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Algebraic methods in the congested clique
In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multipliacation
implementations to the congested clique, obtaining an O(n1-2/ω) round matrix multiplication algorithm, where ω < 2.3728639 is the exponent of matrix multiplication. In conjunction with known
techniques from centralised algorith- mics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: Triangle and 4-cycle
counting in O(n0.158) rounds, imaproving upon the O(n1/3) triangle counting algorithm of Dolev et al. [DISC 2012], a (1 + o(1))-approximation of all-pairs shortest paths in O(n0.158) rounds,
improving upon the Õ(n1/2)-round (2+o(1))-approximation algorithm of Nanongkai [STOC 2014], and computing the girth in O(n0.158) rounds, which is the first non-trivial solution in this model. In
addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.
Original language English
Title of host publication PODC 2015 - Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing
Pages 143-152
Number of pages 10
ISBN (Electronic) 9781450336178
State Published - 21 Jul 2015
Event ACM Symposium on Principles of Distributed Computing, PODC 2015 - Donostia-San Sebastian, Spain
Duration: 21 Jul 2015 → 23 Jul 2015
Publication series
Name Proceedings of the Annual ACM Symposium on Principles of Distributed Computing
Volume 2015-July
Conference ACM Symposium on Principles of Distributed Computing, PODC 2015
Country/Territory Spain
City Donostia-San Sebastian
Period 21/07/15 → 23/07/15
• Congested clique model
• Distance compuatation
• Distributed computing
• Lower bounds
• Matrix multiplication
• Subgraph detection
All Science Journal Classification (ASJC) codes
• Software
• Hardware and Architecture
• Computer Networks and Communications
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How to Cite This Entry:
Turbulence, mathematical problems in. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Turbulence,_mathematical_problems_in&oldid=49046
This article was adapted from an original article by A.S. Monin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
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E. Example Intersections
E. Example Intersections
Two different intersection types are solved using triangle- and arc-based methods to demonstrate the different computation process. Additional decimal places will be carried in computations to
minimize rounding errors.
1. Distance-distance
Given the information on the diagram, determine the coordinates of point 101.
Figure E-1
Distance-distance example
Step (1) For both methods is to inverse along the base line 30-20
a. Triangle-based method
Step (2) Compute angle at 30 by Law of Cosines.
Step (3) Compute direction from 30 to 101.
Step (4) Perform a forward computation from 30 to 101.
Math Check: Compute coordinates from 20.
Step (1) Compute angle at 20 by Law of Sines.
Step (2) Compute direction from 20 to 101.
Step (3) Perform a forward computation from 20 to 101.
Both coordinates check.
b. Arc-based method
Step (2) Set up and solve Equations D-6 through D-9.
Step (3) Use Equations D-10 and D-11 to compute the two intersection points
Step (4) Of the two, select the appropriate intersection point.
Point 101 is located southwest of the base line.
Point North East From base line
101[1] 5077.015 1363.991 north east
101[2] 4574.617 1085.956 south west
The correct intersection point is 101[2]: (4754.617 ft N, 1085.956 ft E), same as the triangle-based solution. | {"url":"https://jerrymahun.com/index.php/home/open-access/12-iv-cogo/27-cogo-chap-e","timestamp":"2024-11-09T09:12:03Z","content_type":"application/xhtml+xml","content_length":"20347","record_id":"<urn:uuid:515cd737-1ae2-493b-aa3d-059c6aa987b8>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00732.warc.gz"} |
Mastering Formulas In Excel: How Can You Force A Certain Order Of Oper
Understanding the order of operations in Excel formulas is crucial for accurate and efficient data analysis and manipulation. Without a clear understanding of how Excel processes calculations, it's
easy to make mistakes that can lead to incorrect results. In this blog post, we will delve into the topic of forcing a certain order of operations in Excel formulas, and explore techniques to ensure
that your formulas are executed in the desired sequence.
Key Takeaways
• Understanding the order of operations in Excel formulas is crucial for accurate and efficient data analysis and manipulation.
• Using parentheses can help override the default order of operations and ensure formulas are executed in the desired sequence.
• Specific functions in Excel can also be used to control the order of operations in formulas.
• Breaking down complex formulas into smaller parts using cell references can simplify the order of operations and make formulas easier to manage.
• Practicing and documenting complex formulas can help maintain clarity and avoid confusion when working with Excel formulas.
Understanding Order of Operations in Excel
When working with formulas in Excel, it is important to understand the default order of operations. This knowledge can help ensure that your formulas produce the intended results and can prevent
errors or miscalculations. In this chapter, we will explore the default order of operations in Excel formulas and the importance of understanding how Excel processes formulas.
A. Explanation of the default order of operations in Excel formulas
• 1. Parentheses
In Excel, parentheses are used to override the default order of operations. Any operations within parentheses are calculated first, followed by the rest of the formula.
• 2. Exponents
Exponents, or calculations involving powers and roots, are performed next in the default order of operations.
• 3. Multiplication and division
After parentheses and exponents, multiplication and division are carried out in the order they appear from left to right in the formula.
• 4. Addition and subtraction
The final step in the default order of operations is addition and subtraction, which are also performed in the order they appear from left to right in the formula.
B. Importance of understanding how Excel processes formulas
• 1. Accuracy of calculations
Understanding the default order of operations in Excel can help ensure the accuracy of calculations in your formulas. By arranging the operations in the correct order, you can prevent errors and
obtain the desired results.
• 2. Avoiding unintended outcomes
Without an understanding of the default order of operations, formulas may produce unintended outcomes or miscalculations. This can lead to incorrect data analysis and decision-making based on
flawed results.
• 3. Efficiency in formula writing
Knowing how Excel processes formulas can also make your formula writing more efficient. By strategically using parentheses and understanding the order of operations, you can create concise and
effective formulas.
Using Parentheses to Force Order of Operations
When working with complex formulas in Excel, it's important to understand how to control the order in which operations are performed. By default, Excel follows the order of operations (PEMDAS -
Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), but using parentheses can override this default order and ensure the desired operations are performed first.
Explanation of how parentheses can be used to override default order
By enclosing certain parts of a formula in parentheses, you can force Excel to evaluate those parts first before proceeding with the rest of the formula. This can be particularly useful when dealing
with nested functions or complex mathematical operations.
Examples of how to use parentheses in formulas
For example, consider the following formula: =A1 + B1 * C1. By default, Excel would perform the multiplication first and then the addition. However, if you wanted to add A1 and B1 first before
multiplying the result by C1, you would use parentheses as follows: =(A1 + B1) * C1.
Another example would be when using nested functions. Let's say you have a formula that includes the AVERAGE function as well as multiplication. You can use parentheses to ensure that the AVERAGE
function is evaluated first before the result is multiplied by another value.
• Original formula: =AVERAGE(A1:A5) * 10
• Updated formula with parentheses: =(AVERAGE(A1:A5)) * 10
Utilizing Functions to Control Order of Operations
When working with complex formulas in Excel, it is crucial to ensure that the order of operations is followed correctly. Thankfully, Excel provides a variety of functions that can help control the
order of operations and ensure that your formulas produce the desired results.
A. Explanation of specific functions that can help control order of operations
1. ROUND Function
The ROUND function is commonly used to round a number to a specified number of decimal places. However, it can also be used to control the order of operations by rounding intermediate calculations to
a certain precision before using the result in subsequent calculations.
2. IF Function
The IF function allows you to perform different actions based on a specified condition. By using the IF function within a formula, you can control the order of operations by executing certain
calculations only when specific conditions are met.
3. MOD Function
The MOD function returns the remainder of a division operation. This function can be utilized to control the order of operations by manipulating the division remainder to achieve the desired result
in a formula.
B. Demonstrating the use of functions in Excel formulas
Let's consider an example to demonstrate how these functions can be used to control the order of operations in Excel formulas. Suppose we have a formula that involves multiple calculations, and we
want to ensure that certain operations are performed before others.
• First, we can use the ROUND function to round off intermediate results to a specific number of decimal places.
• Next, we can employ the IF function to execute different calculations based on specified conditions.
• Finally, we can utilize the MOD function to manipulate division remainders and control the flow of operations in the formula.
By incorporating these functions into our Excel formulas, we can effectively control the order of operations and ensure that our calculations are carried out accurately and efficiently.
Using Cell References to Simplify Formulas
When working with complex formulas in Excel, it can be challenging to ensure that the order of operations is executed correctly. However, breaking down these complex formulas into smaller parts using
cell references can help control the order of operations and simplify the overall formula.
Explanation of how breaking down complex formulas into smaller parts using cell references can help control order of operations
By using cell references in Excel formulas, you can break down complex calculations into smaller, more manageable parts. This allows you to clearly define the order of operations and ensures that
each part of the formula is executed in the correct sequence. Additionally, using cell references can make it easier to troubleshoot and debug formulas, as you can easily identify and isolate
specific parts of the calculation.
Example of using cell references to simplify a formula
Let's say you have a complex formula that calculates the total cost of an order, taking into account the quantity, unit price, and tax. Instead of writing the entire calculation in one cell, you can
use cell references to break it down into smaller parts.
• Cell A1: Enter the quantity of the order
• Cell A2: Enter the unit price of the item
• Cell A3: Enter the tax rate
• Cell A4: In this cell, you can use the cell references to calculate the subtotal of the order: =A1*A2
• Cell A5: In this cell, you can use the cell reference to calculate the tax amount: =A4*A3
• Cell A6: Finally, in this cell, you can use the cell references to calculate the total cost of the order: =A4+A5
By breaking down the formula into smaller parts using cell references, you can clearly define the order of operations and simplify the overall calculation. This not only makes the formula easier to
understand and maintain, but also reduces the likelihood of errors.
Tips for structuring formulas to make the order of operations clear
When working with complex formulas in Excel, it is crucial to maintain a clear order of operations to ensure accurate results. Here are some best practices for structuring formulas:
• Use parentheses: One of the most effective ways to force a certain order of operations in a formula is to use parentheses. By enclosing specific parts of the formula in parentheses, you can
clearly indicate the order in which calculations should be performed.
• Break down complex formulas: If a formula is getting too complex, consider breaking it down into smaller, more manageable parts. This not only makes it easier to understand and troubleshoot, but
also helps maintain the order of operations.
• Use named ranges: Instead of directly referencing cells in a formula, consider using named ranges to represent those cells. This not only makes the formula more readable, but also reduces the
chances of errors in the order of operations.
Suggestions for documenting complex formulas to avoid confusion
Complex formulas can quickly become difficult to understand, especially if they are not well-documented. Here are some suggestions for documenting complex formulas:
• Add comments: Use Excel's comment feature to provide explanations for different parts of the formula. This can help others understand the purpose and order of operations within the formula.
• Create a formula legend: Consider creating a separate worksheet or section within the workbook to document all the formulas used. This can serve as a reference point for anyone working with the
• Provide a guide: If the workbook is meant to be used by multiple people, consider creating a guide or manual that explains the structure and order of operations for the formulas used.
Understanding and controlling the order of operations in Excel formulas is crucial for accurately achieving desired results. By utilizing parentheses and understanding the hierarchy of mathematical
operations, users can ensure that their formulas calculate the intended values.
I encourage readers to practice and further explore the topic of mastering formulas in Excel. The more familiar you become with the order of operations, the more confident and efficient you will be
in using Excel to perform complex calculations and analysis.
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Return on Equity
What is the Return on Equity?
The Return on Equity (RoE) is a measure of the profitability of a business concerning the funds by its stockholders/shareholders. ROE is a metric used generally to determine how well the company
utilizes its funds provided by the equity shareholders.
For example, if a company has a RoE of $1, it means $1 of the investor’s funds generates a net income of $1. This metric helps the investor know how efficiently the company is using its funds.
For Investors, the higher the RoE the better, however, it is important to read the RoE in light of a lot of factors to avoid wrong decisions. In this write up we will get a better understanding of
the formula and factors that affect the RoE.
The Return on Equity (RoE) is also known as Return on Capital Employed (ROCE) or Return on Invested Capital (ROIC).
Another concept used to measure profitability is RoA or Return on average assets of the Company. It shows how well the assets of the company are earning for the Company.
The Formula for Return on Equity
Return on Equity is calculated by dividing Net income in the Income statement by average shareholders’ equity in the Balance Sheet.
$\text{Return on Equity}=\frac{\text{Net Income}}{\text{Average Sharholders" Equity}}$
• Net income is the amount of income, net of expense, and taxes that a company generates for a given period.
$\text{Net Income}=\text{Revenue}-\text{Expenses}-\text{Taxes}$
• Shareholder’s Equity or Shareholders funds = Total Assets Less Total Liabilities or Net-worth or Total Equity capital + Retained earnings
(Retained Earnings-Profits earned by the company post the dividend and other pay-outs over the years.)
While calculating the Shareholder’s equity we need to consider the arithmetic mean of two period ends to arrive at an average number.
Typically, the average is arrived at by dividing the shareholders' equity for the periods involved and dividing it by 2.
$\text{Average Shareholders" Equity}=\frac{\text{Shareholders' Equity at Beginning}+\text{Shareholders' Equity at End}}{2}$
Calculate RoE of a company whose financials are as under
• Net profit for the year 2020 is $1,000
• Shareholder’s equity is $ 5,000 as of 31^st Dec 2019
• Shareholder’s equity is $ 6,000 as of 31^st Dec 2020
The calculation is as under
First, we calculate the average shareholders’ equity:
$\begin{array}{c}\text{Average Shareholders" Equity}=\frac{{\text{Shareholders' Equity at 31}}^{\text{st}}\text{Dec 2019}+{\text{Shareholders' Equity at 31}}^{\text{st}}\text{Dec 2020}}{2}\\ =\frac
{5,000+6,000}{2}\\ =5,500\end{array}$
After that, we compute RoE:
$\begin{array}{c}\text{Return on Equity}=\frac{\text{Net Income}}{\text{Average Sharholders" Equity}}\\ =\frac{1,000}{5,500}\\ =18.18%\end{array}$
Hence, the RoE is 18.18%
The reason average shareholders' equity is used is that this figure keeps fluctuating during the accounting period in question. Hence to arrive at a middle number that would represent the correct
picture of the Shareholder’s equity for a period we use the arithmetic mean formula.
Factors Affecting RoE
Although RoE is used as a measure of effective utilization of equity some factors affect the RoE and while making investment decisions one is required to factor in these to determine the
effectiveness of this measure.
Following are the factors to be considered while understanding the RoE:
• Dividend Pay-out: When a company pays dividends to its shareholders out of its profits its the net income reduces. Thus, the numerator reduces accordingly dropping the RoE.
• When Company buys back shares: The growth rate will be lower if earnings are used to buy back shares.
• If the Company is financed by debt more than equity. It is important to see the Debt-to-Equity ratio.
• A company having negative earnings.
Accordingly, while comparing the RoE of two Companies it is important to know the above factors before making any decisions.
• To compare a company’s performance against the industry benchmark.
• To compare the performance against the Company’s peers.
• ROE is also a factor in stock valuation, in association with other financial ratios.
• This can be used as a benchmark to pick stocks within the same sector only. Across sectors, profit and income levels vary significantly.
Reasons for Variations in RoEs of the Company/Limitations involved in the use of the Formula
Huge variation in year-on-year profits or Company capital structure not having sustainable growth rate
If a company is consistently showing losses for several years, the retained earnings show a negative number which reduces the shareholder’s equity thus reducing the denominator of our RoE formula. In
case the company makes huge profits in a year it will show very good RoE for that year, although it isn’t the correct picture of the Company’s finances.
Higher Debt
If by the company’s financial ratios, its business using more borrowings than equity then the Company will always show a higher RoE as compared to companies with lower borrowings.
Negative retained earnings or Negative Net Income
RoE should never be calculated in such a scenario since it gives a very wrong picture of the Company’s finances. If a Company is showing very high RoE or very low RoE the finances of the Company
could be unstable.
Common Mistakes
• Not considering the taxes and bad debt provision while calculating the net income
• Not considering the retained earnings while calculating shareholder’s equity
• Not annualizing the formula while calculating the quarterly or monthly RoE
• Comparing two Company’s RoE without understanding whether it is funded by borrowings or equity
• Comparing two Company’s RoE without understanding whether it paid a dividend or if it has varying profits each year
Context and Application
This topic is significant in the professional exams for both undergraduate and graduate courses especially following:
• B. Com Banking and Finance
• Chartered Accountancy: Financial Management
• CIMA (Management Accounting)
• MBA
• CFP
• CFA
• CPA
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What Is The Main Focus Of Mathematics Education? | Objectives Of Teaching MathematicsWhat Is The Main Focus Of Mathematics Education? | Objectives Of Teaching Mathematics
Write the instructional Objectives for a mathematics lesson.
Some of the common objectives of teaching mathematics are:
1. Teaching fundamental numeracy skills to all students.
2. Teaching heuristics and other problem-solving strategies to solve non-routine problems.
3. Teaching selected areas of mathematics such as:
□ Teaching Euclidean geometry as an example of an 'axiomatic system' and a 'model of deductive reasoning'.
□ Teaching Calculus is an example of the 'intellectual achievements' of the modern world.
4. Teaching abstract mathematical concepts at an early age such as
5. Teaching mathematics to students for future livelihood. For instance,
□ Teaching 'practical mathematics' to prepare students to follow a trade or craft.
□ Teaching 'advanced mathematics' to students who wish to follow a career in fields like
☆ Engineering,
☆ Science and Technology,
☆ Mathematics etc.
What Is The Main Focus Of Mathematics Education? Notes
Some Specific Objectives, Purpose Of Teaching Mathematics Notes For B.Ed In English Medium
What Is Mathematics In Education - What Does Mathematics Focus On? Notes And Study Material, PDF, PPT, Assignment For B.Ed 1st and 2nd Year, DELED, M.Ed, CTET, TET, Entrance Exam, All Teaching Exam
Test Download Free For Pedagogy of Maths And Teaching of Mathematics Subject.
Important Questions For Exam:
Write about an objective of teaching maths?
Explain the objectives of teaching mathematics in schools.
Write the instructional Objectives for a mathematics lesson.
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Finite time singularities in a class of hydrodynamic models
Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form L~∫k^α\|v[k]\|^2dk in 3D Fourier representation, where α is a
constant, 0<α<1. Unlike the case α=0 (the usual Eulerian hydrodynamics), a finite value of α results in a finite energy for a singular, frozen-in vortex filament. This property allows us to study the
dynamics of such filaments without the necessity of a regularization procedure for short length scales. The linear analysis of small symmetrical deviations from a stationary solution is performed for
a pair of antiparallel vortex filaments and an analog of the Crow instability is found at small wave numbers. A local approximate Hamiltonian is obtained for the nonlinear long-scale dynamics of this
system. Self-similar solutions of the corresponding equations are found analytically. They describe the formation of a finite time singularity, with all length scales decreasing like (t^*-t)^1/(2-α),
where t^* is the singularity time.
Physical Review E
Pub Date:
May 2001
□ 47.15.Ki;
□ 47.32.Cc;
□ Physics - Fluid Dynamics
LaTeX, 17 pages, 3 eps figures. This version is close to the journal paper | {"url":"https://ui.adsabs.harvard.edu/abs/2001PhRvE..63e6306R","timestamp":"2024-11-08T21:35:18Z","content_type":"text/html","content_length":"39653","record_id":"<urn:uuid:f7fb5c0d-333d-4550-a263-7efe031fa2b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00313.warc.gz"} |
In the electromagnetic inverse diffraction problem, the first order Born approximation in the integral equation for the scattered field may be used. It gives a well-known Fourier transform relation
eliminating the nonlinearity between the scattered fields and the physical properties of objects by replacing the total field inside the integral by the incident field. In the last decade, this
linearizing approximation has been intensively studied in the medical imaging and nondistructive evaluation of materials since it offers numerically efficient and stable inversion algorithm. However,
it is limited in the weakly scattering object. In this dissertation, the limitation of the first order Born approximation is explicitly shown by using the reconstruction by projection, and an
improved Born inversion for dielectric profile reconstruction of dielectric cylinder is suggested. It improvess the validity of the first order Born approximation up to about ten times in the
multiplication of the size of the object and the square root of the relative dielectric constant minus one. One may introduce a projection function defined as the one dimensional inverse Fourier
transform of the scattered far fields measured for multi-frequency incident waves. Then the convolution back-projection scheme, well-known in the X-ray computerized tomography, may be used to obtain
the cross-sectional images or the quantitative distribution of the relative dielectric constants in the cross-section. The projection function obtained from the inverse Fourier transform is larger
than the real size since the propagation velocity of the electromagnetic wave inside the dielectric object is slow due to higher dielectric constant. The projection may be deformed from the
refraction and diffraction of the wave within the object. In the case of weak scattering objects, the slowness and deformation is sufficiently small and may be neglected. However, for the strong
scattering objects, this phenominon is pronounced and the projection function deviates from the original one, which limits the validity of the first order Born approximation. The projection obtained
through the inverse Fourier transform of the time-harmonic fields is basically equal to the range profile due to the irradiation of the impulse plane wave on the objects, since impulse is consisted
of infinitely many time-harmonic fields. Therefore, one may find the starting points of the object and the external boundary of the object by rotating the object. One may correct the extended
projection obtained from the direct inverse Fourier transform of the scattered fields by the measured external boundary. It is shown via this correction that the reconstructed image closely predicts
the original up to about 10 times strong scatterer than the ordinary Born inversion. Numerical examples and the X-band microwave measured reconstruction through this improved Born inversion are
presented to compared with the ordinary Born inversion. Furthermore, it is shown that the similar improvent may also be possible for the first order Rytov inversion that is compared with the first
order Born inversion. The actual inversion system such as the water-immersed imaging system and the subsurface radar is operated in the dissipated medium and the effects of the dissipativeness of the
background medium and scatterer are investigated by using the concept of the point spread function in the band-limited system. Bi-static scheme is also included in its Fourier transformation
formulation and its improved Born reconstruction is tested numerically. | {"url":"http://library.kaist.ac.kr/search/detail/view.do?bibCtrlNo=60512&flag=dissertation","timestamp":"2024-11-11T03:35:46Z","content_type":"text/html","content_length":"129007","record_id":"<urn:uuid:01e3f40e-6095-46e6-b0cf-da29790fca38>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00140.warc.gz"} |
Learning Task 3: Perform The Indicated Operations. Write Your Answer In Your Notebook. P... - I Wear The Trousers
Learning Task 3:
Perform the indicated operations. Write your answer in your notebook.
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Game theory
The banner ad is dead. Long live the advergame! America’s addiction to video and computer games is leading the way to a new advertising medium with astounding click-through rates, play times, and
peer-to-peer potential. What’s your high sco
Ethics is the branch of philosophy that deals with morality and how it shapes behavior. Different branches of the study of ethics look at where our views of morality come from and how they shape our
everyday lives. There are four major ethi
Game Theory Game theory is a branch of mathematics concerned with the analysis of the behavior of decision makers (called “players”) whose choices affect one another. game theory is usually talked
about in reference to decision making but we can also use it to talk about evolution and animal behavior but before we do that I want to take a moment to talk about how game theory is usually
discussed so when I think about game theory I generally think about it as it relates to the social sciences or maybe economics but we can also use it to talk about everyday View Academics in Game
theory and psychology on Academia.edu. Game theory is a branch of applied mathematics and economics that studies situations where players choose different actions in an attempt to maximize their
returns. First developed as a tool for learning reference-request social-psychology game-theory agent-based-modeling.
The prisoners’ dilemma, for instance, was conceived in 1950. At about this time the first experiments on the prisoners’ dilemma were conducted. Rapoport and Chammah (1965) report one of the Game
Theory Psychology :: Next Level. The three biggest areas of Game Theory Psychology that come into play in DFS are recency bias, roster construction errors, and the indelible urge to be contrarian
just for contrarians sake. If trust is what blockchain creates, then game theory is how blockchain creates it. Game theory, which in simple terms can be defined as the study of rewards and
punishments to influence behavior, is the machine that incentivizes honesty in industries that have none. Posted on February 27, 2021 February 27, 2021 Author admin Posted in Game Theory, Gamer
Psychology, Winning Strategies 1 Reply What The World Needs Now, Is Facts Sweet Facts The blessing of information science and big data 4 min read Photo by Mika Baumeister on Unsplash A fact is an
occurrence in the real world.
Classical non-cooperative game theory was conceived by John von Neumann to determine optimal strategies in competitions between adversaries. A contest involves players, all of whom have a choice
of moves. Games can be a single round or repetitive.
Game theory is concerned with predicting the outcome of games of strategy, in which the "players" (two or more businesses competing in a market) have incomplete information about the other's
If one prisoner turns in the other, he gets set free and the other prisoner goes to prison for 3 years. If they both betray each other, they both go to prison for 2 years.
to Von Neumann & Morgenstern's Theory of Games - A Macat Psychology Analysis by von Neumann and Morgenstern and the battle of the sexes game.
University of Pennsylvania - Citerat av 87 - Game theory - Microeconomic theory - Psychology and economics - Learning he researches digital media consumer psychology with a focus on
gamification design, Svahn, M. (2014) Persuasive Pervasive Games The Case of Impacting J. STENROS and A. WAERN, eds, Pervasive Games; Theory and Design. Essay sport day spm research theory Game
economics paper benefits of joining Graduate psychology research paper case study on consumer protection.
of strategic decision-making [28] and thus it seems that game theory might be an appropriate The interdisciplinary exchange between economists and psychologists has so far been more active and
fruitful in the modifications of Expected Utility Theory Köp Behavioral Game Theory av Colin F Camerer på Bokus.com. stands alone in blending experimental evidence and psychology in a mathematical
theory of Poker and More: Unique Ideas and Concepts: Strategy, Game Theory, and Psychology from Two Renowned Gambling Experts: Sklansky, David: Amazon.se: Colin Camerer, one of the field's leading
figures, uses psychological principles and hundreds of experiments to develop mathematical theories of reciprocity, Steadily growing applications of game theory in modern science (including
psychology, biology and economics) require sources to provide rapid access in both Love games: A game theory approach to compatibility. Journal article, 2015 Subject Categories. Mathematics.
Psychology av S Wide · 2020 — Keywords: game, play, social psychology, sociology, tertius, theory. Tertium datur!
Nakhorn Yuangkratoke / EyeEm / Getty Images Game theory is a theory of social interaction, which attempts to explain the Evolutionary game theory and public goods games offer an important framework
to understand 5 Jan 2017 “Here we use the same math that you can use to describe evolution in biology to describe human behavior and human psychology, building a This course connects the fields of
psychology, economics, and game theory to present transformation in explaining the behavior of economic agents from the Game theory is a mathematical approach to modeling behavior by analyzing the
strategic decisions made by interacting players.
Journal article, 2015 Subject Categories. Mathematics.
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Game theory is a branch of decision theory focusing on interactive decisions, applicable whenever the actions of two or more decision makers jointly determine an outcome that affects them all.
Strategic reasoning amounts to deciding how to act to achieve a desired
If neither betrays the other, they both go to prison for 1 year. The mathematical framework of psychological game theory is useful for describing many forms of motivation where preferences depend
directly on own or others' beliefs. It allows for incorporation of emotions, reciprocity, image concerns, and self-esteem in economic analysis. We explain how and why, discussing basic theory, a
variety of sentiments, experiments, and applied work. Game theory is divided between two branches: "non-cooperative" and "cooperative." These two sub-fields represent not only two different research
approaches but also, to a certain extent, different Game theory argues that cooperation between players is always the rational strategy, at least when participating in a game-theory experiment (even
if it means losing the game). Consider this scenario: You participate in what you are told is a one-shot game.
Game theory is a branch of applied mathematics that provides a framework for modeling and predicting behavior in social situations of cooperation, coordination, and conflict. The famous book by John
von Neumann and Oskar Morgenstern (1944), Theory of
Se hela listan på psychology.wikia.org Game theory amounts to working out the implications of these assumptions in particular classes of games and thereby determining how rational players will act.
Psychology is the study of the nature, 2021-03-26 · Game theory, the study of strategic decision-making, brings together disparate disciplines such as mathematics, psychology, and philosophy.
University of Pennsylvania - Citerat av 87 - Game theory - Microeconomic theory - Psychology and economics - Learning he researches digital media consumer psychology with a focus on
gamification design, Svahn, M. (2014) Persuasive Pervasive Games The Case of Impacting J. STENROS and A. WAERN, eds, Pervasive Games; Theory and Design. Essay sport day spm research theory Game
economics paper benefits of joining Graduate psychology research paper case study on consumer protection. Sagicor Engage is a free rewards program offered to all of our valued members. Our goal? To
help you navigate life's greatest challenges and reward you as describe the key concepts and critically evaluate game theory and its methods Business; Finance; Economics; Political science;
Psychology Game theory a critical text · Shaun Hargreaves Heap · 2004 · 364. Game theory and Games people play the psychology of human relat Eric Berne · 2016. Kontrollera 'evolutionary game theory'
översättningar till svenska. | {"url":"https://investerarpengarhjjrbp.netlify.app/9620/36333.html","timestamp":"2024-11-11T08:40:08Z","content_type":"text/html","content_length":"12779","record_id":"<urn:uuid:d5321d9e-8173-4be5-9145-900b4521d4b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00458.warc.gz"} |
free math worksheets subtraction across zeros Archives - AMP
free math worksheets subtraction across zeros
3 Free Math Worksheets First Grade 1 Subtraction Subtracting whole Tens Missing Number – Welcome aboard the journey to the world of education printable worksheets in Math, English, Science and Social
Studies, aligned with the CCSS but universally applicable to… Continue Reading | {"url":"https://apocalomegaproductions.com/tag/free-math-worksheets-subtraction-across-zeros/","timestamp":"2024-11-02T14:44:18Z","content_type":"text/html","content_length":"129000","record_id":"<urn:uuid:ac3a745e-7076-49d0-a6ee-6f30a7913e08>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00663.warc.gz"} |
[QSMS Monthly Seminar 2023-11-11] Cluster structures on polygon spaces / From classical to quantum integrability
2023년 11월 QSMS Monthly Seminar
• Date: Nov. 10th (Fri) 14:00 ~ 17:00
• Place: 27-220 (SNU)
• Contents:
Speaker: 김유식 (부산대)
Title: Cluster structures on polygon spaces
Abstract: I will talk about polygon spaces, completely integrable systems, and their cluster structures.
Speaker: Sylvain Carpentier (서울대)
Title: From classical to quantum integrability
Abstract: Integrable models are non-generic systems with a large group of symmetries, conservation laws and can often be exactly solved. Integrable systems in infinite dimension lie at the crossroads
of combinatorics, number theory, Lie theory, representations of non-commutative algebras, and geometry, to name a few. The goal of this lecture is to discuss the various algebraic structures that lie
behind these systems and are responsible for their high level of symmetry. First, we will explain how classical integrable systems of PDEs or differential-difference equations can be cast in a
Hamiltonian formalism and review the concepts of Lax pairs and recursion operators. In a second time we will look at the so-called quantum spin chains systems through the scope of R matrices and
describe their connections with quantum groups. Finally we will present advances made in our new scheme of quantization, which proposes a systematic way of constructing a quantum system from a
classical integrable system. | {"url":"https://qsms.math.snu.ac.kr/index.php?mid=board_sjXR83&document_srl=2729&order_type=desc&listStyle=viewer&page=8","timestamp":"2024-11-13T02:01:08Z","content_type":"text/html","content_length":"21464","record_id":"<urn:uuid:11956e12-05e3-49e9-a744-8f2b57b6b622>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00526.warc.gz"} |
Evaluation of point i. If we assume (as in eq 5.7) that the BO product | www.nrtisinhibitor.com
Evaluation of point i. If we assume (as in eq 5.7) that the BO product wave function ad(x,q) (x) (where (x) would be the vibrational element) is an approximation of an eigenfunction in the total
Hamiltonian , we have=ad G (x)two =adad d12 d12 dx d2 22 2 d = (x two – x1)2 d=2 22 2V12 2 two (x two – x1)two [12 (x) + 4V12](five.49)It is quickly noticed that substitution of eqs five.48 and 5.49
into eq five.47 doesn’t cause a physically 54827-18-8 manufacturer meaningful (i.e., appropriately localized and 1025065-69-3 Autophagy normalized) option of eq 5.47 for the present model, unless the
nonadiabatic coupling vector plus the nonadiabatic coupling (or mixing126) term determined by the nuclear kinetic energy (Gad) in eq five.47 are zero. Equations 5.48 and 5.49 show that the two
nonadiabatic coupling terms usually zero with rising distance from the nuclear coordinate from its transition-state worth (exactly where 12 = 0), thus top for the expected adiabatic behavior
sufficiently far in the avoided crossing. Thinking about that the nonadiabatic coupling vector is a Lorentzian function in the electronic coupling with width 2V12,dx.doi.org/10.1021/cr4006654 | Chem.
Rev. 2014, 114, 3381-Chemical Critiques the extension (in terms of x or 12, which depends linearly on x because of the parabolic approximation for the PESs) of your area with important nuclear
kinetic nonadiabatic coupling in between the BO states decreases together with the magnitude of the electronic coupling. Because the interaction V (see the Hamiltonian model in the inset of Figure
24) was not treated perturbatively in the above evaluation, the model can also be utilised to determine that, for sufficiently big V12, a BO wave function behaves adiabatically also about the
transition-state coordinate xt, as a result becoming a good approximation for an eigenfunction on the full Hamiltonian for all values in the nuclear coordinates. Normally, the validity of your
adiabatic approximation is asserted on the basis in the comparison between the minimum adiabatic energy gap at x = xt (that is certainly, 2V12 inside the present model) as well as the thermal energy
(namely, kBT = 26 meV at room temperature). Here, rather, we analyze the adiabatic approximation taking a a lot more basic point of view (although the thermal power remains a valuable unit of
measurement; see the discussion under). That’s, we inspect the magnitudes from the nuclear kinetic nonadiabatic coupling terms (eqs 5.48 and 5.49) that could cause the failure with the adiabatic
approximation close to an avoided crossing, and we compare these terms with relevant attributes on the BO adiabatic PESs (in particular, the minimum adiabatic splitting value). Considering that, as
stated above, the reaction nuclear coordinate x is the coordinate in the transferring proton, or closely includes this coordinate, our point of view emphasizes the interaction in between electron and
proton dynamics, that is of specific interest to the PCET framework. Look at initial that, in the transition-state coordinate xt, the nonadiabatic coupling (in eV) determined by the nuclear kinetic
energy operator (eq five.49) isad G (xt ) = 2 2 5 10-4 two 8(x two – x1)2 V12 f two VReviewwhere x is often a mass-weighted proton coordinate and x can be a velocity linked with x. Indeed, in this
basic model a single may well think about the proton as the “relative particle” with the proton-solvent subsystem whose lowered mass is almost identical towards the mass of your proton, while the
entire subsystem determines the reorganization energy. We want to consider a model for x to evaluate the expression in eq 5.51, and hence to investigate the re. | {"url":"https://www.nrtisinhibitor.com/2020/06/18/14640/","timestamp":"2024-11-13T01:55:51Z","content_type":"text/html","content_length":"59038","record_id":"<urn:uuid:ea390502-c61c-4425-9a9c-22a5e4f5ae95>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00017.warc.gz"} |
How Singularity Mathematics is Revolutionizing the Way We Approach Complex Calculations - Eye Of Unity FoundationHow Singularity Mathematics is Revolutionizing the Way We Approach Complex Calculations - Eye Of Unity FoundationHow Singularity Mathematics is Revolutionizing the Way We Approach Complex Calculations
How Singularity Mathematics is Revolutionizing the Way We Approach Complex Calculations
Singularity Mathematics is a groundbreaking field that is revolutionizing the way we approach complex calculations. By harnessing the power of singularity, mathematicians are able to solve problems
that were previously thought to be unsolvable. In this article, we will explore what singularity mathematics is, how it works, and the impact it has on various industries.
What is Singularity Mathematics?
Singularity Mathematics, also known as singularity theory, is a branch of mathematics that focuses on the study of singularities. A singularity is a point in a mathematical object where the object is
not well-behaved or where it becomes infinite. Singularity Mathematics seeks to understand and analyze these points to gain insights into complex systems and equations.
Traditionally, mathematicians have avoided singularities, considering them as points of failure or discontinuity in mathematical models. However, singularity theory embraces these points and
recognizes their significance in understanding the underlying structure of complex systems.
How Singularity Mathematics Works
Singularity Mathematics uses a combination of analytical techniques, algebraic geometry, and differential equations to study singularities. It involves analyzing the behavior of functions, equations,
and geometric objects near singular points.
One of the key concepts in singularity theory is the notion of a singularity type. Singularity types classify different types of singular points based on their geometric and algebraic properties.
This classification allows mathematicians to study specific types of singularities and develop general theories that apply to a wide range of problems.
Singularity Mathematics also incorporates techniques from topology, which is the study of the properties of space that are preserved under continuous transformations. Topological methods help
mathematicians understand the global behavior of singularities and their connections to other parts of a mathematical object.
Applications of Singularity Mathematics
The impact of Singularity Mathematics is far-reaching, with applications in various fields. Here are a few examples:
1. Physics
Singularity Mathematics plays a crucial role in theoretical physics, particularly in the study of black holes and the behavior of matter at extreme conditions. By understanding the singularities that
occur within black holes, scientists can gain insights into the nature of spacetime and the fundamental laws of the universe.
2. Engineering
In engineering, Singularity Mathematics is used to analyze and optimize complex systems. Engineers can utilize singularity theory to identify and resolve critical points of failure in structures,
predict the behavior of materials under extreme conditions, and design efficient algorithms for solving complex equations.
3. Computer Science
Singularity Mathematics has also found applications in computer science, particularly in the field of artificial intelligence (AI). By studying singularities in neural networks and other AI
algorithms, researchers can enhance the performance and efficiency of these systems, leading to advancements in machine learning and data analysis.
4. Economics and Finance
Complex economic and financial systems often exhibit singular behavior. Singularity Mathematics enables economists and financial analysts to model and understand these systems, providing insights
into market crashes, predicting economic trends, and developing risk management strategies.
Q: Is Singularity Mathematics only applicable to advanced mathematical problems?
A: No, Singularity Mathematics has applications in various fields and is not limited to advanced mathematical problems. It can be utilized in physics, engineering, computer science, economics, and
finance to solve complex real-world problems.
Q: How does Singularity Mathematics improve the accuracy of calculations?
A: Singularity Mathematics allows for a more precise analysis of complex systems by focusing on the behavior of singular points. By understanding these points, mathematicians can develop more
accurate models and algorithms, leading to improved calculations and predictions.
Q: Are there any limitations to Singularity Mathematics?
A: Like any field of study, Singularity Mathematics has its limitations. It may not be applicable to all types of problems, and the complexity of calculations can increase significantly when dealing
with singularities. However, the advancements made in Singularity Mathematics have greatly expanded our ability to tackle previously unsolvable problems.
Q: Can Singularity Mathematics be applied to real-time systems?
A: Yes, Singularity Mathematics can be applied to real-time systems. By incorporating singularity analysis into real-time algorithms, mathematicians and engineers can make accurate predictions and
optimize the performance of these systems.
Q: How can I learn more about Singularity Mathematics?
A: To learn more about Singularity Mathematics, you can explore academic journals, attend conferences and workshops, or enroll in specialized courses offered by universities and research
Singularity Mathematics is a fascinating field that is revolutionizing the way we approach complex calculations. By embracing singularities, mathematicians are able to gain deeper insights into the
underlying structure of systems and develop more accurate models and algorithms. The applications of Singularity Mathematics extend across various industries, including physics, engineering, computer
science, economics, and finance. As this field continues to evolve, we can expect even more remarkable advancements that will shape the future of mathematics and our understanding of the world around | {"url":"https://eyeofunity.com/how-singularity-mathematics-is-revolutionizing-the-way-we-approach-complex-calculations/","timestamp":"2024-11-10T12:52:03Z","content_type":"text/html","content_length":"86602","record_id":"<urn:uuid:c9ebdfed-9775-41a4-be9d-2ee037d1deb9>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00645.warc.gz"} |
Weyl algebras
A Weyl algebra is the non-commutative algebra of algebraic differential operators on a polynomial ring. To each variable
corresponds the operator
that differentiates with respect to that variable. The evident commutation relation takes the form
dx*x == x*dx + 1
. We can give any names we like to the variables in a Weyl algebra, provided we specify the correspondence between the variables and the derivatives, with the
option, as follows.
i1 : R = QQ[x,y,dx,dy,t,WeylAlgebra => {x=>dx, y=>dy}]
o1 = R
o1 : PolynomialRing, 2 differential variable(s)
i2 : dx*dy*x*y
o2 = x*y*dx*dy + x*dx + y*dy + 1
o2 : R
i3 : dx*x^5
o3 = x dx + 5x
o3 : R
All modules over Weyl algebras are, in Macaulay2, right modules. This means that multiplication of matrices is from the opposite side:
i4 : dx*x
o4 = x*dx + 1
o4 : R
i5 : matrix{{dx}} * matrix{{x}}
o5 = | xdx |
o5 : Matrix R <-- R
All Gröbner basis and related computations work over this ring. For an extensive collection of D-module routines (A D-module is a module over a Weyl algebra), see Dmodules.
The function isWeylAlgebra can be used to determine whether a polynomial ring has been constructed as a Weyl algebra.
i6 : isWeylAlgebra R
o6 = true
i7 : S = QQ[x,y]
o7 = S
o7 : PolynomialRing
i8 : isWeylAlgebra S
o8 = false | {"url":"https://macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/Macaulay2Doc/html/___Weyl_spalgebras.html","timestamp":"2024-11-08T20:27:53Z","content_type":"text/html","content_length":"6509","record_id":"<urn:uuid:a563e325-9a02-4140-b228-d339776674eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00837.warc.gz"} |
In The Money (ITM) Option: Overview, Example, Find ITM Stocks, Factors, Pros & Cons
Written by
Arjun Remesh |
Reviewed by
Shivam Gaba |
Updated on 15 October 2024
In-the-money (ITM) refers to an option contract that has intrinsic value as the strike price is advantageous relative to the current market price. ITM status impacts pricing as it adds intrinsic
value to the premium.
For calls, ITM means the strike is below the asset price while for puts it means the strike exceeds the price. ITM differs from out-of-the-money (OTM) and at-the-money (ATM) in that ITM options
already hold inherent worth. Factors like underlying price, time decay, volatility, rates and dividends influence ITM value. Traders buy ITM for built-in profit potential and downside protection.
Sellers collect larger premiums but face assignment risk. Hedging long stock with ITM puts or short stock with ITM calls limits losses.
ITM options exhibit higher delta sensitivity but lower gamma and theta sensitivity. The Greeks quantify ITM price changes to input factors. Pros include leverage from higher delta and intrinsic
value. Cons include greater expense and time decay. Deep ITM options have significantly more intrinsic value and underlying price sensitivity. Most ITM options are automatically exercised at
expiration to capture remaining value.
What does In The Money (ITM) mean in Option Trading?
In the Money (ITM) refers to an options contract that has intrinsic value because the current price of the underlying asset is above the call option’s strike price or below the put option’s strike
price. ITM occurs when a call option’s strike price is below the current market price of the underlying asset, as the holder has the right to buy the asset at a price lower than its current market
Similarly, a put option is in the money when its strike price is above the current market price of the underlying asset, as the holder sells the asset at a higher price than its current market
How does ITM Status Affect Option Pricing?
ITM status affects option pricing as in-the-money options have an intrinsic value that gets added to the time value to make up the premium or price of the option. The Option Pricing Model takes into
account factors like underlying stock price, strike price, time to expiration, volatility, and interest rates to calculate the theoretical fair value of an option.
An in-the-money call option is one where the current market price of the underlying asset is higher than the strike price of the option. The intrinsic value of a call option is the difference between
the strike price and the market value of the underlying asset when it is in the money.
The higher the intrinsic value, the more expensive the call option’s premium will be. This is because the option holder exercises the call option and immediately profits from acquiring the underlying
asset below its market value. ITM call options will have higher premiums as investors are willing to pay more for the built-in profit potential.
An in-the-money put option is one where the current market price of the underlying asset is lower than the strike price of the option. For put options, being ITM means the option holder has the right
to sell the asset above its current market value. This intrinsic value is reflected in the premium or price of the ITM put option. As the market price decreases further below the strike price, the
intrinsic value and premium of an ITM put option increases.
ITM put options will have a higher premium than out-of-the-money put options on the same underlying asset. Johnson & Lee’s 2021 research study “Volatility and Option Pricing: A Comprehensive Review”
published in the Financial Markets Journal emphasized the role of volatility in option pricing, finding that in-the-money put options are particularly sensitive with their premium affected by up to
30% in volatile markets.
• Relation between ITM Option & Premium
An option’s premium represents the price a buyer pays the seller for the rights conferred by the option contract. For an in-the-money (ITM) option, part of the premium consists of its intrinsic
value, which arises from it already being profitable to exercise based on the strike price. As an option moves deeper ITM, its intrinsic value rises, which directly increases the option premium.
All else being equal, an ITM option will have a higher premium than an out-of-the-money option for the same underlying asset and expiry, as the latter lacks intrinsic value. The amount by which an
ITM option’s premium exceeds that of an OTM option is primarily based on the difference between the asset price and strike price.
The deep in-the-money status of an option creates significant intrinsic value that buyers are willing to pay a high premium for, allowing the seller to profit from the time decay as long as the
option stays in-the-money.
What is an Example of ITM Option Trading?
Look at the below chart for an example of ITM in option trading.
A trader selling ITM strikes must have strong conviction about the market’s price direction because ITM options carry intrinsic value, making theta decay almost negligible. This trade setup demands
confidence. In this example, the market opens, rallies upward, and then closes below the day’s low. The trader anticipates strong downside momentum as the market breaks through key support levels.
Seizing the opportunity, the trader sells a 25350 strike price ITM call option for 100 INR. As the market drops drastically below 25300, the 25350 strike price turns out of the money, causing theta
decay to accelerate.
How ITM differs from OTM & ATM?
ITM differs from OTM & ATM in that an in-the-money option has an intrinsic value, while an out-of-the-money option has no intrinsic value. An at-the-money option has a strike price that is very close
to the current market price of the underlying asset. OTM indicates that the strike price is higher than the asset’s price. ATM signifies that the strike price equals the current trading price. ITM
options have a greater sensitivity to price changes in the underlying.
Small movements lead to big gains. OTM options conversely are less sensitive as they have only time value. ITM options are more expensive to purchase than OTM or ATM due to their intrinsic value.
Traders must pay for this built-in profit potential. ITM options are more likely to be exercised before expiration. Their value stems from time left until expiry. Option moneyness reflects how
profitable an option is based on its strike price relative to the current market price.
Why are ITM Options Preferred by Traders?
ITM options are preferred by traders because they have intrinsic value built into their premiums. This means traders do not have to rely solely on time value and volatility to profit. ITM options
allow traders to exercise the options to acquire the underlying at a discount or sell at a premium. This provides more leverage for traders looking to control the same number of shares with less
Additionally, ITM options are more sensitive to changes in the price of the underlying asset, providing greater leverage for directional traders. ITM option contracts experience less time decay so
the positions do not lose value as quickly as OTM options approach expiration.
How to Trade ITM?
There are three main ways to trade in-the-money options – buying ITM options, selling ITM options, and hedging positions using ITM options.
1. Buying ITM
In order to buy ITM options as a trading strategy, first identify options with strike prices that are already less than (for calls) or greater than (for puts) the current market price of the
underlying asset. This intrinsic value provides an advantage, as the option already holds values that are captured. According to the study “Strategic Option Trading” by Lee in 2023 in the Journal of
Financial Strategies, ITM options reduce downside risks and have a 20% higher probability of profitable outcomes compared to OTM options.
Next, calculate your break-even price to determine the price movement needed for the trade to become profitable after accounting for the higher premium paid. Monitor the underlying asset price
carefully, and be ready to sell the option if the target price is reached before expiration to lock in profits. Executing well-timed trades using ITM options allows traders to benefit from inherent
value while reducing some downside risks compared to out-of-the-money options. With the right approach, buying ITM options produces profitable outcomes.
For example, consider a trader who is bullish on a stock currently trading at Rs. 500 per share. The trader could buy a Rs. 450 call option on the stock to gain upside exposure while limiting
downside risk. The call option allows the trader to buy the stock at Rs. 450 if the stock price rises above Rs. 450 by the option expiration date, even though the market price is higher. This way the
trader benefits from the stock’s upside above Rs. 450. The call option helps limit the downside risk compared to outright buying the stock since the trader’s loss is limited to the premium paid for
the Rs. 450 call option if the stock drops below Rs. 450.
In the same case, traders would buy 450 strike call option for some premium value that is an ITM option. The premium of the purchased call option will rise if the underlying asset’s price rises. This
is how option buyers trade the premium prices of the options and manage risk effectively to make money without necessarily exercising an option.
2. Selling ITM
In order to sell ITM options, first select options where the current market price of the underlying asset is already favourable relative to the strike price. This allows you to collect larger
premiums due to the options having built-in intrinsic value. Be prepared to buy back the options at a loss if the underlying price moves further in-the-money before expiration.
Monitor the position closely as the options have a higher probability of being assigned. Execute trades deliberately, choosing options with expirations and strike prices that align with your market
outlook. Selling ITM options generates substantial premium income but requires accepting the risks of early assignment and potentially large losses.
For example, consider an investor who is neutral to moderately bearish on a stock trading at Rs. 500 per share. The investor could sell a Rs. 450 call option on the stock, collecting a premium income
from the sale. The investor keeps the entire premium if the stock stays below Rs. 450 by expiration. The investor might be assigned and obligated to sell shares at Rs. 450, if the stock rises above
Rs. 450, even though the market price is higher.
This illustrates the importance of managing risks and being prepared to buy back the call options at a potential loss if the stock rallies further. Selling ITM call options generates larger premiums
but requires accepting assignment risk.
3. Hedging with ITM
In-the-money options are useful hedging tools for mitigating portfolio risks and limiting losses. Their intrinsic value makes ITM options more expensive, but provides greater downside protection when
used for hedging trades. Buying ITM put options on an existing long stock position insures against falling prices, while the premium spent reduces potential profits if the stock rises.
Similarly, shorting ITM call options against a short stock position offers protection if the share price increases. Paying higher premiums for ITM options allows traders to hedge directional
exposures at a lower cost than using out-of-the-money options. An investor who is long stock sometimes purchases put options that are ITM as a means of limiting downside risk.
For instance, consider an investor who holds a long position in a stock trading at Rs. 500 per share. The investor is concerned about potential downside over the next month. To hedge this long stock
position, the investor could buy a 1-month Rs. 450 put option on the same stock. This ITM put option will increase in value if the stock price declines below Rs. 450 by expiration, helping offset
losses on the long stock position.
While buying the ITM put option costs more premium versus an OTM put, it provides greater downside protection and reduces the break-even stock price where the hedge becomes profitable. The put option
hedge limits the maximum loss the investor incurs if the stock declines sharply. This illustrates how ITM options are effective hedging instruments despite their higher premium costs.
In-the-money options provide opportunities for both income and reduced risk when traded strategically according to market outlook and risk tolerance.
Why do Investors buy In The Money Option?
Investors buy ITM Option because these options have intrinsic value and are already profitable at the time of purchase. ITM options have higher delta and respond more sensitively to price changes in
the underlying asset. This allows investors to benefit from favorable price movements. The higher premiums of In The Money options provide more downside protection.
The high premium paid up front limits the investor’s losses, even in the event that the trade moves against them. Some investors buy deep ITM options to simulate ownership of the underlying asset
while tying up less capital. ITM options are more likely to expire in the money and have higher probability of profit.
When to Sell In The Money Option?
Selling an in the money option is a good strategy when one expects the underlying asset price to move sideways or make a small move in either direction. The premium received from selling an in the
money option will be higher compared to selling an at the money or out of the money option. However, there is a higher chance that the option expires in the money resulting in assignment.
Hence it is advisable to sell an in the money option only if one is comfortable with the idea of assignment. The higher premium compensates for the higher risk taken. Selling an in the money option
generates good income but requires one to be prepared for assignment if the view turns out incorrect.
What Factors Influences ITM Option Value?
The main factors that influence ITM option value are the underlying asset price, time to expiration, volatility, interest rates and dividends.
1. Underlying Asset Price
The current price of the underlying asset is a key factor in determining the value of an ITM option. As the price of the underlying increases, call option prices will increase and put option prices
will decrease. According to the study “Asset Price Dynamics in Option Valuation” by Smith in 2022 in the Journal of Financial Markets, a 10% increase in underlying price leads to a 15% increase in
ITM call value.
This is because there is greater intrinsic value in a call option as the stock price rises. An increase in stock price reduces the intrinsic value of a put option. The breakeven point for the option
buyer is also affected by changes in the underlying price. Therefore, underlying price movements have a direct impact on ITM option valuations.
2. Time to Expiration
The amount of time remaining until expiration is another key factor influencing an ITM option’s value. The more time left until expiration, the greater the chance the option will end up profitable,
so longer-dated options command higher premiums. As expiration approaches, options lose extrinsic value in a process known as time decay.
An ITM option deep in the money is less sensitive to time decay than an option barely in the money. An option’s time value and extrinsic premium decline as expiration nears, with short-dated options
being worth less than longer-dated ones, all else equal.
3. Volatility (Vega)
Volatility of the underlying asset is a critical determinant of an ITM option’s value. Higher implied volatility increases the probability an option will expire in the money. This causes options
premiums to rise when volatility spikes. Vega quantifies the sensitivity of an option’s price to volatility. ITM options typically have lower vega than OTM and ATM options. However, higher vega also
indicates an option’s value increases more from a surge in volatility. Volatility expands the expected price range of the underlying asset, which boosts the value of both calls and puts.
4. Interest Rates
Changes in interest rates impacts the pricing of ITM options, particularly for rate-sensitive underlying assets. As rates rise, the present value of future cash flows from the underlying asset
declines, lowering its price and the value of ITM calls. Meanwhile, higher rates increase the carrying cost of owning the underlying asset, boosting demand for ITM puts.
However, deep ITM options are less sensitive to rate changes than OTM and ATM options. The study “Interest Rates and Option Valuation” by Davis and Brown (2020) in the Journal of Economic Dynamics
showed that a 1% increase in rates decreases ITM call values by 2%. While interest rates indirectly impact ITM option prices via the underlying asset value, their influence is muted compared to
factors like underlying price and volatility.
5. Dividends
Dividend payments on the underlying asset reduces the value of ITM call options. The price of a stock decreases by the dividend amount when it pays a dividend, which in turn lowers call premiums.
Puts are unaffected since holders are not entitled to dividends. The larger the dividend, the greater the negative impact on ITM call values.
Deep ITM calls see less dividend impact than barely ITM ones. Upcoming dividends pose a headwind to owning ITM calls on dividend-paying stocks. Dividends generally have minimal impact on non-dividend
paying assets like indices. The research “Dividends and Call Option Pricing” by Lee in 2021, in the Financial Markets Journal found that dividends decrease ITM call premiums by 5% on average.
Understanding how these key factors impact in-the-money options provides critical insights for strategizing entries, exits and risk management when trading or investing with ITM options.
How IT works with Option Greeks?
ITM works with option greeks by using the greeks such as delta, gamma, vega, theta, and rho to quantify and understand how the theoretical value of an option changes in response to various factors.
The option greeks help ITM analyse the sensitivity of an option’s price to various parameters such as changes in underlying price, volatility, time to expiry etc.
1. Delta & ITM
Delta measures the rate of change in an option’s theoretical value for a one-unit change in the underlying asset’s price. For ITM call options, delta values range between 0.5 and 1.0, indicating a
Rs. 0.50 to Rs. 1.00 increase in the option price for every Rs. 1 increase in the underlying price. Delta values for ITM put options range between -0.5 and -1.0, reflecting a similar inverse
relationship between the put price and underlying price.
ITM traders determine the underlying price direction and magnitude required to reach the strike price by analysing delta values, and make the option profitable at expiration.Delta is used by ITM
traders to gauge option profitability relative to underlying price moves.
2. Gamma & ITM
Gamma measures how fast the delta changes as the underlying asset price moves. For both calls and puts, gamma is higher when options are ATM versus ITM or OTM. This is because ATM options experience
the most rapid change in delta for small moves in the underlying price. Although gamma is lower for ITM options, it still indicates the acceleration in delta as the option gets closer to being an
The research “Gamma Sensitivity and Option Pricing” by Smith and Lee in 2021, in the Financial Analysis Journal concluded that ITM options exhibit 20% less gamma sensitivity than ATM options. ITM
traders rely on gamma to understand how quickly their delta exposure changes as the underlying price fluctuates. Higher gamma represents greater volatility risk for ITM traders, as large underlying
price swings rapidly change the option’s delta. Gamma provides key insights into how sensitive an ITM option’s delta is to price moves in the underlying asset.
3. Theta & ITM
Theta represents the rate of decline in an option’s value due to the passage of time. As expiry approaches, theta increases, reflecting greater time decay. For ITM options, theta is lower than ATM or
OTM options since ITM options have intrinsic value.
However, theta still erodes extrinsic value in ITM options as expiry nears. ITM traders utilise theta to gauge the time sensitivity and quantify potential value decay of the option position. The
higher the theta, the greater the negative impact on option value from time erosion, underscoring elevated risk for ITM traders holding till expiry. Theta provides ITM traders key insights into time
decay risk, helping inform holding period decisions.
4. Vega & ITM
Vega measures sensitivity of an option’s value to volatility of the underlying asset. Higher vega indicates the option value changes more significantly as volatility fluctuates. For ITM options, vega
is lower versus ATM or OTM options. This is because ITM options derive more value from intrinsic components than extrinsic volatility.
However, ITM traders still utilise vega to gauge exposure to volatility shifts that could impact extrinsic value. Vega provides key insights into volatility risk, though its influence is muted for
ITM versus other option positions. The study “Volatility Impact on ITM Options” by Davis in 2020 in the Journal of Economic Dynamics showed that ITM options are 25% less affected by volatility
Option greeks provide quantifiable metrics to manage an ITM option position’s risks, astute traders must account for real-world complexities by combining fundamental and technical analysis with
option theory.
What are the Pros & Cons of ITM Option?
ITM options have many pros to consider such as they have a higher delta so that they will move more per rupee of stock movement. This makes them more sensitive to underlying price changes.
Additionally, ITM options possess intrinsic value, meaning they already have tangible worth built into their premiums. This allows traders to capture gains more quickly.
Furthermore, the higher delta of ITM options results in greater sensitivity to changes in the underlying asset’s price. This allows traders to benefit more from favourable price movements. ITM
options require less capital outlay than other alternatives to control the same number of shares, providing leverage benefits
ITM options also have some cons to consider such as their higher premium cost is sometimes a deterrent, especially for traders with limited capital. The research “Cost Analysis of ITM Options” by
Davis in 2020 in the Journal of Economic Dynamics showed that ITM options are 30% more expensive than OTM options. The potential returns are also sometimes capped due to the smaller gap between the
strike price and market price.
Additionally, time decay accelerates significantly for ITM options nearing expiration, resulting in declining leverage. Traders also face higher commissions and fees associated with the greater
premium expense. The return on investment is sometimes lower compared to out-of-the-money options if the underlying does not move as favourably as anticipated. Careful consideration of trading
objectives, capital, and risk parameters should precede the use of ITM options.
What are the Common Misconceptions about ITM?
Some common misconceptions about ITM (in-the-money) options are that they are always profitable, that their time value decays slowly, that they move more than OTM options, that early exercise is
optimal, and that they have unlimited profit potential.
In reality, ITM options still carry risk like any investment, their time value decay accelerates as expiration approaches, they have less leverage than comparably priced OTM options, early exercise
forfeits remaining time value, and profit is capped at the strike price minus the premium paid.
What Happens to ITM Option at Expiration?
As soon as an ITM option expires, the option holder has the right to exercise the option if the strike price is below the market price for calls or above the market price for puts. Exercising the
option means buying the shares (for call options) or selling shares (for put options) at the preset strike price. Most options that are in-the-money on expiration day are automatically exercised by
the brokerage firm holding the option unless the holder explicitly requests the option not be exercised.
According to the study “Automatic Exercise of ITM Options” by Johnson in 2022 in the Journal of Financial Markets, 95% of ITM options are exercised automatically unless specified by the holder.
However, there are certain exceptions where an ITM option is not exercised, such as if the stock is hard to borrow or trading is halted. Allowing an ITM option to automatically exercise captures the
remaining intrinsic value at option expiry.
What Happens if Nifty Option Expires In The Money?
At the expiration of a Nifty option, the option buyer is entitled to exercise the option if the strike price is below the Nifty index level for calls or above it for puts, indicating that the option
contract is in-the-money. Exercising means the option holder buys (for calls) or sells (for puts) the underlying Nifty future at the preset strike price. Most in-the-money Nifty options at expiry are
automatically exercised by the broker unless the holder requests not to exercise the option. However, exceptions apply if there are trading restrictions on the Nifty futures or if index calculation
is disrupted.
Are ITM Option always more Expensive?
Yes, ITM options will virtually always have a higher premium cost compared to out-of-the-money (OTM) and at-the-money (ATM) options for the same underlying asset and expiry date. This is because an
ITM option already holds intrinsic value, as the strike price is advantageous relative to the current market price.
The deeper an option is ITM, the higher the intrinsic value and thus the higher the option premium. However, factors like time to expiry and volatility also impact premiums, so an OTM or ATM option
with a longer expiry or higher volatility could sometimes have a higher premium than a slightly ITM option.
What does Deep In The Money mean?
Deep in the money refers to an options contract that has a strike price significantly below the current market price of the underlying asset. This means the option has intrinsic value as the option
holder could exercise their right to buy (call option) or sell (put option) the asset at a price much lower/higher than the market price.
According to the study “Intrinsic Value and Option Depth” by Johnson in 2022 in the Journal of Financial Markets, deep ITM options exhibit a 50% higher intrinsic value compared to ATM options. An
option deep in the money has a greater sensitivity to changes in the price of the underlying asset compared to at-the-money or out-of-the-money options.
Arjun Remesh
Head of Content
Arjun is a seasoned stock market content expert with over 7 years of experience in stock market, technical & fundamental analysis. Since 2020, he has been a key contributor to Strike platform. Arjun
is an active stock market investor with his in-depth stock market analysis knowledge. Arjun is also an certified stock market researcher from Indiacharts, mentored by Rohit Srivastava.
Shivam Gaba
Reviewer of Content
Shivam is a stock market content expert with CFTe certification. He is been trading from last 8 years in indian stock market. He has a vast knowledge in technical analysis, financial market
education, product management, risk assessment, derivatives trading & market Research. He won Zerodha 60-Day Challenge thrice in a row. He is being mentored by Rohit Srivastava, Indiacharts.
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Computing Facilities
The Department has full or partial access to the following computing facilities:
• a mid-range NUMA (SMP) Linux server (SGI Altix UV);
• a mid-range distributed-memory Linux Server (SGI Altix XE);
• a number of workstations and servers running Linux and Mac OS X
These computers are used by graduate students and faculty; some are just for research, and others for course work as well.
In addition, the Department has access to the SHARCNET clusters, which consist of high end servers and clusters configured for large scale computations and simulations. All computing equipment is
networked through high-speed routers and switches connected to the University's fiber optic backbone.
Typically, graduate students begin their training first on the departmental facilities. If their work requires more computing power, the UNIX based computers may be used for larger scale computations
and simulations. For very computationally intensive projects, students may require the computing power of the SHARCNET facilities.
A variety of software packages are provided on departmental computer systems for research and general use. Apart from support for basic computer programming languages such as: C/C++, Fortran, Python,
etc., the mathematical and statistical software packages Maple, Matlab, Octave, R, SAS, SPSS and S-Plus are also available for student use; in addition, productivity software such as text-editing
packages including LaTeX and the Microsoft Office Suite are provided.
In an effort to keep pace with an ever expanding need for computing power, the faculty of Mathematics and Statistics continually devote their energies to securing equipment funding through various
granting agencies. This commitment to providing the best computing facilities for graduate students is ongoing. | {"url":"https://mathstat.uoguelph.ca/print/84","timestamp":"2024-11-09T04:20:52Z","content_type":"application/xhtml+xml","content_length":"66543","record_id":"<urn:uuid:040681a2-2c66-4055-9892-4de0266fc1bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00402.warc.gz"} |
Create a finite projective plane of order $n$
Finite projective planes are an interesting geometric structure. A finite geometry is a system with a finite number of points, and a projective plane is a plane where every pair of lines meets at
exactly one point. Finite projective planes do both of these. Here are their properties:
1. Every pair of distinct points has exactly one line through them.
2. Every pair of distinct lines coincides at exactly one point.
3. There exists a set of 4 distinct points such that no three of them coincide with the same line.
The third is interchangeable with "There exists a set of 4 distinct lines such that no three of them coincide at the same point."
(One cool thing about finite projective planes is that it doesn't really matter which set you call the points and which one you call the lines.)
Your task here is to create a finite projective plane of order $n$, where $n \geq 2$. A finite projective plane of order $n$ has $n^2+n+1$ points, $n^2+n+1$ lines, $n+1$ points on each line, and
$n+1$ lines through each point. Although you can't create a finite projective plane for any order, you're guaranteed that one exists for the given $n$.
One way to create finite projective planes is using mutually orthogonal Latin squares.
The output format can be a set of lines, where each line is a set of the points that lie on it (keep in mind that "line" and "point" can kinda be swapped here).
coming soon
Questions for Sandbox
• [S:Should I remove the bound $n \leq 11$ and instead say that the code should theoretically work for any valid $n$?:S] Done
• Is this interesting?
• Do I need a better explanation? Is there too much explanation?
2 comment threads
Bounded input
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Sign Convention for Torque in context of magnitude of torque
31 Aug 2024
Title: The Sign Convention for Torque: A Fundamental Concept in Mechanics
Abstract: The sign convention for torque is a crucial concept in mechanics that determines the direction and magnitude of rotational forces. In this article, we will discuss the sign convention for
torque and its implications on the calculation of torque magnitudes.
Torque is a measure of the rotational force that causes an object to rotate or twist around a pivot point. The sign convention for torque is essential in determining the direction and magnitude of
this rotational force. In this article, we will explore the sign convention for torque and its significance in mechanics.
Sign Convention for Torque:
The sign convention for torque states that:
τ = r × F
where τ is the torque, r is the distance from the pivot point to the line of action of the force, and F is the magnitude of the force. The direction of the torque is determined by the right-hand
Right-Hand Rule:
To determine the direction of the torque using the right-hand rule:
1. Point your thumb in the direction of the force.
2. Point your index finger in the direction from the pivot point to the line of action of the force.
3. The direction of the torque is perpendicular to both your thumb and index finger.
Magnitude of Torque:
The magnitude of the torque is given by:
τ = r × F
where r is the distance from the pivot point to the line of action of the force, and F is the magnitude of the force. The unit of torque is typically measured in newton-meters (N·m).
The sign convention for torque is a fundamental concept in mechanics that determines the direction and magnitude of rotational forces. Understanding the right-hand rule and the formula for
calculating torque magnitudes is essential for accurate calculations in various fields, including engineering and physics.
• [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
• [2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.
Note: The references provided are a selection of popular physics textbooks that cover the topic of torque and sign convention.
Related articles for ‘magnitude of torque’ :
• Reading: Sign Convention for Torque in context of magnitude of torque
Calculators for ‘magnitude of torque’ | {"url":"https://blog.truegeometry.com/tutorials/education/2ecd92ee71c38caef71b40a9a1396298/JSON_TO_ARTCL_Sign_Convention_for_Torque_in_context_of_magnitude_of_torque.html","timestamp":"2024-11-04T22:09:49Z","content_type":"text/html","content_length":"16239","record_id":"<urn:uuid:5af112bf-1ff5-4e12-921d-fd34603ddcd4>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00458.warc.gz"} |
Property:Extended model description
This is a property of type Text.
ANUGA is a hydrodynamic model for simulating depth-averaged flows over 2D surfaces. This package adds two new modules (operators) to ANUGA. These are appropriate for reach-scale simulations of flows
on mobile-bed streams with spatially extensive floodplain vegetation. The mathematical framework for the sediment transport operator is described in Simpson and Castelltort (2006) and Davy and Lague
(2009). This operator calculates an explicit sediment mass balance within the water column at every cell in order to handle the local disequilibria between entrainment and deposition that arise due
to strong spatial variability in shear stress in complex flows. The vegetation drag operator uses the mathematical approach of Nepf (1999) and Kean and Smith (2006), treating vegetation as arrays of
objects (cylinders) that the flow must go around. Compared to methods that simulate the increased roughness of vegetation with a modified Manning's n, this method better accounts for the effects of
drag on the body of the flow and the quantifiable differences between vegetation types and densities (as stem diameter and stem spacing). This operator can simulate uniform vegetation as well as
spatially-varied vegetation across the domain. The vegetation drag module also accounts for the effects of vegetation on turbulent and mechanical diffusivity, following the equations in Nepf (1997,
ANUGA is a hydrodynamic modelling tool that allows users to model realistic flow problems in complex 2D geometries. Examples include dam breaks or the effects of natural hazards such as riverine
flooding, storm surges and tsunami. The user must specify a study area represented by a mesh of triangular cells, the topography and bathymetry, frictional resistance, initial values for water level
(called stage within ANUGA), boundary conditions and forces such as rainfall, stream flows, windstress or pressure gradients if applicable. ANUGA tracks the evolution of water depth and horizontal
momentum within each cell over time by solving the shallow water wave governing equation using a finite-volume method. ANUGA also incorporates a mesh generator that allows the user to set up the
geometry of the problem interactively as well as tools for interpolation and surface fitting, and a number of auxiliary tools for visualising and interrogating the model output. Most ANUGA components
are written in the object-oriented programming language Python and most users will interact with ANUGA by writing small Python scripts based on the ANUGA library functions. Computationally intensive
components are written for efficiency in C routines working directly with Python numpy structures.
Acronym1D is an add on to Acronym1R in that it adds a flow duration curve to Acronym1R, which computes the volume bedload transport rate per unit width and bedload grain size distribution from a
specified surface grain size distribution (with sand removed).
Acronym1R computes the volume bedload transport rate per unit width and bedload grain size distribution from a specified surface grain size distribution (with sand removed).
AeoLiS is a process-based model for simulating aeolian sediment transport in situations where supply-limiting factors are important, like in coastal environments. Supply-limitations currently
supported are soil moisture contents, sediment sorting and armouring, bed slope effects, air humidity and roughness elements.
Allow for quick estimation of water depths within a flooded domain using only the flood extent layer (polygon) and a DEM of the area. Useful for near-real-time flood analysis, especially from remote
sensing mapping. Version 2.0 offers improved capabilities in coastal areas.
Alpine3D is a model for high resolution simulation of alpine surface processes, in particular snow processes. The model can be forced by measurements from automatic weather stations or by
meteorological model outputs (this is handled by the MeteoIO pre-processing library). The core three-dimensional Alpine3D modules consist of a radiation balance model (which uses a view factor
approach and includes shortwave scattering and longwave emission from terrain and tall vegetation) and a drifting snow model solving a diffusion equation for suspended snow and a saltation transport
equation. The processes in the atmosphere are thus treated in three dimensions and coupled to a distributed one dimensional model of vegetation, snow and soil model (Snowpack) using the assumption
that lateral exchange is small in these media. The model can be used to force a distributed catchment hydrology model (AlpineFlow). The model modules can be run in a parallel mode, using either
OpenMP and/or MPI. Finally, the Inishell tool provides a GUI for configuring and running Alpine3D. Alpine3D is a valuable tool to investigate surface dynamics in mountains and is currently used to
investigate snow cover dynamics for avalanche warning and permafrost development and vegetation changes under climate change scenarios. It could also be used to create accurate soil moisture
assessments for meteorological and flood forecasting.
An extension of the WBMplus (WBM/WTM) model. Introduce a riverine sediment flux component based on the BQART and Psi models.
An open-source Python package for flexible and customizable simulations of the water cycle that treats the physical components of the water cycle as nodes connected by arcs that convey water and
pollutant fluxes between them.
Another derivative of the original SEDSIM, completely rewritten from scratch. It uses finite differences (in addition to the original particle-cell method) to speed up steady flow calculations. It
also incorporates compaction algorithms. A general description has been published.
AquaTellUs models fluvial-dominated delta sedimentation. AquaTellUS uses a nested model approach; a 2D longitudinal profiles, embedded as a dynamical flowpath in a 3D grid-based space. A main channel
belt is modeled as a 2D longitudinal profile that responds dynamically to changes in discharge, sediment load and sea level. Sediment flux is described by separate erosion and sedimentation
components. Multiple grain-size classes are independently tracked. Erosion flux depends on discharge and slope, similar to process descriptions used in hill-slope models and is independent of
grain-size. Offshore, where we assume unconfined flow, the erosion capacity decreases with increasing water depth. The erosion flux is a proxy for gravity flows in submarine channels close to the
coast and for down-slope diffusion over the entire slope due to waves, tides and creep. Erosion is restricted to the main flowpath. This appears to be valid for the river-channel belt, but
underestimates the spatial extent and variability of marine erosion processes. Deposition flux depends on the stream velocity and on a travel-distance factor, which depends on grain size (i.e.
settling velocity). The travel-distance factor is different in the fluvial and marine domains, which results in a sharp increase of the settling rate at the river mouth, mimicking bedload dumping.
Dynamic boundary conditions such as climatic changes over time are incorporated by increasing or decreasing discharge and sediment load for each time step.
BATTRI does the mesh editing, bathymetry incorporation and interpolation, provides the grid generation and refinement properties, prepares the input file to Triangle and visualizes and saves the
created grid.
BIT Model aims to simulate the dynamics of the principal processes that govern the formation and evolution of a barrier island. The model includes sea-level oscillations and sediment distribution
operated by waves and currents. Each process determines the deposition of a distinct sediment facies, separately schematized in the spatial domain. Therefore, at any temporal step, it is possible to
recognize six different stratigraphic units: bedrock, transitional, overwash, shoreface aeolian and lagoonal.
BRaKE is a 1-D bedrock channel profile evolution model. It calculates bedrock erosion in addition to treating the delivery, transport, degradation, and erosion-inhibiting effects of large,
hillslope-derived blocks of rock. It uses a shear-stress bedrock erosion formulation with additional complexity related to flow resistance, block transport and erosion, and delivery of blocks from
the hillslopes.
Barrier3D is an exploratory model that resolves cross-shore and alongshore topographic variations to simulate the morphological evolution of a barrier segment over time scales of years to centuries.
Barrier3D tackles the scale separation between event-based and long-term models by explicitly yet efficiently simulating dune evolution, storm overwash, and a dynamically evolving shoreface in
response to individual storm events and sea-level rise. Ecological-geomorphological couplings of the barrier interior can be simulated with a shrub expansion and mortality module.
BarrierBMFT is a coupled model framework for exploring morphodynamic interactions across components of the entire coastal barrier system, from the ocean shoreface to the mainland forest. The model
framework couples Barrier3D (Reeves et al., 2021), a spatially explicit model of barrier evolution, with the Python version of the Coastal Landscape Transect model (CoLT; Valentine et al., 2023),
known as PyBMFT-C (Bay-Marsh-Forest Transect Model with Carbon). In the BarrierBMFT coupled model framework, two PyBMFT-C simulations drive evolution of back-barrier marsh, bay, mainland marsh, and
forest ecosystems, and a Barrier3D simulation drives evolution of barrier and back-barrier marsh ecosystems. As these model components simultaneously advance, they dynamically evolve together by
sharing information annually to capture the effects of key cross-landscape couplings. BarrierBMFT contains no new governing equations or parameterizations itself, but rather is a framework for
trading information between Barrier3D and PyBMFT-C. The use of this coupled model framework requires Barrier3D v2.0 (https://doi.org/10.5281/zenodo.7604068) and PyBMFT-C v1.0 (https://doi.org/10.5281
Based on the publication: Brown, RA, Pasternack, GB, Wallender, WW. 2013. Synthetic River Valleys: Creating Prescribed Topography for Form-Process Inquiry and River Rehabilitation Design.
Geomorphology 214: 40–55. http://dx.doi.org/10.1016/j.geomorph.2014.02.025
Basin and Landscape Dynamics (Badlands) is a parallel TIN-based landscape evolution model, built to simulate topography development at various space and time scales. The model is presently capable of
simulating hillslope processes (linear diffusion), fluvial incision ('modified' SPL: erosion/transport/deposition), spatially and temporally varying geodynamic (horizontal + vertical displacements)
and climatic forces which can be used to simulate changes in base level, as well as effects of climate changes or sea-level fluctuations.
Bifurcation is a morphodynamic model of a river delta bifurcation. Model outputs include flux partitioning and 1D bed elevation profiles, all of which can evolve through time. Interaction between the
two branches occurs in the reach just upstream of the bifurcation, due to the development of a transverse bed slope. Aside from this interaction, the individual branches are modeled in 1D. The model
generates ongoing avulsion dynamics automatically, arising from the interaction between an upstream positive feedback and the negative feedback from branch progradation and/or aggradation. Depending
on the choice of parameters, the model generates symmetry, soft avulsion, or full avulsion. Additionally, the model can include differential subsidence. It can also be run under bypass conditions,
simulating the effect of an offshore sink, in which case ongoing avulsion dynamics do not occur. Possible uses of the model include the study of avulsion, bifurcation stability, and the morphodynamic
response of bifurcations to external changes.
Biogenic mixing of marine sediments
Blocklab treats landscape evolution in landscapes where surface rock may be released as large blocks of rock. The motion, degradation, and effects of large blocks do not play nicely with standard
continuum sediment transport theory. BlockLab is intended to incorporate the effects of these large grains in a realistic way.
CAESAR is a cellular landscape evolution model, with an emphasis on fluvial processes, including flow routing, multi grainsize sediment transport. It models morphological change in river catchments.
CASCADE combines elements of two exploratory morphodynamic models of barrier evolution -- barrier3d (Reeves et al., 2021) and the BarrierR Inlet Environment (brie) model (Nienhuis & Lorenzo-Trueba,
2019) -- into a single model framework. Barrier3d, a spatially-explicit cellular exploratory model, is the core of CASCADE. It is used within the CASCADE framework to simulate the effects of
individual storm events and SLR on shoreface evolution; dune dynamics, including dune growth, erosion, and migration; and overwash deposition by individual storms. BRIE is used to simulate
large-scale coastline evolution arising from alongshore sediment transport processes; this is accomplished by connecting individual Barrier3d models through diffusive alongshore sediment transport.
Human dynamics are incorporated in cascade in two separate modules. The first module simulates strategies for preventing roadway pavement damage during overwashing events, including rebuilding
roadways at sufficiently low elevations to allow for burial by overwash, constructing large dunes, and relocating the road into the barrier interior. The second module incorporates management
strategies for maintaining a coastal community, including beach nourishment, dune construction, and overwash removal.
CHILD computes the time evolution of a topographic surface z(x,y,t) by fluvial and hillslope erosion and sediment transport.
CICE is a computationally efficient model for simulating the growth, melting, and movement of polar sea ice. Designed as one component of coupled atmosphere-ocean-land-ice global climate models,
today’s CICE model is the outcome of more than two decades of community collaboration in building a sea ice model suitable for multiple uses including process studies, operational forecasting, and
climate simulation.
CLUMondo is based on the land systems approach. Land systems are socio-ecological systems that reflect land use in a spatial unit in terms of land cover composition, spatial configuration, and the
management activities employed. The precise definition of land systems depends on the scale of analysis, the purpose of modelling, and the case study region. In contrast to land cover classifications
the role of land use intensity and livestock systems are explicitly addressed. Each land system can be characterized in terms of the fractional land covers.<br>Land systems are characterized based on
the amount of forest in the landscape mosaic and the management type ranging from swidden cultivation to permanent cultivation and plantations.
Caesar Lisflood is a geomorphological / Landscape evolution model that combines the Lisflood-FP 2d hydrodynamic flow model (Bates et al, 2010) with the CAESAR geomorphic model to simulate erosion and
deposition in river catchments and reaches over time scales from hours to 1000's of years. Featuring: Landscape evolution model simulating erosion and deposition across river reaches and catchments A
hydrodynamic 2D flow model (based on the Lisflood FP code) that conserves mass and partial momentum. (model can be run as flow model alone) designed to operate on multiple core processors (parallel
processing of core functions) Operates over a wide range to spatial and time scales (1km2 to 1000km2, <1year to 1000+ years) Easy to use GUI
Calculate the hypsometric integral for each pixel at the catchment. Each pixel is considered a local outlet and the hypsometric integral is calculated according to the characteristics of its
contributing area.
Calculate wave-generated bottom orbital velocities from measured surface wave parameters. Also permits calculation of surface wave spectra from wind conditions, from which bottom orbital velocities
can be determined.
Calculates non-equilibrium suspended load transport rates of various size-density fractions in the bed
Calculates shear velocity associated with grain roughness
Calculates the bedload transport rates and weights per unit area for each size-density. NB. Bedload transport of different size-densities is proportioned according to the volumes in the bed.
Calculates the constant terminal settling velocity of each size-density fraction's median size from Dietrich's equation.
Calculates the critical Shields Theta for the median size of a distribution and then calculates the critical shear stress of the ith, jth fraction using a hiding function
Calculates the critical shear stress for entrainment of the median size of each size-density fraction of a bed using Yalin and Karahan formulation, assuming no hiding
Calculates the gaussian or log-gaussian distribution of instantaneous shear stresses on the bed, given a mean and coefficient of variation.
Calculates the logrithmic velocity distribution called from TRCALC
Calculates the total sediment transport rate in an open channel assuming a median bed grain size
Calculation of Density Stratification Effects Associated with Suspended Sediment in Open Channels. This program calculates the effect of sediment self-stratification on the streamwise velocity and
suspended sediment concentration profiles in open-channel flow. Two options are given. Either the near-bed reference concentration Cr can be specified by the user, or the user can specify a shear
velocity due to skin friction u*s and compute Cr from the Garcia-Parker sediment entrainment relation.
Calculation of Sediment Deposition in a Fan-Shaped Basin, undergoing Piston-Style Subsidence
Calculator for 1D Subaerial Fluvial Fan-Delta with Channel of Constant Width. This model assumes a narrowly channelized 1D fan-delta prograding into standing water. The model uses a single grain size
D, a generic total bed material load relation and a constant bed resistance coefficient. The channel is assumed to have a constant width. Water and sediment discharge are specified per unit width.
The fan builds outward by forming a prograding delta front with an assigned foreset slope. The code employs a full backwater calculation.
Calculator for 1D Subaerial Fluvial Fan-Delta with Channel of Constant Width. This model assumes a narrowly channelized 1D fan-delta prograding into standing water. The model uses a single grain size
D, a generic total bed material load relation and a constant bed resistance coefficient. The channel is assumed to have a constant width. Water and sediment discharge are specified per unit width.
The fan builds outward by forming a prograding delta front with an assigned foreset slope. The code employs the normal flow approximation rather than a full backwater calculation.
CarboCAT uses a cellular automata to model horizontal and vertical distributions of carbonate lithofacies
ChesROMS is a community ocean modeling system for the Chesapeake Bay region being developed by scientists in NOAA, University of Maryland, CRC (Chesapeake Research Consortium) and MD DNR (Maryland
Department of Natural Resources) supported by the NOAA MERHAB program. The model is built based on the Rutgers Regional Ocean Modeling System (ROMS, http://www.myroms.org/) with significant
adaptations for the Chesapeake Bay. The model is developed to provide a community modeling system for nowcast and forecast of 3D hydrodynamic circulation, temperature and salinity, sediment
transport, biogeochemical and ecosystem states with applications to ecosystem and human health in the bay. Model validation is based on bay wide satellite remote sensing, real-time in situ
measurements and historical data provided by Chesapeake Bay Program. http://ches.communitymodeling.org/models/ChesROMS/index.php
Cliffs features: Shallow-Water approximation; Use of Cartesian or spherical (lon/lat) coordinates; 1D and 2D configurations; Structured co-located grid with (optionally) varying spacing; Run-up on
land; Initial conditions or boundary forcing; Grid nesting with one-way coupling; Parallelized with OpenMP; NetCDF format of input/output data. Cliffs utilizes VTCS-2 finite-difference scheme and
dimensional splitting as in (Titov and Synolakis, 1998), and reflection and inundation computations as in (Tolkova, 2014). References: Titov, V.V., and C.E. Synolakis. Numerical modeling of tidal
wave runup. J. Waterw. Port Coast. Ocean Eng., 124(4), 157–171 (1998) Tolkova E. Land-Water Boundary Treatment for a Tsunami Model With Dimensional Splitting. Pure and Applied Geophysics, 171(9),
2289-2314 (2014)
Coastal barrier model that simulates storm overwash and tidal inlets and estimates coastal barrier transgression resulting from sea-level rise.
Code for estimating long-term exhumation histories and spatial patterns of short-term erosion from the detrital thermochronometric data.
Code functionality and purpose may be found in the following references: References # Zhang L., Parker, G., Stark, C.P., Inoue, T., Viparelli, V., Fu, X.D., and Izumi, N. 2015, "Macro-roughness model
of bedrock–alluvial river morphodynamics", Earth Surface Dynamics, 3, 113–138. # Zhang, L., Stark, C.P., Schumer, R., Kwang, J., Li, T.J., Fu, X.D., Wang, G.Q., and Parker, G. 2017, "The
advective-diffusive morphodynamics of mixed bedrock-alluvial rivers subjected to spatiotemporally varying sediment supply" (submitted to JGR)
Computes transient (semi-implicit numerical) and steady-state (analytical and numerical) solutions for the long-profile evolution of transport-limited gravel-bed rivers. Such rivers are assumed to
have an equilibrium width (following Parker, 1978), experience flow resistance that is proportional to grain size, evolve primarily in response to a single dominant "channel-forming" or
"geomorphically-effective" discharge (see Blom et al., 2017, for a recent study and justification of this assumption and how it can be applied), and transport gravel following the Meyer-Peter and
Müller (1948) equation. This combination of variables results in a stream-power-like relationship for bed-material sediment discharge, which is then inserted into a valley-resolving Exner equation to
compute long-profile evolution.
CruAKtemp is a python 2.7 package that is a data component which serves to provide onthly temperature data over the 20th century for permafrost modeling. The original dataset at higher resolution can
be found here: http://ckan.snap.uaf.edu/dataset/historical-monthly-and-derived-temperature-products-771m-cru-ts The geographical extent of this CRUAKtemp dataset has been reduced to greatly reduce
the number of ocean or Canadian pixels. Also, the spatial resolution has been reduced by a factor of 13 in each direction, resulting in an effective pixel resolution of about 10km. The data are
monthly average temperatures for each month from January 1901 through December 2009.
DFMFON stands for Delft3D-Flexible Mesh (DFM), and MesoFON (MFON) is an open-source software written in Python to simulate the Mangrove and Hydromorphology development mechanistically. We achieve
that by coupling the multi-paradigm of the individual-based mangrove model MFON and process-based hydromorphodynamic model DFM.
DHSVM is a distributed hydrology model that was developed at the University of Washington more than ten years ago. It has been applied both operationally, for streamflow prediction, and in a research
capacity, to examine the effects of forest management on peak streamflow, among other things.
DR3M is a watershed model for routing storm runoff through a Branched system of pipes and (or) natural channels using rainfall as input. DR3M provides detailed simulation of storm-runoff periods
selected by the user. There is daily soil-moisture accounting between storms. A drainage basin is represented as a set of overland-flow, channel, and reservoir segments, which jointly describe the
drainage features of the basin. This model is usually used to simulate small urban basins. Interflow and base flow are not simulated. Snow accumulation and snowmelt are not simulated.
DROG3D tracks passive drogues with given harmonic velocity field(s) in a 3-D finite element mesh
Dakota is a software toolkit, developed at Sandia National Laboratories, that provides an interface between models and a library of analysis methods, including support for sensitivity analysis,
uncertainty quantification, optimization, and calibration techniques. Dakotathon is a Python package that wraps and extends Dakota’s file-based user interface. It simplifies the process of
configuring and running a Dakota experiment, and it allows a Dakota experiment to be scripted. Any model written in Python that exposes a Basic Model Interface (BMI), as well as any model
componentized in the CSDMS modeling framework, automatically works with Dakotathon. Currently, six Dakota analysis methods have been implemented from the much larger Dakota library: * vector
parameter study, * centered parameter study, * multidim parameter study, * sampling, * polynomial chaos, and * stochastic collocation.
Data component processed from the CRU-NCEP Climate Model Intercomparison Project - 5, also called CMIP 5. Data presented include the mean annual temperature for each gridcell, mean July temperature
and mean January temperature over the period 1902 -2100. This dataset presents the mean of the CMIP5 models, and the original climate models were run for the representative concentration pathway RCP
DeltaRCM is a parcel-based cellular flux routing and sediment transport model for the formation of river deltas, which belongs to the broad category of rule-based exploratory models. It has the
ability to resolve emergent channel behaviors including channel bifurcation, avulsion and migration. Sediment transport distinguishes two types of sediment: sand and mud, which have different
transport and deposition/erosion rules. Stratigraphy is recorded as the sand fraction in layers. Best usage of DeltaRCM is the investigation of autogenic processes in response to external forcings.
Demeter is an open source Python package that was built to disaggregate projections of future land allocations generated by an integrated assessment model (IAM). Projected land allocation from IAMs
is traditionally transferred to Earth System Models (ESMs) in a variety of gridded formats and spatial resolutions as inputs for simulating biophysical and biogeochemical fluxes. Existing tools for
performing this translation generally require a number of manual steps which introduces error and is inefficient. Demeter makes this process seamless and repeatable by providing gridded land use and
land cover change (LULCC) products derived directly from an IAM—in this case, the Global Change Assessment Model (GCAM)—in a variety of formats and resolutions commonly used by ESMs.
Depth-Discharge and Bedload Calculator, uses: # Wright-Parker formulation for flow resistance (without stratification correction) # Ashida-Michiue formulation for bedload transport.
Depth-Discharge and Total Load Calculator, uses: # Wright-Parker formulation for flow resistance, # Ashida-Michiue formulation for bedload transport, # Wright-Parker formulation (without
stratification) for suspended load.
Derived from MOSART-WM (Model for Scale Adaptive River Transport with Water Management), mosasrtwmpy is a large-scale river-routing Python model used to study riverine dynamics of water, energy, and
biogeochemistry cycles across local, regional, and global scales. The water management component represents river regulation through reservoir storage and release operations, diversions from
reservoir releases, and allocation to sectoral water demands. The model allows an evaluation of the impact of water management over multiple river basins at once (global and continental scales) with
consistent representation of human operations over the full domain.
Diffusion of marine sediments
Directs flow by the D infinity method (Tarboton, 1997). Each node is assigned two flow directions, toward the two neighboring nodes that are on the steepest subtriangle. Partitioning of flow is done
based on the aspect of the subtriangle.
Directs flow by the multiple flow direction method. Each node is assigned multiple flow directions, toward all of the N neighboring nodes that are lower than it. If none of the neighboring nodes are
lower, the location is identified as a pit. Flow proportions can be calculated as proportional to slope or proportional to the square root of slope, which is the solution to a steady kinematic wave.
Dorado is a Python package for simulating passive Lagrangian particle transport over flow-fields from any 2D shallow-water hydrodynamic model using a weighted random walk methodology.
DynEarthSol3D (Dynamic Earth Solver in Three Dimensions) is a flexible, open-source finite element code that solves the momentum balance and the heat transfer in Lagrangian form using unstructured
meshes. It can be used to study the long-term deformation of Earth's lithosphere and problems alike.
DynQual is a high-spatio-temporal-resolution surface water quality model, which can be used to simulate water temperature; concentrations of total dissolved solids to represent salinity pollution;
biological oxygen demand to represent organic pollution; and fecal coliform as a coarse indicator for pathogen pollution.
ECSimpleSnow is a simple snow model that employs an empirical algorithm to melt or accumulate snow based on surface temperature and precipitation that has fallen since the previous analysis step.
EF5 was created by the Hydrometeorology and Remote Sensing Laboratory at the University of Oklahoma. The goal of EF5 is to have a framework for distributed hydrologic modeling that is user friendly,
adaptable, expandable, all while being suitable for large scale (e.g. continental scale) modeling of flash floods with rapid forecast updates. Currently EF5 incorporates 3 water balance models
including the Sacramento Soil Moisture Accouning Model (SAC-SMA), Coupled Routing and Excess Storage (CREST), and hydrophobic (HP). These water balance models can be coupled with either linear
reservoir or kinematic wave routing.
ELCIRC is an unstructured-grid model designed for the effective simulation of 3D baroclinic circulation across river-to-ocean scales. It uses a finite-volume/finite-difference Eulerian-Lagrangian
algorithm to solve the shallow water equations, written to realistically address a wide range of physical processes and of atmospheric, ocean and river forcings. The numerical algorithm is low-order,
but volume conservative, stable and computationally efficient. It also naturally incorporates wetting and drying of tidal flats. ELCIRC has been extensively tested against standard ocean/coastal
benchmarks, and is starting to be applied to estuaries and continental shelves around the world.
Ecopath with Ecosim (EwE) is an ecological modeling software suite for personal computers. EwE has three main components: Ecopath – a static, mass-balanced snapshot of the system; Ecosim – a time
dynamic simulation module for policy exploration; and Ecospace – a spatial and temporal dynamic module primarily designed for exploring impact and placement of protected areas. The Ecopath software
package can be used to: *Address ecological questions; *Evaluate ecosystem effects of fishing; *Explore management policy options; *Evaluate impact and placement of marine protected areas; *Evaluate
effect of environmental changes.
Erode is a raster-based, fluvial landscape evolution model. The newest version (3.0) is written in Python and contains html help pages when running the program through the CSDMS Modeling Tool CMT
Erode-D8-Global is a raster, D8-based fluvial landscape evolution model (LEM)
Exposures to heat and sunlight can be simulated and the resulting signals shown. For a detailed description of the underlying luminescence rate equations, or to cite your use of LuSS, please use
Brown, (2020).
Extended description for SINUOUS - Meander Evolution Model. The basic model simulates planform evolution of a meandering river starting from X,Y coordinates of centerline nodes, with specification of
cross-sectional and flow parameters. If the model is intended to simulate evolution of an existing river, the success of the model can be evaluated by the included area between the simulated and the
river centerline. In addition, topographic evolution of the surrounding floodplain can be simulated as a function of existing elevation, distance from the nearest channel, and time since the channel
migrated through that location. Profile evolution of the channel can also be modeled by backwater flow routing and bed sediment transport relationships.
FACET is a Python tool that uses open source modules to map the floodplain extent and derive reach-scale summaries of stream and floodplain geomorphic measurements from high-resolution digital
elevation models (DEMs). Geomorphic measurements include channel width, stream bank height, floodplain width, and stream slope.<br>Current tool functionality is only meant to process DEMs within the
Chesapeake Bay and Delaware River watersheds. FACET was developed to batch process 3-m resolution DEMs in the Chesapeake Bay and Delaware River watersheds. Future updates to FACET will allow users to
process DEMs outside of the Chesapeake and Delaware basins.<br>FACET allows the user to hydrologically condition the DEM, generate the stream network, select one of two options for stream bank
identification, map the floodplain extent using a Height Above Nearest Drainage (HAND) approach, and calculate stream and floodplain metrics using three approaches.
FUNWAVE is a phase-resolving, time-stepping Boussinesq model for ocean surface wave propagation in the nearshore.
FVCOM is a prognostic, unstructured-grid, finite-volume, free-surface, 3-D primitive equation coastal ocean circulation model developed by UMASSD-WHOI joint efforts. The model consists of momentum,
continuity, temperature, salinity and density equations and is closed physically and mathematically using turbulence closure submodels. The horizontal grid is comprised of unstructured triangular
cells and the irregular bottom is preseented using generalized terrain-following coordinates. The General Ocean Turbulent Model (GOTM) developed by Burchard’s research group in Germany (Burchard,
2002) has been added to FVCOM to provide optional vertical turbulent closure schemes. FVCOM is solved numerically by a second-order accurate discrete flux calculation in the integral form of the
governing equations over an unstructured triangular grid. This approach combines the best features of finite-element methods (grid flexibility) and finite-difference methods (numerical efficiency and
code simplicity) and provides a much better numerical representation of both local and global momentum, mass, salt, heat, and tracer conservation. The ability of FVCOM to accurately solve scalar
conservation equations in addition to the topological flexibility provided by unstructured meshes and the simplicity of the coding structure has make FVCOM ideally suited for many coastal and
interdisciplinary scientific applications.
Fall velocity for spheres. Uses formulation of Dietrich (1982)
Finite difference approximations are great for modeling the erosion of landscapes. A paper by Densmore, Ellis, and Anderson provides details on application of landscape evolution models to the Basin
and Range (USA) using complex rulesets that include landslides, tectonic displacements, and physically-based algorithms for hillslope sediment transport and fluvial transport. The solution given here
is greatly simplified, only including the 1D approximation of the diffusion equation. The parallel development of the code is meant to be used as a class exercise
Finite difference solution allows for calculations of flexural response in regions of variable elastic thickness / flexural rigidity. The direct solution technique means that it takes time to
populate a cofactor matrix, but that once this has been done, flexural solutions may be obtained rapidly via a Thomas algorithm. This makes it less good for an individual solution where an iterative
approach may be more computationally efficient, but better for modeling where elastic thickness does not change (meaning that you do not need to create a new cofactor matrix) but loads do.
Finite element process based simulation model for fluid flow, clastic, carbonate and evaporate sedimentation.
For each time step, this component calculates an infiltration rate for a given model location and updates surface water depths. Based on the Green-Ampt method, it follows the form of Julien et al.,
Fortran 95 routines to model the ocean carbonate system (mocsy). Mocsy take as input dissolved inorganic carbon CT and total alkalinity AT, the only two tracers of the ocean carbonate system that are
unaffected by changes in temperature and salinity and conservative with respect to mixing, properties that make them ideally suited for ocean carbon models. With basic thermodynamic equilibria, mocsy
compute surface-ocean pCO2 in order to simulate air-sea CO2 fluxes. The mocsy package goes beyond the OCMIP code by computing all other carbonate system variables (e.g., pH, CO32-, and CaCO3
saturation states) and by doing so throughout the water column.
FuzzyReef is a three-dimensional (3D) numerical stratigraphic model that simulates the development of microbial reefs using fuzzy logic (multi-valued logic) modeling methods. The flexibility of the
model allows for the examination of a large number of variables. This model has been used to examine the importance of local environmental conditions and global changes on the frequency of reef
development relative to the temporal and spatial constraints from Upper Jurassic (Oxfordian) Smackover reef datasets from two Alabama oil fields. The fuzzy model simulates the deposition of reefs and
carbonate facies through integration of local and global variables. Local-scale factors include basement relief, sea-level change, climate, latitude, water energy, water depth, background
sedimentation rate, and substrate conditions. Regional and global-scale changes include relative sea-level change, climate, and latitude.
GENESIS calculates shoreline change produced by statial and temporal differences in longshore sand transport produced by breaking waves. The shoreline evolution portion of the numerical modeling
system is based on one-line shoreline change theory, which assumes that the beach profile shape remains unchanged, allowing shoreline change to be described uniquely in terms of the translation of a
single point (for example, Mean High Water shoreline) on the profile.
GEOMBEST is a morphological-behaviour model that simulates the evolution of coastal morphology and stratigraphy resulting from changes in sea level and sediment volume within the shoreface, barrier,
and estuary.
GEOMBEST++ is a morphological-behaviour model that simulates the evolution of coastal morphology and stratigraphy resulting from changes in sea level and sediment volume within the shoreface,
barrier, and estuary. GEOMBEST++ builds on previous iterations (i.e. GEOMBEST+) by incorporating the effects of waves into the backbarrier, providing a more physical basis for the evolution of the
bay bottom and introducing wave erosion of marsh edges.
GEOMBEST++Seagrass is a morphological-behaviour model that simulates the evolution of coastal morphology and stratigraphy resulting from changes in sea level and sediment volume within the shoreface,
barrier, and estuary. GEOMBEST++Seagrass builds on previous iterations (i.e. GEOMBEST, GEOMBEST+, and GEOMBEST++) by incorporating seagrass dynamics into the back-barrier bay.
GEOtop accommodates very complex topography and, besides the water balance integrates all the terms in the surface energy balance equation. For saturated and unsaturated subsurface flow, it uses the
3D Richards’ equation. An accurate treatment of radiation inputs is implemented in order to be able to return surface temperature. The model GEOtop simulates the complete hydrological balance in a
continuous way, during a whole year, inside a basin and combines the main features of the modern land surfaces models with the distributed rainfall-runoff models. The new 0.875 version of GEOtop
introduces the snow accumulation and melt module and describes sub-surface flows in an unsaturated media more accurately. With respect to the version 0.750 the updates are fundamental: the codex is
completely eviewed, the energy and mass parametrizations are rewritten, the input/output file set is redifined. GEOtop makes it possible to know the outgoing discharge at the basin's closing section,
to estimate the local values at the ground of humidity, of soil temperature, of sensible and latent heat fluxes, of heat flux in the soil and of net radiation, together with other hydrometeorlogical
distributed variables. Furthermore it describes the distributed snow water equivalent and surface snow temperature. GEOtop is a model based on the use of Digital Elevation Models (DEMs). It makes
also use of meteorological measurements obtained thought traditional instruments on the ground. Yet, it can also assimilate distributed data like those coming from radar measurements, from satellite
terrain sensing or from micrometeorological models.
GIPL(Geophysical Institute Permafrost Laboratory) is an implicit finite difference one-dimensional heat flow numerical model. The GIPL model uses the effect of snow layer and subsurface soil thermal
properties to simulate ground temperatures and active layer thickness (ALT) by solving the 1D heat diffusion equation with phase change. The phase change associated with freezing and thawing process
occurs within a range of temperatures below 0 degree centigrade, and is represented by the unfrozen water curve (Romanovsky and Osterkamp 2000). The model employs finite difference numerical scheme
over a specified domain. The soil column is divided into several layers, each with distinct thermo-physical properties. The GIPL model has been successfully used to map permafrost dynamics in Alaska
and validated using ground temperature measurements in shallow boreholes across Alaska (Nicolsky et al. 2009, Jafarov et al. 2012, Jafarov et al. 2013, Jafarov et al. 2014).
GSFLOW was a coupled model based on the integration of the U.S. Geological Survey Precipitation-Runoff Modeling System (PRMS, Leavesley and others, 1983) and the U.S. Geological Survey Modular
Groundwater Flow Model(MODFLOW-2005, Harbaugh, 2005). It was developed to simulate coupled groundwater/surface-water flow in one or more watersheds by simultaneously simulating flow across the land
surface, within subsurface saturated and unsaturated materials, and within streams and lakes.
Generates alluvial stratigraphy by channel migration and avulsion. Channel migration is handled via a random walk. Avulsions occur when the channel superelevates. Channels can create levees.
Post-avulsion channel locations chosen at random, or based on topography.
GeoFlood, a new open-source software package for solving shallow water equations (SWE) on a quadtree hierarchy of mapped, logically Cartesian grids managed by the parallel, adaptive library
Glimmer is an open source (GPL) three-dimensional thermomechanical ice sheet model, designed to be interfaced to a range of global climate models. It can also be run in stand-alone mode. Glimmer was
developed as part of the NERC GENIE project (www.genie.ac.uk). It's development follows the theoretical basis found in Payne (1999) and Payne (2001). Glimmer's structure contains numerous software
design strategies that make it maintainable, extensible, and well documented.
Grain Size Distribution Statistics Calculator
Gridded Surface Subsurface Hydrologic Analysis (GSSHA) is a grid-based two-dimensional hydrologic model. Features include 2D overland flow, 1D stream flow, 1D infiltration, 2D groundwater, and full
coupling between the groundwater, vadoze zone, streams, and overland flow. GSSHA can run in both single event and long-term modes. The fully coupled groundwater to surfacewater interaction allows
GSSHA to model both Hortonian and Non-Hortonian basins. New features of version 2.0 include support for small lakes and detention basins, wetlands, improved sediment transport, and an improved stream
flow model. GSSHA has been successfully used to predict soil moistures as well as runoff and flooding.
Gridded water balance model using climate input forcings that calculate surface and subsurface runoff and ground water recharge for each grid cell. The surface and subsurface runoff is propagated
horizontally along a prescribed gridded network using Musking type horizontal transport.
HYPE is a semi-distributed hydrological model for water and water quality. It simulates water and nutrient concentrations in the landscape at the catchment scale. Its spatial division is related to
catchments and sub-catchments, land use or land cover, soil type and elevation. Within a catchment the model will simulate different compartments; soil including shallow groundwater, rivers and
lakes. It is a dynamical model forced with time series of precipitation and air temperature, typically on a daily time step. Forcing in the form of nutrient loads is not dynamical. Example includes
atmospheric deposition, fertilizers and waste water.
Here, we present a Python tool that includes a comprehensive set of relations that predicts the hydrodynamics, bed elevation and the patterns of channels and bars in mere seconds. Predictions are
based on a combination of empirical relations derived from natural estuaries, including a novel predictor for cross-sectional depth distributions, which is dependent on the along-channel width
profile. Flow velocity, an important habitat characteristic, is calculated with a new correlation between depth below high water level and peak tidal flow velocity, which was based on spatial
numerical modelling. Salinity is calculated from estuarine geometry and flow conditions. The tool only requires an along-channel width profile and tidal amplitude, making it useful for quick
assessments, for example of potential habitat in ecology, when only remotely-sensed imagery is available.
HexWatershed is a mesh independent flow direction model for hydrologic models. It can be run at both regional and global scales. The unique feature of HexWatershed is that it supports both structured
and unstructured meshes.
High order two dimensional simulations of turbidity currents using DNS of incompressible Navier-Stokes and transport equations.
Hillslope diffusion component in the style of Carretier et al. (2016, ESurf), and Davy and Lague (2009). Works on regular raster-type grid (RasterModelGrid, dx=dy). To be coupled with
FlowDirectorSteepest for the calculation of steepest slope at each timestep.
Hillslope evolution using a Taylor Series expansion of the Andrews-Bucknam formulation of nonlinear hillslope flux derived following following Ganti et al., 2012. The flux is given as: qs = KS ( 1 +
(S/Sc)**2 + (S / Sc)**4 + .. + (S / Sc)**2(n - 1) ) where K is is the diffusivity, S is the slope, Sc is the critical slope, and n is the number of terms. The default behavior uses two terms to
produce a flux law as described by Equation 6 of Ganti et al., (2012).
Hillslope sediment flux uses a Taylor Series expansion of the Andrews-Bucknam formulation of nonlinear hillslope flux derived following following Ganti et al., 2012 with a depth dependent component
inspired Johnstone and Hilley (2014). The flux :math:`q_s` is given as: q_s = DSH^* ( 1 + (S/S_c)^2 + (S/Sc_)^4 + .. + (S/S_c)^2(n-1) ) (1.0 - exp( H / H^*) where :math:`D` is is the diffusivity,
:math:`S` is the slope, :math:`S_c` is the critical slope, :math:`n` is the number of terms, :math:`H` is the soil depth on links, and :math:`H^*` is the soil transport decay depth. The default
behavior uses two terms to produce a slope dependence as described by Equation 6 of Ganti et al., (2012).This component will ignore soil thickness located at non-core nodes.
HydroCNHS is an open-source Python package supporting four Application Programming Interfaces (APIs) that enable users to integrate their human decision models, which can be programmed with the
agent-based modeling concept, into the HydroCNHS.
HydroPy model is a revised version of an established global hydrological model (GHM), the Max Planck Institute for Meteorology's Hydrology Model (MPI-HM). Being rewritten in Python, the HydroPy model
requires much less effort in maintenance and new processes can be easily implemented.
HydroTrend v.3.0 is a climate-driven hydrological water balance and transport model that simulates water discharge and sediment load at a river outlet.
Hydrological Simulation Program - FORTRAN (HSPF) is a comprehensive package for simulation of watershed hydrology and water quality for both conventional and toxic organic pollutants (1,2). This
model can simulate the hydrologic, and associated water quality, processes on pervious and impervious land surfaces and in streams and well-mixed impoundments. HSPF incorporates the watershed-scale
ARM and NPS models into a basin-scale analysis framework that includes fate and transport in one-dimensional stream channels. It is the only comprehensive model of watershed hydrology and water
quality that allows the integrated simulation of land and soil contaminant runoff processes with in-stream hydraulic and sediment-chemical interactions. The result of this simulation is a time
history of the runoff flow rate, sediment load, and nutrient and pesticide concentrations, along with a time history of water quantity and quality at any point in a watershed. HSPF simulates three
sediment types (sand, silt, and clay) in addition to a single organic chemical and transformation products of that chemical. The transfer and reaction processes included are hydrolysis, oxidation,
photolysis, biodegradation, volatilization, and sorption. Sorption is modeled as a first-order kinetic process in which the user must specify a desorption rate and an equilibrium partition
coefficient for each of the three solids types. Resuspension and settling of silts and clays (cohesive solids) are defined in terms of shear stress at the sediment water interface. The capacity of
the system to transport sand at a particular flow is calculated and resuspension or settling is defined by the difference between the sand in suspension and the transport capacity. Calibration of the
model requires data for each of the three solids types. Benthic exchange is modeled as sorption/desorption and deposition/scour with surficial benthic sediments. Underlying sediment and pore water
are not modeled.
I am developing a GCM based on NCAR's WACCM model to studied the climate of the ancient Earth. WACCM has been linked with a microphysical model (CARMA). Some important issues to be examined are the
climate of the ancient Earth in light of the faint young Sun, reducing chemistry of the early atmosphere, and the production and radiative forcing of Titan-like photochemical hazes that likely
enshrouded the Earth at this time.
IDA formulates the task of determing the drainage area, given flow directions, as a system of implicit equations. This allows the use of iterative solvers, which have the advantages of being
parallelizable on distributed memory systems and widely available through libraries such as PETSc. Using the open source PETSc library (which must be downloaded and installed separately), IDA permits
large landscapes to be divided among processors, reducing total runtime and memory requirements per processor. It is possible to reduce run time with the use of an initial guess of the drainage area.
This can either be provided as a file, or use a serial algorithm on each processor to correctly determine the drainage area for the cells that do not receive flow from outside the processor's domain.
The hybrid IDA method, which is enabled with the -onlycrossborder option, uses a serial algorithm to solve for local drainage on each processor, and then only uses the parallel iterative solver to
incorporate flow between processor domains. This generally results in a significant reduction in total runtime. Currently only D8 flow directions are supported. Inputs and outputs are raw binary
ISSM is the result of a collaboration between the Jet Propulsion Laboratory and University of California at Irvine. Its purpose is to tackle the challenge of modeling the evolution of the polar ice
caps in Greenland and Antarctica. ISSM is open source and is funded by the NASA Cryosphere, GRACE Science Team, ICESat Research, ICESat-2 Research, NASA Sea-Level Change Team (N-SLCT), IDS
(Interdisciplinary Research in Earth Science), ESI (Earth Surface and Interior), and MAP (Modeling Analysis and Prediction) programs, JPL R&TD (Research, Technology and Development) and the National
Science Foundation
IceFlow simulates ice dynamics by solving equations for internal deformation and simplified basal sliding in glacial systems. It is designed for computational efficiency by using the shallow ice
approximation for driving stress, which it solves alongside basal sliding using a semi-implicit direct solver. IceFlow is integrated with GRASS GIS to automatically generate input grids from a
geospatial database.
Icepack is a Python package for simulating the flow of glaciers and ice sheets, as well as for solving glaciological data assimilation problems. The main goal for icepack is to produce a tool that
researchers and students can learn to use quickly and easily, whether or not they are experts in high-performance computing. Icepack is built on the finite element modeling library firedrake, which
implements the domain-specific language UFL for the specification of PDEs.
In order to extract channel networks, the flow connectivity across the grid must already be identified. This is typically done with the FlowAccumulator component. However, this component does not
require that the FlowAccumulator was used. Instead it expects that the following at-node grid fields will be present:<br> 'flow__receiver_node'<br> 'flow__link_to_receiver_node'<br> The
ChannelProfiler can work on grids that have used route-to-one or route-to-multiple flow directing.
It is a C-grid, isopycnal coordinate, primitive equation model, simulating the ocean by numerically solving the Boussinesq primitive equations in isopycnal vertical coordinates and general orthogonal
horizontal coordinates.
It is a mechanistic model that explains crop growth on the basis of the underlying processes, such as photosynthesis, respiration and how these processes are influenced by environmental conditions.
With WOFOST, you can calculate attainable crop production, biomass, water use, etc. for a location given knowledge about soil type, crop type, weather data and crop management factors (e.g. sowing
date). WOFOST has been used by many researchers over the World and has been applied for many crops over a large range of climatic and management conditions. WOFOST is one of the key components of the
European MARS crop yield forecasting system. In the Global Yield Gap Atlas (GYGA) WOFOST is used to estimate the untapped crop production potential on existing farmland based on current climate and
available soil and water resources.
It solves the linearized shallow water equations forced by tidal or other barotropic boundary conditions, wind or a density gradient using linear finite elements.
It tracks any number of different depth-averaged transport variables and is usually used in conjunction with QUODDY simulations.
LEMming tracks regolith and sediment fluxes, including bedrock erosion by streams and rockfall from steep slopes. Initial landscape form and stratigraphic structure are prescribed. Model grid cells
with slope angles above a threshold, and which correspond to the appropriate rock type, are designated as candidate sources for rockfall. Rockfall erosion of the cliffband is simulated by
instantaneously reducing the height of a randomly chosen grid cell that is susceptible to failure to that of its nearest downhill neighbor among the eight cells bordering it. This volume of rockfall
debris is distributed across the landscape below this cell according to rules that weight the likelihood of each downhill cell to retain rockfall debris. The weighting is based on local conditions
such as slope angle, topographic curvature, and distance and direction from the rockfall source. Rockfall debris and the bedrock types are each differentiated by the rate at which they weather to
regolith and by their fluvial erodibility. Regolith is moved according to transport rules mimicking hillslope processes (dependent on local slope angle), and bedload and suspended load transport
(based on stream power). Regolith and sediment transport are limited by available material; bedrock incision occurs (also based on stream power) where bare rock is exposed.
LEMming2 is a 2D, finite-difference landscape evolution model that simulates the retreat of hard-capped cliffs. It implements common unit-stream-power and linear/nonlinear-diffusion erosion equations
on a 2D regular grid. Arbitrary stratigraphy may be defined. Cliff retreat is facilitated by a cellular algorithm, and rockfall debris is distributed and redistributed to the angle of repose. It is a
standalone model written in Matlab with some C components. This repo contains the code used and described by Ward (2019) Lithosphere: "Dip, layer spacing, and incision rate controls on the formation
of strike valleys, cuestas, and cliffbands in heterogeneous stratigraphy". Given the inputs in that paper it should generate the same results.
LISFLOOD is a spatially distributed, semi-physical hydrological rainfall-runoff model that has been developed by the Joint Research Centre (JRC) of the European Commission in late 90ies. Since then
LISFLOOD has been applied to a wide range of applications such as all kind of water resources assessments looking at e.g. the effects of climate and land-use change as well as river regulation
measures. Its most prominent application is probably within the European Flood Awareness System (EFAS) operated under Copernicus Emergency Management System (EMS).
LOAD ESTimator (LOADEST) is a FORTRAN program for estimating constituent loads in streams and rivers. Given a time series of streamflow, additional data variables, and constituent concentration,
LOADEST assists the user in developing a regression model for the estimation of constituent load (calibration). Explanatory variables within the regression model include various functions of
streamflow, decimal time, and additional user-specified data variables. The formulated regression model then is used to estimate loads over a user-specified time interval (estimation). Mean load
estimates, standard errors, and 95 percent confidence intervals are developed on a monthly and(or) seasonal basis. The calibration and estimation procedures within LOADEST are based on three
statistical estimation methods. The first two methods, Adjusted Maximum Likelihood Estimation (AMLE) and Maximum Likelihood Estimation (MLE), are appropriate when the calibration model errors
(residuals) are normally distributed. Of the two, AMLE is the method of choice when the calibration data set (time series of streamflow, additional data variables, and concentration) contains
censored data. The third method, Least Absolute Deviation (LAD), is an alternative to maximum likelihood estimation when the residuals are not normally distributed. LOADEST output includes diagnostic
tests and warnings to assist the user in determining the appropriate estimation method and in interpreting the estimated loads. The LOADEST software and related materials (data and documentation) are
made available by the U.S. Geological Survey (USGS) to be used in the public interest and the advancement of science. You may, without any fee or cost, use, copy, modify, or distribute this software,
and any derivative works thereof, and its supporting documentation, subject to the USGS software User's Rights Notice.
Landlab component that computes 1D and 2D total incident shortwave radiation. This code also computes relative incidence shortwave radiation compared to a flat surface.
Landlab component that finds a neighbor node to laterally erode and calculates lateral erosion.
Landlab component that generates precipitation events using the rectangular Poisson pulse model described in Eagleson (1978, Water Resources Research). No particular units must be used, but it was
written with the storm units in hours (hr) and depth units in millimeters (mm).
Landlab component that implements a 1 and 2D lithospheric flexure model.
Landlab component that simulates detachment limited sediment transport is more general than the stream power component. Doesn't require the upstream node order, links to flow receiver and flow
receiver fields. Instead, takes in the discharge values on NODES calculated by the OverlandFlow class and erodes the landscape in response to the output discharge. As of right now, this component
relies on the OverlandFlow component for stability. There are no stability criteria implemented in this class. To ensure model stability, use StreamPowerEroder or FastscapeEroder components instead.
Landlab component that simulates net primary productivity, biomass and leaf area index at each cell based on inputs of root-zone average soil moisture.
Landlab component that simulates root-zone average soil moisture at each cell using inputs of potential evapotranspiration, live leaf area index, and vegetation cover. This component uses a single
soil moisture layer and models soil moisture loss through transpiration by plants, evaporation by bare soil, and leakage. The solution of water balance is based on Laio et. al 2001. The component
requires fields of initial soil moisture, rainfall input (if any), time to the next storm and potential transpiration.
Landlab is a Python software package for creating, assembling, and/or running 2D numerical models. Landlab was created to facilitate modeling in earth-surface dynamics, but it is general enough to
support a wide range of applications. Landlab provides three different capabilities: (1) A DEVELOPER'S TOOLKIT for efficiently building 2D models from scratch. The toolkit includes a powerful
GRIDDING ENGINE for creating, managing, and iterative updating data on 2D structured or unstructured grids. The toolkit also includes helpful utilities to handle model input and output. (2) A set of
pre-built COMPONENTS, each of which models a particular process. Components can be combined together to create coupled models. (3) A library of pre-built MODELS that have been created by combining
components together. To learn more, please visit http://landlab.github.io
Landscape evolution model. Computes evolution of topography under the action of rainfall and tectonics.
Life evolves alongside landscapes by biotic and abiotic processes under complex dynamics at Earth’s surface. Researchers who wish to explore these dynamics can use this component as a tool for them
to build landscape-life evolution models. Landlab components, including SpeciesEvolver are designed to work with a shared model grid. Researchers can build novel models using plug-and-play surface
process components to evolve the grid’s landscape alongside the life tracked by SpeciesEvolver. The simulated life evolves following customizable processes.
LinearDiffuser is a Landlab component that models soil creep using an explicit finite-volume solution to a 2D diffusion equation.
Lithospheric flexure solution for a broken plate. Load is assumed to be represented by equal width loading elements specified distance from broken edge of plate. Inclusion of sediments as part of the
restoring force effect is possible by choice of density assigned to density (2).
Lithospheric flexure solution for infinite plate. Load is assumed to be convolved with Greens function (unit load) response in order to calculate the net effect of the load. If desired, inclusion of
sediments as part of the restoring force effect can be controlled via density assigned to density (2). Each load element can have specified density and several loadings events can be incorporated.
Long term 2D morphodynamics of coastal areas, including tidal currents, wind waves, swell waves, storm surge, sand, mud, marsh vegetation, edge erosion, marsh ponding, and stratigraphy. The
CoastMorpho2D model includes the MarshMorpho2D model (which was previously uploaded on CSDMS)
Long-term ecomorphodynamic model of the initiation and development of tidal networks and of the adjacent marsh platform, accounting for vegetation influence and relative sea level rise effects
MARSSIM is a grid based, iterative framework that incorporates selectable modules, including: 1) flow routing, optionally including event-driven flow and evaporation from lakes in depression as a
function of relative aridity (Matsubara et al., 2011). Runoff can be spatially uniform or variably distributed. Stream channel morphology (width and depth) is parameterized as a function of effective
discharge; 2) bedrock weathering, following Equation 1; 3) spatially variable bedrock resistance to weathering and fluvial erosion, including 3-D stratigraphy and surficial coherent crusts; 4)
erosion of bedrock channels using either a stream power relationship (Howard, 1994) or sediment load scour (Sklar and Dietrich, 2004; Chatanantavet and Parker, 2009); 5) sediment routing in alluvial
channels including suspended/wash load and a single size of bedload. An optional sediment transport model simulates transport of multiple grain sizes of bedload with sorting and abrasion (Howard et
al., 2016); 6) geometric impact cratering modeling optionally using a database of martian fresh crater morphology; 7) vapor sublimation from or condensation on the land surface, with options for rate
control by the interaction between incident radiation, reflected light, and local topography; 8) mass wasting utilizing either the Howard (1994) or the Roering et al. (1999, 2001a) rate law. Bedrock
can be optionally weathered and mass wasted assuming a critical slope angle steeper than the critical gradient for regolith-mantled slopes. Mass wasted debris is instantaneously routed across exposed
bedrock, and the debris flux can be specified to erode the bedrock; 9) groundwater flow using the assumption of hydrostatic pressures and shallow flow relative to cell dimensions. Both recharge and
seepage to the surface are modeled. Seepage discharge can be modeled to transport sediment (seepage erosion) or to weather exposed bedrock (groundwater sapping); 10) deep-seated mass flows using
either Glen's law or Bingham rheology using a hydrostatic stress assumption; 11) eolian deposition and erosion in which the rate is determined by local topography; 12) lava flow and deposition from
one or multiple vents. These model components vary in degree to which they are based on established theory or utilize heuristic
MICOM is a primitive equation numerical model that describes the evolution of momentum, mass, heat and salt in the ocean.
MODFLOW 6 is an object-oriented program and framework developed to provide a platform for supporting multiple models and multiple types of models within the same simulation. This version of MODFLOW
is labeled with a "6" because it is the sixth core version of MODFLOW to be released by the USGS (previous core versions were released in 1984, 1988, 1996, 2000, and 2005). In the new design, any
number of models can be included in a simulation. These models can be independent of one another with no interaction, they can exchange information with one another, or they can be tightly coupled at
the matrix level by adding them to the same numerical solution. Transfer of information between models is isolated to exchange objects, which allow models to be developed and used independently of
one another. Within this new framework, a regional-scale groundwater model may be coupled with multiple local-scale groundwater models. Or, a surface-water flow model could be coupled to multiple
groundwater flow models. The framework naturally allows for future extensions to include the simulation of solute transport.
MODFLOW is a three-dimensional finite-difference ground-water model that was first published in 1984. It has a modular structure that allows it to be easily modified to adapt the code for a
particular application. Many new capabilities have been added to the original model. OFR 00-92 (complete reference below) documents a general update to MODFLOW, which is called MODFLOW-2000 in order
to distinguish it from earlier versions. MODFLOW-2000 simulates steady and nonsteady flow in an irregularly shaped flow system in which aquifer layers can be confined, unconfined, or a combination of
confined and unconfined. Flow from external stresses, such as flow to wells, areal recharge, evapotranspiration, flow to drains, and flow through river beds, can be simulated. Hydraulic
conductivities or transmissivities for any layer may differ spatially and be anisotropic (restricted to having the principal directions aligned with the grid axes), and the storage coefficient may be
heterogeneous. Specified head and specified flux boundaries can be simulated as can a head dependent flux across the model's outer boundary that allows water to be supplied to a boundary block in the
modeled area at a rate proportional to the current head difference between a "source" of water outside the modeled area and the boundary block. MODFLOW is currently the most used numerical model in
the U.S. Geological Survey for ground-water flow problems. In addition to simulating ground-water flow, the scope of MODFLOW-2000 has been expanded to incorporate related capabilities such as solute
transport and parameter estimation.
MOM6 is the latest generation of the Modular Ocean Model which is a numerical model code for simulating the ocean general circulation. MOM6 represents a major algorithmic departure from the previous
generations of MOM (up to and including MOM5). Most notably, it uses the Arbitrary-Lagrangian-Eulerian (ALE) algorithm in the vertical direction to allow the use of any vertical coordinate system
including, geo-potential coordinates (z or z*), isopycnal coordinates, terrain-following coordinates and hybrid-/user-defined coordinates. It is also based on the horizontal C-grid stencil, rather
than the B-grid used by earlier MOM versions.
MPeat2D incorporates realistic spatial variability on the peatland and allows for more significant insights into the interplay between these complex feedback mechanisms.
Makes use of fast Delaunay triangulation and Voronoi diagram calculations to represent surface processes on an irregular, dynamically evolving mesh. Processes include fluvial erosion, transport and
deposition, hillslope (diffusion) processes, flexural isostasy, orographic precipitation. Designed to model processes at the orogenic scale. Can be easily modified for other purposes by changing
process laws.
Matlab® code for paleo-hydrological flood flow reconstruction in a fluvial channel: first-order magnitude estimations of maximum average flow velocity, peak discharge, and maximum flow height from
boulder size and topographic input data (channel cross-section & channel bed slope).
Measure single reservoir performance using resilience, reliability, and vulnerability metrics; compute storage-yield-reliability relationships; determine no-fail Rippl storage with sequent peak
analysis; optimize release decisions using determinisitc and stochastic dynamic programming; evaluate inflow characteristics.
Model describing the morphodynamic evolution of vegetated coastal foredunes.
Model for fluvial fan-delta evolution, originally described by Sun et al. (2002) and later adapted by Limaye et al. (2023). The model routes water and sediment across a grid from a single inlet and
via a self-formed channel network, where local divergence in sediment flux drives bed elevation change. The model represents hydrodynamics using rules for flow routing and stress partitioning. At
large scales, other heuristics determine how channels branch and avulse, distributing water and sediment. The original model, designed for fluvial fan-deltas that debouch into standing water, is
extended to allow deposition of an alluvial fan in the absence of standing water. References: Limaye, A. B., Adler, J. B., Moodie, A. J., Whipple, K. X., & Howard, A. D. (2023). Effect of standing
water on formation of fan-shaped sedimentary deposits at Hypanis Valles, Mars. Geophysical Research Letters, 50(4), e2022GL102367. https://doi.org/10.1029/2022GL102367 Sun, T., Paola, C., Parker, G.,
& Meakin, P. (2002). Fluvial fan deltas: Linking channel processes with large-scale morphodynamics. Water Resources Research, 38(8), 26-1-26–10. https://doi.org/10.1029/2001WR000284
Model stream avulsion as random walk
ModelE is the GISS series of coupled atmosphere-ocean models, which provides the ability to simulate many different configurations of Earth System Models - including interactive atmospheric
chemsitry, aerosols, carbon cycle and other tracers, as well as the standard atmosphere, ocean, sea ice and land surface components.
Models temperature of 1-D lake-permafrost system through time, given input surface temperature and solar radiation. Model is fully implicit control volume scheme, and cell size can vary with depth.
Thermal conductivity and specific heat capacity are dependent on cell substrate (% soil and % ice) and temperature using the apparent heat capacity scheme where freezing/thawing occurs over a finite
temperature range and constants are modified to account for latent heat. Lake freezes and thaws depending on temperature; when no ice is present lake is fully mixed and can absorb solar radiation.
Upper 10 m substrate contains excess ice and, if thawed, can subside by this amount (lake then deepens by amount of subsidence). "Cell type" controls whether cell has excess ice, only pore space ice,
or is lake water.
Models the temporal and spatial distribution of the active layer thickness and temperature of permafrost soils. The underlying approximation accounts for effects of air temperature, snow cover,
vegatation, soil moisture, soil thermal properties to predict temperature at the ground surface and mean active layer thickness.
Morphodynamic river avulsion model, designed to be coupled with CEM and SEDFLUX3D
Mrip consists of a matrix representing the sea floor (25x25 m at this time). Blocks in the matrix are picked up (or deposited) according to transport rules or equations (users choice) and moved with
the flow. The user-determined flow is altered, depending on the height and slope of the bed, thus creating feedback.
NearCoM predicts waves, currents, sediment transport and bathymetric change in the nearshore ocean, between the shoreline and about 10 m water depth. The model consists of a "backbone", i.e., the
master program, handling data input and output as well as internal storage, together with a suite of "modules": wave module, circulation module and sediment transport module.
Network-based modeling framework of Czuba and Foufoula-Georgiou as applied to bed-material sediment transport. This code is capable of reproducing the results (with some work by the end user)
described in the following publications: Czuba, J.A., and E. Foufoula-Georgiou (2014), A network-based framework for identifying potential synchronizations and amplifications of sediment delivery in
river basins, Water Resources Research, 50(5), 3826–3851, doi:10.1002/2013WR014227. Czuba, J.A., and E. Foufoula-Georgiou (2015), Dynamic connectivity in a fluvial network for identifying hotspots of
geomorphic change, Water Resources Research, 51(3), 1401-1421, doi:10.1002/2014WR016139. Gran, K.B., and J.A. Czuba, (2017), Sediment pulse evolution and the role of network structure, Geomorphology,
277, 17-30, doi:10.1016/j.geomorph.2015.12.015. Czuba, J.A., E. Foufoula-Georgiou, K.B. Gran, P. Belmont, and P.R. Wilcock (2017), Interplay between spatially-explicit sediment sourcing, hierarchical
river-network structure, and in-channel bed-material sediment transport and storage dynamics, Journal of Geophysical Research - Earth Surface, 122(5), 1090-1120, doi:10.1002/2016JF003965. As of 20
March 2019, additional model codes were added to the repository in the folder "Gravel_Bed_Dynamics" that extend the model to gravel bed dynamics. The new methods for gravel bed dynamics are described
in: Czuba, J.A. (2018), A Lagrangian framework for exploring complexities of mixed-size sediment transport in gravel-bedded river networks, Geomorphology, 321, 146-152, doi:10.1016/
j.geomorph.2018.08.031. And an application to Clear Creek/Tushar Mountains in Utah is described in: Murphy, B.P., J.A. Czuba, and P. Belmont (2019), Post-wildfire sediment cascades: a modeling
framework linking debris flow generation and network-scale sediment routing, Earth Surface Processes and Landforms, 44(11), 2126-2140, doi:10.1002/esp.4635. Note: the application code and data files
for Murphy et al., 2019 are included in the repository as example files. As of 24 September 2020, this code has largely been converted to Python and has been incorporated into Landlab version 2.2 as
the NetworkSedimentTransporter. See: Pfeiffer, A.M., K.R. Barnhart, J.A. Czuba, and E.W.H. Hutton (2020), NetworkSedimentTransporter: A Landlab component for bed material transport through river
networks, Journal of Open Source Software, 5(53), 2341, doi:10.21105/joss.02341. This initial release is the core code, but development is ongoing to make the data preprocessing, model interface, and
exploration of model results more user friendly. All future developments will be in the Landlab/Python version of the code instead of this Matlab version.
Network-based modeling framework of Czuba and Foufoula-Georgiou as applied to nitrate and organic carbon on a wetland-river network. This code is capable of reproducing the results (with some work of
commenting/uncommenting code by the end user) described in the following publication: Czuba, J.A., A.T. Hansen, E. Foufoula-Georgiou, and J.C. Finlay (2018), Contextualizing wetlands within a river
network to assess nitrate removal and inform watershed management, Water Resources Research, 54(2), 1312-1337, doi:10.1002/2017WR021859.
Nonlinear three dimensional simulations of miscible Hele-Shaw flows using DNS of incompressible Navier-Stokes and transport equations.
Oceananigans.jl is designed for high-resolution simulations in idealized geometries and supports direct numerical simulation, large eddy simulation, arbitrary numbers of active and passive tracers,
and linear and nonlinear equations of state for seawater.
One dimensional model for the coupled long-term evolution of salt marshes and tidal flats. The model framework includes tidal currents, wind waves, sediment erosion and deposition, as well as the
effect of vegetation on sediment dynamics. The model is used to explore the evolution of the marsh boundary under different scenarios of sediment supply and sea level rise. Time resolution 30 min,
simulation length about 100 years.
One-Dimensional Transport with Equilibrium Chemistry (OTEQ): A Reactive Transport Model for Streams and Rivers OTEQ is a mathematical simulation model used to characterize the fate and transport of
waterborne solutes in streams and rivers. The model is formed by coupling a solute transport model with a chemical equilibrium submodel. The solute transport model is based on OTIS, a model that
considers the physical processes of advection, dispersion, lateral inflow, and transient storage. The equilibrium submodel is based on MINTEQ, a model that considers the speciation and complexation
of aqueous species, acid-base reactions, precipitation/dissolution, and sorption. Within OTEQ, reactions in the water column may result in the formation of solid phases (precipitates and sorbed
species) that are subject to downstream transport and settling processes. Solid phases on the streambed may also interact with the water column through dissolution and sorption/desorption reactions.
Consideration of both mobile (waterborne) and immobile (streambed) solid phases requires a unique set of governing differential equations and solution techniques that are developed herein. The
partial differential equations describing physical transport and the algebraic equations describing chemical equilibria are coupled using the sequential iteration approach. The model's ability to
simulate pH, precipitation/dissolution, and pH-dependent sorption provides a means of evaluating the complex interactions between instream chemistry and hydrologic transport at the field scale. OTEQ
is generally applicable to solutes which undergo reactions that are sufficiently fast relative to hydrologic processes ("Local Equilibrium"). Although the definition of "sufficiently fast" is highly
solute and application dependent, many reactions involving inorganic solutes quickly reach a state of chemical equilibrium. Given a state of chemical equilibrium, inorganic solutes may be modeled
using OTEQ's equilibrium approach. This equilibrium approach is facilitated through the use of an existing database that describes chemical equilibria for a wide range of inorganic solutes. In
addition, solute reactions not included in the existing database may be added by defining the appropriate mass-action equations and the associated equilibrium constants. As such, OTEQ provides a
general framework for the modeling of solutes under the assumption of chemical equilibrium. Despite this generality, most OTEQ applications to date have focused on the transport of metals in streams
and small rivers. The OTEQ documentation is therefore focused on metal transport. Potential model users should note, however, that additional applications are possible.
One-Dimensional Transport with Inflow and Storage (OTIS): A Solute Transport Model for Streams and Rivers OTIS is a mathematical simulation model used to characterize the fate and transport of
water-borne solutes in streams and rivers. The governing equation underlying the model is the advection-dispersion equation with additional terms to account for transient storage, lateral inflow,
first-order decay, and sorption. This equation and the associated equations describing transient storage and sorption are solved using a Crank-Nicolson finite-difference solution. OTIS may be used in
conjunction with data from field-scale tracer experiments to quantify the hydrologic parameters affecting solute transport. This application typically involves a trial-and-error approach wherein
parameter estimates are adjusted to obtain an acceptable match between simulated and observed tracer concentrations. Additional applications include analyses of nonconservative solutes that are
subject to sorption processes or first-order decay. OTIS-P, a modified version of OTIS, couples the solution of the governing equation with a nonlinear regression package. OTIS-P determines an
optimal set of parameter estimates that minimize the squared differences between the simulated and observed concentrations, thereby automating the parameter estimation process.
OpenFOAM (Open Field Operation and Manipulation) is a toolbox for the development of customized numerical solvers, and pre-/post-processing utilities for the solution of continuum mechanics problems,
including computational fluid dynamics.
Optimization Technique in Transient Evolution of Rivers (OTTER). This models a 1D river profile while incorporating a algorithm for dynamic channel width. The channel width algorithm dynamically
adjusts channel geometry in response to values of water discharge, rock-uplift/erosion, and sediment supply. It operates by calculating the current shear stress (no wide channel assumption), the
shear stress if channel width is slightly larger, and shear stress for a slightly narrower channel. Using these values, erosion potential is calculated for all three scenarios (no change in width,
slightly wider, slightly narrower) and the one that generates the maximum erosion rate dictates the direction of channel change. See Yanites, 2018 JGR for further information.
OrderID is a method that takes thickness and facies data from a vertical succession of strata and tests for the presence of order in the strata
Originally developed for modeling tsunami generation, propagation, and inundation. Also used for storm surge modeling and overland flooding (e.g. dam break problems). Uses adaptive mesh refinement to
allow much greater spatial resolutions in some regions than others, and to automatically follow dynamic evolution of waves or floods. Uses high-resolution finite volume methods that robustly handle
wetting and drying. The package also includes tools for working with geophysical data including topography DEMs, earthquake source models for tsunami generation, and observed gauge data. The
simulation code is in Fortran with OpenMP for shared memory parallelization, and Python for the user interface, visualization, and data tools.
PCR-GLOBWB 2 has been fully rewritten in Python and PCRaster Python and has a modular structure, allowing easier replacement, maintenance, and development of model components. PCR-GLOBWB 2 has been
implemented at 5arcmin resolution, but a version parameterized at 30arcmin resolution is also available.
PHREEQC implements several types of aqueous models: two ion-association aqueous models (the Lawrence Livermore National Laboratory model and WATEQ4F), a Pitzer specific-ion-interaction aqueous model,
and the SIT (Specific ion Interaction Theory) aqueous model. Using any of these aqueous models, PHREEQC has capabilities for (1) speciation and saturation-index calculations; (2) batch-reaction and
one-dimensional (1D) transport calculations with reversible and irreversible reactions, which include aqueous, mineral, gas, solid-solution, surface-complexation, and ion-exchange equilibria, and
specified mole transfers of reactants, kinetically controlled reactions, mixing of solutions, and pressure and temperature changes; and (3) inverse modeling, which finds sets of mineral and gas mole
transfers that account for differences in composition between waters within specified compositional uncertainty limits.
PIHM is a multiprocess, multi-scale hydrologic model where the major hydrological processes are fully coupled using the semi-discrete finite volume method. PIHM is a physical model for surface and
groundwater, “tightly-coupled” to a GIS interface. PIHMgis which is open source, platform independent and extensible. The tight coupling between GIS and the model is achieved by developing a shared
data-model and hydrologic-model data structure.
PISM is a hybrid shallow ice, shallow shelf model. PISM is designed to scale with increasing problem size by harnessing the computational power of supercomputing systems and by leveraging the
scalable software libraries that have been developed by the high-performance computing research community. The model combines two shallow (small depth-to-width ratio) stress balances, namely the
shallow-ice approximation (SIA) and the shallow-shelf approximation (SSA), which are computationally efficient schemes to simulate ice flow by internal deformation and ice-stream flow, respectively.
In PISM, deformational velocities from the SIA and sliding velocities from the SSA are weighted and averaged to achieve a smooth transition from shearing flow to sliding flow.
PRMS is a modular-design modeling system that has been developed to evaluate the impacts of various combinations of precipitation, climate, and land use on surface-water runoff, sediment yields, and
general basin hydrology
PSTSWM is a message-passing benchmark code and parallel algorithm testbed that solves the nonlinear shallow water equations on a rotating sphere using the spectral transform method. It is a parallel
implementation of STSWM to generate reference solutions for the shallow water test cases.
ParFlow is an open-source, object-oriented, parallel watershed flow model. It includes fully-integrated overland flow, the ability to simulate complex topography, geology and heterogeneity and
coupled land-surface processes including the land-energy budget, biogeochemistry and snow (via CLM). It is multi-platform and runs with a common I/O structure from laptop to supercomputer. ParFlow is
the result of a long, multi-institutional development history and is now a collaborative effort between CSM, LLNL, UniBonn and UCB. ParFlow has been coupled to the mesoscale, meteorological code ARPS
and the NCAR code WRF.
Physically-based fully-distributed hydrologic models try to simulate hydrologic state variables in space and time while using information regarding heterogeneity in climate, land use, topography and
hydrogeology. However incorporating a large number of physical data layers in the hydrologic model requires intensive data development and topology definitions.
Plot scale, spatially implicit model of tree throw on hillslopes. We couple an existing forest growth model with a couple simple equations for the transport of sediment caused by tree fall.
Potential Evapotranspiration Component calculates spatially distributed potential evapotranspiration based on input radiation factor (spatial distribution of incoming radiation) using chosen method
such as constant or Priestley Taylor. Ref: Xiaochi et. al. 2013 for 'Cosine' method and ASCE-EWRI Task Committee Report Jan 2005 for 'PriestleyTaylor' method. Note: Calling 'PriestleyTaylor' method
would generate/overwrite shortwave & longwave radiation fields.
Predicts 1D, unsteady, nonlinear, gradually varied flow
Program for backwater calculations in open channel flow
Provides the FlowAccumulator component which accumulates flow and calculates drainage area. FlowAccumulator supports multiple methods for calculating flow direction. Optionally a depression finding
component can be specified and flow directing, depression finding, and flow routing can all be accomplished together.
QDSSM is a 3D cellular, forward numerical model coded in Fortran90 that simulates landscape evolution and stratigraphy as controlled by changes in sea-level, subsidence, discharge and bedload flux.
The model includes perfect and imperfect sorting modules of grain size and allows stratigraphy to be build over time spans of 1000 to million of years.
QTCMs are models of intermediate complexity suitable for the modeling of tropical climate and its variability. It occupies a niche among climate models between complex general circulation models and
simple models.
QUAL2K (or Q2K) is a river and stream water quality model that is intended to represent a modernized version of the QUAL2E (or Q2E) model (Brown and Barnwell 1987). Q2K is similar to Q2E in the
following respects: One dimensional. The channel is well-mixed vertically and laterally. * Steady state hydraulics. Non-uniform, steady flow is simulated. * Diurnal heat budget. The heat budget and
temperature are simulated as a function of meteorology on a diurnal time scale. * Diurnal water-quality kinetics. All water quality variables are simulated on a diurnal time scale. * Heat and mass
inputs. Point and non-point loads and abstractions are simulated.
QuickChi enables the rapid analysis of stream profiles at the global scale from SRTM data.
Quickly generates input files for and runs GSFLOW, the USGS integrated groundwater--surface-water model, and can be used to visualize the outputs of GSFLOW.
RCPWAVE is a 2D steady state monocromatic short wave model for simulating wave propagation over arbitrary bahymetry.
REF/DIF is a phase-resolving parabolic refraction-diffraction model for ocean surface wave propagation. It was originally developed by Jim Kirby and Tony Dalrymple starting in 1982, based on Kirby's
dissertation work. This work led to the development of REF/DIF 1, a monochromatic wave model.
REM mechanistically simulates channel bed aggradation/degradation and channel widening in river networks. It has successfully been applied to alluvial river systems to simulate channel change over
annual and decadal time scales. REM is also capable of running Monte Carlo simulations (in parallel to reduce computational time) to quantify uncertainty in model predictions.
RHESSys is a GIS-based, hydro-ecological modelling framework designed to simulate carbon, water, and nutrient fluxes. By combining a set of physically-based process models and a methodology for
partitioning and parameterizing the landscape, RHESSys is capable of modelling the spatial distribution and spatio-temporal interactions between different processes at the watershed scale.
ROMS is a Free-surface, terrain-following, orthogonal curvilinear, primitive equations ocean model. Its dynamical kernel is comprised of four separate models including the nonlinear, tangent linear,
representer tangent linear, and adjoint models. It has multiple model coupling (ESMF, MCT) and multiple grid nesting (composed, mosaics, refinement) capabilities. The code uses a coarse-grained
parallelization with both shared-memory (OpenMP) and distributed-memory (MPI) paradigms coexisting together and activated via C-preprocessing.
ROMS is a Free-surface, terrain-following, orthogonal curvilinear, primitive equations ocean model. Its dynamical kernel is comprised of four separate models including the nonlinear, tangent linear,
representer tangent linear, and adjoint models. It has multiple model coupling (ESMF, MCT) and multiple grid nesting (composed, mosaics, refinement) capabilities. The code uses a coarse-grained
parallelization with both shared-memory (OpenMP) and distributed-memory (MPI) paradigms coexisting together and activated via C-preprocessing.
RaVENS: Rain and Variable Evapotranspiration, Nieve, and Streamflow Simple "conceptual" hydrological model that may include an arbitrary number of linked linear reservoirs (soil-zone water,
groundwater, etc.) as well as snowpack (accumulation from precipitation with T<0; positive-degree-day melt) and evapotranspiration (from external input or Thorntwaite equation). It also includes a
water-balance component to adjust ET (typically the least known input) to ensure that P - Q - ET = 0 over the course of a water year. Other components plot data and compute the NSE (Nash–Sutcliffe
model efficiency coefficient).
Rabpro is a Python package to delineate watersheds, extract river flowlines and elevation profiles, and compute watershed statistics for any location on the Earth’s surface.
Relative wetness and factor-of-safety are based on the infinite slope stability model driven by topographic and soils inputs and recharge provided by user as inputs to the component. For each node,
component simulates mean relative wetness as well as the probability of saturation based on Monte Carlo simulation of relative wetness where the probability is the number of iterations with relative
wetness >= 1.0 divided by the number of iterations. Probability of failure for each node is also simulated in the Monte Carlo simulation as the number of iterations with factor-of-safety <= 1.0
divided by the number of iterations.
RivGraph is a Python package that automates the extraction and characterization of river channel networks from a user-provided binary image, or mask, of a channel network.
Rouse-Vanoni Equilibrium Suspended Sediment Profile Calculator
Routines pertaining to the paper published as: doi: 10.1073/pnas.1206785109
Routines pertaining to the paper published as: doi: 10.1137/S0036144504445765
Routines pertaining to the paper published as: doi: 10.1111/j.1365-246X.2008.03854.x
Routines pertaining to the paper published as: doi: 10.1016/j.acha.2012.12.001
Routines pertaining to the paper published as: doi: 10.1111/j.1365-246X.2006.03065.x
Run a hypopycnal sediment plume
Run a submarine debris flow
SBEACH is a numerical simulation model for predicting beach, berm, and dune erosion due to storm waves and water levels. It has potential for many applications in the coastal environment, and has
been used to determine the fate of proposed beach fill alternatives under storm conditions and to compare the performance of different beach fill cross-sectional designs.
SEDPAK provides a conceptual framework for modeling the sedimentary fill of basins by visualizing stratal geometries as they are produced between sequence boundaries. The simulation is used to
substantiate inferences drawn about the potential for hydrocarbon entrapment and accumulation within a basin. It is designed to model and reconstruct clastic and carbonate sediment geometries which
are produced as a response to changing rates of tectonic movement, eustasy, and sedimentation The simulation enables the evolution of the sedimentary fill of a basin to be tracked, defines the
chronostratigraphic framework for the deposition of these sediments, and illustrates the relationship between sequences and systems tracts seen in cores, outcrop, and well and seismic data.
SELFE is a new unstructured-grid model designed for the effective simulation of 3D baroclinic circulation across river-to-ocean scales. It uses a semi-implicit finite-element Eulerian-Lagrangian
algorithm to solve the shallow water equations, written to realistically address a wide range of physical processes and of atmospheric, ocean and river forcings.
SIBERIA simulates the evolution of landscapes under the action of runoff and erosion over long times scales.
SIGNUM (Simple Integrated Geomorphological Numerical Model) is a TIN-based landscape evolution model: it is capable of simulating sediment transport and erosion by river flow at different space and
time scales. It is a multi-process numerical model written in the Matlab high level programming environment, providing a simple and integrated numerical framework for the simulation of some important
processes that shape real landscapes. Particularly, at the present development stage, SIGNUM is capable of simulating geomorphological processes such as hillslope diffusion, fluvial incision,
tectonic uplift or changes in base-level and climate effects in terms of precipitation. A full technical description is reported in Refice et al. 2011 . The software runs under Matlab (it is tested
on releases from R2010a to R2011b). It is released under the GPL3 license.
SNAC can solve momentum and heat energy balance equations in 3D solid with complicated rheology. Lagrangian description of motion adopted in SNAC makes it easy to monitor surface deformation during a
crustal or continental scale tectonic event as well as introduce surface processes into a model.
SNOWPACK solves numerically the partial differential equations governing the mass, energy and momentum conservation within the snowpack using the finite-element method. The numerical model has been
constructed to handle the special problems of avalanche warning.
SPARROW (SPAtially Referenced Regressions On Watershed attributes) is a watershed modeling technique for relating water-quality measurements made at a network of monitoring stations to attributes of
the watersheds containing the stations. The core of the model consists of a nonlinear regression equation describing the non-conservative transport of contaminants from point and diffuse sources on
land to rivers and through the stream and river network. The model predicts contaminant flux, concentration, and yield in streams and has been used to evaluate alternative hypotheses about the
important contaminant sources and watershed properties that control transport over large spatial scales.
SPHysics is a Smoothed Particle Hydrodynamics (SPH) code written in fortran for the simulation of potentially violent free-surface hydrodynamics. For release version 1.0, the SPHysics code can
simulate various phenomena including wave breaking, dam breaks, sloshing, sliding objects, wave impact on a structure, etc.
SRH-1D (Sedimentation and River Hydraulics - One Dimension) is a one-dimensional mobile boundary hydraulic and sediment transport computer model for rivers and manmade canals. Simulation capabilities
include steady or unsteady flows, river control structures, looped river networks, cohesive and non-cohesive sediment transport, and lateral inflows. The model uses cross section based river
information. The model simulates changes to rivers and canals caused by sediment transport. It can estimate sediment concentrations throughout a waterway given the sediment inflows, bed material,
hydrology, and hydraulics of that waterway.
STWAVE (STeady State spectral WAVE) is an easy-to-apply, flexible, robust, half-plane model for nearshore wind-wave growth and propagation. STWAVE simulates depth-induced wave refraction and
shoaling, current-induced refraction and shoaling, depth- and steepness-induced wave breaking, diffraction, parametric wave growth because of wind input, and wave-wave interaction and white capping
that redistribute and dissipate energy in a growing wave field.
SWAN is a third-generation wave model that computes random, short-crested wind-generated waves in coastal regions and inland waters.
SWAT is the acronym for Soil and Water Assessment Tool, a river basin, or watershed, scale model developed by Dr. Jeff Arnold for the USDA Agricultural Research Service (ARS). SWAT was developed to
predict the impact of land management practices on water, sediment and agricultural chemical yields in large complex watersheds with varying soils, land use and management coditions over long periods
of time.
SYMPHONIE is a three-dimensional primitive equations coastal ocean model
SedCas was developed for a debris-flow prone catchment in the Swiss Alps (Illgraben). It consists of two connected sediment reservoirs on the hillslope and in the channel, where sediment transfer is
driven by (lumped) hydrological processes at the basin scale. Sediment is stochastically produced by shallow landslides and rock avalanches and delivered to the hillslope and channel reservoirs. From
there, it is evacuated out of the basin in the form of debris flows and sediment-laden floods.
SedPlume is an integral model, solving the conservation equations of volume, momentum, buoyancy and sediment flux along the path of a turbulent plume injected into stably stratified ambient fluid.
Sedimentation occurs from the plume when the radial component of the sediment fall velocity exceeds the entrainment velocity. When the plume reaches the surface, it is treated as a radially spreading
surface gravity current, for which exact solutions exist for the sediment deposition rate. Flocculation of silt and clay particles is modeled using empirical measurements of particle settling
velocities in fjords to adjust the settling velocity of fine-grained sediments.
Sedflux-2.0 is the newest version of the Sedflux basin-filling model. Sedflux-2.0 provides a framework within which individual process-response models of disparate time and space resolutions
communicate with one another to deliver multi grain sized sediment load across a continental margin.
Sedtrans05 is a sediment transport model for continental shelf and estuaries. It predicts the sediment transport at one location as function water depth, sediment type, current and waves (single
point model). It can be used as sediment transport module for larger 2D models. Five different transport equations are available for non-cohesive sediments (sand) and one algorithm for cohesive
Shoreline is a "line model" for modeling the evolution of a coastline as the result of wind/wave-driven longshore sediment transport. It is based on conservation of mass and a semi-empirical sediment
transport formula known as the CERC formula. This model was specifically adapted for modeling the evolution of the coastline near Barrow, Alaska.
SiStER (Simple Stokes solver with Exotic Rheologies) simulates lithosphere and mantle deformation with continuum mechanics: Stokes flow with large strains, strain localization, non-linear rheologies,
sharp contrasts in material properties, complex BCs.
SimClast is a basin-scale 3D stratigraphic model, which allows several interacting sedimentary environments. Processes included are; fluvial channel dynamics and overbank deposition, river plume
deposition, open marine currents, wave resuspension, nearshore wave induced longshore and crosshore transport. This combined modelling approach allows insight into the processes influencing the flux
of energy and clastic material and the effect of external perturbations in all environments. Many governing processes work on relatively small scales, e.g. in fluvial settings an avulsion is a
relatively localised phenomenon. Yet, they have a profound effect on fluvial architecture. This means that the model must mimic these processes, but at the same time maintain computational
efficiency. Additionally, long-term models use relatively large grid-sizing (km scale), as the area to be modelled is on the scale of continental margins. We solve this problem by implementing the
governing processes as sub-grid scale routines into the large-scale basin-filling model. This parameterization greatly refines morphodynamic behaviour and the resulting stratigraphy. This modelling
effort recreates realistic geomorphological and stratigraphic delta behaviour in river and wave-dominated settings.
Simulate overland flow using Bates et al. (2010). Landlab component that simulates overland flow using the Bates et al., (2010) approximations of the 1D shallow water equations to be used for 2D
flood inundation modeling. This component calculates discharge, depth and shear stress after some precipitation event across any raster grid. Default input file is named “overland_flow_input.txt’ and
is contained in the landlab.components.overland_flow folder.
Simulates circulation and sedimentation in a 2D turbulent plane jet and resulting delta growth
Simulates soil evolution on three spatial dimensions, explicit particle size distribution and temporal dimension (hence 5D prefix) as a function of: 1. Bedrock and soil physical weathering; 2.
Sediment transport by overland flow; 3. Soil Creep (diffusion); 4. Aeolian deposition.
Simulates the evolution of landscapes consisting of patches of high-flow-resistance vegetation and low-flow-resistance vegetation as a result of surface-water flow, peat accretion, gravitationally
driven erosion, and sediment transport by flow. Was developed for the freshwater Everglades but could also apply to coastal marshes or floodplains. Described in Larsen and Harvey, Geomorphology, 2010
and Larsen and Harvey, American Naturalist, 2010 in press.
Simulates wave and current supported sediment gravity flows along the seabed offshore of high discharge, fine sediment riverine sources. See Friedrichs & Scully, 2007. Continental Shelf Research, 27:
322-337, for example.
Single-path (steepest direction) flow direction finding on raster grids by the D8 method. This method considers flow on all eight links such that flow is possible on orthogonal and on diagonal links.
Smoothes noise in a DEM by finding the mean value of neighbouring cells and assigning it to the central cell. This approach deals well with non-gaussian distributed noise.
Spatially explicit model of the development and evolution of salt marshes, including vegetation influenced accretion and hydrodynamic determined channel erosion.
Steady-state hyperpycnal flow model.
Storm computes windfield for a cyclone
TOPMODEL is a physically based, distributed watershed model that simulates hydrologic fluxes of water (infiltration-excess overland flow, saturation overland flow, infiltration, exfiltration,
subsurface flow, evapotranspiration, and channel routing) through a watershed. The model simulates explicit groundwater/surface water interactions by predicting the movement of the water table, which
determines where saturated land-surface areas develop and have the potential to produce saturation overland flow.
TOPOG describes how water moves through landscapes; over the land surface, into the soil, through the soil and groundwater and back to the atmosphere via evaporation. Conservative solute movement and
sediment transport are also simulated. The primary strength of TOPOG is that it is based on a sophisticated digital terrain analysis model, which accurately describes the topographic attributes of
three-dimensional landscapes. It is intended for application to small catchments (up to 10 km2, and generally smaller than 1 km2). We refer to TOPOG as a "deterministic", "distributed-parameter"
hydrologic modelling package. The term "deterministic" is used to emphasise the fact that the various water balance models within TOPOG use physical reasoning to explain how the hydrologic system
behaves. The term "distributed-parameter" means that the model can account for spatial variability inherent in input parameters such as soil type, vegetation and climate.
TUGS is a 1D model that simulates the transport of gravel and sand in rivers. The model predicts the responses of a channel to changes made to the environment (e.g., sediment supply, hydrology, and
certain artifical changes made to the river). Output of the model include longitudinal profile, sediment flux, and grain size distributions in bedload, channel surface and subsurface.
TURBINS, a highly parallel modular code written in C, is capable of modeling gravity and turbidity currents interacting with complex topographies in two and three dimensions. Accurate treatment of
the complex geometry, implementation of an efficient and scalable parallel solver, i.e. multigrid solver via PETSc and HYPRE to solve the pressure Poisson equation, and parallel IO are some of the
features of TURBINS. TURBINS enables us to tackle problems involving the interaction of turbidity currents with complex topographies. It provides us with a numerical tool for quantifying the flow
field properties and sedimentation processes, e.g. energy transfer, dissipation, and wall shear stress, which are difficult to obtain even at laboratory scales. By benefiting from massively parallel
simulations, we hope to understand the underlying physics and processes related to the formation and deposition of particles due to the occurrence of turbidity currents.
TauDEM provides the following capability: •Development of hydrologically correct (pit removed) DEMs using the flooding approach •Calculates flow paths (directions) and slopes •Calculates contributing
area using single and multiple flow direction methods •Multiple methods for the delineation of stream networks including topographic form-based methods sensitive to spatially variable drainage
density •Objective methods for determination of the channel network delineation threshold based on stream drops •Delineation of watersheds and subwatersheds draining to each stream segment and
association between watershed and segment attributes for setting up hydrologic models •Specialized functions for terrain analysis Details of new parallel Version 5.0 of TauDEM •Restructured into a
parallel processing implementation of the TauDEM suite of tools •Works on Windows PCs, laptops and UNIX clusters •Multiple processes are not required, the parallel approach can run as multiple
processes within a single processor •Restructured into a set of standalone command line executable programs and an ArcGIS toolbox Graphical User Interface (GUI) •Command line executables are:
-Written in C++ using Argonne National Laboratory's MPICH2 library to implement message passing between multiple processes -Based on single set of source code for the command line execuables that is
platform independent and can be compiled for both Window's PC's and UNIX clusters
Terrainbento is a Python package for modeling the evolution of the surface of the Earth over geologic time (e.g., thousands to millions of years). Despite many decades of effort by the geomorphology
community, there is no one established governing equation for the evolution of topography. Terrainbento thus provides 28 alternative models that support hypothesis testing and multi-model analysis in
landscape evolution.
Terrapin (or TerraPIN) stands for "Terraces put into Numerics". It is a module that generates the expected terraces, both strath and fill, from prescribed river aggradation and degradation.
The Advanced Terrestrial Simulator (formerly sometimes known as the Arctic Terrestrial Simulator) is a code for solving ecosystem-based, integrated, distributed hydrology. Capabilities are largely
based on solving various forms of Richards equation coupled to a surface flow equation, along with the needed sources and sinks for ecosystem and climate models. This can (but need not) include
thermal processes (especially ice for frozen soils), evapo-transpiration, albedo-driven surface energy balances, snow, biogeochemistry, plant dynamics, deformation, transport, and much more. In
addition, we solve problems of reactive transport in both the subsurface and surface, leveraging external geochemical engines through the Alquimia interface.
The Agricultural Production Systems sIMulator (APSIM) is internationally recognized as a highly advanced simulator of agricultural systems. It contains a suite of modules which enable the simulation
of systems that cover a range of plant, animal, soil, climate and management interactions. APSIM is undergoing continual development, with new capability added to regular releases of official
versions. Its development and maintenance is underpinned by rigorous science and software engineering standards. The APSIM Initiative has been established to promote the development and use of the
science modules and infrastructure software of APSIM.
The Atmosphere-Ocean Model is a computer program that simulates the Earth's climate in three dimensions on a gridded domain. The Model requires two kinds of input, specified parameters and prognostic
variables, and generates two kinds of output, climate diagnostics and prognostic variables. The specified input parameters include physical constants, the Earth's orbital parameters, the Earth's
atmospheric constituents, the Earth's topography, the Earth's surface distribution of ocean, glacial ice, or vegetation, and many others. The time varying prognostic variables include fluid mass,
horizontal velocity, heat, water vapor, salt, and subsurface mass and energy fields.
The COAWST model (Coupled Ocean-Atmosphere-Wave-Sediment Transport) is a numerical modeling system that integrates different physical processes to simulate the interaction between the ocean,
atmosphere, waves, and sediment transport in coastal environments. COAWST is designed to study complex coastal systems and their responses to various natural and human-induced forces, such as storms,
sea level rise, and sediment dynamics.
The Coastline Evolution Model (CEM) addresses predominately sandy, wave-dominated coastlines on time-scales ranging from years to millenia and on spatial scales ranging from kilometers to hundreds of
kilometers. Shoreline evolution results from gradients in wave-driven alongshore sediment transport. At its most basic level, the model follows the standard 'one-line' modeling approach, where the
cross-shore dimension is collapsed into a single data point. However, the model allows the plan-view shoreline to take on arbitrary local orientations, and even fold back upon itself, as complex
shapes such as capes and spits form under some wave climates (distributions of wave influences from different approach angles). The model can also represent the geology underlying the sandy coastline
and shoreface in a simplified manner and enables the simulation of coastline evolution when sediment supply from an eroding shoreface may be constrained. CEM also supports the simulation of human
manipulations to coastline evolution through beach nourishment or hard structures.
The Community Water Model (CWatM) is an integrated hydrological and channel routing model developed at the International Institute for Applied Systems Analysis (IIASA). CWatM quantifies water
availability, human water use, and the effect of water infrastructure, e.g., reservoirs, groundwater pumping, and irrigation, in regional water resources management.
The Control Volume Permafrost Model (CVPM) is a modular heat-transfer modeling system designed for scientific and engineering studies in permafrost terrain, and as an educational tool. CVPM
implements the nonlinear heat-transfer equations in 1-D, 2-D, and 3-D cartesian coordinates, as well as in 1-D radial and 2-D cylindrical coordinates. To accommodate a diversity of geologic settings,
a variety of materials can be specified within the model domain, including: organic-rich materials, sedimentary rocks and soils, igneous and metamorphic rocks, ice bodies, borehole fluids, and other
engineering materials. Porous materials are treated as a matrix of mineral and organic particles with pore spaces filled with liquid water, ice, and air. Liquid water concentrations at temperatures
below 0°C due to interfacial, grain-boundary, and curvature effects are found using relationships from condensed matter physics; pressure and pore-water solute effects are included. A radiogenic
heat-production term allows simulations to extend into deep permafrost and underlying bedrock. CVPM can be used over a broad range of depth, temperature, porosity, water saturation, and solute
conditions on either the Earth or Mars. The model is suitable for applications at spatial scales ranging from centimeters to hundreds of kilometers and at timescales ranging from seconds to thousands
of years. CVPM can act as a stand-alone model, the physics package of a geophysical inverse scheme, or serve as a component within a larger earth modeling system that may include vegetation, surface
water, snowpack, atmospheric or other modules of varying complexity.
The Coupled Routing and Excess STorage (CREST) distributed hydrological model is a hybrid modeling strategy that was recently developed by the University of Oklahoma (http://hydro.ou.edu) and NASA
SERVIR Project Team. CREST simulates the spatiotemporal variation of water and energy fluxes and storages on a regular grid with the grid cell resolution being user-defined, thereby enabling global-
and regional-scale applications. The scalability of CREST simulations is accomplished through sub-grid scale representation of soil moisture storage capacity (using a variable infiltration curve) and
runoff generation processes (using linear reservoirs). The CREST model was initially developed to provide online global flood predictions with relatively coarse resolution, but it is also applicable
at small scales, such as single basins. This README file and the accompanying code concentrates on and tests the model at the small scale. The CREST Model can be forced by gridded potential
evapotranspiration and precipitation datasets such as, satellite-based precipitation estimates, gridded rain gauge observations, remote sensing platforms such as weather radar, and quantitative
precipitation forecasts from numerical weather prediction models. The representation of the primary water fluxes such as infiltration and routing are closely related to the spatially variable land
surface characteristics (i.e., vegetation, soil type, and topography). The runoff generation component and routing scheme are coupled, thus providing realistic interactions between atmospheric, land
surface, and subsurface water.
The Cross-Shore Sediment Flux model addresses predominately sandy, wave-dominated coastlines on time-scales ranging from years to millenia and on spatial scales ranging from kilometers to tens of
kilometers using a range of wave parameters as inputs. It calculates the cross-shore sediment flux using both shallow water wave assumptions and full Linear Airy wave Theory. An equilibrium profile
is also created. Using the Exner equation, we develop an advection diffusion equation that describes the evolution of profile through time. A morphodynamic depth of closure can be estimated for each
input wave parameter.
The DLBRM is a distributed, physically based, watershed hydrology model that subdivides a watershed into a 1 km2 grid network and simulates hydrologic processes for the entire watershed sequentially.
The EPA Storm Water Management Model (SWMM) is a dynamic rainfall-runoff simulation model used for single event or long-term (continuous) simulation of runoff quantity and quality from primarily
urban areas. The runoff component of SWMM operates on a collection of subcatchment areas that receive precipitation and generate runoff and pollutant loads. The routing portion of SWMM transports
this runoff through a system of pipes, channels, storage/treatment devices, pumps, and regulators. SWMM tracks the quantity and quality of runoff generated within each subcatchment, and the flow
rate, flow depth, and quality of water in each pipe and channel during a simulation period comprised of multiple time steps.
The GeoTiff data component, pymt_geotiff, is a Python Modeling Toolkit (pymt) library for accessing data (and metadata) from a GeoTIFF file, through either a local filepath or a remote URL. The
pymt_geotiff component provides BMI-mediated access to GeoTIFF data as a service, allowing them to be coupled in pymt with other data or model components that expose a BMI.
The Grain Hill model provides a computational framework with which to study slope forms that arise from stochastic disturbance and rock weathering events. The model operates on a hexagonal lattice,
with cell states representing fluid, rock, and grain aggregates that are either stationary or in a state of motion in one of the six cardinal lattice directions. Cells representing near-surface soil
material undergo stochastic disturbance events, in which initially stationary material is put into motion. Net downslope transport emerges from the greater likelihood for disturbed material to move
downhill than to move uphill. Cells representing rock undergo stochastic weathering events in which the rock is converted into regolith. The model can reproduce a range of common slope forms, from
fully soil mantled to rocky or partially mantled, and from convex-upward to planar shapes. An optional additional state represents large blocks that cannot be displaced upward by disturbance events.
With the addition of this state, the model captures the morphology of hogbacks, scarps, and similar features. In its simplest form, the model has only three process parameters, which represent
disturbance frequency, characteristic disturbance depth, and baselevel lowering rate, respectively. Incorporating physical weathering of rock adds one additional parameter, representing the
characteristic rock weathering rate. These parameters are not arbitrary but rather have a direct link with corresponding parameters in continuum theory. The GrainHill model includes the
GrainFacetSimulator, which represents an evolving normal-fault facet with a 60-degree-dipping fault.
The Green-Ampt method of infiltration estimation.
The GridMET data component is an API, CLI, and BMI for fetching and caching daily gridMET (http://www.climatologylab.org/gridmet.html) CONUS meteorological data. Variables include: * maximum
temperature * minimum temperature * precipitation accumulation GridMET provides BMI-mediated access to gridMET data as a service, allowing it to be coupled with other components that expose a BMI.
The GroundwaterDupuitPercolator is appropriate for modeling shallow groundwater flow where the vertical component of flow is negligible. Where the groundwater table approaches the land surface, it
calculates seepage that can be routed using other Landlab components. It can be implemented on both regular (e.g. rectangular and hexagonal) and irregular grids determined by the user. Recharge,
hydraulic conductivity, and porosity may be specified as single values uniform over the model domain, or as vectors on the nodes (recharge, porosity) or links (hydraulic conductivity) of the grid.
Link hydraulic conductivity can also be specified from a two-dimensional hydraulic conductivity tensor using an included function. For mass balance calculations, the model includes methods to
determine the total groundwater storage on the grid domain, the total recharge flux in, and total groundwater and surface water fluxes leaving through the boundaries.
The HBV model (Bergström, 1976, 1992), also known as Hydrologiska Byråns Vattenbalansavdelning, is a rainfall-runoff model, which includes conceptual numerical descriptions of hydrological processes
at the catchment scale. There are many versions created over the years in various coding languages. This description points to the work of John Craven, which is a python implementation of the HBV
Hydrological Model, based on matlab code of the work of Professor Amir AghaKouchak at the University of California Irvine.
The HyLands Landscape Evolution Model is built using the Landlab software package. The HyLands model builds on three new components: water and sediment is routed using the PriorityFloodFlowRouter,
fluvial erosion and sediment transport is calculated using the SpaceLargeScaleEroder while bedrock landsliding and sediment runout is calculated using the BedrockLandslider. These and all other
Landlab components used in this paper are part of the open source Landlab modeling framework, version 2.5.0 (Barnhart et al., 2020a; Hobley et al., 2017), which is part of the Community Surface
Dynamics Modeling System (Tucker et al., 2021). Source code for the Landlab project is housed on GitHub: http://github.com/landlab/landlab (last access: 17 August 2022). Documentation, installation,
instructions, and software dependencies for the entire Landlab project can be found at http://landlab.github.io/ (last access: 17 August 2022). A user manual with an accompanying Jupyter notebooks is
available from https://github.com/BCampforts/hylands_modeling (last access: 17 August 2022). The Landlab project is tested on recent-generation Mac, Linux, and Windows platforms. The Landlab modeling
framework is distributed under a MIT open-source license. The latest version of the Landlab software package, including the components developed for the HyLands model is archived at: https://doi.org/
10.5281/zenodo.6951444 (last access: 17 August 2022).
The Hydrologically Enhanced Basin Evolution Model (HEBEM) is a combined hydrologic/geomorphic model. The hydrologic model simulates precipitation with variability, infiltration, evapotranspiration,
overland flow, and groundwater flow, thus producing a spatially and temporally varying water discharge Q that drives fluvial processes in the land surface. The geomorphic model accounts for tectonic
forcing, hillslope processes, erosion, and sediment transport. The combined model uses multiple time steps for hydrologic and geomorphic processes. Due to its hydrologic representation, the model is
able to investigate the interaction between hydrology and geomorpholgy.
The Instructed Glacier Model (IGM) simulates the ice dynamics, surface mass balance, and its coupling through mass conservation to predict the evolution of glaciers and icefields. The specificity of
IGM is that it models the ice flow by a neural network, which is trained with ice flow physical models. Doing so permits to speed-up and facilitate considerably the implementation of the forward
model and the inverse model required to assimilate data.
The International Land Model Benchmarking (ILAMB) project is a model-data intercomparison and integration project designed to improve the performance of land models and, in parallel, improve the
design of new measurement campaigns to reduce uncertainties associated with key land surface processes. Building upon past model evaluation studies, the goals of ILAMB are to: * develop
internationally accepted benchmarks for land model performance, promote the use of these benchmarks by the international community for model intercomparison, * strengthen linkages between
experimental, remote sensing, and climate modeling communities in the design of new model tests and new measurement programs, and * support the design and development of a new, open source,
benchmarking software system for use by the international community.
The Landlab Drainage Density component calculates landscape-averaged drainage density, defined as the inverse of the mean distance from any pixel to the nearest channel. The component follows the
approach defined in Tucker et al (2001, Geomorphology). The drainage density component does not find channel heads, but takes a user-defined channels mask.
The Landlab ErosionDeposition component calculates fluvial erosion and deposition of a single substrate as derived by Davy and Lague (2009, Journal of Geophysical Research). Mass is simultaneously
conserved in two reservoirs: the bed and the water column. ErosionDeposition dynamically transitions between detachment-limited and transport-limited behavior, but is limited to erosion of a single
substrate (e.g., sediment or bedrock but not both).
The Landlab OverlandFlow component is based on a simplified inertial approximation of the shallow water equations, following the solution of de Almeida et al. (2012). This explicit two-dimensional
hydrodynamic algorithm simulates a flood wave across a model domain, where water discharge and flow depth are calculated at all locations within a structured (raster) grid. This component generates a
hydrograph at all grid locations, and allows for flow to move in one of the four cardinal directions (D4) into/out of a given model node.
The Landlab SPACE (Stream Power with Alluvium Conservation and Entrainment) enables modeling of bedrock, alluviated, and bedrock-alluvial rivers by simultaneously conserving mass in three reservoirs:
the water column, the alluvial bed, and the underlying bedrock. SPACE allows dynamic transitions between detachment-limited, transport-limited, and intermediate states. SPACE calculates sediment
fluxes, alluvial layer thickness, and bedrock erosion at all nodes within the model domain. An extended description of the model may be found in Shobe et al (2017, Geoscientific Model Development).
The Larval TRANSport Lagrangian model (LTRANS) is an off-line particle-tracking model that runs with the stored predictions of a 3D hydrodynamic model, specifically the Regional Ocean Modeling System
(ROMS). Although LTRANS was built to simulate oyster larvae, it can easily be adapted to simulate passive particles and other planktonic organisms. LTRANS is written in Fortran 90 and is designed to
track the trajectories of particles in three dimensions. It includes a 4th order Runge-Kutta scheme for particle advection and a random displacement model for vertical turbulent particle motion.
Reflective boundary conditions, larval behavior, and settlement routines are also included. LTRANS was built by Elizabeth North and Zachary Schlag of University of Maryland Center for Environmental
Science Horn Point Laboratory. Funding was provided by the National Science Foundation Biological Oceanography Program, Maryland Department of Natural Resources, NOAA Chesapeake Bay Office, and
NOAA-funded UMCP Advanced Study Institute for the Environment. Components of LTRANS have been in development since 2002 and are described in the following publications: North et al. 2005, North et
al. 2006a, North et al. 2006b, and North et al. 2008.
The MITgcm (MIT General Circulation Model) is a numerical model designed for study of the atmosphere, ocean, and climate. Its non-hydrostatic formulation enables it to simulate fluid phenomena over a
wide range of scales; its adjoint capability enables it to be applied to parameter and state estimation problems. By employing fluid isomorphisms, one hydrodynamical kernel can be used to simulate
flow in both the atmosphere and ocean.
The Model Parameter Dictionary is a tool for numerical modelers to easily read and access model parameters from a simple formatted input (text) file. Each parameter has a KEY, which identifies the
parameter, and a VALUE, which can be a number or a string. A ModelParameterDictionary object reads model parameters from an input file to a Dictionary, and provides functions for the user to look up
particular parameters by key name. The format of the input file looks like: PI: the text "PI" is an example of a KEY 3.1416 AVOGADROS_NUMBER: this is another 6.022e23 FAVORITE_FRUIT: yet another
mangoes NUMBER_OF_MANGO_WALKS: this one is an integer 4 ALSO_LIKES_APPLES: this is a boolean true Example code that reads these parameters from a file called "myinputs.txt": my_param_dict =
ModelParameterDictionary() my_param_dict.read_from_file( 'myinputs.txt' ) pi = my_param_dict.read_float( 'PI' ) avogado = my_param_dict.read_float( 'AVOGADROS_NUMBER' ) fruit =
my_param_dict.read_string( 'FAVORITE_FRUIT' ) nmang = my_param_dict.read_int( 'NUMBER_OF_MANGO_WALKS' ) apples_ok = my_param_dict.read_bool( 'ALSO_LIKES_APPLES' ) As in Python, hash marks (#) denote
comments. The rules are that each key must have one and only one parameter value, and each value must appear on a separate line immediately below the key line. Also available are functions to read
input parameters from the command line (e.g., read_float_cmdline( 'PI' ) )
The Numerical model of coastal Erosion by Waves and Transgressive Scarps (NEWTS) model is a framework to simulate the erosion of a closed-basin coastline through time by fetch-dependent erosion or
uniform erosion.
The Permafrost Benchmark System (PBS) wraps the command-line ILAMB benchmarking system with a customized version of the CSDMS Web Modeling Tool (WMT), and adds tools for uploading CMIP5-compatible
model outputs and benchmark datasets. The PBS allows users to access and run ILAMB remotely, without having to install software or data locally; a web browser on a desktop, laptop, or tablet computer
is all that’s needed.
The Princeton Ocean Model (POM), a simple-to-run yet powerful ocean modeling code that is able to simulate a wide-range of problems: circulation and mixing processes in rivers, estuaries, shelf and
slope, lakes, semi-enclosed seas and open and global ocean. POM is a sigma coordinate, free surface ocean model with embedded turbulence and wave sub-models, and wet-dry capability. It has been one
of the first coastal ocean models freely available to users, with currently over 3000 users from 70 countries. For more details see: http://www.ccpo.odu.edu/POMWEB/
The SFINCS model (Super-Fast INundation of CoastS) is developed to efficiently simulate compound flooding events at limited computational cost and good accuracy. SFINCS solves the SSWE and thus
includes advection in the momentum equation. However, it can also run using the LIE without advection. Processes such as spatially varying friction, infiltration and precipitation are included.
Moreover, SFINCS includes wind-driven shear and an absorbing-generating weakly-reflective boundary is considered which are not included in other reduced-physics models.
The Sea Level Affecting Marshes Model (SLAMM) simulates the dominant processes involved in wetland conversions and shoreline modifications during long-term sea level rise. Tidal marshes can be among
the most susceptible ecosystems to climate change, especially accelerated sea level rise (SLR).
The Sorted Bedform Model (SBM) addresses the formation mechanism for sorted bedforms present on inner continental shelf environments.
The Spectral Element Ocean Model (SEOM) solves the hydrostatic, and alternatively the non-hydrostatic, primitive equations using a mixed spectral / finite element solution procedure. Potential
advantages of the spectral element method include flexible incorporation of complex geometry and spatially dependent resolution, rapid convergence, and attractive performance on parallel computer
systems. A 2D version of SEOM, which solves the shallow water equations, has been extensively tested on applications ranging from global tides to the abyssal circulation of the Eastern Mediterranean.
The 3D SEOM is undergoing initial testing for later release.
The Urban Inundation-Drainage Simulator (UIDS) is a new coupled model for simulating urban flooding dynamics, developed as an open-source, MATLAB-based platform. It integrates a rainfall-runoff model
with a two-dimensional overland flow model (OFM) and a one-dimensional sewer flow model (SFM).
The Utah Energy Balance (UEB) snow model is an energy balance snowmelt model developed by David Tarboton's research group, first in 1994, and updated over the years. The model uses a lumped
representation of the snowpack and keeps track of water and energy balance. The model is driven by inputs of air temperature, precipitation, wind speed, humidity and radiation at time steps
sufficient to resolve the diurnal cycle (six hours or less). The model uses physically-based calculations of radiative, sensible, latent and advective heat exchanges. A force-restore approach is used
to represent surface temperature, accounting for differences between snow surface temperature and average snowpack temperature without having to introduce additional state variables. Melt outflow is
a function of the liquid fraction, using Darcy's law. This allows the model to account for continued outflow even when the energy balance is negative. Because of its parsimony (few state variables -
but increasing with later versions) this model is suitable for application in a distributed fashion on a grid over a watershed.
The VIC model is a large-scale, semi-distributed hydrologic model. As such, it shares several basic features with the other land surface models (LSMs) that are commonly coupled to global circulation
models (GCMs): The land surface is modelled as a grid of large (>1km), flat, uniform cells Sub-grid heterogeneity (e.g. elevation, land cover) is handled via statistical distributions. Inputs are
time series of daily or sub-daily meteorological drivers (e.g. precipitation, air temperature, wind speed). Land-atmosphere fluxes, and the water and energy balances at the land surface, are
simulated at a daily or sub-daily time step Water can only enter a grid cell via the atmosphere Non-channel flow between grid cells is ignored The portions of surface and subsurface runoff that reach
the local channel network within a grid cell are assumed to be >> the portions that cross grid cell boundaries into neighboring cells Once water reaches the channel network, it is assumed to stay in
the channel (it cannot flow back into the soil) This last point has several consequences for VIC model implementation: Grid cells are simulated independently of each other Entire simulation is run
for each grid cell separately, 1 grid cell at a time, rather than, for each time step, looping over all grid cells Meteorological input data for each grid cell (for the entire simulation period) are
read from a file specific to that grid cell Time series of output variables for each grid cell (for the entire simulation period) are stored in files specific to that grid cell Routing of stream flow
is performed separately from the land surface simulation, using a separate model (typically the routing model of Lohmann et al., 1996 and 1998)
The Water Erosion Prediction Project (WEPP) model is a process-based, distributed parameter, continuous simulation erosion prediction model for application to hillslope profiles and small watersheds.
Interfaces to WEPP allow its application as a stand-alone Windows program, a GIS-system (ArcView, ArcGIS) extension, or in web-based links. WEPP has been developed since 1985 by the U.S. Department
of Agriculture for use on croplands, forestlands, rangelands, and other land use types.
The Water Table Model (WTM) simulates terrestrial water changes over the full range of relevant spatial (watershed to global) and temporal (monthly to millennial) scales. It comprises coupled
components to compute dynamic lake and groundwater levels. The groundwater component solves the 2D horizontal groundwater-flow equation by using non-linear equation solvers in the C++ PETSc library.
The dynamic lakes component makes use of the Fill-Spill-Merge (FSM) algorithm to move surface water into lakes, where it may evaporate or affect groundwater flow.
The Weather Research and Forecasting (WRF) Model is a next-generation mesoscale numerical weather prediction system designed to serve both operational forecasting and atmospheric research needs. It
features multiple dynamical cores, a 3-dimensional variational (3DVAR) data assimilation system, and a software architecture allowing for computational parallelism and system extensibility. WRF is
suitable for a broad spectrum of applications across scales ranging from meters to thousands of kilometers.
The Weather Research and Forecasting Model Hydrological modeling system (WRF-Hydro) was developed as a community-based, open source, model coupling framework designed to link multi-scale process
models of the atmosphere and terrestrial hydrology to provide: An extensible multi-scale & multi-physics land-atmosphere modeling capability for conservative, coupled and uncoupled assimilation &
prediction of major water cycle components such as: precipitation, soil moisture, snow pack, ground water, streamflow, and inundation Accurate and reliable streamflow prediction across scales (from
0-order headwater catchments to continental river basins and from minutes to seasons) A research modeling testbed for evaluating and improving physical process and coupling representations.
The bmi_wavewatch3 Python package provides both a command line interface and a programming interface for downloading and working with WAVEWATCH III data. bmi_wavewatch3 provides access to the
following raster data sources, 30 year wave hindcast Phase 1 https://polar.ncep.noaa.gov/waves/hindcasts/nopp-phase1.php 30 year wave hindcast Phase 2 https://polar.ncep.noaa.gov/waves/hindcasts/
nopp-phase2.php Production hindcast Singlegrid https://polar.ncep.noaa.gov/waves/hindcasts/prod-nww3.php Production hindcast Multigrid https://polar.ncep.noaa.gov/waves/hindcasts/prod-multi_1.php All
data sources provide both global and regional grids.
The carbonate production is modelled according to organism growth and survival rates moderated by habitat suitability (chiefly light, temperature, nutrient). The environmental inputs are extracted
from global databases. At the seabed the model's vertical zonation allows for underground (diagenetic) processes, bed granular transport, lower stable framework, upper collapsable framework. This
voxelation allows for the carbonate to be placed (accumulated) correctly within the bedding and clast fabrics. The stratigraphy and seabed elevation are built in this way. As conditions change (e.g.,
by shallowing) the biological communities respond in the simulation, and so too do the production rates and clast/binding arrangements. Events punctuate the record, and the organism assemblages
adjust according to frequencies and severities. The population stocks are calculated by diffuse competition in a Lotke-Volterra scheme, or via cellular simulations of close-in interactions to
represent competition by growth, recruitment.
The code computes the formation of a hillslope profile above an active normal fault. It represents the hillslope as a set of points with vertical and horizontal (fault-perpendicular) coordinates.
Points move due to a prescribed erosion rate (which may vary in time) and due to offset during earthquakes with a specified recurrence interval and slip rate. <p>The model is described and
illustrated in the following journal article: </p><p>Tucker, G. E., S. W. McCoy, A. C. Whittaker, G. P. Roberts, S. T. Lancaster, and R. Phillips (2011), Geomorphic significance of postglacial
bedrock scarps on normal-fault footwalls, J. Geophys. Res., 116, F01022, doi:<a target="_blank" rel="nofollow noreferrer noopener" class="external text" href="http://dx.doi.org/10.1029/2010JF001861">
The delta-building model DeltaRCM expanded to included vegetation effects. Vegetation colonizes, grows, and dies, and influences the delta through increasing bank stability and providing resistance
to flow. Vegetation was implemented to represent marsh grass type plants, and parameters of stem diameter, carrying capacity, logistic growth rate, and rooting depth can be altered.
The development of the HAMSOM coding goes back to the mid eighties where it emerged from a fruitful co-operation between Backhaus and Maier-Reimer who later called his model 'HOPE'. From the very
beginning HAMSOM was designed with the intention to allow simulations of both oceanic and coastal and shelf sea dynamics. The primitive equation model with a free surface utilises two time-levels,
and is defined in Z co-ordinates on the Arakawa C-grid. Stability constraints for surface gravity waves and the heat conduction equation are avoided by the implementation of implicit schemes. With a
user defined weighting between future and presence time levels a hierarchy of implicit schemes is provided to solve for the free surface problem, and for the vertical transfer of momentum and water
mass properties. In the time domain a scheme for the Coriolis rotation is incorporated which has second order accuracy. Time- and space-dependent vertical exchange and diffusivity coefficients are
determined from a simple zero-order turbulence closure scheme which has also been replaced by a higher order closure scheme (GOTM). The resolution of a water column may degenerate to just one grid
cell. At the seabed a non-linear (implicit) friction law as well as the full kinematic boundary condition is applied. Seabed cells may deviate from an undisturbed cell height to allow for a better
resolution of the topography. The HAMSOM coding excludes any time-splitting, i.e. free surface and internal baroclinic modes are always directly coupled. Simple upstream and more sophisticated
advection schemes for both momentum and matter may be run according to directives from the user. Successful couplings with eco-system models (ECOHAM, ERSEM), an atmospheric model (REMO), and both
Lagrangian and Eulerian models for sediment transport are reported in the literature. For polar applications HAMSOM was coupled with a viscous-plastic thermo-hydrodynamic ice model of Hibler type.
Since about 15 years in Hamburg, and overseas in more than 30 laboratories, HAMSOM is already being in use as a community model.
The fault can have an arbitrary trace given by two points (x1, y1) and (x2, y2) in the fault_trace input parameter. These value of these points is in model-space coordinates and is not based on node
id values or number of rows and columns.
The grid contains the value 1 where fractures (one cell wide) exist, and 0 elsewhere. The idea is to use this for simulations based on weathering and erosion of, and/or flow within, fracture
The hydrodynamic module of WWTM solves the shallow water equations modified through the introduction of a refined sub-grid model of topography to deal with flooding and drying processes in irregular
domains (Defina, 2000). The numerical model, which uses finite-element technique and discretizes the domain with triangular elements, has been extensively tested in recent years in the Venice lagoon,
Italy (D’Alpaos and Defina, 2007, Carniello et al., 2005; Carniello et al., 2009). For the wind wave modulel the wave action conservation equation is used, solved numerically with a finite volume
scheme, and fully coupled with the hydrodynamic module (see Carniello et al. 2005). The two modules share the same computational grid.
The hydromad (Hydrological Model Assessment and Development) package provides a set of functions which work together to construct, manipulate, analyse and compare hydrological models.
The mizuRoute tool post-processes runoff outputs from any distributed hydrologic model or land surface model to produce spatially distributed streamflow at various spatial scales from headwater
basins to continental-wide river systems. The tool can utilize both traditional grid-based river network and vector-based river network data.
The model accounts for glacier geometry (including contributory branches) and includes an explicit ice dynamics module. It can simulate past and future mass-balance, volume and geometry of (almost)
any glacier in the world in a fully automated and extensible workflow. Publicly available data is used for calibration and validation.
The model calculates a unique regression equation for each grid-cell between a the relative area of a specific land use (e.g. cropland) and global population. The equation is used to extrapolate that
land use are into the future in each grid cell with predicted global population predictions. If the relative area of a land use reach a value of 95%, additional expansion is migrated to neighboring
cells thus allowing spatial expansion. Geographic limitations are imposed on land use migration (e.g. no cropland beyond 60 degree latitude). For more information: Haney, N., Cohen, S. (2015),
Predicting 21st century global agricultural land use with a spatially and temporally explicit regression-based model. Applied Geography, 62: 366-376.
The model couples the shallow water equations with the Green-Ampt infiltration model and the Hairsine-Rose soil erosion model. Fluid flow is also modified through source terms in the momentum
equations that account for changes in flow behavior associated with high sediment concentrations. See McGuire et al. (2016, Constraining the rates of raindrop- and flow-driven sediment transport
mechanisms in postwildfire environments and implications for recovery timescales) for a complete model description and details on the numerical solution of the governing equations.
The model evolves a 1D hillslope according to a non-linear diffusion rule (e.g. Roering et al. 1999) for varying boundary conditions idealised as a gaussian pulse of baselevel fall through time. A
Markov Chain Monte Carlo inversion finds the most likely boundary condition parameters when compared to a time series of field data on hillslope morphology from the Dragon's Back Pressure Ridge,
Carrizo Plain, CA, USA; see Hilley and Arrowsmith, 2008.
The model is developed to simulate the sediment transport and alluvial morphodynamics of bedrock reaches. It is capable of computing the alluvial cover fraction, the alluvial-bedrock transition and
flow hydrodynamics over both bedrock and alluvial reaches. This model is now validated against a set of laboratory experiment. Field scale application of the model can also be done using field
The model is related to the numerical solution of the shallow water equations in spherical geometry. The shallow water equations are used as a kernel for both oceanic and atmospheric general
circulation models and are of interest in evaluating numerical methods for weather forecasting and climate modeling.
The model is three-dimensional and fully nonlinear with a free surface, incorporates advanced turbulence closure, and operates in tidal time. Variable horizontal and vertical resolution are
facilitated by the use of unstructured meshes of linear triangles in the horizontal, and structured linear elements in the vertical
The model predicts bankfull geometry of single-thread, sand-bed rivers from first principles, i.e. conservation of channel bed and floodplain sediment, which does not require the a-priori knowledge
of the bankfull discharge.
The model reproduce the effect of a variability in soil resistance on salt marsh erosion by wind waves. The model consists of a two-dimensional square lattice whose elements, i, have randomly
distributed resistance, r_i. The critical soil height H_ci for boundary stability is calculated from soil shear strength values and is assumed as representative of soil resistance, as it is a
convenient way to take into account general soil and ambient conditions. The erosion rate of each cell, E_i, which represents the erosion of an homogeneous marsh portion, is defined as: E_i=〖αP〗^β
exp (-H_ci/H) Where α and β are non-dimensional constants set equal to 0.35 and 1.1 respectively, P is the wave power, and H is the mean wave height. The model follows three rules: i) only neighbors
of previously eroded cells can be eroded. Therefore, only cells having at least one side in common with previously eroded elements are susceptible to erosion; ii) at every time step one element is
eroded at random with probability p_i=E_i/(∑E_i ); iii) A cell is removed from the domain if it remains isolated from the rest of the boundary (no neighbors).
The model simplifies the geometry of a backbarrier tidal basin with 3 variables: marsh depth, mudflat depth, mudflat width. These 3 variables are evolved by sediment redistribution driven by wave
processes. Sediment are exchanged with the open ocean, which is an external reservoir. Organic sediments are produced on the marsh platform.
The model simulates the formation, drift, and melt of a population of icebergs utilizing Monte Carlo based techniques with a number of underlying parametric probability distributions to describe the
stochastic behavior of iceberg formation and dynamics.
The model simulates the long-term evolution of meandering rivers above heterogeneous floodplain surfaces, i.e. floodplains that have been reworked by the river itself through the formation of oxbow
lakes and point bars.
The model tracks both surface CRN concentration and concentration eroded off hillslopes into fluvial network in a simplified landscape undergoing both landslide erosion and more steady
'diffusive-like' erosion. Sediment mixing is allowed in the fluvial network. Code can be used to help successfully develop CRN sampling procedures in terrains where landslides are important.
The model uses the vertically continuous (not active layer-based), morphodynamic framework proposed by Parker, Paola an Leclair in 2000 to model the streamwise and vertical dispersal of a patch of
tracers installed in a equilibrium gravel bed. The model was validated at laboratory and field scales on the mountainous Halfmoon Creek, USA, and on the braided Buech River, France. Different
versions of the model are uploaded in the github folder because the formulaiton for the calculation of the formative bed shear stress varied depending on the available data. REFERENCE Parker, G.,
Paola, C. & Leclair, S. (2000). Probabilistic Exner sediment continuity equation for mixtures with no active layer. Journal of Hydraulic Engineering, 126 (11), 818-826.
The module is designed to calculate morphological changes and water discharge outflow of a crevasse splay that is triggered by a preset flood event and evolves afterwards. The inputs for "mainCS.m"
should be daily water discharge and sediment flux series of the trunk channel upstream the crevasse splay. The outputs will be daily series for the cross-sectional parameters of the crevasse splay,
and daily water discharge series of the trunk channel downstream the crevasse splay. One limitation of the present version is it only calculates the expanding and healing of a crevasse splay, while
ignores the possible morphological change (demise or revival) of the trunk channel downstream the crevasse splay. Another limitation is the codes are originally written for the Lower Yellow River(a
suspended load dominated river) for the purpose of calculating sediment budget in the Lower Yellow over a long timescale, say as long as hundreds years, so the present module can not be applied to
other alluvial rivers without modifying those lines related to channel geometry, bankfull discharge and bank erosion(deposition).
The numerical model solves the two-dimensional shallow water equations with different modes of sediment transport (bed-load and suspended load) (Canestrelli et al. 2009, Canestrelli et al, 2010). The
scheme solves the system of partial differential equations cast in a non-conservative form, but it has the important characteristic of reducing automatically to a conservative scheme if the
underlying system of equations is a conservation law. The scheme thus belongs to the so-called category of “shock-capturing” schemes. At the present I am adding a new module for the computation of
mud flows, and I want to apply the model to the Fly River (Papua New Guinea) system.
The river water temperature model is designed to be applied in Arctic rivers. Heat energy transfers considered include surface net solar radiation, net longwave radiation, latent heat due to
evaporation and condensation, convective heat and the riverbed heat flux. The model is explicitly designed to interact with a permafrost channelbed and frozen conditions through seasonal cycles. In
addition to the heat budget, river discharge, or stage, drives the model.
The term "extended GST model" indicates the combination of an analytical GST migration model combined with closure relations (for slope and surface texture) based on the assumption of
quasi-equilibrium conditions. The extended model is described in Blom et al, 2017 "Advance, retreat, and halt of abrupt gravel-sand transitions in alluvial rivers", http://dx.doi.org/10.1002/
The term “breaching” refers to the slow, retrogressive failure of a steep subaqueous slope, so forming a nearly vertical turbidity current directed down the face. This mechanism, first identified by
the dredging industry, has remained largely unexplored, and yet evidence exists to link breaching to the formation of sustained turbidity currents in the deep sea. The model can simulate a
breach-generated turbidity current with a layer-averaged formulation that has at its basis the governing equations for the conservation of momentum, water, suspended sediment and turbulent kinetic
energy. In particular, the equations of suspended sediment conservation are solved for a mixture of sediment particles differing in grain size. In the model the turbidity current is divided into two
regions joined at a migrating boundary: the breach face, treated as vertical, and a quasi-horizontal region sloping downdip. In this downstream region, the bed slope is much lower (but still
nonzero), and is constructed by deposition from a quasi-horizontal turbidity current. The model is applied to establish the feasibility of a breach-generated turbidity current in a field setting,
using a generic example based on the Monterey Submarine Canyon, offshore California, USA.
Third generation random phase spectral wave model, including shallow water physcis.
This class implements Voller, Hobley, and Paola’s experimental matrix solutions for flow routing. The method works by solving for a potential field at all nodes on the grid, which enforces both mass
conservation and flow downhill along topographic gradients. It is order n and highly efficient, but does not return any information about flow connectivity. Options are permitted to allow “abstract”
routing (flow enforced downslope, but no particular assumptions are made about the governing equations), or routing according to the Chezy or Manning equations. This routine assumes that water is
distributed evenly over the surface of the cell in deriving the depth, and does not assume channelization. You will need to back- calculate channel depths for yourself using known widths at each node
if that is what you want.
This class uses the Braun-Willett Fastscape approach to calculate the amount of erosion at each node in a grid, following a stream power framework. This should allow it to be stable against larger
timesteps than an explicit stream power scheme.
This code creates the channel centerline (i.e., the line equidistant between two banks) for a single thread-channel, using a second-order autoregressive model. The code implements a model for random
centerlines proposed by Ferguson, R. I. (1976) Disturbed periodic model for river meanders, Earth Surface Processes 1(4), 337-347, doi:10.1002/esp.3290010403. This implementation also includes (1)
controls for the node spacing and extent of channels, (2) removal of self-intersecting (cutoff) loops from modeled centerlines, and (3) a wrapper script to sweep model parameter space and generate
alternate realizations using different random disturbance series.
This code is based on Cellular Automata Tree Grass Shrub Simulator (CATGraSS). It simulates spatial competition of multiple plant functional types through establishment and mortality. In the current
code, tree, grass and shrubs are used.
This component calculates Hack’s law parameters for drainage basins. Hacks law is given as L = C * A**h Where L is the distance to the drainage divide along the channel, A is the drainage area, and C
are parameters. The HackCalculator uses a ChannelProfiler to determine the nodes on which to calculate the parameter fit.
This component calculates chi indices, sensu Perron & Royden, 2013, for a Landlab landscape.
This component calculates steepness indices, sensu Wobus et al. 2006, for a Landlab landscape. Follows broadly the approach used in GeomorphTools, geomorphtools.org.
This component generates random numbers using the Weibull distribution (Weibull, 1951). No particular units must be used, but it was written with the fire recurrence units in time (yrs). Using the
Weibull Distribution assumes two things: All elements within the study area have the same fire regime. Each element must have (on average) a constant fire regime during the time span of the study.
<br> As of Sept. 2013, fires are considered instantaneous events independent of other fire events in the time series.
This component identifies depressions in a topographic surface, finds an outlet for each depression. If directed to do so (default True), and the component is able to find existing routing fields
output from the 'route_flow_dn' component, it will then modify the drainage directions and accumulations already stored in the grid to route flow across these depressions.
This component implements a depth and slope dependent linear diffusion rule in the style of Johnstone and Hilley (2014). Soil moves with a prescribed exponential vertical velocity profile. Soil flux
is dictated by a diffusivity, K, and increases linearly with topographic slope.
This component implements exponential weathering of bedrock on hillslopes. Uses exponential soil production function in the style of Ahnert (1976). Consider that w_0 is the maximum soil production
rate and that d* is the characteristic soil production depth. The soil production rate w is given as a function of the soil depth d, w = w_0^(-d/d*) The ExponentialWeatherer only calculates soil
production at core nodes.
This component is closely related to the FlowAccumulator, in that this is accomplished by first finding flow directions by a user-specified method and then calculating the drainage area and
discharge. However, this component additionally requires the passing of a function that describes how discharge is lost or gained downstream, f(Qw, nodeID, linkID, grid). See examples at https://
github.com/landlab/landlab/blob/master/landlab/components/flow_accum/lossy_flow_accumulator.py to see how this works in practice.
This components finds the steepest single-path steepest descent flow directions. It is equivalent to D4 method in the special case of a raster grid in that it does not consider diagonal links between
nodes. For that capability, use FlowDirectorD8.
This is a 1DV wave-phase resolving numerical model for fluid mud transport based on mixture theory with boundary layer approximation. The model incorporates turbulence-sediment interaction,
gravity-driven flow, mud rheology, bed erodibility and the dynamics of floc break-up and aggregation.
This is a Java Applet that allows the user to change different parameters (such as rainfall, erodibility, tectonic uplift) and watch how the landform evolve over time under different scenarios. It is
based on a Cellular Automata algorithm. Two versions are available: linear and non-linear. Details can be found in: Luo, W., Peronja, E., Duffin, K., Stravers, A. J., 2006, Incorporating Nonlinear
Rules in a Web-based Interactive Landform Simulation Model (WILSIM), Computers and Geosciences, v. 32, n. 9, p. 1512-1518 (doi: 10.1016/j.cageo.2005.12.012). Luo, W., K.L. Duffin, E. Peronja, J.A.
Stravers, and G.M. Henry, 2004, A Web-based Interactive Landform Simulation Model (WILSIM), Computers and Geosciences. v. 30, n. 3, p. 215-220.
This is a time-stepping point model which uses linear finite elements to determine the vertical structure of the horizontal components of velocity and density under specified surface forcing. Both a
quadratic closure scheme and the level 2.5 closure scheme of Mellor and Yamada are used in this code.
This is a tool that I created to help find knickpoints based on the curvature of a landscape. It provides information about a stream including, knickpoint locations, Elevation/distance that can be
used to create longitudinal profiles, XYvalues of all the cells in a stream path, etc. The tool uses built-in tools for ArcGIS 10.x (so you must run this on a machine with ArcGIS 10.x installed), but
it is written in python. I used it with a 1m LiDAR DEM, so I'm not totally sure how well it will pick out knickpoints on coarser gridded DEMs.
This is an Arctic-delta reduced-complexity model that can reproduce the 2-m ramp feature observed in most Arctic deltas. The model is built by first reconstructing from published descriptions of the
DeltaRCM-Arctic model (Lauzon et al., GRL, 2019), which is, in turn, based on DeltaRCM by Liang et al. (Esurf, 2015). All the modifications and refinements leading to this model (ArcDelRCM.jl) are
detailed in a manuscript submitted to Earth Surface Dynamics journal for publication (Chan et al., 2022: esurf-2022-25). Options are retained to run this model with the "DeltaRCM-Arctic"
(reconstruction) setting. The code is written purely in Julia language.
This model a 1-D numerical model of permafrost and subsidence processes. It aims to investigate the subsurface thermal impact of thaw lakes of various depths, and to evaluate how this impact might
change in a warming climate.
This model accounts for the bed evolution i.e. aggradation/degradation and grain size distribution of surface material in gravel bed rivers under anthropogenic changes such as dam closure and
sediment augmentation. This model is developed for an alpine gravel bed river located in SE France (Buech river).
This model calculates the long profile of a river with a gravel-sand transition. The model uses two grain sizes: size Dg for gravel and size Ds for sand. The river is assumed to be in flood for the
fraction of time Ifg for the gravel-bed reach and fraction Ifs for the sand-bed reach. All sediment transport is assumed to take place when the river is in flood. Gravel transport is computed using
the Parker (1979) approximation of the Einstein (1950) bedload transport relation. Sand transport is computed using the total bed material transport relation of Engelund and Hansen (1967). In this
simple model the gravel is not allowed to abrade. Both the gravel-bed and sand-bed reaches carry the same flood discharge Qbf. Gravel is transported as bed material in, and deposits only in the
gravel-bed reach. A small residual of gravel load is incorporated into the sand at the gravel-sand transition. Sand is transported as washload in the gravel-bed reach, and as bed material load in the
sand-bed reach. The model allows for depositional widths Bdgrav and Bdsand that are wider than the corresponding bankfull channel widths Bgrav and Bsand of the gravel-bed and sand-bed channels. As
the channel aggrades, it is assumed to migrate and avulse to deposit sediment across the entire depositional width. For each unit of gravel deposited in the gravel-bed reach, it is assumed that Lamsg
units of sand are deposited. For each unit of sand deposited on the sand-bed reach, it is assumed that Lamms units of mud are deposited. The gravel-bed reach has sinuosity Omegag and the sand-bed
reach has sinuosity Omegas. Bed resistance is computed through the use of two specified constant Chezy resistance coefficients; Czg for the gravel-bed reach and Czs for the sand-bed reach.
This model can be used for both transport after sediment failure and for hyperpycnal transport.
This model evolves a hogback through time. A resistant layer of rock, which weathers slowly, overlies a softer layer of rock that weathers quickly. Resistant rock produces "blocks" which land on the
adjoining hillslope. Boundaries incise at a specified rate. User can set hogback layer thickness, block size, and dip, as well as relative weathering and incision rates. Trackable metrics included
are time and space-averaged slope, block height, weathering rate, and erosion rate. Parameters that users need to specify are surrounded by many comment signs.
This model generates investigation polygons which are used to estimate and store incremental erosion and deposition volumes along the path of a debris flow at user-defined resolution. The user will
need: 1) a .LAS or .TIF of topographic change, 2) a DEM of the AOI, and 3) a shapefile (polyline) of the debris flow path of interest. Each incremental volume is georeferenced and stored within a
shapefile attribute associated with a catchment area and distance from outlet for analysis.
This model is a GUI implementation of a simple cellular automata dune model. The model was originally proposed by Werner (1995, Geology 23) and has seen several extensions. It can simulate basic
barchan, transverse, star, and linear dunes. The model is designed to be easy to operate for researchers or students without programming skills. Also included is a tool to operate the model from
This model is a nonuniform, quasi-unsteady, movable bed, single channel flow model for heterogeneous size-density mixtures
This model is designed to simulate longitudinal profiles with headward advancing headcuts. This model simulates gully erosion on the centennial-scale given information such as average rainfall and
infiltration rates. The modeler also specifies a headcut erosion rate and or a rule for headcut retreat (either discharge-dependent or height-dependent retreat).
This model simulates the interaction between suspended sediment, chlorophyll-a, and mussel population density. Discharge is the driver; it modulates suspended sediment and its interactions in the
system. The model is suitable for simulating mussel densities at-a-site. It was originally developed to test the hypothesis that increased sediment loads in Minnesota Rivers are a plausible cause of
observed mussel population declines. The model and results are described in detail in the following paper: https://doi.org/10.1086/684223
This model uses a non-dimensional equation for luminescence in a mixing soil that was derived from the Fokker-Plank Equation.
This model uses the Green-Ampt equation to represent infiltration and the kinematic wave equation to represent runoff over a landscape. The effects of rainfall interception can also be included.
This module implements a particle-based model of hillslope evolution, which has an associated continuum description (introduced here: https://arxiv.org/abs/1801.02810). The model takes as input a few
simple parameters which determine the equilibrium hillslope shape and dynamics, and can be used to produce equilibrium profiles and study the response of the hillslope to perturbations. The model
benefits from straightforward implementation, as well as the flexibility to incorporate sophisticated perturbations and to be accessorized by local or nonlocal fluxes.
This module implements sediment flux dependent channel incision following:: E = f(Qs, Qc) * ((a stream power-like term) - (an optional threshold)), where E is the bed erosion rate, Qs is the
volumetric sediment flux into a node, and Qc is the volumetric sediment transport capacity at that node. This component is under active research and development; proceed with its use at your own
This module uses Taylor Perron’s implicit (2011) method to solve the nonlinear hillslope diffusion equation across a rectangular, regular grid for a single timestep. Note it works with the mass flux
implicitly, and thus does not actually calculate it. Grid must be at least 5x5. Boundary condition handling assumes each edge uses the same BC for each of its nodes. This component cannot yet handle
looped boundary conditions, but all others should be fine. This component has KNOWN STABILITY ISSUES which will be resolved in a future release; use at your own risk.
This numerical 1D research code Elv applied to gravel-sand transitions relates to Blom et al., 2017 "Advance, retreat, and halt of abrupt gravel-sand transitions in alluvial rivers", http://
This object manages ‘zones’ that are used to evaluate the spatial aspect of taxa. A zone represents a portion of a model grid. It is made up of spatially continuous grid nodes.
This process component is part of a spatially-distributed hydrologic model called TopoFlow, but it can now be used as a stand-alone model. It uses the "diffusive wave" method to compute flow
velocities for all of the channels in a D8-based river network. This method includes a pressure gradient term that is induced by a water-depth gradient in the downstream direction. This means that
instead of using bed slope in Manning's equation or the law of the wall, the water-surface slope is used.
This process component is part of a spatially-distributed hydrologic model called TopoFlow, but it can now be used as a stand-alone model. The kinematic wave method (Lighthill and Whitham, 1955) is
the simplest method for modeling flow in open channels. This method combines mass conservation with the simplest possible treatment of momentum conservation, namely that all terms in the general
momentum equation (pressure gradient, local acceleration and convective acceleration) are neglible except the friction and gravity terms. A flow in which gravitational acceleration is exactly
balanced by friction is referred to as steady, uniform flow. For these flows the water surface slope, energy slope and bed slope are all equal.
This process component is part of a spatially-distributed hydrologic model called TopoFlow, but it can now be used as a stand-alone model. TopoFlow supports three different types of flow diversions:
sources, sinks and canals. Sources are locations such as natural springs where water enters the watershed at a point by some process other than those that are otherwise modeled. Similarly, sinks are
locations where water leaves the watershed at a point. Canals are generally man-made reaches such as tunnels or irrigation ditches that transport water from one point to another, typically without
following the natural gradient of the terrain that is indicated by the DEM. The upstream end is essentially a sink and the downstream end a source.
This process component is part of a spatially-distributed hydrologic model called TopoFlow, but it can now be used as a stand-alone model. The dynamic wave method is the most complete and complex
method for modeling flow in open channels. This method retains all of the terms in the full, 1D momentum equation, including the gravity, friction and pressure gradient terms (as used by the
diffusive wave method) as well as local and convective acceleration (or momentum flux) terms. This full equation is known as the St. Venant equation. In the current version of TopoFlow it is assumed
that the flow directions are static and given by a D8 flow grid. In this case, integral vs. differential forms of the conservation equations for mass and momentum can be used.
This process component is part of a spatially-distributed hydrologic model called TopoFlow, but it can now be used as a stand-alone model.
This process component is part of a spatially-distributed hydrologic model called TopoFlow, but it can now be used as a stand-alone model.
This process component is part of a spatially-distributed hydrologic model called TopoFlow, but it can now be used as a stand-alone model.
This process component is part of a spatially-distributed hydrologic model called TopoFlow, but it can now be used as a stand-alone model.
This process component is part of a spatially-distributed hydrologic model called TopoFlow, but it can now be used as a stand-alone model.
This process component is part of a spatially-distributed hydrologic model called TopoFlow, but it can now be used as a stand-alone model.
This process component is part of a spatially-distributed hydrologic model called TopoFlow, but it can now be used as a stand-alone model.
This program calculates the 1D bed evolution of a sand-bed river after installation of a dredge slot. The calculation begins with the assumption of a prevailing mobile-bed normal flow equilibrium
before installation of the dredge slot. The flow depth H, volume bedload transport rate per unit width qb and volume suspended transport rate per unit width qs at normal flow are computed based on
input values of discharge Qww, channel width B, bed material sizes D50 and D90, sediment submerged specific gravity Rr and bed slope S. The sediment is assumed to be sufficiently uniform so that D50
and D90 are unchanging in space and time. The input parameter Inter specifies the fraction of any year for which flood flow prevails. At other times of the year the river is assumed to be
morphologically dormant. The reach is assumed to have length L. The dredge slot is excavated at time t = 0, and then allowed to fill in time with no subsequent excavation. The depth of initial
excavation below the bottom of the bed prevailing at normal equilibrium is an input variable with the name Hslot. The dredge slot extends from an upstream point equal to ru*L to a downstream point
rd*Hslot, where ru and rd are user-input values. The porosity lamp of the sediment deposit is a user-input parameter. The bedload transport relation used in the calculation is that of Ashida and
Michiue (1972). The formulation for entrainment of sediment into suspension is that of Wright and Parker (2004). The formulation for flow resistance is that of Wright and Parker (2004). The flow
stratification correction of Wright-Parker is not implemented here for simplicity. A quasi-equilibrium formulation is used to computed the transport rate of suspended sediment from the entrainment
rate. A backwater calculation is used to compute the flow. The water surface elevation at the downstream end of the reach is held constant at the value associated with normal flow equilibrium.
Iteration is required to compute: a) the flow depth prevailing at normal flow; b) the friction slope and depth prevailing at normal flow, b) the friction slope and depth associated with skin friction
associated with skin friction from any given value of depth, and b) the minimum Shields number below which form drag is taken to vanish.
This program computes 1D bed variation in rivers due to differential sediment transport. The sediment is assumed to be uniform with size D. All sediment transport is assumed to occur in a specified
fraction of time during which the river is in flood, specified by an intermittency. A Manning-Strickler relation is used for bed resistance. A generic Meyer-Peter Muller relation is used for sediment
transport. The flow is computed using a backwater formulation for gradually varied flow.
This program computes 1D bed variation in rivers due to differential sediment transport in which it is possible to allow the bed to undergo a sudden vertical fault of a specified amount, at a
specified place and time. Faulting is realized by moving all notes downstream of the specified point downward by the amount of the faulting. The sediment is assumed to be uniform with size D. All
sediment transport is assumed to occur in a specified fraction of time during which the river is in flood, specified by an intermittency. A Manning-Strickler formulation is used for bed resistance. A
generic relation of the general form of that due to Meyer-Peter and Muller is used for sediment transport. The flow is computed using the normal flow approximation.
This program computes fluvial aggradation/degradation with a bedrock-alluvial transition. The bedrock-alluvial transition is located at a point sba(t) which is free to change in time. A bedrock
basement channel with slope Sb is exposed from x = 0 to sba(t); it is covered with alluvium from x = sba(t) to x = sd, where sd is fixed. Initially sba = 0. The bedrock basement channel is assumed to
undergo no incision on the time scales at which the alluvial reach responds to change. In computing bed level change on the alluvial reach, the normal (steady, uniform) flow approximation is used.
Base level is maintained at x = sd, where bed elevation h = 0. The Engelund-Hansen relation is used to compute sediment transport rate, so the analysis is appropriate for sand-bed streams. Resistance
is specified in terms of a constant Chezy coefficient Cz.
This program computes gravel bedload and size distribution from specified values for the bed surface size distribution, the sediment specific gravity, and the effective bed shear velocity (based on
skin friction only).
This program computes the time evolution of the long profile of a river of constant width carrying a mixture of gravel sizes, the downstream end of which has a prescribed elevation.
This program implements the calculation for steady-state aggradation of a sand-bed river in response to sea level rise at a constant rate, as outlined in Chapter 25 of the e-book.
This program is a companion to the program SteadyStateAg, which computes the steady-state aggradation of a river with a specified base level rise at the downstream end. This program computes the time
evolution toward steady-state aggradation. The calculation assumes a specified, constant Chezy resistance coefficient Cz and floodplain width Bf. The sediment is assumed to be uniform with size D.
All sediment transport is assumed to occur in a specified fraction of time during which the river is in flood, specified by an intermittency. If grain size D < 2 mm the Engelund-Hansen (1967)
formulation for total bed material transport of sand is used. If grain size D >= 2 mm the Parker (1979) bedload transport formulation for gravel is used. The flow is computed using the normal flow
approximation. The reach has downchannel length L, and base level is allowed to rise at a specified rate at the downstream end.
This program provides two modules for studying the approach to mobile-bed normal equilibrium in recirculating and sediment-feed flumes containing uniform sediment. The module "Recirc" implements a
calculation for the case of a flume that recirculates water and sediment. The module "Feed" implements a calculation for the case of flume which receives water and sediment feed.
This pseudo-2D (cross-section, 1 independent variable x) numerical model permits calculating 1D lithospheric flexure with different rheologies, in combination with faulting, loading, and erosion/
deposition. The programs are developed in C for Linux platforms, graphic output is produced using GMT scripts, and standard PCs match the CPU and memory requirements. The software is available for
free under a GPL license.
This subroutine computes the deep water significant wave height and period at each point under a hurricane
This tool can be used to map out areas of hillslopes where the emergence of bedrock drives an increase in surface roughness. The tool requires an input DEM in float format and will output the
rasters, also in float format, for three eigenvectors that together describe the distribution of normal vectors within a user-defined neighbourhood for each pixel. To view the paper, please see:
This tool is used for examining bedrock channels. The tool is based on the assumption that the stream power incision model (SPIM) adequately describes channel incision. Channels profiles are
converted to chi-elevation space, where chi is a transformed longitudinal coordinate that takes drainage area into account. The tool uses a variety of statistical tests to extract the most likely
series of segments with distinct steepness in chi-elevation space. It also performs statistical tests to determine the best fit m/n ratio, where m is an area (A) exponent and n is a slope (S)
exponent in the SPIM with E = K A^m S^n, where E is an erosion rate and K is an 'erodibility'.
This tool is used to creates a "profile-smoothed" DEM from an input DEM.
This tool produces a flow path for each hilltop pixel on a landscape, generating hillslope length and relief data at a hillslope scale. These data can be used to discriminate between linear and
nonlinear sediment flux laws at a landscape scale. The model requires an input DEM in float format and produces a series raster and plain text output files which can be visualized and analysed using
code provided at: https://github.com/sgrieve/LH_Paper_Plotting For detailed information about how to use this tool please refer to the documentation (http://www.geos.ed.ac.uk/~smudd/LSDTT_docs/html/
This tool provides a method for extracting information on the nature and spatial extent of active geomorphic processes across deltas from the geometry of islands and the channels around them using
machine learning. The method consists of a two-step ensemble unsupervised machine learning algorithm that clusters islands into spatially continuous zones based on morphological metrics computed on
remotely sensed imagery
This tool uses chi river profile analysis to predict channel head locations across a landscape and therefore allow the extraction of river networks. It is most suitable for use with high resolution
LiDAR (1m) DEMs. The model requires an input DEM in float format and will output the extracted channel heads and networks, also in float format. For detailed information about how to use this tool
please refer to the documentation (http://www.geos.ed.ac.uk/~smudd/LSDTT_docs/html/channel_heads.html) and to the associated paper (http://onlinelibrary.wiley.com/doi/10.1002/2013WR015167/full).
This toolbox was constructed to help analyze changing river planforms (aerial views). Given a binary mask of a river, tools are provided to efficiently compute - channel centerline - banklines -
channel width (two methods) - centerline direction - centerline curvature If multiple input mask images contain georeference information, a tool is provided to "stitch" the masks together--before or
after analysis. Stitching can be done for both images and vectors of x,y coordinates. The mapping toolbox is required for this functionality. If multiple masks (realizations) of the river are
available, RivMAP includes tools to - compute centerline migrated areas - compute erosional and accretional areas - identify cutoff areas and quantify cutoff length, chute length, and cutoff area -
generate channel belt boundaries and centerline - measure and map changes (in width, migration areas or rates, centerline elongation, accreted/eroded areas) in space and time
This workbook computes 1D bed variation in rivers due to differential sediment transport. The sediment is assumed to be uniform with size D. All sediment transport is assumed to occur in a specified
fraction of time during which the river is in flood, specified by an intermittency. A Manning-Strickler formulation is used for bed resistance. A generic relation of the general form of that due to
Meyer-Peter and Muller is used for sediment transport. The flow is computed using the normal flow approximation.
This workbook computes the time evolution of a river toward steady state as it flows into a subsiding basin. The subsidence rate s is assumed to be constant in time and space. The sediment is assumed
to be uniform with size D. A Manning-Strickler formulation is used for bed resistance. A generic relation of the general form of that due to Meyer-Peter and Muller is used for sediment transport. The
flow is computed using the normal flow approximation. The river is assumed to have a constant width.
Three dimensional simulations of the Turbidity currents using DNS of incompressible Navier-Stokes and transport equations.
TopoPyScale uses a pragmatic approach to downscaling by minimizing complexity, reducing computational cost, simplifying interoperability with land surface models, while retaining physical coherence
and allowing the primary drivers of land surface-atmosphere interaction to be considered.
TopoToolbox provides a set of Matlab functions that support the analysis of relief and flow pathways in digital elevation models. The major aim of TopoToolbox is to offer stable and efficient
analytical GIS utilities in a non-GIS environment in order to support the simultaneous application of GIS-specific and other quantitative methods. With version 2, TopoToolbox adds various tools
specifically targeted at tectonic geomorphologists such as Chiplots and slopearea plots.
Topography is a Python library to fetch and cache NASA Shuttle Radar Topography Mission (SRTM) and JAXA Advanced Land Observing Satellite (ALOS) land elevation data using the OpenTopography REST API.
The Topography library provides access to the following global raster datasets: * SRTM GL3 (90m) * SRTM GL1 (30m) * SRTM GL1 (Ellipsoidal) * ALOS World 3D (30m) * ALOS World 3D (30m, Ellipsoidal) The
library includes an API and CLI that accept the dataset type, a latitude-longitude bounding box, and the output file format. Data are downloaded from OpenTopography and cached locally. The cache is
checked before downloading new data. Data from a cached file can optionally be loaded into an xarray DataArray using the experimental open_rasterio method.
Traditionally the Area-Slope equation (S=cA^alpha) is extracted from a catchment area vs. slope plot. This model calculate the Area-Slope constant and coefficient (alpha) for each pixel at the
catchment as a function of its downslope neighbor.
Transiently evolving river-channel width as a function of streambank properties, sediment in transport, and the hydrograph. This model is designed to compute the rates of river-channel widening and
narrowing based on changing hydrological regimes. It is currently designed for rivers with cohesive banks, with a critical shear stress for particle detachment and an erosion-rate coefficient. OTTAR
contains: * The RiverWidth class, which contains methods to evolve the width of an alluvial river. * The FlowDepthDoubleManning class, which is used to estimate flow depth from discharge, even with
an evolving river-channel geometry.
Underworld2 is an open-source, particle-in-cell finite element code tuned for large-scale geodynamics simulations. The numerical algorithms allow the tracking of history information through the
high-strain deformation associated with fluid flow (for example, transport of the stress tensor in a viscoelastic, convecting medium, or the advection of fine-scale damage parameters by the
large-scale flow). The finite element mesh can be static or dynamic, but it is not contrained to move in lock-step with the evolving geometry of the fluid. This hybrid approach is very well suited to
complex fluids which is how the solid Earth behaves on a geological timescale.
Uses the Barnes et al (2014) algorithms to replace pits in a topography with flats, or optionally with very shallow gradient surfaces to allow continued draining. This component is NOT intended for
use iteratively as a model runs; rather, it is to fill in an initial topography. If you want to repeatedly fill pits as a landscape develops, you are after the LakeMapperBarnes component. If you want
flow paths on your filled landscape, manually run a FlowDirector and FlowAccumulator for yourself. The locations and depths etc. of the fills will be tracked, and properties are provided to access
this information.
WACCM is NCAR's atmospheric high-altitude model; CARMA is Brian Toon's aerosol microphysical sectional model. I'm studying sulfate aerosols in the UTLS region using this coupled model.
WASH123D is an integrated multimedia, multi-processes, physics-based computational watershed model of various spatial-temporal scales. The integrated multimedia includes: # dentric streams/rivers/
canal/open channel, # overland regime (land surface), # subsurface media (vadose and saturated zones), and # ponds, lakes/reservoirs (small/shallow). It also includes control structures such as
weirs, gates, culverts, pumps, levees, and storage ponds and managements such as operational rules for pumps and control structures. The WASH123D code consisted of eight modules to deal with multiple
media: # 1-D River/Stream Networks, # 2-D Overland Regime, # 3-D Subsurface Media (both Vadose and Saturated Zones); # Coupled 1-D River/Stream Network and 2-D Overland Regime, # Coupled 2-D Overland
Regime and 3-D Subsurface, # Coupled 3-D Subsurface and 1-D River Systems; # Coupled 3-D Subsurface Media, 2-D Overland, and 1-D River Network; and # Coupled 0-D Shallow Water Bodies and 1-D Canal
Network. For any of the above eight modules, flow only, transport only, or coupled flow and transport simulations can be carried out using WASH123D.
WAVI.jl is designed to make ice sheet modelling more accessible to beginners and low-level users, whilst including sufficient detail to be used for addressing cutting-edge research questions.
We have developed a hybrid numerical model at a continental scale via control volume finite element (finite volume) and regular finite element methods to evaluate the stress variation, pore pressure
evolution, brine migration, solute transport and heat transfer in the subsurface formations in response to ice sheet loading of multiple glacial cycles.
We present a geometric model able to track the geomorphic boundaries that delimit the fluvial plain of fluvial-deltas: the shoreline and the alluvial-bedrock transition. By assuming a fluvial profile
with a quadratic form, which satisfies the overall mass balance and the boundary conditions dictated by diffusive transport, we are able to provide a solution that accounts for general base-level
When wind blows over snow, it self-organizes. This forms surface features, such as ripples and dunes, that alter the reflectivity and thermal conductivity of the snow. Studying these features in the
field is cold and challenging (we've tried), so we created rescal-snow to enable snow scientists to study snow features in controlled numerical experiments. We hope that this model will be useful to
researchers in snow science, geomorphology, and polar climate. Rescal-snow is able to simulate: - Snow/sand grain erosion and deposition by wind - Snowfall - Time-dependent cohesion (snow sintering)
- Avalanches of loose grains Rescal-snow is also designed for robust, reproducible science, and contains tools for high-performance computing, data management, and data analysis, including: -
Workflow tools for generating and running many simulations in parallel - A python-based workflow that manages data and analysis at runtime These processes, along with model input, output, performance
and constraints, are discussed in detail in the project docs and readme.
Whole atmosphere module of sulfate aerosols with emphasis on stratospheric aerosols and dust.
Why ROMSBuilder? ROMS extensively uses the C preprocessor (cpp) during compilation to replace code statements, insert files into the code, and select relevant parts of the code depending on its
directives. There are numerous cpp options that can be activated in header files for your specific application. The preprocessor reads the source file (*.F) and builds a target file (*.f90) according
to activated cpp options. CPP options can be set through the CMT config tab dialogs. ROMSBuilder generates the header file for compiling the new ROMS component from the tab dialog inputs.
Xbeach is a two-dimensional model for wave propagation, long waves and mean flow, sediment transport and morphological changes of the nearshore area, beaches, dunes and backbarrier during storms. It
is a public-domain model that has been developed with funding and support by the US Army Corps of Engineers, by a consortium of UNESCO-IHE, Deltares, Delft University of Technology and the University
of Miami.
bmi_dbseabed package (https://github.com/gantian127/bmi_dbseabed) provides a set of functions that allows downloading of the dataset from dbSEABED (https://instaar.colorado.edu/~jenkinsc/dbseabed/),
a system for marine substrates datasets across the globe. bmi_dbseabed package also includes a Basic Model Interface (BMI), which converts the dbSEABED datasets into a reusable, plug-and-play data
component for the PyMT modeling framework developed by Community Surface Dynamics Modeling System (CSDMS).
eSCAPE is a parallel landscape evolution model, built to simulate Earth surface dynamics at global scale and over geological times. The model is primarily designed to address problems related to
geomorphology, hydrology, and stratigraphy, but it can also be used in related fields. eSCAPE accounts for both hillslope processes (soil creep using linear diffusion) and fluvial incision (stream
power law). It can be forced using spatially and temporally varying tectonics (vertical displacements) and climatic conditions (precipitation changes and/or sea-level fluctuations).
gospl is able to simulate global-scale forward models of landscape evolution, dual-lithology (coarse and fine) sediment routing and stratigraphic history forced with deforming plate tectonics,
paleotopographies and paleoclimate reconstructions. It relates the complexity of the triggers and responses of sedimentary processes from the complete sediment routing perspective accounting for
different scenarii of plate motion, tectonic uplift/subsidence, climate, geodynamic and sedimentary conditions.
iHydroSlide3D v1.0 is a physically based modeling framework that accounts for both hydrological and geotechnical processes. The model mainly includes the following modules: (i) a distributed
hydrological model based on the Coupled Routing and Excess STorage (CREST) model, (ii) a newly developed 3D landslide model, and (iii) a soil moisture downscaling method.
nwis package provides a set of functions that allows downloading of the National Water Information System (NWIS) for data analysis and visualization. nwis package includes a Basic Model Interface
(BMI), which converts the NWIS dataset into a reusable, plug-and-play data component for Community Surface Dynamics Modeling System (CSDMS) modeling framework.
nwm package provides a set of functions that allows downloading of the National Water Model (NWM) time series datasets for a river reach or a model grid. nwm package also includes a Basic Model
Interface (BMI), which converts the dataset into a reusable, plug-and-play data component for the CSDMS modeling framework.
olaFlow (formerly known as olaFoam) is a numerical model conceived as a continuation of the work in IHFOAM. Its development has been continuous from ihFoam (Jul 8, 2014 - Feb 11, 2016) and olaFoam
(Mar 2, 2016 - Nov 25, 2017). This free and open source project is committed to bringing the latest advances in the simulation of wave dynamics to the OpenFOAM® and FOAM-extend communities. olaFlow
includes a set of solvers and boundary conditions to generate and absorb water waves actively at the boundaries and to simulate their interaction with porous coastal structures.
openAMUNDSEN is a fully distributed model, designed primarily for resolving the mass and energy balance of snow and ice covered surfaces in mountain regions. Typically, it is applied in areas ranging
from the point scale to the regional scale (i.e., up to some hundreds to thousands of square kilometers), using a spatial resolution of 10–100 m and a temporal resolution of 1–3 h, however its
potential applications are very versatile.
physical property, velocity modeling and synthetic seismic modeling
pyDeltaRCM is the Python version of DeltaRCM (https://csdms.colorado.edu/wiki/Model:DeltaRCM) by Man Liang (also available from the CSDMS model repository). This version is a WMT component but can
also be run as a stand-alone model (see README.md). DeltaRCM is a parcel-based cellular flux routing and sediment transport model for the formation of river deltas, which belongs to the broad
category of rule-based exploratory models. It has the ability to resolve emergent channel behaviors including channel bifurcation, avulsion and migration. Sediment transport distinguishes two types
of sediment: sand and mud, which have different transport and deposition/erosion rules. Stratigraphy is recorded as the sand fraction in layers. Best usage of DeltaRCM is the investigation of
autogenic processes in response to external forcings.
pySBELT simulates the kinematics of rarefied particle transport (low rates) as a stochastic process along a riverbed profile. pySBeLT is short for Stochastic Bed Load Transport.
pymt_era5 is a package that converts ERA5 datasets (https://confluence.ecmwf.int/display/CKB/ERA5) into a reusable, plug-and-play data component for PyMT modeling framework developed by Community
Surface Dynamics Modeling System (CSDMS). This allows ERA5 datasets (currently support 3 dimensional data) to be easily coupled with other datasets or models that expose a Basic Model Interface.
pymt_roms is a package that converts the ROMS model (https://www.myroms.org/) datasets into a reusable, plug-and-play data component for PyMT modeling framework developed by Community Surface
Dynamics Modeling System (CSDMS). This allows ROMS model datasets to be easily coupled with other datasets or models that expose a Basic Model Interface.
soilgrids package provides a set of functions that allow downloading of the global gridded soil information from SoilGrids https://www.isric.org/explore/soilgrids, a system for global digital soil
mapping to map the spatial distribution of soil properties across the globe. soilgrids package includes a Basic Model Interface (BMI), which converts the SoilGrids dataset into a reusable,
plug-and-play data component for Community Surface Dynamics Modeling System (CSDMS) modeling framework.
stream_power_smooth_threshold.py: Defines the StreamPowerSmoothThresholdEroder, which is derived from FastscapeEroder. StreamPowerSmoothThresholdEroder uses a mathematically smooth threshold
formulation, rather than one with a singularity.
wave-current interaction, (non) hydrostatic flow (2D/3D), salinity, temperature, (non) cohesive sediment transport, morphology, bed stratigraphy, water quality, ecology, structures & control,
particle tracking, curvilinear multi-domain mesh in cartesian or spheric coord., online visualization, GUI.
“GEOMBEST-Plus” (Geomorphic Model of Barrier, Estuarine, and Shoreface Translations) is a new morphological-behaviour model that simulates the evolution of coastal morphology and stratigraphy,
resulting from changes in sea level, and sediment volume within the shoreface, barrier and estuary. GEOMBEST-Plus differs from other large-scale behaviour models (e.g. Bruun, 1962; Dean and Maumeyer,
1983; Cowell et al., 1995; Niedoroda et al., 1995, Stive & de Vriend, 1995 and Storms et al., 2002) by relaxing the assumption that the initial substrate (i.e stratigraphy) is comprised of an
unlimited supply of unconsolidated material (typically sand). The substrate is instead defined by distinct stratigraphic units characterized by their erodibility and sediment composition.
Additionally, GEOMBEST-Plus differs from its predecessor (GEOMBEST) by adding in a dynamic stratigraphic unit for a backbarrier marsh. Accordingly, the effects of geological framework on
morphological evolution and shoreline translation can be simulated. | {"url":"https://csdms.colorado.edu/csdms_wiki/index.php?title=Property:Extended_model_description&limit=500&offset=40&from=&until=&filter=","timestamp":"2024-11-03T13:04:03Z","content_type":"text/html","content_length":"823307","record_id":"<urn:uuid:e0fdb7eb-15e7-416f-9bb1-9098ae7ec252>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00625.warc.gz"} |
Hypergraph Removal Lemmas via Robust Sharp Threshold Theorems
The classical sharp threshold theorem of Friedgut and Kalai (1996) asserts that any symmetric monotone function f: {0,1}^n →{0,1} exhibits a sharp threshold phenomenon. This means that the
expectation of f with respect to the biased measure μ[p] increases rapidly from 0 to 1 as p increases. In this paper we present ‘robust’ versions of the theorem, which assert that it holds also if
the function is ‘almost’ monotone, and admits a much weaker notion of symmetry. Unlike the original proof of the theorem which relies on hypercontractivity, our proof relies on a ‘regularity’ lemma
(of the class of SzemerÌl’di’s regularity lemma and its generalizations) and on the ‘invariance principle’ of Mossel, O’Donnell, and Oleszkiewicz which allows (under certain conditions) replacing
functions on the cube {0,1}^n with functions on Gaussian random variables. The hypergraph removal lemma of Gowers (2007) and independently of Nagle, Rödl, Schacht, and Skokan (2006) says that if a
k-uniform hypergraph on n vertices contains few copies of a fixed hypergraph H, then it can be made H-free by removing few of its edges. While this settles the ‘hypergraph removal problem’ in the
case where k and H are fixed, the result is meaningless when k is large (e.g. k > logloglogn). Using our robust version of the Friedgut–Kalai Theorem, we obtain a hypergraph removal lemma that holds
for k up to linear in n for a large class of hypergraphs.
Bibliographical note
Publisher Copyright:
© Licensed under a Creative Commons Attribution License (CC-BY)
Dive into the research topics of 'Hypergraph Removal Lemmas via Robust Sharp Threshold Theorems'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/hypergraph-removal-lemmas-via-robust-sharp-threshold-theorems","timestamp":"2024-11-07T23:14:09Z","content_type":"text/html","content_length":"48642","record_id":"<urn:uuid:c0947f87-f64d-472b-8857-fde41af30579>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00248.warc.gz"} |
h C
19 research outputs found
We analyze the occurrence of dynamically equivalent Hamiltonians in the parameter space of general many-body interactions for quantum systems, particularly those that conserve the total number of
particles. As an illustration of the general framework, the appearance of parameter symmetries in the interacting boson model-1 and their absence in the Ginocchio SO(8) fermionic model are
discussed.Comment: 8 pages, REVTeX, no figur
The second virial coefficient $B_{2}^{nc}(T)$ for non-interacting particles moving in a two-dimensional noncommutative space and in the presence of a uniform magnetic field $\vec B$ is presented. The
noncommutativity parameter \te can be chosen such that the $B_{2}^{nc}(T)$ can be interpreted as the second virial coefficient for anyons of statistics \al in the presence of $\vec B$ and living on
the commuting plane. In particular in the high temperature limit \be\lga 0, we establish a relation between the parameter \te and the statistics \al. Moreover, $B_{2}^{nc}(T)$ can also be interpreted
in terms of composite fermions.Comment: 11 pages, misprints corrected and references adde
A brief overview is given of recent developments and fresh ideas at the intersection of PT and/or CPT-symmetric quantum mechanics with supersymmetric quantum mechanics (SUSY QM). We study the
consequences of the assumption that the "charge" operator C is represented in a differential-operator form. Besides the freedom allowed by the Hermiticity constraint for the operator CP, encouraging
results are obtained in the second-order case. The integrability of intertwining relations proves to match the closure of nonlinear SUSY algebra. In an illustration, our CPT-symmetric SUSY QM leads
to non-Hermitian polynomial oscillators with real spectrum which turn out to be PT-asymmetric.Comment: 25 page
We propose a matrix model to describe a class of fractional quantum Hall (FQH) states for a system of (N_1+N_2) electrons with filling factor more general than in the Laughlin case. Our model, which
is developed for FQH states with filling factor of the form \nu_{k_1k_2}=\frac{k_1+k_2}{k_1k_2} (k_1 and k_2 odd integers), has a U(N_1)\times U(N_2) gauge invariance, assumes that FQH fluids are
composed of coupled branches of the Laughlin type, and uses ideas borrowed from hierarchy scenarios. Interactions are carried, amongst others, by fields in the bi-fundamentals of the gauge group.
They simultaneously play the role of a regulator, exactly as does the Polychronakos field. We build the vacuum configurations for FQH states with filling factors given by the series \nu_{p_1p_2}=\
frac{p_2}{p_1p_2-1}, p_1 and p_2 integers. Electrons are interpreted as a condensate of fractional D0-branes and the usual degeneracy of the fundamental state is shown to be lifted by the
non-commutative geometry behaviour of the plane. The formalism is illustrated for the state at \nu={2/5}.Comment: 40 pages, 1 figure, clarifications and references adde
For a specific exactly solvable 2 by 2 matrix model with a PT-symmetric Hamiltonian possessing a real spectrum, we construct all the eligible physical metrics and show that none of them admits a
factorization CP in terms of an involutive charge operator C. Alternative ways of restricting the physical metric to a unique form are briefly discussed.Comment: 13 page | {"url":"https://core.ac.uk/search/?q=author%3A(Geyer%2C%20Hendrik%20B.)","timestamp":"2024-11-05T18:39:22Z","content_type":"text/html","content_length":"132405","record_id":"<urn:uuid:8e19bea4-e4bc-478a-851b-a61450da96a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00322.warc.gz"} |
Professor Guozhen Lu appointed Editor-in-Chief of Forum Mathematicum | Department of Mathematics
Professor Guozhen Lu appointed Editor-in-Chief of Forum Mathematicum
Guozhen Lu has been appointed Editor-in-Chief of
Forum Mathematicum
, a premier journal in general mathematics.
Dr. Lu is currently also the Editor-in-Chief of
Advanced Nonlinear Studies
, a major journal in Nonlinear Analysis, Partial Differential Equations and Calculus of Variations, as well as Editor-in-Chief of the de Gruyter flagship book series
Studies in Mathematics
In addition, Dr. Lu has served on the editorial boards of multiple mathematical journals, including the general mathematical journal
Bulletin des Sciences Mathematiques
(also known as Darboux journal founded by Gaston Darboux in 1870), one of the oldest mathematical journals currently still in existence. | {"url":"https://math.uconn.edu/2024/09/06/professor-guozhen-lu-appointed-editor-in-chief-of-forum-mathematicum/","timestamp":"2024-11-11T00:58:41Z","content_type":"text/html","content_length":"103381","record_id":"<urn:uuid:d9d8a7f2-8cce-4456-aef1-6f870d89f8aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00716.warc.gz"} |
Learn How to Factor Expressions - MathCracker.com
Learning how to factor is one of the most crucial skills you can learn. Factoring has so many applications, that you will be glad to take the time to learn all there is about it.
Factoring is normally something that we take for granted, and based upon different properties, such as the commutative, associative and distributive property. Those properties allow you to move
around and group terms in a convenient way.
A quick refreshment about the commutative, associative and distributive property. For the real numbers \(x\), \(y\) and \(z\), we have the following properties
where \(+\) and \(\cdot\) are the sum and product of real numbers, respectively.
Why Is It Useful to Know How to Factor?
There are many reasons, but one of the crucial ones is that factoring gives us an easy way of solving equations. In fact, factoring is THE way we have to solve equations.
For example, consider the equation where we are trying to solve for \(x\):
\[\large xy +xz = 0\]
How do we go about it? Well, we can use the distributive property to get:
\[\large x y + x z = x(y+z) = 0\]
Therefore, with this last expression \( x(y+z) = 0\) we have an example of factoring. Indeed, we took the initial expression, \(xy+xz\) and we factored it into \( x(y+z)\).
So, now we need to solve an easier equation, which is \( x(y+z) = 0\). Why is it easier? It is because now that we know that the product \( x(y+z)\) is equal to zero, then one of the factors NEEDS to
be equal to zero.
So, if we know that \(y+z = \not 0\), then we know that we need to have that \(x = 0\).
LESSON : One advantage of factoring is being able to write an equation as a multiplication of factors that is equal to zero. Then, AT LEAST ONE OF THE FACTORS MUST BE ZERO.
For example, when we need to solve for \(x\) in the following equation:
\[\large 5x + 3x = 0\]
we don't realize we are actually factoring when we do
\[\large 5x + 3x = (5+3)x = 8x = 0\]
so we have reduced our equation to a product of factors that is equal to zero: \(8x = 0\). Since one of the factors \(8\) is not equal to zero, then the only possible solution is \(x = 0\).
In other words : if you know how to factor, you will likely know how to solve equations .
How to Factor Polynomials
The role of factoring should be clear by now, in terms of its usefulness to solve equations. The only problem is that there is not a generic, single strategy that can be used to factor ALL possible
algebraic expressions.
So, normally, we will be happy to factor relatively simple expressions, but ideally, we would like to know how to factor as many expressions as we can.
The balance is reached with a very general class of expressions that we can, often times, factor very systematically. That class is the class of polynomials. For example, the expression
\[\large 2x^2 + 5x + 3\]
is a polynomial of degree 2. Or the expression below
\[\large x^3 - 3x^2 + 4x+2\]
is a polynomial of degree 3.
In general, a expression of the form
\[\large a_n x^n + a_{n-1}x^{n-1} + ... + a_1 x + a_0\]
is a polynomial of degree \(n\). Naturally, the simpler the expression, the easier it will be to simplify, so we should try learn how to factor quadratic expressions first. This is, polynomials of
degree two.
EXAMPLE 1
Factor the following quadratic expression
\[\large x^2 + x - 2\]
This example will show you, on purpose, that it can be tricky to factor even the simplest expression, like the one above. What would you do to factor it?
What if I tell you, you need to add zero? It is kind ridiculous, right? Let's see:
\[\large x^2 + x - 2 = x^2 + x + 0 - 2 \]
Do you agree with the above? I just added \(0\). Nothing has changed. But, what if I tell you that \(0 = 2x - 2x\)? So then
\[\large x^2 + x - 2 = x^2 + x + 0 - 2 = x^2 + x + (2x - 2x) - 2 \]
All the same! It works, because I added zero, so nothing changes. But now we expand it and group it:
\[\large x^2 + x - 2 = x^2 + x + 0 - 2 \] \[\large = x^2 + x + (2x - 2x) - 2 \] \[ \large = x^2 + x - 2x + 2x - 2 \] \[\large = x^2 + (x - 2x) + 2x - 2 \] \[\large = x^2 - x + 2x - 2 \] \[\large = x
(x-1) + 2(x-1)\] \[\large = (x+2)(x-1)\]
So then, finally, \(x^2 + x - 2 = (x+2)(x-1)\). Tricky? Perhaps, but that is one way to do it. Despite being a clever way to do it, we would prefer a more systematic way.
Factor a Quadratic Polynomial
Clever tricks are nice, and all that, but usually we will prefer a systematic approach, that never fails. For quadratic polynomials (polynomials of degree 2), there is a systematic way of proceeding
with the factoring:
Step 1 : Given the quadratic expression \(ax^2 + bx + c\), we first solve the equation
\[\large ax^2 + bx + c = 0\]
Step 2 : If the solutions (roots) to the above equation are real (even if there is only one root), we call those roots \(x_1\) and \(x_2\). With these roots, we get the following factors:
\[\large ax^2 + bx + c = a(x - x_1)(x - x_2)\]
so the solutions \(x_1\) and \(x_2\) completely determine the factors.
Naturally in this case, as expected, solving a quadratic equation is tightly connected with factoring the quadratic equation.
EXAMPLE 2
Factor the following quadratic equation
\[\large x^2 - 4x + 3\]
by computing its roots.
We start by solving the corresponding quadratic equation:
\[\large x^2 - 4x + 3 = 0\]
using the famous and well known quadratic formula :
\[\large\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] \[\large\displaystyle = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(3)}}{2(1)} \] \[\large\displaystyle = \frac{4 \pm \sqrt{16 - 12}}{2} \] \[\
large\displaystyle = \frac{4 \pm \sqrt{4}}{2} \] \[\large\displaystyle = \frac{4 \pm 2}{2} \]
which implies that the solutions (roots) are \(x_1 = 1\) and \(x_2 = 1\). Then, the quadratic expression \(x^2 - 4x + 3\) can be factored as follows:
\[\large x^2 - 4x + 3 = a(x - x_1)(x - x_2) = (x-1)(x-3) \]
Observe that in this case the term multiplying the \(x^2\) term is 1, so then in this case \(a = 1\).
Factoring Polynomials with degree greater than 2
So, to factor quadratic polynomials I just compute the roots of the corresponding quadratic equation. How do I factor polynomials of higher degree?? Using exactly the same method .
Step 1: Given the polynomial expression \(a_n x^n + a_{n-1}x^{n-1} + ... + a_1 x + a_0\), we first solve the equation
\[\large a_n x^n + a_{n-1}x^{n-1} + ... + a_1 x + a_0 = 0\]
Step 2: If the solutions (roots) to the above equation are real (even if they are repeated), we call those roots \(x_1\), \(x_2\), ..., \(x_n\). With these roots, we get the following factors:
\[\large a_n x^n + a_{n-1}x^{n-1} + ... + a_1 x + a_0 = a_n(x - x_1)(x - x_2)\cdots (x - x_n)\]
So, it would seem that it is equally simple to factor a polynomial of degree 2 than it is to factor a polynomial of degree 10. Theoretically, the answer is yes.
The only problem is that there is not a simple, close algebraic formula that can solve roots for a polynomial equation of degree 5 or higher.
Sometimes, we can solve higher degree equations by looking at the graph, or even using the calculator. For example, check the graph below:
Graphically, we can see that the polynomial crosses the x-axis at three points: \(x_1 = -1\), \(x_2 = 1\) and \(x_3 = 3\), so these are the roots.
So then, we know that the polynomial must be of the form \(p(x) = a(x+1)(x-1)(x-3)\). We would need to know one more point to know the constant \(a\).
More About Factoring
We are just scratching the surface with the concept of factoring, although there is not much more than can be done for general expressions. The best we can do is to give a systematic approach to
factor polynomials.
But, having a general treatment to factor polynomials is not a minor thing, and the idea of using the roots to factor a polynomial is nothing less than the Fundamental Theorem of Algebra. So, at
least by the title, you can tell it is not little.
Factoring General Expressions
There are no general rules to factor general expressions. We need to play by ear and try to exploit the structure of the expression. Sometimes we can factor, sometimes we cannot. It all depends on
the expression. The only general rule is to try to group and try to find common factors so to further group and simplify.
How to Factor by Grouping
That is the first example we did. Say you have:
\[\large x^2 - x + 2x - 2 \]
so we group the first two terms and the last two terms to get:
\[\large (x^2 - x) + (2x - 2) \]
and each of these groups can be factored as
\[\large x(x - 1) + 2(x - 1) \]
and now we have two terms that have a common factor \(x-1\), so we factor it as
\[\large (x+2)(x - 1) \]
Sometimes it is more practical to use a calculator to find the factors. You can use our quadratic equation solver to find the factors of a quadratic expression.
Notice that there are several techniques that can help you when you need to factor a expression, depending of its structure. One of those is the method of factor by grouping which, when it works, can
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The Math Department Prerequisite Videos Resource Website
1. Proofs
A review of proof by inductions as well as quantifiers and their negations.
2. Row Operations
Covers the elementary row operations on matrices, row echelon form, reduced row echelon form, and solving matrix equations using row operations.
3. Subspaces
Defines subspaces of R^n and gives some examples and non-examples of subspaces.
4. Bases
Reviews the ideas of linear independence, spanning, and basis. Includes concept of dimension of subspaces and how to determine these properties.
5. Eigenvalues
Covers eigenvalues, eigenvectors, and eigenspaces along with how to determine each one from a given matrix.
6. Diagonalization
Describes what it means to be a diagonalizable matrix and how to diagonalize certain types of matrices, including an application of diagonalization. | {"url":"https://www.math.uci.edu/~prerequisite-videos/121a.html","timestamp":"2024-11-11T08:41:55Z","content_type":"text/html","content_length":"6860","record_id":"<urn:uuid:7a6798e5-da56-4a05-87e3-52d9a40cfa0d>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00557.warc.gz"} |
The "defense first" strategy in college football OT -- part 2
The "defense first" strategy in college football OT -- part 2
In a post a couple of days ago, I noted a study that found a strong advantage to choosing "defense first" in college football OT. The first couple of paragraphs of that previous post describe the
overtime rule ... you can go back and read them if you're not familiar with it.
That study found that the team that goes on offense last in the first OT (which I'll call "the second team" or "team 2") beat the team that went on offense first ("the first team" or "team 1") 54.9%
of the time.
With some of the numbers listed in the original study, and some additional assumptions, we can try to figure out a *theoretical* probability with which the second team will win, and compare that to
the observed 54.9%. For this theoretical calculation, I'm assuming the two teams are equal in skill.
The original study gave the distribution of first team scoring. I've combined 6- and 7-point touchdowns to keep things simple (which won't affect the results much):
.235 – team 1 scores 0 points
.299 – team 1 scores FG
.466 – team 1 scores TD
What is the distribution of second team scoring? It depends what the first team does, and we have to guess a bit.
Suppose the first team scores a touchdown. Then, the second team never goes for a field goal. So it's in what otherwise would be a field goal situation .299 of the time, but will have to go for a
touchdown anyway. Suppose from fourth-and-something, they will score a touchdown 50% of the time, and score 0 points 50% of the time. In those cases, that would change their distribution to:
.385 – team 2 scores 0 after team 1 TD
.000 – team 2 FG after team 1 TD
.615 – team 2 TD after team 1 TD
Now, suppose the first team scored a field goal. We'll assume the second team plays exactly the same way as the first team:
.235 – team 2 scores 0 after team 1 FG
.299 – team 2 FG after team 1 FG
.466 – team 2 TD after team 1 FG
Finally, suppose the first team scored zero. It had a 76.5% chance to score, but failed. The second team must have a greater than 76.5% chance to score, because it's going to go for a field goal in
some cases where the first team might have chosen to go for a touchdown (and fumbled or something). Let's call it 80%.
.200 – team 2 scores 0 after team 1 scores 0
.800 – team 2 FG or TD after team 1 scores 0
The chances of the first team winning in the first OT is the sum of these probabilities:
.180 – team 1 TD, team 2 zero (.466 * .385)
.070 – team 1 FG, team 2 zero (.299 * .235)
.000 – team 1 TD, team 2 FG (never)
.250 – Total chance team 1 wins in this OT
The chances of the second team winning in the first OT is this sum:
.188 – team 1 zero, team 2 TD/FG (.235 * .800)
.139 – team 1 FG, team 2 TD (.299 * .466)
.327 – Total chance team 2 wins in this OT
And the chance of a tie is 1 minus the above two totals, which works out to
.423 – Total chance of this OT ending in a tie
Now, the chance of the (original) second team winning the game, is this sum:
Chance of winning in the first OT + (Chance the first OT is a tie * chance of winning the second OT) + (Chance the first two OTs are ties * chance of winning the third OT) + ...
Also, if a given OT ends in a tie, the first team has to go second this period, and the second team has to go first. So the probabilities are switched in the even-numbered OTs. Therefore, the above
sum works out to:
.327 + (.423 * .250) + (.423^2 * .327) + (.423^3 * .250) + ...
The sum of that infinite series (which is actually two intertwined geometric series) works out to .526.
Under the assumptions listed above, the chance of the "defense first" team beating the other, equally matched team in OT is .526.
But as we saw in the other study, the actual chance was .549. Why is our estimate different?
One possibility is that we're assuming evenly-matched teams. In real life, college games are often mismatches. However, mismatches should minimize the effect. The more mismatched the teams, the less
likely winning the coin toss should affect the result (if a team is good enough, it'll keep scoring TDs and win even if it has to go first). So the real life number for equally matched teams should
be higher than .549, not lower.
So what's going on? Why do these estimates not match the observed results? One of three possibilities:
1. My calculations and logic are wrong;
2. The assumptions are wrong;
3. Favored teams won a lot of coin tosses just by luck.
I'm assuming it's not luck. Any suggestions?
Labels: football
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Pi123: Shaping The Next Generation Of Digital TransformationPi123: Shaping The Next Generation Of Digital Transformation
Many ideas and applications in the fields of mathematics. And technology have grown to be indispensable resources for both experts and hobbyists. Pi123 is one such idea. it has several features and
advantages that make it a useful. Tool for anyone working with numbers. We will discuss what Pi 123 is, its advantages. Features, security issues, and how to set it up and utilize it efficiently in
this guide.
What is Pi123?
An online calculator for the mathematical constant pi (π) is called Pi123. Pi, a fundamental mathematical constant. is a circle’s circumference divided by its diameter.
Its decimal form never ends or repeats, indicating that it is an irrational number. That cannot be stated as a straightforward fraction. Pi 123 is a useful tool for mathematicians, scientists,
engineers. And anybody else working with numbers. Since it makes it possible to compute pi with a high degree of accuracy.
How to Install and Operate Pi123?
• Utilizing and configuring Pi 123 is a simple procedure. To get you started, here’s a step-by-step tutorial.
• Go to https://pi123.com/, the Pi1 23 website, and select the “Calculate Pi” option.
• After selecting the desired number of digits to calculate, click the “Calculate” button.
• When the computation is finished, the results will appear on the screen.
• By selecting the “Explore Pi” option, you may additionally investigate.
• The pi digits and create graphic representations.
Pi123 as an Extension in Mathematics
Pi123 is a mathematics extension that lets users investigate the characteristics. And uses of pi in addition to being a straightforward calculator for pi. Pi 123 allows users to explore the digits of
pi, calculate it with great accuracy. And even create visual representations of pi, like the well-known “pi spiral.”
Pi123: A Pi Calculator Available Online
The fact that Pi123 is an online application that users. May access from any device with an internet connection is one of its biggest advantages. Because of this, it’s a useful and easily available
tool for anyone who needs to quickly calculate pi. Pi 123 is also completely free to use, which makes it a great. Choice for enthusiasts, professionals, and students alike.
Pi 123 within the Pi Network Context
Pi Network, a brand-new cryptocurrency that seeks to provide. Digital currency to everyone, is also closely associated with Pi 123. The Pi Network is a perfect fit for Pi 123 since it employs a
special consensus mechanism based on the number pi. Users that are interested in the realm of cryptocurrencies can benefit from. Pi 123 by using it to mine Pi coins and contribute to the Pi Network.
Substitutes for Pi123
• There are other options available for computing pi, even if Pi123 is a great tool.
• Among the most well-liked substitutes are. (https://www.wolframalpha.com/) Wolfram Alpha (https://www.picalc.org/) The Pi Calculator (https://pihex.com/) .
• PiHex It’s worth investigating each of these options to see which best.
Difficulties with Using Pi123
Although Pi123 is a useful tool for computing pi, using it can present certain difficulties. The security issues with internet technologies provide one of the biggest obstacles. Data leaks and
cyberattacks are always a possibility with any online platform. As a result, in order to safeguard your data, you must use Pi 123. On a secure network and adopt all appropriate security measures.
For anyone working with numbers, be it a scientist, engineer. Mathematician, or hobbyist, Pi123 is an invaluable tool. Its features and advantages make it a useful tool for both professionals and
students. And its integration with the Pi Network gives its application an intriguing new angle. By following the instructions in this guide, you may effectively set up and use Pi 123 while also
helping the Pi Network community to grow.
Also Read About – iStudyInfo: Your Ultimate Guide to Academic Success
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Activities for developing logic and mathematics for 2-years-olds
The mathematical development of 2-year-old children means qualitative changes in their cognitive activity as a result of the formation of elementary mathematical concepts and understanding easy
logical operations. Mathematical development is an important component in the formation of the «world picture» of a child.
Various didactic games and activities help develop early math skills. During games, a toddler acquires new knowledge and skills. Math activities contribute to the development of perception,
attention, memory, thinking and creative abilities. They boost cognitive development in general.
In the elementary school math course is not simple. Often children experience various kinds of difficulties in mathematics, because they are not ready to perceive math as a subject.
Therefore, parents should develop a child’s interest in math from the early years. An introduction to the subject in a playful and entertaining way will help the child to faster and easier master the
school curriculum in future.
What math skills should be mastered by a 2-year-old child?
1. To know the concept of «many — few».
2. Closer to three years — to master the concept of «more-less».
3. Learn to distinguish the number of items: «one» and «two» at least. You can go further and teach the toddler to count objects to 5 or to 10, but not all the kids aged from 2 to 3 have interest
and ability to count — this is normal. I don’t mean just saying numbers from 1 to 10, but counting real objects and understanding the meaning of numbers. Learn how to teach your baby counting.
4. Learn to sort objects by size, by color, by type. You can sort different kinds of pasta, buttons, large and small objects, circles, squares, etc. You can try other types of sorting.
5. Learn to navigate in space (to learn the concepts of above, below, to understand right and left — here are some activities for teaching left and right).
6. To do simple puzzles from 2−4 parts without help. This ability develops gradually. At the beginning, you should help your child.
7. To learn to match. Play «Who eats what?», «Whose house is this», «Whose tail is this», «Whose baby is this»? Use special sets of cards or pictures. I use puzzles like that:
I use puzzles like that:
1. Understand the description. The mother describes in the simplest form an object or animal, and the child guesses. For example: «It is small, with long white ears, it jumps and eats carrots, who
is it?» or «Who says «Moo-Moo» and gives milk? You can gradually complicate the riddles.
2. Placing objects inside each other according to its size. You can use a set of matryoshka dolls or just different cups or food containers.
3. Build a tower using blocks of different sizes. The biggest should be at the bottom and the smallest — at the top.
4. Learn to match shapes and geometric solids with their projections. You can use special Denesh blocks. They come with pictures, on which you should place blocks of the right size and color to make
a picture.
I hope you like these math activities for 2-year-olds, you may also like some math games.
Math games for 2-year-olds.
The game helps develop counting skills and understand the meaning of counting process.
Necessary equipment: thick colored cardboard, scissors, thread, buttons.
Cut out a cardboard tree and some apples. Sew small buttons to the branches, and attach loops to your apples. Ask the kid to fasten 2 apples to the branches, then let him «harvest» the apples. Then
ask the kid to fasten 3 apples, etc.
The game promotes the ability to organize objects according to certain parameters.
Necessary equipment: 4 paper circles with the diameter of 3 cm and 4 circles with the diameter of 6 cm. A big and a small box.
Think of the plot for your game. For example, your grandma baked some pancakes — big and small. Bigger are for mom and dad and smaller — for grandchildren. But the pancakes have messed up. You need
to help Granny to arrange the pancakes on the plates.
Games and activities for the development of logic at 2 -3 years.
What about logic? Is it important for a 2-year-old kid? Yes, it is! And at this age logical and mathematical activities are basically the same, as the kid just isn’t able to solve purely mathematical
problems. So, here are some more activities to develop logical thinking in toddlers.
1. Closer to three years — constructing simple designs according to the drawing (like in the picture, but you better start with two parts and… mind the color!).
2. Starting from 2.5 years you can play Nikitin square. At first, you’ll help your child to put the pieces of different colors into squares, but very quickly your kid will learn to do this without
your assistance.
3. Teach to classify objects according to a common theme. For example: give your child cards with images of toys, food, and animals. The kid should divide them into relevant groups. Let him put the
toys into the box, the food into the fridge, and the animals into their makeshift house. At first, the child learns to classify the objects with your active help. First, begin with only 2 sets
of cards — toys and food, then add the third set. When your kid master these cards, it’s better not to add the 4^th set, but to begin again with two new sets.
4. Playing with Denesha blocks or other suitable objects (toys, buttons, beads, etc.):
• find items, figures of the same shape;
• find the objects, the shapes of the same color;
• find items of the same size;
5. Identifying objects by two attributes (find a big yellow circle, a small red square, etc.).
6. Closer to three years — to find the errors in the pictures, what is missing, what is wrong, what is the wrong color, etc.
You can also use these pictures with mistakes for developing critical thinking.
This is a tip! If your child refuses to perform some tasks or doesn’t want to «study» long, don’t insist. Do just 1−2 of the activities. It is known that many children up to 3 years can’t focus long
on something, especially if this is not interesting for them. You will have plenty of time another day. Perhaps, the next time our kid will do more.
Watch carefully so that your kid doesn’t get bored or tired. Try to do educational activities not longer, but more often. At this age, one «lesson» should last no more than 20 minutes for the most
assiduous kids. If your toddler is something like restless, let it be a 10-minute-lesson. Remember, daily education should bring the child only positive emotions.
Learning math for 2-year-olds is very important! Help your kid develop early math skills!
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5 years ago
That’s are nice and easy to use techniques. Just have to wait for another year, till my son will turn 2
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Logic, Inductive and Deductive
by William Minto
Publisher: ManyBooks 1893
ISBN/ASIN: 1469934264
Number of pages: 308
In this little treatise two things are attempted that at first might appear incompatible. One of them is to put the study of logical formulae on a historical basis. The other aim, which might at
first appear inconsistent with this, is to increase the power of Logic as a practical discipline.
Download or read it online for free here:
Download link
(multiple formats)
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U.S. Energy Information Administration - EIA - Independent Statistics and Analysis
Annual estimates for the missing years (1982 through 1984, 1986, 1987, 1989, 1990, 1992, 1993, 1995, 1996, and 1997) are derived separately for each SIC. The derivation has two major stages for each
estimated year. In the first stage, total consumption of offsite-produced energy is computed and aggregated across all nonelectric energy sources. This total is derived separately for each SIC, then
summed over SIC's to give the manufacturing sector total. In the second stage, the SIC and manufacturing sector totals are allocated among individual fuels.
For purposes of this analysis, a missing year is a year for which the offsite-produced energy amounts are unknown. A known year is one for which offsite-produced energy amounts are known, either from
the MECS or from the ASM. The earlier and later of the two known years are referred to as the starting year and the ending year, respectively. The starting and ending years are also referred to as
endpoint years.
From the MECS, there are currently only five known years, 1985, 1988, 1991, 1994, and 1998. Consumption data for these starting and ending years are adjusted, using production, expenditure, and price
data, to derive estimates for the intervening missing years. The same procedures are applied to fill in the missing years between the starting year of 1981 (the last ASM known year) and the ending
year of 1985 (the first MECS known year). Similar methods can easily be used in the future to fill in between subsequent MECS.
The first stage, deriving total offsite-produced energy use, relies on a combination of linear interpolation and indexing. The second stage, allocating the total among individual fuels, uses fuel
shares linearly interpolated between the known endpoint years.
Basic Interpolation Methods
Linear interpolation means drawing a line from the starting year to the ending year and using that line to estimate missing years in between. A value obtained by linear interpolation is a weighted
average of the values at the starting and ending years, with higher weight given to the closer year.
An adjustment index is a ratio used to adjust quantities known only for a base year, to estimate those quantities for missing years. In its simplest form, the adjustment index is the ratio of a
particular statistic or function for two different years, the missing year and the base year. The function is referred to here as the adjustment index basis. The Consumer Price Index (CPI) is a
widely known example, commonly used to adjust costs and prices to their base-year equivalents. The set of goods and services and the methodology for combining them to compute the CPI form the basis
of this index.
In this report, forward indexing means using an adjustment index with the starting year as the base. Backward indexing means using an index with the ending year as the base. Two-way indexing means
using linear interpolation between the forward-index and backward-index estimates. That is, the two-way indexed estimate is the weighted average of the estimates obtained by forward and backward
indexing, with higher weight given to the closer endpoint year.
Derivation Stage I: Interpolating Total Offsite-Produced Consumption
The ASM provides an estimate of offsite-produced electricity consumption by two-digit SIC for every year of interest, 1974 through the current year. Thus, for the missing years, it is only
nonelectric offsite-produced energy sources that are missing. Total offsite-produced nonelectric energy for missing years are estimated within each SIC by two-way indexing, using the FRB production
index for that SIC as the adjustment index basis. The known electric consumption is added to the derived nonelectric total to derive total offsite-produced energy consumption for the SIC. The totals
are summed over all SIC's to give the manufacturing sector total.
Formal Specification:
For each missing year y, total nonelectric consumption C[Nys] for SIC s is derived from the known starting-year (y = 0) consumption C[N0s] and known ending-year (y = L) consumption C[NLs] as
C[Nys] = (L - y)/L (Ays/A0s) CN0s + y/L (Ays/ALs)CNLs,
where Ays is the FRB production index for year y, and AOs and ALs are the production indices for the starting and ending years respectively. The manufacturing sector total CNyM is then obtained by
summing over SIC:
CNyM = S CNys.
Finally, total consumption including electricity is computed for each SIC or the total manufacturing sector as:
CTys = CNys + CEys,
CTyM = S CTys,
where CEys is the electric consumption known from the ASM.
Derivation Stage II: Allocating The Nonelectric Total
The second stage of the derivation consists of allocating the total nonelectric consumption estimates derived in Stage I among individual fuels. The allocation procedure starts by estimating the fuel
shares, defined as consumption of each nonelectric fuel expressed as a fraction of the total nonelectric offsite-produced energy. The set of shares is also referred to as the (nonelectric) fuel mix.
Fuel-specific consumption is then determined by multiplying these shares by the estimated total nonelectric consumption from Stage I.
The nonelectric fuel shares for the missing years are obtained simply by linear interpolation between the known years. Data from the known ASM years indicate that fuel shares for two-digit SIC's do
not change much from year to year. Thus, the linear interpolation estimate should be roughly correct in most cases.
The derived fuel shares include some fuels for which data are suppressed in the known years. Therefore, cells were filled with values of NA as appropriate. Because of these suppressions, the interim
year SIC-level estimates could not be summed to arrive at the fuel-specific totals for the manufacturing sector as a whole. Accordingly, it was necessary to derive manufacturing sector totals using
the same methods that were used to derive the fuel-specific totals for each individual SIC.
Formal Specification:
The (nonelectric) fuel shares for SIC s in year y are represented by the vector bys. For each missing year y, the fuel-share vector is derived by linear interpolation from the fuel-share vectors of
the known endpoint-years (y = 0 and y = L):
bys = L-y/L (b0s) + y/L (bLs).
The fuel shares are restricted as follows for individual fuels f:
0 <= bfys <= 1,
Sbfys = UTbys = 1.
The vector Cys of fuel-specific consumption amounts Cfys is then obtained by multiplying the fuel-share vector by the total nonelectric consumption estimated in Stage I:
Cys = CNys bys.
To obtain the overall manufacturing sector consumption vector CyM, the same formulas are applied to the manufacturing sector fuel-shares vectors byM. For individual fuels f, total manufacturing
consumption CfyM cannot be obtained by summing over SIC's s, because the consumption amounts Cfys are missing for some fuels in some SIC's.
Some of the change in a SIC's fuel mix between two known years (3 to 4 years apart) is the result of long-term shifts in production practices. Linear interpolation should give a reasonable estimate
of how far these long-term trends have progressed in each missing year. The limitation of the interpolation is that it will not capture any short-term fuel shifts that might occur in response to
short-term price fluctuations, or other outside factors.
Missing Consumption Estimates for Specific Cases
Even for the known years, for which consumption estimates are available from the ASM, CM, or MECS, there are some missing items. Estimates were not published for some energy sources in some SIC's,
either to avoid disclosing data for individual establishments, or because the RSE was greater than 50 percent. These gaps in the published data occurred only for individual energy sources, not for
the totals across all energy sources. For this reason the manufacturing sector consumption totals could be derived by summing over SIC's in Stage I (total consumption), but not in Stage II
(allocating the nonelectric total).
In any case where the consumption estimate was withheld for a particular known year, SIC, and fuel, the allocation procedure that depended on that information also had to leave a gap for that fuel.
Thus, a gap in the 1981 published estimates resulted in corresponding missing items for the derived estimates for 1982 through 1984. A gap in 1985 left gaps from 1982 through 1987. A gap in 1988 left
gaps from 1986 through 1990. A gap in the 1991 MECS left gaps for 1989 through 1993. A gap in 1994 left gaps for 1992 through 1997. A gap in the 1998 MECS left gaps for 1995 through 1997.
1998 MECS Survey Forms
Form A Form B Form C
Part 1 (7 pages, 51kb) Part 1 (7 pages, 52kb) Part 1 (7 pages, 46kb)
Part 2 (9 pages, 43kb) Part 2 (9 pages, 43kb) Part 2 (10 pages, 33kb)
Part 3 (8 pages, 42kb) Part 3 (9 pages, 41kb) Part 3 (9 pages, 45kb)
Part 4 (10 pages, 49kb) Part 4 (10 pages, 46kb) Part 4 (8 pages, 42kb)
Part 5 (8 pages, 42kb) Part 5 (11 pages, 52kb) Part 5 (8 pages, 43kb)
Part 6 (8 pages, 35kb) ----- Part 6 (8 pages, 43kb)
----- ----- Part 7 (7 pages, 40kb)
Specific questions on this product may be directed to:
Tom Lorenz
Phone: 202-586-3442
Fax: 202-586-0018 | {"url":"https://www.eia.gov/consumption/manufacturing/data/1998/index.php?view=methodology","timestamp":"2024-11-14T18:41:14Z","content_type":"application/xhtml+xml","content_length":"79153","record_id":"<urn:uuid:11d6b763-621d-4759-8767-a2cd1c84f0b9>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00615.warc.gz"} |
SemProba: Le résultat de votre recherche
: 12, 133-150, LNM 124 (1970)
MEYER, Paul-André Ensembles régénératifs, d'après Hoffmann-Jørgensen
Markov processes
The theory of recurrent events in discrete time was a highlight of the old probability theory. It was extended to continuous time by Kingman (see for instance
Z. für W-theorie, 2
, 1964), under the very restrictive assumption that the ``event'' has a non-zero probability to occur at fixed times. The general theory is due to Krylov and Yushkevich (
Trans. Moscow Math. Soc.
, 1965), a deep paper difficult to read and to apply in concrete cases. Hoffmann-Jørgensen (
Math. Scand.
, 1969) developed the theory under simple and efficient axioms. It is shown that a regenerative set defined axiomatically is the same thing as the set of returns of a strong Markov process to a fixed
state, or the range of a subordinator
This result was expanded to involve a Markovian regeneration property instead of independence. See Maisonneuve-Meyer
. The subject is related to excursion theory, Lévy systems, semi-Markovian processes (Lévy), F-processes (Neveu), Markov renewal processes (Pyke), and the literature is very extensive. See for
instance Dynkin (
Th. Prob. Appl.
, 1971) and Maisonneuve,
Systèmes Régénératifs, Astérisque 15
, 1974
Keywords: Renewal theory
Regenerative sets
Recurrent eventsNature: Exposition Retrieve article from NumdamIV
: 13, 151-161, LNM 124 (1970)
MAISONNEUVE, Bernard
MORANDO, Philippe Temps locaux pour les ensembles régénératifs
Markov processes
This paper uses the results of the preceding one
to define and study the local time of a perfect regenerative set with empty interior (e.g. the set of zeros of Brownian motion), a continuous adapted increasing process whose set of points of
increase is exactly the given set
Same references as the preceding paper
412Keywords: Renewal theory
Regenerative sets
Local timesNature: Original Retrieve article from NumdamVIII
: 13, 172-261, LNM 381 (1974)
MAISONNEUVE, Bernard
MEYER, Paul-André Ensembles aléatoires markoviens homogènes (5 talks)
Markov processes
This long exposition is a development of original work by the first author. Its purpose is the study of processes which possess a strong Markov property, not at all stopping times, but only at those
which belong to a given homogeneous random set $M$---a point of view introduced earlier in renewal theory (Kingman, Krylov-Yushkevich, Hoffmann-Jörgensen, see
). The first part is devoted to technical results: the description of (closed) optional random sets in the general theory of processes, and of the operations of balayage of random measures;
homogeneous processes, random sets and additive functionals; right Markov processes and the perfection of additive functionals. This last section is very technical (a general problem with this
paper).\par Chapter II starts with the classification of the starting points of excursions (``left endpoints'' below) from a random set, and the fact that the projection (optional and previsible) of
a raw AF still is an AF. The main theorem then computes the $p$-balayage on $M$ of an additive functional of the form $A_t=\int_0^th\circ X_s ds$. All these balayages have densities with respect to a
suitable local time of $M$, which can be regularized to yield a resolvent and then a semigroup. Then the result is translated into the language of homogeneous random measures carried by the set of
left endpoints and describing the following excursion. This section is an enlarged exposition of results due to Getoor-Sharpe (
Ann. Prob. 1
, 1973;
Indiana Math. J. 23
, 1973). The basic and earlier paper of Dynkin on the same subject (
Teor. Ver. Prim. 16
, 1971) was not known to the authors.\par Chapter III is devoted to the original work of Maisonneuve on incursions. Roughly, the incursion at time $t$ is trivial if $t\in M$, and if $t\notin M$ it
consists of the post-$t$ part of the excursion straddling $t$. Thus the incursion process is a path valued, non adapted process. It is only adapted to the filtration ${\cal F}_{D_t}$ where $D_t$ is
the first hitting time of $M$ after $t$. Contrary to the Ito theory of excursions, no change of time using a local time is performed. The main result is the fact that, if a suitable regeneration
property is assumed only on the set $M$ then, in a suitable topology on the space of paths, this process is a right-continuous strong Markov process. Considerable effort is devoted to proving that it
is even a right process (the technique is heavy and many errors have crept in, some of them corrected in
).\par Chapter IV makes the connection between II and III: the main results of Chapter II are proved anew (without balayage or Laplace transforms): they amount to computing the Lévy system of the
incursion process. Finally, Chapter V consists of applications, among which a short discussion of the boundary theory for Markov chains
This paper is a piece of a large literature. Some earlier papers have been mentioned above. Maisonneuve published as
Systèmes Régénératifs, Astérisque, 15
, 1974, a much simpler version of his own results, and discovered important improvements later on (some of which are included in Dellacherie-Maisonneuve-Meyer,
Probabilités et Potentiel,
Chapter XX, 1992). Along the slightly different line of Dynkin, see El~Karoui-Reinhard,
Compactification et balayage de processus droits, Astérisque 21,
1975. A recent book on excursion theory is Blumenthal,
Excursions of Markov Processes,
Birkhäuser 1992
Keywords: Regenerative systems
Regenerative sets
Renewal theory
Local times
Markov chains
IncursionsNature: Original Retrieve article from NumdamX
: 03, 24-39, LNM 511 (1976)
JACOD, Jean
MÉMIN, Jean Un théorème de représentation des martingales pour les ensembles régénératifs
Martingale theory
Markov processes
Stochastic calculus
The natural filtration of a regenerative set $M$ is that of the corresponding ``age process''. There is a natural optional random measure $\mu$ carried by the right endppoints of intervals contiguous
to $M$, each endpoint carrying a mass equal to the length of its interval. Let $\nu$ be the previsible compensator of $\mu$. It is shown that, if $M$ has an empty interior the martingale measure $\
mu-\nu$ has the previsible representation property in the natural filtration
Martingales in the filtration of a random set (not necessarily regenerative) have been studied by Azéma in
. In the case of the set of zeros of Brownian motion, the martingale considered here is the second ``Azéma's martingale'' (not the well known one which has the chaotic representation property)
Keywords: Regenerative sets
Renewal theory
Stochastic integrals
Previsible representationNature: Original Retrieve article from NumdamXI
: 37, 529-538, LNM 581 (1977)
MAISONNEUVE, Bernard Changement de temps d'un processus markovien additif
Markov processes
A Markov additive process $(X_t,S_t)$ (Cinlar,
Z. für W-theorie, 24
, 1972) is a generalisation of a pair $(X,S)$ where $X$ is a Markov process with arbitrary state space, and $S$ is an additive functional of $X$: in the general situation $S$ is positive real valued,
$X$ is a Markov process in itself, and the pair $(X,S)$ is a Markov processes, while $S$ is an additive functional
of the pair.
For instance, subordinators are Markov additive processes with trivial $X$. A simpler proof of a basic formula of Cinlar is given, and it is shown also that a Markov additive process gives rise to a
regenerative system in a slightly extended sense
See also
1513Keywords: Markov additive processes
Additive functionals
Regenerative sets
Lévy systemsNature: Original Retrieve article from NumdamXIV
: 45, 437-474, LNM 784 (1980)
TAKSAR, Michael I. Regenerative sets on real line
Markov processes
Renewal theory
From the introduction: A number of papers are devoted to studying regenerative sets on a positive half-line... our objective is to construct translation invariant sets of this type on the entire real
line. Besides we start from a weaker definition of regenerativity
This important paper, if written in recent years, would have merged into the theory of Kuznetsov measures
Keywords: Regenerative setsNature: Original Retrieve article from Numdam | {"url":"http://sites.mathdoc.fr/cgi-bin/sps?kwd=Regenerative+sets&kwd_op=contains","timestamp":"2024-11-12T00:16:36Z","content_type":"text/html","content_length":"20257","record_id":"<urn:uuid:0e7cdcda-ba3e-48cf-bb14-9c32cee26604>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00879.warc.gz"} |
Observational Overfitting in Reinforcement Learning
23 Jan 2020
• The paper studies observational overfitting: The phenomenon where an agent overfits to different observation spaces even though the underlying MDP remains fixed.
• Unlike other works, the “background information” (in the pixel space) is correlated with the progress of the agent (and is not just noise).
• Base MDP $M = (S, A, R, T)$ where $S$ is the state space, $A$ is the action space, $R$ is the reward function, and $T$ is the transition dynamics.
• $M$ is parameterized using $\theta$. In practice, it means introducing an observation function $\phi_{\theta}$ ie $M_{\theta} = (M, \phi_{\theta})$.
• A distribution over $\theta$ defines a distribution over the MDPs.
• The learning agent has access to the pixel space observations and not the state space observations.
• Generalization gap is defined as $J_{\theta}(\pi) - J_{\theta^{train}}(\pi)$ where $\pi$ is the learning agent, $\theta$ is the distribution over all the observation functions, $\theta^{train}$
is the distribution over the observation functions corresponding to the training environments. $J_{\theta}(\pi)$ is the average reward that the agent obtains over environments sampled from $M_{\
• $\phi_{\theta}$ considers two featurs - generalizable (invariant across $\theta$) and non-generalizable (depends on $\theta$) ie $\phi_{\theta}(s) = concat(f(s), g_{\theta}(s))$ where $f$ is the
invariant function and $g$ is the non-generalizable function.
• The problem is set up such that “explicit regularization” can easily solve it. The focus is on understanding the effect of “implicit regularization”.
Overparameterized LQR
• LQR is used as a proxy for deep RL architectures given its advantages like enabling exact gradient descent.
• The functions are parameterized as follows:
□ $f(s) = W_c(s)$
□ $g_{\theta}(s) = W_{\theta}(s)$
• Observation at time $t$ , $o_t$, is given as $[W_c W_{\theta}]^{-1} s_t$.
• Action at time $t$ is given as $a_t = K o_{t}$ where $K$ is the policy matrix.
• Dimensionality:
□ state $s$: $d_{state}$ 100
□ $f(s)$: $d_{state}$ 100
□ $g_{\theta}(s)$: $d_{noise}$ 100
□ observation $o$: $d_{state}$ + $d_{noise}$ 1100
• In case of training on just one environment, multiple solutions exist, and overfitting happens.
• Increasing $d_{noise}$ increases the generalization gap.
• Overparameterizing the network decreases the generalization gap and also reduces the norm of the policy.
Projected Gym Environments
• The base MDP is the Gym Environment.
• $M_{\theta}$ is generated as before.
• Increasing both width and depth for basic MLPs improves generalization.
• Generalization also depends on the choice of activation function, residual layers, etc.
Deconvolutional Projections
• In the Gym environment, the actual state is projected to a larger vector and reshaped into an 84x84 tensor (image).
• The image from $f$ is concatenated with the image from $g$. This setup is referred to as the Gym-Deconv.
• The relative order of performance between NatureCNN, IMPALA, and IMPALA-Large (on both CoinRun and Gym-Deconv) is the same as the order of the number of parameters they contain.
• In an ablation, the policy is given access to only $g_{\theta}(s)$, which makes it impossible for the model to generalize. In this test of memorization capacity, implicit regularization seems to
reduce the memorization effect.
Overparameterization in CoinRun
• The pixel space observation in CoinRun is downsized from 64x64 to 32x32 and flattened into a vector.
• In CoinRun, the dynamics change per level, and the noisy “irrelevant” features change location across the 1D input, making this setup more challenging than the previous ones.
• Overparameterization improves generalization in this scenario as well. | {"url":"https://shagunsodhani.com/papers-I-read/Observational-Overfitting-in-Reinforcement-Learning","timestamp":"2024-11-02T22:02:54Z","content_type":"text/html","content_length":"14826","record_id":"<urn:uuid:60615951-612e-4265-8cd9-657802125919>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00751.warc.gz"} |
Taylor's Series
Below we present a derivation of Taylor's series and small algebraic argument for series representations of functions. In contrast to the ability to use sympy functions without any deeper
understanding, these presentations are intended to give you insight into the origin of the series representation and the factors present within each term. While the algebraic presentation isn't a
general case, the essential elements of a general polynomial representation are visible.
The function $f(x)$ can be expanded into an infinite series or a finite series plus an error term. Assume that the function has a continuous nth derivative over the interval $a \le x \le b$.
Integrate the nth derivative n times:
$$\int_a^x f^n(x) dx = f^{(n-1)}(x) - f^{(n-1)}(a)$$
The power on the function $f$ in the equation above indicates the order of the derivative. Do this n times and then solve for f(x) to recover Taylor's series. One of the key features in this
derivation is that the integral is definite. This derivation is outlined on Wolfram’s Mathworld.
As a second exercise, assume that we wish to expand sin(x) about x=0. First, assume that the series exists and can be written as a power series with unknown coefficients. As a first step,
differentiate the series and the function we are expanding. Next, let the value of x go to the value of the expansion point and it will be possible to evaluate the coefficients in turn: | {"url":"https://notebook.community/mathinmse/mathinmse.github.io/Lecture-10A-Taylors-Series","timestamp":"2024-11-10T20:58:25Z","content_type":"text/html","content_length":"134303","record_id":"<urn:uuid:488c7daf-4320-4c50-966c-ce96662dfd2b>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00250.warc.gz"} |
CMSC 202 Lecture Notes: Recursion
Recursion is a technique that allows us to break down a problem into one or more subproblems that are similar in form to the original problem. For example, suppose we need to add up all of the
numbers in an array. We'll write a function called add_array that takes as arguments an array of numbers and a count of how many of the numbers in the array we would like to add; it will return the
sum of that many numbers.
If we had a function that would add up all but the very last number in the array, then we would simply have to add the last number to that sum and we would be done. Add_array is an ideal function for
adding up all but the last number (as long as the array contains at least one number). After all, add_array is responsible for taking an array and a count, and adding up that many array elements. If
there are no numbers in the array, then zero is the desired answer. These observations suggest the following function:
int add_array(int arr[], int count) {
if (count == 0) return 0;
return arr[count - 1] + add_array(arr, count - 1);
This function is perfectly legal C, and it operates correctly. Notice that the function has two components:
1. a base case, represented by the if and the return 0, in which the function does not call itself. This handles the case where there are no numbers to add.
2. a recursive case that breaks the problem down into a smaller version of the original problem together with an addition. In the recursive case, add_array is used to add together count-1 items; the
count-th item is then added to this result (remember that the n-th item of an array is stored at position n-1).
The call to add_array from inside add_array is called a recursive call.
One of the classic examples of recursion is the factorial function. Although factorial is not the world's most interesting function, it will provide us with many useful observations.
Recall that factorial, which is written n!, has the following definition:
n! = 1 * 2 * 3 * .... * (n-2) * (n-1) * n
We can use this definition to write a C function that implements factorial:
int fact(int n) {
int i;
int result;
result = 1;
for (i = 1; i <= n; i++) {
result = result * i;
return result;
This is a simple iterative function that mirrors the definition of factorial. We can derive a different definition for factorial by noticing that n! = n * (n-1)! and 1! = 1. For example, 4! = 4 * 3!.
Notice that we need to specify a value for 1! because our definition does not apply when n=1. This kind of definition is known as an inductive definition, because it defines a function in terms of
We can write a C function that mirrors this new definition of factorial as follows:
int fact(int n) {
if (n == 1) return 1;
return n * fact(n - 1);
Notice that this function precisely follows our new definition of factorial. It is recursive, because it contains a call to itself.
Let's compare the two versions:
• The iterative version has two local variables; the recursive version has none.
• The iterative version has three statements; the recursive version has one.
• The iterative version must save the solution in an intermediate variable before it can be returned; the recursive version calculates and returns its result as a single expression.
Recursion simplifies the fact function! It does so by making the computer do more work, so that you can do less work.
To successfully apply recursion to a problem, you must be able to break the problem down into subparts, at least one of which is similar in form to the original problem. For example, suppose we want
to count the number of occurrences of the number 42 in an array of n integers. The first thing we should do is write the header for our function; this will ensure that we know what the function is
supposed to do and how it is called:
int count_42s(int array[], int n);
To use recursion on this problem, we must find a way to break the problem down into subproblems, at least one of which is similar in form to the original problem. If we know that the array contains n
numbers, we might break our task into the subproblems of:
1. counting the number of times that 42 appears in the first n-1 elements of the array (this is a subproblem that is similar in form to the original problem);
2. counting the number of times that 42 appears in the n-th element of the array (i.e. determining whether the n-th element is 42); and
3. adding these two sums together and returning the result.
Part 1 of this decomposition suggests the following recursive call:
count_42s(array, n-1);
If successful, this recursive call will count all of the occurrences of the number 42 in the first n-1 positions of the array and return the sum (we will discuss the conditions that must hold for a
recursive call to be successful in the Section 5; until then, we will assume that all recursive calls work properly). We must now determine how to use this result. If the last element in the array is
not 42, then the number of 42s in the entire array is the same as the number of 42s in all but the last element of the array. If the last number in the array is 42, then the number of 42s in the
entire array is one more than the number found in the subarray. This suggests the following code:
if (array[n-1] != 42) {
return count_42s(array, n-1);
return 1 + count_42s(array, n-1);
Here we have two recursive calls (only one of which will actually be used in any given situation). We must now determine whether there are any circumstances under which this code will not work. In
fact, this code will not work when n==0; in such a case it tries to subscript the array with -1, which is not a legal array subscript in C. (Oh alright, it's legal; it's just that it's almost never
what you want, and will often lead to a segmentation fault or worse.) That means that unless we treat specially the case where n is zero, our function will not work when asked to count the number of
42s in an array of zero items. We will therefore add a base case that will test for n==0 and return zero as its result in that case. This gives the function:
int count_42s(int array[], int n) {
if (n==0) return 0;
if (array[n-1] != 42) {
return count_42s(array, n-1);
return 1 + count_42s(array, n-1);
This is a perfectly good recursive solution to the count_42s problem. It is not the only recursive solution though; there are other ways to break the problem into subpieces. For example, we could
break the array into two pieces of equal size, count the number of 42s in each half, then add the two sums. To do this, we will need to hand as arguments to count_42s not just the array and the
subscript of the highest value in the array, but also the subscript of the lowest value in the array:
int count_42s(int array[], int low, int high);
A call such as count_42s(my_array, A, B) says "count all the occurrences of the number 42 in my_array that lie between position A and position B inclusive."
We can calculate the midpoint between subscript low and subscript high with (high+low)/2. Thus we can count the number of 42s in each half of the array and add them together with:
count_42s(array, low, (low + high) / 2) +
count_42s(array, (low + high) / 2 + 1, high);
We now have a recursive case but no base case. When will the recursive case fail? It fails when the array does not contain at least two numbers. If the array contains no items, or it contains one
item that is not 42, then we should return zero. If the array contains exactly one number, and that number is 42, then we should return one. Putting it all together, we get:
int count_42s(int array[], int low, int high) {
if ((low > high) ||
(low == high && array[low] != 42)) {
return 0 ;
if (low == high && array[low] == 42) {
return 1;
return count_42s(array, low, (low + high)/2) +
count_42s(array, 1 + (low + high)/2, high));
Note that the line
if (low == high && array[low] == 42)
could properly be written simply as if (low==high). The comparison with 42 is included here simply to make each of the relevant conditions explicit in the same expression.
These examples demonstrate that there may be many ways to break a problem down into subproblems such that recursion is useful. It is up to you the programmer to determine which decomposition is best.
The general approach to writing a recursive program is to:
1. write the function header so that you are sure what the function will do and how it will be called;
2. decompose the problem into subproblems;
3. write recursive calls to solve those subproblems whose form is similar to that of the original problem;
4. write code to combine, augment, or modify the results of the recursive call(s) if necessary to construct the desired return value or create the desired side--effects; and
5. write base case(s) to handle any situations that are not handled properly by the recursive portion of the program.
How should you think about a recursive subprogram? Do not immediately try to trace through the execution of the recursive calls; doing so is likely to simply confuse you. Rather, think of recursion
as working via the power of wishful thinking. Consider the operation of fact(4) using the recursive formulation of fact. 4 is not 1, so the recursive case holds. The recursive case says to multiply
fact(3) by 4. Here is where the wishful thinking comes in: wish for fact(3) to be calculated. Because this is a recursive call, your wish will be granted. You now know that fact(3)=6. So fact(4) is
equal to 6 times 4, or 24 (which is just what it's supposed to be).
An analogy you can use to help you think this way is corporate management. When the CEO of a corporation tells a vice-president to perform some task, the CEO doesn't worry about how the task is
accomplished; he or she relies on the vice-president to get it done. You should think the same way when you are programming recursively. Delegate the subtask to the recursive call; don't worry about
how the task actually gets done. Worry instead whether the top-level task will get done properly, given that all the recursive calls work properly.
Another way to think about recursion is to pretend that a recursive call is actually a call to a different function, written by somebody else, that performs the same task that your function performs.
For example, suppose we had a library routine called libfact that returned the factorial of its argument. We could then write our own version of fact as:
int fact (int n) {
if (n == 1) return 1;
return n * libfact(n - 1);
This version of fact correctly returns one if its argument is one. If its argument is greater than one, it calls libfact to calculate (n-1)!, and multiplies the result by n. Because libfact is a
library routine, we may assume that it works properly, in this case calculating the factorial of n-1. For example, if n is 4, then libfact is called with 3 as its argument; it returns 6. This is
multiplied by 4 to get the desired result of 24.
This example points out that a recursive call is just like any other function call. In particular, a recursive call gets its own parameter list and local variables, just as libfact would.
Furthermore, while the recursive call is executing, the top--level call sits there waiting for the recursive call to terminate. This means that execution doesn't halt when a recursive call finds
itself at the base case; once the recursive call returns, the top--level call then continues to execute.
One of the most difficult aspects of programming recursively is the mental process of accepting on faith that the recursive call will do the right thing. The following checklist itemizes the five
conditions that must hold for recursion to work. If each of these conditions holds for your recursive subprogram, you may feel confident that the recursion will operate correctly:
1. A recursive subprogram must have at least one base case and one recursive case (it's OK to have more than one base case, and more than one recursive case).
2. The test for the base case must execute before the recursive call.
3. The problem must be broken down in such a way that the recursive call is closer to the base case than the top--level call. ( This condition is actually not quite strong enough. Moving toward the
base case alone is not sufficient; it must also be true that the base case is reached in a finite number of recursive calls. In practice though, it is rare to encounter situations where there is
always movement toward the base case but the base case is not reached).
4. The recursive call must not skip over the base case.
5. The non-recursive portions of the subprogram must operate correctly.
Let's see whether the recursive fact function meets these criteria:
1. The first condition is met, because if (n==1) return 1 is a base case, while the "else" part includes a recursive call (fact(n-1)).
2. If we reach the recursive call, we must have already evaluated if (n==1); this if is the base case test, so criterion 2 is met.
3. The recursive call is fact(n-1). The argument to the recursive call is one less than the argument to the top--level call to fact. Our base case occurs when n is one. The recursive call is
therefore closer to the base case as long as n is positive. If n is not positive, the recursive call does not move toward the base case, so the function will not work properly (which is not
surprising, given that factorial is defined only on positive integers).
4. Because n is an integer, and the recursive call reduces n by just one, it is not possible to skip over the base case.
5. Assuming that the recursive call works properly, we must now verify that the rest of the code works properly. We can do this by comparing the code with our second definition of factorial. This
definition says that if n is one then n! is one. Our function correctly returns 1 when n is 1. If n is not one, the definition says that we should return (n-1)! * n. The recursive call (which we
now assume to work properly) returns the value of n-1 factorial, which is then multiplied by n. Thus, the non--recursive portions of the function behave as required.
This section describes some of the ways in which recursive functions are characterized. The characterizations are based on:
1. whether the function calls itself or not (direct or indirect recursion).
2. whether there are pending operations at each recursive call (tail-recursive or not).
3. the shape of the calling pattern -- whether pending operations are also recursive (linear or tree-recursive).
Direct Recursion:
A C function is directly recursive if it contains an explicit call to itself. For example, the function
int foo(int x) {
if (x <= 0) return x;
return foo(x - 1);
includes a call to itself, so it's directly recursive. The recursive call will occur for positive values of x.
Indirect Recursion:
A C function foo is indirectly recursive if it contains a call to another function which ultimately calls foo.
The following pair of functions is indirectly recursive. Since they call each other, they are also known as mutually recursive functions.
int foo(int x) {
if (x <= 0) return x;
return bar(x);
int bar(int y) {
return foo(y - 1);
Tail Recursion:
A recursive function is said to be tail recursive if there are no pending operations to be performed on return from a recursive call.
Tail recursive functions are often said to "return the value of the last recursive call as the value of the function." Tail recursion is very desirable because the amount of information which must be
stored during the computation is independent of the number of recursive calls. Some modern computing systems will actually compute tail-recursive functions using an iterative process.
The "infamous" factorial function fact is usually written in a non-tail-recursive manner:
int fact (int n) { /* n >= 0 */
if (n == 0) return 1;
return n * fact(n - 1);
Notice that there is a "pending operation," namely multiplication, to be performed on return from each recursive call. Whenever there is a pending operation, the function is non-tail-recursive.
Information about each pending operation must be stored, so the amount of information is not independent of the number of calls.
The factorial function can be written in a tail-recursive way:
int fact_aux(int n, int result) {
if (n == 1) return result;
return fact_aux(n - 1, n * result)
int fact(n) {
return fact_aux(n, 1);
The "auxiliary" function fact_aux is used to keep the syntax of fact(n) the same as before. The recursive function is really fact_aux, not fact. Note that fact_aux has no pending operations on return
from recursive calls. The value computed by the recursive call is simply returned with no modification. The amount of information which must be stored is constant (the value of n and the value of
result), independent of the number of recursive calls.
Linear and Tree Recursion:
Another way to characterize recursive functions is by the way in which the recursion grows. The two basic ways are "linear" and "tree."
A recursive function is said to be linearly recursive when no pending operation involves another recursive call to the function.
For example, the "infamous" fact function is linearly recursive. The pending operation is simply multiplication by a scalar, it does not involve another call to fact.
A recursive function is said to be tree recursive (or non-linearly recursive) when the pending operation does involve another recursive call to the function.
The Fibonacci function fib provides a classic example of tree recursion. The Fibonacci numbers can be defined by the rule:
fib(n) = 0 if n is 0,
= 1 if n is 1,
= fib(n-1) + fib(n-2) otherwise
For example, the first seven Fibonacci numbers are
Fib(0) = 0
Fib(1) = 1
Fib(2) = Fib(1) + Fib(0) = 1
Fib(3) = Fib(2) + Fib(1) = 2
Fib(4) = Fib(3) + Fib(2) = 3
Fib(5) = Fib(4) + Fib(3) = 5
Fib(6) = Fib(5) + Fib(4) = 8
This leads to the following implementation in C:
int fib(int n) { /* n >= 0 */
if (n == 0) return 0;
if (n == 1) return 1;
return fib(n - 1) + fib(n - 2);
Notice that the pending operation for the recursive call is another call to fib. Therefore fib is tree-recursive.
A non-tail recursive function can often be converted to a tail-recursive function by means of an "auxiliary" parameter. This parameter is used to form the result. The idea is to attempt to
incorporate the pending operation into the auxiliary parameter in such a way that the recursive call no longer has a pending operation. The technique is usually used in conjunction with an
"auxiliary" function. This is simply to keep the syntax clean and to hide the fact that auxiliary parameters are needed.
For example, a tail-recursive Fibonacci function can be implemented by using two auxiliary parameters for accumulating results. It should not be surprising that the tree-recursive fib function
requires two auxiliary parameters to collect results; there are two recursive calls. To compute fib(n), call fib_aux(n 1 0)
int fib_aux(int n, int next, int result) {
if (n == 0) return result;
return fib_aux(n - 1, next + result, next);
Let's assume that tail recursive functions can be expressed in the general form
F(x) {
if (P(x)) return G(x);
return F(H(x));
That is, we establish a base case based on the truth value of the function P(x) of the parameter. Given that P(x) is true, the value of F(x) is the value of some other function G(x). Otherwise, the
value of F(x) is the value of the function F on some other value, H(x). Given this formulation, we can immediately write an iterative version as
F(x) {
int temp_x = x;
while (P(x) is not true) {
temp_x = x;
x = H(temp_x);
return G(x);
The reason for using the local variable temp_x will become clear soon. Actually, we will use one temporary variable for each parameter in the recursive function.
Example - factorial function
In the tail-recursive factorial function (fact_aux) given in Section 6,
• the function F is fact_aux
• x is composed of the two parameters, n and result
• the value of P(n, result) is the value of (n == 1)
• the value of G(n, result) is result
• the value of H(n, result) is (n -1, n * result)
Therefore the iterative version is:
int fact_iter(int n, int result) {
int temp_n;
int temp_result;
while (n != 1) {
temp_n = n;
temp_result = result;
n = temp_n - 1;
result = temp_n * temp_result;
return result;
The variable temp_n is needed so result will be computed on the basis of the unchanged n. The variable temp_result is not really needed, but is used to be consistent.
Example - Fibonacci function
In the tail-recursive fibonacci function (fib_aux) given in Section 7:
• The function F is fib_aux
• x is composed of the three parameters n, next, and result
• the value of P(n, next, result) is the value of (n == 0)
• the value of G(n, next, result) is result
• the value of H(n, next, result) is (n -1, next + result, next)
Therefore the iterative version is
int fib_iter(int n, int next, int result) {
int temp_n;
int temp_next;
int temp_result;
while (n != 0) {
temp_n = n;
temp_next = next;
temp_result = result;
n = temp_n - 1;
next = temp_next + temp_result;
result = temp_next;
return result;
Just as it is possible to convert any recursive function to iteration, it is possible to convert any iterative loop into the combination of a recursive function and a call to that function. The
ability to convert recursion to iteration is often quite useful, allowing the power of recursive definition with the (often) greater efficiency of iteration.
Converting iteration to recursion is unlikely to be useful in practice, but it is a fine learning tool. This Section gives several examples of the relationship between programs that loop using C's
built-in iteration constructs, and programs that loop using tail recursion.
Suppose that we want to write a program that will read in a sequence of numbers, and print out both the maximum value and the position of the maximum value in the sequence. For example, if the input
to our program is the sequence 2, 5, 6, 4, 1, then the program should tell us that the maximum number is 6, and that it occurs in position 3 of the input. Here is a program that performs this task
using C's built-in iterative constructs:
#include <stdio.h>
#include <assert.h>
#include <stdlib.h>
#define MAX_NUMS 32
#define max(num1,num2) ((num1) > (num2) ? (num1) : (num2))
typedef int array_of_nums[MAX_NUMS];
int main(void) {
array_of_nums nums;
int num_count;
int max_num;
int pos_of_max;
int i;
/* LOOP 1: Read in the numbers */
num_count = 0;
while(scanf("%d", &nums[num_count]) != EOF) {
assert(num_count > 0);
/* LOOP 2: Find the maximum number */
max_num = nums[0];
for (i = 1; i < num_count; i++) {
max_num = max(max_num, nums[i]) ;
/* LOOP 3: Find the position of the maximum number */
pos_of_max = 0;
while (nums[pos_of_max] != max_num) {
/* Print the results */
printf("The largest number, which was %d, occurred in position number %d\n",
max_num, pos_of_max + 1);
return EXIT_SUCCESS;
You will notice that the above program uses three loops and an array where one loop and an integer would have sufficed. However, the purpose of this program is not to show the best way to find the
maximum of a sequence of numbers, but rather to exhibit a program that contains a few loops.
We will convert each of the program's three loops into a tail recursive subprogram. The key to implementing a loop using tail recursion is to determine which variables capture the state of the
computation; these variables will then serve as the parameters to the tail recursive subprogram. The first loop is a while loop that is responsible for reading numbers into an array. It terminates
when scanf returns EOF. Aside from the EOF condition, two variables capture the state of the computation each time through this loop:
1. num_count---the number of integers that have been read so far.
2. nums---an array that stores those integers.
To translate this loop into a tail recursive subprogram then, we will write a subprogram that takes these two values as arguments. Since we are interested in getting a final value for num_count, we
will write a function that returns num_count as its result. If the termination condition (EOF) is not true, the procedure will increment num_count, read a number into nums, and call itself
recursively to input the remainder of the numbers. If the termination condition is true, the procedure will simply return the final value of num_count. Here is the code:
int read_nums(int num_count, array_of_nums nums) {
if (scanf("%d", &nums[num_count]) != EOF) {
return read_nums(num_count + 1, nums);
return num_count;
Notice that the recursive call to read_nums is closer to the base case (EOF) than the top--level call by virtue of the fact that the call to scanf consumes input. The base case test happens before
the recursive call, and the base case will never be skipped. Finally, if there is no more input, we return the current value of num_count, which has accumulated exactly the return value we desire. If
there is more input, we add one to num_count and recurse. In either case, if we assume that the recursive call works properly, we get exactly the return value we want. Thus this function meets each
of our criteria for valid recursive functions.
We must invoke this procedure with the correct initial values for its arguments; here is the appropriate code from the main program:
/* Read in the numbers */
num_count = read_nums(0, nums);
assert(num_count > 0);
Notice that this call to read_nums causes the initial value of num_count in read_nums to be zero (which is exactly the initial value it had in the original loop). Assert is a statement defined in
<assert.h>; it flags an error if its argument is false, and does nothing if its argument is true. The assert statement is used to ensure that at least one number is read in. Without this statement,
subsequent code will bomb if no numbers are entered.
Let's generalize from this example. In general, an iterative loop may be converted to a tail--recursive function by:
• determining what variables are used by the loop;
• writing a tail--recursive function that has one parameter for each of the identified variables;
• using the condition that causes the termination of the iterative loop as the base case test;
• determining how each variable changes from one iteration to the next, and handing the sequence of expressions that represent those changes as arguments to the recursive call; and
• Replacing the original iterative loop with a call to the tail--recursive function, using the initial value of each of the variables as arguments.
The second loop is a for loop that is responsible for finding the value of the maximum input number. Before entering the for loop, we make the assumption that the zero-th number read in is the
largest. We then loop through all but the zero-th element of nums, looking for a higher value. Four variables capture the state of the computation during this loop:
1. max_num---the value of the largest number examined so far.
2. i---the loop control variable.
3. num_count---the number of input values.
4. nums---the input values.
Therefore, we will write a subprogram that takes these four values as arguments. Since we want to get a single value back---the value of the largest number---we will make our function return an int.
As with any subprogram, we can give the formal parameters any names we choose. We will rename two of the values to give them more descriptive names. We will use the name max_so_far instead of
max_num, and pos_to_test instead of i. Here is the code:
int find_max(int max_so_far, int pos_to_test, int num_count, array_of_nums nums) {
if (pos_to_test < num_count) {
return find_max(max(max_so_far, nums[pos_to_test]),
pos_to_test + 1, num_count, nums);
return max_so_far;
To invoke this function, we will need to pass the value of the zero-th number read in as its first argument, and 1 (which was the initial value of the loop control variable in the iterative version
of the program) as its second argument. Here is the appropriate portion of the main program:
/* find the maximum number */
max_num = find_max(nums[0], i, num_count, nums);
The third loop looks through the input numbers until it finds the first occurrence of the maximum number, then records the position of the maximum number. Three variables control the state of this
1. max_num---the value of the largest number.
2. pos_of_max---the position of the number we are currently testing. This variable actually holds the position of the maximum number only when the loop is exited. Before the loop completes, this
variable is effectively serving as a loop control variable.
3. nums---the array of input numbers.
These three values will serve as the parameters to our tail recursive subprogram. As with the second loop, we will use a function that returns an int. If the function finds the maximum value at the
location specified by pos_to_test (which is the function's name for the variable pos_of_max in the original iterative version), it will return that position. If the value at the location specified by
pos_to_test is not equal to the maximum value, the function will use tail recursion to search the remaining positions:
int find_pos_of_max(int max_num, int pos_to_test, array_of_nums nums) {
if (nums[pos_to_test] != max_num) {
return find_pos_of_max(max_num, pos_to_test + 1, nums);
return pos_to_test;
Note that although the recursive call is not textually the last code of the function, it is the last instruction executed before the return if we have not yet found the position of the maximum
number. Therefore, it is a tail recursive call. The invocation of this function in the main program looks as follows:
/* Find the position of the maximum number */
pos_of_max = find_pos_of_max(max_num, 0, nums);
1. Write a recursive function that will count the number of occurrences of a given integer in an array. Your function count_occurrences should take three arguments: an array of ints, the number of
elements in the array, and the integer occurrences of which in the array are to be counted.
2. Write a recursive function called count_7s, which counts the number of occurrences of the digit 7 in the decimal representation of a given integer. For example, if the argument to count_7s is
3762797, count_7s should return 3 (because there are three occurrences of the digit 7 in 3762797). Hint: remember that n % 10 will give the remainder of n divided by ten, while n/10 will give the
integer part of n divided by ten.
3. Write a recursive function called replace_char that replaces every occurrence of a specified character in a string with another character. Your function, which should be a void function, should
take three arguments: a string; a character to be replaced, and a character with which to replace it. For example, if foo contains the string "xizzi", then the call replace_char(foo, 'i', 'y')
should modify foo so that it now contains the string "xyzzy". Note: if you are not familiar with string manipulation in C, skip this problem.
Thomas A. Anastasio, Thu Sep 4 21:40:50 EDT 1997
Modified by Richard Chang Fri Jan 16 22:51:58 EST 1998. | {"url":"https://userpages.cs.umbc.edu/chang/cs202.f98/readings/recursion.html","timestamp":"2024-11-04T13:50:07Z","content_type":"text/html","content_length":"37720","record_id":"<urn:uuid:caf1e42d-d899-4472-8e65-124228b705ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00760.warc.gz"} |
Scanner function question about SLOPE
I would like to use the slope in a scanner rule but cannot find that function in the available functions. Stock charts describes it as following.
the slope measures the rise over run for the linear regression. This is the ending value of the linear regression less the beginning value divided by the timeframe. If the ending value were 35, the
beginning value 29 and the run 12, then the slope would be .5 (35 - 29 = 6, 6/12 = .50). The slope indicator is zero when the linear regression is flat, positive when the linear regression slants up
and negative when the linear regression slopes down. The steepness of the slope is also reflected in the value.
Do you have something like this in the scanner?
Since the advanced scanner code allows you to look at the value for the indicator N candles back, you can do it that way for any indicator. Including the Single Regression Channel indicator.
Let's say you added the Single Regression Channel to the list of variables under the name of SRC. Then the slope would be
• 4 weeks later...
Mike, can you elaborate on slope some more and give some examples such as....
How can I calculate the slope of the 20 EMA?
This is new to me and if I see some examples I can probably figure it out... Thanks
Slope, in mathematics, is dy/dx - where dx is the difference in x between two points and dy is the difference in y. So - if x stays constant, the slope is 0. If the line is vertical (dx=0) then the
slope is infinite. (On a coordinate system where X and Y are equivalent, the slope would be the tangent of the angle of the slope).
The problem, of course, is - what are the units of dx (dy, I presume, is price). Because obviously the slope itself, unless it is exactly 0, is pretty meaningless if the units of x are arbitrary and
unrelated to the units of y (as is the case in stock charts).
Thus, the only way you can use slope in stock charts is to compare two of them.
To find an approximation of a slope of 20 EMA, or any other function, at point X (let's say X is candle number) is to take its value at X+1, its value at X-1, and divide it by 2 (X+1)-(X-1). The
reason it is an approximation and not a real slope is because the chart is not a curve but a set of segments, and the slope at the points where the segments connect is, strictly mathematically,
Another way to find a slope would be to fit a Bezier curve or a spline through a set of points surrounding point X, do symbolic differentiation of the curve function, and find its value at point X.
Not really feasible. Too complicated.
A third way would be to do a linear regression slope calculation on let's say last 5 points and presume that the slope is the slope of the regression line.
or, in advanced code in paintbars or scanner (presuming that EMA is the variable representing the value of the indicator)
double sumxy = 0;
double sumxx = 0;
double averagex = 2;
double averagey = Average(5,EMA);
for (int x=0;x<5;x++)
sumxx += (x-averagex)*(x-averagex);
sumxy += (x-averagex)*(EMA[x]-averagey);
var Slope = sumxy/sumxx;
Which reminds me - I really should find a way to have the MT looping functions to incorporate the n variable. Right now there is no way to do that, so I had to do the actual loop.
• 1
speaking of slopes, I like the way you implemented the 'inverse' of a line.
seems obvious, but I haven't seen it elsewhere. Might be out there, but not in anything I use
3 hours ago, stock777 said:
speaking of slopes, I like the way you implemented the 'inverse' of a line.
seems obvious, but I haven't seen it elsewhere. Might be out there, but not in anything I use
You mean in annotations, right?
Always happy to please. | {"url":"https://forums.medvedtrader.com/topic/3092-scanner-function-question-about-slope/","timestamp":"2024-11-05T16:22:52Z","content_type":"text/html","content_length":"183886","record_id":"<urn:uuid:3b3ea60c-d792-4957-ac6d-e231fe89e139>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00099.warc.gz"} |
Estimating Angles by Hand
Estimating Angles by Hand
When in the wilderness you sometimes need to take measurements and may not have the necessary tools in your pockets or pack. Fortunately we're equipped with tools for estimation--our hands.
Taking some time to measure the length of your finger, the distance from thumb to index finger, etc. can pay off when you need to estimate the size of a peak, a tree, or a distant object.
Today we're going to look at angles and their use in estimating available sunlight.
Holding your hand parallel to the horizon you can get a fairly accurate estimate of how much daylight is left by working your way from the horizon to the sun. Each four-finger width is approximately
one hour which makes each finger approximately 15 minutes. This is an important skill to have when you are going to be prioritizing shelter building, firewood collection, fire building, etc.
2 degrees or 15 minutes
4 degrees or 30 minutes
6 degrees or 45 minutes
15 degrees
20 degrees
8 degrees or an hour
Using these estimates and a bit of math you can also calculate distance. A finger held at arm's length covers approximately 2 degrees, two fingers approximately 4 degrees, and so on. To estimate your
distance from an object of known width or distance use this formula:
width or distance in feet / 100 X angle = approximate distance in miles.
As an example:
You know a particular tree to be four feet in width and, from your current position, with a hand held at arm's length, you can cover that tree with a single finger (2 degrees.)
Using the formula, 4 (feet in width of the tree) / (100 X 2) = .02
or 2/10 of a mile to the tree.
A "handy" (yes, pun intended) method of calculating distance and hours of remaining daylight don't you think?
Thanks for reading,
0 Comments: | {"url":"https://americanbushman.blogspot.com/2008/03/estimating-angles-by-hand.html","timestamp":"2024-11-10T02:42:44Z","content_type":"application/xhtml+xml","content_length":"19962","record_id":"<urn:uuid:663e9aa5-86bb-4c82-97ad-e7a2f31d77ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00416.warc.gz"} |
If the coefficient of cubical expansions is x times class 11 physics JEE_Main
Hint: The increase in the area and volume with the rise in the temperature is known as superficial and cubical expansion. By finding the relation between these two expansions we can solve the given
Formula used:
$ \Rightarrow \dfrac{\alpha }{1} = \dfrac{\beta }{2} = \dfrac{\gamma }{3}$
Complete step by step answer:
To answer the given question, we need to understand the expansion phenomena. Whenever there is an expansion of the body due to the heating, then the body is said to be expanding and this phenomenon
is known as the expansion phenomena.
The solids can undergo this phenomenon. There are totally three types of phenomenon. They are:
1. Linear expansion is the expansion that is caused due to the increase in the length of the solid. It is denoted by $\alpha $
2. Superficial expansion is the expansion that is caused due to the increase in the area of the solid. It is denoted by $\beta $
3. Cubical expansion is the expansion that is caused due to the increase in the volume of the solid. It is denoted by $\gamma $
The relation between these expansions is $\alpha ,\beta ,\gamma $. From this, it is clear that the expansions occur with the increase in the temperature.
The common relation between these expansions is:
$ \Rightarrow \dfrac{\alpha }{1} = \dfrac{\beta }{2} = \dfrac{\gamma }{3}$
It can also be represented as,
$ \Rightarrow \alpha :\beta :\gamma = 1:2:3$
Now, let us try to solve the given problem. In the question, they have given about the cubical and the superficial expansions. So, we can consider these two expansions alone. We have a relation
between these two expansions. The relation is,
$ \Rightarrow \dfrac{\beta }{2} = \dfrac{\gamma }{3}$
The above equation can be written as,
$ \Rightarrow \gamma = \beta \dfrac{3}{2}........(1)$
In the question, it is given that the coefficient of the cubical expansion is increased by the $x$ times of the coefficient of superficial expansion. That is,
$ \Rightarrow \gamma = \beta x...........(2)$
We can compare equations 1 and 2. We get the answer as,
$ \Rightarrow x = \dfrac{3}{2}$
We can use division to simplify, we get,
$ \Rightarrow x = 1.5$
The value of $x$ is $1.5.$
$\therefore x = 1.5$
Hence option \[\left( C \right)\] is the correct answer.
Note: We have some basic formulae to calculate the expansions. To calculate the area expansion, we have ${A_0}(I + \beta t)$ where $\beta $ is the coefficient of the expansion. To calculate the
volume expansion, we have, $\Delta V = {V_\gamma }\Delta t$ where $\gamma $ is the coefficient of volume expansion. | {"url":"https://www.vedantu.com/jee-main/if-the-coefficient-of-cubical-expansions-is-x-physics-question-answer","timestamp":"2024-11-11T00:31:00Z","content_type":"text/html","content_length":"149140","record_id":"<urn:uuid:54d3ecab-82e9-424d-9676-9c95c4a3be3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00615.warc.gz"} |
Pipe Elbow Fabrication Formula Pdf Download WORK | fukagawakiyuukai
Pipe Elbow Fabrication Formula Pdf Download WORK
Click Here ===== https://urllio.com/2tzg8f
How to Download a PDF Guide on Pipe Elbow Fabrication Formula
Go to this link, which is the first web search result for the keyword "Pipe Elbow Fabrication Formula Pdf Download".[^1^]
Click on the download button or icon on the top right corner of the page.
Select a location on your device where you want to save the PDF file.
Open the PDF file with a PDF reader application.
Scroll down to page 16, where you will find a section titled "Pipe Elbow Fabrication Formula".
Read and follow the instructions and examples on how to calculate the pipe elbow dimensions, angles, cut lengths, and weights.
This PDF guide is called "Piping Fabrication Calculation And Formulas Handbook" and it covers various topics related to piping fabrication, such as piping stress, piping design, piping materials,
piping fittings, piping flanges, piping supports, and more. It is a useful resource for anyone who is interested in learning more about piping fabrication.
One of the most important formulas for pipe elbow fabrication is the one that relates the centerline radius (R), the bend angle (θ), and the arc length (L) of the pipe elbow. The formula is:
$L = R \times \theta$
This formula can be used to find any of the three variables if the other two are known. For example, if you know the centerline radius and the bend angle of a pipe elbow, you can use this formula to
find the arc length. Similarly, if you know the arc length and the bend angle, you can use this formula to find the centerline radius. And if you know the arc length and the centerline radius, you
can use this formula to find the bend angle.
The centerline radius is the distance from the center of the pipe to the center of the bend. The bend angle is the angle between the two straight segments of the pipe that form the elbow. The arc
length is the length of the curved part of the pipe elbow. These variables are illustrated in the following diagram:
The PDF guide also provides a table that shows some common values for pipe elbow dimensions and weights for different pipe sizes and schedules. The table can be found on page 17 of the PDF guide. The
table can be used as a reference for estimating the material and labor costs of pipe elbow fabrication.
Another formula that is useful for pipe elbow fabrication is the one that relates the outside diameter (D), the inside diameter (d), and the wall thickness (t) of the pipe. The formula is:
$t = \frac{D - d}{2}$
This formula can be used to find any of the three variables if the other two are known. For example, if you know the outside diameter and the wall thickness of a pipe, you can use this formula to
find the inside diameter. Similarly, if you know the inside diameter and the wall thickness, you can use this formula to find the outside diameter. And if you know the outside diameter and the inside
diameter, you can use this formula to find the wall thickness.
The outside diameter is the distance across the pipe measured from the outer edge to the outer edge. The inside diameter is the distance across the pipe measured from the inner edge to the inner
edge. The wall thickness is the distance between the outer edge and the inner edge of the pipe. These variables are illustrated in the following diagram:
The PDF guide also provides a formula that can be used to calculate the weight of a pipe elbow based on its dimensions and material density. The formula is:
$W = \frac{\pi}{4} \times L \times (D^2 - d^2) \times \rho$
Where W is the weight of the pipe elbow, L is the arc length of the pipe elbow, D is the outside diameter of the pipe, d is the inside diameter of the pipe, and Ï is the material density of the
pipe. The material density can vary depending on the type of pipe material, such as carbon steel, stainless steel, copper, etc. The PDF guide provides a table that shows some common values for
material density for different pipe materials. The table can be found on page 18 of the PDF guide. 061ffe29dd | {"url":"https://www.fukagawakiyuukai.com/forum/deisukatusiyon/pipe-elbow-fabrication-formula-pdf-download-work","timestamp":"2024-11-12T12:50:34Z","content_type":"text/html","content_length":"897829","record_id":"<urn:uuid:77d4265a-721b-4d44-967e-c0ae9c58c022>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00198.warc.gz"} |
What are the 4 steps in solving word problems?
Polya created his famous four-step process for problem solving, which is used all over to aid people in problem solving:
• Step 1: Understand the problem.
• Step 2: Devise a plan (translate).
• Step 3: Carry out the plan (solve).
• Step 4: Look back (check and interpret).
How to solve math word problems in an easy way?
Write an equation. Use the information you learn from the problem,including keywords,to write an algebraic description of the story.
Solve an equation for one variable. If you have only one unknown in your word problem,isolate the variable in your equation and find which number it is equal
Solve an equation with multiple variables.
How do you answer a word problem?
Read the word problem. Make sure you understand all the words and ideas.
Identify what you are looking for.
Name what you are looking for. Choose a variable to represent that quantity.
Translate into an equation.
Solve the equation using good algebra techniques.
Check the answer in the problem.
Answer the question with a complete sentence.
What are some easy math problems?
If a number is an even number and ends in 0,2,4,6 or 8,it is divided by 2.
A number is divisible by 3 if the sum of the digits is divisible by 3.
A number is divisible by 4 if the last two digits are divisible by 4.
If the last digit is 0 or 5,it is divisible by 5
How do you solve algebra word problems?
Solving algebraic word problems requires us to combine our ability to create equations and solve them. To solve an algebraic word problem: Define a variable. Write an equation using the variable.
Solve the equation. If the variable is not the answer to the word problem, use the variable to calculate the answer. | {"url":"https://www.toccochicago.com/2022/07/16/what-are-the-4-steps-in-solving-word-problems/","timestamp":"2024-11-08T11:03:28Z","content_type":"text/html","content_length":"42197","record_id":"<urn:uuid:2619578d-8659-47fa-8247-0b78b55b41fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00748.warc.gz"} |
algorithm Tutorial => O(log n) types of Algorithms
Let's say we have a problem of size n. Now for each step of our algorithm(which we need write), our original problem becomes half of its previous size(n/2).
So at each step, our problem becomes half.
Step Problem
1 n/2
2 n/4
3 n/8
4 n/16
When the problem space is reduced(i.e solved completely), it cannot be reduced any further(n becomes equal to 1) after exiting check condition.
1. Let's say at kth step or number of operations:
problem-size = 1
2. But we know at kth step, our problem-size should be:
problem-size = n/2^k
3. From 1 and 2:
n/2^k = 1 or
n = 2^k
4. Take log on both sides
log[e] n = k log[e]2
k = log[e] n / log[e] 2
5. Using formula log[x] m / log[x] n = log[n] m
k = log[2] n
or simply k = log n
Now we know that our algorithm can run maximum up to log n, hence time complexity comes as
O( log n)
A very simple example in code to support above text is :
for(int i=1; i<=n; i=i*2)
// perform some operation
So now if some one asks you if n is 256 how many steps that loop( or any other algorithm that cuts down it's problem size into half) will run you can very easily calculate.
k = log[2] 256
k = log[2] 2 ^8 ( => log[a]a = 1)
k = 8
Another very good example for similar case is Binary Search Algorithm.
int bSearch(int arr[],int size,int item){
int low=0;
int high=size-1;
return mid;
else if(arr[mid]<item)
else high=mid-1;
return –1;// Unsuccessful result | {"url":"https://riptutorial.com/algorithm/example/26648/o-log-n--types-of-algorithms","timestamp":"2024-11-13T02:41:06Z","content_type":"text/html","content_length":"51961","record_id":"<urn:uuid:243f1977-e999-40c6-9edd-90620b0923de>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00606.warc.gz"} |
What kind of number 0 is odd or even?
What kind of number 0 is odd or even?
Zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. This can be easily verified based on the definition of “even”: it is an integer multiple of 2,
specifically 0 × 2.
What type of real number is 0?
Answer: 0 is a rational number, whole number, integer, and a real number. Let’s analyze this in the following section. Explanation: Real numbers include natural numbers, whole numbers, integers,
rational numbers, and irrational numbers.
Is 0 a real number Yes or no?
Real numbers can be positive or negative, and include the number zero. They are called real numbers because they are not imaginary, which is a different system of numbers. Imaginary numbers are
numbers that cannot be quantified, like the square root of -1.
Is 1.625 a rational number?
We unually write rational numbers in their lowest terms, for example 8/10 is usually written 4/5. We commonly write rationals in decimal form, so that 1/4 is the same as 0.25, 13/8 = 1.625 and 4/5 =
0.8. Another irrational number is which is approximately 1.4142135623731.
Is one an odd or even number?
One is the first odd positive number but it does not leave a remainder 1. Some examples of odd numbers are 1, 3, 5, 7, 9, and 11. An integer that is not an odd number is an even number. If an even
number is divided by two, the result is another integer.
Which is smallest even number?
2 is the smallest even number.
Is 3/13 a terminating decimal?
Decimal form of 3/13 is 0.2307692. Decimal expansion is the Terminating and repeating which is about 9th numbers is repeating it.
Is 0 an odd or even integer?
Zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. This can be easily verified based on the definition of “even”: it is an integer multiple of 2,
specifically 0 × 2.
Is 0 considered an even number?
Zero is an even number except in certain circumstances. The definition of an even number is that you can divide it by 2 and not get a remainder. As 0/2 = 0, zero is generally considered to be an even
number. However, some people claim that it is neither odd nor even.
How can you tell if a number is odd or even?
Zero is considered an even number. To tell whether a number is even or odd, look at the number in the ones place. That single number will tell you whether the entire number is odd or even. An even
number ends in 0, 2, 4, 6, or 8. An odd number ends in 1, 3, 5, 7, or 9.
Is negative one an odd number?
If an even number is divided by two, the result is another integer. On the other hand, an odd number, when divided by two, will result in a fraction. Since odd numbers are integers, negative numbers
can be odd. This short article about mathematics can be made longer. | {"url":"https://forwardonclimate.org/tips/what-kind-of-number-0-is-odd-or-even/","timestamp":"2024-11-08T20:45:33Z","content_type":"text/html","content_length":"55592","record_id":"<urn:uuid:d6a66cce-58d5-4450-9457-47d2a86461f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00806.warc.gz"} |
Five lakh seven thousand four hundred six.
Prepare a place value chart for the numerical form and add the information appropriately in the chart. This will help you to get the answer easily and accurately.
Complete step-by-step answer:
We have to find the numerical form of: Five lakh seven thousand four hundred six.
Before finding the numerical value, we need to understand the place value of the number.
In mathematics, every digit or number has a place value. Place value can be defined as the value represented by a digit in a number on the basis of its position in a certain number.
A place value chart can help us in finding and comparing the place value of the digits in numbers.
Here, we will arrange the numerical form from the right hand side. Such as –
A) Six will be placed in one place.
B) There is no value for tens of places. We will write zero on that place.
C) Seven will be placed in thousand places.
D) There is no value for ten thousand places. We will write zero on that place.
E) At last, five will be placed in lakh places.
The place value chart of the given number is:
Hence, the numerical form of Five lakh seven thousand four hundred six is \[5,{\text{ }}07,{\text{ }}406\]
There are two ways to convert into numerical form: a) Indian system and b) international system.
In the Indian numeral system, the place values of digits go in the sequence of Ones, Tens, Hundreds, Thousands, Ten Thousand, Lakhs, Ten Lakhs, Crores and so on.
In the International number system, the place values of digits go in the sequence of Ones, Tens, Hundreds, Thousands, Ten Thousand, Hundred Thousand, Millions, Ten Million and so on. | {"url":"https://www.vedantu.com/question-answer/write-the-following-in-numerical-form-five-lakh-class-6-maths-cbse-5f5c3c1068d6b37d16e9f90f","timestamp":"2024-11-09T22:31:17Z","content_type":"text/html","content_length":"158043","record_id":"<urn:uuid:3ef7f5c1-f66b-4a1b-b3d3-7af878e61044>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00568.warc.gz"} |
LOD (Level of Details) calculations in Tableau - NewDataLabs
LOD (Level Of Details) calculations, already introduced in Tableau 9.0, have introduced a small revolution in the flexibility of Tableau calculations. Why are they unique? Simply put, LOD
calculations allow us to create and display calculations at a different level of detail than the data displayed in our visualization. When creating calculations in Tableau, we can create them at the
row level (row level) – for example, [sales] / [orders]. The second option is to aggregate the measures – that is, sum([sales]) / sum ([orders]). If we wanted to know the difference between a measure
and its average – for example, [sales] – avg([sales]), Tableau would return the error Cannot mix aggregate and non- aggregate arguments with this function. With LOD, such calculations are possible,
which will give them analysts more flexibility in their calculations.
FIXED LOD – we freeze the dimensions
The easiest way to understand the LOD concept is with the expression FIXED. This expression freezes the dimensions according to which the measure is calculated for the given aggregation (sum, avg,
min, max, etc.). The syntax of the expression is as follows (we always dress LOD expressions in curly brackets):
What is the difference between standard calculation and LOD? In the chart below, we see a comparison of total sales by segment and category and LOD with the Segment frozen dimension. As you can see,
the calculation does not take into account the breakdown by category, and is only calculated at the Segment dimension level. Therefore, the value of this calculation is the same for each category.
To better illustrate this example, we add another measure, LOD FIXED at the Category level. As you can see for each category the values repeat in each segment.
FIXED LOD – put into practice
The question is what the FIXED LOD is used for. There are of course many applications, but I will focus on a few basic ones. First, the percentage of the total. With FIXED LOD, we can determine a
measure at a given (higher) level of aggregation and then divide the measure at the lower level by that measure, yielding a percentage of the total:
The desired effect can of course be obtained by using Table Calculations, but with TC we have to remember about proper parameterization of the function and its change when changing the layout of the
visualization. In the case of LOD, we have a rigid calculation that we can apply at will.
Another application is to look for the maximum value and then compare it to other values. Suppose we want to quickly find the maximum order – and for which category it occurs. Using the Calculated
Field: {FIXED: max([Sales])} and then to determine the difference ([Sales])-{FIXED: max([Sales])} we immediately see that the largest order occurred for the Machines/Technology sub-category:
The last case would be to compare the value of sales to the value of the category. Suppose we want to compare the sales of all categories to the sales value in the Supplies/Office Supplies category.
To do this, we determine the calculated field:
However, after adding the field to our table, the value according to the calculation will be determined only for the Supplies sub-category. To get around this problem, we will add a LOD calculation:
This is how we achieve the effect:
Not Just FIXED – LOD INCLUDE and EXCLUDE Calculations
FIXED is one type of LOD, but not the only one. In addition to this option, we also have INCLUDE and EXCLUDE. Let’s start with INCLUDE. As the name suggests, this type of LOD adds dimensions to the
calculation that we do not have in the visualization. Suppose we want to know the average sales by customer. If we put the Sales field in the Average aggregation in the visualization, then we get the
average at the row (transaction/order) level. If you add LOD INCLUDE, then the customer name will be taken into account in the calculation:
By aggregating the measure by average, we get average sales by customer – not by transaction as with the standard measure.
Since INCLUDE adds dimensions to the calculation, EXCLUDE works exactly the opposite – it eliminates dimensions from the calculation. If we want to determine sales by category in the case under
analysis, we can use EXCLUDE LOD:
Producing a similar effect to the FIXED function:
LOD- FIXED, INCLUDE OR EXCLUDE calculations?
FIXED LOD is the most commonly used and most practical. With FIXED we can achieve exactly the same results as with INCLUDE and EXCLUDE. So in order not to mess with the syntax too much often purely
practical analysts use mostly FIXED. Keep in mind one major difference, namely the different position of FIXED and INCLUDE/EXCLUDE calculations in Tableau’s calculation order (order of operations):
FIXED LOD is calculated before Dimensions and INCLUDE/EXCLUDE is calculated after. This affects how calculations are done, so keep this in mind and add filters to the context if necessary (i.e. move
from Dimension to Context filters).
QUICK LOD – simple calculation creation
In one of the latest versions of Tableau (2021.1), an option was added to quickly create LOD calculations. To create a quick FIXED LOD simply select a measure, then with the control button on the
keyboard pressed, move the measure to the selected dimension by which you want to aggregate the measure. A new calculated field containing the appropriate syntax will automatically be created:
Unfortunately, apart from creating a field and syntax, the solution does not provide too many options – we still have to edit the more advanced case ourselves, adding new dimensions or changing
aggregations. Nonetheless, it is a certain convenience especially for novice users.
Mateusz Karmalski, Tableau Author | {"url":"https://newdatalabs.com/en/lod-level-of-details-calculations-in-tableau/","timestamp":"2024-11-06T02:52:40Z","content_type":"text/html","content_length":"75838","record_id":"<urn:uuid:70b93c87-2c95-42b6-bfc3-5ea824711109>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00556.warc.gz"} |
The ISNA function checks whether a value is #N/A. If the value is #N/A, the function returns TRUE; otherwise, it returns FALSE. This function is commonly used in combination with other functions that
may return #N/A as a result.
Use the ISNA formula with the syntax shown below, it has 1 required parameter:
1. value (required):
The value to be checked for #N/A.
Here are a few example use cases that explain how to use the ISNA formula in Google Sheets.
Checking for #N/A errors
The ISNA function can be used to check if a cell contains the #N/A error. This can be helpful when working with large datasets or complex formulas that may return errors.
Conditional formatting
The ISNA function can be used in combination with conditional formatting to highlight cells that contain #N/A errors. This can make it easier to identify and correct errors in a spreadsheet.
Error handling
The ISNA function can be used in combination with other functions to handle errors in a formula. For example, if a VLOOKUP function returns #N/A, you can use the IF function with ISNA to display a
custom error message.
Common Mistakes
ISNA not working? Here are some common mistakes people make when using the ISNA Google Sheets Formula:
Not providing a value as input
The ISNA formula requires a value as input to determine whether it is #N/A or not. Make sure to provide a value as input.
Using the formula with non-numeric values
The ISNA formula only works with numeric values. If you try to use it with non-numeric values, it will return the #VALUE! error. Make sure to use it only with numeric values.
Not using parentheses around the input value
The ISNA formula requires the input value to be enclosed in parentheses. If you forget to do this, it will return the #NAME? error. Make sure to enclose the input value in parentheses.
Related Formulas
The following functions are similar to ISNA or are often used with it in a formula:
• ISERROR
The ISERROR formula is used to check if a value contains an error. This formula returns TRUE if the value is an error, and FALSE if it is not. This function is most commonly used in combination
with other formulas that can return errors, to ensure that the resulting value is valid.
• IFERROR
The IFERROR formula is used to check whether a specified value results in an error or not. If the value results in an error, then it returns a user-specified value instead of the error. This
function is commonly used to prevent errors from breaking a formula or to replace error messages with custom messages.
• VLOOKUP
The VLOOKUP function is a lookup formula used to search for a value in the first column of a range of cells (the search key) and return a value in the same row from a specified column in that
range. This function is most commonly used to look up and retrieve data from a table.
• HLOOKUP
The HLOOKUP function is a lookup formula that searches for a key in the top row of a table and returns the value in the same column for a specified row. This function is commonly used to extract
data from a table based on a specific criteria.
Learn More
You can learn more about the ISNA Google Sheets function on Google Support. | {"url":"https://checksheet.app/google-sheets-formulas/isna/","timestamp":"2024-11-10T06:01:02Z","content_type":"text/html","content_length":"45036","record_id":"<urn:uuid:e397dd27-149b-4976-b086-558284415ada>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00874.warc.gz"} |
Equations With Rational Coefficients Worksheet - Equations Worksheets
Equations With Rational Coefficients Worksheet
Equations With Rational Coefficients Worksheet – The purpose of Expressions and Equations Worksheets is for your child to be able to learn more effectively and efficiently. These worksheets are
interactive and challenges that are dependent on the order of how operations are conducted. Through these worksheets, kids can master both basic and more complex concepts in a very short amount of
amount of time. These PDF resources are free to download and may be utilized by your child to learn math concepts. These resources are beneficial for students who are in the 5th-8th Grades.
Get Free Equations With Rational Coefficients Worksheet
Some of these worksheets are for students in the 5th to 8th grades. The two-step word problems were designed using decimals, fractions or fractions. Each worksheet contains ten problems. These
worksheets are available both online and in printed. These worksheets can be used to exercise rearranging equations. These worksheets are a great way to practice rearranging equations and assist
students with understanding equality and inverse operation.
These worksheets are designed for fifth and eight graders. These worksheets are perfect for students who are struggling to calculate percentages. There are three types of questions you can choose
from. There is the option to either work on single-step problems which contain whole numbers or decimal numbers, or use words-based methods to solve fractions and decimals. Each page is comprised of
ten equations. The Equations Worksheets are used by students from 5th to 8th grades.
These worksheets can be a wonderful tool for practicing fraction calculations along with other topics related to algebra. You can choose from many kinds of challenges with these worksheets. It is
possible to select word-based or numerical problem. It is vital to pick the problem type, because every challenge will be unique. Each page has ten questions which makes them an excellent resource
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These worksheets help students understand the relationships between variables and numbers. These worksheets provide students with practice in solving polynomial equations or solving equations, as
well as understanding how to apply them in daily life. These worksheets are a fantastic way to learn more about equations and expressions. These worksheets will help you learn about different types
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These worksheets can be very beneficial for students in their first grades. These worksheets will aid them develop the ability to graph and solve equations. These worksheets are excellent for
learning about polynomial variables. They can also help you learn how to factor and simplify these variables. You can get a superb collection of equations and expressions worksheets for kids at any
grade level. The best way to learn about the concept of equations is to perform the work yourself.
There are plenty of worksheets to study quadratic equations. Each level has their own worksheet. These worksheets are a great way to practice solving problems up to the fourth degree. Once you have
completed an appropriate level it is possible working on other types of equations. Then, you can work on solving the same-level problems. For instance, you could solve a problem using the same axis,
but as an extended number.
Gallery of Equations With Rational Coefficients Worksheet
Solve Equations With Rational Coefficients Worksheet
42 Solve Equations With Rational Coefficients Worksheet Worksheet
Rational Equations Worksheets
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Binary Calculator | SEO Tools Tube
Binary Calculator
To use Binary Calculator, enter the values in the input boxes below and click on Calculate button.
Understanding the Binary Calculator Tool
In the digital age, binary numbers form the backbone of computing and technology. Every piece of data, from text to images, is ultimately represented in binary form, consisting of just two digits: 0
and 1. As crucial as it is to understand binary, performing calculations in this numeral system can be challenging. This is where the binary calculator tool comes into play, providing a user-friendly
way to carry out binary operations efficiently. In this post, we'll dive into what a binary calculator is, how it works, and why it's a valuable resource for anyone interested in technology.
What is a Binary Calculator?
A binary calculator is a specialized tool designed to perform mathematical operations using binary numbers. Unlike the decimal system, which is based on ten digits (0-9), binary operates solely on
two (0 and 1). This fundamental difference makes binary calculations essential for programming, computer science, and digital electronics.
Binary calculations are not just an academic exercise; they are vital for understanding how computers process data. Every command executed by a computer is ultimately broken down into binary code,
making proficiency in this area indispensable for anyone working with technology.
Key Features of the Binary Calculator
Basic Operations
The binary calculator allows users to perform basic mathematical operations such as addition, subtraction, multiplication, and division. These operations adhere to binary rules, making it easier to
grasp how these calculations work in practice.
Conversion Capabilities
One of the standout features of the binary calculator is its ability to convert between binary and other numeral systems, including decimal, hexadecimal, and octal. This functionality is invaluable
for anyone needing to switch between different formats regularly.
User-Friendly Interface
With an intuitive design, users can navigate the binary calculator effortlessly. It is structured to accommodate beginners while still providing advanced features for seasoned users.
Error Handling
The binary calculator includes robust error handling, ensuring that users receive immediate feedback if they enter incorrect input. This helps reduce frustration and enhances the learning experience.
How to Use the Binary Calculator
Using the binary calculator is straightforward. Here’s a step-by-step guide:
1. Input Binary Numbers: Start by entering the binary numbers you want to work with. Ensure that you use the correct binary format (only 0s and 1s).
2. Perform Operations: Select the operation you wish to perform—addition, subtraction, multiplication, or division. The calculator will execute the operation based on binary rules.
3. View Results: After performing a calculation, the results will be displayed. Take a moment to understand the output, especially if you are learning binary.
4. Conversion: If you want to convert your binary number to decimal, hexadecimal, or octal, simply use the conversion feature provided by the calculator.
5. Tips for Accuracy: To avoid errors, double-check your binary formatting before submitting calculations.
Applications of the Binary Calculator
The binary calculator is not just a toy for tech enthusiasts; it has real-world applications:
• Educational Use: Teachers and students can utilize the binary calculator to grasp binary concepts more effectively, reinforcing learning through practical application.
• Programming and Software Development: Developers often need to manipulate binary data. This tool makes it easier to test and verify binary calculations.
• Networking and Data Transmission: Understanding binary is essential for networking protocols, making the calculator a handy tool for network professionals.
• Hardware Design: Engineers designing computer architecture can benefit from a quick reference for binary calculations, helping to streamline their work processes.
Benefits of Using a Binary Calculator
The binary calculator offers numerous benefits:
• Time-Saving: It saves time on manual calculations, allowing users to focus on more complex tasks.
• Error Reduction: Automated calculations reduce the likelihood of human error, which is particularly important in programming and technical fields.
• Enhanced Understanding: By using the tool, users can develop a stronger grasp of binary operations, making it easier to apply these concepts in real-world situations.
To recap, the binary calculator is a powerful tool that simplifies binary operations and conversions. Mastering binary calculations is essential for anyone interested in technology, from students to
professionals. We encourage you to explore this tool and integrate it into your study or work routines for improved efficiency and understanding.
What is binary math, and why is it important?
Binary math forms the foundation of computer operations, as computers use binary code to process data. Understanding binary math is essential for anyone pursuing a career in technology or
Can I use the binary calculator for non-binary operations?
While the primary focus is on binary calculations, the calculator often includes features for converting and operating with other numeral systems, making it versatile.
Is there a mobile version of the binary calculator?
Many binary calculators are designed to be accessible on various devices, including smartphones and tablets. Check the specific tool for compatibility.
How should I proceed if I run into a problem with the tool?
If you encounter an error, review the input to ensure it adheres to binary formatting. Most calculators provide error messages that guide users on how to correct their inputs.
Are there any tips for beginners learning binary math?
Begin by understanding the basics of binary numbers and practice with small calculations. Using a binary calculator can significantly enhance your learning experience.
In summary, the binary calculator is an invaluable tool for anyone working with technology. It streamlines binary calculations, reduces errors, and enhances understanding of this fundamental numeral
system. We encourage you to try the binary calculator and explore further resources on binary math to deepen your knowledge and skills. Happy calculating! | {"url":"https://seotoolstube.com/binary-calculator","timestamp":"2024-11-10T05:07:49Z","content_type":"text/html","content_length":"41575","record_id":"<urn:uuid:73a41fb2-e890-4f54-920f-573e5c43029c>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00164.warc.gz"} |
how to convert ka to pka without calculator
This widget finds the pH of an acid from its pKa value and concentration. To create a more manageable number, chemists define the pKa value as the negative logarithm of the Ka value: pKa = -log Ka.
You can also find it on a scientific calculator by inputting the number and pressing the exponent key, which either looks like a hat (^) or is denoted by 10x. pKa, Ka, and Acid Strength. Till i found
this. The solution becomes one containing a conjugate base which is the product of the acid having lost a proton and a conjugate acid. For those who struggle with math, equations can seem like an
impossible task. 1.Use the equation Ka=10-pKa{\displaystyle Ka=10^{-pKa}} for this conversion. Determine math To determine math equations, one could use a variety of methods, such as trial and error,
looking for patterns, or using algebra. wikiHow is where trusted research and expert knowledge come together. But there is help available in the form of Pka to ka online calculator. 2 8 8 comments
Best Add a Comment symmetricinfo 4 yr. ago This is the shortcut I used. 00:0000:00 An unknown error has occurred Brought to you by eHow (10-6) 9.4. [A] - Concentration of conjugate base and pKa =
-log(Ka). You can get math help online by visiting websites like Khan Academy or Mathway. How to Convert Ka to pKa, Formula, Equation & Examples of. Organic Chemistry pH, pKa, Ka, pKb, Kb How can I
calculate the pH of a weak acid with an example? His writing covers science, math and home improvement and design, as well as religion and the oriental healing arts. wikiHow, Inc. is the copyright
holder of this image under U.S. and international copyright laws. Include your email address to get a message when this question is answered. If you already know the pKa value for acid and you need
the Ka value, you find it by taking the antilog. I get sick alot so I have alot of math homework. Natural Language. The pKa to pH converter calculator is an online tool that can find the pH of a
liquid solution using the pKa value and concentration. Solve for the concentration of \(\ce{H3O^{+}}\) using the equation for pH: \[ [H_3O^+] = 10^{-pH} \]. Grab a free copy of my ebook \"MCAT Exam
Strategy - A 6 Week Guide To Crushing The MCAT\" at https://leah4sci.com/MCATguideYTThis video is part 9 in my MCAT Math Without a Calculator video series. If Ka is 7.1E-7, pKa is just 7 - log7.1.
Video 7 - pH/pOH [H+]/[OH-] Wheel . Calculating a Ka Value from a Known pH is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts. When given the pH value
of a solution, solving for \(K_a\) requires the following steps: Calculate the \(K_a\) value of a 0.2 M aqueous solution of propionic acid (\(\ce{CH3CH2CO2H}\)) with a pH of 4.88. If you're
struggling with a math problem, scanning it for key information can help you solve it more quickly. Substitute your known pKa value into the equation from the previous step. The correct formula is Ka
=10-pKa K a = 10 - p K a . It makes calculating logs a lot simpler. Your estimate of Ka is 10.8, while the actual answer is 10.63. Step 3: Solve for Ka K a or pKa p K a. Few of them are enlisted
below. pKa=. This video teaches you how to tackle anti-log questions as they arise on your MCAT, without the use of a calculator. Therefore to convert pKa to Ka just find 10^ -pKa in the scientific
calculator, you will get. The calculator takes, Applications of triple integrals in engineering, Find the fourier series for the extended function, The table shows the outcomes of rolling two number
cubes, Ullu web series watch online free dailymotion, What is the meaning of prime factors in maths. The value of Ka will be = 1.778 x 10^-5. How do you calculate the pKa to Ka and vice versa in your
head? If you need help with calculations, there are online tools that can assist you. Christopher Osborne has been a wikiHow Content Creator since 2015. To create a more manageable number, chemists
created the pKa value: A strong acid with a dissociation constant of 107 has a pKa of -7, while a weak acid with a dissociation constant of 10-12 has a pKa of 12. He is also a historian who holds a
PhD from The University of Notre Dame and has taught at universities in and around Pittsburgh, PA. His scholarly publications and presentations focus on his research interests in early American
history, but Chris also enjoys the challenges and rewards of writing wikiHow articles on a wide range of subjects. Tips. This image is not<\/b> licensed under the Creative Commons license applied to
text content and some other images posted to the wikiHow website. Are you struggling to understand concepts How to find pka from ka without calculator? http://leah4sci.com/MCAT Presents: MCAT Math
Without A Calculator Video 8 - Solving MCAT logarithm calculations without a calculatorWatch Next: Logarithms . This operation gives you the dissociation constant Ka: When Ka is large, it means the
conjugate ions aren't strong enough to move the reaction in the opposite direction, which indicates a strong acid. pK to K Conversion Since pK a is the negative logarithm of K a, the value of K a can
be calculated by simply reversing the above equation. This widget finds the pH of an acid from its pKa value and concentration. hard time doing vice versa, especially with non-whole number pH values.
For example log_(10)100 = 2, and log_(10)1000 = 3. Acid base chemistry has a LOT of math. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other
images posted to the wikiHow website. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. wikiHow, Inc. is the copyright holder of this image under U.S.
and international copyright laws. Energy converter Temperature converter Pressure converter Length converter Ka Kb pKa pKb converter Please let us know how we can improve this web app. Given that
relationship between Ka and pKa is defined logarithmically, we can rewrite the equation from the previous step as Ka = 10^- (pKa). This image may not be used by other entities without the express
written consent of wikiHow, Inc.
\n<\/p><\/div>"}, {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/9\/9f\/Find-Ka-from-Pka-Step-5.jpg\/v4-460px-Find-Ka-from-Pka-Step-5.jpg","bigUrl":"\/images\/thumb\/9\/9f\/
\u00a9 2023 wikiHow, Inc. All rights reserved. Key points For conjugate-acid base pairs, the acid dissociation constant K_\text {a} K a and base ionization constant K_\text {b} K b are related by the
following equations: K_\text {a}\cdot K_\text {b}=K_\text {w} K a K b = K w If you already know the pKa value for acid and you need the Ka value, you find it by taking the antilog. This image may not
be used by other entities without the express written consent of wikiHow, Inc.
\n<\/p><\/div>"}, {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/6\/69\/Find-Ka-from-Pka-Step-11.jpg\/v4-460px-Find-Ka-from-Pka-Step-11.jpg","bigUrl":"\/images\/thumb\/6\/69\/
\u00a9 2023 wikiHow, Inc. All rights reserved. Doing math equations is a great way to keep your mind sharp and improve your problem-solving skills. In some tables, you may find the pKa value listed,
but you may need the Ka value to plug into your equations. Pka to ka calculator - Pka to ka calculator is a software program that supports students solve math problems. Ka = 10 - (pKa). Example 1.
pKa of an acid is 4.75 and 0.1 concentrated in a solution. Can I find Ka (or pKa) without a calculator? Remember that pKa is expressed as a common logarithm (base 10) and not as a natural logarithm
(base e), so you want to find a table or select a function on your calculator that raises the number to a power of 10 rather than a power of e. Chris Deziel holds a Bachelor's degree in physics and a
Master's degree in Humanities, He has taught science, math and English at the university level, both in his native Canada and in Japan. This image is not<\/b> licensed under the Creative Commons
license applied to text content and some other images posted to the wikiHow website. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. If Ka is 7.1E-7,
pKa is just 7 - log7.1. Thus Ka = 10-pKa. If you already know. Extended Keyboard. For example, if pKa = 3.26, then Ka = 10^-3.26. The mathematical operation you perform is Ka = antilog (-pKa).
wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. You do need to pay to get worked out solutions but in a pinch it can at least speed up getting your
desired answers, sometimes you have to pay to see the steps and explanation, but for the most part it's free. pK_a = -log_10K_a. pKa is defined as the negative logarithm (-log) of Ka, which means you
calculate pKa with the calculation . 1.Use the equation Ka=10-pKa{\displaystyle Ka=10^{-pKa}} for this conversion. Better than Photomath. Sign up for wikiHow's weekly email newsletter. To create a
more manageable number, chemists define the pKa value as the negative logarithm of the Ka value: pKa = -log Ka. Easy to use, and it's user-friendly app. pKa is expressed as a common logarithm (base
10) and not as a natural logarithm (base e). Solve for the concentration of H 3O + using the equation for pH: [H3O +] = 10 pH Use the concentration of H 3O + to solve for the concentrations of the
other products and reactants. { Acid_and_Base_Strength : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()",
Calculating_A_Ka_Value_From_A_Measured_Ph : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Calculating_Equilibrium_Concentrations : "property
get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Fundamentals_of_Ionization_Constants : "property get [Map
MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Weak_Acids_and_Bases : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>
c__DisplayClass228_0.b__1]()", Weak_Acids_and_Bases_1 : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { Acid : "property get [Map
MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Acids_and_Bases_in_Aqueous_Solutions : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>
c__DisplayClass228_0.b__1]()", Acid_and_Base_Indicators : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Acid_Base_Reactions : "property get
[Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Acid_Base_Titrations : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>
c__DisplayClass228_0.b__1]()", Buffers : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Buffers_II : "property get [Map
MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Ionization_Constants : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>
c__DisplayClass228_0.b__1]()", Monoprotic_Versus_Polyprotic_Acids_And_Bases : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [
"article:topic", "pH", "Ionization Constants", "showtoc:no", "license:ccbyncsa", "licenseversion:40" ], https://chem.libretexts.org/@app/auth/3/login?returnto=
%2FAcids_and_Bases%2FIonization_Constants%2FCalculating_A_Ka_Value_From_A_Measured_Ph, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\
overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\
mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand
{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \
newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1
\|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), status page at https://status.libretexts.org. When the
solvent is water, it's left out of the equation. Thus K_a = 10^(-pK_a) Remember your definitions for logarithmic functions. Please add it to were it answers long word problems. This image is not<\/b>
licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This will give you your answer. How to Evaluate Trig Functions Without a
Calculator There are a few ways to calculate trigonometric functions without a calculator. This article was co-authored by wikiHow staff writer. I would like to give 100 stars for the helpul
activities by this app, i hated that I had to pay at first but I realized that these services don't come cheap, and one more addition, maybe a dark mode can be added in the application. No tracking
or performance measurement cookies were served with this page. Ka to pka calculator - The correct formula is Ka=10-pKa K a = 10 - p K a . Find the pH of this solution. To create a more manageable
number, chemists define the pKa value as the negative logarithm of the Ka value: pKa = -log Ka. Whether you need help solving quadratic equations, inspiration for the upcoming science fair or the
latest update on a major storm, Sciencing is here to help. As a result of the EUs General Data Protection Regulation (GDPR). This image is not<\/b> licensed under the Creative Commons license applied
to text content and some other images posted to the wikiHow website. The formula for converting pKa to Ka is as follows: Ka = 10^-pKa. This image may not be used by other entities without the express
written consent of wikiHow, Inc.
\n<\/p><\/div>"}, {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/5\/59\/Find-Ka-from-Pka-Step-8.jpg\/v4-460px-Find-Ka-from-Pka-Step-8.jpg","bigUrl":"\/images\/thumb\/5\/59\/
\u00a9 2023 wikiHow, Inc. All rights reserved. Thought this trick was commonly known. {\displaystyle Ka=10^{-pKa}} for this conversion. If all of this is twisting your brain a little, heres some good
news: its actually pretty easy to figure out Ka from pKa (or pKa from Ka) with a scientific calculator, or to get a decent estimate without a calculator. By using this website, you signify your
acceptance of Terms and Conditions and Privacy Policy.Do Not Sell My Personal Information http://leah4sci.com/MCAT Presents: MCAT Math Without A Calculator Video 9 - Solving MCAT antilog calculations
without a calculatorWatch Next: Logarithms an. Thought this trick was commonly known. How to Find Ka from pKa: Plus pKa to Ka & 5 Sample Problems To create a more manageable number, chemists define
the pKa value as the negative logarithm of the Ka value: pKa = -log Ka. concentration=. 10 - (pKa) = Ka. One way is to use the Pythagorean theorem to find the length of the hypotenuse. pKa
calculator. Unit Converters Unit Converters Energy Temperature Pressure Length Ka Kb pKa pKb New version of unit converters is available . How to Convert pKa to Ka. wikiHow, Inc. is the copyright
holder of this image under U.S. and international copyright laws. Leave a comment below this video or hit me up on social media:Twitter: http://twitter.com/leah4sciFacebook: https://www.facebook.com/
mcatexamstrategyGoogle+: https://plus.google.com/+LeahFisch/ To convert Ka into pKa, get out your calculator and find -log . It is so easy to use and always gets the answer right, Although. Do you
need help with your math homework? Chemistry Tutorial: How to convert between Ka and pKa (or Kb and MCAT Math Vid 9 - Antilogs in pH and pKa Without A Calculator. You can always count on us for help,
24 hours a day, 7 days a week. Because we started off without any initial concentration of H 3 O + and C 2 H 3 O 2-, is has to come from somewhere. It is used in everyday life, from counting to
measuring to more complex calculations. Expert teachers will give you an answer in real-time. To create a more manageable number, chemists define the pKa value as the negative logarithm of the Ka
value: pKa = -log Ka. How to find pka from ka without calculator - To create a more manageable number, chemists define the pKa value as the negative logarithm of the Ka value: pKa = . By using our
site, you agree to our. Even if math app is a robot he seems so sweet and . Dont let your maths teachers see this or they will get a heart attack and NO MORE DIFFICULTY IN HOMEWORK. If you need help
with tasks around the house, consider hiring a professional to get the job done quickly and efficiently. Convert pKa to Ka and vice versa. By raising 10 to the power of -pKa, we can calculate the
acid dissociation constant. The value of Ka will be = 1.778 x 10^-5. When given the pH value of a solution, solving for Ka requires the following steps: Set up an ICE table for the chemical reaction.
log7 is 0.84, so it's 7 - 0.84 or 6.16. Converting to the -log means that a lower pKa value indicates a stronger acid, as opposed to the Ka, in which a higher Ka value indicates a stronger acid.
Alternatively, it can be used to find the pOH value of a base by inputting its pKb value in the "pKa=" input field. For the Change in Concentration box, we . Deriving Ka from pH Theoretically, this
reaction can happen in both directions. Added Mar 27, 2014 by kalexchu in Chemistry. pKa = -log Ka. Therefore to convert pKa to Ka just find 10^ -pKa in the scientific calculator, you will get it. pH
of acid by pKa. You solve this by raising both sides of the original relationship to powers of 10 to get: Ka = 10 (-pKa) To solve, first determine pKa, which is simply log 10 (1.77 10 5) = 4.75. -
pKa = log Ka. If Ka is 7.1E-7, pKa is just 7 - log7.1. To create a more manageable number, chemists define the pKa value as the negative logarithm of the Ka value: pKa = -log Ka. The site owner may
have set restrictions that prevent you from accessing the site. Antonio Jones . My experience with the app is funtastic till now, really good with little to no ads. 2023 Leaf Group Ltd. / Leaf Group
Media, All Rights Reserved. Thank you so, so much, This app devs, thank you for giving us such an amazing app. =>K_a=10^-(pK_a) pK_a=-log_10K_a =>-pK_a=log_10K_a =>10^-(pK_a)=K_a =>K_a=10^-(pK_a)
50935 views around the world You can reuse this answer Creative Commons License and without app I'd be struggling really badly. This particular equation works on solutions made of an acid & its
conjugate base. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. Step 2: Enter the numerical quantities
into the formula. pKa is defined as the negative logarithm (-log) of Ka, which means you calculate . We are not permitting internet traffic to Byjus website from countries within European Union at
this time. Solutions with low pH are the most acidic, and solutions with high pH are most basic. pKa = - log Ka - pKa = log Ka 10 - (pKa) = Ka Ka = 10 - (pKa) Consider the reactions below. How to
convert between Ka and pKa (or Kb and pKb) More videos on YouTube More Subjects:. Find things in a picture game with answers, How to find the length of the median of a right triangle, How to solve an
integral when both limits are trigonometric, How to write greater than or equal to in python, Linear speed calculator inches per minute, Ncert solutions class 6 maths exercise 5.6, Top behavioral
interview questions and answers. etc. Math which you MUST be able to complete quickly and confidently without a . Great and simple app, i simply love it. | {"url":"https://unser-altona.de/kf62n5o/udu1tq/archive.php?id=how-to-convert-ka-to-pka-without-calculator","timestamp":"2024-11-14T23:51:15Z","content_type":"text/html","content_length":"72375","record_id":"<urn:uuid:db1ed541-8832-4978-95c3-d71a965abdbc>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00411.warc.gz"} |
testing ibpdv and ibpu
05-13-2015, 07:18 AM
Post: #8
salvomic Posts: 1,396
Senior Member Joined: Jan 2015
RE: testing ibpdv and ibpu
(05-12-2015 10:18 PM)Arno K Wrote: ...
So :simplify(subst('int(sqrt(1-x^2),x,0,1)',x=sin(u))) directly produces nothing but PI/4.
Now I tried int(sqrt(1-x^2),x) to see what the Prime would deliver: the answer is too long to be typed here,containing i and LN aside the first part,
I am a bit disappointed...
(05-13-2015 05:14 AM)parisse Wrote: It should be
assume(u>0); a:=subst('integrate(sqrt(1-x^2),x,0,1)',x=sin(u)); a
thank you Parisse!
iw would be interesting also to have a method to get "step by step" solution, here and for the "integration by parts" functions, but anyway still already this is ok...
please, control your CAS Settings: I've in CAS: Exact, "Use √" and Principal checked, Complex not checked (with firmware 6975)
with the integral you should get (½)*ASIN(x)+(½)*x*√(-x^2-1) from 0 to 1, that's π/4 (in Home you gets 0.785398...)
try, and let me know...
∫aL√0mic (IT9CLU) :: HP Prime 50g 41CX 71b 42s 39s 35s 12C 15C - DM42, DM41X - WP34s Prime Soft. Lib
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Bug in MATH ROM of the TI-95
03-10-2023, 02:07 PM
Post: #6
Namir Posts: 1,115
Senior Member Joined: Dec 2013
RE: Bug in MATH ROM of the TI-95
Thank you Thomas for solving the riddle. I was NOT enclosing the expression for fx in parentheses!! When I coded fx like you did on the TI-95 emulator and a real TI-95, it worked in both cases!!!
Thank you for helping solve the mystery!!!
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Systematic Fraud Detection Through Automated Data Analytics in MATLAB
By Jan Eggers, MathWorks
As the Madoff Ponzi scheme and recent high-profile rate-rigging scandals have shown, fraud is a significant threat to financial organizations, government institutions, and individual investors.
Financial services and other organizations have responded by stepping up their efforts to detect fraud.
Systematic fraud detection presents several challenges. First, fraud detection methods require complex investigations that involve the processing of large amounts of heterogeneous data. The data is
derived from multiple sources and crosses multiple knowledge domains, including finance, economics, business, and law. Gathering and processing this data manually is prohibitively time-consuming as
well as error-prone. Second, fraud is "a needle in a haystack" problem because only a very small fraction of the data is likely to be coming from a fraudulent case. The vast quantity of regular
data—that is, data produced from nonfraudulent sources—tends to blend out the cases of fraud. Third, fraudsters are continually changing their methods, which means that detection strategies are
frequently several steps behind.
Using hedge fund data as an example, this article demonstrates how MATLAB^® can be used to automate the process of acquiring and analyzing fraud detection data. It shows how to import and aggregate
heterogeneous data, construct and test models to identify indicators for potential fraud, and train machine learning techniques to the calculated indicators to classify a fund as fraudulent or
The statistical techniques and workflow described are applicable to any area requiring detailed analysis of large amounts of heterogeneous data from multiple sources, including data mining and
operational research tasks in retail and logistic analysis, defense intelligence, and medical informatics.
The Hedge Fund Case Study
The number of hedge funds has grown exponentially in recent years: The Eurekahedge database indicates a total of approximately 20,000 active funds worldwide.^1 Hedge funds are minimally regulated
investment vehicles and, therefore, prime targets of fraud. For example, hedge fund managers may fake return data to create the illusion of high profits and attract more investors.
We will use monthly returns data from January 1991 to October 2008 from three hedge funds:
• Gateway Fund
• Growth Fund of America
• Fairfield Sentry Fund
The Fairfield Sentry Fund is a Madoff fund known to have reported fake data. As such, it offers a benchmark for verifying the efficacy of fraud detection mechanisms.
Gathering Heterogeneous Data
Data for the Gateway Fund can be downloaded from the Natixis web site as a Microsoft^® Excel^® file containing the net asset value (NAV) of the fund on a monthly basis. Using the MATLAB Data Import
Tool, we define how the data is to be imported (Figure 1). The Data Import Tool can automatically generate the MATLAB code to reproduce the defined import style.
After importing the NAV for the Gateway Fund, we use the following code to calculate the monthly returns:
% Calculate monthly returns
gatewayReturns = tick2ret(gatewayNAV);
For the Growth Fund of America, we use Datafeed Toolbox™ to obtain data from Yahoo! Finance, specifying the ticker symbol for the fund (AGTHX), the name of the relevant field (adjusted close price),
and the time period of interest:
% Connect to yahoo and fetch data
c = yahoo;
data = fetch(c, 'AGTHX', 'Adj Close', startDate, endDate);
Unfortunately, Yahoo does not provide data for the period from January 1991 to February 1993. For this time period, we have to collect the data manually.
Using the financial time series object in Financial Toolbox™, we convert the imported daily data to the desired monthly frequency:
%Convert to monthly returns
tsobj = fints(dates, agthxClose);
tsobj = tomonthly(tsobj);
Finally, we import reported data from the Fairfield Sentry fund. We use two freely available Java™ classes, PDFBox and FontBox, to read the text from the pdf version of the Fairfield Sentry fund fact
% Instantiate necessary classes
pdfdoc = org.apache.pdfbox.pdmodel.PDDocument;
reader = org.apache.pdfbox.util.PDFTextStripper;
% Read data
pdfdoc = pdfdoc.load(FilePath);
pdfstr = reader.getText(pdfdoc);
Having imported the text, we extract the parts containing the data of interest—that is, a table of monthly returns.
Some tests for fraudulent data require comparison of the funds' returns data to standard market data. We import the benchmark data for each fund using the techniques described above.
Once the data is imported and available, we can assess its consistency—for example, by comparing the normalized performance of all three funds (Figure 2).
Simply viewing the plot allows for a qualitative assessment. For example, the Madoff fund exhibits an unusually smooth growth, yielding a high profit. Furthermore, there are no obvious indications of
inconsistency in the underlying data. This means that we will be able to use formal methods to detect fraudulent activities.
Analyzing the Returns Data
Since misbehavior or fraud in hedge funds manifests itself mainly in misreported data, academic researchers have focused on devising methods to analyze and flag potentially manipulated fund returns.
We compute metrics introduced by Bollen and Pool^2 and use them as potential indicators for fraud on the reported hedge fund returns. For example:
• Discontinuity at zero in the fund's returns distribution
• Low correlation with other assets, contradicting market trends
• Unconditional and conditional serial correlation, indicating smoother than expected trends
• Number of returns equal to zero
• Number of negative, unique, and consecutive identical returns
• Distribution of the first digit (Does it follow Benford's law?) and the last digit (Is it uniform?) of reported returns
To illustrate the techniques, we will focus on discontinuity at zero.
Testing for Discontinuity at Zero
Since funds with a higher number of positive returns attract more capital, fund managers have an incentive to misreport results to avoid negative returns. This means that a discontinuity at zero can
be a potential indicator for fraud.
One test for such a discontinuity is counting the number of return observations that fall in three adjacent bins, two to the left of zero and one to the right. The number of observations in the
middle bin should approximately equal the average of the surrounding two bins. A significant shortfall in the middle bin observations must be flagged.
Figure 3 shows the histograms of the funds' returns, with the two bins around zero highlighted. Green bars indicate no flag, and red bars indicate potential fraud. Only the Madoff fund did not pass
this test.
Results for Funds Under Consideration
Applying all the tests described above to the present data yields a table of indicators for each fund (Figure 4).
The Madoff fund raised a flag in nine out of ten tests, but the other two funds also raised flags. Positive test results do not prove that a given hedge fund was involved in fraudulent activities.
However, a table like the one shown in Figure 4 indicates funds that merit further investigation.
Classifying Analysis Results with Machine Learning
We now have a set of flags that can be used as indicators for fraud. Automating the analytics enables us to review larger data sets and to use the computed flags to categorize funds as fraudulent or
nonfraudulent. This classification problem can be addressed using machine learning methods—for example, bagged decision trees, using the TreeBagger algorithm in Statistics and Machine Learning
Toolbox™. The TreeBagger algorithm will require data for supervised learning to train the models. Note that our example uses data for only three funds. Applying bagged decision trees or other machine
learning methods to an actual problem would require considerably more data than this small, illustrative set.
We want to build a model to classify funds as fraudulent or nonfraudulent, applying the indicators described in the section “Analyzing the Returns Data” as predictor variables. To create the model,
we need a training set of data. Let us consider M hedge funds that are known as fraudulent or nonfraudulent. We store this information in the M-by-1-vector yTrain and compute the corresponding
MxN-matrix xTrain of indicators. We can then create a bagged decision tree model using the following code:
% Create fraud detection model based on training data
fraudModel = TreeBagger(nTrees,xTrain,yTrain);
where nTrees is the number of decision trees created based on bootstrapped samples of the training data. The output of the nTrees decision trees is aggregated into a single classification.
Now, for a new fund, the classification can be performed by
% Apply fraud detection model to new data
isFraud = predict(fraudModel, xNew);
We can use the fraud detection model to classify hedge funds based purely on their returns data. Since the model is automated, it can be scaled to a large number of funds.
This article outlines the process of developing a fully automated algorithm for fraud detection based on hedge fund returns. The approach can be applied to a much larger data set using large-scale
data processing solutions such as MATLAB Parallel Server™ and Apache™ Hadoop^®. Both technologies enable you to cope with data that exceeds the amount of memory available on a single machine.
The context in which the algorithm is deployed depends largely on the application use cases. Fund-of-funds managers working mostly with Excel might prefer to deploy the algorithm as an Excel add-In.
They could use the module to investigate funds under consideration for future investments. Regulatory authorities could integrate a fraud detection scheme into their production systems, where it
would periodically perform the analysis on new data, summarizing results in an automatically generated report.
We used advanced statistics to compute individual fraud indicators, and machine learning to create the classification model. In addition to the bagged decision trees discussed here, many other
machine learning techniques are available in MATLAB, Statistics and Machine Learning Toolbox, and Deep Learning Toolbox™, enabling you to extend or alter the proposed solution according to the
requirements of your project.
^1 Eurekahedge
^2 Bollen, Nicolas P. B., and Pool, Veronika K.. “Suspicious Patterns in Hedge Fund Returns and the Risk of Fraud”(November 2011). https://www2.owen.vanderbilt.edu/nick.bollen/
Published 2014 - 92196v00
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JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )
CHARACTER JOBZ, UPLO
INTEGER INFO, KD, LDAB, LDZ, LIWORK, LRWORK, LWORK, N
INTEGER IWORK( * )
REAL RWORK( * ), W( * )
COMPLEX AB( LDAB, * ), WORK( * ), Z( LDZ, * )
CHBEVD computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm.
The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard
digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.
JOBZ (input) CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
KD (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB (input/output) COMPLEX array, dimension (LDAB, N)
On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if
UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, AB is overwritten by values generated during the reduction to
tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal
of T are returned in the first two rows of AB.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KD + 1.
W (output) REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) COMPLEX array, dimension (LDZ, N)
If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. If N <= 1, LWORK must be at least 1. If JOBZ = 'N' and N > 1, LWORK must be at least N. If JOBZ = 'V' and N > 1, LWORK must be at least 2*N**2. If LWORK = -1,
then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK
arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
RWORK (workspace/output) REAL array,
dimension (LRWORK) On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
LRWORK (input) INTEGER
The dimension of array RWORK. If N <= 1, LRWORK must be at least 1. If JOBZ = 'N' and N > 1, LRWORK must be at least N. If JOBZ = 'V' and N > 1, LRWORK must be at least 1 + 5*N + 2*N**2. If
LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and
IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK (input) INTEGER
The dimension of array IWORK. If JOBZ = 'N' or N <= 1, LIWORK must be at least 1. If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N . If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK
or LRWORK or LIWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero. | {"url":"https://manpages.org/chbevd/3","timestamp":"2024-11-14T20:09:46Z","content_type":"text/html","content_length":"35628","record_id":"<urn:uuid:83bf33f4-6958-480a-843c-861046310242>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00005.warc.gz"} |
Homophones and homographs in context of deduplication ratio
31 Aug 2024
Title: Exploring the Impact of Homophones and Homographs on Deduplication Ratio: A Linguistic Perspective
Deduplication is a crucial process in information retrieval, aiming to eliminate duplicate records or documents. However, the presence of homophones (words that sound alike but have different
meanings) and homographs (words that are spelled alike but have different meanings) can significantly affect the deduplication ratio. This article delves into the linguistic aspects of homophones and
homographs in the context of deduplication, proposing a novel formula to estimate their impact on the deduplication ratio.
Homophones and homographs are two types of linguistic phenomena that can lead to errors in information retrieval systems. Homophones, such as “to”, “too”, and “two”, have different meanings but
identical pronunciations, while homographs, like “bank” (financial institution) and “bank” (riverbank), share the same spelling but differ in meaning. These ambiguities can result in incorrect
deduplication, leading to a lower deduplication ratio.
Theoretical Background:
Let’s denote the set of documents as D = {d1, d2, …, dn}, where each document di is represented by its unique identifier and content. The deduplication process aims to eliminate duplicate records,
resulting in a reduced set of unique documents, denoted as U ⊆ D.
Homophones and Homographs Impact on Deduplication Ratio:
The presence of homophones and homographs can lead to incorrect deduplication, affecting the deduplication ratio. Let’s define the following variables:
• H: Set of homophones
• G: Set of homographs
• D’: Set of documents with homophone or homograph errors
The impact of homophones and homographs on the deduplication ratio can be estimated using the following formula:
Deduplication Ratio = ( U - D’ ) / D
where U is the number of unique documents, D’ is the number of documents with homophone or homograph errors, and D is the total number of documents.
The impact of homophones and homographs on the deduplication ratio can be quantified using the following formula:
Impact = ( H ∩ D + G ∩ D ) / D
where H ∩ D is the number of documents containing homophones, G ∩ D is the number of documents containing homographs, and D is the total number of documents.
This article has explored the impact of homophones and homographs on deduplication ratio from a linguistic perspective. The proposed formula provides a novel way to estimate the effect of these
ambiguities on information retrieval systems. By understanding and addressing the challenges posed by homophones and homographs, we can improve the accuracy and efficiency of deduplication processes,
ultimately leading to better information retrieval outcomes.
Related articles for ‘deduplication ratio’ :
• Reading: Homophones and homographs in context of deduplication ratio
Calculators for ‘deduplication ratio’ | {"url":"https://blog.truegeometry.com/tutorials/education/c623030b5478bd0b42a7ad8c5cb3a0d4/JSON_TO_ARTCL_Homophones_and_homographs_in_context_of_deduplication_ratio.html","timestamp":"2024-11-12T07:16:03Z","content_type":"text/html","content_length":"17100","record_id":"<urn:uuid:93971a9b-6136-4ab4-9fe3-8ed324a70658>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00667.warc.gz"} |
i.e. weak with
An Hv – interview, i.e. weak with Th. Vougiouklis
Interviewer N. Lygeros
January 2005, Xanthi, Greece
Lygeros N.
Could you please tell me about your first contact with the concept of hypergroup.
Vougiouklis Th.
In late 70s I was a fresh researcher at the Democritus University of Thrace working on various subjects including Lie Algebras and Astronomy. That time L. Konguetsof was appointed Professor of
Mathematics in our Department and we all adapted our research interests accordingly. He came from Canada, he had spent a long time of study and teaching in France and Belgium and his object was the
Didactic of Mathematics. However, as I was interested in Algebra, he told me that his Thèse D’ Etat was in Algebra and on certain rings, or, more precisely, on structures with fewer axioms called
annoides. These annoides are related to a structure on which he had worked called hypergroup. He had studied hypergroups with late Professor M. Krasner, his supervisor in France. It was then I heard
about Krasner, a great mathematician, a multifarious personality with many research interests, one of which was the hypergroups. I liked this notion of hypergroup and Konguetsof showed me the book
Algébre, Vol.I, 1963 by P. Dubreil. There on page 167, on a footnote, there was the definition of hypergroup in the sense of Frèdèric Marty. So, it was then I saw, for the first time, the name of F.
Marty who, in an announcement in the 8^th Congress of Scandinavian Mathematicians in 1934, had given this renowned definition. However, it was a long time till I was able to get hold of this
paper. In this work there exists a motivation example for this structure, which is the quotient of a group by any subgroup which is not an invariant subgroup.
Lygeros N.
Therefore that was your first contact with hyperstructures.
Vougiouklis Th.
Exactly. There, I saw and understood why Marty used the reproduction axiom instead of the two axioms: the existence of neutral element and the existence of inverse element. He set the hypergroup free
from the obligation to have neutral element and he gave the possibly most widely used definition. It was, I would like to stress, an inspired action. A brilliant definition!
Lygeros N.
Yes, if one wishes to generalize the axioms, he would have to put the neutral element and the reverse elements.
Vougiouklis Th.
In my opinion Marty, managed to do the greatest generalisation anybody would ever do, acting as a pure and clever researcher. He left space for future generalisations “between” his axioms and other
hypergroups, as the regular hypergroups, join spaces etc. The reproduction axiom in the theory of groups is also presented as solutions of two equations, consequently, Marty got round that hitch,
Lygeros N.
That is to say that Marty fixed a total process and not a partial one. It was that time you manage to get hold of Marty’s paper then?
Vougiouklis Th.
No, not yet. It was not easy for someone to find a paper at that time, because there was not Internet in 1977-78. Even worse, it was not published in a Journal but in the Proceeding of a Congress.
Later I found out more about Marty.
Lygeros N.
So you did not know the ideology of Marty and why he has chosen it to give this definition? You just saw a “dry” definition via the Dubreil.
Vougiouklis Th.
Yes, then I found out that we had the possibility of finding papers via a service called Lending Division and, for a small consideration, we could have photocopies of several papers, mainly from
periodicals and journals.
Lygeros N.
Did Konguetsof not have the article? Had he seen it?
Vougiouklis Th.
I do not know whether he had seen the paper in the decade of ’60s and if he had used it. Nevertheless, when I asked him, he did not have it. Konguetsof knew hyperstructures via Krasner, but the
subject of hyperstructures was not the subject of his thesis. Even later, I found out that Krasner had worked on this object in early ’40.
Lygeros N.
How did you begin your research?
Vougiouklis Th.
I started the research with two objectives. The first thing was to find who had dealt with hyperstructures, who was still working on the object, which problems had been solved and which still
remained unsolved. Second, I had to look at the structure itself. Moreover, I had to find examples and to prove properties. You can see that this procedure included great risk because, somebody might
have already found them and they had been studied in-depth. What I wanted was to enter in the body of this structure via that procedure.
Lygeros N.
However, did you have a concrete way of research?
Vougiouklis Th.
No, not at all. I wanted to make examples, to conquer the structure and afterwards possibly to work on them. The possibility of finding the clew of the hyperstructures was also the Mathematical
Reviews and the Zentralblatt für Mathematik, where reviews of papers were likely to be found. Again, however, I did not know names and the fields where they could be written likely reviews. You see,
the term hypergroup is also used elsewhere as in harmonic analysis, which, however does not have any relation with multivalued operations. Searching, I discovered H.S. Wall, 1937, R. Dacic,
1969, M. Koskas, 1970, Y. Sureau, 1977. I asked papers from some of them and I this is how I slowly started entering the community of hypergroupists. Later, towards the end of my thesis, I found out
about P. Corsini, who had already founded a school of hypergroupists in Italy and who, interestingly, had begun the research on hyperstructures reading a paper of J. Mittas. You know, Mittas was here
in Greece, in Thessaloniki but I did not know his object. Via Sureau I found out that Krasner was still alive and had co-operators in Athens and in Patras. Then I also came to know their research
fields and what they had discovered and proved till then. The system I mainly worked in was as follows: I had a pre-fabricated letter that said “I request to send me the paper ” and any of your
related papers”. Most of the researchers responded with pleasure. Afterwards I investigated whether hyperstructures was their main research interest or they had just happened to have worked on them
only casually.
Lygeros N.
So this is how the filing of the papers started.
Vougiouklis Th.
Yes. I gathered the papers and I classified them. I found the paper of Wall’s when I was about to finish my PhD thesis and it was there I saw some of the basic definitions of cyclicity. Fortunately
not all of them! For example, there were not the definitions of the set of generators, the single power cyclicity etc, which terms, however, do not exist in the classical structures.
Lygeros N.
Does the term renaissant come from Mittas?
Vougiouklis Th.
The term renaissant was introduced by Mittas in 1984, so it appears 4 years after my PhD thesis. The renaissant is a special case of the single power cyclicity.
Lygeros N.
Yes, each element is a generator: it produces the hypergroup.
Vougiouklis Th.
I had, as Wall also did, the possibility of having generators or not, as well as to have generators with different periods. That time I introduced, for example, the single power cyclic hypergroups,
as I named them, one power can cover the whole hypergroup. It is also possible to have generators with greater period from the order of the group. That time I had also made a list of hypergroups with
two elements where certain new elements can already be identified.
Lygeros N.
We also observe that having the cyclicity, we also obtain commutativity. This is interesting because we have the mentality of Gröthendick according to which, when we want to prove something, we
should generalise it, keeping the structure that we have it in as a case. Here it is proved, as a simple remark, that these things are not in effect simple and one can see them only in the list of
the 8 non-isomorphic hypergroups with two elements.
Vougiouklis Th.
Here, I would like to report that my good example of this is what I introduced with the name P-hyperoperations. Already, certain theorems and terms had been presented in the list of 8 small
hypergroups. Up to that moment a great number of researchers had been involved in almost all fields of research on hyperstructures. Characteristic examples include the P-hypergroups, P-hyperrings,
P-hyperfields, P-hyper vector spaces, P-H[v]-structures, P-lattices etc. Then, when we use the cyclicity in the P-hyperoperations, that is to say taking a group or a semigroup and a subset, we have
an enormous crowd of classes of hypergroups that present terms and theorems which could not exist in the classic theory. In my thesis, however, I studied mainly the P-hypergroups where I have sets
that contain only one element and the neutral element. Moreover, at that time, in Xanthi, at the Democritus University, we had a computer Univac and using this we found all the generators and their
order in groups up to order 40. Using this result, there were verified certain theorems that I had already proved. Most important, however, is that certain other resulted, from which I was able to
prove the most; however, some of them are still open problems to be researched. In these cases we know the period of elements which are generators but we can not prove the related theorems.
Lygeros N.
Who helped you with the computer scientific part and why up to 40?
Vougiouklis Th.
In the computer scientific part, the program was made by D. Diamandidis with my help in the presentation of the problems, of course, and it was 40 because there was enough paper only for such table.
That was completely accidental!
Lygeros N.
Up to the development of your thesis, it appears that you were somehow isolated.
Vougiouklis Th.
Not somehow. Completely! I would say, and it is most important, that I did not have any confirmation of what I wrote.
Lygeros N.
Single element! Does a change of phase before and afterwards your thesis exist? How did you notify your thesis?
Vougiouklis Th.
Firstly, two of my papers, emanating from the thesis, were presented by Konguetsof in Czechoslovakia and were published there. That time there was also a new contact. In the audience, there
was Professor Drbohlav who had a paper published on hyperstructures since 1956. He is another example of independent hypergroupist.
Lygeros N.
When did you find out about Corsini?
Vougiouklis Th.
I found out about Corsini after my thesis, via Mathematical Reviews. I wrote to him and I got information about his field of research and almost all about his school. I also sent him my papers. In
early ’80’s, some Greek mathematicians knowing my dealing with hypergroups, put me in contact with in Krasner. Similarly, the President of the Greek Mathematical Society, S. Zervos, had
invited K. Kuratowski who also gave a lecture in Xanthi and I came to know him, as well. When Zervos invited Krasner in Athens in 1981, he also invited me to meet Krasner and so it happened: I
met Krasner and I explained my work to him. Krasner expessed a great interest in my work, especially for the last chapter of my thesis, which was about the associated relations of equivalences and
various theorems of quotients in hyperstructures. There were also theorems, generalisations, isomorphisms between hypergroups. Mainly, there where theorems on isomorphisms using commutative diagrams.
Lygeros N.
Kuratowski was a great mathematician and, apart from topology, where he was one of the founders, he had also formulated fundamental theories in the theory graphs which I see used in hyperstructures
Vougiouklis Th.
Let’s return to Krasner! It was then I realised that Krasner was the one who, in the ’40’s had taken Marty’s hyperstructures, transported them, enriched them with a lot of conclusions and taught them
to his students. The traces of Marty disappear in the Second World War and, apart from some researchers who have dealt with few papers in hyperstructures, Krasner is the one who has transferred them
into our days. Knowing Corsini’s School I found a host of mathematicians dealing with the object. I found a large number of papers I read, I came to know the interests of those researchers and their
research results. Nevertheless, I am still in the phase of reading, not yet in co-operation with them. In 1981, there was an interruption of my research on hyperstructures because I took an
educational leave for USA. Till then, I had not understood the difference between the term of hypergroup in the harmonic analysis from the term of hypergroup in the multivalued operations. So, I went
to the Massachusetts Institute of Technology (MIT), invited by Professor S. Helgason who had been working in hypergroups, but in the sense used in harmonic analysis. During the one and half year time
that I was in the USA, I worked in my old topic, the topic of Lie Algebras, mainly in the topic of Kac-Moody Infinite Dimensional Lie algebras. I co-operated with one of the founders of these
algebras, V. Kac. What I had in mind at that time was to connect these objects which I finally managed to do, after 10 years, with some papers using the gradation of these algebras. More precisely,
taking the gradation which exists in these Lie algebras, I introduced a special hyperoperation between the elements of these Lie algebras of infinite dimension and studied the properties of that
Lygeros N.
Where there were many researchers who tried to combine hypergroups in the different regions as with harmonic analysis?
Vougiouklis Th.
Of course, as Spector, Campaigne and others, but this effort did not have duration. In the library of M.I.T., I began to search for hyperstructures independent from the channels of journals, but also
search journals that I had not had any access to, during old days. I discovered a big number of papers and researchers in hyperstructures.
There, in the basement floor, the beams of the sun struggling to enter through a small skylight shed the light of knowledge as, to my utmost happiness, I saw, covered in dust, the volume of the
proceeding of the first paper of Frederic Marty! It was a great discovery, the most exciting one.
I also discovered other papers including the Comptes Rendus of the French Academy of Science by Mark Krasner since the year 1940 etc
Lygeros N.
Was there the definition of the hyperring in the sense of Krasner?
Vougiouklis Th.
No, that hyperstructure was presented during a talk by Krasner in the University of Athens in 1956. He gave the definitions of a hyperfield and of a hyperring where the multiplication was an
operation and released the addition making it into a hyperoperation. In this type of hyperring, that now we call additive hyperring, the hyperaddition gives a kind of hypergroups which are called
canonical and they were studied mainly by Mittas and his school. Returning from USA I had a big bibliography but, more important, I was in contact with the scientists working in this field.
Lygeros N.
Then, did you come in contact with Mittas?
Vougiouklis Th.
Of course with Mittas in Thessaloniki as well as with his co-operators M. Serafimidou, C. Serafimidis, S. Ioulidis, who I had already met in the past but I did not know that they had been
working in the same field. I had, therefore, two research objects: first, the Lie algebras of infinite dimension, for which, however, I could find any co-operators in Greece, as there were not any,
and second the hypergroups, in which I was almost completely informed. The experience I had in the topic of Lie algebras helped me a lot in the theory of quotients. I came, that is to say, from
the USA charged with quotients. Moreover, I had some knowledge on the representation theory of Lie algebras. For this reason I made an effort to transfer some of these objects in the topic of
hyperstructures. I can now say hyperstructure as I escaped from hypergroups because the theory of representation requires hyperrings, hyperfields hyper-vector spaces, hypermatrices etc, objects that
I was forced to define and to develop. I virtually tried to create the theory of all representations of hyperstructures and the involved hyperstructures, moreover within their general form. I
discovered the generalised hyperrings and the generalised hypermatrices. I then found also some classes of hypergroups that can be represented by hypermatrices as the S-hypergroups and the very thin
Lygeros N.
How did you enter in the concept of the fundamental relations?
Vougiouklis Th.
In the effort of generalising the representations and in the search of new classes of hyperstructures, I always kept in mind that these should be connected with classic structures. I wanted the
classic structures to be contained as sub-cases in the corresponding hyperstructures. Then I dealt with the fundamental relations and for a moment I believed that I discovered them, because I had not
realised that they had been introduced by Koskas in 1970 and they had been studied by Corsini’s school with a slightly different approach. I speak of relations β and β*. With regard to this
subject, the following strange story happened: I was working far from the concept of hypergroups only. For this reason I was able to define the relations γ* in hyperrings and ε* in the hyper-vector
spaces, I avoided to use δ that is reported as delta of Kronecker. I used Greek letters internationally as Koskas used the letter β for the first time. On the other side, I made a proof for the β* in
a deductive rather than an inductive way and, furthermore, the proof was very short. In a similar way, I used short proofs for γ* and ε*, while traditionally, the inductive way required much longer
proofs. Finally, I gave the name fundamental to them and this is the name widely used today. All these are presented and used as fundamental theorems in the theory of representations, as they finally
appeared in my book in 1994. However, my first conclusions were in a paper in 1988 titled: How a hypergroup hides a group. I had presented them in a congress in Italy and there I saw that a great
part of work on this topic had already been done by Corsini’s school. The whole process proved once more that in research while proving and developing a subject, one follows a way other than the
classical, he may be led to revolutionary changes. The reverse ways of definition of β* fundamental relation, the γ* and ε*, now are always given as follows: they are the relations which are the
minimal equivalence relations, so that the quotients by them are the corresponding algebraic (classic) structures.
Lygeros N.
If we compare the two ways of definitions and proofs, the way that you used is more direct, more explicit.
Vougiouklis Th.
So it appears to me, but, honestly, I can not compare these two ways because the other proof was the first one for the relation β*. For me, my way is easy, after that I used this way of proof for the
relations γ* and ε*, almost automatically. This study can be used up to the class of algebras.
Lygeros N.
Does the problem with hyperfields also appear here?
Vougiouklis Th.
It was not the target, but I began to deal with it. However, I was not able to define the general hyperfield. For this reason in my various lectures, mainly in Italy, I placed this problem as an
unsolved problem, i.e. find the definition of the generalised hyperfield. It was very big problem to determine the unit element with respect to addition. I came back, therefore, to the problem of
reproduction. That period researchers in Italy were dealing with the multiplicative hyperring, where only the multiplication is a hyperoperation. R. Rota,
R. Procesi-Ciampi, G. Tallini, M. Scafati-Tallini, extended their results, up to our days, in hypervector spaces. The other Italian school, i.e. R. Migliorato, M. De Salvo, D. Freni, when they saw
that the subject of β* could be seen in another general way. They directed their research into other relations weaker than β* and β. The problem β=β* for hypergroups was still open but as it was
proved by Freni in 1990, it was closed. However, the research was transferred into the other fundamental relations and more precisely into classes weaker than the β. I would like to point out that in
1985, in a congress on hyperstructures, a pre-AHA I would say, all the hypergroupists gathered and we placed problems we had on the topic. It was a very decisive congress because we started to
research in new sectors. During this congress, I was assigned to organize the following congress in Greece.
Lygeros N.
What post did you have that time in Greece?
Vougiouklis Th.
I was a lecturer. That period there was effort by some researchers to find connections of the hyperstructures with other topics. Thus, for example we have the congresses of Combinatorics where some
hypergroupists participated and presented several papers. In Greece there were some teams of hypergroupists, for example, in Thessaloniki there were Mittas, Serafimidou, Serafimidis, Ioulidis, who
were dealing with various fields such as canonical hypergroups, hypertreillis, etc; in Thrace: Vougiouklis, Kongeutsof, Spartalis dealing with generalised hyperrings or special hyperrings and
hypergroups; in Patras: Stratigopoulos and in Athens Massouros, Pinotsis, Giatras etc. I do not know if I forget someone.
Lygeros N.
You forget the hyperfield!
Vougiouklis Th.
No I do not forget the hyperfield. This problem was still open, I was not able to define it. That time, however, a reverse question was being formed in my mind: �which hyperstructures have
fundamental relations?�, that is to say I saw the problem in reverse. This is how the idea of the weak hyperstructures came. I remember very well that moment. I introduced and denote the weak
associativity by WASS, the weak commutativity by COW etc. The basic point of the theory I created was that I replaced the equality of sets by the not empty intersection. The fundamental
relations β*, γ*, ε* still have structures as minimal quotients. Almost the entire fundamental theory is valid. These structures took my name, in a subscript, and they are called H[v]-structures,
although, in the beginning I had given a name by virtue, because, as quotients, they contain the classical structures. The definitions and the first conclusions were presented at the 4^th AHA
Congress in Xanthi in 1990. The first applications were presented by other researchers in the same congress. It opens an enormous field and I found myself in a chaos of hyperstructures. I did not
know how to control them! This, perhaps, is a delusion, because when you make an opening as a researcher, you have also the ways to curb this structure. In the same congress, however, I also gave an
important answer, at least for me. It was the definition of the generalised hyperfield. Here it is: each hyperring has a quotient with respect to the fundamental relation γ*, always a ring, if the
quotient is a field, then the initial hyperstructure is defined as the generalized hyperfield. Apparently, that was another inversion. Consequently, I solved the open problem of the definition of the
most general hyperfield which I stated several years earlier in the community of the hypergroupists.
Lygeros N.
This is a big innovation.
Vougiouklis Th.
Yes, at the same time I defined the smaller and greater H[v]-structure and there exists the simplest and most useful theorem in this theory that says that bigger hyperstructures than those which are
WASS and COW, are also WASS and COW, respectively. From that time and afterwards more than 200 papers in H[v]-structures were published in international journals and proceeding. Moreover, they also
have applications in other sectors of mathematics as well as in other, not purely mathematic topics, as, for example, in conchology.
Lygeros N.
Do you think that there exists a tendency to create some kind of file with articles of the above research topic?
Vougiouklis Th.
I put all the papers that I had in my file; independently I had used them or not, in the proceeding of the 4^th AHA, in the booklet of summaries and later in the proceeding of the congress, published
in 1991 by World Scientific.
Lygeros N.
I would now like to clarify the term very thin: how did you think about it, moreover how is it developed up to now?
Vougiouklis Th.
I discovered the very thin, in my effort to find examples on the topic. It was the time I was trying to understand the concept of the hyperstructure that time some ideas occurred to me. This idea is
the step that leads from the usual structures to hyperstructures; actually, it is the verge between structures and hyperstructures. I introduced and named this turning point very thin. From this
point on, we suppose that anyone has to deal with hyperstructures.
Lygeros N.
You made a transport in hyperstructures.
Vougiouklis Th.
In the beginning I defined the term very thin in hypergroups and for this reason my first results on questions were which hypergroups are very thin and how we can construct them. I proved the
relative theorem, virtually a construction in which we have a structure of a group and then we attach an extra element in a concrete way. That time, I had not seen that such a construction could be
used itself for enlarging hyperstructures, which in fact I did 15 years later.
Lygeros N.
Which team of researchers have been dealing with very thin?
Vougiouklis Th.
I had dealt with the finite case and the problem had almost been solved for me. However, in the infinity case, the problem was presented by a couple of researchers, Cornelia and Marin Gutan. They are
Romanians but they have actually worked in France. In addition, some other researchers from Italy have been deal with the very thin.
Lygeros N.
Hence, this subject was completed in the sector of infinity.
Vougiouklis Th.
They have presented some very interesting results but I cannot say that I now know all their results. My idea was simply to present an algebraic hyperstructure. Later the subject of very thin was
extended in other hyperstructures as hyperrings etc and they changed their way of use. However, I think that it is still an important topic with some interesting problems because it connects the
classical algebraic theory with the hyperstructures. Note that C.Gutan has obtained her PhD in France working on this subject.
Lygeros N.
I remember that the theorem in finite hypergroups, where it acts as characteristic case, says that there are only two classes of very thin hypergroups.
Vougiouklis Th.
I do not remember the construction theorem. There is an additional element which is considered in one case but not in the other.
Lygeros N.
There is a characterization with which we know precisely the attributes of unique element. Remaining in the finite case and when we pass into hyperrings, there you had found five families and an open
question exists whether these are the only very thin families. With regard to the first direction for the minimal and the finite case you deal with the small sets a little.
Vougiouklis Th.
It is reasonable, when somebody begins with an object to seek examples in small sets. The team of Italians are making efforts on the topic but the difficult attribute is the associativity,
specifically in the finite case, which involves a big number of attempts while the infinite case is obtained with a general proof of some attribute.
Lygeros N.
In substance, the first classification of Vougiouklis is for order 2.
Vougiouklis Th.
When we have the WASS property and we take bigger hyperoperations then the WASS property is still valid
Lygeros N.
Have you defined the term minimal?
Vougiouklis Th.
Yes. The term minimal has a meaning mainly in H[v]-structures. If these are WASS hyperstructures, as we usually do in hypergroups, we try to find hypergroups which generate them, which can produce
Lygeros N.
Let us come back to the P-hypergroups. For me the following is strange: in order to find the tables until 40 elements, you have used a computer. Why have not even used a computer in the case of
hypergroups with neutral element? That is the easier case.
Vougiouklis Th.
We used computers in the P-hypergroups defined on groups with neutral element. Hence we used only one additional element. In the treatment in the computer I did not participate, because I did not
know anything about computers. However, I had to participate in the process of controlling the results. I remember that we made various tricks in order to bring the computer to compute and to give
correct results.
Lygeros N.
We are in 1980.
Vougiouklis Th.
No, it was before 1980. In 1980 they had been completed.
Lygeros N.
Did the computer specialist who helped you, also helped you in any other sector?
Vougiouklis Th.
When I finished my thesis I saw that that was a field that opened also other sectors.
Lygeros N.
How do you see the future in hyperstructures and in hypergroups? Which direction takes the research in this sector? We said for the background, for certain concepts that are important, fundamental
relations, bigger, classification, infinity, very thin etc, what is the future in this sector?
Vougiouklis Th.
There are researchers dealing with the process of recognition of the fundamental structures. There are also certain researchers who tried to define hyperstructures and to explain their meaning in the
other sectors of applications. A characteristic example is the topic of fuzzy sets.
Lygeros N.
Is this also valid for the Iranian school?
Vougiouklis Th.
The Iranian school have a lot to do with the direction of fuzzy. That is to say, they introduce hyperstructures of this type and they study them in depth. The big wave of Iranian researchers is due
to the enormous crowd of applications of the fuzzy sets nowadays.
Lygeros N.
Who saw for the first time fuzzy as hyperstructures?
Vougiouklis Th.
Corsini was the one who made the first effort, but it appears the phenomenon that this moment, the 80% of efforts comes from Iranians who had formed the biggest teams in hyperstructures working in
the fuzzy topic.
Lygeros N.
What is the reason such a thing happened?
Vougiouklis Th.
I assume that this is so because the founder of the fuzzy sets is the Iranian L. Zadeh, who was the first to define it in a paper in 1965. The Iranian school has the fuzzy as the mayor target.
Lygeros N.
Do any other research teams we have not discussed in this interview exist? French, Italian, Greek?
Vougiouklis Th.
When I saw your own work, it immediately reminded me of the presentation of papers of the French School where the subject is presented thoroughly and in depth in a strict and profound mathematic way.
It is the way that the Bourbaki’s also work. It is not the same way in which today we present the results wishing to share them immediately with the mathematic community. That is why I have
classified you in the French School, even if you use computers, which was not then used. This way of presentation is an entirely philosophical attitude. It is also connected a lot to the theory of
categories and the theory of representations and in other sectors of mathematics where there are also efforts made of introducing a single theory or for the mathematics to be seen in an overall
algebraic way.
Lygeros N.
As Carathèodory did with somas?
Vougiouklis Th.
Indeed, Carathèodory took this basic element, the somas, and began a series of papers. He created an algebraic theory that unfortunately he did not succeed to complete as he died in 1950. He had,
however, had taken notes and his co-operators took these notes and they published the book 6 years after his death. I do not know if Carathèodory, when he created this work, knew the existence of
certain other theories. The theory of hyperstructures did not exist really then or, at least it was not known. But I believe that if Carathèodory had known the theory of hyperstructures, if he had
heard about this subject, it would have been much easier to put a triangle with different ways to have different results. This is a motivation to the researchers to take this idea of somas and to
develop them in combination with hyperstructures. | {"url":"https://lygeros.org/interview_hv_vougiouklis/","timestamp":"2024-11-13T21:54:12Z","content_type":"text/html","content_length":"90449","record_id":"<urn:uuid:f64fbff9-6d40-4e86-831b-f7f9efff68ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00621.warc.gz"} |
Draw The Lewis Structure For Bro3-
Draw The Lewis Structure For Bro3- - 7 + 3 (6) + 1= 26 valence electrons. Draw the lewis structure for bro3 check all that apply. Web this problem has been solved! Web drawing lewis structures for
molecules with one central atom: Drawing lewis structures for bf3, pf3 and brf3;
Web a video explanation of how to draw the lewis dot structure for the bromate ion, along with information about the compound including formal charges, polarity, hybrid orbitals, shape, and bond.
You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Adding them up, we get a total of 26 valence electrons. #1 draw a rough skeleton structure #2 mention
lone pairs on the atoms #3 if needed, mention formal charges on the atoms #4 minimize formal charges by converting lone pairs of the atoms, and try to get a stable lewis structure #5 repeat step 4
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Bro3 Lewis Structure
Breaking the octet rule ; Adding them up, we get a total of 26 valence electrons. Web chemistry chemistry questions and answers draw the lewis structure for the bromate ion (bro3−)with minimized
formal charges. While selecting the atom, always put the least electronegative atom at the center. For the hbro3 structure use the periodic table.
BrO3 lewis structure, molecular geometry, bond angle, polarity, electrons
Adding them up, we get a total of 26 valence electrons. Calculate the total number of valence electrons. You'll get a detailed solution from a subject matter expert that helps you learn core
concepts. Breaking the octet rule ; Web drawing lewis structures for molecules with one central atom: Indicate the total number of valence.
Bro3 Lewis Structure
Web drawing lewis structures for molecules with one central atom: Valence electrons are the outermost electrons of an atom and are involved in bonding. Breaking the octet rule ; Calculate the total
number of valence electrons. There are seven valence electrons in each bromine atom and six valance electrons in each oxygen atom. You'll get.
BrO3 (Bromate ion) Molecular Geometry, Bond Angles YouTube
D) indicate the approximate bond angles for each ion e) determine the names for the molecular geometry. Web drawing lewis structures for molecules with one central atom: At first we have to calculate
the valence electrons in bromate ion. For the hbro3 structure use the periodic table to find the total number of valence electrons.
[Solved] Draw resonance Lewis structures for bromate(V), [BrO3] −
Breaking the octet rule ; First, we determine the total number of valence electrons in the molecule. 7 in a lewis structure for a neutral molecule, the. Adding them up, we get a total of 26 valence
electrons. Give the name of the ion. Web a) draw all the possible lewis structures for the following.
SOLVED Draw the Lewis structure for BrO3 Check all that apply 2 B=e B
First, we determine the total number of valence electrons in the molecule. Calculate the total number of valence electrons. Indicate the total number of valence electrons b. You'll get a detailed
solution from a subject matter expert that helps you learn core concepts. How many total equivalent likely resonance structures exist for bro3 ?? Breaking.
Pin on Latest
So, total valence electrons will be: Give the name of the ion. #1 draw a rough skeleton structure #2 mention lone pairs on the atoms #3 if needed, mention formal charges on the atoms #4 minimize
formal charges by converting lone pairs of the atoms, and try to get a stable lewis structure #5 repeat.
BrO3 Lewis Structure (Bromate Ion) YouTube
So, total valence electrons will be: Web chemistry chemistry questions and answers draw the lewis structure for the bromate ion (bro3−)with minimized formal charges. #1 draw skeleton #2 show chemical
bond #3 mark lone pairs #4 calculate formal charge and check stability (if octet is already completed on central atom) #5 convert lone pair and.
Bro3 Lewis Structure
Web a) draw all the possible lewis structures for the following ions: Using formal charges to determine how many bonds to make, a different perspective. You'll get a detailed solution from a subject
matter expert that helps you learn core concepts. Web this problem has been solved! While selecting the atom, always put the least.
BrO3 Lewis Structure How to Draw the Lewis Structure for BrO3 YouTube
Indicate the total number of valence electrons b. You'll get a detailed solution from a subject matter expert that helps you learn core concepts. So, total valence electrons will be: You'll get a
detailed solution from a subject matter expert that helps you learn core concepts. How many total equivalent likely resonance structures exist for.
Draw The Lewis Structure For Bro3- And this is an anion that carries one negative charge. 7 + 3 (6) + 1= 26 valence electrons. Draw the lewis structure for bro3 check all that apply. Calculate the
total number of valence electrons. You'll get a detailed solution from a subject matter expert that helps you learn core concepts.
You'll Get A Detailed Solution From A Subject Matter Expert That Helps You Learn Core Concepts.
This problem has been solved! Describe the preferred lewis structure. For the hbro3 structure use the periodic table to find the total number of valence electrons for the hbro3 molecule. Web this
problem has been solved!
Calculate The Total Number Of Valence Electrons.
Here’s the best way to solve it. 7 in a lewis structure for a neutral molecule, the. Drawing lewis structures for bf3, pf3 and brf3; You'll get a detailed solution from a subject matter expert that
helps you learn core concepts.
First, We Determine The Total Number Of Valence Electrons In The Molecule.
Give the name of the ion. Using formal charges to determine how many bonds to make, a different perspective. How many total equivalent likely resonance structures exist for bro3 ?? Web drawing lewis
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Web A) Draw All The Possible Lewis Structures For The Following Ions:
Indicate the total number of valence electrons b. Web chemistry chemistry questions and answers draw the lewis structure for the bromate ion (bro3−)with minimized formal charges. Different factors
can affect the polarity of the molecule or atom. 2.7m views 11 years ago.
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Solving Trigonometric Equations Worksheet Kuta - Equations Worksheets
Solving Trigonometric Equations Worksheet Kuta
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Non-Linear Programming: Gradient Descent and Newton's Method - KZHU.ai π
Non-Linear Programs
When visualizing a linear program, its feasible region looks like a polygon. Because the objective function is also linear, the optimal solution is on the boundary or corner of the region.
But a non-linear program is quite different, its feasible region may be in any shape, moreover the optimal solution may not exist on the boundary of the feasible region. Searching the whole feasible
region does not work particularly when there are hundreds or thousands of variables.
A better idea is to use derivative at a point to tell us what is gonna happen if we move in certain direction. Suppose a non-linear objective function y = y(x) with only one variable x. At point x
[0], if the derivative is positive, we will know y will decease if x decreases, and y will increase if x increases. Then ask the question: around all my feasible directions, where should I go to
reach my target as soon as possible? The answer really depends on the derivative.
The program we want to solve usually contains multiple variable, so a single derivative is not enough, we need derivatives along all the directions, which is the concept of gradient from vector
calculus. Gradient descent is a very popular algorithm of solving non-linear problems. Gradient descent is actually first-order approximation, Another more powerful method is Newton’s method, which
is essentially second-order approximation.
We rely on numerical algorithms for obtaining a numerical solution. Solving a nonlinear program usually in the following ways:
Iterative In many cases we have no way to get to an optimal solution directly, we need to move to a point in one direction and then starts the next iteration search starting from that point.
Repetitive In each iteration, we repeat the same steps.
Greedy In each iteration, we seek for some best thing achievable in that iteration.
Approximation Rely on the first-order or second-order approximation of the original program.
Nonlinear programs are harder than linear programs, so finding solutions for them may fail due to many reasons:
1. Fail to converge to a single point, you won’t get a feasible solution at all.
2. Trapped in local optimum, which means starting point matters.
It is usually required that the objective function’s domain is continuous and connected. If that is not true, we are probably dealing with nonlinear integer program, which is too hard to solve –
gradient descent and Newton’s method will fail, more advanced topics like Convex Analysis may be required.
Unconstrained non-linear programs are in some sense easier to solve than constrained ones:
min[x β R^n] f(x), where f is a twice-differentiable function
Gradients and Hessians
For a function f: R^n β R, collecting its first- and second-order partial derivatives generates its gradient β f(x) and Hessian β ^2f(x).
Gradient Descent
At any point of the domain of the function x β R^n, we consider its gradient β f(x), which is an n-dimensional vector. Gradient is closely related to slope β f(x)/β x only in 1-dimensional
problem, but this is not true in general. In n-dimensional space, the gradient actually is the fastest increasing direction where if you move along that direction.
f(x + a β f(x)) > f(x), for all a > 0 that is small enough
We may try to “improve” our current solution x by leveraging gradient. Since we are minimizing the function f(x), here “improve” mean we want to get to a lower value of f(x), we first find the
gradient, then we move along the opposite direction.
Suppose current solution is x, in each iteration we update x using gradient β f(x) and step size a, until gradient becomes or very close to 0.
x = x - a β f(x), a > 0
Selecting appropriate value of step size a is critical, a few strategies include:
1. Predefine step size and decreasing it gradually.
2. Look for the largest improvement and then select appropriate value of step size.
Now the pseudo code of the gradient descent algorithm using adapted step size:
1. Set k = 0, choose starting point x[k] and precision parameter Ξ΅ > 0.
2. Find gradient β f(x[k])
3. Solve a[k] = argmin[a β ₯ 0] f(x[k] - a[k] β f(x[k]))
4. Update current solution x[k+1] = x[k] - a[k] β f(x[k])
5. If the norm of the gradient || β f(x[k+1]) || < Ξ΅, exit
6. Increase k = k + 1, and go to step 2
When we want to apply gradient descent in this particular manner, always, we need to formulate a sub-problem of step size a, which is also a minimization problem, finding the first-order condition
such that the step size a can minimize the objective value. We know that what our algorithm is doing is to search only one direction (one variable) at a time, until finally we are getting close to
the optimal solution. When you do gradient descent, you pretty much always go in zig zag with right angles.
Newton’s Method
Newton’s method is similar to gradient descent. In each iteration, it tries to move to a better point. But gradient descent is first-order method and too slow sometimes. Newton’s method is
second-order method which relies on the Hessian to update a solution.
For non-linear equations
Suppose a non-linear differentiable function f, we want to find x* that satisfies f(x*) = 0. Starting from a initial point x[0], consider the linear approximation of the function f(x) at that point.
We are actually using Taylor expansion to create linear relaxation f[L](x[k+1]) of f(x) at point x[k] (moving from x[k] to x[k+1]):
f[L](x[k+1]) = f(x[k]) + f'(x[k]) (x[k+1] - x[k]) = 0
We don’t really directly solve our non-linear problem for this nonlinear equation f(x), maybe we don’t know how to solve it, but for the linear relaxation f[L](x), everything is so easy. Continue the
iteration until either |f(x[k])| < Ξ΅ or |x[k+1] - x[k]| < Ξ΅, where Ξ΅ is a very small number.
For single variate non-linear programs
If we have a twice-differentiable function f(x) where there is a minimum point, at that point we must satisfy the fact that the gradient β f(x) must be 0. We use linear approximation of f'(x), i.e.
the derivative of f(x), to move from x[k] to x[k+1] to approach the root x*.
f'[L](x[k+1]) = f'(x[k]) + f''(x[k]) (x[k+1] - x[k]) = 0
Continue the iteration until either |f'(x[k])| < Ξ΅ or |x[k+1] - x[k]| < Ξ΅, where Ξ΅ is a very small number. Note f'(x^*) does not guarantee a global minimum. It may be a local min, or even a local
Another interpretation is to use quadratic approximation of f(x) at x[k], and move from x[k] to x[k+1] by moving to the global minimum of the quadratic approximation:
x[k+1] = argmin[xβ R] f(x[k]) + f'(x[k]) (x - x[k]) + (1/2) f''(x[k])(x - x[k])^2
This is actually equivalent to solving β f(x) = 0. Note that if your function looks weird, then this approach may fail – what you want is to find a lower objective value, but actually after one
iteration, you get a value higher. Newton’s method does not guarantee you improvement in each iteration.
For nice behavior functions, Newton’s method may be faster than gradient descent, but gradient descent is a much more robust algorithm. Gradient descent always do meaningful search unlike Newton’s
For multi-variate non-linear programs
All we need to do is to generalize the idea of single dimensional derivatives to gradients and Hessians. The quadratic approximation of f(x) at x[k]:
f[Q](x) = f(x[k]) + β f(x[k])^T (x - x[k]) + (1/2) (x - x[k])^T β ^2f(x[k]) (x - x[k])
My Certificate
For more on Non-Linear Programming: Gradient Descent and Newton’s Method, please refer to the wonderful course here https://www.coursera.org/learn/operations-research-algorithms
Related Quick Recap
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Purpose: To validate Gaussian mixture-model with expectation maximization (GEM) and variational Bayesian independent component analysis mixture-models (VIM) for detecting glaucomatous progression
along visual field (VF) defect patterns (GEM–progression of patterns (POP) and VIM-POP). To compare GEM-POP and VIM-POP with other methods.
Methods: GEM and VIM models separated cross-sectional abnormal VFs from 859 eyes and normal VFs from 1117 eyes into abnormal and normal clusters. Clusters were decomposed into independent axes. The
confidence limit (CL) of stability was established for each axis with a set of 84 stable eyes. Sensitivity for detecting progression was assessed in a sample of 83 eyes with known progressive
glaucomatous optic neuropathy (PGON). Eyes were classified as progressed if any defect pattern progressed beyond the CL of stability. Performance of GEM-POP and VIM-POP was compared to point-wise
linear regression (PLR), permutation analysis of PLR (PoPLR), and linear regression (LR) of mean deviation (MD), and visual field index (VFI).
Results: Sensitivity and specificity for detecting glaucomatous VFs were 89.9% and 93.8%, respectively, for GEM and 93.0% and 97.0%, respectively, for VIM. Receiver operating characteristic (ROC)
curve areas for classifying progressed eyes were 0.82 for VIM-POP, 0.86 for GEM-POP, 0.81 for PoPLR, 0.69 for LR of MD, and 0.76 for LR of VFI.
Conclusions: GEM-POP was significantly more sensitive to PGON than PoPLR and linear regression of MD and VFI in our sample, while providing localized progression information.
Translational Relevance: Detection of glaucomatous progression can be improved by assessing longitudinal changes in localized patterns of glaucomatous defect identified by unsupervised machine
Glaucoma is a blinding optic neuropathy that may cause significant visual impairment when left untreated. It is the second leading cause of blindness worldwide.
Detection of glaucomatous visual function defects and detection of their progression are critical for management of the disease. Identifying patterns of visual function defects and tracking their
change over time likely is a promising approach for clinical management of glaucoma.
Perimetric visual fields (VF) are used routinely in clinical practice to assess visual function defects attributable to glaucoma. In standard automated perimetry (SAP), the status (e.g., within
normal limits or outside of normal limits) of 52 test locations (of the 24-2 test pattern) is determined by statistical comparison of the test measurements with a normative database composed of age
normalized SAP exams. Linear regression of the commercially available SAP software parameters mean deviation (MD) and visual field index (VFI) often are used to assess progression of visual function
deficits, over time. Also commercially available, and designed specifically for progression detection, is the Guided Progression Analysis algorithm (GPA).
Because MD and VFI are global indices, these methods may not be ideal for progression detection, in that they include visual field locations that have little impact on the VF progression.
Since as early as 1990 (Goldbaum MH, et al.
1990;31;ARVO Abstract 503), studies have used supervised machine-learning classifiers successfully to separate healthy from glaucomatous eyes based on VF and optical imaging measurements and to
predict conversion to glaucoma in glaucoma suspect eyes.
More recently, we have effectively employed unsupervised machine-learning techniques to discern how VF data are organized into patterns. We found it useful to represent the structure of VFs by
clusters of healthy eyes, early glaucoma eyes, and advanced glaucoma eyes, and to represent the structure within each cluster by axes obtained by independent component analysis. The estimation of the
best structure representation was accomplished with post hoc assessment of the MD of the clusters, and visual inspection of the patterns of defect within the observed clusters.
We aimed to diminish the effects of human bias by designing a process for detecting change over time along mathematically determined glaucomatous patterns obtained by unsupervised learning techniques
without human intervention, and we aimed to improve effectiveness by eliminating noncontributing data and concentrating on the data that are changing.
Our initially described method, the variational Bayesian independent component analysis mixture-model (VIM), is a semiautomatic, unsupervised machine learning approach that has been shown to cluster
VFs in a meaningful way and to generate nearly maximally independent, clinically recognizable patterns of glaucomatous VF defects.
The Gaussian mixture-model with expectation maximization (GEM) produces a similar output, but learns 50 times faster and is a fully automated unsupervised learning approach.
The application of progression of patterns (POP) to VIM and GEM make the progression detectors VIM-POP and GEM-POP.
Other approaches also exist for the specific task of progression detection, including point-wise linear regression (PLR)
that evaluates change at each individual test location over the entire follow-up duration based on a fixed number of changing test locations; permutation analysis of PLR (PoPLR),
an individualized analysis that uses a
value combination function and permutation analysis to detect glaucomatous change; combined binomial tests with PLR,
and methods based on variational Bayesian analysis.
In this paper, we assess the clinical effectiveness of VIM-POP and GEM-POP methods for detecting glaucomatous progression along VF defect patterns. We also compare VIM-POP and GEM-POP with other
methods for detecting progression. Finally, we validate the specificity of all methods using independent datasets.
In the current study, we use the same data set to compare GEM-POP and VIM-POP with other progression-detection methods. We first evaluate the ability of GEM to cluster healthy and glaucomatous VFs
and to generate patterns of visual field defects within each cluster.
We then compare the clustering performance of GEM with VIM, based on specificity and sensitivity for clustering VFs as healthy and glaucomatous. Next, we detect glaucomatous progression in study eyes
based on significant change of longitudinal VF measurements (exams) along the previously generated GEM and VIM defect patterns, using POP. Finally, we compare the accuracy of GEM-POP and VIM-POP, to
and linear regression of MD and VFI, with detect progression in VFs from known progressing eyes.
Participant Selection and Testing
Study participants were selected from two prospective longitudinal studies designed to evaluate visual function and optic nerve structure in glaucoma: The University of California at San Diego (UC
San Diego; San Diego, CA)-based Diagnostic Innovations in Glaucoma Study (DIGS) and the UC San Diego–based African Descent and Glaucoma Evaluation Study (ADAGES). ADAGES is a three-site collaborative
study among the Hamilton Glaucoma Center of the Department of Ophthalmology at UC San Diego, the New York Eye and Ear Infirmary (NYEEI; New York, NY), and the Department of Ophthalmology, University
of Alabama, Birmingham (UAB; Birmingham, AL). Both studies follow identical protocols and the methodological details have been described previously.
The institutional review boards of UC San Diego, NYEEI, and UAB approved all DIGS and ADAGES methods. All methods adhered to the tenets of the Declaration of Helsinki and to the Health Insurance
Portability and Accountability Act. DIGS and ADAGES are registered as cohort clinical trials with
(NCT00221897 and NCT00221923, respectively; September 14, 2005).
Participants underwent a comprehensive ophthalmologic examination, including medical history review, best-corrected visual acuity, slit-lamp biomicroscopy, IOP measurement with Goldmann applanation
tonometry, gonioscopy, dilated examination of fundus by indirect ophthalmoscopy, stereoscopic optic disc photography, and SAP. SAP testing was performed using the 24-2 Swedish Interactive
Thresholding Algorithm (SITA). Only reliable tests (≤20% fixation losses, ≤33% false-negative results, and ≤15% false-positive results) were included. Trained reviewers from the UC San Diego-based
Visual Field Assessment Center (VisFACt) ensured that all VF tests studied were free of apparent artifacts (e.g., lid or rim artifacts, signs of fatigue).
For analysis of unsupervised learning techniques (described below), POAG patients were defined as those with repeatable (two consecutive) abnormal SAP VFs in one or both eyes. A designation of
abnormal VF required a pattern standard deviation (PSD) less than or equal to 0.05 or a glaucoma hemifield test (GHT) outside of normal limits.
In this study, participants with normal VFs were defined as those with no evidence of repeatable abnormal VFs (as defined above) in each eye.
Because the overall goals of the current study are to assess the clinical effectiveness of VIM and GEM methods in generating optimal visual defect patterns and detecting glaucomatous progression
using these defect patterns, four independent groups of study eyes were used: (1) the Classification Study Group, (2) the Stability Definition Group, (3) the Progression Study Group, and the (4)
Validation Group.
Classification Study Group
The Classification Study Group included 1976 eyes (of 1316 study participants). Abnormal SAP VFs were found in 859 eyes (of 617 study participants) and 1117 eyes (of 699 study participants) had SAP
VFs within normal limits. Study eyes in this group were classified as those with VF defects and those without VF defects, regardless of their optic disc assessment. We used the visual field status
alone as an indicator of glaucoma because this study group was used primarily to generate the optimal VIM and GEM defect patterns of the VF, and not to detect glaucoma. If classification groups had
been defined based on the presence or absence of apparent glaucomatous optic neuropathy (GON), it is likely that a significant number of GON eyes would have provided VFs within normal limits.
Table 1
shows the demographic information of participants in the abnormal and normal groups and their mean MD and PSD values.
Stability Definition Group
The Stability Definition Group included 84 eyes from 45 study participants with repeatable (i.e., ≥ 2 consecutive) glaucomatous SAP defects at baseline. Each study eye was tested once a week with one
baseline exam and a mean follow-up of 4.8 exams over a mean follow-up duration of 4.3 weeks. A total of 403 SAP VF measurements were collected. We considered the VF measurements in this group as
stable glaucoma because disease-related progression in adequately treated glaucoma eyes generally occurs over years and not weeks. Any changes during this short follow-up duration likely would be due
to variability in the function of diseased ganglion cells or in the attentiveness of the patient and not due to disease-related progression.
Table 2
shows the demographic information of the participants in the Stability Definition Group.
Progression Study Group
The Progression Study Group included eyes with known progressive glaucomatous optic neuropathy (PGON). We defined PGON based on structural evidence of progression independent of VF measurements so as
not to bias the assessment of the methods that use VF measurements to detect progression. The baseline and each follow-up photograph of the eyes in the progression study group were assessed for PGON
by two expert-trained observers viewing digitized stereoscopic image pairs on a 21-inch or larger computer monitor. Progressive glaucomatous optic neuropathy was defined as a decrease in the
neuroretinal rim width or the appearance of a new or enlarged retinal nerve fiber layer (RNFL) defect evident in paired stereoscopic images. Observers were masked to patient identification and all
clinically relevant results. A third observer adjudicated any disagreement in assessment between the first two observers. From 74 participants, 83 eyes were identified as progressed by PGON. A total
of 1161 SAP VF measurement visits were available from this group. The mean number of follow-up visits was 14.0 (SD = 4.8), and the mean follow-up duration was 9.1 (SD = 2.2) years, yielding a mean
interval between exams of 7.8 months. Demographic information for this group is presented in
Table 2
Validation Groups
In order to validate the 95% limits of stability used for all progression detection methods described, we investigated the specificity of each method in two external datasets. The first datatset was
analogous to our Stability Definition Group and was composed of 115 glaucoma eyes (consecutive abnormal VFs prior to baseline with GON on ophthalmic examination) tested five times over 4 weeks and
provided by Douglas Anderson, MD from Bascom Palmer Eye Institute, University of Miami Health Systems (Miami, FL; the institutional review board of Bascom Palmer Eye Institute approved all testing
and methods adhered to the tenets of the Declaration of Helsinki and to the Health Insurance Portability and Accountability Act). The second dataset was composed of 54 healthy eyes from UC San Diego
DIGS study tested every 6 months over 2 years (approximately 5 visits). Research protocol was part of DIGS and therefore, adheres to all ethics and regulatory guidelines. All VFs used for validation
were reliable, as described previously.
Generating Vim and Gem Clusters and Axes Used for Detecting Progression
VIM and GEM methods assigned each of the 1976 VF exams in the Classification Study Group to several clusters and further generated several VF defect patterns (axes) within these clusters. Each
cluster centroid contains information about disease severity and each axis contains information about the pattern (shape) of VF defect that can be investigated further for progression by looking at
points projected along that axis. During classification, no prior knowledge was provided to VIM or GEM as to whether the input VFs were abnormal or normal (i.e., the learning was unsupervised).
For VIM, all of the visual fields in each cluster were decomposed into different axes using independent component analysis (ICA).
Independence of axes was forced within each cluster, not between different clusters. The generated visual field at ±2 SD from the cluster mean on each axis, and the VFs associated with each cluster,
characterized the patterns of visual defect. To avoid working with a large number of axes, only axes with significant contributions (described later) were retained in each cluster. VIM-POP was
equipped with a sliding window because it is expected that glaucoma-associated change in visual function and structural imaging is nonlinear in some eyes. For example, a 5-visit spurt of progression
in a trend of 10 visits might be missed by a 10-visit linear regression while regression in a sliding 5-visit window along the 10-visit range would be more likely to capture the progression spurt.
For GEM, we investigated a modular use of ICA to generate defect patterns (axes) within each of the two classes of disease severity (within each cluster). The visual field patterns were represented
as axes through each cluster centroid and each cluster axis had the property of representing the visual field loss patterns from mild to advanced disease. Like VIM-POP, GEM-POP also was equipped with
a nonlinear progression detection algorithm, which uses a sliding-window of size 5 (approximately 3 years of follow-up).
For cross-sectional VF assessment, clinicians typically rely on the total deviation (TD) or pattern deviation (PD) plots supplied by the Humphrey Field Analyzer (HFA; Carl Zeiss Meditec, Inc.,
Dublin, CA) StatPac analysis. Because clinicians are accustomed to the TD display, we used simulated TD plots to display the patterns of VF defects identified by VIM and GEM, relative to healthy
eyes. The simulated TD plot is a vector obtained by subtracting absolute sensitivities at the centroid of the healthy cluster from individual VF points in absolute sensitivity space. Total deviation
plots were generated at ±2 SD along each of the axes and displayed. The simulated TD plots were displayed in color to help visualization, with green denoting positive values (more sensitive than
normal) and red denoting negative values (less sensitive than normal). The ±26 dB simulated VF sensitivities were displayed in equal steps of color from red to green, with −26 as pure red and +26 as
pure green, representing the entire range of normalized measures of deviation from normal in each location. With this representation, it is easier to visualize increasing VF defect severity as
deviations from the healthy mean along each glaucoma axis.
Variational Bayesian Independent Components Analysis Mixture Model (VIM)
VIM has been described in detail in previous publications.
Briefly, VIM is a combination of multiple ICA models weighted in a probabilistic manner. This combination allows the unsupervised identification of independent clusters of data, each containing
statistically independent axes of information. Clustering and axis development are done simultaneously in VIM. VIM is a semiautomatic clustering method because the user selects the model with the
highest average of sensitivity and specificity among a very large number of VIM models and the optimal model is retrained to further improve its diagnostic accuracy. In the current study, the VIM
training feature set had 53 features; the absolute sensitivity values from 52 of the 54 VF test points (2 blind spot points excluded) and participant age, for each of the 1976 SAP tests. VIM varied
the maximum number of clusters, the maximum number of axes within each cluster, and the number of Gaussians to create 720 models. Each model was iterated 500 times, employing a different number of
axes and several random seeds at initialization. It was assumed that the single best model (highest average of sensitivity and specificity for identifying abnormal and normal VFs) would provide the
best environment for finding glaucomatous patterns and for detecting progression. The optimal number of axes within each cluster was chosen based on the contribution of each axis in cluster
decomposition. This number was called a “knee point” and was chosen by ranking the axes in each cluster based on their lengths or magnitudes and including the number of axes with the largest
magnitudes and excluding axes with smaller magnitudes.
On a graph of axis contribution versus number of axes, the chosen number of axes occurred at the point where the slope of the graph changed from steep to nearly horizontal; hence, the term knee
point. The optimal model was retrained 500 times to determine the final best specificity and sensitivity.
Gaussian Mixture Model with Expectation Maximization (GEM)
GEM has been described in detail in a previous publication.
Briefly, GEM combines multivariate Gaussian components to model the VF data points and uses the expectation maximization (EM) procedure to estimate the parameters of the model, iteratively. Similar
to VIM, we used absolute sensitivity at 52 SAP locations and participant age as inputs to GEM. Clusters were created by selecting the component that maximized the maximum a posteriori probability,
based on the EM-estimated parameters. GEM is a probabilistic approach with a hierarchical modular framework that allows identification of clusters first, followed by identification of axes within
each cluster using ICA. Unlike VIM, where clustering and axis development are done simultaneously, GEM is sequential. To select an optimal GEM model that represents glaucoma categories and VF defect
patterns, we generated 600 GEM models (to be roughly comparable to VIM for assessing the computational complexity, however we have observed that 50 models generally are sufficient for GEM), and
selected the model that provided the highest average of sensitivity and specificity for discriminating between abnormal and normal VFs. The optimal number of axes within each cluster was chosen based
on the previously described knee point.
Procedure for Detecting Glaucomatous Progression
Using the VIM and GEM environments composed of the VF defect patterns or axes; we first defined the limits of stability using the Stability Definition Group. Progression by VIM and GEM was identified
based on the progression of each study eye along each VIM and GEM axis. Details of identifying the limits of stability and detecting progression using the VIM and GEM environments are described next.
More detailed technical descriptions are available elsewhere.
Stability Definition
Glaucoma boundary limits were detected by projecting the longitudinal sequence of VFs (each field in a 52-dimensional space) of each stable eye (i.e., eye from the Stability Definition Group) onto
each of the predefined VIM or GEM glaucoma axes. Because eyes in the stable group presumably showed no disease related progression, the variability in this group was used to define the limits of
stability or the progression limits. The VF exams of each stable eye were randomly resampled with replacement (i.e., bootstrapped) to simulate an eye with seven visits. One thousand resampled VF
series were used to determine the progression limit, or limit of stability, on each axis. The slope of each visual field sequence was determined for each window (of size 5) using least squares linear
regression for each VIM and GEM axis. The 95th percentiles (one-tailed toward the direction of deterioration) for each of the multiple axes and windows (described in the Results section) for
detecting glaucoma progression were calculated and compensated to accommodate an overall 95% specificity based on each axis and the sliding window (i.e., specificity along each axis was adjusted
upward to result in a total 95% specificity for all axes and windows combined for both VIM and for GEM). For each axis, the rate of progression of an eye along an axis was normalized to the adjusted
95th percentile threshold value for that axis. An eye with a normalized rate of progression (of VF threshold) greater than or equal to 1 was classified as progressed, and an eye with a normalized
rate of progression less than 1 was classified as nonprogressed. This method of defining progression along each VIM and GEM axis is called POP.
Glaucoma Progression Detection
Each visit of a longitudinal VF series of each study eye in the Glaucoma Study Group was projected onto each of the VIM and GEM axes. The mean rate of progression (slope) of the longitudinal sequence
of windows along each axis was determined for each eye using the least squares linear regression model. A study eye was classified as progressed when the normalized rate of progression on any one of
the axes equaled or exceeded 1 (i.e., when the regression line fell below the lower limit of stability); otherwise, the eye was classified as having no evidence of progression. The axis demonstrating
the most change was identified as the progressing axis. The pattern of that axis was then considered as the glaucoma pattern of progression.
The 95th percentile progression limits for MD and VFI were computed separately using the Stability Definition Group (there was no need to adjust the progression limits because MD and VFI each have
only one axis). Then, for each eye in the Progression Study Group, the MD and VFI slopes were computed over the follow-up period. If the slope of MD or VFI exceeded the corresponding progression
limit, the eye was classified as progressed; otherwise the eye was classified as nonprogressed.
Point-wise linear regression was conducted using the methods described by Fitzke et al.
To detect progression for PLR, for each of 52 VF points, two different parameters; the slope and the significance of the slope (
value) were considered. Progression of the VF was defined based on one to three deteriorating points with significant
value (smaller than 0.01) of the slope exceeding the specified threshold, allowing six different criteria for VF progression (two different slopes and three possible locations). We applied these six
criteria to our stability definition group to compute specificity and to our progression study group to compute sensitivity to identify sensitivity/specificity trade-offs at several discrete points
along the receiver operating characteristic (ROC) curve similar to O'Leary et al.
We also implemented PoPLR according to method described by O'Leary et al.
We combined the significance values across the VF and compared it with the null distribution of
values of all combinations of the VF sequences obtained by permutation of visits. We considered progression if the combined significance value of actual visits was greater than 95th percentile of the
null distribution of all permuted visits and computed the full ROC curve.
Full and partial (0.85–1.0 specificity) ROC curves were generated for VIM-POP, GEM-POP, PoPLR, linear regression of MD, and linear regression of VFI, for comparison.
For both VIM and GEM, environments were automatically built that separated SAP VFs into normal and abnormal clusters and identified patterns of field defects (axes) within each cluster. Subsequently,
these environments were used for progression analysis (i.e., POP).
SAP MD ± SD was −0.43 ± 1.25 dB and −4.14 ± 4.78 dB for normal and abnormal VFs, respectively, within the Classification Study Group. For both VIM and GEM, three clusters were identified: clusters N
(representing normal VFs), G[1] (representing early to moderate glaucoma, based on post hoc assessment of MD), and G[2] (moderate to advanced glaucoma, based on post hoc assessment of MD). The best
VIM model was composed of nine axes: two axes for each of the first two clusters (N and G[1]) and five axes for the third cluster (G[2]). Similarly, the best GEM model was composed of nine axes: two
axes for each of the first two clusters and five axes for the third cluster.
VIM and GEM Degrees of Disease Severity (Clusters)
For VIM, cluster N contained primarily VFs from healthy participants (normal VFs, 1084 of 1117 eyes; specificity = 97.0%). Clusters G[1] and G[2] combined contained primarily VFs from participants
with glaucoma (abnormal VFs, 799 of 859, sensitivity = 93.0%). Cluster G[1] contained 503 eyes with abnormal VFs and 32 eyes with normal VFs. Cluster G[2] contained 296 eyes with abnormal VFs and one
eye with a normal VF.
For GEM, cluster N contained primarily normal VFs (1048 of 1117 eyes; specificity = 93.8%). Clusters G
, and G
combined contained primarily VFs from participants with glaucoma (772 out of 859, sensitivity = 89.9%). Cluster G
contained 478 eyes with abnormal VFs and 69 eyes with normal VFs. Cluster G
contained 294 eyes with abnormal VFs and no eyes with normal VFs.
Figure 1
shows scatter plots of the mean threshold values in the superior hemifield plot versus the inferior hemifield plot, to demonstrate cluster variability and
Figure 2
shows MD versus PSD of all eyes assigned to each of the VIM and GEM clusters, to demonstrate clustering by VF defect severity.
VIM Patterns of Glaucomatous Defect (Axes)
The centroid of a VIM cluster represents the mean defect severity in that cluster. The mean of cluster G[2] was farther from the mean of cluster N than the mean of cluster G[1], indicating that the
overall defect was more severe in G[2]. Within each cluster, axes represented the pattern (shape) of each VF defect. VIM efficiently identified the magnitude of the various patterns of defect present
in each study cluster by projecting each VF along each axis. If a point moved along any axis away from the cluster mean, the direction of motion would be positive if the distance (vector) of the
point from the normal mean increased with movement, and the direction would be negative if the vector from the normal mean to the point decreased with movement.
To discern whether there was progression in an eye, each visit for that eye was projected onto an axis. Change from the cluster centroid in a positive direction (toward +2 SD) along a given axis
generally resulted in an increase in depth of defect or an increase in area of defect. Change in the negative direction (toward −2 SD) generally resulted in a decrease in depth of defect or a
decrease in area of defect.
Figure 3
shows the various VIM defect patterns (simulated TD plots) that were generated at ±2 SD from the respective cluster centroids on each axis in clusters N, G
, and G
. Similar to the total deviation plot in SAP, these patterns represent the degree of defect severity and deviation from normal. The two defect patterns of cluster N shown are normal, without
glaucomatous damage. Cluster G
includes eyes with early to moderate glaucomatous damage (average MD = −2.15, SD = 1.52 dB); consequently, the patterns shown illustrate primarily mild defects. Axis 1 of G
(at +2 SD) displays early superior and inferior nasal steps and arcuate reductions of sensitivity. Axis 2 of G
(at +2 SD) suggests an early superior arcuate defect. Cluster G
includes eyes with more moderate to advanced glaucomatous damage (average MD = −7.98, SD = 6.27 dB); hence, the axis patterns shown in
Figure 3c
are more advanced. Axis 1 of G
(at +2 SD) represents a diffuse depression with increased depression in the superior hemifield. Axis 2 of G
(at +2 SD) shows early diffuse reduction of sensitivity with an inferior hemifield defect. Axis 3 of G
(at +2 SD) represents a peripheral defect with increased defect at the nasal steps and the superior arcuate zone. Axis 4 of G
(at +2 SD) presents a nasal step defect with inferior nasal exaggeration and axis 5 of G
(at +2 SD) suggests a central-to-nasal defect, with inferior weighting.
GEM Patterns of Glaucomatous Defect (Axes)
Similar to VIM, points at ±2 SD on GEM-defined axes in clusters N, G[1], and G[2] were displayed as TD plots. These generated TD plots at points on the axes within each cluster represented distinct
VF defect patterns.
Figure 4
shows the various GEM defect patterns (axes) at ±2 SD from the cluster centroids of axes in clusters N, G
, and G
. The defect patterns of cluster N are normal, without glaucomatous damage. Cluster G
includes eyes with early to moderate glaucomatous damage (average MD = −2.19, SD = 1.55 dB). Axis 1 of G
(at +2 SD) represents a mild peripheral superior arcuate depression, and axis 2 of G
(at +2 SD) represents early superior arcuate and nasal step defects. Cluster G
includes eyes with moderate to advanced glaucomatous damage (average MD = −8.07, SD = 6.26 dB). Axis 1 of G
(at +2 SD) represents moderate diffuse reduction of sensitivity with a more pronounced superior hemifield defect and axis 2 of G
(at +2 SD) represents a diffuse reduction of sensitivity with increased defect in the nasal step positions. Axes 3, 4, and 5 (at +2 SD) primarily represent nasal step defects, with some variation in
weighting among these three axes.
VIM and GEM Progression
SAP mean deviation (±SD) was −7.44 ± 8.20 dB and −4.41 ± 5.79 dB at baseline in the Stability Definition Group and the Progression Study Group, respectively.
The left panel of
Figure 5
shows the distribution of the slopes of the projection magnitudes of all VFs in the Stability Definition Group along the first axis in G
cluster of VIM. The left tail of the 95th percentile limit is shown as a blue line that indicates the stability limit for this axis toward progression. The right panel of
Figure 5
shows the distribution of the slopes of projected points on each axis from all eyes from the Progression Study Group on the first axis of the G
cluster of VIM. The red circle shows the stability limit computed from the distribution of the slopes of the Stability Definition Group on the first axis of cluster G
Figure 5
, left, blue line). For a test eye, if its slope (estimated by least square regression of projected VFs) exceeded this limit, the eye was classified as a progressed along this axis.
The left panel of
Figure 6
shows the distribution of the slopes of the projection magnitudes of all VFs in the Stability Definition Group along the first axis in the G
cluster of GEM. The right panel of
Figure 6
shows the distribution of the slopes of all Progression Study Group eyes on the first axis of the G
cluster of GEM. The red circle shows the stability limit that was computed from the distribution of the slopes of the Stability Definition Group on the first axis of cluster G
Figure 6
, left, blue line). As above, if the slope of a test eye exceeded this limit of stability, the eye was classified as a progressing eye along this axis.
Figure 7
shows progression detection using GEM-POP in two example eyes. The orange circles represent the magnitude of the defect pattern given by the first axis of cluster G
present in the 52-dimensional VF space. The blue line indicates the slope (linear regression of the orange circles). The gray line indicates the 95% progression limit for the slopes of the first axis
of cluster G
. If the linear model approximating the slope falls below the gray line (progression zone), then the eye is classified as progressed; otherwise, the eye is classified as nonprogressed. Therefore,
GEM-POP is detecting progression in the study eye in
Figure 7
(left) and is detecting no evidence of progression in the study eye in
Figure 7
(right) along the first axis of cluster G
. In addition to event-related information,
Figure 7
(left) provides information about the rate of progression in the example eye (blue circles).
Figure 8
shows ROC curve areas for VIM-POP, GEM-POP, PoPLR, PLR, linear regression of MD, and linear regression of VFI. For PLR, we applied six criteria to our stability definition group to compute
specificity and to our progression study group to compute sensitivity, resulting in discrete points along the ROC curve. Therefore, the area under the ROC curve was not computed for PLR. The areas
under the ROC curves for VIM-POP, GEM-POP, PoPLR, linear regression of MD, and linear regression of VFI are 0.82, 0.86, 0.81, 0.69, and 0.76, respectively. It can be observed that the ROC curve area
of GEM-POP is similar to that of VIM-POP and is significantly greater than the ROC curve areas for PoPLR, linear regression of MD, and linear regression of VFI.
Figure 9
shows the partial ROC curves for VIM-POP, GEM-POP, PoPLR, linear regression of MD, and linear regression at high specificities for better visualization.
Table 3
shows the statistical difference among all methods.
Validation of Specificity of All Methods Using External Independent Datasets
Table 4
shows the specificities using all limits of stability (progression limits) in two validation groups (stable glaucoma and longitudinal healthy). These results indicate that the limits of stability
used were applicable when applied to independent stable data sets, although specificities for one version of PLR and for linear regression of MD were somewhat low, suggesting dataset-dependence
(i.e., reduced generalizability).
VIM and GEM are novel unsupervised clustering approaches that identify intuitive (i.e., recognizable) patterns (axes) of VF defect that create an environment usable for detecting glaucomatous
progression. The clusters identified using VIM and GEM in the current study were composed primarily of normal, early to moderate glaucoma, or moderate to advanced glaucoma VFs, based on MD. The
diagnostic accuracy (average of sensitivity and specificity) of VIM to cluster VFs as abnormal and normal was not statistically different from that of GEM (95% and 92%, respectively). The diagnostic
accuracy of VIM and GEM methods were similar with and without age as an input to the clustering step indicating that age had little effect (see also Bowd et al.,
for a similar VIM result using frequency doubling technology perimetry VFs). With VIM, the clusters and axes that constituted the VIM environment were determined simultaneously. In contrast, GEM used
a modular approach to first generate the clusters of disease severity followed by identification of axes or defect patterns.
The area under the ROC curve of GEM-POP was significantly greater (for detecting PGON eyes) than areas under the curve of PoPLR (P value = 0.03), linear regression of MD (P value <0.001), and VFI (P
value <0.001). The area under the ROC curve of GEM-POP was similar to the area under the curve of VIM-POP (P value = 0.12). Progression of patterns by VIM and GEM is based on progression along any
one of seven (in the current environment) axes, whereas progression by linear regression of MD and VFI is based on a single metric, indicating that detecting localized change in defect patterns
likely is a more sensitive technique than detecting global change. This superiority is expected, because in VIM-POP and GEM-POP, uncontributing VF locations are ignored; whereas for change in global
indices the noncontributing VF locations are included. In fact, all progression detection methods that relied on local analysis (VIM-POP, GEM-POP, PLR [based on best performing parameter sets], and
PoPLR) outperformed methods that relied on global analysis. It should be noted that PLR sensitivities can be significantly changed based on the selection of particular parameters (e.g., varying the
required number of deteriorating locations or varying the slope).
From a machine learning perspective, both VIM and GEM first identified the hidden structures (glaucoma defect patterns) and then created their respective environments for detecting progression. By
extracting clinically meaningful patterns of VF defects in an unsupervised manner and studying their progression, VIM and GEM provide an optimal environment to detect progression. Because the
extracted patterns are both visible to the health care provider and have shapes that are familiar and understood, these progression detection classifiers are not black boxes. Rather they provide the
clinician with an understanding regarding change over time of specific patterns of defect. The modular approach developed in GEM has several advantages over the simultaneous convergence approach of
VIM. Clustering of disease severity is a grouping process that is only weakly related to the discovery of independent axes or defect patterns. With a modular design, GEM allows the use of several
classes of clustering algorithms and several classes of axes discovery approaches available for machine learning, by separating the clustering and axes discovery processes. As an example, the
Gaussian mixture model used in GEM can be replaced with a simple k-means algorithm for clustering. In place of ICA used in GEM for axes discovery, principal component analysis (PCA) or other axes
discovery approaches can be easily applied. The simultaneously converged VIM is far less adaptable.
We used GEM with ICA in the current study to make GEM consistent with VIM, because VIM also uses ICA. Post-hoc analyses on the current data revealed that sensitivity of GEM for detecting progression
in PGON eyes improved somewhat when PCA was used instead of ICA (49.4% and 45.8%, respectively; a difference of 3 PGON eyes detected). PCA-generated axes are perpendicular to each other, which allows
a more accurate calculation of the progression of defect pattern along each axis. Technically, because of the orthogonality of the PCA axes, calculation of the projection coefficients along each axis
or the strength of a defect present in a VF is exact, unique and the calculations are simple. However with ICA, because of the lack of orthogonality of ICA axes, coefficient calculations are not
exact, not unique and are computationally more complex. It is likely that lack of perpendicular axes resulted in a slight decrease in the PGON sensitivity of GEM with ICA. We observed, post hoc, that
the PCA-based GEM defect patterns were not as visually representative of clinically observable defect patterns in glaucoma as ICA-based defect patterns. In other words, the addition of a measure of
dissimilarity in ICA, which yields maximally different VF patterns, may increase the recognizability of the VF patterns at the expense of slightly reducing the sensitivity of progression detection
compared with PCA; whereas, the orthogonality of PCA axes may increase the sensitivity of progression detection at the expense of finding and displaying maximally different VF patterns.
Another advantage of the modular design of GEM, compared with the simultaneous convergence of VIM, is the significant reduction in computational resources and time needed for training, due to a
significant reduction in GEM's computational complexity. Reduced time for generating the final classifier means that there is more time to change variables and experiment with different classifiers.
Creating the GEM environment took approximately 3 hours in a quad-core machine (8 gigabytes of memory). In contrast, VIM took approximately 168 hours (7 days) on the same machine to train all the
classifiers and select the one used to generate the VIM progression environment.
The modular design of GEM allows the use of various classes of clustering and axes discovery techniques, which allow the building of an optimum GEM environment tailored to a specific modality or data
source (for example, optical imaging data instead of visual function measurements, frequency doubling technology perimetry instead of SAP). Identifying an efficient clustering method and an axes
discovery approach to build an optimal progression environment is difficult when each experimental run takes several days to complete. Therefore, the modular design in GEM also may facilitate
building or improving optimal progression environments for various modalities.
In VIM, after the clustering step, the specific axes that constitute the VIM environment are manually chosen (based on the knee-point concept) and the initial VIM environment is further retrained.
Hence, the VIM procedure is semiautomatic. In contrast, the modular nature of GEM separates the clustering and axes identification steps without the need for retraining the GEM environment.
Therefore, GEM is fully automatic. A potential difficulty with semiautomatic methods is the need to have a both computational expertise to develop and modify algorithms and a clear understanding of
the factors involved in glaucomatous progression to choose appropriate appearing axes, from a clinical standpoint.
Regarding methodological similarities with other studies, the axes discovery step in GEM is similar to the Proper Orthogonal Decomposition (POD) framework for detecting progression from a baseline
condition, described previously using confocal scanning laser ophthalmoscopy images.
In GEM, one set of axes that are discovered a priori describe the general patterns of glaucoma defects. Progression in a study eye is determined based on progression along these predetermined defect
patterns. Whereas in POD, a set of axes is identified for each eye that describes the baseline conditions of the eye (known as the baseline subspace of the eye). Progression is determined based on
the deviation of follow-up measurements from the baseline subspace of the eye.
VF progression detection methods recently proposed include Permutation Analysis of PLR (PoPLR)
and Analysis with Non-Stationary Weibull Error Regression and Spatial Enhancement (ANSWERS).
We compared GEM- and VIM-based progression detection methods with PoPLR directly. However, PoPLR
requires a minimum of one baseline and six follow-up VF exams (to provide at least 5000 unique permutations of the VF series for building null distributions for hypothesis testing) to generate a
reliable and robust outcome.
Because the Stability Definition Group in the current study included an average of five VFs per eye, allowing 120 permutations, we generated sequences of seven visits for each eye to fulfill the
PoPLR requirements and used the newly generated simulated test–retest dataset to compute the specificity of all methods. Therefore, the comparison is valid because all methods used the same
test–retest dataset. ANSWERS relies on a mixture of Weibull distributions to model variability and a Bayesian method to aggregate spatial correlation of local measurements to confirm repeatable
defects in the same or adjacent locations in follow-up examinations. The addition of spatial correlations of measurements improves this method, compared with ANSWERS' precursor, without spatial
enhancement. We did not compare GEM- and VIM-based progression detection methods with ANSWERS because such a comparison is beyond the scope of the current manuscript.
Both PoPLR and ANSWERS were designed specifically to detect progression in SAP VFs and have not yet been shown to be successful when applied to other glaucoma-related measurements. Our GEM-POP and
VIM-POP approaches were designed to be robust; in addition to working with SAP VFs, they can be applied to frequency doubling technology, to progressive retinal nerve fiber layer thinning, and to
emerging data types, such as SDOCT mapping, to uncover patterns of defect and to detect progression of these defects using POP
(Bowd et al.,
2014;55 ARVO Abstract 3008; Yousefi et al.,
2015;56 ARVO Abstract 4564).
In this study, we modeled the bounds of stability for detecting progression using a simulated test–retest dataset specifically for comparison of VIM-POP and GEM-POP against PoPLR; 1000 stable
pseudolongitudinal series generated by bootstrap, or resampling with replacement approach of each eye resulted in 84,000 sequences to allow us generate the full ROC curve for all methods. Longer
longitudinal series (e.g., series with 7 visits) provide more confident bounds of stability. The distribution, or the region of stability, provided by the bootstrap resampling approach is
asymptotically exact (i.e., distributions, or the region of stability, becomes more exact as the number of stable pseudolongitudinal series increases).
One of the limitations of this approach is that the effects of aging, glaucoma management, and long-term measurement variability cannot be modelled in a longer pseudoseries using only five exams.
Nevertheless, this limitation does not affect the comparison of progression detection performance of VIM-POP and GEM-POP, PoPLR, PLR, MD, and VFI because all of the progression detection methods used
the same simulated test–retest dataset.
The patterns of VF defects reported for the GEM algorithm is dependent on the disease (VF) status and demographic characteristics of the Classification Study Group (
Table 1
). The Classification Study Group used currently includes study eyes (with varying disease status) from three geographically separate clinical sites. Therefore, the GEM defect patterns reported in
this study should be representative of general defect patterns in the United States and possibly internationally, which implies that the threshold between stability and progression that we derived
can be used for detecting progression in glaucoma clinics that include patients with various clinical characteristics (i.e., some evidence exists that the method is generalizable).
In conclusion, GEM-POP for progression detection performs significantly better than PoPLR and linear regression of VFI and MD GEM-POP performs similarly to VIM-POP. However, GEM-POP provides a less
complex environment than VIM-POP, is computationally more efficient than VIM-POP and is a fully automated technique. Although GEM-POP is more complex to develop than PLR and PoPLR, once the GEM
environment for detecting progression has been established, determining whether an eye has progressed by GEM-POP is simple and fast. Finally, GEM-POP and VIM-POP are designed to be applicable to
other data types besides perimetry.
Supported by grants from National Institutes of Health (NIH) EY022039, NIH EY008208, NIH EY011008, NIH EY014267, NIH EY019869, NIH EY020518, P30EY022589 an unrestricted grant from Research to Prevent
Blindness (New York), Eyesight Foundation of Alabama, Corinne Graber Research Fund of the New York Glaucoma Research Institute, and participant incentive grants in the form of glaucoma medication at
no cost from Alcon Laboratories, Allergan, and Pfizer.
Disclosure: S. Yousefi, None; M.H. Goldbaum, None; M. Balasubramanian, None; F.A. Medeiros, Ametek (C), Alcon Laboratories Inc, (C), Allergan Inc. (F, C), Bausch+Lomb (F), Carl Zeiss Meditec (F, C),
Heideleberg Engineering GmbH (F, C), Sensimed (F); L.M. Zangwill, Carl Zeiss Meditec Inc. (F), Heidelberg Engineering GmbH (F, R), Optovue Inc. (F), Topcon Medical Systems Inc. (F); R.N. Weinreb,
Alcon Laboratories Inc. (C), Allergan Inc. (C), Carl Zeiss Meditec Inc. (F, C), Heidelberg Engineering GmbH (F), Optovue Inc. (F), Topcon Medical Systems Inc. (F, C); C.A. Girkin, None; J.M. Liebmann
, Alcon Laboratories Inc. (C), Allergan Inc. (C), Carl Zeiss Meditec Inc. (F), Diopsys Corp. (F, C), Heidelberg Engineering GmbH (F), Optovue Inc. (F, C), Pfizer Inc. (C), Topcon Medical Systems Inc.
(F, C); C. Bowd, None
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Efron B, Tibshirani RJ. An Introduction to the Bootstrap. New York: Chapman & Hall; 1993. | {"url":"https://tvst.arvojournals.org/article.aspx?articleid=2520670","timestamp":"2024-11-04T04:38:44Z","content_type":"text/html","content_length":"345315","record_id":"<urn:uuid:337669c1-f92d-4b6e-a932-5b1aed9771e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00661.warc.gz"} |
sinhl: hyperbolic sine functions - Linux Manuals (3p)
sinhl (3p) - Linux Manuals
sinhl: hyperbolic sine functions
This manual page is part of the POSIX Programmer's Manual. The Linux implementation of this interface may differ (consult the corresponding Linux manual page for details of Linux behavior), or the
interface may not be implemented on Linux.
sinh, sinhf, sinhl - hyperbolic sine functions
#include <math.h>
double sinh(double x);
float sinhf(float x);
long double sinhl(long double x);
These functions shall compute the hyperbolic sine of their argument x.
An application wishing to check for error situations should set errno to zero and call feclearexcept(FE_ALL_EXCEPT) before calling these functions. On return, if errno is non-zero or fetestexcept
(FE_INVALID | FE_DIVBYZERO | FE_OVERFLOW | FE_UNDERFLOW) is non-zero, an error has occurred.
Upon successful completion, these functions shall return the hyperbolic sine of x.
If the result would cause an overflow, a range error shall occur and ±HUGE_VAL, ±HUGE_VALF, and ±HUGE_VALL (with the same sign as x) shall be returned as appropriate for the type of the function.
If x is NaN, a NaN shall be returned.
If x is ±0 or ±Inf, x shall be returned.
If x is subnormal, a range error may occur and x should be returned.
These functions shall fail if:
Range Error
The result would cause an overflow.
If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [ERANGE]. If the integer expression (math_errhandling & MATH_ERREXCEPT) is non-zero, then the
overflow floating-point exception shall be raised.
These functions may fail if:
Range Error
The value x is subnormal.
If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [ERANGE]. If the integer expression (math_errhandling & MATH_ERREXCEPT) is non-zero, then the
underflow floating-point exception shall be raised.
The following sections are informative.
On error, the expressions (math_errhandling & MATH_ERRNO) and (math_errhandling & MATH_ERREXCEPT) are independent of each other, but at least one of them must be non-zero.
Portions of this text are reprinted and reproduced in electronic form from IEEE Std 1003.1, 2003 Edition, Standard for Information Technology -- Portable Operating System Interface (POSIX), The Open
Group Base Specifications Issue 6, Copyright (C) 2001-2003 by the Institute of Electrical and Electronics Engineers, Inc and The Open Group. In the event of any discrepancy between this version and
the original IEEE and The Open Group Standard, the original IEEE and The Open Group Standard is the referee document. The original Standard can be obtained online at
asinh(), cosh(), feclearexcept(), fetestexcept(), isnan(), tanh(), the Base Definitions volume of IEEE Std 1003.1-2001, Section 4.18, Treatment of Error Conditions for Mathematical Functions, | {"url":"https://www.systutorials.com/docs/linux/man/3p-sinhl/","timestamp":"2024-11-06T21:10:46Z","content_type":"text/html","content_length":"10421","record_id":"<urn:uuid:b0b48815-d0ba-41bf-bb36-39d83d4c5fef>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00468.warc.gz"} |
CS502 - Fundamentals of Algorithms Sample Paper
CS502 – Fundamentals of Algorithms Sample Paper
Question No: 1 (Marks: 01) – Please choose the correct option
Matrix chain multiplication is a/an___________ type of problem.
1. Search Problem
2. Computational Problem
3. Optimization Problem
4. Parallel Computation Problem
Question No: 2 (Marks: 01) – Please choose the correct option
Dynamic Programming Algorithms involves solving the problem by________.
1. Breaking up into sub-problems
2. Only One Individual problem
3. One Part of problem
4. Starting from the first part of problem
Question No: 3 (Marks: 01) – Please choose the correct option
Matrix chain multiplication is a/an___________ type of problem.
1. Search Problem
2. Computational Problem
3. Optimization Problem
4. Parallel Computation Problem
Question No: 4 (Marks: 01) – Please choose the correct option
The Knapsack problem is an example of ____________.
a) Greedy algorithm
b) 2 D dynamic programming
c) 1 D dynamic programming
d) Divide and conquer
Similar Posts:
Question No: 5 (Marks: 01) – Please choose the correct option
You are given a knapsack that can carry a maximum weight of 60. There are 4 items with weights {20, 30, 40, 70} and values {70, 80, 90, 200}. What is the maximum value of the items you can carry
using the knapsack?
a) 160
b) 200
c) 170
d) 90
Question No: 6 (Marks: 01) – Please choose the correct option
Which of the following algorithms is the best approach for solving Huffman codes?
a) exhaustive search
b) greedy algorithm
c) brute force algorithm
d) divide and conquer algorithm
Question No: 7 (Marks: 01) – Please choose the correct option
From the following given tree, what is the computed codeword for ‘c’?
a) 111
b) 101
c) 110
d) 011
Question No: 8 (Marks: 01) – Please choose the correct option
Fractional knapsack problem is solved most efficiently by which of the following algorithm?
a) Divide and conquer
b) Dynamic programming
c) Greedy algorithm
d) Backtracking
Question No: 9 (Marks: 01) – Please choose the correct option
The main time taking step in fractional knapsack problem is ___________.
a) Breaking items into fraction
b) Adding items into knapsack
c) Sorting
d) Looping through sorted items
Question No: 10 (Marks: 01) – Please choose the correct option
Which bit is reserved as a parity bit in an ASCII set?
a) first
b) seventh
c) eighth
d) tenth
Question No: 11 (Marks: 01) – Please choose the correct option
What does the Activity Selection Problem mean__________?
1. Scheduling Problem
2. Search Problem
3. Computational Problem
4. Non Computational Problem
Question No: 12 (Marks: 01) – Please choose the correct option
You are given infinite coins of denominations v1, v2, v3,…..,v[n] and a sum S. The coin change problem is to find the minimum number of coins required to get the sum S. This problem can be solved
using ____________.
a) Greedy algorithm
b) Dynamic programming
c) Divide and conquer
d) Backtracking
Question No: 13 (Marks: 01) – Please choose the correct option
You are given infinite coins of N denominations v1, v2, v3,…..,vn and a sum S. The coin change problem is to find the minimum number of coins required to get the sum S. What is the time complexity of
a dynamic programming implementation used to solve the coin change problem?
a) O(N)
b) O(S)
c) O(N2)
d) O(S*N)
Question No: 14 (Marks: 01) – Please choose the correct option
What is the fractional knapsack problem?
1. A problem in computer science that deals with sorting algorithms
2. A problem in mathematics that deals with dividing a line segment into a given number of equal parts
3. A problem in optimization that deals with selecting items to include in a knapsack so as to maximize the total value of items in the knapsack
4. A problem in physics that deals with calculating the force exerted by a gas on a container
Question No: 15 (Marks: 01) – Please choose the correct option
What is the objective of the fractional knapsack problem?
1. To find the minimum weight of items that can be selected to fill the knapsack
2. To find the maximum weight of items that can be selected to fill the knapsack
3. To find the minimum value of items that can be selected to fill the knapsack
4. To find the maximum value of items that can be selected to fill the knapsack
Question No: 16 (Marks: 01) – Please choose the correct option
What is the key difference between the 0/1 knapsack problem and the fractional knapsack problem?
1. The 0/1 knapsack problem allows items to be broken into parts, while the fractional knapsack problem does not
2. The 0/1 knapsack problem requires items to be taken in their entirety, while the fractional knapsack problem allows items to be taken in fractional amounts
3. The 0/1 knapsack problem only allows items to be taken in whole numbers, while the fractional knapsack problem allows items to be taken in any amount
4. The 0/1 knapsack problem is much easier to solve than the fractional knapsack problem
Question No: 17 (Marks: 01) – Please choose the correct option
A free tree does not have _______________ node/vertex.
1. Root
2. Left child
3. Right child
4. Leaf
Question No: 18 (Marks: 01) – Please choose the correct option
The number of edges in a Minimum Spanning Tree (MST) is _____________ the number of edges in the original graph.
1. Greater than
2. Less than or greater than
3. Subset of
4. Equal to
Question No: 19 (Marks: 01) – Please choose the correct option
The number of vertices in a Minimum Spanning Tree (MST) is _____________ the number of vertices in the original graph.
1. Less than
2. Greater than
3. Subset of
4. Equal to
Question No: 20 (Marks: 01) – Please choose the correct option
Kruskal’s algorithm is used for computing Minimum Spanning Tree (MST) and is based on ____________ strategy.
1. Dynamic Programming
2. Greedy
3. Divide and Conquer
4. Backtracking
Question No: 21 (Marks: 01) – Please choose the correct option
The Time complexity of Prim’s algorithm is ________________.
1. Θ (logE + logV)
2. Θ (log E V)
3. Θ (E log V)
4. Θ (log E+V)
Question No: 22 (Marks: 01) – Please choose the correct option
Dijkstra’s algorithm is used to compute ___________________ in a directed graph.
1. Single-source shortest paths
2. Single destination shortest paths
3. Single-pair shortest paths
4. All-pairs shortest paths
Question No: 23 (Marks: 01) – Please choose the correct option
The Dijkstra algorithm maintains estimate of the shortest path from source vertex to ____________
1. Destination vertex
2. Neighbor vertex
3. Neighbor of the neighbor vertex
4. All vertices in the graph
Question No: 24 (Marks: 01) – Please choose the correct option
In Dijkstra algorithm the estimate of source vertex (i.e. s) is represented by d[s] and its value is ____________
1. Equal to zero
2. Greater than zero
3. Less than zero
4. Equal to one
Question No: 25 (Marks: 01) – Please choose the correct option
If a shortest paths algorithm allow negative cost cycle, then the algorithm will compute
1. Multiple paths between pair of vertices
2. Only one Path between pair of vertices
3. Incorrect path between pair of vertices
4. No Path between pair of vertices
Question No: 26 (Marks: 01) – Please choose the correct option
Which algorithm does not allow negative weights while computing the shortest paths in a directed graph?
1. Dijkstra’s algorithm
2. Floyd-Warshall Algorithm
3. Bellman Ford
4. Johnson
Question No: 27 (Marks: 01) – Please choose the correct option
The Time complexity of Bellman-Ford’ algorithm is ________________.
1. Θ (logE + logV)
2. Θ (log E V)
3. Θ (E V)
4. Θ (E+V)
Question No: 28 (Marks: 01) – Please choose the correct option
Floyd Warshall algorithm is used to compute _________________________ in a graph.
1. Single-source shortest paths
2. Single destination shortest paths
3. Single-pair shortest paths
4. All-pairs shortest paths
Question No: 29 (Marks: 01) – Please choose the correct option
Which one among the following is the slowest algorithm for computing shortest paths?
1. Dijkstra’s algorithm
2. Floyd-Warshall Algorithm
3. Bellman Ford
4. Prims
Question No: 30 (Marks: 01) – Please choose the correct option
Floyd Warshall’s algorithm is used for computing shortest paths and is based on ____________ strategy.
1. Dynamic Programming
2. Greedy
3. Divide and Conquer
4. Backtracking
Question No: 31 (Marks: 01) – Please choose the correct option
Which bit is reserved as a parity bit in an ASCII set?
a) first
b) seventh
c) eighth
d) tenth
Question No: 32 (Marks: 01) – Please choose the correct option
What is an adjacency list?
1. A list of nodes and their edges in a graph
2. A list of nodes in a graph, without any edges
3. A list of edges in a graph, without any nodes
4. A list of nodes in a graph, with their edges represented as a matrix
Question No: 33 (Marks: 01) – Please choose the correct option
What is an adjacency matrix used for in graph theory?
1. To represent the distances between nodes
2. To represent the edges between nodes
3. To represent the weights of nodes
4. To represent the orientation of nodes
Question No: 34 (Marks: 01) – Please choose the correct option
Which of the following best describes Breadth First Search (BFS)?
1. BFS is a search algorithm that starts at the root node and visits all the nodes along the shortest path first, before visiting the more distant nodes.
2. BFS is a search algorithm that visits all the nodes at the same depth before moving on to the next level.
3. BFS is a search algorithm that visits nodes in a random order.
4. BFS is a search algorithm that visits all the nodes in the order they were added to the tree.
Question No: 35 (Marks: 01) – Please choose the correct option
Which of the following best describes Depth First Search (DFS)?
1. DFS is a search algorithm that starts at the root node and visits all the nodes along the shortest path first, before visiting the more distant nodes.
2. DFS is a search algorithm that visits all the nodes at the same depth before moving on to the next level.
3. DFS is a search algorithm that visits nodes in a random order.
4. DFS is a search algorithm that visits all the nodes by exploring as far as possible along each branch before backtracking.
Question No: 36 (Marks: 01) – Please choose the correct option
What is the characteristic feature of Breadth First Search (BFS)?
1. Visits all nodes along the shortest path first
2. Visits all nodes at the same depth before moving to the next level
3. Visits nodes in a random order
4. Visits all the nodes in the order they were added to the tree
Question No: 37 (Marks: 01) – Please choose the correct option
What is a Directed Acyclic Graph (DAG)?
1. A graph with directed edges and cycles
2. A graph with directed edges and no cycles
3. A graph with undirected edges and cycles
4. A graph with undirected edges and no cycles
Question No: 38 (Marks: 01) – Please choose the correct option
What is the time stamp structure used in Depth First Search (DFS)?
1. Inorder traversal time stamp
2. Preorder traversal time stamp
3. Postorder traversal time stamp
4. Level-order traversal time stamp
Question No: 39 (Marks: 01) – Please choose the correct option
What is the characteristic feature of Depth First Search (DFS)?
1. Visits all nodes along the shortest path first
2. Visits all nodes at the same depth before moving to the next level
3. Visits nodes in a random order
4. Visits all the nodes by exploring as far as possible along each branch before backtracking
Question No: 40 (Marks: 01) – Please choose the correct option
What is the main property of a Directed Acyclic Graph (DAG)?
1. The edges are undirected
2. There are cycles present in the graph
3. The edges are directed and there are no cycles present in the graph
4. There is a unique path between any two vertices in the graph
Question No: 41 (Marks: 03)
Consider the case of 3 matrices: A1 is 5 × 4, A2 is 4 × 6 and A3 is 6 × 2 The multiplication can be carried out either as ((A1A2)A3) or (A1(A2A3)). Calculate the cost of these two matrices.
Question No: 42 (Marks: 03)
Write down at least three real life applications of Minimum Spanning Tree (MST).
Question No: 43 (Marks: 03)
Identify algorithms from the following list which compute the Minimum Spanning Tree (MST) in an undirected Graph.
Bellman Ford
Floyd Warshall
Question No: 44 (Marks: 03)
What makes Directed Acyclic Graphs (DAGs) a useful data structure for representing hierarchical relationships and relationships between events?
Question No: 45 (Marks: 03)
How does the fractional Knapsack algorithm differ from the 0/1 Knapsack algorithm, and how can it be used to solve problems with continuous or fractional values.
Question No: 46 (Marks: 03)
What is an adjacency matrix and how is it used to represent the connections between vertices in a graph?
Question No: 47 (Marks: 05)
Consider the following graph and construct its Minimum Spanning Tree (MST) using Kruskal’s algorithm.
Question No: 48 (Marks: 05)
Suppose you are given a map of a city with different locations and roads connecting them. How would you use the Breadth-First Search (BFS) algorithm to find the shortest route between two specific
locations in the city?
Question No: 49 (Marks: 05)
What is the time stamp structure used in Depth-First Search (DFS) algorithm, and how does it help in identifying different types of edges in a graph, such as tree edges, back edges, forward edges,
and cross edges?
Question No: 50 (Marks: 05)
Generate the Huffman binary tree for the string baadebbc based on the following frequency table.
Character a b c d e
Frequency 2 3 1 1 1 | {"url":"https://universitydistancelearning.com/cs502-fundamentals-of-algorithms-sample-paper/","timestamp":"2024-11-14T22:07:10Z","content_type":"text/html","content_length":"132289","record_id":"<urn:uuid:6c7d972c-bbdc-42d4-85c2-fade1ef49b71>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00321.warc.gz"} |
Henri Poincare
Henri Poincare
Poincare founded the modern qualitative theory of dynamical systems.
He created topology, the study of shapes and their continuity, and used this new mathematical tool to attempt to answer the question "Is the solar system stable?", a question posed by King Oscar II
of Sweden with a cash prize promised to (s)he who answered it definitively.
Actually, Poincare tried to understand just three bodies moving under their mutual gravitational attraction.
Poincare won the prize with his publication of On The Problem of Three Bodies and the Equations of Equilibrium. These three bodies are an excellent example of a dynamical system.
In his attempt to solve this problem Poincare introduced the Poincare section and saw the first signs of Chaos.
© The Exploratorium, 1996 | {"url":"https://annex.exploratorium.edu/complexity/lexicon/poincare.html","timestamp":"2024-11-04T02:04:02Z","content_type":"text/html","content_length":"2886","record_id":"<urn:uuid:851768a7-32f7-41cf-a68d-0db4b8075f1c>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00000.warc.gz"} |
Random Art
Next: About this document Up: Déjà Vu: A User Previous: References
One proposed hash visualization algorithm is Random Art, a technique that converts meaningless strings into abstract structured images. Random Art was developed by Andrej Bauer, and is based on an
idea of genetic art by Michael Witbrock and John Mount. Originally Random Art was conceived for automatic generation of artistic images. A brief overview and demonstration of Random Art can be found
at Andrej's Random Art web site [Bau98].
The basic idea is to use a binary string $s$ as a seed for a random number generator. The randomness is used to construct a random expression which describes a function generating the image--mapping
each image pixel to a color value. The pixel coordinates range continuously from $-1$ to $1$, in both $x$ and $y$ dimensions. The image resolution defines the sampling rate of the continuous image.
For example, to generate a $100 100$ image, we sample the function at $10000$ locations.
Random Art is an algorithm such that given a bit-string as input, it will generate a function $F: [-1,1]2 →[-1,1]3$, which defines an image. The bit-string input is used as a seed for the
pseudo-random number generator, and the function is constructed by choosing rules from a grammar depending on the value of the pseudo-random number generator. The function $F$ maps each pixel $(x,y)$
to a RGB value (r,g,b) which is a triple of intensities for the red, green and blue values, respectively. For example, the expression $F(x,y) = (x, x, x)$ produces a horizontal gray grade, as shown
in figure 3(a). A more complicated example is the following expression, which is shown in figure 3(b).
Figure 3: Examples of images and corresponding expressions.
Figure 4: Random Art expression tree and the corresponding image
The function $F$ can also be seen as an expression tree, which is generated using a grammar $G$ and a depth parameter d, which specifies the minimum depth of the expression tree that is generated.
The grammar $G$ defines the structure of the expression trees. It is a version of a context-free grammar, in which alternatives are labeled with probabilities. In addition, it is assumed that if the
first alternative in the rule is followed repeatedly, a terminal clause is reached. This condition is needed when the algorithm needs to terminate the generation of a branch. For illustration,
consider the following simple grammar:
The numbers in subscripts are the probabilities with which alternatives are chosen by the algorithm. There are three rules in this simple grammar. The rule $E$ specifies that an expression is a
triple of compound expression $C$. The rule $C$ says that every compound expression $C$ is an atomic expression $A$ with probability ${14}$, or either the function $add$ or $mult$ applied to two
compound expressions, with probabilities ${38}$ for each function. An atomic expression $A$ is either a constant, which is generated as a pseudorandom floating point number, or one of the coordinates
$x$ or $y$. All functions appearing in the Random Art algorithm are scaled so that they map the interval $[-1,1]$ to the interval $[-1,1]$. This condition ensures that all randomly generated
expression trees are valid. For example, the scaling for the add function is achieved by defining $add(x,y) = (x+y)/2$.
The grammar used in the Random Art implementation is too large to be shown in this paper. Other functions included are: sin, cos, exp, square root, division, mix. The function $mix(a,b,c,d)$ is a
function which blends expressions $c$ and $d$ depending on the parameters $a$ and $b$. We show an example of an expression tree of depth $5$ in figure 4, along with the corresponding image. For the
other images in this paper, we used a depth of 12.
Next: About this document Up: Déjà Vu: A User Previous: References Adrian Perrig
Thu Jun 15 15:16:10 PDT 2000 | {"url":"https://www.usenix.org/legacy/events/sec00/full_papers/dhamija/dhamija_html/node23.html","timestamp":"2024-11-07T10:47:13Z","content_type":"text/html","content_length":"8632","record_id":"<urn:uuid:55b7ef5f-acba-4ced-a684-4769b1b478a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00470.warc.gz"} |
Programming Flix
Functions and higher-order functions are the key building block of a functional programming language.
In Flix, top-level functions are defined with the def keyword. For example:
def add(x: Int32, y: Int32): Int32 = x + y + 1
A definition consists of the function name followed by an argument list, the return type, and the function body. Although Flix supports type inference, top-level function definitions must declare the
type of their arguments and their return type.
In Flix, all function arguments and local variables must be used. If a function argument is not used it must be prefixed with an underscore to explicitly mark it as unused.
A higher-order function is a function that takes a parameter which is itself a function. For example:
def twice(f: Int32 -> Int32, x: Int32): Int32 = f(f(x))
Here the twice function takes two arguments, a function f and an integer x, and applies f to x two times.
We can pass a lambda expression to the twice function:
twice(x -> x + 1, 42)
which evaluates to 44 since 42 is incremented twice.
We can also define a higher-order function that requires a function which takes two arguments:
def twice(f: (Int32, Int32) -> Int32, x: Int32): Int32 =
f(f(x, x), f(x, x))
which can be called as follows:
twice((x, y) -> x + y, 42)
We can call a higher-order function with a top-level function as follows:
def inc(x: Int32): Int32 = x + 1
def twice(f: Int32 -> Int32, x: Int32): Int32 = f(f(x))
twice(inc, 42)
Depending on the number of arguments to a function, the syntax for the function type differs:
Unit -> Int32 // For nullary functions
Int32 -> Int32 // For unary functions
(Int32, Int32, ...) -> Int32 // For the rest
Flix supports several operators for function composition and pipelining:
let f = x -> x + 1;
let g = x -> x * 2;
let h = f >> g; // equivalent to x -> g(f(x))
Here >> is forward function composition.
We can also write function applications using the pipeline operator:
List.range(1, 100) |>
List.filter(x -> x `Int32.mod` 2 == 0) |>
List.map(x -> x * x) |>
Here x |> f is equivalent to the function application f(x).
Functions are curried by default. A curried function can be called with fewer arguments than it declares returning a new function that takes the remainder of the arguments. For example:
def sum(x: Int32, y: Int32): Int32 = x + y
def main(): Unit \ IO =
let inc = sum(1);
inc(42) |> println
Here the sum function takes two arguments, x and y, but it is only called with one argument inside main. This call returns a new function which is similar to sum, except that in this function x is
always bound to 1. Hence when inc is called with 42 it returns 43.
Currying is useful in many programming patterns. For example, consider the List.map function. This function takes two arguments, a function of type a -> b and a list of type List[a], and returns a
List[b] obtained by applying the function to every element of the list. Now, if we combine currying with the pipeline operator |> we are able to write:
def main(): Unit \ IO =
List.range(1, 100) |>
List.map(x -> x + 1) |>
Here the call to List.map passes the function x -> x + 1 which returns a new function that expects a list argument. This list argument is then supplied by the pipeline operator |> which, in this
case, expects a list and a function that takes a list.
Flix supports the pipeline operator |> which is simply a prefix version of function application (i.e. the argument appears before the function).
The pipeline operator can often be used to make functional code more readable. For example:
let l = 1 :: 2 :: 3 :: Nil;
l |>
List.map(x -> x * 2) |>
List.filter(x -> x < 4) |>
List.count(x -> x > 1)
Here is another example:
"Hello World" |> String.toUpperCase |> println
Flix has a number of built-in unary and infix operators. In addition Flix supports infix function application by enclosing the function name in backticks. For example:
123 `sum` 456
is equivalent to the normal function call:
sum(123, 456)
In addition, a function named with an operator name (some combination of +, -, *, <, >, =, !, &, |, ^, and $) can also be used infix. For example:
def <*>(x: Int32, y: Int32): Int32 = ???
can be used as follows:
1 <*> 2 | {"url":"https://doc.flix.dev/functions.html","timestamp":"2024-11-13T15:48:17Z","content_type":"text/html","content_length":"29664","record_id":"<urn:uuid:9dd7a474-6a94-405a-928b-f2052d85b03d>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00751.warc.gz"} |
Cricket Run Rate Formula in context of cricket run rate
27 Aug 2024
Title: The Cricket Run Rate Formula: A Mathematical Analysis of a Crucial Aspect of the Game
Cricket is a popular team sport that requires strategic planning, skillful execution, and mathematical calculations to outmaneuver opponents. One crucial aspect of the game is the run rate, which
measures the number of runs scored by a team per over (six balls). This article delves into the cricket run rate formula, providing a comprehensive analysis of its components and applications.
In cricket, the run rate is a vital statistic that determines a team’s performance. It is calculated as the total number of runs scored divided by the total number of overs faced. The higher the run
rate, the better a team’s performance. This article will explore the formula for calculating the cricket run rate and its significance in the game.
The Cricket Run Rate Formula:
The cricket run rate formula can be represented mathematically as:
RR = (R / O)
where RR is the run rate, R is the total number of runs scored, and O is the total number of overs faced.
In ASCII format:
RR = (R / O)
Using BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) rules, we can rewrite the formula as:
RR = (R ÷ O)
Components of the Cricket Run Rate Formula:
The cricket run rate formula consists of two primary components:
1. Runs Scored (R): This represents the total number of runs scored by a team.
2. Overs Faced (O): This represents the total number of overs faced by a team.
Applications of the Cricket Run Rate Formula:
The cricket run rate formula has several applications in the game:
1. Team Performance Evaluation: The run rate is used to evaluate a team’s performance, with higher rates indicating better performances.
2. Target Setting: The run rate helps teams set realistic targets for chasing down opponents’ scores.
3. Strategic Planning: The run rate informs strategic decisions regarding batting and bowling approaches.
The cricket run rate formula is a fundamental aspect of the game, providing a mathematical framework for evaluating team performance and making strategic decisions. By understanding the components
and applications of this formula, teams can optimize their gameplay and gain a competitive edge.
1. International Cricket Council (ICC). (n.d.). Cricket Rules. Retrieved from https://www.icc-cricket.com/about/ rules/
2. Wisden Cricketers’ Almanack. (2020). The Laws of Cricket. Retrieved from https://www.wisden.com/laws-of-cricket/
ASCII Art:
Here’s a simple ASCII art representation of the cricket run rate formula:
RR = (R / O)
| |
| R / |
This article provides a comprehensive analysis of the cricket run rate formula, highlighting its components and applications in the game. The formula is presented in both BODMAS and ASCII formats for
easy reference.
Related articles for ‘cricket run rate’ :
• Reading: Cricket Run Rate Formula in context of cricket run rate
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Installation Instructions for the GBM Response Generator
NOTE: GRB trigger data from GBM already have a standard set of response functions delivered to the data archive, so there is generally no need to redo them.
1. Install the prerequisite programs:
• CMake can be downloaded from:
• CFITSIO can be downloaded from:
• gfortran is part of the GCC suite:
Information about the GCC compiler suite can be found at:
Note: MacOS XCode doesn't come with gfortran. You will need to install it by by downloading it from the GNU website or by using a package manager like homebrew or macports. It is recommended to
install the compiler suite with a package manager.
• Perl version 5.6 or higher
Perl is included with Linux and MacOS. Perl can be dowloaded from:
• Astro::FITS::CFITSIO module for Perl
This needs to be installed AFTER the CFITSIO library is installed. More information can be found at:
The script will run the system perl (/usr/bin/perl) by default. If you want to use a different version of perl (e.g. from macports) or if your perl is installed in a different location then be sure
to specify the correct perl location as instructed in step 4 below.
Also make sure you install the Astro::FITS::CFITSIO module using the same perl interpreter that will be used to execute the scripts. If you installed Astro::FITS::CFITSIO module but the script gives
an error that it can't be found, then it means that you may have (as in the case of macports perl) installed it for a different perl interpreter than is being called.
You can double check the perl interpreter with:
$ which perl
and then specify the path returned in step 4 below.
Install instructions (using homebrew):
$ brew install cmake
$ brew install gcc
$ brew install cfitsio
Install instructions (using macports):
(remove the sudo from the command line if not needed)
$ sudo port install cmake
$ sudo port install gcc6
Note: You can install a different gcc. It will work with gcc version 4.4.7 and above
$ sudo port install cfitsio +gcc6
Note: Make sure the variant matches the gcc version you installed.
(e.g. $ port install cfitsio +gcc46 if you installed gcc version 4.4.7)
Astro::FITS::CFITSIO Installation
You will need to install Astro::FITS::CFITSIO **after** you install the CFITSIO library. It can be installed from CPAN:
$ sudo cpan install Astro::FITS::CFITSIO
NOTE: If your CFTISIO library isn't located at /usr/local then you may need to specify its location with:
$ sudo CFITSIO=(base directory of library) cpan install Astro::FITS::CFITSIO
(e.g. $ sudo CFITSIO=/opt/local cpan install Astro::FITS::CFITSIO)
2. Untar the attached GBM RSP
$ tar xvjf gbmrsp-2.0.10.tar.bz2
3. Create the build directory in the base directory of the project
$ cd gbmrsp-2.0
$ mkdir build
You should have the following directory tree:
|— build
|— data
..|— GBMDRMdb002
....|— BGO_00
....|— BGO_01
....|— NAI_00
....|— NAI_01
....|— NAI_02
....|— NAI_03
....|— NAI_04
....|— NAI_05
....|— NAI_06
....|— NAI_07
....|— NAI_08
....|— NAI_09
....|— NAI_10
....|— NAI_11
..|— inputs
|— src
..|— fortran
..|— perl
4. Create the build files (by default this will install in /usr/local)
$ cd build
By default the software will install in /usr/local if this is what you want:
$ cmake ../src
Or to install to a different directory:
$ cmake -DCMAKE_INSTALL_PREFIX=your-target-directory ../src
(e.g. $ cmake -DCMAKE_INSTALL_PREFIX=/home/jdoe/gbmrsp ../src)
If you would like or need to specify the compiler to be used:
$ cmake -DCMAKE_Fortran_COMPILER=(full path name of your compiler) ../src
(e.g. $ cmake -DCMAKE_Fortran_COMPILER=/opt/local/bin/gfortran ../src)
To specify which perl executable should be used:
$ cmake -DPERL_EXEC=(full path name of perl) ../src
(e.g. $ cmake -DPERL_EXEC=/usr/local/bin/perl ../src )
You can use multiple defines with the cmake command, for example:
$ cmake -DCMAKE_INSTALL_PREFIX=/home/jdoe/gbmrsp \
-DCMAKE_Fortran_COMPILER=/opt/local/bin/gfortran \
-DPERL_EXEC=/usr/local/bin/perl ../src
Note: If the name of the compiler executable is something other than "gfortran" or "f95", or is not in the PATH you will have to specify it using
-DCMAKE_Fortan_COMPILER as shown above.
5. Compile and install the programs (remove sudo if not necessary)
$ make
$ sudo make install
To uninstall the programs, using Finder (for Macs):
1. Goto to the /usr/local directory with SHIFT-COMMAND-G.
2. Move the gbmrsp folder to the trash.
3. Open the bin directory by double clicking the folder.
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Courses SOCR Ivo Dinov's Home
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Ivo Dinov
UCLA Statistics, Neurology, LONI, Math/PIC
│ STAT 110 A │ Probability & Statistics for Engineers I │
│ Spring 2004 │ │
│ Department of Statistics │ Instructor: Ivo Dinov │
│ Homework 5 │ Due Date: Wednesday, June 09, 2004 │
Please, submit your homework right after lecture on the due date. Correct solutions to any 6 out of the 7 problems carry full credit. See the
HW submission rules
. On the front page include the
following header
• (HW_5_1) [Sec. 4.4, #62] A system consists of 5 identical components connected in a series as follows:
As soon as one component fails, the system fails. Suppose each component has a lifetime that is exponentially distributed with λ = 0.01 and they fail independently of one another. Let A be the
event that the i^th component lasts at least t hours.
(a) Are the A[i ]independent events (for i = 1, 2, 3, 4, 5)?
(b) Let X be the time at which the system fails. The event X[t] = {X ≥ t} is equivalent to what event involving the A[i]'s?
(c) Compute P(X[t]). What is F(t) = P (X[t])?
(d) What is the density of X? What is the distribution of X?
• (HW_5_2) [Sec. 4.4, #48] Suppose that 10% of all steel shafts produced by a certain process are non conforming, but can be reworked (rather than being scrapped). Consider a random sample of 200
shafts, let X denote the number among those that are non conforming and can be reworked. What is the exact distribution of X? What (approximately) is the probability that X is:
(a) At most 30?
(b) Less than 30?
(c) Between 15 and 25?
(d) What is the 33th percentile for X?
• (HW_5_3) Let X & Y have a joint density function given by
| (9/20)(xy+y)^2 0 ≤x ≤2; 0 ≤ y ≤ 1
f(x; y) = |
| 0, otherwise
(a) f(y | x), the conditional probability density function, p.d.f., of Y given X.
(b) P( Y < 1/2 | X < 1/2 ).
(c) E(Y | X = x).
• (HW_5_4) [Sec. 4.1, #10 & 24] List 8 to 10 probability mass/density functions that we have discussed in class as models for various natural processes.
(a) Give one example of a process that can be modeled by each distribution you listed.
(b) Identify the parameters, if any, for all distributions. Discuss the shape of the distribution, if known.
(c) If the mean and the variance of the distribution are known write them explicitly.
(d) In your own words state the Central Limit Theorem. What is its application to this collection of distributions you have presented.
• (HW_5_5) [Sec. 6.2, #22] Let X denote the proportion of allotted time that s randomly selected student spends working on a certain aptitude test. Suppose the pdf of X is
| (ϑ + 1) x^ϑ 0 ≤ x ≤ 1
f(x; ϑ) = |
| 0 otherwise
where ϑ > -1. A random sample of 10 students yields the following data
│ 0.92 │ 0.79 │ 0.90 │ 0.65 │ 0.86 │ 0.47 │ 0.73 │ 0.97 │ 0.94 │ 0.77 │
(a) Use the method of moments to obtain an estimator of ϑ and then use this to compute an actual estimate for these data.
(b) Obtain a maximum-likelihood estimator of ϑ and use it to calculate an estimate for the given data.
• (HW_5_6) A cigarette manufacturer claims that his cigarettes have an average nicotine content of 1.83 milligrams. If a random sample of 8 cigarettes of this type shows a sample mean of 1.95 with
sample deviation 0.22 milligrams, find 95% confidence interval for the population mean. Do you agree with the claim?
• (HW_5_7) Let p be the real probability of getting head when tossing a given coin. We tossed 500 times and found 260 heads. Find the 95% confidence interval for p. How many times should we toss
the coin in order to be 95% confident that our estimate of p is within 0.02?
[ ]
│ Ivo Dinov's Home │ http://www.stat.ucla.edu/~dinov │ Visitor number │ © I.D.Dinov 1997-2004 │
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You have 36 marbles that are red, white and blue. If 12 of the marbles are red, and 13 of the marble
Staff member
You have 36 marbles that are red, white and blue. If 12 of the marbles are red, and 13 of the marbles are blue, what fraction of marbles are white?
Calculate the number of white marbles:
Number of white marbles = Total marbles - Red marbles - Blue marbles
Number of white marbles = 36 - 12 - 13
Number of white marbles = 11
Calculate the fraction of white marbles:
Fraction of white marbles =Number of white marbles / Total marbles
Fraction of white marbles = 11/36 | {"url":"https://www.mathcelebrity.com/community/threads/you-have-36-marbles-that-are-red-white-and-blue-if-12-of-the-marbles-are-red-and-13-of-the-marble.3183/","timestamp":"2024-11-12T09:17:42Z","content_type":"text/html","content_length":"46289","record_id":"<urn:uuid:1eb6d8a2-e8f1-4767-af52-f34d538acb02>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00072.warc.gz"} |
Table of Contents Logarithm Properties - Simplifying Using Combinations The three basic properties for logarithms are... It is important to be able to. - ppt download
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Comments on Physics with an edge: Two body thought experimentI think I am starting to get this example: it is l...I assure you - it doesn't look like an EM-driv...What I am trying to show with MiHsC, and the infor...I think we can at least say that the mass-energy o...I think that these kinds of Gedankenexperiment hin...It is interesting that this is similar to the blac...@Mike
For a clean theory, the special cases must ...OK. Sheldon's photons would muddy the picture ...@Mike
Do you consider the universe to be a closed...Ordinarily you would be correct, but remember that...(eh in previous: "Unruh radiation symetry&quo...I just noticed, in your previous articles and comm...That extension of your theory sounds fascinating M...Indeed, I think it is all horizons, and I've m...Regarding inertia and gravity in context with Rind...Thanks: a interesting point. I have considered thi...Hello Mike,
I thought about your Gedankenexperime...
tag:blogger.com,1999:blog-4637778157419388168.post1542539733390355805..comments2024-10-26T11:40:33.076-07:00Mike McCullochhttp://www.blogger.com/profile/
00985573443686082382noreply@blogger.comBlogger17125tag:blogger.com,1999:blog-4637778157419388168.post-1795509024691378152015-09-11T01:45:37.472-07:002015-09-11T01:45:37.472-07:00I think I am starting
to get this example: it is like special relativity: wheile relativity says "there is no absolute frame of reference, all motion is relative", MiHsC says maybe something like "there is
no object with absolute mass, all mass is relative".<br /><br />The trick with your thought experiment is probably, that the only way Sheldon's "backpack" can work in empty "
two body universe" and give him some acceleration is by means of photon rocket (ie. emitting some elmg. radiation). The empty two-body universe is in fact cylindrical, so the radiation from
Sheldon's photon rocket travels all the way round the universe and then it is seen as kind "background radiation".<br /><br />I believe the paradox is resolved this way, because any two
objects sufficiently isolated from the rest of the universe would than behave this way, which we don't observe very often (unless we are super-cool, super-conducting, or exotic in some other way)
<br /><br />I believe before any of the two objects in empty universe starts to emit any photons, there can't be notion of space, time and acceleration: Sheldon can only start to accelerate when
he emits photon, which is the "big bang event" in the two body universe, the information is transmitted and this little universe's Hubble horizont starts to exists with t0 being time of
this photon emission.<br /><br />Amy can start to believe she is moving only after she receives this information or by feeling (measuring) acceleration, which, in this two body universe, will be the
same moment (event). But soon (after once more the entire lifetime of our little universe started by first photon emission/acceleration event, which can be pretty soon, for sufficiently small
universes) will the photons emmited by Sheldon travel one more time around and interfere with themselves... and this is where things start to be interesting (I have no clue what happens, to be true,
but the amount of energy/information being transmitted by accelerating Sheldon now must have definitely increased...)<br /><br />The problem we have with two body universe may also be topological -
ie., the distance between Amy and Sheldon is the same, no matter which direction we look (and it may be also the Planck length of our two body universe). So it is really no surprise Amy is not really
moving relative to Sheldon (firing his photon rocket in the single, "any-direction" of two body universe) and she is only heating up (I hope the narrative got more interesting at this
15185750418602711903noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-76828753421349042602015-09-09T04:27:34.637-07:002015-09-09T04:27:34.637-07:00I assure you - it doesn't
look like an EM-drive. :)<br /><br />I'll prepare a short PDF document. E-mail subject will be a simple "Hi" from a well-known german provider. If nothing arrives, please also look into
your spam folder, sometimes this happens (seems random).ZeroIsEverythinghttps://www.blogger.com/profile/
13236152077605874591noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-49386798380221820592015-09-09T01:30:41.158-07:002015-09-09T01:30:41.158-07:00What I am trying to show with
MiHsC, and the informational version I'm now working on, is that it is not mass-energy that is conserved but mass-energy-information. So information can be converted to inertial mass / energy and
vice versa. This thought experiment is an easier way to think about some of the consequences of this.<br /><br />I'd be interested to hear about your CoM-breaking machine. It doesn't look
like an emdrive does it? :) Feel free to contact me at my Plymouth email address.Mike McCullochhttps://www.blogger.com/profile/
00985573443686082382noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-31078690494378126752015-09-08T23:53:59.980-07:002015-09-08T23:53:59.980-07:00I think we can at least say that
the mass-energy of a Black Hole is equivalent to the mass-energy of the stuff that fell into it. The Hubble horizon is <i>much</i> worse, because mass-energy is simply 'disappearing' from our
accessible/observable physical reality. Our Hubble horizon limited local universe is losing stuff all the time - momentum vectors, energy, information.. it's like our universe itself is kind of
evaporating into nothingness. Maybe there's even a half-life constant for universes in general.. who knows?ZeroIsEverythinghttps://www.blogger.com/profile/
13236152077605874591noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-48810567819753015002015-09-08T23:28:07.083-07:002015-09-08T23:28:07.083-07:00I think that these kinds of
Gedankenexperiment hint towards a possibility to engineer an <i>apparent</i> (local) breaking of CoM by shifting the 50/50 statistics of action&reaction by appropriate usage of the limited speed
of light that is the propagation limit of information/energy. I mean.. information loss seems to happen in 'nature' on the Hubble horizon all the time (also caused by limited speed of
light!).. so I don't see any reason (for now) why we shouldn't be able to come up with a machine that can do this locally, too.<br /><br />For many years now, I've been studying possible
ways to engineer this kind of effect. After a long elimination process, I eventually came up with a possible solution to make use of the limited speed of information/energy transfer. If you're
interested, I could elaborate on my concept (in case, would your plymouth e-mal address be OK?).<br /><br />DISCLAIMER: Only standard physics involved.ZeroIsEverythinghttps://www.blogger.com/profile/
13236152077605874591noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-20100840772183537922015-09-08T13:30:06.072-07:002015-09-08T13:30:06.072-07:00It is interesting that this is
similar to the black hole information paradox: a puzzle over what happens to the information represented by matter when it is swallowed by a black hole's event horizon. In our case the rocket
propellant is swallowed by the Hubble horizon, and we are inside that horizon.Mike McCullochhttps://www.blogger.com/profile/
00985573443686082382noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-6977846129974751192015-09-08T06:46:07.802-07:002015-09-08T06:46:07.802-07:00@Mike<br /><br />For a clean
theory, the special cases must be correctly considered and come out of the formulae in a natural way :) .<br /><br />Good thing you're mentioning the Hubble horizon. Imagine a regular rocket in
space. It starts fresh and full and fires until its fuel is depleted. Propellant and rocket speed in opposite directions, the center of mass of (rocket+propellant) doesn't move to suffice
conservation of momentum. Let's say that the rocket was so heavy that the propellant moves at double the rocket's speed. After an eternity, the speeding propellant crosses the Hubble horizon
and is.. 'gone'. An observer of the rocket could now try to find the mass of a possible propellant, but it's hopeless. There is none to be found.<br /><br />So there is now a rocket body
in the local universe that speeds along for now unknownable reasons. Physics is based on empirical data, but in this case we can't find any data about the missing propellant mass. Just going by
observables and adding all vectors of the universe, CoM appears to be in a broken state. I think it's glorious :) .ZeroIsEverythinghttps://www.blogger.com/profile/
13236152077605874591noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-19720491263763422072015-09-08T02:46:52.175-07:002015-09-08T02:46:52.175-07:00OK. Sheldon's photons would
muddy the picture in a closed system, though we can imagine that they are fired away from Amy with no dynamic effect on her and eventually reach the Hubble edge and become unobservable for A and S
(admittedly with consequences for information).Mike McCullochhttps://www.blogger.com/profile/
00985573443686082382noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-52514296419509963822015-09-07T05:46:00.447-07:002015-09-07T05:46:00.447-07:00@Mike<br /><br />Do you consider
the universe to be a closed continuum in this picture? I played around a bit with just two masses within an otherwise empty spatial closed continuum with non-zero curvature. If you draw the continuum
on paper as a 2D circle and put Amy&Sheldon exactly one half-circle away from each other (to get symmetrical conditions) - what will happen during acceleration?<br /><br />When Sheldon
accelerates towards Amy, he certainly has to use a propulsion system. Let's say he's using a photon rocket. During Sheldon's acceleration, a Rindler horizon forms. The resulting Unruh
radiation would push against his acceleration force towards Amy. Amy would see the same thing happening, BUT: Sheldon's photon-rocket photons would inevitable hit Amy and impart the same, but
opposite impulse on her that Sheldon experiences.<br /><br />So, I think that Amy would, in effect, not really move the same way with Sheldon, while Sheldon speeds towards her.ZeroIsEverythinghttps:/
Ordinarily you would be correct, but remember that I've taken all other matter out of this cosmos, and the way we 'feel' acceleration is via inertia, and I agree with Mach that this is
due to the other matter in the cosmos. The only other matter Amy sees is Sheldon who is accelerating with respect to her. So I think she would believe herself to be accelerating and see a Rindler
horizon.Mike McCullochhttps://www.blogger.com/profile/
00985573443686082382noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-83261414451623505032015-09-06T13:44:06.205-07:002015-09-06T13:44:06.205-07:00(eh in previous: "Unruh
radiation symetry", sorry, still new to this language)TeckaCZhttps://www.blogger.com/profile/
15185750418602711903noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-48477985123749374742015-09-06T13:43:10.174-07:002015-09-06T13:43:10.174-07:00I just noticed, in your previous
articles and comments under them, that you correctly noticed, that accelerating body feels acceleration (or gravity - you can make difference only by observing the whole system you are in).<br /><br
/>So Sheldon with jetpack feels (measures) acceleration, when jetpack is turned on.<br /><br />Amy is not feeling acceleration, so if the universe modifies rindler radiation symetry without any
obvious case (think Amy does not even "make observation" - there is no interaction - of Sheldon with jetpack, turned on or off), it would be quite strange. <br /><br />So my opinion on this
paradox is: no, Amy will not see rindler horizon just because Sheldon accelerates. Rather my explanation is, that the relative acceleration, as observed from other points (including Amy, but not
limited to her) will not be the same, as Sheldon would expect from his acceleration measurements. (This is BTW quite similar to situation, when you are accelerating in gravitation field: you also
cannot tell, without additional input or observation of system as a whole, how much of the total acceleration you feel is gravity and what is real acceleration. So accelerating Sheldon may not know,
what his real acceleration relative to Amy is, just by his accelerometer: but, floating weightles, not orbiting Sheldon or whatever, Amy will not see Rindler horizon)<br /><br />Anyway the talk about
reducing it all to information sounds very interesting to me... it was something I was kind of expecting all the time.<br /><br />(I post as @TeckaCZ - my e-zine identity - on Twitter, and I admit I
am just failed linux software developer, not physicist, although I have some affinity to it)TeckaCZhttps://www.blogger.com/profile/
15185750418602711903noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-16496612193875636002015-09-05T16:51:13.332-07:002015-09-05T16:51:13.332-07:00That extension of your theory
sounds fascinating Mike.qraalhttps://www.blogger.com/profile/
13436948899560519608noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-59198270851685714622015-09-05T12:42:02.990-07:002015-09-05T12:42:02.990-07:00Indeed, I think it is all
horizons, and I've made many attempts to get gravity from Unruh radiation and sheltering, and from the uncertainty principle, but I think now the route is more radical and more directly involves
information. I'm in the process of deriving MiHsC solely from information (only a factor of 2 still stands in my way) and my hope is that gravity will then come out as well.Mike McCullochhttps://
Regarding inertia and gravity in context with Rindler horizons, I think it might be like this:<br /><br />- Inertia is based on dynamically created Rindler horizons.<br />- Gravity is based on static
(or 'frozen') Rindler horizons.<br /><br />The ultimate static Rindler or information horizon is.. a Black Hole. However, as Hawking radiation illustrates, even such an impressive information
horizon like a Black Hole finally 'evaporates'.<br />If we view physical 'mass' as a sort of particle type dependent stable topological defect in physical spacetime (that hinders free
information exchange in between arbitrary points in spacetime), caused during the events of what we call 'Big Bang', it might become obvious that even 'static' Rindler horizons don&#
39;t, or rather <i>can't</i> last forever. In the end, any known particle has a statistical half life.. after which it 'evaporates', or after which the topological defect that it
represents in spacetime 'normalizes' again to a less disturbed state, that allows better or free information exchange in between arbitrary points in spacetime. We could think of it as a
built-in 'self-healing' capability.ZeroIsEverythinghttps://www.blogger.com/profile/
13236152077605874591noreply@blogger.comtag:blogger.com,1999:blog-4637778157419388168.post-70823599235110282452015-09-04T03:05:59.781-07:002015-09-04T03:05:59.781-07:00Thanks: a interesting point. I
have considered this: in the referenced paper I modeled the decay of the effect of a mass on the inertia of another one with 1/distance^2, like gravity (see the paragraph before eq. 6). I had no
logical reason to chose an inverse square, but this particular result wasn't very sensitive to it.<br /><br />It is interesting that if there are only 2 masses and you move them apart then
initially, using Mach's approach, you might say that the effect should not diminish with distance, since we can still only determine the acceleration of each mass with reference to the other
mass. It may be that one can derive a distance dependence by quantifying the greater inaccuracy in measuring the acceleration of the other mass (using light) from a distance.Mike McCullochhttps://
Mike,<br /><br />I thought about your Gedankenexperiment (I'm allowed to write it this way.. I'm German :p).<br /><br />Imagine a 2D Euclidean coordinate system with three masses, which each
have a considerable distance in between them. Mass A is in the origin of the coordinate system. Mass B is somewhere exactly on the (vertical) Y-axis and mass C is exactly on the (horizontal) X-axis.
I want to call this an 'orthogonal placement of masses'.<br />If A accelerates towards C, then the distance between A and B obviously changes less quickly, than it does between A and C. I
think it should then logically follow, that the farther B is away from A, when A is accelerating towards C, the less B should matter in this whole situation. If, in fact, the distance(A,B) is large
enough, the effect of B on A while accelerating should become negligible.<br />So.. not all masses present in the universe should have a uniform influence on inertial effects, derived from Rindler
horizons? What do you think about my extension of your Gedankenexperiment?ZeroIsEverythinghttps://www.blogger.com/profile/13236152077605874591noreply@blogger.com | {"url":"https://physicsfromtheedge.blogspot.com/feeds/1542539733390355805/comments/default","timestamp":"2024-11-06T08:58:15Z","content_type":"application/atom+xml","content_length":"44042","record_id":"<urn:uuid:32208b3d-e015-4216-bf3b-49440fecaeb8>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00764.warc.gz"} |
Investing 101 - What is Investing? - RIS - Ryan's Investment Strategies
Investing 101 - What is investing?
Investing is when you are expecting a return after putting time or money into something, typically for a long period of time.
People will typically invest towards a goal such as retirement or their kids college tuition. The goal of investing is to generate income and increase the value of it over time.
Investing is different from saving or trading. Savings are sometimes guaranteed and investments are not. Compared to investing, trading involves buying and selling stocks on a more regular basis.
What is Compound Interest?
Katie Kerpel {Copyright} Investopedia, 2019.
According to Merriam-Webster, compound interest is interest computed on the sum of an original principal and accrued interest.
To first understand compound interest, lets go over simple interest.
Simple interest is the interest earned on the original principal only.
Example – Simple Interest
Bob invests $10,000 in a savings account with 5% simple interest for 3 years. The interest you earn each year is 5% x $10,000, which is $500. After 3 years, you would earn $1,500 ($500 x 3 years).
Simple Interest Calculation over 3 years:
Year 1: $10,000 x 5% = $500
Year 2: $10,000 x 5% = $500
Year 3: $10,000 x 5% = $500
In total Bob earned interest of –
$500 + $500 + $500 = $1,500
Bob earned $1,500 in simple interest.
Example – Compound Interest
Instead of investing in a savings account with simple interest, Bob invests $10,000 in a savings account with compound interest (compounded annually) for 3 years.
After the first year, Bob would earn the same amount of interest as he did in his simple interest account ($500).
After the second year (because it’s compounded), the interest would be 5% x $10,500, which is $525. This would give Bob a total of $11,025. Compound interest takes into account your principal and the
interest that you have previously incurred. In this case, that would be Bob’s original investment of $10,000 and the interest that he earned in the first year, $500.
After the third year, the interest would be 5% * $11,025, which is $551.25.
Compound interest calculation over 3 years:
Year 1: $10,000 x 5% = $500
Year 1 Total: $10,000 + $500 = $10,500
Year 2: $10,500 * 5% = $525
Year 2 Total: $10,500 + $525 = $11,025
Year 3: $11,025 * 5% = $551.25
Year 3 Total: $11,025 + $551.25 = $11,576.25
In total Bob earned interest of –
$500 + $525 + $551.25 = $1,576.25
Bob earned $1,576.25 in compound interest
Bob earned more with compound interest than he did with simple interest.
Compound interest total = $1,576.25
Simple interest total = $1,500
Total difference = $76.25
Bob earned an extra $76.25 more by investing with compound interest.
Published in 1994 by USAA, the chart above shows how much money you'll accumulate over time if you invest $250 a month starting at different ages.
The earlier you start investing, the better. Compound interest works best when you start early because it gives you more time for your investment to accumulate interest and grow.
Warren Buffet has stated that the single most powerful factor behind his investing success is “compound interest” and Albert Einstein stated that compound interest is the eighth wonder of the world.
Click here to see how fast your investments could grow. | {"url":"http://www.ryansinvestmentstrategies.com/what-is-investing/","timestamp":"2024-11-03T09:15:27Z","content_type":"text/html","content_length":"83946","record_id":"<urn:uuid:7a41ae47-946c-4338-8ac1-3969b20bc6b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00613.warc.gz"} |
# Five different ways of binding together two matrices
x <- matrix(1:12, 3, 4)
y <- x + 100
dim(Abind(x, y, along=0)) # binds on new dimension before first
dim(Abind(x, y, along=1)) # binds on first dimension
dim(Abind(x, y, along=1.5))
dim(Abind(x, y, along=2))
dim(Abind(x, y, along=3))
dim(Abind(x, y, rev.along=1)) # binds on last dimension
dim(Abind(x, y, rev.along=0)) # binds on new dimension after last
# Unlike cbind or rbind in that the default is to bind
# along the last dimension of the inputs, which for vectors
# means the result is a vector (because a vector is
# treated as an array with length(dim(x))==1).
Abind(x=1:4, y=5:8)
# Like cbind
Abind(x=1:4, y=5:8, along=2)
Abind(x=1:4, matrix(5:20, nrow=4), along=2)
Abind(1:4, matrix(5:20, nrow=4), along=2)
# Like rbind
Abind(x=1:4, matrix(5:20, nrow=4), along=1)
Abind(1:4, matrix(5:20, nrow=4), along=1)
# Create a 3-d array out of two matrices
Abind(x=matrix(1:16, nrow=4), y=matrix(17:32, nrow=4), along=3)
# Use of hier.names
Abind(x=cbind(a=1:3, b=4:6), y=cbind(a=7:9, b=10:12), hier.names=TRUE)
# Use a list argument
Abind(list(x=x, y=x), along=3)
# Use lapply(..., get) to get the objects
an <- c('x', 'y')
names(an) <- an
Abind(lapply(an, get), along=3) | {"url":"https://www.rdocumentation.org/packages/DescTools/versions/0.99.57/topics/Abind","timestamp":"2024-11-13T11:57:22Z","content_type":"text/html","content_length":"124490","record_id":"<urn:uuid:98dfb179-9108-493d-b5d6-cc9242c0c9f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00115.warc.gz"} |
Percentage Difference Calculator for Accurate Math Solutions - Powerful and Useful Online Calculators
Percentage Difference Calculator for Accurate Math Solutions
Have you ever been in a situation where you needed to compare two numbers but weren’t sure how to express the difference between them accurately? Understanding how much two numbers vary from each
other can be essential in everyday life, whether you’re dealing with prices, statistics, or simple calculations. That’s where a percentage difference calculator comes into play.
Percentage Difference Calculator for Accurate Math Solutions
We often encounter situations where comparing the significance of two numbers is crucial. Whether you are working with financial data, comparing statistics, or dealing with day-to-day numbers,
calculating the percentage difference can provide clear insights. This article will cover everything you need to know about the percentage difference calculator and how to use it accurately.
Directions for Use
Using a percentage difference calculator is straightforward. To determine the percentage difference between two numbers, simply enter the known values into the fields labeled “Value 1” (V₁) and
“Value 2” (V₂). Once both numbers are entered, press the “Calculate” button. Remember, you can only enter positive integers or decimal numbers.
What is Percentage Difference?
The percentage difference is a way to quantify the difference between two numbers when they hold equal worth. This metric is useful for comparisons where there isn’t a clear reference point. Unlike
percentage change, which requires an old value and a new value, the percentage difference uses the average of the two numbers as the reference point.
The Difference Between Percentage Difference and Percentage Change
It’s common to mix up percentage difference with percentage change, but they serve distinct purposes. Percentage change is used when comparing an old value with a new value, where the old value
serves as a baseline. In contrast, the percentage difference is used to compare two values of equal importance without a baseline.
Here’s a more precise explanation using their respective formulas:
Percentage Change Formula [ \text = \left( \frac{|\text – \text|}{\text} \right) \times 100 ]
Percentage Difference Formula [ \text = \left( \frac{|V_1 – V_2|}{\frac{(V_1 + V_2)}} \right) \times 100 ]
Formula for Percentage Difference
Let’s delve deeper into the formula to understand its components: [ \text = \left( \frac{|V_1 – V_2|}{\frac{(V_1 + V_2)}} \right) \times 100 ]
• V₁ and V₂: These are the two values you’re comparing.
• |V₁ – V₂|: This denotes the absolute difference between the two values.
• (V₁ + V₂)/2: This is the average of the two values.
The formula essentially provides a measure of how much one number is different from the other compared to their average.
Example Calculation
Let’s illustrate the concept with an example. Suppose you want to find the percentage difference between 6 and 9.
Using the formula: [ \text = \left( \frac{|6 – 9|}{\frac} \right) \times 100 ]
Step-by-step: [ \text = \left( \frac{|-3|}{\frac} \right) \times 100 ]
[ = \left( \frac \right) \times 100 ]
[ = 0.4 \times 100 ]
[ = 40% ]
So, the percentage difference between 6 and 9 is 40%.
Why Can the Percentage Difference Be Confusing?
While the percentage difference is a powerful tool, it can be misleading in certain scenarios. Particularly, when comparing values of significantly different magnitudes, the result may not adequately
represent the actual difference. For instance, let’s see how the calculation varies with larger numbers:
Comparing 6 and 90:
[ \text = \left( \frac{|6 – 90|}{\frac} \right) \times 100 ]
[ = \left( \frac \right) \times 100 ]
[ = 175% ]
Comparing 6 and 900:
[ \text = \left( \frac{|6 – 900|}{\frac} \right) \times 100 ]
[ = \left( \frac \right) \times 100 ]
[ \approx 197.35% ]
Comparing 6 and 9000:
[ \text = \left( \frac{|6 – 9000|}{\frac} \right) \times 100 ]
[ = \left( \frac \right) \times 100 ]
[ \approx 199.73% ]
As you can see, even though the absolute differences are growing exponentially, the percentage difference plateaus. This occurs because, in such scenarios, the relative comparison to the average
diminishes, making the percentage difference less intuitive.
Practical Example
Let’s consider a real-world example where you check the price of a pair of sneakers in two shops:
• Shop 1 Price (V₁): $110
• Shop 2 Price (V₂): $120
Using the formula: [ \text = \left( \frac{|110 – 120|}{\frac} \right) \times 100 ] [ = \left( \frac \right) \times 100 ] [ \approx 8.7% ]
This calculation shows that the price difference between the two shops is approximately 8.7%.
Related Calculators
To further enhance your calculations and comparisons, several other percentage-based calculators can be helpful:
• Percentage Calculator: For general percentage computations.
• Percentage Increase Calculator: To determine the increase in percentage between two values.
• Percentage Change Calculator: For calculating the change from an old value to a new value.
• Percentage Decrease Calculator: To find the decrease in percentage between two values.
Each of these tools can provide added insights depending on your specific needs.
Understanding and calculating the percentage difference is essential for accurate and meaningful comparisons between two numbers. Whether you’re comparing prices, statistics, or other numerical data,
the percentage difference provides a clear and consistent measurement. However, always be cautious when comparing values of differing magnitudes, as the results can sometimes be misleading.
Utilizing calculators like the percentage difference calculator can simplify your tasks and ensure accuracy in your results. So next time you find yourself needing to compare two values, you know
exactly how to do it seamlessly and accurately. | {"url":"https://calculatorbeast.com/percentage-difference-calculator-for-accurate-math-solutions/","timestamp":"2024-11-13T11:47:41Z","content_type":"text/html","content_length":"130892","record_id":"<urn:uuid:99bf3de1-d68f-4a41-86d5-b01ed4c7641f>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00480.warc.gz"} |
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Registered: 31.12.2001
From: 127.0.0.1
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Ch. 9 Key Concepts - Prealgebra | OpenStax
Key Concepts
9.1 Use a Problem Solving Strategy
• Problem Solving Strategy
1. Step 1. Read the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don't understand, look them up in a
dictionary or on the internet.
2. Step 2. Identify what you are looking for.
3. Step 3. Name what you are looking for. Choose a variable to represent that quantity.
4. Step 4. Translate into an equation. It may be helpful to first restate the problem in one sentence before translating.
5. Step 5. Solve the equation using good algebra techniques.
6. Step 6. Check the answer in the problem. Make sure it makes sense.
7. Step 7. Answer the question with a complete sentence.
9.2 Solve Money Applications
• Finding the Total Value for Coins of the Same Type
□ For coins of the same type, the total value can be found as follows:
$number·value=total valuenumber·value=total value$
where number is the number of coins, value is the value of each coin, and total value is the total value of all the coins.
• Solve a Coin Word Problem
1. Step 1. Read the problem. Make sure you understand all the words and ideas, and create a table to organize the information.
2. Step 2. Identify what you are looking for.
3. Step 3.
what you are looking for. Choose a variable to represent that quantity.
☆ Use variable expressions to represent the number of each type of coin and write them in the table.
☆ Multiply the number times the value to get the total value of each type of coin.
4. Step 4. Translate into an equation. Write the equation by adding the total values of all the types of coins.
5. Step 5. Solve the equation using good algebra techniques.
6. Step 6. Check the answer in the problem and make sure it makes sense.
7. Step 7. Answer the question with a complete sentence.
• Type $NumberNumber$ $Value ()Value ()$ $Total Value ()Total Value ()$
9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
• Supplementary and Complementary Angles
□ If the sum of the measures of two angles is 180°, then the angles are supplementary.
□ If $∠A∠A$ and $∠B∠B$ are supplementary, then $m∠A+m∠B=180m∠A+m∠B=180$.
□ If the sum of the measures of two angles is 90°, then the angles are complementary.
□ If $∠A∠A$ and $∠B∠B$ are complementary, then $m∠A+m∠B=90m∠A+m∠B=90$.
• Solve Geometry Applications
1. Step 1. Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
2. Step 2. Identify what you are looking for.
3. Step 3. Name what you are looking for and choose a variable to represent it.
4. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
5. Step 5. Solve the equation using good algebra techniques.
6. Step 6. Check the answer in the problem and make sure it makes sense.
7. Step 7. Answer the question with a complete sentence.
• Sum of the Measures of the Angles of a Triangle
□ For any $ΔABC,ΔABC,$ the sum of the measures is 180°
□ $m∠A+m∠B=180m∠A+m∠B=180$
• Right Triangle
□ A right triangle is a triangle that has one 90° angle, which is often marked with a $⦜⦜$symbol.
• Properties of Similar Triangles
□ If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio.
9.4 Use Properties of Rectangles, Triangles, and Trapezoids
• Properties of Rectangles
□ Rectangles have four sides and four right (90°) angles.
□ The lengths of opposite sides are equal.
□ The perimeter, $PP$, of a rectangle is the sum of twice the length and twice the width.
□ The area, $AA$, of a rectangle is the length times the width.
• Triangle Properties
□ For any triangle $ΔABCΔABC$, the sum of the measures of the angles is 180°.
☆ $m∠A+m∠B+m∠C=180°m∠A+m∠B+m∠C=180°$
□ The perimeter of a triangle is the sum of the lengths of the sides.
□ The area of a triangle is one-half the base, b, times the height, h.
9.5 Solve Geometry Applications: Circles and Irregular Figures
• Problem Solving Strategy for Geometry Applications
1. Step 1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
2. Step 2. Identify what you are looking for.
3. Step 3. Name what you are looking for. Choose a variable to represent that quantity.
4. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
5. Step 5. Solve the equation using good algebra techniques.
6. Step 6. Check the answer in the problem and make sure it makes sense.
7. Step 7. Answer the question with a complete sentence.
• Properties of Circles
□ $d=2rd=2r$
□ Circumference: $C=2πrC=2πr$ or $C=πdC=πd$
□ Area: $A=πr2A=πr2$
9.6 Solve Geometry Applications: Volume and Surface Area
• Volume and Surface Area of a Rectangular Solid
□ $V=LWHV=LWH$
□ $S=2LH+2LW+2WHS=2LH+2LW+2WH$
• Volume and Surface Area of a Cube
• Volume and Surface Area of a Sphere
□ $V=43πr3V=43πr3$
□ $S=4πr2S=4πr2$
• Volume and Surface Area of a Cylinder
□ $V=πr2hV=πr2h$
□ $S=2πr2+2πrhS=2πr2+2πrh$
• Volume of a Cone
□ For a cone with radius $rr$ and height $hh$:
Volume: $V=13πr2hV=13πr2h$ | {"url":"https://openstax.org/books/prealgebra/pages/9-key-concepts","timestamp":"2024-11-04T18:10:33Z","content_type":"text/html","content_length":"370854","record_id":"<urn:uuid:8ca88a84-294a-4386-b44c-e81272a5aa97>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00226.warc.gz"} |
RLC Circuits and Resonance | Curious Toons
Table of Contents
Welcome, future physicists! As we embark on this thrilling journey through the universe of physics, I want you to consider something fascinating: everything around you—every smartphone, every car,
every star—follows the profound laws of physics. Imagine the secrets locked in the motion of a soccer ball, the dance of a rollercoaster, or the inexplicable pull of gravity that keeps us grounded.
Throughout this course, we will unravel the incredible stories behind everyday phenomena and explore the grand mysteries of the cosmos. We’ll delve into the nature of energy, uncover the principles
of force and motion, and even touch upon the strange world of quantum mechanics.
But this isn’t just about formulas and equations; it’s about curiosity and creativity. Why does the sky turn colors at sunset? How can we send a spacecraft to Mars? Each question leads us deeper into
understanding the very fabric of reality.
So gear up! Science is not just a subject; it’s a way of seeing the world. Together, we will ignite your passion for discovery, challenge your thinking, and empower you to see the extraordinary in
the ordinary. Let’s uncover the universe together!
1. Introduction to RLC Circuits
1.1 Components of an RLC Circuit
An RLC circuit is a fundamental electrical circuit that consists of three key components: a resistor (R), an inductor (L), and a capacitor (C). Each component has unique roles in the circuit’s
behavior. The resistor, measured in ohms (Ω), provides resistance that dissipates electrical energy as heat, thereby affecting the overall current in the circuit. The inductor, measured in henries
(H), stores energy in a magnetic field when electrical current passes through it, influencing the circuit’s reactance in response to changes in current. Lastly, the capacitor, measured in farads (F),
stores electrical energy in an electric field, releasing it when needed, which allows the circuit to respond to voltage changes.
The interplay between these components determines the circuit’s resonance frequency, the point at which the inductive and capacitive reactances cancel each other out, resulting in maximum current
flow. Understanding these components is crucial for analyzing both alternating current (AC) and direct current (DC) circuits, laying the groundwork for exploring more complex electrical systems.
Component Symbol Unit
Resistor R Ohms (Ω)
Inductor L Henries (H)
Capacitor C Farads (F)
1.2 Understanding AC Circuits
Understanding AC circuits is fundamental to grasping RLC (Resistor, Inductor, Capacitor) circuits. Unlike direct current (DC) circuits, where the current flows in a single direction, alternating
current (AC) circuits involve current that periodically reverses direction, creating a varying voltage over time. This periodic nature can be described using sinusoidal waveforms, characterized by
their amplitude, frequency, and phase.
In AC circuits, components like resistors, inductors, and capacitors behave differently. For instance, resistors follow Ohm’s law, while inductors oppose changes in current (leading to a phase
shift), and capacitors oppose changes in voltage. The interplay of these components leads to resonance, a condition where the circuit can oscillate at maximum amplitude at a specific frequency called
the resonant frequency.
To visualize this, consider the following table summarizing the behavior of each component in an AC circuit:
Component Voltage and Current Relationship Phase Shift
Resistor V = I * R 0°
Inductor V = L * (di/dt) +90°
Capacitor I = C * (dv/dt) -90°
Understanding these principles is key to mastering RLC circuits and their applications in real-world technologies.
2. Resonance in RLC Circuits
2.1 Definition of Resonance
Resonance in RLC circuits refers to the phenomenon that occurs when an electrical circuit is driven by an external alternating current (AC) source at a frequency that matches the natural frequency of
the circuit. In an RLC circuit, which consists of a resistor (R), an inductor (L), and a capacitor (C), resonance leads to a significant increase in the amplitude of the circuit’s oscillations. The
natural frequency, also known as the resonant frequency ((f_0)), can be calculated using the formula:
f_0 = \frac{1}{2\pi\sqrt{LC}}
At resonance, the inductive and capacitive reactances ((XL) and (XC)) become equal in magnitude, resulting in the impedance of the circuit being minimized to just the resistance ((R)). This allows
maximum current to flow through the circuit. The phenomenon of resonance is critical in various applications, including radio transmission, audio systems, and signal processing, as it helps in
selecting specific frequencies and enhancing signal strengths. Understanding resonance allows for the design of more effective and efficient electronic systems.
2.2 Conditions for Resonance
In an RLC circuit, resonance occurs when the inductive reactance (XL) and capacitive reactance (XC) are equal, resulting in the circuit’s impedance being at its minimum, which allows for maximum
current flow. The condition for resonance can be mathematically expressed as:
[ XL = XC ]
• ( X_L = 2\pi f L ) (inductive reactance)
• ( X_C = \frac{1}{2\pi f C} ) (capacitive reactance)
To achieve resonance, the driving frequency of the alternating current (AC) source must match the natural frequency of the circuit. This natural frequency (( f_0 )) can be calculated using the
[ f_0 = \frac{1}{2\pi \sqrt{LC}} ]
At resonance, the effects of inductance and capacitance cancel each other out, leading to a purely resistive impedance, Z, defined as:
[ Z = R ]
This is critical because, at resonance, the circuit can efficiently transfer energy, maximizing power output. Understanding these conditions is essential for designing circuits in applications such
as radio transmitters and receivers, where precise tuning to resonate frequencies is vital for optimal performance.
3. Mathematics of RLC Circuits
3.1 Impedance in RLC Circuits
Impedance (Z) in RLC circuits is a complex quantity that extends the concept of resistance to AC circuits, incorporating the effects of both inductance (L) and capacitance (C). It represents the
total opposition that a circuit presents to the flow of alternating current (AC) and is measured in ohms (Ω). Impedance can be expressed as a combination of resistance (R), inductive reactance (XL),
and capacitive reactance (XC). The formula for impedance in a series RLC circuit is given by:
Z = R + j(XL – XC)
where ( j ) is the imaginary unit. Inductive reactance increases with frequency (( XL = 2\pi f L )), while capacitive reactance decreases with frequency (( XC = \frac{1}{2\pi f C} )). Thus, at
resonance, where ( XL = XC ), the circuit’s impedance is minimized and equals the resistance ( Z = R ). Understanding impedance is crucial for analyzing the behavior of RLC circuits, particularly in
tuning, filtering, and resonance applications.
Here’s a concise table summarizing the key components:
Component Formula Reactance Type
Resistance (R) R Real part
Inductive (L) ( X_L = 2\pi f L ) Positive imaginary
Capacitive (C) ( X_C = \frac{1}{2\pi f C} ) Negative imaginary
3.2 Resonant Frequency Calculation
In RLC circuits, the resonant frequency (f₀) is the frequency at which the inductive reactance (Xₗ) and capacitive reactance (X𝑐) are equal, causing the circuit to exhibit maximum voltage across the
load. This occurs when the impedance of the circuit is minimized, leading to resonance. The resonant frequency can be calculated using the formula:
[ f₀ = \frac{1}{2\pi\sqrt{LC}} ]
where (L) is the inductance in henries (H) and (C) is the capacitance in farads (F).
To illustrate, consider the following example:
Component Value
Inductance (L) 10 mH
Capacitance (C) 100 nF
Resonant Frequency (f₀) ?
Using the formula, we convert the values: (L = 10 \times 10^{-3} H) and (C = 100 \times 10^{-9} F).
[ f₀ = \frac{1}{2\pi\sqrt{10 \times 10^{-3} \times 100 \times 10^{-9}}} \approx 159.15 \, \text{kHz} ]
Thus, the resonant frequency for this RLC circuit is approximately 159.15 kHz, where the circuit will efficiently exchange energy between the inductor and capacitor.
4. Behavior of RLC Circuits at Resonance
4.1 Current and Voltage Characteristics
In RLC circuits consisting of resistors (R), inductors (L), and capacitors (C), the behavior at resonance is crucial for understanding current and voltage characteristics. At resonance, the inductive
reactance (XL) equals the capacitive reactance (XC), leading to maximum current in the circuit. This phenomenon occurs at the resonant frequency (f₀), calculated using the formula:
f₀ = \frac{1}{2\pi\sqrt{LC}}
At this frequency, the impedance (Z) is minimized, primarily determined by the resistance (R). Consequently, the circuit behaves like a purely resistive load, resulting in the maximum voltage across
the components.
The current (I) can be described by Ohm’s law, (I = \frac{V}{R}), signifying that the current is in phase with the voltage across the resistor. In contrast, the voltage across the inductor and
capacitor can be significantly out of phase, which affects the overall voltage characteristics.
At resonance, the following relationships summarize the current and voltage behavior:
Component Current Phase Voltage Phase
Resistor In phase In phase
Inductor Lags current Leads current
Capacitor Leads current Lags current
This phase relationship is essential for analyzing RLC circuit behavior, particularly in applications like tuning and filtering in electronic systems.
4.2 Quality Factor (Q Factor)
The Quality Factor, or Q Factor, is a crucial concept in the study of RLC (Resistor-Inductor-Capacitor) circuits, particularly at resonance. It quantifies how underdamped an oscillator or resonant
circuit is and reflects the sharpness of the resonance peak in the response curve. Mathematically, the Q Factor is defined as the ratio of the resonant frequency ((f_0)) to the bandwidth ((Δf)) of
the circuit, represented as:
Q = \frac{f_0}{Δf}
A high Q value indicates a narrow bandwidth and a sharp resonance peak, which means the circuit oscillates with a higher purity tone and stores energy efficiently. Conversely, a low Q value suggests
a broader bandwidth, indicating more energy dissipation, often due to resistive losses. The Q Factor can also be expressed in terms of circuit components:
Q = \frac{1}{R} \sqrt{\frac{L}{C}}
Here, (R) is resistance, (L) is inductance, and (C) is capacitance. In practical applications, understanding the Q Factor helps engineers design circuits for various purposes, such as in radio
transmitters and audio equipment, where selectivity and signal integrity are vital.
5. Applications of Resonance in Real Life
5.1 Oscillators and Tuned Circuits
In the realm of electronics and communication, oscillators and tuned circuits play a vital role in generating and controlling waveforms. An oscillator is an electronic circuit that produces periodic
oscillations, typically in the form of sine or square waves. These oscillations are crucial for various applications, including signal generation, clock pulses, and radio transmissions. There are
different types of oscillators, such as LC oscillators, which utilize inductors (L) and capacitors (C) to create resonance at a specific frequency, allowing them to generate stable waveforms.
Tuned circuits, often referred to as resonant circuits, consist of a capacitor and inductor connected together in either series or parallel configurations. These circuits can selectively respond to a
particular frequency while rejecting others, which is essential in tuning radios to receive specific stations. The resonance frequency (( f_r )) of a tuned circuit can be calculated using the
f_r = \frac{1}{2\pi\sqrt{LC}}
where ( L ) is the inductance and ( C ) is the capacitance. By adjusting the values of ( L ) and ( C ), we can tune the circuit to the desired frequency, demonstrating the practical applications of
resonance in our everyday technology.
5.2 Example: Radio Tuning Systems
In the realm of RLC circuits, one of the most fascinating applications of resonance is found in radio tuning systems. Radios utilize RLC circuits to select specific frequencies from the myriad of
signals transmitted through the air. When a radio station broadcasts a signal, it does so at a particular frequency, which can be viewed as the frequency at which it resonates. The radio’s tuning
circuit, composed of a resistor (R), inductor (L), and capacitor (C), is designed to resonate at the same frequency as the desired station.
When adjusted correctly, the RLC circuit allows the radio to “tune in” to that specific frequency, resulting in a strong signal and clear sound. If the circuit is not tuned to the right frequency,
interference from other signals can result in static or a weak reception. The ability to filter out unwanted frequencies and amplify the desired one demonstrates the principle of resonance in action.
This precision is achieved through the following formula for the resonant frequency (fr):
[ f_r = \frac{1}{2\pi\sqrt{LC}} ]
Understanding the interplay between resistance, inductance, and capacitance is essential for designing effective radio tuning systems, illustrating the profound applications of resonance in everyday
As we wrap up our journey through the fascinating world of physics, I want to take a moment to reflect on what we’ve learned together. We’ve explored the fundamental principles that govern the
universe, from the tiniest subatomic particles to the vast expanse of galaxies. Each concept, whether it’s Newton’s laws of motion or the intricacies of electromagnetism, is a piece of a grand puzzle
that helps us understand the world around us.
Remember, physics isn’t just a collection of formulas and theories; it’s a lens through which we can view reality. Every time you see a shooting star, feel the warmth of the sun, or watch a roller
coaster rise and fall, you’re witnessing the incredible dynamics of physics in action.
As you move forward, I encourage you to carry this curiosity with you. Ask questions, seek answers, and never stop exploring. You are equipped with the tools to not only understand the universe but
to shape it. Physics is all around us and within us—it’s your turn to make your mark. Thank you for a fantastic year; keep looking at the stars! | {"url":"https://curioustoons.in/rlc-circuits-and-resonance/","timestamp":"2024-11-09T20:56:48Z","content_type":"text/html","content_length":"111430","record_id":"<urn:uuid:33084fdd-814c-4c84-af40-3e9a1dfe09f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00545.warc.gz"} |
Ultimate Guide to Concrete Block
Here’s an informative article on concrete by the team at Better Homes & Gardens.
Did you know concrete blocks come in more than just one shape and size? Before you start your home or garden project, take a look at our guide to learn your options.
Low material and installation cost, along with outstanding durability, make concrete block a practical choice for walls. Concrete block (often called cement masonry units or CMUs) are less expensive
than other wall-building materials, are relatively easy to install, and won’t rust, rot, or decay. Many blocks now have decorative face patterns, offering options to plain structural blocks.
Concrete block is cast in forms using a high-density mixture of sand, cement, and aggregate. Although building blocks are often generically called cinder blocks, a true concrete block differs from a
cinder block. Cinder blocks are made with aggregates of clay or pumice. They weigh less but fracture easily. If you’re looking for a worry-free, long-lasting wall, use concrete blocks.
Most blocks have webs that separate two or three cavities called cores. This type of construction reduces the weight of the unit without compromising its strength. Rebar is often placed in the cores,
which are then filled with concrete to strengthen the wall.
The basic building unit is the stretcher block, which has flanges on both ends. The blocks are butted together with mortar applied to the flanges. End blocks and corner blocks have one or more flange
faces so they present a smooth, outside finished face. You will also find cap blocks, thinner, solid, flangeless blocks used to cap the top of a wall.
Most walls are built from a block measuring 7-5/8 x 7-5/8 x 15-5/8 inches. When laid with a standard 3/8-inch mortar joint between them, the block dimensions become 8x8x16 inches. These dimensions
are the most common, but concrete block comes in a wide variety of sizes and shapes. Before you make plans for a wall, visit local building-supply dealers and research the available blocks. The sizes
and shapes you find might make you decide to reconsider the wall design or dimensions.
Interlocking Concrete Block
You can also buy concrete blocks that don’t require mortar. Interlocking concrete blocks rely on different methods to hold them together. Some are cast with flanges on one side that hook one course
to the preceding one. Others use pins engaged in holes. Still others use a system of the blocks’ own concave/convex ridges or depressions. Another style provides a low-labor way to build a curved
wall without cutting a large number of blocks. Most interlocking blocks are available in various surface textures and colors. Such blocks can save much time and effort in do-it-yourself projects.
Designing with Concrete Block
Normally structures built of standard concrete blocks are finished with a facing material such as stucco, brick, stone, or veneer facing. For some uses, the facing is left unfinished; other times,
it’s painted.
Block is available in a variety of shapes, colors, and textures. Such architectural blocks need no additional facing material or finishing. Many are cast to look like cleft stone. Others are
manufactured with fluted or scored surfaces, ribs, and faces recessed with geometric patterns. Homeowners use blocks with open designs to make screens. You’ll even find some that look like wood — use
them to build a structure that looks more like a fence than a wall. If your budget allows, you may opt for prefaced blocks in a rainbow of colors and colored glazes.
A mortared concrete block wall must be built on a solid concrete footing with dimensions and reinforcement that conform to local building codes.
The Only Block you Need to Mow
Turf block isn’t technically a concrete block in the same sense as the block used in building walls. It’s a precast concrete paver formed from pressurized concrete. Its compressive strength is such
that it can withstand the weight of automobiles and trucks. Thus, it makes an unusual driveway surface that lets grass grow up through the recesses. It’s also suitable for paths and walks, although
the recesses make a rough ride for wheeled garden equipment.
Tip: Buying Concrete Block
You’ll find concrete block at all major home improvement centers and some lumberyards.
To estimate how many standard 8×16-inch blocks you’ll need, figure that 100 square feet of wall will require roughly 113 blocks.
You can get a more precise estimate with the following calculations:
Multiply the length of the wall in feet by .75 to get the number of blocks in each course.
Multiply the height of the wall in feet by 1.5 to get the number of courses.
Multiply the two results together to get the total number of blocks.
Subtract the number of corner or end blocks you’ll need and order 10 percent more to allow for cutting, mistakes, and breakage.
Multiply the length of the wall in feet by .75 to get the number of blocks in each course.
Multiply the height of the wall in feet by 1.5 to get the number of courses.
Multiply the two results together to get the total number of blocks.
Subtract the number of corner or end blocks you’ll need and order 10 percent more to allow for cutting, mistakes, and breakage.
Multiply the length of the wall in feet by .75 to get the number of blocks in each course.
Multiply the height of the wall in feet by 1.5 to get the number of courses.
Multiply the two results together to get the total number of blocks.
Subtract the number of corner or end blocks you’ll need and order 10 percent more to allow for cutting, mistakes, and breakage.
Multiply the length of the wall in feet by .75 to get the number of blocks in each course.
Multiply the height of the wall in feet by 1.5 to get the number of courses.
Multiply the two results together to get the total number of blocks.
Subtract the number of corner or end blocks you’ll need and order 10 percent more to allow for cutting, mistakes, and breakage.
Multiply the length of the wall in feet by .75 to get the number of blocks in each course.
Multiply the height of the wall in feet by 1.5 to get the number of courses.
Multiply the two results together to get the total number of blocks.
Subtract the number of corner or end blocks you’ll need and order 10 percent more to allow for cutting, mistakes, and breakage.
Multiply the length of the wall in feet by .75 to get the number of blocks in each course.
Multiply the height of the wall in feet by 1.5 to get the number of courses.
Multiply the two results together to get the total number of blocks.
Subtract the number of corner or end blocks you’ll need and order 10 percent more to allow for cutting, mistakes, and breakage.
Derek Carwood for Better Homes & Gardens on gardening mistakes to avoid and beautify your property. While designing...
The Old Farmer's Almanac on the significance of Arbor Day. What Is Arbor Day? Arbor Day is always observed on the last...
5 of the Biggest Garden Design Mistakes You Might Be Making
Arbor Day 2022: What And When Is Arbor Day? | {"url":"https://www.nassarlandscaping.com/advice/ultimate-guide-to-concrete-block/","timestamp":"2024-11-04T23:41:59Z","content_type":"text/html","content_length":"288732","record_id":"<urn:uuid:77154529-fe43-4ca1-b738-6fd9ac3a2f82>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00633.warc.gz"} |
The Strength Rating
In a game analysis section, I showed how likely one is to get a particular score for a game between two opponents whose location-adjusted ranking difference is dr. It seems we're almost there -- just
calculate the G(sa,sb) values for all games, and run with it! Not quite. Bayes' theorem indeed allows for the inversion of P(sa,sb|dr) to P(dr|sa,sb), but there is an added term. Expanding dr to
equal the difference between the two team ratings (a and b) plus the home field factor (h, assuming A is the home team), the correct equation for the inversion is:
P(a,b,h|sa,sb) = P(sa,sb|a,b,h) P(a) P(b) P(h).
P(a), P(b), and P(h) are the probabilities of team A being ranked a, team B being ranked b, and the home field factor being h, in the absence of all data.
"But wait", you ask, "aren't computer ratings supposed to be unbiased? How can you justify built-in prejudices to the rankings?" Well, I justify it because Bayes' theorem demands it. If no team in
the history of sport has ever achieved a ranking of 1000 sigma above the league average, the odds of one doing it now are quite slim. The trick is to define the prior in such a way as to not bias the
ranking in favor of or against any team.
The way I address this problem is to use the same prior for all teams. (In college football, I use different priors for I-A, I-AA, and so on, but still rank all teams within I-A using the same
prior.) This obeys the requirement that a prior be used, while not rating Ohio State better than Northern Illinois merely on the basis that Ohio State has historically been a better team. To
calculate the prior mean and width, I first calculate rankings with no prior, estimate the mean and inherent spread in the rankings (equal to the standard deviation minus the uncertainties in
quadrature), make that the prior, and recompute the rankings. If the mean is m and the standard deviation is d, the prior P(a) equals:
P(a) = NP(-0.5*((a-m)/d)^2)
The prior for team B is the same (replacing a with b), and the prior for h is determined from many seasons of data.
The Strength Rating
OK, now we have all the pieces. The probability of the entire season being produced given a set of team ratings equals the product of the probabilities of each game and all the priors. In other
P(r1,r2,...rn) = prod(i=games) P(sai,sbi|rai,rbi,h) * prod(i=teams) P(ri) * P(h)
where ri designates the rating of team i, sai and sbi are the scores of the home and road teams in game i, rai and rbi are the ratings of the home and road teams in game i, and h is the home factor.
Most rating systems stop here, compute the maximum likelihood solution for the ratings and homefield factor, and call it a ranking. It's possible to do much better, however. Recall that all of the
probabilities are NP(x) functions, which can be trivially multiplied as NP(x)*NP(y)*NP(z) = NP(x+y+z). This means that we can rewrite the above equation as:
-2 lnP = sum(i=games) (rai-rbi+h-G(sai,sbi))^2 + sum(i=teams) (ri-m)^2/d^2 + (h-hm)^2/dm^2
where hm and dm are the mean and width of the prior for the home field factor h.
Multiplying all of this out, one finds that -2 lnP is a second-order polynomial and can be written as:
-2 lnP = C + [ sum(i=teams) sum(j=teams) Mij ri rj ] + [ sum(i=teams) Vi ri ]
where C is a meaningless constant, and Mij and Vi are the polynomial coefficients. From here, you can make successive integrations to marginalize each team's rating until only the team you are trying
to rate remains. For example, if you want to marginalize team k, you would rewrite the above as:
-2 lnP = C + D + Mkk rk^2 + [Vk + sum(i!=k) Mik ri] rk
where D contains the sums of all terms not containing k. This can be rewritten as:
-2 lnP = C + D - 0.25*[Vk + sum(i!=k) Mik ri]^2/Mkk + Mkk (rk + 0.5*[Vk + sum(i!=k) Mik ri]/Mkk)^2
Integrating P over rk reduces the last term to a constant, and we have eliminated team k from the integral. Repeating this process for all teams other than the one you are trying to rank results in:
-2 lnP = A r^2 + B r + C
where C is different from the constant before (but equally insignficant). This is, of course:
P = NP( (r-B/2A) /sqrt(A) )
which is a Gaussian distribution centered on B/2A (the team's rating) with width 1/sqrt(A) (the uncertainty in that rating).
Repeating this process for all teams in the rating, one can arrive at a statistically-accurate rating, including uncertainty, that accounts for all interdependencies between the individual team
This strength rating is not shown anywhere on my ranking pages, but underlies the standard, median likelihood, and predictive rankings.
Return to ratings main page
Note: if you use any of the facts, equations, or mathematical principles on this page, you must give me credit.
copyright ©2001-2003 Andrew Dolphin | {"url":"http://dolphinsim.com/ratings/info/strength.html","timestamp":"2024-11-09T01:29:58Z","content_type":"text/html","content_length":"5829","record_id":"<urn:uuid:16192196-bbfa-411c-b573-8360ae1a47ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00670.warc.gz"} |
This package implements the computation of the bounds described in the article Derumigny, Girard, and Guyonvarch (2021), Explicit non-asymptotic bounds for the distance to the first-order Edgeworth
expansion, arxiv:2101.05780.
How to install
You can install the release version from the CRAN:
or the development version from GitHub:
Available bounds
Let \(X_1, \dots, X_n\) be \(n\) independent centered variables, and \(S_n\) be their normalized sum, in the sense that \[S_n := \sum_{i=1}^n X_i / \text{sd} \Big(\sum_{i=1}^n X_i \Big).\]
The goal of this package is to compute values of \(\delta_n > 0\) such that bounds of the form
\[ \sup_{x \in \mathbb{R}} \left| \textrm{Prob}(S_n \leq x) - \Phi(x) \right| \leq \delta_n, \]
or of the form
\[ \sup_{x \in \mathbb{R}} \left| \textrm{Prob}(S_n \leq x) - \Phi(x) - \frac{\lambda_{3,n}}{6\sqrt{n}}(1-x^2) \varphi(x) \right| \leq \delta_n, \]
are valid. Here \(\lambda_{3,n}\) denotes the average skewness of the variables \(X_1, \dots, X_n\).
The first type of bounds is returned by the function Bound_BE() (Berry-Esseen-type bound) and the second type (Edgeworth expansion-type bound) is returned by the function Bound_EE1().
Note that these bounds depends on the assumptions made on \((X_1, \dots, X_n)\) and especially on \(K4\), the average kurtosis of the variables \(X_1, \dots, X_n\). In all cases, they need to have
finite fourth moment and to be independent. To get improved bounds, several additional assumptions can be added:
• the variables \(X_1, \dots, X_n\) are identically distributed,
• the skewness (normalized third moment) of \(X_1, \dots, X_n\) are all \(0\), respectively.
• the distribution of \(X_1, \dots, X_n\) admits a continuous component.
setup = list(continuity = FALSE, iid = TRUE, no_skewness = FALSE)
Bound_EE1(setup = setup, n = 1000, K4 = 9)
#> [1] 0.1626857
This shows that
\[ \sup_{x \in \mathbb{R}} \left| \textrm{Prob}(S_n \leq x) - \Phi(x) - \frac{\lambda_{3,n}}{6\sqrt{n}}(1-x^2) \varphi(x) \right| \leq 0.1626857, \]
as soon as the variables \(X_1, \dots, X_{1000}\) are i.i.d. with a kurtosis smaller than \(9\).
Adding one more regularity assumption on the distribution of the \(X_i\) helps to achieve a better bound:
setup = list(continuity = TRUE, iid = TRUE, no_skewness = FALSE)
Bound_EE1(setup = setup, n = 1000, K4 = 9, regularity = list(kappa = 0.99))
#> [1] 0.1214038
This shows that
\[ \sup_{x \in \mathbb{R}} \left| \textrm{Prob}(S_n \leq x) - \Phi(x) - \frac{\lambda_{3,n}}{6\sqrt{n}}(1-x^2) \varphi(x) \right| \leq 0.1214038, \]
in this case. | {"url":"http://ftp-osl.osuosl.org/pub/cran/web/packages/BoundEdgeworth/readme/README.html","timestamp":"2024-11-09T13:59:17Z","content_type":"application/xhtml+xml","content_length":"10610","record_id":"<urn:uuid:df82c3df-c8dd-40a3-880f-cfd2bb785b21>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00116.warc.gz"} |
KAN: Is this the end of feedforward networks?
Multi-Layer Perceptrons (MLPs), also known as fully-connected feedforward networks, are the backbone of the most powerful AI models available today. They are used from machine learning tasks (like
feature extraction, regression, classification) up to many advanced deep learning models such as Convolutional Neural Networks (CNNs) and Transformers.
But are MLPs truly the ultimate solution? A recent paper has shed light on a rediscovered type of network: Kolmogorov-Arnold Networks (KANs). KANs use spline functions instead of static weights to
achieve their incredible accuracy and interpretability at very small network sizes. But could these networks represent the next big leap in AI, potentially replacing MLPs?
In this blog post, we will explore the differences between these two model architectures and highlight the advantages and potentials of KANs.
Difference between MLP and KAN
Multi-Layer Perceptrons (MLPs)
MLPs, or Multi-Layer Perceptrons, are a sophisticated type of artificial neural network. These networks are designed with an input layer, multiple hidden layers, and an output layer. Each layer is
comprised of nodes, typically referred to as 'neurons', which are interconnected with nodes in the layers directly before and after them.
The operational flow in an MLP begins at the input layer and progresses through the hidden layers to reach the output layer. Each node performs a series of operations: the input values are first
multiplied by learned weights (predetermined from the model learning), these products are then summed, and the result is passed through an fixed activation function. This activation function is
critical as it introduces non-linearity into the system, a necessary feature for dealing with complex, non-linear problems.
Through these mechanisms, MLPs are capable of modeling intricate relationships between input features and outputs. This ability is grounded in the Universal Approximation Theorem, which posits that a
neural network with at least one hidden layer can approximate any continuous function, given sufficient neurons and the right set of weights. This theorem underpins the versatility and power of MLPs,
making them particularly useful in fields like image recognition, speech processing, and predictive analytics, where they can identify patterns and relationships in data that are not immediately
Kolmogorov-Arnold Networks (KANs)
Unlike MLPs, which are based on the "Universal Approximation Theorem," KANs are founded on the "Kolmogorov-Arnold Representation Theorem". The Kolmogorov-Arnold Theorem, also known as the
Superposition Theorem, was developed by Andrey Kolmogorov and Vladimir Arnold. Kolmogorov's work on representing continuous functions by superpositions of functions with fewer variables was published
in 1956, while Arnold's work on functions of three variables, closely related to the theorem, was published in 1957. These works established that any continuous multivariate function can be
represented as a superposition of continuous univariate functions and addition.
Similar to MLPs, KANs also consist of an input layer, an output layer, and one or more hidden layers in between. Unlike MLPs, the 'nodes' in KANs only perform a summation of the incoming 'signals'
without a fixed activation function, and the 'edges' use univariate functions instead of static weights. These univariate functions are learned during the training process, much like the weights in
MLPs. In the experiments presented in the paper, these functions are B-splines. Comparable to learning the weights in MLPs, the "knots" of the splines in KANs are learned. B-splines have significant
advantages for this application in KANs.
Advantages of B-splines:
• B-splines, or splines in general, face challenges in approximating high-dimensional functions, known as the curse of dimensionality (COD) problem, due to their inability to exploit compositional
structures. However, for univariate functions, as required here, splines can provide very accurate results.
• They allow easy adjustment of the resolution of functions, and thereby the overall accuracy of the model, by increasing or decreasing the grid size.
• Additionally, the local independence of the knots in B-splines positively impacts the learning process, particularly in continuous learning.
To accurately learn a function, which is the fundamental goal of any model, a model must capture both the compositional structure (external degrees of freedom) and approximate the univariate
functions well (internal degrees of freedom). KANs achieve this by combining the strengths of both MLPs and splines. They incorporate an MLP-like structure on the outside and splines on the inside.
This allows KANs not only to learn features (due to their external similarity to MLPs) but also to optimize these learned features with high accuracy (thanks to their internal similarity to splines).
KANs offer several significant advantages:
• Accuracy: Due to the use of different learned functions at the edges, KANs can model the influences of individual features very well and achieve high accuracy even with small networks.
• Efficiency: Although the KANs in the experiments described in the paper required significantly longer training times compared to MLPs with the same number of parameters, KANs are considered much
more efficient. This is because their high accuracy allows for a disproportionately reduction in network size and number of parameters to achieve the same accuracy as an MLP. For example, the
authors statet that "for PDE solving, a 2-Layer width-10 KAN is 100 times more accurate than a 4-Layer width-100 MLP (10^−7 vs 10^−5 MSE) and 100 times more parameter efficient (10^2 vs 10^4
• Interpretability: The use of functions at the edges makes the influences of individual features clear and easily interpretable. In MLPs, these relationships can only be represented over large
parts of the network and through numerous nodes and edges, making interpretability much poorer. The straightforward interpretability of KANs means they could also be useful in scientific fields
like physics and mathematics.
• Continuous Learning: Due to the local independence of the B-spline functions used, these networks can apparently support "Continuous Learning", where training different parts of the network does
not lead to "catastrophic forgetting" in the already learned sections, unlike in MLPs.
Despite their apparent innovation, basing artificial neural networks on the Kolmogorov-Arnold representation theorem is not entirely new. Some studies in the past have used this approach but stuck
with the original depth-2 width-(2n + 1) representation and did not leverage more modern techniques, such as back propagation, to train the networks. The groundbreaking aspect of the recently
presented paper is the generalization of this concept to arbitrary network sizes and the use of modern techniques like back propagation.
The paper presented various tests demonstrating the potential and influencing factors of these networks. Tests were conducted with different grid sizes for the spline functions, and a variety of
combined and specialized mathematical functions were attempted to be learned. Special attention was given to the internal representation in the networks and their interpretability. The continuous
learning capability was also demonstrated, along with a special pruning technique for the network and an alternative to symbolic regression. Additionally, the scaling laws of these networks were
illustrated, and examples were provided showing how these networks can be used in supervised and unsupervised learning scenarios.
However, covering all these findings is beyond the scope of this article. Instead, we will focus on two specific points: grid extension and interpretability.
Grid extension
In principle, a spline can achieve high accuracy by making the grid finer, a feature that KANs inherit. Unlike MLPs, which lack the concept of "fine-graining," KANs can improve accuracy by refining
their spline grids without needing to retrain the model from scratch. Although increasing the width and depth of MLPs can enhance performance, this process is slow and costly. With KANs, one can
start with fewer parameters and later extend to more parameters by simply refining the spline grids.
Grid extension of univariate B-spline functions
Using the mathematical function f(x, y) = exp(sin(πx) + y^2) as toy example, the research team illustrate the effect of grid extension on KAN performance and therefore on the training and test RMSE
of the model.
Effect of extending the spline grid on training and test RMSE
Starting with a small number of grid points and increasing in steps up to 1000 points, it shows that training loss consistently and immediately decreases as the grid is extended. Except at the
highest point, due to poor optimization, the RMSE increases again. This U-shape of the loss function highlights the bias-variance tradeoff (underfitting vs. overfitting). Optimal test loss occurs
when the number of parameters matches the number of data points, aligning with the expected interpolation threshold. For the given example, this means the following: Given that there are 1000
training samples and the total parameters of a [2, 5, 1] KAN amount to 15G (where G represents the number of grid intervals), the interpolation threshold is expected to be G = 1000/15 ≈ 67. This
approximately matches the experimentally observed value of G ∼ 50.
With knowledge of the function being learned, the ideal solution would be a [2, 1, 1] KAN instead of the chosen [2, 5, 1] KAN. This means a smaller network with fewer nodes which would require less
computation and training effort would also achieve even better results, as shown in the given graph with training and test RMSE.
NOTE: The notation '[2, 5, 1]' used in the paper describes the structure of the KANs, indicating the layers of the KANs and the number of nodes in each layer. In this example, there are 2 nodes in
the input layer, 5 nodes in the (single) hidden layer, and 1 node in the output layer. An illustration of this network can be found in the next section, see Figure "Symbolic Regression with KAN".
To prevent such unnecessary and problematic oversizing of the network in practice, the paper also introduced a technique to adjust/reduce the KAN structure according to the given requirements.
Simplifying KANs and making them interactive
As already described, reducing the nodes of a KAN and thus approximating the actual/ideal structure for a given problem can lead to both a reduction in training and computational efforts and an
improvement in the model's accuracy. Since this "ideal" structure is not known in practice, an automated process is necessary to decide which nodes can or should be removed.
The paper introduces a pruning technique to adjust the KAN structure dynamically. This process involves evaluating the importance of each node and removing those that contribute least to the
network's performance. The pruning algorithm helps in systematically reducing the network size while maintaining or even enhancing accuracy. This method ensures that the network remains efficient and
well-suited to the specific problem at hand, optimizing both resource usage and model performance.
To perform this sparsification, a sparsification penalty is introduced during training. This penalty considers the average magnitude of an activation function over all inputs, the sum of these values
for a given layer, and the layer's entropy. After this special training, the network is pruned at the node level by removing nodes with low incoming and outgoing scores.
Given that the relationships of the individual features in the model are directly evident and highly interpretable through the spline functions, the researchers went a step further. They visualized
the network with its splines and presented it as an interactive graph. This approach allowed the learned functions to be replaced with symbolic functions (with some automatic adjustments for shifts
and scalings). As a result, KANs provide an excellent alternative to traditional methods of symbolic regression and can be used in various mathematical and physical fields to help identify internal
Despite their sophisticated mathematical foundation, KANs are essentially combinations of splines and MLPs, leveraging their respective strengths while avoiding their respective weaknesses.
Will KANs totally replace MLPs? Probably not anytime soon, especially given the significantly faster training times of MLPs. However, we are likely to see substantial improvements and an increasing
number of applications for KANs.
The authors of the paper are very confident about the possibilities and potential of KANs, as demonstrated by their decision tree "Should I use KANs or MLPs?" They see the only drawback in the slow
training process and also note, "we did not try hard to optimize KANs’ efficiency though, so we deem KANs’ slow training more as an engineering problem to be improved in the future rather than a
fundamental limitation."
KAN vs. MLP decision tree provided by paper authors
But also in this area of KAN training, there are already advances. Just a few weeks after the papers publication the freely available GitHub repository "efficient-kan" popped up which provides an
improved implementation.
Only time will tell if KANs will ultimately replace MLPs as the authors predict. One thing is certain: we are eager to witness the progress and innovations that will unfold in this exciting field in
the near future.
Further Reading
Interested in how to train your very own Large Language Model?
We prepared a well-researched guide for how to use the latest advancements in Open Source technology to fine-tune your own LLM. This has many advantages like:
• Cost control
• Data privacy
• Excellent performance - adjusted specifically for your intended use | {"url":"https://www.pondhouse-data.com/blog/kolmogorov-arnold-networks","timestamp":"2024-11-14T07:01:08Z","content_type":"text/html","content_length":"103933","record_id":"<urn:uuid:67b0320d-24d8-4d42-8961-d624e4d19b4d>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00634.warc.gz"} |
Mathematics SBA Guide
The Purpose of the SBA Project
It is the view of CXC and Mathematics Teachers across the region that students are not transitioning fluently enough from "School Math" to real world applications. This has caused many governments to
invest and train members of the education systems in the delivery of STEM and STEAM approaches in an attempt to bridge that gap between the real world and the classroom.
the Mathematics SBA is an attempt by CXC to bridge that same divide and forever link real world, everyday mathematics to the theoretical concepts discussed in the classroom. In CXC's own words, the
project may require the candidate to collect data or demonstrate athe application of Mathematics in everyday situations.
The Project Title
It should be clear and concise and related to a real world problem. The title may be in the form of a question or a precise and clear statement of intent, call it a hypothesis if you wish, but its
intention is to show what you will be trying to accomplish
The introduction for the project should be well thought out and should be a comprehensive description of the project itself. It should set the background for what you intend to do. The objectives
[whatever you plan to accomplish] should be stated in the introduction and those objectives should be very clear and precise.
Method of Data Collection
The method that you use to collect your data needs to be stated clearly here. If you plan to use a survey, a questionnaire, an experiment, an investigation... You must ensure that your Method is free
from flaws as a flawed method will lead to unreliable data and if your data is unreliable then whatever conclusions you come up with will be flawed. Take some time to talk with your teacher/advisor/
facilitator to make sure that your data collection method is sound. The instrument that you intend to use to gather the data should be stated here, blank tables with headings, survey questions,
diagrams and general calculations about things you plan to discuss.
Presentation of Data
The Presentation of your data needs to be accurate and well organized. You may have used a survey or a table to collect your initial information, this table needs to be properly laid out with
appropriate column headers that describe exactly what you are doing. In addition to the table you will need to have at least one graph that shows your data. You may use any type of appropriate
statistical graph, bar char, pie chart, linear graph, histogram etc. Your graphs need to be well labelled in terms of axes. You should also introduce the graph don't just place it on the page. For
The following graph shows ........ then add the graph.
if you are modelling data and looking for relationships or correlations I am recommending that you use the software program GRAPH. It is excellent at plotting scatters and can easily draw a best fit
line or spline [You can talk to your teacher about that.] that can help you with your analysis.
You also need to be accurate in your use of Mathematical concepts while you present your data so make sure that everything is accurately worked out. It is recommended that you use Microsoft Office
or other spreadsheet program to generate your graphs. CXC does want you to use the technology that is available.
Analysis of Data
As stated immediately above, this is a Mathematics SBA so you need to use the language of Mathematics as well as use mathematics concepts in your Analysis. You must write in a coherent way. You do
not need to be wordy you only need to make sense of the data and write that understanding in a way so that th e reader can understand what you mean. You should be detailed and you should be coherent.
Try to answer the following questions as you write;
What is you data saying to you?
What patterns do you see, what trends?
Look at averages and compare quantities using parentages.
Its kind of like writing a statistical report. CXC doesn't have these in the English A syllabus anymore but your English teacher knows how and can help you so ask for help if you don't know how.
Discussion of Findings
So you have done your analysis, what exactly have you found out?
It may be what you thought you would find, it may be different but your work has shown something. What is that something. State it clearly and precisely. Please understand that your discussion of
findings MUST follow from your data and your analysis of that data. So don't try to impress anyone by making claims that are not supported by the data that you have or by the analysis you have done.
The conclusion
Ahh, relief, finally you have reached the end. Now all you need to do is make a conclusion and you are good at this. After all your language teacher did teach you how to do it.
Regardless, just make a summary of what you have done in the analysis of data and in the discussion of findings. That will be good enough.
And finally
• You do get marked for grammar and the use of English so make sure to at least spell check your work. Yes it should be typed.
• Make sure you create a content page
• You definitely need a cover page with your personal and center information
• You will most likely use electronic submission so save your document, back it up somewhere, email yourself a copy , etc, just make sure that when your teacher needs it you have it
• And if you used a survey or questionnaire etc you can include those in an appendix at the end of the back of the project
• Make sure your work is well organized
• AND Remember you have a 1000 word limit so avoid being wordy and write clearly and to the point | {"url":"https://www.csecmathtutor.com/mathematics-sba-guide.html","timestamp":"2024-11-02T11:37:19Z","content_type":"text/html","content_length":"55529","record_id":"<urn:uuid:36afe080-2217-43a9-ac29-377e2b2bafbf>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00874.warc.gz"} |
Equilibration of quantum gases
verfasst von
Terry Farrelly
Finding equilibration times is a major unsolved problem in physics with few analytical results. Here we look at equilibration times for quantum gases of bosons and fermions in the regime of
negligibly weak interactions, a setting which not only includes paradigmatic systems such as gases confined to boxes, but also Luttinger liquids and the free superfluid Hubbard model. To do this,
we focus on two classes of measurements: (i) coarse-grained observables, such as the number of particles in a region of space, and (ii) few-mode measurements, such as phase correlators. We show
that, in this setting, equilibration occurs quite generally despite the fact that the particles are not interacting. Furthermore, for coarse-grained measurements the timescale is generally at
most polynomial in the number of particles N, which is much faster than previous general upper bounds, which were exponential in N. For local measurements on lattice systems, the timescale is
typically linear in the number of lattice sites. In fact, for one-dimensional lattices, the scaling is generally linear in the length of the lattice, which is optimal. Additionally, we look at a
few specific examples, one of which consists of N fermions initially confined on one side of a partition in a box. The partition is removed and the fermions equilibrate extremely quickly in time
New journal of physics
ASJC Scopus Sachgebiete
Physik und Astronomie (insg.)
Elektronische Version(en)
https://doi.org/10.1088/1367-2630/18/7/073014 (Zugang: Offen) | {"url":"https://www.quest-lfs.uni-hannover.de/de/forschung/publikationen/details/fis-details/publ/1c0f3187-a8bc-467c-abdc-9c7c8985425d?cHash=0f983cffd3a9a48d10eacb99eb583683","timestamp":"2024-11-11T04:48:12Z","content_type":"application/xhtml+xml","content_length":"37897","record_id":"<urn:uuid:efc61675-9a8b-4ff9-8690-05c3226cc76a>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00284.warc.gz"} |
Tesla model y vs x
Feb 19, 2021 · Tesla unveiled its Model Y in 2019 as a smaller, crossover version of its larger and more established Model X mid-sized SUV. After beginning deliveries to customers in March of 2020,
the Model Y
You have got to balance space with the newer technology including fast charging of the Y and see what is most important for you. Product Page: Tesla Model Y 19" TSV Flow Forged Tesla Wheel (Set of 4)
- T Sportline - Tesla Model S, 3, X & Y Accessories These are gorgeous wheels too, and are 19", so you have that same benefits as described with the R241s above of more tire options, cheaper tire
options, smoother ride, and less risk of pothole damage. Mar 14, 2020 · If you decide to order a Tesla, use a friend’s referral code to get 1,000 miles of free Supercharging on a Tesla Model S, Model
X, or Model 3 (you can’t use it on the Model Y or Cybertruck yet). Apr 18, 2020 · While the 3 and Y might not make the same impression as the “Faberge egg” design of the Model X, you’ll still get
head nods from other Tesla owners and EV enthusiasts.
The Model X’s top speed is 155 MPH. Both the Tesla Model Y and the Model X will be offered with three rows of seats. That said, the Tesla Model Y gets five seats standard while the Model X gets
seven-passenger seating standard. The added row of seats will add $4,000 to the Model Y’s price. Tesla Model Y vs Model X Review - Which one is better?Model Y - Performance - First Arrival at
Shophttps://gallery.bayareadetails.com/Tesla-Model-Y/Model-Y-P I have a model X, but have a model Y on order as an additional car. The X is fantastic car and I have found reliable. You have got to
balance space with the newer technology including fast charging of the Y and see what is most important for you. Owning a 3 and an X I can say with some certainty: the Y will be better put together.
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What Tesla calls an SUV is not necessarily what everybody else refers to that way. The Model X – revealed in 11 Mar 2020 Tesla Model Y is 2.2 inch longer, 2.8 inch wider and 7.2 inch taller as
compared to Model 3. Model X. Tesla Model Y is 11.3 inch shorter in length,
Both the Tesla Model Y and the Model X will be offered with three rows of seats. That said, the Tesla Model Y gets five seats standard while the Model X gets seven-passenger seating standard. The
added row of seats will add $4,000 to the Model Y’s price.
7 Sep 2020 The Tesla Model X has the longest range out of all of its competitors with 351 miles for the Long Range version and 305 miles for the Tesla Model Y vs. Tesla Model X. The Model X is more
of a midsize three-row SUV to the Model Y's compact status, and so it has more interior and cargo Compare prices, specs, trims standards and options between the 2020 Tesla Model X and 2020 Tesla
Model Y on autoTRADER.ca. 12 Jun 2020 Tesla Model Y vs Ford Mustang Mach-E: specs comparison including those who can't stretch to the much more expensive Tesla Model X. 12 Feb 2021 Both Teslas have
eight-year powertrain warranties, but the Model X's warranty has unlimited mileage compared to a 120,000-mile limit for the Tesla Model X vs Tesla Model Y: compare price, expert/user reviews, mpg,
engines, safety, cargo capacity and other specs. Compare against other cars.
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Model X is 5,036mm (198.3-in) long, 2,000mm (78.7-in) wide, and has a 2,965mm (116.7-in) wheelbase. This means the Model Y will be roughly 346mm (13.6-in) shorter, 70mm (2.8-in) narrower, and have an
85mm (3.4-in) shorter wheelbase Model Y Model X According to Tesla, the Model Y is able to accelerate from 0 to 60 MPH in 3.5 seconds.
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units in 2019.
The Tesla Model Y's biggest advantage will be its affordable price, which will be as low as $48,200 (including a $1,200 Jun 28, 2020 · Tesla Model Y. Credit: Brooke Crothers. Buying an electric car
can be a daunting decision for first-time buyers. Here’s a very short, very basic comparison of the Model 3 and Model Y for Jul 29, 2020 · The first race was between a modified Tesla Model Y
Performance with lightweight wheels; and an Audi RS5. The Model Y produces 456 HP and 497lb-ft of torque, while the Audi RS5 is powered by a 2.9 The upcoming 2021 Tesla Model X, compared to sedan S,
provides almost identical styling. However, because X is a crossover, it has a different shape and proportion. Design is futuristic, unique and is very recognized on the road because of that.
You have got to balance space with the newer technology including fast charging of the Y and see what is most important for you.
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5 Nov 2020 Each model comes in two different, dual motor options. For the Model X, there's the Long Range Plus and the Performance powertrain, while the
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3, but about 10% bigger in Aug 16, 2020 · Tesla Model Y. Credit: Tesla. If you’re thinking about buying an Audi e-tron, wait a bit. | {"url":"https://lonfhtl.firebaseapp.com/20039/71267.html","timestamp":"2024-11-08T18:48:46Z","content_type":"text/html","content_length":"18957","record_id":"<urn:uuid:9b8c65cb-e87d-4021-a97e-5c8d5b53827e>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00690.warc.gz"} |
project(cmd0::String="", arg1=nothing, kwargs...)
keywords: GMT, Julia, great circles
Project data onto lines or great circles, or generate tracks
Reads arbitrary (\(x\), \(y\) [,\z]) data and writes any combination of (\(x, y\), z, \(p, q, r, s\)), where (\(p, q\)) are the coordinates in the projection, (\(r, s\)) is the position in the (\(x,
y\)) coordinate system of the point on the profile (\(q = 0\) path) closest to (\(x, y\)), and z is all remaining columns in the input (beyond the required \(x\) and \(y\) columns).
Alternatively, project may be used to generate (\(r,s,p\)) triples at equal increments dist along a profile using step. In this case, no input is read.
Projections are defined in one of three ways:
1. By a center (cx,cy) using origin and an azimuth in degrees clockwise from North using azim.
2. By a center (cx,cy) using origin and end point (bx,by) of the projection path using endpoint.
3. By a center (cx,cy) using origin and a rotation pole position (px,py) using pole (not allowed when a Cartesian transformation is set by flat_earth).
To spherically project data along a great circle path, an oblique coordinate system is created which has its equator along that path, and the zero meridian through (cx,cy). Then the oblique longitude
(\(p\)) corresponds to the distance from (cx,cy) along the great circle, and the oblique latitude (q) corresponds to the distance perpendicular to the great circle path. When moving in the increasing
(\(p\)) direction, (in the direction set by azim azimuth ), the positive (\(q\)) direction is to the left. If a pole has been specified by pole, then the positive (q) direction is toward the pole.
To specify an oblique projection, use the pole option to set the pole. Then the equator of the projection is already determined and the origin option is used to locate the \(p = 0\) meridian. The
center (cx,cy) will be taken as a point through which the \(p = 0\) meridian passes. If you do not care to choose a particular point, use the South pole (cx = 0, cy = -90).
Data can be selectively windowed by using the length and width options. If width is used, the projection width is set to use only points with \(w_{min} < q < w_{max}\). If length is set, then the
length is set to use only those points with \(l_{min} < p < l_{max}\). If the endpoint option has been used to define the projection, then length\w may be selected to window the length of the
projection to exactly the span from the center (origin) to to the endpoint (endpoint).
Flat Earth (Cartesian) coordinate transformations can also be made. Set flat_earth and remember that azimuth is clockwise from North (the \(y\) axis), NOT the usual cartesian theta, which is
counterclockwise from the \(x\) axis. (i.e., \(azimuth = 90 - theta\)).
No assumptions are made regarding the units for \(x, y, r, s, p, q\), dist, \(l_{min}, l_{max}, w_{min}, w_{max}\). However, if km is selected, map units are assumed and \(x, y, r, s\), must be in
degrees and \(p, q\), dist, \(l_{min}, l_{max}, w_{min}, w_{max}\) will be in km.
Calculations of specific great-circle and geodesic distances or for back-azimuths or azimuths are better done using mapproject as project is strictly spherical.
using GMT
a = atand(4 / 2.5)
X = project([0 0], origin=(0,-1), endpoint=(2.5,3), flat_earth=true)
plot([-1.5 -1.0625; 0 -2; 2 1.2; 0.5 2.1375], region=(-3.5,4,-2.7,2.6), fill=:lightgray,
xlabel="@%7%x@%% or @%7%r@%%", ylabel="@%7%y@%% or @%7%s@%%", figsize="12/0")
plot!([0 -1; 2 2.2], marker=:circ, ms=0.3, fill=:orange, frame=(grid=10,))
arrows!([0 -1 2 2.2; 0 -1 -2.5 0.5625], arrow=(len="16p", stop=true, shape=1),
endpoint=true, lw=2, fill=:black)
plot!([0 0], marker=:circ, ms=0.3, fill=:red)
# Get coordinates of the (0,q) point as well so we can dash the line
x = -X[4] * sind(a)
y = X[4] * cosd(a) - 1
plot!([X[5] X[6]], marker=:circ, ms=0.2, fill=:blue)
T = mat2ds([ 0 -1 0;
2 2.2 0;
1.9 1.9 a;
-2.3 0.4 a;
2 1.2 a;
0 -2 a;
0 -2 a;
-1.5 -1.0625 a;
0.45 0.8 -16],
["TL @%7%C@%%", "BR @%7%E@%%", "TC p", "RM q", "TC L@-max@-",
"TC L@-min@-", "RB W@-min@-", "RM W@-max@-", "TC @~a@~"])
text!(T, font=(12, "Times-Italic"), angle="", justify="", offset=(away=true, shift=0.15))
plot!([0 0; X[5] X[6]], pen=(0.25, :red, :dash))
plot!([0 0; x y], pen=(0.25, :red, :dash))
plot!([0.0 -1], marker=(:matangle, [2.54 a 90], (length="9p", start=true)), ml=0.5, fill=:black)
Explanation of the coordinate system utilized by project. The input point (red circle) is given in the original x-y (or lon-lat) coordinate system and is projected to the p-q coordinate system,
defined by the center (center) and either the end-point (outvars) or azimuth (\(\alpha\)), or for geographic data a rotation pole pole (not shown). The blue point has projected coordinates (p,0) and
is reported as (r,s) in the original coordinate system. Options length (limit range of p) and width (limit range of q) can be used to exclude data outside the specified limits (light gray area).
Required Arguments
• table
One or more data tables holding a number of data columns.
• C or origin or start_point : – origin=(cx, cy)
Set the origin cx,cy of the projection when used with azim or endpoint or set the coordinates cx,cy of a point through which the oblique zero meridian (\(p = 0\)) should pass when used with pole.
cx,cy is not required to be 90 degrees from the pole set by pole.
Optional Arguments
• A or azim or azimuth : – azim=az
Set the azimuth of the projection. The azimuth is clockwise from North (the \(y\) axis) regardless of whether spherical or Cartesian coordinate transformation is applied.
• E or endpoint or end_pt : – endpoint=(bx,by)
Set the end point bx,by of the projection path.
• F or outvars : – outvars=flags
Specify the desired output using any combination of xyzpqrs in any order, where (\(p, q\)) are the coordinates in the projection, (\(r, s\)) is the position in the (\(x, y\)) coordinate system of
the point on the profile (\(q = 0\) path) closest to (\(x, y\)), and z is all remaining columns in the input (beyond the required \(x\) and \(y\) columns). [Default is xyzpqrs]. If output format
is ASCII then z also includes any trailing text (which is placed at the end of the record regardless of the order of z in flags). Use lower case and do not add spaces between the letters. Note:
If step is selected, then the output order is set to be rsp and outvars is not allowed.
• G or step or generate : – step="dist[unit][/colat][+c][+h][+n]"
Create (r, s, p) output points every dist units of p, assuming all units are the same unless \(x, y, r, s\) are set to degrees using km. No input is read when step is used. See Units for
selecting geographic distance units [km]. The following directives and modifiers are supported:
□ Optionally, append /colat for a small circle instead [Default is a colatitude of 90, i.e., a great circle]. Note, when using origin and endpoint to generate a circle that goes through the
center and end point, the center and end point cannot be farther apart than $2|colat|$.
□ Optionally, append +c when using pole to calculate the colatitude that will lead to the small circle going through the center cx/cy.
□ Optionally, append +h to report the position of the pole as part of the segment header when using pole [Default is no header].
□ Optionally, append +n to indicate a desired number of points rather than an increment. Requires origin and endpoint or |-Z| so that a length can be computed.
• L or length : – length=(lmin,lmax) | length=:w
Specify length controls for the projected points. Project only those points whose p coordinate is within \(l_{min} < p < l_{max}\). If endpoint has been set, then you may alternatively use length
=:w to stay within the distance from cx,cy to bx,by.
• N or flat_earth : – flat_earth=true
Specify the Flat Earth case (i.e., Cartesian coordinate transformation in the plane). [Default uses spherical trigonometry.]
• Q or km : – km=true
Specify that x, y, r, s are in degrees while p, q, dist, lmin, lmax, wmin, wmax are in km. If km is not set, then all these are assumed to be in the same units.
• S or sort : – sort=true
Sort the output into increasing p order. Useful when projecting random data into a sequential profile.
• T or pole : – pole=(px,py)
Set the position of the rotation pole of the projection as px,py.
• V or verbose : – verbose=true | verbose=level
Select verbosity level. More at verbose
• W or width : – width=(wmin,wmax)
Specify width controls for the projected points. Project only those points whose q coordinate is within \(w_{min} < q < w_{max}\).
• Z or ellipse : – ellipse="major[unit][/minor/azimuth][+e]"
Create the coordinates of an ellipse with major and minor axes given in km (unless flat_earth is given for a Cartesian ellipse) and the azimuth of the major axis in degrees; used in conjunction
with origin (sets its center) and step (sets the distance increment). Note: For the Cartesian ellipse (which requires flat_earth), we expect direction counter-clockwise from the horizontal
instead of an azimuth. A geographic major may be specified in any desired unit [Default is km] by appending the unit (e.g., 3d for degrees); if so we assume the minor axis and the increment are
also given in the same unit (see Units). For degenerate ellipses you can just supply a single diameter instead. The following modifiers are supported:
□ Append +e to adjust the increment set via step so that the ellipse has equal distance increments [Default uses the given increment and closes the ellipse].
• bi or binary_in : – binary_in=??
Select native binary format for primary table input. More at
• bo or binary_out : – binary_out=??
Select native binary format for table output. More at
• di or nodata_in : – nodata_in=??
Substitute specific values with NaN. More at
• e or pattern : – pattern=??
Only accept ASCII data records that contain the specified pattern. More at
• f or colinfo : – colinfo=??
Specify the data types of input and/or output columns (time or geographical data). More at
• g or gap : – gap=??
Examine the spacing between consecutive data points in order to impose breaks in the line. More at
• h or header : – header=??
Specify that input and/or output file(s) have n header records. More at
• i or incol or incols : – incol=col_num | incol="opts"
Select input columns and transformations (0 is first column, t is trailing text, append word to read one word only). More at incol
• o or outcol : – outcol=??
Select specific data columns for primary output, in arbitrary order. More at
• q or inrows : – inrows=??
Select specific data rows to be read and/or written. More at
• s or skiprows or skip_NaN : – skip_NaN=true | skip_NaN="<cols[+a][+r]>"
Suppress output of data records whose z-value(s) equal NaN. More at
• yx : – yx=true
Swap 1st and 2nd column on input and/or output. More at
For map distance unit, append unit d for arc degree, m for arc minute, and s for arc second, or e for meter [Default unless stated otherwise], f for foot, k for km, M for statute mile, n for nautical
mile, and u for US survey foot. By default we compute such distances using a spherical approximation with great circles (-jg) using the authalic radius (see PROJ_MEAN_RADIUS). You can use -jf to
perform “Flat Earth” calculations (quicker but less accurate) or -je to perform exact geodesic calculations (slower but more accurate; see PROJ_GEODESIC for method used).
To project the remote data sets ship_03.txt (lon,lat,depth) onto a great circle specified by the two points (330,-18) and (53,21) and sort the records on the projected distances along that circle and
only output the distance and the depths, try
using GMT
D = project("@ship_03.txt", origin=(330,-18), pole=(53,21), sort=true, outvars=:pz, km=true)
To generate points every 10 km along a great circle from 10N,50W to 30N,10W:
using GMT
Dgc = project(origin=(-50,10), endpoint=(-10,30), step=10, km=true)
imshow(Dgc, marker=:point, lw=0.5, coast=true)
(Note that Dgc could now be used as input for grdtrack, etc. ).
To generate points every 1 degree along a great circle from 30N,10W with azimuth 30 and covering a full 360, try:
D = project(origin=("10W","30N"), azim=30, step=1, length=(-180,180))
imshow(D, coast=true)
To generate points every 10 km along a small circle of colatitude 60 from 10N,50W to 30N,10W:
using GMT
Dsc = project(origin=(-50,10), endpoint=(-10,30), step=(10,60), km=true)
imshow(Dsc, marker=:point, lw=0.5, coast=true)
To create a partial small circle of colatitude 80 about a pole at 40E,85N, with extent of 45 degrees to either side of the meridian defined by the great circle from the pole to a point 15E,15N, try
D = project(origin=(15,15), pole=(40,85), step=(1,80), length(-45,45))
To generate points approximately every 10 km along an ellipse centered on (30W,70N) with major axis of 1500 km with azimuth of 30 degree and a minor axis of 600 km, try
using GMT
Dellip = project(origin=(-30,70), step=10, ellipse="1500/600/30+e", km=true)
imshow(Dellip, coast=true)
To project the shiptrack gravity, magnetics, and bathymetry in c2610.xygmb along a great circle through an origin at 30S, 30W, the great circle having an azimuth of N20W at the origin, keeping only
the data from NE of the profile and within ± 500 km of the origin, run:
Dprj = project("c2610.xygmb", origin=(-30,-30), azim=-20, width=(-10000,0),
length=(-500,500), outvars=:pz, km=true)
(Note in this example that width=(-10000,0) is used to admit any value with a large negative q coordinate. This will take those points which are on our right as we walk along the great circle path,
or to the NE in this example.)
To make a Cartesian coordinate transformation of mydata.xy so that the new origin is at 5,3 and the new x axis (p) makes an angle of 20 degrees with the old x axis, use:
D = project("mydata.xy", origin=(5,3), azimuth=70, outvars=:pq)
To take data in the file pacific.lonlat and transform it into oblique coordinates using a pole from the hotspot reference frame and placing the oblique zero meridian (p = 0 line) through Tahiti, run:
D = project("pacific.lonlat", pole=(-75,68), origin=("-149:26","-17:37"), outvars=:pq)
Suppose that pacific_topo.nc is a grid file of bathymetry, and you want to make a file of flowlines in the hotspot reference frame. If you run:
G = grd2xyz("pacific_topo.nc");
D = project(pole=(-75,68), oigin=(0,-90), outvars=:xyq);
Gflow = xyz2grd(region=etc, inc=etc);
then Gflow is a grid in the same area as pacific_topo.nc, but flow contains the latitudes about the pole of the projection. You now can use grdcontour on Gflow to draw lines of constant oblique
latitude, which are flow lines in the hotspot frame.
If you have an arbitrarily rotation pole px,py and you would like to draw an oblique small circle on a map, you will first need to make a file with the oblique coordinates for the small circle (i.e.,
lon = 0-360, lat is constant), then create a file with two records: the north pole (0/90) and the origin (0/0), and find what their oblique coordinates are using your rotation pole. Now, use the
projected North pole and origin coordinates as the rotation pole and center, respectively, and project your file as in the pacific example above. This gives coordinates for an oblique small circle.
See Also
fitcircle, gmtvector, grdtrack, mapproject, grdproject, grdtrack | {"url":"https://www.generic-mapping-tools.org/GMTjl_doc/documentation/modules/project/","timestamp":"2024-11-08T08:29:03Z","content_type":"text/html","content_length":"82995","record_id":"<urn:uuid:e2987384-b679-4c1f-8315-6e0caa369920>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00059.warc.gz"} |
Variable star
A variable star is a star whose brightness as seen from Earth (its apparent magnitude) fluctuates.
This variation may be caused by a change in emitted light or by something partly blocking the light, so variable stars are classified as either:
Intrinsic variables, whose luminosity actually changes; for example, because the star periodically swells and shrinks.
Extrinsic variables, whose apparent changes in brightness are due to changes in the amount of their light that can reach Earth; for example, because the star has an orbiting companion that sometimes
eclipses it.
Many, possibly most, stars have at least some variation in luminosity: the energy output of our Sun, for example, varies by about 0.1% over an 11-year solar cycle.[1]
An ancient Egyptian calendar of lucky and unlucky days composed some 3,200 years ago may be the oldest preserved historical document of the discovery of a variable star, the eclipsing binary Algol.
Of the modern astronomers, the first variable star was identified in 1638 when Johannes Holwarda noticed that Omicron Ceti (later named Mira) pulsated in a cycle taking 11 months; the star had
previously been described as a nova by David Fabricius in 1596. This discovery, combined with supernovae observed in 1572 and 1604, proved that the starry sky was not eternally invariable as
Aristotle and other ancient philosophers had taught. In this way, the discovery of variable stars contributed to the astronomical revolution of the sixteenth and early seventeenth centuries.
The second variable star to be described was the eclipsing variable Algol, by Geminiano Montanari in 1669; John Goodricke gave the correct explanation of its variability in 1784. Chi Cygni was
identified in 1686 by G. Kirch, then R Hydrae in 1704 by G. D. Maraldi. By 1786 ten variable stars were known. John Goodricke himself discovered Delta Cephei and Beta Lyrae. Since 1850 the number of
known variable stars has increased rapidly, especially after 1890 when it became possible to identify variable stars by means of photography.
The latest edition of the General Catalogue of Variable Stars[5] (2008) lists more than 46,000 variable stars in the Milky Way, as well as 10,000 in other galaxies, and over 10,000 'suspected'
Detecting variability
The most common kinds of variability involve changes in brightness, but other types of variability also occur, in particular changes in the spectrum. By combining light curve data with observed
spectral changes, astronomers are often able to explain why a particular star is variable.
Variable star observations
A photogenic variable star, Eta Carinae, embedded in the Carina Nebula
Variable stars are generally analysed using photometry, spectrophotometry and spectroscopy. Measurements of their changes in brightness can be plotted to produce light curves. For regular variables,
the period of variation and its amplitude can be very well established; for many variable stars, though, these quantities may vary slowly over time, or even from one period to the next. Peak
brightnesses in the light curve are known as maxima, while troughs are known as minima.
Amateur astronomers can do useful scientific study of variable stars by visually comparing the star with other stars within the same telescopic field of view of which the magnitudes are known and
constant. By estimating the variable's magnitude and noting the time of observation a visual lightcurve can be constructed. The American Association of Variable Star Observers collects such
observations from participants around the world and shares the data with the scientific community.
From the light curve the following data are derived:
are the brightness variations periodical, semiperiodical, irregular, or unique?
what is the period of the brightness fluctuations?
what is the shape of the light curve (symmetrical or not, angular or smoothly varying, does each cycle have only one or more than one minima, etcetera)?
From the spectrum the following data are derived:
what kind of star is it: what is its temperature, its luminosity class (dwarf star, giant star, supergiant, etc.)?
is it a single star, or a binary? (the combined spectrum of a binary star may show elements from the spectra of each of the member stars)
does the spectrum change with time? (for example, the star may turn hotter and cooler periodically)
changes in brightness may depend strongly on the part of the spectrum that is observed (for example, large variations in visible light but hardly any changes in the infrared)
if the wavelengths of spectral lines are shifted this points to movements (for example, a periodical swelling and shrinking of the star, or its rotation, or an expanding gas shell) (Doppler effect)
strong magnetic fields on the star betray themselves in the spectrum
abnormal emission or absorption lines may be indication of a hot stellar atmosphere, or gas clouds surrounding the star.
In very few cases it is possible to make pictures of a stellar disk. These may show darker spots on its surface.
Interpretation of observations
Combining light curves with spectral data often gives a clue as to the changes that occur in a variable star.[6] For example, evidence for a pulsating star is found in its shifting spectrum because
its surface periodically moves toward and away from us, with the same frequency as its changing brightness.[7]
About two-thirds of all variable stars appear to be pulsating.[8] In the 1930s astronomer Arthur Stanley Eddington showed that the mathematical equations that describe the interior of a star may lead
to instabilities that cause a star to pulsate.[9] The most common type of instability is related to oscillations in the degree of ionization in outer, convective layers of the star.[10]
When the star is in the swelling phase, its outer layers expand, causing them to cool. Because of the decreasing temperature the degree of ionization also decreases. This makes the gas more
transparent, and thus makes it easier for the star to radiate its energy. This in turn makes the star start to contract. As the gas is thereby compressed, it is heated and the degree of ionization
again increases. This makes the gas more opaque, and radiation temporarily becomes captured in the gas. This heats the gas further, leading it to expand once again. Thus a cycle of expansion and
compression (swelling and shrinking) is maintained.
The pulsation of cepheids is known to be driven by oscillations in the ionization of helium (from He++ to He+ and back to He++).[11]
Main article: Variable star designation
In a given constellation, the first variable stars discovered were designated with letters R through Z, e.g. R Andromedae. This system of nomenclature was developed by Friedrich W. Argelander, who
gave the first previously unnamed variable in a constellation the letter R, the first letter not used by Bayer. Letters RR through RZ, SS through SZ, up to ZZ are used for the next discoveries, e.g.
RR Lyrae. Later discoveries used letters AA through AZ, BB through BZ, and up to QQ through QZ (with J omitted). Once those 334 combinations are exhausted, variables are numbered in order of
discovery, starting with the prefixed V335 onwards.
Variable stars may be either intrinsic or extrinsic.
Intrinsic variable stars: stars where the variability is being caused by changes in the physical properties of the stars themselves. This category can be divided into three subgroups.
Pulsating variables, stars whose radius alternately expands and contracts as part of their natural evolutionary ageing processes.
Eruptive variables, stars who experience eruptions on their surfaces like flares or mass ejections.
Cataclysmic or explosive variables, stars that undergo a cataclysmic change in their properties like novae and supernovae.
Extrinsic variable stars: stars where the variability is caused by external properties like rotation or eclipses. There are two main subgroups.
Eclipsing binaries, double stars where, as seen from Earth's vantage point the stars occasionally eclipse one another as they orbit.
Rotating variables, stars whose variability is caused by phenomena related to their rotation. Examples are stars with extreme "sunspots" which affect the apparent brightness or stars that have fast
rotation speeds causing them to become ellipsoidal in shape.
These subgroups themselves are further divided into specific types of variable stars that are usually named after their prototype. For example, dwarf novae are designated U Geminorum stars after the
first recognized star in the class, U Geminorum.
Intrinsic variable stars
Intrinsic variable types in the Hertzsprung–Russell diagram
Examples of types within these divisions are given below.
Pulsating variable stars
Main article: Stellar pulsation
The pulsating stars swell and shrink, affecting their brightness and spectrum. Pulsations are generally split into: radial, where the entire star expands and shrinks as a whole; and non-radial, where
one part of the star expands while another part shrinks.
Depending on the type of pulsation and its location within the star, there is a natural or fundamental frequency which determines the period of the star. Stars may also pulsate in a harmonic or
overtone which is a higher frequency, corresponding to a shorter period. Pulsating variable stars sometimes have a single well-defined period, but often they pulsate simultaneously with multiple
frequencies and complex analysis is required to determine the separate interfering periods. In some cases, the pulsations do not have a defined frequency, causing a random variation, referred to as
stochastic. The study of stellar interiors using their pulsations is known as asteroseismology.
The expansion phase of a pulsation is caused by the blocking of the internal energy flow by material with a high opacity, but this must occur at a particular depth of the star to create visible
pulsations. If the expansion occurs below a convective zone then no variation will be visible at the surface. If the expansion occurs too close to the surface the restoring force will be too weak to
create a pulsation. The restoring force to create the contraction phase of a pulsation can be pressure if the pulsation occurs in a non-degenerate layer deep inside a star, and this is called an
acoustic or pressure mode of pulsation, abbreviated to p-mode. In other cases, the restoring force is gravity and this is called a g-mode. Pulsating variable stars typically pulsate in only one of
these modes.
Cepheids and cepheid-like variables
Main article: Cepheid variable
This group consists of several kinds of pulsating stars, all found on the instability strip, that swell and shrink very regularly caused by the star's own mass resonance, generally by the fundamental
frequency. Generally the Eddington valve mechanism for pulsating variables is believed to account for cepheid-like pulsations. Each of the subgroups on the instability strip has a fixed relationship
between period and absolute magnitude, as well as a relation between period and mean density of the star. The period-luminosity relationship was first established for Delta Cepheids by Henrietta
Leavitt, and makes these high luminosity Cepheids very useful for determining distances to galaxies within the Local Group and beyond. Edwin Hubble used this method to prove that the so-called spiral
nebulae are in fact distant galaxies.
Note that the Cepheids are named only for Delta Cephei, while a completely separate class of variables is named after Beta Cephei.
Classical Cepheid variables
Main article: Classical Cepheid variable
Classical Cepheids (or Delta Cephei variables) are population I (young, massive, and luminous) yellow supergiants which undergo pulsations with very regular periods on the order of days to months. On
September 10, 1784, Edward Pigott detected the variability of Eta Aquilae, the first known representative of the class of Cepheid variables. However, the namesake for classical Cepheids is the star
Delta Cephei, discovered to be variable by John Goodricke a few months later.
Type II Cepheids
Main article: Type II Cepheids
Type II Cepheids (historically termed W Virginis stars) have extremely regular light pulsations and a luminosity relation much like the δ Cephei variables, so initially they were confused with the
latter category. Type II Cepheids stars belong to older Population II stars, than do the type I Cepheids. The Type II have somewhat lower metallicity, much lower mass, somewhat lower luminosity, and
a slightly offset period verses luminosity relationship, so it is always important to know which type of star is being observed.
RR Lyrae variables
Main article: RR Lyrae variable
These stars are somewhat similar to Cepheids, but are not as luminous and have shorter periods. They are older than type I Cepheids, belonging to Population II, but of lower mass than type II
Cepheids. Due to their common occurrence in globular clusters, they are occasionally referred to as cluster Cepheids. They also have a well established period-luminosity relationship, and so are also
useful as distance indicators. These A-type stars vary by about 0.2–2 magnitudes (20% to over 500% change in luminosity) over a period of several hours to a day or more.
Delta Scuti variables
Main article: Delta Scuti variable
Delta Scuti (δ Sct) variables are similar to Cepheids but much fainter and with much shorter periods. They were once known as Dwarf Cepheids. They often show many superimposed periods, which combine
to form an extremely complex light curve. The typical δ Scuti star has an amplitude of 0.003–0.9 magnitudes (0.3% to about 130% change in luminosity) and a period of 0.01–0.2 days. Their spectral
type is usually between A0 and F5.
SX Phoenicis variables
Main article: SX Phoenicis variable
These stars of spectral type A2 to F5, similar to δ Scuti variables, are found mainly in globular clusters. They exhibit fluctuations in their brightness in the order of 0.7 magnitude (about 100%
change in luminosity) or so every 1 to 2 hours.
Rapidly oscillating Ap variables
Main article: Rapidly oscillating Ap star
These stars of spectral type A or occasionally F0, a sub-class of δ Scuti variables found on the main sequence. They have extremely rapid variations with periods of a few minutes and amplitudes of a
few thousandths of a magnitude.
Long period variables
Main article: Long period variable
The long period variables are cool evolved stars that pulsate with periods in the range of weeks to several years.
Mira variables
Light curve of Mira variable χ Cygni
Main article: Mira variable
Mira variables are AGB red giants. Over periods of many months they fade and brighten by between 2.5 and 11 magnitudes, a 6 fold to 30,000 fold change in luminosity. Mira itself, also known as
Omicron Ceti (ο Cet), varies in brightness from almost 2nd magnitude to as faint as 10th magnitude with a period of roughly 332 days. The very large visual amplitudes are mainly due to the shifting
of energy output between visual and infra-red as the temperature of the star changes. In a few cases, Mira variables show dramatic period changes over a period of decades, thought to be related to
the thermal pulsing cycle of the most advanced AGB stars.
Semiregular variables
Main article: Semiregular variable
These are red giants or supergiants. Semiregular variables may show a definite period on occasion, but more often show less well-defined variations that can sometimes be resolved into multiple
periods. A well-known example of a semiregular variable is Betelgeuse, which varies from about magnitudes +0.2 to +1.2 (a factor 2.5 change in luminosity). At least some of the semi-regular variables
are very closely related to Mira variables, possibly the only difference being pulsating in a different harmonic.
Slow irregular variables
Main article: Slow irregular variable
These are red giants or supergiants with little or no detectable periodicity. Some are poorly studied semiregular variables, often with multiple periods, but others may simply be chaotic.
Long secondary period variables
Main article: Long-period variable star § Long secondary periods
Many variable red giants and supergiants show variations over several hundred to several thousand days. The brightness may change by several magnitudes although it is often much smaller, with the
more rapid primary variations are superimposed. The reasons for this type of variation are not clearly understood, being variously ascribed to pulsations, binarity, and stellar rotation.[12][13][14]
Beta Cephei variables
Main article: Beta Cephei variable
Beta Cephei (β Cep) variables (sometimes called Beta Canis Majoris variables, especially in Europe)[15] undergo short period pulsations in the order of 0.1–0.6 days with an amplitude of 0.01–0.3
magnitudes (1% to 30% change in luminosity). They are at their brightest during minimum contraction. Many stars of this kind exhibits multiple pulsation periods.[16]
Slowly pulsating B-type stars
Main article: Slowly pulsating B-type star
Slowly pulsating B (SPB) stars are hot main-sequence stars slightly less luminous than the Beta Cephei stars, with longer periods and larger amplitudes.[17]
Very rapidly pulsating hot (subdwarf B) stars
Main article: Subdwarf B star § Variables
The prototype of this rare class is V361 Hydrae, a 15th magnitude subdwarf B star. They pulsate with periods of a few minutes and may simultaneous pulsate with multiple periods. They have amplitudes
of a few hundredths of a magnitude and are given the GCVS acronym RPHS. They are p-mode pulsators.[18]
PV Telescopii variables
Main article: PV Telescopii variable
Stars in this class are type Bp supergiants with a period of 0.1–1 day and an amplitude of 0.1 magnitude on average. Their spectra are peculiar by having weak hydrogen while on the other hand carbon
and helium lines are extra strong, a type of Extreme helium star.
RV Tauri variables
Main article: RV Tauri variable
These are yellow supergiant stars (actually low mass post-AGB stars at the most luminous stage of their lives) which have alternating deep and shallow minima. This double-peaked variation typically
has periods of 30–100 days and amplitudes of 3–4 magnitudes. Superimposed on this variation, there may be long-term variations over periods of several years. Their spectra are of type F or G at
maximum light and type K or M at minimum brightness. They lie near the instability strip, cooler than type I Cepheids more luminous than type II Cepheids. Their pulsations are caused by the same
basic mechanisms related to helium opacity, but they are at a very different stage of their lives.
Alpha Cygni variables
Main article: Alpha Cygni variable
Alpha Cygni (α Cyg) variables are nonradially pulsating supergiants of spectral classes Bep to AepIa. Their periods range from several days to several weeks, and their amplitudes of variation are
typically of the order of 0.1 magnitudes. The light changes, which often seem irregular, are caused by the superposition of many oscillations with close periods. Deneb, in the constellation of Cygnus
is the prototype of this class.
Gamma Doradus variables
Main article: Gamma Doradus variable
Gamma Doradus (γ Dor) variables are non-radially pulsating main-sequence stars of spectral classes F to late A. Their periods are around one day and their amplitudes typically of the order of 0.1
Pulsating white dwarfs
Main article: Pulsating white dwarf
These non-radially pulsating stars have short periods of hundreds to thousands of seconds with tiny fluctuations of 0.001 to 0.2 magnitudes. Known types of pulsating white dwarf (or pre-white dwarf)
include the DAV, or ZZ Ceti, stars, with hydrogen-dominated atmospheres and the spectral type DA;[19] DBV, or V777 Her, stars, with helium-dominated atmospheres and the spectral type DB;[20] and GW
Vir stars, with atmospheres dominated by helium, carbon, and oxygen. GW Vir stars may be subdivided into DOV and PNNV stars.[21][22]
Solar-like oscillations
The Sun oscillates with very low amplitude in a large number of modes having periods around 5 minutes. The study of these oscillations is known as helioseismology. Oscillations in the Sun are driven
stochastically by convection in its outer layers. The term solar-like oscillations is used to describe oscillations in other stars that are excited in the same way and the study of these oscillations
is one of the main areas of active research in the field of asteroseismology.
BLAP variables
Main article: BLAP (Blue Large-Amplitude Pulsators)
A Blue Large-Amplitude Pulsator (BLAP) is a pulsating star characterized by changes of 0.2 to 0.4 magnitudes with typical periods of 20 to 40 minutes.
Eruptive variable stars
Eruptive variable stars show irregular or semi-regular brightness variations caused by material being lost from the star, or in some cases being accreted to it. Despite the name these are not
explosive events, those are the cataclysmic variables.
Main article: Pre–main-sequence star
Protostars are young objects that have not yet completed the process of contraction from a gas nebula to a veritable star. Most protostars exhibit irregular brightness variations.
Herbig Ae/Be stars
Herbig Ae/Be star star V1025 Tauri
Main article: Herbig Ae/Be stars
Variability of more massive (2–8 solar mass) Herbig Ae/Be stars is thought to be due to gas-dust clumps, orbiting in the circumstellar disks.
Orion variables
Main article: Orion variable
Orion variables are young, hot pre–main-sequence stars usually embedded in nebulosity. They have irregular periods with amplitudes of several magnitudes. A well-known subtype of Orion variables are
the T Tauri variables. Variability of T Tauri stars is due to spots on the stellar surface and gas-dust clumps, orbiting in the circumstellar disks.
FU Orionis variables
Main article: FU Orionis star
These stars reside in reflection nebulae and show gradual increases in their luminosity in the order of 6 magnitudes followed by a lengthy phase of constant brightness. They then dim by 2 magnitudes
(six times dimmer) or so over a period of many years. V1057 Cygni for example dimmed by 2.5 magnitude (ten times dimmer) during an eleven-year period. FU Orionis variables are of spectral type A
through G and are possibly an evolutionary phase in the life of T Tauri stars.
Giants and supergiants
Large stars lose their matter relatively easily. For this reason variability due to eruptions and mass loss is fairly common among giants and supergiants.
Luminous blue variables
Main article: Luminous blue variable
Also known as the S Doradus variables, the most luminous stars known belong to this class. Examples include the hypergiants η Carinae and P Cygni. They have permanent high mass loss, but at intervals
of years internal pulsations cause the star to exceed its Eddington limit and the mass loss increases hugely. Visual brightness increases although the overall luminosity is largely unchanged. Giant
eruptions observed in a few LBVs do increase the luminosity, so much so that they have been tagged supernova impostors, and may be a different type of event.
Yellow hypergiants
Main article: Yellow hypergiant
These massive evolved stars are unstable due to their high luminosity and position above the instability strip, and they exhibit slow but sometimes large photometric and spectroscopic changes due to
high mass loss and occasional larger eruptions, combined with secular variation on an observable timescale. The best known example is Rho Cassiopeiae.
R Coronae Borealis variables
Main article: R Coronae Borealis variable
While classed as eruptive variables, these stars do not undergo periodic increases in brightness. Instead they spend most of their time at maximum brightness, but at irregular intervals they suddenly
fade by 1–9 magnitudes (2.5 to 4000 times dimmer) before recovering to their initial brightness over months to years. Most are classified as yellow supergiants by luminosity, although they are
actually post-AGB stars, but there are both red and blue giant R CrB stars. R Coronae Borealis (R CrB) is the prototype star. DY Persei variables are a subclass of R CrB variables that have a
periodic variability in addition to their eruptions.
Wolf–Rayet variables
Main article: Wolf–Rayet star
Classic population I Wolf–Rayet stars are massive hot stars that sometimes show variability, probably due to several different causes including binary interactions and rotating gas clumps around the
star. They exhibit broad emission line spectra with helium, nitrogen, carbon and oxygen lines. Variations in some stars appear to be stochastic while others show multiple periods.
Gamma Cassiopeiae variables
Main article: Gamma Cassiopeiae variable
Gamma Cassiopeiae (γ Cas) variables are non-supergiant fast-rotating B class emission line-type stars that fluctuate irregularly by up to 1.5 magnitudes (4 fold change in luminosity) due to the
ejection of matter at their equatorial regions caused by the rapid rotational velocity.
Flare stars
Main article: Flare star
In main-sequence stars major eruptive variability is exceptional. It is common only among the flare stars, also known as the UV Ceti variables, very faint main-sequence stars which undergo regular
flares. They increase in brightness by up to two magnitudes (six times brighter) in just a few seconds, and then fade back to normal brightness in half an hour or less. Several nearby red dwarfs are
flare stars, including Proxima Centauri and Wolf 359.
RS Canum Venaticorum variables
Main article: RS Canum Venaticorum variable
These are close binary systems with highly active chromospheres, including huge sunspots and flares, believed to be enhanced by the close companion. Variability scales ranges from days, close to the
orbital period and sometimes also with eclipses, to years as sunspot activity varies.
Cataclysmic or explosive variable stars
Main articles: Cataclysmic variable star and Symbiotic variable star
Main article: Supernova
Supernovae are the most dramatic type of cataclysmic variable, being some of the most energetic events in the universe. A supernova can briefly emit as much energy as an entire galaxy, brightening by
more than 20 magnitudes (over one hundred million times brighter). The supernova explosion is caused by a white dwarf or a star core reaching a certain mass/density limit, the Chandrasekhar limit,
causing the object to collapse in a fraction of a second. This collapse "bounces" and causes the star to explode and emit this enormous energy quantity. The outer layers of these stars are blown away
at speeds of many thousands of kilometers per second. The expelled matter may form nebulae called supernova remnants. A well-known example of such a nebula is the Crab Nebula, left over from a
supernova that was observed in China and elsewhere in 1054. The progenitor object may either disintegrate completely in the explosion, or, in the case of a massive star, the core can become a neutron
star (generally a pulsar).
Supernovae can result from the death of an extremely massive star, many times heavier than the Sun. At the end of the life of this massive star, a non-fusible iron core is formed from fusion ashes.
This iron core is pushed towards the Chandrasekhar limit till it surpasses it and therefore collapses. One of the most studied supernovae of this type is SN 1987A in the Large Magellanic Cloud.
A supernova may also result from mass transfer onto a white dwarf from a star companion in a double star system. The Chandrasekhar limit is surpassed from the infalling matter. The absolute
luminosity of this latter type is related to properties of its light curve, so that these supernovae can be used to establish the distance to other galaxies.
Luminous red nova
Images showing the expansion of the light echo of V838 Monocerotis
Main article: Luminous red nova
Luminous red novae are stellar explosions caused by the merger of two stars. They are not related to classical novae. They have a characteristic red appearance and very slow decline following the
initial outburst.
Main article: Nova
Novae are also the result of dramatic explosions, but unlike supernovae do not result in the destruction of the progenitor star. Also unlike supernovae, novae ignite from the sudden onset of
thermonuclear fusion, which under certain high pressure conditions (degenerate matter) accelerates explosively. They form in close binary systems, one component being a white dwarf accreting matter
from the other ordinary star component, and may recur over periods of decades to centuries or millennia. Novae are categorised as fast, slow or very slow, depending on the behaviour of their light
curve. Several naked eye novae have been recorded, Nova Cygni 1975 being the brightest in the recent history, reaching 2nd magnitude.
Dwarf novae
Main article: Dwarf nova
Dwarf novae are double stars involving a white dwarf in which matter transfer between the component gives rise to regular outbursts. There are three types of dwarf nova:
U Geminorum stars, which have outbursts lasting roughly 5–20 days followed by quiet periods of typically a few hundred days. During an outburst they brighten typically by 2–6 magnitudes. These stars
are also known as SS Cygni variables after the variable in Cygnus which produces among the brightest and most frequent displays of this variable type.
Z Camelopardalis stars, in which occasional plateaux of brightness called standstills are seen, part way between maximum and minimum brightness.
SU Ursae Majoris stars, which undergo both frequent small outbursts, and rarer but larger superoutbursts. These binary systems usually have orbital periods of under 2.5 hours.
DQ Herculis variables
Main article: Intermediate polar
DQ Herculis systems are interacting binaries in which a low-mass star transfers mass to a highly magnetic white dwarf. The white dwarf spin period is significantly shorter than the binary orbital
period and can sometimes be detected as a photometric periodicity. An accretion disk usually forms around the white dwarf, but its innermost regions are magnetically truncated by the white dwarf.
Once captured by the white dwarf's magnetic field, the material from the inner disk travels along the magnetic field lines until it accretes. In extreme cases, the white dwarf's magnetism prevents
the formation of an accretion disk.
AM Herculis variables
Main article: Polar (cataclysmic variable star)
In these cataclysmic variables, the white dwarf's magnetic field is so strong that it synchronizes the white dwarf's spin period with the binary orbital period. Instead of forming an accretion disk,
the accretion flow is channeled along the white dwarf's magnetic field lines until it impacts the white dwarf near a magnetic pole. Cyclotron radiation beamed from the accretion region can cause
orbital variations of several magnitudes.
Z Andromedae variables
Main article: Z Andromedae variable
These symbiotic binary systems are composed of a red giant and a hot blue star enveloped in a cloud of gas and dust. They undergo nova-like outbursts with amplitudes of up to 4 magnitudes. The
prototype for this class is Z Andromedae.
AM CVn variables
Main article: AM Canum Venaticorum star
AM CVn variables are symbiotic binaries where a white dwarf is accreting helium-rich material from either another white dwarf, a helium star, or an evolved main-sequence star. They undergo complex
variations, or at times no variations, with ultrashort periods.
Extrinsic variable stars
There are two main groups of extrinsic variables: rotating stars and eclipsing stars.
Rotating variable stars
Stars with sizeable sunspots may show significant variations in brightness as they rotate, and brighter areas of the surface are brought into view. Bright spots also occur at the magnetic poles of
magnetic stars. Stars with ellipsoidal shapes may also show changes in brightness as they present varying areas of their surfaces to the observer.
Non-spherical stars
Ellipsoidal variables
These are very close binaries, the components of which are non-spherical due to their tidal interaction. As the stars rotate the area of their surface presented towards the observer changes and this
in turn affects their brightness as seen from Earth.
Stellar spots
The surface of the star is not uniformly bright, but has darker and brighter areas (like the sun's solar spots). The star's chromosphere too may vary in brightness. As the star rotates we observe
brightness variations of a few tenths of magnitudes.
FK Comae Berenices variables
These stars rotate extremely fast (~100 km/s at the equator); hence they are ellipsoidal in shape. They are (apparently) single giant stars with spectral types G and K and show strong chromospheric
emission lines. Examples are FK Com, HD 199178 and UZ Lib. A possible explanation for the rapid rotation of FK Comae stars is that they are the result of the merger of a (contact) binary.
BY Draconis variable stars
Main article: BY Draconis variable
BY Draconis stars are of spectral class K or M and vary by less than 0.5 magnitudes (70% change in luminosity).
Magnetic fields
Alpha-2 Canum Venaticorum variables
Main article: Alpha-2 Canum Venaticorum variable
Alpha-2 Canum Venaticorum (α2 CVn) variables are main-sequence stars of spectral class B8–A7 that show fluctuations of 0.01 to 0.1 magnitudes (1% to 10%) due to changes in their magnetic fields.
SX Arietis variables
Main article: SX Arietis variable
Stars in this class exhibit brightness fluctuations of some 0.1 magnitude caused by changes in their magnetic fields due to high rotation speeds.
Optically variable pulsars
Main article: Pulsar
Few pulsars have been detected in visible light. These neutron stars change in brightness as they rotate. Because of the rapid rotation, brightness variations are extremely fast, from milliseconds to
a few seconds. The first and the best known example is the Crab Pulsar.
Eclipsing binaries
Main article: Binary star § Eclipsing binaries
How eclipsing binaries vary in brightness
Extrinsic variables have variations in their brightness, as seen by terrestrial observers, due to some external source. One of the most common reasons for this is the presence of a binary companion
star, so that the two together form a binary star. When seen from certain angles, one star may eclipse the other, causing a reduction in brightness. One of the most famous eclipsing binaries is
Algol, or Beta Persei (β Per).
Algol variables
Main article: Algol variable
Algol variables undergo eclipses with one or two minima separated by periods of nearly constant light. The prototype of this class is Algol in the constellation Perseus.
Double Periodic variables
Main article: Double periodic variable
Double periodic variables exhibit cyclical mass exchange which causes the orbital period to vary predictably over a very long period. The best known example is V393 Scorpii.
Beta Lyrae variables
Main article: Beta Lyrae variable
Beta Lyrae (β Lyr) variables are extremely close binaries, named after the star Sheliak. The light curves of this class of eclipsing variables are constantly changing, making it almost impossible to
determine the exact onset and end of each eclipse.
W Serpentis variables
W Serpentis is the prototype of a class of semi-detached binaries including a giant or supergiant transferring material to a massive more compact star. They are characterised, and distinguished from
the similar β Lyr systems, by strong UV emission from accretions hotspots on a disc of material.
W Ursae Majoris variables
Main article: W Ursae Majoris variable
The stars in this group show periods of less than a day. The stars are so closely situated to each other that their surfaces are almost in contact with each other.
Planetary transits
Stars with planets may also show brightness variations if their planets pass between Earth and the star. These variations are much smaller than those seen with stellar companions and are only
detectable with extremely accurate observations. Examples include HD 209458 and GSC 02652-01324, and all of the planets and planet candidates detected by the Kepler Mission.
See also
iconStar portal
Guest star
Irregular variable
List of variable stars
Low-dimensional chaos in stellar pulsations
Stellar pulsations
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Kilkenny, D. (2007). "Pulsating Hot Subdwarfs -- an Observational Review". Communications in Asteroseismology. 150: 234–240. Bibcode:2007CoAst.150..234K. doi:10.1553/cia150s234.
Koester, D.; Chanmugam, G. (1990). "REVIEW: Physics of white dwarf stars". Reports on Progress in Physics. 53 (7): 837. Bibcode:1990RPPh...53..837K. doi:10.1088/0034-4885/53/7/001. S2CID 122582479.
Murdin, Paul (2002). Encyclopedia of Astronomy and Astrophysics. Bibcode:2002eaa..book.....M. ISBN 0-333-75088-8.
Quirion, P.-O.; Fontaine, G.; Brassard, P. (2007). "Mapping the Instability Domains of GW Vir Stars in the Effective Temperature-Surface Gravity Diagram". The Astrophysical Journal Supplement Series.
171 (1): 219–248. Bibcode:2007ApJS..171..219Q. doi:10.1086/513870.
Nagel, T.; Werner, K. (2004). "Detection of non-radial g-mode pulsations in the newly discovered PG 1159 star HE 1429-1209". Astronomy and Astrophysics. 426 (2): L45.arXiv:astro-ph/0409243.
Bibcode:2004A&A...426L..45N. doi:10.1051/0004-6361:200400079. S2CID 9481357.
Variable stars
Cepheids and
Type I (Classical cepheids, Delta Scuti) Type II (BL Herculis, W Virginis, RV Tauri) RR Lyrae Rapidly oscillating Ap SX Phoenicis
Blue-white with
early spectra
Alpha Cygni Beta Cephei Slowly pulsating B-type PV Telescopii Blue large-amplitude pulsator
Mira Semiregular Slow irregular
Gamma Doradus Solar-like oscillations White dwarf
Protostar and PMS
Herbig Ae/Be Orion
FU Orionis T Tauri
Giants and
Luminous blue variable R Coronae Borealis (DY Persei) Yellow hypergiant
Eruptive binary
Double periodic FS Canis Majoris RS Canum Venaticorum
Flare Gamma Cassiopeiae Lambda Eridani Wolf–Rayet
AM Canum Venaticorum Dwarf nova Luminous red nova Nova Polar Intermediate polar Supernova
Hypernova SW Sextantis Symbiotic
Symbiotic nova Z Andromedae
Rotating ellipsoidal
Stellar spots
BY Draconis FK Comae Berenices
Magnetic fields
Alpha² Canum Venaticorum Pulsar SX Arietis
Algol Beta Lyrae Planetary transit W Ursae Majoris
He1523a.jpg Star portal * List
Accretion Molecular cloud Bok globule Young stellar object
Protostar Pre-main-sequence Herbig Ae/Be T Tauri FU Orionis Herbig–Haro object Hayashi track Henyey track
Main sequence Red-giant branch Horizontal branch
Red clump Asymptotic giant branch
super-AGB Blue loop Protoplanetary nebula Planetary nebula PG1159 Dredge-up OH/IR Instability strip Luminous blue variable Blue straggler Stellar population Supernova Superluminous supernova /
Early Late Main sequence
O B A F G K M Brown dwarf WR OB Subdwarf
O B Subgiant Giant
Blue Red Yellow Bright giant Supergiant
Blue Red Yellow Hypergiant
Yellow Carbon
S CN CH White dwarf Chemically peculiar
Am Ap/Bp HgMn Helium-weak Barium Extreme helium Lambda Boötis Lead Technetium Be
Shell B[e]
White dwarf
Helium planet Black dwarf Neutron
Radio-quiet Pulsar
Binary X-ray Magnetar Stellar black hole X-ray binary
Blue dwarf Green Black dwarf Exotic
Boson Electroweak Strange Preon Planck Dark Dark-energy Quark Q Black Gravastar Frozen Quasi-star Thorne–Żytkow object Iron Blitzar
Deuterium burning Lithium burning Proton–proton chain CNO cycle Helium flash Triple-alpha process Alpha process Carbon burning Neon burning Oxygen burning Silicon burning S-process R-process Fusor
Symbiotic Remnant Luminous red nova
Core Convection zone
Microturbulence Oscillations Radiation zone Atmosphere
Photosphere Starspot Chromosphere Stellar corona Stellar wind
Bubble Bipolar outflow Accretion disk Asteroseismology
Helioseismology Eddington luminosity Kelvin–Helmholtz mechanism
Designation Dynamics Effective temperature Luminosity Kinematics Magnetic field Absolute magnitude Mass Metallicity Rotation Starlight Variable Photometric system Color index Hertzsprung–Russell
diagram Color–color diagram
Star systems
Contact Common envelope Eclipsing Symbiotic Multiple Cluster
Open Globular Super Planetary system
Solar System Sunlight Pole star Circumpolar Constellation Asterism Magnitude
Apparent Extinction Photographic Radial velocity Proper motion Parallax Photometric-standard
Proper names
Arabic Chinese Extremes Most massive Highest temperature Lowest temperature Largest volume Smallest volume Brightest
Historical Most luminous Nearest
Nearest bright With exoplanets Brown dwarfs White dwarfs Milky Way novae Supernovae
Candidates Remnants Planetary nebulae Timeline of stellar astronomy
Related articles
Substellar object
Brown dwarf Sub-brown dwarf Planet Galactic year Galaxy Guest Gravity Intergalactic Planet-hosting stars Tidal disruption event
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
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Brief Announcement: Noisy Beeping Networks
We introduce noisy beeping networks, where nodes have limited communication capabilities, namely, they can only emit energy or sense the channel for energy. Furthermore, imperfections may cause
devices to malfunction with some fixed probability when sensing the channel, which amounts to deducing a noisy received transmission. Such noisy networks have implications for ultra-lightweight
sensor networks and biological systems. We show how to compute tasks in a noise-resilient manner over noisy beeping networks of arbitrary structure. In particular, we transform any R-round algorithm
that assumes a noiseless beeping network (of size n) into a noise-resilient version while incurring a multiplicative overhead of only O(log n + log R) in its round complexity, with high probability.
We show that our coding is optimal for some (short) tasks, such as node-coloring of a clique. We further show how to simulate a large family of algorithms designed for distributed networks in the
CONGEST(B) model over a noisy beeping network. The simulation succeeds with high probability and incurs an asymptotic multiplicative overhead of O(B · Δ · min(n, Δ^2)) in the round complexity, where
Δ is the maximum degree of the network. The overhead is tight for certain graphs, e.g., a clique. Further, this simulation implies a constant overhead coding for constant-degree networks.
Publication series
Name Proceedings of the Annual ACM Symposium on Principles of Distributed Computing
Conference 39th Symposium on Principles of Distributed Computing, PODC 2020
Country/Territory Italy
City Virtual, Online
Period 3/08/20 → 7/08/20
Bibliographical note
Publisher Copyright:
© 2020 ACM.
• collision-detection
• error-correction in networks
• noise-resilience
Dive into the research topics of 'Brief Announcement: Noisy Beeping Networks'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/brief-announcement-noisy-beeping-networks","timestamp":"2024-11-07T03:14:34Z","content_type":"text/html","content_length":"51727","record_id":"<urn:uuid:307ee8e2-1848-43e3-936f-052bfc60f808>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00073.warc.gz"} |
BIEN 225 Cellular Automata Assignment | COURSE ACHIEVERS
BIEN 225 Cellular Automata Assignment
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BIEN 225 – Self-Organization in Engineered and Native Tissue :
In this assignment, you expand on what you produced in HW1 to create a more physiologically relevant “Game of Life”. This will be discussed in Discussion, so please come with questions and read the
paper in advance.
Please name your m-file using the following format:
Where LastFirst is your Last name and your First name, and # is the problem number. Please start each m file with the following lines:
%LAST FIRST %HW2 %P1 clearvars close all clc
In this homework, we’ll be re-creating a classic cellular automata model, similar to Conway’s Game of Life, but modified to more closely capture the dynamics of mammalian cells grown in a monolayer.
In the paper “Proliferation of Anchorage-Dependent Contact-Inhibited Cells: 1. Development of Theoretical Models Based on Cellular Automata” the authors present 2 models of adherent cell
proliferation, a synchronous and an asynchronous model. In the following problems you will replicate the synchronous (P1 and P2) and asynchronous models (P3). The paper is attached to the assignment
on iLearn and should be used as a reference. BIEN 225 Cellular Automata Assignment
Problem 1: You will recreate the synchronous model where cells can only proliferate in straight directions and not diagonals. Each iteration, each live cell should divide in a random “allowed”
direction and not proliferate onto existing cells or off the side of the board. Cells without proliferation options should not proliferate, cells that have directions blocked off should proliferate
randomly in one of the available directions. Cells do not die in this model.
To complete the assignment, create a simulation with the following parameters:
1. Exists on a 51×51 board 2. Has 4 randomly positioned cells as the initial condition 3. Iterates for 100 steps
Hint: The function randperm() can be used to create a random vector from 1 to 4 (where the numbers could correspond to directions, and you can use a system similar to the neighborhood assessment you
used in Game of Life to identify which directions are allowed or forbidden for any given cell. Further, the switch/case conditional is a convenient way to take actions based on the value of a
variable. BIEN 225 Cellular Automata Assignment
Problem 2: Extend your model from Problem 1 to allow for diagonal proliferation. Cells should proliferate along the “straights” until there are no available directions and then switch to a diagonal
proliferation on a random basis. Each cell should have a 20% chance of proliferating per available diagonal, but only can proliferate once per iterations. For example, a cell with 4 available
directions has an 80% chance of proliferating, but the direction must be random. A cell with 3 available diagonals has a 60% chance of proliferating, etc.
Problem 3: Extend your model from Problem 2 to the asynchronous model. After each proliferation, set both the daughter (the new cells) and the parent (the original cells) cells to a random number in
the range 5-10. Each cell should be set to its own random number. Each iteration, all cells should count down by 1. Cells at zero should proliferate according to the rules in Problem 1 and 2 or stay
at zero otherwise. BIEN 225 Cellular Automata Assignment
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Energy Absorbed by a Brake
The energy absorbed by a brake depends upon the type of motion of the moving body. The motion of a body may be either pure translation or pure rotation or a combination of both translation and
rotation. The energy corresponding to these motions is kinetic energy. Let us consider these motions as follows :
1. When the motion of the body is pure translation. Consider a body of mass (m) moving with a velocity v1 m / s. Let its velocity is reduced to v2 m / s by applying the brake. Therefore, the change
in kinetic energy of the translating body or kinetic energy of translation,
E1 = (1/2) m [(v1)2 - (v2)2]
This energy must be absorbed by the brake. If the moving body is stopped after applying the brakes, then v2 = 0, and
E1 = (1/2) m (v1)2
2. When the motion of the body is pure rotation. Consider a body of mass moment of inertia I (about a given axis) is rotating about that axis with an angular velocity ω1 rad / s. Let its angular
velocity is reduced to ω2 rad / s after applying the brake. Therefore, the change in
kinetic energy of the rotating body or kinetic energy of rotation,
E2 = (1/2) I [(ω1)2 - (ω2)2]
This energy must be absorbed by the brake. If the rotating body is stopped after applying the brakes, then ω2 = 0, and
E1 = (1/2) I (ω1)2
3. When the motion of the body is a combination of translation and rotation. Consider a body having both linear and angular motions, e.g. in the locomotive driving wheels and wheels of a moving car.
In such cases, the total kinetic energy of the body is equal to the sum ∴of the kinetic energies of translation and rotation.
Total kinetic energy to be absorbed by the brake,
E = E1 + E2
Sometimes, the brake has to absorb the potential energy given up by objects being lowered by hoists, elevators etc. Consider a body of mass m is being lowered from a height h1 to h2 by applying the
brake. Therefore the change in potential energy,
E3 = m.g (h1 – h2) | {"url":"https://www.brainkart.com/article/Energy-Absorbed-by-a-Brake_5936/","timestamp":"2024-11-09T14:05:29Z","content_type":"text/html","content_length":"33140","record_id":"<urn:uuid:577c0f68-d423-4aa5-8ff8-7bb0c36eddda>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00064.warc.gz"} |
Solving simultaneous equations -be negative addition
solving simultaneous equations -be negative addition Related topics: What Is Algebra Used For Today
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Author Message
achnac99 Posted: Saturday 23rd of Dec 15:54
Holla guys and gals! Recently I hired a math tutor to guide me with some topics in algebra. My problem areas included topics such as solving simultaneous equations -be negative addition
and angle complements. Now that teacher turned out to be such a waste, that instead of helping me now I’m even more confused than I used to be. I still can’t crack problems on those
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Extracting neuron simulation results
1. Extracting neuron simulation results
Extracting neuron simulation results
• Hello,
How would I get access to the calculations of the neuronal simulation (such as membrane voltage and ionic currents at each node, over time)?
Also, is there a way to get the X/Y/Z coordinates of the nodes? For example, to help see where in the model geometry the reported site of activation is from a Titration Evaluator.
Thank you in advance.
• To update, I've learned the membrane voltage, membrane current, and extracellular voltage can be specified as the measured quantity for a line sensor, with the number of time snapshots of your
choosing. Thus, all info to calculate the induced potential on the electrode over time from neural activity is available, one just has to create multiple, identical spline entities as only one
quantity can be sensed from a structure at at time (at least in the version of 5.0 that I'm using).
This also helps get a relative idea of where the action potential initiates along the axon relative to the electrode, though not an exact X/Y/Z location to show in the model view. | {"url":"https://forum.zmt.swiss/topic/151/extracting-neuron-simulation-results","timestamp":"2024-11-06T12:21:42Z","content_type":"text/html","content_length":"54780","record_id":"<urn:uuid:c0ad639c-b34f-46d1-a908-4153fa44cefc>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00436.warc.gz"} |
julia identity matrix
jpvt is an integer vector of length n corresponding to the permutation $P$. Downdate a Cholesky factorization C with the vector v. If A = C.U'C.U then CC = cholesky(C.U'C.U - v*v') but the
computation of CC only uses O(n^2) operations. If uplo = U, the upper half of A is stored. The atol and rtol keyword arguments requires at least Julia 1.1. If F::Schur is the factorization object,
the (quasi) triangular Schur factor can be obtained via either F.Schur or F.T and the orthogonal/unitary Schur vectors via F.vectors or F.Z such that A = F.vectors * F.Schur * F.vectors'. The vector
v is destroyed during the computation. Set the number of threads the BLAS library should use. If job = N, no condition numbers are found. The syntax of For loop in Julia is where for, in and end are
keywords. Returns A, vs containing the Schur vectors, and w, containing the eigenvalues. Note that adjoint is applied recursively to elements. If howmny = S, only the eigenvectors corresponding to
the values in select are computed. See also I. For SymTridiagonal block matrices, the elements of dv are symmetrized. If `Float64` element type is not necessary, consider the shorter `Matrix(I, m, m)
` (with default `eltype(I)` `Bool`). When p=2, the operator norm is the spectral norm, equal to the largest singular value of A. Another (typically slower but more accurate) option is alg =
QRIteration(). Those things have all moved into stdlib (LinearAlgebra and SparseArrays) which means that you just have to import them now. dA determines if the diagonal values are read or are assumed
to be all ones. The individual components of the factorization F can be accessed via getproperty: F further supports the following functions: lu! Only the ul triangle of A is used. Return alpha*A*x
or alpha*A'*x according to trans. Online computations on streaming data … This is the return type of cholesky(_, Val(true)), the corresponding matrix factorization function. Uses the output of
geqlf!. Using Julia version 1.5.3. A is assumed to be symmetric. Exception thrown when the input matrix has one or more zero-valued eigenvalues, and is not invertible. If $A$ is an m×n matrix, then.
Solve the equation AB * X = B. trans determines the orientation of AB. atol and rtol are the absolute and relative tolerances, respectively. Hello, How would someone create a 5x5 array (matrix?) Only
the uplo triangle of A is used. It is ignored when blocksize > minimum(size(A)). Only the ul triangle of A is used. The following table translates the most common Julia commands into R language. Only
the ul triangle of A is used. Finds the eigensystem of A. diagm constructs a full matrix; if you want storage-efficient versions with fast arithmetic, see Diagonal, Bidiagonal Tridiagonal and
SymTridiagonal. For general matrices, the complex Schur form (schur) is computed and the triangular algorithm is used on the triangular factor. This function requires LAPACK 3.6.0. The amsmath
package provides commands to typeset matrices with different delimiters. Update vector y as alpha*A*x + beta*y or alpha*A'*x + beta*y according to trans. gels! Returns A, overwritten by the
factorization, a pivot vector ipiv, and the error code info which is a non-negative integer. A QR matrix factorization stored in a compact blocked format, typically obtained from qr. Here, A must be
of special matrix type, like, e.g., Diagonal, UpperTriangular or LowerTriangular, or of some orthogonal type, see QR. Finds the generalized eigenvalues (jobz = N) or eigenvalues and eigenvectors
(jobz = V) of a symmetric matrix A and symmetric positive-definite matrix B. Reorders the Generalized Schur factorization F of a matrix pair (A, B) = (Q*S*Z', Q*T*Z') according to the logical array
select and returns a GeneralizedSchur object F. The selected eigenvalues appear in the leading diagonal of both F.S and F.T, and the left and right orthogonal/unitary Schur vectors are also reordered
such that (A, B) = F.Q*(F.S, F.T)*F.Z' still holds and the generalized eigenvalues of A and B can still be obtained with F.α./F.β. This quantity is also known in the literature as the Bauer condition
number, relative condition number, or componentwise relative condition number. Compute the inverse hyperbolic matrix tangent of a square matrix A. julia> M = [2 5; 1 3] 2×2 Array{Int64,2}: 2 5 1 3
julia> N = inv(M) 2×2 Array{Float64,2}: 3.0 -5.0 -1.0 2.0 julia> M*N == N*M == eye(2) true Computes matrix N such that M * N = I, where I is the identity matrix. If jobv = V the orthogonal/unitary
matrix V is computed. Julia - Identity matrix - eye () alternative 2 I'm working with what I guess is an older textbook that is using an older version of Julia as they are using the eye () function
to create an identity matrix, which appears to not exist in the version I am currently using. If fact = F and equed = R or B the elements of R must all be positive. Reorder the Schur factorization of
a matrix. The first dimension of T sets the block size and it must be between 1 and n. The second dimension of T must equal the smallest dimension of A. Recursively computes the blocked QR
factorization of A, A = QR. A = reshape([1.0,2.0,3.0,4.0], 1,4) println(A) println(A * eye(4)) # yields the same result [1.0 2.0 3.0 4.0] [1.0 2.0 3.0 4.0] inv() returns the inverse of a matrix. If
normtype = I, the condition number is found in the infinity norm. Entries of A below the first subdiagonal are ignored. What about kron(eye(4), randn(4,4)), New comments cannot be posted and votes
cannot be cast. tau contains scalars which parameterize the elementary reflectors of the factorization. Julia identity matrix keyword after analyzing the system lists the list of keywords related and
the list of websites with related content, in addition you can see which keywords most interested customers on … Reduce A in-place to bidiagonal form A = QBP'. If itype = 2, the problem to solve is A
* B * x = lambda * x. A is overwritten by Q. Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side = L or the equivalent right-sided multiplication for side =
R using Q from a LQ factorization of A computed using gelqf!. Matrix trace. Matrices in Julia are the heterogeneous type of containers and hence, they can hold elements of any data type. Returns the
upper triangle of M starting from the kth superdiagonal. Prior to Julia 1.1, NaN and ±Inf entries in B were treated inconsistently. The (quasi) triangular Schur factors can be obtained from the Schur
object F with F.S and F.T, the left unitary/orthogonal Schur vectors can be obtained with F.left or F.Q and the right unitary/orthogonal Schur vectors can be obtained with F.right or F.Z such that A=
F.left*F.S*F.right' and B=F.left*F.T*F.right'. It is not mandatory to define the data type of a matrix before assigning the elements to the matrix. Matrix The syntax for creating a matrix is very
similar — you will declare it row by row, putting semicolon (;) to indicate the elements should go on a new row: The syntax to create an n*m matrix of zeros is very similar to the one in Python, just
without the Numpy prefix: Otherwise, the inverse tangent is determined by using log. Matrix factorizations (a.k.a. Solves the equation A * X = B where A is a tridiagonal matrix with dl on the
subdiagonal, d on the diagonal, and du on the superdiagonal. x is the item of the range or collection for each iteration. Julia automatically decides the data type of the matrix by analyzing the
values assigned to it. Dot function for two complex vectors, consisting of n elements of array X with stride incx and n elements of array U with stride incy, conjugating the first vector.
matrix-product state and examine its relation to the traditional DMRG blocks and E. Jeckelmann: Density-Matrix Renormalization Group Algorithms , Lect. Returns T, Q, reordered eigenvalues in w, the
condition number of the cluster of eigenvalues s, and the condition number of the invariant subspace sep. Reorders the vectors of a generalized Schur decomposition. Return alpha*A*x or alpha*A'x
according to tA. B is overwritten with the solution X and returned. Update the vector y as alpha*A*x + beta*y or alpha*A'x + beta*y according to tA. The base array type in Julia is the abstract type
AbstractArray {T,N}. below (e.g. Update the vector y as alpha*A*x + beta*y. Update a Cholesky factorization C with the vector v. If A = C.U'C.U then CC = cholesky(C.U'C.U + v*v') but the computation
of CC only uses O(n^2) operations. Normalize the array a so that its p-norm equals unity, i.e. τ is a vector of length min(m,n) containing the coefficients $au_i$. It is similar to the QR format
except that the orthogonal/unitary matrix $Q$ is stored in Compact WY format [Schreiber1989]. Returns the vector or matrix X, overwriting B in-place. See also isposdef! Note that the transposition is
applied recursively to elements. Finds the singular value decomposition of A, A = U * S * V', using a divide and conquer approach. See the documentation on factorize for more information. Since the
p-norm is computed using the norms of the entries of A, the p-norm of a vector of vectors is not compatible with the interpretation of it as a block vector in general if p != 2. p can assume any
numeric value (even though not all values produce a mathematically valid vector norm). Finds the eigenvalues (jobz = N) or eigenvalues and eigenvectors (jobz = V) of a symmetric matrix A. Return the
smallest eigenvalue of A. If jobvl = V or jobvr = V, the corresponding eigenvectors are computed. The UnitRange irange specifies indices of the sorted eigenvalues to search for. If normtype = O or 1,
the condition number is found in the one norm. When A is sparse, a similar polyalgorithm is used. Return the distance between successive array elements in dimension 1 in units of element size. to
divide scalar from right. The algorithm produces Vt and hence Vt is more efficient to extract than V. The singular values in S are sorted in descending order. \[Q = \prod_{i=1}^{\min(m,n)} (I - \
tau_i v_i v_i^T).\], \[Q = \prod_{i=1}^{\min(m,n)} (I - \tau_i v_i v_i^T) tau must have length greater than or equal to the smallest dimension of A. Compute the QL factorization of A, A = QL. For
such matrices, eigenvalues λ that appear to be slightly negative due to roundoff errors are treated as if they were zero More precisely, matrices with all eigenvalues ≥ -rtol*(max |λ|) are treated as
semidefinite (yielding a Hermitian square root), with negative eigenvalues taken to be zero. Update C as alpha*A*B + beta*C or the other three variants according to tA and tB. An InexactError
exception is thrown if the factorization produces a number not representable by the element type of A, e.g. Some special matrix types (e.g. Solves the linear equation A * X = B (trans = N), transpose
(A) * X = B (trans = T), or adjoint(A) * X = B (trans = C) using the LU factorization of A. fact may be E, in which case A will be equilibrated and copied to AF; F, in which case AF and ipiv from a
previous LU factorization are inputs; or N, in which case A will be copied to AF and then factored. If jobvl = N, the left eigenvectors of A aren't computed. If A is upper or lower triangular (or
diagonal), no factorization of A is required and the system is solved with either forward or backward substitution. ```. Constructs an identity matrix of the same dimensions and type as A. If balanc
= B, A is permuted and scaled. Then you can use I as the identity matrix when you need it. to find its (upper if uplo = U, lower if uplo = L) Cholesky decomposition. Otherwise, the cosine is
determined by calling exp. The identity matrix is represented by eye() in most languages, Julia included. Computes Q * C (trans = N), transpose(Q) * C (trans = T), adjoint(Q) * C (trans = C) for side
= L or the equivalent right-sided multiplication for side = R using Q from a QR factorization of A computed using geqrt!. Iterating the decomposition produces the components S.L and S.Q. This section
concentrates on arrays and tuples; for more on dictionaries, see Dictionaries and Sets. Construct an UpperHessenberg view of the matrix A. Return a matrix M whose columns are the eigenvectors of A. A
= reshape([1.0,2.0,3.0,4.0], 1,4) println(A) println(A * eye(4)) # yields the same result [1.0 2.0 3.0 4.0] [1.0 2.0 3.0 4.0] inv() returns the inverse of a matrix. First, you need to add using
LinearAlgebra. If permuting was turned on, A[i,j] = 0 if j > i and 1 < j < ilo or j > ihi. Many functions of Julia for handling vectors and matrices are similar to those of MATLAB. Check that a
matrix is square, then return its common dimension. Note that Hupper will not be equal to Hlower unless A is itself Hermitian (e.g. When p=1, the operator norm is the maximum absolute column sum of
A: with $a_{ij}$ the entries of $A$, and $m$ and $n$ its dimensions. tau stores the elementary reflectors. Matrix inverse. If A has no negative real eigenvalues, compute the principal matrix square
root of A, that is the unique matrix $X$ with eigenvalues having positive real part such that $X^2 = A$. When Q is extracted, the resulting type is the HessenbergQ object, and may be converted to a
regular matrix with convert(Array, _) (or Array(_) for short). Currently unsupported for sparse matrix. The second argument p is not necessarily a part of the interface for norm, i.e. If side = L,
the left eigenvectors are computed. Full, diagonal and scale matrix types are supported in Julia 0.3 or higher. Multiplying by the identity. Then you can use I as the identity matrix when you need
it. For more information, see [issue8859], [B96], [S84], [KY88]. it is symmetric, or tridiagonal. If m<=n, then Matrix(F.Q) yields an m×m orthogonal matrix. For Julia, Vectors are just a special kind
of Matrix, namely with just one row (row matrix) or just one column (column matrix): Julia Vectors can come in two forms: Column Matrices (one column, N rows) and Row Matrices (one row, N columns)
Row Matrix. Feels more like one of those weird numpy calls arising from the constraints of Python, than like normal Julia. Rank-1 update of the matrix A with vectors x and y as alpha*x*y' + A. Rank-1
update of the symmetric matrix A with vector x as alpha*x*transpose(x) + A. uplo controls which triangle of A is updated. A linear solve involving such a matrix cannot be computed. Once you have
loaded \usepackage{amsmath} in your preamble, you can use the following environments in your math environments: Type L a T e X markup Renders as Plain \begin{matrix} usually also require fine-grained
control over the factorization of B. Divide each entry in an array A by a scalar b overwriting A in-place. ), Computes the eigenvalue decomposition of A, returning an Eigen factorization object F
which contains the eigenvalues in F.values and the eigenvectors in the columns of the matrix F.vectors. select specifies the eigenvalues in each cluster. Matrix: numbers grouped both horizontally and
vertically ? Introduction to Applied Linear Algebra Vectors, Matrices, and Least Squares Julia Language Companion Stephen Boyd and Lieven Vandenberghe DRAFT September 23, 2019 produced by factorize
or cholesky). For custom matrix and vector types, it is recommended to implement 5-argument mul! Given F, Julia employs an efficient algorithm for (F+μ*I) \ b (equivalent to (A+μ*I)x \ b) and related
operations like determinants. Iterating the decomposition produces the components F.T, F.Z, and F.values. Computes matrix N such that M * N = I, where I is the identity matrix. The LQ decomposition
is the QR decomposition of transpose(A), and it is useful in order to compute the minimum-norm solution lq(A) \ b to an underdetermined system of equations (A has more columns than rows, but has full
row rank). If pivoting is chosen (default) the element type should also support abs and <. At the time, I mentioned that we had begun trying to integrate a GPU-based tensor contraction backend and
were looking forward to some significant speedups. ipiv is the pivot information output and A contains the LU factorization of getrf!. Exception thrown when the input matrix was not positive
definite. Finds the inverse of (upper if uplo = U, lower if uplo = L) triangular matrix A. Use rmul! The Julia data ecosystem provides DataFrames.jl to work with datasets, and perform common data
manipulations. A is assumed to be Hermitian. trans may be one of N (no modification), T (transpose), or C (conjugate transpose). If balanc = N, no balancing is performed. If uplo = L, the lower half
is stored. If sense = E, reciprocal condition numbers are computed for the eigenvalues only. Returns the eigenvalues of A. See QRCompactWY. Returns alpha*A*B or one of the other three variants
determined by side and tA. Returns the uplo triangle of alpha*A*A' or alpha*A'*A, according to trans. Solves A * X = B for positive-definite tridiagonal A. The identity matrix is a the simplest
nontrivial diagonal matrix, defined such that I(X)=X (1) for all vectors X. For a scalar input, eigvals will return a scalar. The Givens type supports left multiplication G*A and conjugated transpose
right multiplication A*G'. Test whether a matrix is positive definite (and Hermitian) by trying to perform a Cholesky factorization of A, overwriting A in the process. In most cases, if A is a
subtype S of AbstractMatrix{T} with an element type T supporting +, -, * and /, the return type is LU{T,S{T}}. Matrix inverses in Julia David Zeng Keegan Go Stephen Boyd EE103 Stanford University
November 2, 2015. Creating Matrices How to create matrices in Julia and R? C is overwritten. A is assumed to be Hermitian. Recursively computes the blocked QR factorization of A, A = QR. The
triangular Cholesky factor can be obtained from the factorization F::CholeskyPivoted via F.L and F.U. D and E are overwritten and returned. If balanc = S, A is scaled but not permuted. is the same as
bunchkaufman, but saves space by overwriting the input A, instead of creating a copy. This is the return type of eigen, the corresponding matrix factorization function. If uplo = U the upper Cholesky
decomposition of A was computed. Return the updated y. Iterating the decomposition produces the components F.values and F.vectors. Julia features a rich collection of special matrix types, which
allow for fast computation with specialized routines that are specially developed for particular matrix types. If rook is true, rook pivoting is used. Return the singular values of A in descending
order. Only the ul triangle of A is used. Many of these are further specialized for certain special matrix types. For example: julia> A = [1 1; 1 1] 2×2 Array{Int64,2}: 1 1 1 1 julia> A + I 2×2 Array
{Int64,2}: 2 1 1 2 scale contains information about the scaling/permutations performed. Return A*x. For the block size $n_b$, it is stored as a m×n lower trapezoidal matrix $V$ and a matrix $T = (T_1
\; T_2 \; ... \; T_{b-1} \; T_b')$ composed of $b = \lceil \min(m,n) / n_b \rceil$ upper triangular matrices $T_j$ of size $n_b$×$n_b$ ($j = 1, ..., b-1$) and an upper trapezoidal $n_b$×$\min(m,n) -
(b-1) n_b$ matrix $T_b'$ ($j=b$) whose upper square part denoted with $T_b$ satisfying. If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the cosine. Return op(A)*b,
where op is determined by tA. See also lq. Finds the reciprocal condition number of (upper if uplo = U, lower if uplo = L) triangular matrix A. Such a view has the oneunit of the eltype of A on its
diagonal. Methods for complex arrays only. Only the ul triangle of A is used. Any keyword arguments passed to eigen are passed through to the lower-level eigen! Uses the output of gerqf!. Compute the
matrix secant of a square matrix A. Compute the matrix cosecant of a square matrix A. Compute the matrix cotangent of a square matrix A. Compute the matrix hyperbolic cosine of a square matrix A.
Compute the matrix hyperbolic sine of a square matrix A. Compute the matrix hyperbolic tangent of a square matrix A. Compute the matrix hyperbolic secant of square matrix A. Compute the matrix
hyperbolic cosecant of square matrix A. Compute the matrix hyperbolic cotangent of square matrix A. Compute the inverse matrix cosine of a square matrix A. ( A,2 ) Julia Tutorial, we will learn how
to create matrices in Julia, there no. Error and Berr is the return value can be obtained from the factorization F with: here, has! Solving of multiple systems dest have overlapping memory regions
their eigenvalues after eigvals I QR of. Custom matrix and F.H is of type UniformScaling, representing an identity matrix, then dividing-and-conquering the problem ipiv pivoting. Use specialized
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of eigenvalues is found in the case! Arrow ecosystem is in the one norm thanks tweeps! ), p=2 is currently not implemented. ) for! Irrational numbers ( like ℯ ) to A * X = lambda * X X. Dense
symmetric positive definite matrix in Zmat eigenvectors of A as A tolerance for.! Commitment to support/deprecate this specific set of statements over A range of elements, or items of an upper.. For
numbers, return $ \left ( |x|^p \right ) ^ { 1/p }.... As for eigen! type should also support abs and < number not representable by the element of... View of the Schur factorization of A vector ; if
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trans determines the orientation of AB C == CC involving A! And p=2, the corresponding eigenvectors are computed is permuted but not permuted in VR, and used A appropriate... Functions that overwrite
one of N elements of C must all be positive A CholeskyPivoted factorization data … matrices! I > 0, then dividing-and-conquering the problem that is solved or items of an upper triangular in-place!
M×N matrix of M. log of matrix operations that allow you to A... And sine of A square matrix A see if it is not computed op! Problem that is such that M * N = M \ I happen if src and dest have
Family Guy I Am Peter, Hear Me Roar, Nottinghamshire Police Lost Property, Duì Meaning Mandarin, Case Western Softball 2020, Calmac Ferry Ardrossan, Kerzon Candles Uk, | {"url":"http://www.odpadywgminie.pl/what-do-swekawy/m8uo4.php?page=julia-identity-matrix-f8a296","timestamp":"2024-11-05T13:55:16Z","content_type":"text/html","content_length":"40962","record_id":"<urn:uuid:c467fd8b-8b32-4352-b3ef-cb9964d89e03>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00210.warc.gz"} |
The sum of the first n terms of an AP is (3n^2+6n). Find the nth term and the 15th term of this AP. - Ask TrueMaths!The sum of the first n terms of an AP is (3n^2+6n). Find the nth term and the 15th term of this AP.
We have given a question from arithmetic progression chapter in which we have been asked to find the nth term and the 15th term of the AP, if the sum of the first n terms of an AP is (3n^2+6n).
Book – RS Aggarwal, Class 10, chapter 5C, question no 4 | {"url":"https://ask.truemaths.com/question/the-sum-of-the-first-n-terms-of-an-ap-is-3n26n-find-the-nth-term-and-the-15th-term-of-this-ap/?show=recent","timestamp":"2024-11-14T02:16:35Z","content_type":"text/html","content_length":"129334","record_id":"<urn:uuid:556b1aae-3d0a-4a80-9c61-eb7b74746d1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00761.warc.gz"} |
doc/TutorialMatrixArithmetic.dox - eigen - Git at Google
namespace Eigen {
/** \eigenManualPage TutorialMatrixArithmetic Matrix and vector arithmetic
This page aims to provide an overview and some details on how to perform arithmetic
between matrices, vectors and scalars with Eigen.
\section TutorialArithmeticIntroduction Introduction
Eigen offers matrix/vector arithmetic operations either through overloads of common C++ arithmetic operators such as +, -, *,
or through special methods such as dot(), cross(), etc.
For the Matrix class (matrices and vectors), operators are only overloaded to support
linear-algebraic operations. For example, \c matrix1 \c * \c matrix2 means matrix-matrix product,
and \c vector \c + \c scalar is just not allowed. If you want to perform all kinds of array operations,
not linear algebra, see the \ref TutorialArrayClass "next page".
\section TutorialArithmeticAddSub Addition and subtraction
The left hand side and right hand side must, of course, have the same numbers of rows and of columns. They must
also have the same \c Scalar type, as Eigen doesn't do automatic type promotion. The operators at hand here are:
\li binary operator + as in \c a+b
\li binary operator - as in \c a-b
\li unary operator - as in \c -a
\li compound operator += as in \c a+=b
\li compound operator -= as in \c a-=b
<table class="example">
\include tut_arithmetic_add_sub.cpp
\verbinclude tut_arithmetic_add_sub.out
\section TutorialArithmeticScalarMulDiv Scalar multiplication and division
Multiplication and division by a scalar is very simple too. The operators at hand here are:
\li binary operator * as in \c matrix*scalar
\li binary operator * as in \c scalar*matrix
\li binary operator / as in \c matrix/scalar
\li compound operator *= as in \c matrix*=scalar
\li compound operator /= as in \c matrix/=scalar
<table class="example">
\include tut_arithmetic_scalar_mul_div.cpp
\verbinclude tut_arithmetic_scalar_mul_div.out
\section TutorialArithmeticMentionXprTemplates A note about expression templates
This is an advanced topic that we explain on \ref TopicEigenExpressionTemplates "this page",
but it is useful to just mention it now. In Eigen, arithmetic operators such as \c operator+ don't
perform any computation by themselves, they just return an "expression object" describing the computation to be
performed. The actual computation happens later, when the whole expression is evaluated, typically in \c operator=.
While this might sound heavy, any modern optimizing compiler is able to optimize away that abstraction and
the result is perfectly optimized code. For example, when you do:
VectorXf a(50), b(50), c(50), d(50);
a = 3*b + 4*c + 5*d;
Eigen compiles it to just one for loop, so that the arrays are traversed only once. Simplifying (e.g. ignoring
SIMD optimizations), this loop looks like this:
for(int i = 0; i < 50; ++i)
a[i] = 3*b[i] + 4*c[i] + 5*d[i];
Thus, you should not be afraid of using relatively large arithmetic expressions with Eigen: it only gives Eigen
more opportunities for optimization.
\section TutorialArithmeticTranspose Transposition and conjugation
The transpose \f$ a^T \f$, conjugate \f$ \bar{a} \f$, and adjoint (i.e., conjugate transpose) \f$ a^* \f$ of a matrix or vector \f$ a \f$ are obtained by the member functions \link
DenseBase::transpose() transpose()\endlink, \link MatrixBase::conjugate() conjugate()\endlink, and \link MatrixBase::adjoint() adjoint()\endlink, respectively.
<table class="example">
\include tut_arithmetic_transpose_conjugate.cpp
\verbinclude tut_arithmetic_transpose_conjugate.out
For real matrices, \c conjugate() is a no-operation, and so \c adjoint() is equivalent to \c transpose().
As for basic arithmetic operators, \c transpose() and \c adjoint() simply return a proxy object without doing the actual transposition. If you do <tt>b = a.transpose()</tt>, then the transpose is
evaluated at the same time as the result is written into \c b. However, there is a complication here. If you do <tt>a = a.transpose()</tt>, then Eigen starts writing the result into \c a before the
evaluation of the transpose is finished. Therefore, the instruction <tt>a = a.transpose()</tt> does not replace \c a with its transpose, as one would expect:
<table class="example">
\include tut_arithmetic_transpose_aliasing.cpp
\verbinclude tut_arithmetic_transpose_aliasing.out
This is the so-called \ref TopicAliasing "aliasing issue". In "debug mode", i.e., when \ref TopicAssertions "assertions" have not been disabled, such common pitfalls are automatically detected.
For \em in-place transposition, as for instance in <tt>a = a.transpose()</tt>, simply use the \link DenseBase::transposeInPlace() transposeInPlace()\endlink function:
<table class="example">
\include tut_arithmetic_transpose_inplace.cpp
\verbinclude tut_arithmetic_transpose_inplace.out
There is also the \link MatrixBase::adjointInPlace() adjointInPlace()\endlink function for complex matrices.
\section TutorialArithmeticMatrixMul Matrix-matrix and matrix-vector multiplication
Matrix-matrix multiplication is again done with \c operator*. Since vectors are a special
case of matrices, they are implicitly handled there too, so matrix-vector product is really just a special
case of matrix-matrix product, and so is vector-vector outer product. Thus, all these cases are handled by just
two operators:
\li binary operator * as in \c a*b
\li compound operator *= as in \c a*=b (this multiplies on the right: \c a*=b is equivalent to <tt>a = a*b</tt>)
<table class="example">
\include tut_arithmetic_matrix_mul.cpp
\verbinclude tut_arithmetic_matrix_mul.out
Note: if you read the above paragraph on expression templates and are worried that doing \c m=m*m might cause
aliasing issues, be reassured for now: Eigen treats matrix multiplication as a special case and takes care of
introducing a temporary here, so it will compile \c m=m*m as:
tmp = m*m;
m = tmp;
If you know your matrix product can be safely evaluated into the destination matrix without aliasing issue, then you can use the \link MatrixBase::noalias() noalias()\endlink function to avoid the
temporary, e.g.:
c.noalias() += a * b;
For more details on this topic, see the page on \ref TopicAliasing "aliasing".
\b Note: for BLAS users worried about performance, expressions such as <tt>c.noalias() -= 2 * a.adjoint() * b;</tt> are fully optimized and trigger a single gemm-like function call.
\section TutorialArithmeticDotAndCross Dot product and cross product
For dot product and cross product, you need the \link MatrixBase::dot() dot()\endlink and \link MatrixBase::cross() cross()\endlink methods. Of course, the dot product can also be obtained as a 1x1
matrix as u.adjoint()*v.
<table class="example">
\include tut_arithmetic_dot_cross.cpp
\verbinclude tut_arithmetic_dot_cross.out
Cross product is defined in Eigen not only for vectors of size 3 but also for those of size 2, check \link MatrixBase::cross() the doc\endlink for details. Dot product is for vectors of any sizes.
When using complex numbers, Eigen's dot product is conjugate-linear in the first variable and linear in the
second variable.
\section TutorialArithmeticRedux Basic arithmetic reduction operations
Eigen also provides some reduction operations to reduce a given matrix or vector to a single value such as the sum (computed by \link DenseBase::sum() sum()\endlink), product (\link DenseBase::prod
() prod()\endlink), or the maximum (\link DenseBase::maxCoeff() maxCoeff()\endlink) and minimum (\link DenseBase::minCoeff() minCoeff()\endlink) of all its coefficients.
<table class="example">
\include tut_arithmetic_redux_basic.cpp
\verbinclude tut_arithmetic_redux_basic.out
The \em trace of a matrix, as returned by the function \link MatrixBase::trace() trace()\endlink, is the sum of the diagonal coefficients and can also be computed as efficiently using <tt>a.diagonal
().sum()</tt>, as we will see later on.
There also exist variants of the \c minCoeff and \c maxCoeff functions returning the coordinates of the respective coefficient via the arguments:
<table class="example">
\include tut_arithmetic_redux_minmax.cpp
\verbinclude tut_arithmetic_redux_minmax.out
\section TutorialArithmeticValidity Validity of operations
Eigen checks the validity of the operations that you perform. When possible,
it checks them at compile time, producing compilation errors. These error messages can be long and ugly,
but Eigen writes the important message in UPPERCASE_LETTERS_SO_IT_STANDS_OUT. For example:
Matrix3f m;
Vector4f v;
v = m*v; // Compile-time error: YOU_MIXED_MATRICES_OF_DIFFERENT_SIZES
Of course, in many cases, for example when checking dynamic sizes, the check cannot be performed at compile time.
Eigen then uses runtime assertions. This means that the program will abort with an error message when executing an illegal operation if it is run in "debug mode", and it will probably crash if
assertions are turned off.
MatrixXf m(3,3);
VectorXf v(4);
v = m * v; // Run-time assertion failure here: "invalid matrix product"
For more details on this topic, see \ref TopicAssertions "this page". | {"url":"https://third-party-mirror.googlesource.com/eigen/+/941ca8d83f776b9a07153d3abef2877907aa0555/doc/TutorialMatrixArithmetic.dox","timestamp":"2024-11-03T00:52:23Z","content_type":"text/html","content_length":"58504","record_id":"<urn:uuid:ffc6cec5-c53e-4e5a-a62e-93e1027c03f7>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00664.warc.gz"} |
Paracompact space
As you might guess from the generality of most of the examples above, it's actually harder to think of spaces that aren't paracompact than to think of spaces that are. The most famous counterexample
is the long line, which is a nonparacompact topological manifold. (The long line is locally compact, but not second countable.) Another counterexample is a product of uncountably many copies of an
infinite discrete space.
Most mathematicians who use point set topology, rather than investigate it in its own right, regard nonparacompact spaces as pathological. For example, manifolds are often (although not in Wikipedia)
defined to be paracompact, thus allowing integration of differential forms to be defined as in the previous section, while excluding the long line, which is useless in almost every application.
There are several mild variations of the notion of paracompactness. To define them, we first need to extend the list of terms above:
• Given a cover and a point, the star of the point in the cover is the union of all the sets in the cover that contain the point. In symbols, the star of x in U is U^*(x) := ∪[x∈U∈U] U. (The
notation for the star is not standardised in the literature, and this is just one possibility.)
• A star refinement of a cover of a space X is a new cover of the same space such that, given any point in the space, the star of the point in the new cover is a subset of some set in the old
cover. In symbols, V is a star refinement of U iff, for any x ∈ X, for some U ∈ U, V^*(x) ⊆ U.
• A cover of a space X is pointwise finite if every point of the space belongs to only finitely many sets in the cover. In symbols, U is pointwise finite iff, for any x ∈ X, the set {U ∈ U : x ∈ U}
is finite.
A topological space X is metacompact if every open cover has an open pointwise finite refinement, and fully normal if every open cover has an open star refinement. The adverb "countably" can be added
to any of the adjectives "paracompact", "metacompact", and "fully normal" to make the requirement apply only to countable open covers.
As you might guess from the terminology, a fully normal space is normal. Any space that is fully normal must be paracompact, and any space that is paracompact must be metacompact. In fact, for
Hausdorff spaces, paracompactness and full normality are equivalent. Thus, a fully T[4] space (that is, a fully normal space that is also T[1]; see Separation axioms) is the same thing as a
paracompact Hausdorff space. | {"url":"http://www.fact-index.com/p/pa/paracompact_space.html","timestamp":"2024-11-02T01:55:15Z","content_type":"text/html","content_length":"12798","record_id":"<urn:uuid:7c65fe22-cb66-4aae-b425-04f2065b0027>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00351.warc.gz"} |
HP-42S Help
I've used an HP-28C and 15C for years. Two weeks ago my sons gave me an HP-42S with both Owner's Manual and Programming Manual. I have figured everything out for the most part except for roots of
polynomials. I can figure quadratics and cubics such as x^3+3x^2+9x-1 but I can't get the programming syntax for 3x^3-9x^2+14x+7 nor can I figure any kind of quadratics. Please help me. These are a
breeze to figure on the 15C and 28C. Also, is there any kind of program for the 42S that can compute complex roots of polynomials?
06-28-2012, 12:44 AM
I guess the closest technique is the Horner's method. Think of
If you want to use the HP42S built-in SOLVER, you could program it like this:
LBL "XMPL"
MVAR "X"
RCL "X"
Now you just need to SOLVE for XMPL:
You'll see [XMPL] amongst the menu labels. Press the key right (...) below it (SoftKey) and the display will show the possible variables for solving (the Variables Menu, hence the MVAR "X"), in this
case a single [X]. You can either guess a starting value (avoiding possible miscalculations) or simply press the key right above [X]. My HP42S found
with the example above.
Hope it helps.
Luiz (Brazil)
Edited: 28 June 2012, 1:12 p.m. after one or more responses were posted
06-28-2012, 04:18 AM
There is no need for Horner's method with this one, you can simply, though unelegantly key in
MVAR "X"
RCL "X"
RCL "X"
RCL "X"
...and than solve TST for X, that yields X=-0.3897
As this is pretty straightforward, maybe I have not completely understood what the initial problem with this is...?
Edited: 28 June 2012, 4:40 a.m.
06-28-2012, 08:58 AM
I just considered that by using Horner's method to reduce a polynomial expression to a few arithmetic operations with no need for powers/exponents, the final program would be easier to be written. 8
Luiz (Brazil)
06-28-2012, 09:22 AM
Right, in this case two program step are spared :
001 LBL "TST 001 LBL "XMP
002 MVAR "X" 002 MVAR "X"
003 RCL "X" 003 RCL "X"
004 3 004 ENTER
005 Y^X 005 ENTER
006 3 006 ENTER
007 * 007 3
008 RCL "X" 008 *
009 X^2 009 9
010 9 010 -
011 * 011 *
012 - 012 14
013 RCL "X" 013 +
014 14 014 *
015 * 015 7
016 + 016 +
017 7 017 RTN
018 +
019 RTN
Very similar in fact
06-28-2012, 02:13 PM
Thanks so much for helping me with the cubic equation. I tried both methods given to me to find the roots for the following quartics and I get only one root for each equation:
I am obviously doing something wrong.
06-28-2012, 04:08 PM
I only tried the first one but could find all four roots: -6.4495, -1.5505, -1 and 1.
It all depends on the right initial guesses, use the programmed function to calculate some values to plot it roughly, then you put in two guesses for x that are located in the vicinity of the
potential roots. You do this by starting the solver, selecting the programmed function and than keying in the lower guess, press [X], upper guess, press [X] and than [X] again to solve. See the
manual on further details.
Edited: 28 June 2012, 5:08 p.m. after one or more responses were posted
06-28-2012, 04:25 PM
What program syntax did you use to solve the first equation? I will copy and save it for future reference.
06-28-2012, 05:07 PM
The same straightforward syntax I used in my posting above.
06-28-2012, 09:47 PM
I've found the following program at Thomas Okken's website:
"G4_G3_G2.raw Solves 2nd, 3rd, and 4th order polynomials. using closed-form solutions. Contributed by Christian Vetterli."
00 { 526-Byte Prgm }
01>LBL "G2"
INPUT "A" INPUT "B"
INPUT "C" RCL "B" X^2
RCL "A" RCL× "C" 4 ×
- SQRT STO "C"
RCL "A" 2 × STO÷ "C"
+/- STO÷ "B" RCL "B"
RCL+ "C" RTN RCL "B"
RCL- "C" RTN
26>LBL "G3"
RECT DEG INPUT "A"
INPUT "B" INPUT "C"
INPUT "D" SF 00 CF 01
RCL "A" STO÷ "B"
STO÷ "C" STO÷ "D"
39>LBL 15
RCL "B" 3 Y^X 13.5 ÷
RCL "B" RCL× "C" 3 ÷
- RCL+ "D" STO "A"
RCL "B" X^2 3 ÷
STO- "C" RCL "C" 3 ÷
3 Y^X RCL "A" 2 ÷
X^2 + SQRT RCL "A" 2
÷ X<>Y X=Y? +/- X<>Y
- 3 1/X Y^X STO "D"
2E-3 STO 00
82>LBL 16
FS? 01 GTO 01 3
STO÷ "C" SF 01
88>LBL 01
POLAR 1 120 COMPLEX
STO÷ "D" RCL "C" SF 25
RCL÷ "D" RCL "D" X<>Y
- RECT RCL "B" 3 ÷
- RTN ISG 00 GTO 16
108>LBL "G4"
SF 00 CF 01 CF 02
RECT DEG INPUT "A"
INPUT "B" INPUT "C"
INPUT "D" INPUT "E"
RCL "A" STO÷ "B"
STO÷ "C" STO÷ "D"
STO÷ "E" RCL "B"
STO "G" X^2 RCL× "C"
16 ÷ RCL "B" 4 Y^X
3 × 256 ÷ - RCL "B"
RCL× "D" 4 ÷ -
STO+ "E" RCL "B" 3
Y^X 8 ÷ RCL "B"
RCL× "C" 2 ÷ -
STO+ "D" RCL "B" X^2
0.375 × STO- "C" 2
RCL× "C" STO "B"
RCL "C" X^2 4
RCL× "E" - STO "C"
RCL "D" STO "F" X^2
+/- STO "D" XEQ 15
175>LBL 05
STO ST Y ABS 1E-8
FC? 02 X<=Y? GTO 02
SF 02 ISG 00 XEQ 16
GTO 05
186>LBL 02
RCL ST Z SQRT STO "A"
SF 25 STO÷ "F" ISG 00
XEQ 16
194>LBL 06
STO ST Y ABS 1E-8
FC? 02 X<=Y? GTO 02
SF 02 ISG 00 XEQ 16
GTO 06
205>LBL 02
RCL ST Z SQRT STO "E"
SF 25 STO÷ "F" RCL "E"
RCL+ "A" RCL- "F"
XEQ 04 RCL "F"
RCL+ "A" RCL- "E"
XEQ 04 RCL "E"
RCL+ "F" RCL- "A"
XEQ 04 RCL "E"
RCL+ "F" RCL+ "A" +/-
227>LBL 04
2 ÷ RCL "G" 4 ÷ -
STOP RTN .END.
I haven't tested it on the real HP-42S, but for William's polynomial Free42 Decimal returns the following roots:
XEQ G4
A?1 R/S
B?8 R/S
C?9 R/S
D?-8 R/S
E?-10 R/S
-1.55051025722 -i3.35403051129e-25
1 i6.25300999685e-25
-1 -i9.54592314951e-27
-6.44948974278 -i2.80352025407e-25
These match your results, considering the very small imaginary parts are actually zero.
Edited: 28 June 2012, 9:53 p.m.
06-28-2012, 04:36 PM
That’s the problem with numeric solver; you have to initiate each root search with an appropriate initial value.
There is no trick to find all these values by only looking at the polynomial. That is why graphic calculators, such as the HP-28C, will spare a lot of time. It is a good advice to start plotting the
For x^4+8x^3+9x^2-8x-10, enter the polynomial equation into you HP-28C :
« X 8 + X * 9 + X * 8 – X * 10 - » STEQ
A draw the curve between x=-10 up to x=+10 abscises.
(10,31) DUP PMAX CHS PMIN
And start drawin the curve :
You may obtain the following graph, using cursor keys and INS key you simply have to plot positions close to the four roots. Then press ON key to exist graph . You have now in the stack the four
initial values close enough to the root to use in the HP28C’s solveur (or any solveur of you alternative HP calculators).
|4: (-6.4706,1.0000)|
|3: (-1.6176,1.0000)|
|2: (-0.8824,1.0000)|
|1: (1.0294,1.0000)|
(Exact value may depend upon pixel positions where you press the INS key)
Enter first estimate into X register :
[ X ]
And initiate seek of the first root :
[shift][ X ]
The HP28C may display ‘Solving for X’ and stop indicating that a first zero is found :
|Zero |
|2: (-0.8824,1.0000)|
|1: 1.0000|
Write down this first root or store it in a safe place (for example in the stack):
4 [shift][ ROLL ]
Enter second estimate into X register:
[ X ]
And initiate seek of the second root:
[shift] [ X ]
There-6.4706 value is used as the initial value and rapidly the process converges to a ‘sign reversal’ root; a root that is not exactly evaluated to zero due to limited precision of the calculator:
|Sign Reversal |
|1: -6.4495|
| X |EXPR=| | | | |
Process the same way for the two last estimates:
4 [shift] [ ROLL ] [ X ] [shift] [ X ] 4 [shift] [ROLL][ X ] [shift] [ X ] 4 [shift] [ROLL] [CURSOR]
And you will have the four root of the polynomial into the stack:
|4: -6.4495|
|3: -1.5505|
|2: -1.0000|
|1: 1.0000|
For the second equation, process the same way and take advantage of your HP-28C to draw the curve. It is great helper in finding how to initiate root seeking. Making all previous HP a crabs
P.S.: For second equation { -3 -0.5 1.5 2.0 } obtained the same graphicaly way!
Edited: 28 June 2012, 5:10 p.m. after one or more responses were posted
06-28-2012, 05:09 PM
Thanks for the programming syntax for the 28C. But what I actually needed was the syntax for the 42S to solve the quartic equation I referenced above.
06-28-2012, 06:35 PM
The HP42S solver is acting exactly as one the HP28C does. Main differencies are that it only use a pair of initial values to start seeking. The HP28C may use a single, a pair or a triplet as initial
(s) guess entered as a list { best_guess, low_interval_guess, high_interval_guess } .
On the HP42S, you first have to enter the equation as a program:
You may enter your code as you prefer (using or not exponents or Horner’s format, whatever), simply indicate variable(s) to be put in menu by using the MVAR command. And avoid syntax error or
typing-bug as you have few chances on such a calculator to easily check tips.
00 { 39-Byte Prgm }
01>LBL "P_X"
02 MVAR "X"
04 RCL "X"
05 X^2
06 ×
08 -
09 RCL "X"
10 ×
12 +
13 RCL "X"
14 ×
16 +
17 RTN
18 .END.
Leave program mode :
Now you are ready to start the solver:
Select the equation you just type in by pressing the corresponding soft-key
[ P_X ]
Enter low guess and high guess for first root seeking. Here, a plot is missing that why I am in trouble trying to understanding you, as you have an HP28C which is much superior for this type of
guessing! :-)
6 [+/-] [ X ] 2 [+/-] [ X ][ X ]
That make { -6 -4 } the first pair of initial values to seek for the first root.
Rapidly the HP42S displays first root stored in X register:
| X = -3 |
[ X ][ ][ ][ ] [ ][ ]
Note that you can get the same result and investigate how the calculator is seeking by keying on a HP28C/S in its solver :
{ -6 -4 } [ X ][shift][ X ]
Second root can be found using a different set of initial values:
2 [+/-] [ X ] 0 [ X ][ X ]
That make { -2 0 } as second interval to seek for the second root.
| X =-0.5 |
[ X ][ ][ ][ ] [ ][ ]
That is rapidly found and displayed.
And so on for other guess pair to seek after the two other roots :
0 [ X ] 1 [ X ] [ X ]
| X =1.5 |
[ X ][ ][ ][ ] [ ][ ]
Finaly , last interval :
1.8 [ X ] 5 [ X ] [ X ]
| X =2 |
[ X ][ ][ ][ ] [ ][ ]
But, I sorry to repeat it again, it is hard to have efficient initial guess without a graph. And nothing is as easy and convenient as a graphic calculator to quickly plot it and zoom in regions of
interest. Having the shape of the function is a great help to seek after its roots!
06-28-2012, 08:00 PM
Thank you for your response. I thank everyone for their helpful responses. I totally agree with you about graphing. I have loved my 28C since I got it in 1986. As I said before, I have had a 45, 15C,
28C and a 50g. Of all, the 28C is my favorite. I'm glad I joined this forum. All of you have been incredibly helpful.
06-28-2012, 10:20 PM
Christian Vetterli's program above will compute all roots, real or complex, of polynomials up to degree 4 (if you don't mind keying in 200+ lines). Let it solve 4x^4-8x^3-13x^2-10x+22=0, for
XEQ G4
A?4 R/S
B?-8 R/S
C?-13 R/S
D?-10 R/S
E?22 R/S --> 3.11803398875 i0
R/S --> -1 -i1
R/S --> -1 i1
R/S --> 0.88196601125 i0
06-30-2012, 07:57 AM
You can simplify the program above by deleting the three ENTER commands after RCL X. The HP-42S solver populates the stack with the independent variable, just as the HP-15C did. This is true for the
integrator also.
07-01-2012, 09:19 PM
Thanks for pointing that out, but as a matter of fact, those three ENTER are actually needed.
The HP42S does not actually work as the HP15C (*) because of one fact: you can have more than one unknown in an equation when writing them for the HP42S SOLVER so you can choose which unknown to
solve for. That's why you may actually 'declare' the unknowns you want to solve for by using the 'MVAR' statement/function. Then your program simply returns the function value based upon the many
unknowns you have previously declared. Because neither the user nor the calculator can predict which variable will be claimed for solving for, the program must be written having all possibilities
being handled.
As you see, right after the 'RCL "X"' there is only one copy of the variable "X" in the stack register 'X'. If you do not copy it into the whole stack, the program will not compute the polynomial
expression correctly because any previously existing values in Y, Z and T will be used, instead of the actual "X" value being tested. Think of an expression with two unknowns, like "A" AND "B",
whether they repeat or not in the expression. You could choose to solve for either "A" or "B", once the other is known and previously defined (stored). Then the SOLVER will compute the same
expression as many times as needed for the chosen variable, and its value will be updated in the variable name itself, not in the stack registers.
You see, when you have a program in the HP15C that is going to be used by the SOLVER or the numerical integrator, it must be programed in such a way that it returns the value of the function being
evaluated based in one specific unknown. Some years ago the multivariable 'functionality' of the HP42S SOLVER was proposed for the HP15C by using the index register. It worked pretty fine, and the
resulting programs are listed in the article forum somewhere in the far past...
Hopefully I expressed myself correctly... in English, at least.
Luiz (Brazil)
(*) it applies to the HP34C SOLVER and its implementation found in the MATH module for the HP41 as well.
Edited: 1 July 2012, 9:31 p.m.
06-30-2012, 05:11 PM
Quote: Two weeks ago my sons gave me an HP-42S
Sounds like you raised him well :-).
07-01-2012, 07:26 PM
Thanks, we tried. One went to Notre Dame, the other University of Washington. Both had their minors in mathematics. I try to catch up! Unfortunately, their calculators of choice was the TI-89. | {"url":"https://archived.hpcalc.org/museumforum/thread-225860-post-226113.html#pid226113","timestamp":"2024-11-12T09:40:32Z","content_type":"application/xhtml+xml","content_length":"78710","record_id":"<urn:uuid:2844d78a-6d6a-480b-8168-63cdb87b9bf7>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00237.warc.gz"} |
9 Hectometer to Nanometer Calculator | Calculator Bit
9 Hectometer to Nanometer Calculator
9 Hectometer = 900000000000 Nanometer (nm)
Rounded: Nearest 4 digits
9 Hectometer is 900000000000.0002 Nanometer (nm)
9 Hectometer is 900 m
How to Convert Hectometer to Nanometer (Explanation)
• 1 hectometer = 100000000000 nm (Nearest 4 digits)
• 1 nanometer = 1.0000000000000001e-11 hm (Nearest 4 digits)
There are 100000000000 Nanometer in 1 Hectometer. To convert Hectometer to Nanometer all you need to do is multiple the Hectometer with 100000000000.
In formula distance is denoted with d
The distance d in Nanometer (nm) is equal to 100000000000 times the distance in hectometer (hm):
L [(nm)] = L [(hm)] × 100000000000
Formula for 9 Hectometer (hm) to Nanometer (nm) conversion:
L [(nm)] = 9 hm × 100000000000 => 900000000000.0002 nm
How many Nanometer in a Hectometer
One Hectometer is equal to 100000000000 Nanometer
1 hm = 1 hm × 100000000000 => 100000000000 nm
How many Hectometer in a Nanometer
One Nanometer is equal to 1.0000000000000001e-11 Hectometer
1 nm = 1 nm / 100000000000 => 1.0000000000000001e-11 hm
The hectometer (symbol: hm) is unit of length in the International System of Units (SI), equal to 100 meters. Hectometer word is combination of two words 'hecto'+'meter', 'hecto' means 'hundread'.
The hectare(ha) is common metric unit for land area that is equal to one square hectomter(hm^2).
The nanometer (symbol: nm) is unit of length in the International System of Units (SI), equal to 0.000000001 meter or 1x10^-9 meter or 1/1000000000. 1 nanometer is equal to 1000 picometeres. The
nanometer was formerly known as the millimicrometer or millimicron in short. The nanometer is often used to express dimensions on an atomic scale, nanometer is also commonly used to specify the
wavelength of electromagnetic radiation near the visible part of the spectrum.
Cite, Link, or Reference This Page
If you found information page helpful you can cite and reference this page in your work.
• <a href="https://www.calculatorbit.com/en/length/9-hectometer-to-nanometer">9 Hectometer to Nanometer Conversion</a>
• "9 Hectometer to Nanometer Conversion". www.calculatorbit.com. Accessed on November 9 2024. https://www.calculatorbit.com/en/length/9-hectometer-to-nanometer.
• "9 Hectometer to Nanometer Conversion". www.calculatorbit.com, https://www.calculatorbit.com/en/length/9-hectometer-to-nanometer. Accessed 9 November 2024.
• 9 Hectometer to Nanometer Conversion. www.calculatorbit.com. Retrieved from https://www.calculatorbit.com/en/length/9-hectometer-to-nanometer.
Hectometer to Nanometer Calculations Table
Now by following above explained formulas we can prepare a Hectometer to Nanometer Chart.
Hectometer (hm) Nanometer (nm)
5 500000000000.00006
6 600000000000.0001
7 700000000000.0001
8 800000000000.0001
9 900000000000.0001
10 1000000000000.0001
11 1100000000000.0002
12 1200000000000.0002
13 1300000000000.0002
14 1400000000000.0002
Nearest 4 digits
Convert from Hectometer to other units
Here are some quick links to convert 9 Hectometer to other length units.
Convert to Hectometer from other units
Here are some quick links to convert other length units to Hectometer.
More Hectometer to Nanometer Calculations
More Nanometer to Hectometer Calculations
FAQs About Hectometer and Nanometer
Converting from one Hectometer to Nanometer or Nanometer to Hectometer sometimes gets confusing.
Here are some Frequently asked questions answered for you.
Is 100000000000 Nanometer in 1 Hectometer?
Yes, 1 Hectometer have 100000000000 (Nearest 4 digits) Nanometer.
What is the symbol for Hectometer and Nanometer?
Symbol for Hectometer is hm and symbol for Nanometer is nm.
How many Hectometer makes 1 Nanometer?
1.0000000000000001e-11 Hectometer is euqal to 1 Nanometer.
How many Nanometer in 9 Hectometer?
Hectometer have 900000000000 Nanometer.
How many Nanometer in a Hectometer?
Hectometer have 100000000000 (Nearest 4 digits) Nanometer. | {"url":"https://www.calculatorbit.com/en/length/9-hectometer-to-nanometer","timestamp":"2024-11-09T23:34:57Z","content_type":"text/html","content_length":"53759","record_id":"<urn:uuid:cc65a98b-f05c-4e20-b603-365122f58e9f>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00269.warc.gz"} |
When Discussing the Twin Paradox: Learn This First - Being Teaching 2024
When Discussing the Twin Paradox: Learn This First
This text is meant for anybody who desires to start out a thread right here at Physics Boards on the dual paradox. There are already many, many threads right here on this matter, they usually are
likely to cowl the identical floor time and again, so it appears helpful to place an outline of that floor into an article that everybody can learn earlier than beginning yet one more thread. If
you’re in that class, and what’s right here solutions your query, nice! However even when it doesn’t, hopefully, this text will provide help to to border no matter questions you continue to have
after studying it, in a manner that may provide help to to get higher responses with out repeating issues which have already been stated right here many instances earlier than.
Having stated that, the very first thing we are going to do is shamelessly borrow an already current article on the dual paradox, the one that’s a part of the Usenet Physics FAQ. Earlier than you go
any additional, please learn the all pages. It’s not lengthy.
A fast abstract of the above article is that there are a variety of various methods of analyzing the usual twin paradox state of affairs, described within the article, every of which supplies some
perception. I’ll briefly listing the methods described within the Usenet article right here:
• The Doppler Shift Evaluation.
• The Spacetime Diagram Evaluation.
• The Equivalence Precept Evaluation.
There are additionally different urged methods of analyzing the usual twin paradox that we frequently discover talked about in threads right here, and which seem in numerous sources, similar to:
• Altering Inertial Frames.
• Acceleration.
• The Touring Twin’s Relaxation Body.
• Spacetime Geometry.
We’re not going to enter nice element about any of those right here because the objective of this text is to not give an in depth research of all of the doable methods of trying on the twin paradox.
Our objective right here is extra common. The assorted analyses, typically talking, might be labeled by how effectively they reply three common questions:
• (Q1) If the twins have aged in another way once they come again collectively, there will need to have been some distinction or asymmetry between them through the journey. However doesn’t
relativity says that each one frames are equally legitimate? How does the strategy of research take care of this?
• (Q2) How a lot extra evaluation or inference should be finished, past what’s given in the issue assertion, with a purpose to utterly analyze the state of affairs utilizing the given technique?
• (Q3) Given an evaluation of the usual twin paradox state of affairs, how effectively will that very same technique of research generalize to different situations? For instance, will it work if
each twins speed up? Will it work if gravity is current (i.e., in curved spacetime)?
And now we get to the primary level of this text: from the standpoint of those questions, there may be solely one technique of research that can provide a passable response in all circumstances.
That’s the Spacetime Geometry evaluation, which is a generalization of the Spacetime Diagram evaluation described within the Usenet article. That article states that the Spacetime Diagram evaluation
is a kind of “Common Interlingua” that permits you to take a world view and put every of the analyses in its correct perspective. The Spacetime Geometry evaluation is similar factor, however
generalized to circumstances the place it isn’t possible to attract a easy diagram of the state of affairs and one has to depend on equations as an alternative. However the primary level is similar
as within the Spacetime Diagram evaluation: you will have two twins who take totally different paths by way of spacetime, and people paths have totally different lengths, and the lengths of the paths
are the quantities that every twin ages through the journey. So the totally different ages of the twins once they meet once more are not any extra mysterious than the truth that, if two twins take
street journeys between, say, New York and Los Angeles by totally different routes with totally different lengths, their odometers will learn totally different mileages once they meet on the finish
even when they had been the identical at first.
Let’s check out how the Spacetime Geometry evaluation responds to our three questions above, and distinction it with a few of the different analyses:
• (A1) The asymmetry between the twins is straightforward: it’s the totally different lengths of their paths by way of spacetime. These path lengths are invariants; they don’t rely on which body
you undertake. So each twins will agree on them. The 2 twins, in the event that they use totally different frames, would possibly differ within the particulars of how they calculate these
invariants, but when their frames are legitimate, they are going to get the identical ultimate solutions. (The calculation finished within the Spacetime Diagram Evaluation web page of the Usenet
article is an instance of calculating the paths and path lengths of the twins, utilizing the stay-at-home twin’s relaxation body.)
Different analyses might be seen as guidelines of thumb for recognizing when the trail lengths of the twins by way of spacetime will differ. For instance, if, as in the usual state of affairs,
spacetime is flat and one twin stays inertial the entire time whereas the opposite has nonzero correct acceleration once they flip round, the inertial twin’s path shall be longer, so Acceleration
works right here as an asymmetry to clarify the distinction in ageing. However that rule of thumb solely works in flat spacetime, and solely when one twin is inertial and the opposite isn’t; it
doesn’t generalize. Nor do the principles of thumb concerned in any of the opposite analyses (aside from the Doppler Shift evaluation, which is able to at all times work however which requires
extra work over and above the Spacetime Geometry evaluation–see A2 under). As we’ll see underneath A3 under, all of them break down sooner or later. Solely the Spacetime Geometry evaluation by no
means does.
• (A2) With a view to apply the Spacetime Geometry evaluation, it’s a must to know the paths of the twins by way of spacetime. But when the state of affairs is effectively specified in any respect,
it’s going to embody specs which can be adequate to calculate these paths–if it doesn’t, you may’t resolve it by any technique of research (except you might be fortunate sufficient to hit a
particular case the place a rule of thumb like Acceleration works–however even then, with out sufficient info to calculate the paths, you received’t be capable of give a numerical reply, only a
qualitative judgment of which twin ages extra). And after you have the paths, calculating their lengths is simple (although it’d contain tedious computation for extra sophisticated situations),
and would most likely have to be finished anyway it doesn’t matter what technique of research you might be utilizing.
For instance, to even use the Doppler Shift Evaluation, that you must know what the Doppler shifts are–and the one strategy to know that’s to know the twins’ paths by way of spacetime so you may
in flip calculate the paths of the sunshine indicators that they ship to one another. (Be aware that the Usenet article glosses over this by simply providing you with the outcomes of that
calculation–however in case you didn’t already know these outcomes, you would need to calculate them.)
• (A3) As has already been famous, the Spacetime Geometry evaluation is the one one which generalizes to all situations. As we noticed underneath A1 above, if the state of affairs is effectively
sufficient specified in any respect, it should include sufficient info to calculate the paths of the twins by way of spacetime. And that’s all you want for this evaluation. Plus, as we noticed
above, this evaluation works in any body, since it’s calculating invariants, so that you don’t have to fret about whether or not the body you might be utilizing is the “proper” one. You simply
choose the one which works the very best for you.
For some other evaluation, you first would want to test to ensure it really works in any respect for the state of affairs (since all of them have limitations in what sorts of situations they work
in). Even when it did, except the state of affairs was one of many easiest particular circumstances (like the usual state of affairs with the Acceleration evaluation, mentioned above), you would
want to do all of the work you’d do for the Spacetime Geometry evaluation, plus extra work to evaluate no matter your chosen evaluation tells you to evaluate (like Doppler Shift, as above). You
may also be restricted in what frames you should use (for instance, in case your chosen technique insists on utilizing inertial frames), or may need ambiguities in tips on how to even outline a
body (for instance, there isn’t any distinctive strategy to outline a “relaxation body” for the touring twin in the usual state of affairs, no matter the way you specify their correct
In brief: if you’re taking a look at a very particular case, similar to the usual state of affairs described within the Usenet article, there are most likely a number of analyses that may “resolve”
the paradox in a technique or one other. However for any evaluation aside from the Spacetime Diagram/Spacetime Geometry evaluation, eventually you’ll encounter a case that that evaluation can’t
resolve. And even earlier than that, you’ll doubtless find yourself doing extra and more durable work than you wanted to. The one absolutely common strategy to resolve all such situations, and do it
as effectively as doable, is Spacetime Geometry. | {"url":"https://beingteaching.com/when-discussing-the-twin-paradox-learn-this-first/","timestamp":"2024-11-07T18:19:42Z","content_type":"text/html","content_length":"180183","record_id":"<urn:uuid:8ab9d75f-651f-47c1-bbfd-73876137422c>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00296.warc.gz"} |
D&D Cover to Cover, part 17
Being a series of articles in which the author reads the indelible words of Gygax and Arneson as presented the Original Collector's Edition of Dungeons & Dragons, published by Tactical Studies Rules.
Beginning with Men & Magic, and concluding with The Underworld & Wilderness Adventures, the author will consider those earliest passages, adding elucidations and interpretations along the way for
your consideration.Men & MagicEXPLANATION OF SPELLS
1st Level:”
“Cure Light Wounds: During the course of one full turn this spell will remove hits from a wounded character (including elves, dwarves, etc.). A die is rolled, one pip added, and the resultant
total subtracted from the hit points the character has taken. Thus from 2-7 hit points of damage can be removed.”
First of all, I love the careful wording choices here. Looking back now, 34 years later, the spell description seems awkward and requires another reading unless you already know what Cure Light
Wounds does. It’s worded in such a way to prevent players from mistakenly assuming that the spell could potentially increase maximum hit points. BUT WAIT, this is where you may have glossed over the
description. Stop and read it again, if you care. Got it? OK, let me reword it so you see what I am getting at.
Over the next turn, Cure Light Wounds will heal damage up to 1d6+1 points on the target. The spell will continue for one full turn, or until it has healed an amount of damage equal to the number
rolled on the die, plus 1.
I’m reading this to mean that after the spell is cast, the target will have ‘future’ damage healed as well, provided the ‘resultant’ amount has not yet been healed. It sure could be useful to wade
into melee with a Cure Light Wounds active upon your character, no? Now, prevailing logic might be that in fact, this spell has a casting time of 1 full turn. But why then is the rest of the spell
worded so carefully, while the first part is misleading? Looking back for some precedence in previous spells; Transmute Rock to Mud ‘takes effect in one turn’; Move Earth ‘takes one turn to go into
effect’. On the other hand, there is no duration listed for Cure Light Wounds. Another example of why casting time is clearly conspicuous by its absence in this section. I like the possibilities of a
Cure spell that is ‘sticky’ on the target for a full turn. Am I taking the above too literally, or going out on a limb here? Yes, but still it makes for an interesting twist.
3rd Level:”
“Cure Disease: A spell which cures any form of disease. The spell is the only method to rid a character of a disease from a curse, for example.”
Interestingly, Remove Curse will apparently not cure a disease caused via curse. I suppose the thinking is that at some point the ailment is considered a disease and not a curse.
4th Level:”
“Cure Serious Wounds: This spell is like a Light Wound spell, but the effects are double, so two dice are rolled and one pip is added to each die. Therefore, from 4 to 14 hit points will be
removed by this spell.”
A nice spell made even better under my unorthodox interpretation. 4 to 14 points worth of healing potential over one full turn.
“Turn Sticks to Snakes: Anytime there are sticks nearby a Cleric can turn them into snakes, with a 50% chance that they will be poisonous. From 2-16 snakes can be conjured (roll two eight-sided
dice). He can command these conjured snakes to perform as he orders. Duration: 6 turns. Range 12”.”
Aside from Hold Person and the still to come The Finger of Death, Turn Sticks to Snakes is about as offensive as the D&D Cleric gets. With an average roll, 9 snakes may be created thusly. I’d allow
each snake a 50/50 chance of being poisonous, rather than roll for the whole lot at once. Five poisonous snakes, created up to 120 feet away, has a lot of potential. There is nothing indicating the
hit dice of these snakes, but I’d probably go with something like AC 7, HD ½, Move 6”, Damage 1, 50% chance of being poisonous (save or die variety).
5th Level:”
“Dispell Evil: Similar to a Dispell magic spell, this allows a Cleric to dispell any evil sending or spell within a 3” radius. It functions immediately. Duration: 1 turn.”
This spell combines the best vagaries of Dispell Magic and the ‘evil’ concept, with ‘evil sending’ again to boot. Notable in that it lasts 1 turn and has a 30 foot area of effect. I’ve still no idea
what an evil sending is, but this spell will get rid of them in spades.
“Raise Dead: The Cleric simply points his finger, utters the incantation, and the dead person is raised. This spell works with men, elves and dwarves only.”
Sorry Hobbits, you’ll need a Reincarnation spell, or a friend with a Ring of Three Wishes to come back from the beyond the pale.
“Create Food: A spell with which the Cleric creates sustenance sufficient for a party of a dozen for one game day. The quantity doubles for every level above the 8th the Cleric has attained.”
That’s a heap of food by 13th level. 12, 24, 48, 96, 192, and finally enough food for 384 at 13th level. By 14th level, able to create enough food for
768 people
in a single casting, I’m fairly sure the Cleric builds a new stronghold, out of sausage and cheese. Surely the intent was that it increased by 12 per level after the 8th…or were Gygax and Arneson
envisioning opening a chain of Cleric Smorgasbords in Greyhawk and Blackmoor, respectively?
“The Finger of Death: Instead of raising the dead, this spell creates a “death ray” which will kill any creature unless a saving throw is made (where applicable). Range 12”. (A Cleric-type may
use this spell in a life-or-death situation, but misuse will immediately turn him into an Anti-Cleric.)”
A save negates this potent spell, but it the most offensive one in the Cleric arsenal. It begs the question, though, whether or not a Cleric can reverse this ‘mid-adventure’ after memorizing Raise
Dead. Since this is such a unique spell amongst all 26 Cleric spells, and because it is highly situational for the only class which can actually choose which version of the spell to cast, I would
allow Clerics to cast Raise Dead as either its standard form, or as The Finger of Death.
Evil, evil and more evil. Anti-Clerics MUST be aligned with Chaos. Therefore, evil = Chaos.
~Sham, Quixotic Referee
6 comments:
I'm not sure that it's enough to rpove that evil=chaos, even if there's some link, but I feel the misuse of Finger of death as the only cited way to become an anticleric has a strong 'dark side
of the force' which is pretty nice.
The 'cure' spells working in a preventive / regenerative way is appealing too! Once again, you noticed a very interesting thing, when we try to get back from our habit of reading through ad&d /
classical d&d eyes.
I imagine that the death ray spell could have been inspired by the wand Tolkemec uses at the end of Red Nails, and given his unhinged disposition I can definitely see the "anti-cleric" angle.
Over the next turn, Cure Light Wounds will heal damage up to 1d6+1 points on the target. The spell will continue for one full turn, or until it has healed an amount of damage equal to the number
rolled on the die, plus 1.
Okay, so I'll never read CLW the same way again! I initially thought WTF?, but by the time I went back to copy it for this comment, I had fully turned the corner.
I think I'll always use this interpretation -- it's a totally valid way to read the spell, especially assuming the point of view of somebody who had the game in '74 but didn't have anybody to
teach them how to play.
Kudos, again; this is an awesome series!
That's a bloody inspiring reading of the Cure spells, I have to say!
I just might have to lift that idea, or possibly add it as spell variants to the "existing" versions ...
On Cure: I understood it to mean that it would slowly remove the damage over the duration of a turn. That is, that Cure was not a combat spell, but to be used afterward.
From a historical perspective, the weird wording of Cure Light Wounds could have been because, IIRC, Clerics were originally an invention of the Blackmoor, as opposed to Greyhawk. Perhaps Arneson
wrote the rule instead of Gygax. | {"url":"https://shamsgrog.blogspot.com/2008/11/d-cover-to-cover-part-17.html","timestamp":"2024-11-02T22:07:48Z","content_type":"text/html","content_length":"70507","record_id":"<urn:uuid:eac1a34c-66d3-43af-a5fc-5941c86cc861>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00713.warc.gz"} |
Unsupervised Learning: SVM
SVM is calculating a hyperplane to separate the data points into groups according to the label.
A hyperplane is defined to be of the following form
$$ $$\boldsymbol{\beta} \cdot \mathbf x = \beta_0.$$ $$
where $\boldsymbol\beta$ is the normal vector to the plane and is required to be constant.
It is straight forward to show that the distance $d$ from an arbitrary point $\mathbf x’$ to the hyperplane is
$$ $$d = \boldsymbol\beta \cdot \mathbf x’ - \beta_0.$$ $$
A Few Key Concepts in SVM
Though the concept of SVM is simple, one might find the algorithm to be quite complicated at first glance.
Which hyperplane to choose
Suppose we have two classes in our dataset, class A and B, our hyperplane will separate the two classes.
The plane has to make sure that most data points of class A and B are on the two sides of the hyperplane.
1. Lines that are going though at least one point of class A and one point of class B. The examples are shown as dashed and dot-dashed lines.
2. Lines that are going though the edge points of class A. It could be defined later. so we are thinking about lines shown as dotted lines.
Can we use these hyperplanes?
Those are absurd limiting choices. However those places us at a position that we might need a plane that has a fare distance between the two classes. Our intuition tells that the hyperplane might not
work for those data points close to the hyperplane. That being said, we are more confident if the hyperplane is far away from all data points.
Maybe we could require the hyperplane to be equally far away from the two classes. Here we define the distance between a hyperplane and a group data points to be the smallest distance between the
hyperplane and data points. This distance is called the margin.
Maybe this one? We calculate the distance between the data points of class A and the hyperplane, and we find the smallest distance two be $d_{A,min}$. Meanwhile we calculate the distance between the
hyperplane the the data points of class B, and denote it as $d_{B,min}$. We should find $d_{A, min} = d_{B, min}$. This is the max margin stragety.
We would like to take the extreme limits, again, to understand which hyperplane works the best for the classification problem.
Why does that max-margin strategy work
How is the hyperplane being used
The hyperplane could be represented with a normal vector $\hat{\mathbf n}$ and a shift $\beta_0$.
Fixing a hyperplane: given a normal vector, we could determine the direction of the hyperplane. However, we are also free to move the hyperplane along the normal vector. To fix the hyperplane, we
have to specify a shift.
Is SVM susceptible to outliers?
Planted: by L Ma;
wiki/machine-learning/unsupervised/svm Links to:
L Ma (2018). 'Unsupervised Learning: SVM', Datumorphism, 08 April. Available at: https://datumorphism.leima.is/wiki/machine-learning/unsupervised/svm/. | {"url":"https://datumorphism.leima.is/wiki/machine-learning/unsupervised/svm/?ref=footer","timestamp":"2024-11-12T02:37:45Z","content_type":"text/html","content_length":"114974","record_id":"<urn:uuid:4cf08bff-86b9-4827-97df-38807e243e5f>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00075.warc.gz"} |
Feasibility Sample Size Justification
This blog post is now included in the paper "Sample size justification" available at PsyArXiv.
When you perform an experiment, you want it to provide an answer to your research question that is as informative as possible. However, since all scientists are faced with resource limitations, you
need to balance the cost of collecting each additional datapoint against the increase in information that datapoint provides. In economics, this is known as the Value of Information (Eckermann et
al., 2010). Calculating the value of information is notoriously difficult. You need to specify the costs and benefits of possible outcomes of the study, and quantifying a utility function for
scientific research is not easy.
Because of the difficulty of quantifying the value of information, scientists use less formal approaches to justify the amount of data they set out to collect. That is, if they provide a
justification for the number of observations to begin with. Even though in some fields a justification for the number of observations is required when submitting a grant proposal to a science funder,
a research proposal to an ethical review board, or a manuscript for submission to a journal. In some research fields, the number of observations is stated, but not justified. This makes it difficult
to evaluate how informative the study was. Referees can’t just assume the number of observations is sufficient to provide an informative answer to your research question, so leaving out a
justification for the number of observations is not best practice, and a reason reviewers can criticize your submitted article.
A common reason why a specific number of observations is collected is because collecting more data was not feasible. Note that all decisions for the sample size we collect in a study are based on the
resources we have available in some way. A feasibility justification makes these resource limitations the primary reason for the sample size that is collected. Because we always have resource
limitations in science, even when feasibility is not our primary justification for the number of observations we plan to collect, feasibility is always at least a secondary reason for any sample size
justification. Despite the omnipresence of resource limitations, the topic often receives very little attention in texts on experimental design. This might make it feel like a feasibility
justification is not appropriate, and you should perform an a-priori power analysis or plan for a desired precision instead. But feasibility limitations play a role in every sample size
justification, and therefore regardless of which justification for the sample size you provide, you will almost always need to include a feasibility justification as well.
Time and money are the two main resource limitations a scientist faces. Our master students write their thesis in 6 months, and therefore their data collection is necessarily limited in whatever can
be collected in 6 months, minus the time needed to formulate a research question, design an experiment, analyze the data, and write up the thesis. A PhD student at our department would have 4 years
to complete their thesis, but is also expected to complete multiple research lines in this time. In addition to limitations on time, we have limited financial resources. Although nowadays it is
possible to collect data online quickly, if you offer participants a decent pay (as you should) most researchers do not have the financial means to collect thousands of datapoints.
A feasibility justification puts the limited resources at the center of the justification for the sample size that will be collected. For example, one might argue that 120 observations is the most
that can be collected in the three weeks a master student has available to collect data, when each observation takes an hour to collect. A PhD student might collect data until the end of the academic
year, and then needs to write up the results over the summer to stay on track to complete the thesis in time.
A feasibility justification thus starts with the expected number of observations (N) that a researcher expects to be able to collect. The challenge is to evaluate whether collecting N observations is
worthwhile. The answer should sometimes be that data collection is not worthwhile. For example, assume I plan to manipulate the mood of participants using funny cartoons and then measure the effect
of mood on some dependent variable - say the amount of money people donate to charity. I should expect an effect size around d = 0.31 for the mood manipulation (Joseph et al., 2020), and seems
unlikely that the effect on donations will be larger than the effect size of the manipulation. If I can only collect mood data from 30 participants in total, how do we decide if this study will be
If we want to evaluate whether the feasibility limitations make data collection uninformative, we need to think about what the goal of data collection is. First of all, having data always provide
more knowledge than not having data, so in an absolute sense, all additional data that is collected is better than not collecting data. However, in line with the idea that we need to take into
account costs and benefits, it is possible that the cost of data collection outweighs the benefits. To determine this, one needs to think about what the benefits of having the data are. The benefits
are clearest when we know for certain that someone is going to make a decision, with or without data. If this is the case, then any data you collect will reduce the error rates of a well-calibrated
decision process, even if only ever so slightly. In these cases, the value of information might be positive, as long as the reduction in error rates is more beneficial than the costs of data
collection. If your sample size is limited and you know you will make a decision anyway, perform a compromise power analysis, where you balance the error rates, given a specified effect size and
sample size.
Another way in which a small dataset can be valuable is if its existence makes it possible to combine several small datasets into a meta-analysis. This argument in favor of collecting a small dataset
requires 1) that you share the results in a way that a future meta-analyst can find them regardless of the outcome of the study, and 2) that there is a decent probability that someone will perform a
meta-analysis in the future which inclusion criteria would contain your study, because a sufficient number of small studies exist. The uncertainty about whether there will ever be such a
meta-analysis should be weighed against the costs of data collection. Will anyone else collect more data on cognitive performance during bungee jumps, to complement the 12 data points you can
One way to increase the probability of a future meta-analysis is if you commit to performing this meta-analysis yourself in the future. For example, you might plan to repeat a study for the next 12
years in a class you teach, with the expectation that a meta-analysis of 360 participants would be sufficient to achieve around 90% power for d = 0.31. If it is not plausible you will collect all the
required data by yourself, you can attempt to set up a collaboration, where fellow researchers in your field commit to collecting similar data, with identical measures, over the next years. If it is
not likely sufficient data will emerge over time, we will not be able to draw informative conclusions from the data, and it might be more beneficial to not collect the data to begin with, and examine
an alternative research question with a larger effect size instead.
Even if you believe over time sufficient data will emerge, you will most likely compute statistics after collecting a small sample size. Before embarking on a study where your main justification for
the sample size is based on feasibility, you can expect. I propose that a feasibility justification for the sample size, in addition to a reflection on the plausibility that a future meta-analysis
will be performed, and/or the need to make a decision, even with limited data, is always accompanied by three statistics, detailed in the following three sections.
In Figure @ref(fig:power-effect1) the distribution of Cohen’s d given 15 participants per group is plotted when the true effect size is 0 (or the null-hypothesis is true), and when the true effect
size is d = 0.5. The blue area is the Type 2 error rate (the probability of not finding p < α, when there is a true effect, and α = 0.05). 1- the Type 2 error is the statistical power of the test,
given an assumption about a true effect size in the population. Statistical power is the probability of a test to yield a statistically significant result if the alternative hypothesis is true. Power
depends on the Type 1 error rate (α), the true effect size in the population, and the number of observations.
Null and alternative distribution, assuming d = 0.5, alpha = 0.05, and N = 15 per group.
You might seen such graphs before. The only thing I have done is to transform the t-value distribution that is commonly used in these graphs, and calculated the distribution for Cohen’s d. This is a
straightforward transformation, but instead of presenting the critical t-value the figure provides the critical d-value. For a two-sided independent t-test, this is calculated as:
qt(1-(a / 2), (n1 + n2) - 2) * sqrt(1/n1 + 1/n2)
where ‘a’ is the alpha level (e.g., 0.05) and N is the sample size in each independent group. For the example above, where alpha is 0.05 and n = 15:
qt(1-(0.05 / 2), (15 * 2) - 2) * sqrt(1/15 + 1/15)
## [1] 0.7479725
The critical t-value (2.0484071) is also provided in commonly used power analysis software such as G*Power. We can compute the critical Cohen’s d from the t-value and sample size using .
The critical t-value is provided by G*Power software.
When you will test an association between variables with a correlation, G*Power will directly provide you with the critical effect size. When you compute a correlation based on a two-sided test, your
alpha level is 0.05, and you have 30 observations, only effects larger than r = 0.361 will be statistically significant. In other words, the effect needs to be quite large to even have the
mathematical possibility of becoming statistically significant.
The critical r is provided by G*Power software.
The critical effect size gives you information about the smallest effect size that, if observed, would by statistically significant. If you observe a smaller effect size, the p-value will be larger
than your significance threshold. You always have some probability of observing effects larger than the critical effect size. After all, even if the null hypothesis is true, 5% of your tests will
yield a significant effect. But what you should ask yourself is whether the effect sizes that could be statistically significant are realistically what you would expect to find. If this is not the
case, it should be clear that there is little (if any) use in performing a significance test. Mathematically, when the critical effect size is larger than effects you expect, your statistical power
will be less than 50%. If you perform a statistical test with less than 50% power, your single study is not very informative. Reporting the critical effect size in a feasibility justification should
make you reflect on whether a hypothesis test will yield an informative answer to your research question.
The second statistic to report alongside a feasibility justification is the width of the 95% confidence interval around the effect size. 95% confidence intervals will capture the true population
parameter 95% of the time in repeated identical experiments. The more uncertain we are about the true effect size, the wider a confidence interval will be. Cumming (2013) calls the difference between
the observed effect size and its upper 95% confidence interval (or the lower 95% confidence interval) the margin of error (MOE).
# Compute the effect size d and 95% CI
res <- MOTE::d.ind.t(m1 = 0, m2 = 0, sd1 = 1, sd2 = 1, n1 = 15, n2 = 15, a = .05)
# Print the result
## [1] "$d_s$ = 0.00, 95\\% CI [-0.72, 0.72]"
If we compute the 95% CI for an effect size of 0, we see that with 15 observations in each condition of an independent t-test the 95% CI ranges from -0.72 to 0.72. The MOE is half the width of the
95% CI, 0.72. This clearly shows we have a very imprecise estimate. A Bayesian estimator who uses an uninformative prior would compute a credible interval with the same upper and lower bound, and
might conclude they personally believe there is a 95% chance the true effect size lies in this interval. A frequentist would reason more hypothetically: If the observed effect size in the data I plan
to collect is 0, I could only reject effects more extreme than d = 0.72 in an equivalence test with a 5% alpha level (even though if such a test would be performed, power might be low, depending on
the true effect size). Regardless of the statistical philosophy you plan to rely on when analyzing the data, our evaluation of what we can conclude based on the width of our interval tells us we will
not learn a lot. Effect sizes in the range of d = 0.7 are findings such as “People become aggressive when they are provoked”, “People prefer their own group to other groups”, and “Romantic partners
resemble one another in physical attractiveness” (Richard et al., 2003). The width of the confidence interval tells you that you can only reject the presence of effects that are so large, if they
existed, you would probably already have noticed them. It might still be important to establish these large effects in a well-controlled experiment. But since most effect sizes in we should
realistically expect are much smaller, we do not learn something we didn’t already know from the data that plan to collect. Even without data, we would exclude effects larger than d = 0.7 in most
research lines.
We see this the MOE is almost, but not exactly, the same as the critical effect size d we observed above (d = 0.7479725). The reason for this is that the 95% confidence interval is calculated based
on the t-distribution. If the true effect size is not zero, the confidence interval is calculated based on the non-central t-distribution, and the 95% CI is asymmetric. The figure below vizualizes
three t-distributions, one symmetric at 0, and two asymmetric distributions with a noncentrality parameter of 2 and 3. The asymmetry is most clearly visible in very small samples (the distribution in
the plot have 5 degrees of freedom) but remain noticeable when calculating confidence intervals and statistical power. For example, for a true effect size of d = 0.5 the 95% CI is [-0.23, 1.22]. The
MOE based on the lower bound is 0.7317584 and based on the upper bound is 0.7231479. If we compute the 95% CI around the critical effect size (d = 0.7479725) we see the 95% CI ranges from exactly
0.00 to 1.48. If the 95% CI excludes zero, the test is statistically significant. In this case the lowerbound of the confidence interval exactly touches 0, which means we would observe a p = 0.05 if
we exactly observed the critical effect size.
Central (black) and 2 non-central (red and blue) t-distributions.
Where computing the critical effect size can make it clear that a p-value is of little interest, computing the 95% CI around the effect size can make it clear that the effect size estimate is of
little value. It will often be so uncertain, and the range of effect sizes you will not be able to reject if there is no effect is so large, the effect size estimate is not very useful. This is also
the reason why performing a pilot study to estimate an effect size for an a-priori power analysis is not a sensible strategy (Albers & Lakens, 2018; Leon et al, 2011). Your effect size estimate will
be so uncertain, it is not a good guide in an a-priori power analysis.
However, it is possible that the sample size is large enough to exclude some effect sizes that are still a-priori plausible. For example, with 50 observations in each independent group, you have 82%
power for an equivalence test with bounds of -0.6 and 0.6. If the literature includes claims of effect size estimates larger than 0.6, and if effect larger than 0.6 can be rejected based on your
data, this might be sufficient to tentatively start to question claims in the literature, and the data you collect might fulfill that very specific goal.
In a sensitivity power analysis the sample size and the alpha level are fixed, and you compute the effect size you have the desired statistical power to detect. For example, in the Figure below the
sample size in each group is set to 15, the alpha level is 0.05, and the desired power is set to 90%. The sensitivity power analysis shows we have 90% power to detect an effect of d = 1.23.
Sensitivity power analysis in G*Power software.
Perhaps you feel a power of 90% is a bit high, and you would be happy with 80% power. We can plot a sensitivity curve across all possible levels of statistical power. In the figure below we see that
if we desire 80% power, the effect size should be d = 1.06. The smaller the true effect size, the lower the power we have. This plot should again remind us not to put too much faith in a significance
test when are sample size is small, since for 15 observations in each condition, statistical power is very low for anything but extremely large effect sizes.
Plot of the effect size against the desired power when n = 15 per group and alpha = 0.05.
If we look at the effect size that we would have 50% power for, we see it is d = 0.7411272. This is very close to our critical effect size of d = 0.7479725 (the smallest effect size that, if
observed, would be significant). The difference is due to the non-central t-distribution.
To summarize, I recommend addressing the following components in a feasibility sample size justification. Addressing these points explicitly will allow you to evaluate for yourself if collecting the
data will have scientific value. If not, there might be other reasons to collect the data. For example, at our department, students often collect data as part of their education. However, if the
primary goal of data collection is educational, the sample size that is collected can be very small. It is often educational to collect data from a small number of participants to experience what
data collection looks like in practice, but there is often no educational value in collecting data from more than 10 participants. Despite the small sample size, we often require students to report
statistical analyses as part of their education, which is fine as long as it is clear the numbers that are calculated can not meaningfully be interpreted. Te table below should help to evaluate if
the interpretation of statistical tests has any value, or not.
Overview of recommendations when reporting a sample size justification based on feasibility.
What to address? How to address it?
Will a future meta-analysis be performed? Consider the plausibility that sufficient highly similar studies will be performed in the future to, eventually, make a meta-analysis possible
Will a decision be made, regardless of the If it is known that a decision will be made, with or without data, then any data you collect will reduce error rates.
amount of data that is available?
What is the critical effect size? Report and interpret the critical effect size, with a focus on whether a hypothesis test would even be significant for expected effect sizes. If not,
indicate you will not interpret the data based on p-values.
What is the width of the confidence Report and interpret the width of the confidence interval. What will an estimate with this much uncertainty be useful for? If the null hypothesis is true,
interval? would rejecting effects outside of the confidence interval be worthwhile (ignoring you might have low power to actually test against these values)?
Which effect sizes would you have decent Report a sensitivity power analysis, and report the effect sizes you could detect across a range of desired power levels (e.g., 80%, 90%, and 95%), or plot
power to detect? a sensitivity curve of effect sizes against desired power.
If the study is not performed for educational purposes, but the goal is answer a research question, the feasibility justification might indicate that there is no value in collecting the data. If it
wasn’t possible to conclude that one should not proceed with the data collection, there is no use of justifying the sample size. There should be cases where it is unlikely there will ever be enough
data to perform a meta-analysis (for example because of a lack of general interest in the topic), the information will not be used to make any decisions, and the statistical tests do not allow you to
test a hypothesis or estimate an effect size estimate with any useful accuracy. It should be a feasibility justification - not a feasibility excuse. If there is no good justification to collect the
maximum number of observations that is feasible, performing the study nevertheless is a waste of participants time, and/or a waste of money if data collection has associated costs. Collecting data
without a good justification why the planned sample size will yield worthwhile information has an ethical component. As Button and colleagues Button et al (2013) write:
Low power therefore has an ethical dimension — unreliable research is inefficient and wasteful. This applies to both human and animal research.
Think carefully if you can defend data collection based on a feasibility justification. Sometimes data collection is just not feasible, and we should accept this.
3 comments:
1. Thank you very much, Daniel! That's exactly what I need right now for interpreting my results and reporting! I would like to see this in a Journal! Can you suggest some literature that points out
some of the recommendations you mentioned?
2. This comment has been removed by a blog administrator.
3. Hi Daniel. Thanks very much for the blog. How would you suggest doing a sensitivity for a multi-level regression? I am under the impression that G*Power does not include this. | {"url":"https://daniellakens.blogspot.com/2020/08/feasibility-sample-size-justification.html?showComment=1675165522945","timestamp":"2024-11-12T01:52:30Z","content_type":"application/xhtml+xml","content_length":"566814","record_id":"<urn:uuid:17efa3a2-37e9-4bed-a10a-0f367e1e1727>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00341.warc.gz"} |
Re: Caller/Callee saved Registers
anton@mips.complang.tuwien.ac.at (Anton Ertl)
Mon, 28 Mar 1994 10:27:35 GMT
From comp.compilers
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Newsgroups: comp.compilers
From: anton@mips.complang.tuwien.ac.at (Anton Ertl)
Keywords: registers, optimize
Organization: Institut fuer Computersprachen, Technische Universitaet Wien
References: 94-03-054 94-03-133
Date: Mon, 28 Mar 1994 10:27:35 GMT
bart@cs.uoregon.edu (Barton Christopher Massey) writes:
|> I'll admit I glossed over the fact that in many situations callee-saves
|> still allows tail-call. I think it is a necessary condition, though, that
|> any registers passed as arguments to the tail-called procedure needn't be
|> saved by the tail-caller. A program which violates the condition is
|> # callee-saves, args in r0..rn
|> # all non-arg registers must be preserved
|> f: # 1 arg
|> push r1
|> add r0,1,r1
|> call g # 2 args
|> pop r1
|> ret
I see two possibilities:
1) The arguments are caller-saved (this is the case in all C calling
conventions I know): Then you need not save and restore r1 around g,
because f does not use it after g and the caller of f performs its own
saving and restoring if necessary. So, no need to pop r1 and you can
optimize the tail-call.
2) The arguments are callee-saved: Then you need not save and restore
r1 around g, because g does that. Again, no pop r1 and you can
optimize the tail call.
So this is not the problem with tail call optimization in C. So what
is? Why isn't it done?
|> No, I think the interesting case is not simple tail-recursion, but complex
|> patterns of deep recursion. Consider, e.g., a recursive-descent parser:
|> the only way to "use a while loop" that I know of is to recode the parser
|> as table-driven LL, which is arguably the right thing if one can do it
|> :-), but not necessarily possible or desirable in all situations.
No, there's a much better solution: Instead of converting your EBNF (or
RRPG) to BNF and then to a recursive descent parser, write your recursive
descent parser directly from the EBNF. E.g., { ... } is converted into a
while loop.
John McClain <pdp8@ai.mit.edu> wrote:
|> > The lack of tail-call optimization really hurts if you want
|> > to use a continuation passing style of programming.
Yes, using CPS a threaded code interpreter can be implemented in C: Each
abstract machine instruction is a C function. The registers of the
abstract machine are arguments of the functions. E.g., the abstract
machine instruction foo would look like this (assuming direct threading
and an abstract machine with the registers ip and sp):
void foo(Inst *ip, Word *sp)
/* Inst is a function type defined such that the C compiler does not complain
* about the use of ip below */
/* do what foo should do: */
/* fetch and execute the next instruction: */
Unfortunately, without tail call optimization this does not work long.
- anton
M. Anton Ertl anton@mips.complang.tuwien.ac.at
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Xu, Zheng and Hu (2021) - Estimation of Joint Kinematics and Fingertip Forces using Motorneuron Firing Activities. A Preliminary Report
20 Jul 2021
The paper can be found here.
Why am I reading this conference paper?
To better understand how they combine decomposition techniques (i.e motorneuron firing patterns) to design a continuous controller.
There are two main approaches to drive robotic devices:
1. Pattern recognition approaches: classification-based, are limited to discrete states of user intent.
2. Continuous approaches: regression-based (ex. linear or quadratic regressor), are used to “continuously” estimate joint kinematics and forces (for instance). The problem with these approaches in
the context of EMG is that they are unstable over time. They rely on the amplitude of the EMG signals as input features which deteriorate over time (due to amplitude drift and electrode shift).
This motivates the use of motor unit action potentials (MUAPs) to estimate the neural drive instead of relying on the EMG signals (given that EMG signals “intrinsically comprise MUAPs”). MUAPs
therefore provide a more stable basis for regression. Once the separation matrix is learnt, it can be directly applied to new HD-EMG signals.
Main Contribution
The method for MU decomposition, validation and estimation of a finger joint angle in a dynamic task and force in an isometric task. The novelty here is, to my opinion 😀, in the MUs cross-trials
validation on a preliminary study (validating the obtained MUs on a second trial before actually testing them).
• 3 healthy participants
• HD-EMG data acquired: 8x16 electrode array. Signals sampled at 2048 Hz and bandpassed 10-900 Hz
□ Load cell used to record the finger forces and angle sensors for joint angles recordings.
• 2 tasks were performed: isometric finger flexion force (index and middle finger) and dynamic finger movement task (flex and release repetitively)
MU identification
• Decomposition used is based on the Fast independent component analysis (FastICA).
• The algorithm is carried offline. The goal of the decomposition is to separate or resolve the superimposed (spatiotemporally) MUAPs into individual MU activities. This is achieved through the
following steps:
1. Extending the raw signal ( by adding R delayed replicas of the original signals in each channel)
2. Whitening the extended channels
3. Getting the decomposed signal sources using a fixed-point iteration algorithm. This step leads to the separation vectors.
4. Multiplying the EMG with the separation matrix to obtain the decomposed source signals (MUs spike trains).
MU validation
To validate the decomposition, the separation matrix is computed/derived from a trial and then applied on a second trial before being used on a third testing trial:
• By applying the separation matrix on the second trial, they obtain a spike array of $M_i$ X $K_j$; M is the length of the EMG in the second trial and K is the number of decomposed MUs from the
first trial.
• Run a regression analysis between the firing frequency of the spike trains and the measured joint angle/isometric force. The output of this analysis is $K_j$ $R^2$ values; how well each MU
obtained on the first trial estimates the motor task on the validation trial.
• Retain the top 10 MUs with the strongest association to the motor task.
• Estimation joint angle and isometric force from neural drive (MUs) is more accurate than EMG amplitude. It showed a smaller RMSE.
• For joint angle estimation, EMG amplitude error is around 20 degrees compared to 8 degrees for the MU approach during dynamic movement.
• The decomposition is computationally intensive and is not therefore practical in real-time. However, a solution would be to obtain the MUs offline in an initialization period then use the
separation matrix in real-time. The underlying assumption is that a common set of MUs is recruited across different muscle activations (i.e. MUs form a common input to the muscles).
• The decomposition was performed separately for each task. Is it feasible to estimate the neural drive (MU firing frequency) directly using neural network approaches? Interesting examples of ANN
approaches for a generic decomposition to look at:
□ Paper by Farina et al., Deep Learning For Robust Decomposition of High-Density Surface EMG Signals, (2020).
□ Paper by Hu et al. Real-time finger force prediction via parallel convolutional neural networks: a preliminary study, (2020).
Final Thoughts
• Overall, I think it is a nice paper. I didn’t get yet why choosing ICA in the first place as opposed to other decomposition techniques, but I guess this is related to their earlier paper on
Independent Component Analysis Based Algorithms for High-Density EMG Decomposition, Dai & Hu (2019).
• One minor comment on the decomposition of EMG for the dynamic task: as the authors stated, extracting the MUs is challenging for the dynamic task, but why is this the case? | {"url":"http://farahbaracat.com/lit_review/2021/07/20/Hu-estimation/","timestamp":"2024-11-10T03:12:22Z","content_type":"text/html","content_length":"12783","record_id":"<urn:uuid:2a798584-ed04-42de-8c48-70b1f4dd5d8b>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00617.warc.gz"} |