders of magnitude.
Fig. 2.
Internal profiles of the Debye length as a function of the fractional radius. All models represented were computed with Z=0.02. Left panel: Changes in the Debye length profiles for models with
masses in the range 0.7−1.4M[⊙]. Central panel: Debye length profiles for models with 0.7 and 1.4M[⊙] at two different stages of evolution: at the ZAMS and at midlife. Right panel: Internal
profiles of the total energy density of electrostatic interactions as a function of the fractional radius. All the models in this figure are at the IAMS, meaning they are in the middle
(intermediate age) of their evolutionary path on the main sequence.
Screening effects, and in particular the screening potential (Equation (5)), have a significant influence on the microscopic and macroscopic properties of stellar interiors (e.g., Basu & Antia 2008;
Bahcall et al. 2004). One important consequence is the reduction of pressure in the stellar medium as a result of screening effects. This occurs because a charge produces a surrounding cloud of
radius λ[D] containing an excess of charges with opposite sign. The system formed by the charge and the cloud is bound and electrically neutral, and has a negative energy as it is necessary to
provide energy to separate it. This negative energy leads to a pressure decrease in the stellar interior. The total energy density of electrostatic interactions depends on the Debye length, and can
be written as (Maeder 2009):
$u ES = − 1 2 e 2 λ D ∑ i n i Z i 2 .$(7)
This energy density is plotted for different stellar masses in the right panel of Figure 2. All models represented in Figure 2 were computed at Z=0.02 metallicity and have evolved up to the MS
halftime, with the exception of two models in the central panel, represented by dashed lines, which were computed at zero age main sequence (ZAMS).
3.2. The plasma coupling parameter
The occurrence of the Debye shielding effect is usually a first example of a plasma collective behavior. It is easy to see from Equation (6) that, as density increases, the Debye length decreases.
Furthermore, as the thermal kinetic energy, kT, of the particles increases, the Debye length increases as well. As a consequence, the properties and the particular characteristics of the stellar
plasma will depend on two quantities: the electrostatic potential energy, and the thermal energy of the particles that constitute the stellar material (electrons and ions).
An important ratio that allows us to characterize the electrostatic properties of stellar interiors is the plasma coupling parameter. This parameter, represented here by Γ[i], is defined as the ratio
of Coulomb potential energy to the thermal energy. Considering the ionic case, Γ[i] can be written as
$Γ i = ( Z i ¯ e ) 2 a i k T ,$(8)
where the Wigner-Steitz radius, a[i], is given by
$a i = ( 3 Z i ¯ 4 π n e ) 1 / 3 ,$(9)
and represents the ion sphere radius, or in another words, the mean inter-ion distance (e.g., Paquette et al. 1986; Maeder 2009). Here, e is the elementary charge, k the Boltzmann constant, T the
local temperature, $Z i ¯$ the mean ionic charge, and n[e] the electron density. The higher the value of the plasma coupling parameter, the more important the Coulomb interactions between ions. When
Coulomb and thermal energies are balanced, the plasma coupling parameter takes values of around unity. For cases where electrostatic interactions dominate over thermal energies, the plasma coupling
parameter becomes greater than one. It is well known that, for low-mass stars, Coulomb effects should be considered in the equation of state (e.g., Christensen-Dalsgaard 2021). Moreover, a recent
study (Brito & Lopes 2021) showed the particular significance of electrostatic interactions in the ionization zones of the most abundant elements for low-mass stellar models.
Figure 3 displays the ionic plasma coupling parameter plotted for three of our theoretical stellar models (0.7,1.0 and 1.4M[⊙]), with all three computed at Z=0.02. We note the clear increase in
the importance of electrostatic effects as the mass of the stellar model decreases, with Γ[i] peaking near the surface for all models, a known behavior of the plasma parameter (e.g.,
Christensen-Dalsgaard 2021). The lower panel in Figure 3 shows that the maximum values of the plasma coupling parameter are reached within the temperature interval where the two helium ionizations
occur. This figure highlights the fact that specific aspects of the stellar structure, particularly the underlying microphysics of the outer convective zones, are crucial for understanding the
electrostatic properties of low-mass stellar interiors. In Figure 3, convective zones are represented in light red, while radiative zones are shown in light blue.
Fig. 3.
Ionic plasma coupling parameter, Γ[i], illustrated for three models with varying masses at Z=0.02. The upper panel shows Γ[i] as a function of the fractional radius, while the lower panel shows Γ
[i] as a function of temperature (log T). Regions with higher values of Γ[i] indicate a greater influence of electrostatic effects within the stellar interior. Across these three models, the
significance of Coulomb effects is more pronounced in the outer stellar layers. The helium ionization zones are indicated with a purple double arrow, where the plasma coupling parameter reaches its
maximum value. Additionally, three regions of equal length in radius are shown: Region A corresponds to the innermost third of the model, Region B to the central third, and Region C to the
outermost third of the stellar interior. In all the plots, light blue regions correspond to radiative zones, and light red regions to convective regions. Finally, all three models in this figure
are at IAMS, meaning they are in the middle of their evolutionary path on the MS.
3.3. Global stellar plasma parameter
The plasma coupling parameter is a local variable that characterizes the balance between electrostatic and thermal energies at each location in the stellar interior. Nevertheless, by considering a
mean value of the parameter Γ[i], defined by the relation
$Γ ¯ = 1 M ∫ 0 M Γ i d M ( r ) ,$(10)
$d M ( r ) = 4 π r 2 ρ d r ,$(11)
it is possible to define a new variable $Γ ¯$, which carries a global character. This new global variable can be used to characterize a stellar model from the viewpoint of the importance of Coulomb
interactions. The greater the value of $Γ ¯$, the greater the significance of the Coulomb interactions for the star as a whole. We call this new variable, $Γ ¯$, the global mean plasma parameter, or
simply, the global plasma parameter. From Figure 3, we are able to see that we can use this global variable to study the importance of electrostatic effects in specific regions within the star.
For example, the stellar interior can be divided in three parts: a deeper region that includes the stellar core (the inner 33.33% of the stellar radius), a central radiative region (the central
33.33% of the stellar radius), and an upper region that contains the bulk of the convective zone of the stellar models (the external 33.33% of the stellar radius). All our models have an outer
convective zone and a radiative interior, with 16 models including a convective core. The outer convective zones of the models range in depth from 3% to 39% of the stellar outer layer, while the
convective cores vary in thickness from 4.5% to 7.5% of the innermost layer. A natural choice for dividing the stellar structure would be to consider the radiative and convective regions. However,
because more massive stars have a very thin convective zone, we considered the possibility that such a division could introduce a bias in the comparison of mean plasma parameter values. We also note
that Region C (see Figure 3), the outermost region, includes the convective upper zone of all models and could thus be considered an approximation of a region containing the convective zone.
Additionally, we were interested in examining a region that includes the core of the model (not just the convective core), where energy is generated through nuclear reactions. Given these
considerations, we ultimately decided to divide the structure into three regions of equal length for this initial study of the electrostatic properties of stellar interiors.
Figure 4 shows the dependence of $Γ ¯$ on the effective temperature for all our models described in Section 2. We observe a clear scaling of $Γ ¯$ that results in three almost distinct clusters of
stars according to their $Γ ¯$ value. As expected, cooler stars have higher values of $Γ ¯$, whereas hotter stars exhibit lower values of $Γ ¯$. The cluster of stars with intermediate T[eff] values
correspond to stars with the middle $Γ ¯$ values. This behavior of the $Γ ¯$ – T[eff] relation mirrors the observational behavior of the P[rot] – T[eff] dependence, in the sense that cooler stars
have higher values of the rotation period, whereas hotter stars exhibit lower values of the rotation period, with the faster-rotating hot stars usually being those above the Kraft break. The Kraft
break (Kraft 1967) is an observational feature characterized by a steep variation in stellar rotational velocities occurring over a small mass (or temperature) range. The approximate location of this
transition is around 6200–6300 K (e.g., Krishnamurthi et al. 1997; van Saders & Pinsonneault 2013; van Saders et al. 2016; Spada & Lanzafame 2020; Mathur et al. 2023; Rebassa-Mansergas et al. 2023),
and it is represented in Figure 4 by a vertical gray bar.
Fig. 4.
Value of $Γ ¯$ as a function of the effective temperature for all the stellar models. Upper panel: Value of $Γ ¯$ computed throughout the entire stellar interior. The green color highlights the
less massive models (0.7, 0.8, and 0.9 M[⊙]), whereas the gray color highlights the more massive models (1.2, 1.3, and 1.4 M[⊙]). The orange color corresponds to models with 1.0 or 1.1 M[⊙]. The
vertical dashed line indicates the approximate location of the Kraft break. Lower panel: Value of $Γ ¯$ computed for three different regions (A, B, and C) within the stellar interiors. Here, we
show the computed values as a function of the effective temperature for all stellar models. Regions A, B, and C are described in Figure 3. The green, gray, and orange colors have the same meaning
as in the upper panel. The small black circles in all the plots indicate the presence of a model with a convective core.
As we know, a higher value of $Γ ¯$ indicates that the electrostatic interactions are more important, and thus collective effects are more significant. These are theoretical models that represent
stars located below the Kraft break, which are stars that experience spin down along the MS according to the experimental data. Instead, hotter stars can be associated with lower values of $Γ ¯$. The
fact that $Γ ¯$ has a low value means that the electrostatic interactions are less important, making collective effects also less significant. In this case, the theoretical models represent stars
that are above the Kraft break, and thus do not experience spin down on the MS. These stars do not lose angular momentum and retain the rapid rotation rates typical of the ZAMS. Therefore, we can say
that the properties of the global plasma parameter correlate with the observed rotational dependence on mass and effective temperature.
The bottom panel of the three plots in Figure 4 also shows $Γ ¯$ as a function of effective temperature for all the stellar models used in this study, but in this case, the global plasma parameter is
computed for three specific regions within the star as illustrated in Figure 3. These plots are interesting as they show that the different regions of the stellar interior appear to contribute
differently to the total value of the global plasma parameter: while the upper layers of the stellar models contribute strongly to the linear scaling obtained in Figure 4, the inner regions are
responsible for a dispersion around the linear scaling.
The global plasma parameter, which serves as an indicator of the average Coulomb coupling strength within the stellar interior, emerges as a characteristic of the interior that could result in
observable effects. From a physical standpoint, the connection between the global plasma parameter and observable rotational patterns appears logical. This is because the plasma coupling parameter
quantifies the extent to which many-body interactions influence the dynamics of the plasma. Consequently, it acts as a bridge between the microphysics of the stellar interior and the observable
characteristics of the star.
3.4. The electron number density, the electron degeneracy parameter, and the mean molecular weight
In relation to the results of Figure 4, it would be useful to find out whether or not the dependence of $Γ ¯$ on the effective temperature is merely a consequence of stellar structure and evolution
that can be replicated with other fundamental physical parameters. Alternatively, the dependence of $Γ ¯$ on the effective temperature may exhibit a unique character that reveals a significant
connection between the interiors of stars and the observed rotational properties of low-mass MS stars.
In an attempt to decipher between these two possibilities, we also computed global mean values for the electron number density (n[e]), the electron degeneracy parameter (η), and the mean molecular
weight (μ). Definitions of these quantities can be found in, for example, Kippenhahn et al. (2013). The global mean values were calculated in exactly the same way as the global plasma coupling
parameter. Namely,
$n e ¯ = 1 M ∫ 0 M n e d M ( r ) ,$(12)
$η ¯ = 1 M ∫ 0 M η d M ( r ) ,$(13)
$μ ¯ = 1 M ∫ 0 M μ d M ( r ) ,$(14)
where dM(r) is again given by Equation 11 .
The plots for the dependence of these three quantities on the effective temperature are shown in Figure 5. The global electron density dependence on the effective temperature (left panel of Figure 5)
does not share the scaling properties of the global plasma parameter. In this case, we cannot distinguish a clear scaling of $n e ¯$ with the effective temperature, as the groups of stars with masses
of 1.0,1.1,1.2,1.3, and 1.4 M[⊙] have mean electron densities that span the entire possible range of values. Similarly, the dependence of the global mean molecular weight on effective temperature
does not exhibit the strong scaling properties of $Γ ¯$. Nevertheless, cooler stars generally have lower values of $μ ¯$ than hotter stars. As is well known, the mean molecular weight is sensitive to
the composition of the stellar material. We show in the next section that the global plasma parameter strongly depends on metallicity. Finally, as a consequence of the Pauli exclusion principle,
electrons in the interiors of stars become degenerate. The electron degeneracy parameter, η, measures the degree of degeneracy, with larger values of η indicating more significant degeneracy. The
central panel of Figure 5 shows the global degeneracy parameter, $η ¯$, as a function of effective temperature. Here, unlike the other two cases discussed above ($n e ¯$ and $μ ¯$), we notice that $η
¯$ shares similar scaling properties with the global plasma parameter. However, the dispersion of $η ¯$ is much more pronounced than that of $Γ ¯$. From the perspective of the importance of nonideal
effects in stellar interiors, particularly in the cooler upper layers of low-mass stars, this is interesting because two of the main contributors to the nonideal character of the equation of state in
this region are pressure –due to electrostatic interactions– and electron degeneracy (e.g., Clayton 1968).
Fig. 5.
Values of $n e ¯$, $η ¯$, and $μ ¯$ as a function of the effective temperature for all the stellar models. The green, gray, and orange colors have the same meaning as in Fig. 4, as do the vertical
dashed line and the small black circles.
Regarding the question we outline at the beginning of this subsection, we can now conclude that the dependence of $Γ ¯$ on the effective temperature is not merely a consequence of stellar structure
and evolution and cannot be easily replicated with other fundamental physical parameters. In the following section, we continue our investigation of the properties of $Γ ¯$, namely of its dependence
on age and metallicity.
4. Dependence of the global plasma coupling parameter on age and metallicity
4.1. Age
The evolution of the global plasma parameter, $Γ ¯$, throughout the entire MS lifetime is shown in Figures 6 and 7 for different stellar masses and metallicities. The ages at ZAMS and at TAMS for the
models at Z=0.02 metallicity represented in Figures 6 and 7 are listed in Table 2. We used the central mass fraction of hydrogen to define the TAMS; in our models, the TAMS is reached when the
hydrogen mass fraction drops below 10^−9.
Fig. 6.
Variation of the global plasma parameter, $Γ ¯$, as a function of the normalized age for stellar models with different masses. All the models represented in this figure were computed with Z=0.02.
Lower-mass stellar models exhibit higher values of the global plasma parameter throughout the MS lifetime. The dashed black line represents the approximate location of the Kraft break.
Fig. 7.
Global plasma parameter, $Γ ¯$, as a function of age for different metallicities. The solid lines show the same models as those represented in Figure 6, i.e., models computed at Z=0.02
metallicity. Dashed lines represent high-metallicity models (Z=0.03), whereas dotted lines represent low-metallicity models (Z=0.01).
Table 2.
Ages of the models at Z=0.02 metallicity.
Figure 6 focuses on comparing the global plasma parameter for stellar models with different masses but the same metallicity (all models in Figure 6 were computed at Z=0.02 metallicity). This figure
demonstrates that the relevance of electrostatic interactions throughout the MS lifetime –for the same metallicity– is entirely mass dependent. Specifically, as the mass of the star decreases, the
value of the global parameter increases. This supports the expected result that as the mass of the star decreases, the Coulomb interactions between particles become increasingly significant in the
thermodynamics of stellar interiors. From Figure 7, we also observe that the values of the global plasma parameter decrease from ZAMS up to approximately 75% of the MS lifetime, and then begin to
increase until the end of the MS. From the behavior of $Γ ¯$ as a function of age, and knowing that slower rotators are associated with higher values of $Γ ¯$ while rapid rotators are associated with
lower values of $Γ ¯$, it is natural to infer that as the values of $Γ ¯$ decrease over approximately three-quarters of the MS lifetime, the stellar spin-down should weaken (again, because $Γ ¯$ is
decreasing). Interestingly, in recent years, some observational studies (e.g., van Saders et al. 2016; Curtis et al. 2020; Hall et al. 2021) have proposed a scenario of weakened magnetic braking that
deviates from the standard stellar spin-down laws. We have known about the link between the rotation rate of a star and its age since the seminal works of Kraft (1967) and Skumanich (1972).
Specifically, based on observational data, Skumanich derived that for low-mass stars, v[eq]∝t^−1/2, where v[eq] is the equatorial rotational velocity and t is the stellar age. In the weakened
magnetic braking scenario, the star deviates from the standard Skumanich law and appears to be stalled at intermediate or older ages in the MS (e.g., Saunders et al. 2024). The underlying physical
mechanism that leads to this weakened magnetic braking remains unknown. Nevertheless, we want to emphasize that the theoretical properties of the global plasma parameter, particularly its dependence
on age, can be related to the weakened magnetic braking scenario. Moreover, Figure 7 shows that in the last quarter of the MS, the global plasma parameter increases. If $Γ ¯$ can indeed be related to
stellar rotation rates, this would suggest that older MS stars might resume spin down after a period of stalling. This behavior of resumed spin down at older ages was reported for stars in the
Ruprecht 147 cluster by Curtis et al. (2020).
4.2. Metallicity
The effect of metallicity on the global plasma parameter is clearly marked, with high-metallicity stellar models exhibiting higher values of $Γ ¯$. For all the different categories of stellar masses,
the metallicity has a large impact on the global plasma parameter throughout the entire MS evolution. Figure 7 unambiguously demonstrates the impact of metallicity on the global plasma parameter.
This means that, during the MS lifetimes of all the represented stellar models, the higher the metallicity, the greater the significance of electrostatic effects in these stellar interiors. The
relationships between rotation, magnetic activity, and metallicity are still poorly understood. However, some recent studies have unveiled several insights into how metallicity might affect stellar
rotation and activity (Amard et al. 2020; Simonian et al. 2020; See et al. 2021; Avallone et al. 2022). Specifically, the study by Amard et al. (2020), which considered thousands of Kepler stars with
masses ranging from 0.85 to 1.3M[⊙] –nearly the mass range addressed in the present study–found a correlation between metallicity and rotation: metal-rich stars tend to rotate slower than metal-poor
stars. Another very recent study identified a link between slow rotation and high metallicity values for a specific group of Kepler stars (Santos et al. 2023). Here again, we can link the properties
of the global plasma parameter with the observed rotation rates, that is, the higher the metallicity of the stellar model, the higher the value of the global plasma parameter, which in turn can be
related to lower rotation rates.
5. Electrostatic effects and the stellar rotation rates of low-mass stars on the MS
It is well known that one-dimensional stellar models include several approximations that allow us to solve the set of differential equations plus the corresponding boundary conditions. These
approximations are particularly impactful if rotation and magnetic fields are not considered, because spherical symmetry is conserved. Nonetheless, it is still possible to obtain revealing insights
by looking at the physical ingredients that can be linked to rotation and magnetism. One of these ingredients is the relevance of electrostatic interactions, because in a stellar plasma, all
particles experience the Lorentz force, and the Lorentz force can in turn be related to mechanisms of angular momentum transport in stellar interiors (e.g., Aerts et al. 2019)
The discovery of the solar wind by Parker (1958) boosted the studies of stellar magnetized winds. It is thought that these winds exert a breaking torque that removes angular momentum from the stars
(e.g., Schatzman 1962; Weber & Davis 1967; Mestel 1968; Kawaler 1988; Krishnamurthi et al. 1997; Sills et al. 2000; Spada et al. 2011). Taking into consideration all the accumulated empirical data
and also the theoretical predictions, stellar rotation appears to depend strongly on mass, age, and metallicity. In the previous sections of this work we study the properties of a global plasma
parameter, $Γ ¯$, which allows us to compare the importance of electrostatic effects among a group of stellar models. We also explore some connections of this global plasma parameter with the
observed patterns for rotation in the MS. In the present section, we again use our set of theoretical models described in Section 2 to further investigate how the properties of the global plasma
parameter relate to the rotational observational data for low-mass MS stars.
The sample of stars on which we base our models (see Figure 2) is a sample of stars with asteroseismically measured rotation periods (Hall et al. 2021). This means that the values obtained for the
rotation periods do not depend on the starspot variations. To investigate further whether the electrostatic properties of stellar interiors correlate with their observed rotational trends, we plot
the observational rotation periods versus the (also observed) effective temperature, superimposed with the variation of the global plasma parameter as a function of effective temperature obtained
from our models. These plots are shown in Figure 8, where the global plasma parameter is computed for different regions within the interiors of the stellar models. Red circles stand for observational
data, whereas blue circles indicate theoretical data from models. We note that the behavior of $Γ ¯$ as a function of effective temperature unexpectedly follows a scaling, with the slope of this line
being very similar to the scaling observed in the relationship between rotation periods and effective temperature. Moreover, the outer layers of the stars appear to be related to the slope of the
linear scaling, whereas the core region of the star introduces a dispersion around the global trend. The fact that the outer layers have a marked contribution should not be a surprise as the base of
the convective zone, located in the upper 30% of the stellar radius, is thought to be responsible for a dynamo mechanism of magnetic field amplification (e.g., Lopes & Passos 2009; Passos & Lopes
2012; Lopes et al. 2014). Moreover, Brito & Lopes (2019) also found a correlation between a structural feature occurring in the outer convective layers and the rotational behaviors of a group of
Kepler stars.
Fig. 8.
Rotation periods as a function of effective temperature for a group of 66 low-mass MS stars in the Kepler field (the same stars that are represented in Figure 1 with red color, and that serve as
references for the theoretical models). Red circles represent observational data. Blue circles represent theoretical data. Specifically, they denote the $Γ ¯$ values computed for different regions
within the star, as indicated on the right y-axis of each subplot. Also plotted in this figure is a linear regression model fit to each set of data. The shaded area around the regression line
represents a 95% confidence interval for the regression estimate.
To better understand these trends, we fitted a linear regression model to the data using the least squares method. The fits are also represented in Figure 8. Because the observational data are
limited to 66 low-mass MS stars in the Kepler field, we aim to investigate whether the similarity between the two trends (the observational trend and the theoretical trend shown in Figure 8) is a
feature of this particular group of Kepler stars, or if the same correlation can be obtained when considering other groups of stars. For this purpose, we used the Kepler sample published in Santos et
al. (2021a) to compare with our theoretical model results. From this large catalog of observational data, we selected all stars within the T[eff] window of our models. This choice results in a
dataset of 38987 Kepler stars. In order to compare all of the groups of stars, that is, the 66 stars from Hall et al. (2021), the large sample from Santos et al. (2021a), and our theoretical results
for the global plasma parameter, we normalized all three datasets to their maximum values. The results for the linear regression model fits for these three datasets are shown in Figure 9, and the
regression outputs are given in Table 3. The results evidenced by the regression models in Figure 9 reveal a strong correlation between the properties of $Γ ¯$ and the observed rotational trends.
Based on our analysis of the results for the regression lines, we conclude that the correlation between the theoretical trend of the global plasma parameter with the effective temperature, and the
observational trend of the stellar rotation period and the effective temperature, is not a consequence of using a small dataset (with 66 Kepler stars). This correlation between the two trends is even
closer when we use a large observational dataset of Kepler stars.
Fig. 9.
Rotation periods as a function of effective temperature for a group of 38 987 low-mass MS stars in the Kepler field, compared with a smaller group of 66 Kepler low-mass main-sequence stars and with
theoretical data from models. Data represented in blue and red are the same data shown in Figure 8. In this plot, we add a third regression model fitted to the observational data, for a set of
38987 stars from the catalog by Santos et al. (2021b).
6. Conclusions
The connections between the internal thermodynamic behavior of the stellar plasma and stellar rotation represent an understudied research field. From the observational data, we know that rotation is
an intricate function of mass, age, and metallicity. In a previous work, we demonstrated that Coulomb interactions can significantly impact the sound-speed gradient in the outer layers of lower-mass
stars (Brito & Lopes 2021). In the present paper, we focus on describing the electrostatic effects comprehensively across a set of low-mass theoretical stellar models, and correlate these effects
with the observed main rotational trends for low-mass stars on the MS.
By studying the electrostatic properties of stellar interiors across a set of theoretical MS stellar models with masses ranging from 0.7 to 1.4M[⊙], we have discovered that this type of microphysics
in stellar interiors can be directly linked to observed rotational behaviors in stars within this mass range. Our main conclusions can be summarized as follows:
1. The linear scaling of the global plasma parameter with effective temperature shows a very high level of similarity to the linear scaling of rotation periods with effective temperature. Slow
rotators can be linked to stars where electrostatic effects are more significant, whereas rapid rotators can be linked to stars where these effects are less significant. This similarity is not merely
a consequence of stellar evolution, as it cannot be replicated for other stellar parameters.
2. The significance of electrostatic effects, as measured by $Γ ¯$, decreases in stellar interiors as stars age on the MS; it reaches a minimal value at approximately 75% of the MS lifetime and then
starts to increase again. This behavior can be linked to an observational phenomenon known as weakened magnetic braking (e.g., van Saders et al. 2016; Curtis et al. 2020).
3. High-metallicity stellar models exhibit higher values of the global plasma parameter compared to low-metallicity models, indicating that metallicity significantly impacts electrostatic
interactions within stellar interiors. The observational data linking rotation and metallicity is limited. Some studies suggest that high-metallicity stars rotate more slowly than low-metallicity
stars (e.g., Amard et al. 2020). The properties of the global plasma parameter can be related to this scenario.
In this initial study, we use a set of nonrotating low-mass stellar models to explore the relationships between the electrostatic properties of stellar interiors and the observed rotation rates for
stars in this mass range. However, given the impact that metallicity has on these electrostatic properties (see subsection 4.2) and previous studies with rotating models that demonstrated the
influence of metallicity on internal angular momentum transport and the rotation periods of low-mass stars (e.g., Amard et al. 2016, 2019), we believe that further investigation of electrostatic
properties using rotating models will significantly improve our knowledge regarding this topic.
It is also well known that the observed rotation rates of red giant cores are in disagreement with theoretical predictions. In stellar interiors, the fluid constituting the stellar material
experiences Lorentz forces, which are known to play an important role in angular momentum transport. For example, the torque generated by the so-called Tayler-Spruit dynamo (Spruit 2002) could
represent an important step forward in understanding this disagreement. The implementation of the Tayler-Spruit dynamo in the MESA stellar evolution code by Cantiello et al. (2014) for low-mass stars
showed that models with this implementation are in better agreement with the observed core rotation rates. Subsequently, a revised formulation of the Tayler-Spruit dynamo, known as the Fuller
formalism (Fuller et al. 2019), led to an even more efficient transport of angular momentum in the cores of red giants and consequently to better agreement with observational data. Moreover, a
mechanism of angular momentum transport based on the Tayler instability was also successful in explaining the uniform rotation observed in the solar radiative zone (Eggenberger et al. 2005). Although
the Tayler-Spruit mechanism is being used in stellar evolution codes, it still has physical deficiencies, as pointed out in several works (e.g., Zahn et al. 2007; Braithwaite 2006; Goldstein et al.
2019). Therefore, understanding the properties of electrostatic interactions in stellar interiors that can be related to the transport of angular momentum by magnetic fields (e.g., MacGregor &
Charbonneau 1999; MacGregor 2000) could significantly improve our comprehension of the mechanisms behind the redistribution of angular momentum in low-mass stars.
We thank the anonymous referee for the valuable comments and remarks, which have significantly improved the clarity and accuracy of the manuscript. The authors A. Brito and I. Lopes also thank the
Fundação para a Ciência e Tecnologia (FCT), Portugal for the financial support to the Center for Astrophysics and Gravitation–CENTRA, Instituto Superior Tecnico, Universidade de Lisboa, through the
Project No. UIDB/00099/2020. and grant No. PTDC/FISAST/ 28920/2017.
All Tables
Table 1.
Basic statistics (theoretical models vs observational sample).
Table 2.
Ages of the models at Z=0.02 metallicity.
All Figures
Fig. 1.
Kiel diagram of the sample of 66 Kepler stars that serve as a basis for the theoretical models. These stars are represented in red. The theoretical data points, representing stellar model values,
are colored according to age in the top panel and to metallicity in the bottom panel. The location of the Sun is also represented with its usual symbol for reference.
In the text
Fig. 2.
Internal profiles of the Debye length as a function of the fractional radius. All models represented were computed with Z=0.02. Left panel: Changes in the Debye length profiles for models with
masses in the range 0.7−1.4M[⊙]. Central panel: Debye length profiles for models with 0.7 and 1.4M[⊙] at two different stages of evolution: at the ZAMS and at midlife. Right panel: Internal
profiles of the total energy density of electrostatic interactions as a function of the fractional radius. All the models in this figure are at the IAMS, meaning they are in the middle
(intermediate age) of their evolutionary path on the main sequence.
In the text
Fig. 3.
Ionic plasma coupling parameter, Γ[i], illustrated for three models with varying masses at Z=0.02. The upper panel shows Γ[i] as a function of the fractional radius, while the lower panel shows Γ
[i] as a function of temperature (log T). Regions with higher values of Γ[i] indicate a greater influence of electrostatic effects within the stellar interior. Across these three models, the
significance of Coulomb effects is more pronounced in the outer stellar layers. The helium ionization zones are indicated with a purple double arrow, where the plasma coupling parameter reaches its
maximum value. Additionally, three regions of equal length in radius are shown: Region A corresponds to the innermost third of the model, Region B to the central third, and Region C to the
outermost third of the stellar interior. In all the plots, light blue regions correspond to radiative zones, and light red regions to convective regions. Finally, all three models in this figure
are at IAMS, meaning they are in the middle of their evolutionary path on the MS.
In the text
Fig. 4.
Value of $Γ ¯$ as a function of the effective temperature for all the stellar models. Upper panel: Value of $Γ ¯$ computed throughout the entire stellar interior. The green color highlights the
less massive models (0.7, 0.8, and 0.9 M[⊙]), whereas the gray color highlights the more massive models (1.2, 1.3, and 1.4 M[⊙]). The orange color corresponds to models with 1.0 or 1.1 M[⊙]. The
vertical dashed line indicates the approximate location of the Kraft break. Lower panel: Value of $Γ ¯$ computed for three different regions (A, B, and C) within the stellar interiors. Here, we
show the computed values as a function of the effective temperature for all stellar models. Regions A, B, and C are described in Figure 3. The green, gray, and orange colors have the same meaning
as in the upper panel. The small black circles in all the plots indicate the presence of a model with a convective core.
In the text
Fig. 5.
Values of $n e ¯$, $η ¯$, and $μ ¯$ as a function of the effective temperature for all the stellar models. The green, gray, and orange colors have the same meaning as in Fig. 4, as do the vertical
dashed line and the small black circles.
In the text
Fig. 6.
Variation of the global plasma parameter, $Γ ¯$, as a function of the normalized age for stellar models with different masses. All the models represented in this figure were computed with Z=0.02.
Lower-mass stellar models exhibit higher values of the global plasma parameter throughout the MS lifetime. The dashed black line represents the approximate location of the Kraft break.
In the text
Fig. 7.
Global plasma parameter, $Γ ¯$, as a function of age for different metallicities. The solid lines show the same models as those represented in Figure 6, i.e., models computed at Z=0.02
metallicity. Dashed lines represent high-metallicity models (Z=0.03), whereas dotted lines represent low-metallicity models (Z=0.01).
In the text
Fig. 8.
Rotation periods as a function of effective temperature for a group of 66 low-mass MS stars in the Kepler field (the same stars that are represented in Figure 1 with red color, and that serve as
references for the theoretical models). Red circles represent observational data. Blue circles represent theoretical data. Specifically, they denote the $Γ ¯$ values computed for different regions
within the star, as indicated on the right y-axis of each subplot. Also plotted in this figure is a linear regression model fit to each set of data. The shaded area around the regression line
represents a 95% confidence interval for the regression estimate.
In the text
Fig. 9.
Rotation periods as a function of effective temperature for a group of 38 987 low-mass MS stars in the Kepler field, compared with a smaller group of 66 Kepler low-mass main-sequence stars and with
theoretical data from models. Data represented in blue and red are the same data shown in Figure 8. In this plot, we add a third regression model fitted to the observational data, for a set of
38987 stars from the catalog by Santos et al. (2021b).
In the text
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Quantum Microwave Engineering: Key Skills for Every Quantum Hardware Engineer
The expansive realm of quantum information science and technology (QIST) encompasses quantum computing, communication, sensing and simulation. These groundbreaking technologies are poised to rapidly
reshape our world. Aspects like secure quantum communication, quantum internet and advanced quantum sensors are on the horizon, set to become integral parts of our daily lives. Quantum computing,
however, poses significantly greater challenges and requires more time to realize its anticipated potential. Figure 1 illustrates a superconducting quantum computer.
The QIST field is still emerging. As the discipline continues to develop, it will present numerous technical challenges alongside the demand for a highly skilled workforce. A significant part of this
challenge lies in cultivating a multidisciplinary workforce that combines excellent analytical skills with specialized expertise in engineering and science.
Hughes et al. highlight the quantum industry’s urgent demand for skilled professionals in Assessing the Needs of the Quantum Industry.^1 Conventional quantum engineering training tends to be a
prolonged process, posing a challenge. However, by prioritizing essential real-world technical skills crucial for quantum research and development, the training process can be accelerated.
Furthermore, there is a significant opportunity to upskill current professionals, including hardware and software engineers, to meet the increasing demands of the QIST field. To expedite this
transition, having resources that facilitate rapid training is crucial. Recognizing this need, companies like Quaxys are specializing in practice-oriented courses for specialized microwave and
quantum hardware training. These efforts have also spawned books, such as Microwave Techniques in Superconducting Quantum Computers^2, that aim to impart essential skills for quantum hardware
engineers involved in the development of semiconductor quantum platforms, covering topics like superconducting, spin and topological qubits.
Quantum computing platforms fall into two categories: natural systems like atom and ion qubits and artificial systems like superconducting, spin and topological qubits. Each category demands specific
skills, some unique to the platform and others shared. The following sections investigate the four primary skill sets: microwave engineering, cryogenic engineering, nanofabrication and data
acquisition and measurement that are crucial for quantum hardware engineers and companies working on semiconductor qubits to enable success in this field.
The realm of quantum technologies, particularly semiconductor qubits, has opened numerous possibilities for hardware engineers specializing in microwave and embedded hardware engineering. As a
starting point, it is instructive to explore some of the fundamentals. These include the definition of a qubit and the rationale behind utilizing microwave frequencies for superconducting qubits.
A qubit is a fundamental unit of information in quantum computing, representing a two-level quantum system. The encoding of information occurs through the ground state, represented as “0” and the
excited state, represented as “1.”
In the following, we conceptually describe how a superconducting qubit can be implemented as a nonlinear LC circuit to mimic the behavior of an atom. Figure 2a illustrates an atom relaxing from a
higher-energy level to a lower one. The characteristic of this transition is the emission of a photon with a certain frequency corresponding to the difference in energy levels, as shown in the
diagram. In an atom with more than two levels, such as the hydrogen atom, a series of spectral emission lines result from an electron transitioning from a higher-energy orbit to a lower-energy orbit.
These are called Balmer series transitions of a hydrogen atom and are shown in Figure 2b. The n values are integers corresponding to the principal quantum numbers involved in the transition. Each
transition has a unique frequency, as shown. It’s possible to mimic this atomic behavior using a superconducting circuit.
A superconducting LC circuit can be shown to be a quantum harmonic oscillator with equidistant energy levels, as illustrated in Figure 2c. Consequently, there is no unique transition frequency
between energy levels, making it impossible to isolate and address only two energy levels to build a qubit.
By adding non-linearity to the LC circuit using a Josephson junction, we can build an anharmonic quantum oscillator with non-equidistant energy levels, as shown in Figure 2d. This results in unique
transition energies between energy levels, similar to a real atom. This type of nonlinear superconducting circuit provides a two-level system with a unique transition frequency, enabling the
construction of a qubit.
Consider the scenario where the energy difference between the ground and excited states is low, allowing thermal energy in the qubit’s environment to cause the qubit to transition from the ground
state to the first excited state. This uncontrolled thermal transition is undesirable since we seek precise external control over the qubit’s state.
Semiconductor qubits are housed within a dilution refrigerator for various reasons. Foremost among these reasons is the need to mitigate uncontrolled excitation caused by thermal energy. The thermal
energy can be calculated in Equation 1:
k[B] = Boltzmann’s constant
T = temperature
A rough calculation provides insight into the possible effects of unwanted thermal energy. The thermal energy calculated in Equation 1 will have an associated frequency. The Planck-Einstein
relationship of E = hf can be used to determine the frequency (f[th]) associated with thermal energy in a dilution refrigerator. This is shown in Equation 2:
h = Planck’s constant
A typical dilution refrigerator can achieve a temperature of 20 mK, corresponding to a frequency of f[th] = 0.4 GHz. To minimize the occurrence of thermal excitations from the ground to the excited
state of the qubit, it is essential for the qubit’s transition energy (E[01]) to be much higher than the thermal energy, i.e., E[01] >> E[th]. The term “much higher” implies at least 10x greater,
which leads to a transition frequency f[01] = 10f[th] = 4 GHz.
Most superconducting charge qubits, such as the transmon, operate in the microwave frequency range of 2 to 10 GHz. Microwave frequencies are particularly appealing since they are sufficiently high to
leverage standard cryogenic and well-established microwave techniques and components commonly used in the telecommunications industry. While higher frequencies may offer certain advantages, they pose
challenges regarding component cost, design complexity and fabrication capabilities. Therefore, to effectively design and operate superconducting qubit hardware, it is necessary to have a solid
understanding of microwave systems and components.
Microwave engineering involves the generation, transmission, processing and detection of microwave signals as shown, generically, in Figure 3. Depending on the application, one or more of the four
functional areas described in Figure 3 may be involved. In certain applications, such as microwave heating, detection may not be necessary. On the other hand, detection is essential in fields like
radio astronomy. However, in applications like a communication link or a superconducting quantum computer, all four areas play a crucial role. To enhance the understanding of these concepts, they
will be expressed in the context of a microwave link in the next section. | {"url":"https://www.microwavejournal.com/articles/42451-quantum-microwave-engineering-key-skills-for-every-quantum-hardware-engineer?page=1","timestamp":"2024-11-14T00:45:42Z","content_type":"text/html","content_length":"70470","record_id":"<urn:uuid:bd15555c-8244-4933-8c74-84589e4dc5fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00633.warc.gz"} |
David P. DiVincenzo's research works | RWTH Aachen University and other places
Classical chaos in quantum computers
August 2024
13 Reads
3 Citations
Physical Review Research
The development of quantum computing hardware is facing the challenge that current-day quantum processors, comprising 50–100 qubits, already operate outside the range of quantum simulation on
classical computers. In this paper we demonstrate that the simulation of limits can be a potent diagnostic tool for the resilience of quantum information hardware against chaotic instabilities
potentially mitigating this problem. As a testbed for our approach we consider the transmon qubit processor, a computing platform in which the coupling of large numbers of nonlinear quantum
oscillators may trigger destabilizing chaotic resonances. We find that classical and quantum simulations lead to similar stability metrics (classical Lyapunov exponents vs quantum wave function
participation ratios) in systems with O ( 10 ) transmons. However, the big advantage of classical simulation is that it can be pushed to large systems comprising up to thousands of qubits. We exhibit
the utility of this classical toolbox by simulating all current IBM transmon chips, including the 433-qubit processor of the Osprey generation, as well as devices with 1 121 qubits (Condor
generation). For realistic system parameters, we find a systematic increase of Lyapunov exponents with system size, suggesting that larger layouts require added efforts in information protection.
Published by the American Physical Society 2024 | {"url":"https://www.researchgate.net/scientific-contributions/David-P-DiVincenzo-7051608","timestamp":"2024-11-03T01:50:29Z","content_type":"text/html","content_length":"397718","record_id":"<urn:uuid:941fd31e-6e3e-483f-be62-265104f94eba>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00206.warc.gz"} |
Insertion Sort in C & C++ - Program & Algorithm
Insertion Sort in C & C++ – Program & Algorithm
In this tutorial I will explain about algorithm for insertion sort in C and C++ using program example. The insertion sort inserts each element in proper place. The strategy behind the insertion sort
is similar to the process of sorting a pack of cards. You can take a card, move it to its location in sequence and move the remaining cards left or right as needed.
In insertion sort, we assume that first element A[0] in pass 1 is already sorted. In pass 2 the next second element A[1] is compared with the first one and inserted into its proper place either
before or after the first element. In pass 3 the third element A[2] is inserted into its proper place and so on.
Algorithm for Insertion Sort in C & C++
Let ARR is an array with N elements
1. Read ARR
2. Repeat step 3 to 8 for I=1 to N-1
3. Set Temp=ARR[I]
4. Set J=I-1
5. Repeat step 6 and 7 while Temp<ARR[J] AND J>=0
6. Set ARR[J+1]=ARR[J] [Moves element forward]
7. Set J=J-1
[End of step 5 inner
8. Set ARR[J+1]=Temp [Insert element in proper place]
[End of step 2 outer
9. Exit
Program for Insertion Sort in C
int main()
int i,j,n,temp,a[30];
printf("Enter the number of elements:");
printf("\nEnter the elements\n");
a[j+1]=a[j]; //moves element forward
a[j+1]=temp; //insert element in proper place
printf("\nSorted list is as follows\n");
printf("%d ",a[i]);
return 0;
Program for Insertion Sort in C++
using namespace std;
int main()
int i,j,n,temp,a[30];
cout<<"Enter the number of elements:";
cout<<"\nEnter the elements\n";
a[j+1]=a[j]; //moves element forward
a[j+1]=temp; //insert element in proper place
cout<<"\nSorted list is as follows\n";
cout<<a[i]<<" ";
return 0;
Complexity of Insertion Sort in C & C++
There is 1 comparison during pass 1 for proper place. There are 2 comparisons during pass 2 for proper place. There are 3 comparisons during pass 3 for proper place, and so on accordingly.
F(n) = 1 + 2 + 3 + . . . . + (n-1) = n (n-1)/2 = O(n^2)
Hence complexity for insertion sort program in C and C++ is O(n^2).
23 thoughts on “Insertion Sort in C & C++ – Program & Algorithm”
You should change while statement
while((temp=0)) —> while ((j >= 0) && (temp<arr[j]))
no bro
thank you! It helps me a lot!
it’s gud ,but you have to explain more
THANKS A LOT !!!!!!! IT HELPS ME VERY MUCH
please can you modify it add 2 more things.
1. duplicates values are not allowed.
2.if you enter array size greater than current size of an array it display a message that you enter large size array and again we have to choose the size unless we have chose right.
suppose you decleare an array a[12] and you chose array size 13 it is an error .
you can store the array size dynamically.
You mean I have to create a heap to store the sorted elements ?
sorry i didnt say anything because i have fever and headche
How floating number sorted in insertion sort algorithm. …????
Please help me. ..
Please describe the best case algo
thanks it helps me.
why is not working?
because its not in the mood to work
from the sorted list is not working?
it says cant build project (i use code blocks)
how to print each iteration?
Bro, I need merge sort also..
can we write a[i]=a[j]; inside the while loop instead of a[j+1]=a[j];
if not ? why,,,,
please explain
I have an error at (i=0;i<n;i++)
The last part
Why there is j=j-1?
could you please implement Insertion Sort and Merge Sort in a single source file. You are required to note the time Merge Sort and Insertion Sort takes. Actually this is my assignment. I’ll be
very thankful to you.
How to specify space between the elements
Leave a Comment | {"url":"https://www.thecrazyprogrammer.com/2014/12/insertion-sort.html","timestamp":"2024-11-09T01:18:37Z","content_type":"text/html","content_length":"178176","record_id":"<urn:uuid:246bf9f4-5f42-46ab-b3fa-3f99b6581a34>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00102.warc.gz"} |
Co-Existence of Local Limit Cycles from Degenerate and Weak Foci in Cubic Systems
Co-Existence of Local Limit Cycles from Degenerate and Weak Foci in Cubic Systems ()
1. Degenerate Focus
We begin our investigation of local limit cycles by considering a planar cubic system of the following form:
where A, B, C, D, F, K, L, M, N, Q, and R are real constants. We note here that the origin is a degenerate focus as the linearization about the origin is nilpotent but nonzero, and the other
necessary conditions, as given in Perko ( [1] , p. 173), are also met. To find local limit cycles of this system, we build a Liapunov function in the fashion outlined in Blows [2] , an extension of a
result developed by Andreev, Sadovskii, and Tsikalyuk [3] . This function takes the form:
from Blows [2] , it follows that:
and so forth. These calculations quickly become tedious by hand, so the use of Mathe- matica, or a similar program capable of symbolic computation, is absolutely necessary to continue.
We are able to construct
(see [2] ). Provided that the first nonzero [i], called Liapunov numbers, are ordered as they arise in the construction of
We compute the first several
Before going any further, we note that
We say the origin of (1) is said to be a degenerate focus of odd order k if
We continue by considering the equation
Now, setting
after setting
From here, we set
as proved in [2] . This is not desirable, so we choose
and, in summary, with the constraints below, the origin is a center:
Thus, we will have a degenerate focus at the origin of the highest order taking:
2. Coexisting Weak Focus
We continue our investigation of our planar cubic system (1). We have already estab- lished criteria for this system to have three local limit cycles near the origin. Here, we wish to consider the
condition that this system has a weak focus at
It is easy to calculate the necessary constraints on this system for
Since we further require that this fixed point is a weak focus, we need that the Jacobian matrix of the system evaluated at
Entering these results into the system gives us the following:
Next, we add in the values previously determined that give us the highest odd order degenerate focus whilst simultaneously preserving the constraints for the weak focus. Altogether, we have:
We then apply a transformation to take the weak focus onto the origin and write this in canonical form. This gives:
To analyze behavior in a neighborhood of the origin, we apply a familiar method. See Blows and Lloyd [4] for example. Recall that we may use a Liapunov function of the form:
As is well known, in this case we are able to construct
The sign of the first nonzero
If we set this to zero and solve for M, our only non-imaginary choice is
is disallowed, as is
we have established
Hence, it follows from the
3. Results
Theorem 1
The system:
Proof. This result follows from the work carried out in the prior two sections above, and it is clear that both foci have the stability of
Theorem 2
The system:
Proof. We begin by noting if
local limit cycle about the origin, since
duces a second local limit cycle about the origin, since
Remark: Although we only considered the case for | {"url":"https://scirp.org/journal/paperinformation?paperid=71269","timestamp":"2024-11-02T15:18:05Z","content_type":"application/xhtml+xml","content_length":"93986","record_id":"<urn:uuid:4b3d4578-da0a-47a0-943f-6e8ef0a96d64>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00412.warc.gz"} |
Create graphs from adjacency matrices — graph_from_adjacency_matrix
Create graphs from adjacency matrices
graph_from_adjacency_matrix() is a flexible function for creating igraph graphs from adjacency matrices.
mode = c("directed", "undirected", "max", "min", "upper", "lower", "plus"),
weighted = NULL,
diag = TRUE,
add.colnames = NULL,
add.rownames = NA
A square adjacency matrix. From igraph version 0.5.1 this can be a sparse matrix created with the Matrix package.
Character scalar, specifies how igraph should interpret the supplied matrix. See also the weighted argument, the interpretation depends on that too. Possible values are: directed, undirected,
upper, lower, max, min, plus. See details below.
This argument specifies whether to create a weighted graph from an adjacency matrix. If it is NULL then an unweighted graph is created and the elements of the adjacency matrix gives the number of
edges between the vertices. If it is a character constant then for every non-zero matrix entry an edge is created and the value of the entry is added as an edge attribute named by the weighted
argument. If it is TRUE then a weighted graph is created and the name of the edge attribute will be weight. See also details below.
Logical scalar, whether to include the diagonal of the matrix in the calculation. If this is FALSE then the diagonal is zerod out first.
Character scalar, whether to add the column names as vertex attributes. If it is ‘NULL’ (the default) then, if present, column names are added as vertex attribute ‘name’. If ‘NA’ then they will
not be added. If a character constant, then it gives the name of the vertex attribute to add.
Character scalar, whether to add the row names as vertex attributes. Possible values the same as the previous argument. By default row names are not added. If ‘add.rownames’ and ‘add.colnames’
specify the same vertex attribute, then the former is ignored.
Passed to graph_from_adjacency_matrix().
The order of the vertices are preserved, i.e. the vertex corresponding to the first row will be vertex 0 in the graph, etc.
graph_from_adjacency_matrix() operates in two main modes, depending on the weighted argument.
If this argument is NULL then an unweighted graph is created and an element of the adjacency matrix gives the number of edges to create between the two corresponding vertices. The details depend on
the value of the mode argument:
The graph will be directed and a matrix element gives the number of edges between two vertices.
This is exactly the same as max, for convenience. Note that it is not checked whether the matrix is symmetric.
An undirected graph will be created and max(A(i,j), A(j,i)) gives the number of edges.
An undirected graph will be created, only the upper right triangle (including the diagonal) is used for the number of edges.
An undirected graph will be created, only the lower left triangle (including the diagonal) is used for creating the edges.
undirected graph will be created with min(A(i,j), A(j,i)) edges between vertex i and j.
undirected graph will be created with A(i,j)+A(j,i) edges between vertex i and j.
If the weighted argument is not NULL then the elements of the matrix give the weights of the edges (if they are not zero). The details depend on the value of the mode argument:
The graph will be directed and a matrix element gives the edge weights.
First we check that the matrix is symmetric. It is an error if not. Then only the upper triangle is used to create a weighted undirected graph.
An undirected graph will be created and max(A(i,j), A(j,i)) gives the edge weights.
An undirected graph will be created, only the upper right triangle (including the diagonal) is used (for the edge weights).
An undirected graph will be created, only the lower left triangle (including the diagonal) is used for creating the edges.
An undirected graph will be created, min(A(i,j), A(j,i)) gives the edge weights.
An undirected graph will be created, A(i,j)+A(j,i) gives the edge weights.
g1 <- sample(
x = 0:1, size = 100, replace = TRUE,
prob = c(0.9, 0.1)
) %>%
matrix(ncol = 10) %>%
g2 <- sample(
x = 0:5, size = 100, replace = TRUE,
prob = c(0.9, 0.02, 0.02, 0.02, 0.02, 0.02)
) %>%
matrix(ncol = 10) %>%
graph_from_adjacency_matrix(weighted = TRUE)
#> [1] 5 2 3 1 4 4 5 5 3 4 4 3
## various modes for weighted graphs, with some tests
non_zero_sort <- function(x) sort(x[x != 0])
adj_matrix <- matrix(runif(100), 10)
adj_matrix[adj_matrix < 0.5] <- 0
g3 <- graph_from_adjacency_matrix(
(adj_matrix + t(adj_matrix)) / 2,
weighted = TRUE,
mode = "undirected"
g4 <- graph_from_adjacency_matrix(
weighted = TRUE,
mode = "max"
expected_g4_weights <- non_zero_sort(
pmax(adj_matrix, t(adj_matrix))[upper.tri(adj_matrix, diag = TRUE)]
actual_g4_weights <- sort(E(g4)$weight)
all(expected_g4_weights == actual_g4_weights)
#> [1] TRUE
g5 <- graph_from_adjacency_matrix(
weighted = TRUE,
mode = "min"
expected_g5_weights <- non_zero_sort(
pmin(adj_matrix, t(adj_matrix))[upper.tri(adj_matrix, diag = TRUE)]
actual_g5_weights <- sort(E(g5)$weight)
all(expected_g5_weights == actual_g5_weights)
#> [1] TRUE
g6 <- graph_from_adjacency_matrix(
weighted = TRUE,
mode = "upper"
expected_g6_weights <- non_zero_sort(adj_matrix[upper.tri(adj_matrix, diag = TRUE)])
actual_g6_weights <- sort(E(g6)$weight)
all(expected_g6_weights == actual_g6_weights)
#> [1] TRUE
g7 <- graph_from_adjacency_matrix(
weighted = TRUE,
mode = "lower"
expected_g7_weights <- non_zero_sort(adj_matrix[lower.tri(adj_matrix, diag = TRUE)])
actual_g7_weights <- sort(E(g7)$weight)
all(expected_g7_weights == actual_g7_weights)
#> [1] TRUE
g8 <- graph_from_adjacency_matrix(
weighted = TRUE,
mode = "plus"
halve_diag <- function(x) {
diag(x) <- diag(x) / 2
expected_g8_weights <- non_zero_sort(
halve_diag(adj_matrix + t(adj_matrix)
)[lower.tri(adj_matrix, diag = TRUE)])
actual_g8_weights <- sort(E(g8)$weight)
all(expected_g8_weights == actual_g8_weights)
#> [1] TRUE
g9 <- graph_from_adjacency_matrix(
weighted = TRUE,
mode = "plus",
diag = FALSE
zero_diag <- function(x) {
diag(x) <- 0
expected_g9_weights <- non_zero_sort((zero_diag(adj_matrix + t(adj_matrix)))[lower.tri(adj_matrix)])
actual_g9_weights <- sort(E(g9)$weight)
all(expected_g9_weights == actual_g9_weights)
#> [1] TRUE
## row/column names
rownames(adj_matrix) <- sample(letters, nrow(adj_matrix))
colnames(adj_matrix) <- seq(ncol(adj_matrix))
g10 <- graph_from_adjacency_matrix(
weighted = TRUE,
add.rownames = "code"
#> IGRAPH 9662ecf DNW- 10 56 --
#> + attr: name (v/c), code (v/c), weight (e/n) | {"url":"https://r.igraph.org/reference/graph_from_adjacency_matrix.html","timestamp":"2024-11-11T12:47:01Z","content_type":"text/html","content_length":"37430","record_id":"<urn:uuid:87ffeaa8-5450-4b48-952b-b0b51ce8ed61>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00633.warc.gz"} |
Two ships are there in the sea on either side of a light house in such a way that the ships and the light house are in the same straight line. The angles of depression of two ships as observed from the top of the light house are 60∘ and 45∘. If the height of the light house is 200 m, find the distance between the two ships. [Use √3=1.73]
Two ships are there in the sea on either side of a light house in such a way that the ships and the light house are in the same straight line. The angles of depression of two ships as observed from
the top of the light house are 60∘ and 45∘. If the height of the light house is 200 m, find the distance between the two ships. [Use √3=1.73] | {"url":"https://byjus.com/question-answer/two-ships-are-there-in-the-sea-on-either-side-of-a-light-house-in/","timestamp":"2024-11-13T11:19:14Z","content_type":"text/html","content_length":"215784","record_id":"<urn:uuid:59bbaf5e-2d0d-490b-81ad-9f955be55fcd>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00768.warc.gz"} |
Elevator Buttons
In a building’s lobby, some number (N) of people get on an elevator that goes to some number (M) of floors. There may be more people than floors, or more floors than people. Each person is equally
likely to choose any floor, independently of one another. When a floor button is pushed, it will light up.
What is the expected number of lit buttons when the elevator begins its ascent? | {"url":"http://www.brainteasers.io/brain-teasers/elevator-buttons/question.html","timestamp":"2024-11-04T14:24:11Z","content_type":"text/html","content_length":"11990","record_id":"<urn:uuid:ddfddd31-90f6-4247-a25e-0234b25978c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00641.warc.gz"} |
Should You Bet On It? The Mathematics of Gambling
On November 9th, 2008, 22-year-old professional poker player Peter Eastgate defeated 6,843 other gamblers and became the youngest player to win the Main Event at the World Series of Poker. For his
achievement, Eastgate earned $9,152,416 in cash and a spot on the list of the highest earning poker players.
Eastgate did not reach his number one spot simply through chance and speculation, however. On the contrary, casino games involve probabilities and statistics that skilled players use to guide their
gambling decisions.
Three basic principles underlie casino games: definite probabilities, expected value, and volatility index. Understanding these concepts elucidates how these games work and how people like Eastgate
beat their competition.
All events in gambling games have absolute probabilities that depend on sample spaces, or the total number of possible outcomes. For example, if you toss a six-sided die, the sample space is six,
with the probability of landing on any particular side one in six. Games with huge sample spaces, like poker, have events with small probabilities. For instance, in five card poker, the probability
of drawing four of a kind is 0.000240, while the chance of drawing a royal flush, the rarest hand, is a mere 0.00000154.
Skilled poker players understand the sample spaces of the game and probabilities associated with each hand. Thus, estimating the odds of a particular hand will guide their gambling choices.
Adept players are interested not only in probabilities, but also in how much money they can theoretically win from a game or event. The average amount you can expect to win is aptly called the
expected value (EV), and it is mathematically defined as the sum of all possible probabilities multiplied by their associated gains or losses.
For example, if a dealer flips a coin and pays a gambler $1.00 for every time the gambler flips heads, but takes away $1.00 when the gambler flips tails, the expected value would be zero since the
probability of a heads occurring is equivalent to that of a tails occurring (EV = 0.5*$1.00 + 0.5*(-$1.00) = 0). This is considered a “fair” game because the players have no monetary advantage or
disadvantage if they play the game many times.
However, if the dealer gives $1.50 for every time the gambler flips heads, then the EV would be $0.25 (EV = 0.5*$1.50 +0.5*- $1.00 = $0.25). If this game were played 100 times, the gambler would
expect to walk away with $25.
The concept of EV is important in gambling because it tells players how much money they could expect to earn or lose overall. Interestingly, all casino games have negative EVs in the long run. More
commonly known as “house advantage,” negative EVs explain how casinos profit from gamblers.
Why, then, do professional gamblers, cognizant of house advantage, continue to gamble if the casino is mathematically engineered to win? Additionally, how are players still able to make tens of
thousands of dollars in a single game?
Though luck may be the answer for some, the mathematical answer resides in the nuanced difference between expected and actual values. EVs dictate how much a player should expect to gain in the long
run, an arbitrary length of time that most gamblers do not play for. Instead, players are more interested in the actual values of each hand and the fluctuation from its EV.
The volatility index, a technical term for standard deviation, tells a player the chance of earning more or less than the EV. Using the earlier coin example, after 100 games, the player has a 68%
chance of leaving the game with between -$10 and $10 and a 95% of leaving with between -$20 and $20.
Volatility index thus quantifies luck by telling players their odds of earning more than the expected value for a specific number of rounds played. High volatility games or hands have a larger
variation between the expected and actual outcomes and therefore, a greater possibility of winning above the EV. This possibility of earning above the EV is ultimately what attracts gamblers to
Generally, skilled gamblers assess the risk of each round based on the mathematical properties of probability, odds of winning, expected value, volatility index, length of play, and size of bet.
These factors paint a numerical picture of risk and tell the player whether a bet is worth pursuing.
Still, gambling involves far more than simple mathematical properties. Gamblers use a great deal of social psychology to read their fellow players. The ability to decipher bodily cues, for instance,
helps discern fellow players’ mental states and possibly gives a clue to the statistics of their hands.
Gambling is an art and a science; only the best players can synthesize the two to reap millions.
Further Reading
• Hannum, Robert C. (2005). Practical Casino Math. Reno, NV: Trace Publication.
• Packel, Edward W. (2006). The mathematics of games and gambling. Washington, DC: Mathematical Association of America.
• Thorp, Edward (1985). The Mathematics of Gambling. New York, NY: Gambling Times. | {"url":"https://www.yalescientific.org/2010/02/should-you-bet-on-it-the-mathematics-of-gambling/","timestamp":"2024-11-13T02:29:35Z","content_type":"text/html","content_length":"68716","record_id":"<urn:uuid:9ea180ac-327e-4d2d-9568-409f005f9130>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00468.warc.gz"} |
Post-stratification or non-response adjustment? | Published in Survey Practice
Elevator Version
Post-stratification means that the weights are adjusted so that the weighted totals within mutually exclusive cells equal the known population totals. This term is misused most of the time, as in
practice it is often attributed to weighting methods for which only margins of a multivariate contingency table are known, but not the higher order cells. These methods should be referred to as
calibration, and the specific raking algorithm is usually used. While calibrated weights are based on achieving the alignment between the sample and the known population figures, non-response
adjusted weights are based on achieving the alignment between the responding sample and the original sample.
A Round of Golf Version Follows
Before we move on to discuss the different types of weighting adjustments, let us check our gear to make sure we are on the same foot. The target population of a survey is a (hopefully, well-defined)
group of social, economic, etc. units, such as persons, households, or businesses. It can be large (e.g., the decennial U.S. Census has a target population of everybody in the country) to pretty
narrow. (National Immunization Survey has the target population of households with children 19–35 months of age.) The frame of the survey is a technical method by which the units from the target
population are found and enrolled into the sample. The frame can be explicit, as is the case with list samples – e.g., schools in a district with known contact information; or implicit, as is the
case with random digit dialing (RDD) samples, where no full list of phone numbers in the United States may exist, but the ways to generate phone numbers that are likely to lead to a real person are
known (Brick and Tucker 2007). In some cases, an explicit frame may have to be constructed by the survey organization – e.g., by listing housing units in area face-to-face samples. The original
sample (which is a somewhat nonstandard term) is the list of units to be approached with the survey request. This list must contain the unit identifier and contact information (which is often one and
the same – e.g., the phone number in RDD surveys and the household address in area and mail surveys) and may contain additional information (e.g., a sample of hospital patients is likely taken from
the hospital records and may have variables such as age, gender, length of stay and diagnosis). The responding sample (also a somewhat non-standard term) is the final list of units that are
considered completed interviews, i.e., have a final disposition code of 1.1 (AAPOR 2015).
Now that we checked our clubs, let us move to the course. I usually do not like the term “post-stratification.” It has a very specific technical meaning (Holt and Smith 1979), which is to say that
you adjusted the weights so that the weighted totals agree with the known counts or proportions of nonoverlapping, mutually exclusive cells. (I prefer to think in terms of totals rather than
proportions; as the sampling theory books explain (Thompson 1997), totals work better for certain technical reasons.) It is a close relative of stratification, and in some cases, the standard errors
formulae that you can use for stratification also work for post-stratification. (Although these days, all these formulas are hidden behind the software packages interfaces.) Stratification works by
splitting the population into non-overlapping groups (strata), with the purpose of drawing samples independently between strata. Stratification can utilize a single variable (example: area,
address-based sampling [ABS] or RDD designs in which you can only stratify by geography, somewhat approximately on RDD) or several variables (example: establishment surveys, in which the strata are
usually defined as a cross-classification of industry, defined by the first digits of the North American Industry Classification System (NAICS), geography, and categories of the establishment size,
measured by revenue or employment).
The crucial feature of stratification is that it relies on the frame data available before sampling. So when somebody says that his or her general population survey was stratified on race, age,
gender, education, and income, then the literal meaning is that they knew these characteristics for every unit on the frame before sampling. In the United States this is impossible to achieve in
general population surveys, because no general population frame, like RDD or ABS, has this sort of data. (Although in some European countries that have population registers, this is entirely
plausible; also, list samples of specialized populations, like beneficiaries in medical programs or students in a university, may have this sort of demographic variables available through the
administrative channels, so the samples can be properly stratified on them.) If this was not the case, then the survey methods report should have stated something different – e.g., that the sample
was post-stratified on race, age, gender, education, and income.
Unlike stratification, post-stratification relies on the data obtained in the survey itself that were not available before sampling, and adjusts the weights so that the totals in each group are equal
to the known population totals. It still needs the post-stratification cells to be mutually exclusive and cover the whole population. The post-stratified weight for unit i in group g is:
The superscript PS stands for post-stratified; SRC, for source (which could be the base weight of the survey, or frame-integrated weight if multiple frames are being used, non-response weights as
explained below, or some other intermediate weight that serves as the input to post-stratification). All weights within a group are increased proportionately so that the sum of post-stratified
weights equals the known population total. If as a result of the random sampling error and/or non-response, a group becomes relatively underrepresented compared to the population, post-stratification
aligns the representation of that group to match that of the population.
Like in stratification, the post-stratification cells are mutually exclusive, and cover the whole population. So when somebody says that his or her sample was post-stratified on race, age, gender,
education and income (we already learned that we cannot stratify an RDD sample on these variables), it technically means that he or she obtained a five-way table of population counts and adjusted the
weights in each cell. While I could buy somebody doing a two-way cross-classification of demographics, I think anything beyond that is a big, BIG stretch for most types of our data unless the sample
sizes run well into thousands and tens of thousands.
So when somebody said that his or her weight was post-stratified by race, age, gender, education, and income, he or she probably mean that his or her weight was calibrated on these variables.
Calibration (Deville and Sarndal 1992) means that the weights were made to agree with the known population totals for each margin, i.e., the weighted totals in the groups defined by race are equal to
the known population totals; the weighted totals in the groups defined by gender are equal to the known population totals; etc. However, it is not guaranteed that the totals agree in finer cells,
such as race-by-age or education-by-gender-by-age. They are likely to be close, but calibration does not attempt to perfectly align them with the population totals unless these interactions are
themselves entered explicitly as calibration targets. The reasons why these interactions may not be practical are that these totals may not be known, to begin with; and that the sample sizes in the
higher-order crossed cells may be small leading to unstable weights.
Specific implementations of calibration vary. European agencies like to use linear calibration, in which the adjustment factor is a linear combination of the calibration variables. (With a stretch of
technical accuracy, we can say that the linearly calibrated weights are obtained as predictions in a linear regression where instead of equating the normal equations to zero, you equate them to the
negative of discrepancy between the original weighted total for that variable and the known population total.) Most of the U.S. agencies rely on a form of iterative proportional fitting, or raking,
in which each of the margins is adjusted at a time in turn. First, you adjust the margin of race, so that each of the weighted totals of race categories aligns with the known population total. (This
is precisely post-stratification on race). Then you post-stratify on age, then on gender, then on education, then on income. By the time you are done with income, in all likelihood, your weighted
totals on race disagree with the population figures, so you cycle back and post-stratify on race again, and then keep cycling until you get close enough on all variables. Implementations in SAS
Institute (Battaglia, Hoaglin, and Frankel 2009) and Stata (Kolenikov 2014) are available online. Both papers describe practical aspects of implementation like stopping criteria and trimming.
In view of calibration/raking, post-stratification is simply calibration with one margin (although it may be a complicated margin with two- or three-way interactions). I implement the true
post-stratification simply by running my Stata program with one single target.
This has been a long enough post already, and yet I have not touched non-response adjustments (NRA). Non-response is whatever stands between you receiving your original sample (general population
samples from the vendor such as Marketing System Group (MSG) or Survey Sampling International (SSI) for RDD or ABS samples or list samples for specialized clients that have the full population
somewhere in their systems, e.g., a list of patients of a hospital) and you closing the study with your final responding sample that is only a fraction of the original sample. Non-response is a
complicated confluence of noncontacts (your mail was never opened by the intended recipient who just tossed the envelope; your calls on the phone always went to the answering machine), refusals (the
intended respondent explicitly told you that he or she is not interested in continuing the survey), unknown eligibility (it is unclear from the message on the phone if the phone number is active),
If the original sample came with auxiliary information, this information can be used to scale the responding sample back to resemble the original sample more closely:
With an abuse of notation, SRC subscript is recycled to denote the use of weights that serve as input to non-response adjustments (base weights, frame integrated weights, etc.), NRA stands for
non-response adjustment, and is the non-response adjustment factor. Since you know the relation between your original sample and the population, provided by the base weights, the non-response
adjusted weights will align your sample with the population. List samples from client populations often have the variables necessary for such adjustments. For example, in surveys of university
alumni, the information about the year of the degree and the major is typically available, and often date of birth, gender, and race/ethnicity can be found as well. Even if the data on the population
counts are not provided by the client university, and all you receive is the sample of the alumni names, contact information, and the minimal demographics such as above, you can still try to adjust
the responding sample back to the original sample.
There are several methods of non-response adjustment. Let me highlight two of them.
Non-response adjustment can be carried out within cells defined by the existing sample information on both respondents and non-respondents, very much like in post-stratification. You can break the
full sample into the cells defined by cross-classification of cohorts and majors, and define non-response adjustment factor as the inverse of the plain or weighted response rate:
Another popular method of non-response adjustment is by modeling response propensity (Little 1986). Defining the dependent variable as 0 for non-response and 1 for response, you can run a logistic
(or probit, although it is less frequently used) regression model using the auxiliary variables available on the full sample (graduation year, major, demographics) as the explanatory variables. Then
you can define the non-response adjustment factor as the inverse of the predicted response propensity where are coefficient estimates from the logistic regression, or as the inverse of the mean
response rate within a group of units with similar response propensities. A common approach is to break the sample into five groups by the response propensity, either as groups of equal size, in
terms of number of responding units or sampled units, or as intervals of equal length; and then using these groups as non-response adjustment cells to be used in expression (3) or (4).
A word of caution is applicable to all of post-stratification, calibration, and cell non-response adjustment: make sure that each of the cells has at least 50 respondents. Otherwise, you may be
adjusting on something too noisy, and the weights may blow up leading to undesirably high design effects.
Non-response adjustments and calibration are not mutually exclusive, and when the data allow, can and should be used together. There is evidence (Krueger and West 2014) that performing just the
calibration/raking is insufficient without properly accounting for non-response processes.
Let me provide an illustrative example. Let us say that we are taking a sample of a university alumni. The alumni records provide breakdown by cohort and degree, and the sample that is drawn from
these records additionally has the alumni gender. An informative response was simulated to produce higher response rates for females, for holders of graduate degrees, and for more recent graduates,
reflecting the known demographic trends for non-response, and a likely better contact information for recent graduates. The counts are given in Table 1.
Let me construct four different weights, each of which can be considered reasonable, and in some circumstances, the only feasible.
1. Weight 1, the non-response adjusted weight: a logistic regression with main effects of cohort, degree and gender was fit to the data. No attempt to model interactions was made, although the
regression fit poorly as shown by an “almost significant” p-value of the Hosmer-Lemeshow (Hosmer, Lemeshow, and Sturdivant 2013) goodness-of-fit χ2 statistic (p=0.056). The NRA weight was
produced as the base weight divided by the estimated response propensity. No aggregation of respondents was done, and response propensities were used as is. This would be the only feasible weight
if the population totals in column 4 of Table 1 were not known.
2. Weight 2, the post-stratified weight: post-stratification (1) was based on the four cells of cohort and degree. This would be the only feasible weight if the original sample did not have any
additional information on top of the existing population information (although in this case, gender is additionally available), and population counts in all cells are known.
3. Weight 3, raked to margins only: this weight is constructed using the raking algorithm, with population targets being the 22,000 Bachelor vs. 6,500 graduate degrees, and 13,000 vs. 15,500
graduates in a cohort. This would be the only feasible weight if these counts were known, but not the counts of cohort-by-degree cells.
4. Weight 4, non-response-adjusted, post-stratified weight: the non-response adjusted Weights 1 were taken and further post-stratified in the same way as Weights 2 were. This weight uses as much
information about both the population and the response process as possible, and thus is likely to be the most accurate one.
Table 2 reports the values and the summaries of the weights, and Table 3 provides the estimated totals. Note that the true population counts are only known for the total rows of gender. All other
total entries are statistical estimates.
The base weights reflect the stratification by cohort and degree used in the sampling plan. They are identical across gender since gender was not used at the sampling stage. To produce entries in
Table 3, the base weights were scaled up to sum up to the population size. The total estimates reflect the non-response biases in the sample: the graduates from the later cohort and the graduate
degree holders are clearly over-represented when compared to the known totals. The base weights have the lowest degree of variability, which translates to the lowest apparent unequal weighting design
effect of 1.054.
NRA Weight 1 combines the base weights and the response propensities. Since the latter varied across the eight cells with known demographics, the resulting weights demonstrate these differences, as
well. However, since these weights do not make any attempts to equate the totals with the population figures, the latter are off in the total rows. As these weights incorporate both the four
different level of the base weights and eight cell-specific non-response adjustment factors, they demonstrate a higher variability and a higher apparent DEFF of 1.112.
Post-stratified Weight 2 explicitly aligns the weighted totals with those of the population, and hence reproduced them exactly. These weights removed the main effects of cohort and degree in the
non-response process, adjusting up the representation of the earlier cohorts and BA graduates. However, this weight is agnostic to gender, which is a covariate of non-response, and we see in Table 2
that post-stratified weights do not differ by gender. Hence non-response associated with gender (after accounting for cohort and degree) remains in the sample estimates. Unequal weighting effects are
between those of the base weights and the non-response adjusted weights: post-stratification effectively uses only four non-response effective adjustment factors, compared to eight incorporated into
the NRA Weight 2.
While post-stratified Weight 2 effectively uses four known totals (in each cohort-by-gender cell), raked Weight 3 only uses three known totals (overall total, total for cohort 2006, total for BA; the
totals for cohort 2012 and graduate degrees can be obtained as the balance from the overall total). It does not use information on gender, producing identical weights across gender in all
cohort-by-degree cells in Table 2, just as the base weights and post-stratified weights did. The raked weights failed to reproduce the (otherwise known) totals in the cohort-by-degree cells. In other
aspects, these weights seemed to be closer to the post-stratified Weight 2 than to any other weight, although this is only an artifact of the non-response simulation model that did not feature very
strong interaction.
The non-response adjusted, post-stratified Weight 4 utilized both non-response adjustment and post-stratification steps. While Weight 2 took the base weights as the input weights to
post-stratification, Weight 4 took the non-response adjusted Weight 1 as the input. It shows some face validity in both demonstrating variation across gender, and matching the known totals for
cohort-by-degree cells. While this weight arguably removes the greatest fraction of non-response bias compared to other weights, it correspondingly demonstrates the greatest unequal weighting effect:
the trade-off between robustness and efficiency is a very typical one. Weight 4 starts off with four levels of base weight, uses the eight cell-specific factors incorporated into Weight 1 to bring
the responding sample back to the original sample, and then further incorporated four additional cell-specific cohort-by-degree factors used in post-stratification.
Based on both methodological considerations and evidence presented, the most accurate weight, in the sense of having the potential to produce the estimates with the least amount of non-response bias,
is the weight that is based on the combination of the explicit non-response adjustment and post-stratification to the most detailed known figures (cells of degree and cohort).
A stylized flow of the survey data from frame to sample to the final data set, as well as the steps that the survey statistician can undertake to balance the sample back to the population, are
represented on Figure 1 below.
• Do not say “post-stratification”/“post-stratified weight” unless the weight adjustment cells were mutually exclusive. Say “calibration”/“calibrated weight” or “raking”/“raked weight” instead,
especially when unsure.
• Calibration is applicable when you have population totals. For general population surveys, they typically come from the ACS (demographics; http://www.census.gov/acs/www/), NHIS (phone use; http:/
/www.cdc.gov/nchs/nhis.htm) or CPS (detailed labor force participation and economics characteristics; http://www.bls.gov/cps/).
• Non-response adjustments are applicable when your sample comes with auxiliary variables available for both respondents and non-respondents. Then you can adjust your responding sample to make it
closer to the original sample on these variables.
• Non-response adjustment and calibration are not mutually exclusive. If the existing frame, sample and population data permit, both can be used to enhance the sample quality: a non-response
adjustment can be made to align the responding sample with the original sample, and calibration can further be applied to align the resulting sample with the population.
Non-response adjustments and calibration are two of potentially many steps in creating survey weights. This short tutorial provides but a cursory look at these two steps. Other components of weights
may include frame integration, eligibility adjustments, multiplicity adjustments, etc. For more discussion, see Kalton and Flores Cervantes (1998), Valliant et al. (2013) and Lavallee and Beaumont | {"url":"https://www.surveypractice.org/article/2809-post-stratification-or-non-response-adjustment","timestamp":"2024-11-11T19:36:04Z","content_type":"text/html","content_length":"317913","record_id":"<urn:uuid:f9beeb97-1182-4f36-a60f-0ef62528ca64>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00181.warc.gz"} |
Modal testing and analysis of high-rise laminated timber building
Xu, Q., Zou, H., Wang, Z., He, Y., Qi, L., and Wang, J. (2024). "Modal testing and analysis of high-rise laminated timber building," BioResources 19(4), 9616–9630.
To enhance the design and research work on dynamic characteristics of high-rise laminated timber buildings, this paper carried out a modal analysis study on one of the largest laminated timber
buildings in China. The finite element calculating modal analysis was carried out using SAP2000 software, and the experimental modal analysis of the building was carried out via environmental
excitation. The calculating modal results and the experimental modal results showed good agreement. The calculating modal frequency values were generally lower than the experimental modal frequency
values. The natural frequencies obtained by the two methods appeared in the Y-direction first-order bending mode and had values of 2.03 and 2.5 Hz, respectively. The corresponding frequencies of the
first-order torsional mode were 2.82 and 3.25 Hz, respectively. The distribution of the CLT core tube along the length direction of the building has an impact on the vibration mode. The six-story
part shows the second-order bending form, while the four-story section only shows the first-order bending form. The above work provides a case study and reference for the simulation and modal
analysis of high-rise laminated timber buildings, demonstrating the critical role of the core tube structure in such wooden buildings. This insight contributes to a better understanding of structural
performance and design considerations in similar projects.
Download PDF
Full Article
Modal Testing and Analysis of High-rise Laminated Timber Building
Qiyun Xu,^a Hongyan Zou,^c Zheng Wang,^a,* Yuhang He,^b Liang Qi,^c and Jun Wang ^d
To enhance the design and research work on dynamic characteristics of high-rise laminated timber buildings, this paper carried out a modal analysis study on one of the largest laminated timber
buildings in China. The finite element calculating modal analysis was carried out using SAP2000 software, and the experimental modal analysis of the building was carried out via environmental
excitation. The calculating modal results and the experimental modal results showed good agreement. The calculating modal frequency values were generally lower than the experimental modal frequency
values. The natural frequencies obtained by the two methods appeared in the Y-direction first-order bending mode and had values of 2.03 and 2.5 Hz, respectively. The corresponding frequencies of the
first-order torsional mode were 2.82 and 3.25 Hz, respectively. The distribution of the CLT core tube along the length direction of the building has an impact on the vibration mode. The six-story
part shows the second-order bending form, while the four-story section only shows the first-order bending form. The above work provides a case study and reference for the simulation and modal
analysis of high-rise laminated timber buildings, demonstrating the critical role of the core tube structure in such wooden buildings. This insight contributes to a better understanding of structural
performance and design considerations in similar projects.
DOI: 10.15376/biores.19.4.9616-9630
Keywords: Laminated timber; Natural frequency; Dynamic characteristics; Calculating modal; Experimental modal
Contact information: a: College of Materials Science and Engineering, Nanjing Forestry University, Nanjing, Jiangsu, China; b: College of Civil Engineering, Southeast University, Nanjing, Jiangsu,
China; c: College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing, Jiangsu, China; d: College of Information Science and Technology, Nanjing Forestry University,
Nanjing, Jiangsu, China; *Corresponding author: wangzheng63258@163.com
As a type of building material, glulam (glued-laminated timber) has been widely used in the construction industry and is favored for its unique advantages. Glulam features high strength, good
stability, and moisture resistance. However, its drawbacks include a higher cost and the potential for reduced performance in extreme environments due to the effects of the adhesive. Nevertheless,
glulam not only enhances the stability and durability of the wood, but also improves its mechanical properties, making it ideal for large structures and complex designs. The glulam structure has been
applied to emporiums, gymnasiums, schools, restaurants, libraries, and other large public buildings (Sueyoshi 2008; Hayashi and Miyatake 2015), such as the Tacoma Dome gymnasium (USA) and the Bullitt
Center in Seattle (USA). In China, glulam buildings have accounted for 16% of the whole wood buildings.
Modern wooden structure buildings are showing a rapid development trend, and significant progress has been made in research on building materials, component connections, fire resistance, and
anti-corrosion capabilities (Jiang et al. 2022; Quintero et al. 2022; Wang and Ghanem 2023). The traditional structural design and structural analysis theory mainly considers the strength, stiffness,
and stability of the structure. However, in most cases, the results that cause structural failure are most affected by dynamic loads. Therefore, in addition to obtaining the static characteristics of
the building structure, scholars also focus on the dynamic characteristics of the building structure under the action of dynamic loads. Reynolds et al. (2015) took a seven-story cross-laminated
timber (CLT) wood structure building as an object, measured the dynamic characteristics of the building at two construction stages on site using the environmental excitation method, and extracted the
modal parameters of the structure from the acceleration response using random subtraction method and Ibrahim time-domain method. Hafeez et al. (2019) conducted environmental excitation tests on 41
light wood structures in different regions of Canada and obtained the dynamic characteristic parameters of the buildings through finite element numerical simulation. Zhang et al. (2021) developed
twenty-two three-dimensional finite element models with different connection combinations to study the effects of fasteners between CLT plates, and shear stiffness on the dynamic characteristics and
seismic performance of CLT buildings. These studies have conducted dynamic characteristics research on both heavy and light wooden structures, as well as on wooden structure building nodes, which
have high reference values for improving the level of research and design on dynamic characteristics of wooden structure buildings.
When previous researchers conducted research on the dynamic characteristics of building structures, they usually took the modal shapes of the building as the starting point. Modal analysis is a
principal technique utilized to investigate the dynamic properties of structures. Through modal analysis, the natural vibration characteristics of the structure can be obtained. For large wooden
structures, the energy generated by human excitation is difficult to transmit within the structure, so it is necessary to use environmental excitation methods to analyze the building structure. At
present, research projects carried out on the dynamic characteristics of multi-story laminated wood structures often are excessively simplistic, lacking research results that combine experimental
modal analysis with experimental modal analysis. Most of them only obtain the experimental modal shapes and modal parameters of the building structure through testing. However, a single experimental
modal analysis will mask the influence of factors, such as the structural characteristics of the building itself and material defects. The theoretical basis is often insufficient, which is not
conducive to researchers’ and designers’ understanding and mastery of the local and overall aspects of the building. At the same time, the characteristic values of the nine independent elastic
constants of the materials used by a large number of researchers in the finite element calculation of wood structure buildings usually refers to the existing literature, rather than the measured
data, so that the simulation results are not supported by data and their reliability needs to be verified. Therefore, this work involved a modal analysis on a large glulam structure building in
China. Through the calculating modal analysis and experimental modal analysis of the building, the characteristic parameters, such as vibration mode and natural frequency of the structure are
expected to be mastered. The finite element simulation was carried out with nine independent elastic constants of the material, which were measured accurately. The modal and frequency values of the
experimental modal were guided by the calculated modal results, and the accuracy of the calculated modal results was verified by the experimental modal results. The two results were analyzed and
discussed. This work has a high value for engineering projects and practical significance for optimizing the design of large timber structures, including glulam buildings, as well as engineering
inspection and research on their dynamic characteristics.
Research Object
The research object is the R&D center building of Shandong Dingchi Wood Industry Group, located in Penglai District, Yantai City, Shandong Province, with a length of 66.9 m, a width of 17.4 m, a
maximum height of 24.7 m, and a total construction area of 4778.5 m^2. The building is a glulam frame shear wall structure. The main structure is made of glulam, and the lamination is SPF
(Spruce-pine-fir). The stairwell walls are made of CLT shear walls. The remaining walls are mainly light wood shear walls. Building structural materials were produced in Canada and processed by
Shandong Dingchi Wood Industry Group Co., LTD. The R&D center building is mainly composed of a beam-column frame system consisting of one type of glued timber column and four types of glued timber
beams. The first and sixth floors of the building are 4.15 m and 4.7 m high, respectively, and the height of the other floors is 3.4 m. The effective length of each layer of light wood shear wall is
the same, and the edge wall bone column is SPF specification material. The building is shown in Fig. 1.
Fig. 1. Shandong Dingchi Wood Industry Group R&D center building
Test and Results of Building Structure Material Parameters
More than 85% of the structural materials of the R&D center building are spruce-pine-fir (SPF) species. In the macro-analysis of wood structure buildings, it is assumed that wood is a continuous and
uniform orthotropic material with flat texture and no growth defects and has 9 independent elastic constants, namely 3 elastic moduli, 3 shear moduli, and 3 Poisson’s ratio (Wang et al. 2014; Wang et
al. 2022).
The wood has three main directions and three sections, and the three main directions are the direction of grain, radial, and chord, respectively, which are represented by the letters L, R, and T. The
three sections are radial section (LR or RL), tangential section (LT or TL), and cross-section (RT or TR). The nine independent elastic constants of SPF timber are expressed as 3 elastic moduli E[L],
E[T], and E[R], 3 shear moduli G[LR], G[LT], and G[RT], and 3 Poisson ratios μ[LR], μ[LT], μ[RT] or μ[TR], respectively.
In this paper, a series of SPF specifications provided by Shandong Dingchi Wood Industry were tested. Through the CRAS dynamic signal acquisition and analysis system, the free board transient
excitation method (Wang et al. 2015; Gao et al. 2016; Wang et al. 2018) was used to measure the elastic modulus E and shear modulus G of all sides upward of the SPF specimen, and the cantilever board
transient excitation method (Wang et al. 2016) was used to obtain the Poisson ratio μ of all sides upward of SPF timber. The specifications and quantities of test specimens used are shown in Table 1.
The average air-dry density of SPF specimens was 420 kg/m^3, and the moisture content was 7 to 10%. The symmetrical four-point bending method and the asymmetric four-point bending method (Wang et al.
2019, 2023) of beam specimens were also used to verify the accuracy of dynamic test results. The test results of SPF specimens are shown in Table 2, and the dynamic and static errors of the nine
elastic constants did not exceed 10%. Therefore, the results of elastic modulus, shear modulus, and Poisson’s ratio of SPF timbers tested by dynamic free plate and cantilever plate in this study were
judged to be reliable and were applied to the numerical analysis of finite element calculating modal analysis.
Table 1. Specifications and Quantity Table of Board and Beam Specimens
Table 2. Dynamic and Static Test Results of SPF Specimen
Finite Element Simplification of the Building Structure
By utilizing the elastic modulus and thickness of the laminated wood, the elastic modulus of glulam beams and columns can be calculated based on the sum of the torque generated by the normal stress
on the cross-section to the neutral layer as the bending moment received. The laminates of the glulam used in this study are of equal thickness and have the same assemble patterns. According to the
above relationship, the elastic modulus of glulam is approximately equal to that of laminates. Furthermore, according to the analysis of symmetrically laminated glulam in ASTM D3737-09 (2009), the
elastic modulus of the glulam used in this building is also approximately the elastic modulus of the laminated board.
For the analysis of the overall dynamic characteristics of the building, the CLT wall is simulated with orthotropic thin-shell elements. The balsa shear wall is modeled as a diagonal spring element
with the ends hinged to the beams and columns, taking into account only its resistance to lateral forces. Figure 2 illustrates the simplified structure.
Fig. 2. Constructing a simplified model
This paper refers to the calculation method of lateral stiffness of balsa wood shear wall involved in the technical regulations for light wood structure buildings (DG/T J08-2059 2009), and considers
the bending deformation of wall bone columns, shear deformation of cladding panels, and vertical deformation of the bottom of the shear wall, and calculates the equivalent horizontal lateral
stiffness K’ per unit length of single-sided cladding shear wall with Eq. 1, Eq. 2, and Eq. 3:
In Eq. 1, f[vd] is the shear strength of the unit length shear wall (kN/m); h[w] is the height of the wall limb of the monolithic shear wall (mm); L[w] is the length of the shear wall limb parallel
to the load direction (m); A is the cross-sectional area of the wall bone column at the end of the shear wall (mm^2); E is the elastic modulus of the wall bone column at the edge of the shear wall (N
/mm^2); G[a] is the equivalent shear stiffness of wood-based structural panels (kN/mm); dn is the vertical deformation at the bottom of one side of the shear wall when the shear capacity f[vd] is
reached (mm); and k[d] is the horizontal lateral stiffness of the unit length of the shear wall (kN/mm/m). In Eq. 2, L[w] is the length of the shear wall limb parallel to the direction of load (m); γ
[1] is the adjustment factor for the use environment; γ[2] is the adjustment coefficient of shear wall height-to-width ratio; γ[3] is the stiffness adjustment coefficient; K is the horizontal lateral
stiffness (kN/mm/m); and cosθ is the cosine of the angle of the spring element to the horizontal plane.
The effective lengths of the light wood shear wall are the same, and the edge studs are all made of SPF specification materials, and their elastic modulus along grain adopts the test data E[L] in
Table 2. The experimental data f[vd ]= 7.48 kN/m was obtained from Enchun Zhu et al. (2010) of Harbin Institute of Technology as the unit length shear strength of shear walls. The unit length
horizontal lateral stiffness of all shear walls in the building and the equivalent spring element stiffness were calculated according to the above equations and applied to the finite element
Main Steps of Finite Element Model Establishment
Two orthotropic materials, glulam and CLT, were added into SAP2000 software, and the elastic modulus, shear modulus, and Poisson ratio of the measured glulam and CLT materials in Table 2 were
recorded in the software. SAP2000 is a powerful structural analysis and design software that supports both linear and nonlinear analysis. It features an intuitive user interface that allows users to
easily create and modify models, while integrating various international design codes. The version used in this article was V18.2, which operates on a 64-bit Windows operating system. Glulam beams,
and columns are defined as frame sections, while floors and CLT walls are defined as shell sections. The light wood shear wall is defined as an equivalent spring element and the equivalent stiffness
value of the light wood shear wall obtained according to Eq. 3 is used as its stiffness coefficient.
According to the structural layout of the building, the whole glulam structure building was modeled by each layer. The support type of the glulam structure building was rigid-pinned (Polastri et al.
2019), the beam-and-column nodes were pinned and the CLT wall was holed to simulate the doors and windows. The overall building model is shown in Fig. 3. The CLT wall was divided into cell grids, and
all CLT shell cells were selected according to the surface section properties. The nodes were selected where the floors were located and the nodes restraint type were specified as a diaphragm.
Finally, dead load was added and modal analysis was carried out under modal conditions.
Fig. 3. 3D view of the building as a whole
To obtain the actual dynamic characteristics of the six-story laminated timber building in this study and verify the accuracy of the structural modal analysis, the experimental modal analysis was
carried out on the R&D Center building of Shandong Dingchi Wood Industry Group (Fig. 1). The first two-order bending modal and torsional modal in the X and Y directions (Building length direction is
X-direction, building width direction is Y-direction, building height direction is Z-direction) and their corresponding frequency values were obtained.
Principle of Modal Testing Based on Environmental Excitation
The excitation methods of experimental modal analysis were mainly divided into transient excitation, steady excitation, and environmental excitation. Given the properties and effects of transient
excitation and steady-state excitation, it is more reasonable and effective to apply the environmental excitation method to the modal testing of multi-story laminated timber buildings. The external
environment, such as ground pulsation, wind load, and traffic load, is used to stimulate the building to vibrate, and the response signals were collected by an accelerometer or velocity meter and
processed to obtain its dynamic characteristic parameters. Experimental modal analysis based on environmental excitation cannot establish a strict dynamic model because the input of the system and
the transfer functions of the system are not available. Given the premise that the structural vibration system, which possesses n degrees of freedom, is stimulated by uniform white noise, the
frequency response function of the input white noise on n degrees of freedom is:
This expression specifies either a column or a row in the system’s mode matrix. The frequency response function for the i and j degrees of freedom when subjected to stochastic excitation can be
denoted by Eq. 5:
In Eqs. 5 and 6, F is a spectrum, and is its conjugate. Because the input force signal cannot be measured, to identify the vibration mode under natural environment excitation, the vibration of the
first reference degree of freedom i is replaced by F of Eq. 5 and Eq. 6, and the transfer function of the j degree of freedom to the i degree of freedom is defined as:
This transfer function is also known as the Operational Deformation State Frequency Response Function (ODS FRF). Its amplitude is,
where φ[j] = ( j=1, 2, …, n) forms the mode vector. T[ji] is the phase difference between X[j] and X[i]. The modal frequency of each order under environmental excitation is the peak point frequency ω
[k] on the amplitude-frequency curve of the origin transfer function T[ji](ω), which is selected on the total average curve of the vibration power spectrum set of all measurement points.
Test Instrument and Setup
In this experiment, four 941B vibration pickups (56 mm × 56 mm × 77 mm, weight 0.75 kg) and four SYV50-3-1 cables produced by Shanghai Qiyue High-Temperature Cable Co., Ltd. were used to collect
signals. The collected vibration signals were processed and fitted by vibration and dynamic signal acquisition analysis system (including modal analysis instrument and software MaCras, instrument and
software developed by Nanjing Anzheng Software Engineering Co., Ltd.) and computer.
The MaCras intermediate frequency response function test excitation method was set to environmental excitation, and the number of collector channels was set to 4 channels (respectively collected as
reference points X and Y signals and moving points X and Y signals). The analysis frequency was 50 Hz and the FFT block length was 1024.
Main Steps of Experimental Modal Testing
To preliminarily estimate the range of the building’s natural frequency and determine the parameter settings in modal analysis software, preliminary experiments were conducted based on the
calculating modal results. Because the vibration of buildings is similar to that of vertical cantilever beams, a simple model of vertical cantilever beams was established through MaCras, and the
natural frequency of Y-direction was determined within 10 Hz through environmental excitation. The calculating modal frequency also satisfied the range. To eliminate the interference of
high-frequency background noise, the analysis frequency range was set within 50 Hz. Due to the low natural frequency of high-rise buildings, it is not appropriate to use accelerometers suitable for
obtaining medium to high frequencies. Therefore, in the overall test, the accelerometer gear of the vibration pickup will be adjusted to the speed gear for testing.
In addition, to avoid interference from human movement on the collected signal, the whole experimental modal test was carried out during non-working time at night. According to the actual size of the
multi-story laminated timber structure, the model was built in MaCras software. The exterior walls of the building were meshed and test points were arranged on the ground of each floor. The test
model is shown in Fig. 4.
Fig. 4. Framework test model
The total number of geometric nodes in the test model was 292, among which 72 points in the first layer were model constraint points that were not involved in the test. To obtain a clearer vibration
pattern, point No. 72 on the 6^th floor of the building was selected as the reference point for this test, and the remaining 220 test points were tested as moving points. The X and Y direction
velometers of the reference point were placed at point 72, and the X and Y direction velometers of the moving points were placed in the remaining 220 test points for vibration pickup. The velometers
at the reference points and the moving points were facing the same direction. After preliminary parameter estimation, curve fitting, modal normalization, and other steps, the modal shape was
obtained. Figure 5 shows the environmental incentive test site at point 244.
Fig. 5. Environmental incentive test at point No. 244
Results and Discussion of Calculating Modal
The calculating modal shapes and frequency values of each order of the building are shown in Fig. 6. In Fig. 6(a), the Y-direction first-order bending amplitude of the six-story part (the tested
building is of asymmetric construction, with the highest part having six floors, which is referred to as the six-story part in this article) is larger than that of the four-story part. This is
because the building itself is not symmetrical and there is a height difference in the direction of its length. In addition, the amplitude of vibration modes at different positions of the six-story
part is inconsistent. The CLT wall in the middle of the building has a smaller amplitude than the stairwell at the end. This is because the whole CLT wall separates the four-story part from the
six-story part, and its rigidity is relatively large, which limits its Y-direction first-order bending amplitude.
In Fig. 6 (a) (b), there is a certain overlap between the Y-direction and X-direction bending modal shapes, because the CLT wall arranged in the Y-direction is more than that in the X-direction,
which weakens the stiffness advantage provided by the glulam frame and the light wood shear wall in the X-direction, narrowing the first-order bending frequency of the X and Y-direction. The
difference between the two is only 5.1%. In the X-direction first-order bending modal analysis shown in Fig. 6 (b), the amplitude of the six-story part is also larger than that of the four-story
part. In Fig. 6 (e) (d), the six-story part has an obvious second-order bending mode, while the four-story part only has a first-order bending mode, which is caused by the large height difference
between different parts of the building. The higher the number of building floors, the more easily the high-order modes will appear. In Fig. 6 (c), the center of the torsional mode is not located in
the geometric center of the building but shifts toward the middle of the six-story part of the building. This is because the rigidity of the two CLT core tubes in this area is relatively large, and
it is easy to form a torsional center between the two core tubes when torsional vibration occurs.
Fig. 6. Calculating modal shapes
According to the frequency results of the calculating modal, it can be seen that the bending amplitude in the Y-direction is the smallest, indicating that the multi-story laminated timber structure
is more prone to damage under the action of Y-direction bending. The first-order bending frequencies in the X and Y directions are both higher than the first-order torsion frequencies, indicating
that the building is more prone to bending failure than torsion. This is because the three CLT core tubes of the building are distributed at both ends and in the middle of the length direction of the
building, which plays a role in strengthening the local stiffness, thus making it difficult for the building to undergo torsion deformation.
Experimental Modal Results and Discussion of Experimental Modal
The modal test preferably obtained first-order bending, second-order bending, and first-order torsional vibration modal shapes. Among them, the first-order bending frequency in the X-direction was
2.625 Hz, the second-order bending frequency in the X-direction was 6.25 Hz, the first-order bending frequency in the Y-direction was 2.5 Hz, and the first-order torsion frequency in the X-direction
was 3.25 Hz. The experimental modal shapes are shown in Fig. 7.
Fig. 7. Experimental modal shapes
Due to the limited energy distribution range of environmental excitation, this experiment did not measure the third and higher-order modal shapes and frequencies of high-rise glulam structures. As
can be seen from the frequency parameters in Fig. 7, the first-order bending frequency in the Y-direction is the lowest, and the vibration energy is more likely to concentrate on the first-order
bending modal, so the building is more likely to be damaged by the first-order bending in the Y-direction. The first-order torsional frequency measured in this study occurs in the X-direction of the
building, which is larger than the first-order torsional frequency in the X and Y directions. This is because the building has three large-volume CLT core tubes, which are approximately evenly
distributed in the length direction, increasing the difficulty of building torsional frequency.
It can be seen from the Y-direction first-order bending modal in Fig. 7 (a) that the amplitude of the six-story part in the Y-direction first-order bending is larger than that of the four-story part.
Similar to the cantilever beam, this is because the higher the building is, the greater the deviation is more likely to occur under the external dynamic load, and the part away from the fixed end
will have a larger amplitude. The inner wall in the middle of the building has a lower amplitude because the entire wall is made of CLT, and the high rigidity in the surface limits the amplitude of
the building in the first-order bending modal in the Y-direction.
As for Fig. 7(d), the six-story part shows an obvious X-direction second-order bending modal, while the four-story part only shows an X-direction first-order bending mode, showing the same trend. The
second-order bending mode in the middle of the six-story part of the building is the most obvious because there is less arrangement of light wood shear walls and a large area of glass curtain wall,
which has low rigidity in the X-direction and is more prone to large amplitude.
The experimental modal analysis identifies the first-order torsional modal shape in the X-direction, but not in the Y-direction. This phenomenon may be because the X-direction of the building tends
to be symmetrical, while the Y-direction torsion shows a difference in height between the top floors. In addition, two of the three CLT cores are on the sixth floor of the building, and the entire
CLT wall is also on this part, resulting in a more uneven stiffness distribution in the Y-direction, making torsional vibration patterns complex and difficult to identify.
Comparative Analysis of Calculating Modal and Experimental Modal
From Table 3, it is evident that the results from the calculated modal analysis were generally lower than those from the experimental modal analysis. This discrepancy arises because the complex
structure of the test object necessitates an equivalent and simplified approach for the components and nodes in the finite element model. Consequently, this simplification results in a reduced
stiffness of the building structure, leading to modal frequency values that are lower than those obtained from field tests. In the torsional modal, the difference is about 13.2%, and in the bending
modal, the difference is between 18% and 19%. In the second-order bending modal, due to the difference in height of different parts of the building and the influence of the location of the core
cylinder, the Y-direction experimental modal is difficult to identify. The difference in the X-direction second-order bending frequency between the two modal analyses is about 14.6%. Due to the large
variety and quantity of components covered by this research object, the equivalence and simplified processing in the calculation of modal analysis will inevitably have a certain impact on the dynamic
characteristics analysis. Therefore, the difference between calculation results and test results is acceptable within 20%. According to the results of the calculating modal analysis and the
experimental modal analysis, the first three modal categories of the two modal analysis results are consistent. They are first-order bending mode in the Y-direction, first-order bending mode in the
X-direction, and first-order torsional mode.
Table 3. Parameter Comparison
From the first-order bending mode in the X and Y directions in the calculating modal results, the X-direction first-order bending modal analysis has the characteristics of the first-order bending
mode in the Y-direction, and the Y-direction also contains the X-direction partial mode, which is due to the close proximity of the frequencies corresponding to the two modal shapes. The experimental
modal analysis is tested separately in the X and Y directions, so the recombination of the first-order bending mode in the two directions is not reflected. The calculating modal analysis and
experimental modal analysis are consistent in frequency and vibration mode. The difference in natural frequency is 18.8%, and the overall frequency difference is within 20%. The experimental modal
analysis model has certain reference and feasibility.
The comparison of experimental and calculated modal results indicates that the model simplification method used in this study is feasible and applicable for simulating various types of large timber
structures. Additionally, both results demonstrate the significant impact of the CLT core tube on the overall natural frequency of the building. Therefore, incorporating the core tube structure
effectively can lower the natural frequency, enhancing the overall comfort and safety of timber building designs.
1. The model simplification and analysis methods discussed in this article are suitable for modal analysis of this type of building, allowing for the attainment of relatively accurate results. The
calculating modal shows that the first-order bending frequencies in the X and Y directions are approximate, and the modal shapes of the two coincide with each other. The six-story part has a
greater bending amplitude than the four-story part. The torsional center of the first-order torsional modal is offset to the middle of the six-story part. The natural frequency of buildings
occurs in the first-order bending modal in the Y-direction, and buildings are more susceptible to bending failure than torsion.
2. The point-testing method employed in this study effectively captures the modal information of glued-laminated timber structures. The experimental modal shows that the Y-direction first-order
bending frequency of the building is the lowest value in the measured data, indicating that the probability of the failure of the building will occur in the form of first-order bending failure.
At the same time, the six-story part has a greater amplitude than the four-story part in the first-order bending in the Y-direction. The first-order torsional frequency value of the X-direction
is large, and the building is less prone to X-direction torsional damage. The torsional modal in the Y-direction is difficult to identify due to the difference in the height of the top layer of
the torsional part of the Y-direction and the uneven stiffness distribution.
3. The calculated modal frequencies align well with the experimental modal frequencies, although the calculated values are generally lower. Specifically, both analyses identify the natural
frequencies in the first-order bending mode in the Y-direction, with an error of 18.8% between the two. The smallest error is observed in the first-order torsional mode, at 13.2%. In the
X-direction, the frequency errors for the first-order bending and second-order bending modes are approximately 18.5% and 14.6%, respectively. Regarding modal shapes, the distribution of the
architectural CLT core along the length significantly affects the modal shape. The first three modal categories from both analyses are consistent. In the first-order bending mode in the
Y-direction, the amplitude in the six-story section is greater than that in the four-story section.
Conflict of Interest
The authors have no competing interests to declare that are relevant to this article.
Funding Declaration
This project is funded by the 2024 Forestry Science and Technology Innovation and Extension Project in Jiangsu Province (LYKJ[2024]05).
Date and Code Availability
All relevant data are within the paper.
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Article submitted: August 7, 2024; Peer review completed: September 21, 2024; Revised version received: September 25, 2024; Accepted: October 18, 2024; Published: October 28, 2024.
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Engineering Reference — EnergyPlus 8.0
Solar Collectors[LINK]
Solar collectors are devices that convert solar energy into thermal energy by raising the temperature of a circulating heat transfer fluid. The fluid can then be used to heat water for domestic hot
water usage or space heating. Flat-plate solar collectors using water as the heat transfer fluid, Integral-Collector Storage solar collectors using water and unglazed transpired solar collectors
using air are currently the only types of collector available in EnergyPlus.
Flat-Plate Solar Collectors[LINK]
The input object SolarCollector:FlatPlate:Water provides a model for flat-plate solar collectors that are the most common type of collector. Standards have been established by ASHRAE for the
performance testing of these collectors (ASHRAE 1989; 1991) and the Solar Rating and Certification Corporation (SRCC) publishes a directory of commercially available collectors in North America (SRCC
The EnergyPlus model is based on the equations found in the ASHRAE standards and Duffie and Beckman (1991). This model applies to glazed and unglazed flat-plate collectors, as well as banks of
tubular, i.e. evacuated tube, collectors.
Solar and Shading Calculations[LINK]
The solar collector object uses a standard EnergyPlus surface in order to take advantage of the detailed solar and shading calculations. Solar radiation incident on the surface includes beam and
diffuse radiation, as well as radiation reflected from the ground and adjacent surfaces. Shading of the collector by other surfaces, such as nearby buildings or trees, is also taken into account.
Likewise, the collector surface can shade other surfaces, for example, reducing the incident radiation on the roof beneath it.
Thermal Performance[LINK]
The thermal efficiency of a collector is defined as the ratio of the useful heat gain of the collector fluid versus the total incident solar radiation on the gross surface area of the collector.
q = useful heat gain
A = gross area of the collector
I[solar] = total incident solar radiation
Notice that the efficiency is only defined for I[solar] > 0.
An energy balance on a solar collector with double glazing shows relationships between the glazing properties, absorber plate properties, and environmental conditions.
[g1] = transmittance of the first glazing layer
[g2] = transmittance of the second glazing layer
[abs] = absorptance of the absorber plate
R[rad] = radiative resistance from absorber to inside glazing
R[conv] = convective resistance from absorber to inside glazing
R[cond] = conductive resistance from absorber to outdoor air through the insulation
T[abs] = temperature of the absorber plate
T[g2] = temperature of the inside glazing
T[air] = temperature of the outdoor air
The equation above can be approximated with a simpler formulation as:
F[R] = an empirically determined correction factor
() = the product of all transmittance and absorptance terms
U[L] = overall heat loss coefficient combining radiation, convection, and conduction terms
T[in] = inlet temperature of the working fluid
Substituting this into Equation ,
A linear correlation can be constructed by treating F[R]() and -F[R]U[L] as characteristic constants of the solar collector:
Similarly, a quadratic correlation can be constructed using the form:
Both first- and second-order efficiency equation coefficients are listed in the Directory of SRCC Certified Solar Collector Ratings.
Incident Angle Modifiers[LINK]
As with regular windows the transmittance of the collector glazing varies with the incidence angle of radiation. Usually the transmittance is highest when the incident radiation is normal to the
glazing surface. Test conditions determine the efficiency coefficients for normal incidence. For off-normal angles, the transmittance of the glazing is modified by an incident angle modifier
Additional testing determines the incident angle modifier as a function of incident angle . This relationship can be fit to a first-order, linear correlation:
or a second-order, quadratic correlation:
The incident angle modifier coefficients b[0] and b[1] are usually negative, although some collectors have a positive value for b[0]. Both first- and second-order incident angle modifier equation
coefficients are listed in the Directory of SRCC Certified Solar Collector Ratings.
The SRCC incident angle modifier equation coefficients are only valid for incident angles of 60 degrees or less. Because these curves can be valid yet behave poorly for angles greater than 60 degree,
the EnergyPlus model cuts off collector gains for incident angles greater than 60 degrees.
For flat-plate collectors, the incident angle modifier is generally symmetrical. However, for tubular collectors the incident angle modifier is different depending on whether the incident angle is
parallel or perpendicular to the tubes. These are called bi-axial modifiers. Some special flat-plate collectors may also exhibit this asymmetry. The current model cannot yet handle two sets of
incident angle modifiers. In the meantime it is recommended that tubular collectors be approximated with caution using either the parallel or perpendicular correlation.
Incident angle modifiers are calculated separately for sun, sky, and ground radiation. The net incident angle modifier for all incident radiation is calculated by weighting each component by the
corresponding modifier.
For sky and ground radiation the incident angle is approximated using Brandemuehl and Beckman’s equations:
where is the surface tilt in degrees.
The net incident angle modifier is then inserted into the useful heat gain equation :
Equation is also modified accordingly.
Outlet Temperature[LINK]
Outlet temperature is calculated using the useful heat gain q as determined by Equation , the inlet fluid temperature T[in], and the mass flow rate available from the plant simulation:
c[p] = specific heat of the working fluid
Solving for T[out],
If there is no flow through the collector, T[out] is the stagnation temperature of the fluid. This is calculated by setting the left side of Equation to zero and solving for T[in] (which also equals
T[out] for the no flow case).
ASHRAE. 1989. ASHRAE Standard 96-1980 (RA 89): Methods of Testing to Determine the Thermal Performance of Unglazed Flat-Plate Liquid-Type Solar Collectors. Atlanta: American Society of Heating,
Refrigerating and Air-Conditioning Engineers, Inc.
ASHRAE. 1991. ASHRAE Standard 93-1986 (RA 91): Methods of Testing to Determine the Thermal Performance of Solar Collectors. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning
Engineers, Inc.
Duffie, J. A., and Beckman, W. A. 1991. Solar Engineering of Thermal Processes, Second Edition. New York: Wiley-Interscience.
Solar Rating and Certification Corporation. 2004. Directory of SRCC Certified Solar Collector Ratings, OG 100. Cocoa, Florida: Solar Rating and Certification Corporation.
Integral-collector-storage (ICS) Solar Collector[LINK]
Solar collectors with integral storage unit models use SolarCollector:IntegralCollectorStorage object, and the characteristics parameter inputs of this collector are provided by the
SolarCollectorPerformance:IntegralCollectorStorage object. This model is based on detailed Energy Balance equations of solar collectors that integrates storage in it. This model has two options to
represent the collector bottom outside boundary conditions: AmbientAir, and OtherSideConditionsModel. AmbientAir simply applies outside air temperature using combined convection and radiation
conductance, and the OtherSideConditionsModel applies combined radiation and convection models that exiats in a naturally ventilated cavity to represent the collector bottom outside boundary
condition. The later boundary condition accounts for the shading of the collector on the underlying surface, hence, the ICS collector can be assumed as an integral part of the building envelope.
Schamtic diagram of a rectangular ICS solar collector is shown in Figure 273 below:
Solar and Shading Calculations[LINK]
The solar collector object uses a standard EnergyPlus surface in order to take advantage of the detailed solar and shading calculations. Solar radiation incident on the surface includes beam and
diffuse radiation, as well as radiation reflected from the ground and adjacent surfaces. Shading of the collector by other surfaces, such as nearby buildings or trees, is also taken into account.
Likewise, the collector surface shades the roof surface beneath it, hence no direct solar radiation incident on the roof surface. The collector and the roof outside boundary conditions should be
specified as OtherSideConditionModel to account for solar collector shading impact on the roof surface.
Mathematical Model[LINK]
The integral-collector-storage (ICS) solar collector is represented using two transient energy balance equations shown below. These equations represent the energy balance equation for the absorber
plate, and the water in the collector.
m[p]C[p]= thermal capacity of the absorber surface, J/°C
A= collector gross area, m^2
()[e]= transmittance-absorptance product of the absorber plate and cover system
I[t] = total solar irradiation, (W/m^2)
h[pw]= convective heat transfer coefficient from absorber plate to water, (W/m2°K)
U[t]= overall heat loss coefficient from absorber to the ambient air, (W/m2 °K)
T[p]= absorber plate average temperature, (°C)
T[w]= collector water average temperature, (°C)
T[a]= ambient air temperature, (°C)
m[w]C[pw] = thermal capacity of the water mass in the collector, (J/°C)
U[s]= area-weighted conductance of the collector side insulation, (W/m^2°K)
U[b]= conductance of the collector bottom insulation, (W/m^2°K)
T[osc]=outside temperature of bottom insulation determined from the other side condition model, (°C)
T[wi]= Entering makeup or mains water temperature, (°C)
The other side condition model boundary condition represented by the T[osc], allows us to apply a realistic outside boundary condition for a collector mounted on a building roof. This also accounts
for the shading impact of the collector on the under-laying surface (roof). On the other hand if ambient air boundary condition is specified, then the collector does not shade the underlying surface
it is mounted on.
The two energy balance equation can be written as non-homogeneous first order DE with constant coefficients. The initial conditions for these equations are the absorber plate average temperature and
the collector water average temperature at previous time steps.
The two coupled first order differential equation are solved analytically. Auxiliary equation of the the coupled homogeneous differential equation is given by:
This auxiliary quadratic equation has always two distinct real roots ([1] and [2]) hence the solution of the homogeneous equation is exponential, and the general solutions of the differential
equations are given by:
The constant terms A and B are the particular solution of the non-homogeneous differential equations, the coefficients of the exponential terms (c[1], c[2], r[1], and r[2]) are determined from the
initial conditions of the absorber and collector water temperatures (T[p0], T[w0]) and are given by:
Thermal Network Model:[LINK]
The thermal network model requires energy balance for each of the collector covers as well. The heat balance equation of the collector covers is assumed to obey steady state formulation by ignoring
their thermal mass. The thermal-network representation of the ICS collector is shown in Figure 274. Also, the heat balance at each cover surface requires knowledge of the amount of solar fraction
absorbed, which is determined from the ray tracing analysis. For the thermal network model shown above the overall top heat loss coefficient is determined from combination of the resistances in
series as follows:
The convection and radiation heat transfer coefficients in equation above are calculated based on temperatures at the previous time step and determined as described in the Heat Transfer Coefficients
Collector Cover Heat Balance
Ignoring the thermal mass of the collector cover, a steady state heat balance equations are formulated for each cover that allow us to determine cover temperatures. The cover surface heat balance
representation is shown in Figure 275 below.
The steady state cover heat balance equation is given by:
Linearizing the longwave radiation exchange and representing the convection terms using the classical equation for Newton’s law of cooling, the equations for the temperatures of covers 1 and 2 are
given by:
[c] = the weighted average solar absorptance of covers 1 and 2, (-)
h[r,c1-a]= adjusted radiation heat transfer coefficient between cover 1 and the ambient air, (W/m^2K)
h[c,c1-a]= convection heat transfer coefficient between cover 1 and the ambient, (W/m^2K)
h[r,c2-c1]= radiation heat transfer coefficient between covers 1 and 2, (W/m^2K)
h[c,c2-c1]= convection heat transfer coefficient between covers 1 and 2, (W/m^2K)
h[r,p-c2]= radiation heat transfer coefficient between covers 2 and the absorber plate, (W/m^2K)
h[c,p-c2]= convection heat transfer coefficient between covers 2 and the absorber plate, (W/m^2K)
q[LWR,1]= longwave radiation exchange flux on side 1 of the collector cover, (W/m^2)
q[CONV,1]= convection heat flux on side 1 of the collector cover, (W/m^2)
q[LWR,2]= longwave radiation exchange flux on side 2 of the collector cover, (W/m^2)
q[CONV,2]= convection heat flux on side 2 of the collector cover, (W/m^2)
q[solar,abs]= net solar radiation absorbed by the collector cover, (W/m^2)
R= thermal resistance for each section along the heat flow path, (m^2K/W)
Other Side Condition Model[LINK]
ICS Solar Collectors are commonly mounted on building heat transfer surfaces hence the collectors shade the underlying heat transfer surface and require a unique boundary condition that reflects the
air cavity environment created between the bottom of the collector surface and the underlying surface. The other side condition model that allows us to estimate the other side temperature, T[osc],
can be determined based on steady state heat balance using the known collector water temperature at the previous time step.
Ignoring thermal mass of the collector bottom insulation, steady state surface heat balance can be formulated on the outer plane of the collector bottom surface facing the cavity as shown in Figure
4. The heat balance equation on the outer plane of the collector bottom surface is given by:
Substituting the equations for each term in the above equation yields:
Simplifying yields the bottom insulation other side condition temperature:
The cavity air temperature is determined from cavity air heat balance as follows:
h~r, cav~= linearized radiation coefficient for underlying surface in the cavity, (W/m^2K)
h~c, cav~= convection coefficient for underlying surface in the cavity, (W/m^2K)
T[so]= the outside face temperature of the underlying heat transfer surface, (ºC)
q[cond]= conduction heat flux though the insulation bottom and, (W/m^2)
q[conv,cav]= convection heat flux between the collector bottom outside surface and the cavity air, (W/m^2)
q[rad,cav]=longwave radiation exchange flux between the collector bottom outside surface and the outside surface of the underlying surface, (W/m^2)
The cavity air temperature is determined from the cavity air energy balance. The air heat balance requires the ventilated cavity air natural ventilation rates. The calculation of the ventilation rate
is described else where in this document. The SurfaceProperty:ExteriorNaturalVentedCavity, object is required to describe the surface properties, the characteristics of the cavity and opening for
natural ventilation.
Heat Transfer Coefficients[LINK]
The equations used to determine for the various heat transfer coefficients in the absorber and water heat balance equations are given below. The absorbed solar energy is transferred to the water by
convection. Assuming natural convection dominated heat transfer for a hot surface facing down and a clod surface facing down the following correlation for Nusselt number by Fujii and Imura (1972).
The Nusselt number for hot surface facing down ward is given by:
The Nusselt number for hot surface upward and cold surface facing down is given by:
g = gravitation force constant, 9.806 (m/s^2)
T[r]= reference properties where the thermo-physical properties are calculated, (°C)
L[c]= characteristic length for the absorber plate, (m)
k= thermal conductivity of water at reference temperature, (W/mK)
= kinematic viscosity of water at reference temperature, (m^2/s)
= thermal diffusivity of water at reference temperature, (m^2/s)
β[v]= volumetric expansion coefficient evaluated at Tv, Tv =Tw+0.25(Tp-Tw), (K-1)
Nu= Nusselt number calculated for water properties at the reference temperature, (-)
Gr= Grashof number calculated for water properties at the reference temperature, (-)
Pr= Prandtle number calculated for water properties at the reference temperature, (-)
The various radiation and convection heat transfer coefficients are given by the following equations. The convection heat transfer coefficients between the covers and the absorber plate are estimated
from the empirical correlation for the Nusselt number for air gap between two parallel plates developed by Hollands et al. (1976) is:
The long wave radiation exchange coefficients between the outer collector cover and the sky and ground referencing the ambient air temperature for mathematical simplification are given.
The convection heat transfer coefficient from the outer cover to the surrounding air is given by:
When the bottom surface boundary condition is AmbientAir, the combined conductance from the outer cover to the surrounding is calculated from the equation below (Duffie and Beckman, 1991).
The overall loss coefficient through the bottom and side of the collector-storage is estimated as follows:
[c1] = thermal emissivity of collector cover 1, (-)
[c2] = thermal emissivity of collector cover 2, (-)
F[s] = view factor from the collector to the sky, (-)
F[g] = view factor from the collector to the ground, (-)
T[c1] = temperature of collector cover 1, (K)
T[c2] = temperature of collector cover 2, (K)
T[s] = sky temperature, (K)
T[g]= ground temperature, (K)
k = thermal conductivity of air, (W/m K)
L = air gap between the covers, (m)
β = inclination of the plates or covers to the horizontal, (radian)
V[w] = wind speed, (m/s)
U[Lb]= user specified bottom heat loss conductance, W/m^2K
U[Ls]= user specified side heat loss conductance, W/m^2K
A[b]= collector bottom heat transfer area, m^2
A[s]= collector side area, m^2
h[comb]= combined conductance from the outer cover to the ambient air, W/m^2K
Transmittance-Absorptance Product
The transmittance-absorptance product of solar collector is determined using ray tracing method for any incident angle (Duffie and Beckman, 1991). This requires optical properties of the cover and
absorber materials and the the transmittance-absorptance product for any incident angle is given by:
The transmittance of the cover system for single and two cover are given by:
The effective transmittance, reflectance and absorptance of a single cover are given by:
The transmittance of the cover system with absorption only considered [a], is defined as:
The reflectance of un-polarized radiation on passing from medium 1 with reflective index n[1] to medium 2 with reflective index n[2] is given by:
The sky and ground reflected diffuse radiations average equivalent incident angles are approximated by Brandemuehl and Beckman correlation (Duffie and Beckman, 1991) as follows:
**= transmittance of the cover system, (-)
[1] = transmittance of the cover 1, (-)
[2] = transmittance of the cover 2, (-)
**= absorptance of the absorber plate, (-)
[d]= diffuse reflectance of the inner cover, (-)
L = thickness of a cover material, (m)
K = extinction coefficient of a cover material, (m^-1)
[1] = angle of incidence, degree
[2]= angle of refraction, degree
**= slope of the collector, degree
[sd] = equivalent incident angle for sky diffuse solar radiation, degree
[gd]= equivalent incident angle for ground diffuse solar radiation, degree
The integral collector storage unit thermal performance parameters are calculated as follows:
Duffie, J.A., and W.A. Beckman. 1991. Solar Engineering of Thermal Processes, 2d ed. New York: John Wiley & Sons.
Kumar, R. and M.A. Rosen. Thermal performance of integrated collector storage solar water heater with corrugated absorber surface. Applied Thermal Engineering: 30 (2010) 1764–1768.
Fujii, T., and H. Imura. Natural convection heat transfer from aplate with arbitrary inclination. International Journal of Heat and Mass Transfer: 15(4), (1972), 755-764.
Photovoltaic Thermal Flat-Plate Solar Collectors[LINK]
Photovoltaic-Thermal solar collectors (PVT) combine solar electric cells and thermal working fluid to collect both electricity and heat. Athough there are currently comparatively few commercial
products, PVT research has been conducted for the past 30 years and many different types of collectors have been studied. Zondag (2008) and Charalambous et. al (2007) provide reviews of the PVT
literature. Because PVT is much less commercially-mature, there are no standards or rating systems such as for thermal-only, hot-water collectors. EnergyPlus currently has one simple model based on
user-defined efficiencies but a more detailed model based on first-principles and a detailed, layer-by-layer description is under development.
The PVT models reuse the PV models for electrical production. These are described elsewhere in this document in the section on Photovoltaic Arrays-Simple Model
Simple PVT Thermal Model[LINK]
The input object SolarCollector:FlatPlate:PhotovoltaicThermal provides a simple PVT model that is provided for quick use during design or policy studies. The user simply provides values for a thermal
efficiency and the incident solar heats the working fuild. The model also includes a cooling mode for air-based systems where a user-provided surface emmittance is used to model cooling of the
working fluid to the night sky (water-based cooling will be made available once a chilled water storage tank is available). No other details of the PVT collector’s construction are required as input
The simple model can heat either air or liquid. If it heats air, then the PVT is part of HVAC air system loop with air nodes connected to an air system. If it heats liquid, then the PVT is part of
plant loop with nodes connected to a plant loop and the plant operating scheme determines flows.
Air-system-based PVT modeling include a modulating bypass damper arrangement. Control logic decides if the air should bypass the collector to better meet setpoint. The model requires a drybulb
temperature setpoint be placed on the outlet node. The model assume the collector is intended and available for heating when the incident solar is greater than 0.3 W/m^2 and otherwise it is intended
for cooling. The inlet temperature is compare to the setpoint on the outlet node to determine if cooling or heating are beneficial. If they are, then the PVT thermal models are applied to condition
the air stream. If they are not beneficial, then the PVT is completely bypassed and the inlet node is passed directly to the outlet node to model a completely bypassed damper arrangement. A report
variable is available for bypass damper status.
Plant-based PVT do not include a bypass (although one could be used in the plant loop). The collector requests its design flow rate but it otherwise relies on the larger plant loop for control.
When the PVT themal collector is controlled to be “on,” in heating mode, and working fluid is flowing, the model calculates the outlet temperature based on the inlet temperature and the collected
heat using the following equations.
For air-based systems, the value of
When the PVT themal collector is controlled to be “on,” in cooling mode, and working fluid is flowing, the model calculates the outlet temperature based on the inlet temperature and the heat radiated
and convected to the ambient using a heat balance on the outside face of the collector:
The simple model assumes that the effective collector temperature,
Substituting and solving for
Then the outlet temperature can be calculated and heat losses determined. However, the model allows only sensible cooling of the air stream and limits the outlet temperature to not go below the
dewpoint temperature of the inlet.
PVT collectors have a design volume flow rate for the working fluid that is autosizable. For air-based systems used as pre-conditioners, the volume flow rate is sized to meet the maximum outdoor air
flow rate. For water-based systems on the supply side of a plant loop, each of the PVT collectors are sized to the overall loop flow rate. For water-based systems on the demand side of a plant loop,
the collectors are sized using a rule-of-thumb for typical flow rates per unit of collector area. This rule-of-thumb is based on a constant factor of 1.905x10^-5 m^3/s-m^2^^that was developed by
analyzing SRCC data set for conventional solar collectors (see data set SolarCollectors.idf) and averaging the ratio for all 171 different collectors.
Charalambous, P.G., Maidment, G.G., Kalagirou, S.A., and Yiakoumetti, K., Photovoltaic thermal (PV/T) collectors: A review. Applied Thermal Engineering 27 (2007) 275-286.
Zondag, H.A. 2008. Flat-plate PV-Thermal collectors and systems: A review. Renewable and Sustainable Energy Reviews 12 (2008) 891-959.
Unglazed Transpired Solar Collectors[LINK]
The input object SolarCollector:UnglazedTranspired provides a model for transpired collectors that are perhaps one of the most efficient ways to collect solar energy with demonstrated instantaneous
efficiencies of over 90% and average efficiencies of over 70%. They are used for preheating outdoor air needed for ventilation and processes such as crop drying.
In EnergyPlus, an unglazed transpired solar collector (UTSC) is modeled as a special component attached to the outside face of a heat transfer surface that is also connected to the outdoor air path.
A UTSC affects both the thermal envelope and the HVAC air system. From the air system’s point of view, a UTSC is heat exchanger and the modeling needs to determine how much the device raises the
temperature of the outdoor air. From the thermal envelope’s point of view, the presence of the collector on the outside of the surface modifies the conditions experienced by the underlying heat
transfer surfaces. EnergyPlus models building performance throughout the year and the UTSC will often be “off” in terms of forced airflow, but the collector is still present. When the UTSC is “on”
there is suction airflow that is assumed to be uniform across the face. When the UTSC is “off” the collector acts as a radiation and convection baffle situated between the exterior environment and
the outside face of the underlying heat transfer surface. We distinguish these two modes of operation as active or passive and model the UTSC component differently depending on which of these modes
it is in.
Heat Exchanger Effectiveness[LINK]
The perforated absorber plate is treated as a heat exchanger and modeled using a traditional effectiveness formulation. The heat exchanger effectiveness,
Kutscher Correlation[LINK]
Kutscher’s (1994) correlation encompasses surface convection between the collector and the incoming outdoor air stream that occurs on the front face, in the holes, and along the back face of the
collector. The correlation uses a Reynolds number based on the hole diameter as a length scale and the mean velocity of air as it passes through the holes as the velocity scale:
The correlation is a function of Reynolds number, hole geometry, the free stream air velocity, and velocity through the holes:
The Nusselt number is formulated as:
^2·K], and
The heat exchanger effectiveness is:
Kutscher’s relation was formulated for triangular hole layout, but based on Van Decker et al. (2001) we allow using the correlation for square hole layout and scale
Van Decker, Hollands, and Brunger Correlation[LINK]
Van Decker et. al. extended Kutscher’s measurements to include a wider range of collector parameters including plate thickness, pitch, suction velocities, and square hole patterns. Their model
formulation differs from Kutscher’s in that the model was built up from separate effectiveness models for the front, back, and holes of the collector. Their published correlation is:
Heat Exchanger Leaving Temperature[LINK]
Using either of the correlations above allows determining the heat exchanger effectiveness from known values. By definition the heat exchanger effectiveness is also:
By rewriting equation to solve for
Collector Heat Balance[LINK]
The collector is assumed to be sufficiently thin and high-conductivity so that it can be modeled using a single temperature (for both sides and along its area). This temperature
Observe that for the passive case, we do not use the heat exchanger relations to directly model the interaction of ventilating air with the collector. This is because these relations are considered
to not apply when the UTSC is in passive mode. They were developed for uni-directional flow (rather than the balanced-in-and-out flow expected from natural forces) and for specific ranges of suction
face velocity. Therefore, this heat transfer mechanism is handled using classical surface convection models (as if the collector was not perforated). (Air exchanges are modeled as ventilation in the
plenum air heat balance but do not interact with the hole edges in the collector surface.)
When the UTSC is active, the heat balance on the collector surface control volume is:
While the heat balance on the passive collector surface control volume is:
All terms are positive for net flux to the collector except the heat exchanger term, which is taken to be positive in the direction from collector to incoming air stream. Each of these heat balance
components is introduced briefly below.
External SW Radiation[LINK]
External LW Radiation[LINK]
External Convection[LINK]
[co](T[air] - T[o]) where h[co], is the convection coefficient. This coefficient will differ depending on whether or not the UTSC is active or passive. When the UTSC is passive, h[co] is treated in
the same way as an outside face with ExteriorEnvironment conditions. When the UTSC is active, the special suction airflow situation of a transpired collector during operation means that h[co]is often
zero because the suction situation can eliminate mass transport away from the collector. However when the winds are high, the strong turbulence and highly varying pressures can cause the suction flow
situation to breakdown. Therefore, we include the
Heat Exchanger[LINK]
Plenum LW Radation[LINK]
Plenum Convection[LINK]
[cp](T[air] - T[o]) where h[cp], is the convection coefficient. This coefficient is taken as zero when the UTSC is operating because of the suction airflow situation. When the UTSC is off, the value
for h[cp] is obtained from correlations used for window gaps from ISO (2003) standard 15099.
Substituting models into and solving for
and substituting into yields the following equation when the UTSC is passive (“off”):
^2·K], and
Plenum Heat Balance[LINK]
The plenum is the volume of air located between the collector and the underlying heat transfer surface. The plenum air is modeled as well-mixed. The uniform temperature of the plenum air,
Note that we have formulated the control volumes with slight differences for the active and passive cases. For the active case, the suction air situation and heat exchanger effectiveness formulations
dictate that the collector surface control volume encompass part of the air adjacent to both the front and back surfaces of the collector. However for the passive case, the collector surface control
volume has no air in it and the plenum air control volume extends all the way to the surface of the collector.
When the UTSC is active, the heat balance on the plenum air control volume is:
When the UTSC is passive, the heat balance on the plenum air control volume is:
Substituting into and solving for
And substituting into yields the following equation when the UTSC is passive:
The literature on UTSC does not appear to address the passive mode of operation and no models for
^3], and
Mass continuity arguments lead to modeling the area of the openings as one half of the total area of the holes, so we have:
If the UTSC is horizontal and
Underlying Heat Transfer Surface[LINK]
The UTSC is applied to the outside of a heat transfer surface. This surface is modeled using the usual EnergyPlus methods for handling heat capacity and transients – typically the CTF method. These
native EnergyPlus Heat Balance routines are used to calculate
Solar and Shading Calculations[LINK]
The transpired collector object uses a standard EnergyPlus surface in order to take advantage of the detailed solar and shading calculations. Solar radiation incident on the surface includes beam and
diffuse radiation, as well as radiation reflected from the ground and adjacent surfaces. Shading of the collector by other surfaces, such as nearby buildings or trees, is also taken into account.
Local Wind Speed Calculations[LINK]
The outdoor wind speed affects terms used in modeling UTSC components. The wind speed in the weather file is assumed to be measured at a meteorological station located in an open field at a height of
10 m. To adjust for different terrain at the building site and differences in the height of building surfaces, the local wind speed is calculated for each surface.
The wind speed is modified from the measured meteorological wind speed by the equation (ASHRAE 2001):
where z is the height of the centroid of the UTSC, z[met] is the height of the standard metereological wind speed measurement, and a and are terrain-dependent coefficients. is the boundary layer
thickness for the given terrain type. The values of a and are shown in the following tables:
Terrain-Dependent Coefficients (ASHRAE 2001).
Terrain Description Exponent, a Layer Thickness, (m)
1 Flat, open country 0.14 270
2 Rough, wooded country 0.22 370
3 Towns and cities 0.33 460
4 Ocean 0.10 210
5 Urban, industrial, forest 0.22 370
The UTSC can be defined such that it has multiple underlying heat transfer surfaces. The centroid heights for each surface are area-weighted to determine the average height for use in the local wind
Convection Coefficients[LINK]
UTSC modeling requires calculating up to three different coefficients for surface convection heat transfer. These coefficients are defined in the classic way by:
Window5 subroutine “nusselt”.
Radiation Coefficients[LINK]
UTSC modeling requires calculating up to four different linearized coefficients for radiation heat transfer. Whereas radiation calculations usually use temperature raised to the fourth power, this
greatly complicates solving heat balance equations for a single temperature. Linearized radiation coefficients have the same units and are used in the same manner as surface convection coefficients
and introduce very little error for the temperature levels involved.
The radiation coefficient,
all temperatures are converted to Kelvin,
The three other coefficients,
Bypass Control[LINK]
The UTSC is assumed to be arranged so that a bypass damper controls whether or not air is drawn directly from the outdoors or through the UTSC. The control decision is based on whether or not it will
be beneficial to heat the outdoor air. There are multiple levels of control including an availability schedule, whether or not the outdoor air is cooler than the mixed air setpoint, or whether or not
the zone air temperature is lower than a so-called free heating setpoint.
Sizing Warnings[LINK]
Although the design of the transpired collector is left to the user, the program issues warnings when the suction airflow velocity falls outside the range 0.003 to 0.08 m/s.
Overall Efficiency[LINK]
The overall thermal efficiency of the UTSC is a useful output report and is defined as the ratio of the useful heat gain of the entire system versus the total incident solar radiation on the gross
surface area of the collector.
Note that the efficiency
Collector Efficiency[LINK]
The thermal efficiency of the collector is a useful output report and is defined as the ratio of the useful heat gain of the collector fluid versus the total incident solar radiation on the gross
surface area of the collector.
Note that the efficiency
Kutscher, C.F. 1994. Heat exchange effectiveness and pressure drop for air flow through perforated plates with and without crosswind. Journal of Heat Transfer. May 1994, Vol. 116, p. 391. American
Society of Mechanical Engineers.
Van Decker, G.W.E., K.G.T. Hollands, and A.P. Brunger. 2001. Heat-exchange relations for unglazed transpired solar collectors with circular holes on a square of triangular pitch. Solar Energy. Vol.
71, No. 1. pp 33-45, 2001.
ISO. 2003. ISO 15099:2003. Thermal performance of windows, doors, and shading devices – Detailed calculations. International Organization for Standardization. | {"url":"https://bigladdersoftware.com/epx/docs/8-0/engineering-reference/page-101.html","timestamp":"2024-11-07T17:32:45Z","content_type":"text/html","content_length":"105801","record_id":"<urn:uuid:c0f268b3-c1e5-44f9-9c6f-1c12515dc457>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00128.warc.gz"} |
Mastering Clustering Techniques with Scikit-Learn
Written on
Chapter 1: Understanding Clustering
Imagine wandering through a vast library filled with numerous books. Each book contains distinct information, and your task is to categorize them based on shared characteristics. As you navigate the
shelves, you realize that certain books have common themes or subjects. This process of grouping related books is known as clustering.
In the realm of Data Science, clustering serves to group similar instances, unveiling patterns, hidden structures, and intrinsic relationships within a dataset. In this introductory guide, I will
present a formal overview of clustering in Machine Learning, covering the most widely used clustering models and demonstrating how to implement them practically using Scikit-Learn. With a hands-on
approach, you'll encounter ample code examples and visualizations to enhance your understanding of clustering, an essential tool for every data scientist.
This section will result in an indented block of text, typically used for quoting other text.
Section 1.1: The Basics of Clustering
Clustering is a key unsupervised learning task in Machine Learning, differing from supervised learning due to the absence of labeled data. While classification algorithms like Random Forest or
Support Vector Machines rely on labeled data points for training, clustering algorithms operate on unlabeled data, aiming to reveal the structures and patterns within the dataset.
To illustrate this concept, let’s consider a synthetic dataset representing three different species of flowers.
In this scatter plot, each flower species is depicted in a unique color. If the dataset includes labels for each data point, we can use a classification algorithm, such as Random Forest or SVM.
However, in many real-world scenarios, the data we collect may lack labels.
In such cases, classification algorithms become ineffective. Instead, clustering algorithms excel at identifying groups of data points that share similar characteristics.
Identifying similarities and differences among data points can sometimes be straightforward; for instance, the cluster of points in the bottom-left corner is noticeably distinct from others. Yet, it
can be challenging to separate the remaining instances into coherent groups, particularly when the number of classes in the dataset is unknown.
Moreover, clustering algorithms outperform human analysts significantly in segregating data classes, as they can evaluate multiple dimensions efficiently, leveraging all data features. In contrast,
humans are limited to visualizing only two or occasionally three dimensions.
The table below summarizes the primary differences between clustering and classification approaches.
If you're interested in how the synthetic data above was generated, here’s a simple code snippet:
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
# Set random seed for reproducibility
# Number of data points per cluster
num_points = 50
spread = 0.5
# Generate data for cluster 1
cluster1_x = np.random.normal(loc=1.5, scale=spread, size=num_points)
cluster1_y = np.random.normal(loc=0.5, scale=spread, size=num_points)
# Generate data for cluster 2
cluster2_x = np.random.normal(loc=4, scale=spread, size=num_points)
cluster2_y = np.random.normal(loc=1.2, scale=spread, size=num_points)
# Generate data for cluster 3
cluster3_x = np.random.normal(loc=6, scale=spread, size=num_points)
cluster3_y = np.random.normal(loc=2, scale=spread, size=num_points)
# Concatenate data from all clusters
x = np.concatenate([cluster1_x, cluster2_x, cluster3_x])
y = np.concatenate([cluster1_y, cluster2_y, cluster3_y])
# Plot 1
fig, ax = plt.subplots(figsize=(16, 8))
plt.scatter(cluster1_x, cluster1_y, color=sns.color_palette("hls", 24)[1], alpha=.9, s=140)
plt.scatter(cluster2_x, cluster2_y, color=sns.color_palette("hls", 24)[7], alpha=.9, s=140)
plt.scatter(cluster3_x, cluster3_y, color=sns.color_palette("hls", 24)[15], alpha=.9, s=140)
plt.legend(labels=['Flower A', 'Flower B', 'Flower C'], loc='lower right')
plt.title('Synthetic Data with 3 Clusters: Labels Available')
plt.xlabel('Petal length')
plt.ylabel('Petal width')
# Plot 2
fig, ax = plt.subplots(figsize=(16, 8))
plt.scatter(x, y, color='k', alpha=.9, s=140)
plt.title('Synthetic Data with 3 Clusters: Labels Not Available')
plt.xlabel('Petal length')
plt.ylabel('Petal width')
Section 1.2: Applications of Clustering
Clustering plays a vital role in various domains within Machine Learning and Data Science. Here are some notable applications:
1. Customer Segmentation: Commonly used in e-commerce and financial applications, clustering techniques categorize customers based on purchasing behaviors, preferences, or demographics.
2. Anomaly Detection: Clustering is a robust tool for identifying anomalies in fields like cybersecurity and finance. By clustering normal data patterns, outliers can be swiftly identified and
3. Genomic Clustering: In bioinformatics, clustering algorithms analyze genomic data to find similarities or differences in genetic material, aiding in the classification of genes into functional
In summary, clustering algorithms are essential for extracting meaningful patterns from unlabeled data.
Chapter 2: Popular Clustering Algorithms
In this chapter, we will explore some of the most prominent clustering algorithms in Machine Learning, including:
1. K-Means Clustering
2. Hierarchical Clustering
3. DBSCAN
Section 2.1: K-Means Clustering
K-means clustering is a well-known unsupervised Machine Learning algorithm designed to partition data into K distinct clusters. The algorithm iteratively assigns data points to the nearest cluster
centroid and updates the centroids based on the mean of the assigned points. This process continues until convergence, where the centroids change minimally or a predefined number of iterations is
K-means aims to minimize the within-cluster sum of squares (WCSS), also referred to as inertia. Mathematically, the objective function can be defined as follows:
text{Inertia} = sum_{i=1}^{K} sum_{x in C_i} |x - mu_i|^2
where (C_i) represents the data points assigned to cluster (i), and (mu_i) denotes the centroid of cluster (i).
To effectively minimize inertia, K-means follows these steps:
1. Initialization: Randomly select K centroids.
2. Assignment: Assign each data point to the nearest centroid.
3. Update Centroids: Recalculate the centroids based on the mean of the assigned points.
4. Repeat: Continue steps 2 and 3 until convergence or a maximum number of iterations is reached.
Although K-means converges mathematically, it may not always reach the optimal solution due to local optima. To mitigate this issue, you can either manually choose initial centroids if you have a
rough idea of their locations or run the K-means algorithm multiple times with different random initializations and keep the best result.
A critical aspect of K-means is determining the number of clusters, K. Various methods can assist in this decision, with the Elbow method being the most common. This method involves plotting WCSS
against the number of clusters, K. As K increases, WCSS typically decreases; however, the rate of decrease slows down at a certain point, creating an "elbow" in the plot. The optimal number of
clusters is usually identified at this elbow point.
Another advanced method is Silhouette analysis, which assesses clustering quality based on the cohesion and separation of clusters. Each data point has a silhouette score that indicates how similar
it is to its cluster compared to others, ranging from -1 to 1.
Now, let’s shift gears and explore some hands-on coding to better understand K-means clustering. We will generate synthetic data representing five clusters and train a K-means model using the KMeans
class from Scikit-Learn.
from sklearn.datasets import make_blobs
from sklearn.cluster import KMeans
import matplotlib.pyplot as plt
# Generate synthetic data
X, _ = make_blobs(n_samples=300, centers=5, cluster_std=0.4, random_state=42, center_box=(-4, 4))
# Apply K-means clustering
kmeans = KMeans(n_clusters=5, random_state=42)
# Plot data points and cluster centroids
fig, ax = plt.subplots(figsize=(16, 8))
plt.scatter(X[:, 0], X[:, 1], c=kmeans.labels_, cmap='gist_rainbow', alpha=0.7, s=180)
plt.scatter(kmeans.cluster_centers_[:, 0], kmeans.cluster_centers_[:, 1], marker='x', s=200, c='k', label='Centroids')
plt.title('K-Means Clustering')
plt.xlabel('Feature 1')
plt.ylabel('Feature 2')
The results of the model illustrate its effectiveness in capturing the five clusters and assigning centroids correctly.
Next, let’s apply the Elbow method to determine the optimal number of clusters.
This video titled "Unsupervised Machine Learning - Flat Clustering with KMeans with Scikit-learn and Python" provides an in-depth explanation of K-means clustering and its applications.
Despite its simplicity, K-means clustering does have limitations, such as its sensitivity to initial centroid placement and its assumption of spherical clusters, which can hinder performance on
non-linear or irregular shapes.
Section 2.2: Exploring DBSCAN
DBSCAN (Density-Based Spatial Clustering of Applications with Noise) addresses one of K-means' main limitations. Unlike K-means, which struggles with irregularly shaped clusters, DBSCAN employs a
density-based approach to identify clusters of arbitrary shapes. It does not require a predefined number of clusters and is robust to noise and outliers.
DBSCAN relies on two parameters: epsilon (ε) and min_samples. Epsilon defines the maximum distance within which two points are considered neighbors, while min_samples specifies the minimum number of
points required to form a dense region.
The DBSCAN process involves the following steps:
1. Initialization: Start with a random data point and determine its neighborhood (points within ε reach).
2. Core Point Identification: Identify core points with at least min_samples neighbors.
3. Cluster Expansion: Expand clusters by recursively adding reachable points.
4. Outlier Detection: Label points not included in any cluster as outliers.
Choosing appropriate values for epsilon and min_samples is critical for DBSCAN's effectiveness. Epsilon should be determined based on dataset density, and min_samples influences minimum cluster size.
Let’s see how to tune these parameters through a practical example:
from sklearn.datasets import make_moons
from sklearn.cluster import DBSCAN
import matplotlib.pyplot as plt
X, _ = make_moons(n_samples=1000, noise=0.05, random_state=42)
dbscan_1 = DBSCAN(eps=0.05, min_samples=5)
dbscan_2 = DBSCAN(eps=0.2, min_samples=5)
In this example, I generated synthetic data shaped like two half-circles using Scikit-Learn's make_moons function. The first instance of DBSCAN, with a smaller epsilon, detected seven clusters with
numerous outliers, while increasing epsilon to 0.2 yielded a more appropriate result.
Let’s further examine DBSCAN’s performance with another synthetic dataset:
from sklearn.datasets import make_circles
from sklearn.cluster import DBSCAN
import matplotlib.pyplot as plt
X, _ = make_circles(n_samples=1000, noise=0.05, random_state=42, factor=.5)
dbscan_1 = DBSCAN(eps=0.05, min_samples=5)
dbscan_2 = DBSCAN(eps=0.2, min_samples=5)
As demonstrated, a smaller epsilon results in an excessive number of clusters, while a larger epsilon causes all data points to merge into one cluster. Hence, DBSCAN parameters must be carefully
adjusted and combined with data visualization.
You can find the complete code for generating simulations and visualizations in my GitHub repository.
Section 2.3: Hierarchical Clustering
The last algorithm we will explore is Hierarchical Clustering, which operates by grouping data points into a hierarchy of clusters. This algorithm merges or divides clusters based on a distance
metric until either a single cluster containing all data points is formed or a predefined number of clusters is reached.
There are two primary approaches to Hierarchical Clustering:
1. Agglomerative Clustering: Begins with each data point as its own cluster and iteratively merges the closest pairs.
2. Divisive Clustering: Starts with all data points in a single cluster and iteratively splits it into smaller clusters.
The steps followed by Hierarchical Clustering include:
1. Initialization: Treat each point as a cluster (agglomerative) or all points as one cluster (divisive).
2. Merge or Divide Clusters.
3. Distance Calculation.
4. Update Hierarchy.
Let’s illustrate this with a synthetic dataset generated using the make_blobs function:
from sklearn.datasets import make_blobs
from sklearn.cluster import AgglomerativeClustering
# Generate synthetic data
X, _ = make_blobs(n_samples=300, centers=5, cluster_std=0.4, random_state=42, center_box=(-4, 4))
# Apply agglomerative hierarchical clustering
agg_clustering = AgglomerativeClustering(n_clusters=5, linkage='ward')
Upon fitting the Hierarchical Clustering model, we can visualize the resulting clusters.
We can also examine the model's dendrogram, which displays the entire hierarchy. Each leaf node represents a data point, and the height of each branch indicates the distance at which clusters are
From the dendrogram, it is evident that at a distance of 5, exactly five clusters are formed, representing the optimal clustering for this dataset.
Wrap Up
This post has provided a comprehensive overview of the most utilized clustering algorithms: K-means, DBSCAN, and Hierarchical Clustering. Through numerous examples, we have established that no single
clustering model is universally superior. Each algorithm has unique characteristics that yield better results in specific scenarios and with particular datasets.
K-means is straightforward, computationally efficient, and interpretable, making it suitable for large datasets. However, it is sensitive to initial centroid placement and requires prior knowledge of
the number of clusters.
DBSCAN excels at identifying clusters of various shapes while being robust to noise and outliers. However, its effectiveness hinges on appropriate parameter selection, which can be challenging in
datasets with varying densities.
Hierarchical Clustering offers flexibility by not requiring a predefined number of clusters, but it can be computationally intensive and complex to interpret.
In conclusion, understanding and practicing with these clustering algorithms is crucial for selecting the most suitable model based on your dataset and project objectives.
If you found this article valuable, consider following me for updates on my future projects and articles!
This video titled "Intro to scikit-learn (I), SciPy2013 Tutorial, Part 1 of 3" provides foundational insights into using Scikit-Learn for Machine Learning. | {"url":"https://nepalcargoservices.com/mastering-clustering-techniques-sklearn.html","timestamp":"2024-11-07T09:32:36Z","content_type":"text/html","content_length":"26745","record_id":"<urn:uuid:f6c78472-06cf-44de-b8a9-fc1fae620d57>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00537.warc.gz"} |
Integrating and plotting an orbit in an NFW potential
Integrating and plotting an orbit in an NFW potential#
We first need to import some relevant packages:
>>> import astropy.units as u
>>> import matplotlib.pyplot as plt
>>> import numpy as np
>>> import gala.integrate as gi
>>> import gala.dynamics as gd
>>> import gala.potential as gp
>>> from gala.units import galactic
In the examples below, we will work use the galactic UnitSystem: as I define it, this is: \({\rm kpc}\), \({\rm Myr}\), \({\rm M}_\odot\).
We first create a potential object to work with. For this example, we’ll use a spherical NFW potential, parametrized by a scale radius and the circular velocity at the scale radius:
>>> pot = gp.NFWPotential.from_circular_velocity(v_c=200*u.km/u.s,
... r_s=10.*u.kpc,
... units=galactic)
As a demonstration, we’re going to first integrate a single orbit in this potential.
The easiest way to do this is to use the integrate_orbit method of the potential object, which accepts a set of initial conditions and a specification for the time-stepping. We’ll define the initial
conditions as a PhaseSpacePosition object:
>>> ics = gd.PhaseSpacePosition(pos=[10,0,0.] * u.kpc,
... vel=[0,175,0] * u.km/u.s)
>>> orbit = gp.Hamiltonian(pot).integrate_orbit(ics, dt=2., n_steps=2000)
This method returns a Orbit object that contains an array of times and the (6D) position at each time-step. By default, this method uses Leapfrog integration to compute the orbit (LeapfrogIntegrator
), but you can optionally specify a different (more precise) integrator class as a keyword argument:
>>> orbit = gp.Hamiltonian(pot).integrate_orbit(ics, dt=2., n_steps=2000,
... Integrator=gi.DOPRI853Integrator)
We can integrate many orbits in parallel by passing in a 2D array of initial conditions. Here, as an example, we’ll generate some random initial conditions by sampling from a Gaussian around the
initial orbit (with a positional scale of 100 pc, and a velocity scale of 1 km/s):
>>> norbits = 128
>>> new_pos = np.random.normal(ics.pos.xyz.to(u.pc).value, 100.,
... size=(norbits,3)).T * u.pc
>>> new_vel = np.random.normal(ics.vel.d_xyz.to(u.km/u.s).value, 1.,
... size=(norbits,3)).T * u.km/u.s
>>> new_ics = gd.PhaseSpacePosition(pos=new_pos, vel=new_vel)
>>> orbits = gp.Hamiltonian(pot).integrate_orbit(new_ics, dt=2., n_steps=2000)
We’ll now plot the final positions of these orbits over isopotential contours. We use the plot_contours() method of the potential object to plot the potential contours. This function returns a Figure
object, which we can then use to over-plot the orbit points:
>>> grid = np.linspace(-15,15,64)
>>> fig,ax = plt.subplots(1, 1, figsize=(5,5))
>>> fig = pot.plot_contours(grid=(grid,grid,0), cmap='Greys', ax=ax)
>>> fig = orbits[-1].plot(['x', 'y'], color='#9ecae1', s=1., alpha=0.5,
... axes=[ax], auto_aspect=False)
(Source code, png, pdf) | {"url":"https://gala-astro.readthedocs.io/en/latest/tutorials/integrate-potential-example.html","timestamp":"2024-11-04T02:48:03Z","content_type":"text/html","content_length":"37051","record_id":"<urn:uuid:1e8db844-a7c4-45bb-9d7e-30da893620c1>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00218.warc.gz"} |
How do you find the parametrization of a plane?
How do you find the parametrization of a plane?
To find a parametrization, we need to find two vectors parallel to the plane and a point on the plane. Finding a point on the plane is easy. We can choose any value for x and y and calculate z from
the equation for the plane. Let x=0 and y=0, then equation (1) means that z=18−x+2y3=18−0+2(0)3=6.
What does it mean to parameterize a model?
Parameterization in a weather or climate model in the context of numerical weather prediction is a method of replacing processes that are too small-scale or complex to be physically represented in
the model by a simplified process.
What do parameterized means?
“To parameterize” by itself means “to express in terms of parameters”. Parametrization is a mathematical process consisting of expressing the state of a system, process or model as a function of some
independent quantities called parameters.
What is a parameterized surface?
A parametrization of a surface is a vector-valued function r(u, v) = 〈x(u, v), y(u, v), z(u, v)〉 , where x(u, v), y(u, v), z(u, v) are three functions of two variables. Because two parameters u and
v are involved, the map r is also called uv-map. A parametrized surface is the image of the uv-map.
Why do we use parametrization?
Most parameterization techniques focus on how to “flatten out” the surface into the plane while maintaining some properties as best as possible (such as area). These techniques are used to produce
the mapping between the manifold and the surface.
What is parametrization of curve?
A parametrization of a curve is a map r(t) = from a parameter interval R = [a, b] to the plane. The functions x(t), y(t) are called coordinate functions. As t varies, the end point of this vector
moves along the curve. The parametrization contains more information about the curve then the curve alone.
What is natural parametrization?
The natural parametric equations of a curve are parametric equations that represent the curve in terms of a coordinate-independent parameter, generally arc length , instead of an arbitrary variable
like . For example, while the usual parametric equations for circle of radius centered at the origin are given by. (1) (2)
How do you parameterize?
56 second suggested clip0:366:34How to Parametrize a Curve – YouTubeYouTubeStart of suggested clipEnd of suggested clipAnything and then just take all the X’s out in your function. And replace them
with what you’veMoreAnything and then just take all the X’s out in your function. And replace them with what you’ve chosen X to be and then you’ve got a your parametrizations for the y part.
How do you write a parametrization?
We usually write this condition for x being on the line as x=tv+a. This equation is called the parametrization of the line, where t is a free parameter that is allowed to be any real number. The idea
of the parametrization is that as the parameter t sweeps through all real numbers, x sweeps out the line.
How do you find the parametrization of a vector?
61 second suggested clip0:062:13Vector Parameterization of a Line – YouTubeYouTube
What is a parameterized or generic type?
A parameterized type is an instantiation of a generic type with actual type arguments. A generic type is a reference type that has one or more type parameters. These type parameters are later
replaced by type arguments when the generic type is instantiated (or declared ).
What’s another word for parameters?
What is another word for parameter?
boundary framework
limit limitation
constant criterion
guideline restriction
specification variable
What is the parametric equation of a plane?
This gives us the following parametric equation of a plane hx;y;zi= h1;3;0i+ th2; 4; 1i+ sh2; 3;7i To nd the implicit formula, we must nd a vector orthogonal/normal to the plane.
How to plot a plane?
To graph complex numbers, you simply combine the ideas of the real-number coordinate plane and the Gauss or Argand coordinate plane to create the complex coordinate plane. In other words, given a
complex number A+Bi, you take the real portion of the complex number (A) to represent the x-coordinate, and you take the imaginary portion ]
Can you have a plane within a plane in geometry?
Since it is not contained within the outline of the plane, you can imagine that it is floating above the plane. You can picture it as if the blue plane is the floor of a room and point S is a soap
What is the intersection of a sphere and a plane?
When a plane intersects a sphere at more than two points, it is a circle (given). Let x^2+y^2+z^2=1 be a sphere S, and P be a plane that intersects S to make a circle (called C). | {"url":"https://www.meatandsupplyco.com/how-do-you-find-the-parametrization-of-a-plane/","timestamp":"2024-11-03T04:00:36Z","content_type":"text/html","content_length":"55195","record_id":"<urn:uuid:b8e1ae97-52e9-42cf-aefa-2892d89afab1>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00093.warc.gz"} |
Python Square Root Without Math Module - ** or Newton's Method
In Python, the easiest way to find the square root of a number without the math module is with the built in exponentiation operator **.
sqrt_of_10 = 10**(1/2)
When working with numeric data in Python, one calculation which is valuable is finding the square root of a number.
We can find the square root of a number easily with the math module, but sometimes we don’t want to import modules to our code.
We can also use the built in ** to find exponents in Python. To find a square root with the ** operator, we just put “(1/2)” after **.
Below are some examples of how to use the Python built in ** operator to find square roots.
import math
Finding the Square Root of a Number Without the Python math Module
We can also estimate the square root of a number without the Python math module. To compute the square root in Python without the Python math module, we can employ the help of Newton’s Method.
Newton’s method is a root finding algorithm which can help us find an approximation of a function.
We can use Newton’s method to find the square root of a number in Python.
Below is a function which you can use to utilize Newton’s method to find an approximation for the square root of a number to precision level “a”. For a comparison, we will also use the sqrt()
function from the Python math module.
import math
def newton_sqrt(n,a):
x = n
root = 0.5*(x+(n/x))
if (abs(root-x) < a):
x = root
return root
As shown above, Newton's method allows us to get a pretty good approximation of the square root of a number without using the math module.
Hopefully this article has been beneficial for you to learn how to find the square root of a number without the math module in Python. | {"url":"https://daztech.com/python-square-root-without-math/","timestamp":"2024-11-09T03:53:18Z","content_type":"text/html","content_length":"242093","record_id":"<urn:uuid:1a9de978-465b-4553-a568-ebcd07527138>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00383.warc.gz"} |
Chapter 6: Internal Forces
In the last chapter we looked at the normal (axial) force running through beams joined into trusses by analyzing either the joints or a whole section of the truss.
In this chapter, we look at what happens along a single beam. We will look at three types of internal forces and moments. Note that when we say ‘internal forces’, we really mean ‘internal forces and
moments’. Inside a beam, we will calculate the normal and shear forces as well as the bending moment at any point in the beam.
For this chapter: the shear force and bending moment change throughout the beam because additional transverse forces are applied. However, the normal force usually stays the same, because it’s
uncommon to have applied axial forces along the beam.
Here are the sections in this Chapter:
Here are the important equations for this chapter: | {"url":"https://pressbooks.library.upei.ca/statics/part/chapter-6-internal-forces/","timestamp":"2024-11-05T02:53:28Z","content_type":"text/html","content_length":"82092","record_id":"<urn:uuid:2bc5556f-0c17-4027-9d1e-7343dc6e90fa>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00808.warc.gz"} |
Classification - Data Mining
Classification is the process of learning a model that describes different classes of data. The classes are predetermined. For example, in a banking application, customers who apply for a credit card
may be classified as a poor risk, fair risk, or good risk. Hence this type of activity is also called supervised learning. Once the model is built, it can be used to classify new data. The first
step—learning the model—is accomplished by using a training set of data that has already been classified. Each record in the training data contains an attribute, called the class label, which
indicates which class the record belongs to. The model that is produced is usually in the form of a decision tree or a set of rules. Some of the important issues with regard to the model and the
algorithm that produces the model include the model’s ability to predict the correct class of new data, the computational cost associated with the algorithm, and the scalability of the algorithm.
We will examine the approach where our model is in the form of a decision tree. A decision tree is simply a graphical representation of the description of each class or, in other words, a
representation of the classification rules. A sample decision tree is pictured in Figure 28.5. We see from Figure 28.5 that if a customer is married and if salary >= 50K, then they are a good risk
for a bank credit card. This is one of the rules that describe the class good risk. Traversing the decision tree from the root to each leaf node forms other rules for this class and the two other
classes. Algorithm 28.3 shows the procedure for constructing a decision tree from a training data set. Initially, all training samples are at the root of the tree. The samples are partitioned
recursively based on selected attributes. The attribute used at a node to partition the samples is the one with the best splitting criterion, for example, the one that maximizes the information gain
Algorithm 28.3. Algorithm for Decision Tree Induction
Input: Set of training data records: R[1], R[2], ..., R[m] and set of attributes: A[1], A[2], ..., A[n]
Output: Decision tree
procedure Build_tree (records, attributes);
create a node N;
if all records belong to the same class, C then return N as a leaf node with class label C;
if attributes is empty then
return N as a leaf node with class label C, such that the majority of records belong to it;
select attribute A[i] (with the highest information gain) from attributes; label node N with A[i];
for each known value, v[j], of A[i] do
add a branch from node N for the condition A[i] = v[j]; S[j] = subset of records where A[i] = v[j];
if S[j] is empty then
add a leaf, L, with class label C, such that the majority of records belong to it and return L
else add the node returned by Build_tree(S[j], attributes – A[i]);
Before we illustrate Algorithm 28.3, we will explain the information gain measure in more detail. The use of entropy as the information gain measure is motivated by the goal of minimizing the
information needed to classify the sample data in the resulting partitions and thus minimizing the expected number of conditional tests needed to classify a new record. The expected information
needed to classify training data of s samples, where the Class attribute has n values (v[1], ..., v[n]) and s[i] is the number of samples belonging to class label v[i], is given by
where p[i] is the probability that a random sample belongs to the class with label v[i]. An estimate for p[i] is s[i] /s. Consider an attribute A with values {v[1], ..., v[m]} used as the test
attribute for splitting in the decision tree. Attribute A partitions the samples into the subsets S[1], ..., S[m] where samples in each S[j] have a value of v[j] for attribute A. Each S[j] may
contain samples that belong to any of the classes. The number of samples in S[j] that belong to class i can be denoted as s[ij]. The entropy associated with using attribute A as the test attribute is
defined as
I(s[1j], ..., s[nj]) can be defined using the formulation for I(s[1], ..., s[n]) with p [i] being replaced by p[ij] where p[ij] = s[ij] /s[j]. Now the information gain by partitioning on attrib-ute A
, Gain(A), is defined as I(s[1], ..., s[n]) – E(A). We can use the sample training data from Figure 28.6 to illustrate the algorithm.
The attribute RID represents the record identifier used for identifying an individual record and is an internal attribute. We use it to identify a particular record in our example. First, we compute
the expected information needed to classify the training data of 6 records as I(s[1], s[2]) where there are two classes: the first class label value cor-responds to yes and the second to no. So,
I(3,3) = − 0.5log[2] 0.5 − 0.5log[2] 0.5 = 1.
Now, we compute the entropy for each of the four attributes as shown below. For
Married = yes, we have s[11] = 2, s[21] = 1 and I(s[11], s[21]) = 0.92. For Married = no, we have s[12] = 1, s[22] = 2 and I(s[12], s[22]) = 0.92. So, the expected information needed to classify
a sample using attribute Married as the partitioning attribute is
E(Married) = 3/6 I(s[11], s[21]) + 3/6 I(s[12], s[22]) = 0.92.
The gain in information, Gain(Married), would be 1 – 0.92 = 0.08. If we follow simi-lar steps for computing the gain with respect to the other three attributes we end up with
E(Salary) = 0.33 and Gain(Salary) = 0.67
E(Acct_balance) = 0.92 and Gain(Acct_balance) = 0.08
E(Age) = 0.54 and Gain(Age) = 0.46
Since the greatest gain occurs for attribute Salary, it is chosen as the partitioning attribute. The root of the tree is created with label Salary and has three branches, one for each value of Salary
. For two of the three values, that is, <20K and >=50K, all the samples that are partitioned accordingly (records with RIDs 4 and 5 for <20K
and records with RIDs 1 and 2 for >=50K) fall within the same class loanworthy no and loanworthy yes respectively for those two values. So we create a leaf node for each. The only branch that needs
to be expanded is for the value 20K...50K with two samples, records with RIDs 3 and 6 in the training data. Continuing the process using these two records, we find that Gain(Married) is 0, Gain(
Acct_balance) is 1, and Gain(Age) is 1.
We can choose either Age or Acct_balance since they both have the largest gain. Let us choose Age as the partitioning attribute. We add a node with label Age that has two branches, less than 25, and
greater or equal to 25. Each branch partitions the remaining sample data such that one sample record belongs to each branch and hence one class. Two leaf nodes are created and we are finished. The
final decision tree is pictured in Figure 28.7. | {"url":"https://www.brainkart.com/article/Classification---Data-Mining_11618/","timestamp":"2024-11-02T05:11:18Z","content_type":"text/html","content_length":"60906","record_id":"<urn:uuid:c7f3dde1-2034-42d7-a062-5b6793ea1182>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00167.warc.gz"} |
Eureka Math Grade 3 Module 7 End of Module Assessment Answer Key
Engage NY Eureka Math 3rd Grade Module 7 End of Module Assessment Answer Key
Eureka Math Grade 3 Module 7 End of Module Assessment Task Answer Key
Question 1.
Katy and Jane construct a four-sided wall to surround their castle. The wall has a perimeter of 100 feet. One side measures 16 feet. A different side measures 16 feet. A third side measures 34 feet.
a. Draw and label a diagram of the wall. Use a letter to represent the unknown side length.
b. What is the unknown side length? Show your work, or explain how you know.
c. Katy and Jane build a square fence around the castle’s pool. It has a perimeter of 36 feet. What is the area that the fence encloses? Use a letter to represent the unknown. Show your work.
Question 2.
Each shape has a missing side length labeled with a letter. The perimeter of the shape is labeled inside. Find the unknown side length for each shape.
Question 3.
Suppose each
a. Find the area and perimeter of each shape.
b. John says, “If two shapes have the same area, they must also have the same perimeter.” Is John correct? Use your answer from part (a) above to explain why or why not.
Question 4.
Mr. Jackson’s class finds all possible perimeters for a rectangle composed of 36 centimeter tiles. The chart below shows how many students found each rectangle.
┃Perimeter │Number of Students ┃
┃24 cm │6 ┃
┃26 cm │9 ┃
┃30 cm │5 ┃
┃40 cm │7 ┃
┃74 cm │4 ┃
a. Check the students’ work. Did they find all the possible perimeters? How do you know?
b. Use the chart. Estimate to construct a line plot of how many students found each perimeter.
Number of Students Who Found Each Perimeter
Question 5.
The square to the right has an area of 16 square centimeters.
a. What is the length of each side? Explain how you know.
b. Draw copies of the square above to make a figure with a perimeter of 32 centimeters.
c. Write a number sentence to show that your figure has the correct perimeter of 32 centimeters.
Leave a Comment | {"url":"https://bigideasmathanswers.com/eureka-math-grade-3-module-7-end-of-module-assessment/","timestamp":"2024-11-13T12:32:19Z","content_type":"text/html","content_length":"138224","record_id":"<urn:uuid:51690902-88a3-4a26-8413-f4b893363d4e>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00521.warc.gz"} |
Understanding Mathematical Functions: Exploring Linear, Quadratic, and Exponential Relationships
Introduction to Discrete Mathematics
Discrete mathematics is a branch of mathematics that deals with mathematical structures that are fundamentally discrete rather than continuous. It focuses on objects that can only take on distinct,
separate values. One important concept in discrete mathematics is mathematical functions.
What are Mathematical Functions?
In mathematics, a function is a relation between a set of inputs (called the domain) and a set of outputs (called the codomain or range), such that each input is associated with exactly one output.
Functions are often represented by equations, graphs, or tables.
Mathematical functions are used to describe relationships between different quantities. They can be used to model real-world phenomena, solve problems, and make predictions. Functions are an
essential tool in various fields, including computer science, engineering, and economics.
Examples of Mathematical Functions
1. Linear Functions
A linear function is a function that can be represented by a straight line on a graph. It has the form:
f(x) = mx + b
where m is the slope of the line and b is the y-intercept. The slope determines how steep the line is, and the y-intercept is the point where the line crosses the y-axis.
For example, let’s consider the function f(x) = 2x + 3. This function represents a line with a slope of 2 and a y-intercept of 3. By plugging in different values for x, we can determine the
corresponding y-values and plot the points on a graph to visualize the function.
2. Quadratic Functions
A quadratic function is a function that can be represented by a parabolic curve on a graph. It has the form:
f(x) = ax^2 + bx + c
where a, b, and c are constants. The graph of a quadratic function is a U-shaped curve called a parabola. The coefficient a determines whether the parabola opens upwards (a > 0) or downwards (a < 0).
For example, let’s consider the function f(x) = x^2 – 4. This function represents a parabola that opens upwards and has its vertex at the point (0, -4). By plugging in different values for x, we can
determine the corresponding y-values and plot the points on a graph to visualize the function.
3. Exponential Functions
An exponential function is a function in which the variable appears in the exponent. It has the form:
f(x) = a^x
where a is a constant. Exponential functions grow or decay at a constant rate. If a > 1, the function represents exponential growth, and if 0 < a < 1, the function represents exponential decay.
For example, let’s consider the function f(x) = 2^x. This function represents exponential growth with a base of 2. As x increases, the function values double. By plugging in different values for x,
we can determine the corresponding y-values and plot the points on a graph to visualize the function.
Mathematical functions are a fundamental concept in discrete mathematics. They allow us to describe relationships between different quantities and solve problems in various fields. In this article,
we explored examples of linear functions, quadratic functions, and exponential functions. These examples demonstrate the versatility and applicability of mathematical functions in real-world | {"url":"https://tutoline.com/discrete-mathematics-mathematical-functions/","timestamp":"2024-11-03T12:51:13Z","content_type":"text/html","content_length":"202391","record_id":"<urn:uuid:d752fc72-b005-4efb-89a1-1531d97198e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00654.warc.gz"} |
Open CASCADE notes: Topology and Geometry in Open CASCADE. Part 3
Topology and Geometry in Open CASCADE. Part 3
OK, let's continue to eat our elephant bit by bit. The next bit is edge. Hope you won't get difficulties with it.
Edge is a topological entity that corresponds to 1D object – a curve. It may designate a face boundary (e.g. one of the twelve edges of a box) or just a ‘floating' edge not belonging to a face
(imagine an initial contour before constructing a prism or a sweep). Face edges can be shared by two (or more) faces (e.g. in a stamp model they represent connection lines between faces) or can only
belong to one face (in a stamp model these are boundary edges). I'm sure you saw all of these types – in default viewer, in wireframe mode, they are displayed in red, yellow and green respectively.
Edge contains several geometric representations (refer to the diagram in Part1):
- Curve C(t) in 3D space, encoded as Geom_Curve. This is considered as a primary representation;
- Curve(s) P(t) in parametric 2D space of a surface underlying each face the edge belongs to. These are often called pcurves and are encoded as Geom2d_Curve;
- Polygonal representation as an array of points in 3D, encoded as Poly_Polygon3D;
- Polygonal representation as an array of indexes in array of points of face triangulation, encoded as Poly_PlygonOnTriangulation.
The latter two are tessellation analogues of exact representations with the help of former two.
These representations can be retrieved using already mentioned BRep_Tool, for instance:
Standard_Real aFirst, aLast, aPFirst, aPLast;
Handle(Geom_Curve) aCurve3d = BRep_Tool::Curve (anEdge, aFirst, aLast);
Handle(Geom2d_Curve) aPCurve = BRep_Tool::CurveOnSurface (anEdge, aFace, aPFirst, aPLast);
The edge must have pcurves on all surfaces, the only exception is planes where pcurves can be computed on the fly.
The edge curves must be coherent, i.e. go in one direction. Thus, a point on the edge can be computed using any representation - as C(t), t from [first, last]; S1 (P1x (u), P1y(u)), u from [first1,
last1], where Pi – pcurve in parametric space of surface Si
Edge flags
Edge has two special flags:
- "same range" (BRep_Tool::SameRange()), which is true when first = first_i and last = last_i, i.e. all geometric representations are within the same range;
- "same parameter" (BRep_Tool::SameParameter()), which is true when C(t) = S1(P1x(t), P1y(t)), i.e. any point along the edge corresponds to the same parameter on any of its curves.
Many algorithms assume that they are both set, therefore it is recommended that you ensure that these conditions are respected and flags are set.
Edge's tolerance is a maximum deviation between its 3D curve and any other representation. Thus, its geometric meaning is a radius of a pipe that goes along its 3D curve and encompass curves restored
from all representations.
Special edge types
There are two kinds of edges that are distinct from others. These are:
- seam edge – one which is shared by the same face twice (i.e. has 2 pcurves on the same surface)
- degenerated edge – one which lies on a surface singularity that corresponds to a single point in 3D space.
The sphere contains both of these types. Seam-edge lies on pcurves corresponding to surface U iso-lines with parameters 0 and 2*PI. Degenerated edges lies on North and South poles and correspond to V
iso-lines with parameters –PI/2 and PI/2.
Other examples - torus, cylinder, and cone. Torus has two seam-edges – corresponding to its parametric space boundaries; cylinder has a seam-edge. Degenerated edge represents on a cone apex.
To check if the edge is either seam or degenerated, use BRep_Tool::IsClosed(), and BRep_Tool::Degenerated().
Edge orientation
Forward edge orientation means that its logical direction matches direction of its curve(s). Reversed orientation means that logical direction is opposite to curve's direction. Therefore, seam-edge
always has 2 orientations within a face – one reversed and one forward.
To be continued...
P.S. As usual, many thanks to those who voted and sent comments. Is this series helpful ?
16 comments
1. Very useful article even for non-beginners. Roman, keep going and go deeper into the geometry/topology stuff.
Beside this, the rate buttons don't work on Opera. I voted in IE.
2. Are you kidding me?
Don't doubt more if it is useful or not. This is ESSENTIAL.
I am very faithful to your blog.
Come on, let's go through the faces because I am getting several issues to solve!
As far as possible, you are getting my project alive!
Thank you Roman
3. OK, folks, thanks for continued support. Yes, face is a next target. Stay tuned ;-)
4. Great and interesting articles! With such background theory get easier to read throught the OCC code!
5. Roman, thanks for all of this. I've understood OCC for years, but now I start to really understand it.
6. Ever thought about writing "Open CASCADE for Dummies" :)
7. Great article Roman, much enjoying the topology series.
A little surprised that the face is the following article, I thought Wire was the node up the topology graph? Thanks for your efforts!
8. Great blog, thanks for posting all this fun stuff.
It would be great if you could elaborate on orientation. It seems to me, that sometimes, the orientation attribute of a TopoDS_Edge refers the orientation of the TopoDS_Edge with respect to the
underlying TopoDS_TEdge (or BRepCurve) and other times the orientation of the TopoDS_Edge refers to the orientation of the TopoDS_Edge with respect to the TopoDS_Face that it bounds. I am always
confused as to which one I actually have at any given moment.
9. Thanks for continuous feedback, guys, and high ratings. Appreciate your interest and support.
OK to include more details on wires and orientation.
We are working hard here at Intel to release new Beta Update for Parallel Amplifier and Inspector (www.intel.com/go/parallel, fighting with remaining bugs and polishing GUI. In addition to other
personal projects, this makes it more challenging to quickly issue posts. But I will keep on, I promise...
Thanks again.
10. As productive as you are, how the heck did you manage to escape OpenCascade? It's a wonder that they didn't break out handcuffs when you announced you were leaving! :)
A Dummies book is not a bad idea. If not, could you at least take your blog posts and put them in a PDF some time? A PDF would be easier if someone wants to print it out.
11. Hi Roman,
first of all thanks for the series - it's really essential!
However, there's an issues I didn't get. It's about orientation of seam-edges. Why do these always have two different orientations? If there are two separate pcurves for each seam-edge it is
possible to decide how those are parameterized, isn't it? Additionally the rule you introduced in part 4 (material on the left for forward edges or on the right for reversed edges) will not be
violated if the edges are all forward (or reversed).
If the solution comes in one of the next articles you mentioned in part 4 don't bother answering this post.
12. Mark, I left the company quite gracefully and continue to maintain relationships with many folks there. This is a great team.
Pawel, remember that curve representations (3D and all pcurves) must be consistent – see Part3. Thus, in 2D parametric space they will be parallel and co-directional (see sphere pcurves in
figures in Parts 3 and 4). Thus, to keep material on the left on the one and on the right of the other, one must be forward and the other – reversed.
13. Ok I finally got it! For some reason I misinterpreted the arrows as the edge orientation (not the parameterization direction!).
15. Hi Roman,
Say you have a TopoDS_Compound that contains multiple TopoDS_Edge - how would you pick each individual edge through IVtkTools_ShapePicker? For example, say I've constructed a box out of 12
TopoDS_Edge, and combined them using BRep_Builder into a single TopoDS_Compound. I then want to, using the picker, select an edge of the box. How would this be done?
16. Hi Mark,
I've never used the OCC-VTK bundle so cannot say for the ShapePicker. I would start with IVtkDraw which demonstrates the bundle in action (though perhaps on X11 only).
Good luck! | {"url":"https://opencascade.blogspot.com/2009/02/topology-and-geometry-in-open-cascade_12.html","timestamp":"2024-11-05T21:32:00Z","content_type":"application/xhtml+xml","content_length":"276794","record_id":"<urn:uuid:ea4c6a1c-f351-417a-ae7f-60b057bbddaa>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00378.warc.gz"} |
Stephen Hawking
Stephen William Hawking was an English theoretical physicist, cosmologist, and author who was director of research at the Centre for Theoretical Cosmology at the University of Cambridge at the time
of his death. He was the Lucasian Professor of Mathematics at the University of Cambridge between 1979 and 2009. Hawking was born in Oxford into a family of doctors. He began his university education
at University College, Oxford, in October 1959 at the age of 17, where he received a first-class BA degree in physics. He began his graduate work at Trinity Hall, Cambridge, in October 1962, where he
obtained his PhD degree in applied mathematics and theoretical physics, specialising in general relativity and cosmology in March 1966. In 1963, Hawking was diagnosed with an early-onset
slow-progressing form of motor neurone disease that gradually paralysed him over the decades. | {"url":"https://atomo.relevanpress.com/ent/stephen-hawking--2669/","timestamp":"2024-11-04T14:22:02Z","content_type":"text/html","content_length":"76465","record_id":"<urn:uuid:5eaf1532-76f7-4e51-bc2e-595f2eb089c1>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00743.warc.gz"} |
I'm loving it
When the user measure targets of the same type, it would be cool if the machine would remember the first target and compare the rest of them to it and automatically find the interested edges.
Image registration is employed to find the correspondence between the first target (template) and the rest of them.
For VMM, the transform is apparently rigid, not even affine. So it should be an easy task, but two types of targets should be considered.
(1) targets with much texture information, like printed PCB board and colored particles.
(2) targets with only edge information, such as plugins and little mechanic parts.
I'm now testing two strategys, the first one is used in my MSc thesis to estimate camera motion, i.e. identifying some lanmark points, match them and then compute the affine transformation; the
second strategy is based on Chamfer Matching (borgerfors, 1988).
The first strategy proves to be very accurate on PCB board images, but has poor performance on the second type of targets. The algorithms include:
(1) detect landmarkers (something like cvGoodFeaturesToTrack)
(2) match the landmarkers by neighborhood correlation
(3) compute the affine transformation by iterated outlier removing.
To speed up the computation, I did this on a image piramid with different image resolution. Firstly compute the affine model on the coasest image, then localize step(2) according to the model on a
finer image and so on.
The average registration error ( defined in many literatures, it's not convinient to input formulas in a blog) decreases each level on the piramid. In the example below, the average error is 60, 53,
26 for each layer in the herachical structure respectively.
fig. layer 3 source image
fig. layer 3 dst image
fig. layer 2 dist image
fig.layer 2 source image
BTW: some literatures prefer methods in a optical flow fashion, but these ideas are unsuitable for VMM: the fundamental hypothesis that the image sequence would be 'continuous' doesn't hold here.
When a user puts on targets on the platform, a minimum offset would cause the image shift a lot, so the source and target images won't be close spatially at all. For optical flow computation, a very
large scan window should be used here, which is undesirable both in time and accuracy.
Things become ugly when they are zoomed very large, curves can be zig-zag and straight lines are not straight anymore.
Hough algorithm won't solve all problems. They are good at identifying parametric shapes, even the shapes are broken or not complete.
But when the shape itself cannot be approximated by parametric equations, Hough algorithms would find many scattered local maxima in the parameter space.
So when the images become ugly, we gotta fit something instead of detect a nice shape from the image.
To cope with the noise and to prune adjuncted edges from what we want, an iterated fitting strategy should be used to remove those points with large residuals with decreasing thresholds.
The steps are:
(1) track the edges
(2) fit it to line/circle/ellipse
(3) prune the points that has larger residual than current threshold
(4) decrease the threshold and goto (2)
fig. A plug zoom 30 times with a line fitted.
I'm now working on a Visual Measuring Machine. It's something like an automatic microscope.
With a powerful optical system, it can zoom the image for many times. After snapping a picture, a user would point out which edge he/she wants to measure, then the software system needs to recognize
the simple geometries like lines and circles in the picture and locate their end points.
After obtaining these metrics in pixels, its real length could be computed using the scale factor.
The user can, of course, manually point out the line on this picture, but that would be a boring task. As the targets being measured are usually small and are manufactured according to CAD scripts.
So, a vectorization of the picture is often desired and eligible.
The development has 2 stages:
1. recognize lines and circles. To aid user with recognition power.
2. implement a full-scaled vectorization feature and using CAD data to check the vectorization result automatically. This would be a very useful feature. After a product is manufatured, its every
edge and surface is reconstructed with the system and compared with its original design for deficiency.
The first stage is almost done. It's relatively easy, anyway, here's the pictures.
1. the source image
2. after canny filtering
3. edge tracking ( to extract edge points chosen by the user)
4. The result edge with endpoints
After taking a ferocious screening test, the Heracles of Software Industry, Microsoft, offered me a free 3 days round trip to Beijing for visit to Microsoft Research Asia.
It's the first time for me to take a flight. ^__^
And I'm gonna meet so many geeks there!
The process are:
(1) recognize the green spots.
(2) solve the correspondence problem between the current frame and the registered global senario. ref: veenjman
(3) find out the Affine transform model by solving a linear equation
(4) transform the registered global senario and update their current positions.
Some of the frames:
The global view and result of counting:
It's a common sense that RHT is faster than HT. For, RHT makes several samples each term to calculate one point in the parameter space in a determined fashion, but RT use only one sample and compute
a whole buntch of parameters( a curve or a surface in the parameter space). The higher the dimension is, the slower HT will be. But as the experiments shows that the "randomized" characteristics of
the RHT seems to be problematic if there are many sample points. Because, the samples generated by computer is not random enough! So, when I detect long curves using RHT, the sample points tends to
be localized on some part of the curve, and the precision of the algorithm is greatly penalized.
So, how to solve this problem?
One IEEE papar proposes a recursive RHT algorithm:
run RHT once, then narrow down the samples and the parameter space and run RHT again using higher precision.
This approach, which I tested by the tile quality checking system below, is still not good enough, although its execution time is very short.
My method is:
run RHT first , narrowing down the parameter space enough.
Then use the brutal HT on it.
Because the RHT, HT can be very fast, and retain its precision.
This optimization is thoroughly discussed in my paper.
Snap of the digits.
Picture after preprocessing it to extract the red channel.
After segmentation
Hoorah! The result.
These are steel bars running on the product chain at speed 5m/s. We use ccd camera to recognize and track them. When the number of passed steel bars reached to the predefined one, we stop the product
chain and indicating the workers to wrap them.
Our algorithm has realtime execution speed(processing 25 frames in 1 second). But has one big problem: it might lose tracking of some steel bar when they vibrate sharply. Although the chance is very
low (once per day), it is still annoying. We are working on this problem.
After the counting is done, we will work out a plan to wrap the steel bars by robot hand. This require very good tracking algorithm and control strategy. We gotta simulate human hand to do this job.
Above is a ceramic tile with one of its border broken. Below is the broken edge that the system detects. The system can also detects the spots on the ceramic tile and check the quality of the ceramic
tile automatically.
All the algorithms were worked out and have realtime response speed. But, embarrassingly, I gotta problem to find a buyer now.
This project is to measuring steel pipes on the product line. The key point is to get accurate length of the pipe, then with the weighting system and the diameter of the pipe is known, we can
caculate the thickness of these pipes.
We use Computer Vision to solve the problem. But, before explaining our solution, we shall get some background of the project.
The oldest solution belongs to a German company. They use laser device to get the reflection from the end of the pipe and after measuring the time of the reflection, they can caculate the distance
from the light source to the end point of a steel pipe. The plan works well for thick pipes, but virtually useless for the smaller one, because, when the pipes get smaller, and thinner, one can
hardly point a laster emitter to the end of the pipe accurately, so the reflection becomes impossible.
Then another company came, they use line scan cameras fixed on each side of the pipe, after measuring each end of the pipe, they simply add the distance between the two cameras and get the whole
length. This plan proves to be effective for pipes with any diameter, yet, in the meantime, has big problems even worse than the German idea.
For one thing, in the factory, vibrations caused by all kinds of machinary is inevitable, these subtle vibrations make the line scan cameras vibrate too and the output of the cameras change
drastically just like in a earthquake, so you can image how bad the result might be. And for another, when the scan line get some noise on it, it's very hard to get rid of them, because all you
depend is a single line of data. You got no more data to recover from the noise.
So, here comes our rescue, we use ordinary CCD cameras.
As the picture shows, we use 11 cameras. The first one pointing to the head of a pipe and the other 10 cameras are divided into 5 groups, so we can choose one group of them to measuring pipes with
particular length.
Like the second solution, we measuring the end and rear of the pipe, then add the distance between them to get the total length.
But how to cope with vibration? The point is that we fix 11 standard boxes on the manufacturing line. When the pipe vibrates, the boxes vibrates as well. And the dimensions of the standard boxes are
known, we can caculate the ratio of the pixel number to the real-world length. So, taking the standard boxes as references, we make the things simpler, we even don't need to calibrate the cameras by
hand and even if the cameras are blurred, we can still get a accurate length. The picture below demonstrate how we archieve this:
On this picture, the standard box and the pipe all get blurred due to the defocus of the camera. Then on the scanline, the trasition of gray scale of the pixels will not be so sharp, instead, it's a
slope which I draw with a green curve, so we cannot get the edge point using Gaussian operator, otherwise, we could introduce bigger error.
First we average all these green curves of the standard box to get another curve, the black one. This curve describes how the gray scale change on the edge of the standard box, Then we can use this
curve as a template to find the horizontal displacement of the pipe's edge to the box's edge.
Then, we do the same thing to the pipe and get its edge curve.
Now, we can shift the first curve from left to the right and find a best matching point, on which the two black curves fits perfect, that point, is where the 'true' edge of the pipe lies.
Other hard stuff include capturing 11 frames in a round-around way from 2 capture board and get weight data from a RS232 cable.
The project goes well and is in the tuning phrase.
This is the screen shot of our system(in Chinese), you can see the standard boxes and a pipe at the top. We got the accuracy around 5 mm, which is much better than expected. Vibrations are not
problem at all, even you kick the camera (joke), or tap it by your hand, the system still works well.
This is the preview system for tuning the 11 cameras, some of these cameras are not switched on yet.
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Comments on 'Limits of Econometrics' by David Freedman
Yıl 2009, Cilt: 1 Sayı: 1, 28 - 32, 01.04.2009
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• Zellner, A. (1979). Causality and Econometrics. In Three Aspects of Policy and Policymaking: Knowledge, Data and Institutions, eds. K. Brunner and A.H. Meltzer. Amsterdam: North-Holland
Publishing Co, 9-54.
• Zellner, A. (1984). Basic Issues in Econometrics. Chicago: U. of Chicago Press.
• Zellner, A. (1988). Optimal Information Processing and Bayes’ Theorem. American Statistician, 42, 278-284 with discussion by E.T. Jaynes, B.M. Hill, J.M. Bernardo and S. Kullback and the author’s
• Zellner, A. (1989). Bayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys. Reprint of 1980 edition, North-Holland: Amsterdam.
• Zellner, A., Kuezenkamp, H. and M. McAleer (2001). Simplicity, Inference and Modeling: Keeping it Sophisticatedly Simple. Cambridge: Cambridge U. Press.
• Zellner, A. and Chen, B. (2001), Bayesian Modeling of Economies and Data Requirements. Macroeconomic Dynamics, 5, 673-700.
• Zellner, A. and F.C. Palm (2004). The Structural Econometric Modeling. Time Series Analysis (SEMTSA) Approach, Cambridge: Cambridge U. Press.
• Zellner, A. and G. Israilevich (2005). The Marshallian Macroeconomic Model: A Progress Report. Macroeconomic Dynamics, 9, 220-243 and reprinted in International Journal of Forecasting, 21,
627-645 with discussion by A. Espasa, Carlos III U. of Madrid.
Yıl 2009, Cilt: 1 Sayı: 1, 28 - 32, 01.04.2009
• Christ, C.F. (1951). A Test of an Econometric Model for the United States, 1921-1947. In Conference on Business Cycles with comments by Milton Friedman. New York: National Bureau of Economic
Research, 35-107.
• Feigl, H. (1953), Notes on Causality. In Readings in the Philosophy of Science, ed. H. Feigl and M. Brodbeck. New York: Appleton-Century-Crofts, 408-418.
• Geisser, S. (1989), The Contributions of Sir Harold Jeffreys to Bayesian Inference. In Bayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys, ed. A. Zellner,
reprint of 1980 edition. North-Holland: Amsterdam, 13-20.
• Good, I.J. (1962). Theory of Probability by Harold Jeffreys. Journal of the Royal Statistical Society A, 125, 487–489.
• Hadamard, J. (1945). The psychology of invention in the mathematical field. New York, NY: Dover Publications
• Jeffreys, H. (1998). Theory of Probability. In Oxford Classic Texts in the Physical Sciences, first published in 1939 with subsequent editions in 1948, 1961, 1981 and 1983. Oxford: Oxford U.
• Lindley, D.V. (1989), Jeffreys’s Contribution to Modern Statistical Thought. In Bayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys, ed. A. Zellner, reprint of
1980 edition. North-Holland: Amsterdam, 35-39.
• Ngoie, J.K. and A. Zellner (2008). The Effects of Freedom Reforms on the Growth Rate of the South African Economy. Working Paper, H.G.B. Alexander Research Foundation, U. of Chicago.
• Pearson, K. (1938). The Grammar of Science. London: Everyman Edition.
• Robert, C.P., Chopin, N. and J. Rousseau (2008). Harold Jeffreys’ Theory of Probability Revisited. arXiv:0804.3173v6. in Statistical Science with invited discussion
• Zellner, A. (1979). Causality and Econometrics. In Three Aspects of Policy and Policymaking: Knowledge, Data and Institutions, eds. K. Brunner and A.H. Meltzer. Amsterdam: North-Holland
Publishing Co, 9-54.
• Zellner, A. (1984). Basic Issues in Econometrics. Chicago: U. of Chicago Press.
• Zellner, A. (1988). Optimal Information Processing and Bayes’ Theorem. American Statistician, 42, 278-284 with discussion by E.T. Jaynes, B.M. Hill, J.M. Bernardo and S. Kullback and the author’s
• Zellner, A. (1989). Bayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys. Reprint of 1980 edition, North-Holland: Amsterdam.
• Zellner, A., Kuezenkamp, H. and M. McAleer (2001). Simplicity, Inference and Modeling: Keeping it Sophisticatedly Simple. Cambridge: Cambridge U. Press.
• Zellner, A. and Chen, B. (2001), Bayesian Modeling of Economies and Data Requirements. Macroeconomic Dynamics, 5, 673-700.
• Zellner, A. and F.C. Palm (2004). The Structural Econometric Modeling. Time Series Analysis (SEMTSA) Approach, Cambridge: Cambridge U. Press.
• Zellner, A. and G. Israilevich (2005). The Marshallian Macroeconomic Model: A Progress Report. Macroeconomic Dynamics, 9, 220-243 and reprinted in International Journal of Forecasting, 21,
627-645 with discussion by A. Espasa, Carlos III U. of Madrid.
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t-Test, Chi-Square, ANOVA, Regression, Correlation...
Kaplan Meier Curve
Load example data
The Kaplan-Meier curve is commonly used to analyze time-to-event data, such as the time until death or the time until a specific event occurs. For this, the Kaplan Meier curve graphically represent
the survival rate or survival function. Time is plotted on the x-axis and the survival rate is plotted on the y-axis.
Survival rate
The first question is what is the survival rate. Let's look at this with an example. Suppose you're a dental technician and you want to study the "survival time" of a filling in a tooth.
So your start time is the moment when a person goes to the dentist for a filling, and your end time, the event, is the moment when the filling breaks. The time between these two events is the focus
of your study.
You can now see how likely it is that a filling will last longer than a certain point in time by looking at the Kaplan-Meier curve.
Thus the horizontal axis represents time, usually measured in months or years. The vertical axis represents the estimated probability.
For example, you may be interested in the probability that your filling will last longer than 5 years. To do this, you read off the value at 5 years on the graph, which is the survival rate. At 5
years, the Kaplan-Meier curve gives you a value of 0.7. So there is a 70% chance that your filling will last longer than 5 years.
Interpreting the Kaplan-Meier curve
The Kaplan-Meier curve shows the cumulative survival probabilities.
A steeper slope indicates a higher event rate (death rate) and therefore a worse survival prognosis. A flatter slope indicates a lower event rate and therefore a better survival prognosis. The curve
may have plateaus or flat areas, indicating periods of relatively stable survival.
If there are multiple curves representing different groups, you can compare their shapes and patterns. If the curves are parallel, it suggests that the groups have similar survival experiences. If
the curves diverge or cross, it indicates differences in survival between the groups.
At specific time points, you can estimate the survival probability by locating the time point on the horizontal axis and dropping a vertical line to the curve. Then, read the corresponding survival
probability from the vertical axis.
Calculating the Kaplan-Meier curve
To create a Kaplan-Meier curve, you first need the data for your subjects. Let's say the filling lasted 3 years for the first subject, 4 years for the second subject, 4 years for the third subject,
and so on.
Let's assume that none of the cases are "censored". The data are already arranged so that the shortest survival time is at the top and the longest at the bottom.
Now we create a second table that we can use to draw the Kaplan-Meier curve. To do this, we look at the time points in the left table and add the time zero. So we have the time points 0, then 3, 4,
6, 7, 8 11 and 13. In total we have 10 subjects.
Now we look at how many fills break out at each time. We enter this in the column m. So at time 0, no fillings were broken out. After 3 years, there was one broken filling, after 4 years there were
two, after 6 years there was one. We now do the same for all the other times.
Next, we look at the number of cases that have survived to the time plus the number of cases where the event occurs at the exact time. We enter this in column n.
So n is the number of cases that survived to that point, plus the people who dropped out at that exact point.
After zero years we still have all 10 people. After 3 years, we get 10 for n, 9 people still have their fill intact, and one person's fill broke out exactly after 3 years.
The easiest way to get n is to take the previous n value and subtract the previous m value. So we get 10 - 1 equals 9. Then 9 minus 2 equals 7, 7 - 1 equals 6... and so on and so forth.
From column n we can now calculate the survival rates. To do this, we simply divide n by the total number, i.e. 10.
So 10 divided by 10 is equal to 1, 9 divided by 10 is equal to 0.9, 7 divided by 10 is equal to 0.7. Now we do the same for all the others.
Draw Kaplan Meier curve
We can now plot the Kaplan-Meier curve. At time 0 we have a value of 1, after 3 years we have a value of 0.9 or 90%. After 4 years we get 0.7, after 6 years 0.6 and so on and so forth.
From the Kaplan-Meier curve, we can now see what percentage of the filling has not broken out after a certain time.
Censored data
Next, we look at what to do when censored data is present. For this purpose, censored data has been added to the example in these three places. If you're not sure what censored data is, see the
survival analysis tutorial.
We now need to enter this data into our Kaplan-Meier curve table. We do this as follows: We create our m exactly as we did before, looking at how many cases failed at each time point.
Now we add a column q, in which we enter how many cases were censored at each time.
Note that the time at which each censored case occurred does not get its own row, but is assigned to the previous time.
Let's look at this case. The censoring took place at time 9. In this table, however, there is no event with nine years and we also don't add it. The person is added at time 8.
We can now re-calculate the values for the survival curve. If we have censored data, this is a little more complex.
For this, we write down the values in the first step. We get these values by calculating n-m/n. In the third row, for example, we get the value 10/12 with 12-2 by 12.
The calculation of the real value is iterative. To do this, we multiply the result from the previous row by the value we have just calculated.
So, in the first row we get 1, now we calculate 12/13 times 1, which is equal to 0.923. In the next row we calculate 10/12 times 0.923 and get a value of 0.769. We take this value again for the next
We do this for all the rows. We can then plot the Kaplan-Meier curve with this data in the same way as before.
Comparing different groups
If you are comparing several groups or categories (e.g. treatment groups), the Kaplan-Meier curve consists of several lines, each representing a different group. Each line shows the estimated
survival rate for that particular group. To test whether there is a statistically significant difference between the groups, the log-rank test can be used.
Kaplan-Meier curve assumptions
Random or Non-informative censoring: This assumption states that the occurrence of censoring is unrelated to the likelihood of experiencing the event of interest. In other words, censoring should be
random and not influenced by factors that affect the event outcome. If censoring is not non-informative, the estimated survival probabilities may be biased.
Independence of censoring: This assumption assumes that the censoring times of different individuals are independent of each other. This means that the occurrence or timing of censoring for one
participant should not provide any information about the censoring times for other participants.
Survival probabilities do not change over time: The Kaplan-Meier curve assumes that the survival probabilities estimated at each time point remain constant over time. This assumption may not be valid
if there are time-varying factors or treatments that can influence survival probabilities.
No competing risks: The Kaplan-Meier curve assumes that the event of interest is the only possible outcome and there are no other competing events that could prevent the occurrence of the event being
studied. Competing events can include other causes of death or events that render the occurrence of the event of interest impossible.
Create Kaplan Meier curve with DATAtab
Load example data
To create the Kaplan Meier curve with DATAtab, simply go to the statistics calculator on datatab.net and copy your own data into the table.
Now click on "Plus" and select Survival Analysis. Here you can create the Kaplan Meier curve online. If you select the variable "Time" DATAtab will create the Kaplan Meier curve and you will get the
survival table. If you do not click on a status, Datatab assumes that the data is not censored. If this is not the case, click also on the variable that contains the information which case is
censored and which is not. One stands for event occurred and 0 stands for censored. Now you will get the appropriate results.
Statistics made easy
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• ideal for exams and theses
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"It could not be simpler"
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Roof Height Calculator - CivilGang
What is a Roof Height Calculator?
A roof height calculator is a tool used to estimate the height of a roof based on the roof pitch (slope) and the horizontal span.
It is commonly used in construction and roofing to determine the overall height of a roof.
Why Use a Roof Height Calculator?
Using a roof height calculator is important for construction and design purposes.
It helps architects, builders, and roofing professionals estimate the height of a roof, which is crucial for planning and material calculations.
Roof Height Calculator
A roof height calculator is a tool used to estimate the height of a roof based on the roof pitch (slope) and the horizontal span. It requires input values for the pitch and span, providing an
estimate of the roof height. | {"url":"https://civil-gang.com/roof-height-calculator/","timestamp":"2024-11-05T22:13:09Z","content_type":"text/html","content_length":"90353","record_id":"<urn:uuid:aef89772-fc3c-4717-ae59-875e31922c36>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00489.warc.gz"} |
What Is Net Operating Income (NOI) In Real Estate?
Net operating income (NOI) is a vital real estate profitability metric to help you calculate an investment property’s potential revenue. The NOI provides valuable data to determine whether to
purchase a property, predict potential rental income, or raise rents to boost cash flow.
The NOI formula is straightforward. You deduct the operating expenses from the gross operating income (GOI) to calculate a property’s potential profitability. In other words, net operating income is
the difference between how much the property costs to operate and the amount of revenue it generates.
Net operating income doesn’t use expenses like debt payments, mortgage payments, depreciation, or capital expenditures in the calculation. However, it helps you compare properties when buying or
selling real estate. As a result, some investors consider this to be the most critical metric in real estate investing.
This article explains why NOI is important when making real estate investment decisions.
What Is Net Operating Income (NOI)?
Net operating income is an easy formula for calculating the potential rental revenue from an income-generating property. The formula uses two metrics — projected rental income and all expenses. The
net operating income figure is a property’s total income minus the operating expenses.
How Net Operating Income (NOI) Relates to Real Estate
The beauty of using the NOI formula is that it’s a simple calculation to determine a property’s operating performance. Here is what it means in real estate terms:
• NOI and real estate investment: Calculating the difference between gross operating income and operating expenses is vital when evaluating different properties. You can easily estimate the revenue
potential from single-family homes, condos, and multifamily properties. Putting the figures side-by-side helps you evaluate the best investment.
• NOI and your rental portfolio: The net operating income formula is also helpful in assessing the profitability of your current investments. For example, a simple analysis could show that the NOI
has changed since you purchased the property. This could mean that you must start looking for ways to find additional revenue. Or you may decide to sell the rental unit due to revenue losses.
NOI Formula
The net operating income formula is this:
Net Operating Income = Gross Operating Income – Operating Expenses
Here are some helpful explanations to help break down the formula:
• Gross: The total amount of revenue before deducting fees, expenses, taxes, or commissions.
• Net: Your “take home” amount after paying all related expenses.
Suppose the annual operating revenue from a single-family rental unit is $21,600. This is gross operating income. However, say that ongoing expenses amount to $4,800 annually. That means your net
income after operating expenses is $16,800.
How to Figure Out Net Operating Income (NOI)
Although the NOI formula is straightforward, there are several variables you must consider to get an accurate picture of a property’s potential profitability. For example, you must allow for vacancy
rates, all operating expenses, and additional income sources to figure out NOI. The calculation also includes potential income fluctuations.
Here are some of the variables to consider regarding expenses and income.
Gross Operating Income (GOI)
Ideally, you could calculate GOI as monthly rent multiplied by 12 to determine the gross annual income. However, it’s vital to remember that income can fluctuate depending on vacancy rates and
potential sources of extra income. Also, you face the real possibility of a tenant not paying rent.
Here are factors to consider when working out gross operating income:
• Vacancy rates: An empty rental unit affects your potential cash flow. Therefore, factor vacancy rates using metrics from comparable properties or information from the current rental property
• Credit loss: It’s wise to factor in occasions when a tenant doesn’t pay their rent. Like financial loss through vacancies, credit loss impacts your bottom line.
• Additional income: Does the condo or multifamily property have additional sources of income? Here are some examples:
□ Venning machines
□ Laundry services like a coin laundry machine
□ Parking fees
Related: How to increase rental income.
Operating Expenses
It is important not to confuse income with cash flow. Therefore, knowing which expenditures to include and which to omit in the net operating income formula is vital.
Here are the operating and non-operating expenses included in the NOI calculation:
• Property maintenance and repair costs
• Landlord insurance
• Property management fees
• Other landlord-related expenses like accounting and attorney fees
• Property taxes
Because NOI is used to assess a property’s ongoing revenue, capital expenditures are not included. Therefore, you do not have the following capital expenses:
• Income taxes
• Property depreciation
• Capital expenditures like installing a new roof
• Property depreciation
• Mortgage payments
Why are mortgage payments excluded from operating expenses? After all, paying a mortgage may be your largest monthly expenditure. This is because mortgage payments depend on individual investors, not
the property’s overall health.
Net Operating Income vs. Gross Operating Income
The difference between net operating income (NOI) and gross operating income (GOI) is how expenses affect the outcome.
Gross operating income is the potential total income from a property, considering vacancy and credit losses. It’s also vital to include additional income sources not included in rent.
Net operating income is the revenue when day-to-day expenses and fees are considered. NOI is the amount of cash you have left over after the costs of owing the rental property are deducted.
Net Income vs. Operating Income?
The primary difference between net operating income and net income is the type of expenses included. The operating income only refers to the revenue minus the day-to-day running costs of owning a
rental property. Net income is your bottom line. It factors all debts, mortgage payments, operating costs, and additional income streams.
In short — operating income is the rental property’s profitability and is the most important metric when comparing individual investment properties.
Examples of Net Operating Income
Net operating income measures the potential income stream from real estate investments. Typically, you calculate the figure annually because of variations in month-to-month income and expenditure.
Here is an example of calculating NOI using the formula “GOI – Operating Expenses = NOI.”
Let’s say you are considering an investment property — a small multifamily property with five rental units. Here are some figures:
• Monthly rent for each unit — $1,600
• Potential annual rental income — $96,000 ($1,600 x 5 x 12)
• Annual income from the coin laundry machine — $1,200
We must also factor in vacancy losses to figure out a realistic GOI. The average is 10% for the area. The calculation is $96,000 x 10% = $9,600. Therefore, our Gross Operating Income is:
• $87,600 ($96,000 + $1,200 – $9,600)
The current owner’s accounts show that annual property expenses for the previous year were $16,500.
Here is our Net Operating Income calculation:
• $87,600 – $16,500 = $71,100
This real estate metric can compare the property with other potential investments. Additionally, you can work out if you can cover your mortgage payments and calculate the property’s value.
The NOI calculation also helps you ascertain the total return on investment — the capitalization rate.
NOI and Capitalization Rate
The net operating income calculation helps determine other metrics like the capitalization rate. Also called cap rate enables you to decide on your potential return on investment (ROI).
Here is how to use NOI and the Cap Rate formula:
Capitalization rate = Net operating income ÷ purchase price
For example, let’s say that the five-unit property in our example has a listing price of $460,000. Then, we can use the cap rate formula to calculate the rate of return on investment.
Therefore, you can use NOI and capitalization rate to determine your annual return and if the investment is solid.
Related: How to calculate cap rate.
What is an Ideal Net Operating Income Percentage?
A common question in real estate investing is what is the best NOI percentage? Net operating income is not expressed as a percentage. Instead, it is a number you get when deducting operating expenses
from gross operating income.
Most investors use loans or finance for real estate investing. Therefore, it’s necessary to factor in the cost of financing when assessing properties, calculating the cap rate, and working out your
business cash flow.
Generally, it would be best to look for properties with higher net operating income figures when compared to the property price. Most real estate investors agree that margins and operating incomes
should be above 15% of the investment cost.
In Summary
Calculating net operating income is invaluable when comparing real estate investments. And the good news is that NOI is easy to calculate and helps you quickly identify potential profitable
investments. A higher NOI usually indicates a better investment opportunity.
Note By BiggerPockets: These are opinions written by the author and do not necessarily represent the opinions of BiggerPockets.
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1.3.5 Continuous Probability Models
Consider a scenario where your sample space $S$ is, for example, $[0,1]$. This is an uncountable set; we cannot list the elements in the set. At this time, we have not yet developed the tools needed
to deal with continuous probability models, but we can provide some intuition by looking at a simple example.
Your friend tells you that she will stop by your house sometime after or equal to $1$ p.m. and before $2$ p.m., but she cannot give you any more information as her schedule is quite hectic. Your
friend is very dependable, so you are sure that she will stop by your house, but other than that we have no information about the arrival time. Thus, we assume that the arrival time is completely
random in the $1$ p.m. and $2$ p.m. interval. (As we will see, in the language of probability theory, we say that the arrival time is "uniformly" distributed on the $[1,2)$ interval). Let $T$ be the
arrival time.
a. What is the sample space $S$?
b. What is the probability of $P(1.5)$? Why?
c. What is the probability of $T \in [1,1.5)$?
d. For $1 \leq a \leq b \leq 2$, what is $P(a \leq T \leq b)=P([a,b])$?
• Solution
a. Since any real number in $[1,2)$ is a possible outcome, the sample space is indeed $S=[1,2)$.
b. Now, let's look at $P(1.5)$. A reasonable guess would be $P(1.5)=0$. But can we provide a reason for that? Let us divide the $[1,2)$ interval to $2N+1$ equal-length and disjoint
intervals, $[1,1+\frac{1}{2N+1}), [1+\frac{1}{2N+1}, 1+\frac{2}{2N+1}), \cdots, [1+\frac{N}{2N+1}, 1+\frac{N+1}{2N+1}),\cdots,[1+\frac{2N}{2N+1}, 2)$. See Figure 1.18. Here, $N$ could be
any positive integer.
Fig.1.18 - Dividing the interval $[1,2)$ to $2N+1$ equal-length intervals.
The only information that we have is that the arrival time is "uniform" on the $[1,2)$ interval. Therefore, all of the above intervals should have the same probability, and since their
union is $S$ we conclude that
$$P\left(\Big[1,1+\frac{1}{2N+1}\Big)\right)=P\left(\Big[1+\frac{1}{2N+1}, 1+\frac{2}{2N+1}\Big)\right)=\cdots\\$$ $$\cdots=P\left(\Big[1+\frac{N}{2N+1}, 1+\frac{N+1}{2N+1}\Big)\right)=\
cdots\\$$ $$\cdots=P\left(\Big[1+\frac{2N}{2N+1},2\Big)\right)=\frac{1}{2N+1}.$$
In particular, by defining $A_N=\left[1+\frac{N}{2N+1}, 1+\frac{N+1}{2N+1}\right)$, we conclude that $$P(A_N)=P\left(\Big[1+\frac{N}{2N+1}, 1+\frac{N+1}{2N+1}\Big)\right)=\frac{1}
{2N+1}.$$ Now note that for any positive integer $N$, $1.5 \in A_N$. Thus, $\{1.5\} \subset A_N$, so $$P(1.5) \leq P(A_N)=\frac{1}{2N+1}, \hspace{20pt} \textrm{for all }N \in \mathbb
{N}.$$ Note that as $N$ becomes large, $P(A_N)$ approaches $0$. Since $P(1.5)$ cannot be negative, we conclude that $P(1.5)=0$. Similarly, we can argue that $P(x)=0$ for all $x \in [1,2)
c. Next, we find $P([1,1.5))$. This is the first half of the entire sample space $S=[1,2)$ and because of uniformity, its probability must be $0.5$. In other words, $$P([1,1.5))=P([1.5,2)) \
hspace{20pt} \textrm{(by uniformity),}$$ $$P([1,1.5))+P([1.5,2))=P(S)=1.$$ Thus $$P([1,1.5))=P([1.5,2))=\frac{1}{2}.$$
d. The same uniformity argument suggests that all intervals in $[1,2)$ with the same length must have the same probability. In particular, the probability of an interval is proportional to
its length. For example, since $$[1,1.5)=[1,1.25) \cup [1.25, 1.5).$$ Thus, we conclude
$$P\big([1,1.5)\big)=P\big([1,1.25)\big)+ P\big([1.25, 1.5)\big)\\$$ $$=2P\big([1,1.25)\big).$$
And finally, since $P\big([1,2)\big)=1$, we conclude $$P([a,b])=b-a, \hspace{20pt} \textrm{for } 1\leq a \leq b \lt 2.$$
The above example was a somewhat simple situation in which we have a continuous sample space. In reality, the probability might not be uniform, so we need to develop tools that help us deal with
general distributions of probabilities. These tools will be introduced in the coming chapters.
Discussion: You might ask why $P(x)=0$ for all $x \in [1,2)$, but at the same time, the outcome of the experiment is always a number in $[1,2)$? We can answer this question from different points of
view. From a mathematical point of view, we can explain this issue by using the following analogy: consider a line segment of length one. This line segment consists of points of length zero.
Nevertheless, these zero-length points as a whole constitute a line segment of length one. From a practical point of view, we can provide the following explanation: our observed outcome is not all
real values in $[1,2)$. That is, if we are observing time, our measurement might be accurate up to minutes, or seconds, or milliseconds, etc. Our continuous probability model is a limit of a discrete
probability model, when the precision becomes infinitely accurate. Thus, in reality we are always interested in the probability of some intervals rather than a specific point $x$. For example, when
we say, "What is the probability that your friend shows up at $1:32$ p.m.?", what we may mean is, "What is the probability that your friend shows up between $1:32:00$ p.m. and $1:32:59$ p.m.?" This
probability is nonzero as it refers to an interval with a one-minute length. Thus, in some sense, a continuous probability model can be looked at as the "limit" of a discrete space. Remembering from
calculus, we note that integrals are defined as the limits of sums. That is why we use integrals to find probabilities for continuous probability models, as we will see later.
The print version of the book is available on Amazon.
Practical uncertainty: Useful Ideas in Decision-Making, Risk, Randomness, & AI | {"url":"https://www.probabilitycourse.com/chapter1/1_3_5_continuous_models.php","timestamp":"2024-11-09T12:38:06Z","content_type":"text/html","content_length":"16277","record_id":"<urn:uuid:7678d0c4-dc3f-4058-bee8-cba171743278>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00704.warc.gz"} |
Quantum principle of sensing gravitational waves: From the zero-point fluctuations to the cosmological stochastic background of spacetime
We carry out a theoretical investigation on the collective dynamics of an ensemble of correlated atoms, subject to both vacuum fluctuations of spacetime and stochastic gravitational waves. A general
approach is taken with the derivation of a quantum master equation capable of describing arbitrary confined nonrelativistic matter systems in an open quantum gravitational environment. It enables us
to relate the spectral function for gravitational waves and the distribution function for quantum gravitational fluctuations and to indeed introduce a new spectral function for the zero-point
fluctuations of spacetime. The formulation is applied to two-level identical bosonic atoms in an off-resonant high-Q cavity that effectively inhibits undesirable electromagnetic delays, leading to a
gravitational transition mechanism through certain quadrupole moment operators. The overall relaxation rate before reaching equilibrium is found to generally scale collectively with the number N of
atoms. However, we are also able to identify certain states of which the decay and excitation rates with stochastic gravitational waves and vacuum spacetime fluctuations amplify more significantly
with a factor of N2. Using such favorable states as a means of measuring both conventional stochastic gravitational waves and novel zero-point spacetime fluctuations, we determine the theoretical
lower bounds for the respective spectral functions. Finally, we discuss the implications of our findings on future observations of gravitational waves of a wider spectral window than currently
accessible. Especially, the possible sensing of the zero-point fluctuations of spacetime could provide an opportunity to generate initial evidence and further guidance of quantum gravity.
Bibliographical note
© 2017 American Physical Society
• gravitational waves
• open quantum systems
• quantum gravity
Dive into the research topics of 'Quantum principle of sensing gravitational waves: From the zero-point fluctuations to the cosmological stochastic background of spacetime'. Together they form a
unique fingerprint. | {"url":"https://research.brighton.ac.uk/en/publications/quantum-principle-of-sensing-gravitational-waves-from-the-zero-po","timestamp":"2024-11-08T04:05:17Z","content_type":"text/html","content_length":"59009","record_id":"<urn:uuid:6e312010-6866-48cc-a719-6fb17ccc7db5>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00028.warc.gz"} |
calculate relative frequency | Excelchat
Create a frequency distribution for the data using 7 classes, beginning at 115 lbs, with a class width of 10 lbs. List upper and lower class limits on the graph. With the frequency distribution
develop the following charts o Frequency Histogram o Relative frequency distribution o Relative frequency histogram
Solved by A. F. in 24 mins | {"url":"https://www.got-it.ai/solutions/excel-chat/excel-help/how-to/calculate/calculate-relative-frequency","timestamp":"2024-11-02T06:06:02Z","content_type":"text/html","content_length":"342144","record_id":"<urn:uuid:1c64a21a-2eff-4c89-896b-11e2c768dc53>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00674.warc.gz"} |
Cake Cutting
Problem C
Cake Cutting
It’s SoCCat’s birthday, and they have baked a delicious cake for the occasion. Unfortunately, there are too many students in SoC who is attending the party and interested in eating the cake. It is
not feasible to cut the cake into sectors, otherwise the angle of each sector would be too small.
We model the cake on the Cartesian coordinate plane as a circle with radius $10000$ (in some unknown unit) with center $(0, 0)$. $N$ SoC students are invited to cut the cake, and each of them will be
making a single straight cut. The $i$-th student will make a cut on the straight line from point $(x_ i, y_ i)$ to point $(x’_ i, y’_ i)$, where $(x_ i, y_ i)$ and $(x’_ i, y’_ i)$ are distinct
points lying on the circumference of the circle.
Obviously, the cake will be divided into several pieces by the cuts. SoCCat wants to know the number of pieces that the cake will be divided into after all the cuts are made, in order to call the
correct number of students to eat the cake.
The first line of input contains an integer $N$ ($1 \leq N \leq 30$), the number of cuts by the students.
The next $N$ lines each contain four integers $x_ i$, $y_ i$, $x’_ i$, $y’_ i$ ($-10000 \leq x_ i, y_ i, x’_ i, y’_ i \leq 10000$), the coordinates of the two endpoints of the cut made by the $i$-th
It is guaranteed that the cuts are distinct, i.e. there are no two cuts which entirely overlap.
Output a single integer, the number of pieces that the cake will be divided into after all the cuts are made.
Sample Input 1 Sample Output 1
10000 0 -10000 0 4
0 10000 0 -10000
Sample Input 2 Sample Output 2
8000 6000 -8000 -6000 6
10000 0 -10000 0
0 10000 0 -10000 | {"url":"https://nus.kattis.com/courses/CS3233/CS3233_S2_AY2324/assignments/io3nhf/problems/cakecutting","timestamp":"2024-11-14T00:22:40Z","content_type":"text/html","content_length":"29247","record_id":"<urn:uuid:f2d4a9da-c1ed-4044-83f6-e1ff32e5f43f>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00679.warc.gz"} |
Python - BirdBrain Technologies
In this last lesson, you will explore recursion with the Finch. Recursion is when a function calls itself. In this lesson, you will use recursion to draw Koch fractals.
The Koch fractal, also called the Koch snowflake, can be drawn with increasing levels of complexity. The level of complexity is called the order of the Koch fractal. A Koch fractal of order 0 is just
a straight line with some length L. Starting from order 0, which we call the base case, we can build Koch fractals with larger orders. To create the order 1 fractal, we make four copies of the order
0 fractal; each copy has a length of L/3. We put them together so that the first and last copies lie along a straight line, while the two middle ones protrude from the line and make a 60° angle. We
continue this process to make the order 2 fractal. We make four copies of the order 1 fractal where each copy is 1⁄3 the size of the original. Then we put these four copies together so that the two
copies in the middle form a 60° angle. You can keep going in this way to make a fractal for any positive order n! As n approaches infinity, the length of the Koch fractal also approaches infinity
because the length of each order of the fractal is 4/3 times the length of the previous order. As you zoom in on a portion of the Koch snowflake, it is self-similar, meaning that each smaller portion
looks like the larger whole. The Koch snowflake is a fractal because a simple algorithm generates a complex, self-similar pattern.
In the description of Koch fractals above, we started with order 0 and then worked our way up to larger orders. To draw a fractal with your Finch, you actually need to do the opposite. You start with
some number n , and you want to draw a Koch fractal with that order. Luckily, there is an algorithm for this! We will use the algorithm given in How to Think Like a Computer Scientist: Learning with
Python 3 by Wentworth, Elkner, Downey, and Meyers. The steps below can be used to draw a Koch fractal of order n:
You are going to use this algorithm to write a program that uses the Finch to draw a Koch fractal of a given order and size.
When the order is not 0, your function will need to call itself (recursion!). If the parameters of drawFractal() are order and distance, what parameters should you use in the recursive calls to
drawFractal()? Remember, you are implementing the algorithm given above Exercise 1. Fill in the blanks below.
Now implement your plan from Exercise 2. Remember, drawFractal() should only call itself when the order is greater than 0. When the order is 0, the Finch should move in a straight line. You have
already implemented that part!
When testing your program, start by testing with order 1, and then work your way up to larger orders. This will make it easier to see if your program is working as you expect. Once your program is
working, use a marker to draw some beautiful pictures of fractals! If you want to make a Koch Snowflake, as shown below, draw three Koch fractals with 120° turns between them.
Note: We highly recommend using a brush tip marker with the Finch. These markers work well. If you use a marker with a harder tip, the friction of the marker may make your drawing less accurate. | {"url":"https://learn.birdbraintechnologies.com/finch/python/program/lesson-15-finch-fractals","timestamp":"2024-11-05T17:04:45Z","content_type":"text/html","content_length":"139753","record_id":"<urn:uuid:a9eb4fa4-ea29-4061-8745-ca10501464cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00099.warc.gz"} |
Optimizing the Performance of Open-Type Refrigerated Display Cabinets: Block Schemes and Key Tasks
Department of Mechanical and Materials Engineering, Vilnius Gediminas Technical University, 10105 Vilnius, Lithuania
Power Engineering and Engineering Thermophysics
Volume 3, Issue 2, 2024
Pages 134-147
Received: 05-08-2024,
Revised: 06-15-2024,
Accepted: 06-22-2024,
Available online: 06-29-2024
View Full Article|Download PDF
The performance of open-type refrigerated display cabinets has been rigorously examined through the development and application of two comprehensive block schemes, which integrate numerical
simulations with experimental research. Central to these schemes is the use of a simplified two-dimensional, time-dependent computational fluid dynamics (CFD) model, designed to evaluate and optimize
airflow patterns, thermal behavior, and energy efficiency within the cabinets. The numerical simulations, validated against experimental data, demonstrate that the strategic design and configuration
of air curtains and internal components significantly mitigate the impact of ambient air, thereby reducing temperature fluctuations that are critical for maintaining food quality and safety. The
application of these block schemes has been shown to enhance energy efficiency and reduce electrical consumption, contributing to operational cost savings. The strong correlation between CFD results
and experimental findings underscores the reliability of these models for accurately representing real-world conditions. Future investigations could benefit from exploring additional geometric
configurations and incorporating more advanced CFD techniques to further refine the performance of refrigerated display systems. This integrated approach offers a robust framework for improving the
operational effectiveness and food preservation capabilities of open-type refrigerated display cabinets.
Keywords: Open-type refrigerated display cabinet, Air curtain, Air velocity, Computational fluid dynamics simulations, Honeycomb, Heat transfer, Temperature
Cite this:
APA Style
IEEE Style
BibTex Style
MLA Style
Chicago Style
Vengalis, T. (2024). Optimizing the Performance of Open-Type Refrigerated Display Cabinets: Block Schemes and Key Tasks. Power Eng. Eng Thermophys., 3(2), 134-147. https://doi.org/10.56578/peet030205
©2024 by the author(s). Published by Acadlore Publishing Services Limited, Hong Kong. This article is available for free download and can be reused and cited, provided that the original published
version is credited, under the
CC BY 4.0 license
Figure 1. Two-dimensional side view of (a) The Ordc-1 refrigerated cabinet and (b) The corresponding geometric model [18] | {"url":"https://www.acadlore.com/article/PEET/2024_3_2/peet030205","timestamp":"2024-11-04T21:01:59Z","content_type":"text/html","content_length":"297052","record_id":"<urn:uuid:c52a9c83-504d-49f3-b569-28d5025b7def>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00170.warc.gz"} |
Solution assignment 10 Integration by parts
Return to Assignments Integration by parts
Assignment 10
In assignment 3 it was asked to calculate the following integral and we applied integration by parts. Is there another way and if so, which?
An important formula from trigonometry is:
Or, written differently:
This integral can be calculated using standard functions and substitution:
Return to Assignments Integration by parts | {"url":"https://4mules.nl/en/integration-by-parts/assignments/solution-assignment-10-integration-by-parts/","timestamp":"2024-11-06T21:02:00Z","content_type":"text/html","content_length":"38872","record_id":"<urn:uuid:7c65b651-83f8-4eff-977f-513bd3b7803c>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00106.warc.gz"} |
Trigonometric Ratios Worksheet Pdf
Trigonometric Ratios Worksheet Pdf
The trigonometric ratios with right angled triangles having three sides it is possible to have 6 ratios. The emphasis is on choosing the correct trigonometric ratio.
Geometry Worksheets Trigonometry Worksheets Trigonometry Worksheets Geometry Worksheets Trigonometry
Trignometry is one of the major section of advance mathematics for different exams including competitive exams trignometry study materials pdf with practice questions worksheet is available here to
download in english and hindi language.
Trigonometric ratios worksheet pdf. 7 sin 62 8 sin 14 9 cos 60 10 cos 31 11 tan 79 12 tan 25. 7 10 hw. M worksheet by kuta software llc kuta software infinite geometry name trigonometric ratios date
period find the value of each trigonometric ratio.
Opposite adjacent opposite hypotenuse adjacent hypotenuse and adjacent opposite hypotenuse opposite hypotenuse adjacent there are basically 3 unique ratios. Use trigonometric ratios to find missing
lengths of a right triangle pgs. Trigonometry study materials pdf with practice questions worksheet.
1 tan a 16 34 30 a b c 1 8750 2 cos c 12 9 15. 1 4 hw. Law of sines and cosines worksheet this sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well
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1 tan z 28 21 35 z y x 3 4 2 cos c 16 34 30 c b a 8 17 3 sin c 21 28 35 c b a 4 5 4 tan x 24 32 40 x y z 4 3 5 cos a 30 16 34 a b c 15 17 6 sin a 24 32 40 a c b 4 5 7 sin z 32. Calculate the length
of a side a right triangle using the pythagorean theorem pgs. The 3 trig ratios commonly used are given the names sine cosine.
Sine cosine tangent worksheets. This video covers the first of the application videos in which we use the trigonometric ratios to determine the length of a side in a right angled triangle. Students
can use math worksheets to master a math skill through practice in a study group or for peer tutoring.
K s fmqaxdze w tw qiltyh b hijn vfqisnoi xt mef ganlvgqehbdrsa2 k1 9 5 worksheet by kuta software llc kuta software infinite algebra 1 name finding trigonometric ratios date period find the value of
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worksheet. C fkeuktmao lsboefjt ywiamrwed 6lrlgcq v 7 vajl7lw drwiygghmtwsc drceystejrzvmetd9 i i im la fdxeu qwti ztmhn riun8fxixn5i3t0eh 8a9lbguebbdruao s1r i worksheet by kuta software llc find
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Trigonometry worksheet t2 sine cosine tangent values give the value of each of the following. Table of contents day 1. Find the three basic trigonometric ratios in a right triangle pgs.
Ambiguous case of the law of sines. 11 12 day 3. 5 6 day 2.
Using trigonometric to calculate an angle this video covers the second of the application videos in which we use the. The size of the pdf file is 43287 bytes.
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viewmtx (MATLAB Functions)
MATLAB Function Reference
viewmtx computes a 4-by-4 orthographic or perspective transformation matrix that projects four-dimensional homogeneous vectors onto a two-dimensional view surface (e.g., your computer screen).
T = viewmtx(az,el) returns an orthographic transformation matrix corresponding to azimuth az and elevation el. az is the azimuth (i.e., horizontal rotation) of the viewpoint in degrees. el is the
elevation of the viewpoint in degrees. This returns the same matrix as the commands
but does not change the current view.
T = viewmtx(az,el,phi) returns a perspective transformation matrix. phi is the perspective viewing angle in degrees. phi is the subtended view angle of the normalized plot cube (in degrees) and
controls the amount of perspective distortion.
┃ Phi │ Description ┃
┃ 0 degrees │ Orthographic projection ┃
┃ 10 degrees │ Similar to telephoto lens ┃
┃ 25 degrees │ Similar to normal lens ┃
┃ 60 degrees │ Similar to wide-angle lens ┃
You can use the matrix returned to set the view transformation with view(T). The 4-by-4 perspective transformation matrix transforms four-dimensional homogeneous vectors into unnormalized vectors of
the form (x,y,z,w), where w is not equal to 1. The x- and y-components of the normalized vector (x/w, y/w, z/w, 1) are the desired two-dimensional components (see example below).
T = viewmtx(az,el,phi,xc) returns the perspective transformation matrix using xc as the target point within the normalized plot cube (i.e., the camera is looking at the point xc). xc is the target
point that is the center of the view. You specify the point as a three-element vector, xc = [xc,yc,zc], in the interval [0,1]. The default value is xc = [0,0,0].
A four-dimensional homogenous vector is formed by appending a 1 to the corresponding three-dimensional vector. For example, [x,y,z,1] is the four-dimensional vector corresponding to the
three-dimensional point [x,y,z].
Determine the projected two-dimensional vector corresponding to the three-dimensional point (0.5,0.0,-3.0) using the default view direction. Note that the point is a column vector.
Vectors that trace the edges of a unit cube are
Transform the points in these vectors to the screen, then plot the object.
Use a perspective transformation with a 25 degree viewing angle:
Transform the cube vectors to the screen and plot the object:
See Also
view, hgtransform
Controlling the Camera Viewpoint for related functions
Defining the View for more information on viewing concepts and techniques
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New Riddles - Riddles.com
One afternoon, Cara came home and found that her favorite vase had been shattered. The woman questioned her three kids—Ali, Gia, and Joe. Ali said, "It was Gia!". Gia kept silent, and so did Joe.
Assuming that the culprit tells the truth, who shattered Cara's vase?
Answer: It was Joe. If Ali is telling the truth, then he's the culprit. But that would make Gia the culprit, too, which would then create a paradox. Therefore, Ali is lying, and Joe is the culprit by | {"url":"https://www.riddles.com/riddles?sort=new&page=3","timestamp":"2024-11-02T17:33:14Z","content_type":"text/html","content_length":"146467","record_id":"<urn:uuid:3ca72ee8-6830-4539-9a52-466dc8955014>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00826.warc.gz"} |
DeMorgan's theorem
A logical theorem which states that the complement of a conjunction is the disjunction of the complements or vice versa. In symbols:
not (x and y) = (not x) or (not y)
not (x or y) = (not x) and (not y)
E.g. if it is not the case that I am tall and thin then I am either short or fat (or both). The theorem can be extended to combinations of more than two terms in the obvious way. The same laws also
apply to sets, replacing logical complement with set complement, conjunction ("and") with set intersection, and disjunction ("or") with set union. A (
) programmer might use this to re-write
if (!foo && !bar) ...
if (!(foo || bar)) ...
thus saving one operator application (though an
optimising compiler
should do the same, leaving the programmer free to use whichever form seemed clearest).
Last updated: 1995-12-14
Nearby terms:
Demon Internet Ltd. ♦ DeMorgan's theorem ♦ demo version ♦ Denis Howe
Try this search on Wikipedia, Wiktionary, Google, OneLook. | {"url":"https://foldoc.org/DeMorgan's+theorem","timestamp":"2024-11-13T05:56:00Z","content_type":"text/html","content_length":"9594","record_id":"<urn:uuid:589343e4-60c7-4551-b4ff-6106b1aa1a85>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00710.warc.gz"} |
Y. Find the circumcentre of the triangle whose vertices are (1,... | Filo
Question asked by Filo student
Y. Find the circumcentre of the triangle whose vertices are and . 7. Find the values of , if the angle between the straight lines and is . 8. Find the equation of the straight line passing through
the origin and also through the point of intersection of the lines and . 9. Find the equation of the straight line parallel to the line and passing through the point of intersection of the lines and
. 10. Find the equation of the straight line perpendicular to the line and passing through the point of intersection of the lines and . 11. Find the equation of the straight line making non-zero
equal intercepts on the coordinate axes and passing through the point of intersection of the lines and 12. Find the length of the perpendicular drawn from the point of intersection of the lines and
to the straight line . 13. Find the value of ' ' if the distances of the points and from the straight line are equal. 14. Find the circumcenter of the triangle formed by the straight lines and 15. If
is the angle between the lines and , find the value of when . [1. 1. Find the equations of the straight lines passing through the point and making an angle with the line such that . 2. Find the
equations of the straight lines passing through the point and making an angle of with the line . 3. The base of an equilateral triangle is and the opposite vertex is . Find the equations of the
remaining sides. 1. Find the orthocenter the triangle with the following vertices (i) and (ii) and 5. Find the circumcenter of the triangle whose vertices are given below (i) and (ii) and 6. Let be
the median of the triangle with vertices and . Find the equation
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Y. Find the circumcentre of the triangle whose vertices are and . 7. Find the values of , if the angle between the straight lines and is . 8. Find the equation of the straight line passing
through the origin and also through the point of intersection of the lines and . 9. Find the equation of the straight line parallel to the line and passing through the point of intersection
of the lines and . 10. Find the equation of the straight line perpendicular to the line and passing through the point of intersection of the lines and . 11. Find the equation of the straight
line making non-zero equal intercepts on the coordinate axes and passing through the point of intersection of the lines and 12. Find the length of the perpendicular drawn from the point of
Question intersection of the lines and to the straight line . 13. Find the value of ' ' if the distances of the points and from the straight line are equal. 14. Find the circumcenter of the triangle
Text formed by the straight lines and 15. If is the angle between the lines and , find the value of when . [1. 1. Find the equations of the straight lines passing through the point and making an
angle with the line such that . 2. Find the equations of the straight lines passing through the point and making an angle of with the line . 3. The base of an equilateral triangle is and the
opposite vertex is . Find the equations of the remaining sides. 1. Find the orthocenter the triangle with the following vertices (i) and (ii) and 5. Find the circumcenter of the triangle
whose vertices are given below (i) and (ii) and 6. Let be the median of the triangle with vertices and . Find the equation
Updated Nov 3, 2022
Topic Coordinate Geometry
Subject Mathematics
Class Class 11
Answer Video solution: 1
Upvotes 110
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Mathematical friend from Gargi
Hi Gladis I would like to contribute towards Maths Glossary project. But I dont know how to use Mathematical tools while editing the pages. Will you pl. guide me. Thanks and regards
Hi Promila, Thanks for the message. The main Math tool we need is Latex, which is the required markup language for inserting o rendering math symbols.
Wayne Mackintosh told me about the posibility of schedulling a Latex Tutorial in WE. Meantime please:
1. Read complete the Math Glossary Main Page
2. Fill in your name at the bottom of the page in section Join the Project
3. Open a new window and click here: Displaying Special Characters. Study the page.
4. Check up the page Absolute Value to see your contribution now converted to math symbols. Switch to edit mode to study the codes.
5. Main codes for writing special and math symbols are: <math></math> which are automaticaly written for you by clicking the which is located in the menu bar on editing mode.
6. Look for more math symbols in math pages, switch to edit mode and study the codes.
7. Practice math notation in your sandbox. Please let me know so we can practice together.
Hope this can help Gladys --chela5808 03:39, 15 January 2009 (UTC)
Hi Chela,
Thanks for editing the definition of binary numbers.I would like to learn this skill as well. Thanks again.
Hi Gladys Thanks for the suggestion of deleting the page.I did that. --Promilakumar 16:20, 6 February 2009 (UTC) | {"url":"https://wikieducator.org/Thread:Mathematical_friend_from_Gargi_(3)","timestamp":"2024-11-08T17:26:35Z","content_type":"text/html","content_length":"25907","record_id":"<urn:uuid:446f54bd-6f62-4d61-9456-ac13a9f3c4a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00532.warc.gz"} |
MINIFS function
Returns the minimum of the values of cells in a range that meets multiple criteria in multiple ranges.
MINIFS(Func_Range; Range1; Criterion[; Range2; Criterion2][; … ; [Range127; Criterion127]])
Func_Range – required argument. A range of cells, a name of a named range or a label of a column or a row containing values for calculating the minimum.
Ús simple
Calculates the minimum of values of the range B2:B6 that are lower than or equal to 20. Returns 17.
Calculates the minimum of values of the range C2:C6 that are lower than 90 and correspond to cells of the B2:B6 range with values greater than or equal to 20. Returns 190.
Ús d'expressions regulars i funcions imbricades
Calculates the minimum of values of the range C2:C6 that correspond to all values of the range B2:B6 except its minimum and maximum. Returns 65.
Calculates the minimum of values of the range C2:C6 that correspond to all cells of the A2:A6 range ending with "book" and to all cells of the B2:B6 range except its minimum. Returns 190.
Referència a una cel·la com a criteri
If you need to change a criterion easily, you may want to specify it in a separate cell and use a reference to this cell in the condition of the MINIFS function. For example, the above function can
be rewritten as follows:
If E2 = "book", the function returns 180, because the reference to the cell is substituted with its content. | {"url":"https://help.libreoffice.org/latest/ca/text/scalc/01/func_minifs.html","timestamp":"2024-11-02T23:14:11Z","content_type":"text/html","content_length":"26064","record_id":"<urn:uuid:ffc4392a-a2d4-414d-a890-5f8063aa0e2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00282.warc.gz"} |
Letter Boxed Answers Archives - Page 2 of 32 - Learn With Shikha
Here are the answers for the Letter Boxed November 03, 2024 Answers, from New York Times Games. Our solutions and answers are completely correct. We recommend trying to figure out the game by
yourself before checking our website for help. Sides of this Letter Box are: ARU IHL ZMB EOS Answers: HUMOROUS SIZABLE
Letter Boxed November 02, 2024 Answers
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yourself before checking our website for help. Sides of this Letter Box are: WNT KYO ARH LVE Answers: WALKATHON NERVY
Letter Boxed November 01, 2024 Answers
Here are the answers for the Letter Boxed November 01, 2024 Answers, from New York Times Games. Our solutions and answers are completely correct. We recommend trying to figure out the game by
yourself before checking our website for help. Sides of this Letter Box are: ASL OIU EHR QCG Answers: QUESO OLIGARCH
Letter Boxed October 31, 2024 Answers
Here are the answers for the Letter Boxed October 31, 2024 Answers, from New York Times Games. Our solutions and answers are completely correct. We recommend trying to figure out the game by yourself
before checking our website for help. Sides of this Letter Box are: CLU STJ XNK EIO Answers: JUICE EXOSKELETON
Letter Boxed October 30, 2024 Answers
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before checking our website for help. Sides of this Letter Box are: RPM GBK NAE ICY Answers: PAYBACK KINGMAKER
Letter Boxed October 29, 2024 Answers
Here are the answers for the Letter Boxed October 29, 2024 Answers, from New York Times Games. Our solutions and answers are completely correct. We recommend trying to figure out the game by yourself
before checking our website for help. Sides of this Letter Box are: CTO BDI AFP REL Answers: CLIPBOARD DEFT
Letter Boxed October 28, 2024 Answers
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before checking our website for help. Sides of this Letter Box are: TIA UWL DBY RMO Answers: LIMBO OUTWARDLY
Letter Boxed October 27, 2024 Answers
Here are the answers for the Letter Boxed October 27, 2024 Answers, from New York Times Games. Our solutions and answers are completely correct. We recommend trying to figure out the game by yourself
before checking our website for help. Sides of this Letter Box are: SLQ XTI ENO FUA Answers: EXFOLIANT TOQUES
Letter Boxed October 26, 2024 Answers
Here are the answers for the Letter Boxed October 26, 2024 Answers, from New York Times Games. Our solutions and answers are completely correct. We recommend trying to figure out the game by yourself
before checking our website for help. Sides of this Letter Box are: NWT EOL RAV BYI Answers: WROTE ENVIABLY
Letter Boxed October 25, 2024 Answers
Here are the answers for the Letter Boxed October 25, 2024 Answers, from New York Times Games. Our solutions and answers are completely correct. We recommend trying to figure out the game by yourself
before checking our website for help. Sides of this Letter Box are: WLU ZBK DSI AHE Answers: BUSHWA ALKALIZED | {"url":"https://learnwithshikha.com/category/letter-boxed-answers/page/2/","timestamp":"2024-11-13T15:41:05Z","content_type":"text/html","content_length":"161992","record_id":"<urn:uuid:7ad3a3b3-2fa2-4fee-b380-0e3ac491b16d>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00780.warc.gz"} |
558 Sign/Square Month to Radian/Square Hour
Sign/Square Month [sign/month2] Output
558 sign/square month in degree/square second is equal to 2.4205274469854e-9
558 sign/square month in degree/square millisecond is equal to 2.4205274469854e-15
558 sign/square month in degree/square microsecond is equal to 2.4205274469854e-21
558 sign/square month in degree/square nanosecond is equal to 2.4205274469854e-27
558 sign/square month in degree/square minute is equal to 0.0000087138988091473
558 sign/square month in degree/square hour is equal to 0.03137003571293
558 sign/square month in degree/square day is equal to 18.07
558 sign/square month in degree/square week is equal to 885.39
558 sign/square month in degree/square month is equal to 16740
558 sign/square month in degree/square year is equal to 2410560
558 sign/square month in radian/square second is equal to 4.2246173584787e-11
558 sign/square month in radian/square millisecond is equal to 4.2246173584787e-17
558 sign/square month in radian/square microsecond is equal to 4.2246173584787e-23
558 sign/square month in radian/square nanosecond is equal to 4.2246173584787e-29
558 sign/square month in radian/square minute is equal to 1.5208622490523e-7
558 sign/square month in radian/square hour is equal to 0.00054751040965884
558 sign/square month in radian/square day is equal to 0.31536599596349
558 sign/square month in radian/square week is equal to 15.45
558 sign/square month in radian/square month is equal to 292.17
558 sign/square month in radian/square year is equal to 42072.21
558 sign/square month in gradian/square second is equal to 2.6894749410949e-9
558 sign/square month in gradian/square millisecond is equal to 2.6894749410949e-15
558 sign/square month in gradian/square microsecond is equal to 2.6894749410949e-21
558 sign/square month in gradian/square nanosecond is equal to 2.6894749410949e-27
558 sign/square month in gradian/square minute is equal to 0.0000096821097879415
558 sign/square month in gradian/square hour is equal to 0.034855595236589
558 sign/square month in gradian/square day is equal to 20.08
558 sign/square month in gradian/square week is equal to 983.76
558 sign/square month in gradian/square month is equal to 18600
558 sign/square month in gradian/square year is equal to 2678400
558 sign/square month in arcmin/square second is equal to 1.4523164681912e-7
558 sign/square month in arcmin/square millisecond is equal to 1.4523164681912e-13
558 sign/square month in arcmin/square microsecond is equal to 1.4523164681912e-19
558 sign/square month in arcmin/square nanosecond is equal to 1.4523164681912e-25
558 sign/square month in arcmin/square minute is equal to 0.00052283392854884
558 sign/square month in arcmin/square hour is equal to 1.88
558 sign/square month in arcmin/square day is equal to 1084.15
558 sign/square month in arcmin/square week is equal to 53123.27
558 sign/square month in arcmin/square month is equal to 1004400
558 sign/square month in arcmin/square year is equal to 144633600
558 sign/square month in arcsec/square second is equal to 0.0000087138988091473
558 sign/square month in arcsec/square millisecond is equal to 8.7138988091473e-12
558 sign/square month in arcsec/square microsecond is equal to 8.7138988091473e-18
558 sign/square month in arcsec/square nanosecond is equal to 8.7138988091473e-24
558 sign/square month in arcsec/square minute is equal to 0.03137003571293
558 sign/square month in arcsec/square hour is equal to 112.93
558 sign/square month in arcsec/square day is equal to 65048.91
558 sign/square month in arcsec/square week is equal to 3187396.4
558 sign/square month in arcsec/square month is equal to 60264000
558 sign/square month in arcsec/square year is equal to 8678016000
558 sign/square month in sign/square second is equal to 8.0684248232846e-11
558 sign/square month in sign/square millisecond is equal to 8.0684248232846e-17
558 sign/square month in sign/square microsecond is equal to 8.0684248232846e-23
558 sign/square month in sign/square nanosecond is equal to 8.0684248232846e-29
558 sign/square month in sign/square minute is equal to 2.9046329363824e-7
558 sign/square month in sign/square hour is equal to 0.0010456678570977
558 sign/square month in sign/square day is equal to 0.60230468568826
558 sign/square month in sign/square week is equal to 29.51
558 sign/square month in sign/square year is equal to 80352
558 sign/square month in turn/square second is equal to 6.7236873527371e-12
558 sign/square month in turn/square millisecond is equal to 6.7236873527371e-18
558 sign/square month in turn/square microsecond is equal to 6.7236873527371e-24
558 sign/square month in turn/square nanosecond is equal to 6.7236873527371e-30
558 sign/square month in turn/square minute is equal to 2.4205274469854e-8
558 sign/square month in turn/square hour is equal to 0.000087138988091473
558 sign/square month in turn/square day is equal to 0.050192057140689
558 sign/square month in turn/square week is equal to 2.46
558 sign/square month in turn/square month is equal to 46.5
558 sign/square month in turn/square year is equal to 6696
558 sign/square month in circle/square second is equal to 6.7236873527371e-12
558 sign/square month in circle/square millisecond is equal to 6.7236873527371e-18
558 sign/square month in circle/square microsecond is equal to 6.7236873527371e-24
558 sign/square month in circle/square nanosecond is equal to 6.7236873527371e-30
558 sign/square month in circle/square minute is equal to 2.4205274469854e-8
558 sign/square month in circle/square hour is equal to 0.000087138988091473
558 sign/square month in circle/square day is equal to 0.050192057140689
558 sign/square month in circle/square week is equal to 2.46
558 sign/square month in circle/square month is equal to 46.5
558 sign/square month in circle/square year is equal to 6696
558 sign/square month in mil/square second is equal to 4.3031599057518e-8
558 sign/square month in mil/square millisecond is equal to 4.3031599057518e-14
558 sign/square month in mil/square microsecond is equal to 4.3031599057518e-20
558 sign/square month in mil/square nanosecond is equal to 4.3031599057518e-26
558 sign/square month in mil/square minute is equal to 0.00015491375660706
558 sign/square month in mil/square hour is equal to 0.55768952378543
558 sign/square month in mil/square day is equal to 321.23
558 sign/square month in mil/square week is equal to 15740.23
558 sign/square month in mil/square month is equal to 297600
558 sign/square month in mil/square year is equal to 42854400
558 sign/square month in revolution/square second is equal to 6.7236873527371e-12
558 sign/square month in revolution/square millisecond is equal to 6.7236873527371e-18
558 sign/square month in revolution/square microsecond is equal to 6.7236873527371e-24
558 sign/square month in revolution/square nanosecond is equal to 6.7236873527371e-30
558 sign/square month in revolution/square minute is equal to 2.4205274469854e-8
558 sign/square month in revolution/square hour is equal to 0.000087138988091473
558 sign/square month in revolution/square day is equal to 0.050192057140689
558 sign/square month in revolution/square week is equal to 2.46
558 sign/square month in revolution/square month is equal to 46.5
558 sign/square month in revolution/square year is equal to 6696 | {"url":"https://hextobinary.com/unit/angularacc/from/signpm2/to/radph2/558","timestamp":"2024-11-04T11:42:22Z","content_type":"text/html","content_length":"113244","record_id":"<urn:uuid:8e39ef54-172a-40d5-bd9f-b0a39e31b7c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00112.warc.gz"} |
Science Break: Practical Applications of Science
Perusing my recent articles, I realized I’ve been touching on some rather somber topics lately, such as mental illness, psychopathy, prion disease, Nazi death camps, etc. So this month I decided to
lighten things up and cover some cheerful examples of how science can be applied in day-to-day ways. Astute readers will see through this explanation, and quickly assume I am running short of topics.
It may be a bit of that, but the truth probably lies somewhere in between – it’s the dog days of winter and dredging up the energy to research and properly cover a complex topic just seems so
daunting compared to watching TV or doing crosswords. I’ll be off to a beach vacation in a couple of weeks, and I expect that will re-energize me for next month’s article on corals.
Alcohol and Winter
You’re not Canadian if you haven’t been faced with this dilemma – “Is it too cold to leave my beer or wine in the car overnight?” Well this is the sort of situation where science can be put to a
practical use for a change. I mean really, is my quality of life going to be affected if they prove the existence of the Higgs boson? But if I can confidently leave my drinks in the car to chill
before tomorrow’s trip to my imaginary multi-million dollar ski chalet at Blue “Mountain”, and save myself the trouble of lugging them to the refrigerator and back, then I’m a happy man. We all know
that alcohol has a lower freezing point, so beer and wine can be stored at temperatures lower than 0 °C without freezing, but how much lower?
It turns out to be not entirely straightforward, because alcoholic beverages are a mix of water, alcohol, and other components. As the temperature drops below freezing, the water freezes but the
alcohol doesn’t, creating a form of slush. As more and more of the water freezes, the remaining liquid has a higher and higher alcohol content which lowers its freezing point. Incidentally, this is a
poor man’s alternative to distillation, and is also used at a commercial level – that is how ice beers achieve a higher alcohol content. This has been done for centuries; for example, in Germany
there is a longstanding tradition of beers known as Eisbock. The beer is chilled when it is close to fully fermented, and ice crystals form. The remaining liquid is drawn off, creating a beer that is
up to 10% alcohol, twice the strength of regular beer. I intend to try this with cider, maybe this weekend. But I digress…
Let’s assume our drinks are just water and alcohol – that will allow us to get ballpark estimates, which should be good enough for our purposes. What we’re going to use is the chemistry concept of
freezing point depression, which applies to any situation where there is a solution formed by the mix of a solvent (water in this case) and a solute (alcohol in this case), and the freezing point is
required. The formula is:
ΔT[f] = T[f] (pure solvent) – T[f] (solution)
ΔT[f] = K[f]m
ΔT[f] is the freezing point depression, and T[f] denotes the temperature at which freezing occurs,
K[f] , the cryoscopic constant is solvent dependent, (for water K[f] = 1.853 K*kg/mol);
m is the molality (moles solute particles per kg of solution).
Let’s try it out for a New Zealand Sauvignon Blanc with 12% alcohol content, and assume we have 1000g of wine – that means 120g of ethyl alcohol (C[2]H[6]O) and 880g of water (H[2]O).
To determine the molality, first we need to calculate the number of moles of C[2]H[6]O in the wine. C[2]H[6]O has a molar mass of 46.07g/mol, so,
Moles of C[2]H[6]O in the bottle of wine
= (Mass C[2]H[6]O) / (Molecular Weight C[2]H[6]O)
= (120g C[2]H[6]O) / (46.07g/mol C[2]H[6]O)
= 2.60 mol C[2]H[6]O
Molality of wine = (Moles Solute) / (Mass Solvent)
= (2.60 mol CH[3]OH) / (0.88kg H[2]O) = 2.96m
With that done, we can now plug the result into the freezing point depression equation.
ΔT[f] = K[f]m
= (1.86m/°C)*(2.96m)
= 5.5 °C
ΔT[f] = T[f] (pure solvent) –T[f](solution)
T[f] (solution) = T[f] (pure solvent) – ΔT[f]
T[f] (solution) = 0 °C – 5.5 °C = – 5.5 °C
So we’d expect our wine to freeze at -5.5 °C. Going through the same exercise for hard liquor (40% alcohol) and beer (5% alcohol) gives freezing points of -26.9 °C and -0.2 °C respectively. The
presence of other substances in these drinks (sugar, sulphides, etc.) pulls the freezing point down a bit more, and the slushy effect I mentioned above does as well. The handy chart in Figure 1 shows
the freezing point curve of a water / alcohol solution, and I’ve superimposed freezing point estimates, which are beer at ~-2.5 °C, wine ~-5.0 °C, and liquor ~-30.0 °C. The labels on bottles show the
percent alcohol volume, whereas the equations and the graph are based on percent by mass, so that explains the difference in numbers somewhat.
Figure 1. Freezing point of alcohol solutions.
The eutectic point is interesting as applied to another winter topic – the use of salt to prevent ice on roads and sidewalks. In Toronto they dump mountains of the stuff everywhere. I’m aghast and
can only speculate what it’s doing to the environment. The eutectic point represents the temperature at the lowest melting point for a substance made up of two or more components. Adding salt to
water lowers the melting point, which is desired to prevent slippery conditions, but the eutectic point for that mix is -18.0 °C. In Calgary, where winter temperatures are often lower than that, sand
and gravel become the preferred method for dealing with ice. In Toronto where winter temperatures aren’t as cold, salt is used extensively, and the lower range is extended a bit by the additional use
of liquid antifreeze agents.
It borders on insulting to present this topic to a readership made up largely of seismologists, who should be familiar with the velocity of sound in air. But let’s make the reasonable assumption that
unless a person is exposed to a scientific concept or fact on a routine basis, they will forget it as soon as the last exam on the subject is handed in. Seismic processors are familiar with air blast
noise trains on raw shot records (Fig. 2). These are generally caused by hissing noises given off by the hydraulic systems in Vibroseis trucks. The sound waves travel through the air and are picked
up by the geophones. The slope of these noise trains gives the known velocity of sound waves in air, which is 343.2 m/sec in dry air at 20 °C. Given that, it’s easy to estimate the proximity of a
lightning strike, if you can estimate the time between seeing the flash of lightning and hearing the thunder. Why this is of any use is beyond me, but as a boy I was very impressed by people who
could do it. I think a good rule of thumb would be to say that each 3 seconds of time represents about 1 km. For the time estimate, I recommend using the “Mississippi one, Mississippi two, …” count
Figure 2. 2D seismic shot record showing air blast velocity.
Estimating Heights
When I was a Cub Scout, I remember getting a badge for learning some impossibly complicated way of estimating the height of an object, like a tree or cliff. As I recall it involved estimating angles,
which right there makes it unreliable in my mind. Since then I have learned a much easier way of doing this.
First, find something like a pencil or straight stick. Hold it vertically upright in your hand with your arm straight ahead of you. Close one eye and squint in a manly fashion (or womanly as the case
may be) and line up the top of the feature being measured with the top of the pencil. Slide your thumb so it lines up with the bottom of the feature. Now rotate the pencil 90 ° holding your thumb in
place so that the pencil is horizontal and your thumb is still at the base of the object. Make a mental note of some object that lines up exactly with the tip of the pencil. Now simply walk over to
the base of the object and pace off the distance from there to the object you identified – this distance will reasonably represent the height of the object. This is just simple geometry – you’ve
scribed a simple quarter circle with your thumb as its axis, and the radius is the height of the object. I’ve tried to depict this method in a picture (Fig. 3), if you can imagine holding the pencil
up so it lines up with the Eiffel Tower.
Figure 3. Estimating the height of the Eiffel Tower using a pencil.
The key to this technique is to accurately pace off the distance between the two points you’ve identified in your mind. To achieve this you’ll have to know what your typical stride length is,
something that is second nature to golfers, who routinely pace off the distance between fixed distance markers and where their ball lies, in order to estimate the distance to the hole.
A quick and dirty alternative to this approach is to keep splitting the unknown object’s height in half (visually) until you can compare your divided height with an object at the base of known
height. Each time you split the height in half means you have to multiply your known object height by another factor of 2 to get your estimate. In other words, you multiply your known object’s height
by 2 raised to the power of the times you’ve divided the unknown object’s height in half. For example, the Eiffel Tower is 324 m high (but you don’t know that). If you visually divide it in half 8
times, you’re down to 1.27 m (324 / 162 / 81 / 40.5 / 20.25 / 10.125 / 5.06 / 2.53 / 1.27). Now say Toulouse Lautrec is standing at the bottom and your Eiffel Tower-divided-by-8 height visually seems
about the same height as him, and you happen to know his height is about 1.3 m. Just take that 1.3 m and multiply it by 2^8.
H = 1.3 * 2^8 = 1.3 * 256 = 333 m
Going the other way, it’s common to want to know how far down something is, like the bottom of a well, or the distance from the top of a cliff to the bottom. The good thing about these situations is
that you’re in control of the experiment, which involves dropping an object and timing how long it takes to get to the bottom, plus nowadays everyone has a timer and a calculator on their phone. This
is all grade 10 physics, so in other words, impossibly difficult for most of us without a refresher. The velocity of a body under constant acceleration, starting from rest, at time t, you’ll vaguely
remember is given by,
V = a * t
Where a is the constant force of acceleration. On earth, at sea level, with no air resistance, this value is 9.80665 m/s^2. Integrating the formula for velocity gives the corresponding formula for
distance travelled, for the same body under constant acceleration, starting from rest.
D = ½ a * t^2
This is the equation we need to use, and given it’s a crude experiment designed to come up with an estimate only, we’ll ignore the effects of air resistance, and round the 9.80665 to 10. We can
minimize the air resistance error by choosing an object with low air resistance, such as a round stone or maybe a jellybean (versus, say, a piece of chewed gum). If we’re going to use sound as the
method for determining the time when the object hits bottom (versus a visual of the angry man with gum on his head looking up), then there is an error there as well. You now know sound travels ~343 m
/s, but to save you the trouble of coming up with a recursive ballpark correction, I’ve put together the following little chart of time corrections to account for the time it takes for the sound to
get back to you – simply locate your recorded time in the “In air” time column, roughly interpolate the “Correction”, then subtract this correction from your recorded time.
Time (seconds)
Height (m) In vacuum In air Correction
50 3.19 3.34 0.15
100 4.52 4.81 0.29
200 6.39 6.97 0.58
300 7.82 8.70 0.87
400 9.03 10.20 1.17
500 10.10 1156 1.46
600 11.06 12.81 1.75
700 11.95 13.99 2.04
800 12.77 15.11 2.33
900 13.55 16.17 2.62
1000 14.28 17.20 2.92
So all that’s required is to time the dropping object, subtract the correction if using sound, then square the time and multiply by 5 ( ½ * 10) and you’ve got your estimate. For example, the Eiffel
Tower observation deck is 276 m above ground (but we don’t know that). We drop our piece of gum, and the shouts of l’homme en colère reach us 8.3 seconds later. Glancing at the chart we apply a rough
correction of 0.8 seconds, giving a corrected time of 7.5 seconds. Our crude estimate for height above ground is given by,
H = ½ * 10 * 7.5^2 = 281 m
Most people are subjected to a jellybean guessing contest at some point in life, or something similar, especially if they have children. Usually there’s a jar of jellybeans, and the person who comes
closest to guessing the actual number of jellybeans in the jar wins the jellybeans. How exciting is that?! Now I will ask readers to make a large leap of faith, which is to assume there actually
exist among us some who would want the jellybeans badly enough to prepare ahead of time a method to more accurately estimate jellybean numbers.
What is required to win the jellybeans actually involves three estimates – the volume of the container, the volume of an average jellybean, and the ratio of pore space (air) to matrix (jellybean).
To estimate the jar size, it helps to realize that most jars and containers come in standard sizes. However, given the use of both metric and imperial sizes here in Canada, this is a dangerous game,
and as a scientist you should be self-reliant – use those simple geometrical formulae you thought you’d never need! As you’ll recall, the volume of a cylinder is given by the product of its height
and the area of its base, so
V[cyl] = h * (πr^2)
That should suffice to give an estimate of the jar’s volume, but remember not to mix up radius and diameter! Reduce your estimate appropriately if it has a curved top and bottom as in Figure 4. If
it’s a rectangle, then volume is even easier,
V[rect] = h * w * d
And for a sphere,
V[sph] = 4/3 * (πr^3)
To estimate the volume of a jellybean, it can be approximated by a cylinder as well. Lastly, what percentage of the jar’s volume is made up of air? This is similar to approximating pore space
percentage from a thin section, an area in which geologists may have an advantage. What I can tell you is that I have researched this extensively, and the jellybean geology experts agree that 20% is
a very good number to use. So to put it all together, calculate the volume of the jar, multiply that by 0.8, and then divide that by the volume you’ve calculated for one jellybean. That will give you
a very good estimate of the number of jellybeans in the jar. Good luck!
Figure 4. A jar of jellybeans.
Eggs – Soft or Hard?
There are a couple of simple science-based tests to determine if an egg is raw or cooked. To be honest, I have totally forgotten all the physics surrounding rotating bodies, bearing credence to my
earlier assumption that most knowledge is quickly forgotten after university. And as per my earlier comment, I am in no mood to read up on the physics of rotating bodies of different rigidities.
However, I am totally confident in the knowledge that a rigid rotating body will spin longer and faster than one of equal size and mass that is deformable. For example, a singles squash ball, which
is very squishy and deformable, loses its spin very quickly, whereas a doubles squash ball, which is hard, spins a lot and retains that spin even after bouncing off two or three walls, with baffling
effect. With the soft squash ball, most of the energy of the spin is converted to deformation, and this acts as a damper on the spin.
A soft egg is a deformable body with a rigid shell, whereas a hard-boiled egg is hard throughout. This difference is manifested in the ways the two kinds of egg spin. If you have an egg of unknown
hardness, place it on a flat hard surface like a table, and give it a spin. If it spins fast and smooth, it’s likely hard-boiled; if it wobbles as it spins and slows down quickly, it’s likely raw. If
it falls somewhere in between, it’s likely soft-boiled. Further, if you stop the spinning egg by gently placing enough pressure with a finger from above, a hard-boiled egg will stay stopped, whereas
a soft egg will actually start to spin slightly again. This is because the soft insides have not stopped moving, and this internal movement causes the egg to start spinning a bit again. If you take a
cooking egg out of the boiling water, a hard-boiled egg will dry off very quickly (~10 seconds), whereas a soft-boiled one will stay wet for a while longer (~20 seconds). This is because the
hard-boiled egg holds greater residual heat.
Figure 5. Hard-boiled egg spinning on its end (RedStateEclectic, 2010).
I found one fun reference to egg spinning (RedStateEclectic, 2010) which contains a poem, a picture, and a video clip, describing how a hard-boiled egg will end up spinning on one of its ends. Start
the egg spinning horizontally at a rate of at least around 10 rotations per second. If it is hard-boiled it will quite quickly stand up on end and spin faster, just the way figure skaters spin faster
when they pull their arms and legs in. This probably has something to do with polar moment of inertia or some equally baffling concept, so I say let’s just finish off with this spinning egg poem, and
move on to next month!
“Place a hard-boiled egg on a table,
And spin it as fast as you’re able;
It will stand on one end
With vectorial blend Of precession and spin that’s quite stable.”
AUS-e-TUTE. (2014, August 6). Boiling Point Elevation and Freezing Point Depression. Retrieved February 16, 2015, from AUS-e-TUTE: http://www.ausetute.com.au/freezing.html
Lee, R. J. (2014, January 2). Chemistry of Beer, Part II: Freezing Point Depression and Fractional Freezing. Retrieved February 9, 2015, from TheMadScienceBlog: http://www.themadscienceblog.com/2014/
RedStateEclectic. (2010, January 30). Eggsperiment. Retrieved February 17, 2015, from RedStateEclectic: http://redstateeclectic.typepad.com/redstate_commentary/2010/01/eggsperiment.html
Wikimedia Foundation, Inc. (2014, December 4). Freezing-point depression. Retrieved February 16, 2015, from Wikipedia: https://en.wikipedia.org/wiki/Freezing-point_depression | {"url":"https://csegrecorder.com/columns/view/science-break-201504","timestamp":"2024-11-03T15:36:52Z","content_type":"text/html","content_length":"39632","record_id":"<urn:uuid:38eb7703-2923-4ea8-9943-593994fbc671>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00417.warc.gz"} |
NVIDIA Jetson Xavier AGX
Build All CUDA Samples
1. Go to the samples path
cd /usr/local/cuda/samples
2. Construct the samples using the makefile
sudo make
CUDA Samples
All the samples are in:
Simple Samples
Path Sample Description
/0_Simple/asyncAPI asyncAPI This sample uses CUDA streams and events to overlap execution on CPU and GPU.
/0_Simple/cdpSimplePrint cdpSimplePrint This sample demonstrates simple printf implemented using CUDA Dynamic Parallelism. This sample requires devices with compute capability 3.5 or
/0_Simple/ cdpSimpleQuicksort This sample demonstrates simple quicksort implemented using CUDA Dynamic Parallelism. This sample requires devices with compute capability 3.5 or
cdpSimpleQuicksort higher.
/0_Simple/clock clock This example shows how to use the clock function to measure the performance of a block of threads of a kernel accurately.
This example demonstrates how to integrate CUDA into an existing C++ application, i.e. the CUDA entry point on the host side is only a function
/0_Simple/cppIntegration cppIntegration which is called from C++ code, and only the file containing this function is compiled with nvcc. It also demonstrates that vector types can be used
from cpp.
/0_Simple/cppOverload cppOverload This sample demonstrates how to use C++ function overloading on the GPU.
/0_Simple/cudaOpenMP cudaOpenMP This sample demonstrates how to use OpenMP API to write an application for multiple GPUs.
/0_Simple/ fp16ScalarProduct Calculates scalar product of two vectors of FP16 numbers.
/0_Simple/inlinePTX inlinePTX A simple test application that demonstrates a new CUDA 4.0 ability to embed PTX in a CUDA kernel.
/0_Simple/matrixMul matrixMul This sample implements matrix multiplication which makes use of shared memory to ensure data reuse, the matrix multiplication is done using the
tiling approach.
/0_Simple/matrixMulCUBLAS matrixMulCUBLAS This sample implements matrix multiplication. To illustrate GPU performance for matrix multiply, this sample also shows how to use the new CUDA 4.0
interface for CUBLAS to demonstrate high-performance performance for matrix multiplication.
/0_Simple/matrixMulDrv matrixMulDrv This sample implements matrix multiplication and uses the new CUDA 4.0 kernel launch Driver API.
/0_Simple/simpleAssert simpleAssert This CUDA Runtime API sample is a very basic sample that implements how to use the assert function in the device code. Requires Compute Capability
/0_Simple/ simpleAtomicIntrinsics A simple demonstration of global memory atomic instructions. Requires Compute Capability 2.0 or higher.
/0_Simple/simpleCallback simpleCallback This sample implements multi-threaded heterogeneous computing workloads with the new CPU callbacks for CUDA streams and events introduced with CUDA
/0_Simple/ simpleCooperativeGroups This sample is a simple code that illustrates the basic usage of cooperative groups within the thread block.
/0_Simple/ simpleCubemapTexture Simple example that demonstrates how to use a new CUDA 4.1 feature to support cubemap Textures in CUDA C.
/0_Simple/ simpleCudaGraphs A demonstration of CUDA Graphs creation, instantiation, and launch using Graphs APIs and Stream Capture APIs.
/0_Simple/ simpleLayeredTexture Simple example that demonstrates how to use a new CUDA 4.0 feature to support layered Textures in CUDA C.
/0_Simple/simpleMPI simpleMPI Simple example demonstrating how to use MPI in combination with CUDA.
/0_Simple/simpleMultiCopy simpleMultiCopy This sample illustrates the usage of CUDA streams to achieve overlapping of kernel execution with data copies to and from the device.
/0_Simple/simpleMultiGPU simpleMultiGPU This application demonstrates how to use the new CUDA 4.0 API for CUDA context management and multi-threaded access to run CUDA kernels on
/0_Simple/simpleOccupancy simpleOccupancy This sample demonstrates the basic usage of the CUDA occupancy calculator and occupancy-based launch configurator APIs by launching a kernel with
the launch configurator and measures the utilization difference against a manually configured launch.
/0_Simple/ simplePitchLinearTexture Use of Pitch Linear Textures
/0_Simple/simplePrintf simplePrintf This CUDA Runtime API sample is a very basic sample that implements how to use the printf function in the device code.
/0_Simple/ simpleSeparateCompilation This sample demonstrates a CUDA 5.0 feature, the ability to create a GPU device static library and use it within another CUDA kernel. This example
simpleSeparateCompilation demonstrates how to pass in a GPU device function (from the GPU device static library) as a function pointer to be called.
/0_Simple/simpleStreams simpleStreams This sample uses CUDA streams to overlap kernel executions with memory copies between the host and a GPU device.
/0_Simple/ simpleSurfaceWrite Simple example that demonstrates the use of 2D surface references (Write-to-Texture).
/0_Simple/simpleTemplates simpleTemplates This sample is a templatized version of the template project. It also shows how to correctly templatize dynamically allocated shared memory arrays.
/0_Simple/simpleTexture simpleTexture Simple example that demonstrates use of Textures in CUDA.
/0_Simple/ simpleTextureDrv Simple example that demonstrates the use of Textures in CUDA. This sample uses the new CUDA 4.0 kernel launch Driver API.
/0_Simple/ simpleVoteIntrinsics Simple program which demonstrates how to use the Vote (any, all) intrinsic instruction in a CUDA kernel.
/0_Simple/simpleZeroCopy simpleZeroCopy This sample illustrates how to use Zero MemCopy, kernels can read and write directly to pinned system memory
/0_Simple/template template A trivial template project that can be used as a starting point to create new CUDA projects.
/0_Simple/ UnifiedMemoryStreams This sample demonstrates the use of OpenMP and streams with Unified Memory on a single GPU.
/0_Simple/vectorAdd vectorAdd This CUDA Runtime API sample is a very basic sample that implements element by element vector addition.
/0_Simple/vectorAddDrv vectorAddDrv This Vector Addition sample is a basic sample that is implemented element by element.
Utilities Samples
Path Sample Description
/1_Utilities/ bandwidthTest This is a simple test program to measure the memcopy bandwidth of the GPU and memcpy bandwidth across PCI-e.
/1_Utilities/ deviceQuery This sample enumerates the properties of the CUDA devices present in the system.
/1_Utilities/ deviceQueryDrv This sample enumerates the properties of the CUDA devices present using CUDA Driver API calls.
/1_Utilities/ p2pBandwidthLatencyTest This application demonstrates the CUDA Peer-To-Peer (P2P) data transfers between pairs of GPUs and computes latency and bandwidth.
/1_Utilities/ UnifiedMemoryPerf This sample demonstrates the performance comparison using matrix multiplication kernel of Unified Memory with/without hints and other types of memory
UnifiedMemoryPerf like zero-copy buffers, pageable, page locked memory performing synchronous and Asynchronous transfers on a single GPU.
Graphics Samples
Path Sample Description
/2_Graphics/ bindlessTexture This example demonstrates use of cudaSurfaceObject, cudaTextureObject, and MipMap support in CUDA.
/2_Graphics/ Mandelbrot This sample uses CUDA to compute and display the Mandelbrot or Julia sets interactively. It also illustrates the use of "double single" arithmetic to improve
Mandelbrot precision when zooming a long way into the pattern.
/2_Graphics/ marchingCubes This sample extracts a geometric isosurface from a volume dataset using the marching cubes algorithm. It uses the scan (prefix sum) function from the Thrust
marchingCubes library to perform stream compaction.
/2_Graphics/simpleGL simpleGL Simple program which demonstrates interoperability between CUDA and OpenGL. The program modifies vertex positions with CUDA and uses OpenGL to render the
/2_Graphics/ simpleGLES Demonstrates data exchange between CUDA and OpenGL ES (aka Graphics interop). The program modifies vertex positions with CUDA and uses OpenGL ES to render
simpleGLES the geometry.
/2_Graphics/ simpleGLES_EGLOutput Demonstrates data exchange between CUDA and OpenGL ES (aka Graphics interop). The program modifies vertex positions with CUDA and uses OpenGL ES to render
simpleGLES_EGLOutput the geometry, and shows how to render directly to the display using the EGLOutput mechanism and the DRM library.
/2_Graphics/ simpleTexture3D Simple example that demonstrates use of 3D Textures in CUDA.
/2_Graphics/ volumeFiltering This sample demonstrates 3D Volumetric Filtering using 3D Textures and 3D Surface Writes.
/2_Graphics/ volumeRender This sample demonstrates basic volume rendering using 3D Textures.
Imaging Samples
Path Sample Description
/3_Imaging/bicubicTexture bicubicTexture This sample demonstrates how to efficiently implement a Bicubic B-spline interpolation filter with CUDA texture.
/3_Imaging/ bilateralFilter Bilateral filter is an edge-preserving non-linear smoothing filter that is implemented with CUDA with OpenGL rendering. It can be used in image
bilateralFilter recovery and denoising. Each pixel is weight by considering both the spatial distance and color distance between its neighbors.
/3_Imaging/boxFilter boxFilter Fast image box filter using CUDA with OpenGL rendering.
/3_Imaging/ convolutionFFT2D This sample demonstrates how 2D convolutions with very large kernel sizes can be efficiently implemented using FFT transformations.
/3_Imaging/ convolutionSeparable This sample implements a separable convolution filter of a 2D signal with a gaussian kernel.
/3_Imaging/ convolutionTexture Texture-based implementation of a separable 2D convolution with a gaussian kernel.
/3_Imaging/dct8x8 dct8x8 This sample demonstrates how Discrete Cosine Transform (DCT) for blocks of 8 by 8 pixels can be performed using CUDA: a naive implementation by
definition and a more traditional approach used in many libraries.
/3_Imaging/dwtHaar1D dwtHaar1D Discrete Haar wavelet decomposition for 1D signals with a length which is a power of 2.
/3_Imaging/dxtc dxtc High-Quality DXT Compression using CUDA. This example shows how to implement an existing computationally-intensive CPU compression algorithm in
parallel on the GPU, and obtain an order of magnitude performance improvement.
/3_Imaging/ EGLStream_CUDA_CrossGPU Demonstrates CUDA and EGL Streams interop, where consumer's EGL Stream is on one GPU and producer's on other and both consumer-producer are
EGLStream_CUDA_CrossGPU different processes.
/3_Imaging/ EGLStreams_CUDA_Interop Demonstrates data exchange between CUDA and EGL Streams.
/3_Imaging/ EGLSync_CUDAEvent_Interop Demonstrates interoperability between CUDA Event and EGL Sync/EGL Image using which one can achieve synchronization on GPU itself for GL-EGL-CUDA
EGLSync_CUDAEvent_Interop operations instead of blocking CPU for synchronization.
/3_Imaging/histogram histogram This sample demonstrates the efficient implementation of 64-bin and 256-bin histograms.
/3_Imaging/HSOpticalFlow HSOpticalFlow Variational optical flow estimation example. Uses textures for image operations. Shows how a simple PDE solver can be accelerated with CUDA.
/3_Imaging/imageDenoising imageDenoising This sample demonstrates two adaptive image denoising techniques: KNN and NLM, based on the computation of both geometric and color distance
between texels.
/3_Imaging/postProcessGL postProcessGL This sample shows how to post-process an image rendered in OpenGL using CUDA.
/3_Imaging/ recursiveGaussian This sample implements a Gaussian blur using Deriche's recursive method.
/3_Imaging/simpleCUDA2GL simpleCUDA2GL This sample shows how to copy a CUDA images back to OpenGL using the most efficient methods.
/3_Imaging/SobelFilter SobelFilter This sample implements the Sobel edge detection filter for 8-bit monochrome images.
/3_Imaging/ stereoDisparity A CUDA program that demonstrates how to compute a stereo disparity map using SIMD SAD (Sum of Absolute Difference) intrinsics.
Finance Samples
Path Sample Description
/4_Finance/binomialOptions binomialOptions This sample evaluates fair call price for a given set of European options under the binomial model.
/4_Finance/BlackScholes BlackScholes This sample evaluates fair call and put prices for a given set of European options by Black-Scholes formula.
/4_Finance/ MonteCarloMultiGPU This sample evaluates fair call price for a given set of European options using the Monte Carlo approach, taking advantage of all CUDA-capable GPUs
MonteCarloMultiGPU installed in the system.
/4_Finance/ quasirandomGenerator This sample implements Niederreiter Quasirandom Sequence Generator and Inverse Cumulative Normal Distribution functions for the generation of Standard
quasirandomGenerator Normal Distributions.
/4_Finance/SobolQRNG SobolQRNG This sample implements Sobol Quasirandom Sequence Generator.
Simulations Samples
Path Sample Description
/5_Simulations/fluidsGL fluidsGL An example of fluid simulation using CUDA and CUFFT, with OpenGL rendering.
/5_Simulations/ fluidsGLES An example of fluid simulation using CUDA and CUFFT, with OpenGLES rendering.
/5_Simulations/nbody nbody This sample demonstrates the efficient all-pairs simulation of a gravitational n-body simulation in CUDA.
/5_Simulations/ nbody_opengles This sample demonstrates the efficient all-pairs simulation of a gravitational n-body simulation in CUDA. Unlike the OpenGL nbody sample, there is no user
nbody_opengles interaction.
/5_Simulations/oceanFFT oceanFFT This sample simulates an Ocean height field using CUFFT Library and renders the result using OpenGL.
/5_Simulations/ particles This sample uses CUDA to simulate and visualize a large set of particles and their physical interaction. Adding "-particles=<N>" to the command line will allow
particles users to set # of particles for simulation.
/5_Simulations/ smokeParticles Smoke simulation with volumetric shadows using half-angle slicing technique.
Advanced Samples
Path Sample Description
/6_Advanced/ alignedTypes A simple test, showing huge access speed gap between aligned and misaligned structures.
/6_Advanced/ cdpAdvancedQuicksort This sample demonstrates an advanced quicksort implemented using CUDA Dynamic Parallelism.
/6_Advanced/ cdpBezierTessellation This sample demonstrates bezier tessellation of lines implemented using CUDA Dynamic Parallelism.
/6_Advanced/cdpQuadtree cdpQuadtree This sample demonstrates Quad Trees implemented using CUDA Dynamic Parallelism.
/6_Advanced/ concurrentKernels This sample demonstrates the use of CUDA streams for concurrent execution of several kernels on devices of computing capability 2.0 or higher. Devices
concurrentKernels of computing capability 1.x will run the kernels sequentially.
/6_Advanced/eigenvalues eigenvalues This sample demonstrates a parallel implementation of a bisection algorithm for the computation of all eigenvalues of a tridiagonal symmetric matrix of
arbitrary size with CUDA.
/6_Advanced/ fastWalshTransform Naturally(Hadamard)-ordered Fast Walsh Transform for batching vectors of arbitrary eligible lengths that are the power of two in size.
/6_Advanced/FDTD3d FDTD3d This sample applies a finite differences time domain progression stencil on a 3D surface.
/6_Advanced/ FunctionPointers This sample illustrates how to use function pointers and implements the Sobel Edge Detection filter for 8-bit monochrome images.
/6_Advanced/interval interval Interval arithmetic operators example.
/6_Advanced/lineOfSight lineOfSight This sample is an implementation of a simple line-of-sight algorithm: Given a height map and a ray originating at some observation point, it computes
all the points along the ray that are visible from the observation point.
/6_Advanced/ matrixMulDynlinkJIT This sample revisits matrix multiplication using the CUDA driver API. It demonstrates how to link to CUDA driver at runtime and how to use JIT
matrixMulDynlinkJIT (just-in-time) compilation from PTX code.
/6_Advanced/mergeSort mergeSort This sample implements a merge sort (also known as Batcher's sort), algorithms belonging to the class of sorting networks.
/6_Advanced/newdelete newdelete This sample demonstrates dynamic global memory allocation through device C++ new and delete operators and virtual function declarations available with
CUDA 4.0.
/6_Advanced/ptxjit ptxjit This sample uses the Driver API to just-in-time compile (JIT) a Kernel from PTX code. Additionally, this sample demonstrates the seamless
interoperability capability of the CUDA Runtime and CUDA Driver API calls.
/6_Advanced/ radixSortThrust This sample demonstrates a very fast and efficient parallel radix sort that uses the Thrust library. The included RadixSort class can sort either
radixSortThrust key-value pairs (with a float or unsigned integer keys) or keys only.
/6_Advanced/reduction reduction A parallel sum reduction that computes the sum of a large array of values.
/6_Advanced/scalarProd scalarProd This sample calculates scalar products of a given set of input vector pairs.
/6_Advanced/scan scan This example demonstrates an efficient CUDA implementation of parallel prefix sum, also known as "scan". Given an array of numbers, scan computes a new
array in which each element is the sum of all the elements before it in the input array.
/6_Advanced/ segmentationTreeThrust This sample demonstrates an approach to the image segmentation trees construction. This method is based on Boruvka's MST algorithm.
/6_Advanced/shfl_scan shfl_scan This example demonstrates how to use the shuffle intrinsic __shfl_up to perform a scan operation across a thread block.
/6_Advanced/ simpleHyperQ This sample demonstrates the use of CUDA streams for concurrent execution of several kernels on devices that provide HyperQ (SM 3.5). Devices without
simpleHyperQ HyperQ (SM 2.0 and SM 3.0) will run a maximum of two kernels concurrently.
/6_Advanced/ sortingNetworks This sample implements bitonic sort and odd-even merge sort (also known as Batcher's sort), algorithms belonging to the class of sorting networks.
sortingNetworks While generally subefficient, for large sequences compared to algorithms with better asymptotic algorithmic complexity (i.e. merge sort or radix sort).
/6_Advanced/ threadFenceReduction This sample shows how to perform a reduction operation on an array of values using the thread Fence intrinsic to produce a single value in a single
threadFenceReduction kernel.
/6_Advanced/ threadMigration Simple program illustrating how to the CUDA Context Management API and uses the new CUDA 4.0 parameter passing and CUDA launch API. CUDA contexts can
threadMigration be created separately and attached independently to different threads.
/6_Advanced/transpose transpose This sample demonstrates Matrix Transpose.
/6_Advanced/ warpAggregatedAtomicsCG This sample demonstrates how using Cooperative Groups (CG) to perform warp aggregated atomics, a useful technique to improve performance when many
warpAggregatedAtomicsCG threads atomically add to a single counter.
CUDALibraries Samples
Path Sample Description
/7_CUDALibraries/ batchCUBLAS A CUDA Sample that demonstrates how using batched CUBLAS API calls to improve overall performance.
/7_CUDALibraries/BiCGStab BiCGStab A CUDA Sample that demonstrates Bi-Conjugate Gradient Stabilized (BiCGStab) iterative method for nonsymmetric and symmetric positive definite
(s.p.d.) linear systems using CUSPARSE and CUBLAS.
/7_CUDALibraries/ An NPP CUDA Sample that demonstrates using nppiLabelMarkers to generate connected region segment labels in an 8-bit grayscale image then
boundSegmentsNPP boundSegmentsNPP compressing the sparse list of generated labels into the minimum number of uniquely labeled regions in the image using
nppiCompressMarkerLabels. Finally, a boundary is added surrounding each segmented region in the image using nppiBoundSegments.
/7_CUDALibraries/ boxFilterNPP A NPP CUDA Sample that demonstrates how to use NPP FilterBox function to perform a Box Filter.
/7_CUDALibraries/ cannyEdgeDetectorNPP An NPP CUDA Sample that demonstrates the recommended parameters to use with the nppiFilterCannyBorder_8u_C1R Canny Edge Detection image filter
cannyEdgeDetectorNPP function.
/7_CUDALibraries/ conjugateGradient This sample implements a conjugate gradient solver on GPU using CUBLAS and CUSPARSE library.
/7_CUDALibraries/ cuSolverDn_LinearSolver A CUDA Sample that demonstrates cuSolverDN's LU, QR, and Cholesky factorization.
/7_CUDALibraries/cuSolverRf cuSolverRf A CUDA Sample that demonstrates cuSolver's refactorization library - CUSOLVERRF.
/7_CUDALibraries/ cuSolverSp_LinearSolver A CUDA Sample that demonstrates cuSolverSP's LU, QR, and Cholesky factorization.
/7_CUDALibraries/ cuSolverSp_LowlevelCholesky A CUDA Sample that demonstrates Cholesky factorization using cuSolverSP's low-level APIs.
/7_CUDALibraries/ cuSolverSp_LowlevelQR A CUDA Sample that demonstrates QR factorization using cuSolverSP's low-level APIs.
/7_CUDALibraries/ This NPP CUDA Sample demonstrates how any border version of an NPP filtering function can be used in the most common mode (with border control
FilterBorderControlNPP FilterBorderControlNPP enabled), can be used to duplicate the results of the equivalent non-border version of the NPP function, and can be used to enable and disable
border control on various source image edges depending on what portion of the source image is being used as input.
/7_CUDALibraries/ freeImageInteropNPP A simple CUDA Sample demonstrate how to use FreeImage library with NPP.
/7_CUDALibraries/ histEqualizationNPP This CUDA Sample demonstrates how to use NPP for histogram equalization for image data.
/7_CUDALibraries/jpegNPP jpegNPP This sample demonstrates a simple image processing pipeline. First, a JPEG file is Huffman decoded and inverse DCT transformed and dequantized.
Then the different plances are resized. Finally, the resized image is quantized, forward DCT transformed and Huffman encoded.
/7_CUDALibraries/ MC_EstimatePiInlineP This sample uses Monte Carlo simulation for Estimation of Pi (using inline PRNG). This sample also uses the NVIDIA CURAND library.
/7_CUDALibraries/ MC_EstimatePiInlineQ This sample uses Monte Carlo simulation for Estimation of Pi (using inline QRNG). This sample also uses the NVIDIA CURAND library.
/7_CUDALibraries/ MC_EstimatePiP This sample uses Monte Carlo simulation for Estimation of Pi (using batch PRNG). This sample also uses the NVIDIA CURAND library.
/7_CUDALibraries/ MC_EstimatePiQ This sample uses Monte Carlo simulation for Estimation of Pi (using batch QRNG). This sample also uses the NVIDIA CURAND library.
/7_CUDALibraries/ MC_SingleAsianOptionP This sample uses Monte Carlo to simulate Single Asian Options using the NVIDIA CURAND library.
/7_CUDALibraries/ MersenneTwisterGP11213 This sample demonstrates the Mersenne Twister random number generator GP11213 in cuRAND.
/7_CUDALibraries/randomFog randomFog This sample illustrates pseudo- and quasi- random numbers produced by CURAND.
/7_CUDALibraries/ simpleCUBLAS Example of using CUBLAS using the new CUBLAS API interface available in CUDA 4.0.
/7_CUDALibraries/ simpleCUBLASXT Example of using CUBLAS-XT library.
/7_CUDALibraries/ simpleCUFFT Example of using CUFFT. In this example, CUFFT is used to compute the 1D-convolution of some signal with some filter by transforming both into
simpleCUFFT the frequency domain, multiplying them together, and transforming the signal back to the time domain.
/7_CUDALibraries/ simpleCUFFT_2d_MGPU Example of using CUFFT. In this example, CUFFT is used to compute the 2D-convolution of some signal with some filter by transforming both into
simpleCUFFT_2d_MGPU the frequency domain, multiplying them together, and transforming the signal back to the time domain on Multiple GPU.
/7_CUDALibraries/ simpleCUFFT_MGPU Example of using CUFFT. In this example, CUFFT is used to compute the 1D-convolution of some signal with some filter by transforming both into
simpleCUFFT_MGPU the frequency domain, multiplying them together, and transforming the signal back to the time domain on Multiple GPU.
For more information about CUDA, go to: Xavier/JetPack_4.1/Components/Cuda | {"url":"https://developer.ridgerun.com/wiki/index.php/Xavier/Processors/GPU/CUDA","timestamp":"2024-11-03T03:34:25Z","content_type":"text/html","content_length":"91677","record_id":"<urn:uuid:05f30efb-cfcd-4bb7-98e0-38c9e1b681e5>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00392.warc.gz"} |
Can someone provide guidance on quantum algorithms for solving problems in quantum health and medical applications for my computer science assignment? Computer Science Assignment and Homework Help By CS Experts
Can someone provide guidance on quantum algorithms for solving problems in quantum health and medical applications for my computer science assignment? I was just looking at the top of the most recent
guidebook on quantum algorithms for solving problem-solving problems. The search for a correct solution using quantum computers in the early 1990’s wasn’t out of the question and nobody seemed to
know where to look for it, so I’ll be asking for guidance. So let’s take a look at the previous pages, along with the rest of the list. Did I make a mistake? The problem in quantum physics is that
systems of particles can be predicted from the quantum state space Hilbert space. This state space contains all real-valued, even one dimensional states of particles — particles together with some of
the interactions that naturally constrain them. The problem is not that we can’t do that, but that there are so many more possibilities. But we all have entangled particles, so it makes sense for us
to conjecture that, for small constants, the quantum state space doesn’t contain any kind of entangled particle. We know by Quantum mechanics that particles can be separated from one another by a
small constant. There are some other interesting examples, of course, that we can’t just assume, and even we can’t prove. Many of those examples fail because there are also more entangled particles.
That is why it is really important to experiment with more entangled particles. What’s I have to say that while the quantum state space is a lot less hard to deal with than our physics computers, it
may be possible to figure out how to find the correct answer using the quantum bit and any number of numbers as inputs. This kind of search may start in 2003, when the information structure within
the quantum bits was quite close to the knowledge structure within the classical bit, and takes two-by-one time. (I don’t have time to type that in, but maybe that’s a hintCan someone provide
guidance on quantum algorithms for solving problems in quantum health and medical applications for my computer science assignment? To make these questions more specific, I think there’s some
importance for this class; I have seen the algorithms at the top of the list (see examples) and found that they look promising and depend on the nature of their algorithms. I also think there’s too
much information in there. To accomplish this, I’m going to start using the algorithms as we find value in algorithms and as I’m writing this assignment, I need to ensure that my computer science
computer science assignment is accurate and that the algorithms that I’ve already researched are the ones that the academic computer science classes are working on. 3) How does it work? One of the
methods that I’m inspired to use in the creation of this assignment is from the Stanford Physics lab; our computational capabilities are already better than anyone in the world but I often come up
with a number of mathematical expressions to determine what ought to be observed in a laboratory setting. We’re going to first try to build our computational algorithms to produce confidence ratings;
they’re pretty easy to measure and work well, but we’ll not be teaching any new concepts or writing the first paper on how to use them. The goal is to get the confidence ratings from the experiment
and use them a bit and then, as we continue to work out the algorithm, we’ll see what the weights are and what the probabilities are. There will be periods where a one-to-one resemblance between the
experimental results is useful, if not required since the properties of the statistical properties of these words are usually difficult to deduce and know.
You Can’t Cheat With Online Classes
The test functions used in these equations are either very good (very slow) or excellent (very fast) but they’re weak compared to other commonly used weighted indicators by virtue of the fact that,
although our code is correct, there are differences in weights that make their calculation even harder. The algorithmCan someone provide guidance on quantum algorithms for solving problems in quantum
health and medical applications for go to this website computer science assignment? I am an undergraduate student in quantum mechanics and mathematics at the Massachusetts Institute of Technology
(MIT). What I am learning from physics at MIT is that although quantum mechanics approaches are far from clear from top to bottom, that the highest attainable quantum level is yet to be achieved. In
this scenario, we achieve quantum-level quantum-level accuracy and gain some new insights into the consequences of this level. Abstract In this article, I describe my postdoctoral work and my
approach of trying to develop a mathematical method for programming questions using mathematical facts and techniques. Thus my postdoctoral work closely employs quantum physics concepts and the
method of ‘entering the source’, rather than quantum mechanics. The method can be read as a practical argument for quantum algorithms using quantum physics concepts, as shown in Figure 1. 1The idea
of postdoctoral work is to teach my students new basic principles of quantum mechanics. That is, some of the basic principles of quantum mechanics are present for the first time on their computers.
Figure 2 displays some implementation of quantum physics principles in four different virtual platforms. Users, starting with novice students, might be able to use a computer ‘partly hidden’ in the
virtual platform. By ‘inputting’ the virtual platform using a different programming language or using quantum physics concepts for inputting, understanding the basics of physics, the rules of
calculus and integrals, the concept of a quantum circuit, and a computer programmatic approach, we can quickly confirm and compare values obtained by different ‘part of hidden’ virtual platforms, and
gain some new insights into how physical measurements can access information about the true state of the classical system. We can also use it for experiment itself, if you happen to be one such
person. 2In ‘inputting’ the virtual platform, students are given the physical states of the system; specifically, each element of the quantum-machined system can be inputted into | {"url":"https://csmonsters.com/can-someone-provide-guidance-on-quantum-algorithms-for-solving-problems-in-quantum-health-and-medical-applications-for-my-computer-science-assignment","timestamp":"2024-11-10T02:02:26Z","content_type":"text/html","content_length":"86183","record_id":"<urn:uuid:dd7ecb99-3f88-420d-b1ce-78eedc4c80f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00435.warc.gz"} |