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Integral of
Introduction to integral of sinh(ax)
In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is
categorized into two parts, definite integral and indefinite integral. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function.
Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to a trigonometric function sine. You will
also understand how to compute sinh(ax) integral by using different integration techniques.
What is the integral of sinh(ax)?
The integral of sinh(ax) is an antiderivative of sine function which is equal to cosh(ax)/a. It is also known as the reverse derivative of sine function which is a hyperbolic function. By definition,
a hyperbolic function is a combination of two exponential functions e^x and e^-x. Mathematically, the sinh formula is;
$$\sinh ax=\frac{e^{ax}-e^{-ax}}{2}$$
Integral of sinh(ax) formula
The formula of integral of sinh ax contains integral sign, coefficient of integration and the function as sine. It is denoted by ∫(sinh(ax))dx. In mathematical form, the integral of sinh(ax) is:
$∫\sinh(ax)dx = \frac{\cosh(ax)}{a}+c$
Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral.
How to calculate the integral of sinh(ax)?
$The integral of sinh(ax) is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of sine by using:
1. Derivatives
2. Substitution method
3. Definite integral
Integral of sinh(ax) by using derivatives
The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. Therefore, we can use the derivative to calculate the integral
of a function. Let’s discuss calculating the integral of sinh(ax) by using derivatives.
Proof of integral of sinh(ax) by using derivatives
Since we know that the integration is the reverse of the derivative. Therefore, we can calculate the integral of sinh(ax) by using its derivative. For this, we have to look for some derivatives
formulas or a formula that gives sinh(ax) as the derivative of any function.
In derivative, we know that,
$\frac{d}{dx}(\cosh (ax))=a\sinh(ax)$
It means that the derivative of cos x gives us sinh(ax). Now by using integral, the integral of sinh(ax) is:
$∫\sinh(ax)dx=\frac{\cosh (ax)}{a}+c$
Hence the integral of sinh(ax) is equal to the cosh(ax)/a.
Integral of sinh(ax) by using substitution method
The u-substitution calculator involves many trigonometric formulas. We can use these formulas to verify the integrals of different trigonometric functions such as sine, cosine, tangent, etc. Let’s
understand how to prove the integral of sin by using the substitution method.
Proof of Integral of sinh(ax) by using substitution method
To proof the integral of sinh(ax) by using substitution method, suppose that:
Differentiating with respect to x,
To calculate integral, we can write the above equation as:
$dy = a\cosh(ax)dx$
By trigonometric substitution identities, we know that cosh ax = √1 + sinh²(ax). Then the above equation becomes,
$dy = a\sqrt{1+ \sinh^2(ax)}.dx$
Now, substituting the value of sinh2 x, such as:
$dy=a\sqrt{1 + y^2}.dx$
Multiplying both sides by sinh(ax),
$\frac{\sinh(ax) dy}{a\sqrt{1+y^2}}=\sinh(ax)dx$
Again substitute sinh(ax) = y on the left side.
Integrating on both sides by applying integral,
$∫\frac{y dy}{a\sqrt{1 + y^2}}=∫\sinh(ax)dx$
Let 1+ y² = u. Then 2y dy = du (or) y dy = 1/2 du.
Then the above left-hand side integral becomes,
Since the power rule of integration is
Therefore, by using this formula we get,
Again substituting u = 1 + y², we get
$\frac{(1 + y^2)^{1/2}}{a}+C=∫\sinh(ax) dx$
And again Substitute y = sinh(ax) here,
$\frac{(1 + \sinh^2x)^{1/2}}{a}+C=∫\sinh(ax)dx$
$\frac{\cosh ax}{a}+C=∫\sinh(ax)dx$
Hence the integral of sinh(ax) is cosh x.
Integral of sinh(ax) by using definite integral
The definite integral is a type of integral that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral can be written as:
$∫^b_a f(x) dx = F(b) – F(a)$
Let’s understand the verification of the integral of sinh(ax) by using the definite integral finder.
Proof of integral of sinh(ax) by using definite integral
To compute the integral of sinh(ax) by using a definite integral, we can use the interval from 0 to π or 0 to π/2. Let’s compute the integral of sinh(ax) from 0 to π. For this we can write the
integral as:
$∫^π_0 \sinh(ax)dx=-\left|\frac{\cosh ax}{a}\right|^π_0$
Now, substituting the limit in the given function.
$∫^π_0 \sinh(ax)dx=\frac{\cosh (aπ)}{a}-\frac{\cosh a(0)}{a}$
Since cos 0 is equal to 1 and cos π is equal to -1, therefore,
$∫^π_0 \sinh(ax)dx=-\frac{1}{a}-\frac{1}{a}=-\frac{2}{a}$
Which is the calculation of the definite integral of sinh(ax). Now to calculate the integral of sinh(ax) between the interval 0 to π/2, we just have to replace π by π/2. Therefore,
$∫^{\frac{π}{2}}_0 \sinh(ax)dx=-\left|\frac{\cosh ax}{a}\right|^{\frac{π}{2}}_0$
$∫^{\frac{π}{2}}_0 \sinh(ax)dx=-\frac{\cosh a(π/2)}{a}+\frac{\cosh a(0)}{a}$
Since cos 0 is equal to 1 and cos π/2 is equal to 0, therefore,
$∫^{\frac{π}{2}}_0 \sinh(ax)dx =0 -1/a=-1/a$
$Therefore, the definite integral of sinh(ax) is equal to -1/a. | {"url":"https://calculator-integral.com/integral-of-sinh-ax","timestamp":"2024-11-03T00:53:41Z","content_type":"text/html","content_length":"50673","record_id":"<urn:uuid:f998d88a-2765-4b4b-9287-7e0c1394c7a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00322.warc.gz"} |
Mathematics for Class ViII
5. The inside perimeter of a r... | Filo
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Mathematics for Class ViII 5. The inside perimeter of a running track (shown in Fig. 20.24) is . The length of each of the straight portion is and the ends are semi-circles. If track is everywhere
wide, find the area of the track. Also, find the length of the outer running track. Fig.20.24 Fig.
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Mathematics for Class ViII 5. The inside perimeter of a running track (shown in Fig. 20.24) is . The length of each of the straight portion is and the ends are semi-circles. If track is
Question Text everywhere wide, find the area of the track. Also, find the length of the outer running track. Fig.20.24 Fig.
Updated On Dec 9, 2022
Topic All topics
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 60
Avg. Video 7 min | {"url":"https://askfilo.com/user-question-answers-mathematics/mathematics-for-class-viii-5-the-inside-perimeter-of-a-33323535313537","timestamp":"2024-11-05T03:19:59Z","content_type":"text/html","content_length":"320292","record_id":"<urn:uuid:76592567-bc6d-4c09-a2d5-fdbe05f66151>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00639.warc.gz"} |
Quick Ratio
Quick Ratio (Acid Test Ratio) – an indicator of a firm’s short-term liquidity measuring how well company can meet its short-term obligations with its highly liquid assets, such as cash and
equivalents, marketable securities and receivables. This ratio is similar to current ratio, as both of them measure the short-term solvency of a firm. However, quick ratio does it more rigorously,
excluding the less liquid current assets and potential sources of loss from calculation. Reason for excluding these (for example, slow-moving or obsolete inventory) is need of estimating more
immediate liquidity of a company than current ratio does.
The normative value of the quick ratio depends on the firm's industry and peculiarities. Most commonly quick ratio value of 1 is considered to be normal. It means that the company has as much assets
with relatively good liquidity, as its current liabilities. However, to make an accurate conclusion the ratio should be compared with the dynamics of values from past periods, and with industry
averages as well. Some industries find quick ratio below 1 acceptable, while for some industries only ratio above 1 is adequate. Businesses selling goods or services primarily for cash with no
involvement of accounts receivable can still be on a good level of liquidity even with quick ratio moderately below 1. Conversely, if a firm has an active policy of providing buyers with consumer
loans, and the accounts receivalbe turnover is slow, creating big amount of bad quality receivables, company can demonstrate unsatisfactory liquidity even with quick ratio values much greater than 1.
Resolving the problems with the quick ratio exceeding the normative range:
If the quick ratio value is lower than normative, the firm should take some actions to increase the amount of highly liquid assets and decrease the amount of current assets.
Quick Ratio = (Cash Equivalents + Marketable Securities + Net Receivables) ÷ Current Liabilities
Quick Ratio = (Cash + Marketable Securities + Accounts and Notes Receivable) ÷ Current Liabilities
Quick Ratio = (Current assets - Inventory) ÷ Current liabilities
Quick Ratio (Year 1) = (51 + 25 + 34) ÷ 464 = 0,23
Quick Ratio (Year 2) = (13 + 0 + 47) ÷ 911 = 0,06
The quick ratio for year 1 was 0,23, meaning that there was $0,23 of the most liquid assets for every $1,00 of current liabilities. The decline of the ratio to 0,06 in year 2 witnessed, that the
company’s short term-liquidity has become worse and it most likely would face problems paying its short-term obligations.
Quick ratio helps an analyst to estimate firm's short-term solvency and measure the amount of highly liquid assets available to cover its current liabilities, if needed. Optimal is the situation when
this ratio equals 1, or close to it. | {"url":"https://finstanon.com/ratios-dictionary/57-quick-ratio","timestamp":"2024-11-04T17:50:16Z","content_type":"text/html","content_length":"6886","record_id":"<urn:uuid:f5a71026-2929-4c07-9638-b2c08fc58b65>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00184.warc.gz"} |
Replace with "[\U\1]" always make word missing inside squared bracket
Replace with "[\U\1]" always make word missing inside squared bracket
As the title say when I type “[\U\1]” it always made the word missing.
I am somewhat of a beginner when it comes to regular expressions, but I think your problem is with the “find what” section that don’t group the dr.
You could try this:
Find what: (dr)
Replace with: [\U\1]
This is if you want the result to be [DR] if you are expecting the result DR you just replace with \U\1 instead.
@Pierre-Åberg Actually I want to be [Dr Knox]
Given the picture I would assume that it’s now [dr. knox]
If this assumption is right you could try this:
Find what: (\<dr\>)
Replace with \u\1
This will replace the lowercase d with an uppercase d.
Updated the post just in case you have dr somewhere in the middle of a word, such as android.
This will make sure that it looks for the word dr and not the letter d directly followed by the letter r. | {"url":"https://community.notepad-plus-plus.org/topic/24751/replace-with-u-1-always-make-word-missing-inside-squared-bracket","timestamp":"2024-11-02T14:47:56Z","content_type":"text/html","content_length":"69202","record_id":"<urn:uuid:3966ad1c-cb1f-42bc-9d16-3417a7d4e448>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00343.warc.gz"} |
Algorithm Design Techniques
So far, we have been concerned W~i~th the efficient implementation of algorithms. We have seen that when an algorithm is given, the actual data structures need not be specified. It is up to the
programmer to choose the approriate data structure in order to make the running time as small as possible.
In this chapter, we sW~i~tch our attention from the implementation of algorithms to the design of algorithms. Most of the algorithms that we have seen so far are straightforward and simple. Chapter 9
contains some algorithms that are much more subtle, and some require an argument (in some cases lengthy) to show that they are indeed correct. In this chapter, we W~i~ll focus on five of the common
types of algorithms used to solve problems. For many problems, it is quite likely that at least one of these methods W~i~ll work. Specifically, for each type of algorithm we W~i~ll
• See the general approach.
• Look at several examples (the exercises at the end of the chapter provide many more examples).
• Discuss, in general terms, the time and space complex~i~ty, where appropriate. | {"url":"http://gurukulams.com/books/csebooks/ds-algorithms/algorithm-design-techniques/","timestamp":"2024-11-03T10:22:10Z","content_type":"text/html","content_length":"26863","record_id":"<urn:uuid:c9e4ff19-f85e-453a-91f4-a3c56a09aab1>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00144.warc.gz"} |
Design Like A Pro
Solving Equations Worksheet With Answers
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Chapter 205: Mathematical Education And Science In The Chechen Republic: State And Perspective
Mathematics education and mathematical science in the USSR were the level which other countries aimed to achieve. The development of mathematical education and mathematical science in the Chechen
Republic has a history of almost a century, but still does not have an analysis of the education system, functioning, results achieved, evaluation of successes, consideration of shortcomings, there
is no elaborated line for further development of mathematical education and mathematical science in the republic. The main research works analysis of scientists-mathematicians in the republic, the
analysis of the state and conduct of research works, as in the field of general secondary education in the direction of mathematics, vocational education and mathematical science development, has not
been carried out. It follows that all work should be studied, systematized and further ways of improvement should be determined. The article attempts to carry out such work on the basis of the
available factual material obtained from various sources. The task of the study is to consider issues related to the history of the system formation of mathematical education and mathematical science
in the Chechen Republic in the twentieth and early twenty-first century, basic statistical data, problems that need to be solved and prospects; to determine the reasons for the unsatisfactory state
of affairs in mathematics teaching in the schools of the republic, as well as in the institutions of general and vocational education in the republic; and some ways to problem solution have been
Keywords: Mathematical education historystaffingcandidates of physical and mathematical sciencesdoctors of physical and mathematical sciencesUSE resultslanguage instruction problems
Mathematics is the universal language of all sciences, which determines its special position. The attitude toward mathematics, understanding its role and significance show the future development of
society in the modern world. Other sciences are also important, but they all converge on mathematics. Mathematics is the central nerve, aorta and artery of human civilization and the basis of any
Mathematical education has always been a problem. Today it seems to be a serious task on a national scale. First we will give a few quotes.
“The quality of schoolchildren’s knowledge is terrifying, and there is not even hope for improvement. This fact has recently been recognized by top managers, for example, the Deputy Minister V.
Bolotov” (Sadovnichy, 2010, p. 4).
The report of the rector of Moscow State University named after M.V. Lomonosov at the All-Russian Congress of Mathematics Teachers at Moscow State University is said “Mathematical education is going
through hard times” (Sadovnichy, 2010).
The level of regional mathematics education in a multinational country, which is Russia, is an integral and essential component of the single country preservation; the factor affecting national
security. Practice shows that high-quality mathematical education is usually the privilege of large cities - Moscow, St. Petersburg, Novosibirsk, etc. Besides other factors, modern technologies give
the opportunity to achieve the same level in the regions.
Naturally, the development of regional mathematics education requires special attention and separate analysis. As a rule, the regions where mathematical science and education are put on the proper
level are powerful industrial centers. These two factors do supplement each other.
In this regard, the development of mathematics and the formulation of mathematical education in the national subjects of the Russian Federation are of great interest.
The USSR education system, and, in the Chechen Republic as its constituent, can be the subject of analysis, beginning in 1930, when compulsory primary education was introduced by law.
Studying of the Chechen Republic history of mathematical science formation and development, as well as the system of mathematical education as a whole, being insufficiently studied, can permit to
build models for its improvement, to organize research work on improving indicators.
Problem Statement
The problem of the history of the formation and development of mathematical education and mathematical science in the Chechen Republic has not been studied, and has not even been systematized. The
article provides some factual material that allows assessing the state of general secondary, vocational education and mathematical science.
Research Questions
The subject of the research is the state and history description of the formation and development of mathematical science and mathematical education in the Chechen Republic.
Purpose of the Study
The purpose of the work is to update the problems of mathematical science and mathematical education development in the Chechen Republic based on the consideration of some issues related to the
history of the emergence and development of mathematical science and mathematical education in the region, systematization of available data, analysis of the reasons for the low quality of general
secondary and professional mathematical education.
Research Methods
The analysis of pedagogical, educational, methodical literature, statistical data and other factual material available on the topic was carried out by general logical methods (analysis, synthesis,
deduction, etc.).
For the state review of the mathematical education and mathematical science level in the Chechen Republic, we take the following parameters as a basis:
А) USE results in mathematics; Б) study in SESC named after A.N. Kolmogorov or the similar, including in the area; В) participation results in mathematical competitions and other competitive events;
Г) study on this specialty in the leading universities of the country; Д) the amount of candidates and doctors of physical and mathematical sciences, the republic representatives.
With obligatory participation in the exam, since 2009, the percentage of students who did not pass mathematics exam ranges within 10-15. This result remains the lowest in the country.
Relatively objective USE results in the republic can only be called from 2015. Without doing a more detailed analysis, we present only the following data on the USE results. An average score of those
who passed the profile mathematics did not reach 29, who passed – 38:
The majority of those who received positive results are the result of tutors work. The school does not provide an adequate level and therefore a need to attract tutors appears.
There is information about only 2 students studying in SESC named after A.N. Kolmogorov for all the time of its existence. In the entire history in 1976, one student from the ChIASSR became the
winner of the All-Russian Olympiad. Here is one remark: “The results at the international level mathematical Olympiad speak about the general level of education development in the country and the
readiness of these countries to create and reproduce new technologies” (Agakhanov & Podlipsky, 2008). Graduates from the republic in the entire history of education studied in leading universities:
14 at Moscow State University, 4 - at Leningrad State University, 1 - MIPT. Studying on mechanical and mathematical, mathematical and mechanical faculties is meant.
According to the available data, in the country and abroad, the owners of candidate and the doctor of physical and mathematical sciences scientific degrees among the representatives of the indigenous
nationality on mathematics and mechanics are about 30 people. Moreover, this number includes those who have passed away. Two barbaric military campaign have had a huge negative impact on the
situation in the education and science sphere of the Chechen Republic, past on its territory in the last decade of the twentieth century and at the beginning of new twenty-first century.
All data speak about serious problems in mathematics education both in general and in vocational education. Among the positive facts, for the first time in the history of the republic it seems to be
essential to note the mathematical school opening in 2015, which should set the grade both in the mathematical education level of schoolchildren and in the organization of mathematics teaching.
Education in Chechnya, as a direction in the social sphere, began to take shape as a system in the early twenties of the twentieth century.
The mathematics teaching at the national school was conducted in the Chechen language at that time. In 1927, by the decision of the Bolshevik authorities, the writing in the schools of Chechnya, as
well as in other national republics, was changed, that is to say the graphic base from the Arabic script was completely translated into Latin. This translation was only used in the national
republics. Talking about the serious results of the Vainakhs mathematical education (Vainakhs - the name of the Chechens and Ingushes) is hardly appropriate in those years.
This was the period of the education system formation with its shifts, with the change of alphabets, the content of school programs in mathematics, the introduction of ideological components, etc.
Additional difficulties were associated with the acute shortage of mathematics teachers. The Republic has not passed examples similar to the well-known anecdotal cases about the sum of fractions ½
and 1/3 “equal” to 2/5.
At the beginning of the 30 s no data is found on the presence of both scientists, mathematicians, representatives of indigenous nationality, and people with higher education in this specialty.
Shamsadov M.M. (1913-1960) is the first student who entered the Faculty of Mechanics and Mathematics of the country leading Moscow State University named after Pokrovsky (now M.V. Lomonosov) on the
specialty of Mathematics. He studied at MSU in 1933-1938 and graduated with a second degree diploma (MSU Archive).
In the 1920s, in the republic, as in many regions of the country, so-called working faculties appeared which were aimed at eradicating illiteracy, training primary-level teachers, which graduate was
M.M. Shamsadov mentioned above.
The teacher institute was opened in Grozny in 1938. Prior to the eviction of the Chechens and Ingushes in 1944, according to available data, the following Chechens representatives graduated from the
Physics and Mathematics Faculty of this institute, who later worked at schools of the republic as mathematics teachers, heads of schools, and education departments for all their lives: Magomadov
Sh.M. (1919- 1983), Denilkhanov S.M. (1919-1997), Khoguyev M.U., Ibragimov N.Kh. (1916-2001). Ibragimov N.Kh. was awarded the title of School Honored Teacher of the RSFSR by Decree of the Supreme
Soviet Presidium of the RSFSR on May 25, 1960. Perhaps he was the first representative among the Chechens, who had such a high rank. But in any case he was the first representative among Chechen
mathematicians in this rank.
Thus, the available information shows the presence of several specialists, mathematics teachers among the indigenous population of Chechnya. Accordingly, nonresident teachers, or people who have
attended the course of working faculties work mainly at schools. This continues until the 1944 eviction. The Chechens, abandoned for extinction in Kazakhstan and Kyrgyzstan steppes, were cut off from
education and science for almost 13 years. Although individuals, despite numerous obstacles, were able to get a higher education there.
According to available information, with a higher mathematical education the following people returned from Kazakhstan: Elimbayev P.Kh. (1931-1998), Mutsaev A.M. (1931-2004), Janar-Aliev A. Ya.
(1932-2000), Ilyasov A.A. (1929-1997), Yandarov V.O. (1937-2014) and Israilov S.V. (born in 1936). Yandarov V.O. and Israilov S.V. devoted themselves to mathematical science. Moreover, Israilov S.V.
became the first Chechen who got a degree in Physics and Mathematics.
The return of the evicted peoples in the late 50s of the twentieth century to their historic homeland set the task of restoring the education system and its mathematical component.
After the Chechen-Ingush autonomy restoration at the end of the 50s, the learning process was organized in Russian. The knowledge of mathematics is foremost the ability to solve issues. Children,
first of all, should understand the content of the text tasks, what is given and what is required from them. One of the goals of school education in mathematics is to contribute to the development of
students' logical thinking. Nevertheless, logical thinking is impossible without an elementary understanding of the task content.
Stress arising in children, particularly of primary school age, in the perception of educational material presented in a non-native language, leads to many negative consequences, primarily, reducing
interest in the subject and learning in general. If the child catches even shades of sounds, syllables, aside from words in the native language, then learning non-native language, this does not occur
even in relation to the whole sentence. A lot of examples can be cited where wording of the sentence in Russian does not cause any reaction; a literal translation may even cause a smile, but the
semantic translation involuntarily puts the student in need of a conscious perception of the sentence.
A number of studies show that studying a non-native language reduces the intellectual and mental development of children by 15-30 percent. In other studies, this figure reaches 60%.
Studies show that children for whom and in the language of whom mathematics textbooks were written used today in most schools of the republic, in particular, written by M.I. Moro (Moro, 2008), and
others, go to school, having the vocabulary from 3,000 to 7,000 units. Children, for instance, the Chechens, had the vocabulary of 20-30 units corresponding to the language of the used mathematics
textbook at the beginning of the zero years, and the first lesson material of mathematics contains more than 120 words. When we bring them, these words, to one root, we will get about 90. What should
a teacher do in this situation? Should he explain the semantics of words in the mathematical tasks formulation or mathematical knowledge formation, thinking? The second cannot be done without the
first one, the first will be done or will have to be done, but at the expense of the second, moreover, being confident in the ineffectiveness of their actions, because the words with which vocabulary
work is conducted at the lesson do not become an active stock of schoolchildren due to the lack of support in the form of communication environment. The teacher has the opportunity to use an
understandable native language in the visual-illustrative setting of tasks.
ICT terminology use in the mathematical tasks formulation at primary school, as the most common, would allow the teacher to reduce the time for vocabulary work.
The studies conducted in 2008-2018 in preschool institutions of Grozny and Urus-Martan showed that even younger children understood the meaning of many ICT words without problems. The children
accurately knew the semantics of more than 20% of words out of 500 terms presented. These words are almost absent in the math textbook for the 1st grade of 2007 (Moro, 2008). For the first time, a
few words of this terminology appeared in the task formulation in the mathematics textbook for the 1st grade of 2016 (Moro, 2016).
There has been a tendency to increase the text in the task formulation in mathematics textbooks, which creates additional difficulties for the teacher in the national school, who needs to explain the
semantics of these words at the expense of lesson time, and the student who needs to understand these words.
The math textbook author L.G. Peterson explains “All textual wording increases are exclusively connected with the requirements of the RAS experts about "mathematical accuracy and uniqueness" of the
texts. These are their corrections” (from correspondence with the author). For comparison: lexical material available to the author do not exceed 20-30 words at the initial period in mathematics
textbooks for 1 grade in the UK, Norway schools (Heineman Mathematics, 1981; Pederson, Anderson & Johansson, 2006; Broadbent, 2004; Haanæs & Dahle, 2000).
Initially, our children have unequal conditions for getting education. If we consider the first class, then they have a higher workload by the number of hours in the curricula, they need to
additionally study the Chechen alphabet, containing one and a half times more letters than in the Russian language, 49 instead of 33, etc. The Chechen alphabet learning ends only in May of the school
year, while the Russian - in February. The foundation for success is basically laid in the first class.
The well-known statement of P.K. Uslar, the first enlightener and organizer of Chechnya schools about the need to educate children at the initial stage in their native language is being ignored in
the educational process organization in the republic, aside from the classic statement “Native word - D.K. Ushinsky said - is the basis of all mental development and the treasury of all knowledge”. (
Ulsar, 1887, p. 23).
When mathematics teaching in the early 60s, after the restoration of Chechen-Ingush ASSR autonomy, began to give negative results and problems with the language as the main reason became obvious,
there was a need to solve it.
“An empirical study of the most effective forms and methods of teaching non-Russian (Chechen and Ingush) children and the structure of primary education in schools of the republic was conducted in
The expediency of arithmetic (mathematics) textbook translating in the primary school into the native (Chechen and Ingush) language of students and others.
“The question of whether it is advisable to translate arithmetic (mathematics) primary school textbooks into their native language (Chechen and Ingush), got a negative answer by all teachers:
“Such a translation would be a brake on the development of Russian spoken language” (Grozny region). (The Russian-speaking population prevailed here - A.Ya.). (Umarov, 1982, p. 68)
“There is no need to translate arithmetic (mathematics) textbooks into their native language. Arithmetic (mathematics) itself contributes to the development of Russian spoken language, the
development of thinking in Russian” (Achkhoy-Martanovsky region). (Umarov, 1982, p. 69)
“It is inexpedient to translate arithmetic (mathematics) textbooks into the Chechen language, since almost every school has children of non-Chechen nationality in the region” (Nadterechny region). (
Umarov, 1982, p. 71).
The latter opinion requires a special analysis: “there are children of non-Chechen nationality at school” - the presence of children of non-Chechen nationality, not even in the classroom, but at the
school, is an argument for all children of the school to study in a language that is not native for them. That is the fact that some children cannot (and are not likely to want) study in a foreign
language, so all the other children have to be taught in their native language. The argument is deadly “convincing”. But it was the “iron” logic of state policy in the field of education of that
period. The “experiment” to translate the mathematics teaching into the Chechen language was again launched in 2008, after numerous appeals and discussions, in 47 schools of the republic, involving
almost a thousand and a half children. By this time, the translation of M.I. Moro textbook had been privately done. Then this translation was approved by the educational and methodological council of
the Ministry, despite official warnings about poor-quality translation, about serious shortcomings. We will give an example of translation. Task No.9, p.22, part 2. (Moro , 2008).
“Make up one task according to the drawing, in the condition of which there is a word more, and another in the question of which there is a word less . Solve these tasks”.
This task translation into the Chechen language is given in the following form.
“Сизкепаца хIоттаде цхьа хьесап шен хаамехь дукхах дош а долуш, шолгIа хьесап — хаттарехь кIезгах дош а долуш. И хьесапаш кхочушде”.
We gave this translation in this view to 4 doctors of sciences, including philological ones, to more than 10 candidates of sciences who graduated from MSU, RSU and GOI, and parents. They read the
task up to 4-5 times. Nevertheless, we could not achieve an understanding of the two components: what is given, and what is required.
In addition, this task is for students just starting their studies.
The translation included a huge amount of archaisms and neologisms. Learning tasks for mathematical knowledge formation were replaced by the form. Mathematics, as an academic discipline, had far more
problems and obstacles than the language, that is to say, there was a language, but there was no mathematics. No doubt, that at the expense of mathematics, the language problems could not be solved.
А поскольку язык был, то за его счет проблемы математики еще можно было решать. При всех недостатках перевода, довольно существенных, результативность использования этих учебников, как показала
практика посещенных уроков и справка по итогам преподавания по переводному учебнику, составленная Министерством образования и науки ЧР, была гораздо выше. Перевод был сделан людьми, ни одного дня не
проработавшими в школе.
A multilingual educational model, completely copied from North Ossetia, was launched in 2010 by the leadership of the Ministry of Education and Science, consisting of people who did not have
pedagogical experience even for one year. The model was “valid” for 5 years, and then also collapsed. Pupils from 96 schools took part.
In practice, it turned out that an experiment was being conducted ... on electronic textbooks or textbooks, published subsequently, after the end of the school year. Two math textbooks were published
for the first grade, but in 2011, when the school year was over. The first (edition of 1560 copies) was almost in Russian, with the exception of some words. The second (460 copies) was vice versa (
Adamova, Bushueva, & Sultanova, 2010). How to use and how to distribute these textbooks among students, teachers, methodologists, etc. is unclear. Although the model is more favorable for the
educational process of children in schools with a national composition of children. It allows us to use some capabilities of the native language.
To judge the level of these textbooks, we will give examples from the mathematics electronic textbook for grade 4. (Sultanova, 2014)
Dada and Askhab decided to lay the floor with square ceramic tiles in the kitchen. The length of the kitchen is 6 m and the width is 9 m. How many ceramic tiles are required if the side of a tile is
3 dm?
Remember, 1m^2=100 dm^2
Length is less than width.
Aslan and Dima went simultaneously in opposite directions from Red Square to Georgy Konstantinovich Zhukov monument. After 5 min the distance between them was 715 m. Alan's speed is 65m/min. Define
Dima’s speed.
Multiculturalism is “respected”, Aslan is Chechen, Dima is Russian. Everything can be allowed, but Aslan is unlikely (we may assume that the author means him) would go to the same monument in the
opposite direction. He will have to go around the globe.
Such pearls were characteristic for the entire textbook used in the “experiment”.
The study language is paid the main attention in the article as the most important factor in improving the quality of the educational process. Especially at the initial level, where the study
foundations are laid both at the school and after it. If there are no qualitative results in the initial level, then the followers are engaged in “working on errors”, trying to fill gaps in
knowledge. The middle ranking works for primary, the senior for middle...
Another serious problem of modern mathematics education in the republic is personnel. New schools being built in the republic require new mathematics teachers. Teachers of mathematics are people with
different basic qualifications.
The teaching of mathematics has the following picture in one of the regions of the Chechen Republic. More than 130 people are teachers of mathematics. 15 people have a basic economic education among
them, engineers - 3, a lawyer, a chemist, a specialist in tourism – 3, without education or students - 11 people, that is to say, more than a quarter of teachers do not have the appropriate
qualification. In the same area in one of the schools, three teachers of mathematics do not have the appropriate qualifications. A 2nd year student, without an experience, has mathematics workload of
30 hours in the school of the region; here the workload of another teacher is 40 hours. There is a school where mathematics teaching the 11th grade is conducted by a teacher without job seniority.
This is only one subject. And if we analyze the teaching staff, we can get a completely different picture.
Lawyers, economists, and engineers work at school as directors in the area, making up almost 20% of managers. The practice of appointing heads of schools - directors and head teachers, which took
place in the Soviet school, doesn’t apply to the humanities and natural scientists. Thus if the humanist was appointed as the director, then the head teacher, as a rule, was a natural scientist and
vice versa. Naturally, these leaders are not “teachers of teachers,” as they used to be called in the Soviet period. The level of leadership on the basis of the principle “a teacher lives on our
street, I communicate with her, I know the problems of the school and the ways to solve them”, which took place in the republic, cannot solve educational problems. A teacher “who lives on the
official’s street”, for example, will never tell him that she made mistakes in preparing for the lesson, in explaining the material, in polling, in consolidating, etc.
Thus, the improvement of the state of affairs in mathematical education and science direction in the republic requires a set of measures.
1. The tasks of preparing and raising the professional level of teaching and managing educational institutions, organizing the educational process taking into account the language situation are
significant for the Chechen Republic.
2. It is shown that one of the main problems of mathematical education poor quality in the Chechen Republic is unequal conditions for obtaining general education. They are associated with the
instruction language, especially in primary schools, with an increased academic load during the entire period of study. The solution of the language problem should be based on the principle: at the
present stage of the national school development, the instruction language should assist to improve the quality of studying mathematics, but not to language issues at the expense of mathematics.
3. The lexical units of a mathematics textbook, especially in primary schools, should contain a large number of ICT terminology as the most understandable for children, regardless of their
nationality. To sum up, mathematics textbooks for primary classes need to be improved.
Measures may be different.
1. Agakhanov, N.Kh., Podlipsky, O.K. (2008). Maths. All-Russian Olympiad. Moscow: Enlightenment
2. Adamova, L.A., Bushueva, R.B., Sultanova, Kh.Z. (2010). Maths. 1 class. Mines: Publishing house LLC " Informsvyaz ".Kirillova, I.A. (2007). Reducing the level of mathematical knowledge. Their
causes and ways to overcome. Retrieved from : http : //torkyurok.rf/% D 1% 81% D 1% 82% D 0% B 0% D 1% 82% D 1% 8 C % D 0% B 8/513010 /
3. Broadbent, P. (2004). Premier Math. Ages 7-8. Key stage 2. London: Letts Education LTD.
4. Haanæs, M., Dahle, A.B. (2000). Pluss. Matematikk. Grunnbok. Tangen, NKS FORLAGET.
5. Heineman Mathematics, (1981) Name. Share workbook. R. 16, Scottish Primary Mathematics Group, United Kingdom.
6. Moro, M.I., Volkova, S.I., Stepanova, S.V. (2008). Mathematics - 1 class. Part 1, 2. Moscow: Enlightenment.
7. Moro, M.I., Volkova, S.I., Stepanova, S.V. (2016). Mathematics - 1 class. Part 1, 2. Moscow: Enlightenment.
8. Pederson, B.B., Anderson, K., Johansson, E. (2006). Abakus. matematikk for barnetrinnet. grunnbok 3a, H.Ashehhoug & Co ( W.Nygaard ).
9. Sadovnichy, V.A. (2010). About mathematics and its teaching in school. Collection of theses of the All-Russian congress of teachers of mathematics (p.15). Moscow: MAKS Press Publishers.
10. Sultanova, Kh.Z. (2014). Mathematics - 4th grade. Retrieved from: http://mon95.ru/
11. Umarov, M.U. (1982). On the way to universal secondary education. Grozny: Chechen-Ingush book publishing house
12. Uslar P.R. (1887) Etnography of Causasus. Vladikavkaz: Department of Governor.
About this article
Publication Date
29 March 2019
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eBook ISBN
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Sociolinguistics, linguistics, semantics, discourse analysis, science, technology, society
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Yakubov, A. V. (2019). Mathematical Education And Science In The Chechen Republic: State And Perspective. In D. K. Bataev (Ed.), Social and Cultural Transformations in the Context of Modern
Globalism, vol 58. European Proceedings of Social and Behavioural Sciences (pp. 1764-1772). Future Academy. https://doi.org/10.15405/epsbs.2019.03.02.205 | {"url":"https://www.europeanproceedings.com/article/10.15405/epsbs.2019.03.02.205","timestamp":"2024-11-11T14:07:23Z","content_type":"text/html","content_length":"68570","record_id":"<urn:uuid:ab621ab1-3400-4e25-9f91-8d745dc29f46>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00092.warc.gz"} |
Perform T-Tests in R I Types and Assumptions
As a data analyst with a Ph.D. in data science and five years of freelance experience, I often think about the intricacies of statistical tests. One such test that has always intrigued me is the
Have you ever wondered how researchers determine whether there is any statistically significant difference between two groups?
Or how they make confident decisions based on data?
The answer lies in the T-Test, a statistical hypothesis test that allows us to compare means and assess whether the studied groups are distinct.
Table of Contents
Key Points
• The t-test is a widely used statistical method for assessing whether there is a significant difference in means between two groups or samples.
• It is a parametric test that considers the variability within each group when comparing means.
• Three main types of t-tests are Independent (also known as two-sample), Paired (also known as Dependent), and One-Sample T-test; the two-sample t-test is often utilized when you want to test
unequal or var.equal conditions.
• Key assumptions include random sampling, independence of observations, normal distribution, and homogeneity of variances.
• Larger sample sizes enhance accuracy and the ability to detect significant differences.
What is a T-test?
The t-test is a statistical test used to determine if there is any statistically significant difference present in the means of two groups. It is a parametric test that compares means while
considering variability within each group; read more.
It helps researchers analyze the means of the different groups considerably from one another while considering variability within each group. It is extensively used in scientific research, social
sciences, and business analytics.
Different types of t-tests in r
│Type of T-Test │ Description │ Formula │ Use Case │
│Independent or │Compares means of two independent groups to determine if they differ │t = (mean1 - mean2) / sqrt((s1^2 / n1)│Used when comparing two distinct and unrelated groups, e.g., │
│Two-Sample │significantly. Also known as the between-subjects test. │+ (s2^2 / n2)) │assessing the impact of different treatments on separate groups. │
│T-Test │ │ │ │
│Paired or │Compares means of two related samples to assess if they differ │t = (mean of the differences) / │Appropriate for connected observations, like before-and-after │
│Dependent │significantly. Also used for pre-and post-intervention measurements on the│(standard deviation of the differences│measurements on the same subjects, such as in medical studies. │
│T-Test │same individuals. │/sqrt (n)) │ │
│One-Sample │Compares the mean of a single sample to a known or predicted value. │t = (sample mean - hypothesized mean) │Useful when evaluating if a sample mean significantly deviates from│
│T-Test │Determines if the sample mean significantly differs from the expected │/ (sample standard deviation/sqrt (n))│a predetermined value, e.g., comparing a sample mean to a │
│ │value. │ │population mean. │
How to select the appropriate T-test
Paired t-test: When to use it and how it works
The paired t-test is used to compare two related samples. It computes the mean and standard deviation of the differences between the paired observations and then performs a one-sample t-test on the
mean difference.
One-sample / Unpaired: Meaning and usage
The one-sample t-test is a statistical test used to compare a sample mean to a known or expected value. It compares the model mean to the anticipated value to see if the difference is statistically
Paired vs. unpaired: Choosing the appropriate test
A paired t-test is used when observations are connected or related, such as pre- and post-treatment measures. An unpaired t-test is appropriate when the observations are unpaired and independent. The
choice of test depends on the nature of the data and the study approach.
ANOVA vs. t-test: A comparison
ANOVA (Analysis of Variance) and t-tests compare means but in different conditions. ANOVA is used when comparing means across three or more groups or samples. Meanwhile, t-tests are appropriate for
comparing means between two groups or samples.
How to perform t.test in R
1. Define the hypothesis and the research question.
2. Collect data from the two groups or samples of interest.
3. Examine the assumptions of the t-test (normality, independence, and variance homogeneity).
4. Calculate the test statistic (t-value) using the appropriate formula.
5. Determine the degree of freedom and its crucial value.
6. Calculate the p-value using the t-value and degrees of freedom (df).
7. Compare the p-value to the desired significance level (e.g., 0.05) to determine the null hypothesis.
8. Based on the results, make conclusions.
Assumptions and Sample Size Considerations
The following assumptions should be met to perform:
• Random sample: Data should be collected using a random sample method.
• Independence: The observations in the sample should be independent of one another.
• Normality: The population distribution should be normally distributed.
• Variance homogeneity: The variance of the populations under consideration should be the same.
• Larger sample sizes yield more accurate results and improve the result's capacity to detect statistically significant differences.
Assumption of T-test
To check the assumption of the t-test, read this comprehensive article, Parametric Tests in R : Your Guide to Powerful Statistical Analysis.
Perform t-test in R
R is a popular computer language for statistical analysis. R includes several methods and statistical analysis functions and packages. You can program and include these tools in your data analysis
Before We start, Make sure you read the following:
T.test Function
In R, the t.test function performs t-tests for comparing means. It takes two numeric vectors as input, conducting various types of t-tests, such as independent, paired, or one-sample tests. The
function returns test statistics, p-values, and confidence intervals, aiding statistical inference and hypothesis testing;
Load the data set
#T test Using iris data set
data(iris) ## Load the iris dataset
dim(iris) # dimension of the data
head(iris,10) # top ten rows of the data
# Subset the data for the two species you want to compare
# In actual data set have three species for t test we are using only two.
# Subset the data set
setosa <- iris$Sepal.Length[iris$Species == "setosa"]
versicolor <- iris$Sepal.Length[iris$Species == "versicolor"]
Want to explore the iris data set, I suggest you read these articles :
How to perform an Independent t-test in R
The independent t-test compares the means of two groups to determine if they significantly differ. It assesses the impact of different treatments or conditions on unrelated groups, providing
statistical evidence of significant differences in means; Read more.
Null hypothesis (H0): the mean sepal length of setosa species equals the mean sepal length of the versicolor species.
Alternative Hypothesis ( Ha): Setosa species have different mean sepal lengths than versicolor species.
Facing problems in formulating a hypothesis, here you can read a comprehensive article: Hypothesis Test: Step-by-Step Guide for Students & Researchers.
Perform independent t-test in R.
t.test(setosa, versicolor)
Interpretation of t-test results
The independent t-test has a t-value of -10.521 and a degree of freedom (df) of 86.538. The p-value is incredibly low (2.2e-16), much below the commonly used significance level of 0.05. As a result,
the null hypothesis is rejected.
These results support the alternative hypothesis. It implies that the mean sepal length of iris flowers from the species setosa and versicolor differs statistically considerably. Setosa species have
5.006 average sepal lengths, while versicolor species have 5.936 average sepal lengths.
Furthermore, the 95% confidence interval (-0.7542926, -1.1057074) eliminates 0. These values represent a range of plausible values for the actual difference in means. The absence of 0 confirms that
the sepal lengths of the two species differ significantly.
How to perform a Paired t-test
• Null hypothesis ( H0): the mean difference in sepal lengths between setosa and versicolor species equals zero.
• Alternative hypothesis ( Ha), The mean difference in sepal lengths between setosa and versicolor species is greater than zero.
Paired sample t.test in R
# Perform paired t-test
t.test(setosa, versicolor, paired = TRUE)
Interpretation of paired sample t-test
A paired t-test result of -10.146 and a degree of freedom (df) of 49 is obtained. The p-value is 1.242e-13, which is very little. The null hypothesis is rejected because the p-value is less than the
commonly used significance level of 0.05.
It demonstrates that there is substantial evidence to support the alternative theory. It exhibits statistically significant differences in the mean sepal length of iris flowers from the setosa and
versicolor species. The negative mean difference of -0.93 indicates that versicolor species have shorter sepal lengths on average than setosa species.
Furthermore, the 95% confidence interval (CI) (-1.114203, -0.745797) has no zeros. CI provides a range of values for the true mean difference. The absence of 0 confirms the conclusion that the sepal
lengths of the two species differ significantly.
Based on the results, we may say that strong evidence supports the alternative hypothesis. Because results suggest that iris flowers' mean sepal length differs significantly between the setosa and
versicolor species.
How to perform One sample t-test in R
The one-sample t-test evaluates whether the mean of a single sample significantly deviates from a known or predicted value. It assesses if observed data substantially differs from a specified
expectation, aiding researchers in making statistically supported conclusions about the population mean based on a representative sample.
• Null hypothesis ( H0): the population's mean sepal length is 5.5.
• Alternative hypothesis ( Ha): The mean sepal length of the population is not 5.5.
Perform one Sample t-test
# Perfrom one sample t test
t.test(iris$Sepal.Length, mu = 5.5)
Interpretation of one Sample t-test
The test calculations are 5.078 and a degree of freedom (df) of 149. The p-value is 1.123e-06, which is very small. We reject the null hypothesis because the p-value is less than the commonly used
significance level of 0.05.
The results significantly support the alternative hypothesis. It means that there is a statistically significant difference between the mean sepal length of the population. The hypothesized value is
5.5, and the sample's mean sepal length (mean of x) is 5.843333.
Furthermore, the 95% confidence interval (5.709732, 5.976934) contains no 5.5. The absence of 5.5 validates the conclusion that there is a substantial gap between the mean and postulated sepal
What is the Significance level?
The significance level (alpha) is a criterion for determining statistical significance. When the null hypothesis is true, it can be rejected (Type I error).
The most frequent significance level is 0.05. It indicates that the null hypothesis is rejected if the p-value is less than 0.05. The significance threshold, however, can be adjusted depending on the
unique study environment and desired level of assurance.
Problems with solutions
Consider the following problems and solutions to gain a better understanding:
Problem: Is there a big difference in the average earnings of employees at two different companies?
Solution: Conduct an unbiased t-test on employee earnings data from both firms.
Problem: Does a new teaching method considerably improve students' test scores?
Solution: Use a paired t-test to compare students' test scores before and after implementing the new teaching method.
Finally, the t-test is a useful statistical tool for comparing mean values across groups or samples. It allows researchers to assess the significance of observed differences and form conclusions
about the populations they represent. If researchers understand the types of tests, formulas, assumptions, and applications. They can reliably assess data and derive relevant conclusions.
The t-test is helpful in a wide range of disciplines, including psychology, biology, medicine, economics, and social sciences. It helps researchers evaluate the effectiveness of interventions,
compare group means, and investigate research problems. However, to achieve reliable and valid findings, it is necessary to ensure that the t-test assumptions are met.
Frequently Asked Questions (FAQs)
Why is it called a t-test and used for?
To determine whether or not there is a significant difference between the means of two groups or samples. When the data have a normal distribution, and the populations' variances arither the same or
different, it is applied relatively frequently in the data analysis.
What are the three main types of t-tests and their differences?
Three main types of tests are Independent, Paired, and one sample test.
What is the difference between a t-test and an ANOVA?
While both tests are used to compare means, and the differences are:T-test It reaches means between two groups or samples. ANOVA: It compares means between multiple groups or samples simultaneously.
It assesses if there are any significant differences among the means of the groups but does not identify which specific group means differ from each other.
What does the t-test value tell you, and what are the T score, t-value, and p-value?
A statistic, the t-test value, is used to evaluate the degree of disparity between the means of two groups concerning the amount of variance within the groups.
What is the minimum sample size for a t-test, and is parametric or nonparametric?
The required power of the test, the effect size, and the significance level are some criteria that can influence the minimum sample size that should be used. Generally, a bigger sample size yields
more accurate results. The t-test is a parametric test, which means it makes specific assumptions about the population distribution, such as that it is normal and the variances are all the same.
How do you analyze t-test results and interpret them?
When analyzing results, looking at the determined t-value and the corresponding p-value is common practice. If the p-value is lower than the significance level you have set (for example, 0.05), this
indicates a statistically significant difference between the two means. You would conclude that there is evidence of a substantial difference and reject the hypothesis that there is no change.
However, you cannot reject the null hypothesis if the p-value exceeds the significance level. This indicates that there is not enough evidence to conclude that there is a significant difference.
Why do we reject the null hypothesis in a t-test, and how do we do it?
When the p-value is lower than the significance level that we've established, we disregard the null hypothesis as invalid. This suggests that the discrepancy in the means could not have resulted from
pure random chance alone; instead, it is highly implausible. There is adequate evidence to suggest a significant difference between the groups after comparing the p-value to the significance level
(for example, 0.05). This allows us to conclude that the null hypothesis should be rejected.
What is a good score in a t-test, and what does 95% represent for a T score?
The setting and the particular research topic both play a role in determining what constitutes a good score on a t-test. In most cases, a meaningful difference between the groups being compared can
be inferred from a significant t-score. A T score with a confidence level of 95% indicates that if the experiment were to be Using a car dataset, we can illustrate the application of the two sample
t-test, assuming unequal variances, where welch or pooled variance solutions could be considered.ried out more than once, we would anticipate that 95% of the calculated T scores would fall within the
critical region, resulting in the rejection of the null hypothesis.
Do you need help with a data analysis project? Let me assist you! With a PhD and ten years of experience, I specialize in solving data analysis challenges using R and other advanced tools. Reach out
to me for personalized solutions tailored to your needs.
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Braided Rota-Baxter algebras, quantum quasi-shuffle algebras and braided dendriform algebras
Rota-Baxter algebras and the closely related dendriform algebras have important physics applications, especially to renormalization of quantum field theory. Braided structures provide effective ways
of quantization such as for quantum groups. Continuing recent study relating the two structures, this paper considers Rota-Baxter algebras and dendriform algebras in the braided contexts. Applying
the quantum shuffle and quantum quasi-shuffle products, we construct free objects in the categories of braided Rota-Baxter algebras and braided dendriform algebras, under the commutativity condition.
We further generalize the notion of dendriform Hopf algebra to the braided context and show that quantum shuffle algebra gives a braided dendriform Hopf algebra. Enveloping braided commutative
Rota-Baxter algebras of braided commutative dendriform algebras are obtained.
All Science Journal Classification (ASJC) codes
• Algebra and Number Theory
• Applied Mathematics
• Quantum shuffle algebra
• Rota-Baxter algebra
• Yang-Baxter equation
• braided Rota-Baxter algebra
• braided dendriform Hopf algebra
• braided dendriform algebra
• dendriform algebra
• quantum quasi-shuffle algebra
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What is called zero error?
Zero error is defined as the condition where a measuring instrument records a reading when no reading is required. In case of Vernier calipers it occurs when a zero on the main scale does not
coincide with a zero on Vernier scale it is called zero error for Vernier.
What is a zero error in physics?
zero error Any indication that a measuring system gives a false reading when the true value of a measured quantity is zero, eg the needle on an ammeter failing to return to zero when no current
flows. A zero error may result in a systematic uncertainty.
What is a zero error in physics ammeter?
When a circuit is not connected, the ammeter or voltmeter’s pointer is exactly at zero, and the instrument is said to have no zero error. The zero error is negative if the pointer is above the zero
mark. Subtract the number of divisions if it is above the zero point from the reading.
What is zero reading in physics?
In cases where the reading falls exactly on a scale division, the estimated figure would be 0; e.g. 48.50 cm, indicating that you know the reading more accurately than 48.5 cm.
What is least count and zero error?
Least count gives the resolution of the instrument. Ammeter or Voltmeter is said to have zero error if their pointer doesn’t read zero when it is supposed to read i.e. when not connected to the
What is zero error and correction?
Zero correction is the error that does not return zero when the instrument is kept at the rest. Also, say that zero error is the calculation that removes the zero error to get an error-free result.
How do you write a zero error?
Zero error = – (10 – 6) ร L.C. Correction: To get correct measurement with vernier callipers having a zero error, the zero error with its proper sign is always subtracted from the observed reading.
What causes a zero error?
Zero errors are caused by faulty equipment that doesn’t reset to zero properly. Check before you start measuring that the measuring instruments read zero for zero input. A zero error would affect
every reading you take.
How do you find the zero error in physics?
What are the types of zero errors?
There are two types of zero error when it indicates behind the zero position then negative zero error and opposite when indicated on the other side of zero.
What is the error of Vernier?
While measuring the zero error of the vernier we see that zero of vernier scale is to the right of the zero marking of the main scale with 6th vernier division coinciding with main scale divisions.
The actual reading for length measurement is 4.3 cm with 2nd vernier division coinciding with main scale graduation.
What is the zero error of voltmeter?
When a circuit is not connected, the pointer of the ammeter or voltmeter is exactly at zero, thus said to be the instrument has zero error. If the pointer is above the zero mark, the zero error is
Why do we use zero correction in physics?
Zero correction is used in an instrument for removing the zero error. Stay tuned with BYJU’S to learn more about other concepts such as the vernier caliper. Was this answer helpful?
What is zero error as shown in the figure?
Since, least count is the smallest and accurate magnitude of any measuring physical quantity and zero error means when the zero of main scale and measuring scale like Vernier calliper do not coincide
and there is lag of some value is called zero error.
What is random error in physics?
In physics, there are two kinds of errors: Random Errors: When repeated measurements of the quantity yield different results under the same conditions, this is referred to as random error. This
random error occurs for unknown reasons.
What is the SI unit of least count?
1 mm is the lowest main scale division according to the SI units.
What is the unit of least count?
The least count of an instrument is the smallest measurement that can be taken accurately with it. The least count of meter scale is 1mm, and that of the ammeter shown below is 2 ampere.
What is least count error definition?
Least Count Error The smallest value that can be measured by the measuring instrument is called its least count. Measured values are good only up to this value. The least count error is the error
associated with the resolution of the instrument.
What is the value of zero error?
Its zero correction is (take L.C. = 0.01 cm) A.
What is called error correction?
Error correction is the process of detecting errors in transmitted messages and reconstructing the original error-free data. Error correction ensures that corrected and error-free messages are
obtained at the receiver side.
What is zero error and parallax error?
Zero Error is due to non zero reading. When the actual reading should be zero. This error is usually due to the fact that the pointer of the instrument doesn’t return to zero when it is not being
used. Parallax error is the random. error due to incorrect positioning of the eye when taking the reading of a measurement.
Is zero error is positive or negative?
Zero is neither positive nor negative.
Why is zero error positive or negative?
When the zeroth division on the vernier scale appears on the right side of zero of the main scale , the error is called positive zero error. Was this answer helpful?
Where would a zero error occur?
A zero error on a caliper can be positive or negative in direction. A positive zero error occurs when the caliper jaws are closed, but the readout has some positive value, whereas a negative zero
error occurs when the caliper jaws are closed, but the readout has some negative value.
What are the 3 types of errors in science?
Three general types of errors occur in lab measurements: random error, systematic error, and gross errors. Random (or indeterminate) errors are caused by uncontrollable fluctuations in variables that
affect experimental results. | {"url":"https://physics-network.org/what-is-called-zero-error/","timestamp":"2024-11-05T19:20:33Z","content_type":"text/html","content_length":"306156","record_id":"<urn:uuid:09ac63c6-7fae-46c4-a513-490893b2b7bd>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00227.warc.gz"} |
How to do logarithmsHow to do logarithms 🚩 Logarithms examples of solutions 🚩 Math.
You will need
• Calculator, Handbook of elementary mathematics
First you need to clearly grasp the essence of logarithms. The logarithm is the inverse operation of exponentiation. Repeat the theme "exponentiation of natural numbers". It is especially important
to repeat the properties of the degree (
, private, bachelor degree).
Every logarithm consists of two numeric parts. The lower index is called a base. The upper index is the number that happens during the construction of the base to the power equal to the logarithm.
Have irrational logarithms, calculate that is not necessary. If the logarithm gives the response of a finite natural number, it must be calculated.
When solving examples with logarithms you should always be aware of the limitations of the region of the allowed values. The base is always greater than 0 and not equal to one. There are also special
types of logarithms lg (common logarithm) and ln (natural logarithm). Common logarithm has base 10 and the natural number e (approximately equal to 2.7).
For the solution of the logarithmic examples to learn the basic properties of logarithms. In addition to the basic logarithmic identities need to know the formula of sum and difference of logarithms.
Table of basic logarithmic properties are shown in Fig.
Using the properties of logarithms, we can solve any logarithmic example. We just need to bring all the logarithms to one base, then to reduce them to a single logarithm, which is easy to calculate
using a calculator. | {"url":"https://eng.kakprosto.ru/how-23038-how-to-do-logarithms","timestamp":"2024-11-05T22:32:44Z","content_type":"text/html","content_length":"35155","record_id":"<urn:uuid:6d514a82-bab6-408a-9537-fc60493d5673>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00541.warc.gz"} |
Excel Functions by category
Functions alphabetically | Functions by category
Functions are sorted by category, click a category name to find Excel functions in that category.
Array manipulation
DROP function
removes a given number of rows or columns from a 2D cell range or array.
EXPAND function
increases a cell range or array by a specified number of columns and rows.
HSTACK function
combines cell ranges or arrays. Joins data to the first blank cell to the right of a cell range or array (horizontal stacking)
TAKE function
returns a given number of rows or columns from a 2D cell range or array.
TOCOL function
rearranges values in 2D cell ranges to a single column.
TOROW function
rearranges values from a 2D cell range to a single row.
VSTACK function
combines cell ranges or arrays. Joins data to the first blank cell at the bottom of a cell range or array (vertical stacking)
WRAPCOLS function
rearranges values from a single row to a 2D cell range based on a given number of values per column.
WRAPROWS function
rearranges values from a single row to a 2D cell range based on a given number of values per column.
BETADIST function
calculates the beta distribution, it represents outcomes in the form of probabilities.
CHIINV function
calculates the inverse probability of the chi-squared distribution.
CHITEST function
calculates the test for independence, the value returned from the chi-squared statistical distribution and the correct degrees of freedom. Use this function to check if hypothesized results are
CONFIDENCE function
calculates the confidence interval for a population mean, using a normal distribution.
CRITBINOM function
calculates the minimum value for which the binomial distribution is equal to or greater than a given threshold value.
EXPONDIST function
calculates the exponential distribution representing an outcome in the form of probability.
FDIST function
calculates the F probability of the right-tailed distribution for two tests.
FLOOR function
rounds a number down, toward zero, to the nearest multiple of significance.
FORECAST function
calculates a value based on existing x and y values using linear regression. Use this function to predict linear trends.
FTEST function
calculates the value from an F-test. The value shows if the variances from two data sets are not significantly different.
GAMMADIST function
calculates the gamma often used in queuing analysis (probability statistics).
LOGNORMDIST function
calculates the cumulative lognormal distribution of argument x, based on a normally distributed ln(x) with the arguments of mean and standard_dev.
MODE function
calculates the most frequent number in a cell range.
QUARTILE function
returns the quartile of a data set, use the QUARTILE function to divide data into groups.
RANK function
calculates the rank of a specific number compared to a list of numbers.
STDEV function
calculates the standard deviation of a group of values.
DAVERAGE function
calculates an average based on values in a list or database that meet specific conditions.
DCOUNT function
counts cells containing numbers and that meet a condition or criteria.
DCOUNTA function
counts nonempty cells in a column you specify, in a database where records also meet a condition or criteria.
DGET function
fetches a value from a column in a database whose records meet a condition or criteria.
DMAX function
extracts the maximum number from a column in a database whose records match a condition or criteria.
DMIN function
extracts the smallest number from a column in a database whose records match a condition or criteria.
DPRODUCT function
multiplies numbers that match a condition or criteria in a database.
DSTDEV function
calculates an estimation of the standard deviation based on a sample of a population. The function also allows you to specify criteria applied to a database.
DSTDEVP function
calculates the standard deviation based on a population. The function also allows you to specify criteria applied to a database.
DSUM function
adds numbers in a database/list that meets a condition or criteria.
DVARP function
returns the variance of an entire population. The numbers are in a column of records in a dataset or database that meets a given condition or criteria.
Date and Time
DATE function
returns a number that acts as a date in the Excel environment.
DATEDIF function
returns the number of days, months, or years between two dates. The DATEDIF function exists in order to ensure compatibility with Louts 1-2-3.
DATEVALUE function
returns an Excel date value (serial number) based on a date stored as text.
DAY function
extracts the day as a number from an Excel date.
EDATE function
returns a date determined by a start date and a number representing how many months.
EOMONTH function
returns an Excel date for the last day of a given month using a number and a start date.
HOUR function
returns an integer representing the hour of an Excel time value.
MINUTE function
returns a whole number representing the minute based on an Excel time value. The returned number is ranging from 0 to 59.
NETWORKDAYS function
returns the number of working days between two dates, excluding weekends. It also allows you to ignore a list of holiday dates that you can specify.
SECOND function
returns an integer representing the second based on an Excel time value
TIME function
returns a decimal value between 0 (zero) representing 12:00:00 AM and 0.99988426 representing 11:59:59 P.M.
TODAY function
returns the Excel date (serial number) of the current date.
WEEKNUM function
calculates a given date's week number based on a return_type parameter that determines which day the week begins.
WORKDAY function
returns a date based on a start date and a given number of working days (nonweekend and nonholidays).
YEAR function
converts a date to a number representing the year in the date.
YEARFRAC function
returns the fraction of the year based on the number of whole days between a start date and an end date.
BITLSHIFT function
calculates a number whose binary representation is shifted left by a specified number of bits.
BITRSHIFT function
calculates the number where the binary equivalent is shifted right by a specified number of bits and then converted back to a number.
BITXOR function
calculates a decimal number that is a result of a bitwise comparison "XOR" of two numbers.
IMABS function
calculates the absolute value (modulus) of a complex number in x + yi or x + yj text format.
IMAGINARY function
calculates the imaginary value of a complex number in x + yi or x + yj text format.
IMARGUMENT function
calculates theta θ which is an angle displayed in radians based on complex numbers in rectangular form.
IMCONJUGATE function
calculates the complex conjugate of a complex number in x + yi or x + yj text format.
IMCOS function
calculates the cosine of a complex number in x + yi or x + yj text format.
IMCOSH function
calculates the hyperbole cosine of a complex number in x + yi or x + yj text format.
IMCOT function
calculates the cotangent of a complex number in x + yi or x + yj text format.
IMCSC function
calculates the cosecant of a complex number in x + yi or x + yj text format.
IMCSCH function
calculates the hyperbolic cosecant of a complex number in x + yi or x + yj text format.
IMDIV function
calculates the quotient of two complex numbers in x + yi or x + yj text format.
IMEXP function
calculates the exponential of a complex number in x + yi or x + yj text format.
IMLN function
calculates the natural logarithm of a complex number in x + yi or x + yj text format.
IMLOG10 function
calculates the base 10 logarithm of a complex number in x + yi or x + yj text format.
IMLOG2 function
calculates the base 2 logarithm of a complex number in x + yi or x + yj text format.
IMPOWER function
calculates a complex number raised to a given power in x + yi or x + yj text format.
IMPRODUCT function
calculates the product of complex numbers in x + yi or x + yj text format.
IMREAL function
calculates the real coefficient of a complex number in x + yi or x + yj text format.
IMSEC function
calculates the secant of a complex number in x + yi or x + yj text format.
IMSECH function
calculates the hyperbolic secant of a complex number in x + yi or x + yj text format.
IMSIN function
calculates the sine of a complex number in x + yi or x + yj text format.
IMSINH function
calculates the hyperbolic sine of a complex number in x + yi or x + yj text format.
IMSQRT function
calculates the square root of a complex number in x + yi or x + yj text format.
IMSUB function
calculates the difference between two complex numbers in x + yi or x + yj text format.
IMSUM function
calculates the total of two or more complex numbers in x + yi or x + yj text format.
IMTAN function
calculates the tangent of a complex number in x + yi or x + yj text format.
ACCRINT function
calculates the accrued interest for a security that pays periodic interest.
ACCRINTM function
calculates the accrued interest for a security that pays interest at maturity.
CUMPRINC function
calculates the accumulated principal based on a start and end period on a loan.
DB function
calculates the depreciation of an asset for a given period using the fixed-declining balance method.
DDB function
calculates the depreciation of an asset for a given period using the double-declining balance method or based on user input.
EFFECT function
calculates the effective annual interest rate, given the nominal annual interest rate and the number of compounding periods per year.
FV function
returns the future value of an investment based on a constant interest rate.
IPMT function
calculates the interest payment for a specific period for an investment based on repeated constant payments and a constant interest rate.
ISPMT function
calculates the interest paid during a specific period of an investment.
NOMINAL function
calculates the nominal annual interest rate based on the effective rate and the number of compounding periods per year.
NPER function
calculates the number of periods for an investment based on periodic, constant payments and a fixed interest rate.
PDURATION function
calculates how many periods required by an investment to reach a given amount based on a rate in percentage.
PMT function
returns the payment needed for borrowing a fixed sum of money based on constant payments (annuity) and interest rate.
PPMT function
calculates the principal payment for a specific period for an investment based on repeated constant payments and a constant interest rate.
PRICEMAT function
calculates the price per $100 nominal value of a bond that pays interest at maturity.
PV function
calculates the net present value for an investment or loan.
RATE function
returns the interest rate per period of an annuity.
RRI function
calculates the growth of an investment in percent per period.
SYD function
calculates the yearly asset depreciation of a given year.
VDB function
calculates the depreciation of an asset for a given period using the double-declining balance method or based on user input, you may use partial periods in this function.
XNPV function
calculates net present value for cash flows that may or may not be periodic
YIELD function
calculates the yield for a security that pays interest. The YIELD function is designed to calculate the bond yield.
CELL function
gets information about the formatting, location, or the contents of a cell.
INFO function
returns information about the current operating environment, file path, number of active worksheets, Excel version etc.
ISBLANK function
returns TRUE if the argument is an empty cell, returns FALSE if not.
ISERR function
returns TRUE if a cell returns an error, except error value #N/A.
ISFORMULA function
returns TRUE if a cell contains a formula, FALSE if text, number or boolean value.
ISLOGICAL function
returns TRUE if value is boolean. A boolean value is either TRUE or FALSE.
ISODD function
returns TRUE if a cell contains an odd number, FALSE if even number.
N function
returns a value converted into a number.
NA function
returns the error value #N/A meaning "value is not available".
TYPE function
use TYPE to find out what type of data is returned by a function or formula.
AND function
perform a logical test in each argument and if all arguments return TRUE the AND function returns TRUE.
BYCOL function
passes all values in a column based on an array to a LAMBDA function, the LAMBDA function calculates new values based on a formula you specify.
BYROW function
puts values from an array into a LAMBDA function row-wise.
IF function
returns one value if the logical test is TRUE and another value if the logical test is FALSE.
IFERROR function
if the value argument returns an error, the value_if_error argument is used. If the value argument does NOT return an error, the IFERROR function returns the value argument.
IFNA function
handles #N/A errors only, it returns a specific value if the formula returns a #N/A error.
IFS function
checks whether one or more conditions are met and returns a value that corresponds to the first TRUE condition.
MAKEARRAY function
returns an array with a specific number of rows and columns calculated by applying a LAMBDA function.
MAP function
passes all values in an array to a LAMBDA function, the LAMBDA function calculates new values based on a formula you specify. It then returns an array with the same size as the original array.
NOT function
returns the boolean opposite to the given argument.
OR function
evaluates a logical expression in each argument and if at least one argument returns TRUE the OR function returns TRUE. If all arguments return FALSE the OR function also returns FALSE.
REDUCE function
shrinks an array to an accumulated value, a LAMBDA function is needed to properly accumulate each value in order to return a total.
SCAN function
passes values in an array to a LAMBDA function, the LAMBDA function calculates new values based on a formula you specify. It then returns an array with the same size as the original array using an
accumulator parameter.
SWITCH function
returns a given value determined by an expression and a list of values.
XOR function
calculates the logical exclusive OR meaning if at least one of the arguments evaluates to TRUE then the XOR returns TRUE. All arguments must be evaluated to FALSE for the XOR function to return
Lookup and reference
ADDRESS function
returns the address of a specific cell, you need to provide a row and column number.
AREAS function
returns the number of cell ranges and single cells in a reference.
COLUMN function
returns the column number of the top-left cell of a cell reference.
HLOOKUP function
searches the top row in a data range for a value and return another value on the same column in a row you specify.
INDEX function
returns a value or reference from a cell range or array, you specify which value based on a row and column number.
INDIRECT function
returns the cell reference based on a text string and shows the content of that cell reference.
LOOKUP function
find a value in a cell range and return a corresponding value on the same row.
MAKEARRAY function
returns an array with a specific number of rows and columns calculated by applying a LAMBDA function.
MATCH function
returns the relative position of an item in an array that matches a specified value in a specific order.
OFFSET function
returns a reference to a range that is a given number of rows and columns from a given reference.
SORTBY function
sorts a cell range or array based on values in a corresponding range or array.
VLOOKUP function
lets you search the leftmost column for a value and return another value on the same row in a column you specify.
XLOOKUP function
search one column for a given value, and return a corresponding value in another column from the same row.
XMATCH function
searches for an item in an array or cell range and returns the relative position.
Math and trigonometry
AVEDEV function
calculates the average of the absolute deviations of data points from their mean.
AVERAGEA function
returns the average of a group of values. Text and boolean value FALSE evaluates to 0. TRUE to 1.
AVERAGEIF function
returns the average of cell values that are valid for a given condition.
AVERAGEIFS function
returns the average of cell values that evaluates to TRUE for multiple criteria.
BINOM.INV function
calculates the minimum value for which the binomial distribution is equal to or greater than a given threshold value.
CHISQ.DIST function
calculates the probability of the chi-squared distribution, cumulative distribution or probability density.
CHISQ.INV function
calculates the inverse of the left-tailed probability of the chi-squared distribution.
CHISQ.INV.RT function
calculates the inverse of the right-tailed probability of the chi-squared distribution.
CHISQ.TEST function
calculates the test for independence, the value returned from the chi-squared statistical distribution and the correct degrees of freedom. Use this function to check if hypothesized results are
CONFIDENCE.T function
calculates the confidence range for a population mean using a Student's t distribution.
COUNTIFS function
calculates the number of cells across multiple ranges that equals all given conditions.
COVARIANCE.P function
calculates the covariance meaning the average of the products of deviations for each pair in two different datasets.
COVARIANCE.S function
calculates the sample covariance meaning the average of the products of deviations for each pair in two different datasets.
EXPON.DIST function
calculates the exponential distribution representing an outcome in the form of probability.
FREQUENCY function
calculates how often values occur within a range of values and then returns a vertical array of numbers.
GAMMA.DIST function
calculates the gamma often used in queuing analysis (probability statistics) that may have a skewed distribution.
INTERCEPT function
returns a value representing the y-value where a line intersects the y-axis.
LARGE function
calculates the k-th largest value from an array of numbers.
LINEST function
returns an array of values representing the parameters of a straight line based on the "least squares" method.
LOGEST function
returns an array of values representing the parameters of an exponential curve that fits your data, based on the "least squares" method.
LOGNORM.DIST function
calculates the lognormal distribution of argument x, based on a normally distributed ln(x) with the arguments of mean and standard_dev.
MAXIFS function
calculates the highest value based on a condition or criteria.
MEDIAN function
calculates the median based on a group of numbers. The median is the middle number of a group of numbers.
MINA function
returns the smallest number. Text values and blanks are ignored, boolean value TRUE evaluates to 1 and FALSE to 0 (zero).
MINIFS function
calculates the smallest value based on a given set of criteria.
MODE.MULT function
returns the most frequent number in a cell range. It will return multiple numbers if they are equally frequent.
NORM.DIST function
calculates the normal distribution for a given mean and standard deviation.
NORM.INV function
calculates the inverse of the normal cumulative distribution for a given mean and standard deviation.
PERMUT function
returns the number of permutations for a set of elements that can be selected from a larger number of elements.
PERMUTATIONA function
returns the number of permutations for a specific number of elements that can be selected from a larger group of elements.
PHI function
calculates a number of the density function for a standard normal distribution.
PROB function
calculates the probability that values in a range are between a given lower and upper limit.
QUARTILE.INC function
returns the quartile of a data set, based on percentile values from 0..1, inclusive.
RANK.EQ function
calculates the rank of a number in a list of numbers, based on its position if the list were sorted.
SLOPE function
calculates the slope of the linear regression line through coordinates.
SMALL function
returns the k-th smallest value from a group of numbers.
STANDARDIZE function
calculates a normalized value from a distribution characterized by mean and standard_dev.
STDEV.S function
returns standard deviation based on a sample of the entire population.
STDEVPA function
returns the standard deviation based on the entire population, including text and logical values.
VAR.P function
returns the variance based on the entire population. The function ignores logical and text values.
VAR.S function
the VAR.S function tries to estimate the variance based on a sample of the population. The function ignores logical and text values.
CHAR function
converts a number to the corresponding ANSI character determined by your computers character set.
CLEAN function
deletes the first 32 nonprinting characters in 7-bit ASCII code in your argument.
CODE function
returns the corresponding number for the first character based on your computers character set. (PC- ANSI)
EXACT function
checks if two values are precisely the same, it returns TRUE or FALSE. The EXACT function also considers upper case and lower case letters.
FIND function
returns the position of a specific string in another string, reading left to right. Note, the FIND function is case-sensitive.
LEFT function
extracts a specific number of characters always starting from the left.
LEN function
returns the number of characters in a cell value.
MID function
returns a substring from a string based on the starting position and the number of characters you want to extract.
REPLACE function
substitutes a part of a text string based on the number of characters and length with a text string you provide.
RIGHT function
extracts a specific number of characters always starting from the right.
SEARCH function
returns the number of the character at which a specific character or text string is found reading left to right (not case-sensitive)
T function
returns a text value if the argument is a text value.
TEXT function
converts a value to text in a specific number format.
TRIM function
deletes all blanks or space characters except single blanks between words in a cell value.
VALUE function
converts a text string that represents a number to a number.
VALUETOTEXT function
returns a value in text form. Text values are unmodified and non-text values are converted to text.
Excel Function Basics
The sidebar (or click the hamburger icon if you are on mobile) shows a list of Excel functions based on category. Press CTRL + F in your web browser to quickly search the sidebar for the function you
are looking for.
If you rather prefer an alphabetically sorted list then this link takes you to Excel functions sorted from A to Z.
Table of Contents
How to enter a function in Excel
You are now shown the arguments for the chosen function.
Enter the values for each argument, some arguments display a list of options if the function requires you to enter one of many predetermined parameters, see image below.
Use the arrow keys to select an argument and press TAB key to choose the selected argument. Type a closing parentheses when all arguments have been entered.
Lastly press ENTER or CTRL + SHIFT + ENTER if the function requires you to.
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How do I insert a function?
1. Select a cell.
2. Click the "fx" button next to the formula bar, a dialog box appears.
3. The dialog box allows you to search for a function. You can also browse functions based on category or recently used functions.
4. The dialog box also specifies the arguments of the selected function which is great if you are not familiar with the selected function.
5. At the very bottom of the dialog box is a link that takes you to Microsofts support page based on the function you selected.
6. Click the OK button.
This takes you to another dialog box that guides you through each argument in the selected function.
The dialog box describes each argument and lets you select cell(s) in your worksheet in order to complete the function.
Click the OK button when all arguments are specified.
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How many arguments in a function can I use?
It depends on the function, some lets you use up to 255 arguments and others have none. For example, the SUM function allows you to have up to 255 arguments in one function where as the TODAY
function has no arguments.
An argument may contain a cell reference to a single cell or a cell reference to multiple cells, constants, arrays or structured references.
What are the most used functions of Excel?
Excel provides you functions to perform mathematical, statistical and financial calculations.
The most used functions are probably:
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What are the most useful functions?
It depends who ask, users may have different goals. In general, if you have a task that is tedious and time-consuming there is a great chance there is a feature or function that can help you out.
Start by doing a internet search on what you want to accomplish in Excel, most likely there are others that have had the same problem/question.
Many consider the VLOOKUP function or INDEX - MATCH functions to be immensely powerful. They allow you to do lookups and return another value on the same row as the matching value.
In fact, there are som many functions that most problems you encounter can be solved. Excel lets you also, in most cases, combine several functions in order to build a formula that is exactly what
you are looking for.
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How many functions are there in Excel?
It depends on your Excel version, older versions have fewer functions. There are at least 467 different functions in the latest version.
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Are the functions categorized?
Yes, there are at least 12 different categories.
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What are the latest Excel functions?
The following six functions are the latest in the Office 365 subscription:
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What functions are outdated?
Some functions have been replaced by newer better functions, you can find them in the Compatibility category. They are still working, however, they may be removed in future updates. I recommend that
you replace the outdated functions with their newer equivalents.
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What is an Excel formula?
A formula calculates a value based on the function or functions used. It always begins with a equal sign so Excel can distinguish it from a text or numerical value.
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What is an array formula?
In general, an array formula calculates multiple values simultaneously and sometimes but not always return multiple values.
To enter an array formula, type the formula in a cell then press and hold CTRL + SHIFT simultaneously, now press Enter once. Release all keys.
The formula bar now shows the formula with a beginning and ending curly bracket telling you that you entered the formula successfully. Don't enter the curly brackets yourself.
This article demonstrates this in greater detail: How to enter an array formula
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What functions are you required to enter as an array formula?
In general, functions that return more than one value requires you to enter them as an array formula. You must enter the following functions as an array formula in order to make it work properly.
You can also use regular functions in an array formula and when you start doing that you are beginning to discover all the really powerful stuff that is possible with Excel.
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What is a volatile function?
A volatile function is a function that is recalculated every time the worksheet is calculated. What this means is that it may slow down your workbook considerably.
For example, the TODAY function returns the date each time the worksheet is recalculated to make sure the current day is displayed. This is not cpu intensive if entered in a single cell, however,
hundreds or thousands of cells then you may notice a difference.
The following functions are volatile:
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How do I troubleshoot a function/formula?
Excel has a built-in feature that allows you to see each step in the formula calculation, this makes it easier to see which function is giving you trouble.
1. Select the cell containing the formula you want to troubleshoot.
2. Go to tab "Formulas" on the ribbon.
3. Click "Evaluate Formula" button, a dialog box appears.
4. Click the "Evaluate" button to see each calculation step.
You can also right-click on the cell to see a context menu. This menu allows you to investigate the error further, you can see a descriptive text of the error message.
The second line allows you to see the support page of this error, the third line opens the "Evaluate formula" tool that I demonstrated above.
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What is a cell reference?
A cell reference is an address to a specific cell in your workbook. Each column is labeled from A to XFD and rows are numbered from 1 to 1048576, a cell reference contains the column letter and the
row number like coordinates.
Example, here is a cell reference: Sheet1!G34. Sheet1 is the name of the worksheet the cell is located on. G is the column which is the seventh column from the left, 34 is the row number.
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What is an array?
Most functions lets you enter arrays in one or more arguments. An array contains multiple values with a comma and/or semicolon as delimiting characters.
The array begins with a curly bracket { and ends with a curly bracket } Here is an example of an array: {"One", 2, "Three"}. Text values must begin an end with double quotes or Excel thinks they are
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Square Inch to Square Centimeter Converter (in² to cm²)
Square Inch to Square Centimeter Converter
“Expand Your Measurements: Convert Square Inches to Square Centimeters Instantly.”
How to Convert Square Inches to Square Centimeters?
To convert area from Square Inches (in²) to Square Centimeters (cm²), use the formula:
Square Centimeters (cm²) = Square Inches (in²) × 6.4516
This formula is based on the conversion factor where 1 square inch equals approximately 6.4516 square centimeters.
Steps to Convert Square Inches to Square Centimeters:
To convert an area from Square Inches to Square Centimeters:
1. Multiply the Square Inches value by 6.4516.
Example Conversion:
To convert 10 Square Inches (in²) to Square Centimeters (cm²):
10 in² × 6.4516 = 64.516 square centimeters
So, 10 Square Inches is equal to approximately 64.516 Square Centimeters.
Square Inch to Square Centimeter Conversion Formula
To convert square inches to square centimeters, you can use the following formula:
Square centimeters = Square inches × 6.4516
Example of Square Inch to Square Centimeter Conversion
Example 1: 10 Square Inch to Square Centimeter Conversion
For example, if you have 10 square inches, the conversion to square centimeters would be:
10 square inches = 10 × 6.4516 square centimeters = 64.516 square centimeters
Example 2: 1 Square Inch to Square Centimeter Conversion
For example, if you have 1 square inch, the conversion to square centimeters would be:
1 square inch = 1 × 6.4516 square centimeters = 6.4516 square centimeters
Example 3: 283 Square Inch to Square Centimeter Conversion
For example, if you have 283 square inches, the conversion to square centimeters would be:
283 square inches = 283 × 6.4516 square centimeters = 1825.4848 square centimeters
Square Inch to Square Centimeter Conversion Table
Here’s a conversion table for square inches to square centimeters for the first 20 entries:
Square Inches Square Centimeters
1 6.4516
5 32.258
10 64.516
15 96.774
20 129.032
25 161.29
30 193.548
35 225.806
40 258.064
45 290.322
50 322.58
55 354.838
60 387.096
65 419.354
70 451.612
75 483.87
80 516.128
85 548.386
90 580.644
95 612.902
100 645.16
Square Centimeter to Square Inch Converter FAQs
How do I convert square inches to square centimeters?
To convert square inches to square centimeters, multiply the number of square inches by 6.4516. This is because 1 square inch equals approximately 6.4516 square centimeters.
Why is the conversion factor for square inches to square centimeters approximately 6.4516?
The conversion factor is derived from the relationship between inches and centimeters. Since 1 inch equals approximately 2.54 centimeters, squaring this value (2.54 × 2.54) gives approximately 6.4516
square centimeters per square inch.
Can I use an online calculator to convert square inches to square centimeters?
Yes, there are many online calculators available where you can input the value in square inches, and it will automatically convert it to square centimeters. These tools are convenient for quick
Is the conversion from square inches to square centimeters consistent across all contexts?
Yes, the conversion factor of approximately 6.4516 square centimeters per square inch remains the same in all contexts.
How many square centimeters are in 10 square inches?
To convert 10 square inches to square centimeters, multiply by approximately 6.4516. The result is approximately 64.516 square centimeters.
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Square inch to square centimeter converter online, 1 square inch to inch, 1 square to cm, 1 square inch to mm, 283 square inches to square centimeters, Square inch to square m, Sq cm to sq ft, Square
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Long Division Calculator (2024)
home / math / long division calculator
Division is one of the basic arithmetic operations, the others being multiplication (the inverse of division), addition, and subtraction. The arithmetic operations are ways that numbers can be
combined in order to make new numbers. Division can be thought of as the number of times a given number goes into another number. For example, 2 goes into 8 4 times, so 8 divided by 4 equals 2.
Division can be denoted in a few different ways. Using the example above:
8 ÷ 4 = 2
8/4 = 2
In order to more effectively discuss division, it is important to understand the different parts of a division problem.
Components of division
Generally, a division problem has three main parts: the dividend, divisor, and quotient. The number being divided is the dividend, the number that divides the dividend is the divisor, and the
quotient is the result:
One way to think of the dividend is that it is the total number of objects available. The divisor is the desired number of groups of objects, and the quotient is the number of objects within each
group. Thus, assuming that there are 8 people and the intent is to divide them into 4 groups, division indicates that each group would consist of 2 people. In this case, the number of people can be
divided evenly between each group, but this is not always the case. There are two ways to divide numbers when the result won't be even. One way is to divide with a remainder, meaning that the
division problem is carried out such that the quotient is an integer, and the leftover number is a remainder. For example, 9 cannot be evenly divided by 4. Instead, knowing that 8 ÷ 4 = 2, this
can be used to determine that 9 ÷ 4 = 2 R1. In other words, 9 divided by 4 equals 2, with a remainder of 1. Long division can be used either to find a quotient with a remainder, or to find an
exact decimal value.
How to perform long division?
To perform long division, first identify the dividend and divisor. To divide 100 by 7, where 100 is the dividend and 7 is the divisor, set up the long division problem by writing the dividend under a
radicand, with the divisor to the left (divisorvdividend), then use the steps described below:
1. Starting from left to right, divide the first digit in the dividend by the divisor. If the first digit cannot be divided by the divisor, write a 0 above the first digit of the divisor. 7 cannot
be divided into 1, so:
2. Continue the problem by dividing the divisor into the number formed by the combination of the previous and subsequent digit of the dividend. In this case, the next number formed is 10, which 7
can be divided into once, so write a 1 above the 2^nd digit of the dividend, and a 7 below.
3. Subtract, then bring down the following digit in the original dividend to determine the new dividend.
4. Determine the number of times the divisor goes into the new dividend; in this case, the number of times 7 goes into 30. Write this value above the radicand and write the product of the divisor
and this value below, then subtract. 7 goes into 30 a total of 4 times, and the product of 7 and 4 is 28.
This is the stopping point if the goal is to find a quotient with a remainder. In this case, the quotient is 014 or 14, and the remainder is 2. Thus, the solution to the division problem is:
100 ÷ 7 = 14 R2
To continue the long division problem to find an exact value, continue the same process above, adding a decimal point after the quotient, and adding 0s to form new dividends until an exact solution
is found, or until the quotient to a desired number of decimal places is determined.
5. Add a decimal point after the quotient and a 0 to the new dividend, and continue the same process as above.
6. Continue this process to the desired number of decimal places. In some cases, long division will reveal that a problem has a solution that is a repeating decimal. In other cases, the problem may
result in a terminating decimal or a non-terminating decimal. 100 ÷ 7 results in a non-terminating decimal eventually, or it can closely be estimated by the mixed number | {"url":"https://wanaquerepublicans.com/article/long-division-calculator","timestamp":"2024-11-04T01:48:18Z","content_type":"text/html","content_length":"67390","record_id":"<urn:uuid:72951e28-60e0-455c-b2b2-d2a3570aeff0>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00092.warc.gz"} |
Information About GATE Question Paper 2020
A Virtual Scientific Calculator will be available on the computer screen during the examination. Candidates have to use the same during the examination. Personal calculators, wristwatches, mobile
phones or any other electronic devices are NOT allowed inside the examination hall. Candidates should not bring any charts/tables/papers into the examination hall. GATE officials will not be
responsible for the safe-keep of the candidates’ personal belongings.
Question’s Pattern
GATE Examination 2020 would contain questions of two different types in all the papers:
Examination’s duration:
The candidates will provide Scribble pads for any rough work. The candidate has to write his/her name and registration number on the scribble pad before he/she starts using it. The scribble the pad
must be returned to the invigilator at the end of the examination.
All the papers of the GATE 2020 examination will be for 3 hours duration and they consist of 65 questions for a total of 100 marks. Since the examination is an ONLINE computer-based test, at the end
of the stipulated time (3-hours), the computer will automatically close the screen inhibiting any further action.
Information About GATE Question Paper 2020
Candidates will be permitted to occupy their allotted seats 40 minutes before the scheduled start of the examination. Candidates can log in and start reading the instructions 20 minutes before the
start of the examination.
Candidates will NOT be permitted to enter the examination hall after 10:00 hours in the forenoon session and after 15:00 hours in the afternoon session. Candidates will NOT be permitted to leave the
examination hall before the end of the examination.
Multiple Choice Questions:
carrying 1 or 2 marks each in all the papers and sections. These questions are objective in nature, and each will have a choice of four answers, out of which the candidate has to select (mark) the
correct answer.
Negative Marking for Wrong Answers: For a wrong answer chosen in an MCQ, there will be negative marking. For 1-mark MCQ, 1/3 mark will be deducted for a wrong answer. Likewise, for 2-mark MCQ, 2/3
mark will be deducted for a wrong answer.
Numerical Answer Type:
Questions carrying 1 or 2 marks each in all the papers and sections. For these questions, the answer is a signed real number, which needs to be entered by the candidate using the virtual numeric
keypad on the monitor (keyboard of the computer will be disabled). No choices will be shown for these types of questions. The answer can be a number such as 10 or -10 (an integer only). The answer
may be in decimals as well, for example, 10.1 (one decimal) or 10.01 (two decimals) or -10.001 (three decimals).
These questions will be mentioned with, up to which decimal places, the candidates need to present the answer. Also, for some NAT type problems an appropriate range will be considered while
evaluating the numerical answer type questions so that the candidate is not unduly penalized due to the usual round-off errors. Wherever required and possible, it is better to give NAT answer up to a
maximum of three decimal places.
Example: If the wire diameter of a compressive helical spring is increased by 2%, the change in spring stiffness (in %) is _ (correct to two decimal places).
Note: There is NO negative marking for a wrong answer in NAT questions.
Marks and Question’s Distribution:
In all the papers, there will be a total of 65 questions carrying 100 marks, out of which 10 questions carrying a total of 15 marks will be on General Aptitude (GA), which is intended to test the
Language and Analytical Skills. In the papers bearing the codes AE, AG, BM, BT, CE, CH, CS, EC, EE, IN, ME, MN, MT, PE, PI, TF, and XE.
Gate Marks & Question Information
Engineering Mathematics will carry around 15% of the total marks, the General Aptitude section will carry 15% of the total marks and the remaining around 70% of the total marks is devoted to the
subject of the paper. In the papers bearing the codes AR, CY, EY, GG, MA, PH, ST and XL, the General Aptitude section will carry 15% of the total marks and the remaining 85% of the total marks is
devoted to the subject of the paper.
General Aptitude Questions:
In all papers, GA questions carry a total of 15 marks. The GA section includes 5 questions carrying 1-mark each (sub-total 5 marks) and 5 questions carrying 2-marks each (sub-total 10 marks).
Question Papers other than GG, XE, and XL:
These papers would contain 25 questions carrying 1-mark each (sub-total 25 marks) and 30 questions carrying 2-marks each (sub-total 60 marks) consisting of both the MCQ and NAT Questions.
Geology and Geophysics Paper:
Apart from the General Aptitude (GA) section, the GG question paper consists of two parts: Part A and Part B. Part A are compulsory for all the candidates. Part B contains two sections: Section 1
(Geology) and Section 2 (Geophysics). Candidates will have to attempt questions in Part A and questions in either Section 1 or Section 2 of Part B.
Part A consists of 25 questions carrying 1-mark each (sub-total 25 marks and some of these may be numerical answer type questions). Either section of Part B (Section 1 and Section 2) consists of 30
questions carrying 2-marks each (sub-total 60 marks and some of these may be numerical answer type questions).
XE Paper (Engineering Sciences):
A candidate appearing in the XE paper has to answer the following:
GA – General Aptitude:
Carrying a total of 15 marks.
Section A– Engineering Mathematics (Compulsory):
This section contains 11 questions carrying a total of 15 marks: 7 questions carrying 1-mark each (sub-total 7 marks), and 4 questions carrying 2-marks each (sub-total 8 marks). Some questions may be
of numerical answer type.
Any two of XE Sections B to H:
The choice of two sections from B to H can be made during the examination after viewing the questions. Only TWO optional sections can be answered at a time. A candidate wishing to change midway of
the examination to another optional section must first choose to deselect one of the previously chosen optional sections (B to H).
Each of the optional sections of the XE paper (Sections B through H) contains 22 questions carrying a total of 35 marks: 9 questions carrying 1-mark each (sub-total 9 marks) and 13 questions carrying
2-marks each (sub-total 26 marks). Some questions may be of numerical answer type.
XL Paper (Life Sciences):
A candidate appearing in the XL paper has to answer the following:
GA – General Aptitude:
Carrying a total of 15 marks.
Section P– Chemistry (Compulsory):
This section contains 15 questions carrying a total of 25 marks: 5 questions carrying 1-mark each (sub-total 5 marks) and 10 questions carrying 2-marks each (sub-total 20 marks). Some questions may
be of numerical answer type.
Any two of XL Sections Q to U:
The choice of two sections from Q to U can be made during the examination after viewing the questions. Only TWO optional sections can be answered at a time. A candidate wishing to change midway of
the examination to another optional section must first choose to deselect one of the previously chosen optional sections (Q to U).
Each of the optional sections of the XL paper (Sections Q through U) contains 20 questions carrying a total of 30 marks: 10 questions carrying 1-mark each (sub-total 10 marks) and 10 questions
carrying 2-marks each (sub-total 20 marks). Some questions may be of numerical answer type.
Question’s Design:
The questions in a paper may be designed to test the following abilities:
These are based on facts, principles, formulae or laws in the discipline of the paper. The candidate is expected to be able to obtain the answer either from his/her memory of the subject or at most
from a one-line computation.
Success Ted
Q. During machining, maximum heat is produced:
(A) in the flank face (B) in rake face
(C) in the shear zone (D) due to friction between chip and tool.
These questions will test the candidate’s understanding of the basics of his/her field, by requiring him/her to draw simple conclusions from fundamental ideas.
Q. A DC motor requires a starter in order to
(A) develop a starting torque (B) compensate for auxiliary field ampere turns
(C) limit armature current at starting (D) provide regenerative braking.
In these questions, the candidate is expected to apply his/her knowledge either through computation or by logical reasoning.
Q. The sequent depth ratio of a hydraulic jump in a rectangular channel is 16.48. The Froude number at the beginning of the jump is: (A) 5.0 (B) 8.0 (C)10.0 (D) 12.0
The questions based on the above logics may be a mix of single standalone statement/phrase/data type questions, the combination of option codes type questions or match items type questions.
Analysis and Synthesis:
In these questions, the candidate is presented with data, diagrams, images, etc. that require analysis before a question can be answered. A Synthesis question might require the candidate to compare
two or more pieces of information. Questions in this category could, for example, involve candidates in recognizing unstated assumptions or separating useful information from irrelevant information.
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Causal Decision Theory
First published Sat Oct 25, 2008; substantive revision Thu Oct 24, 2024
Causal decision theory adopts principles of rational choice that attend to an act’s consequences. It maintains that an account of rational choice must use causality to identify the considerations
that make a choice rational.
Given a set of options constituting a decision problem, decision theory recommends an option that maximizes utility, that is, an option whose utility equals or exceeds the utility of every other
option. It evaluates an option’s utility by calculating the option’s expected utility. It uses probabilities and utilities of an option’s possible outcomes to define an option’s expected utility. The
probabilities depend on the option. Causal decision theory takes the dependence to be causal rather than merely evidential.
This essay explains causal decision theory, reviews its history, describes current research in causal decision theory, and surveys the theory’s philosophical foundations. The literature on causal
decision theory is vast, and this essay covers only a portion of it.
1. Expected Utility
Suppose that a student is considering whether to study for an exam. He reasons that if he will pass the exam, then studying is wasted effort. Also, if he will not pass the exam, then studying is
wasted effort. He concludes that because whatever will happen, studying is wasted effort, it is better not to study. This reasoning errs because studying raises the probability of passing the exam.
Deliberations should take account of an act’s influence on the probability of its possible outcomes.
An act’s expected utility is a probability-weighted average of its possible outcomes’ utilities. Possible states of the world that are mutually exclusive and jointly exhaustive, and so form a
partition, generate an act’s possible outcomes. An act-state pair specifies an outcome. In the example, the act of studying and the state of passing form an outcome comprising the effort of studying
and the benefit of passing. The expected utility of studying is the probability of passing if one studies times the utility of studying and passing plus the probability of not passing if one studies
times the utility of studying and not passing. In compact notation,
\[ \textit{EU} (S) = P(P \mbox{ if } S) \util (S \amp P) + P({\sim}P \mbox{ if } S) \util (S \amp{\sim}P). \]
Each product specifies the probability and utility of a possible outcome. The sum is a probability-weighted average of the possible outcomes’ utilities.
How should decision theory interpret the probability of a state \(S\) if one performs an act \(A\), that is, \(P(S \mbox{ if }A)\)? Probability theory offers a handy suggestion. It has an account of
conditional probabilities that decision theory may adopt. Decision theory may take \(P(S \mbox{ if }A)\) as the probability of the state conditional on the act. Then \(P(S \mbox{ if }A)\) equals \(P
(S\mid A)\), which probability theory defines as \(P(S \amp A)/P(A)\) when \(P(A) \ne 0\). Some theorists call expected utility computed using conditional probabilities conditional expected utility.
I call it expected utility tout court because the formula using conditional probabilities generalizes a simpler formula for expected utility that uses nonconditional probabilities of states. Also,
some theorists call an act’s expected utility its utility tout court because an act’s expected utility appraises the act and yields the act’s utility in ideal cases. I call it expected utility
because a person by mistake may attach more or less utility to a bet than its expected utility warrants. The equality of an act’s utility and its expected utility is normative rather than
Expected utilities obtained from conditional probabilities steer the student’s deliberations in the right direction.
\[\textit{EU} (S) = P(P\mid S)\util (S \amp P) + P({\sim}P\mid S)\util (S \amp{\sim}P), \]
\[\textit{EU} ({\sim}S) = P(P\mid {\sim}S)\util ({\sim}S \amp P) + P({\sim}P\mid {\sim}S)\util ({\sim}S \amp{\sim}P). \]
Because of studying’s effect on the probability of passing, \(P(P\mid S) \gt P(P\mid {\sim}S)\) and \(P({\sim}P\mid S) \lt P({\sim}P\mid {\sim}S)\). So \(\textit{EU} (S) \gt \textit{EU} ({\sim}S)\),
assuming that studying’s increase in the probability of passing compensates for the effort of studying. Maximization of expected utility recommends studying.
The handy interpretation of the probability of a state if one performs an act, however, is not completely satisfactory. Suppose that one tosses a coin with an unknown bias and obtains heads. This
result is evidence that the next toss will yield heads, although it does not causally influence the next toss’s result. An event’s probability conditional on another event indicates the evidence that
the second event provides for the first. If the two events are correlated, the second may provide evidence for the first without causally influencing it. Causation entails correlation, but
correlation does not entail causation. Deliberations should attend to an act’s causal influence on a state rather than an act’s evidence for a state. A good decision aims to produce a good outcome
rather than evidence of a good outcome. It aims for the good and not just signs of the good. Often efficacy and auspiciousness go hand in hand. When they come apart, an agent should perform an
efficacious act rather than an auspicious act.
Consider the Prisoner’s Dilemma, a stock example of game theory. Two people isolated from each other may each act either cooperatively or uncooperatively. They each do better if they each act
cooperatively than if they each act uncooperatively. However, each does better if he acts uncooperatively, no matter what the other does. Acting uncooperatively dominates acting cooperatively.
Suppose, in addition, that the two players are psychological twins. Each thinks as the other thinks. Moreover, they know this fact about themselves. Then if one player acts cooperatively, he
concludes that his counterpart also acts cooperatively. His acting cooperatively is good evidence that his counterpart does the same. Nonetheless, his acting cooperatively does not cause his
counterpart to act cooperatively. He has no contact with his counterpart. Because he is better off not acting cooperatively whatever his counterpart does, not acting cooperatively is the better
course. Acting cooperatively is auspicious but not efficacious.
To make expected utility track efficacy rather than auspiciousness, causal decision theory interprets the probability of a state if one performs an act as a type of causal probability rather than as
a standard conditional probability. In the Prisoner’s Dilemma with twins, consider the probability of one player’s acting cooperatively given that the other player does. This conditional probability
is high. Next, consider the causal probability of one player’s acting cooperatively if the other player does. Because the players are isolated, this probability equals the probability of the first
player’s acting cooperatively. It is low if that player follows dominance. Using conditional probabilities, the expected utility of acting cooperatively exceeds the expected utility of acting
uncooperatively. However, using causal probabilities, the expected utility of acting uncooperatively exceeds the expected utility of acting cooperatively. Switching from conditional to causal
probabilities makes expected-utility maximization yield acting uncooperatively.
Michael Titelbaum (2022) introduces the conceptual apparatus of causal decision theory, including subjective probability taken as degree of belief or credence, and devotes a chapter to decision
theory. Brian Hedden (2023), following points in Christopher Hitchcock (2013), shows that the slogan characterization of causal decision theory as favoring options with good effects is not completely
accurate. The theory’s guiding idea is, instead, to promote options that would have good outcomes if they were realized. Consequently, Hedden suggests renaming causal decision theory as
counterfactual decision theory, but J. Dimitri Gallow (2024a) maintains that the name change is unnecessary. Some theorists argue for causal decision theory by showing that it has desirable features
that distinguish it from other decision theories. Bacon (2022) shows that only following causal decision theory maximizes the expectation of actual value, the expectation he takes to be the basic
action-guiding quantity. Nielsen (2024) shows that only causal decision theory respects the value of information.
2. History
This section tours causal decision theory’s history and along the way presents various formulations of the theory.
2.1 Newcomb’s Problem
Robert Nozick (1969) presented a dilemma for decision theory. He constructed an example in which the standard principle of dominance conflicts with the standard principle of expected-utility
maximization. Nozick called the example Newcomb’s Problem after the physicist, William Newcomb, who first formulated the problem.
In Newcomb’s Problem an agent may choose either to take an opaque box or to take both the opaque box and a transparent box. The transparent box contains one thousand dollars that the agent plainly
sees. The opaque box contains either nothing or one million dollars, depending on a prediction already made. The prediction was about the agent’s choice. If the prediction was that the agent will
take both boxes, then the opaque box is empty. On the other hand, if the prediction was that the agent will take just the opaque box, then the opaque box contains a million dollars. The prediction is
reliable. The agent knows all these features of his decision problem.
Figure 1 displays the agent’s options and their outcomes. A row represents an option, a column a state of the world, and a cell an option’s outcome in a state of the world.
Prediction Prediction
one-boxing two-boxing
Take one box \(\$M\) \(\$0\)
Take two boxes \(\$M + \$T\) \(\$T\)
Figure 1. Newcomb’s Problem
Because the outcome of two-boxing is better by \(\$T\) than the outcome of one-boxing given each prediction, two-boxing dominates one-boxing. Two-boxing is the rational choice according to the
principle of dominance. Because the prediction is reliable, a prediction of one-boxing has a high probability given one-boxing. Similarly, a prediction of two-boxing has a high probability given
two-boxing. Hence, using conditional probabilities to compute expected utilities, one-boxing’s expected utility exceeds two-boxing’s expected utility. One-boxing is the rational choice according to
the principle of expected-utility maximization.
Decision theory should address all possible decision problems and not just realistic decision problems. However, if Newcomb’s problem seems untroubling because unrealistic, realistic versions of the
problem are plentiful. The essential feature of Newcomb’s problem is an inferior act’s correlation with a good state that it does not causally promote. In realistic, medical Newcomb problems, a
medical condition and a behavioral symptom have a common cause and are correlated although neither causes the other. If the behavior is attractive, dominance recommends it although expected-utility
maximization prohibits it. Also, Allan Gibbard and William Harper (1978: Sec. 12) and David Lewis (1979) observe that a Prisoner’s Dilemma with psychological twins, a case Section 1 mentions, poses a
Newcomb problem for each player. For each player, the other player’s act is a state affecting the outcome. Acting cooperatively is a sign, but not a cause, of the other player’s acting cooperatively.
Dominance recommends acting uncooperatively, whereas expected utility computed with conditional probabilities recommends acting cooperatively. In some realistic instances of the Prisoner’s Dilemma,
the players’ anticipated similarity of thought creates a conflict between the principle of dominance and the principle of expected-utility maximization. Arif Ahmed (2018) collects essays by several
authors on Newcomb’s problem, and Kenny Easwaran (2021) distinguishes Newcomb-like problems according to opportunities for causal intervention.
2.2 Stalnaker’s Solution
Robert Stalnaker (1968) presented truth conditions for subjunctive conditionals. A subjunctive conditional is true if and only if in the nearest antecedent-world, its consequent is true. (This
analysis is understood so that a subjunctive conditional is true if its antecedent is true in no world.) Stalnaker used analysis of subjunctive conditionals to ground their role in decision theory
and in a resolution of Newcomb’s problem.
In a letter to Lewis, Stalnaker (1972) proposed a way of reconciling decision principles in Newcomb’s problem. He suggested calculating an act’s expected utility using probabilities of conditionals
in place of conditional probabilities. Accordingly,
\[ \textit{EU} (A) = \sum_i P(A \gt S_i)\util (A \amp S_i), \]
where \(A \gt S_i\) stands for the conditional that if \(A\) were performed then \(S_i\) would obtain. Thus, instead of using the probability of a prediction of one-boxing given one-boxing, one
should use the probability of the conditional that if the agent were to pick just one box, then the prediction would have been one-boxing. Because the agent’s act does not cause the prediction, the
probability of the conditional equals the probability that the prediction is one-boxing. Also, consider the conditional that if the agent were to pick both boxes, then the prediction would have been
one-boxing. Its probability similarly equals the probability that the prediction is one-boxing. The act the agent performs does not affect any prediction’s probability because the prediction occurs
prior to the act. Consequently, using probabilities of conditionals to compute expected utility, two-boxing’s expected utility exceeds one-boxing’s expected utility. Therefore, the principle of
expected-utility maximization makes the same recommendation as does the principle of dominance.
Gibbard and Harper (1978) elaborated and made public Stalnaker’s resolution of Newcomb’s problem. They distinguished causal decision theory, which uses probabilities of subjunctive conditionals, from
evidential decision theory, which uses conditional probabilities. Because in decision problems probabilities of subjunctive conditionals track causal relations, using them to calculate an option’s
expected utility makes decision theory causal.
Gibbard and Harper distinguished two types of expected utility. One type they called value and represented with \(V\). It indicates news-value or auspiciousness. The other type they called utility
and represented with \(U\). It indicates efficacy in attainment of goals. A calculation of an act’s expected value uses conditional probabilities, and a calculation of its expected utility uses
probabilities of conditionals. They argued that expected utility, calculated with probabilities of conditionals, yields genuine expected utility.
As Gibbard and Harper introduce \(V\) and \(U\), both rest on an assessment \(D\) (for desirability) of maximally specific outcomes. Instead of adopting a formula for expected utility that uses an
assessment of outcomes neutral with respect to evidential and causal decision theory, this essay follows Stalnaker (1972) in adopting a formula that uses utility to evaluate outcomes.
2.3 Variants
Consider a conditional asserting that if an option were adopted, then a certain state would obtain. Gibbard and Harper assume, to illustrate the main ideas of causal decision theory, that the
conditional has a truth-value, and that, given its falsity, if the option were adopted, then the state would not obtain. This assumption may be unwarranted if the option is flipping a coin, and the
relevant state is obtaining heads. It may be false (or indeterminate) that if the agent were to flip the coin, he would obtain heads. Similarly, the corresponding conditional about obtaining tails
may be false (or indeterminate). Then probabilities of conditionals are not suitable for calculating the option’s expected utility. The relevant probabilities do not sum to one (or do not even
exist). To circumvent such impasses, some theorists calculate causally-sensitive expected utilities without probabilities of subjunctive conditionals. Causal decision theory has many formulations.
Brian Skyrms (1980: Sec IIC; 1982) presented a version of causal decision theory that dispenses with probabilities of subjunctive conditionals. His theory separates factors that the agent’s act may
influence from factors that the agent’s act may not influence. It lets \(K_i\) stand for a possible full specification of factors that an agent may not influence and lets \(C_j\) stand for a possible
(but not necessarily full) specification of factors that the agent may influence. The set of \(K_i\) forms a partition, and the set of \(C_j\) forms a partition. The formula for an act’s expected
utility first calculates its expected utility using factors the agent may influence, with respect to each possible combination of factors outside the agent’s influence. Then it computes a
probability-weighted average of those conditional expected utilities. An act’s expected utility calculated this way is the act’s \(K\)-expectation, \(\textit{EU}_k(A)\). According to Skyrms’s
\[\textit{EU}_k(A) = \sum_i P(K_i)\sum_j P(C_j \mid K_i \amp A)\util (C_j \amp K_i \amp A).\]
Skyrms holds that an agent should select an act that maximizes \(K\)-expectation.
Lewis (1981) presented a version of causal decision theory that calculates expected utility using probabilities of dependency hypotheses instead of probabilities of subjunctive conditionals. A
dependency hypothesis for an agent at a time is a maximally specific proposition about how the things the agent cares about do and do not depend causally on his present acts. An option’s expected
utility is its probability-weighted average utility with respect to a partition of dependency hypotheses \(K_i\). Lewis defines the expected utility of an option \(A\) as
\[ \textit{EU} (A) = \sum_i P(K_i)\util (K_i \amp A) \]
and holds that to act rationally is to realize an option that maximizes expected utility. His formula for an option’s expected utility is the same as Skyrms’s assuming that \(U(K_i \amp A)\) may be
expanded with respect to a partition of factors the agent may influence, using the formula
\[ U(K_i \amp A) = \sum_j P(C_j\mid K_i \amp A)\util (C_j \amp K_i \amp A). \]
Skyrms’s and Lewis’s calculations of expected utility dispense with causal probabilities. They build causality into states of the world so that causal probabilities are unnecessary. In cases such as
Newcomb’s problem, their calculations yield the same recommendations as calculations of expected utility employing probabilities of subjunctive conditionals. The various versions of causal decision
theory make equivalent recommendations when cases meet their background assumptions. Adam Bales (2016) compares versions in special cases that do not meet the background assumptions.
2.4 Representation Theorems
Decision theory often introduces probability and utility with representation theorems. These theorems show that if preferences among acts meet certain constraints, such as transitivity, then there
exist a probability function and a utility function (given a choice of scale) that generate expected utilities agreeing with preferences. David Krantz, R. Duncan Luce, Patrick Suppes, and Amos
Tversky (1971) offer a good, general introduction to the purposes and methods of constructing representations theorems. In Section 3.1, I discuss the theorems’ function in decision theory.
Richard Jeffrey ([1965] 1983) presented a representation theorem for evidential decision theory, using its formula for expected utility. Brad Armendt (1986, 1988a) presented a representation theorem
for causal decision theory, using its formula for expected utility. James Joyce (1999) constructed a very general representation theorem that yields either causal or evidential decision theory
depending on the interpretation of probability that the formula for expected utility adopts.
2.5 Objections
The most common objection to causal decision theory is that it yields the wrong choice in Newcomb’s problem. It yields two-boxing, whereas one-boxing is correct. Terry Horgan (1981 [1985]), Paul
Horwich (1987: Chap. 11), and Caspar Hare and Brian Hedden (2016) for example, promote one-boxing. The main rationale for one-boxing is that one-boxers fare better than do two-boxers. Causal decision
theorists respond that Newcomb’s problem is an unusual case that rewards irrationality. One-boxing is irrational even if one-boxers prosper. Bales (2018) rejects the argument that two-boxing is
Some theorists hold that one-boxing is plainly rational if the prediction is completely reliable. They maintain that if the prediction is certainly accurate, then choice reduces to taking \(\$M\) or
taking \(\$T\). This view oversimplifies. If an agent one-boxes, then that act is certain to yield \(\$M\). However, the agent still would have done better by taking both boxes. Dominance still
recommends two-boxing. Making the prediction certain to be accurate does not change the character of the problem. Efficacy still trumps auspiciousness, as Howard Sobel (1994: Chap. 5) argues.
A way of reconciling the two sides of the debate about Newcomb’s problem acknowledges that a rational person should prepare for the problem by cultivating a disposition to one-box. Then whenever the
problem arises, the disposition will prompt a prediction of one-boxing and afterwards the act of one-boxing (still freely chosen). Causal decision theory may acknowledge the value of this
preparation. It may conclude that cultivating a disposition to one-box is rational although one-boxing itself is irrational. Hence, if in Newcomb’s problem an agent two-boxes, causal decision theory
may concede that the agent did not rationally prepare for the problem. It nonetheless maintains that two-boxing itself is rational. Although two-boxing is not the act of a maximally rational agent,
it is rational given the circumstances of Newcomb’s problem.
Causal decision theory may also explain that it advances a claim about the evaluation of an act given the agent’s circumstances in Newcomb’s problem. It asserts two-boxing’s conditional rationality.
Conditional and nonconditional rationality treat mistakes differently. In contrast with conditional rationality, nonconditional rationality does not grant past mistakes. It evaluates an act taking
account of the influence of past mistakes. However, conditional rationality accepts present circumstances as they are and does not discredit an act because it stems from past mistakes. Causal
decision theory maintains that two-boxing is rational, granting the agent’s circumstances and so ignoring any mistakes leading to those circumstances, such as irrational preparation for Newcomb’s
Another objection to causal decision theory concedes that two-boxing is the rational choice in Newcomb’s problem but rejects causal principles of choice that yield two-boxing. It seeks noncausal
principles that yield two-boxing. Positivism is a source of aversion to decision principles incorporating causation. Some decision theorists shun causation because no positivist account specifies its
nature. Without a definition of causation in terms of observable phenomena, they prefer that decision theory avoid causation. Causal decision theory’s response to this objection is both to discredit
positivism and also to clarify causation so that puzzles concerning it no longer give decision theory any reason to avoid it.
Evidential decision theory has weaker metaphysical assumptions than has causal decision theory, even if causation has impeccable metaphysical credentials. Some decision theorists do not omit
causation because of metaphysical scruples but for conceptual economy. Jeffrey ([1965] 1983, 2004), for the sake of parsimony, formulates decision principles that do not rely on causal relations.
Ellery Eells (1981, 1982) contends that evidential decision theory yields causal decision theory’s recommendations but, more economically, without reliance on causal apparatus. In particular,
evidential decision theory yields two-boxing in Newcomb’s problem. An agent’s reflection on his evidence makes conditional probabilities support two-boxing.
A noncontentious elaboration of Newcomb’s problem posits that the agent’s choice and its prediction have a common cause. The agent’s choice is evidence of the common cause and evidence of the
choice’s prediction. Once an agent acquires the probability of the common cause, he may put aside the evidence his choice provides about the prediction. That evidence is superfluous. Given the
probability of the common cause, the probability of a prediction of one-boxing is constant with respect to his options. Similarly, the probability of a prediction of two-boxing is constant with
respect to his options. Because the probability of a prediction is the same conditional on either option, the expected utility of two-boxing exceeds the expected utility of one-boxing according to
evidential decision theory. Horgan (1981 [1985]) and Huw Price (1986) make similar points.
Suppose that an event \(S\) is a sign of a cause \(C\) that produces an effect \(E\). For the probability of \(E\), knowing whether \(C\) holds makes superfluous knowing whether \(S\) holds.
Observation of \(C\) screens off the evidence that \(S\) provides for \(E\). That is, \(P(E\mid C \amp S) = P(E\mid C)\). In Newcomb’s problem, assuming that the agent is rational, his beliefs and
desires are a common cause of his choice and the prediction. So his choice is a sign of the prediction’s content. For the probability of a prediction of one-boxing, knowing one’s beliefs and desires
makes superfluous knowing the choice that they yield. Knowledge of the common cause screens off evidence that the choice provides about the prediction. Hence, the probability of a prediction of
one-boxing is constant with respect to one’s choice, and maximization of evidential expected-utility agrees with the principle of dominance. This defense of evidential decision theory is called the
tickle defense because it assumes that an introspected condition screens off the correlation between choice and prediction.
Eells’s defense of evidential decision theory assumes that an agent chooses according to beliefs and desires and knows his beliefs and desires. Some agents may not choose this way and may not have
this knowledge. Decision theory should prescribe a rational choice for such agents, and evidential decision theory may not do that correctly, as Lewis (1981: 10–11) and John Pollock (2010) argue.
Armendt (1988b: 326–329) and David Papineau (2001: 252–255) concur that the phenomenon of screening off does not in all cases make evidential decision theory yield the results of causal decision
Horwich (1987: Chap. 11) rejects Eells’s argument because, even if an agent knows that her choice springs from her beliefs and desires, she may be unaware of the mechanism by which her beliefs and
desires produce her choice. The agent may doubt that she chooses by maximizing expected utility. Then in Newcomb’s problem her choice may offer relevant evidence about the prediction. Eells (1984a)
constructs a dynamic version of the tickle defense to meet this objection. Sobel (1994: Chap. 2) discusses that version of the defense. He argues that it does not yield evidential decision theory’s
agreement with causal decision theory in all decision problems in which an act furnishes evidence concerning the state of the world. Moreover, it does not establish that an evidential theory of
rational desire agrees with a causal theory of rational desire. He concludes that even in cases where evidential decision theory yields the right recommendation, it does not yield it for the right
Jeffrey (1981) and Eells (1984b) use tickles or metatickles to reconcile evidential and causal decision theory, and Huttegger (2023) elaborates the method of reconciliation in the case of Newcomb’s
problem using deliberational dynamics in the style of Skyrms (1990). He notes, however that the reconciliation makes assumptions that some cases do not meet. The two decision theories may disagree if
the predictor knows more about the agent’s decision-making than does the agent.
Price (2012) proposes a blend of evidential and causal decision theory and motivates it with an analysis of cases in which an agent has foreknowledge of an event occurring by chance. Causal decision
theory on its own accommodates such cases, argue Bales (2016) and Gallow (2024b). Ahmed (2014a) champions evidential decision theory and advances several objections to causal decision theory. His
objections assume some controversial points about rational choice, including a controversial principle for sequences of choices. A common view distinguishes principles for evaluating choices from
principles for evaluating sequences of choices. The principle of utility maximization evaluates an agent’s choice as a resolution of a decision problem only if the agent has direct control of each
option in the decision problem, that is, only if the agent can at will immediately adopt any option in the decision problem. The principle does not evaluate an agent’s sequence of multiple choices
because the agent does not have direct control of such a sequence. She realizes a sequence of multiple choices only by making each choice in the sequence at the time for it; she cannot at will
immediately realize the entire sequence. Rationality evaluates an option in an agent’s direct control by comparing it with alternatives but evaluates a sequence in an agent’s indirect control by
evaluating the directly controlled options in the sequence; a sequence of choices is rational if the choices in the sequence are rational. Adopting this common method of evaluating sequences of
choices fends off objections to causal decision theory that assume rival methods.
3. Current Issues
Decision theory is an active area of research. Current work addresses a number of problems. Causal decision theory’s approach to those problems arises from its nonpositivistic methodology and its
attention to causation. This section mentions some topics on causal decision theory’s agenda.
3.1 Probability and Utility
Principles of causal decision theory use probabilities and utilities. The interpretation of probabilities and utilities is a matter of debate. One tradition defines them in terms of functions that
representation theorems introduce to depict preferences. The representation theorems show that if preferences meet certain structural axioms, then if they also meet certain normative axioms, they are
as if they follow expected utility. That is, preferences follow expected utility calculated using probability and utility functions constructed so that preferences follow expected utility. Expected
utility calculated this way differs from expected utility calculated using probability and utility assignments grounded in attitudes toward possible outcomes. For example, a person confused about
bets concerning a coin toss may have preferences among those bets that are as if he assigns probability 60% to heads, when, in fact, the evidence of past tosses leads him to assign probability 40% to
heads. Consequently, when preferences meet a representation theorem’s structural axioms, the theorem’s normative axioms justify only conformity with expected utility fabricated to agree with
preferences and do not justify conformity with expected utility in the traditional sense. Defining probability and utility using the representation theorems thus weakens the traditional principle of
expected utility. It becomes merely a principle of coherence among preferences.
Instead of using the representation theorems to define probabilities and utilities, decision theory may use them to establish probabilities’ and utilities’ measurability when preferences meet
structural and normative axioms. This employment of the representation theorems allows decision theory to advance the traditional principle of expected utility and thereby enrich its treatment of
rational decisions. Decision theory may justify that traditional principle by deriving it from general principles of evaluation, as in Weirich (2001).
A broad account of probabilities and utilities takes them to indicate attitudes toward propositions. They are rational degrees of belief and rational degrees of desire, respectively. This account of
probabilities and utilities recognizes their existence in cases where they are not inferable from preferences or their other effects but instead are inferable from their causes, such as an agent’s
information about objective probabilities, or are not inferable at all (except perhaps by introspection). The account relies on arguments that degrees of belief and degrees of desire, if rational,
conform to standard principles of probability and utility. Bolstering these arguments is work for causal decision theory.
Besides clarifying its general interpretation of probability and utility, causal decision theory searches for the particular probabilities and utilities that yield the best version of its principle
to maximize expected utility. The causal probabilities in its formula for expected utility may be probabilities of subjunctive conditionals or various substitutes. Versions that use probabilities of
subjunctive conditionals must settle on an analysis of those conditionals. Lewis (1973: Chap. 1) modifies Stalnaker’s analysis to count a subjunctive conditional true if and only if as antecedent
worlds come closer and closer to the actual world, there is a point beyond which the consequent is true in all the worlds at least that close. Joyce (1999: 161–180) advances probability images, as
Lewis (1976) introduces them, as substitutes for probabilities of subjunctive conditionals. The probability image of a state \(S\) under subjunctive supposition of an act \(A\) is the probability of
\(S\) according to an assignment that shifts the probability of \({\sim}A\)-worlds to nearby \(A\)-worlds. Causal relations among an act and possible states guide probability’s reassignment.
A common formula for an act’s expected utility takes the utility for an act-state pair, the utility of the act’s outcome in the state, to be the utility of the act’s and the state’s conjunction:
\[ \textit{EU} (A) = \sum_i P(A \gt S_i)\util (A \amp S_i). \]
Does causal decision theory need an alternative, more causally-sensitive utility for an act-state pair? Weirich (1980) argues that it does. A person contemplating a wager that the capital of Missouri
is Jefferson City entertains the consequences if he were to make the wager given that St. Louis is Missouri’s capital. A rational deliberator subjunctively supposes an act attending to causal
relations and indicatively supposes a state attending to evidential relations, but can suppose an act’s and a state’s conjunction only one way. Furthermore, using the utility of an act’s and a
state’s conjunction prevents an act’s expected utility from being partition-invariant. The next subsection elaborates this point.
3.2 Partition Invariance
An act’s expected utility is partition invariant if and only if it is the same under all partitions of states. Partition invariance is a vital property of an act’s expected utility. If acts’ expected
utilities lack this property, then decision theory may use only expected utilities computed from selected partitions. Expected utility’s partition invariance makes an act’s expected utility
independent of selection of a partition of states and thereby increases expected utility’s explanatory power.
Partition invariance ensures that various representations of the same decision problem yield solutions that agree. Take Newcomb’s problem with Figure 2’s representation.
Right prediction Wrong prediction
Take only one box \(\$M\) $0
Take two boxes \(\$T\) \(\$M + \$T\)
Figure 2. New States for Newcomb’s Problem
Dominance does not apply to this representation. It nonetheless settles the problem’s solution because it applies to a decision problem if it applies to any accurate representation of the problem,
such as Figure 1’s representation of the problem. If expected utilities are partition-sensitive, then acts that maximize expected utility may be partition-sensitive. The principle of expected utility
does not yield a decision problem’s solution, however, if acts of maximum expected-utility change from one partition to another. In that case an act is not a solution to a decision problem simply
because it maximizes expected utility under some accurate representation of the problem. Too many acts have the same credential.
The expected utility principle, using probabilities of conditionals, applies to Figure 2’s representation of Newcomb’s problem. Letting \(P1\) stand for a prediction of one-boxing and \(P2\) stand
for a prediction of two-boxing, the acts’ expected utilities are:
\[ \textit{EU} (1) & = P(1 \gt R)\util (\$M) + P(1 \gt W)0\\ & = P(P1)\util (\$M)\\ \textit{EU} (2) & = P(2 \gt R)\util (\$T) + P(2 \gt W)\util (\$M + \$T)\\ & = P(P2)\util (\$T) + P(P1)\util (\$M +
\$T)\\ \]
Hence \(\textit{EU}(1) \lt EU(2)\). This result agrees with the verdict of causal decision theory given other accurate representations of the problem. Provided that causal decision theory uses a
partition-invariant formula for expected utility, its recommendations are independent of a decision problem’s representation.
Lewis (1981: 12–13) observes that the formula
\[ EU(A) = \sum_i P(S_i)\util (A \amp S_i) \]
is not partition invariant. Its results depend on the partition of states. If a state is a set of worlds with equal utilities, then with respect to a partition of such states every act has the same
expected utility. An element \(S_i\) of the partition obscures the effects of \(A\) that the utility of an outcome should evaluate. Lewis overcomes this problem by using only partitions of dependency
hypotheses. However, causal decision theory may craft a partition-invariant formula for expected utility by adopting a substitute for \(U(A \amp S_i)\).
Sobel (1994: Chap. 9) investigates partition invariance. Putting his work in this essay’s notation, he proceeds as follows. First, he takes a canonical computation of an option’s expected utility to
use worlds as states. His basic formula is
\[ \textit{EU} (A) = \sum_i P(A \gt W_i)\util (W_i). \]
A world \(W_i\) absorbs an act performed in it. Only the worlds in which \(A\) holds contribute positive probabilities and so affect the sum. Next, Sobel searches for other computations, using
coarse-grained states, that are equivalent to the canonical computation. A suitable specification of utilities achieves partition invariance given his assumptions. According to a theorem he proves
(1994: 185),
\[ U(A) = \sum_i P(S_i)\util (A \mbox{ given } S_i) \]
for any partition of states.
Joyce (2000: S11) also articulates for causal decision theory a partition-invariant formula for an act’s expected utility. He achieves partition invariance, assuming that
\[ \textit{EU} (A) = \sum_i P(A \gt S_i)\util (A \amp S_i), \]
by stipulating that \(U(A \amp S_i)\) equals
\[ \sum_{ij} P^A(W_j\mid S_i)\util (W_j), \]
where \(W_j\) is a world and \(P^A\) stands for the probability image of \(A\). Weirich (2001: Secs. 3.2, 4.2.2), as Sobel does, substitutes \(U(A \mbox{ given }S_i)\) for \(U(A \amp S_i)\) in the
formula for expected utility and interprets \(U(A \mbox{ given }S_i)\) as the utility of the outcome that \(A\)’s realization would produce if \(S\) obtains. Accordingly, \(U(A \mbox{ given }S_i)\)
responds to \(A\)’s causal consequences in worlds where \(S_i\) holds. Then the formula
\[ \textit{EU} (A) = \sum_i P(S_i) \util (A \mbox{ given }S_i) \]
is invariant with respect to partitions in which states are probabilistically independent of the act. A more complex formula,
\[ \textit{EU} (A) = \sum_i P(S_i \mbox{ if }A)\util (A \mbox{ given } (S_i \mbox{ if } A)), \]
assuming a causal interpretation of its probabilities, relaxes all restriction on partitions. \(U(A \mbox{ given }(S_i \mbox{ if }A))\) is the utility of the outcome if \(A\) were realized, given
that it is the case that \(S_i\) would obtain if \(A\) were realized.
3.3 Outcomes
One issue concerning outcomes is their comprehensiveness. Are an act’s outcomes possible worlds, temporal aftermaths, or causal consequences? Gibbard and Harper ([1978] 1981: 166–168) mention the
possibility of narrowing outcomes to causal consequences, as practical applicability advocates. The narrowing must be judicious, however, because the expected-utility principle requires that outcomes
include every relevant consideration. For example, if an agent is averse to risk, then each of a risky act’s possible outcomes must include the risk the act generates. Its inclusion tends to lower
each possible outcome’s utility.
In Sobel’s canonical formula for expected utility,
\[ \textit{EU} (A) = \sum_i P(A \gt W_i)\util (W_i). \]
The formula, from one perspective, omits states of the world because the outcomes themselves form a partition. The distinction between states and outcomes dissolves because worlds play the role of
both states and outcomes. States are dispensable means of generating outcomes that are exclusive and exhaustive. According to a basic principle, an act’s expected utility is a probability-weighted
average of possible outcomes that are exclusive and exhaustive, such as the worlds to which the act may lead.
Suppose that a world’s utility comes from realization of basic intrinsic desires and aversions. Granting that the utilities of their realizations are additive, the utility of a world is a sum of the
utilities of their realizations. Then besides being a probability-weighted average of the utilities of worlds to which it may lead, an option’s expected utility is also a probability-weighted average
of the realizations of basic intrinsic desires and aversions. In this formula for its expected utility, states play no explicit role:
\[ \textit{EU} (A) = \sum_i P(A \gt B_i)\util (B_i), \]
where \(B_i\) ranges over possible realizations of basic intrinsic desires and aversions. The formula considers for each basic desire and aversion the prospect of its realization if the act were
performed. It takes the act’s expected utility as the sum of the prospects’ utilities. The formula provides an economical representation of an act’s expected utility. It eliminates states and obtains
expected utility directly from outcomes taken as realizations of basic desires and aversions.
To illustrate calculation of an act’s expected utility using basic intrinsic desires and aversions, suppose that an agent has no basic intrinsic aversions and just two basic intrinsic desires, one
for health and the other for wisdom. The utility of health is 4, and the utility of wisdom is 8. In the formula for expected utility, a world covers only matters about which the agent cares. In the
example, a world is a proposition specifying whether the agent has health and whether he has wisdom. Accordingly, there are four worlds: \[ H \amp W, \\ H \amp {\sim}W, \\ {\sim}H \amp W, \\ {\sim}H
\amp {\sim}W.\\ \]
Suppose that \(A\) is equally likely to generate any world. Using worlds,
\[ \textit{EU} (A) & = P(A \gt(H \amp W))\util (H \amp W) \\ &\qquad + P(A \gt(H \amp{\sim}W))\util (H \amp{\sim}W) \\ &\qquad + P(A \gt({\sim}H \amp W))\util ({\sim}H \amp W) \\ &\qquad + P(A \gt({\
sim}H \amp{\sim}W))\util ({\sim}H \amp{\sim}W) \\ & = (0.25)(12) + (0.25)(4) + (0.25)(8) + (0.25)(0) \\ & = 6.\\ \]
Using basic intrinsic attitudes,
\[ \textit{EU} (A) &= P(A \gt H)\util (H) + P(A \gt W)\util (W) \\ & = (0.5)(4) + (0.5)(8) \\ & = 6. \]
The two methods of computing an option’s utility are equivalent given that, under supposition of an act’s realization, the probability of a basic intrinsic desire’s or aversion’s realization is the
sum of the probabilities of the worlds that realize it.
3.4 Acts
In deliberations, a first-person action proposition represents an act. The proposition has a subject-predicate structure and refers directly to the agent, its subject, without the intermediary of a
concept of the agent. A centered world represents the proposition. Such a world not only specifies individuals and their properties and relations, but also specifies which individual is the agent and
where and when his decision problem arises. Realization of the act is realization of a world with, at its center, the agent at the time and place of his decision problem.
Isaac Levi (2000) objects to any decision theory that attaches probabilities to acts. He holds that deliberation crowds out prediction. While deliberating, an agent does not have beliefs or degrees
of belief about the act that she will perform. Levi holds that Newcomb’s problem, and evidential and causal decision theories that address it, involve mistaken assignments of probabilities to an
agent’s acts. He rejects both Jeffrey’s ([1965] 1983) evidential decision theory and Joyce’s (1999) causal decision theory because they allow an agent to assign probabilities to her acts during
In opposition to Levi’s views, Joyce (2002) argues that (1) causal decision theory need not accommodate an agent’s assigning probabilities to her acts, but (2) a deliberating agent may legitimately
assign probabilities to her acts. Evidential decision theory computes an act’s expected utility using the probability of a state given the act, \(P(S\mid A)\), defined as \(P(S \amp A)/P(A)\). The
fraction’s denominator assigns a probability to an act. Causal decision theory replaces \(P(S\mid A)\) with \(P(A \gt S)\) or a similar causal probability. It need not assign a probability to an act.
May an agent deliberating assign probabilities to her possible acts? Yes, a deliberator may sensibly assign probabilities to any events, including her acts. Causal decision theory may accommodate
such probabilities by forgoing their measurement with betting quotients. According to that method of measurement, willingness to make bets indicates probabilities. Suppose that a person is willing to
take either side of a bet in which the stake for the event is \(x\) and the stake against the event is \(y\). Then the probability the person assigns to the event is the betting quotient \(x/(x + y)
\). This method of measurement may fail when the event is an agent’s own future act. A bet on an act’s realization may influence the act’s probability, as a thermometer’s temperature may influence
the temperature of a liquid it measures.
Joyce (2007: 552–561) considers whether Newcomb problems are genuine decision problems despite strong correlations between states and acts. He concludes that, yes, despite those correlations, an
agent may view her decision as causing her act. An agent’s decision supports a belief about her act independently of prior correlations between states and her act. According to a principle of
evidential autonomy (2007: 557),
A deliberating agent who regards herself as free need not proportion her beliefs about her own acts to the antecedent evidence that she has for thinking that she will perform them.
She should proportion her beliefs to her total evidence, including her self-supporting beliefs about her own acts. Those beliefs provide new relevant evidence about her acts.
How should an agent deliberating about an act understand the background for her act? She should not adopt a backtracking supposition of her act. Standing on the edge of a cliff, she should not
suppose that if she were to jump, she would have a parachute to break her fall. Also, she should not imagine gratuitous changes in her basic desires. She should not imagine that if she were to choose
chocolate instead of vanilla, despite currently preferring vanilla, that she would then prefer chocolate. She should imagine that her basic desires are constant as she imagines the various acts she
may perform, and, moreover, should adopt during deliberations the pretense that her will generates her act independently of her basic desires and aversions.
Christopher Hitchcock (1996) holds that an agent should pretend that her act is free of causal influence. Doing this makes partitions of states yielding probabilities for decisions agree with
partitions of states yielding probabilities defining causal relevance. As a result, probabilities in causal decision theory may form a foundation for probabilities in the probabilistic theory of
causation. Causal decision theory, in particular, the version using dependency hypotheses, grounds theories of probabilistic causation.
Ahmed (2013) argues that causal decision theory goes awry in cases in which the universe is deterministic. In response, Alexander Sandgren and Timothy Luke Williamson (2021), Adam Elga (2022), Boris
Kment (2023), and Williamson and Sandgren (2023) propose amendments to causal decision theory. Joyce (2016), Melissa Fusco (2023), and Calum McNamara (2023) defend causal decision theory against
Ahmed’s objection.
3.5 Generalizing Expected Utility
Problems such as Pascal’s Wager and the St. Petersburg paradox suggest that decision theory needs a means of handling infinite utilities and expected utilities. Suppose that an option’s possible
outcomes all have finite utilities. Nonetheless, if those utilities are infinitely many and unbounded, then the option’s expected utility may be infinite. Alan Hájek and Harris Nover (2006) also show
that the option may have no expected utility. The order of possible outcomes, which is arbitrary, may affect convergence of their utilities’ probability-weighted average and the value to which the
average converges if it does converge. Causal decision theory should generalize its principle of expected-utility maximization to handle such cases.
Also, common principles of causal decision theory advance standards of rationality that are too demanding to apply to humans. They are standards for ideal agents in ideal circumstances (a precise
formulation of the idealizations may vary from theorist to theorist). Making causal decision theory realistic requires relaxing idealizations that its principles assume. A generalization of the
principle of expected-utility maximization, for example, may relax idealizations to accommodate limited cognitive abilities. Weirich (2004, 2021) and Pollock (2006) take steps in this direction.
Appropriate generalizations distinguish taking maximization of expected utility as a procedure for making a decision and taking it as a standard for evaluating a decision even after the decision has
been made.
3.6 Decision Instability
Gibbard and Harper (1978: Sec. 11) present a problem for causal decision theory using an example drawn from literature. A man in Damascus knows that he has an appointment with Death at midnight. He
will escape Death if he manages at midnight not to be at the place of his appointment. He can be in either Damascus or Aleppo at midnight. As the man knows, Death is a good predictor of his
whereabouts. If he stays in Damascus, he thereby has evidence that Death will look for him in Damascus. However, if he goes to Aleppo he thereby has evidence that Death will look for him in Aleppo.
Wherever he decides to be at midnight, he has evidence that he would be better off at the other place. No decision is stable. Decision instability arises in cases in which a choice provides evidence
for its outcome, and each choice provides evidence that another choice would have been better. Reed Richter (1984, 1986) uses cases of decision instability to argue against causal decision theory.
The theory needs a resolution of the problem of decision instability. The problem does not refute causal decision theory but shows that it needs generalization to handle cases of decision
A common analysis of the problem classifies options as either self-ratifying or not self-ratifying. Jeffrey ([1965] 1983) introduced ratification as a component of evidential decision theory. His
version of the theory evaluates a decision according to the expected utility of the act it selects. The distinction between an act and a decision to perform the act grounds his definition of an
option’s self-ratification and his principle to make self-ratifying, or ratifiable, decisions. According to his definition ([1965] 1983: 16),
A ratifiable decision is a decision to perform an act of maximum estimated desirability relative to the probability matrix the agent thinks he would have if he finally decided to perform that
Estimated desirability is expected utility. An agent’s probability matrix is an array of rows and columns for acts and states, respectively, with each cell formed by the intersection of an act’s row
and a state’s column containing the probability of the state given that the agent is about to perform the act. Before performing an act, an agent may assess the act in light of a decision to perform
it. Information the decision carries may affect the act’s expected utility and its ranking with respect to other acts.
Jeffrey used ratification as a means of making evidential decision theory yield the same recommendations as causal decision theory. In Newcomb’s problem, for instance, two-boxing is the only
self-ratifying option. However, Jeffrey (2004: 113n) concedes that evidential decision theory’s reliance on ratification does not make it agree with causal decision theory in all cases. Moreover,
Joyce (2007) argues that the motivation for ratification appeals to causal relations, so that even if it yields correct recommendations using Jeffrey’s formula for expected utility, it still does not
yield a purely evidential decision theory.
Causal decision theory’s account of self-ratification may put aside Jeffrey’s method of evaluating a decision by evaluating the act it selects. Because the decision and the act differ, they may have
different consequences. For example, a decision may fail to generate the act it selects. Hence, the decision’s expected utility may differ from the act’s expected utility. Driving through a flooded
section of highway may have high expected utility because it minimizes travel time to one’s destination. However, the decision to drive through the flooded section may have low expected utility
because for all one knows the water may be deep enough to swamp the car. Using an act’s expected utility to assess a decision to perform the act leads to faulty evaluations of decisions. As, for
example, Hedden (2012) maintains, it is better to evaluate a decision by comparing its expected utility to the expected utilities of rival decisions. A decision’s expected utility depends on the
probability of its execution as well as the expected consequences of the act it selects.
Weirich (1985) and Harper (1986) define ratification in terms of an option’s expected utility given its realization rather than given a decision to realize it. An option is self-ratifying if and only
if it maximizes expected utility given its realization. This account of ratification accommodates cases in which an option and a decision to realize it have different expected utilities. Weirich and
Harper also assume causal decision theory’s formula for expected utility. In the case of Death in Damascus, causal decision theory concludes that the threatened man lacks a self-ratifying option. A
self-ratifying option emerges, however, if the man may flip a coin to make his decision. Adopting the probability distribution for locations is called a mixed strategy, whereas choices of location
are called pure strategies. Assuming that Death cannot predict the coin flip’s outcome, the mixed strategy is self-ratifying.
During deliberations to resolve a decision problem, an agent may revise the probabilities she assigns to pure strategies in light of computations of their expected utilities using earlier probability
assignments. The process of revision may culminate in a stable probability assignment that represents a mixed strategy. Skyrms (1982, 1990) and Eells (1984b) investigate these dynamics of
deliberation. Some open issues are whether adoption of a mixed strategy resolves a decision problem and whether a pure strategy arising from a mixed strategy that constitutes an equilibrium of
deliberations is rational if the pure strategy itself is not self-ratifying.
Andy Egan (2007) argues that causal decision theory yields the wrong recommendation in decision problems with an option that provides evidence concerning its outcome. He entertains the case of an
assassin who deliberates about pulling the trigger, knowing that the option’s realization provides evidence of a brain lesion that ruins his aim. Egan maintains that causal decision theory mistakenly
ignores the evidence that the option provides. However, versions of causal decision theory that incorporate ratification are innocent of the charges. Ratification takes account of evidence an option
provides concerning its outcome.
Any version of the expected utility principle, whether it uses conditional probabilities or probabilities of conditionals, must specify the information that guides assignments of probabilities and
utilities. Principles of nonconditional expected-utility maximization use the same information for all options, and hence exclude information about an option’s realization. The principle of
ratification uses for each option information that includes the option’s realization. It is a principle of conditional expected-utility maximization. Egan’s cases count against nonconditional
expected-utility maximization, and not against causal decision theory. Conditional expected-utility maximization using causal decision theory’s formula for expected utility addresses the cases he
Egan’s examples do not refute causal decision theory but present a challenge for it. Suppose that in a decision problem no self-ratifying option exists, or multiple self-ratifying options exist. How
should a rational agent proceed, granting that a decision principle should take account of information that an option provides? This is an open problem in causal decision theory (and in any decision
theory acknowledging that an option’s realization may constitute evidence concerning its outcome). Ratification analyzes decision instability but is not a complete response to it.
In response to Egan, Frank Arntzenius (2008) and Joyce (2012) argue that in some decision problems an agent’s rational deliberations using freely available information do not settle on a single
option but instead settle on a probability distribution over options. They acknowledge that the agent may regret the option issuing from these deliberations but differ about the regret’s
significance. Arntzenius holds that the regret counts against the option’s rationality, whereas Joyce denies this. Ahmed (2012) and Ralph Wedgwood (2013) reject Arntzenius’s and Joyce’s responses to
Egan because they hold that deliberations should settle on an option. Wedgwood introduces a novel decision principle to accommodate Egan’s decision problems. Ahmed contends that Egan’s analysis of
these decision problems has a flaw because when it is extended to some other decision problems, it declares every option irrational. Anna Kusser and Wolfgang Spohn (1992: 17–18) show that decision
instability may arise because an option’s realization changes utilities, instead of probabilities, of possible outcomes.
Ahmed (2014b) and Jack Spencer (2021) criticize causal decision theory in cases of decision instability, and Spencer and Ian Wells (2019) criticize a principle of causal dominance attributed to
causal decision theory. Gallow (2024c) argues that a decision theory that removes instability also rejects the sure-thing principle (a principle of dominance). Rhys Borchert and Spencer (2024) argue
that solutions to problems with decision instability are hard to reconcile with two-boxing in Newcomb’s problem. To handle cases of decision instability, Benjamin Levinstein and Nate Suares (2020)
advance functional decision theory, even though it permits one-boxing in Newcomb’s Problem. Gallow (2020) and David Barnett (2022) introduce degrees of ratifiability. In a decision problem with two
options A and B, A’s degree of ratifiability is U(A given A) – U(B given A), and similarly for B. They propose that the rationally preferable option has the greater degree of ratifiability. Spencer
(2023) argues that this proposal in some cases forbids an option that an agent will realize and that has better prospects than all other options because of the information that the option’s
realization carries. Joyce (2018), Armendt (2019), Bales (2020), and Greg Lauro and Simon Huttegger (2021), and Williamson (2021) elaborate ways that causal decision theory may respond to decision
Points about ratification in decision problems clarify points about equilibrium in game theory because in games of strategy a player’s choice often furnishes evidence about other players’ choices.
Decision theory underlies game theory because a game’s solution identifies rational choices in the decision problems the game creates for the players. Solutions to games distinguish correlation and
causation, as do decision principles. Because in simultaneous-move games two agent’s strategies may be correlated but not related as cause and effect, solutions to such games do not have the same
properties as solutions to sequential games. Causal decision theory attends to distinctions on which solutions to games depend. It supports game theory’s account of interactive decisions. Joyce and
Gibbard (2016) describe the role of ratification in game theory, and Stalnaker (2018) describes causal decision theory’s place in game theory.
The existence of self-ratifying mixed strategies in decision problems such as Death in Damascus suggests that ratification, as causal decision theory explains it, supports participation in a Nash
equilibrium of a game. Such an equilibrium assigns a strategy to each player so that each strategy in the assignment is a best response to the others. Suppose that two people are playing Matching
Pennies. Simultaneously, each displays a penny. One player tries to make the sides match, and the other player tries to prevent a match. If the first player succeeds, he gets both pennies. Otherwise,
the second player gets both pennies. Suppose that each player is good at predicting the other player, and each player knows this. Then if the first player displays heads, he has reason to think that
the second player displays tails. Also, if the first player displays tails, he has reason to think that the second player displays heads. Because Matching Pennies is a simultaneous-move game, neither
player’s strategy influences the other player’s strategy, but each player’s strategy is evidence of the other player’s strategy. Mixed strategies help resolve decision instability in this case. If
the first player flips his penny to settle the side to display, then his mixed strategy is self-ratifying. The second player’s situation is similar, and she also reaches a self-ratifying strategy by
flipping her penny. The combination of self-ratifying strategies is a Nash equilibrium of the game.
Weirich (2004: Chap. 9) presents a method of selecting among multiple self-ratifying strategies, and hence a method by which a group of players may coordinate to realize a particular Nash equilibrium
when several exist. Although decision instability is an open problem, causal decision theory has resources for addressing it. The theory’s eventual resolution of the problem will offer game theory a
justification for participation in a Nash equilibrium of a game.
4. Related Topics and Concluding Remarks
Causal decision theory has foundations in various areas of philosophy. For example, it relies on metaphysics for an account of causation. It also relies on inductive logic for an account of
inferences concerning causation. A comprehensive causal decision theory treats not only causal probabilities’ generation of options’ expected utilities, but also evidence’s generation of causal
Research concerning causation contributes to the metaphysical foundations of causal decision theory. Nancy Cartwright (1979), for example, draws on ideas about causation to flesh out details of
causal decision theory. Also, some accounts of causation distinguish types of causes. Both oxygen and a flame are metaphysical causes of tinder’s combustion. However, only the flame is causally
responsible for, and so a normative cause of, the combustion. Causal responsibility for an event accrues to just the salient metaphysical causes of the event. Causal decision theory is interested not
only in events for which an act is causally responsible, but also in other events for which an act is a metaphysical cause. Expected utilities that guide decisions are comprehensive.
Judea Pearl (2000) and also Peter Spirtes, Clark Glymour, and Richard Scheines (2000) present methods of inferring causal relations from statistical data. They use directed acyclic graphs and
associated probability distributions to construct causal models. In a decision problem, a causal model yields a way of calculating an act’s effect. A causal graph and its probability distribution
express a dependency hypothesis and yield each act’s causal influence given that hypothesis. They specify the causal probability of a state under supposition of an act. An act’s expected utility is a
probability-weighted average of its expected utility according to the dependency hypotheses that candidate causal models represent, as Weirich (2015: 225–236) explains.
A causal model’s directed graph and probability distribution indicate causal relations among event types. As Pearl (2000: 30) and Sprites et al. (2000: 11) explain, a causal model meets the causal
Markov condition if and only if with respect to its probability distribution each event type in its directed graph is independent of all the event type’s nondescendants, given its parents. Given a
model meeting the condition, knowledge of all an event’s direct causes makes other information statistically irrelevant to the event’s occurrence, except for information about the event and its
effects. Knowledge of an event’s direct causes screens off evidence from indirect causes and independent effects of its causes. Given a typical causal model for Newcomb’s problem, knowledge of the
common cause of a decision and a prediction screens off the correlation between the decision and the prediction.
Directed acyclic graphs present causal structure clearly, and so clarify in decision theory points that depend on causal structure. For example, Eells (2000) observes that choice is not genuine
unless a decision screens off an act’s correlation with states. Joyce (2007: 546) uses a causal graph to depict how this may happen in a Newcomb problem that arises in a Prisoner’s Dilemma with a
psychological twin. He shows that the Newcomb problem is a genuine choice despite correlation of acts and states because a decision screens off that correlation. Spohn (2012) constructs for Newcomb’s
problem a causal model that distinguishes a decision and its execution and argues that given the model causal decision theory recommends one-boxing. An act in a decision problem may constitute an
intervention in the causal model for the decision problem, as Christopher Meek and Clark Glymour (1994) explain. Hitchcock (2016) and Joyce and Gibbard (2016) maintain that treating an act as an
intervention enriches causal decision theory.
Timothy Williamson (2007: Chap. 5) studies the epistemology of counterfactual, or subjunctive, conditionals. He points out their role in contingency planning and decision making. According to his
account, one learns a subjunctive conditional if one robustly obtains its consequent when imagining its antecedent. Experience disciplines imagination. The experience leading to a judgment that a
subjunctive conditional holds may be neither strictly enabling nor strictly evidential so that knowledge of the conditional is neither purely a priori nor purely a posteriori. Williamson claims that
knowledge of subjunctive conditionals is foundational so that decision theory appropriately grounds knowledge of an act’s choiceworthiness in knowledge of such conditionals.
Most texts on decision theory are consistent with causal decision theory. Many do not treat the special cases, such as Newcomb’s problem, that motivate a distinction between causal and evidential
decision theory. For example, Leonard Savage (1954) analyzes only decision problems in which options do not affect probabilities of states, as his account of utility makes clear (1954: 73). Causal
and evidential decision theories reach the same recommendations in these problems. Causal decision theory is the prevailing form of decision theory among those who distinguish causal and evidential
decision theory.
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• MIT Course on Decision Theory, offered by Robert Stalnaker.
• Decision Theory, as of this writing (November 4, 2016), the Wikipedia site has a good overall introduction to decision theory and a list of references.
I thank Christopher Haugen for bibliographical research and Brad Armendt, David Etlin, William Harper, Xiao Fei Liu, Calum McNamara, Brian Skyrms, Howard Sobel, and an anonymous referee for helpful | {"url":"https://plato.stanford.edu/ENTRIES/decision-causal/","timestamp":"2024-11-06T11:05:45Z","content_type":"text/html","content_length":"107261","record_id":"<urn:uuid:33e84b03-2fa0-4b37-b045-64923fd92116>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00790.warc.gz"} |
22 research outputs found
We consider dense rapid shear flow of inelastically colliding hard disks. Navier-Stokes granular hydrodynamics is applied accounting for the recent finding \cite{Luding,Khain} that shear viscosity
diverges at a lower density than the rest of constitutive relations. New interpolation formulas for constitutive relations between dilute and dense cases are proposed and justified in molecular
dynamics (MD) simulations. A linear stability analysis of the uniform shear flow is performed and the full phase diagram is presented. It is shown that when the inelasticity of particle collision
becomes large enough, the uniform sheared flow gives way to a two-phase flow, where a dense "solid-like" striped cluster is surrounded by two fluid layers. The results of the analysis are verified in
event-driven MD simulations, and a good agreement is observed
We investigate shear-induced crystallization in a very dense flow of mono-disperse inelastic hard spheres. We consider a steady plane Couette flow under constant pressure and neglect gravity. We
assume that the granular density is greater than the melting point of the equilibrium phase diagram of elastic hard spheres. We employ a Navier-Stokes hydrodynamics with constitutive relations all of
which (except the shear viscosity) diverge at the crystal packing density, while the shear viscosity diverges at a smaller density. The phase diagram of the steady flow is described by three
parameters: an effective Mach number, a scaled energy loss parameter, and an integer number m: the number of half-oscillations in a mechanical analogy that appears in this problem. In a steady shear
flow the viscous heating is balanced by energy dissipation via inelastic collisions. This balance can have different forms, producing either a uniform shear flow or a variety of more complicated,
nonlinear density, velocity and temperature profiles. In particular, the model predicts a variety of multi-layer two-phase steady shear flows with sharp interphase boundaries. Such a flow may include
a few zero-shear (solid-like) layers, each of which moving as a whole, separated by fluid-like regions. As we are dealing with a hard sphere model, the granulate is fluidized within the "solid"
layers: the granular temperature is non-zero there, and there is energy flow through the boundaries of the "solid" layers. A linear stability analysis of the uniform steady shear flow is performed,
and a plausible bifurcation diagram of the system, for a fixed m, is suggested. The problem of selection of m remains open.Comment: 11 pages, 7 eps figures, to appear in PR
The position of a reaction front, propagating into a metastable state, fluctuates because of the shot noise of reactions and diffusion. A recent theory [B. Meerson, P.V. Sasorov, and Y. Kaplan, Phys.
Rev. E 84, 011147 (2011)] gave a closed analytic expression for the front diffusion coefficient in the weak noise limit. Here we test this theory in stochastic simulations involving reacting and
diffusing particles on a one-dimensional lattice. We also investigate a small noise-induced systematic shift of the front velocity compared to the prediction from the spatially continuous
deterministic reaction-diffusion equation.Comment: 5 pages, 5 figure
We present a discrete stochastic model which represents many of the salient features of the biological process of wound healing. The model describes fronts of cells invading a wound. We have
numerical results in one and two dimensions. In one dimension we can give analytic results for the front speed as a power series expansion in a parameter, p, that gives the relative size of
proliferation and diffusion processes for the invading cells. In two dimensions the model becomes the Eden model for p near 1. In both one and two dimensions for small p, front propagation for this
model should approach that of the Fisher-Kolmogorov equation. However, as in other cases, this discrete model approaches Fisher-Kolmogorov behavior slowly.Comment: 16 pages, 7 figure
The importance of a strict quarantine has been widely debated during the COVID-19 epidemic even from the purely epidemiological point of view. One argument against strict lockdown measures is that
once the strict quarantine is lifted, the epidemic comes back, and so the cumulative number of infected individuals during the entire epidemic will stay the same. We consider an SIR model on a
network and follow the disease dynamics, modeling the phases of quarantine by changing the node degree distribution. We show that the system reaches different steady states based on the history: the
outcome of the epidemic is path-dependent despite the same final node degree distribution. The results indicate that two-phase route to the final node degree distribution (a strict phase followed by
a soft phase) are always better than one phase (the same soft one) unless all the individuals have the same number of connections at the end (the same degree); in the latter case, the overall number
of infected is indeed history-independent. The modeling also suggests that the optimal procedure of lifting the quarantine consists of releasing nodes in the order of their degree - highest
first.Comment: 6 pages, 4 figures, accepted to EPL (Europhysics Letters
In this work, we study the in-vitro dynamics of the most malignant form of the primary brain tumor: Glioblastoma Multiforme. Typically, the growing tumor consists of the inner dense proliferating
zone and the outer less dense invasive region. Experiments with different types of cells show qualitatively different behavior. Wild-type cells invade a spherically symmetric manner, but mutant cells
are organized in tenuous branches. We formulate a model for this sort of growth using two coupled reaction-diffusion equations for the cell and nutrient concentrations. When the ratio of the nutrient
and cell diffusion coefficients exceeds some critical value, the plane propagating front becomes unstable with respect to transversal perturbations. The instability threshold and the full phase-plane
diagram in the parameter space are determined. The results are in a good agreement with experimental findings for the two types of cells.Comment: 4 pages, 4 figure
Many populations in nature are fragmented: they consist of local populations occupying separate patches. A local population is prone to extinction due to the shot noise of birth and death processes.
A migrating population from another patch can dramatically delay the extinction. What is the optimal migration rate that minimizes the extinction risk of the whole population? Here we answer this
question for a connected network of model habitat patches with different carrying capacities.Comment: 7 pages, 3 figures, accepted for publication in PRL, appendix contains supplementary materia
We consider population dynamics on a network of patches, each of which has a the same local dynamics, with different population scales (carrying capacities). It is reasonable to assume that if the
patches are coupled by very fast migration the whole system will look like an individual patch with a large effective carrying capacity. This is called a "well-mixed" system. We show that, in
general, it is not true that the well-mixed system has the same dynamics as each local patch. Different global dynamics can emerge from coupling, and usually must be figured out for each individual
case. We give a general condition which must be satisfied for well-mixed systems to have the same dynamics as the constituent patches.Comment: 4 page | {"url":"https://core.ac.uk/search/?q=authors%3A(Evgeniy%20Khain)","timestamp":"2024-11-09T03:13:02Z","content_type":"text/html","content_length":"123727","record_id":"<urn:uuid:c6fb59a6-f75b-4697-96d3-fc983fd0ca57>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00285.warc.gz"} |
Causality: Probabilities of Causation
Why read this?
Questions of attribution are everywhere: i.e., did \(X=x\) cause \(Y=y\)? From legal battles to personal decision making, we are obsessed by them. Can we give a rigorous answer to the problem of
One alternative to solve the problem of attribution is to reason in the following manner: if there is no possible alternative causal process, which does not involve \(X\),that can cause \(Y=y\), then
\(X=x\) is necessary to produce the effect in question. Therefore, the effect \(Y=y\) could not have happened without \(X=x\). In causal inference, this type of reasoning is studied by computing the
Probability of Necessity (PN)
In the above alternative, however, the reasoning is tailored to a specific event under consideration. What if we are interested in studying a general tendency of a given effect? In this case, we are
asking how sufficient is a cause, \(X=x\), for the production of the effect, \(Y=y\). We answer this question with the Probability of Sufficiency (PS)
In this blogpost, we will give counterfactual interpretations to both probabilities: \(PN\) and \(PS\). Thereby, we will be able to study them in a systematic fashion using the tools of causal
inference. Although we will realize that they are not generally identifiable from a causal diagram and data alone, if we are willing to assume monotonicity, we will be able to estimate both \(PN\)
and \(PS\) with a combination of experimental and observational data.
Finally, we will work through some examples to put what we have learnt into practice. This blogpost follows the notation of Pearl’s Causality, Chapter 9 and Pearl’s Causal Inference in Statistics: A
Counterfactual definitions
Let \(X\) and \(Y\) be two binary variables in causal model. In what follows, we will give counterfactual interpretations to both \(PN\) and \(PS\).
Probability of Necessity
The Probability of Necessity (PN) stands for the probability that the event \(Y=y\) would not have occurred in the absence of event \(X=X\), given that \(x\) and \(y\) did in fact occur.
\[ PN := P(Y_{x'} = false | X = true, Y = true) \]
To gain some intuition, imagine Ms. Jones: a former cancer patient that underwent both a lumpectomy and irradiation. She speculates: do I owe my life to irradiation? We can study this question by
figuring out how necessary was the lumpectomy for the remission to occur:
\[ PN = P(Y_{\text{no irradiation}} = \text{no remission}| X = irradiation, Y = remission) \]
If \(PN\) is high, then, yes, Ms. Jones owes her life to her decision of having irradiation.
Probability of Sufficiency
On the other hand:
The Probability of Sufficiency (PS) measures the capacity of \(x\) to produce \(y\), and, since “production” implies a transition from absence to presence, we condition \(Y_x\) on situations
where x and y are absent
\[ PS := P(Y_x = true | X = false, Y = false) \]
The following example may clarify things. Imagine that, contrary to Ms. Jones above, Mrs. Smith had a lumpectomy alone and her tumor recurred. She speculates on her decision and concludes: I should
have gone through irradiation. Is this regret warranted? We can quantify this by speaking about \(PS\):
\[ PS = P(Y_{irradiation} = remission| X = \text{no irradiation}, Y = \text{no remission}) \] That is, \(PS\) computes the probability that remission would have occurred had Mrs. Smith gone through
irradiation, given that she did not go and remission did not occur. Thus, it measures the degree to which the action not taken, \(X=1\), would have been sufficient for her recovery.
Combining both probabilities
We can compute the probability that the cause is necessary and sufficient thus:
\[ P N S:= P(y_{x}, y'_{x'}) =P(x, y) P N+P\left(x^{\prime}, y^{\prime}\right) P S \]
That is, the contribution of \(PN\) is amplified or reduced by \(P(x, y)\) and likewise for the \(PS\)’s contribution.
In the general case, when we have a causal diagram and observed (and experimental) data, neither \(PN\) nor \(PS\) are identifiable. This happens due to the relationship between the counterfactual’s
antecedent and the fact that we are conditioning on \(Y\). If we wanted to model this relationship, we would need to know the functional relationship between \(X, Pa_y\) and \(Y\).
In practice, if we don’t know the functional relationship, we must at least assume monotonicity of \(Y\) relative to \(X\) to be able to identify them. Otherwise, we must content ourselves with
theoretically sharp bounds on the probabilities of causation.
What is Monotonicity?
Let \(u\) be the unobserved background variables in a SCM, then, \(Y\) is monotonic relative to \(X\) if:
\[ Y_1(u) \geq Y_0 (u) \]
That is, exposure to treatment \(X=1\) always helps to bring about \(Y=1\).
Identifying the probabilities of causation
If we are willing to assume that Y is monotonic relative to X, then both \(PN\) and \(PS\) are identifiable when the causal effects \(P(Y| do(X=1)\) and \(P(Y| do(X=0)\) are identifiable. Whether
this causal effect is identifiable because we have experimental evidence, or because we can identify them by using the Back-Door criterion or other graph-assisted identification strategy, it does not
matter. Then, we can estimate \(PN\) with a combination of do-expressions and observational data thus:
\[ PN = P(y'_{x'} | x, y) = \frac{P(y) - P(y| do (x'))}{P(x, y)} \]
Moreover, if monotonicity does not hold, the above expression becomes a lower bound for \(PN\). The complete bound is the following:
\[ \max \left\{0, \frac{P(y)-P\left(y \mid d o\left(x^{\prime}\right)\right)}{P(x, y)}\right\} \leq P N \leq \min \left\{1, \frac{P\left(y^{\prime} \mid d o\left(x^{\prime}\right)\right)-P\left(x^{\
prime}, y^{\prime}\right)}{P(x, y)}\right\} \]
Equivalently, \(PS\) can be estimated thus:
\[ PS = P(y_x | x', y') = \frac{P(y| do(x)) - P(y)}{P(x', y')} \]
Which becomes the lower bound if we are not willing to assume monotonicity:
\[ \max \left\{\begin{array}{c} 0, \frac{P\left(y | do (x) \right)-P(y)}{P\left(x^{\prime}, y^{\prime}\right)} \end{array}\right\} \leq P S \leq \min \left\{\begin{array}{c} 1, \frac{P\left(y | do(x)
\right)-P(x, y)}{P\left(x^{\prime}, y^{\prime}\right)} \end{array}\right\} \]
Let’s use the estimators and the bounds in the following example.
A first example
A lawsuit is filed against the manufacturer of a drug X that was supposed to relieve back-pain. Was the drug a necessary cause for the the death of Mr. A?
The experimental data provide the estimates [ \[\begin{aligned} P(y \mid d o(x)) &=16 / 1000=0.016 \\ P\left(y \mid d o\left(x^{\prime}\right)\right) &=14 / 1000=0.014 \end{aligned}\] ] whereas the
non-experimental data provide the estimates [ P(y)=30 / 2000=0.015 ] [ \[\begin{array}{l} P(x, y)=2 / 2000=0.001 \\ P(y \mid x)=2 / 1000=0.002 \\ P\left(y \mid x^{\prime}\right)=28 / 1000=0.028 \end
Therefore, assuming that the drug could only cause (but never prevent death), monotonicity holds:
\[ PN = \frac{0.015-0.014}{0.001} = 1 \]
The plaintiff was correct; barring sampling errors, the data provide us with 100% assurance that drug x was in fact responsible for the death of Mr A.
A second example
Remember Ms. Jones? Is she right in attributing her recovery to the irradiation therapy. Suppose she gets her hands on the following data:
\[ \begin{aligned} P\left(y^{\prime}\right) &=0.3 \\ P\left(x^{\prime} \mid y^{\prime}\right) &=0.7 \\ P(y \mid d o(x)) &=0.39 \\ P\left(y \mid d o\left(x^{\prime}\right)\right) &=0.14 \end{aligned}
We can therefore start to bound \(PN\) to figure out whether irradiation was necessary for remission:
\[ \begin{aligned} P N & \geq \frac{P(y)-P\left(y \mid d o\left(x^{\prime}\right)\right)}{P(x, y)} \\ &=\frac{P(y)-P\left(y \mid d o\left(x^{\prime}\right)\right)}{P(y \mid x) P(x)} \\ &=\frac{P(y)-P
\left(y \mid d o\left(x^{\prime}\right)\right)}{\left(1-\frac{P\left(x \mid y^{\prime}\right) P\left(y^{\prime}\right)}{P(x)}\right) P(x)} \\ &=\frac{P(y)-P\left(y \mid d o\left(x^{\prime}\right)\
right)}{P(x)-P\left(x \mid y^{\prime}\right) P\left(y^{\prime}\right)} \end{aligned} \]
We don’t have data for \(P(x)\). However, given that we are interested in a lower bound, we can choose the parametrization that yields the smallest bound: \(P(x) = 1\). Therefore, the bound becomes:
\[ \begin{aligned} P N & \geq \frac{P(y)-P\left(y \mid d o\left(x^{\prime}\right)\right)}{P(x)-P\left(x \mid y^{\prime}\right) P\left(y^{\prime}\right)} \\ & \geq \frac{0.7-0.14}{1-(1-0.7) * 0.3} = \
\ & 0.62 > 0.5 \end{aligned} \]
Therefore, irradiation was more likely than not necessary for her remission.
In this blogpost, we saw how we can analyze attribution problems by giving counterfactual interpretations to the probability that a cause is necessary and/or sufficient. These quantities turn out be
generally not identifiable because they are sensitive to the specific functional relationships that connect \(X\) and \(Y\) in a SCM.
However, we can give theoretically sharp bounds for them by combining experimental and observational data. If we are willing to assume monotonicity, the bounds collapse to give a point estimate for
both probabilities of causation, \(PN\) and \(PS\). | {"url":"https://david-salazar.github.io/posts/causality/2020-08-20-causality-probabilities-of-causation.html","timestamp":"2024-11-01T19:31:48Z","content_type":"application/xhtml+xml","content_length":"27263","record_id":"<urn:uuid:436c5067-12d3-41a6-bb4c-84fcc32692f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00628.warc.gz"} |
Homotopy of Product Systems and K-Theory of Cuntz-Nica-Pimsner Algebras
We introduce the notion of a homotopy of product systems, and show that the Cuntz-Nica-Pimsner algebras of homotopic product systems over N^k have isomorphic K-theory. As an application, we give a
new proof that the K-theory of a 2-graph C^*-algebra is independent of the factorisation rules, and we further show that the K-theory of any twisted 2-graph C^*-algebra is independent of the twisting
2-cocycle. We also explore applications to K-theory for the C^*-algebras of single-vertex k-graphs, reducing the question of whether the K-theory is independent of the factorisation rules to a
question about path-connectedness of the space of solutions to an equation of Yang-Baxter type.
• Cuntz-Nica-Pimsner
• K-theory
• Product system
• higher-rank graph
Dive into the research topics of 'Homotopy of Product Systems and K-Theory of Cuntz-Nica-Pimsner Algebras'. Together they form a unique fingerprint. | {"url":"https://umimpact.umt.edu/en/publications/homotopy-of-product-systems-and-k-theory-of-cuntz-nica-pimsner-al","timestamp":"2024-11-07T00:44:46Z","content_type":"text/html","content_length":"45999","record_id":"<urn:uuid:a3e93a34-7d59-4531-ace1-267308f031e7>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00313.warc.gz"} |
Nonlinear interaction of photons and phonons in electron-positron plasmas
Nonlinear interaction of electromagnetic waves and acoustic modes in an electron-positron plasma is investigated. The plasma of electrons and positrons is quite plastic so that the imposition of
electromagnetic (EM) waves causes depression of the plasma and other structural imprints on it through either the nonresonant or resonant interaction. The theory shows that the nonresonant
interaction can lead to the coalescence of photons and collapse of plasma cavity in higher (greater than or = to 2) dimensions. The resonant interaction, in which the group velocity of EM waves is
equal to the phase velocity of acoustic waves, is analyzed and a set of basic equations of the system is derived via the reductive perturbation theory. New solutions of solitary types were found:
bright solitons, kink solitons, and dark solitons as the solutions to these equations. The computation hints their stability. An impact of the present theory on astrophysical plasma settings is
expected, including the cosmological relativistically hot electron-positron plasma.
Pub Date:
March 1990
□ Electron-Positron Plasmas;
□ Fokker-Planck Equation;
□ Nonlinearity;
□ Pair Production;
□ Phonons;
□ Photons;
□ Shock Wave Propagation;
□ Solitary Waves;
□ Annihilation Reactions;
□ Electromagnetic Wave Transmission;
□ Maxwell Equation;
□ Plasma Waves;
□ Schroedinger Equation;
□ Plasma Physics | {"url":"https://ui.adsabs.harvard.edu/abs/1990nipp.rept.....T/abstract","timestamp":"2024-11-07T16:33:32Z","content_type":"text/html","content_length":"36703","record_id":"<urn:uuid:89b7cef4-a812-40d8-8598-3f574dbae787>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00668.warc.gz"} |
Molecular Biology and Evolution (MBE) LaTeX Template and Example Article
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Some Probability and Topological Logics
Title: Some Probability and Topological Logics
Author: Ikodinović, Nebojša
The thesis is devoted to logics which are applicable in different areas of mathematics (such as topology and probability) and computer sciences (reasoning with uncertainty). Namely, some
extensions of the classical logic, which are either model-theoretical or non-classical, are studied. The thesis consists of three chapters: an introductory chapter and two main parts
(Chapter 2 and Chapter 3). In the introductory chapter of the thesis the well-known notions and properties from extensions of the first order logic and nonclassical logics are presented.
Abstract: Chapter 2 of the thesis is related to logics for topological structures, particularly, topological class spaces (topologies on proper classes). One infinite logic with new quantifiers added
is considered as the corresponding logic. Methods of constructing models, which can be useful for many others similar logics, are used to prove the completeness theorem. A number of
probabilistic logic suitable for reasoning with uncertainty are investigated in Chapter 3. Especially, some ways of incorporation into the realm of logic conditional probability understood
in different ways (in the sense of Kolmogorov or De Finnety) are given. For all these logics the corresponding axiomatizations are given and the completeness for each of them is proved. The
decidability for all these logics is discussed too.
URI: http://hdl.handle.net/123456789/194
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Senior Thesis Projects | Department of Physics
The following options are available to satisfy the writing intensive curriculum (WIC). There is a formal class (PH317), or, students can write a senior thesis with one-on-one guidance from a research
mentor. If working with a research mentor, you will enroll in PH401 (research) and PH403 (thesis).
PH317 - Adv. Phys. Lab
12 available projects
Project information
The advanced physics lab gives students experience with designing and performing physics experiments and writing advanced lab reports. The next offering of PH317 is Winter term, 2025.
David Craig
0 available projects
Continuing students: Jake Bullard ('25), Emmitt Allen ('26)
Project information
Quantum measurements
Canonical structure of loop quantum cosmology/Hamiltonian cosmology
Effective dynamics in loop quantum cosmology
Other topics in quantum theory or gravitation
Paul Emigh
1 available project
Continuing students: new!
Project information
• Physics Education Research: evaluating effectiveness and student attitudes toward new online book; evaluating effectiveness and improving quality of pedagogical representations in introductory
Elizabeth Gire
1 available project
Graduating students: Seghers ('23), Shapton ('24)
Continuing students:
Project information
• Students use of various sensemaking strategies in PH315 and PH335
• Development of course materials and assessments for Paradigms courses.
Matt Graham
0-1 available project
Current students: Logan Winder ('25), Patrick Moret ('25 or '26), Elijah Zacharia ('26, active URSA)
If interested, please consider applying to URSA-ENGAGE advertisement, SURE or other scholarship programs through with lab (http://sites.science.oregonstate.edu/physics/energetics/). End of the Spring
term is the best time to inquire unless applying for a scholarship program (then ask early, ask often).
Project information
Optoelectronics: electronic confinement and current generation in emerging semiconductor materials. Optical laser spectroscopy.
Kathryn Hadley
0 available projects
Graduating students: Jones, C ('24), Jackson, D ('24)
Project information
Computational astrophysics: modeling protostellar systems; Rossby wave instabilities, plasma shocks
Patti Hamerski
1 available project
Continuing Students: New!
Project information
Physics education research — specializing in computational physics education and other topics. Reach out to learn more!!
Jeff Hazboun
1 available project
Graduating students: Peter Orndoff ('24), Maddie Thompson ('24)
Continuing students: Kyle Gourlie ('25), Trevor Le Rarick ('25)
Project information
Gravitational wave astronomy with pulsar timing arrays. Using signal analysis, Bayesian data analysis methods and astrostatistics to understand gravity. Computational astrophysics focused on
supermassive binary blackholes.
Pavel Kornilovich (Hewlett Packard)
0 available projects
Project information
• (computational) Stable knots in nematic liquid crystals.
Davide Lazzati
2 available projects
Graduating students: Ian Busby ('24)
Project information
Computational astrophysics: 1) modeling gamma-ray burst light-cures and spectra and 2) 3D visualization of granular bodies collisions
Yun-Shik Lee
0 available projects for 24-25
Graduating students: Johnson, N ('24), Worrell C ('24)
Project information
• High-Field Terahertz spectroscopy of 2D materials
Yangqiuting (Doris) Li
3 available projects
Continuing students: new!
Project information
My research focuses on enhancing students' physics learning and motivational beliefs in both introductory and advanced physics courses. Specifically, one of my research directions is to investigate
students' sense-making processes in physics learning and assist them in establishing connections among physics concepts and their various representations. Another goal of my research is to reduce
demographic gaps in students’ academic achievement and motivational beliefs by investigating how to create an inclusive and equitable learning environment in which all students can thrive.
Ethan Minot
2 available projects
Graduating students: Bryce Wall ('24)
Continuing students: Ryu Joy ('25), Miller Nelson ('25)
Project information
1. Physics experiments using atomically thin semiconductor crystals. Both application-driven* and hypothesis-driven experiments**.
2. Numerical simulation of topological physics in a solid state device using split-step Fourier method to simulate 1d wave function in a time-dependent potential.
*Application example: single-pixel spectrometer that use voltage tuning and machine learning.
**Hypothesis example: we suspect that van-der-Waals junctions have a new degree of freedom.
Oksana Ostroverkhova
2 available projects
Graduating students: Jason Culley ('24), Claire Swartz ('24)
Continuing students: Madalyn Gragg ('25), Aidan Bagshaw ('25), Corey Cleveland ('26)
Project information
Light-matter interactions in organic microcavities and plasmonic nanostructures
Project 1: organic optoelectronic devices;
Project 2: properties of 2D magnets
Vanessa Polito
1 available project
Graduating students: Gessner, J. ('24), Guillen, R ('24)
Project information
Solar physics research projects will involve analysis of data from NASA satellites to study solar flares or other energetic events on the Sun and/or comparison with computational models.
Weihong Qiu
2 available projects
Graduating students: Owen Williamson (BB, '24)
Continuing students:
Project information
Experimental/Computation Biophysics
• Computational biophysics: modeling the interaction between molecular motors with microtubules;
• Experimental biophysics: determine how molecular motors determine the directionality of movement
Heidi Schellman
0 available projects
Continuing students: Phoebe Andromeda ('25)
Project information
Neutrino physics and large scale data analysis.
Xavier Siemens
Continuing students: Pelletier (?)
Project information
• Using radio telescopes to search for pulsars
• Using radio telescopes to perform gravitational wave observations (pulsar timing)
• Searching pulsar timing data for nanohertz gravitational waves.
Nick Siler
1 available project
Continuing students: Weigel, M ('24)
Project information
How will climate change affect extreme precipitation from atmospheric rivers in western North America? Involves analyzing numerical simulations using Matlab or Python.
Bo Sun
1 available project
Graduating students: Hanna O'Meara ('24), Ty Zuber ('24), Hailey Richter ('24)
Continuing students:
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• computer model of cell migration
Rebecka Tumblin
Project information
• Computational astrophysics.
• Metalicity of stars and planets, data mining, cluster analysis
Graduating students: Vincent Vaughn-Uding ('24)
KC Walsh
0 available projects
Graduating students: Toll, M ('23)
Project information
• Ecampus comparative study of introductory physics using educational data mining and learning analytics.
• Predictive modeling student success using various artificial intelligence methods.
• Language processing student's reflective writing.
Allied-disciplines, OSU advisors:
Chong Fang (Chemistry)
0-1 available projects
Graduating students: Bailey-Darland, S ('23)
Project information
Using steady-state and time-resolved spectroscopic methods to study fluorescence mechanisms of functional molecular systems (e.g., proteins, chromophores, biosensors) in solution. Contact Dr. Fang at
Chong.Fang@oregonstate.edu or visit the group website.
Erin Pettit (CEOAS)
College of Earth, Ocean, and Atmospheric Sciences
Paul Cheong (Chemistry)
Computational Chemistry
Brian Woods (Nuc. Eng.)
Graduating students: Holler ('23)
Continuing students: Miles, H ('24)
Project information
Inertial Electrostatic Confinement Reactor
Vince Remcho (Chemistry)
Graduating students: Chirica ('23)
Allied-disciplines, advisors outside OSU: | {"url":"https://physics.oregonstate.edu/undergraduate/academics/writing-intensive-courses/thesis-project","timestamp":"2024-11-11T04:16:58Z","content_type":"text/html","content_length":"70035","record_id":"<urn:uuid:4200f389-7ee1-4d34-bda3-c3fa5c2a736f>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00244.warc.gz"} |
Q1 Write code in python that performs 1D convolution with inputs A Random input of length 100K or more let it be huge for eg 120KB Random filters of length 3 to 25 or more any rand Archives - Gotit Pro
Q1) Write code in python that performs 1D convolution with inputs: A) Random input of length 100K or more (let it be huge, for eg 120K)B) Random filters of length 3 to 25 or more (any random, eg 5)
need to write two implementations of 1D convolution (one using for loops and other using matrix multiplication) in python as follows:1) A naive for-loop implementation of filter applied on input2)
Matrix multiplication implementations(i) where the input is the column vector, (ii) where the filter is t | {"url":"https://gotit-pro.com/tag/q1-write-code-in-python-that-performs-1d-convolution-with-inputs-a-random-input-of-length-100k-or-more-let-it-be-huge-for-eg-120kb-random-filters-of-length-3-to-25-or-more-any-rand/","timestamp":"2024-11-09T14:23:18Z","content_type":"text/html","content_length":"79228","record_id":"<urn:uuid:36e8ab8f-b9c0-4c0b-b172-594fec54b42a>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00864.warc.gz"} |
The integral
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Certain infinite series can be studied using improper integrals.
In order to study the convergence of a series
$\sum _{k=k_0}^{\infty } a_k$
, our first attempt to determine whether the series converges is to form the sequence of partial sums
since we know that the series
$\sum _{k=k_0}^{\infty } a_k$
converges if and only if
$\lim _{n \to \infty } s_n$
exists. In the case of geometric or telescoping series, we were able to find an explicit formula for
, and analyze
$\lim _{n \to \infty } s_n$
by explicit computation. However, we cannot always find such an explicit formula, and when this is the case, we try to use properties of the terms in the sequence
to determine whether
$\lim _{n \to \infty } s_n$
exists. Our first result was the divergence test, which states
If $\lim _{n \to \infty } a_n eq 0$, then $\sum _{k=k_0} a_k$ diverges.
However, there are still some divergent series that the divergence test does not pick out! We begin this section with such an example that shows how there is a connection between certain special
types of series and improper integrals.
Consider the series
$\sum _{k=1}^{\infty } \frac {1}{k}$
. This series is called the
harmonic series
and will be important in the coming material. Notice that this series
isis not
geometric. It
isis not
telescoping, and
$\lim _{n \to \infty } \frac {1}{n} = \answer {0}$
, so the divergence test
isis not
conclusive. So what should we do?
We have seen that we can graph a sequence as a collection of points in the plane. We consider the harmonic sequence where $a_n = 1/n$, and write out the ordered list that represents it. If we plot
the harmonic sequence, it looks like this.
As it turns out, there is a nice way to visualize the sum $\sum _{k=1}^\infty \frac {1}{k}$ too! One such way is to to make rectangles whose areas are equal to the terms in the sequence.
Note that the height of the $k$-th rectangle is precisely $\frac {1}{k}$ and the width of all of the rectangles is $1$, so the area of the $k$-th rectangle is $\frac {1}{k}$.
Now, in order to conclude whether $\sum _{k=1}^{\infty } a_k$ converges, we must analyze $\lim _{n \to \infty } s_n$. Note that $s_n$ has a nice visual interpretation as the sum of the areas of the
first $n$ rectangles, but since we don’t have an explicit formula for $s_n$ we can try to establish that
• $\{s_n\}$ is bounded and monotonic, and hence $\lim _{n \to \infty } s_n$ exists.
• $\{s_n\}$ is unbounded so $\lim _{n \to \infty } s_n$ does not exist.
How can we establish this? The previous image might remind you of a Riemann Sum and for good reason. This technique lets us visually compare the sum of an infinite series to the value of an improper
integral. For instance, if we add a plot of $1/x$ to our picture above
we see that
Notice that the sum on the righthand side is simply $s_n$. Since $\int _1^{\infty } \frac {1}{x} \d x$ is an improper integral, so we need to determine whether $\lim _{n \to \infty } \left [ \int _1^
n \frac {1}{x} \d x \right ]$ exists. Notice
This means that $\{s_n\}$ is not bounded and hence $\lim _{n \to \infty } s_n = \infty$ and $\sum _{k=1}^{\infty } \frac {1}{k}$ must diverge.
To argue why
is not bounded a little more formally, note that if
were bounded, then
of the terms in the sequence
would be less than some number
. Since
$\lim _{n \to \infty } \left [ \int _1^n \frac {1}{x} \d x \right ] = \infty$
, this means that, for some value of
after which all values of
$\int _1^n \frac {1}{x} \d x > M$
, and since
$\int _1^n \frac {1}{x} \d x < s_n$
for all
, we may conclude that
$s_n >M$
for all
$n\geq N$
and hence
is unbounded.
Now, let’s take a step back and see what we really needed in the previous example.
• We needed to find a function for which the area under the curve over any particular interval $[n,n+1]$ was less than the area of the rectangle whose height is $a_k$ to establish a lower bound for
each $s_n$. Note that we can always do this if $f(x)$ is eventually positive and decreasingincreasing since we may view each $a_k$ as the area of the rectangle that coincides with $f(x)$ at its
lefthandrighthand endpoint.
• We needed the function to be “eventually continuous” so the improper integral $\int _{a}^{\infty } f(x) \d x$ can be computed as the limit of a single definite integral.
By “eventually” above, we really mean that $f(x)$ should be continuous, positive, and decreasing on some interval $[a,\infty )$ for some $a>0$; it doesn’t need to happen right away, but it should
hold for all real large enough $x$-values. This leads us to an interesting observation.
Let $f(x)$ be an eventually continuous, positive, and decreasing function with $a_k = f(k)$. If $\int _1^\infty f(x) \d x$ diverges, so does $\sum _{k=1}^\infty a_k$.
That’s a pretty good observation, but we can do even better.
Consider the sequence
$a_n = \frac {1}{n^2}$
. We can write out a few terms
and try the same idea of visualizing the series $\sum _{k=1}^{\infty } \frac {1}{k^2}$ as an area.
The shaded area above
represents to the sum We again visually compare the sum of an infinite series to the value of an improper integral by adding a plot of
to our picture above.
Note that we have a slight annoyance if we consider $\int _{0}^{\infty } \frac {1}{x^2} \d x$ since $\frac {1}{x^2}$ has a vertical asymptote at $x=0$. This is easily avoided if we instead consider $
\frac {1}{x^2}$ on the interval $[1,\infty )$. This requires that we consider the rectangles on that interval too. We update our picture.
Notice that the sum of the areas of the rectangles now is the series $\sum _{k=0}^{\infty } \frac {1}{k^2}$$\sum _{k=1}^{\infty } \frac {1}{k^2}$$\sum _{k=2}^{\infty } \frac {1}{k^2}$ .
This is not an issue, because we know
The value of the lower index of summation does not affect whether an infinite series converges or diverges.
That is, if we can show $\sum _{k=2}^{\infty } \frac {1}{k^2}$ converges, then $\sum _{k=1}^{\infty } \frac {1}{k^2}$ must also converge. Now to determine whether $\sum _{k=2}^{\infty } \frac {1}{k^
2}$ converges, let $s_n = \sum _{k=2}^n \frac {1}{k^2}$. We must determine if $\lim _{n \to \infty } s_n$ exists. Since we expect $\int _1^{\infty } f(x) \d x$ to converge, we expect $\sum _{k=2}^{\
infty } \frac {1}{k^2}$ to converge. To show this, we should establish $\{s_n\}$ is bounded and monotonic, and hence $\lim _{n \to \infty } s_n$ exists$\{s_n\}$ is unbounded so $\lim _{n \to \infty }
s_n$ does not exist .
Notice that since $\frac {1}{k^2} > 0$ for all $k$, and $s_n = \sum _{k=1}^n$, the sequence $\{s_n\}$ is increasingdecreasing .
To show that it is also bounded, note that for any $n \geq 2$, we can observe from the picture that
In particular, since we shade more area if we increase $n$, we have that $\int _1^n \frac {1}{x^2} \d x \leq \int _1^\infty \frac {1}{x^2} \d x$. By a routine calculation of the latter improper
integral, we can show
and we thus have
Hence $\{s_n\}$ is both bounded and monotonic, so $\lim _{n \to \infty } s_n$ exists, and $\sum _{k=2}^{\infty } \frac {1}{k^2}$ converges. Since $\sum _{k=2}^{\infty } \frac {1}{k^2}$ converges, $\
sum _{k=1}^{\infty } \frac {1}{k^2}$ will also converge since the index where we start the summation does not affect whether the series converges.
Note that from the above, the value of the improper integral isis not the value of the series; indeed, by visualizing the series as the sum of the areas of the rectangles in the image, we see that
the value of the series $\sum _{k=2}^{\infty } \frac {1}{k^2}$ should be less thanequal togreater than the value of the improper integral $\int _2^{\infty } \frac {1}{x^2}$.
Now, let’s take a step back and see what we really needed in the this example.
• We needed to find a function for which the area under the curve over any particular interval $[n,n+1]$ was greater than the area of the rectangle whose height is $a_k$ to establish a lower bound
for each $s_n$. Note that we can always do this if $f(x)$ is eventually positive and decreasingincreasing since we may view each $a_k$ as the area of the rectangle that coincides with $f(x)$ at
its lefthandrighthand endpoint.
• We needed to establish that the sequence of partial sums is eventually increasing. This must happen if all of the $a_k$ are positivenegative .
• We needed the function to be “eventually continuous” so the improper integral $\int _{a}^{\infty } f(x) \d x$ can be computed as the limit of a single definite integral.
By “eventually” above, we really mean that $f(x)$ should be continuous, positive, and decreasing on some interval $[a,\infty )$; it doesn’t need to happen right away, but it should hold for all real
large enough $x$-values. This leads us to an interesting observation.
Let $f(x)$ be an eventually continuous, positive, and decreasing function with $a_k = f(k)$. If $\int _1^\infty f(x) \d x$ converges, so does $\sum _{k=1}^\infty a_k$.
The Integral Test
The observations from the previous examples give us a new convergence test called the integral test:
Integral Test
Suppose that
is a sequence, and suppose that
is an eventually continuous, positive, and decreasing function with
for all
$n \geq N$
, where
is an integer. Then,
either both converge or both diverge
Determine whether
$\sum _{k=1}^{\infty } \frac {k}{(2k^2+1)^2}$
Note that
$\lim _{n \to \infty } \frac {n}{(2n+1)^2} =0$
so the divergence test is inconclusive. Also, this is not a geometric series. Let’s try the integral test with
$f(x) = \frac {x}{(2x+1)^2}$
. Notice that the function
$f(x) = \frac {x}{(2x^2+1)^2}$
is continuous and positive on
$[1,\infty )$
. We can check where it is decreasing part by computing
$f'(x) = \frac {-6x^2+1}{(2x^2+1)^3} < 0$
. So,
is certainly negative for
$x >1$
and hence
is also decreasing on
$[1,\infty )$
Notice that, after performing a substitution if necessary,
so $\lim _{b \to \infty } \eval { -\frac {1}{8x^2+4}}_1^b = \frac {1}{8}$ and hence the improper integral $\int _1^{\infty } \frac {x}{(2x+1)^2} \d x$convergesdiverges . Thus, $\sum _{k=1}^{\infty }
\frac {k}{(2k^2+1)^2}$convergesdiverges .
The next examples synthesizes some concepts we have seen thus far.
$a_n = \frac {1}{n^2+4}$
$n \geq 1$
and let
$s_n = \sum _{k=1}^n a_k$
. Determine if
are bounded and/or monotonic.
Note that
is decreasing (and hence monotonic) because denominator increases as
increases. Hence, the fraction becomes smaller as
grows larger.
$\{a_n\}_{n=1}$ is also bounded since $0 \leq a_n \leq 1$ for all $n$ (i.e. no terms are larger than $1$ or smaller than $0$).
For $s_n$, note that
Since $\frac {1}{n^2+4} >0$, we have that $s_{n+1} > s_n$ for all $n$, so $\{s_n\}_{n=1}$ is increasing (and hence monotonic). To determine if $\{s_n\}_{n=1}$ bounded, we can determine whether $\lim
_{n \to \infty } s_n$ exists. Indeed, since $\{s_n\}_{n=1}$ is increasing, we have that $s_n < \lim _{n \to \infty } s_n$, so if the limit exists, it serves as an upper bound for all of the terms in
the sequence.
We can write out the limit in this case and find
Thus, all we have to do is determine if the infinite series above converges.
To do this, we can apply the integral test. Let $f(x)= \frac {1}{x^2+4}$. Notice that $f(x)$ is positive, continuous, and decreasing on $[1,\infty )$, and
Since the improper integral converges, the series $\sum _{k=1}^{\infty } \frac {1}{k^2+4}$ converges. Hence, $\lim _{n \to \infty }s_n$ exists so $\{s_n\}_{n=1}$ is bounded.
A very important type of series for future sections is $\sum _{k=1}^\infty \frac {1}{k^p}$, where $p>0$. We call a series that can be brought into this form a $p$-series. We want to determine for
which values of $p$ these series converge and diverge.
Notice that in our model examples, both series were $p$-series.
• The harmonic series $\sum _{k=1}^{\infty } \frac {1}{k}$ is a $p$-series with $p = \answer {1}$. It convergesdiverges .
First, recall that we have already dealt with the case that
$p = 1$
, since this case is exactly the harmonic series. We have used the integral test to show that the harmonic series diverges. Now, assume that
$p e 1$
. By the integral test, converges if and only if converges. Let’s check this, write with me: Note that if
$p >1$
, the exponent
is negative so
$\lim _{b\to \infty } \left [\frac {b^{1-p}}{1-p}\right ] = \answer {0}$
. If
$p <1$
, the exponent
is positive so
$\lim _{b\to \infty } \left [\frac {b^{1-p}}{1-p}\right ] = \answer {\infty }$
. Thus,
$\int _1^\infty \frac {1}{x^p} \d x$
only converges when
and thus
$\sum _{k=1}^\infty \frac {1}{k^p}$
converges if and only if
This result is important enough to list as a theorem. | {"url":"https://ximera.osu.edu/mooculus/calculus2/integralTest/digInIntegralTest","timestamp":"2024-11-03T02:39:48Z","content_type":"text/html","content_length":"115809","record_id":"<urn:uuid:97198cd9-43d9-44be-98b1-45d3b101cc5e>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00191.warc.gz"} |
Explaining the Angle Sum Property in Geometry
The Angle Sum Property is one of the most important principles in geometry. It states that the sum of all angles in a triangle is equal to 180 degrees. This property applies to any shape with three
or more sides, such as triangles, quadrilaterals, pentagons, and hexagons. Let’s take a closer look at this essential theorem and how it can be used to solve problems.
How to Use the Angle Sum Property
The Angle Sum Property states that the sum of all angles in a triangle is equal to 180 degrees. This means that if you know two angles in a triangle, you can use the Angle Sum Property to calculate
the third angle. For example, let’s say you have a triangle with two known angles—90 degrees and 50 degrees. Using the Angle Sum Property, you can easily calculate that the third angle must be 40
degrees (180 - 90 - 50 = 40).
In addition to calculating unknown angles in triangles, you can also use this theorem to calculate interior and exterior angles in polygons. For example, if you have a pentagon with four known
interior angles measuring 70°, 50°, 80° and 60° respectively, then you can use the Angle Sum Property to calculate the fifth angle as 130° (180 x 5 – 70 – 50 – 80 – 60 = 130).
You can also use this theorem for segments of polygons. For instance, if you have an octagon with seven known segments measuring 30° each then you can calculate for the eighth segment by subtracting
210 from 360 (360 – 210 = 150). This means that your eighth segment has an angle measure of 150°.
The Angle Sum Property is an essential principle for understanding geometry and solving problems involving shapes with three or more sides. By understanding how this theorem works and being able to
apply it correctly when needed, students will be well prepared for tackling any geometry-related questions they might come across during their studies!
What is angle sum property explain?
The Angle Sum Property states that the sum of all angles in a triangle is equal to 180 degrees. This means that if you know two angles in a triangle, you can use this property to calculate the third
angle. In addition to triangles, this theorem also applies to any shape with three or more sides such as quadrilaterals, pentagons, and hexagons. It can also be used to calculate interior and
exterior angles in polygons, as well as segments of polygons. By understanding this theorem and being able to apply it correctly, students will have a better understanding of geometry problems.
What is angle sum property formula?
The Angle Sum Property states that the sum of all angles in a triangle is equal to 180 degrees. This means that if you know two angles in a triangle, you can use this property to calculate the third
angle by subtracting the sum of the known angles from 180 degrees. For example, if you have a triangle with two known angles measuring 90° and 50° respectively, then the third angle would be 40° (180
- 90 - 50 = 40). This formula can also be used to calculate interior and exterior angles in polygons, as well as segments of polygons.
What is an example of angle sum property?
An example of the Angle Sum Property is a triangle with two known angles measuring 90° and 50° respectively. Using this theorem, the third angle would be 40° (180 - 90 - 50 = 40). This example can
also be applied to any shape with three or more sides such as quadrilaterals, pentagons, hexagons, etc. In addition, this theorem can also be used to calculate interior and exterior angles in
polygons as well as segments of polygons. By understanding this theorem and being able to apply it correctly when needed, students will have a better understanding of geometry problems.
What is angle sum property class 8?
The Angle Sum Property states that the sum of all angles in a triangle is equal to 180 degrees. This theorem applies to any shape with three or more sides such as quadrilaterals, pentagons and
hexagons. It can also be used to calculate interior and exterior angles in polygons, as well as segments of polygons. Class 8 students are expected to understand this theorem and be able to apply it
correctly when needed. By doing so, they will have a better understanding of geometry problems which could help them excel in their studies.
What is angle property?
The Angle Property is a theorem which states that the sum of all angles in a triangle is equal to 180 degrees. This means that if you know two angles in a triangle, then you can calculate for the
third angle by subtracting their sum from 180° (180 - x - y = z). This property also applies to any shape with three or more sides such as quadrilaterals, pentagons and hexagons. Moreover, it can
also be used to calculate interior and exterior angles in polygons as well as segments of polygons. By understanding this theorem and being able to apply it correctly when needed, students will have
a better understanding of geometry problems. | {"url":"https://www.intmath.com/functions-and-graphs/explaining-the-angle-sum-property-in-geometry.php","timestamp":"2024-11-06T15:31:33Z","content_type":"text/html","content_length":"103168","record_id":"<urn:uuid:70ca58be-c0da-46f4-b69b-00b8a75a9aa6>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00760.warc.gz"} |
75 Fun Math Games - Школа по математика София-Мат
75 Fun Math Games & Activities for Middle School Students – Nerd’s Magazine Nerd’s Magazine
Learning mathematics does not have to be frightening for middle school kids. Instead, it can be a fantastic experience filled with fun and …
This selection of math games for middle school students is meant to turn math’ standard numbers and word problems into pleasurable experiences. This collection ranges from strategic math board games
to a coordinate plane and interactive online games that convert equations into thrilling challenges.
Whether you’re a teacher searching for fun math activities or a student looking for an alternative to standard learning, this game list transforms a number line into something more than simply
Online Games
These online math games provide various activities to engage students in a virtual learning journey. Whether mastering logic via math blocks or using a coordinate plane in a virtual realm, these
online math games provide an interactive way for kids to practice their math while navigating the digital world.
1. Sudoku
Sudoku is a challenging game that improves middle school kids’ critical thinking abilities. Players must work with a 9×9 grid to insert numerals ranging from 1 to 9 strategically. As students
painstakingly answer each problem, online math games such as Sudoku build number sense and teach patience.
Photo byBozhin KaraivanovonUnsplash
2. Puzzle Picks
Puzzle Picks, from Math Playground, is one of the best online games that adds an interactive twist to standard math class. Students uncover intriguing graphics by clicking and dragging puzzle pieces
to solve math problems. Puzzle Picks fosters students’ engagement, with several versions catering to different grade levels.
3. Number Bonds
Number Bonds, another game from Math Playground, is one of the best interactive games that encourages children to learn absolute value and number bonds. Students improve their math by selecting a
target sum and utilizing the center number and surrounding balls. This game provides a dynamic platform for middle school students to practice essential math concepts.
4. High-Stakes Heist
High-Stakes Heist requires students to break a safe by solving equations step by step, emphasizing the proper sequence of basic operations. With a risk-and-reward dynamic, students must answer
equations correctly to complete the heist.
5. Nerdle
Nerdle is an exciting derivative of Wordle designed just for math fans. Players get six chances to guess the concealed number code with each new task. As players work to break the code, Nerdle
transforms into an interesting problem-solving exercise, making it a perfect online math game for middle schoolers.
6. Orbit Integers
In Orbit Integers, students compete to add and subtract positive and negative numbers in a space race. This online game incorporates a competitive aspect into integer and basic operations, making
math instruction more exciting while teaching critical math skills.
7. Move Here, Move There
Move Here, Move There is an online game that blends logical thinking with coding components. Students must use reasoning to devise a route from the starting location to the destination tile. This
game provides an immersive experience while building number sense in a digital environment.
Image courtesy ofhttps://www.crazygames.com/game/move-here-move-there
8. Mathle
In Mathle, the goal is to solve the proper addition or subtraction problem in five tries. The mental math necessary to infer the equations benefits students, making Mathle a fun complement to middle
school math exercises.
9. Primel
Primel, another intriguing Wordle offshoot developed for math lovers, challenges players to discover a prime number in six tries. This game teaches a number line, and it is a unique numerical puzzle
for middle schoolers that is enjoyable and instructional.
10. King of Math
King of Math starts with kids as humble farmers, allowing them to advance in society by solving various math problems. As they go, kids confront more complicated word problems, which fosters skill
growth. King of Math provides a gamified learning method, making it a fantastic tool for middle school students looking for an involved math experience.
11. Play Math Tic-Tac-Toe
Play Math Tic-Tac-Toe converts the popular game into a multipurpose online resource appropriate for middle school students. With various grade levels, this game gives a challenging task with rewards.
Students may put their math skills to the test by completing problems to gain a spot on the board.
12. Order of Operations
Order of Operations transforms learning into an exciting heist experience in which students break a safe using the correct sequence of basic operations. The Order of Operations game emphasizes the
vital skill of knowing the sequence of basic operations and provides a thrilling story to inspire students.
13. Prodigy Math
Prodigy Math is an adaptive math game for kids in free and paid editions. The game adapts to complexity depending on individual success, delivering a customized learning experience. It is an
excellent resource for middle school students looking to develop their abilities in an interactive way.
Image courtesy ofhttps://www.prodigygame.com/main-en/
14. Age of the Angles
Age of the Angles is an online game that hones students’ angle measuring and protractor abilities. Students gain skills in analyzing and measuring angles via interactive games. It teaches math
principles and offers a dynamic platform for middle schoolers to practice geometry.
15. Alien Powers
Alien Powers adds a traditional video game element to the learning experience by forcing students to solve exponents before alien spacecraft arrive. This game blends enjoyment with math instruction,
challenging students to answer exponent problems quickly.
16. Death to Decimals
Death to Decimals is one of the best online math games where players remove decimals by picking the correct fraction equivalents. This game is a fun online resource for students who want to learn
more about absolute value, decimals, and fraction connections.
17. Rags to Riches
Rags to Riches turns math practice into a Millionaire quiz-style game where players answer algebraic problems to go from rags to riches. Students tackle progressively challenging equations and math
word problems as the game progresses.
18. Math Word Search
Math Word Search is an engaging online game where middle school students improve their math vocabulary interactively. As players click and drag to circle clusters of letters matching the words on the
list, the game develops visual awareness and attention to detail.
19. Exponents Jeopardy
Exponents Jeopardy is an instructional game that adds a fun twist to the standard Jeopardy game. Players in Exponents Jeopardy explore the world of exponents, answering questions ranging from
elementary to complex. Exponents Jeopardy challenges students to use their knowledge of exponent rules, resulting in a more in-depth comprehension of this topic.
Image courtesy ofhttps://slideplayer.com/slide/12535131/
Classroom Games
With this classroom games list, teachers can transform their standard math class into a math playground. These classroom math games for middle school students have been carefully selected to make
learning an entertaining experience, seamlessly integrating with lesson plans.
From vibrant group exercises emphasizing fast memory of a math fact to inventive challenges, including a coordinate plane and real-world applications, these classroom games make math an exciting
subject that kids look forward to. Teachers can say goodbye to boring classes and enrich their lesson plans with interactive activities.
20. Minecraft
Minecraft, a popular game among kids and adolescents, has been transformed into an innovative tool for teaching mathematics. In addition to building its creative sandbox setting, Minecraft allows
instructors to incorporate math blocks and principles into educational gaming effortlessly.
21. Molkky
Molkky, a popular outdoor game in Europe, is now available as a fun math activity for middle school kids. This game for two or more players combines enjoyment with mental math abilities. This
classroom math game involves knocking over pins numbered 1 to 12 using a hurling stick.
22. Candy Bar Volume
Candy Bar Volume blends the attraction of sweets with a hands-on lesson that requires students to calculate the quantities of genuine candy bars. In this classroom math game, students participate in
experiential learning by picking candy bars depending on volume.
23. Math Football
Math Football turns math class into a dynamic arena of mathematical inquiry. This interactive game divides students into two teams and gives them complete control over the math questions, mixing
sports enthusiasm with math problems. As teams pass and defend the virtual ball, they gain points by properly completing different math problems.
24. Math Facts Race
This game incorporates physical exercise with math fact fluency training. Students are separated into teams and race to complete a grid sheet with solutions to a math fact. Math Facts Race motivates
students as they race to answer math problems and contribute to their team’s success.
25. Chess
A timeless two-player game, chess defies age boundaries and serves as an engaging math practice for middle school students. Chess matches help students refine their math skills, decision-making and
strategic planning abilities.
Photo byPixabay
26. Tang Math Games
Tang Math Games, created by math expert and author Greg Tang, include a variety of game packs and classroom packs that target specific math skills. These fascinating middle school math games create
better knowledge by providing a dynamic and interactive approach to learning.
27. Rubik’s Cube
The popular puzzle Rubik’s Cube transcends its reputation as a classic toy and has become a remarkable tool for teaching math in middle school classes. Solving the Rubik’s Cube requires knowledge of
spatial connections, math blocks, and pattern identification.
28. Slice Some Sandwich Fractions
Slice Some Sandwich Fractions transforms lunchtime into an interactive way to practice proportions and fractions. Students practice converting fractions to decimals and vice versa using sandwiches.
This classroom math game makes complicated math ideas simple to understand.
29. Around the Block
Around the Block involves just a ball and a series of math questions on a particular topic. Students form a circle and pass the ball around while answering math questions loudly. This game encourages
collaboration and allows for adaptability to different math concepts. It is also a great way for students to reinforce their math skills in a collaborative atmosphere.
30. Bouncing Sums
Bouncing Sums brings energy into the whole class, allowing students to strengthen their mental math muscles. To play, students throw a beach ball labeled with positive and negative integers,
decimals, or fractions. Bouncing Sums blends physical exercise with learning, making it a refreshing approach for middle school kids to improve their math skills.
31. Play Smart Dice
Play Smart Dice revolutionizes arithmetic fluency development. Unlike ordinary dice, the Play Smart Dice game focuses on difficult numerical sequences. This fun dice game turns math practice into an
exciting challenge, pushing kids to learn a complicated number line in a great way. The Play Smart Dice game is a wonderful resource for middle school math class, making lessons more effective and
Image courtesy ofhttps://sempersmartgames.com/collections/playsmart-dice
32. Math Facts Bingo
Math Facts Bingo takes math exercises and turns them into an enjoyable bingo game. Students get bingo cards with answers to several multiplication tables. They must rush to complete math problems and
write the correct answers on their bingo cards as the instructor shouts them out.
33. 101 and Out
In 101 and Out, the aim is to get as close as 101 points without going over. Divide the class into two small groups, each must have a dice, paper, and pencil. This dice game promotes strategic
thinking and a fun activity for children to practice math skills in a competitive environment.
34. One-Meter Dash
This game provides a fast and interactive way for kids to enhance their measuring comprehension. Students work in small groups to discover two to four things in the room that sum up to one meter.
This game turns a typical measuring lecture into an engaging experience for middle schoolers.
35. Back-to-Back
Back-to-Back capitalizes on the class’s competitive atmosphere by combining students with comparable abilities and grade levels. In this fun activity, pairs of students stand back-to-back with chalk
in hand, facing away from each other.
When the instructor shouts out math problems, pupils rush to solve them and write the solution on the board. Back-to-Back transforms the whole class into a battleground for mathematics, giving a fun
way for kids to review their math skills.
36. Stand Up, Sit Down
Stand Up, Sit Down is a hands-on exercise that may be tailored to different grade levels. The students form a circle, and the instructor reads equations aloud. This simple game teaches math
principles and encourages rapid thinking.
37. Math Relay Race
The Math Relay Race adds fun to the learning process by transforming a route with various math problems at each station into a race. Students must rush from one station to the next, attempting to
solve math problems before moving on. This game is a great option for middle school kids who want to do math practice before practical exams.
Image courtesy ofQueen of the First Grade Jungle
38. 100s
In this game, students take turns selecting numbers from a set, then adding them aloud clockwise. The catch is that any student who achieves or above 100 is out. Before math lectures, middle school
students may refresh their math knowledge competitively by playing 100s.
Board Games
With these enthralling math board games, you may gather your family for a math-infused game night at home as your math playground. These strategic and fun math games for middle school students
incorporate a splash of a probability game, logic, and critical thinking into the mix.
From conquering territories in Catan to constructing equations in Equate, these math board games are excellent methods for middle school kids to apply math ideas while enjoying the fun of friendly
39. Lost Cities
Lost Cities, a family favorite, is a lovely blend of strategy and stealth math. This fun activity aims to go on expeditions and earn points by playing regular and face cards in increasing sequence.
Players must carefully assess risks and benefits with each step, refining their decision-making skills.
40. Catan
Catan is a probability game where players acquire decision-making skills as they traverse resources and territory. This is a fun probability game that teaches kids about the complexities of
probability in a social context.
41. Quoridor
Quoridor is a fun game for 2-4 players with a basic board and different set of strategy options. The goal is for players to move their piece to the opposing side while obstructing their opponents.
This simple aim reveals a complex tapestry of strategic thought and spatial reasoning.
42. The Genius Star
The Genius Star, a compelling logic game, captivates players with its solo or two-player configuration. The aim is to solve the problem as quickly as possible, with points provided for solving the
game with the golden star.
43. Onitama
In Onitama, players try to capture their opponent’s master or reach their opponent’s temple, like chess. Its appealing design and dynamic gameplay guarantee that pupils are amused and engaged in a
mentally fun activity.
Image courtesy ofAmazon
44. Kanoodle
Kanoodle is a three-dimensional game that pushes players to solve 3-D problems. This game is an engaging solitary gaming experience focusing on logic and spatial abilities. Players adjust colored
puzzle pieces to fit inside the provided limits, stimulating strategic thinking and problem-solving.
45. Monopoly
A traditional board game that many people like, Monopoly, delivers many math exercises while playing. Players buy, sell, and trade properties using real-world basic operations and absolute value.
This probability game fosters bargaining abilities and financial literacy.
46. Math Sprint
Byron’s Games’ award-winning board game Math Sprint focuses on improving math fluency in addition, subtraction, multiplication, division, and absolute value. Players compete against one another to
answer math problems and move along the game board.
47. Clumsy Thief
Clumsy Thief is a fast-paced and amusing card game where players practice their rapid mental math skills with up to three-digit figures. The goal is to amass piles of face cards worth $100 before
opponents can take them away.
48. Equate
Equate turns the popular crossword-style board game into a
mathematical puzzle in which players construct equations instead of words. Players strategically form equations on the game board using a number line and mathematical symbols. This game emphasizes
basic operations and fosters an interactive way for students to increase their comprehension of math concepts.
49. Prime Climb
Prime Climb encourages students to practice the world of factors via an engaging board game in which the aim is to climb quicker than everyone else. Players move the board using basic operations and
strategy, adding complexity to the game.
Image courtesy ofAmazon
50. Outnumbered
Outnumbered is a superhero-themed card game that actively encourages the development of mental math abilities. This game turns math into a thrilling superhero adventure, making it a fun activity for
middle school kids looking for creative ways of learning.
51. Sumoku
Sumoku turns math into a fun game played with number tiles, with many crossword-style gameplay options. This game is appropriate for players of varying grade levels. It provides a fun resource for
students to improve their math ability in a great way.
Card Games
These fast-paced, strategic card games transform a standard deck of cards on your math playground into a dynamic instrument for developing numerical abilities. Whether arranging face cards to make a
precise number or identifying math-related phrases, these math games for middle school students demonstrate that learning can be as simple as a deck of cards, making math fun for everyone.
52. Star Realms
Star Realms is an excellent game for students since it is inexpensive, easy to learn, and engaging. Players obtain new face cards each round and carefully place them to weaken their opponents. Star
Realms encourages critical thinking and strategic planning.
53. Five Crowns
Five Crowns is a very addicting card game that not only entertains but also sharpens the cognitive abilities of middle school kids. The low cost and mobility of this probability game make it an
appealing option for on-the-go entertainment.
54. Sortify: Angles
Sortify: Angles is an online card game where students sort cards by dragging them into bins and labeling them appropriately. This game challenges students to connect cards that make complementary or
supplementary angles.
55. Proof Math Game
Proof Math Game adds excitement to the quest of equations by racing players to be the first to identify a valid equation inside a deck of cards. This fast-paced game tests students’ mental math
abilities in a competitive environment.
Image courtesy ofAmazon
56. Skyjo
This entertaining card game is a wonderful alternative for big families or social events due to its speedy playtime. Skyjo promotes number recall and memorization, making it one of the best
entertaining and approachable middle school math games.
57. Antiquity Quest
Grandpa Beck’s Antiquity Quest exposes players to the exciting realm of acquisition. Players compete to construct the most extraordinary collection by skillfully playing face cards to gain valuable
goods. This is an excellent option for middle school students searching for a thrilling card game.
58. Cover Your Assets
Cover Your Assets is a quick-play fun game where players seek to steal from others while attempting to safeguard their precious goods from being taken by opponents. This game teaches vital skills in
bargaining and resource management.
59. Absolute Zero
Absolute Zero challenges players to mix positive and negative integers to achieve zero. This novel card game strengthens numerical abilities and promotes strategic thinking. Balancing positive and
negative numbers adds another complexity to Absolute Zero, making it an exciting option for middle schoolers.
60. Dutch Blitz
A card game for up to 8 players, Dutch Blitz, uses numeric sequencing throughout play and demands addition and subtraction abilities for scoring. Its adaptability to large and small groups makes it
an excellent option for family evenings or school activities.
61. 24 Game – Integers Card Game
The 24 Game is a simple but adaptable fun game that involves players with integers, addition, subtraction, and division. It accommodates various grade levels by offering many options, such as
factors, positive and negative integers, and even algebra. The 24 Game’s versatility makes it a fantastic opportunity for students looking to practice essential math concepts in a great way.
Image courtesy ofAmazon
62. 31-derful
31-derful requires students to strategically arrange 25 cards from the regular deck such that the rows and columns total up to a specific goal number—31. This card game is a fun approach for middle
school kids to engage with math subjects in a fun and competitive
63. Algebra Taboo
Algebra Taboo is a math-themed version of the popular word-guessing game Taboo. The goal is for players to get their companions to guess word problems without mentioning particular critical phrases
connected to the math topic on the card.
64. Can You Make It
The goal of this game is for kids to create an equation equal to an enormous number using just the supplied single integers. This game develops mental math abilities and turns math practice into a
fascinating game, enabling middle school students to experiment with various math combinations in creative ways.
65. Exponent Game
The Exponent Game reinforces the idea of exponents using a typical deck of playing cards. Players choose cards and use numbers to form exponential expressions in this fun activity. Exponent Game
gives students a hands-on and pleasant approach to learning about exponents.
66. Mathematics Pictionary
This game is played in the same fun way as the conventional version, except the cards are all connected to math subjects. Players sketch representations of math concepts while their teammates try to
guess the correct answers.
Electronic Games – Battery-Operated
In this collection of electronic math games for middle school students, you may use the power of technology to achieve mathematical proficiency. With interactive tasks spanning addition, subtraction,
multiplication, and more, these portable games bring mathematics to life, making math fun and engaging.
67. Multiplication Slam
Multiplication Slam is a fun hand-held game that boosts math fluency via speed and accuracy. Players compete in an exciting race to calculate multiplication solutions, providing diversity and
challenges across many math subjects.
Image courtesy ofAmazon
68. Math Shark
Math Shark, a multipurpose hand-held game, pushes math fluency to new heights by covering various mathematical ideas. As players race against the time to calculate, they improve their mental
arithmetic and thoroughly comprehend numerous math concepts.
69. Math Whiz
Math Whiz is a helpful tool for middle school kids to teach core math concepts and build their confidence in numerical competency, whether as a fun game or a convenient calculator.
70. Math Slam
Math Slam is a dynamic portable game that improves addition and subtraction abilities with five action-packed rounds. Students bash through math problems, hoping to outperform themselves and acquire
mastery, enhancing mental math speed and accuracy.
71. Number Genius
Number Genius is an all-in-one interactive math game that combines fun, memory, and logic challenges to provide an immersive learning experience. Students can improve their math abilities, memory,
and logical thinking using math review games and a sketch pad for hands-on problem-solving. The game’s interactive elements make it an excellent option for students wishing to reinforce an important
concept while having fun.
72. Math Trekker
Math Trekker is a portable game that teaches addition and subtraction concepts from 1 to 12 and varied exercises. The math review games encourage children to beat their times and continue playing.
73. Minute Math Electronic Flash Card
The Minute Math Electronic Flash Card is an innovative gadget that stimulates the practice of basic operations. This game encourages students’ engagement as they work on their math fluency.
Image courtesy ofAmazon
74. Light ‘N’ Strike Math
Light N Strike Math is an arcade-style game designed for self-paced learning and has three degrees of difficulty to suit different grade levels. Students may learn and practice an important concept
in an interactive environment.
75. Multiplication Master
Multiplication Master is a portable flash card game that aims to improve multiplication fluency with numbers 0-12. Math review games like this engage students in a dynamic and interactive
multiplication exercise. | {"url":"https://sofia-math.org/bg/75-fun-math-games/","timestamp":"2024-11-13T14:58:17Z","content_type":"text/html","content_length":"148237","record_id":"<urn:uuid:bdb906b9-11d5-4392-90ab-dfa8992610eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00678.warc.gz"} |
uestion #35
Physics Question #35
Mike Smith, a 17 year old male from the Internet asks on July 24, 1999,
Can you help me with the physics of an amusement park ride called the Magic Carpet? The Magic Carpet is a ride where you sit on a platform (the carpet) and it starts to swing back and forth like a
pendulum. When it gets to the highest point, you don't move for a few moments, then drop back into the swinging. I'm trying to find the apparent weight that you would have at the bottom of the swing.
I know the real mass, which is 70.3 kg. The time of fall is 3.18 sec. Acceleration in Gs is 1; deceleration in Gs is 3.25, and the height of the ride is 6.24 meters. I now need the Centripetal Force
so I could add it onto the real weight but I am unable to find it. Could you help?
viewed 18965 times
The answer
Your variables are: mass m = 70.3 kg, pendulum height d = 6.24 m g = local gravitational constant = 9.8 N/kg.
The "trick" to this problem is to realize that the potential energy you must have at the point of release (the top of the pendulum arc) is entirely converted to kinetic energy at the moment the
pendulum swings through the bottom of its arc. This is the moment of most rapid centripetal acceleration and therefore the moment the passenger on the carpet ride will feel the greatest force. The
potential energy at the point of release = m g d equate this with kinetic energy for a mass of 70.3 kg (0.5 m v^2) i.e., m g d = 0.5 m v^2 to get the maximum velocity. Thus v = square root (2 g d).
Now from this velocity you can determine the centripetal acceleration. You must know the radius of the pendulum itself (this works for roller coasters, too, if you can find the radius of the circle
which matches the curvature of the bends in the track). Centripetal acceleration equals v^2 / r for acceleration along a circle of radius r. If this is also the pendulum height d (i.e., you are
dropped so you are instantaneously in free fall) then the formula is quite simple: centripetal acceleration = (sqrt(2 g d)) ^ 2 / d = 2 g d / d = 2 g As you correctly observed, there's also the extra
g contributed by the Earth (it doesn't turn off) so the participant would feel 3 g of force at the bottom of the ride.
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Is 90 feet “a pick from heaven”?
I’ve started watching Ken Burns’ “Baseball” series while I exercise, and this morning I saw something interesting. In the first episode, Charles McDowell says this about the decision to put the bases
90 feet apart:
That’s so interesting that that would come out 90 feet. That somebody sat down, Mr. Cartwright or whoever said, “Hey it ought to be 90 feet — it just sounds like a logical number.” The fact of
the matter is, in retrospect, if it was 88 feet the game would be very different. Think of the plays at first base. Think of the double plays that wouldnâ??t be completed on an 88 feet first
base, and second base. If it were 94 feet weâ??d be throwing people out all over the place. Batting averages would drop remarkably. So if 90 feet was something somebody said, “Hey, that’s a good
number,” that was a pick from heaven.
I’m not a mathematician, but I was pretty good at math in high school, and I am a computer programmer, which has some mathematical components. Maybe I know just enough to be dangerous, as they say,
but my very first thought on hearing that statement was: “That doesn’t sound right.”
If the bases were only 88 feet apart, then yes, the runners would have two fewer feet to run. But the fielders would be closer together, too, which means shorter throws. On the other hand, it also
means balls that they currently dive to catch would sneak through the infield. But back on the first hand, the holes between the fielders would be slightly smaller and therefore harder for hitters to
I think it mostly comes down to ratios and percentages — as long as the bases are all the same distance from each other, we’re good. Whether that distance is 88 feet or 94 feet or any other number
doesn’t matter a ton, I don’t think. Physical requirements would change, of course — can you imagine David Eckstein or Dee Gordon trying to play shortstop on a field with 110-foot bases? — but
statistically speaking, things would all even out.
The distance from the pitcher’s mound to home plate, though … well, changing that would have a huge impact on the game. And it is telling that THAT distance is the result of trial-and-error, not some
“pick from heaven.” | {"url":"https://www.jeffjsnider.com/archives/is-90-feet-a-pick-from-heaven/","timestamp":"2024-11-14T04:10:55Z","content_type":"text/html","content_length":"29587","record_id":"<urn:uuid:f8a882f6-1985-4e7c-bd0e-6de9a7770697>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00154.warc.gz"} |
rod load calculator calculation for Calculations
02 Mar 2024
Popularity: ⭐⭐⭐
Rod Load Calculator
This calculator provides the calculation of stress, strain, and elongation of a rod subjected to a load.
Calculation Example: A rod load calculator is a tool that helps to determine the stress, strain, and elongation of a rod subjected to a load. These calculations are important in various engineering
applications, such as the design of bridges, buildings, and other structures.
Related Questions
Q: What is the significance of stress and strain in engineering?
A: Stress and strain are important concepts in engineering as they provide insights into the behavior of materials under load. Understanding stress and strain allows engineers to design structures
and components that can withstand the forces they will be subjected to.
Q: How is Young’s modulus used in engineering?
A: Young’s modulus is a measure of the stiffness of a material. It is used in engineering to predict the behavior of materials under load and to design structures that can withstand the forces they
will be subjected to.
| —— | —- | —- |
A Cross-sectional Area m^2
Calculation Expression
Stress: The stress on the rod is given by Stress = P / A
Strain: The strain on the rod is given by Strain = Stress / E
Elongation: The elongation of the rod is given by Elongation = Strain * L
Calculated values
Considering these as variable values: P=1000.0, A=1.0E-4, E=2.0E11, L=1.0, the calculated value(s) are given in table below
| —— | —- |
Similar Calculators
Calculator Apps
Matching 3D parts for rod load calculator calculation for Calculations
App in action
The video below shows the app in action. | {"url":"https://blog.truegeometry.com/calculators/rod_load_calculator_calculation_for_Calculations.html","timestamp":"2024-11-08T10:52:12Z","content_type":"text/html","content_length":"25945","record_id":"<urn:uuid:8f894818-2c45-4755-bb03-6e24189453ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00041.warc.gz"} |
Transactions Online
Jeng-Long LEOU, Jiunn-Ming HUANG, Shyh-Kang JENG, Hsueh-Jyh LI, "Application of Wavelets to Scattering Problems of Inhomogeneous Dielectric Slabs" in IEICE TRANSACTIONS on Communications, vol. E82-B,
no. 10, pp. 1667-1676, October 1999, doi: .
Abstract: In this paper, we apply the discrete wavelet transform (DWT) and the discrete wavelet packet transform (DWPT) with the Daubechies wavelet of order 16 to effectively solve for the
electromagnetic scattering from a one-dimensional inhomogeneous slab. Methods based on the excitation vector and the [Z] matrix are utilized to sparsify an MoM matrix. As we observed, there are no
much high frequency components of the field in the dielectric region, hence the wavelet coefficients of the small scales components (high frequency components) are very small and negligible. This is
different from the case of two-dimensional scattering from perfect conducting objects. In the excitation-vector-based method, a modified excitation vector is introduced to extract dominant terms and
achieve a better compression ratio of the matrix. However, a smaller compression ratio and a tiny relative error are not obtained simultaneously owing to their deletion of interaction between
different scales. Hence, it is inferior to the [Z]-matrix-based methods. For the [Z]-marix-based methods, our numerical results show the column-tree-based DWPT method is a better choice to sparsify
the MoM matrix than DWT-based and other DWPT-based methods. The cost of a matrix-vector multiplication for the wavelet-domain sparse matrix is reduced by a factor of 10, compared with that of the
original dense matrix.
URL: https://global.ieice.org/en_transactions/communications/10.1587/e82-b_10_1667/_p
author={Jeng-Long LEOU, Jiunn-Ming HUANG, Shyh-Kang JENG, Hsueh-Jyh LI, },
journal={IEICE TRANSACTIONS on Communications},
title={Application of Wavelets to Scattering Problems of Inhomogeneous Dielectric Slabs},
abstract={In this paper, we apply the discrete wavelet transform (DWT) and the discrete wavelet packet transform (DWPT) with the Daubechies wavelet of order 16 to effectively solve for the
electromagnetic scattering from a one-dimensional inhomogeneous slab. Methods based on the excitation vector and the [Z] matrix are utilized to sparsify an MoM matrix. As we observed, there are no
much high frequency components of the field in the dielectric region, hence the wavelet coefficients of the small scales components (high frequency components) are very small and negligible. This is
different from the case of two-dimensional scattering from perfect conducting objects. In the excitation-vector-based method, a modified excitation vector is introduced to extract dominant terms and
achieve a better compression ratio of the matrix. However, a smaller compression ratio and a tiny relative error are not obtained simultaneously owing to their deletion of interaction between
different scales. Hence, it is inferior to the [Z]-matrix-based methods. For the [Z]-marix-based methods, our numerical results show the column-tree-based DWPT method is a better choice to sparsify
the MoM matrix than DWT-based and other DWPT-based methods. The cost of a matrix-vector multiplication for the wavelet-domain sparse matrix is reduced by a factor of 10, compared with that of the
original dense matrix.},
TY - JOUR
TI - Application of Wavelets to Scattering Problems of Inhomogeneous Dielectric Slabs
T2 - IEICE TRANSACTIONS on Communications
SP - 1667
EP - 1676
AU - Jeng-Long LEOU
AU - Jiunn-Ming HUANG
AU - Shyh-Kang JENG
AU - Hsueh-Jyh LI
PY - 1999
DO -
JO - IEICE TRANSACTIONS on Communications
SN -
VL - E82-B
IS - 10
JA - IEICE TRANSACTIONS on Communications
Y1 - October 1999
AB - In this paper, we apply the discrete wavelet transform (DWT) and the discrete wavelet packet transform (DWPT) with the Daubechies wavelet of order 16 to effectively solve for the electromagnetic
scattering from a one-dimensional inhomogeneous slab. Methods based on the excitation vector and the [Z] matrix are utilized to sparsify an MoM matrix. As we observed, there are no much high
frequency components of the field in the dielectric region, hence the wavelet coefficients of the small scales components (high frequency components) are very small and negligible. This is different
from the case of two-dimensional scattering from perfect conducting objects. In the excitation-vector-based method, a modified excitation vector is introduced to extract dominant terms and achieve a
better compression ratio of the matrix. However, a smaller compression ratio and a tiny relative error are not obtained simultaneously owing to their deletion of interaction between different scales.
Hence, it is inferior to the [Z]-matrix-based methods. For the [Z]-marix-based methods, our numerical results show the column-tree-based DWPT method is a better choice to sparsify the MoM matrix than
DWT-based and other DWPT-based methods. The cost of a matrix-vector multiplication for the wavelet-domain sparse matrix is reduced by a factor of 10, compared with that of the original dense matrix.
ER - | {"url":"https://global.ieice.org/en_transactions/communications/10.1587/e82-b_10_1667/_p","timestamp":"2024-11-11T19:53:44Z","content_type":"text/html","content_length":"64500","record_id":"<urn:uuid:3f4fbc99-0e2b-4b21-be5c-77e03fdf8c73>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00539.warc.gz"} |
Visualising Pythagoras: ultimate proofs and crazy contortions | Video Summary and Q&A | Glasp
Visualising Pythagoras: ultimate proofs and crazy contortions | Summary and Q&A
Visualising Pythagoras: ultimate proofs and crazy contortions
Pythagoras's theorem is explored, debunking the misconception that it was discovered by Pythagoras himself. The video presents various beautiful and simple proofs of the theorem, as well as its
generalizations in different shapes and dimensions.
Key Insights
• β Pythagoras's theorem was not discovered by Pythagoras but was known to the Babylonians before his time.
• π There are numerous beautiful and simple proofs of Pythagoras's theorem, with the video presenting a few examples, including those involving triangles within a square and parallelograms.
• π Euclid's proof of Pythagoras's theorem, from his book "Elements," is more comprehensive and detailed than other proofs.
• π Pythagoras's theorem can be generalized to other shapes and dimensions, such as through the use of areas instead of distances.
• π ¬π ΅ Several interesting Pythagorean facts were mentioned, including de Gua's theorem and the existence of Pythagorean triples.
• π Pythagoras's theorem has applications beyond right-angled triangles, including in trigonometry and higher-dimensional geometry.
• π € A new book by the Mathologer team, featuring mathematical articles with an Australian theme, has been published by the American Mathematical Society.
Welcome to another Mathologer video A squared plus B squared equals C squared. Forget about Euler's formula and company, Pythagoras's theorem beats them all in just about every conceivable way, at
least in my books. Ok so finally a Mathologer video about THE theorem of theorems. My main mission today is to chase down the all-time greatest, simplest... Read More
Questions & Answers
Q: Was Pythagoras the first to discover Pythagoras's theorem?
No, Pythagoras did not discover the theorem. It was known to the Babylonians before his time, and there is no evidence to support the claim that he produced a rigorous proof for it.
Q: How is Euclid's proof of Pythagoras's theorem different from other proofs?
Euclid's proof, found in his book "Elements," is more rigorous and detailed than other proofs presented in the video. It uses geometric constructions and logical deductions to demonstrate the
theorem's validity.
Q: Can Pythagoras's theorem be generalized to other shapes and dimensions?
Yes, Pythagoras's theorem can be generalized. For example, it can apply to any shape as long as the areas are considered instead of distances. It also holds true in higher dimensions, where it
relates to volumes and hyper volumes.
Q: What are some interesting Pythagorean facts mentioned in the video?
Some fascinating Pythagorean facts include de Gua's theorem, which relates the areas of right-angled triangles and a triangular pyramid's base, and the existence of Pythagorean triples, which are
sets of three positive integers satisfying the theorem.
Summary & Key Takeaways
• Pythagoras's theorem, commonly attributed to Pythagoras, was actually known to the ancient Babylonians before he was born.
• The video presents several elegant proofs of Pythagoras's theorem, including the arrangement of triangles within a square and using parallelograms, as well as Euclid's proof from his book
• The cosine rule, which generalizes Pythagoras's theorem for all triangles, is introduced, along with the concept of Pythagorean triples and higher-dimensional versions of the theorem.
Explore More Summaries from Mathologer π | {"url":"https://glasp.co/youtube/p/visualising-pythagoras-ultimate-proofs-and-crazy-contortions","timestamp":"2024-11-08T11:18:59Z","content_type":"text/html","content_length":"359842","record_id":"<urn:uuid:22d7695c-18e8-4f77-9f72-ecbd3e40b058>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00783.warc.gz"} |
Orthogonal Matrix Test
11-04-2016, 07:31 PM
Post: #1
Eddie W. Shore Posts: 1,614
Senior Member Joined: Dec 2013
Orthogonal Matrix Test
For the square matrix M, it is orthogonal when either of the following conditions are met:
(I) M * M^T = M^T * M = I
(II) M^-1 = M^T
The program presented on this blog entry will use the first test. Since matrices, unfortunately, cannot be directly compared on the Casio graphing calculators, a work around with two FOR loops is
HP Prime Program ORTHOG
EXPORT ORTHOG(m)
// 2016-11-01 EWS
// orthogonal test
LOCAL n,p,s;
IF n==p THEN
RETURN 1;
RETURN 0;
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Jump to navigation Jump to search
This page presents a novelty topic. It features ideas which are less likely to find practical applications in xenharmonic music. It may contain numbers that are impractically large, exceedingly
complex, or chosen arbitrarily. Novelty topics are often developed by a single person or a small group. As such, this page may also feature idiosyncratic terms, notations, or conceptual frameworks.
Interval information
Ratio 68630377364883/68630356164608
Subgroup monzo 2.3.7.23.397 [-30 29 -1 -1 -1⟩
Size in cents 0.00053478714¢
Name Zudilisma
Color name L^4397u23ur-5
FJS name [math]\text{dddd}{-4}_{7,23,397}[/math]
Special properties reduced
Tenney height (log[2] nd) 91.9278
Weil height (log[2] max(n, d)) 91.9278
Wilson height (sopfr (nd)) 574
Harmonic entropy ~1.19982 bits
(Shannon, [math]\sqrt{nd}[/math])
Comma size unnoticeable
open this interval in xen-calc
68630377364883/68630356164608, the Zudilisma, is a 2.3.7.23.397 subgroup ratio which is the difference between 127834/1 and a stack of 29 3/2.
It appears in the sequence of numbers where the fractional part of 1.5^n gets progressively closer to an integer than for any number before it - sequence A1267122 in OEIS. Said sequence was described
by Zudilin, hence the name of the ratio.
If this ratio is taken as a comma to be tempered out, it will produce a temperament that very closely approximates Pythagorean tuning and, in diatonic notation, maps 63917/32768 as C - Cxx. | {"url":"https://en.xen.wiki/w/Zudilisma","timestamp":"2024-11-06T01:30:01Z","content_type":"text/html","content_length":"26375","record_id":"<urn:uuid:931cbecd-a30a-4755-ab84-a10d68ac09c0>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00877.warc.gz"} |
mem_fun1_t<Result, X, Arg>
mem_fun1_t<Result, X, Arg>
Categories: functors, adaptors Component type: type
Mem_fun1_t is an adaptor for member functions. If X is some class with a member function Result X::f(Arg) (that is, a member function that takes one argument of type Arg and that returns a value of
type Result [1]), then a mem_fun1_t<Result, X, Arg> is a function object adaptor that makes it possible to call f as if it were an ordinary function instead of a member function.
Mem_fun1_t<Result, X, Arg>'s constructor takes a pointer to one of X's member functions. Then, like all function objects, mem_fun1_t has an operator() that allows the mem_fun1_t to be invoked with
ordinary function call syntax. In this case, mem_fun1_t's operator() takes two arguments; the first is of type X* and the second is of type Arg.
If F is a mem_fun1_t that was constructed to use the member function X::f, and if x is a pointer of type X* and a is a value of type Arg, then the expression F(x, a) is equivalent to the expression
x->f(a). The difference is simply that F can be passed to STL algorithms whose arguments must be function objects.
Mem_fun1_t is one of a family of member function adaptors. These adaptors are useful if you want to combine generic programming with inheritance and polymorphism, since, in C++, polymorphism involves
calling member functions through pointers or references.
As with many other adaptors, it is usually inconvenient to use mem_fun1_t's constructor directly. It is usually better to use the helper function mem_fun [2] instead.
struct Operation {
virtual double eval(double) = 0;
struct Square : public Operation {
double eval(double x) { return x * x; }
struct Negate : public Operation {
double eval(double x) { return -x; }
int main() {
vector<Operation*> operations;
vector<double> operands;
operations.push_back(new Square);
operations.push_back(new Square);
operations.push_back(new Negate);
operations.push_back(new Negate);
operations.push_back(new Square);
transform(operations.begin(), operations.end(),
ostream_iterator<double>(cout, "\n"),
Defined in the standard header functional, and in the nonstandard backward-compatibility header function.h.
Template parameters
│Parameter│ Description │Default│
│Result │The member function's return type. │ │
│X │The class whose member function the mem_fun1_t invokes. │ │
│Arg │The member function's argument type. │ │
Model of
Adaptable Binary Function
Type requirements
• X has at least one member function that takes a single argument of type Arg and that returns a value of type Result. [1]
Public base classes
binary_function<X*, Arg, Result>
│ Member │ Where defined │ Description │
│first_argument_type │Adaptable Binary Function│The type of the first argument: X* │
│second_argument_type │Adaptable Binary Function│The type of the second argument: Arg │
│result_type │Adaptable Binary Function│The type of the result: Result │
│Result operator()(X* x, Arg a) const │Binary Function │Function call operator. Invokes x->f(a), where f is the member function that was passed to the constructor.│
│explicit mem_fun1_t(Result (X::*f)(Arg)) │mem_fun1_t │See below. │
│template <class Result, class X, class Arg>│mem_fun1_t │See below. │
│mem_fun1_t<Result, X, Arg> │ │ │
│mem_fun(Result (X::*f)(Arg)); [2] │ │ │
New members
These members are not defined in the Adaptable Binary Function requirements, but are specific to mem_fun1_t.
│ Member │ Description │
│explicit mem_fun1_t(Result (X::*f)(Arg)) │The constructor. Creates a mem_fun1_t that calls the member function f. │
│template <class Result, class X, class Arg>│If f is of type Result (X::*)(Arg) then mem_fun(f) is the same as mem_fun1_t<Result, X, Arg>(f), but is more convenient. This is a global function, not a│
│mem_fun1_t<Result, X, Arg> │member function. │
│mem_fun(Result (X::*f)(Arg)); [2] │ │
[1] The type Result is permitted to be void. That is, this adaptor may be used for functions that return no value. However, this presents implementation difficulties. According to the draft C++
standard, it is possible to return from a void function by writing return void instead of just return. At present, however (early 1998), very few compilers support that feature. As a substitute,
then, mem_fun1_t uses partial specialization to support void member functions. If your compiler has not implemented partial specialization, then you will not be able to use mem_fun1_t with member
functions whose return type is void.
[2] This helper function was called mem_fun1 in drafts of the C++ standard, but it is called mem_fun in the final standard. This implementation provides both versions for backward compatibility, but
mem_fun1 will be removed in a future release.
See also
mem_fun_t, mem_fun_ref_t, mem_fun1_ref_t
STL Main Page | {"url":"http://ld2014.scusa.lsu.edu/STL_doc/mem_fun1_t.html","timestamp":"2024-11-11T20:20:52Z","content_type":"text/html","content_length":"11493","record_id":"<urn:uuid:9b7450ff-ca7c-47bb-b250-ba7e4f9266de>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00125.warc.gz"} |
Rossby modes in slowly rotating stars: Depth dependence in distorted polytropes with uniform rotation
Context. Large-scale Rossby waves have recently been discovered based on measurements of horizontal surface and near-surface solar flows. Aims. We are interested in understanding why it is only
equatorial modes that are observed and in modelling the radial structure of the observed modes. To this aim, we have characterised the radial eigenfunctions of r modes for slowly rotating polytropes
in uniform rotation. Methods. We followed Provost et al. (1981, A&A, 94, 126) and considered a linear perturbation theory to describe quasi-toroidal stellar adiabatic oscillations in the inviscid
case. We used perturbation theory to write the solutions to the fourth order in the rotational frequency of the star. We numerically solved the eigenvalue problem, concentrating on the type of
behaviour exhibited where the stratification is nearly adiabatic. Results. We find that for free-surface boundary conditions on a spheroid of non-vanishing surface density, r modes can only exist for
ℓ = m spherical harmonics in the inviscid case and we compute their depth dependence and frequencies to leading order. For quasi-adiabatic stratification, the sectoral modes with no radial nodes
are the only modes which are almost toroidal and the depth dependence of the corresponding horizontal motion scales as rm. For all r modes, except the zero radial order sectoral ones, non-adiabatic
stratification plays a crucial role in the radial force balance. Conclusions. The lack of quasi-toroidal solutions when stratification is close to neutral, except for the sectoral modes without nodes
in radius, follows from the need for both horizontal and radial force balance. In the absence of super- or sub-adiabatic stratification and viscosity, both the horizontal and radial parts of the
force balance independently determine the pressure perturbation. The only quasi-toroidal cases in which these constraints on the pressure perturbation are consistent are the special cases where ℓ =
m and the horizontal displacement scales with rm.
• Methods: analytical
• Stars: interiors
• Stars: oscillations
• Stars: rotation
• Sun: oscillations
ASJC Scopus subject areas
• Astronomy and Astrophysics
• Space and Planetary Science
Dive into the research topics of 'Rossby modes in slowly rotating stars: Depth dependence in distorted polytropes with uniform rotation'. Together they form a unique fingerprint. | {"url":"https://nyuscholars.nyu.edu/en/publications/rossby-modes-in-slowly-rotating-stars-depth-dependence-in-distort","timestamp":"2024-11-14T01:18:32Z","content_type":"text/html","content_length":"54850","record_id":"<urn:uuid:f1c89e12-057b-4947-8871-0d7cbdba6779>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00413.warc.gz"} |
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5-methyl-5-hexen-3-ol. The molecular ion (m/z=114 Da) is not observed under electron impact ionization conditions. The highest mass ion (m/z=85) is due to an alpha-cleavage of ethyl; the other
alpha-cleavage generates m/z=59. The rearrangement cleavage shown here generates the m/z=56 ion. 4,4 18 Z: 7 Z: Answers (6am, 9pm, 12noon, 1am) 3. Convert the following Z times to Central Standard
Time in the US. 0 Z MON: 12 Z WED: 21 Z FRI: 2 Z SAT: Answers (6pm Sunday, 6am Wednesday, 3pm Friday, 8pm Friday) *When it turns to 0Z in Greenwich, England, it is still the previous day's evening in
the U.S. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. See more of Plumbers Local #15 - Minneapolis & St. Cloud -
Minnesota on Facebook.
Log In. Forgot account? or. Create New Account. Not Now. Vendo Casa en zona 15. Real Estate . Community See All. 376 people like this. 384 people follow this.
Then type the @ symbol. Then type x=6. Try it now: 2x+3=15 @ x=6 Clickable Demo Try entering 2x+3=15 @ x=6 into the text box. After you enter the expression, Algebra Calculator will plug x=6 in for
the equation 2x+3=15: 2(6)+3 = 15.
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3.5 J.A.Beachy 3 32. Use the the result in Problem 31 to show that the multiplicative groups Z× 15 and Z 21 are not cyclic groups. Solution: In Z× 15, both [−1]15 and [4]15 are easily checked to have
order 2.
18 is 15% of 120. Steps to solve "18 is 15 percent of what number?" We have, 15% × x = 18; or, 15 / 100 × x = 18; Multiplying both sides by 100 and dividing both sides by 15, we have x = 18 × 100 /
15; x = 120. If you are using a calculator, simply enter 18×100÷15, which will give you the answer. MathStep (Works offline) Z=15, A=31 : Sulphur S Z=16, A=32 : Chlorine Cl Z=17, A=35.5 : Argon Ar Z=
18, A=40 : Potassium K Z=19, A=39 Calcium Ca Z=20, A=40 Scandium Sc Z=21, A=45 Titanium Ti The 15 cm Kanone 18 (15 cm K 18) was a German heavy gun used in the Second World War. Design and history. In
1933 Rheinmetall began development of a new First type the equation 2x+3=15. Then type the @ symbol.
[15pts.] Use the Stokes' Theorem to evaluate F.dr when the vector field is F(x, y, z)= (2y, xz,x+y) and C is the curve of intersection of the plane z= y +2 and the cylinder x2 + y2 = 1. [15pts.] Use
the Stokes' Theorem to evaluate F.dr when the vector field is F(x, y, z)= (2y, xz,x+y) and C is the curve of intersection of the plane z= y +2 and the cylinder x2 + y2 = 1. 26/01/2021 01/04/2018
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Intel Core i7 Options. Price. to. Less =18/1.5 = 12 If you divide your final number by 1.5 (since 150% equals to 1.5), you can get your starting number, which is 12. 18/1.5 = 12 This is your answer
(12). 18 x 18 x 15" Corrugated Boxes. | {"url":"https://valutafxzq.web.app/20019/65860.html","timestamp":"2024-11-03T16:25:39Z","content_type":"text/html","content_length":"18977","record_id":"<urn:uuid:54967fe7-3e8f-44e8-8395-5f97e225f5a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00139.warc.gz"} |
XLOOKUP Strange Rounding Error when Searching Multiple Variables | Microsoft Community Hub
XLOOKUP Strange Rounding Error when Searching Multiple Variables
My colleague and I are attempting to use XLOOKUP to search an array of values for two variables and return a third, rounded up to the next largest value. The formula we came up with for this is:
It looks for the value in cell A2 in the range Table!C2:C241 and then searches for the value in B2 in the range Table!D2:D241. I believe both of these should be looking for an exact match (which will
work every time for the A2 value) or rounding up to the next highest value (which will usually be the case for the B2 value).
This formula works great most of the time, but when it encounters the need to round across a new tens place digit it seems to loop back. For instance, if asked to find 18.21 in the 15.5 section, it
should return the value corresponding to the next highest value (20.72), instead it returns the value corresponding to 2.05. It's as though when increasing the value of the tens place in the rounding
function there's a missing 0 placeholder and it reverts the correct number (2) to the ones place and finds that.
This is all being done on Windows 10 PCs using Office 365 Apps for enterprise. I'd gladly upload the document, it's not large, but there appears to be no functionality for that.
• Noel-T Not sure that what you see as correct results are in fact correct, though merely a coincidence. When you use A2&B2, you are concatenating the two numbers into a text. So, if A2 contains
the number 10.5 and B2 contains 18.21, XLOOKUP will look for the text "10.518.21" in the texts concatenated from C2:C24 & D2:D24. Hence, you are no longer working with numbers. | {"url":"https://techcommunity.microsoft.com/discussions/excelgeneral/xlookup-strange-rounding-error-when-searching-multiple-variables/3052148","timestamp":"2024-11-09T23:18:47Z","content_type":"text/html","content_length":"209535","record_id":"<urn:uuid:8c93e81f-773f-4899-a362-3fbfd4d3f944>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00527.warc.gz"} |
American Mathematical Society
Numerical solution of stochastic differential equations with constant diffusion coefficients
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Math. Comp. 49 (1987), 523-542 Request permission
We present Runge-Kutta methods of high accuracy for stochastic differential equations with constant diffusion coefficients. We analyze ${L_2}$ convergence of these methods and present convergence
proofs. For scalar equations a second-order method is derived, and for systems a method of order one-and-one-half is derived. We further consider a variance reduction technique based on Hermite
expansions for evaluating expectations of functions of sample solutions. Numerical examples in two dimensions are presented. References
• Ludwig Arnold, Stochastic differential equations: theory and applications, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Translated from the German. MR 0443083 S. W.
Benson, The Foundations of Chemical Kinetics, McGraw-Hill, New York, 1980. S. Chandrasekhar, "Stochastic problems in physics and astronomy," Noise and Stochastic Processes (N. Wax, ed.), Dover,
New York, 1954. C. C. Chang, Numerical Solution of Stochastic Differential Equations, Ph.D. Dissertation, University of California, Berkeley, 1985.
• Alexandre Joel Chorin, Hermite expansions in Monte-Carlo computation, J. Comput. Phys. 8 (1971), 472–482. MR 297092, DOI 10.1016/0021-9991(71)90025-8
• Alexandre Joel Chorin, Accurate evaluation of Wiener integrals, Math. Comp. 27 (1973), 1–15; corrigenda, ibid. 27 (1973), 1011. MR 329205, DOI 10.1090/S0025-5718-1973-0329205-7
• Alexandre Joel Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), no. 4, 785–796. MR 395483, DOI 10.1017/S0022112073002016
• Alexandre Joel Chorin, Lectures on turbulence theory, Mathematics Lecture Series, No. 5, Publish or Perish, Inc., Boston, Mass., 1975. MR 0502876
• Kai Lai Chung, A course in probability theory, 2nd ed., Probability and Mathematical Statistics, Vol. 21, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0346858
• L. Fahrmeir, Approximation von stochastischen Differentialgleichungen auf Digitalund Hybridrechnern, Computing 16 (1976), no. 4, 359–371. MR 405789, DOI 10.1007/BF02252084
• Aaron L. Fogelson, A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting, J. Comput. Phys. 56 (1984), no. 1, 111–134. MR 760745, DOI
• Joel N. Franklin, Difference methods for stochastic ordinary differential equations, Math. Comp. 19 (1965), 552–561. MR 193340, DOI 10.1090/S0025-5718-1965-0193340-2
• C. William Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0315898 A. H. Jazwinski, Stochastic Processes and
Filtering Theory, Academic Press, New York, 1970.
• Paul Lévy, Wiener’s random function, and other Laplacian random functions, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California
Press, Berkeley-Los Angeles, Calif., 1951, pp. 171–187. MR 0044774
• E. J. McShane, Stochastic differential equations and models of random processes, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California,
Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 263–294. MR 0402921
• F. H. Maltz and D. L. Hitzl, Variance reduction in Monte Carlo computations using multidimensional Hermite polynomials, J. Comput. Phys. 32 (1979), no. 3, 345–376. MR 544556, DOI 10.1016/
• G. N. Mil′šteĭn, Approximate integration of stochastic differential equations, Teor. Verojatnost. i Primenen. 19 (1974), 583–588 (Russian, with English summary). MR 0356225 G. N. Mil’shtein, "A
method of second-order accuracy integration of stochastic differential equations," Theory Probab. Appl., v. 23, 1978, pp. 396-401.
• E. Platen, Weak convergence of approximations of Itô integral equations, Z. Angew. Math. Mech. 60 (1980), no. 11, 609–614 (English, with German and Russian summaries). MR 614912, DOI 10.1002/
• Eckhard Platen and Wolfgang Wagner, On a Taylor formula for a class of Itô processes, Probab. Math. Statist. 3 (1982), no. 1, 37–51 (1983). MR 715753
• N. J. Rao, J. D. Borwanker, and D. Ramkrishna, Numerical solution of Ito integral equations, SIAM J. Control 12 (1974), 125–139. MR 0343367
• W. Rümelin, Numerical treatment of stochastic differential equations, SIAM J. Numer. Anal. 19 (1982), no. 3, 604–613. MR 656474, DOI 10.1137/0719041
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• © Copyright 1987 American Mathematical Society
• Journal: Math. Comp. 49 (1987), 523-542
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• DOI: https://doi.org/10.1090/S0025-5718-1987-0906186-6
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Plot Two Continuous Variables: Scatter Graph and Alternatives
Scatter plots are used to display the relationship between two continuous variables x and y. In this article, we’ll start by showing how to create beautiful scatter plots in R.
We’ll use helper functions in the ggpubr R package to display automatically the correlation coefficient and the significance level on the plot.
We’ll also describe how to color points by groups and to add concentration ellipses around each group. Additionally, we’ll show how to create bubble charts, as well as, how to add marginal plots
(histogram, density or box plot) to a scatter plot.
We continue by showing show some alternatives to the standard scatter plots, including rectangular binning, hexagonal binning and 2d density estimation. These plot types are useful in a situation
where you have a large data set containing thousands of records.
R codes for zooming, in a scatter plot, are also provided. Finally, you’ll learn how to add fitted regression trend lines and equations to a scatter graph.
1. Install cowplot package. Used to arrange multiple plots. Will be used here to create a scatter plot with marginal density plots. Install the latest developmental version as follow:
2. Install ggpmisc for adding the equation of a fitted regression line on a scatter plot:
3. Load required packages and set ggplot themes:
• Load ggplot2 and ggpubr R packages
• Set the default theme to theme_minimal() [in ggplot2]
theme_minimal() +
theme(legend.position = "top")
4. Prepare demo data sets:
Dataset: mtcars. The variable cyl is used as grouping variable.
# Load data
df <- mtcars
# Convert cyl as a grouping variable
df$cyl <- as.factor(df$cyl)
# Inspect the data
head(df[, c("wt", "mpg", "cyl", "qsec")], 4)
## wt mpg cyl qsec
## Mazda RX4 2.62 21.0 6 16.5
## Mazda RX4 Wag 2.88 21.0 6 17.0
## Datsun 710 2.32 22.8 4 18.6
## Hornet 4 Drive 3.21 21.4 6 19.4
Basic scatter plots
Key functions:
• geom_point(): Create scatter plots. Key arguments: color, size and shape to change point color, size and shape.
• geom_smooth(): Add smoothed conditional means / regression line. Key arguments:
□ color, size and linetype: Change the line color, size and type.
□ fill: Change the fill color of the confidence region.
b <- ggplot(df, aes(x = wt, y = mpg))
# Scatter plot with regression line
b + geom_point()+
geom_smooth(method = "lm")
# Add a loess smoothed fit curve
b + geom_point()+
geom_smooth(method = "loess")
To remove the confidence region around the regression line, specify the argument se = FALSE in the function geom_smooth().
Change the point shape, by specifying the argument shape, for example:
b + geom_point(shape = 18)
To see the different point shapes commonly used in R, type this:
Create easily a scatter plot using ggscatter() [in ggpubr]. Use stat_cor() [ggpubr] to add the correlation coefficient and the significance level.
# Add regression line and confidence interval
# Add correlation coefficient: stat_cor()
ggscatter(df, x = "wt", y = "mpg",
add = "reg.line", conf.int = TRUE,
add.params = list(fill = "lightgray"),
ggtheme = theme_minimal()
stat_cor(method = "pearson",
label.x = 3, label.y = 30)
Multiple groups
• Change point colors and shapes by groups.
• Add marginal rug: geom_rug().
# Change color and shape by groups (cyl)
b + geom_point(aes(color = cyl, shape = cyl))+
geom_smooth(aes(color = cyl, fill = cyl), method = "lm") +
geom_rug(aes(color =cyl)) +
scale_color_manual(values = c("#00AFBB", "#E7B800", "#FC4E07"))+
scale_fill_manual(values = c("#00AFBB", "#E7B800", "#FC4E07"))
# Remove confidence region (se = FALSE)
# Extend the regression lines: fullrange = TRUE
b + geom_point(aes(color = cyl, shape = cyl)) +
geom_rug(aes(color =cyl)) +
geom_smooth(aes(color = cyl), method = lm,
se = FALSE, fullrange = TRUE)+
scale_color_manual(values = c("#00AFBB", "#E7B800", "#FC4E07"))+
ggpubr::stat_cor(aes(color = cyl), label.x = 3)
• Split the plot into multiple panels. Use the function facet_wrap():
b + geom_point(aes(color = cyl, shape = cyl))+
geom_smooth(aes(color = cyl, fill = cyl),
method = "lm", fullrange = TRUE) +
facet_wrap(~cyl) +
scale_color_manual(values = c("#00AFBB", "#E7B800", "#FC4E07"))+
scale_fill_manual(values = c("#00AFBB", "#E7B800", "#FC4E07")) +
• Add concentration ellipse around groups. R function stat_ellipse(). Key arguments:
□ type: The type of ellipse. The default “t” assumes a multivariate t-distribution, and “norm” assumes a multivariate normal distribution. “euclid” draws a circle with the radius equal to
level, representing the euclidean distance from the center.
□ level: The confidence level at which to draw an ellipse (default is 0.95), or, if type=“euclid”, the radius of the circle to be drawn.
b + geom_point(aes(color = cyl, shape = cyl))+
stat_ellipse(aes(color = cyl), type = "t")+
scale_color_manual(values = c("#00AFBB", "#E7B800", "#FC4E07"))
Instead of drawing the concentration ellipse, you can: i) plot a convex hull of a set of points; ii) add the mean points and the confidence ellipse of each group. Key R functions: stat_chull(),
stat_conf_ellipse() and stat_mean() [in ggpubr]:
# Convex hull of groups
b + geom_point(aes(color = cyl, shape = cyl)) +
stat_chull(aes(color = cyl, fill = cyl),
alpha = 0.1, geom = "polygon") +
scale_color_manual(values = c("#00AFBB", "#E7B800", "#FC4E07")) +
scale_fill_manual(values = c("#00AFBB", "#E7B800", "#FC4E07"))
# Add mean points and confidence ellipses
b + geom_point(aes(color = cyl, shape = cyl)) +
stat_conf_ellipse(aes(color = cyl, fill = cyl),
alpha = 0.1, geom = "polygon") +
stat_mean(aes(color = cyl, shape = cyl), size = 2) +
scale_color_manual(values = c("#00AFBB", "#E7B800", "#FC4E07")) +
scale_fill_manual(values = c("#00AFBB", "#E7B800", "#FC4E07"))
# Add group mean points and stars
ggscatter(df, x = "wt", y = "mpg",
color = "cyl", palette = "npg",
shape = "cyl", ellipse = TRUE,
mean.point = TRUE, star.plot = TRUE,
ggtheme = theme_minimal())
# Change the ellipse type to 'convex'
ggscatter(df, x = "wt", y = "mpg",
color = "cyl", palette = "npg",
shape = "cyl",
ellipse = TRUE, ellipse.type = "convex",
ggtheme = theme_minimal())
Add point text labels
Key functions:
• geom_text() and geom_label(): ggplot2 standard functions to add text to a plot.
• geom_text_repel() and geom_label_repel() [in ggrepel package]. Repulsive textual annotations. Avoid text overlapping.
First install ggrepel (ìnstall.packages("ggrepel")), then type this:
# Add text to the plot
.labs <- rownames(df)
b + geom_point(aes(color = cyl)) +
geom_text_repel(aes(label = .labs, color = cyl), size = 3)+
scale_color_manual(values = c("#00AFBB", "#E7B800", "#FC4E07"))
# Draw a rectangle underneath the text, making it easier to read.
b + geom_point(aes(color = cyl)) +
geom_label_repel(aes(label = .labs, color = cyl), size = 3)+
scale_color_manual(values = c("#00AFBB", "#E7B800", "#FC4E07"))
Bubble chart
In a bubble chart, points size is controlled by a continuous variable, here qsec. In the R code below, the argument alpha is used to control color transparency. alpha should be between 0 and 1.
b + geom_point(aes(color = cyl, size = qsec), alpha = 0.5) +
scale_color_manual(values = c("#00AFBB", "#E7B800", "#FC4E07")) +
scale_size(range = c(0.5, 12)) # Adjust the range of points size
Color by a continuous variable
• Color points according to the values of the continuous variable: “mpg”.
• Change the default blue gradient color using the function scale_color_gradientn() [in ggplot2], by specifying two or more colors.
b + geom_point(aes(color = mpg), size = 3) +
scale_color_gradientn(colors = c("#00AFBB", "#E7B800", "#FC4E07"))
Add marginal density plots
The function ggMarginal() [in ggExtra package] (Attali 2017), can be used to easily add a marginal histogram, density or box plot to a scatter plot.
First, install the ggExtra package as follow: install.packages("ggExtra"); then type the following R code:
# Create a scatter plot
p <- ggplot(iris, aes(Sepal.Length, Sepal.Width)) +
geom_point(aes(color = Species), size = 3, alpha = 0.6) +
scale_color_manual(values = c("#00AFBB", "#E7B800", "#FC4E07"))
# Add density distribution as marginal plot
ggMarginal(p, type = "density")
# Change marginal plot type
ggMarginal(p, type = "boxplot")
One limitation of ggExtra is that it can’t cope with multiple groups in the scatter plot and the marginal plots.
A solution is provided in the function ggscatterhist() [ggpubr]:
# Grouped Scatter plot with marginal density plots
iris, x = "Sepal.Length", y = "Sepal.Width",
color = "Species", size = 3, alpha = 0.6,
palette = c("#00AFBB", "#E7B800", "#FC4E07"),
margin.params = list(fill = "Species", color = "black", size = 0.2)
# Use box plot as marginal plots
iris, x = "Sepal.Length", y = "Sepal.Width",
color = "Species", size = 3, alpha = 0.6,
palette = c("#00AFBB", "#E7B800", "#FC4E07"),
margin.plot = "boxplot",
ggtheme = theme_bw()
Continuous bivariate distribution
In this section, we’ll present some alternatives to the standard scatter plots. These include:
• Rectangular binning. Rectangular heatmap of 2d bin counts
• Hexagonal binning: Hexagonal heatmap of 2d bin counts.
• 2d density estimation
1. Rectangular binning:
Rectangular binning is a very useful alternative to the standard scatter plot in a situation where you have a large data set containing thousands of records.
Rectangular binning helps to handle overplotting. Rather than plotting each point, which would appear highly dense, it divides the plane into rectangles, counts the number of cases in each rectangle,
and then plots a heatmap of 2d bin counts. In this plot, many small hexagon are drawn with a color intensity corresponding to the number of cases in that bin.
Key function: geom_bin2d(): Creates a heatmap of 2d bin counts. Key arguments: bins, numeric vector giving number of bins in both vertical and horizontal directions. Set to 30 by default.
2. Hexagonal binning: Similar to rectangular binning, but divides the plane into regular hexagons. Hexagon bins avoid the visual artefacts sometimes generated by the very regular alignment of
Key function: geom_hex()
3. Contours of a 2d density estimate. Perform a 2D kernel density estimation and display results as contours overlaid on the scatter plot. This can be also useful for dealing with overplotting.
Key function: geom_density_2d()
• Create a scatter plot with rectangular and hexagonal binning:
# Rectangular binning
ggplot(diamonds, aes(carat, price)) +
geom_bin2d(bins = 20, color ="white")+
scale_fill_gradient(low = "#00AFBB", high = "#FC4E07")+
# Hexagonal binning
ggplot(diamonds, aes(carat, price)) +
geom_hex(bins = 20, color = "white")+
scale_fill_gradient(low = "#00AFBB", high = "#FC4E07")+
• Create a scatter plot with 2d density estimation:
# Add 2d density estimation
sp <- ggplot(iris, aes(Sepal.Length, Sepal.Width)) +
geom_point(color = "lightgray")
sp + geom_density_2d()
# Use different geometry and change the gradient color
sp + stat_density_2d(aes(fill = ..level..), geom = "polygon") +
scale_fill_gradientn(colors = c("#FFEDA0", "#FEB24C", "#F03B20"))
Zoom in a scatter plot
• Key function: facet_zomm() [in ggforce] (Pedersen 2016).
• Demo data set: iris. The R code below zoom the points where Species == "versicolor".
ggplot(iris, aes(Petal.Length, Petal.Width, colour = Species)) +
geom_point() +
ggpubr::color_palette("jco") +
facet_zoom(x = Species == "versicolor")+
To zoom the points, where Petal.Length < 2.5, type this:
ggplot(iris, aes(Petal.Length, Petal.Width, colour = Species)) +
geom_point() +
ggpubr::color_palette("jco") +
facet_zoom(x = Petal.Length < 2.5)+
Add trend lines and equations
In this section, we’ll describe how to add trend lines to a scatter plot and labels (equation, R2, BIC, AIC) for a fitted lineal model.
1. Load packages and create a basic scatter plot facetted by groups:
# Load packages and set theme
theme_bw() +
theme(legend.position = "top")
# Scatter plot
p <- ggplot(iris, aes(Sepal.Length, Sepal.Width)) +
geom_point(aes(color = Species), size = 3, alpha = 0.6) +
scale_color_manual(values = c("#00AFBB", "#E7B800", "#FC4E07")) +
scale_fill_manual(values = c("#00AFBB", "#E7B800", "#FC4E07"))+
2. Add regression line, correlation coefficient and equantions of the fitted line. Key functions:
□ stat_smooth() [ggplot2]
□ stat_cor() [ggpubr]
□ stat_poly_eq()[ggpmisc]
formula <- y ~ x
p +
stat_smooth( aes(color = Species, fill = Species), method = "lm") +
stat_cor(aes(color = Species), label.y = 4.4)+
aes(color = Species, label = ..eq.label..),
formula = formula, label.y = 4.2, parse = TRUE)
3. Fit polynomial equation:
x <- 1:100
y <- (x + x^2 + x^3) + rnorm(length(x), mean = 0, sd = mean(x^3) / 4)
my.data <- data.frame(x, y, group = c("A", "B"),
y2 = y * c(0.5,2), block = c("a", "a", "b", "b"))
• Fit polynomial regression line and add labels:
# Polynomial regression. Sow equation and adjusted R2
formula <- y ~ poly(x, 3, raw = TRUE)
p <- ggplot(my.data, aes(x, y2, color = group)) +
geom_point() +
geom_smooth(aes(fill = group), method = "lm", formula = formula) +
aes(label = paste(..eq.label.., ..adj.rr.label.., sep = "~~~~")),
formula = formula, parse = TRUE
ggpar(p, palette = "jco")
Note that, you can also display the AIC and the BIC values using ..AIC.label.. and ..BIC.label.. in the above equation.
Other arguments (label.x, label.y) are available in the function stat_poly_eq() to adjust label positions.
For more examples, type this R code: browseVignettes(“ggpmisc”).
1. Create a basic scatter plot:
b <- ggplot(mtcars, aes(x = wt, y = mpg))
Possible layers, include:
• geom_point() for scatter plot
• geom_smooth() for adding smoothed line such as regression line
• geom_rug() for adding a marginal rug
• geom_text() for adding textual annotations
2. Continuous bivariate distribution:
c <- ggplot(diamonds, aes(carat, price))
Possible layers include:
• geom_bin2d(): Rectangular binning.
• geom_hex(): Hexagonal binning.
• geom_density_2d(): Contours from a 2d density estimate | {"url":"http://www.sthda.com/english/articles/32-r-graphics-essentials/131-plot-two-continuous-variables-scatter-graph-and-alternatives/","timestamp":"2024-11-15T03:48:52Z","content_type":"text/html","content_length":"88167","record_id":"<urn:uuid:359c3ebc-b732-47ad-b3a5-28217b73ef8c>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00738.warc.gz"} |
FAQ - Parsum EN
Parsum probes measure the particle size of a single particle, where the statistic chord length acts as a particle attribute. The statistic chord length is obtained by the optical scanning of the
projection area of the particle. The particle size is represented by the symbol “x” in accordance with DIN ISO 9276-1. Alternatively, the symbol “d” can also be used. | {"url":"https://www.parsum.de/en/faq/","timestamp":"2024-11-03T13:56:54Z","content_type":"text/html","content_length":"106843","record_id":"<urn:uuid:03edfdcb-3d4d-41a6-bf61-1ab28a44602c>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00290.warc.gz"} |
Development of Anthropometry-Based Equations for the Estimation of the Total Body Water in Koreans
The measurement of total body water (TBW) is frequently performed to evaluate the body composition and nutritional status. The accurate measurement of TBW is difficult, and it requires isotopic
dilution techniques. Therefore, several indirect equations for estimating the TBW using simple anthropometric variables are commonly employed. However, these equations are largely based on
individuals of the western hemisphere.
The purpose of this study was to develop anthropometry-based TBW equations in Korean and to compare these equations with the other available TBW equations. Since it is difficult to perform isotopic
dilution techniques on a large number of subjects, we used bioelectrical impedance analysis (BIA), which has been shown to accurately and reliably estimate TBW (
). Therefore, we first measured the TBW using BIA (TBW
) in a large study population to develop an anthropometry-based TBW equation. Then to validate this equation, we analyzed the agreement between the TBW
and the TBW derived from anthropometry-based equations in another control group.
A total of 2,943 healthy adults were selected for this study from the 3,781 people visiting the Health Promotion Center (HPC) at Inha University Hospital (IUH) from May to December 2003. The
exclusion criteria were as follows: age <18 yr, a serum creatinine >1.4 mg/dL, positive urine protein, subjects who complained of edema, those with an amputation or who had diabetes mellitus,
congestive heart failure, chronic liver disease, or those subjects who did not allow BIA to be performed. Among them, 2,223 subjects were used for the development of equations. The remaining 720
subjects were used for the validation of equation. This study was approved by the ethical board of IUH.
After 8 hr of fasting, the subjects visited to the HPC at 9 a.m. Their height (Ht) and body weight (BW) were measured to the nearest 0.1 cm and 0.1 kg using a linear height scale and an electronic
weight scale, respectively. The mean values of two measurements were used for data analysis.
BIA (Inbody 3.0, Biospace Co., Seoul, Korea) was performed by a well trained nursing staff. The equipment involves placing eight tactile electrodes on a patient in an upright posture. When the
subject was standing on the sole electrodes and gripping the hand electrodes, the microprocessor was switched on and the impedance analyzer started to measure the segmental resistances of the right
arm, left arm, trunk, right leg, and left leg at four frequencies (5, 50, 250 and 500 kHz), thus measuring a set of 20 segmental resistances for one individual. The mean values of two sets of BIA
measurements were used for analysis. The repeat measured coefficient of variation for TBW was 0.29%, and the day-to-day coefficient of variation of TBW was 1.18%. The accuracy of the 8-point
tactile-electrode impedance method on the measurement of TBW
was validated on healthy subjects (
). The procedure was performed in 3 min or less and the TBW
was automatically calculated from the BIA with equations installed in the instrument's program.
We chose the Watson (
) and Hume-Weyers (
) formulas to compare the accuracy of the newly developed equation:
Watson formula
Male: TBW[W]=2.447-(0.09156×age)+(0.1074×Ht)+(0.3362×BW)
Female: TBW[W]=-2.097+(0.1069×Ht)+(0.2466×BW)
Hume-Weyers formula
Male: TBW[H]=(0.194786×Ht)+(0.296785×BW)-14.012934
Female: TBW[H]=(0.34454×Ht)+(0.183809×BW)-35.270121
Where age in years, Ht in cm, and the BW in kg.
Statistical analysis
The data were expressed as means±SD. Linear regression analysis was performed to develop the anthropometry-based TBW equation. Stepwise selection was employed using entry and exit criteria of
<0.01. TBW
was used as a dependent variable. Sex, age, Ht and BW were used as independent variables. Polynomial terms for continuous variables and multiplicative interaction terms were considered in the model
building process. Pearson's correlation coefficient (r) was used to find the relationship between two variables. To analyze the differences in TBW
and TBWs derived from anthropometry-based equations, one-way analysis of variance (ANOVA) was performed with using the Bonferroni method for the post-hoc test. To assess the agreement, Bland-Altman
plots using the means and differences between TBW
and calculated TBW were used (
). To quantitate the degrees of bias, we compared the correlation coefficients of the respective differences and means. The closer the correlation coefficient of Bland-Altman plot was to zero, the
less the bias. Root mean square error (RMSE) and mean prediction error (ME) were also used. ME was also an indication of bias, but not of accuracy. The RMSE value was used as a measure of the
goodness-of-fit of an equation. If there were more than one equation to fit the data, the one with the smallest RMSE value had the highest precision. The equations used for ME and RMSE are as
A p value less than 0.05 was considered as statistically significant.
Development of anthropometry-based TBW equations
For the 2,223 subjects, the male to female ratio was 1.72:1, the mean age was 45.1±10.9 yr, the mean BW was 64.3±11.0 kg, the mean Ht was 164.9±8.5 cm, and the mean TBW
was 34.9±6.6 L. The simple (TBW
) and complicated (TBW
) TBW equations based on the anthropometric variables were developed by linear regression analysis (
Table 1
). The adjusted R
was 0.908 for TBW
and 0.910 for TBW
Validation of newly developed TBW equations
In another 720 control subjects, the male to female ratio was 1.28:1, the mean age was 47.0±11.1 yr, the mean BW was 63.6±10.5 kg, the mean Ht was 163.8±9.3 cm, and the mean TBW
was 33.6±6.2 L. In males, TBW
showed the highest correlation with TBW
(r=0.951), followed by TBW
(r=0.945), TBW
(r=0.945) and TBW
(r=0.937) (
Table 2
). There were no differences between the TBW
and TBW
or TBW
. However, TBW
and TBW
were significantly larger than the TBW
. There were significant differences between TBW
and TBW
or TBW
and between the TBW
and TBW
. In females, TBW
showed the highest correlation with TBW
(r=0.902), followed by TBW
(r=0.895), TBW
(r=0.890), and TBW
(r=0.887). There were no differences between TBW
and TBW
, TBW
or TBW
. The TBW
was significantly larger than the others.
In males, the TBW
and TBW
showed the lower RMSE (1.58, 1.58, 2.14, and 2.08 for TBW
and TBW
, respectively) and ME (0.526, 0.547, 1.426, and 1.362 for TBW
and TBW
, respectively) than the TBW
and TBW
Table 3
). On the Bland-Altman plot, the correlations between the difference and means were smallest for the TBW
(r= -0.192), followed by the TBW
, TBW
, and TBW
Fig. 1A, C, E, G
). In females, the RMSEs were smallest for the TBW
, followed by the TBW
, TBW
, and TBW
(1.49, 1.50, 1.62, and 1.70 for the TBW
, TBW
, TBW
and TBW
, respectively). The ME was closest to zero for the TBW
, followed by the TBW
, TBW
and TBW
(0.554, 0.556, 0.593, and 0.988 for the TBW
, TBW
, TBW
and TBW
, respectively). The correlation coefficients between the means and differences were highest for the TBW
(r=-0.553), and lowest for the TBW
(r=0.057) (
Fig. 1B, D, F, and H
In this study, we developed two anthropometry-based TBW equations (TBW[K1] and TBW[K2]) for Koreans using TBW[BIA] as a reference. Among them, TBW[K2] showed the highest precision and the smallest
bias for males and a similar precision and the smallest bias for females compared to the TBWs derived from Watson or Hume-Weyers formulas.
Accurate estimation of the TBW is important in many pathophysiologic states, as the clinical symptoms and signs of volume dysregulation complicate a variety of medical and surgical conditions.
Furthermore, the disposition of electrolytes, enteral and parenteral nutrition, and selected drugs largely depends on the size and distribution of the TBW space. As the majority of TBW resides in the
skeletal muscle, TBW may also be used as estimates of somatic protein stores (
The need for an accurate measurement of the TBW is particularly important for dialysis patients, as it equates to the distribution volume of urea (V). In hemodialysis (HD) patients, urea is the
substance that is most often monitored as a surrogate for measurement of dialysis adequacy (
). A dose of HD (prescribed or delivered) is best described as the fractional clearance of urea as a function of its distribution volume (Kt/V) (
). However, in the dialysis centers, it is not easy to measure TBW each time using an accurate method such as BIA. For convenience, the Kt/V is automatically calculated using a computerized program
in which the TBW equations are installed by simply entering height, sex, the pre- and post-HD blood urea nitrogen concentration, the ultrafiltration amount and duration of HD. For the calculation of
V, the Watson and Hume-Weyers formulas are generally recommended (
). However, these TBW equations were mainly derived from the age, gender, height and weight of a western population. These equations have not been validated in a Korean population, nor have their
accuracies been compared with a race-specific formula. In this study, we found that the TBW equations derived from a western population showed greater bias than our formulas. They tended to
overestimate the small TBWs and underestimate large TBWs. Compared to Caucasians, Koreans are smaller with lower body weights and lower values of TBW (
). Therefore, it is natural that systematic errors occur when applying the prediction formula from a reference population to another population under study. Several studies have pointed out that
race-specific TBW equations should be used when applying them to another race with a different body build (
). Considering this background, TBW
may be helpful for assessing the nutritional status and dialysis adequacy more exactly for the Korean healthy control population and the Korean patients with end-stage renal disease.
In this study, TBW
showed a lower RMSE value than the TBW
in females. Therefore, TBW
might have a better accuracy than TBW
, at least in females. However, TBW
showed a greater bias than TBW
, as shown in
Fig. 1B, H
. TBW
had a similar RMSE value and its ME was closer to zero than TBW
. Furthermore, it had the least bias in females. Therefore, TBW
seemed to be more suitable for the estimation of the TBW in Korean females.
In this study, TBWs estimated from the Watson and Hume-Weyers formula showed overestimation in small TBWs and underestimation in large TBWs. The reason for this might be due to the characteristics of
subjects when the Watson and Hume-Weyers formulas were derived. For the Watson formula, the mean TBW was between 36.7 and 44.1 L in males and between 31.4 and 33.2 L in females (
). For the Hume-Weyers formula, the mean TBW was between 35.3 and 46.2 L in males and between 30.2 and 39.8 L in females (
). Therefore, when the TBWs were out of those ranges, the TBWs estimated from the Watson and Hume-Weyer formula seemed to over- or under-estimate the real TBWs.
There are several limitations to this study. First, TBW was estimated using BIA rather than using deuterium oxide or another standard dilution method. However, any methods, even the gold standard
methods, for the assessment of TBW are based upon assumptions that allow for some inherent errors. Furthermore, the gold standard methods are expensive, laborious and hard to apply to a large number
of subjects, as in this study. BIA does have several advantages; it is easy to use, rapid, non-invasive, inexpensive and applicable at the bedside. Several studies have shown that TBW can accurately
and reliably estimated by BIA in normal healthy subjects (
). We used segmental BIA by the eight-polar tactile electrode impedance method. Segmental BIA reduced the errors from whole body BIA estimation (
). The accuracy of the TBW assessment by this method has been validated in control subjects (
). Second, the subjects of this study were not randomly selected from nationwide regions. Therefore, the study subjects may not be representative of the entire Korean population. In spite of this
problem, the number of study subjects was large enough to overcome this drawback. We also validated the accuracy of newly developed equations in another set of subjects. Third, for the males, the
newly developed TBW equations (and even the TBW
) still showed weak correlation between the means and differences in the Bland-Altman plot. Thus, the TBW derived from TBW
might underestimate the real TBW in men with large BW. Fourth, this study was limited to the healthy subjects. Therefore, it should be validated for patients with the volume disorders such as acute
renal failure, liver cirrhosis with ascites, ESRD, congestive heart failure, and nephrotic syndrome.
In summary, our race specific anthropometry-based equation provides superior or at least similar precision of TBW, compared to Watson or Hume-Weyers formula, in Korean subjects, with least bias. This
equation may be useful for the estimation of TBW in a large number of subjects. | {"url":"https://synapse.koreamed.org/articles/1020010","timestamp":"2024-11-07T10:55:51Z","content_type":"application/xhtml+xml","content_length":"61547","record_id":"<urn:uuid:30fdd382-db64-4359-b24d-d7d030a9fef8>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00454.warc.gz"} |
Would you mind explaining Van der Waals force? - Does quantum mechanical tunneling play much of a role in nuclear fusion? - Are you planning on ever
The calculations and discussions presented here have demonstrated or suggested the following: • That quantum tunnelling can be explained without Heisenberg’s uncertainty principle.
The wave function may be 1. An almost complete quantum mechanical picture of photosynthesis is hypothesised, but as of now, only a little of the beginning of the reaction is well explained
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quantum tunneling. Tunneling i Watch quantum "particles" tunnel through barriers. Explore the properties of the wave functions that describe these particles.
Quantum tunneling is a subject that is quite difficult to understand as it is highly mathematical in nature. In this video, I attempt to provide a basic intu — Quantum Tunneling explained. Google
Classroom Facebook Twitter. Email. 2020 Challenge — Winner “Walk through walls” hack in real life?! — Quantum Tunneling explained. This is the currently selected item.
in common with quantum tunneling as the center-of-mass literally goes through the barrier.
9 Dec 2020 Quantum tunnelling refers to the quantum mechanical phenomenon whereby electrons can 'tunnel' through barriers that they theoretically
. .
What is quantum tunnelling? Imagine releasing a quantum mechanical particle, like an electron or proton, into a space on one side of an potential energy hill. Since you’re sure that the particle
Videos you watch may be added to the TV's watch history and influence TV recommendations. Quantum Tunneling : The phenomenon of tunneling, which has no counterpart in classical physics, is an
important consequence of quantum mechanics. Consider a particle with energy E in the inner region of a one-dimensional potential well V(x). Quantum tunneling is a subject that is quite difficult to
understand as it is highly mathematical in nature.
Consider rolling a ball up a hill. If the ball is not given enough Quantum tunneling refers to the general physical phenomenon where a particle tunnels through a barrier that it cannot overcome, i.e.
it takes a shortcut. Tunneling plays an essential role in several physical, chemical, and biological phenomena, such as radioactive decay or the manifestation of large kinetic isotope effects in
chemicals of enzymatic reactions. In short, quantum tunneling seemed to allow faster-than-light travel, a supposed physical impossibility.
Håkan nesser fru
Google Classroom A quantitative analysis of the physical implications of this tunneling effect had to await Erwin Schrödinger's wave mechanics and Max Born's probability Quantum tunnelling. Quantum
tunnelling (or tunneling) is the quantum- mechanical effect of transitioning through a classically-forbidden Tunneling, in physics, passage of minute particles through seemingly impassable physics
and requires instead explanation in terms of quantum mechanics. in common with quantum tunneling as the center-of-mass literally goes through the barrier.
Stars can be plotted on something called Hertzsprung-Russell diagram or HR Diagram.
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TrueDrive: a sophisticated spatial analysis technology
June 15, 2022
Reading time: 16 minute(s)
June 15, 2022
TrueDrive: a sophisticated spatial analysis technology
Reading time: 16 minute(s)
Everybody knows that finding ‘the perfect’ location to open a sales point or to start a building construction is a very nerve-wracking and time-consuming process.
Defining a proposed trade area based on just traditional marketing studies can cause crucial omission errors, lead to poor sales and wrong site selection decisions. First of all you should precisely
evaluate the market environment, the planned site location suitability, the competitive supply concerning local demographics and logistical factors, and many other things!
TrueDrive provides an API for performing the drive time/distance calculation. TrueDrive enables you to develop interactive desktop, server or web-based mapping or market analysis applications for
solving various routing and logistical problems. A smart calculation algorithm requiring low computation resources and memory consumption apart from flexible TrueDrive integration capabilities can be
valuable for performing different marketing analyses.
Based on TrueDrive, it is possible to create a web service using which the network analysis algorithms can be run. The following road network analysis algorithms are currently available:
• Calculation of shortest route between points
• Calculation of service areas from specified points for specific time or distance.
On the left: calculation of shortest route, on the right: calculation of service area
One of the main TrueDrive benefits is the extremely high speed of applied algorithms with deep optimization. This enables you to almost instantly complete tasks that previously required a delayed
start or long wait.
It can help you solve various non-standard problems in the domain of road transportation, enabling you to automate and optimize many work processes: decrease costs, improve safety, etc. The range of
solvable problems is wide and can be tailored further to the exact needs of your project or organization.
Technology benefits
TrueDrive is not yet another software product for creating road network services, but the product with number of benefits:
• High speed of the algorithms work enables you to perform typical calculations on the fly.
• Possibility to use not only standard road parameters for optimization (time and distance), but also specific parameters calculated in advance at the stage of building the road graph index. For
example, you can use the traffic safety criterion (if data is available) or apply the formula for calculating the optimization parameter on the fly, for which the SQL notation is used.
• Possibility to use barriers. Each request may contain a set of spatial restrictions on traffic on certain road segments or zones. If traffic on such sections is impossible or obstructed
(transparent barriers), then this circumstance will be taken into account when calculating the route. Using barriers does not require rebuilding of the road graph index, each calculation request
can contain its own set of barriers.
• When optimizing time, the turn penalties can be used in calculations depending on the turn angle. Thus, in case of the right-hand traffic, turning left will take longer than turning right. This
feature enables you to bring the calculation results closer to real conditions.
• When calculating service areas, multiple centers can be used at a time. The number of such centers is not limited and can reach up to several hundred, thus, for example, per one launch you can
get the distribution of service areas for many stores or logistics centers at once.
• Polygons obtained during the service areas calculation look very detailed, exactly repeating boundaries of passed road network sections. The detailed service areas boundary allows for greater
accuracy in determining the reachability of certain points and calculating overlay parameters (for example, calculation of demographics covered by service area).
• When calculating service areas from multiple centers, a mode is possible in which the output service areas will compete, that is, they will not overlap each other. In this case, the disputed
overlap areas will be divided between competing centers according to the criterion of proximity to such centers. The algorithm is somewhat similar to building of Thyssen polygons, but the
calculation is carried out on the road network.
Key features
Web service key features are as follows:
• availability of ready-made network analysis tools, including:
□ search for the shortest route between specific points,
□ optimization of the order of avoidance of intermediate points of the route,
□ calculation of isolated service areas,
□ calculation of competitive service areas;
• ability to use ready-made data sets for calculation (indices), as well as building new data sets (to order) based on user data;
• support for OS Linux and Windows Server;
• TrueDrive software is developed in Russian federation and belongs to Russian company;
• possibility of deep integration with the CoGIS infrastructure platform enables you to solve complex thematic problems using network analysis within the existing information infrastructure.
Technological features
The major distinguishing features of the TrueDrive technology are as follows.
Optimization types
All the network analysis algorithms available in TrueDrive are provided to optimize specific characteristic. That is, for example, a typical task of finding optimal route between two points on map is
minimization of travel time.
It is also possible to use characteristic of the distance traveled, rather than the time spent. This enables you to get the shortest route, despite the fact that the time used to travel along the
found route may be longer.
In everyday tasks, optimization of time and distance is usually sufficient, but in the workflows of an enterprise or a government agency, calculations may require specific characteristics depending
on the problem. The TrueDrive road network analysis algorithms enable you to use both standard parameters such as time or distance, and any other custom characteristics.
Optimal routes calculated by time and distance
To take advantage of this opportunity, at the stage of preparing a special road graph index based on the source data, it is necessary to set a formula for calculating the user characteristic based on
the source road segments properties.
The ability to work with user characteristics during optimization enables you to solve a wide range of non-standard problems. That is, for example, instead of optimizing time, you can minimize the
amount of used fuel, focusing on the average consumption of the car, which depends on the current speed.
Even more impressive is the use of characteristics determining the accident rate of each road section, taking into account the segments length. This algorithm will help you find the safest route.
The user characteristics can define a variety of parameters for each road segment, and in each case a unique optimization system can be obtained. This approach enables you to solve non-standard
problems of complex enterprise workflows.
Moving types
The use of network analysis algorithms involves moving along the road network. But there are multiple moving types: you can move by car, truck, special vehicle, bicycle or just walk.
TrueDrive technology enables you to determine all types of vehicles at the stage of preparing the road graph index and to set different corrections for each type. This helps you determine which roads
can drive all types of vehicles, and which can drive only cars, which roads can drive special vehicles only, and which can be used by public transport only.
For each moving type it is possible to set your own travel cost (for example, time) for each road segment. This cost, such as the travel time for the certain road segment, can be influenced by the
length of the segment, the quality of the road surface, the slope, and other characteristics.
The moving type is set each time the network analysis algorithm is run. You can run several calculations at once using different settings, including specifying the type of vehicle.
Using barriers
After building the indexed road graph based on the source data, the set of road connectivity characteristics and their parameters are fixed. This enables you to create auxiliary data structures, the
use of which significantly increases the calculation speed. The logical roads graph at the time of creation of the index structure becomes unchanged.
Often there are situations when certain road segments should be ignored, since traffic is prohibited (the road is blocked, repairs are underway, etc.). For such cases, the TrueDrive technology
provides a specific ability of setting barriers.
Recalculation of found route by adding barrier
Barriers can be of point, polyline or polygon type. The point barriers snap to the nearest road segment and build the virtual barrier on that segment. The polyline barrier marks as impassable those
segments that intersect with the given polyline. Same way, the polygon barriers mark those segments that fall completely or partially inside the polygon.
Barriers can be semi-transparent, i.e. not strictly prohibit traffic, but increase the travel time on the given segment. This mechanism enables you to define areas of unwanted presence. As a result,
the network analysis algorithm would try to bypass the marked area until it becomes irrational. If such bypass is not possible or the bypass cost is too high considering the selected optimization,
then the pass with minimal presence in the undesirable area is allowed.
The barrier transparency level is set by the penalty parameter with the following values:
• -1 – absolute barrier (traffic prohibited)
• 0 - instant teleportation (cost travel on segment is ignored)
• 0.5 - movement speed twice as fast
• 1 - no barrier
• 2 - movement speed twice as slow
Thus, setting barrier on the road can either completely prohibit traffic or reduce the passage priority on the given segment.
A special case is setting the penalty parameter in the range from 0 to 1. Values from this range allow, on the contrary, to increase the travel speed along the road segment. This possibility can be
used in situations where road characteristics have been improved, for example, the asphalt pavement has been laid over the gravel embankment.
The use of barriers enables you to decrease dependance on the need to recalculate the road graph index, but with a significant increase in the number of specified barriers, it is worth changing the
source road network data and recalculating the index file.
Peculiarities of passing crossroads
Each crossroad consists of several road segments converging at one point. While driving, the vehicle uses only two segments. The vehicle can go straight, turn right or left or make a U-turn.
The traffic rules may prohibit a certain type of maneuver at a given intersection. Most often, the left turns are prohibited. When building the road graph index, the TrueDrive technology enables you
to set a list of road segments pairs, the passage between which is prohibited. The network analysis algorithm will consider this rule and will not allow you to build the route that includes such
In addition, according to statistics, driving at different angles takes different time to pass the intersection. For example, driving straight ahead on average takes less time than turning left,
since when turning you need to significantly reduce your speed, and only after passing the intersection you can speed up again.
TrueDrive enables you to take into account such turn circumstances. To do this, when building the road graph index, you should specify the desired deceleration parameters depending on angles of
passing the intersection.
Optimal route search algorithm
The optimal route search algorithm is provided to calculate the route connecting several stopping points. As a result, the route will be calculated that enables you to travel from the start point to
the end point in the minimum time/distance, passing through all intermediate points.
Calculation of optimal route with intermediate points
Intermediate points order
There are two optimization modes when specifying multiple stopping points to calculate the optimal route:
• Preserving the initial intermediate stops order
• Optimization of the initial intermediate stops order.
When choosing the mode of preserving the intermediate points order, the algorithm performs a sequential search for optimal routes between each of the neighboring stopping points. The output result is
obtained by collecting all the parts into one route. Optimization of passing the intermediate points order is not done in this case.
If the mode of optimizing the intermediate points order is selected, then the optimal route search algorithm is used, considering all possible order combinations of the given stopping points. It
should be taken into account that the calculation with the enabled mode of optimizing the order of intermediate points may be slower than the calculation with preserving the order of intermediate
Roads hierarchy
In the algorithm for searching the optimal route from one city to another, in most cases, the main highways between cities are used. This is due to the fact that on highways the quality of roads and
the average speed are higher, and the distance is shorter.
Normally, the optimal route search algorithm should process most of the indexed graph of roads connecting the specified stopping points. The analysis of many road branches does not lead to the best
result, and in most cases such maneuvers can be neglected.
Calculation of optimal route without considering roads hierarchy (5-7 seconds)
TrueDrive offers the mode of calculating the optimal route based on the roads hierarchy. In this case, between cities (long sections), the subgraph consisting of only high-speed roads is used, and in
the small radius of stopping points, the complete graph of roads is used.
Calculation of optimal route considering roads hierarchy (1-1,5 seconds)
The road hierarchy mode can significantly reduce the calculation time for long routes and provide more expected result for travel or long-distance transportation, in which traffic between cities
mostly takes place on large highways, even though there may be a better route. Depending on the task and at the user’s discretion, you can turn on or off the mode of considering the roads hierarchy
to find the most optimal route.
Building service areas algorithm
Algorithm of building service areas can be used when you need to define availability of service areas from specified centers (points) based on set time or distance.
The algorithm results can be used in a variety of tasks, for example, to determine the areas of likely presence of an ideal client (who is solvent and interested in the presented goods) or to
determine city areas where emergency vehicles cannot reach within the prescribed standard time.
Building service areas (10, 15, 20 minutes)
Before starting the calculation of service areas, it is necessary to determine the number and threshold values of all the subareas (rings). As a result of the algorithm work, several nested polygons
(service areas) are obtained, each of which corresponds to certain threshold value.
Thus, in one run of the algorithm, several service areas with different threshold values can be obtained at once, which can be analyzed separately or jointly.
Types of comparison of service areas of different points
To calculate service areas, you can specify several centers (points). In this case, the algorithm will build separate service area for each center, which will then be compared in a way determined
before the algorithm launch. The following types of comparison of service areas of different points are available:
• Service areas of different points are independent of each other
• Combination of service areas of different points
• Competitive service areas
Independent service areas, combination of service areas, competitive service areas
When it is required not to mix the calculated service areas of different centers, the first type is used - without comparison. In this case, the user receives service areas for each center
separately, moreover, the service areas of different centers can intersect and have common areas. This type of comparison can be used, for example, to determine the areas of responsibility of various
municipal services.
In case when it is necessary to build a common service area for all centers at once, a combination of different services is used. This kind of comparison is helpful in tasks where the goal is to
determine, for example, what part of a city is available for a service in a specified time.
The type of comparison "competitive zones" is used when in addition to determining the common service area it is necessary to distribute responsibility between different centers according to the
selected criterion (time, distance). The result in this case will be similar to the "no comparison" option, except that the intersecting areas are divided between adjacent service areas in such a way
that each resulting part has the nearest center corresponding to each service area.
Integration possibilities
TrueDrive is the software product using which the web service can be run. Interaction with the TrueDrive services can be implemented via direct network connection over HTTP/HTTPS protocols,
via RESTful.
TrueDrive can be easily integrated to ecosystem of CoGIS infrastructure platform and is supported by all its clients. This enables you to extend functionality of geoinformation system based on CoGIS
via network analysis algorithms.
TrueDrive web service can be integrated to CoGIS platform as following:
• By using TrueDrive services directly from CoGIS clients.
• By using TrueDrive functions as part of geoprocessing models within geoprocessing tools on a server side of the information system.
The external TrueDrive API works over the protocol compatible with Network Analyst Server, which allows to integrate TrueDrive to the ArcGIS Server based information systems.
The TrueDrive development technologies enable high performance and reliability of implemented solutions, do not impose restrictions on use and are cross-platform. For example, the TrueDrive core is
written in C++, and the upper level TrueDrive logic in .NET Core (C#), ASP.NET Core. | {"url":"https://dataeast.com/en/journal/magazineblog/products-and-services/truedrive-a-sophisticated-spatial-analysis-technology/","timestamp":"2024-11-12T04:11:07Z","content_type":"text/html","content_length":"65403","record_id":"<urn:uuid:e0ac0e4d-d45c-4e40-adf4-17a0c174dce4>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00861.warc.gz"} |
System identification of force transducers for dynamic measurements using particle swarm optimization
A method of system identification for force transducers against the oscillation force is developed. In this method, force transducers are equipped with an additional top mass and excited by a
facility with the sine mechanism. Particle swarm optimization (PSO) algorithm is employed to identify the parameters of the derived mathematical models. For improving the convergence speed of PSO,
exponential transformation is introduced to the fitness function. Subsequently, numerical simulations and experiments are carried out, and consistent results demonstrate that the identification
method proposed in this investigation is feasible and efficient for estimating the transfer functions from sinusoidal force calibration measurements.
1. Introduction
As the development of science and technology, the demands of measuring dynamic forces in research and industrial applications, such as model analysis, process monitoring, material testing, dynamic
weighing and crash testing, are more and more strict. In all these fields, considerable measurement uncertainties can occur if only statically calibrated force transducers are used for the
measurements of dynamic forces [1]. And in order to reduce the uncertainties of dynamic measurements, some effort has been devoted to dynamic force calibrations. But procedures for dynamic
calibration of force transducers are still not well established. Until now, there have been three major attempts to develop dynamic calibration methods for force transducers, namely, methods for
calibrating transducers against an impact force, methods for calibrating transducers against a step force, and methods for calibrating transducers against an oscillation force [2]. The three methods
are categorized mainly according to the dynamic input behaviors. And as we know, from dynamic calibrations we not only can get the amplitude characteristics such as sensitivity, nonlinearity,
repeatability, etc., but also can obtain the time domain characteristics and frequency response characteristics. For the methods of calibrating transducers against an oscillation force, Fujii and
Kumme both have proposed their own methods which can be referred to [3, 4] respectively. Fujii proposed the Levitation Mass Method (LMM), and he evaluated the dynamic response of force transducer
against an oscillation force, which was generated with a spring as a connected mass was hit by a hammer manually in [3]. Apart from the LMM, Kumme developed a method with a shaker, in which the
inertial force of an attached mass was directly acting on the force transducer [5]. Kumme and his colleagues mainly focused on the dynamic sensitivity in the calibrations [4-7]. Dynamic sensitivity
is the ratio of the electrical output signal of the force transducer and the acting dynamic force. In addition to the sensitivity, Ch Schlegel et al. also determined the stiffness and damping of the
transducer [7]. And when determining a transient force from the transducer’s output signal, knowledge of the transfer function of a force transducer is required. Thus, Link et al. studied the dynamic
input-output behavior of an employed force transducer with Kumme’s method in [8]. They applied a linear least-squares fit method to estimate the transfer function. But almost all physical systems are
nonlinear to a certain extent and recursive in nature, and nonlinear system identification has attracted attention in the field of science and engineering [9].
Particle Swarm Optimization (PSO) algorithm is a population-based algorithm formed by a set of particles representing potential solutions for a given problem [10]. It has been successfully used in
many identification applications such as IIR system identification problem [11], power system state estimation [12], ship motion model identification [13], etc. Compared with other artificial
intelligence (AI) algorithms, i.e. neural networks (NN), genetic algorithm (GA), PSO can be easily programmed with basic mathematical and logic operation [12]. Unfortunately, PSO suffers from the
premature convergence problem, which is particularly true in complex problems since the interacted information among particles in PSO is too simple to encourage a global search [14]. Researchers also
have attempted various ways to analyze and improve conventional PSO algorithm [15-21]. In this investigation, PSO algorithm will be employed to finish the system identification of force transducers,
and to preserve the fast convergence rate, a small modification is incorporated in the conventional PSO algorithm.
The remainder of this paper is organized as follows. In Section 2, the system set-up of calibration facility is presented and its mathematical model is derived. Section 3 introduces PSO algorithm
into the system identification of force transducer, and algorithm procedure is summarized. In Section 4, PSO algorithm with exponential transformation is validated with numerical simulations. Some
experiments are carried out in Section 5 and the experimental findings are discussed. Finally, Section 6 concludes this paper.
2. Calibration facility for force transducers and its mathematical approximation
2.1. System set-up
The system set-up for dynamic calibrations of force transducers used in this investigation is shown in Fig. 1. This calibration facility is similar to the one developed by Kumme at
Physikalisch-Technische Bundesanstalt (PTB). But different from Kumme’s method, the facility uses a sine mechanism instead of a shaker to generate sinusoidal motion. The sine mechanism has been
successfully applied in the field of testing automobile shock absorber. When the crank disk is rotating, the frame and slider move orthogonally as illustrated in Fig. 1 [22]. With kinematic analysis,
we can get the following functions:
$s=r\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t+{\phi }_{0}\right),$
where $s$ and $a$ are the displacement and acceleration of the frame respectively, $r$ is the rotation radius, $\omega$ is the rotational angular velocity, ${\phi }_{0}$ is the initial angle and $t$
is time. The force transducer to be calibrated is mounted on the frame and moves together with it according to Eq. (1).
Fig. 1System set-up for dynamic calibrations of force transducers
The displacement $s$ can be collected with a displacement transducer, and the acceleration $a$ is assumed to be the same for the frame and the base mass inside force transducer. The dynamic force
could be obtained based on the determination of the inertia force as a known mass mounted on the force transducer [22].
2.1.1. Mathematical model
Force transducer can be described by the Voigt model, consisting of a spring mass system with stiffness ${k}_{f}$ and damping ${b}_{f}$ acting between the end mass ${m}_{e}$ and the base mass ${m}_
{b}$ of the transducer [4]. When considering the influence of the coupling between the force transducer and the load mass ${m}_{l}$, another spring mass system with stiffness ${k}_{c}$ and damping $
{b}_{c}$ could be employed. Therefore, in this investigation, the schematic diagram illustrated in Fig. 2 can be used to represent the force transducer which is mounted in the calibration set-up.
Fig. 2Schematic diagram of a force transducer
And the following differential equations could be established according to Fig. 2:
where ${x}_{l}$, ${x}_{e}$, ${x}_{b}$ are the vertical movements of the load mass, the end mass and the base mass respectively, ${F}_{s}$ is the acting force. Since the measured signals in this
investigation are collected from the force transducer to be calibrated and the displacement transducer connected to the frame, our goal is to obtain the relationship between ${x}_{e}-{x}_{b}$ and $
{x}_{b}$. ${x}_{e}-{x}_{b}$ is proportional to the output signal ${U}_{f}$ of the force transducer to be calibrated, and ${x}_{b}$ can be collected with the displacement transducer.
To simplify system Eq. (3), take ${\stackrel{¨}{X}}_{l}={\stackrel{¨}{x}}_{l}+g$, ${\stackrel{¨}{X}}_{e}={\stackrel{¨}{x}}_{e}+g$, ${\stackrel{¨}{X}}_{b}={\stackrel{¨}{x}}_{b}+g$, ${{Z}_{1}=x}_{l}-
{x}_{e}$ and ${{Z}_{2}=x}_{e}-{x}_{b}$. In addition, the problem can be further considered in the Laplace space by introducing the complex variable $s=j\omega$ to make the substitution $\stackrel{˙}
{x}=sx$, and the abbreviations ${K}_{f}={k}_{f}+s\bullet {b}_{f}$ and ${K}_{c}={k}_{c}+s\bullet {b}_{c}$ will be employed, too [7]. Thus system Eq. (3) can be expressed as following:
${\stackrel{¨}{Z}}_{1}=-\left(\frac{{K}_{c}}{{m}_{l}}+\frac{{K}_{c}}{{m}_{e}}\right)\bullet {Z}_{1}+\frac{{K}_{f}}{{m}_{e}}\bullet {Z}_{2},$
${\stackrel{¨}{Z}}_{2}=\frac{{K}_{c}}{{m}_{e}}\bullet {Z}_{1}-\left(\frac{{K}_{f}}{{m}_{e}}+\frac{{K}_{f}}{{m}_{b}}\right)\bullet {Z}_{2}-\frac{{F}_{s}}{{m}_{b}}.$
Take ${u}_{1}={m}_{l}{m}_{e}/\left({m}_{l}+{m}_{e}\right)\text{,}$${u}_{2}={m}_{e}{m}_{b}/\left({m}_{e}+{m}_{b}\right)\text{,}$$F=-{F}_{s}/{m}_{b}\text{,}$ and transfer the algebraic equation system
Eq. (4) into matrix formalism:
$\left[\begin{array}{cc}{s}^{2}+\frac{{K}_{c}}{{u}_{1}}& -\frac{{K}_{f}}{{m}_{e}}\\ -\frac{{K}_{c}}{{m}_{e}}& {s}^{2}+\frac{{K}_{f}}{{u}_{2}}\end{array}\right]\left[\begin{array}{c}{Z}_{1}\\ {Z}_{2}\
end{array}\right]=\left[\begin{array}{c}0\\ F\end{array}\right].$
The solution of this system can be performed using Cramer’s rule [7]:
$∆=\left({s}^{2}+\frac{{K}_{c}}{{u}_{1}}\right)\left({s}^{2}+\frac{{K}_{f}}{{u}_{2}}\right)-\frac{{K}_{f}{K}_{c}}{{m}_{e}^{2}},{∆}_{1}=\frac{{K}_{f}}{{m}_{e}}\bullet F,{∆}_{2}=\left({s}^{2}+\frac{{K}
_{c}}{{u}_{1}}\right)\bullet F.$
To obtain the relationship between ${x}_{e}-{x}_{b}$ and ${x}_{b}$, take ${Z}_{2}/{\stackrel{¨}{X}}_{b}$ as the target model:
When considering the influence of the coupling between the force transducer and the load mass, by substituting expressions of ${K}_{f}$, $F$, $∆$ and ${∆}_{2}$ into Eq. (7) we can obtain:
In the case where the coupling of the load mass to the force transducer is very stiff, the transfer function Eq. (7) can be simplified using the limits ${k}_{c}\to \infty$ and ${b}_{c}\to 0$ [7]:
$\frac{{Z}_{2}\left(s\right)}{{s}^{2}{X}_{b}\left(s\right)}\approx -\frac{1}{{s}^{2}+{d}_{1}^{"}s+{d}_{2}^{\text{'}}},$
Since the system identification is finished in computer with application of advanced tools from Digital Signal Processing (DSP), we carry out $Z$ transform with bilinear method for $-\rho \bullet {Z}
_{2}\left(s\right)/{\mathrm{s}}^{2}{X}_{b}\left(s\right)$, where the constant $\rho$ realizes the transformation of the displacement ${Z}_{2}$ into a force signal. And corresponding discrete transfer
functions ${H}_{1}\left(z\right)$ and ${H}_{2}\left(z\right)$ for Eqs. (8) and (9) can be obtained as following:
${a}_{4}={d}_{4}{T}^{4}-2{d}_{3}{T}^{3}+4{d}_{2}{T}^{2}-8{d}_{1}T+16,{b}_{0}={n}_{2}\rho {T}^{4}+2{n}_{1}\rho {T}^{3}+4\rho {T}^{2},$
${b}_{1}={4n}_{2}\rho {T}^{4}+4{n}_{1}\rho {T}^{3},{b}_{2}={6n}_{2}\rho {T}^{4}-8\rho {T}^{2},$
${b}_{3}={4n}_{2}\rho {T}^{4}-4{n}_{1}\rho {T}^{3},{b}_{4}={n}_{2}\rho {T}^{4}-2{n}_{1}\rho {T}^{3}+4\rho {T}^{2},$
and $T$ is the sampling time.
${b}_{1}^{"}=2{\rho T}^{2},{b}_{0}^{\text{'}}={b}_{2}^{\text{'}}={b}_{1}^{\text{'}}/2$
3. PSO based system identification
3.1. Schematic for system identification
The main task of system identification is to search iteratively for the parameters of the modeled system such that the input-output relationship matches closely to that of the actual system [11]. The
basic block diagram for system identification is shown in Fig. 3. System input $x\left(k\right)$ is given to both the unknown system to be identified and the modeled system. The output ${y}^{"}\left
(k\right)$ mixed with a noise signal gives the final output $y\left(k\right)$ to the actual system$\mathrm{}$. On the other hand, the modeled system has an output of $\stackrel{^}{y}\left(k\right)$
for the same input. The difference $e\left(k\right)$ between the two output signals is used by the identifier to adjust the parameters [22]. Since traditional algorithms usually have difficulties in
optimizing complex nonlinear systems, the identification of transfer Eqs. (10) and (11) will be finished with particle swarm optimization algorithm in this work.
Fig. 3Block diagram for PSO based system identification
3.2. PSO algorithm
Particle swarm optimization algorithm is initialized with a population of possible solutions called particles. Particles start flying from the initial positions through the search space with
velocities exploring almost all optimal solutions. The velocity of each particle is dynamically adjusted according to its own flying experience and its companions’ flying experience, and the
performance of each particle position is evaluated by a fitness function. During the flights, the best previous experience for each particle is stored in its memory and called the personal best
(Pbest), and the best previous position among all particles is called the global best (Gbest) [10, 12, 22, 23].
Let ${X}_{i}=\left({x}_{i1},{x}_{i2},\cdots ,{x}_{iN}\right)$ be the position of the $i$th particle, for Eq. (10) $N=10$, and for Eq. (11) $N=4$. Similarly, the velocity ${V}_{i}$, the personal best
${X}_{i}^{Pbest}$ and the global best ${X}^{Gbest}$ are represented as ${V}_{i}=\left({v}_{i1},{v}_{i2},\cdots ,{v}_{iN}\right)\text{,}$${X}_{i}^{Pbest}=\left({x}_{i1}^{Pbest},{x}_{i2}^{Pbest},\cdots
,{x}_{iN}^{Pbest}\right)\text{,}$ and ${X}^{Gbest}=\left({x}_{1}^{Gbest},{x}_{2}^{Gbest},\cdots ,{x}_{N}^{Gbest}\right)$ respectively. The particles are manipulated according to the following
equations [10, 24]:
where acceleration coefficients ${c}_{1}\text{,}$${c}_{2}$ are positive constant parameters with the constraint ${c}_{1}+{c}_{2}\le 4$ that control the maximum step size. Typically, ${c}_{1}$, ${c}_
{2}$ are both suggested to be 2.0. In [22] the authors also have studied the influence of acceleration coefficients ${c}_{1}$, ${c}_{2}$ on PSO performance, and the results indicate that parameters $
{c}_{1}=\text{2.0}$ and ${c}_{2}=\text{2.0}$ can get the tradeoff between identification accuracy and convergence speed. Therefore, ${c}_{1}=\text{2.0}$ and ${c}_{2}=\text{2.0}$ will be taken in this
investigation. ${r}_{1}$, ${r}_{2}$ are uniformly distributed random variables in the range [0, 1]. $w$ is the inertia weight and controls the impact of the previous velocity of the particle on its
current one. In general, a bigger $w$ can guarantee the globally search ability, and avoid sinking into a local optimal solution. On the contrary, a smaller $w$ is good for local search, and can
guarantee the convergence of the arithmetic. Thus, a self-adaptive adjusted strategy for inertia weight $w$ as follows can be taken [13]:
where $j$ is the current iteration generation, ${iter}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum iteration times, ${w}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${w}_{\mathrm{m}\mathrm{a}\mathrm
{x}}$ are the minimum inertia weight and maximum inertia weight, respectively. The value of inertia weight $w$ in PSO is linearly decreased according to Eq. (14), and ${w}_{\mathrm{m}\mathrm{a}\
mathrm{x}}$, ${w}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ are initial and final values for the inertia weight [23]. In this way, PSO tends to have more global search ability at the beginning of the run
while having more local search ability near the end of the run [22]. ${w}_{\mathrm{m}\mathrm{a}\mathrm{x}}=\text{0.9}$ and ${w}_{\mathrm{m}\mathrm{i}\mathrm{n}}=\text{0.4}$ are generally taken in
previous studies [14, 22].
In the system identification problem, PSO algorithm tries to minimize the error $e\left(k\right)$ by adjusting the parameters of the modeled system. And mean square error (MSE) of time samples
between the outputs of the actual system and the designed system as given by Eq. (15) is always considered as the fitness function [9, 11]. In mathematical statistics, mean square error is the
mathematical expectation of differences between the estimated values and the truth values, and it is a convenient method to measure the average error:
$\mathrm{M}\mathrm{S}\mathrm{E}=F=\frac{1}{M}\sum _{k=1}^{M}{\left(\stackrel{^}{y}\left(k\right)-y\left(k\right)\right)}^{2},$
where $M$ is the number of data points, $y\left(k\right)$ is the output of the actual system, $\stackrel{^}{y}\left(k\right)=\frac{\stackrel{^}{B}\left({z}^{-1}\right)}{\stackrel{^}{A}\left({z}^{-1}\
right)}x\left(k\right)$ is the output of the modeled system with $\stackrel{^}{A}\left({z}^{-1}\right)={\stackrel{^}{a}}_{0}+{\stackrel{^}{a}}_{1}{z}^{-1}+\cdots +{\stackrel{^}{a}}_{n}{z}^{-n}\text
{,}$$\stackrel{^}{B}\left({z}^{-1}\right)={\stackrel{^}{b}}_{0}+{\stackrel{^}{b}}_{1}{z}^{-1}+\cdots +{\stackrel{^}{b}}_{m}{z}^{-m}$. The smaller the MSE value is, the higher accuracy of the modeled
system is.
In general, the main problems of the PSO algorithm are slow convergence and premature convergence [25]. To deal with the problems, exponential transformation is introduced to the fitness function for
helping quickly search the global optimal solution:
And with Eq. (16), the fitness function can be converted to a positive indicator in the range [0, 1].
3.3. Algorithm procedure
The implementation of PSO algorithm with exponential transformation (PSO+ algorithm) applied to system identification in this work is summarized as follows:
Step 1. Generate $M$ input-output data points to form the system to be identified.
Step 2. Set acceleration coefficients ${c}_{1}$, ${c}_{2}$, the maximum iteration times ${iter}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, and the scope of the inertial factors ${w}_{\mathrm{m}\mathrm{i}\
mathrm{n}}$, ${w}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ that appear in Eqs. (12) and (14).
Step 3. Initialize the position and velocity for each particle. In this investigation, particle position corresponds to the parameters of transfer functions (10) and (11).
Step 4. Remember current particle position as Pbest of each particle. Find the particle with the best value of fitness function and remember its position as Gbest.
Step 5. Generate inertia weight $w$ and random numbers ${r}_{1}$, ${r}_{2}$[]in Eq. (12), and update the velocities and positions according to Eqs. (12) and (13).
Step 6. Evaluate fitness values and update Pbest and Gbest.
Step 7. Return to step 5 until the total number of iterations is achieved.
Step 8. Output the identified parameters namely Gbest and end the program.
4. Numerical simulations
4.1. Validation of PSO+ algorithm
For testing the performance of PSO+ algorithm, two benchmark functions listed in Table 1 which are similar to transfer functions Eqs. (10) and (11) are considered for the case studies. The inputs for
the two studies are both $5\mathrm{s}\mathrm{i}\mathrm{n}\left(2\pi ×4t\right)$. And the two cases are simulated with PSO and PSO+ algorithms in MATLAB with the following parameters: ${c}_{1}={c}_{2}
=\text{2}$, ${w}_{\mathrm{m}\mathrm{i}\mathrm{n}}=\text{0.4}$, ${w}_{\mathrm{m}\mathrm{a}\mathrm{x}}=\text{0.9}$, population size, maximum position and maximum velocity are 150, 2 and 0.1
respectively. For case I, $N=\text{4}$ and the maximum generation is set to 1000. For case II, $N=\text{10}$ and the maximum generation is 2000. The transfer functions and their identification
results are shown in Table 1. The outputs of actual systems and modeled systems for cases I and II are illustrated in Figs. 4 and 7 respectively. Figs. 5 and 6 are the curves of fitness functions for
PSO and PSO+ in case I while Figs. 8 and 9 are those in case II.
From Figs. 4-9 and Table 1 we can see that, identification results of the two cases are in accordance with each other. Identification accuracies of PSO in the two cases are slightly higher than those
of PSO+, but identification errors of PSO+ are very small too, since the MSE values listed in Table 1 are in the 10^-17 order or less. From this point we know that, PSO+ algorithm is effective for
system identification.
When comparing the convergence processes shown in Figs. 5-6 and Figs. 8-9, we can find that the convergence speed of PSO+ algorithm is obviously faster than that of PSO. For case I, PSO takes almost
880 iterations to converge to the global best while PSO+ takes less than 100 iterations. And for case II, PSO takes 1416 iterations to converge to the global best while PSO+ takes only 850
iterations. Consequently, we reach a conclusion that PSO+ algorithm can be used for system identification and it converges faster than conventional PSO.
Fig. 5Fitness value of global best particle for PSO in case I
Fig. 6Fitness value of global best particle for PSO+ in case I
Fig. 7Outputs for case II
Fig. 8Fitness value of global best particle for PSO in case II
Fig. 9Fitness value of global best particle for PSO+ in case II
Table 1Identification results of benchmark systems
Case I Case II
Benchmark function $\frac{0.25+0.5{z}^{-1}+0.25{z}^{-2}}{1-0.3{z}^{-1}+0.4{z}^{-2}}$ $\frac{{z}^{-1}-0.9{z}^{-2}+0.8{1z}^{-3}-0.729{z}^{-4}}{1-0.04{z}^{-1}-0.2775{z}^{-2}+0.2101{z}^{-3}-0.14
PSO $\frac{0.2907+0.5814{z}^{-1}+0.2907{z}^{-2}}{1.1628-0.3488{z}^{-1}+0.4651 $\frac{-2{z}^{-1}+1.8{z}^{-2}-1.62{z}^{-3}+1.458{z}^{-4}}{-2+0.08{z}^{-1}+0.555{z}^{-2}-0.4202{z}^{-3}
Modeled {z}^{-2}}$ +0.28{z}^{-4}}$
PSO+ $\frac{0.3137+0.6274{z}^{-1}+0.3137{z}^{-2}}{1.2547-0.3764{z}^{-1}+0.5019 $\frac{-2{z}^{-1}+1.8{z}^{-2}-1.62{z}^{-3}+1.458{z}^{-4}}{-2+0.08{z}^{-1}+0.555{z}^{-2}-0.4202{z}^{-3}
{z}^{-2}}$ +0.28{z}^{-4}}$
PSO 1.0402e-31 2.5862e-31
PSO+ 9.0450e-18 5.4054e-17
4.2. Performance study of PSO+ in the presence of the additive noise
For studying the ability of PSO+ algorithm to operate in the presence of the additive noise, the benchmark function of case I listed in Table 1 is employed to finish another simulation study. The
input for this study is still $5\mathrm{s}\mathrm{i}\mathrm{n}\left(2\pi ×4t\right)$, and white noises of different signal to noise ratios (SNRs) are added to the output. The parameters of PSO+
algorithm is set as those in case I.
Fig. 10Output in the case of 50 dB SNR
Fig. 11Output in the case of 40 dB SNR
Fig. 12Output in the case of 30 dB SNR
Fig. 13Output in the case of 20 dB SNR
The outputs of actual systems and modeled systems are illustrated in Figs. 10-14 respectively. The MSE values are shown in Fig. 15.
From Figs. 10-14 we can see that, the higher the SNR is, the smoother the output curve is. And in this situation, the identification accuracy of PSO+ algorithm is higher, too. The relationship
between identification accuracy and SNR is shown in Fig. 15. The MSE values in the cases of 10 dB-50 dB SNRs are 80.1880, 7.9021, 0.9229, 0.0921 and 0.0092 respectively. In all these situations, the
outputs of the modeled systems can follow the changes of the actual outputs. But in order to obtain the higher identification accuracy, filtering process for the actual outputs is suggested to be
finished before identifying with PSO+ algorithm.
Fig. 14Output in the case of 10 dB SNR
Fig. 15MSEs in the cases of different SNRs
5. Experiments and analysis
As described in Section 2, the experimental device shown in Fig. 16 is built. The transducer is calibrated with a top mass of 25.717 kg, which is connected to the transducer with a special mechanical
adaption. The Minor KTC type displacement transducer is employed to measure the displacement of the frame. The signal from the force transducer is amplified with a SGA powered signal conditioner. An
industrial computer with a PCI-1716L data acquisition card is used to record the two transducer signals through a PCLD-8710 terminal board [22].
Fig. 16The experimental device
5.1. Experiment I
Firstly, the experiment is carried out with a force transducer of 5 kN. The sine mechanism is operating at a frequency of $4\text{Hz}$ and the transducer signals are sampled at 20 kHz. After
filtering and resampling, PSO+ algorithm is employed to identify the simplified function Eq. (11) (Case III) and function Eq. (10) (Case IV). 200 input-output data points are taken for system
identification and another 400 data points are taken for testing. The parameters of PSO+ algorithm are set as follows: ${c}_{1}={c}_{2}=\text{2}$, ${w}_{\mathrm{m}\mathrm{i}\mathrm{n}}=\text{0.4}$, $
{w}_{\mathrm{m}\mathrm{a}\mathrm{x}}=\text{0.9}$, population size and maximum position are 200 and 100 respectively. When identifying the simplified function Eq. (11), Recursive Least Squares (RLS)
method is also used to do the identification besides the PSO+ algorithm. Alternative form of function Eq. (11) for RLS identification is shown below:
$y\left(k\right)={\mathrm{\Phi }}^{T}\mathrm{\Theta },$
$\mathrm{\Phi }={\left[\begin{array}{ccc}-y\left(k-1\right)& -y\left(k-2\right)& x\left(k\right)+2x\left(k-1\right)+x\left(k-2\right)\end{array}\right]}^{T},\mathrm{\Theta }={\left[\begin{array}{ccc}
\frac{{a}_{1}^{"}}{{a}_{0}^{\text{'}}}& \frac{{a}_{2}^{\text{'}}}{{a}_{0}^{\text{'}}}& \frac{{b}_{0}^{\text{'}}}{{a}_{0}^{\text{'}}}\end{array}\right]}^{T}.$
Fig. 17 shows the identification errors of RLS and PSO+ for cases III and IV, and the convergence processes of fitness values for PSO+ algorithm are illustrated in Fig. 18. Testing results of the
identified models for cases III and IV are shown in Fig. 19, and Fig. 20 shows the testing errors of the identified models with PSO+ algorithm. And for the empirical analysis of the testing results,
MSE is employed as the accuracy evaluation standard which is shown in Table 2.
Fig. 17Identification errors for cases III and IV
Fig. 18Fitness values of PSO+ for cases III and IV
Fig. 19Testing results for cases III and IV
Fig. 20Testing errors of PSO+ for cases III and IV
Table 2Testing results of experiment I
RLS for case III PSO+ for case III PSO+ for case IV
Testing MSE 19.7667 0.0191 0.0186
The results shown in Figs. 17 and 19 indicate that, RLS is not suitable for the identification of function Eq. (11) since big errors exist in the processes of identifying and testing. The testing MSE
values shown in Table 2 also tell this, as MSE of RLS is more than 1000 times bigger than MSE of PSO+. So, when compared with RLS, the identification performance of PSO+ for case III is encouraging.
When comparing the results of PSO+ for cases III and IV shown in Figs. 19-20 and Table 2, we can find that simplification between functions Eqs. (10) and (11) almost has no influence on
identification accuracy. The testing MSE values of PSO+ for case III and IV are 0.0191 and 0.0186 respectively, identification accuracy in case IV is slightly better than that in case III.
Nevertheless, in Fig. 18, we can see that the convergence speed of PSO+ for case III is faster than that for case IV. Therefore, by comprehensively considering the identification accuracy and
convergence speed, we reach a conclusion that the simplification between functions Eq. (10) and (11) is feasible. And with the identified parameters obtained in case III we can calculate out the
stiffness and damping of the force transducer conveniently. In this experiment, the stiffness and damping of the 5 kN force transducer are 0.291×10^8 Nm^-1 and 17136 Nsm^-1 respectively.
5.2. Experiment II
For investigating the generality of this identification method, another force transducer of 12.5 kN is taken to carry out a same experiment. The parameters of PSO+ algorithm is set as those in
experiment I. The identifications of 12.5 kN transducer for function Eq. (11) and function Eq. (10) are denoted as case V and case VI respectively. The identification results are shown in Figs. 21-24
and Table 3.
Fig. 21Identification errors for cases V and VI
Fig. 22Fitness values of PSO+ for cases V and VI
Fig. 23Testing results for cases V and VI
Fig. 24Testing errors of PSO+ for cases V and VI
The results of experiment II are basically in accordance with those of experiment I. Identification error of RLS in Fig. 21 is still very big, and this leads to that the testing performance of RLS
illustrated in Fig. 23 is not acceptable. Its MSE value is up to 0.9304 while MSE of PSO+ is only 0.0020. Then compare the identification performances of PSO+ for cases V and VI. As shown in Fig. 22,
the convergence speed of PSO+ for case V is also faster than that for case VI obviously. The testing accuracies of PSO+ algorithm for cases V and VI are almost the same since MSE values of cases V
and VI are 0.0020 and 0.0017 respectively. And with the identified parameters obtained in case V, the stiffness and damping of 12.5 kN force transducer are calculated as 0.246×10^8 Nm^-1 and 3032 Nsm
^-1 respectively.
Table 3Testing results of experiment II
RLS for case V PSO+ for case V PSO+ for case VI
Testing MSE 0.9304 0.0020 0.0017
6. Conclusions
This paper represents a system set-up for dynamic calibrations of force transducers, and its mathematical model is derived in detail. For identifying the parameters of force transducers, PSO
algorithm is introduced and a small modification is made to improve the convergence speed. Numerical simulations and experiments are carried out, and some conclusions are reached.
1) The sine mechanism described in this investigation is feasible for the dynamic calibration of force transducer.
2) The two experiments both indicate when PSO+ algorithm is employed to identify the simplified Eq. (11) and un-simplified Eq. (10), the identification accuracies are almost the same. Therefore,
simplification between Eqs. (10) and (11) almost has no influence on system identification, and the mathematical approximation is reasonable.
3) PSO+ algorithm is effective for system identification, and with the aid of exponential transformation, the convergence speed of PSO+ is improved obviously.
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About this article
Mechanical vibrations and applications
force transducer
system identification
dynamic calibration
sinusoidal force
This project is supported by National Natural Science Foundation of China (Grant No. 51405437), the Key Laboratory of Intelligent Perception and Systems for High-Dimensional Information (Nanjing
University of Science and Technology), Ministry of Education (No. JYB201609).
Copyright © 2017 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
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What is the radius of convergence of the MacLaurin series expansion for f(x)= sinh x? | HIX Tutor
What is the radius of convergence of the MacLaurin series expansion for #f(x)= sinh x#?
Answer 1
Let's first find the Maclaurin series expansion for #sinhx#:
#f(x)=sinhx=(e^x-e^-x)/2, f(0)=(e^0-e^0)/2=0#
#f'(x)=coshx=(e^x+e^-x)/2, f'(0)=(e^0+e^0)/2=1#
#f''(x)=sinhx, f''(0)=0#
#f'''(x)=coshx, f'''(0)=1#
#f^((4))(x)=sinhx, f^((4))(0)=0#
#f^((5))(x)=coshx, f^((5))(0)=1#
Thus, we observe a fairly regular pattern of ones and zeros alternately. Let's put the series' initial terms in writing:
The expansion of the Maclaurin Series is provided by
Thus, for our purpose, we obtain
When we take out the terms that involve zero, we observe
So, we want odd exponents and odd factorials starting at #1#, so the summation is
We'll use the Ratio Test to determine the radius of convergence, where
Remove certain terms from the larger factorial because we want the factorials to cancel each other out:
#(2n+3)! = (2n+3)(2n+1)(2n+1)!#
#|x^2|lim_(n->oo)1/((2n+3)(2n+2))<1# results in convergence.
The limit goes to #0.# Thus, this quantity is always #0<1# regardless of what we pick for #x#. We have convergence for all real numbers, IE, #R=oo#
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Thermodynamics of Computation
Karpur Shukla
From Thermodynamics of Computation
Biography: I'm currently a PhD student at the School of Engineering at Brown University. My interests lie at the intersection of geometric phenomena in quantum systems, conformal field theory,
quantum thermodynamics, and condensed matter theory. In particular, I'm deeply interested in geometric properties of Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) dynamics and their applications to
condensed matter systems, quantum information processing, and classical information processing. I'm also interested in the properties of conformal field theories (CFTs) out of equilibrium, and the
ways by which nonequilibrium CFT phenomena manifest in condensed matter models. Finally, I'm interested in the consequences that renormalisation group transformations have for resource theories.
At present, my work focuses on applications of Gorini-Kossakowski-Lindblad-Sudarshan (GKSL) dynamics with multiple steady states, resource theories, and shortcuts-to-adiabaticity to physical models
for reversible computing and conformally invariant systems. Reversible computing is a paradigm of computing that relies on preserving and unwinding correlations, which allows us to avoid the energy
cost resulting from irretrievably ejecting information stored in memory devices into the environment. Although systems implementing reversible logic were first proposed as early as 1978 by Fredkin
and Toffoli; designing a model of fast, fully adiabatic, and scalable classical reversible operations remains an ongoing and active area of interest. Here, the language of GKSL dynamics,
shortcuts-to-adiabaticity, resource theories, and quantum speed limits are especially suited to helping us design our desired models for reversible computing. I'm currently working alongside several
others to develop these models.
My other work focuses on the consequences that conformal invariance can have for resource theories, as well as the lessons resource theories can have for conformally invariant systems. Recent results
by Bernamonti et al. for holographic second laws, Guarnieri et al. for relationships between stochastic quantum work techniques and resource theories, and Faist and Renner on new information measures
for the work cost of quantum processes, and Albert et al. on the geometric properties of Lindbladians themselves have substantial implications for systems described by CFTs. My interest here is in
examining what lessons these results have for CFTs: in particular, understanding whether stochastic quantum work techniques can be expressed for CFTs via the holographic second laws, where an
extension to the holographic second laws can be developed using this new information measure, and what lessons we may derive for CFTs out of equilibrium with degenerate steady states.
Before my current appointment, I was a research fellow and visiting faculty member at the Department of Applied Mathematics at Flame University. I received my M.Sci. in physics from Carnegie Mellon
University in 2016, and my B.Sci. in physics from Carnegie Mellon University in 2014. There, I worked under Di Xiao on optoelectronic phenomena on the surfaces of topological insulators, in
particular examining properties of the photogalvanic effect on the surfaces of topological insulators at zero and finite temperature. I also had the brief opportunity to work on curve fitting for
experimental soft condensed matter physics under Stephanie Tristram-Nagle, as well as on analytic analysis of the dynamical RG flow of the Ising model under Robert Swendsen.
Field(s) of Research: General Non-equilibrium Statistical Physics, Stochastic Thermodynamics, Logically Reversible Computing, Quantum Thermodynamics and Information Processing
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Amortization Calculator (2024)
The amortization calculator or loan amortization calculator is a handy tool that not only helps you to compute the payment of any amortized loan, but also gives you a detailed picture of the loan in
question through its amortization schedule. The main strength of this calculator is its high functionality, that is, you can choose between different compounding frequencies (including continuous
compounding), and payment frequencies You can even set an extra payment.
You can also study the loan amortization schedule on a monthly and yearly bases, and follow the progression of the balances of the loan in a dynamic amortization chart. If you read on, you can learn
what the amortization definition is, as well as the amortization formula, with relevant details on this topic. For these reasons, if you would like to get familiar with the mechanism of loan
amortization or would like to analyze a loan offer in different scenarios, this tool will be of excellent help.
If you are more interested in other types of repayment schedule, you may check out our loan repayment calculator, where you can choose balloon payment or an even principal repayment options as well.
In case you would like to compare different loans, you may make good use of the APR calculator as well.
What is amortized loan? - the amortization definition
The repayment of most loans is realized by a series of even payments made on a regular basis. The popular term in finance to describe loans with such a repayment schedule is an amortized loan.
Accordingly, we may phrase the amortization definition as "a loan paid off by equal periodic installments over a specified term". Typically, the details of the repayment schedule are summarized in
the amortization schedule, which shows how the payment is divided between the interest (computed on the outstanding balance) and the principal. The amortization chart might also represent the unpaid
balance at the end of each period. A few examples of loan amortization are automobile loans, home mortgage loans, student loans, and many business loans.
As in general the core concept that governs financial instruments is the time value of money, the loan amortization is similarly strongly connected to the present value and future value of money.
More specifically, there is a concept called the present value of annuity that conforms the most to the loan amortization framework.
💡 You can learn more about these concepts from our time value of money calculator.
To see why, let's consider the following simple example. Suppose you borrow $1,000, which you need to repay in five equal parts due at the end of every year (the amortization term is five years with
a yearly payment frequency). The lender charges you 12 percent interest, that is calculated on the outstanding balance at the beginning of each year (therefore, the compounding frequency is yearly).
The illustration below represents the timeline of this example, where PMT is the yearly payment or installment. To find PMT, we need to find a value such that the sum of their present values equals
the loan amount: $1,000.
The solution of this equation involves complex mathematics (you may check out the IRR calculator for more on its background); so, it's easier to rely on our amortization calculator. After setting the
parameters according to the above example, we get the result for the periodic payment, which is $277.41.
Loan amortization schedule - the amortization table
The specific feature of amortized loans is that each payment is the combination of two parts: the repayment of principal and the interest on the remaining principle . The amortization chart below,
which appears in the calculator as well, represents the payment schedule of the previous example. As you can see, the interest payments are typically high in early periods and decrease over time,
while the reverse is true for the principal payments. The lowering interest amount is matched by the increasing amount of principal so that the total loan payment remains the same over the loan term.
The large unpaid principal balance at the beginning of the loan term means that most of the total payment is interest, with a smaller portion of the principal being paid. Since the principal amount
being paid off is comparably low at the beginning of the loan term, the unpaid balance of the loan decreases slowly. As the loan payoff proceeds, the unpaid balance declines, which gradually reduces
the interest obligations, making more room for a higher principal repayment. Logically, the higher the weight of the principal part in the periodic payment, the higher the rate of decline in the
unpaid balance.
It may be easier to understand this concept if it is displayed as a graph of the relevant balances, which is why this option is also displayed in the calculator.
An amortized loan is a form of credit where the loan is paid off with equal, consecutive payments over a specified period. An amortization schedule shows the structure of these consecutive payments:
the interest paid, the principal repaid, and the unpaid balance at the end of each period, which must reach zero during the amortization term.
What is the amortization formula?
As you have now gained some insight into the logic behind the amortized loan structure, in this section you can learn two basic formulas employed in our amortization calculator:
• Monthly repayment formula
$P = \frac{A \times i}{1-(1 + i)^{-t}}$P=1−(1+i)−tA×i
$B = A \times (1 + i)^t - \frac{P}{i} \times ((1 + i)^t - 1)$B=A×(1+i)t−iP×((1+i)t−1)
• $P$P - monthly payment amount
• $A$A - loan repayment amount
• $i$i - periodic interest rate
• $t$t - number of periods
• $B$B - unpaid balance
For more details and formulas, you may check BrownMath.com, where you can also check the precise derivation of the related equations.
Amortization calculator with extra payments
It is worth knowing that the amortization term doesn't necessarily equal to the original loan term; that is, you may pay off the principal faster than the time estimated with the periodic payments
based on the initial amortization term. An obvious way to shorten the amortization term is to decrease the unpaid principal balance faster than set out in the original repayment plan. You may do so
by a lump sum advance payment, or by increasing the periodic installments.
In this calculator, you can set an extra payment, which raises the regular payment amount. The power of such an extra payment is that its amount is directly allocated to the repayment of the loan
amount. In this way, the principal balance decreases in an accelerating fashion, resulting in a shorter amortization term and a considerably lower total interest burden.
The beneficial effect of extra payments is especially profound when the initial loan term is relatively long, such as most mortgage loans. When you set the extra payment in this calculator, you can
follow and compare the progress of new balances with the original plan on the dynamic chart, and the amortization schedule with extra payment.
Since the shorter repayment period with advance payments mean lower interest earnings to the banks, lenders often try to avert such action with additional fees or penalties. For this reason, it is
always advisable to negotiate with the lender when altering the contractual payment amount.
The results of this calculator, due to rounding, should be considered as just a close approximation financially. For this reason, and also because of possible shortcomings, the calculator is created
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TripZero - Calculating Event Carbon Footprints
Want more?
Get the details on flight footprints.
One can apply a remarkable number of variables in an effort to come up with a "perfect" carbon footprint calculation. For example, take flights. Here are just some of the variables deployed in the
TripZero flight footprint calculation and a brief explanation of why they matter:
Trips. We assume round-trip travel for every trip. For some travelers, this means we’re calculating (and offsetting) more carbon footprint than necessary.
Distance. Our calculation uses the nautical miles between destinations and adds in an appropriate factor to account for route inefficiencies. We do this because planes don’t typically fly directly
between point A and point B; they need to spread out to avoid other flights and avoid restricted airspace (like military installations).
Segments. Planes use a lot of fuel during taxi, takeoff, climbing and landing. Even one stop dramatically increases your carbon footprint. For short and medium flights, we assume direct travel. For
long and international flights our calculation adds an appropriate factor (based on industry averages) to account for connecting flights.
Radiative Forcing. With advance apology to our friends at MIT, we’ll oversimplify this important factor. Scientists agree that when you "deliver" pollution to higher altitudes (as airplanes do) you
magnify its impact on climate change. Relying on a study from the Stockholm Institute of this topic, our calculation applies a factor of 2.0—effectively doubling the standard carbon footprint
calculation for a flight.
Seating Density. Full flights, on the same plane and route, have a lower footprint per passenger than flights that are only half full. Our calculation uses industry averages to compute the effect of
load factor on your flight. Relying on averages means that may understate the impact of one trip and overstate the impact of another. However across all of our customers, we’re confident that
TripZero offsets the right amount of carbon footprint.
These are just a few of the variables we use to compute the carbon footprint for flights. As you might expect, we uses similarly complex models to calculate the footprint for each form of
transportation and hotels. | {"url":"https://www.tripzero.events/carbon-footprints","timestamp":"2024-11-06T13:28:06Z","content_type":"text/html","content_length":"14248","record_id":"<urn:uuid:12cd432d-2f5c-49bc-a5e9-55b611d5fc43>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00468.warc.gz"} |
Photovoltaic Integrated Shunt Active Power Filter with Simpler ADALINE Algorithm for Current Harmonic Extraction
School of Electrical and Electronic Engineering, Engineering Campus, Universiti Sains Malaysia, 14300 Nibong Tebal, Pulau Pinang, Malaysia
Department of Electrical and Electronic Engineering, Faculty of Engineering, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia
UM Power Energy Dedicated Advanced Centre, Wisma R&D, Universiti Malaya, 59990 Kuala Lumpur, Malaysia
Department of Vehicle Engineering, National Taipei University of Technology, No. 1, Sec. 3, Chung-Hsiao E. Road, Taipei 106, Taiwan
Author to whom correspondence should be addressed.
Submission received: 2 April 2018 / Revised: 29 April 2018 / Accepted: 1 May 2018 / Published: 4 May 2018
This manuscript presents a significant work in improving the current harmonics extraction algorithm and indirectly improving the injection current produced by a single-phase Photovoltaic Shunt Active
Power Filter (PV SAPF). Improvement to the existing adaptive linear neuron (ADALINE) technique has been carried out, leading to the formation of a simpler ADALINE; it is expected to perform as fast
as the current harmonics extraction algorithm. Further analysis on the DC link capacitor control algorithm, called “self-charging with step size error cancellation”, was also done to inspect the
performance of the algorithm in a single-phase photovoltaic shunt active power filter system. Both algorithms, configured in single-phase PV SAPF, were simulated in MATLAB/Simulink (R2012b). A
laboratory prototype was developed, and the algorithms were computed on a TMS320F28335 Digital Signal Processing (DSP) board for hardware implementation purposes. From the acquired results, the
simpler ADALINE algorithm has effectively performed with lower total harmonic Distortion (THD) and outstanding compensation. The established algorithm of self-charging with step size error
cancellation works well with single-phase PV SAPF and has shown less overshoot, a fast response time, and minimal energy losses.
1. Introduction
The power quality in a power system is the extensive range of electromagnetic phenomena that describe current and voltage at given locations and times in the system [
]. A supply system has its own frequency that it works on. The integer multiples of this frequency are frequencies where harmonic (sinusoidal voltages or currents) power quality issues arise [
]. Typically, the nonlinear load operations of power electronic devices are the cause of current harmonics. Another cause is when the supply network is injected with applications that are added in
latter stages. The participation of multiple energy source systems with the inclusion of a photovoltaic (PV) grid-connected system in the power system gives rise to these problems [
]. Excessive neutral currents, equipment overheating, motor vibration, and capacitor blowing are among the consequences of current harmonics [
]. Current harmonics can be compensated for by using the shunt active power filter (SAPF), which is a very capable tool compared with a passive filter. This is mainly due to its ability to handle
multiple harmonics simultaneously [
Since photovoltaic (PV) energy is free to harvest, inexhaustible, and much cleaner, it is one of the most renowned renewable energy sources [
]. The integration of renewable energy sources such as PV with SAPF is a possible avenue of exploration in current research works. This integration provides significant advantages by having an
alternative energy source rather than depending on the energy source from the supply grid [
]. However, as PV entirely depends on the availability of irradiance from the sun, SAPF may operate with possible dynamic changes in injection current. Therefore, the injection current must be
controlled suitably with a specific end goal to ensure the effectiveness of the SAPF to compensate for current harmonics. To control the injection current, as further explained in the next sections,
at least three algorithms are actively participating, which include DC voltage, current harmonics extraction, and maximum power point tracking (MPPT) control algorithms. In this work specifically, a
current harmonics extraction algorithm is proposed for further improvements.
There are three groups into which the current algorithms of harmonics extraction can be classified. They include artificial intelligence, time domain, and frequency domain techniques [
]. The frequency domain is always associated with the computation of Fourier coefficients and time delay in sampling, which causes all algorithms in this domain to face difficulties in real-time
applications especially with dynamically varying loads [
]. The time domain basically has better performance in terms of convergence speed compared with the frequency domain [
]. However, it remains doubtful as there are possible sings of existing flickers and noise caused by the conversion of the coordinates from the input signals [
]. The artificial neural network (ANN) is one of the techniques of artificial intelligence. To mitigate harmonic components, it can correctly approximate or extract time-varying fundamental
components in terms of the phase angle and magnitude [
]. Among the various ANN architectures, because of its ability to perform current harmonics extraction simply and well, and because it only consists of a single linear system, Widrow–Hoff (W-H)
adaptive linear neuron (ADALINE) is preferable over others. However, the learning time of the algorithm is altered because it needs to learn multiple harmonic components; this is a weakness [
]. By adding a learning rate in the updating algorithm, an improvement (Modified Widrow–Hoff (W-H) ADALINE) is made to the algorithm’s extraction of the fundamental component [
]. However, further improvements need to be made because of the unnecessary characteristics that still exist within the algorithm. They do not act as the basic requirement of extracting current
harmonics in the power system. These unnecessary characteristics affect the performance of the algorithm because of the slow learning rate and possible large size of the average square error [
]. These two factors are due to the learning factor of the updating algorithm and the cosine component that may affect the response time of the algorithm. As a consequence, there is still the
existence of lag compensation [
]. This algorithm performs with total harmonic distortion (THD) below 5%, accompanied by a response time of 40 ms [
]. The updating algorithm of the Modified W-H ADALINE uses Weight (W) as its learning factor.
The DC link capacitor voltage control algorithm also plays a crucial part in producing a proper injection current for PV SAPF. The main purpose of the algorithm is to control the voltage of the DC
link capacitor. The DC link capacitor functions as a bridge between PV and SAPF. The DC link capacitor voltage, if unstable, will affect the overall injection current of the PV SAPF itself. The
desired DC link voltage and instantaneous voltage can be changed directly to control the DC link capacitor; this is the conventional method used. Nevertheless, this method does not allow for a
desired DC link voltage to be controlled and accurately regulated. As a consequence, voltage that is unclean is produced. Due to the unstable injection current, problems such as high THD and
capacitor blowing may occur, which contribute to the major disadvantages [
]. Therefore, self-charging algorithms have shown a significant increase in usage as an alternative option over the conventional algorithm [
]. Principally, the controlling of the charging and discharging of the DC link capacitor uses the energy conversion law [
]. Subsequently, it provides high accuracy and regulated voltage that is clean with the least noise, and fewest ripples and spikes. In its specific operation, the input value of the self-charging
algorithm (also known as voltage error) must be obtained from the difference between the instantaneous and desired capacitor voltages. Previously, this voltage error was controlled by such algorithms
like proportional–integral (PI) and fuzzy logic techniques. Both techniques were configured to directly process the input value without having a prior understanding of its behavior. Hence, there is
no room for such flexibility whether the voltage error changes or not—it still has to be controlled and processed. The possibility of failure may occur when the operation of the self-charging
algorithm later deals with a varied parameter and nonlinearity. Unfortunately, previous studies on self-charging algorithms for the most part took into account only steady-state operation [
]. No further analysis has been done on dynamic operation [
]. An indirect approach incorporated with fuzzy logic control techniques was introduced [
], called “Self-charging with step size error cancellation”. This indirect approach produces better performance with low overshoot and undershoot (within 0.5–1 V) and fast response time (about 0.5
s), but it is only applicable for shunt active power filter systems and no further testing was done for PV SAPF, especially under various irradiance levels (dynamic operations).
Therefore, this paper presents a significant work in improving the current harmonics extraction algorithm and indirectly improving the injection current produced by a single-phase PV SAPF.
Improvements to the ADALINE technique are carried out to produce what we call the Simpler ADALINE algorithm. It is expected to perform as a fast current harmonics extraction algorithm. Meanwhile,
additional analysis was performed to investigate the performance of the self-charging with step size error cancellation algorithm for a single-phase PV SAPF system under dynamic and steady-state
operations. The proposed single-phase PV SAPF is covered in
Section 2
, while
Section 3
covers the proposed current harmonics extraction algorithm used in the PV SAPF. This is followed by an elaboration of the established self-charging algorithm in
Section 4
Section 5
Section 6
discuss the hardware implementation and simulation work, along with the results. This work is concluded in
Section 7
2. Single-Phase Photovoltaic Shunt Active Power Filter
A block diagram is shown in
Figure 1
to represent operation of the PV SAPF connected at a point of common coupling of the grid source which supplies power to a nonlinear load.
The nonlinear load operation contains load current
, which comprises harmonic component
, within the source current
An injection current
is generated after the SAPF is connected; this compensates for the harmonic current, leaving only the fundamental component
, as shown below.
$I S = I L − I i n j = I 1 + I H − I i n j$
With existence of PV, the injection current
will comprise a combination of the capacitor charging current
, inverter current
, and PV current
$I i n j = I i n v + I P V ± I d c$
For the injection of appropriate current to be accomplished by the SAPF, each current parameter in Equation (2) must be controlled. Hence, we take into consideration the three algorithms. First, the
current harmonics extraction algorithm is used to control the inverter current I[inv]; second, the maximum power point tracking (MPPT) algorithm is used to control the PV current I[PV]; third and
finally, the DC voltage control algorithm is used to control the capacitor charging current I[dc]. For the capacitor charging current I[dc], there is possibility it is supplied from the grid or the
capacitor with both directions. Its value and sign depend on the voltage of the DC link capacitor as when the voltage is overshot, its sign will be positive (discharging) in order to reduce the
voltage, and when the voltage is undershot, its sign will be negative (charging) for voltage to increase. The capacitor charging current I[dc] is equal to zero when the desired voltage is on point.
Figure 2
shows the overall circuit of the single-phase PV SAPF which contains a PV array, DC/DC boost converter, full bridge inverter, DC link capacitor, and its controller. The controller consists of
algorithms such as DC link capacitor voltage control, current harmonics extraction, MPPT, current control, and synchronizer. However, as mentioned earlier, this paper focuses on the current harmonics
extraction and DC link capacitor voltage control algorithms in order to control the inverter current
and the capacitor charging current
respectively. The Adaptive Perturb and Observe (P&O)–Fuzzy algorithm is implemented as the MPPT algorithm [
], due to the fact that it can perform with fast response time and high accuracy. Proportional–integral (PI) was used as the current control algorithm for controlling the steady-state error of the
reference current signal [
]. For this particular research work, the unified ADALINE-based fundamental voltage extraction algorithm is used as the synchronizer [
3. Simpler ADALINE-Based Current Harmonics Extraction
In basic principle, the normal ADALINE algorithm estimates harmonic components based on the principle of sine and cosine components that exist in the electrical system for current harmonics
extraction. The harmonic components and fundamental component for each sample
and sampling period
in digital operation with assigned fundamental frequency
can be depicted by the nonlinear load current
], or
$I L ( k ) = ∑ n = 1 , 2 … N [ W a n sin ( n k ω t s ) − W b n cos ( n k ω t s ) ]$
where the order of the harmonic to
maximum is given by
, and amplitudes of the sine and cosine components are given by
. By rearranging Equation (3) in vector form, the following equations holds:
$I L ¯ ( k ) = W ¯ T X ¯ ( k )$
$W ¯ T = [ w 11 w 21 …… w a n w b n ]$
is the weight matrix and
$X ¯$
describes the cosine and sine vector as
$X ¯ T = [ sin ( k ω t s ) cos ( k ω t s ) …… sin ( n k ω t s ) cos ( n k ω t s ) ]$
. To train
$W ¯ T$
to be equivalent to the value of nonlinear load current
, the algorithm is used. The Widrow–Hoff (W-H) method is used because of its updating algorithm that is the main feature of this extraction algorithm [
]. Weight is used as the learning factor in the Widrow–Hoff (W-H) method. However, to reduce complexity of the normal ADALINE, the Modified W-H ADALINE has been proposed. This only uses the first
order of the harmonic component as opposed to the
-many harmonic components in the normal Windrow–Hoff ADALINE, as depicted in
Figure 3
a [
]. It is independent of number of harmonic orders due to the need to only update the two weights of the fundamental component. However, a learning rate
is introduced as a by-product because of the large average square error
produced in this method [
]. Therefore, the weight updating equation becomes
$W ¯ ( k + 1 ) = W ¯ ( k ) + α e ( k ) Y ¯ ( k ) Y ¯ T ( k ) Y ¯ ( k )$
$W ¯ T = [ w 11 w 21 ]$
$Y ¯ = [ sin ( k ω t s ) cos ( k ω t s ) ]$
, and
is the learning rate. The harmonics current
can be produced as below [
$I H ( k ) = I L ( k ) − W sin ( k ω t s )$
) represents the fundamental sine component multiplied by its weight factor
To reimburse harmonic distortion, the inverter current
is used; this is inversely proportional to the harmonic current
. Although it is capable of reducing THD below 5%, The Modified W-H ADALINE algorithm still has disadvantages that may lead to the current harmonics extraction being slowed. Extracting current
harmonics has basic requirements that it has to fulfil and the unnecessary characteristics that it has do not represent them. Hence, it can be further simplified and improved. The first
simplification is made by discarding the periodic signal cosine component. According to the symmetrical theory of AC power systems, the odd function of periodic signals is of the sine component only.
This is because odd functions are inversely symmetrical about the
axis. In Equation (3), when
is made equal to zero (
= 0), odd functions are of the sine components only. The sum of elements is automatically removed when the cosine component is removed. As a result, the average square error
is removed in large magnitudes, making this the second improvement.
Changing the learning factor of the updating algorithm is the third improvement. The functionality of SAPF is not clearly shown by the weight factor
in Equation (5). Before it is multiplied by the sine component, this learning factor represents the active fundamental current peak value. In simpler terms, it is renamed as the fundamental active
The last improvement is made by replacing the average square error
, as shown in
Figure 3
a, with the negative inverter current −
as further elaborated below.
$e ( k ) = I L ( k ) − I e s t ( k ) I i n v ( k ) = − [ I L ( k ) − I e s t ( k ) ] ∴ e ( k ) = − I i n v ( k )$
A better representation can be seen where the negative inverter current −
is the actual value used later (by removing its negative sign) for the amount of current to be injected by the SAPF. A new updating technique called the Fundamental Active Current (FAC) updating
technique is formed by the average square error to the negative inverter current and weight learning factor to the fundamental active current
, or
$I f ( k + 1 ) = I f ( k ) − α I i n v ( k ) sin ( k ω t s )$
Figure 3
b shows the last form of the simpler ADALINE algorithm. The new harmonic current equation is
$I H ( k ) = I L ( k ) − I f s i n ( k ω t s$
By referring to
Figure 3
b and Equation (10), the active current is generated, depicted by
It is generated by multiplying
with the unity sine function.
4. Self-Charging with Step Size Error Cancellation Algorithm
In principle, the DC link capacitor voltage should be maintained at the desired set point so that the injection current
is properly generated for compensating harmonics. Therefore, the charging capacitor current
is an important factor to be controlled as such change to it may affect the performance of the DC link capacitor. Interestingly,
may also affect the performance of the PV array in supplying
, besides the existing role carried out by the MPPT algorithm. Considering an analysis of the circuit’s connection between the DC link capacitor and the inverter as shown in
Figure 1
, the effective amount of
injected to the grid depends on the condition of
. Therefore, a new parameter named the capacitor–PV current
is introduced, by combining
as follows:
$I C P V = I P V ± I d c$
Any change to I[dc] may affect the delivery of I[PV], which has to be used for keeping the DC link capacitor voltage at a certain level, rather than to be used for injection. Therefore, it is
critical to control I[dc], not only to ensure that the voltage of the DC link capacitor is maintained at its desired value, but also to ensure a fast and optimum supply of I[PV].
As mentioned, the self-charging algorithm is implemented to control
. The charging capacitor current
is determined by the energy conversion law mathematical equation in relation to the DC link capacitor. The DC link capacitor voltage always fluctuates from the desired voltage value during the
charging process. The energy stored in the DC link capacitor is forced to change as a result. Hence, the self-charging equations [
] are
$I d c = 2 C [ ( V d c 2 ) 2 − ( V d c 1 ) 2 ] V T$
where the DC link capacitor value is given by
is the desired DC link capacitor voltage,
is the instantaneous DC link capacitor voltage, and the period of the supply frequency is given by
. The difference between
can be represented as voltage error
, or
$e = ( V d c 2 ) 2 − ( V d c 1 ) 2$
The change in
gives the impacts in determination of
, and, thus, it should be controlled accordingly. To realize control of
, the PI technique has been implemented [
]; with rapid growth of artificial intelligence techniques, especially Fuzzy Logic Control (FLC) [
], better control of voltage error
can be achieved, as shown in
Figure 4
. However, from the figure it is noticeable that the
is controlled directly, where the FLC has to perform and act accordingly based on whatever its value (including zero). Thus, the capability of the self-charging algorithm is still limited, especially
for dynamic operation.
Therefore, to address dynamic operations, modifications and improvements have been made to the direct control approach by arranging it to be an indirect approach, as shown in
Figure 5
. This indirect control in the self-charging algorithm introduced a new parameter which was named step size error ∆
. It is now part of the existing voltage error
to form a new voltage error
], or
$e n e w = e + Δ e e n e w = [ ( V d c 2 ) 2 − ( V d c 1 ) 2 ] + Δ e$
The new voltage error
provides an appropriate value to minimize such drastic changes due to change of the capacitor’s voltage. Thus, the charging current
] becomes
$∴ I d c = 2 C [ [ ( V d c 2 ) 2 − ( V d c 1 ) 2 ] + Δ e ] V T$
The self-charging algorithm is supposed to be able to react smoothly. This is done by giving the algorithm adaptability to abort any change of the voltage error
with respect to overshoot and undershoot. There will be no unnecessary task to perform since the algorithm provides another route to mitigate the voltage error. Even though this approach is
considered stable with good undershoot and overshoot, and with fast response time, this approach was only tested with normal shunt active power filter and no further testing and analysis was done for
the single-phase PV SAPF system. The FLC membership functions are shown in
Figure 6
5. Simulation Results
Matlab-Simulink was used to carry out the simulation works. The proposed single-phase PV SAPF was designed and connected with the test bed made up of a nonlinear load and supply grid source. The
nonlinear load was developed using an H-bridge rectifier (240 mH inductor and 20 Ω resistor connected in series). Matlab-Simulink was used to implement all the algorithms in
Figure 2
together with the proposed single-phase PV SAPF including both proposed algorithms. The simulation work was carried out under dynamic operations to evaluate both algorithms, which covers penetration
of PV through off–on operation and change of irradiance from low to high. The PV irradiances were set at about 200 W/m
(low irradiance), 600 W/m
(medium irradiance), and 1000 W/m
(high irradiance). For comprehensive evaluation, performance of the Simpler ADALINE algorithm in reducing THD level was evaluated with the Modified W-H ADALINE algorithm, by fixing the self-charging
with step size error cancellation algorithm. In addition, the performance of the self-charging with step size error cancellation algorithm was compared with that of the Direct Fuzzy-based
Self-charging algorithm by fixing the simpler ADALINE algorithm. Among the major parameters which were evaluated, besides THD, were overshoot, undershoot, response time, and energy losses. The
simulation sampling time was set at about 6.67 µs, while the learning rate of 0.0001 was configured for both proposed and existing algorithms of current harmonics extraction. The duty cycle to boost
up the PV voltage to 400
was set to 0.46. The PV module used was a SHARP NT-180UI (Sharp Electronics Corporation, Huntington Beach, CA, USA) its characteristics are as shown in
Table 1
Table 2
shows the main parameters and components used in this work. The configuration of the proposed PV SAPF is based on a voltage source inverter (VSI) which is considered as a conventional inductor-based
converter. According to Middlebrook’s extra element theorem [
], to avoid instability, the input impedance of the converter should be much higher than the output impedance of the filter. For the PV SAPF configuration, the switching frequency was set at high
frequency—around 20 kHz. The inductive element inside the PV SAPF increases the input impedance for the high switching frequency; therefore, Middlebrook’s condition is verified and the filter does
not affect the stability of the proposed PV SAPF.
The simulation outcomes of PV SAPF with different irradiances (including without PV) using both Simpler ADALINE and Modified W-H ADALINE current harmonics extraction algorithms are shown in
Figure 7
. It covers source current
, injection current
, voltage
, and load current
for the nonlinear load.
Figure 8
shows the simulation results of harmonic spectra for different irradiances (including without PV) for the Simpler ADALINE algorithm. The simulation results of harmonic spectra for the Modified W-H
ADALINE algorithm are shown in
Figure 9
. From
Figure 7
Figure 8
Figure 9
, the source current
is properly compensated for by both current harmonics extraction algorithms. Specifically, for THD, with irradiance of 0 W/m
, the values are 1.48% and 2.12% for Simpler ADALINE and Modified W-H ADALINE algorithms, respectively. At irradiance of 200 W/m
, the THDs are 1.62% and 2.25% for the Simpler ADALINE and Modified W-H ADALINE algorithms. After irradiance is increased to 600 W/m
, the THD recorded using Simpler ADALINE is 1.93%, while for Modified W-H ADALINE, it is around 2.57%. Lastly, at irradiance of 1000 W/m
, the Simpler ADALINE algorithm gives a THD of about 2.28% and Modified W-H ADALINE algorithm gives a THD of about 2.85%.
Both algorithms give THD values below the 5% benchmark as per the IEEE Standard 519-2014 [
]. According to the findings, there is a slight increase in the THD values when the irradiance of the PV is increased, which subsequently increases the PV source current
. The source current
is decreased with the increase of
where the load is depending more on power from the PV. Another point to take note of from the findings is during the operation of the normal SAPF, when PV is in the off condition and there should be
no additional active power flow to the grid. Only when the PV is in the on condition will additional active power flow to the grid, affecting the injection and source currents. In addition, according
to the previous works on SAPF, operation of the normal SAPF compensates only the reactive component, which means it only has the effect of removing harmonics from the grid [
]. Therefore, it is confirmed that PV is the main source of producing active power. In comparing both algorithms, Simpler ADALINE clearly shows much better performance over the Modified W-H ADALINE.
The significant reduction of THD values shows that the performance of the SAPF is better with the algorithm that is proposed, with or without PV connectivity. Meanwhile, the power factor is improved
from 0.89 to almost unity, which directly confirms the effectiveness of the proposed current harmonics extraction algorithm to perform power correction, too.
In regard to the evaluation of DC link capacitor voltage control, dynamic operation of off–on between PV and SAPF was implemented. This is done by considering the level of irradiance to be set at 600
, as it is considered as medium irradiance in the Malaysia climate [
]. The performances of both DC link capacitor voltage control algorithms and both harmonic extraction algorithms during off–on operation between PV and SAPF are shown in
Figure 10
. Consideration is given to evaluate both current harmonics extraction algorithms which were simulated together with the DC link capacitor voltage control algorithm. The self-charging with step size
error cancellation algorithm performs at a much lower overshoot (0.5 V) and fast response time (0.1 s), as compared to the direct control where it has high overshoot (4.5 V) with slow response time
(1.5 s). Meanwhile, both current harmonics extraction algorithms respond well during off–on dynamic operation, but Simpler ADALINE performs with a much better response time of only 15 ms, as opposed
to the Modified W-H ADALINE algorithm which needs 40 ms to respond.
Another dynamic operation focuses on the change of irradiance from low to high levels.
Figure 11
shows the obtained results. The self-charging with step size error cancellation algorithm produces much better regulated DC link capacitor voltage with a low overshoot of 1 V and fast response time
of 0.2 s compared to the direct control approach with an overshoot and response time of 4 V and 1.6 s, respectively. This is almost similar to the case during off–on dynamic operation. At the same
time, the Simpler ADALINE also has a response time of about 15 ms, performing faster than the Modified W-H ADALINE with 40 ms. Further analyses to be explored are energy losses during dynamic
operations of off–on between PV and SAPF and change of irradiance. Good DC link capacitor voltage control should result in minimal energy losses to the SAPF, especially during integration of PV and
Figure 12
Figure 13
show clearly the energy losses resulting from operation on the DC link capacitor in dynamic operations. During the period of dynamic operation, energy losses on the DC link capacitor
can be calculated as below:
$E l o s s − C d c = ∫ t 1 t 2 P d c d t$
is the steady-state power of the DC link capacitor that should be obtained after change,
is the starting time of change, and
is the end time of change before achieving steady state. However, considering that the change is linear, the calculation of energy can be performed as follows:
$E l o s s − C d c = P d c × ( t 2 − t 1 ) 2$
By referring to
Figure 12
Figure 13
, operation of self-charging with step size error cancellation causes lower energy losses as compared to the direct control approach. For off–on operation, the proposed self-charging algorithm only
causes energy losses of 36 J, whereas the direct control causes loses up to 540 J. Meanwhile for change of irradiance, the self-charging with step size error cancellation algorithm only causes losses
of 112 J; for the direct control, it is 896 J. The results obtained clearly show the capability of the self-charging with step size error cancellation algorithm to significantly minimize the energy
losses of the DC link capacitor.
Table 3
presents THDs obtained from the simulation results of both current harmonics extraction algorithms with different irradiances and
Table 4
shows overall performances of both DC link capacitor control algorithms for both dynamic operations.
6. Experimental Results
An experimental prototype was developed as in
Figure 14
to evaluate the projected algorithms practically (in real time). A model in Matlab-Simulink was used to develop the single-phase PV SAPF. The operated power rating of the PV SAPF is about 536 W with
a DC link capacitor of 4400 µF and output filter inductor of 5 mH. The IGBT IHW15N120R3 with maximum operating current of 15 A and maximum operating voltage of 1200 V was selected. For this
experimental purpose, using a variable transformer, the supply source voltage was configured to 100
. Hence, the voltage was put to 200
which is the desired voltage of the DC link capacitor. To execute all the control strategies for the single-phase PV SAPF, the DSP TMS320F28335 board was programmed and configured. These strategies
include the current harmonics extraction, current control, DC link capacitor voltage control, MPPT, and synchronizer. As in the simulation for dynamic operations, the proposed Simpler ADALINE was
compared with the established Modified Widrow–Hoff ADALINE algorithm by using the self-charging with step size error cancellation algorithm as the DC link capacitor voltage control algorithm. All the
algorithms were evaluated to validate their performance practically in real-time applications. All the measured waveforms were taken by using an oscilloscope Tektronix TBS1000 (Tektronix, Inc.,
Beaverton, OR, USA) with 4 channels, 150 MHz bandwidth, and 1 GS/s sample rate. The PV simulator Chroma 62100H-600S (Chroma Ate Inc., Kuei-Shan Hsiang, Taoyuan, Taiwan) was the main PV source used
for this experiment. It has a voltage range of 0–1.5 kV with output power up to 15 kW.
The experimental outcomes obtained by using the Simpler ADALINE algorithm are shown in
Figure 15
. It includes the source voltage
, injection current
, load current
, and source current
, for different irradiance levels. The outcomes of harmonic spectra with different irradiances (including without PV) for the Simpler ADALINE algorithm are shown in
Figure 16
Figure 17
shows the results of harmonic spectra for the Modified W-H ADALINE algorithm.
Figure 18
shows the experimental results of the self-charging with step size error cancellation algorithm with Simpler ADALINE and Modified Widrow-Hoff ADALINE current harmonics extraction algorithms with
off–on operation between PV and SAPF. The results of the self-charging with step size error cancellation algorithm with Simpler ADALINE and Modified Widrow–Hoff ADALINE current harmonics extraction
algorithm under low to high irradiance operation of PV are shown in
Figure 19
Figure 20
shows the energy losses using the self-charging with step size error cancellation DC link capacitor voltage control algorithm for both dynamic operations.
Figure 15
Figure 16
Figure 17
show that the THDs obtained are 2.3% for irradiance at 0 W/m
, 2.54% at 200 W/m
, 2.8% at 600 W/m
, and lastly 3.2% at 1000 W/m
. The THDs obtained are almost the same as in the simulation work. The proposed current harmonics extraction algorithm is proven to work well to compensate current harmonics to below than 5% THD
under any level of PV irradiance. The power factor has been improved from 0.86 to almost unity. This confirms the productiveness of the proposed algorithm to accomplish power factor corrections as
well. By referring to
Figure 18
Figure 19
, the self-charging with step size error cancellation algorithm performs with low overshoot (0.5 V) for off–on operation between PV and SAPF, and only 1 V for change of irradiance. The self-charging
with step size error cancellation algorithm also produces a fast response time within 0.3 s for the off–on operation between PV and SAPF, and within 0.4 s for the change of irradiance. The same also
applies to the proposed harmonic extraction algorithm where for both dynamic operations, the Simpler ADALINE algorithm achieved a fast response time of only 20 ms. The established Modified
Widrow–Hoff ADALINE algorithm produces a slow response time of about 40 ms—about 20 ms slower than the proposed algorithm. This really shows that the proposed current harmonic extraction algorithm
performs well with good THD values under various irradiances and fast response times under various dynamic operations. Referring to
Figure 20
, under the first dynamic operation which involves off–on operation between the PV and SAPF, the self-charging with step size error cancellation algorithm produces low energy losses of only 54 J. The
self-charging with step size error cancellation algorithm produces an energy loss of around 112 J for the second dynamic operation of the adjustment of irradiance between low and high irradiance
7. Conclusions
A new algorithm in relation to current harmonics extraction for SAPF integrated with a PV source has been presented in this paper. The new current harmonics extraction algorithm is a simpler and
improved version of the established Modified Widrow–Hoff ADALINE algorithm and is called the Simpler ADALINE algorithm. The improvements were made by removing the unnecessary features inside the
existing algorithm such as cosine factor and sum of elements and by rearranging the weight factor. The Simpler ADALINE algorithm shows fast and accurate extraction. Analysis of the self-charging with
step size error cancellation algorithm’s further performance towards a single-phase PV SAPF system has also been presented. Evaluations in terms of dynamic operations have been made to verify the
performances of both algorithms. The analysis under steady-state operation has extensively been used in the past. Hence, dynamic operations such as the change of irradiance level and off–on operation
between PV and SAPF were investigated. These analyses provide the novelty of this work due to the extensive findings and results for further evaluation.
Demonstration of the projected current harmonics extraction algorithm has been successfully accomplished. A comparative evaluation has also been completed with the established algorithm (Modified W-H
ADALINE). As for the DC link capacitor voltage control algorithm, the established self-charging with step size error cancellation was successfully demonstrated and was compared with the established
Direct Fuzzy-based Self-charging algorithm. The simulation and experimental works confirm the much better performance of the projected current harmonics extraction algorithm over the established
algorithms. The Simpler ADALINE algorithm performs with THD values that are low with fast response time and various irradiance levels during both dynamic operations. The self-charging with step size
error cancellation algorithm works well for single-phase PV SAPF and was able to have low overshoot accompanying fast response time in dynamic operations. A vast difference was observed during the
two dynamic operations where the projected Simpler ADALINE algorithm and the established self-charging with step size error cancellation algorithm were able to control any effects from the change of
irradiance level for the PV source and off and on operations between PV and SAPF.
Author Contributions
M.A.A.M.Z. designed and developed the main parts of the research work, including simulation model, experimental set up, and analyses of the obtained results. M.A.A.M.Z. was also mainly responsible
for preparing the paper. M.A.M.R., A.C.S., N.M., and N.A.R. contributed in simulation, experimental, and writing parts. J.T. and C.-M.L. also involved in verifying the work and actively contributed
to finalize the manuscript.
This work is supported by the Putra grant scheme from Universiti Putra Malaysia and key performance index (KPI) grant under the School of Electrical and Electronic Engineering, Universiti Sains
Malaysia, Malaysia.
Conflicts of Interest
The authors declare no conflicts of interest.
ω Angular frequency
α Learning rate
t[s] Sampling period
e Average square error
e(k) Digital time-varying average square error
I[L] Load current
I[L](k) Digital time-varying load current
I[1] Fundamental current
I[S] Source current
I[S](k) Digital time-varying source current
W Weight learning factor
W(k + 1) Matrix of next iteration weight
I[f](k + 1) Matrix of next iteration fundamental active current
W[an] Amplitude of the sine component
W[bn] Amplitude of the cosine component
n Harmonic order
N Maximum harmonic order
Sin (k ωt[s]) Sine function
V[dc] DC link capacitor voltage
V[dc1] Desired DC link capacitor voltage
V[dc2] Instantaneous DC link capacitor voltage
V[s] Source voltage
Y(k) Matrix of sine and cosine function
I[H] Harmonic current
I[H](k) Digital time-varying harmonic current
I[f] Fundamental active current
I[inj] Injection current
I[est](k) Digital time-varying estimation current
I[PV] PV current
I[inv] Inverter current
I[dc] Capacitor charging current
I[CPV] Capacitor–PV current
E[ac] Charging energy of AC
P Real power
t[c] Charging time of the capacitor
V[rms] RMS value of the supply voltage
I[dc,rms] RMS value of the charging capacitor current
V Peak value of the supply voltage
T Period
Ө Phase angle
∆E Energy differential
∆e Step size error
e[new] New voltage error
Figure 3. Block diagrams of (a) Modified Widrow–Hoff adaptive linear neuron (ADALINE) and (b) Simpler ADALINE algorithms.
Figure 5. Indirect control in self-charging algorithm: (a) block diagram and (b) details of Fuzzy Logic Control.
Figure 6. Membership functions for change of voltage error, previous voltage error, and step size error.
Figure 7. Simulation results which cover source voltage V[s], load current I[L], injection current I[inj], and source current I[s] using both current harmonics extraction algorithms for (a) 0 W/m^2
(without PV), (b) 200 W/m^2, (c) 600 W/m^2, and (d) 1000 W/m^2.
Figure 8. Simulation results of harmonic spectra for inductive load using Simpler ADALINE algorithm with (a) no active power filter and at (b) 0 W/m^2 (without PV), (c) 200 W/m^2, (d) 600 W/m^2, and
(e) 1000 W/m^2.
Figure 9. Simulation results of harmonic spectra for inductive load using the Modified W-H ADALINE algorithm at (a) 0 W/m^2 (without PV), (b) 200 W/m^2, (c) 600 W/m^2, and (d) 1000 W/m^2.
Figure 10. Simulation results of DC link capacitor voltage under off–on operation between PV and SAPF by using (a) step size error cancellation and (b) direct control in self-charging algorithms; (c)
performance of current harmonics extraction algorithms.
Figure 11. Simulation results of DC link capacitor voltage under low to high irradiance using (a) step size error cancellation and (b) direct control in self-charging algorithms; (c) performance of
current harmonics extraction algorithms.
Figure 12. Simulation results of DC link capacitor power under off–on operation between PV and SAPF using (a) step size error cancellation and (b) direct control in self-charging algorithms.
Figure 13. Simulation results of DC link capacitor power under low to high irradiance using (a) step size error cancellation and (b) direct control in self-charging algorithms.
Figure 15. Experimental results which cover source voltage V[s] (200 A/div), load current I[L] (5 A/div), injection current I[inj] (5 A/div), and source current I[s] (5 A/div) using the Simpler
ADALINE current harmonics extraction algorithm for (a) 0 W/m^2, (b) 200 W/m^2, (c) 600 W/m^2, and (d) 1000 W/m^2.
Figure 16. Experimental results of harmonic spectra for inductive load using the Simpler ADALINE algorithm at (a) 0 W/m^2 (without PV), (b) 200 W/m^2, (c) 600 W/m^2, and (d) 1000 W/m^2.
Figure 17. Experimental results of harmonics spectra for inductive load using the Modified W-H ADALINE algorithm at (a) 0 W/m^2 (without PV), (b) 200 W/m^2, (c) 600 W/m^2, and (d) 1000 W/m^2.
Figure 18. Experimental results of (a) the self-charging with step size error cancellation DC link capacitor voltage control algorithm under off–on operation between PV and SAPF, with (b) Simpler
ADALINE and (c) Modified Widrow–Hoff ADALINE current harmonics extraction algorithms covering load current I[L] (5 A/div), injection current I[inj] (5 A/div), and source current I[s] (5 A/div).
Figure 19. Experimental results of (a) self-charging with step size error cancellation DC link capacitor voltage control algorithm under change of irradiance of PV, with (b) Simpler ADALINE and (c)
Modified Widrow–Hoff ADALINE current harmonics extraction algorithms covering load current I[L] (5 A/div), injection current I[inj] (5 A/div), and source current I[s] (5 A/div).
Figure 20. Energy loss analyses for experimental results of the self-charging with step size error cancellation DC link capacitor voltage control algorithm under (a) off–on and (b) change of
irradiance operations.
Electrical Characteristics
Maximum power P[max] 180 W
Short circuit current I[sc] 5.60 A
Voltage at maximum power V[mp] 35.86 V
Current at maximum power I[mp] 5.02 A
Open circuit voltage V[oc] 44.8 V
Type Value
Switching frequency 20 kHz
Injection inductor 10 mH
DC link voltage 450 V[dc]
Boost inductor 600 µH
PV voltage 35.86 V[dc] × 8
Line inductor 2 mH
DC link capacitor 1600 µF
Voltage source 230 V[ac]
Table 3. Total Harmonics Distortions (THDs) of current harmonics extraction algorithms in simulation work with different irradiances.
Current Harmonics Extraction Algorithm Total Harmonics Distortion (%)
0 W/m^2 200 W/m^2 600 W/m^2 1000 W/m^2
Simpler ADALINE 1.48 1.62 1.93 2.28
Modified W-H ADALINE 2.12 2.25 2.57 2.85
Table 4. Overall performance of DC link capacitor voltage control algorithms for both dynamic operations.
DC Link Capacitor Control Algorithm Off-On Change of Irradiance
Voltage Overshoot (V) Response Time (s) Energy Losses (J) Voltage Overshoot (V) Response Time (s) Energy Losses (J)
Self-charging with step size error cancellation 0.5 0.1 36 1 0.2 112
Direct fuzzy-based Self-charging 4.5 1.5 540 4 1.6 896
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/
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MDPI and ACS Style
Mohd Zainuri, M.A.A.; Mohd Radzi, M.A.; Che Soh, A.; Mariun, N.; Abd Rahim, N.; Teh, J.; Lai, C.-M. Photovoltaic Integrated Shunt Active Power Filter with Simpler ADALINE Algorithm for Current
Harmonic Extraction. Energies 2018, 11, 1152. https://doi.org/10.3390/en11051152
AMA Style
Mohd Zainuri MAA, Mohd Radzi MA, Che Soh A, Mariun N, Abd Rahim N, Teh J, Lai C-M. Photovoltaic Integrated Shunt Active Power Filter with Simpler ADALINE Algorithm for Current Harmonic Extraction.
Energies. 2018; 11(5):1152. https://doi.org/10.3390/en11051152
Chicago/Turabian Style
Mohd Zainuri, Muhammad Ammirrul Atiqi, Mohd Amran Mohd Radzi, Azura Che Soh, Norman Mariun, Nasrudin Abd Rahim, Jiashen Teh, and Ching-Ming Lai. 2018. "Photovoltaic Integrated Shunt Active Power
Filter with Simpler ADALINE Algorithm for Current Harmonic Extraction" Energies 11, no. 5: 1152. https://doi.org/10.3390/en11051152
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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YOLOX Explanation—SimOTA For Dynamic Label Assignment
YOLOX Explanation — SimOTA For Dynamic Label Assignment
This article is the third in the series where I thoroughly explain how the YOLOX (You Only Look Once X) model works. If you are interested in the code, you can find a link to it below:
This series has 4 parts to fully go over the YOLOX algorithm:
• SimOTA For Dynamic Label Assignment (self)
Label Assignment in Object Detection
Label assignment is a critical task in object detection as it determines what bounding boxes go with what ground truth object during training. As mentioned in the previous article, label assignment
splits anchors into positive and negative groups.
Positive grouped anchors are thought of as good predictions that bound an object while negative grouped anchors are thought of as bad predictions that don’t bound an object.
For example, take a look at the image below:
Bear Box
Notice there are three bounding boxes labeled as A, B, and C. If a human were to go through and label the bounding boxes as positive or negative, they would likely say that B is positive as it bounds
the bear’s head completely while A and C are negative as it doesn’t bound the bear’s head. If the bear’s head was the ground truth object, they might also say that B would go with that ground truth
object while A and C would not go with any ground truth or would just go with the background.
There are many methods that have come about to label bounding boxes as positive or negative and I will go over how YOLOX solves this problem using SimOTA.
Why Use Label Assignment During Training?
SimOTA requires ground truth objects to assign labels. So, it is only used during training and not during inference time.
Label assignment helps the model be more stable as it trains. Instead of optimizing all predictions (in YOLOX the number of predictions is somewhere around 1344 for an input image of 256), we can use
label assignment to get the best predictions. Then, we can optimize the best predictions to make them even better.
Remember that the model optimizes positive predictions for regression and class targets, but optimizes the objectiveness for both pos/neg targets. This way, the model learns to make the predictions
that are already good even better while not worrying about the other predictions. Puning out the bad predictions leads the final model to make much better predictions.
The reason training is more stable is that the model has to deal with fewer gradient updates. As opposed to the model dealing with thousands of gradient updates for the regression and class losses,
it only has to deal with a few gradient updates per image. Since the model updates deal with less outputs, the optimization space is much easier to optimize.
Label Assignment Before SimOTA
Label assignment can be done in many ways. One of the most common ways to assign labels is by finding the highest IoU between a ground truth bounding box and all other bounding boxes. The predictions
with the highest IoU are assigned to that ground truth object.
Other similar methods may be used by other algorithms, but the authors of SimOTA claim that “independently assigning pos/neg samples for each gt (ground truth) without context could be sub-optimal,
just like the lack of context may lead to improper prediction.” To deal with the problem of sub-optimal label assignment, the authors suggest using global context in the image to assign labels rather
than local context which old label assignment algorithms used.
OTA is the proposed method to deal with global context label assignment. OTA treats the label assignment problem as an optimal transport (OT) problem.
The authors of OTA (which are also the authors of YOLOX) define the OT problem as one which has “m supplies and n demanders in a certain area. The i-th supplier holds sᵢ units of goods while the j-th
demanded needs dⱼ units of goods. Transporting cost for each unit of good from one suplier i to demander j is denoted as cᵢⱼ. The goal of the problem is to find a transportation plan 𝝅* according to
which all goods from suppliers can be transported to demanders at a minimal transportation cost.” (page 3)
Essentially, OT has suppliers and demanders and the goal is to find the best plan to transport the supplies to the demanders at a minimal cost.
OTA formulates label assignment as this OT problem where the m suppliers are the gt targets and the n demanders are the predictions or the anchor locations on the image. Remember that each prediction
is assigned to an anchor on the image, so the two can be used synonymously. The gt targets are supplying positive labels to the demanding anchors and the goal is to form the best plan to supply the
positive labels to the anchors. As you can see, the goal is actually to find the most optimal way to label anchors/predictions as pos/neg using the OT problem.
Along with the gts, there is another supplier, the background. This supplier holds all other labels and is shown in step 4 of the algorithm.
Before explaining the OTA algorithm, first let’s define some notation:
• I - The input image
• A - The set of anchors on the input image
• G - The ground truth bounding boxes in image I
• m - The number of ground truth targets
• n - The number of anchors
• k - The number of positive labels each gt can supply
• sᵢ - The supply of the ith gt (sᵢ = k)
• dⱼ - The demand of the jth anchor
• c - Cost to transport one positive label from gtᵢ to anchor aⱼ
• Ø - The background class which is usually denoted as 0
• α - The regression balancing coefficient (usually greater than or equal to 1 to weight the regression loss greater than 1)
• T - The number of iterations to run Sinkhorn-Knopp Iteration
The algorithm to obtain the optimal label assignments is as follows.
1. Assign m and n as the counts of the number of ground truths and number of anchors
2. Get the class predictions Pᶜˡˢ and the regression predictions Pᵇᵒˣ by sending image I through the model.
3. Create the supplying vector, s, which has m + 1 values. Use dynamic k estimation to get the supply of each gt and store it in the vector.
4. s[m+1] = n - sum(s), The background supply at location m + 1 in the supplying vector is equal to the n - sum(s)
5. Initialize the demanding vector, d, as a vector of size n filled with ones.
6. Get the pairwise cls loss between each jth prediction and its corresponding ith ground truth label. cᶜˡˢ = FocalLoss(Pᶜˡˢ, Gᶜˡˢ)
7. Get the pairwise reg loss between each jth prediction and its corresponding ith ground truth label. cʳᵉᵍ = IoULoss(Pᵇᵒˣ, Gᵇᵒˣ)
8. Get the pairwise center prior between each jth anchor and its corresponding ith gt. cᶜᵖ = CenterPrior(Aⱼ, Gᵇᵒˣ)
9. Get the background class cost: cᵇᵍ = FocalLoss(Pᶜˡˢ, Ø)
10. Get the foreground cost: cᶠᵍ = cᶜˡˢ + αcʳᵉᵍ + cᶜᵖ
11. Compute the final cost matrix c by concatenating cᵇᵍ to cᶠᵍ to form a final matrix of shape (m+1, n)
12. Initialize u and v to ones
13. Fill u and v by running Sinkhorn-Knopp Iter for T steps.
14. ^
15. Compute the optimal assignment plan 𝝅* according to Eq.11 in the original paper.
16. return 𝝅*
The paper shows the algorithm as follows.
The algorithm looks like a lot to deal with, but it’s not that bad. SimOTA makes the algorithm much simpler and faster.
The YOLOX authors realized that even though OTA improves the model’s performance, it also makes the model 25% slower which is a lot of time when talking about training a model for about 300
iterations. The authors realized they could make the OTA OT algorithm much faster while retaining a good performance boost by removing the Sinkhorn Iter steps and instead approximating the optimal
assignment plan.
Instead of using Sinkhorn Iteration, SimOTA selects the top kᵢ (or sᵢ) predictions with the lowest cost as the positive samples for the ith ground truth object. Using the SimOTA method of assignment,
a single iteration over all gt objects approximates the assignment instead of using an optimization algorithm to get the most optimal assignment.
The SimOTA algorithm looks like the following:
1. Assign m and n as the counts of the number of ground truths and number of anchors
2. Get the class predictions Pᶜˡˢ and the regression predictions Pᵇᵒˣ by sending image I through the model.
3. Create the supplying vector, s, which has m + 1 values. Use dynamic k estimation to get the supply of each gt and store it in the vector.
4. s[m+1] = n — sum(s), The background supply at location m + 1 in the supplying vector is equal to the n — sum(s)
5. Initialize the demanding vector, d, as a vector of size n filled with ones.
6. Get the pairwise cls loss between each jth prediction and its corresponding ith ground truth label. cᶜˡˢ = FocalLoss(Pᶜˡˢ, Gᶜˡˢ)
7. Get the pairwise reg loss between each jth prediction and its corresponding ith ground truth label. cʳᵉᵍ = IoULoss(Pᵇᵒˣ, Gᵇᵒˣ)
8. Get the pairwise center prior between each jth anchor and its corresponding ith gt. cᶜᵖ = CenterPrior(Aⱼ, Gᵇᵒˣ)
9. Get the background class cost: cᵇᵍ = FocalLoss(Pᶜˡˢ, Ø)
10. Get the foreground cost: cᶠᵍ = cᶜˡˢ + αcʳᵉᵍ + cᶜᵖ
11. Compute the final cost matrix c by concatenating cᵇᵍ to cᶠᵍ to form a final matrix of shape (m+1, n)
12. Iterate over all supply sᵢ in s and get the top sᵢ best predictions with the lowest cost cᵢ. The resulting array should have m values where each mᵢ index in the resulting array has at most sᵢ
13. Return the resulting array
After running SimOTA, the output will be an array of size m where each ith element in the resulting array is a positive labeled anchor/prediction corresponding to the ith ground truth Gᵢ. The rest of
the predictions that are not in the resulting array are considered negative labeled predictions without a gt assignment.
Dynamic k Estimation
k is the supply of each gt object and there are two ways to calculate it. The naive way of calculating k is making it a constant across all gt objects. The problem with this way of assigning supply
to each gt is that not all ground truths should have the same number of anchors assigned to them.
The second proposed way of calculating k is by looking at each gt separately. The authors of OTA propose using Dynamic k Estimation which approximates the supply of each gt. To approximate the supply
for each gt, we can look at all predictions and select the top q predictions according to the IoU values between each prediction and the gt. Then, we sum up the top q IoU values and use that as the k
value for that gt.
Using this method, we estimate the supply, or number of positive labels, for each gt by looking at how accurately each prediction bounds that gt. This way, gts with predictions that are more accurate
are more likely to be assigned to that gt when using the SimOTA algorithm. The authors of OTA state the intuition behind this algorithm is that “the appropriate number of positive anchors for a
certain gt should be positively correlated with the number of anchors that well-regress this gt.”
Note: Although k is no longer a parameter we must change, q is now a parameter that must be tuned. In my code, I use 20 as the q value which seemed to work fine.
Below is my code for dynamic k estimation:
# The supplying vector
s_i = np.ones(m+1, dtype=np.int16)# The sum of all k values
k_sum = 0# Iterate over all ground truth boxes (i = gt_i)
for i in range(0, m): # Get the ith truth value
gt = G_reg[i] # Get the (x, y) coordinates of the intersections
xA = np.maximum(gt[0], P_reg[:, 0])
yA = np.maximum(gt[1], P_reg[:, 1])
xB = np.minimum(gt[0]+gt[2], P_reg[:, 0]+P_reg[:, 2])
yB = np.minimum(gt[1]+gt[3], P_reg[:, 1]+P_reg[:, 3]) # Get the area of the intersections
intersectionArea = np.maximum(0, xB - xA + 1) * np.maximum(0, yB - yA + 1) # Compute the area of both rectangles
areaA = (gt[2]+1)*(gt[3]+1)
areaB = (P_reg[:, 2]+1)*(P_reg[:, 3]+1) # Get the union of the rectangles
union = areaA + areaB - intersectionArea # Compute the intersection over union for all anchors
IoU = intersectionArea/union # Get the q top IoU values (the top q predictions)
# and sum them up to get the k for this gt
k = np.sort(IoU)[-q:].sum() # Add the k value to the total k sum
k_sum += k # Save the k value to the supplying vector
# as an iteger
s_i[i] = int(round(k))
Center Prior
One problem that SimOTA runs into when assigning labels is that a positive label from a gt can be assigned to any anchor prediction. Sometimes a gt doesn’t have very good options to choose from when
assigning positive labels, but since it has to assign k positive labels, the gt must resort to assigning bad positive labels to itself. For example, look at the following image below:
Notice how all bounding boxes on the image don’t cover the bear’s face well. If the bear’s face is a gt object and let’s say it has a supply or k value of 2, then it has to pick two predictions as
positive labels. Clearly, no two predictions are good and the gt object will resort to making two bad predictions positive.
In the early stages of training, the bad prediction problem is very prominent. So, the issue of gts not having a very good prediction to pick from is a big problem.
The solution is to define a radius r and select the r² closest anchors to the center of each gt to not be penalized while penalizing all other anchors outside this r² radius by adding an additional
cost to those predictions. As shown in step 8 in the SimOTA algorithm, a center prior cost is calculated and added to the total cost of all predictions.
The intuition behind selecting the predictions inside the r² radius having a higher chance to be selected as the positive anchor for that gt is that closer anchor predictions are easier to optimize
while further predictions are harder to optimize.
To get the r² closest predictions, we can use the good old distance formula to get the distance between all points and the center of the gt.
To start, we can find the center of the gt, G, by adding half the width to the x coordinate and half the height to the y coordinate as follows:
center = (G[0]+G[2]//2, G[1]+G[3]//2)
Then, we can use the distance formula between all anchors A and gt G:
diff = A-center
dist = np.sqrt(np.sum(diff**2, axis=-1))
Note: center will be a pair of two coordinates. If A are the anchor locations on the (x, y) axis, then we can subtract center from A to get A number of difference pairs. So, the distance formula is
computing the distance along these 2-d coordinate pairs. The result of dist should be equal to the number of anchors in A.
Below is a function I wrote which can be found in my code to calculate the center prior between a single ground truth and all anchors:
# Get the center prior between a gt and the anchor locations
# on the image
# Inputs:
# A - All anchors for a single image
# G_box - A ground truth box for this image
# r - radius used to select anchors in this function
# extraCost - The extra cost to add to those anchors not in
# the r**2 radius
# Output:
# Array with the same number of values as the number of anchors
# where each value is the center prior value of that anchor
def centerPrior(A, G_box, r, extraCost):
## Center Prior selects the r**2 closest anchors according to the
## center distance between the anchors and gts. Those anchors
## that are in the radius are not subject to any extra cost, but those
## anchors outside the radius are assigned extra cost to avoid
## having them be labelled as positive anchors for this gt.
# Get the center location of the ground truth boudning box
center = (G_box[0]+(G_box[2]//2), G_box[1]+(G_box[3]//2))
# Get the difference between the center locations in A and the
# center location of the gt bounding box
diff = A-center
# Use the distance formula to get the distance from the
# gt center location for each anchor
dist = np.sqrt(np.sum(diff**2, axis=-1))
# Get the indices of the distances which are greater
# than r**2 meaning the anchor is outside the radius
idx_neg = np.where(dist > r**2)
# Array of zeros corresponding to the center prior of each anchor
c_cp = np.zeros(A.shape[0])
# Those values not in the r**2 radius are subject to a constant cost
c_cp[idx_neg] = extraCost
return c_cp
Remember how each FPN level has a different number of anchors. On a 256 pixel input image, there will be 64, 256, 1024 anchors on the FPN levels with a stride of 32, 16, and 8 respectively. If the
radius was pretty large and the bear’s head was the gt, then the intersection points on the grid in the blue circle would be the anchor points that are not subject to an extra cost.
The authors of OTA note that picking the r² closest anchors stabilizes training, especially at an early stage of training. Additionally, they claim that due to this extra stability in the model’s
training, the model results in better performance.
YOLOX actually prunes out the labels outside of the r² radius. Instead of giving predictions outside the radius an extra cost, it removes all the predictions outside the r² radius. Sometimes this
will lead to no predictions for that gt, but removing the predictions outside the radius maintains the center prior and helps the stability of the model so the gradients don’t blow up due to a few
bad predictions. I actually found that completely removing the predictions outside the radius hurts performance, but that may be because I only trained on 1000 images.
That wraps up SimOTA. All that’s left to explain is the data augmentations that YOLOX uses which will be explained in the next article.
OTA: https://arxiv.org/abs/2103.14259
YOLOX: https://arxiv.org/abs/2107.08430 | {"url":"https://gmongaras.medium.com/yolox-explanation-simota-for-dynamic-label-assignment-8fa5ae397f76","timestamp":"2024-11-13T19:40:10Z","content_type":"text/html","content_length":"209349","record_id":"<urn:uuid:cfd75ab5-4b0f-4b09-84c5-86f1519d2a4f>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00819.warc.gz"} |
Voltage drop
Voltage drop
The voltage drop is the voltage that is lost in a wire due to its resistance. The voltage drop can be decreased by using shorter or thicker wire, lowering the current or using a metal with a lower
V is the symbol for voltage and is measured in volt (V).
ρ is the symbol for resistivity and is measured in ohm-meter (Ω⋅m).
l is the symbol for length and is measured in meter (m).
I is the symbol for current and is measured in ampere (A).
Resistivity is the property of a metal to resist the flow of current. The resistivity is different for each material and it is measured in ohm-meter (Ω⋅m).
material resistivity ρ 10⁻⁹Ω⋅m
silver 15.9
copper 16.8
aluminium 26.5
tungsten 56
iron 97.1
platinum 106
manganin 482
lead 220
mercury 980
nichrome 1000
constantan 490 | {"url":"https://www.basictables.com/electronics/voltage/voltage-drop","timestamp":"2024-11-07T13:59:36Z","content_type":"text/html","content_length":"12161","record_id":"<urn:uuid:74411f00-ea91-4586-a9db-367691f7fdea>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00403.warc.gz"} |
24 Tablespoons to Cups
24 tbsp to cup conversion result above is displayed in three different forms: as a decimal (which could be rounded), in scientific notation (scientific form, standard index form or standard form in
the United Kingdom) and as a fraction (exact result). Every display form has its own advantages and in different situations particular form is more convenient than another. For example usage of
scientific notation when working with big numbers is recommended due to easier reading and comprehension. Usage of fractions is recommended when more precision is needed.
If we want to calculate how many Cups are 24 Tablespoons we have to multiply 24 by 1 and divide the product by 16. So for 24 we have: (24 × 1) ÷ 16 = 24 ÷ 16 = 1.5 Cups
So finally 24 tbsp = 1.5 cup | {"url":"https://unitchefs.com/tablespoons/cups/24/","timestamp":"2024-11-04T05:41:53Z","content_type":"text/html","content_length":"22893","record_id":"<urn:uuid:855eda0e-185f-4908-bf6a-96739c35674d>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00673.warc.gz"} |
Primer to Numbers and Endianness in Computing
2024-03-07 | 5 minute read | 1007 Words
I’ve been working on my C fluency as of late but taking on a handful of random projects to tackle for when I have time. For one of these projects, I decided to write a chip-8 emulator. I’ll spare you
the details of the actual implementation for now and instead touch on a rather interesting topic on the forefront of my mind as I have been working on this emulator: number systems and endianness.
While the latter isnt really all that important for this project, I think both are a interesting in their own right and I wanted to write a summary on both.
Number Systems in Computing
For most of history, humans have been using a positional number system to represent values. Commonly, we have used a base-10 system to represent our numbers, meaning that 10 becomes the radix, or the
place value, for numbers.
Computers at the lowest level really have no way of representing numbers in base-10, rather since most CPUs are based of digital logic processing ‘on’ and ‘off’ signals, binary, or base 2, has become
the key system for representing numbers. We can abstract “on” and “off” to 1 and 0 respectively in binary. This is the basis of how numbers are represented on computer hardware. In the image above,
our radix is now 2 instead of 10 and we do place value in the exact same way, but we add a new place value every time we reach a multiple of 2.
binary: 11 ( [2^1 * 1] + [2^0 * 1] )
decimal: 3 ( [10^0 * 3] )
The two numbers above are eqivalent representations of the number '3' in decimal
Now this number system is pretty great, but if you do anything low-level with computing, you will see this rarely, instead, most programmers use another number system to represent binary data that
has a lot of unique benefits compared to raw binary. This system is hexadecimal, or base-16. Numbers start from zero and go to F (or 15), place moves over once we hit F0, or 16. This system has a
rather large amount of benefit when dealing with manipulating binary data, particularly when bit widths are considered.
A byte is a binary number consists of eight bits (or places). Thus the number 11111111 or 255 in decimal is a single byte of data, half of this number, 1111 or 15 is called a nibble. What makes hex
so useful is that it is a multiple of 2, 4, and 8, thus we can represent a byte as 0xFF, and a nibble as 0xF (0x is a common prefix for hex numbers). Thus a hex number with 2(n) digits is n bytes
long. It also abstracts the over-verbosity of binary into a format that is both easier to work with and easier to read^.
bcal> c 0xF
(b) 1111
(d) 15
(h) 0xf
bcal> c 0xFF
(b) 11111111
(d) 255
(h) 0xff
Endianness is the property of how multi-byte (remember: multi-byte) values are stored in memory. Chip8 containes 35 opcodes that are all represented as big-endian multi-byte values. So what actually
is the magic property “endianness”? Consider a normal base-10 number 512. We read this number as a some of 10^n powers:
(10^2 * 5) + (10^1 * 1) + (10^0 * 2)
Thus 512 is “broken up” into 3 place parts: 500, 10, and 2. We represent the number ‘512’ buy taking 500 as the most ‘significant’ digit, or the digit with the most weight, and sticking it on the
leftmost side of the number. Then we build on with the remaining digits in descending ‘significance’ based on place value. So endianness is basically how we represent sequences of multi-byte values
in aa similar manner, thus if we have the number 0xAABB as our multi-byte value, we can represent it in two ways based on endinannes:
Given Hex: 0xAABB
In memory:
Big endian: | AA | BB | or (16^1 * AA), (16^0 * BB)
Little endian: | BB | AA | or (16^0 * BB), (16^1 * AA)
Big endianness is when the most significant byte is stored first, Little endianness is when the least significant byte is stored first. I am not entirely sure the reason for using one over the other,
thats research for another day, but it does play a role in some systems. However you must be careful, this is not the method of how arbitrary data is stored. For instance lets say I have a char[4] =
{ 'J', 'o', 'h', 'n' }; in memory, it would NOT be represented as 'n' | 'h' | 'o' | 'J' in memory. Each char is a single 8 bit value, or one byte. Instead if we had an array of multi-bit values, they
would appear as the following in memory:
// in hex: { 0xFA12, 0xCE34 };
int a[4] = { 64018, 52803 };
// In memory:
// Big endian: [ 0xFA | 0x12 ], [ 0xCE | 0x34 ]
// Little endian: [ 0x12 | 0xFA ], [ 0x34 | 0xCE ]
In most systems you don’t notice the difference, because the hardware will convert from little endian to what you expect. Thus given the above example, if you were to inspect the first 2 bytes of the
1st element of array a, you would get 0xFA12 on both a little and big endian system. However if you were only inspecting a single byte of the first element of a, you would get 0xFA on a big endian
system and 0x12 on a little endian system. The actual width of the int depends on multiple factors, but the overall message of the pseudocode is the same.
Overall I find these minute nuances of computers quite interesting, even if I rarely deal with them much day to day. | {"url":"http://theoryware.net/blog/2024/numbers-and-endianness-in-computing/","timestamp":"2024-11-14T22:13:03Z","content_type":"application/xhtml+xml","content_length":"11186","record_id":"<urn:uuid:87f264e2-1c54-47f1-8aeb-407c08a481ca>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00400.warc.gz"} |
Finite Automata Theory and Formal Languages
TMV027/DIT321, LP4 2013
[ News | Prerequisites | Goals | Literature | Assignments | Schedule | Exam | Course Evaluation | Contact Information | Useful Links ]
130829: The results of the re-exams are already reported. Please send me an email to book a time if you want to discuss the correction of your exam.
130822: Solution to the exam on August 21st now available.
130614: The student office of the CSE department is already closed for the summer. You might be able to get hold of your exam if you mail them. Otherwise exams will be in my office (6116 in the
EDIT building) during Tuesday to Friday in week 26 and you can have a look at it provided I am in my office (which I will most of the time but not all the time!).
Tentagransking will take place on Thursday 27th of June 11-12 in my office. (If many of you come at the same time we move to a closed-by bigger room.) If you cannot make any time next week, just
mail me after middle of August so we decide on some time to discuss your exam.
130614: The results of the exams are already reported. Passing rates at both universities were much higher than in previous year (87% at CTH and 72% at GU), I believe due to the introduction of
obligatory assignment.
130528: Solution to the exam on May 28th now available.
130528: Protocol from second evaluation meeting available.
130514: When I wrote exercise 1 of assignment 6, I thought of W as one of the variables in the grammar (as in capital letters for variables, small letters for terminals). I see now that this
doesn't have to be assumed for W and that one could think of it as one of the terminals in the language.
We will correct this exercise with this in mind, that is, that there are those 2 possibilities for W.
Sorry about this ambiguity.
130514: Information about the second evaluation meeting is now available, check course evaluation.
130422: A google group has been created for the course. Please visit the following link to subscribe.
Post your questions to the following email address chalmers-fatfl@googlegroups.com.
130418: Protocol from first evaluation meeting available.
130417: For some reason I cannot explain, it seems that in connection with a change I made on the heading of assignment 2, the text of exercise 3 changed from "... contain the subword 101" to
"... contain the subword 110". Even though I have now changed it back to 101, we will of course, accept solutions to either of both formulations. So don't worry about that and sorry for this
change which I cannot really explain...
130412: Information about students representatives and first evaluation meeting is now available, check course evaluation.
130411: Some of you are having problems with Fire.
If you cannot submit your assignment via Fire do mail it to me and Simon.
As I mentioned in the first lecture, we are testing a beta-version. Please do report any problems with Fire to us so we can repair it.
130325: There was an error in ex.3 of Assignment 1: should be "2 node(t) = st (t)". A corrected version is already in place. Sorry about this.
130322: Instructions on how to submit assignments are now available.
130311: To GU students You have to register online at GU Student portal. The registration is obligatory in order to attend the course.
Note that you need to register yourself in the course the same day as the first lecture, otherwise you will lose your place.
For further information about registration and how to activate your student account, click on this link.
130218: Notice that from VT13 this course is divided into 2 courses elements: 1) obligatory weekly written assignments valid 1.5pts and 2) written exam in an examination hall valid 6pts.
Observe that assignments DO NOT generate bonus points from VT13.
Knowledge in mathematics, including a course in Discrete mathematics, and in programming.
Here you can find some short notes to refresh your knowledge on basic notions like sets, relations and functions. (We would like to thank Einar Steingrímsson for these notes.)
Note: There are 2 typos in the second page on set theory; the right most B in the distributive laws (1) and (2) should be an A.
This course presents both the theory of finite automata and of pushdown automata. If time allows it also includes a short introduction to Turing machines.
Finite automata (and regular expressions) are one of the first and simplest model of computations. Its mathematical theory is quite elegant and simple, and finite automata are widely used in
applications (traffic light, lexical analysis, pattern search algorithm, etc...). Finite automata constitute also a perfect illustration of basic concepts in set theory and discrete structure.
Pushdown automata are finite automata with stacks. The theory is more complex, but has important applications in parsing and analysis of context-free languages which is also a fundamental concept
in computer science.
Turing machines were described by Alan Turing in 1937 and they are a powerful model of computation since they help computer scientists understand the limits of mechanical computation by providing
a precise definition of an 'algorithm' or 'mechanical procedure'.
Learning Outcomes
After completion of this course, the student should be able to:
Knowledge and understanding:
□ Explain and manipulate the different concepts in automata theory and formal languages such as formal proofs, (non-)deterministic automata, regular expressions, regular languages, context-free
grammars, context-free languages, Turing machines;
□ Have a clear understanding about the equivalence between deterministic and non-deterministic finite automata, and regular expressions;
□ Acquire a good understanding of the power and the limitations of regular languages and context-free languages.
Skills and abilities:
□ Prove properties of languages, grammars and automata with rigorously formal mathematical methods;
□ Design automata, regular expressions and context-free grammars accepting or generating a certain language;
□ Describe the language accepted by an automata or generated by a regular expression or a context-free grammar;
□ Simplify automata and context-free grammars;
□ Determine if a certain word belongs to a language;
□ Define Turing machines performing simple tasks.
Judgement and approach:
□ Differentiate and manipulate formal descriptions of languages, automata and grammars with focus on regular and context-free languages, finite automata and regular expressions.
Introduction to Automata Theory, Languages, and Computation, by Hopcroft, Motwani and Ullman. Addison-Wesley. Both second or third edition are fine for the course.
Observe that the web page of the book contains solutions to some of the exercises in the book.
Have a look also at Thierry Coquand's note on the two definitions for the transition function on strings and why they are equivalent.
Here you can find some short notes to refresh your knowledge on basic notions like sets, relations and functions. (We would like to thank Einar Steingrímsson for these notes.)
Note: There are 2 typos in the second page on set theory; the right most B in the distributive laws (1) and (2) should be an A.
There will be obligatory weekly assignments to be done individually. The total amount of points for all assignments will be around 65.
Assignments will NOT generate bonus points from VT2013.
To pass the assignment part of the course you need to individually get at least 50% of the sum of the points of all the weekly assignments together.
For example, if there are 7 assignments during the courses, each with 10pts, you need to have gathered at least 35pts during all your submissions.
Important Note Be aware that assignments are part of the examination. Hence, standard procedure will be followed if cheating is suspected. Please read carefully the page on cheating and its
Who should submit? Chalmers: All of you who are registered in the code TMV027.
GU: All of you who have registered or re-register VT2013.
All other students must not submit. Instead their exam will be 7.5pts and not 6pts like for those who need to do the assignments.
When to submit? Links to the assignments with preliminary deadlines:
□ Deadline Thursday 11/4 23:59: Assignment-1 on Formal proofs (10 pts);
□ Deadline Thursday 18/4 23:59: Assignment-2 on DFA and NFA (10 pts);
□ Deadline Thursday 25/4 23:59: Assignment-3 on epsilon-NFA and RE (9.5 pts);
□ Deadline Friday 3/5 23:59: Assignment-4 on regular languages (10.5 pts);
□ Deadline Monday 13/5 23:59: Assignment-5 on context free-grammars (10 pts);
□ Deadline Monday 20/5 23:59: Assignment-6 on context-free languages (10 pts);
□ Deadline Sunday 26/5 23:59: Assignment-7 on Turing machines (4 pts).
What to submit?
□ You must write your name, personal number and e-mail address in the assignment you submit, even if the submission is done completely electronically.
□ You must have thought and written the solution on your own.
□ You can write either in English or in Swedish.
□ Any non-trivial claim must be supported with a justification/proof or with the proper reference to the book/slides of the course.
□ Solutions should be presented in a coherent and clear manner.
□ If you write your solution by hand, make sure your hand-writing is legibly: what we cannot read we cannot grade!
Failure to comply with any of these requirements invalidates your submission, in part or in full.
How to submit? We will use (a new beta-version of) the Fire system to administrate the submission.
Fire Account: In order to submit any assignment you MUST open an account in Fire by clicking on "register as a student". Fill in the requested information. You will get an email with a
confirmation link. After you have clicked on that link your account is ready and can log-in to it.
This must be done only once, before you submit any assignment.
Submission: EACH submission can be done in two different ways:
□ Electronically using the Fire system. You should attach all necessary files to your submission. Only pdf or plain text are valid formats. Observe that scanned versions of hand-written
solutions are valid electronic solutions.
This is the preferred submission method since it is secure and simple.
□ Personally to any of the teachers of the course in connection to the lectures, exercises or consultation times, or at Ana's office (room 6116 of the D&IT building).
Once the teacher in question gets to his/her office, he/she will inform the Fire system that your submission has been done. You will then get an email confirming your submission.
Important Notes:
☆ You should create a Fire account before you submit any assignment to us personally. If we cannot find you in the system we cannot register your submission and it will be discarded.
☆ Your submission is actually NOT done until the teacher indicates to the system he/she has received your submission.
☆ Contact us immediately if a few hours after you have submitted in this way you have yet NOT received any confirmation mail from the system.
☆ We take NO responsibility for lost submission before they are registered in the system. If you want to make sure no submission is lost please consider making an electronically submission.
☆ If the submission doesn't contain the necessary identification information (see information on "What to submit"), it will automatically be discarded.
☆ If your submission has more than 1 page, then the pages must be stapled, or securely clipped, together, in the top-left corner only (no plastic-pocketed, rolled, folded, taped, or glued
submissions). We take no responsibility for lost pages.
Whatever method you have used for the submission, once the assignment has been corrected you will get an email with the points you have got in it and possible some comments from the grader.
No books or written help during the exam.
Check the studieportalen for the exam's dates.
Any exercise in the assignments of this course is a typical exam question.
During one of the last lectures we will discuss some old exams (Exam 100527, Exam 100827).
See the schedule for further information.
Here you can find some older exams (you may skip exercise 13 of the first exam) and some solutions (ignore the comments on derivatives in the solutions since we have not covered that).
The course should have at least 3-4 students representatives. During the course we should meet a couple of times. In these meetings the student representative should write the minutes. The final
meeting should be done after the exam, sometime during the next study period. Representatives from the IT and GU programmes are usually in this last meeting and take notes.
Students representatives CTH: Johan Andersson, mail: johandf(at)student.chalmers.se
Julia Gustafsson, mail: juliagu(at)student.chalmers.se
Students representatives GU: Anders Bastos Palomino, mail: gusbastan(at)student.gu.se
Björn Norgren, mail: gusinokibj@(at)student.gu.se
First meeting: Tuesday 16th of April at 12:00. Protocol.
Second meeting: Wednesday 15th of May at 15:15. Protocol.
Additional comments: inform from the start about the max. nr of points in assignments, fixed the time for guest lectures before the day so people can choose whether to go or not.
Lecturer: Ana Bove, mail: bove(at)chalmers(dot)se
Assistants: Pablo Buiras, mail: buiras(at)chalmers(dot)se and
Simon Huber, mail: simonhu(at)chalmers(dot)se
Feel free to contact us if you have any further questions, either after the lectures/exercise sessions or via email.
□ Previous year's , 2011, 2010, 2009
□ Short notes on basic notions like sets, relations and functions.
(We would like to thank Einar Steingrímsson for these notes.)
Note: There are 2 typos in the second page on set theory; the right most B in the distributive laws (1) and (2) should be an A. | {"url":"https://www.cse.chalmers.se/edu/year/2013/course/TMV027/","timestamp":"2024-11-06T15:20:15Z","content_type":"text/html","content_length":"19774","record_id":"<urn:uuid:18522584-5e0a-46b2-85f1-b2c15a82da47>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00513.warc.gz"} |
Week 2 Problem Set #2 Quiz Answer
In this article i am gone to share Coursera Course Shortest Paths Revisited, NP-Complete Problems and What To Do About Them Week 2 Problem Set #2 Quiz Answer with you..
Shortest Paths Revisited, NP-Complete Problems and
What To Do About Them Week 2 Quiz Answer
Also visit: Week 1 Problem Set #1 Quiz Answer
Week 2 Problem Set #2 Quiz Answer
Question 1)
Which of the following statements cannot be true, given the current state of knowledge?
• Some NP-complete problems are polynomial-time solvable, and some NP-complete problems are not polynomial-time solvable.
• There is an NP-complete problem that is polynomial-time solvable.
• There is an NP-complete problem that can be solved in O(nlogn) time, where n is the size of the input.
• There is no NP-complete problem that can be solved in O(nlogn) time, where n is the size of the input.
Question 2)
Let TSP1 denote the following problem: given a TSP instance in which all edge costs are positive integers, compute the value of an optimal TSP tour. Let TSP2 denote: given a TSP instance in which all
edge costs are positive integers, and a positive integer T, decide whether or not there is a TSP tour with total length at most T. Let HAM1 denote: given an undirected graph, either return the edges
of a Hamiltonian cycle (a cycle that visits every vertex exactly once), or correctly decide that the graph has no such cycle. Let HAM2 denote: given an undirected graph, decide whether or not the
graph contains at least one Hamiltonian cycle.
• If TSP2 is polynomial-time solvable, then so is TSP1. If HAM2 is polynomial-time solvable, then so is HAM1.
• Polynomial-time solvability of TSP2 does not necessarily imply polynomial-time solvability of TSP1. Polynomial-time solvability of HAM2 does not necessarily imply polynomial-time solvability of
• If TSP2 is polynomial-time solvable, then so is TSP1. But, polynomial-time solvability of HAM2 does not necessarily imply polynomial-time solvability of HAM1.
• Polynomial-time solvability of TSP2 does not necessarily imply polynomial-time solvability of TSP1. But, if HAM2 is polynomial-time solvable, then so is HAM1.
Question 3)
Assume that PENP. Consider undirected graphs with nonnegative edge lengths. Which of the following problems can be solved in polynomial time?
Hint: The Hamiltonian path problem is given an undirected graph with n vertices, decide whether or not there is a (cyclefree) path with n-1 edges that visits every vertex exactly once. You can use
the fact that the Hamiltonian path problem is NP-complete. There are relatively simple reductions from the Hamiltonian path problem to 3 of the 4 problems below.
• For a given source s and destination t, compute the length of a shortest s-t path that has exactly n – 1 edges (+∞ or too, if no such path exists). The path is allowed to contain cycles.
• Amongst all spanning trees of the graph, compute one with the smallest possible number of leaves.
• Amongst all spanning trees of the graph, compute one with the minimum-possible maximum degree. (Recall the degree of a vertex is the number of incident edges.)
• For a given source s and destination t, compute the length of a shortest s-t path that has exactly n – 1 or(+∞ edges or too, if no such path exists). The path is not allowed to contain cycles.
Question 4)
Choose the strongest true statement.
• If the minimum-size vertex cover problem can be solved in time O(Tn)) in bipartite graphs, then the maximumsize independent set problem can be solved in time O(T(n)) in bipartite graphs.
• If the minimum-size vertex cover problem can be solved in time O(T(n) in general graphs, then the maximumsize independent set problem can be solved in time O(T(n) in general graphs.
• If the maximum-size independent set problem can be solved in time (T(n) in general graphs, then the minimum-size vertex cover problem can be solved in time O(T(n)) in general graphs.
• All three of the other assertions are true.
Question 5)
Which of the following statements is true?
• Consider a TSP instance in which every edge cost is the Euclidean distance between two points in the place (just like in Programming Assignment #5). Deleting a vertex and all of its incident
edges cannot increase the cost of the optimal (i.e., minimum sum of edge lengths) tour.
• Consider a TSP instance in which every edge cost is negative. The dynamic programming algorithm covered in the video lectures might not correctly compute the optimal (i.e., minimum sum of edge
lengths) tour of this instance.
• Consider a TSP instance in which every edge cost is negative. Deleting a vertex and all of its incident edges cannot increase the cost of the optimal (i.e., minimum sum of edge lengths) tour.
• Consider a TSP instance in which every edge cost is either 1 or 2. Then an optimal tour can be computed in polynomial time. | {"url":"https://niyander.com/shortest-paths-revisited-np-complete-problems-and-what-to-do-about-them-week-2-quiz-answer/","timestamp":"2024-11-10T01:24:18Z","content_type":"text/html","content_length":"71802","record_id":"<urn:uuid:5df6cbb8-f445-40b8-b4cb-0d037d5f9103>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00321.warc.gz"} |
What is pi? Simple geometry - Math, circles, circumference, diameter - Quatr.us Study GuidesWhat is pi? Simple geometry - Math, circles, circumference, diameter
What is pi? Simple geometry – Math, circles, circumference, diameter
What is pi?
Pi is a number that is just a little bigger than 3.14. It is the number you get if you divide the circumference of any circle by its diameter. It’s the same for all circles. You can approximate pi
for yourself by taking some circular things like the tops of jars and CD’s and frisbees and measuring their diameter and their circumference. When you divide the circumference by the diameter, you’ll
get an answer something like 3.14. It will be the same every time (unless you measured wrong).
Pi is an irrational number
Actually, 3.14 is only approximately equal to pi. That’s because pi is an irrational number. That means that when you write pi as a decimal it goes on forever, never ending and never repeating
itself. The first six digits of pi are 3.14159, and that’s all you need for most practical purposes. In most cases just 3.14 is enough.
A million digits of pi
Check out this webpage with a million digits of pi.
Why is this number called pi? History of pi
Usually in math we write pi with the Greek letter π, which is the letter “p” in Greek. You pronounce it “pie”, like apple pie. It is called pi because π is the first letter of the Greek word
“perimetros” or perimeter. But it was not the Ancient Greeks who first discussed the value of pi. Mathematicians in the Babylonian Empire, about 2000 BC, had already figured out that pi was about 25/
8, or 3.125. By about 1700 BC, in the Middle Kingdom, Egyptian mathematicians calculated pi to be about 3.16. Archimedes calculated that π was a little bigger than 3.1408. People have been gradually
getting closer ever since, with early contributions from mathematicians working in China, India, and the Islamic Empire. By 263 AD, the Chinese mathematician Liu Hui had calculated that pi was 3.141.
Bibliography and further reading about pi:
8 Comments
1. What is the Center Pi? Anyone knows the answer?
2. za
very nice!
3. pi is equal to 3.14
□ No, it’s not. Pi is approximately 3.14, but it is an irrational number that continues infinitely, more and more decimals, forever, without repeating itself. A more accurate number is
3.1415923, but even that is only an approximation good enough for most practical purposes.
4. how do you get the area of pi
5. this was very helpful what sa do
□ I’m glad we could help!
Share This Story, Choose Your Platform! | {"url":"https://quatr.us/math/pi-simple-geometry.htm","timestamp":"2024-11-09T17:04:14Z","content_type":"text/html","content_length":"128032","record_id":"<urn:uuid:1bf35325-2369-450e-a2c0-1565de5ac364>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00809.warc.gz"} |
Computer Class - Week #2
This class continues with the number system (binary, octal) work that we started last week. Because these first few classes are review of material that I covered with these students previously, this
post will fill in some gaps by including material from last spring (when I first introduced binary/octal).
Binary Quote
Before class starts, I write the following quote along the top of the whiteboard (high enough that it's out of the way for the entire class):
"There are 10 types of people in this world, those that understand binary and those that don't."
If anyone asks about it, I say that we'll cover it at the end of class.
[Note: This quote should only be used for the first class that you introduce binary, otherwise it won't be nearly as effective. I used this quote last spring with these students when I introduced
binary, but did not use it in this review class.] Powers of Two
At the start of each class, we begin with a simple exercise. For the first few classes, we cover the powers of 2. I first (jokingly) check to make sure that everyone knows how to multiply by 2 and
then write:
and then go around the room asking the students to multiply the previous value by 2:
8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536
I usually stop at 65536 and then point out 2
(256) and 2
(65536) as values that we'll be encountering later as we learn more about computers.
Review Octal
Even though my goal is to teach binary and hexadecimal, I always start with octal since I believe that it's an easier transition from decimal: when counting in binary, new digits are added rapidly
for the first few number and this can be confusing. And with hexadecimal, we're adding additional symbols into the mix, so it's easier to introduce that once they've already mastered an alternate
number system.
So, what did we cover last week? Numbers, binary, octal. We're going to continue with that this week.
I briefly review place-value number systems and describe base-10 (decimal) and base-8 (octal).
Decimal is based on 10's. It has 10 digits (0-9), and the places are based on 10:
1000's 100's 10's 1's
10x10x10 10x10 10 1
10^3 10^2 10^1 10^0
Octal is based on 8's. It has 8 digits (0-7), and the places are bases on 8:
512's 64's 8's 1's
8x8x8 8x8 8 1
8^3 8^2 8^1 8^0
[Sidenote: I use this opportunity to remind the students that any number raised to the 0^th power is 1. If the students were thinking of powers as "how many times do I multiply this number by itself"
then this result will seem odd (since you'll expect the answer to be 0). Hence, this is a great opportunity to reinforce the correct answer.] Octal Worksheets
To practice a bit with octal, we start with a few worksheets:
Octal counting (1 & 2)
These worksheets have the students count circles that are pre-arranged into groups of eight to make it easy to determine the octal number. In addition to writing the octal number, the students must
also write the corresponding decimal number.
I hand out these worksheets in class and then give the students a few minutes to complete them before going over the answers. The purpose of this worksheet is to make the numbers concrete (by using
the circles) since that helps break the association that (for example) the number '10' always means ten items (it means ten items in decimal, but eight items in octal). In addition, these worksheets
also allow practice with the idea that ten items can be written in multiple ways: '10' in decimal, or '12' in octal.
Octal dots
This is similar to the previous worksheet except that they students are now required to draw the correct number of dots and then convert to decimal.
When reviewing the answers for these worksheets, I point out that the answers can be checked by simply multiplying out the numbers by the place values. So if you have octal 35, you can break it up
= 3 in the eights position, 5 in the ones position
= 3 x 8 + 5 x 1
= 29
This is a direct result of how positional number systems work, and it parallels what we do in decimal, where 29 is really 2 x 10 + 9 x 1.
[Sidenote: Depending on how much you want your students to suffer, you can tell the following octal joke: "Why do computer programmers often confuse Christmas and Halloween?" Answer: Because Oct 31 =
Dec 25 (Octal 31 = Decimal 25).]
Counting in octal
This is the last of the octal worksheets and has the students count from 0 to 63 in octal.
[Sidenote: I make sure that I hand out this worksheet out after I finish all of the octal-decimal conversions that I want to cover in class. One time I had the students do this worksheet first and
some of them kept it around as a reference for the later work. This meant that they were thinking less about the number representation/conversion and doing simple table lookups to get the answer.]
Review Binary
For binary, it follows the same pattern that we saw with decimal and octal:
Here's the decimal information for comparison: Decimal is based on 10's. It has 10 digits (0-9), and the places are based on 10:
1000's 100's 10's 1's
10x10x10 10x10 10 1
10^3 10^2 10^1 10^0
Binary is based on 2's. It has 2 digits (0 and 1), and the places are bases on 2:
32's 16's 8's 4's 2's 1's
2x2x2x2x2 2x2x2x2 2x2x2 2x2 2 1
2^5 2^4 2^3 2^2 2^1 2^0
At this point we practice converting numbers between binary and decimal. I describe the process on the whiteboard and the students practice:
• Converting a decimal number into binary by repeatedly subtracting powers of 2.
• Converting a binary number into decimal by adding the power-of-2 that corresponds to each '1' digit in the binary number.
[Sidenote: I created some worksheets for this, but didn't have them ready in time for this class, so I'll provide more details in my post for next week's class.]
If there is time, I also like to introduce the
Binary Magic Trick
just after I cover how to convert from binary to decimal. Be prepared for the students to be really disappointed when they learn how the trick works - the younger students especially seem to prefer
to live a world where you were able to magically read their mind (rather than simply convert from binary to decimal).
Binary Worksheets
Counting in binary
Similar to the "Counting in octal" worksheet, but this time for binary. The students must count from 0 to 63 in binary ('0' to '111111')
Wrapping up
In summary:
• Binary is for computers, decimal is for humans
• Converting between decimal and binary is not difficult, but it is tedious:
□ Converting binary to decimal requires lots of adding
□ Converting decimal to binary requires lots of repeated subtraction
Since it's annoying to do this conversion all the time (especially for large numbers), we'd like to have an easier way. And we'll cover that next week.
Binary Quote (part II)
At the very end of class, I go back to the binary quote written at the top of the board. I ask a student to read the quote aloud. Invariably, they will start off with "There are ten types of...", and
I'll stop them at that point and write "ten" on the board?
"Does it say 'ten'? I don't see a 'ten' there - I see a '1' and a '0'."
This is usually sufficient to make the connection, but if needed, you can ask the students what '10' is in binary.
I now congratulate the students on their transition from the "those that don't" group into the "those that understand binary" one.
[Sidenote: I once asked a student what he thought the quote was about when he first read it at the beginning of class (before he knew binary). He responded that he thought it was unfinished and that
I would be adding 8 more items later.]
End of class. Next week we'll continue with converting between number systems and introduce hexadecimal. | {"url":"https://cse4k12.blogspot.com/2010/11/computer-class-week-2.html","timestamp":"2024-11-11T14:48:16Z","content_type":"text/html","content_length":"58208","record_id":"<urn:uuid:4e134ecf-ae5d-4773-bf57-425a59700c98>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00653.warc.gz"} |
1/574 as a
Converting 1/574 to 0.002 starts with defining whether or not the number should be represented by a fraction, decimal, or even a percentage. Decimals and Fractions represent parts of numbers, giving
us the ability to represent smaller numbers than the whole. The difference between using a fraction or a decimal depends on the situation. Fractions can be used to represent parts of an object like 1
/8 of a pizza while decimals represent a comparison of a whole number like $0.25 USD. Now, let's solve for how we convert 1/574 into a decimal.
1/574 is 1 divided by 574
The first step of teaching our students how to convert to and from decimals and fractions is understanding what the fraction is telling is. 1 is being divided into 574. Think of this as our
directions and now we just need to be able to assemble the project! Fractions have two parts: Numerators on the top and Denominators on the bottom with a division symbol between or 1 divided by 574.
We use this as our equation: numerator(1) / denominator (574) to determine how many whole numbers we have. Then we will continue this process until the number is fully represented as a decimal. This
is how we look at our fraction as an equation:
Numerator: 1
• Numerators are the top number of the fraction which represent the parts of the equation. With a value of 1, you will have less complexity to the equation; however, it may not make converting any
easier. 1 is an odd number so it might be harder to convert without a calculator. Smaller numerators doesn't mean easier conversions. Now let's explore X, the denominator.
Denominator: 574
• Denominators are located at the bottom of the fraction, representing the total number of parts. 574 is one of the largest two-digit numbers to deal with. And it is nice having an even denominator
like 574. It simplifies some equations for us. Ultimately, don't be afraid of double-digit denominators. So without a calculator, let's convert 1/574 from a fraction to a decimal.
How to convert 1/574 to 0.002
Step 1: Set your long division bracket: denominator / numerator
$$ \require{enclose} 574 \enclose{longdiv}{ 1 } $$
To solve, we will use left-to-right long division. Yep, same left-to-right method of division we learned in school. This gives us our first clue.
Step 2: Extend your division problem
$$ \require{enclose} 00. \\ 574 \enclose{longdiv}{ 1.0 } $$
We've hit our first challenge. 1 cannot be divided into 574! So that means we must add a decimal point and extend our equation with a zero. Even though our equation might look bigger, we have not
added any additional numbers to the denominator. But now we can divide 574 into 1 + 0 or 10.
Step 3: Solve for how many whole groups you can divide 574 into 10
$$ \require{enclose} 00.0 \\ 574 \enclose{longdiv}{ 1.0 } $$
How many whole groups of 574 can you pull from 10? 0 Multiply by the left of our equation (574) to get the first number in our solution.
Step 4: Subtract the remainder
$$ \require{enclose} 00.0 \\ 574 \enclose{longdiv}{ 1.0 } \\ \underline{ 0 \phantom{00} } \\ 10 \phantom{0} $$
If there is no remainder, you’re done! If you have a remainder over 574, go back. Your solution will need a bit of adjustment. If you have a number less than 574, continue!
Step 5: Repeat step 4 until you have no remainder or reach a decimal point you feel comfortable stopping. Then round to the nearest digit.
Sometimes you won't reach a remainder of zero. Rounding to the nearest digit is perfectly acceptable.
Why should you convert between fractions, decimals, and percentages?
Converting between fractions and decimals is a necessity. Remember, they represent numbers and comparisons of whole numbers to show us parts of integers. Same goes for percentages. It’s common for
students to hate learning about decimals and fractions because it is tedious. But each represent values in everyday life! Without them, we’re stuck rounding and guessing. Here are real life examples:
When you should convert 1/574 into a decimal
Speed - Let's say you're playing baseball and a Major League scout picks up a radar gun to see how fast you throw. Your MPH will not be 90 and 1/574 MPH. The radar will read: 90.0 MPH. This
simplifies the value.
When to convert 0.002 to 1/574 as a fraction
Time - spoken time is used in many forms. But we don't say It's '2.5 o'clock'. We'd say it's 'half passed two'.
Practice Decimal Conversion with your Classroom
• If 1/574 = 0.002 what would it be as a percentage?
• What is 1 + 1/574 in decimal form?
• What is 1 - 1/574 in decimal form?
• If we switched the numerator and denominator, what would be our new fraction?
• What is 0.002 + 1/2?
Convert more fractions to decimals
From 1 Numerator From 574 Denominator
What is 1/575 as a decimal? What is 2/574 as a decimal?
What is 1/576 as a decimal? What is 3/574 as a decimal?
What is 1/577 as a decimal? What is 4/574 as a decimal?
What is 1/578 as a decimal? What is 5/574 as a decimal?
What is 1/579 as a decimal? What is 6/574 as a decimal?
What is 1/580 as a decimal? What is 7/574 as a decimal?
What is 1/581 as a decimal? What is 8/574 as a decimal?
What is 1/582 as a decimal? What is 9/574 as a decimal?
What is 1/583 as a decimal? What is 10/574 as a decimal?
What is 1/584 as a decimal? What is 11/574 as a decimal?
What is 1/585 as a decimal? What is 12/574 as a decimal?
What is 1/586 as a decimal? What is 13/574 as a decimal?
What is 1/587 as a decimal? What is 14/574 as a decimal?
What is 1/588 as a decimal? What is 15/574 as a decimal?
What is 1/589 as a decimal? What is 16/574 as a decimal?
What is 1/590 as a decimal? What is 17/574 as a decimal?
What is 1/591 as a decimal? What is 18/574 as a decimal?
What is 1/592 as a decimal? What is 19/574 as a decimal?
What is 1/593 as a decimal? What is 20/574 as a decimal?
What is 1/594 as a decimal? What is 21/574 as a decimal?
Convert similar fractions to percentages
From 1 Numerator From 574 Denominator
2/574 as a percentage 1/575 as a percentage
3/574 as a percentage 1/576 as a percentage
4/574 as a percentage 1/577 as a percentage
5/574 as a percentage 1/578 as a percentage
6/574 as a percentage 1/579 as a percentage
7/574 as a percentage 1/580 as a percentage
8/574 as a percentage 1/581 as a percentage
9/574 as a percentage 1/582 as a percentage
10/574 as a percentage 1/583 as a percentage
11/574 as a percentage 1/584 as a percentage | {"url":"https://www.mathlearnit.com/fraction-as-decimal/what-is-1-574-as-a-decimal","timestamp":"2024-11-07T06:11:43Z","content_type":"text/html","content_length":"33246","record_id":"<urn:uuid:c72a59da-fccd-4b2d-84ed-a6bd9f6b9ae7>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00293.warc.gz"} |
Coefficient of Friction Formula in context of coefficient of friction to acceleration
30 Aug 2024
Journal of Mechanical Engineering
Volume 12, Issue 3, 2023
The Relationship Between Coefficient of Friction and Acceleration: A Theoretical Analysis
The coefficient of friction (μ) is a fundamental concept in the field of tribology, playing a crucial role in determining the motion of objects. In this article, we explore the relationship between μ
and acceleration (a), with a focus on the theoretical framework that governs their interaction. We derive the formula for the coefficient of friction in terms of acceleration, highlighting its
significance in understanding the dynamics of moving objects.
The coefficient of friction is a dimensionless quantity that represents the ratio of the force of friction to the normal force between two surfaces in contact. It is a critical parameter in various
engineering applications, including mechanical design, materials science, and robotics. In this article, we investigate the relationship between μ and acceleration, with a view to providing a deeper
understanding of their interplay.
The coefficient of friction can be expressed as:
μ = F_f / N
where F_f is the force of friction and N is the normal force. When an object is accelerated, its velocity changes over time, resulting in a change in the force of friction. The acceleration (a) of
the object is related to the force of friction by the following formula:
F_f = m * a
where m is the mass of the object.
Substituting this expression for F_f into the equation for μ, we obtain:
μ = m * a / N
This formula highlights the relationship between the coefficient of friction and acceleration. As the acceleration of an object increases, so does its coefficient of friction, assuming that the
normal force remains constant.
In conclusion, this article has explored the theoretical framework governing the relationship between the coefficient of friction and acceleration. The derived formula (μ = m * a / N) provides a
fundamental understanding of how these two quantities interact, with implications for various engineering applications. Further research is needed to investigate the practical consequences of this
relationship in real-world scenarios.
[1] R. S. Lakes, “Tribology: Friction, Lubrication, and Wear,” CRC Press, 2013. [2] J. A. Greenwood, “Friction and Wear,” Butterworth-Heinemann, 2006.
Related articles for ‘coefficient of friction to acceleration’ :
Calculators for ‘coefficient of friction to acceleration’ | {"url":"https://blog.truegeometry.com/tutorials/education/5eae84b9757df1a1527875fff3126fc4/JSON_TO_ARTCL_Coefficient_of_Friction_Formula_in_context_of_coefficient_of_frict.html","timestamp":"2024-11-02T11:41:25Z","content_type":"text/html","content_length":"17064","record_id":"<urn:uuid:2f88d150-3b14-4ce3-a07a-251b582b74bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00369.warc.gz"} |
The zero ring with underlying set ${\displaystyle \{0\}}$ is a semiring called the trivial semiring. This triviality can be characterized via ${\displaystyle 0=1}$ and so when speaking of nontrivial
semirings, ${\displaystyle 0eq 1}$ is often silently assumed as if it were an additional axiom. Now given any semiring, there are several ways to define new ones.
As noted, the natural numbers ${\displaystyle {\mathbb {N} }}$ with its arithmetic structure form a semiring. Taking the zero and the image of the successor operation in a semiring ${\displaystyle R}
$ , i.e., the set ${\displaystyle \{x\in R\mid x=0_{R}\lor \exists p.x=p+1_{R}\}}$ together with the inherited operations, is always a sub-semiring of ${\displaystyle R}$ .
If ${\displaystyle (M,+)}$ is a commutative monoid, function composition provides the multiplication to form a semiring: The set ${\displaystyle \operatorname {End} (M)}$ of endomorphisms ${\
displaystyle M\to M}$ forms a semiring where addition is defined from pointwise addition in ${\displaystyle M}$ . The zero morphism and the identity are the respective neutral elements. If ${\
displaystyle M=R^{n}}$ with ${\displaystyle R}$ a semiring, we obtain a semiring that can be associated with the square ${\displaystyle n\times n}$ matrices ${\displaystyle {\mathcal {M}}_{n}(R)}$
with coefficients in ${\displaystyle R}$ , the matrix semiring using ordinary addition and multiplication rules of matrices. Given ${\displaystyle n\in {\mathbb {N} }}$ and ${\displaystyle R}$ a
semiring, ${\displaystyle {\mathcal {M}}_{n}(R)}$ is always a semiring also. It is generally non-commutative even if ${\displaystyle R}$ was commutative.
Dorroh extensions: If ${\displaystyle R}$ is a semiring, then ${\displaystyle R\times {\mathbb {N} }}$ with pointwise addition and multiplication given by ${\displaystyle \langle x,n\rangle \bullet \
langle y,m\rangle :=\langle x\cdot y+(x\,m+y\,n),n\cdot m\rangle }$ defines another semiring with multiplicative unit ${\displaystyle 1_{R\times {\mathbb {N} }}:=\langle 0_{R},1_{\mathbb {N} }\rangle
}$ . Very similarly, if ${\displaystyle N}$ is any sub-semiring of ${\displaystyle R}$ , one may also define a semiring on ${\displaystyle R\times N}$ , just by replacing the repeated addition in the
formula by multiplication. Indeed, these constructions even work under looser conditions, as the structure ${\displaystyle R}$ is not actually required to have a multiplicative unit.
Zerosumfree semirings are in a sense furthest away from being rings. Given a semiring, one may adjoin a new zero ${\displaystyle 0'}$ to the underlying set and thus obtain such a zerosumfree semiring
that also lacks zero divisors. In particular, now ${\displaystyle 0\cdot 0'=0'}$ and the old semiring is actually not a sub-semiring. One may then go on and adjoin new elements "on top" one at a
time, while always respecting the zero. These two strategies also work under looser conditions. Sometimes the notations ${\displaystyle -\infty }$ resp. ${\displaystyle +\infty }$ are used when
performing these constructions.
Adjoining a new zero to the trivial semiring, in this way, results in another semiring which may be expressed in terms of the logical connectives of disjunction and conjunction: ${\displaystyle \
langle \{0,1\},+,\cdot ,\langle 0,1\rangle \rangle =\langle \{\bot ,\top \},\lor ,\land ,\langle \bot ,\top \rangle \rangle }$ . Consequently, this is the smallest semiring that is not a ring.
Explicitly, it violates the ring axioms as ${\displaystyle \top \lor P=\top }$ for all ${\displaystyle P}$ , i.e. ${\displaystyle 1}$ has no additive inverse. In the self-dual definition, the fault
is with ${\displaystyle \bot \land P=\bot }$ . (This is not to be conflated with the ring ${\displaystyle \mathbb {Z} _{2}}$ , whose addition functions as xor ${\displaystyle \veebar }$ .) In the von
Neumann model of the naturals, ${\displaystyle 0_{\omega }:=\{\}}$ , ${\displaystyle 1_{\omega }:=\{0_{\omega }\}}$ and ${\displaystyle 2_{\omega }:=\{0_{\omega },1_{\omega }\}={\mathcal {P}}1_{\
omega }}$ . The two-element semiring may be presented in terms of the set theoretic union and intersection as ${\displaystyle \langle {\mathcal {P}}1_{\omega },\cup ,\cap ,\langle \{\},1_{\omega }\
rangle \rangle }$ . Now this structure in fact still constitutes a semiring when ${\displaystyle 1_{\omega }}$ is replaced by any inhabited set whatsoever.
The ideals on a semiring ${\displaystyle R}$ , with their standard operations on subset, form a lattice-ordered, simple and zerosumfree semiring. The ideals of ${\displaystyle {\mathcal {M}}_{n}(R)}$
are in bijection with the ideals of ${\displaystyle R}$ . The collection of left ideals of ${\displaystyle R}$ (and likewise the right ideals) also have much of that algebraic structure, except that
then ${\displaystyle R}$ does not function as a two-sided multiplicative identity.
If ${\displaystyle R}$ is a semiring and ${\displaystyle A}$ is an inhabited set, ${\displaystyle A^{*}}$ denotes the free monoid and the formal polynomials ${\displaystyle R[A^{*}]}$ over its words
form another semiring. For small sets, the generating elements are conventionally used to denote the polynomial semiring. For example, in case of a singleton ${\displaystyle A=\{X\}}$ such that ${\
displaystyle A^{*}=\{\varepsilon ,X,X^{2},X^{3},\dots \}}$ , one writes ${\displaystyle R[X]}$ . Zerosumfree sub-semirings of ${\displaystyle R}$ can be used to determine sub-semirings of ${\
displaystyle R[A^{*}]}$ .
Given a set ${\displaystyle A}$ , not necessarily just a singleton, adjoining a default element to the set underlying a semiring ${\displaystyle R}$ one may define the semiring of partial functions
from ${\displaystyle A}$ to ${\displaystyle R}$ .
Given a derivation ${\displaystyle {\mathrm {d} }}$ on a semiring ${\displaystyle R}$ , another the operation "${\displaystyle \bullet }$ " fulfilling ${\displaystyle X\bullet y=y\bullet X+{\mathrm
{d} }(y)}$ can be defined as part of a new multiplication on ${\displaystyle R[X]}$ , resulting in another semiring.
The above is by no means an exhaustive list of systematic constructions.
Derivations on a semiring ${\displaystyle R}$ are the maps ${\displaystyle {\mathrm {d} }\colon R\to R}$ with ${\displaystyle {\mathrm {d} }(x+y)={\mathrm {d} }(x)+{\mathrm {d} }(y)}$ and ${\
displaystyle {\mathrm {d} }(x\cdot y)={\mathrm {d} }(x)\cdot y+x\cdot {\mathrm {d} }(y)}$ .
For example, if ${\displaystyle E}$ is the ${\displaystyle 2\times 2}$ unit matrix and ${\displaystyle U={\bigl (}{\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}{\bigr )}}$ , then the subset of ${\
displaystyle {\mathcal {M}}_{2}(R)}$ given by the matrices ${\displaystyle a\,E+b\,U}$ with ${\displaystyle a,b\in R}$ is a semiring with derivation ${\displaystyle a\,E+b\,U\mapsto b\,U}$ .
A basic property of semirings is that ${\displaystyle 1}$ is not a left or right zero divisor, and that ${\displaystyle 1}$ but also ${\displaystyle 0}$ squares to itself, i.e. these have ${\
displaystyle u^{2}=u}$ .
Some notable properties are inherited from the monoid structures: The monoid axioms demand unit existence, and so the set underlying a semiring cannot be empty. Also, the 2-ary predicate ${\
displaystyle x\leq _{\text{pre}}y}$ defined as ${\displaystyle \exists d.x+d=y}$ , here defined for the addition operation, always constitutes the right canonical preorder relation. Reflexivity ${\
displaystyle y\leq _{\text{pre}}y}$ is witnessed by the identity. Further, ${\displaystyle 0\leq _{\text{pre}}y}$ is always valid, and so zero is the least element with respect to this preorder.
Considering it for the commutative addition in particular, the distinction of "right" may be disregarded. In the non-negative integers ${\displaystyle \mathbb {N} }$ , for example, this relation is
anti-symmetric and strongly connected, and thus in fact a (non-strict) total order.
Below, more conditional properties are discussed.
Any field is also a semifield, which in turn is a semiring in which also multiplicative inverses exist.
Any field is also a ring, which in turn is a semiring in which also additive inverses exist. Note that a semiring omits such a requirement, i.e., it requires only a commutative monoid, not a
commutative group. The extra requirement for a ring itself already implies the existence of a multiplicative zero. This contrast is also why for the theory of semirings, the multiplicative zero must
be specified explicitly.
Here ${\displaystyle -1}$ , the additive inverse of ${\displaystyle 1}$ , squares to ${\displaystyle 1}$ . As additive differences ${\displaystyle d=y-x}$ always exist in a ring, ${\displaystyle x\
leq _{\text{pre}}y}$ is a trivial binary relation in a ring.
Commutative semirings
A semiring is called a commutative semiring if also the multiplication is commutative. Its axioms can be stated concisely: It consists of two commutative monoids ${\displaystyle \langle +,0\rangle }$
and ${\displaystyle \langle \cdot ,1\rangle }$ on one set such that ${\displaystyle a\cdot 0=0}$ and ${\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}$ .
The center of a semiring is a sub-semiring and being commutative is equivalent to being its own center.
The commutative semiring of natural numbers is the initial object among its kind, meaning there is a unique structure preserving map of ${\displaystyle {\mathbb {N} }}$ into any commutative semiring.
The bounded distributive lattices are partially ordered, commutative semirings fulfilling certain algebraic equations relating to distributivity and idempotence. Thus so are their duals.
Ordered semirings
Notions or order can be defined using strict, non-strict or second-order formulations. Additional properties such as commutativity simplify the axioms.
Given a strict total order (also sometimes called linear order, or pseudo-order in a constructive formulation), then by definition, the positive and negative elements fulfill ${\displaystyle 0<x}$
resp. ${\displaystyle x<0}$ . By irreflexivity of a strict order, if ${\displaystyle s}$ is a left zero divisor, then ${\displaystyle s\cdot x<s\cdot y}$ is false. The non-negative elements are
characterized by ${\displaystyle eg (x<0)}$ , which is then written ${\displaystyle 0\leq x}$ .
Generally, the strict total order can be negated to define an associated partial order. The asymmetry of the former manifests as ${\displaystyle x<y\to x\leq y}$ . In fact in classical mathematics
the latter is a (non-strict) total order and such that ${\displaystyle 0\leq x}$ implies ${\displaystyle x=0\lor 0<x}$ . Likewise, given any (non-strict) total order, its negation is irreflexive and
transitive, and those two properties found together are sometimes called strict quasi-order. Classically this defines a strict total order – indeed strict total order and total order can there be
defined in terms of one another.
Recall that "${\displaystyle \leq _{\text{pre}}}$ " defined above is trivial in any ring. The existence of rings that admit a non-trivial non-strict order shows that these need not necessarily
coincide with "${\displaystyle \leq _{\text{pre}}}$ ".
Additively idempotent semirings
A semiring in which every element is an additive idempotent, that is, ${\displaystyle x+x=x}$ for all elements ${\displaystyle x}$ , is called an (additively) idempotent semiring.^[9] Establishing $
{\displaystyle 1+1=1}$ suffices. Be aware that sometimes this is just called idempotent semiring, regardless of rules for multiplication.
In such a semiring, ${\displaystyle x\leq _{\text{pre}}y}$ is equivalent to ${\displaystyle x+y=y}$ and always constitutes a partial order, here now denoted ${\displaystyle x\leq y}$ . In particular,
here ${\displaystyle x\leq 0\leftrightarrow x=0}$ . So additively idempotent semirings are zerosumfree and, indeed, the only additively idempotent semiring that has all additive inverses is the
trivial ring and so this property is specific to semiring theory. Addition and multiplication respect the ordering in the sense that ${\displaystyle x\leq y}$ implies ${\displaystyle x+t\leq y+t}$ ,
and furthermore implies ${\displaystyle s\cdot x\leq s\cdot y}$ as well as ${\displaystyle x\cdot s\leq y\cdot s}$ , for all ${\displaystyle x,y,t}$ and ${\displaystyle s}$ .
If ${\displaystyle R}$ is additively idempotent, then so are the polynomials in ${\displaystyle R[X^{*}]}$ .
A semiring such that there is a lattice structure on its underlying set is lattice-ordered if the sum coincides with the meet, ${\displaystyle x+y=x\lor y}$ , and the product lies beneath the join $
{\displaystyle x\cdot y\leq x\land y}$ . The lattice-ordered semiring of ideals on a semiring is not necessarily distributive with respect to the lattice structure.
More strictly than just additive idempotence, a semiring is called simple iff ${\displaystyle x+1=1}$ for all ${\displaystyle x}$ . Then also ${\displaystyle 1+1=1}$ and ${\displaystyle x\leq 1}$ for
all ${\displaystyle x}$ . Here ${\displaystyle 1}$ then functions akin to an additively infinite element. If ${\displaystyle R}$ is an additively idempotent semiring, then ${\displaystyle \{x\in R\
mid x+1=1\}}$ with the inherited operations is its simple sub-semiring. An example of an additively idempotent semiring that is not simple is the tropical semiring on ${\displaystyle {\mathbb {R} }\
cup \{-\infty \}}$ with the 2-ary maximum function, with respect to the standard order, as addition. Its simple sub-semiring is trivial.
A c-semiring is an idempotent semiring and with addition defined over arbitrary sets.
An additively idempotent semiring with idempotent multiplication, ${\displaystyle x^{2}=x}$ , is called additively and multiplicatively idempotent semiring, but sometimes also just idempotent
semiring. The commutative, simple semirings with that property are exactly the bounded distributive lattices with unique minimal and maximal element (which then are the units). Heyting algebras are
such semirings and the Boolean algebras are a special case.
Further, given two bounded distributive lattices, there are constructions resulting in commutative additively-idempotent semirings, which are more complicated than just the direct sum of structures.
Number lines
In a model of the ring ${\displaystyle {\mathbb {R} }}$ , one can define a non-trivial positivity predicate ${\displaystyle 0<x}$ and a predicate ${\displaystyle x<y}$ as ${\displaystyle 0<(y-x)}$
that constitutes a strict total order, which fulfills properties such as ${\displaystyle eg (x<0\lor 0<x)\to x=0}$ , or classically the law of trichotomy. With its standard addition and
multiplication, this structure forms the strictly ordered field that is Dedekind-complete. By definition, all first-order properties proven in the theory of the reals are also provable in the
decidable theory of the real closed field. For example, here ${\displaystyle x<y}$ is mutually exclusive with ${\displaystyle \exists d.y+d^{2}=x}$ .
But beyond just ordered fields, the four properties listed below are also still valid in many sub-semirings of ${\displaystyle {\mathbb {R} }}$ , including the rationals, the integers, as well as the
non-negative parts of each of these structures. In particular, the non-negative reals, the non-negative rationals and the non-negative integers are such a semirings. The first two properties are
analogous to the property valid in the idempotent semirings: Translation and scaling respect these ordered rings, in the sense that addition and multiplication in this ring validate
• ${\displaystyle (x<y)\,\to \,x+t<y+t}$
• ${\displaystyle (x<y\land 0<s)\,\to \,s\cdot x<s\cdot y}$
In particular, ${\displaystyle (0<y\land 0<s)\to 0<s\cdot y}$ and so squaring of elements preserves positivity.
Take note of two more properties that are always valid in a ring. Firstly, trivially ${\displaystyle P\,\to \,x\leq _{\text{pre}}y}$ for any ${\displaystyle P}$ . In particular, the positive additive
difference existence can be expressed as
• ${\displaystyle (x<y)\,\to \,x\leq _{\text{pre}}y}$
Secondly, in the presence of a trichotomous order, the non-zero elements of the additive group are partitioned into positive and negative elements, with the inversion operation moving between them.
With ${\displaystyle (-1)^{2}=1}$ , all squares are proven non-negative. Consequently, non-trivial rings have a positive multiplicative unit,
Having discussed a strict order, it follows that ${\displaystyle 0eq 1}$ and ${\displaystyle 1eq 1+1}$ , etc.
Discretely ordered semirings
There are a few conflicting notions of discreteness in order theory. Given some strict order on a semiring, one such notion is given by ${\displaystyle 1}$ being positive and covering ${\displaystyle
0}$ , i.e. there being no element ${\displaystyle x}$ between the units, ${\displaystyle eg (0<x\land x<1)}$ . Now in the present context, an order shall be called discrete if this is fulfilled and,
furthermore, all elements of the semiring are non-negative, so that the semiring starts out with the units.
Denote by ${\displaystyle {\mathsf {PA}}^{-}}$ the theory of a commutative, discretely ordered semiring also validating the above four properties relating a strict order with the algebraic structure.
All of its models have the model ${\displaystyle \mathbb {N} }$ as its initial segment and Gödel incompleteness and Tarski undefinability already apply to ${\displaystyle {\mathsf {PA}}^{-}}$ . The
non-negative elements of a commutative, discretely ordered ring always validate the axioms of ${\displaystyle {\mathsf {PA}}^{-}}$ . So a slightly more exotic model of the theory is given by the
positive elements in the polynomial ring ${\displaystyle {\mathbb {Z} }[X]}$ , with positivity predicate for ${\displaystyle p={\textstyle \sum }_{k=0}^{n}a_{k}X^{k}}$ defined in terms of the last
non-zero coefficient, ${\displaystyle 0<p:=(0<a_{n})}$ , and ${\displaystyle p<q:=(0<q-p)}$ as above. While ${\displaystyle {\mathsf {PA}}^{-}}$ proves all ${\displaystyle \Sigma _{1}}$ -sentences
that are true about ${\displaystyle \mathbb {N} }$ , beyond this complexity one can find simple such statements that are independent of ${\displaystyle {\mathsf {PA}}^{-}}$ . For example, while ${\
displaystyle \Pi _{1}}$ -sentences true about ${\displaystyle \mathbb {N} }$ are still true for the other model just defined, inspection of the polynomial ${\displaystyle X}$ demonstrates ${\
displaystyle {\mathsf {PA}}^{-}}$ -independence of the ${\displaystyle \Pi _{2}}$ -claim that all numbers are of the form ${\displaystyle 2q}$ or ${\displaystyle 2q+1}$ ("odd or even"). Showing that
also ${\displaystyle {\mathbb {Z} }[X,Y]/(X^{2}-2Y^{2})}$ can be discretely ordered demonstrates that the ${\displaystyle \Pi _{1}}$ -claim ${\displaystyle x^{2}eq 2y^{2}}$ for non-zero ${\
displaystyle x}$ ("no rational squared equals ${\displaystyle 2}$ ") is independent. Likewise, analysis for ${\displaystyle {\mathbb {Z} }[X,Y,Z]/(XZ-Y^{2})}$ demonstrates independence of some
statements about factorization true in ${\displaystyle \mathbb {N} }$ . There are ${\displaystyle {\mathsf {PA}}}$ characterizations of primality that ${\displaystyle {\mathsf {PA}}^{-}}$ does not
validate for the number ${\displaystyle 2}$ .
In the other direction, from any model of ${\displaystyle {\mathsf {PA}}^{-}}$ one may construct an ordered ring, which then has elements that are negative with respect to the order, that is still
discrete the sense that ${\displaystyle 1}$ covers ${\displaystyle 0}$ . To this end one defines an equivalence class of pairs from the original semiring. Roughly, the ring corresponds to the
differences of elements in the old structure, generalizing the way in which the initial ring ${\displaystyle \mathbb {Z} }$ can be defined from ${\displaystyle \mathbb {N} }$ . This, in effect, adds
all the inverses and then the preorder is again trivial in that ${\displaystyle \forall x.x\leq _{\text{pre}}0}$ .
Beyond the size of the two-element algebra, no simple semiring starts out with the units. Being discretely ordered also stands in contrast to, e.g., the standard ordering on the semiring of
non-negative rationals ${\displaystyle {\mathbb {Q} }_{\geq 0}}$ , which is dense between the units. For another example, ${\displaystyle {\mathbb {Z} }[X]/(2X^{2}-1)}$ can be ordered, but not
discretely so.
Natural numbers
${\displaystyle {\mathsf {PA}}^{-}}$ plus mathematical induction gives a theory equivalent to first-order Peano arithmetic ${\displaystyle {\mathsf {PA}}}$ . The theory is also famously not
categorical, but ${\displaystyle \mathbb {N} }$ is of course the intended model. ${\displaystyle {\mathsf {PA}}}$ proves that there are no zero divisors and it is zerosumfree and so no model of it is
a ring.
The standard axiomatization of ${\displaystyle {\mathsf {PA}}}$ is more concise and the theory of its order is commonly treated in terms of the non-strict "${\displaystyle \leq _{\text{pre}}}$ ".
However, just removing the potent induction principle from that axiomatization does not leave a workable algebraic theory. Indeed, even Robinson arithmetic ${\displaystyle {\mathsf {Q}}}$ , which
removes induction but adds back the predecessor existence postulate, does not prove the monoid axiom ${\displaystyle \forall y.(0+y=y)}$ .
Complete semirings
A complete semiring is a semiring for which the additive monoid is a complete monoid, meaning that it has an infinitary sum operation ${\displaystyle \Sigma _{I}}$ for any index set ${\displaystyle
I}$ and that the following (infinitary) distributive laws must hold:^[10]^[12]
${\displaystyle {\textstyle \sum }_{i\in I}{\left(a\cdot a_{i}\right)}=a\cdot \left({\textstyle \sum }_{i\in I}{a_{i}}\right),\qquad {\textstyle \sum }_{i\in I}{\left(a_{i}\cdot a\right)}=\left
({\textstyle \sum }_{i\in I}{a_{i}}\right)\cdot a.}$
Examples of a complete semiring are the power set of a monoid under union and the matrix semiring over a complete semiring. For commutative, additively idempotent and simple semirings, this property
is related to residuated lattices.
Continuous semirings
A continuous semiring is similarly defined as one for which the addition monoid is a continuous monoid. That is, partially ordered with the least upper bound property, and for which addition and
multiplication respect order and suprema. The semiring ${\displaystyle \mathbb {N} \cup \{\infty \}}$ with usual addition, multiplication and order extended is a continuous semiring.^[14]
Any continuous semiring is complete:^[10] this may be taken as part of the definition.
Star semirings
A star semiring (sometimes spelled starsemiring) is a semiring with an additional unary operator ${\displaystyle {}^{*}}$ ,^[9]^[15] satisfying
${\displaystyle a^{*}=1+aa^{*}=1+a^{*}a.}$
A Kleene algebra is a star semiring with idempotent addition and some additional axioms. They are important in the theory of formal languages and regular expressions.
Complete star semirings
In a complete star semiring, the star operator behaves more like the usual Kleene star: for a complete semiring we use the infinitary sum operator to give the usual definition of the Kleene star:
${\displaystyle a^{*}={\textstyle \sum }_{j\geq 0}{a^{j}},}$
${\displaystyle a^{j}={\begin{cases}1,&j=0,\\a\cdot a^{j-1}=a^{j-1}\cdot a,&j>0.\end{cases}}}$
Note that star semirings are not related to *-algebra, where the star operation should instead be thought of as complex conjugation.
Conway semiring
A Conway semiring is a star semiring satisfying the sum-star and product-star equations:^[9]^[17]
{\displaystyle {\begin{aligned}(a+b)^{*}&=\left(a^{*}b\right)^{*}a^{*},\\(ab)^{*}&=1+a(ba)^{*}b.\end{aligned}}}
Every complete star semiring is also a Conway semiring, but the converse does not hold. An example of Conway semiring that is not complete is the set of extended non-negative rational numbers ${\
displaystyle \mathbb {Q} _{\geq 0}\cup \{\infty \}}$ with the usual addition and multiplication (this is a modification of the example with extended non-negative reals given in this section by
eliminating irrational numbers). An iteration semiring is a Conway semiring satisfying the Conway group axioms,^[9] associated by John Conway to groups in star-semirings.^[19] | {"url":"https://www.knowpia.com/knowpedia/Semiring","timestamp":"2024-11-14T15:30:25Z","content_type":"text/html","content_length":"629116","record_id":"<urn:uuid:fc5a5f20-9237-4f56-845e-b7dbf12cec7e>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00751.warc.gz"} |
Barbed Wire Fence Cost Calculator – Estimate Your Project
This tool calculates the total cost for installing a barbed wire fence based on your specific requirements.
How to Use the Barbed Wire Fence Cost Calculator
To use this calculator, enter the following parameters:
• Fence Length: The total length of the fence in meters.
• Fence Height: The height of the fence in meters.
• Number of Barbed Wires: The number of horizontal barbed wires you plan to use.
• Post Spacing: The distance between adjacent posts in meters.
• Cost per Meter of Wire: The cost per meter for the barbed wire in dollars.
• Cost per Post: The cost of a single post in dollars.
Once you’ve filled in all the fields, click the ‘Calculate’ button to get the estimated total cost of the barbed wire fence.
How the Calculation Works
• Wire Length Calculation: The total length of barbed wire needed is calculated by multiplying the fence length by the number of barbed wires.
• Total Post Cost: The number of posts required is determined by dividing the fence length by the post spacing and adding one for the end post. This total is then multiplied by the cost per post.
• Total Wire Cost: The total wire cost is obtained by multiplying the total length of barbed wire by the cost per meter of wire.
• Total Cost: The total cost is calculated by adding the total post cost and the total wire cost.
The calculator provides an estimate based on average costs and measurements. Actual costs can vary depending on local prices, ground conditions, and specific project requirements. Always consult with
a professional for precise measurements and cost evaluations.
Use Cases for This Calculator
Calculate Total Cost for Barbed Wire Fence Installation
Enter the length of the fence line, the distance between each fence post, the cost of materials, and labor costs to get the total cost for installing a barbed wire fence at your property.
Determine Total Length of Barbed Wire Needed
Specify the area to be fenced, the number of sides to fence, and the length of each side to calculate the total length of barbed wire required for the project. This helps you accurately estimate
material costs.
Estimate Number of Fence Posts Required
Input the length of the fence line and the spacing between each fence post to determine the total number of posts needed for your barbed wire fence. This calculation ensures proper support along the
fence line.
Factor in Gates and Corner Braces
Include the number of gates and corner braces needed for your barbed wire fence to get a comprehensive cost estimate. This feature helps you plan and budget for all components of the fence project.
Adjust Material Costs for Different Wire Types
Choose between various types of barbed wire with different prices per foot to see how each option impacts the overall cost of your fence installation. This feature allows you to compare costs and
select the most suitable wire for your needs.
Consider Terrain and Slope Costs
Factor in additional costs for challenging terrains or slopes that may require extra work during fence installation. Ensure your cost estimate reflects the specific conditions of your property for an
accurate budget forecast.
Account for Permit and Survey Expenses
Add permit fees and survey costs to your overall estimation to cover any legal requirements before installing a barbed wire fence. This step ensures compliance with regulations and prevents
unexpected expenses down the line.
Get Labor Cost Estimate Based on Project Size
Specify the size of your fence project and the average labor cost per foot to calculate the total labor expenses. This helps you understand the labor component of your budget and plan accordingly.
Compare Quotes and Adjust Parameters Easily
Quickly compare cost estimates by adjusting parameters like fence length, material prices, and labor costs until you find the right balance for your budget. This feature allows you to tailor the
calculation to your specific needs and preferences.
Save and Print Detailed Cost Breakdown
After calculating the total cost of your barbed wire fence installation, save and print a detailed breakdown including materials, labor, additional expenses, and total project cost. This summary
helps you keep track of your budget and communicate effectively with contractors or suppliers. | {"url":"https://calculatorsforhome.com/barbed-wire-fence-cost-calculator/","timestamp":"2024-11-10T04:59:59Z","content_type":"text/html","content_length":"149764","record_id":"<urn:uuid:606ee3ee-13c9-4cd5-a6cd-edbab516342a>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00539.warc.gz"} |
Data Science : Regression & its variants - EasyCourses
Data Science is all about analyzing the data, finding patterns, and predicting the future. If the pattern identified is accurate, the prediction is correct. So the struggle is always to do the right
analysis. Though being skilled in Coding is essential for a data scientist, that’s not all. A data scientist needs to have skills in coding, statistics, and critical thinking. Our online course on
Data Science using Python is a complete course from basics to the latest tools and techniques followed.
Regression is a popular statistical technique used in Data Science for the prediction of unknown values in a data set based on the known features. It is used when there is a missing value in a data
set. By analyzing the relationship between the dependent (target) and the independent (predictor) variable we can forecast the nature of the missing variable.
This statistical relationship between the known and unknown can be of different forms based on various factors like the type of predictors, outcomes, the function used to build the relation, etc.
There are innumerable forms of regressions. Three main factors for deciding on the regression model is
• Number of independent variables
• The shape of the regression line
• Type of independent variable
Let’s look at a few commonly used regression forms:
• Linear Regression: This is the most simple, popular regression form. Here there is only 1 dependent value and mostly only 1 independent value. The shape of the regression is linear (straight
• Multiple Regression: Quite similar to linear regression however the difference is that the independent variable is more than 1. Here since there are more independent values the result is expected
to be more accurate.
• Logistic Regression: This regression is used to find out the probability of a class or event. Whether the result will be ‘Pass or Fail’ or ‘Yes or No’. it is widely used for classification
problems. Logistic regression can be binary, ordinal, or multinomial.
• Stepwise Regression: This form of regression helps with high dimensional data sets. It is used when we are working with more than one independent variable. The selection of the independent
variable is an automatic process. In each step, a variable is considered to be added or removed from the set based on specific criteria.
• Ridge Regression: In a data set where the independent variables are highly correlated (multicollinearity) ridge regression is used. Here the L2 regularization tool is used. Ridge regression uses
a type of shrinkage called ‘ridge shrinkage’. It shrinks the value of the coefficients but not to zero. Unlike least square estimates here a degree of bias is added to the regression estimates to
reduce the standard errors.
• LASSO (Least Absolute Shrinkage & Selection Operator) Regression: Here unlike Ridge regression, the value of the coefficient gets shrunk to zero. This regression uses the L1 regularization
technique. LASSO regression provides a subset of predictors which is simple and sparse.
• ElasticNet Regression: This is a combination of Ridge and LASSO regression forms .i.e a hybrid of L1 and L2 regression methods. Though it inherits advantages from both the models it might also
suffer from double shrinkage.
These are a few Regression models you learn and use as a Data Scientist. Data Science is a very interesting and in-demand job. Join our online course on Data Science using Python for the right
understanding of the technology before you plunge into the career. Our course uses Python as the programming language as it the most suitabe for all advanced technologies. Our expert team of trainers
can rightly handhold you with proper explanations, real-life examples, and practical assignments.
Follow our blogs to stay updated in your industry. We continuosly try to bring you all the latest and interesting information in your Industry. | {"url":"https://easycourses.in/blog_details/data-science-regression-its-variants/","timestamp":"2024-11-12T12:53:10Z","content_type":"text/html","content_length":"76546","record_id":"<urn:uuid:3f0a92c4-34e4-4bea-aab2-4f1eb7246e9c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00824.warc.gz"} |
Machine Learning
This section of the math expressions user guide covers machine learning functions.
Feature Scaling
Before performing machine learning operations its often necessary to scale the feature vectors so they can be compared at the same scale.
All the scaling function operate on vectors and matrices. When operating on a matrix the rows of the matrix are scaled.
Min/Max Scaling
The minMaxScale function scales a vector or matrix between a minimum and maximum value. By default it will scale between 0 and 1 if min/max values are not provided.
Below is a simple example of min/max scaling between 0 and 1. Notice that once brought into the same scale the vectors are the same.
let(a=array(20, 30, 40, 50),
b=array(200, 300, 400, 500),
c=matrix(a, b),
This expression returns the following response:
"result-set": {
"docs": [
"d": [
"EOF": true,
"RESPONSE_TIME": 0
The standardize function scales a vector so that it has a mean of 0 and a standard deviation of 1. Standardization can be used with machine learning algorithms, such as Support Vector Machine (SVM),
that perform better when the data has a normal distribution.
let(a=array(20, 30, 40, 50),
b=array(200, 300, 400, 500),
c=matrix(a, b),
This expression returns the following response:
"result-set": {
"docs": [
"d": [
"EOF": true,
"RESPONSE_TIME": 17
Unit Vectors
The unitize function scales vectors to a magnitude of 1. A vector with a magnitude of 1 is known as a unit vector. Unit vectors are preferred when the vector math deals with vector direction rather
than magnitude.
let(a=array(20, 30, 40, 50),
b=array(200, 300, 400, 500),
c=matrix(a, b),
This expression returns the following response:
"result-set": {
"docs": [
"d": [
"EOF": true,
"RESPONSE_TIME": 6
Distance and Distance Measures
The distance function computes the distance for two numeric arrays or a distance matrix for the columns of a matrix.
There are five distance measure functions that return a function that performs the actual distance calculation:
• euclidean (default)
• manhattan
• canberra
• earthMovers
• haversineMeters (Geospatial distance measure)
The distance measure functions can be used with all machine learning functions that support distance measures.
Below is an example for computing Euclidean distance for two numeric arrays:
let(a=array(20, 30, 40, 50),
b=array(21, 29, 41, 49),
c=distance(a, b))
This expression returns the following response:
"result-set": {
"docs": [
"c": 2
"EOF": true,
"RESPONSE_TIME": 0
Below the distance is calculated using Manahattan distance.
let(a=array(20, 30, 40, 50),
b=array(21, 29, 41, 49),
c=distance(a, b, manhattan()))
This expression returns the following response:
"result-set": {
"docs": [
"c": 4
"EOF": true,
"RESPONSE_TIME": 1
Below is an example for computing a distance matrix for columns of a matrix:
let(a=array(20, 30, 40),
b=array(21, 29, 41),
c=array(31, 40, 50),
d=matrix(a, b, c),
This expression returns the following response:
"result-set": {
"docs": [
"e": [
"EOF": true,
"RESPONSE_TIME": 24
K-Means Clustering
The kmeans functions performs k-means clustering of the rows of a matrix. Once the clustering has been completed there are a number of useful functions available for examining the clusters and
The examples below cluster term vectors. The section Text Analysis and Term Vectors offers a full explanation of these features.
Centroid Features
In the example below the kmeans function is used to cluster a result set from the Enron email data-set and then the top features are extracted from the cluster centroids.
let(a=select(random(enron, q="body:oil", rows="500", fl="id, body"),
analyze(body, body_bigram) as terms),
b=termVectors(a, maxDocFreq=.10, minDocFreq=.05, minTermLength=14, exclude="_,copyright"),
c=kmeans(b, 5),
e=topFeatures(d, 5))
Let’s look at what data is assigned to each variable:
1 a: The random function returns a sample of 500 documents from the "enron" collection that match the query "body:oil". The select function selects the id and and annotates each tuple with the
analyzed bigram terms from the body field.
2 b: The termVectors function creates a TF-IDF term vector matrix from the tuples stored in variable a. Each row in the matrix represents a document. The columns of the matrix are the bigram terms
that were attached to each tuple.
3 c: The kmeans function clusters the rows of the matrix into 5 clusters. The k-means clustering is performed using the Euclidean distance measure.
4 d: The getCentroids function returns a matrix of cluster centroids. Each row in the matrix is a centroid from one of the 5 clusters. The columns of the matrix are the same bigrams terms of the term
vector matrix.
5 e: The topFeatures function returns the column labels for the top 5 features of each centroid in the matrix. This returns the top 5 bigram terms for each centroid.
This expression returns the following response:
"result-set": {
"docs": [
"e": [
"enron enronxgate",
"north american",
"energy services",
"conference call",
"power generation"
"financial times",
"chief financial",
"financial officer",
"exchange commission",
"houston chronicle"
"southern california",
"california edison",
"public utilities",
"utilities commission",
"rate increases"
"rolling blackouts",
"public utilities",
"electricity prices",
"federal energy",
"price controls"
"california edison",
"regulatory commission",
"southern california",
"federal energy",
"power generators"
"EOF": true,
"RESPONSE_TIME": 982
Cluster Features
The example below examines the top features of a specific cluster. This example uses the same techniques as the centroids example but the top features are extracted from a cluster rather than the
let(a=select(random(collection3, q="body:oil", rows="500", fl="id, body"),
analyze(body, body_bigram) as terms),
b=termVectors(a, maxDocFreq=.09, minDocFreq=.03, minTermLength=14, exclude="_,copyright"),
c=kmeans(b, 25),
d=getCluster(c, 0),
e=topFeatures(d, 4))
1 The getCluster function returns a cluster by its index. Each cluster is a matrix containing term vectors that have been clustered together based on their features.
2 The topFeatures function is used to extract the top 4 features from each term vector in the cluster.
This expression returns the following response:
"result-set": {
"docs": [
"e": [
"electricity board",
"maharashtra state",
"power purchase",
"state electricity",
"reserved enron"
"electricity board",
"maharashtra state",
"state electricity",
"purchase agreement",
"independent power"
"maharashtra state",
"reserved enron",
"federal government",
"state government",
"dabhol project"
"purchase agreement",
"power purchase",
"electricity board",
"maharashtra state",
"state government"
"investment grade",
"portland general",
"general electric",
"holding company",
"transmission lines"
"state government",
"state electricity",
"purchase agreement",
"electricity board",
"maharashtra state"
"electricity board",
"state electricity",
"energy management",
"maharashtra state",
"energy markets"
"electricity board",
"maharashtra state",
"state electricity",
"state government",
"second quarter"
"EOF": true,
"RESPONSE_TIME": 978
Multi K-Means Clustering
K-means clustering will produce different results depending on the initial placement of the centroids. K-means is fast enough that multiple trials can be performed and the best outcome selected.
The multiKmeans function runs the k-means clustering algorithm for a given number of trials and selects the best result based on which trial produces the lowest intra-cluster variance.
The example below is identical to centroids example except that it uses multiKmeans with 100 trials, rather than a single trial of the kmeans function.
let(a=select(random(collection3, q="body:oil", rows="500", fl="id, body"),
analyze(body, body_bigram) as terms),
b=termVectors(a, maxDocFreq=.09, minDocFreq=.03, minTermLength=14, exclude="_,copyright"),
c=multiKmeans(b, 5, 100),
e=topFeatures(d, 5))
This expression returns the following response:
"result-set": {
"docs": [
"e": [
"enron enronxgate",
"energy trading",
"energy markets",
"energy services",
"unleaded gasoline"
"maharashtra state",
"electricity board",
"state electricity",
"energy trading",
"chief financial"
"price controls",
"electricity prices",
"francisco chronicle",
"wholesale electricity",
"power generators"
"southern california",
"california edison",
"public utilities",
"francisco chronicle",
"utilities commission"
"california edison",
"power purchases",
"system operator",
"term contracts",
"independent system"
"EOF": true,
"RESPONSE_TIME": 1182
Fuzzy K-Means Clustering
The fuzzyKmeans function is a soft clustering algorithm which allows vectors to be assigned to more then one cluster. The fuzziness parameter is a value between 1 and 2 that determines how fuzzy to
make the cluster assignment.
After the clustering has been performed the getMembershipMatrix function can be called on the clustering result to return a matrix describing which clusters each vector belongs to. There is a row in
the matrix for each vector that was clustered. There is a column in the matrix for each cluster. The values in the columns are the probability that the vector belonged to the specific cluster.
A simple example will make this more clear. In the example below 300 documents are analyzed and then turned into a term vector matrix. Then the fuzzyKmeans function clusters the term vectors into 12
clusters with a fuzziness factor of 1.25.
let(a=select(random(collection3, q="body:oil", rows="300", fl="id, body"),
analyze(body, body_bigram) as terms),
b=termVectors(a, maxDocFreq=.09, minDocFreq=.03, minTermLength=14, exclude="_,copyright"),
c=fuzzyKmeans(b, 12, fuzziness=1.25),
e=rowAt(d, 0),
f=precision(e, 5))
1 The getMembershipMatrix function is used to return the membership matrix;
2 and the first row of membership matrix is retrieved with the rowAt function.
3 The precision function is then applied to the first row of the matrix to make it easier to read.
This expression returns a single vector representing the cluster membership probabilities for the first term vector. Notice that the term vector has the highest association with the 12^th cluster,
but also has significant associations with the 3^rd, 5^th, 6^th and 7^th clusters:
"result-set": {
"docs": [
"f": [
"EOF": true,
"RESPONSE_TIME": 2157
K-Nearest Neighbor (KNN)
The knn function searches the rows of a matrix for the k-nearest neighbors of a search vector. The knn function returns a matrix of the k-nearest neighbors.
The knn function supports changing of the distance measure by providing one of these distance measure functions as the fourth parameter:
• euclidean (Default)
• manhattan
• canberra
• earthMovers
The example below builds on the clustering examples to demonstrate the knn function.
let(a=select(random(collection3, q="body:oil", rows="500", fl="id, body"),
analyze(body, body_bigram) as terms),
b=termVectors(a, maxDocFreq=.09, minDocFreq=.03, minTermLength=14, exclude="_,copyright"),
c=multiKmeans(b, 5, 100),
e=rowAt(d, 0),
g=knn(b, e, 3),
h=topFeatures(g, 4))
1 In the example, the centroids matrix is set to variable d.
2 The first centroid vector is selected from the matrix with the rowAt function.
3 Then the knn function is used to find the 3 nearest neighbors to the centroid vector in the term vector matrix (variable b).
4 The topFeatures function is used to request the top 4 featurs of the term vectors in the knn matrix.
The knn function returns a matrix with the 3 nearest neighbors based on the default distance measure which is euclidean. Finally, the top 4 features of the term vectors in the nearest neighbor matrix
are returned:
"result-set": {
"docs": [
"h": [
"california power",
"electricity supply",
"concerned about",
"companies like"
"maharashtra state",
"california power",
"electricity board",
"alternative energy"
"electricity board",
"maharashtra state",
"state electricity",
"houston chronicle"
"EOF": true,
"RESPONSE_TIME": 1243
K-Nearest Neighbor Regression
K-nearest neighbor regression is a non-linear, multi-variate regression method. Knn regression is a lazy learning technique which means it does not fit a model to the training set in advance. Instead
the entire training set of observations and outcomes are held in memory and predictions are made by averaging the outcomes of the k-nearest neighbors.
The knnRegress function prepares the training set for use with the predict function.
Below is an example of the knnRegress function. In this example 10,000 random samples are taken, each containing the variables filesize_d, service_d and response_d. The pairs of filesize_d and
service_d will be used to predict the value of response_d.
let(samples=random(collection1, q="*:*", rows="10000", fl="filesize_d, service_d, response_d"),
filesizes=col(samples, filesize_d),
serviceLevels=col(samples, service_d),
outcomes=col(samples, response_d),
observations=transpose(matrix(filesizes, serviceLevels)),
lazyModel=knnRegress(observations, outcomes , 5))
This expression returns the following response. Notice that knnRegress returns a tuple describing the regression inputs:
"result-set": {
"docs": [
"lazyModel": {
"features": 2,
"robust": false,
"distance": "EuclideanDistance",
"observations": 10000,
"scale": false,
"k": 5
"EOF": true,
"RESPONSE_TIME": 170
Prediction and Residuals
The output of knnRegress can be used with the predict function like other regression models.
In the example below the predict function is used to predict results for the original training data. The sumSq of the residuals is then calculated.
let(samples=random(collection1, q="*:*", rows="10000", fl="filesize_d, service_d, response_d"),
filesizes=col(samples, filesize_d),
serviceLevels=col(samples, service_d),
outcomes=col(samples, response_d),
observations=transpose(matrix(filesizes, serviceLevels)),
lazyModel=knnRegress(observations, outcomes , 5),
predictions=predict(lazyModel, observations),
residuals=ebeSubtract(outcomes, predictions),
This expression returns the following response:
"result-set": {
"docs": [
"sumSqErr": 1920290.1204126712
"EOF": true,
"RESPONSE_TIME": 3796
Setting Feature Scaling
If the features in the observation matrix are not in the same scale then the larger features will carry more weight in the distance calculation then the smaller features. This can greatly impact the
accuracy of the prediction. The knnRegress function has a scale parameter which can be set to true to automatically scale the features in the same range.
The example below shows knnRegress with feature scaling turned on.
Notice that when feature scaling is turned on the sumSqErr in the output is much lower. This shows how much more accurate the predictions are when feature scaling is turned on in this particular
example. This is because the filesize_d feature is significantly larger then the service_d feature.
let(samples=random(collection1, q="*:*", rows="10000", fl="filesize_d, service_d, response_d"),
filesizes=col(samples, filesize_d),
serviceLevels=col(samples, service_d),
outcomes=col(samples, response_d),
observations=transpose(matrix(filesizes, serviceLevels)),
lazyModel=knnRegress(observations, outcomes , 5, scale=true),
predictions=predict(lazyModel, observations),
residuals=ebeSubtract(outcomes, predictions),
This expression returns the following response:
"result-set": {
"docs": [
"sumSqErr": 4076.794951120683
"EOF": true,
"RESPONSE_TIME": 3790
Setting Robust Regression
The default prediction approach is to take the mean of the outcomes of the k-nearest neighbors. If the outcomes contain outliers the mean value can be skewed. Setting the robust parameter to true
will take the median outcome of the k-nearest neighbors. This provides a regression prediction that is robust to outliers.
Setting the Distance Measure
The distance measure can be changed for the k-nearest neighbor search by adding a distance measure function to the knnRegress parameters. Below is an example using manhattan distance.
let(samples=random(collection1, q="*:*", rows="10000", fl="filesize_d, service_d, response_d"),
filesizes=col(samples, filesize_d),
serviceLevels=col(samples, service_d),
outcomes=col(samples, response_d),
observations=transpose(matrix(filesizes, serviceLevels)),
lazyModel=knnRegress(observations, outcomes, 5, manhattan(), scale=true),
predictions=predict(lazyModel, observations),
residuals=ebeSubtract(outcomes, predictions),
This expression returns the following response:
"result-set": {
"docs": [
"sumSqErr": 4761.221942288098
"EOF": true,
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Autodesk Fusion 360 (4th Edition)
Autodesk Fusion 360: A Power Guide for Beginners and Intermediate Users (4th Edition) textbook has been designed for instructor-led courses as well as self-paced learning. It is intended to help
engineers and designers, interested in learning Fusion 360, to create 3D mechanical designs. This textbook is a great help for new Fusion 360 users and a great teaching aid for classroom training.
This textbook consists of 14 chapters, a total of 750 pages covering major workspaces of Fusion 360 such as DESIGN, ANIMATION, and DRAWING. | {"url":"https://cadartifex.com/cad-textbooks/autodesk-fusion-360/autodesk-fusion-360-4th-edition-books","timestamp":"2024-11-10T10:36:46Z","content_type":"text/html","content_length":"62318","record_id":"<urn:uuid:01068f7c-8170-4823-a139-7d6feb7af9ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00280.warc.gz"} |
The misguided idea to do math to numbers rolled on dice
While the “d20 system” as introduced by D&D third edition wasn’t the first game to have this not-so-good feature as a core part of the design (the original “d6” version of Star Wars has the same
problem, requiring you to add dice together), it’s well known and can serve as an example of what I’m talking about.
In 3e, 4E and 5e D&D, you roll a d20, then add a number to that die roll, and then compare to a target number. In its worst iteration, that target number is even kept secret by the Dungeon Master,
but even when rolled against an open target number, this has a couple of problems.
Math all the time
Since the d20 result number is variable, you’re afforded to re-add this for every single strike. If instead it was the target number that changed, then you’d only need to do it once. The most
straight-forward example of that is the house variant of subtracting your to-hit bonus from the enemy AC to “know what you need to roll”. For example, you’re bashing a poor AC 13 skeleton and you
have a +3 to hit? You need to roll ten or higher. And now you know that for every strike against those rattling bones, you need to roll a ten or higher. Math once instead of math all day.
Math when you should be the most excited
The math-all-the-time isn’t even the biggest problem. Some people are happy to do nothing but add add add all day long it seems. And against a weak foe it might be one-hit-one-kiss anyway i.e. you
only needed that one strike.
But the timing of the math is what bugs me the most. You roll and then you add. I wanna have all the math behind me when I roll so I only know what I need to look for! Fortune at the very end of
The unnecessary d20 system
D20 didn’t even change anything because you could already roll, add their armor class to your roll and compare that to THAC0 and you were done. You could already do d20 + small number vs big number.
Not that I like doing that, I hate it, as you could see above.
The two official ways to THAC0
In Rules Cyclopedia, on page 108, it says to subtract the enemy AC from THAC0 so you know what to roll. Good. There’s also a less good version on page 9 where it says to subtract your rolled roll
from your THAC0 to find out what AC you hit. Awful. All the same problems as in 3e but now you’ve got to subtract all day long.
Gee, thanks for the bonus
Even though one of the THAC0 approaches, specifically “subtract their AC from your THAC0 and know that before you roll”, makes sense, what does not make sense is that on top of that you still get
more bonuses sometimes and if so, those bonuses do go on the die roll instead of lowering the target number. I can’t even with this.
Target 20 and other mistakes
There’s an even bigger problem called Target 20, which is for systems that use THAC0 and descending armor class where you need to roll your die and then add add both of those things together and then
compare that to twenty. So it’s all of the worst of the previous add-after-rolling methods but with one more term, congratulations. And unfortunately it’s made its way into many OSR games.
Static targets
I’m not sure there even needs to be three terms. With saving throws, you just roll over a specific number and you’re done. In Apocalypse World the targets numbers are always seven and ten, maybe
that’s fine? But then again you do need to add your Sharp and Hard and stuff to the dice.
Suldokar’s Wake
I accidentally invented a method which I thought was only good for cards, we used them for initiative cards, but turns out it’s good for dice too, I didn’t think of that at all.
One game that uses that method is Christian Mehrstam’s Suldokar’s Wake. Rolling high is good but so is rolling low. With old school stats, roll over your THAC0 or under their AC to hit; with new
school stats, roll over their AC or under your to-hit-bonus to hit. Finally I can speedrun that Castle Ravenloft boardgame which uses 4e-style stats.
There’s also the idea of having a low decreasing attack roll value (like how saves work in the old game) and a small increasing AC value; add them together and roll over that. But that needs
conversion from every system which is no fun. If only that had been the standard, that woulda been my favorite method. | {"url":"https://idiomdrottning.org/dice-math","timestamp":"2024-11-03T15:47:20Z","content_type":"application/xhtml+xml","content_length":"8527","record_id":"<urn:uuid:62363cfd-92d2-441e-a8ca-fc01e4278eb4>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00702.warc.gz"} |
Free Printable Math Worksheets | Math Resource Studio
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The free math worksheets below were generated with Math Resource Studio and provide practice in number operations, number concepts, fractions, numeration, time, measurement, money, algebra and more.
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Feasibility of numerical simulation methods on the Cold Gas Dynamic Spray (CGDS) Deposition process for ductile materials
Issue Manufacturing Rev.
Volume 7, 2020
Article Number 24
Number of page(s) 15
DOI https://doi.org/10.1051/mfreview/2020023
Published online 17 August 2020
Manufacturing Rev.
, 24 (2020)
Research Article
Feasibility of numerical simulation methods on the Cold Gas Dynamic Spray (CGDS) Deposition process for ductile materials
Department of Mechanical Engineering Science, University of Johannesburg, Gauteng, 2006, South Africa
^* e-mail: tjen@uj.ac.za
Received: 24 June 2020
Accepted: 24 July 2020
The techniques of cold gas dynamic spray (CGDS) coating involve the deposition of solid, high speed micron to nano particles onto a substrate. In contrast to a thermal spray, CGDS does not melt
particles to retain their physico-chemical properties. There have been many advantages in developing microscopic analysis of deformation mechanisms with numerical simulation methods. Therefore, this
study focuses on four cardinal numerical methods of analysis which are: Lagrangian, Smoothed Particles Hydrodynamics (SPH), Arbitrary Lagrangian-Eulerian (ALE), and Coupled Eulerian-Lagrangian (CEL)
to examine the Cold Gas Dynamic Spray (CGDS) deposition system by simulating and analyzing the contact/impact problem at deformation zone using ductile materials. The details of these four numerical
approaches are explained with some aspects of analysis procedure, model description, material model, boundary conditions, contact algorithm and mesh refinement. It can be observed that the material
of the particle greatly influences the deposition and the deformation than the material of the substrate. Concerning the particle, a higher-density material such as Cu has a higher initial kinetic
energy, which leads to a larger contact area, a longer contact time and, therefore, better bonding between the particle and the substrate. All the numerical methods studied, however, can be used to
analyze the contact/impact problem at deformation zone during cold gas dynamic spray process.
Key words: Numerical models / deformation / plastic strain / CGDS
© S.T. Oyinbo and T.-C. Jen, Published by EDP Sciences 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The cold gas dynamic spray (CGDS) mechanism is based on the solid-state deposition technique. CGDS is suitable for various engineering applications, which include composites, ceramics, metals and
polymers. The gradual shear and plastic deformation generated by impact velocity of the accelerated particle is accomplished by the expansion of pressurized gases through a nozzle thus, the
metallurgical coalescence is produced between the particle and the substrate [1,2]. Plastic deformation of particles occurs as it impacts on the target surface to form a uniform layer. Only when the
velocity of the sprayed material reaches a pre- defined velocity called critical velocity under given operating conditions can the particles/substrate bonding occur [1,3]. Nanoscale cold spraying is
a potential technology for depositing or coating nanostructured materials on the surface of the substrate without affecting its properties or structure significantly [3–5,38]. The technology finds
enormous applications including metal matrix composite (MMC), metallic, ceramic, and plastic coatings in vital engineering fields [3,5–7]. The suitability of the material for CGDS depends on its
physical and mechanical properties, including density, melting temperature, melting hardness and material hardness [4,5,8]. Relatively low yield material such as zinc, aluminium and copper are
considered desirable because they display comparatively greater softening at high temperature [5,9,10], whereas high resistance materials are not perfect for CGDS, because of the lack of adequate
energy available for deformation.
The technique of finite element analysis has been the focus of many researchers among several models of material impact phenomena developed to investigate the deformation mechanism due to its
modelling capacity of material models, complex geometries and contact algorithms. In cold gas dynamic spraying, the particle/substrate impact can be termed a high-velocity impact process which can be
handled by the Lagrangian method [11]. The Lagrangian numerical algorithm has been the focus of many researchers. 3D and axisymmetric models for particle and substrate impact was first used by Assadi
et al. [12], to establish the impact phenomenon by using ABAQUS/Explicit version 6.2–1. Yin et al. [13], Li et al. [14], and Grujicic et al. [15] also investigated the behaviour of particles and
substrate during impact with the application of Lagrangian analysis model. Li et al. [14] were the first among the researchers that incorporated material damage mechanisms in the model as well as
Lagrangian adaptive mesh domains (Lagrangian-Eulerian method) to control the excessive element distortion and mesh size control respectively. Xie et al. [16], provides the basic understanding of
particle/substrate impact during cold spray by explicitly examined the different numerical model by modeling high-velocity impacts of spherical particles onto a flat substrate under various
conditions. For the first time, they proposed the coupled Eulerian–Lagrangian (CEL) numerical approach as a means of solving the high-strain rate deformation problem.
Research to study the particles and substrate impact behaviour was carried out with Cu as the particle material using Eulerian Formulation, another ABAQUS/Explicit model [17]. At about 290m/s
minimum velocity, a jet was discovered to have formed and had maximum reached plastic strain (PEEQ). Therefore, at a velocity below 290m/s, no jet could occur. This velocity was assumed to be a
critical velocity. Jet formation discontinued and the material splashed at a speed greater than 290–400m/s. A critical velocity could, therefore, be predicted in accordance with the theoretical
analysis of jet morphology as a prediction tool by the Eulerian model [18–20].
Yildirim et al. [21] use several model reference domains in ABAQUS/Explicit package to systematically study the impact of a single particle on the semi-infinite substrate at an initial velocity of
100–700m/s. The 2-dimensional axisymmetric structure was used in the case of Lagrangian and Arbitrary Lagrangian-Eulerian (ALE) model for the impact process, and one-quarter symmetry 3D model was
used to study the material failure mechanism. The temperature was initiated at 293K (room temperature). The fictional formulation with surface-to-surface interaction was also incorporated in the 2D
model and general contact algorithm for the 3D model with the friction coefficient of 0.3 at the particle and substrate interface. It was discovered that interpolation error occurs in the ALE
adaptive remeshing technique with a significant decrease in equivalent plastic strain.
Another numerical approach in investigating the impact behaviour of cold spray particles is Smoothed Particle Hydrodynamics (SPH) [22]. Manap et al. [23] and Yildirim et al. [21] used the SPH
approach for critical velocity prediction during the CS deposition process. Furthermore, the SPH method is appropriate for the multiple-particles impact process due to the appropriate solution
techniques regarding the interface contact problem and its unique meshless feature [24]. However, the problem of tensile instability in SPH approach and the lack of interaction between the particles
can lead to large tensile deformation which is a significant numerical problem.
With the introduction of a modern numerical system and complex representations of finite elements, the damage was induced to aeronautical structures by Smojver and Ivančevic [25] to predict the
damages induced of a bird strike. Coupled Eulerian-Lagrangian (CEL), a modern finite element technique was however used to model and solved the soft body impacts. For modelled bird replacement
material hydrodynamic reactions, material volumetric force and pressure-density ratio by the material equation of state (EOS) were used. The lagrangian bird model and experimental results are used to
verify observations from the bird model with the CEL approach.
Gang et al. [26] explored the potential of the CEL numerical method to address geotechnical problems. Their studies have shown that CEL is capable of resolving difficult problems which FEM considers
difficult to solve. In order to further explore the capacity of the CEL, a pile installation was simulated by a CEL approach and it was discovered that the CEL is better suitable for studying the
pile influence on the soil-preceding structure's relationship, namely that the friction values are high when the simulation results and measuring data are considered. Therefore, they believe that due
to the quality of parallelization, the CEL method achieves successful results.
This study presents four cardinal numerical methods of analysis which are: Lagrangian, Smoothed Particles Hydrodynamics (SPH), Arbitrary Lagrangian-Eulerian (ALE), and Coupled Eulerian-Lagrangian
(CEL) to examine the Cold Gas Dynamic Spray (CGDS) deposition system by simulating and analyzing the contact/impact problem at deformation zone using ductile materials. This was done with the aim of
accomplishing a qualitative understanding of when the particle deforms plastically during the cold gas dynamic spray process and find a way of addressing high-strain-rate dynamic problems of cold
sprayed particles and substrate.
2 Problem description
This study investigates the feasibility of four numerical analysis models to simulate both the single and multiple particle impact on a deformable substrate during the CGDS process. Based on Abaqus
Analysis User's Manual [27], an explicit-finite element analysis program was adopted for the analysis. A 3D model was established for deformable Copper (Cu) particle and deformable Aluminum (Al)
substrate. The 500m/s initial impact velocity used for this simulation of Cu/Al impact is below the critical velocity of copper/aluminum system. Note that 507m/s is the approximated critical
velocity for Cu/Al by using shear localization analysis [15]. Because of SEM observation, the morphology of copper particle was taken to be spherical for the numerical model (Fig. 1) and its
corresponding mean particle size was taken to be 10μm in the CGDS process. The penalty formulation and general contact explicit available in ABAQUS/Explicit FEA program were used to describe the
relative motion between the particle and substrate surface during cold gas dynamic spray process. The coefficient of friction was taken to be 0.3 for all the analyses [10,21]. The initial temperature
is set to be 25°C in all the cases of Cu/Al impacts [28] with a fixed simulation time of 60ns which is enough to study the contact process [10,16,29,30]. The cylindrical model was used for the
substrate, the height and the radius were 8R and 16R respectively, where R is the diameter of the particle.
The Johnson-Cook plasticity model offers a definition of material movement for both the particles and substrate [29]. The flow stress (σ) functions as illustrated in equation (1) are the strain
hardening, strain rate hardening and temperature softening where the working hardening exponent (n) and plastic strain is denoted by $ε$, the dimensionless plastic strain rate is the ratio of $ε / ε
0$, $ε 0$=1.0 s^−1 and the substance constants A, B, C and m are shown in Table 1. T, T[m] and T [0] are the measured, melting and reference temperature respectively$σ = ( A + B ε n ) [ 1 + Cln ( 1
+ ε ε 0 ) ] ( 1 − [ T − T 0 T m − T 0 ] m )$(1)
The thermal reaction study is carried out using the thermal conductivity properties and specific heat. Table 1 shows the properties of the materials used for the analysis [21,31,32]
Fig. 1
Schematic diagram illustrating 3D model used is this study for (a) Lagrangian, SPH and ALE method (b) CEL approach (c) Boundary conditions for the 3D model.
Table 1
Material model for the numerical analysis.
3 Computational procedure
3.1 Lagrangian model description
Abaqus/explicit program [27], was used to model the impact behaviour of Cu particle upon Al substrate in the Lagrangian simulation. The penalty formulation and general contact explicit available in
ABAQUS/Explicit FEA program were used to describe the relative motion between the particle and substrate surface during cold gas dynamic spray process. The following analysis was also used for the
Lagrangian domain; an 8-node thermally coupled brick (C3D8RT), reduced integration, hourglass control, trilinear displacement and temperature [27]. The mesh size of 0.0003mm was used, i.e. the
resolution of 1/100 particle diameter. Hexahedral meshing elements have been used. The application of boundary conditions with respect to the x–z plane involves symmetry, the bottom of the substrate
with zero displacements, and the boundary conditions Lagrangian parts are shown in Figure 1c. Where particle initial velocity (v=500m/s), nodal rotation (r) and nodal displacement (u) are defined
with the coordinates x, y z. The particle and substrate initial temperature of 25°C was used for all the calculations.
The conservative equations outlined in equations (4)–(7) represent the equation of mass, energy and momentum derived by the spatial time derivative approach [33]. Where ρ, σ, u, E and e are the
density, Cauchy stress, material velocity, total energy per unit volume and the internal energy respectively. The addition of the internal energy (e) and kinetic energy gives the total energy E (Eq.
7).$d ρ d t = − ρ ∂ u ∂ x$(2) $d E d t = 1 ρ σ i j ∂ u ∂ x$(3) $d u d t = 1 ∂ ρ ∂ σ y ∂ x$(4) $E = e + 1 2 ρ u . v$(5)
3.2 Smoothed particle hydrodynamics (SPH) model description
Another numerical model used in this study is Smoothed particle hydrodynamics (SPH). These methods belong to the meshless (or mesh-free)- family in which elements and nodes are not defined in the
domain against the normal practice in the analysis of the finite element method; instead, the given body is represented by a collection of points. These nodes are generally called pseudo-particles or
particles in smoothed particle hydrodynamics. Smoothed particle hydrodynamics is a modelling scheme in a fully Lagrangian domain in which continuum equations of a prescribed set are discretized by
directly interpolating the properties over the solution region at a discrete set of points without necessarily define a spatial mesh. All the theory of energy conservation such as conservation of
momentum, conservation of mass and conservation of energy are all defined with the Lagrangian domain. The interaction that occurs between the neighbouring pseudo-particles and current pseudo-particle
over time causes the material to change and the field approximation at each step is done on the basis of locally distributed neighbouring particles. As the word ‘particle’ might suggest, the discrete
particles (spheres) collide in compression with each other or in tension exhibiting cohesive-like behaviour. The SPH method at its core does not base on such phenomenon, rather, a method of
discretization of continuum partial differential equations. In this analysis, the SPH reference frame was used only to model the particle because its deformation is much more than that of the
substrate, whereas the substrate was modelled using the Lagrangian reference frame. The Explicit dynamic stress-displacement analysis was used in this method because the coupled mechanical −thermal
procedure in Lagrangian domain does not support the PC3D element type SPH approach. The element type for the substrate is an 8-node thermally coupled brick (C3D8RT), reduced integration, hourglass
control, trilinear displacement and temperature, while the PC3D unique ABAQUS/Explicit element type was used for the particle in the SPH analysis.
The interpolation theory is the foundation of the smoothed particle hydrodynamics (SPH) numerical approach. The interpolation theory transforms the continuum fluid dynamics conservation laws into an
integral equation. Smoothing kernels (W) can be used at a certain position to obtain the kernel approximation of a function f(R ^′) by integration over the computational domain [34].$f ( R ′ ) = ∫ f
( R ′ ) W ( R − R ′ , h ) d R ′$(6) $∫ W ( R − R ′ , h ) d R ′ = 1$(7) $lim h → 0 W ( R − R ′ , h ) = δ ( R − R ′ )$(8) $W ( r , h ) = 1 N ( δ ) h δ { 1 − 3 2 r 2 + r 3 | r | ≤ 1 1 4 ( 2 − r ) 3 1 <
| r | ≤ 2 0 2 < | r |$(9)where weighting function is W with respect to support scale (h). Equation (10) is the Delta-function property when the limit of smoothing length is tending to zero. Kernel
functions have many possible choices. The third-order B-spline function was selected for this analysis to reduce the code frequency and the number of interactions of the particles. If the relative
displacement (R) is defined between points Rnd R ^′, r=|R−R ^′|/h then equation (11) gives the B-spine function. The function of normalization N(δ) is given to be {3/2, 7/10π, π, 31/5π ^2};δ
=1, ..., 5. Herefore, the fluid flow momentum equation is given by equation (12):$d v a d t = g − ∑ b m b [ ( p a ρ a 2 + p b ρ b 2 ) − ξ ρ a ρ b 4 μ a μ b ( μ a + μ b ) v a b R a b R a b 2 + η 2 ]
× ∇ a W a b .$(10)were the particle velocity, viscosity, density and pressure are respectively represented by v [ a ], μ [ a ], ρ [ a ], and p [ a ] for particle, a. For particle b, the velocity,
viscosity, density, mass and pressure are represented by v [ b ], μ [ b ], ρ [ b ], m [ b ] and p [ b ] respectively. The position vector and interpolation kernel are denoted by R [ ab ]=R [ a ]−
R [ b ] form particle a to b and W [ ab ]=W(R [ ab ], h) respectively. Gravity vector and viscous term factor are denoted by g and ξ. At R [ ab ]=0, smoothing of singularity is η.
3D SPH model was employed in this analysis. There is no special treatment for contact boundary condition in the SPH approach. The bottom surface of substrate is constrained by the PINNED boundary
condition (the substrate is constraint in both x, y and z- direction). The condition of SPH particles geometric proximity is detected automatically to calculate the contact process of particle/
substrate interface. The SPH interact with each other by meeting this condition following the requirement of boundary compatibility.
3.3 Arbitrary Lagrangian-Eulerian (ALE) model description
The Arbitrary Lagrangian-Eulerian (ALE) analysis is used to study large deformations of the transient problem by using Lagrangian adaptive mesh domains. In Figure 1a, the domain of Lagrangian
adaptive mesh is created by the blue line region so that the orientation of the material present in this domain will follow the material flow path, which validates the most structural analyses of
physical interpretation. The computational cost will be reduced by defining the domains of adaptive mesh as a fraction of the entire domain. The material direction is always proportional to the mesh
and normal to the boundary on the Lagrangian domain boundary so that at all times the material domain is covered by the mesh. A smoother, new mesh is made in an adaptive meshing increase by
iteratively sweeping over the domain of the adaptive mesh. Element distortion is reduced as each mesh sweep in the domain by relocating the nodes based on the current location of the neighbouring
nodes as well as the elements. In each increment of adaptive meshing, the adaptive meshing intensity increases when the number of sweeps increases. The sole objective of the mesh smoothing method in
the adaptive mesh domain is to improve element aspect ratios by minimizing mesh distortion expense of diffusing initial mesh gradation. Adaptive meshing robustness in ABAQUS/Explicit is achieved by
adopting an enhanced algorithm based on the geometry of the evolving element.
2D axisymmetric ALE model was employed in this analysis. CAX4RT: a 4-node bilinear displacement, viscoelastic hourglass control, axisymmetric quadrilateral, reduced integration, thermally coupled and
the temperature was used for the ALE domain. Here, 0.0001μm was used as the mesh size. Quadrilateral elements were used for the meshing. The number of mesh sweeps and frequency for this analysis is
5 and 15 respectively.
3.4 Coupled Eulerian-Lagrangian model description
Another numerical model used in this study is the Coupled Eulerian-Lagrangian approach. This method is also used to study large deformations of transient problems. Coupled temperature-displacement
elements for the Eulerian domain (EC3D8RT) and fully coupled thermal-stress analysis for the Lagrangian domain (C3D8RT) are used in this analysis. The volume-of-fluid method is the foundation of
Coupled Eulerian-Lagrangian model for the implementation of the Eulerian part in ABAQUS/Explicit. Within each element, this method computes the Eulerian Volume-Fraction (EVF) as the material flows
and tracked in the mesh. Generally, one is the volume fraction of an element if it is completely filled with material and if the element contains no material, its volume fraction is zero. More than
one material can be present in a Eulerian element simultaneously. If all the volume fractions of the material in an element are sum together and is less than one, automatically, the remaining
elements will be occupied by ‘void’ material. There is no mass and weight for void material. The 3D model is implemented for this numerical approach. The geometry, boundary conditions, analysis
procedure, mesh and interaction are the same as that of the pure Lagrangian method. In the case of Eulerian frame, prescribed velocity along x, y and z-direction are constraint.
4 Results and discussion
4.1 Lagrangian numerical approach for single-particle impact model
At 50m/s initial impact velocity, the equivalent plastic strain evolution (PEEQ) for Cu/Al impact after the simulation time of 30ns is shown in Figure 2. The peak value of PEEQ during the
calculation is found to be at the interfacial zone between the particle and substrate. As predicted by the Lagrangian method, the single spherical particle impacts the flat substrate, becomes
flattens, and generate a crater at the edge of the contact region. At the outer interfacial region, there is an observation of an intensive plastic deformation between the Cu particle and Al
substrate, where the value of PEEQ exceeds 2.0.
The temperature (TEMP) distribution at 500m/s initial impact velocity for Cu/Al impact after the simulation time of 30ns is shown in Figure 3. The process of deformation in cold gas dynamic spray
is purely adiabatic [35]. Temperature distribution of material is dependent on the plastic deformation, and since the deformation of the particle is largely observed with a maximum plastic strain
close to the interfacial region, the maximum TEMP is also found to be located near the interface. The particle and substrate materials formed jet at the interfacial region as the deformation of the
particle and substrate proceeds. The appearance of the deformed particle is now lens-like shape. As the material of the particle and the substrate deform further, then, more jet builds up.
Figures 2a–d and 3a–d shows the evolution of the jet impact time histories of 30ns, and Figure 4a and b displays the TEMP and PEEQ distribution for the corresponding single-particle The material
true time history impact at the impact time of 60ns. The shear strain and temperature evolution is not enough to clearly predict adiabatic shear instability compared with [16,36] from these figures.
As shown in Figure 4, the substrate equivalent plastic strain evolution is higher than that of the particle, thus higher temperature is obtained.
Fig. 2
Equivalent plastic strain time-evolution at 500m/s Cu/Al impact of (a) 5ns (b) 10ns (c) 20ns (d) 30ns by using the Lagrangian numerical modelling.
Fig. 3
The temperature evolution at 500m/s Cu/Al impact of (a) 5ns (b) 10ns (c) 20ns (d) 30ns by using the Lagrangian numerical modelling.
Fig. 4
The time distribution of (a) temperature (b) PEEQ of Cu/Al impact using the Lagrangian numerical modelling at 500m/s impact velocity.
4.2 Smoothed particle hydrodynamics (SPH) numerical approach for single-particle impact model
SPH deformation pattern for Cu/Al impact is different from Lagrangian and ALE deformation pattern of particle/substrate impact. At the edge of the interfacial zone, there is no material jet of a
particle, and the particle penetration in the substrate is deeper.
At 500m/s initial impact velocity, the evolution of equivalent plastic strain (PEEQ) for Cu/Al impact after the simulation time of 30ns is shown in Figure 5. In these figures, the particle aspect
ratio is decreased, with an increase in the collision time of the contact process, and the width and depth of the substrate increase. The plastic strain observed in the substrate increases greatly
after 5ns, whereas, the maximum equivalent plastic strain continues to approach a horizontal asymptote. A viscous-like resistance created by the excessive deformation can hinder the further
deformation process under high strain rate and high impact velocity. This effect can be attributed to the second term Johnson-cook of the stress-strain law which describes the strain rate hardening.
Figure 6b shows the stress evolution at 25ns impact time which is however different from the PEEQ curve in Figure 6a. When the substrate temperature increases, there is low resistance of the
material to shear flow when thermal softening is considered. This means that if any amount of shear stress is applied as the material approaching melting temperature, the shear strength of the
material will be lost, and it will experience excessive deformation. However, this analysis is isotherm, and the temperature is kept constant of the whole system at 25°C. Moreover, material damage
model is not employed in the material. The Dynamic-Explicit procedure used in this analysis produces much higher PEEQ as clearly observed in Figure 6. The conduction of heat is obvious in the case of
Dynamic-Temperature Displacement-Explicit procedure from the impact zone and conducted into the inner part of the particle and substrate. Although for large models' analysis with extremely
discontinuous events and relatively short times response, Dynamic-Explicit procedure is recommended because of its computational efficiency.
Fig. 5
The distribution of PEEQ of various impact time at 500m/s Cu/Al impact (a) 5ns (b) 10ns (c) 20ns (d) 30ns by using the SPH numerical modelling.
Fig. 6
The evolution of (a) PEEQ (b) stress distribution of Cu/Al impact at 500m/s impact velocity using the SPH numerical modelling.
4.3 Arbitrary Lagrangian-Eulerian (ALE) numerical approach for single-particle impact model
The ALE numerical approach is also used to solve the single-impact problem. The equivalent plastic strain (PEEQ) evolution and temperature distribution at 500m/s initial impact velocity for Cu/Al
impact during the simulation time of 30ns are shown in Figures 7 and 8 respectively. Excessive distortion of mesh does not occur by using this numerical method. The material jet formed in this
analysis is smoother at the interfacial region instead of acute and thin. The value of frequency applied in the analysis is 15 and remeshing sweeps per increment are 5. The absence of adhesion model
makes the particle rebound to occur after 30ns of the impact time. It is observed that the temperature at the outer region of the particle and substrate impact is higher than what the inner part
experiences because of the locally formed plastic deformation around the region surrounding the interface.
The evolution of the particle and substrate equivalent plastic strain (PEEQ) are shown in Figure 9b at 60ns impact time. The plastic strain observed in the substrate increases unreasonably after
10ns, whereas, the maximum equivalent plastic strain continues to approach a horizontal asymptote. However, the monotonical increase in the history of PEEQ until it reaches the plateau is another
feature in a pure Lagrangian approach. The sudden decrease of the PEEQ after reaching the peak value is an unrealistic behaviour and can be attributed to errors of interpolation caused in the
remapping point and surrounding interface of high strain gradients by the adaptive meshing algorithm [16,37]. The deviation of material points and integration points is due to the usage of adaptive
meshing with some features similar to CEL approach. Therefore, the adaptive meshing and material motion composite effect is represented by the motion of the interior mesh of an adaptive mesh domain.
Figure 9a shows the temperature (TEMP) distribution at 60ns after impact. Before 10ns, the TEMP slowly increases, but because of the high strain rate, it increases sharply between 10 and 20ns,
thereafter, there is a gradual decrease as the calculation continues. Because the substrate experiences larger plastic deformation, its temperature is higher than that of the particle. Adaptive
meshing frequency and intensity are the key factors that affect the computational cost and simulation results in this analysis.
Fig. 7
Equivalent plastic strain evolution (PEEQ) at 500m/s Cu/Al impact at different impacting times of (a) 5ns (b) 10ns (c) 20ns (d) 30ns by using the ALE numerical modelling.
Fig. 8
The time evolution of temperature at 500m/s Cu/Al impact of (a) 5ns (b) 10ns (c) 20ns (d) 30ns by using the ALE numerical modelling.
Fig. 9
The time distribution of (a) The temperature (b) The PEEQ of Cu/Al impact at 500m/s impact velocity using the ALE numerical modelling.
4.4 Coupled Eulerian-Lagrangian (CEL) numerical approach for single-particle impact model
The problem of mesh excessive distortion and abnormal deformation can be avoided using the Coupled Eulerian-Lagrangian numerical approach. CEL particle model enhances the modelling of the fluid-like
particle as shown in Figure 10. Although there is still an occurrence of the material jet in this analysis and it has no effect on the completion of the analysis. The interpretation of the results
from CEL should be different from that of a Lagrangian method (Figs. 10 and 11). Since the Eulerian part in CEL analysis is rigid and fixed, any nodal displacements result is meaningless. Therefore,
the calculation of PEEQ of all materials within the element is based on volume fraction weighted average (PEEQVAVG). The value of this volume average is significantly lower whether the adaptive
remeshing is used or not than that of the Lagrangian analysis. The same explanation can as well be applied to the particle temperature (TEMPMAVG) distribution as shown in Figure 11.
The equivalent plastic strain (PEEQ) evolution and temperature distribution at 500m/s initial impact velocity for Cu/Al impact during the simulation time of 20ns for particle and substrate are
shown in Figure 12. The PEEQ evolution in the substrate increases rapidly until it reaches the peak value of 5. The PEEQAVG for the substrate is significantly higher than that of the particle. One of
the CEL analysis shortcomings is the inability to trace the materials history behaviour.
Fig. 10
Evolution of PEEQ and volume average equivalent plastic strain (PEEQVAVG) 500m/s Cu/Al impact by using the CEL numerical modelling.
Fig. 11
Evolution of temperature (TEMP) and mass average temperature (TEMPMAVG) at 500m/s Cu/Al impact by using the CEL numerical modelling.
Fig. 12
The evolution of (a) temperature (b) plastic strain of Cu/Al impact at 500m/s impact velocity using the CEL numerical modelling.
4.5 Comparison of four numerical methods
At 500m/s initial impact velocity, the evolution of equivalent plastic strain (PEEQ) for Cu/Al impact after the simulation time of 60ns is shown in Figure 13 calculated by Lagrangian, SPH, ALE, and
CEL numerical model. As predicted by the Lagrangian method (Fig. 13a), the single spherical particle impacts the flat substrate and becomes flattens and generate a crater at the edge of the contact
region. The particle and substrate materials formed jet at the interfacial region as the deformation of the particle and substrate proceeds. The appearance of the deformed particle is now lens-like
shape. As the material of the particle and the substrate deform further, then, more jet builds up. For the SPH numerical method, as indicated in Figure 13b, there is no material jet of the particle
at the interfacial region edge. The PEEQ distribution in this analysis, when compared to the Lagrangian result, was similar. Excessive distortion of mesh does not occur by using ALE numerical method.
The material jet formed in this analysis is smoother at the interfacial region instead of acute and thin (Fig. 13c). CEL particle model enhances the modelling of the fluid-like particle (Fig. 13d).
Although the occurrence of the material jet in this analysis has no effect on the completion of the analysis, and the particle penetration in the substrate is deeper than other numerical methods.
Figure 14a shows the normalised kinetic energy over the period of 60ns impact time using Cu/Al impact for the four numerical approaches under consideration. In the deposition of Cold Spray process,
energy from initial kinetic energy (ALLKE) is converted into energy saved in the particle/substrate (ALLSE), the energy that deforms the material plastically (ALLPD) and energy that propagates the
stress wave. After plastic deformation of the material, the process is irreversible, and the kinetic energy stored, is the energy which can be recovered during the restitution. This energy is known
as rebound kinetic energy. The initial kinetic energy is much higher than the rebound kinetic energy. That means over 98% of the kinetic energy is converted into internal energy and 2% is converted
into rebound kinetic energy. The kinetic energy which is then produced by rebounded particles has not been monotonous, because, after 20ns, the kinetic energy damped periodically. The recoverable
elastic energy matches this energy. The pattern of the normalized kinetic energy in all cases of numerical approaches agrees well except for the CEL approach. The deviation is due to the different
approach used in the analysis.
The removal of hourglass control (singular modes) is usually done by the associated ‘artificial’ strain energy. The strain energy that will control hourglass deformation will be too much if there is
excessive artificial strain energy during the process. Figure 14b shows how to determine whether the artificial strain energy is excessive or not by comparing the internal energy (ALLIE) with that of
artificial strain energy (ALLAE). Generally, the proportion of when ALLAE divides ALLIE should not be up to 5%. The initial ratios close to 0.0s can be neglected because it is basically noise
produced when a very small number is divided by another, all four numerical approaches have a ratio of less than 4%. The problem of the hourglass can be sufficiently prevented by intrinsic hourglass
control. The fine mesh size used in this analysis is enough to stop zero-energy modes propagation which potentially yields inaccurate results.
Table 2 presents a comparison between the four numerical approaches in term of computational costs. The most efficient approach among the four is the pure Lagrangian approach, and the SPH consumes
more time.
Fig. 13
Effective plastic strain evolution of (a) Lagrangian model (b) SPH model (c) ALE model (d) CEL model of a Cu/Al single particle impact at 500m/s.
Fig. 14
Impact time (a) kinetic energy (b) artificial strain energy for Cu/Al impact for the four numerical approaches at 500m/s.
Table 2
Schematic illustration of computational costs by the four numerical modellings.
5 Conclusion
The Finite Element analysis has been carried out using four different numerical approaches - Lagrangian, Smoothed Particles Hydrodynamics (SPH), Arbitrary Lagrangian-Eulerian (ALE), and Coupled
Eulerian-Lagrangian (CEL), to examine the Cold Gas Dynamic Spray (CGDS) deposition system, through simulating and analyzing the contacts/impacts at the deformation zones. It can be observed that the
particle material has greater influences on the deposition process and the deformations than the substrate material does. Regarding the particle, a material with higher densitysuch as Cu has a higher
initial kinetic energy, leading to a larger deformation area and a longer contacttime, and hence, suggesting a better bonding between the particle and the substrate. The study suggested that all the
numerical methods tested could be used to analyze the contact/impact problems at the deformation zones in cold gas dynamic spray processes. The higher computational efficiency of the Lagrangian
approach and its ability to incorporate a complex material model into the simulation, nevertheless, makes it one of the most suitable numerical methods. However, the severe distortion of the mesh
structure in the deformed area could result in non-convergence of simulation and inaccuracy of the calculated result, due to the numerical backslide effects.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
The authors would like to acknowledge the financial support from the National Research Foundation (NRF) and the University Research Committee (URC) of the University of Johannesburg of South Africa.
Cite this article as: Sunday Temitope Oyinbo, Tien-Chien Jen, Feasibility of numerical simulation methods on the cold gas dynamic spray (CGDS) deposition process for ductile materials, Manufacturing
Rev. 7, 24 (2020)
All Tables
Table 1
Material model for the numerical analysis.
Table 2
Schematic illustration of computational costs by the four numerical modellings.
All Figures
Fig. 1
Schematic diagram illustrating 3D model used is this study for (a) Lagrangian, SPH and ALE method (b) CEL approach (c) Boundary conditions for the 3D model.
In the text
Fig. 2
Equivalent plastic strain time-evolution at 500m/s Cu/Al impact of (a) 5ns (b) 10ns (c) 20ns (d) 30ns by using the Lagrangian numerical modelling.
In the text
Fig. 3
The temperature evolution at 500m/s Cu/Al impact of (a) 5ns (b) 10ns (c) 20ns (d) 30ns by using the Lagrangian numerical modelling.
In the text
Fig. 4
The time distribution of (a) temperature (b) PEEQ of Cu/Al impact using the Lagrangian numerical modelling at 500m/s impact velocity.
In the text
Fig. 5
The distribution of PEEQ of various impact time at 500m/s Cu/Al impact (a) 5ns (b) 10ns (c) 20ns (d) 30ns by using the SPH numerical modelling.
In the text
Fig. 6
The evolution of (a) PEEQ (b) stress distribution of Cu/Al impact at 500m/s impact velocity using the SPH numerical modelling.
In the text
Fig. 7
Equivalent plastic strain evolution (PEEQ) at 500m/s Cu/Al impact at different impacting times of (a) 5ns (b) 10ns (c) 20ns (d) 30ns by using the ALE numerical modelling.
In the text
Fig. 8
The time evolution of temperature at 500m/s Cu/Al impact of (a) 5ns (b) 10ns (c) 20ns (d) 30ns by using the ALE numerical modelling.
In the text
Fig. 9
The time distribution of (a) The temperature (b) The PEEQ of Cu/Al impact at 500m/s impact velocity using the ALE numerical modelling.
In the text
Fig. 10
Evolution of PEEQ and volume average equivalent plastic strain (PEEQVAVG) 500m/s Cu/Al impact by using the CEL numerical modelling.
In the text
Fig. 11
Evolution of temperature (TEMP) and mass average temperature (TEMPMAVG) at 500m/s Cu/Al impact by using the CEL numerical modelling.
In the text
Fig. 12
The evolution of (a) temperature (b) plastic strain of Cu/Al impact at 500m/s impact velocity using the CEL numerical modelling.
In the text
Fig. 13
Effective plastic strain evolution of (a) Lagrangian model (b) SPH model (c) ALE model (d) CEL model of a Cu/Al single particle impact at 500m/s.
In the text
Fig. 14
Impact time (a) kinetic energy (b) artificial strain energy for Cu/Al impact for the four numerical approaches at 500m/s.
In the text
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Whats a dozen dozens called?
Asked by: Leopold Pouros Score: 5/5
21 votes
A dozen dozens is known as a “gross.” Any large number can be expressed by using a dozen as a reference.
What is a dozen Dozens called?
Twelve dozen (12^2 = 144) are known as a gross; and twelve gross (12^3 = 1,728, the duodecimal 1,000) are called a great gross, a term most often used when shipping or buying items in bulk. ... A
great hundred, also known as a small gross, is 120 or ten dozen.
Why is it called baker's dozen?
Baker's dozen means 13, instead of 12. The tale behind its origin is that a mediaeval law specified the weight of bread loaves, and any baker who supplied less to a customer was in for dire
punishment. So bakers would include a thirteenth loaf with each dozen just to be safe.
What is bigger than a dozen?
A gross refers to a group of 144 items (a dozen dozen or a square dozen, 12^2). A great gross refers to a group of 1728 items (a dozen gross or a cubic dozen, 12^3). A small gross or a great hundred
refers to a group of 120 items (ten dozen, 10×12).
What is a farmer's dozen?
Akin to a baker's dozen, my Farmer's Dozen is a quantity of a dozen or so questions – a series of questions with fellow designers, authors, tastemakers, friends and Southerners alike.
A Dozen Dozens
44 related questions found
How many is a butcher's dozen?
Encyclopædia Britannica, Inc. Request a dozen eggs from a farmer, a dozen steaks from a butcher, or a dozen pencils from a traveling office supplies salesman, and you will almost certainly receive 12
of your chosen item (counting errors do happen). But a baker's dozen is commonly understood to mean 13.
How many donuts are in a dozen?
If you've ever bought a dozen donuts, you probably had a lot of fun choosing the donuts that you took home in that big rectangular box. If you kept track of the donuts on your fingers, you had to
start over when you ran out of 10 fingers, because a dozen equals 12 delicious donuts.
What does 5 dozen mean?
60 eggs are in five dozens.
Why is 144 called gross?
The use of “gross” as a noun to mean “twelve dozen” (144) of something arose in English in the 15th century, drawn from the French “grosse douzaine” meaning “large dozen.” Interestingly, “gross” in
this sense is always singular; we speak of “sixteen gross of ostrich eggs,” not “grosses.”
What is 12 gross called?
It is equal to 12 dozen, or 144 pieces. In the same manner, a dozen gross, or 1728 items is called great gross. Small gross or great hundred is used to refer to ten dozen items.
How much does 2 dozen equal?
When Todd and Nolan realized they needed two dozen, or 24 eggs, they pushed these to the side and looked at the rest of the eggs.
Why do eggs come in a dozen?
Under a system that came to be known as English units, which was a combination of old Anglo-Saxon and Roman systems of measurement, eggs were sold by the dozen. It made sense to sell them that way
because one egg could be sold for a penny or 12 for a shilling, which was equal to 12 pennies.
Who invented the dozen?
The first to have used the unit were probably the Mesopotamians. 12 dozen (144 items) are a gross. 12 gross (1728 items) are called a great gross.
How many eggs are in a dozen?
There are twelve eggs per dozen.
How many units are in a dozen?
A dozen is a grouping of twelve objects, shapes or numbers. It is abbreviated as doz or dz. Dozen is one of the most primitive customary units of numbers.
What does 2 dozen mean?
Definitions of two dozen. the cardinal number that is the sum of twenty-three and one. synonyms: 24, XXIV, twenty-four. type of: large integer. an integer equal to or greater than ten.
Why is 20 referred to as a score?
score (n.)
late Old English scoru "twenty," from Old Norse skor "mark, notch, incision; a rift in rock," also, in Icelandic, "twenty," from Proto-Germanic *skur-, from PIE root *sker- (1) "to cut." The
connecting notion probably is counting large numbers (of sheep, etc.) with a notch in a stick for each 20.
What is a gross person?
The definition of gross is something that is foul, crude or very bad. ... An example of gross is a person who curses every other word.
How many eggs are in 15 dozen?
Plastic 15-Dozen Egg Case
It can house and transport six of our 30-egg plastic trays, or 15 traditional egg cartons, for a total of 180 eggs.
How many eggs are in 12 dozen?
The word 'dozen' is a very common word that you should know. It means '12. ' If you have a dozen eggs, you have 12 eggs. If you have a dozen hats, you...
What is the price of one dozen banana?
A Grade Fresh Banana, Packaging Size: 1 Dozen, Rs 35 /dozen Godavari Realistic Agro Exports Pvt.
How many is a half dozen donuts?
A half-dozen is exactly six. | {"url":"https://moviecultists.com/whats-a-dozen-dozens-called","timestamp":"2024-11-10T17:23:07Z","content_type":"text/html","content_length":"39155","record_id":"<urn:uuid:31020291-035e-4359-bf53-8c5d4597fecf>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00159.warc.gz"} |
Steady-state Heat Conduction in Planar Walls: Screencast
Shows how to write a thermal circuit for a composite wall with two different materials in series and parallel.
We suggest that after watching this screencast, you list the important points as a way to increase retention.
For a composite wall with no heat generation:
The equation for heat flux is:
\[ q=\frac{T_0\, -\, T_4}{R_{total}}\]
where \(T_0\) = temperature of the left side of the left wall
\(T_4\) = temperature on the right side of the right wall
\(R_{total}\) = total resistance to heat transfer
\[R_{total} = \sum_{i=1}^{4} \frac{x_i}{k_i}\]
where \(x_i\) = thickness of each wall segment (cm)
\(k_i\) = thermal conductivity of the wall’s material (W/[cm K])
The wall temperatures can be calculated from left to right according to:
\[T_i = T_{i-1}\, -\, qR_i = T_{i-1}\, -\, q\frac{x_i}{k_i}\]
where \(R_i\) = thermal resistance
\(T_i\) = temperature | {"url":"https://learncheme.com/quiz-yourself/interactive-self-study-modules/steady-state-heat-conduction-in-planar-walls/steady-state-heat-conduction-in-planar-walls-screencasts/","timestamp":"2024-11-05T20:02:30Z","content_type":"text/html","content_length":"75660","record_id":"<urn:uuid:7786484a-3e02-431f-b4bf-0c197e49c0bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00599.warc.gz"} |
Identifying the Graph of a Dynamic Equilibrium Using the Initial and Equilibrium Concentrations
Question Video: Identifying the Graph of a Dynamic Equilibrium Using the Initial and Equilibrium Concentrations Chemistry • Third Year of Secondary School
A dynamic equilibrium is established between two reactants A and B according to the equation shown: A (aq) ⇌ 2B (aq). Compound A has an initial concentration of 0.8 mol/dm³ that drops to 0.4 mol/dm³
once equilibrium is established. Which graph for this equilibrium is correct?
Video Transcript
A dynamic equilibrium is established between two reactants A and B according to the equation shown: A, aqueous, is in equilibrium with two B, aqueous. Compound A has an initial concentration of 0.8
moles per cubic decimeter that drops to 0.4 moles per cubic decimeter once equilibrium is established. Which graph for this equilibrium is correct?
Dynamic equilibrium is established when the forward and reverse reactions occur at the same nonzero rate. When the forward and reverse reactions have the same rate, the reactant and product
concentrations will remain constant even though both reactions are still occurring. Looking at the graphs, we can determine when equilibrium is established by identifying the point in time where the
concentrations of A and B no longer change.
We are told in the question that compound A has an initial concentration of 0.8 moles per cubic decimeter that drops to 0.4 moles per cubic decimeter once equilibrium is established. Looking at the
graphs, we can see that only graphs (C) and (D) show compound A as having an initial concentration of 0.8 moles per cubic decimeter. So, we can eliminate graphs (A) and (B). Graphs (C) and (D) also
show that the concentration of compound A is 0.4 moles per cubic decimeter once equilibrium is established.
To determine which of these graphs is correct, we’ll need to consider how the concentration of B changes over time. Looking at the balanced chemical equation, we see that for every mole of A that
reacts, two moles of B are produced. Over the course of this reaction, the concentration of A decreases by 0.4 moles per cubic decimeter. So, the concentration of B should increase by twice as much.
We should therefore expect that compound B has an initial concentration of zero moles per cubic decimeter that increases to 0.8 moles per cubic decimeter once equilibrium is established. This is
shown in graph (D).
In conclusion, the graph for the equilibrium between A and two B that is correct is the graph shown in answer choice (D). | {"url":"https://www.nagwa.com/en/videos/752120210309/","timestamp":"2024-11-05T03:11:54Z","content_type":"text/html","content_length":"251508","record_id":"<urn:uuid:f25a7dee-3fb5-498e-8fa6-4067968c92b4>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00567.warc.gz"} |
• u-μP, which improves upon μP by combining it with Unit Scaling:
□ μP ensures that the scale of activations is independent of model size
□ Unit Scaling ensures that activations, weights and gradients begin training with a scale of one.
• you need to divide the residual add to prevent the model from blowing up
Reminder of mup setting
• ABC-parametrizations μP, SP, and the Neural Tangent Kernel (NTK) are all instances of abc-parametrizations. This assumes a model under training where weights are defined as:
□ ,
□ ,
□ with a time-step and the weight update based on previous loss gradients.
• = parameter multiplier
□ For example, in the attention logit calculation where , the factor is a multiplier. It may also be thought of as the parameter multiplier of if we rewrite the attention logit as .
□ Note that parameter multipliers cannot be absorbed into the initialization in general, since they affect backpropagation. Nevertheless, after training is done, parameter multipliers can
always be absorbed into the weight.
• = per-parameter initialization
• = per-parameter learning rate
• A parametrization scheme such as μP is then defined specifying how scalars , , change with model width.
□ This can be expressed in terms of width-dependent factors , , , such that , , .
□ (The scaling defining mup)
• A key property of the abc-parametrization is that one can shift scales between , , in a way that preserves learning dynamics (i.e. the activations computed during training are unchanged). We term
this abc-symmetry. For a fixed , the behavior of a network trained with Adam is invariant to changes of the kind:
□ This means that parametrizations like can be presented in different but equivalent ways. ABC-symmetry is a key component in developing u-. (and why the u-mup scheme is consistent mup and
Spectral mup)
Transferable HPs
• The above terms are defined by the parametrization choice, however there are also the hyperparameters chosen by the user.
• All μTransferable HPs function as multipliers and can be split into three kinds, which contribute to the three (non-HP) multipliers given by the abc-parametrization: , , where , , .
• = operator scaling
• = init scaling
• = per-parameter learning rate scaling
The challenges with mup in practice
• Not all training setups give mu-Transfer
□ Vanilla mup works in overfitting regime but fails to generalize to standard LM training regime (under-fitting)
□ The fix is (1) Removal of trainable parameters from normalization layers (2) Use the independent form of AdamW Independent weight decay
• It’s not clear which hyperparameters to sweep
□ In theory, the search space of μTransferable HPs includes , , for every parameter tensor W in the model
□ there’s coupling between them if you’re not careful
☆ The relative size of a weight update is determined by the ratio (size of update / size of current weight)
☆ Consider the commonly-used global HP. At initialization the activations going into the FFN swish function have , whereas the self-attention softmax activations have . A global HP thus has
a linear effect on the FFN and a quadratic effect on attention, suggesting that this grouping may not be ideal.
• Base shape complicates usage
□ Orignal mup requires an extra “base” model to correctly init the model
• mup struggles with low-precision
□ Low-precision training runs succesfully converging in SP can diverge with mup because of the generally lower init and scaling (underflow of gradients)
The Unit-Scaled Maximal Update Parametrization
The u-mup ABC parametrization
• How they go from mup to u-mup
□ drop the (because unit scaling assumes unit variance)
☆ they can do so by using ABC-parametrization to shift the scale in (i.e. the ) from mup to and (Equation 4 and 5 in the paper)
□ drop the HP with the above trick and also shifting the HPs burden to the unit scaling ops
□ change the input learning rate to .
☆ slight deviation for mup in terms of the math
☆ key change for performance, not much theoretical justification
• It’s important to note that the scaling rules in this table must be combined with the standard Unit Scaling rules for other non-matmul operations.
□ e.g. gated SILU, residual_add, softmax cross-entropy, …
Why it works better than vanilla mup
• We can attribute the difficulties μP has with low precision to the fact that it ignores constant factors (along with weight and gradient-scaling), only ensuring that activations are of order .
The stricter condition of unit scale across all tensors at initialization provides a way of leveraging μP’s rules in order to make low-precision training work.
A principled approach to hyperparameters
• How to sweep HPs has been a mess in mup literature
• We want
□ Minimal cardinality: the use of as few HPs as possible.
□ Minimal interdependency: the optimal value of each HP should not depend on the value of other HPs, simplifying the search space.
□ Interpretability: there should be a clear explanation for what an HP’s value ‘means’ in the context of the model
1. we can drop all as we assume unit scaling (and abc-symmetry allows us to do so), leaving just and
2. Second, several combine linearly with other HPs. It is easier to define things at the operator level instead of the weight level, e.g. . In this instance, it is more natural to use a single
parameter and associate it with
3. Use a single global and group HPs across layers. (This is the best tradeoff between expressivity and cardinality)
• The considered hyperparameters for a Transformer
• How to choose such operator multipliers for an architecture is described in Appendix F and Appendix G
□ The idea is that you want a multiplier at every operation in your computational graph where there’s a non-homogeneous function or a non-unary function (not single input)
☆ i.e. a k-homogeneous function is s.t.
☆ RMS norm is 0-homo., Linear is 1-homo. and QK matmul is 2-homo.
☆ A residual add is non-unary
☆ Sigmoid and cross-entropy loss is non-homogeneous
□ After having settled on needed multipliers, simplify them following the minimal cardinality, expressivity, and interpretability constraints
☆ assuming unit-scaling usually allows for more interpretable mutlipliers
How to do the HPs sweep
• The standard approach to HP search for mu-Transfer is via random sweep over all HPs simultaneously. This is costly
• Due to the inter-dependence criterion applied previously, u-mup supposedly allows for a simpler scheme, called independent search
• The idea is to first sweep the LR, followed by a set of one-dimensional sweeps of the other HPs (which can be run in parallel). The best results from the individual sweeps are combined to from
the final set of HP values.
• The simpler scheme, which only sweeps the LR, leaving other HP values at 1, seems to work well in practice.
Numerical properties
• mup has gradients and weights with low RMS, at risk of FP8 underflow, whereas u-μP starts with RMS ≈ 1.
• Many input activations do not grow RMS during training (due to a preceding non-trainable RMSNorm), however the attention out projection and FFN down projection have unconstrained input
activations that grow considerably during training
• The decoder weight grows during training. Since it is preceded by a RMSNorm, the model may require scale growth in order to increase the scale of softmax inputs. Other weights grow slightly
during training.
• Gradients grow quickly but stabilize, except for attention out projection and FFN down projection, whose gradients shrink as the inputs grow.
• The the main parameter affecting scale growth is learning rate
□ End-training RMS is remarkably stable as width, depth, training steps and batch size are independently increased.
Prerequisites before applying u-mup
• Remove trainable parameters from normalization layers
• Use the independent form of AdamW Independent weight decay
• Ensure training is the under-fitting regime (i.e. avoid excessive data repetition)
A guide to using u-mup
• Being careful of tensors with scale growth (inputs to FFN and self-attention final projections)
□ use E5M2 format to represent the larger scales or apply dynamic rescaling of the matmul input
• apply unit scaling with the correct scale constraints, for new operations, don’t hesitate to fit an empirical model for the scale of the op.
Hyperparameter transfer results
• The setting matters a lot (i.e. number of tokens, model size, sequence length). Should be as representative as possible of the final training.
• At large scale, learning rate and residual attention ratio were the most important HPs. All other HPs can be left at their default value of 1.
• Non-LR HPs also have approximately constant optima across width under u-μP
How to use a good proxy model
• When using a relatively small proxy model with 8 layers and a width of 512 (4 attention heads), the HP-loss landscape is rather noisy. By doubling the width, they are able to discern the optimal
values of the HPs more clearly.
• In general, width is the most reliable feature to transfer. Training steps and batch size also give good transfer, so moderate changes here are permissible. Depth is the least reliable feature
for transfer, so they only recommend modest changes in depth
• Keep the number of warmup steps constant, but always decay to the same final LR when varying the number of steps. | {"url":"https://notes.haroldbenoit.com/ml/general/scaling/mu-transfer/u-mup","timestamp":"2024-11-09T20:34:45Z","content_type":"text/html","content_length":"180275","record_id":"<urn:uuid:217d2f07-48ac-47d2-81e9-143543c4f558>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00720.warc.gz"} |
Various inequivalent definitions of Kleene algebras and related structures have been given in the literature.^[2] Here we will give the definition that seems to be the most common nowadays.
A Kleene algebra is a set A together with two binary operations + : A × A → A and · : A × A → A and one function ^* : A → A, written as a + b, ab and a^* respectively, so that the following axioms
are satisfied.
The above axioms define a semiring. We further require:
It is now possible to define a partial order ≤ on A by setting a ≤ b if and only if a + b = b (or equivalently: a ≤ b if and only if there exists an x in A such that a + x = b; with any definition, a
≤ b ≤ a implies a = b). With this order we can formulate the last four axioms about the operation ^*:
• 1 + a(a^*) ≤ a^* for all a in A.
• 1 + (a^*)a ≤ a^* for all a in A.
• if a and x are in A such that ax ≤ x, then a^*x ≤ x
• if a and x are in A such that xa ≤ x, then x(a^*) ≤ x ^[3]
Intuitively, one should think of a + b as the "union" or the "least upper bound" of a and b and of ab as some multiplication which is monotonic, in the sense that a ≤ b implies ax ≤ bx. The idea
behind the star operator is a^* = 1 + a + aa + aaa + ... From the standpoint of programming language theory, one may also interpret + as "choice", · as "sequencing" and ^* as "iteration".
Notational correspondence between
Kleene algebras and + · ^* 0 1
Regular expressions | not written ^* ∅ ε
Let Σ be a finite set (an "alphabet") and let A be the set of all regular expressions over Σ. We consider two such regular expressions equal if they describe the same language. Then A forms a Kleene
algebra. In fact, this is a free Kleene algebra in the sense that any equation among regular expressions follows from the Kleene algebra axioms and is therefore valid in every Kleene algebra.
Again let Σ be an alphabet. Let A be the set of all regular languages over Σ (or the set of all context-free languages over Σ; or the set of all recursive languages over Σ; or the set of all
languages over Σ). Then the union (written as +) and the concatenation (written as ·) of two elements of A again belong to A, and so does the Kleene star operation applied to any element of A. We
obtain a Kleene algebra A with 0 being the empty set and 1 being the set that only contains the empty string.
Let M be a monoid with identity element e and let A be the set of all subsets of M. For two such subsets S and T, let S + T be the union of S and T and set ST = {st : s in S and t in T}. S^* is
defined as the submonoid of M generated by S, which can be described as {e} ∪ S ∪ SS ∪ SSS ∪ ... Then A forms a Kleene algebra with 0 being the empty set and 1 being {e}. An analogous construction
can be performed for any small category.
The linear subspaces of a unital algebra over a field form a Kleene algebra. Given linear subspaces V and W, define V + W to be the sum of the two subspaces, and 0 to be the trivial subspace {0}.
Define V · W = span {v · w | v ∈ V, w ∈ W}, the linear span of the product of vectors from V and W respectively. Define 1 = span {I}, the span of the unit of the algebra. The closure of V is the
direct sum of all powers of V.
${\displaystyle V^{*}=\bigoplus _{i=0}^{\infty }V^{i}}$
Suppose M is a set and A is the set of all binary relations on M. Taking + to be the union, · to be the composition and ^* to be the reflexive transitive closure, we obtain a Kleene algebra.
Every Boolean algebra with operations ${\displaystyle \lor }$ and ${\displaystyle \land }$ turns into a Kleene algebra if we use ${\displaystyle \lor }$ for +, ${\displaystyle \land }$ for · and set
a^* = 1 for all a.
A quite different Kleene algebra can be used to implement the Floyd–Warshall algorithm, computing the shortest path's length for every two vertices of a weighted directed graph, by Kleene's algorithm
, computing a regular expression for every two states of a deterministic finite automaton. Using the extended real number line, take a + b to be the minimum of a and b and ab to be the ordinary sum
of a and b (with the sum of +∞ and −∞ being defined as +∞). a^* is defined to be the real number zero for nonnegative a and −∞ for negative a. This is a Kleene algebra with zero element +∞ and one
element the real number zero. A weighted directed graph can then be considered as a deterministic finite automaton, with each transition labelled by its weight. For any two graph nodes (automaton
states), the regular expressions computed from Kleene's algorithm evaluates, in this particular Kleene algebra, to the shortest path length between the nodes.^[4]
Zero is the smallest element: 0 ≤ a for all a in A.
The sum a + b is the least upper bound of a and b: we have a ≤ a + b and b ≤ a + b and if x is an element of A with a ≤ x and b ≤ x, then a + b ≤ x. Similarly, a[1] + ... + a[n] is the least upper
bound of the elements a[1], ..., a[n].
Multiplication and addition are monotonic: if a ≤ b, then
• a + x ≤ b + x,
• ax ≤ bx, and
• xa ≤ xb
for all x in A.
Regarding the star operation, we have
• 0^* = 1 and 1^* = 1,
• a ≤ b implies a^* ≤ b^* (monotonicity),
• a^n ≤ a^* for every natural number n, where a^n is defined as n-fold multiplication of a,
• (a^*)(a^*) = a^*,
• (a^*)^* = a^*,
• 1 + a(a^*) = a^* = 1 + (a^*)a,
• ax = xb implies (a^*)x = x(b^*),
• ((ab)^*)a = a((ba)^*),
• (a+b)^* = a^*(b(a^*))^*, and
• pq = 1 = qp implies q(a^*)p = (qap)^*.^[5]
If A is a Kleene algebra and n is a natural number, then one can consider the set M[n](A) consisting of all n-by-n matrices with entries in A. Using the ordinary notions of matrix addition and
multiplication, one can define a unique ^*-operation so that M[n](A) becomes a Kleene algebra.
Kleene introduced regular expressions and gave some of their algebraic laws.^[6]^[7] Although he didn't define Kleene algebras, he asked for a decision procedure for equivalence of regular
expressions.^[8] Redko proved that no finite set of equational axioms can characterize the algebra of regular languages.^[9] Salomaa gave complete axiomatizations of this algebra, however depending
on problematic inference rules.^[10] The problem of providing a complete set of axioms, which would allow derivation of all equations among regular expressions, was intensively studied by John Horton
Conway under the name of regular algebras,^[11] however, the bulk of his treatment was infinitary. In 1981, Kozen gave a complete infinitary equational deductive system for the algebra of regular
languages.^[12] In 1994, he gave the above finite axiom system, which uses unconditional and conditional equalities (considering a ≤ b as an abbreviation for a + b = b), and is equationally complete
for the algebra of regular languages, that is, two regular expressions a and b denote the same language only if a = b follows from the above axioms.^[13]
Generalization (or relation to other structures)
See also
1. ^ Marc Pouly; Jürg Kohlas (2011). Generic Inference: A Unifying Theory for Automated Reasoning. John Wiley & Sons. p. 246. ISBN 978-1-118-01086-0.
2. ^ For a survey, see: Kozen, Dexter (1990). "On Kleene algebras and closed semirings" (PDF). In Rovan, Branislav (ed.). Mathematical foundations of computer science, Proc. 15th Symp., MFCS '90,
Banská Bystrica/Czech. 1990. Lecture Notes Computer Science. Vol. 452. Springer-Verlag. pp. 26–47. Zbl 0732.03047.
3. ^ Kozen (1990), sect.2.1, p.3
4. ^ Gross, Jonathan L.; Yellen, Jay (2003), Handbook of Graph Theory, Discrete Mathematics and Its Applications, CRC Press, p. 65, ISBN 9780203490204.
5. ^ Kozen (1990), sect.2.1.2, p.5
6. ^ S.C. Kleene (Dec 1951). Representation of Events in Nerve Nets and Finite Automata (PDF) (Technical report). U.S. Air Force / RAND Corporation. p. 98. RM-704. Here: sect.7.2, p.52
7. ^ Kleene, Stephen C. (1956). "Representation of Events in Nerve Nets and Finite Automata" (PDF). Automata Studies, Annals of Mathematical Studies. 34. Princeton Univ. Press. Here: sect.7.2,
8. ^ Kleene (1956), p.35
9. ^ V.N. Redko (1964). "On defining relations for the algebra of regular events" (PDF). Ukrainskii Matematicheskii Zhurnal. 16 (1): 120–126. (In Russian)
10. ^ Arto Salomaa (Jan 1966). "Two complete axiom systems for the algebra of regular events" (PDF). Journal of the ACM. 13 (1): 158–169. doi:10.1145/321312.321326. S2CID 8445404.
11. ^ Conway, J.H. (1971). Regular algebra and finite machines. London: Chapman and Hall. ISBN 0-412-10620-5. Zbl 0231.94041. Chap.IV.
12. ^ Dexter Kozen (1981). "On induction vs. ^*-continuity" (PDF). In Dexter Kozen (ed.). Proc. Workshop Logics of Programs. Lect. Notes in Comput. Sci. Vol. 131. Springer. pp. 167–176.
13. ^ Dexter Kozen (May 1994). "A Completeness Theorem for Kleene Algebras and the Algebra of Regular Events" (PDF). Information and Computation. 110 (2): 366–390. doi:10.1006/inco.1994.1037. — An
earlier version appeared as: Dexter Kozen (May 1990). A Completeness Theorem for Kleene Algebras and the Algebra of Regular Events (Technical report). Cornell. p. 27. TR90-1123.
14. ^ Jonathan S. Golan (30 June 2003). Semirings and Affine Equations over Them. Springer Science & Business Media. pp. 157–159. ISBN 978-1-4020-1358-4.
15. ^ ^a ^b ^c Marc Pouly; Jürg Kohlas (2011). Generic Inference: A Unifying Theory for Automated Reasoning. John Wiley & Sons. pp. 232 and 248. ISBN 978-1-118-01086-0.
Further reading
• Kozen, Dexter. "CS786 Spring 04, Introduction to Kleene Algebra".
• Peter Höfner (2009). Algebraic Calculi for Hybrid Systems. BoD – Books on Demand. pp. 10–13. ISBN 978-3-8391-2510-6. The introduction of this book reviews advances in the field of Kleene algebra
made in the last 20 years, which are not discussed in the article above. | {"url":"https://www.knowpia.com/knowpedia/Kleene_algebra","timestamp":"2024-11-13T02:00:17Z","content_type":"text/html","content_length":"108591","record_id":"<urn:uuid:a26c5238-5fec-4062-8b81-6f21b42d7ad3>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00470.warc.gz"} |
GAUSS8 2016 Problem 25
In the table, the numbers in each row form an arithmetic sequence when read from left to right. Similarly, the numbers in each column form an arithmetic sequence when read from top to bottom. What is
the sum of the digits of the value of x?\ (An arithmetic\:sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3,5,7,9 are
the first four terms of an arithmetic sequence.)
Answer Choices
A. 5
B. 2
C. 10
D. 7
E. 13 | {"url":"https://forums.randommath.com/t/gauss8-2016-problem-25/6226","timestamp":"2024-11-03T12:03:57Z","content_type":"text/html","content_length":"20755","record_id":"<urn:uuid:cb6a9b1b-8b2b-410b-8db5-17a8f345a6ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00868.warc.gz"} |
About the new MARS project, how to contribute?
Thank you for this project, I will be watching and happy to contribute.
But I have a couple of doubts, what kind of data exactly do you need?
- What camera, mono, color, both, cooled or not?
- What filters, narrowband, broadband?
- Images with some calibration (darks and flats) or pure .fits?
- Does light pollution matter?
- Any sensor size preference, will a small one like the 533 work for example?
That, for now
Hi Marcelo,
Thank you so much! As described
in the MARS webpage
, your contributed images must meet the following requirements:
• The images can be acquired with mono or color cameras (RGB bands). In the case of mono cameras, we also ask for narrowband H-alpha, [O-III], and [S-II] images.
• In the case of broadband images, they should have been acquired from a sky with a minimum darkness quality (Bortle scale 1 to 4).
• The images must be completely preprocessed master lights with no additional processing. We need your original masters in XISF format without any gradient correction.
• The master lights must be generated with drizzle (even if you use drizzle x1). This is especially true for color cameras since drizzle preserves the validity of the stellar photometry.
Best regards,
Ok, to avoid future confusion I just uploaded* to my google drive my latest master light Ha of the Tarantula Nebula (NGC 2070, southern hemisphere
Is this the kind of data they need?
*I guess for the official image collection there will be a better way to send the images.
Thank you very much for your data. It is better for us to have wider fields of view, but your data is very welcome.
Unfortunatley I will not be able to contribute as live in a bortle 6 + area
but wanted to say I always hoped this could be done this way and will be a huge step forward particularly
for thoese of us who live in the orange goo
really exciting stuff
Hi Vicent
I live in NZ in a Bortle 2-3
I think we can help here for the southern hemisphere
I can do NB / broadband too
was there specific width you need ... it was 3-50 degrees
do you need any particular groups of image wdith / and pixel scale??
Greendale Observatory, Greendale, NZ
I live in a Bortle 2 area in NZ. I have taken over 150 1hr-1.5hr ZWO 2400MC images with Nikon 200mm f/2 [10x 7deg field size] on 7.5x5deg field centres. The data covers about 25% of the Southern Sky
at 6arcsec resolution, with a SNR~40 at 22mag/arcsec^2.
All data have been processed by WBPP [max quality], with no ABE/DBE applied. This is quite a large volume of data, but its all sitting on a DropBox account. I am happy to give you access.
It is part of an attempt to cover the whole sky - initiated as part of an Astrobin community survey. Unfortunately there are only a few active participants. But they will probably be happy to share
their data too. In total we cover about 40% of the Southern Sky and aim to have a complete first pass but July next year.
Full survey field list and observations to date can be found here:
Professor Brian J Boyle
Antimony Observatory and Vineyard
Queenstown RD1
New Zealand.
That is amazing! I have made a thread on
to let everyone know.. I'm from Australia, so hopefully I can get a few peeps to contribute. Do the original masters really need to be in XISF? not everyone uses Pixinsight for stacking in my crew of
Last edited:
PixInsight Staff
Staff member
That is amazing! I have made a thread on
to let everyone know.. I'm from Australia, so hopefully I can get a few peeps to contribute. Do the original masters really need to be in XISF? not everyone uses Pixinsight for stacking in my
crew of astrophotographers.
Thank you for your interest. Images in FITS format are also welcome.
Best regards
Hi all,
Thank you so much for your interest. Certainly, your contribution would be very welcome since we are more limited in acquiring our own data in the southern hemisphere. Any field of view in the range
we specify is needed, though by the method design the critical piece here is the wider fields since they are the seed for a global correction towards longer focal lengths.
The survey data you are collecting in Astrobin would be a very valuable contribution to our project.
We also want to thank the community for this enthusiastic reception.
Best regards,
Hi, a quick question to Vincet et al. I have a few data sets taken with my ASI6200 mono and Canon EF200mm lens that fit the criteria for field of view etc. The FOV is 10d x 7d in this case. I'm also
in Bortle 3 site in the southern hemisphere which helps. My question relates to the drizzle integration requirements. I have the masters that have been prepared by the WBPP script which comes with
the requisite *.xdrz files. To supply these masters as Drizzle x 1 do I need to run the registered files through the drizzle integration process to produce new masters before I post them to a Dropbox
Hello MARS team!
I am interested contributing with my northern hemisphere data obtained with an 135mm lens and a crop sensor DSLR. I visually rate my sky as Bortle 5 but I have a feeling that it might be less than
that. Most data have been taken with care in order to avoid local LP sources and with the objects as high as possible in the local sky.
I have some questions:
• The data have been manually calibrated and integrated without local normalization by using older versions of PI. Is this acceptable?
• What about the licensing of the data? I don't want my dataset being part of some closed-source machine learning training dataset from someone outside Pleiades.
• If possible, I would like to have some feedback regarding the quality of my submissions in order to improve my acquisition procedure for any future submissions.
• Are you planning publishing your efforts in a scientific journal? If not, please consider to do so!
Thanks, and good luck with your project!
Hi, I am interested in contributing if possible. I live in Australia under a Bortle 2 sky. Is there guidance for areas of the sky you especially need data for? Are there areas that you do not need
more data for? cheers
Hi, I am interested in contributing if possible. I live in Australia under a Bortle 2 sky. Is there guidance for areas of the sky you especially need data for? Are there areas that you do not
need more data for? cheers
Hi, thank you so much in the first place! We are just starting this project, so
data is welcome. We can only cover the northern hemisphere (plus a band down to about 25 degrees south in good conditions) with the observing station we are installing in Spain (MARS-pi), so all
contributions with southern data are fantastic.
Hello MARS team!
I am interested contributing with my northern hemisphere data obtained with an 135mm lens and a crop sensor DSLR. I visually rate my sky as Bortle 5 but I have a feeling that it might be less
than that. Most data have been taken with care in order to avoid local LP sources and with the objects as high as possible in the local sky.
Great! Thank you!
□ The data have been manually calibrated and integrated without local normalization by using older versions of PI. Is this acceptable?
Yes, there should be no problems.
□ What about the licensing of the data? I don't want my dataset being part of some closed-source machine learning training dataset from someone outside Pleiades.
Rest assured that the data used to build the MARS databases will be used exclusively by our team and staff. Nobody outside Pleiades Astrophoto will have access to it. MARS is a project based
exclusively on observational data. We are against using generative "AI" techniques for gradient correction so that nobody will use either the raw data or the final MARS databases for these purposes.
However, I want to be crystal clear about this project since the very beginning: MARS databases and the tools that will use them will be closed-source and exclusive to the PixInsight platform. We are
investing significant intellectual and economic resources in this project and will invest much more in the future.
□ f possible, I would like to have some feedback regarding the quality of my submissions in order to improve my acquisition procedure for any future submissions.
Of course. We'll try to inform all contributors about our use of their data. Many aspects of this project are still under development, and we are now working on a dedicated MARS website, where we'll
provide up-to-date (or even real-time) detailed information.
□ Are you planning publishing your efforts in a scientific journal? If not, please consider to do so!
We have yet to think about this, but it's always possible. Thank you for pointing out this.
Thanks, and good luck with your project!
Thank you so much!
Hi, a quick question to Vincet et al. I have a few data sets taken with my ASI6200 mono and Canon EF200mm lens that fit the criteria for field of view etc. The FOV is 10d x 7d in this case. I'm
also in Bortle 3 site in the southern hemisphere which helps. My question relates to the drizzle integration requirements. I have the masters that have been prepared by the WBPP script which
comes with the requisite *.xdrz files. To supply these masters as Drizzle x 1 do I need to run the registered files through the drizzle integration process to produce new masters before I post
them to a Dropbox file?
Hi Rodney, thank you for your interest!
We ask for drizzle-integrated data because we need master images generated without any pixel interpolation. This is important in all cases because pixel interpolation generates aliasing artifacts,
which are large-scale structures that can potentially contaminate the survey. Aliasing also degrades stellar photometry, which we will apply (using Gaia photometric and spectral data) to normalize
all the data. This is particularly critical for color images (DSLR, OSC) because demosaicing artifacts can have a very negative impact on photometry. In other words, we need data as
as possible, and currently only drizzle can guarantee this.
To apply a drizzle integration, just load the .xdrz files with the DrizzleIntegration tool and set Scale = 1. Ensure the original calibrated frames (before registration) are still on the same folders
when WBPP generated the .xdrz files; otherwise, open the Format Hints section and select the appropriate input directory. Apply the process globally (F6, or blue circle) and wait. Let us know if you
need further help with this.
Hi Rodney, thank you for your interest!
We ask for drizzle-integrated data because we need master images generated without any pixel interpolation. This is important in all cases because pixel interpolation generates aliasing
artifacts, which are large-scale structures that can potentially contaminate the survey. Aliasing also degrades stellar photometry, which we will apply (using Gaia photometric and spectral data)
to normalize all the data. This is particularly critical for color images (DSLR, OSC) because demosaicing artifacts can have a very negative impact on photometry. In other words, we need data as
pure as possible, and currently only drizzle can guarantee this.
To apply a drizzle integration, just load the .xdrz files with the DrizzleIntegration tool and set Scale = 1. Ensure the original calibrated frames (before registration) are still on the same
folders when WBPP generated the .xdrz files; otherwise, open the Format Hints section and select the appropriate input directory. Apply the process globally (F6, or blue circle) and wait. Let us
know if you need further help with this.
Excellent, thanks for the heads up! I will run the new process as soon as I get a chance. I have moved the folders since I did the preprocessing so will need to spend a moment to assign the
appropriate directories.
I think this is a really brilliant and exciting project, I am very much looking forward to the first results from this!
I have some integrations taken in bortle 3 with a 50 mm lens and APS-C camera from last year. Maybe they are good enough to contribute them, I will give it a try as soon as possible. They cover the
fields in the northern winter constellations (Orion, Taurus, etc…)
Just some points that come to my mind:
- How are gradients in the 35 mm handled? These will be the initial reference for all the other images, thus gradients still present in them could influence all the other images. Maybe all sky images
from truly dark sky are a solution to correct for gradients in these initial large field images.
- The MARS-pi survey will be captured with 1h/2h integration per field. How will this play out when normalizing gradients in images with very high SNR (>4h integration with similar conditions)? Will
the gradient normalization in our images be limited by the depth of the MARS survey, or is such a limit only of theoretical nature?
- For the sake of this example, let's assume I'm imaging with an unmodified DSLR. All the images from the MARS survey will be captured with Ha sensitive camera (at least that's what I assume). Will
normalizing my images against the survey introduce the brightness variations caused by Ha nebula into my images? I'm asking because the same principle (although with much less intensity) will
probably apply to all differences in spectral sensitivity.
CS Gerrit
Hello Juan and thank you for your elaborate answers.
Rest assured that the data used to build the MARS databases will be used exclusively by our team and staff. Nobody outside Pleiades Astrophoto will have access to it. MARS is a project based
exclusively on observational data. We are against using generative "AI" techniques for gradient correction so that nobody will use either the raw data or the final MARS databases for these
However, I want to be crystal clear about this project since the very beginning: MARS databases and the tools that will use them will be closed-source and exclusive to the PixInsight platform. We
are investing significant intellectual and economic resources in this project and will invest much more in the future.
I don't have anything against ML, as there are directions of research which aim to respect physics (see physics-informed neural networks) and I am open to adopt such solutions, especially if they are
developed from Pleiades.
My concern is about others (and not Pleiades) who can potentially use PI for the purpose of assembling training sets for their "AI" products. This malpractice is already a threat: for example some
can use SPCC in order to create training sets for a commercial, standalone "AI" color correction tool/product outside the PI ecosystem. I am far from being an expert in intellectual property but I
suspect that neither the PI license nor the Gaia DR3/SP license allow for such practices.
Hello, does SQM=20.55 qualify for the MARS survey.
ZWO533MC/135MM F2.8 Samyang.? | {"url":"https://pixinsight.com/forum/index.php?threads/about-the-new-mars-project-how-to-contribute.22075/","timestamp":"2024-11-09T01:48:17Z","content_type":"text/html","content_length":"180634","record_id":"<urn:uuid:389fd963-b7dd-4b05-ac68-e32e92d1a83a>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00742.warc.gz"} |
Correlation Methodologies between Land Use and Greenhouse Gas emissions: The Case of Pavia Province (Italy)
Department of Building Engineering and Architecture, University of Pavia, 27100 Pavia, Italy
Author to whom correspondence should be addressed.
Submission received: 13 February 2024 / Revised: 10 April 2024 / Accepted: 24 April 2024 / Published: 27 April 2024
The authors present an analysis of the correlation between demographic and territorial indicators and greenhouse gas (GHG) emissions, emphasizing the spatial aspect using statistical methods.
Particular attention is given to the application of correlation techniques, considering the spatial correlation between the involved variables, such as demographic, territorial, and environmental
indicators. The demographic data include factors such as population, demographic distribution, and population density; territorial indicators include land use, particularly settlements, and road soil
occupancy. The aims of this study are as follows: (1) to identify the direct relationships between these variables and emissions; (2) to evaluate the spatial dependence between geographical entities;
and (3) to contribute to generating a deeper understanding of the phenomena under examination. Using spatial autocorrelation analysis, our study aims to provide a comprehensive framework of the
territorial dynamics that influence the quantity of emissions. This approach can contribute to formulating more targeted environmental policies, considering the spatial nuances that characterize the
relationships between demographics, territory, and GHGs. The outcome of this research is the identification of a direct formula to obtain greenhouse gas emissions from data about land use starting
from the case study of Pavia Province in Italy. In the paper, the authors highlight different methodologies to compare land use and GHG emissions to select the most feasible correlation formula. The
proposed procedure has been tested and can be used to promote awareness of the spatial dimension in the analysis of complex interactions between anthropogenic factors and environmental impacts.
1. Introduction
In recent years, due to the global population’s gravity, human activities have profoundly altered the natural landscape. These alterations are driven by the demands of urbanization, agriculture, and
industrialization. Simultaneously, emissions of greenhouse gas (GHG) into the atmosphere, water bodies, and soil have escalated. These emissions result from expanding industrial activities and
unsustainable consumption patterns [
]. As a consequence, the intricate web of interactions within ecosystems is under strain, with far-reaching implications for biodiversity, ecosystem services, and human well-being. The delicate
balance between the environment and human activity lies at the heart of growing global concerns regarding the sustainability and conservation of natural resources [
]. In this context, the correlation between GHG emissions and changes in land use emerges as a critical aspect. It requires in-depth analysis to understand the complex interactions and consequences
for the ecosystem [
]. This study aims to explore and quantify the relationship between emissions from different sources and the state of land use. The goal is to provide a scientific basis for environmental and spatial
management strategies. The working assumption is that there exists a close interdependence between land use and emission scenarios
In areas with high urban density, soil plays a crucial role as a regulator of climate and microclimate. Its status as the habitat for green spaces is closely linked to air quality. Understanding the
dynamics associated with land use is crucial for territorial planning. It allows for the interpretation of current conditions as the culmination of past changes while simultaneously monitoring
ongoing transformations and anticipating future ones. Drawing upon interdisciplinary perspectives from ecology, environmental science, geography, and urban planning, the authors examine how much
human-driven alterations to the landscape amplify the impact of GHGs [
The authors’ contribution focuses on analyzing the relationship between land use and GHG emissions using both correlation and spatial correlation methods. Their study specifically examines the
current state of Pavia Province and conducts an in-depth analysis of all its municipalities.
By investigating the interplay between land use patterns and emissions, the authors aim to provide some direct dependencies of cause and effects that usually require extremely complex (and time
consuming) elaborations. This result can furnish valuable insights for environmental management and policy decisions. Stressing a direct formula that interrelates how land use influences GHG
emissions is crucial for sustainable development policies at every decision-making scale [
Study Area: Pavia Province
The study focuses on Pavia Province, located in the Po Valley, one of the most disadvantaged areas in Europe regarding air quality and particularly for its dynamic economy, a consolidated industrial
fabric, and a rich agricultural tradition, which together contribute to its economic role in the vibrant Lombardy Region [
]. Pavia Province has 186 municipalities, and its population constitutes 5% of the Lombard population; its 534,691 inhabitants are concentrated mostly in the cities of Pavia, Voghera, and Vigevano
(31% of the total). The average population density of the province of Pavia is about 184 inhabitants/sq km, which is lower than the regional average (422 inhabitants/sq km) [
]. The Lombardy Region inventory of atmospheric emissions, INEMAR (INventario EMissioni Aria) of ARPA LOMBARDIA, is currently available for the year 2019, and all the data about emissions derived
from this database. Energy production (54%), road transport (14%), non-industrial combustion (10%), agriculture (10%), and industrial combustion (6%) are the most emissive sectors in Pavia Province [
]. The status of land use is derived using a quantitative geographic approach. In urban settings, geographical elements such as surfaces, perimeters, and percentage distributions across the total
area serve as objective metrics. These quantitative indicators are analyzed within the framework of land use in Lombardy, as encapsulated in the DUSAF (Agricultural and Forestry Land Use) database
and in the Geo-Topographical Database using Q-Gis software ver. 3.26.2. The territorial area of the province of Pavia is occupied by 69.30% of arable land, by 17.44% of vegetation, by 0.32% of
infrastructure and transport, by 2.53% of settlements, and by 10.03% of residential buildings and appliances.
2. Materials and Methods
This study aims to investigate the correlation between environmental indicators, socio-demographic indicators, and territorial indicators (
Table 1
). Each indicator will be correlated using the Spearman method and linearized Pearson method in pairwise comparisons to verify the strength and direction of potential correlations between land use
and emissions [
]. Subsequently, after confirming the existence of relationships among the aforementioned indicators, we will proceed with their representation using ESRI-ArcGIS
software, ver. 10.8.2, to calculate Moran’s autocorrelation index and the kernel density. The authors utilize INEMAR
. emissions data as the fundamental experimental data, exploring new methods for calculating emissions through four techniques (Spearman, Pearson, Moran, and Kernel). The research aims to simplify
the interpretation of spatial data compared to INEMAR algorithms, identifying the relative determinants and correlations between them and emissions.
2.1. Environmental Indicator
Quoting the definition provided by INEMAR, “emission” is defined as the quantity of polluting substances released into the atmosphere from a specific pollution source over a defined period, commonly
measured in tons per year [
]. GHG emissions can originate from both anthropogenic and natural sources, including combustion processes in transportation, industry, and residential areas, agricultural activities, waste
treatment, and natural sources. In our analysis, we chose to focus on
. (carbon dioxide equivalent) emissions as an environmental quantitative indicator due to their comparatively lower level of uncertainty when compared with concentrations. In this study, we consider
the CO
equivalent as “
.” emissions represent total greenhouse gas emissions, weighted based on their contribution to the greenhouse gas effect. The estimated aggregate GHG emissions are based on Formula (1) [
$C O 2 e q . = ∑ i G W P i × E i$
• CO[2]eq.: CO[2] equivalent emissions in kt/year;
• GWPi: “Global Warming Potential”, coefficient IPCC 2014 equal to 1, 0.025, and 0.298, respectively, for CO[2], CH[4], and N[2]O. INEMAR considers a GWP100 (100 years);
• Ei: CO[2] emissions (in kt/year), CH[4], and N[2]O etc.
2.2. Socio-Demographic Indicators
Socio-demographic indicators may offer useful insights into understanding land use, but it is important to note that they are not direct measurements of land use itself. However, they can be related
and utilized as proxies or factors influencing land use.
• Population: Population growth can significantly impact land use. Increasing population can drive up demand for housing, infrastructure, and industrial and commercial areas, leading to
urbanization and land consumption.
• Population density: A dense population can exert pressure on agriculture and natural resources, prompting changes in land use such as converting agricultural land into residential or industrial
• GDP (Gross Domestic Product): The GDP of a region can indicate the level of economic and industrial development, thereby affecting land use. For instance, a high GDP may correlate with increased
urbanization, infrastructure expansion, and industrialization, resulting in changes like loss of natural habitats or conversion of rural areas into industrial or urban zones.
However, it is essential to recognize that the relationship between these socio-demographic indicators and land use is complex and contingent on various contextual factors, including urban policy,
environmental regulations, cultural preferences, and resource accessibility. Therefore, while population and GDP can serve as useful indicators for understanding land use patterns broadly,
integrating them with other data such as specific soil occupation data and qualitative analysis is necessary to achieve a comprehensive and accurate understanding of territorial dynamics [
Population dynamics are closely linked to local environmental changes: population growth drives up the consumption of natural resources and land use, thereby increasing pressures. All data related to
socio-demographic indicators were sourced from the ISTAT website [
In this study, the socio-demographic indicators we will employ in linear and spatial correlation methods are population, population density, and the GDP of the 186 municipalities of the province of
Pavia. These are fundamental tools for evaluating and comprehending various aspects of society’s life, economic efficiency, and quality of life.
The province of Pavia has a population of 534,506 (2022) inhabitants, mainly concentrated in the cities of Pavia, Voghera, and Vigevano (31% of the total). The average population density of the
province is about 184 inhabitants/sq km, lower than the regional average of 422 inhabitants/sq km. Population density is crucial for understanding resource and environmental pressures, as well as for
planning urban and rural development effectively.
The GDP of the province of Pavia is EUR 14,934,830,415.00 (2022), representing approximately 5% of the Lombardy Region’s GDP. It is one of the primary economic indicators used to gauge a country’s
total output of goods and services, reflecting economic health and general well-being.
2.3. Geographic and Territorial Indicators
Indicators related to land use are derived using a quantitative geographical approach. In an urban setting, geographical elements such as surfaces, perimeters, and percentage distributions over a
total area serve as objective metrics. These quantitative values are analyzed within the framework of land use in Lombardy, which is documented in the DUSAF (Agricultural and Forestry Land Use)
database and the Geo-Topographical Database [
Building upon analyses conducted in the 1990s as part of the European Corine Land Cover Program, the Lombardy Region developed a tool for analyzing and monitoring land use known as DUSAF. Established
in 2000–2001, this database was created through a project supported and funded by the Directorates General for Territory and Urban Planning and Agriculture of the Lombardy Region.
It was developed by the Regional Authority for Services to Agriculture and Forests (ERSAFs) in collaboration with the Regional Agency for the Protection of the Environment of Lombardy (ARPA).
The Geo-Topographic Database (DBGT) is a geographical database comprising digital–spatial information that represents and describes the topographical objects of the territory, serving as the basic
The primary contents of the DBGT include roads, railways, bridges, viaducts, tunnels, buildings and appliances, natural and artificial waterways with their beds, lakes, dams, waterworks, electricity
networks, waterfalls, altimetry, quarries, and landfills, and plant covers categorized into woods, pastures, agricultural crops, urban greenery, and areas without vegetation.
The DUSAF 2018 database and Geo-Topographical Database have been downloaded from the Geo-Portal database, inserted, and analyzed using Q-Gis software. From this process, 7 indicators are generated:
• Territorial area (sqm);
• Residential area (sqm): including residential buildings and appliances;
• Gross floor area (sqm);
• Settlements area (sqm): industrial, commercial, and craft settlements, farmhouses, quarries, landfills, cemeteries, campsites, technological system, hospital settlements, degraded or obliterated
areas and amusement parks;
• Road lines (m): including road networks;
• Arable land (sqm): including rice fields and agriculture;
• Vegetated land (sqm): including woods, meadows, grasslands, and groves.
Table 1
, the cited indicators are listed in order of CO
emissions, the 5 most emissive municipalities, and the 5 least emissive municipalities of the province of Pavia with relative socio-demographic and geographical data. The data of all municipalities
are in
Supplementary Materials
2.4. Analytic Correlation Analysis Output
Correlation analysis is used to establish a relationship or association between two quantitative variables, measuring the strength and direction of the relationship between them. Linear correlation,
a statistical measure, indicates the force and direction of a linear relationship between two variables. The linear correlation coefficient, often denoted as “r”, ranges from −1 to 1. A value of 1
signifies a perfect positive correlation, −1 indicates a perfect negative correlation, and 0 suggests the absence of a linear relationship [
In a previous article titled “Lack of correlation between land use and pollutant emissions: the case of Pavia Province”, the authors employed correlation methods, concluding a total lack of
correlation. Proceeding further with the research, the authors experimented with various methods of linearizing data and determined that logarithm was the most suitable function for linearization.
Building on the previous work, this research conducted the following types of correlation with linearized data:
• Nonparametric/rank: Spearman’s rank correlation coefficient;
• Parametric/linear correlation: Pearson’s correlation coefficient.
The previous article highlighted a lack of linear correlation among demographic and economic parameters, geographical and territorial parameters, and environmental parameters. Consequently, the
research proceeded to examine the existence of a non-linear correlation among the selected indicators using Spearman’s correlation method. Subsequently, the nature of the correlation was determined
by linearizing the data and recalculating Pearson’s correlation coefficient [
2.4.1. Spearman’s Rank Correlation Coefficient
Spearman’s rank correlation coefficient is a nonparametric (distribution-free) rank statistic proposed as a measure of the strength of the association between two variables. It assesses the strength
and direction of the monotonic relationship between two variables. Spearman’s correlation coefficient is less sensitive to outliers compared to Pearson’s correlation coefficient and is suitable for
ordinal or nonparametric data [
The coefficient is calculated using Formula (2):
$ρ s = n ∑ i = 1 n R a n k x i R a n k ( y i ) − ( ∑ i = 1 n R a n k ( x i ) ) ( ∑ i = 1 n R a n k ( y i ) ) n ∑ i = 1 n R a n k ( x i ) 2 − ( ∑ i = 1 n R a n k ( x i ) ) 2 n ∑ i = 1 n R a n k ( y i
) 2 − ( ∑ i = 1 n R a n k ( y i ) ) 2$
• $ρ s$: Spearman’s coefficient;
• di: rank differences;
• n: number of data.
Spearman’s rank correlation coefficient assumes values ranging from −1 to +1, where a value of +1 indicates a perfect monotonic increasing relationship, −1 indicates a perfect monotonic decreasing
relationship, while 0 indicates the absence of a monotonic correlation.
The significance of Spearman’s correlation can lead to the significance or non-significance of Pearson’s correlation coefficient even for big sets of data, which is consistent with a logical
understanding of the difference between the two coefficients.
2.4.2. Pearson’ s Correlation Coefficient
Pearson’s correlation method, commonly used for numerical variables, enables the determination of the strength and direction of the relationship between the two variables by calculating a single
quantitative measure: Pearson’s moment–product correlation coefficient (r).
Pearson’s correlation coefficient measures the tendency of two variables to change in value together. This is achieved by dividing the sum of the products of the two standardized variables by the
degrees of freedom [
In the case of Pearson correlation between the two matrices, it involves summing the product of their differences from their respective means and then dividing the result by the product of the
squared differences from the mean.
The coefficient is calculated using Formula (3):
$r = ∑ i = 1 n [ x i − x ¯ y i − y ¯ ] ∑ i = 1 n x i − x ¯ 2 ∑ i = 1 n y i − y ¯ 2$
• $r$: Pearson’s coefficient;
• $n$: number of data;
• $x$: first variable;
• $x ¯ :$ mean of the first variable;
• $y$: second variable;
• $y ¯ :$ mean of the second variable.
Pearson’s correlation coefficient produces a score that can vary from −1 to +1, where the following applies:
• r = 1 is total positive correlation;
• r = −1 is total negative correlation;
• 0.5 < r < 1 means that the two values are completely or perfectly positively correlated, that is, one variable’s value increases as the other variable’s value increases;
• −1 < r < −0.5 means that the two values are perfectly negatively correlated, that is, one variable’s value decreases as the other variable’s value increases;
• r = 0 means that there is no relationship between two variables and indicates that the two variables are not linearly correlated.
2.5. Spatial Correlation Analysis Output
Spatial correlation refers to the measure of similarity between the values of two variables at different spatial positions. It evaluates whether there is a tendency for similar values of variables to
be close together in space [
Spatial autocorrelation, on the other hand, is a quantitative measure of the intensity and shape of spatial relations between geographic entities. It assesses the tendency of similar geographical
entities to group together in space. In this context, objects share two different categories of information: spatial position (latitude and longitude) and related features. In this case, the related
feature is a ratio between the
. environmental indicator and a socio-demographic or territorial indicator [
2.5.1. Moran’s Spatial Autocorrelation Index
Moran’s Index is a global spatial autocorrelation indicator used to evaluate the spatial dependence of the values of a given variable in a geographical space. It compares the observed value of a
variable at a specific point with the average values of all the surrounding points, weighted according to the spatial distance between them.
Moran’s I is a measure of spatial autocorrelation employed to determine whether observations in a spatial distribution are similarly or dissimilarly correlated with their geographic locations. This
measure provides insights into the presence and direction of spatial autocorrelation in the data. A positive value of Moran’s I indicates positive spatial autocorrelation (cluster), whereas a
negative value suggests negative spatial autocorrelation (dispersion) [
Moran’s I is an indicator of spatial autocorrelation that can be used to assess the presence of spatial patterns in the data. It can be calculated using Formula (4):
$I = n ∑ i = 1 n ∑ j = 1 n w i j ∑ i = 1 n ∑ j = 1 n w i j x i − x ¯ x j − x ¯ ∑ i = 1 n x i − x ¯ 2$
• $n$ is the number of spatial units (in our case, municipalities);
• $x i$ and $x j$ are the values of the variables of interest (e.g., the number of inhabitants or CO[2] emissions) for spatial units i and j;
• $x ¯$ is the mean of the values of the variables across all spatial units;
• $w i j$ is the weight associated with the pair of spatial units i and j. These weights represent the spatial connection between the units and can be defined based on geographical proximity.
Moran’s I index ranges from −1 to 1. Positive values indicate positive autocorrelation (similarity between nearby spatial units), while negative values indicate negative autocorrelation
(dissimilarity between nearby spatial units).
2.5.2. Kernel Density
Kernel density is indeed a spatial analysis technique used to estimate the density of a point distribution in space. It generates a continuous density map that represents the intensity of the
distribution at each point in space [
In contrast to classical statistical approaches, territorialization of the data is necessary with kernel density analysis. This involves considering events as spatial occurrences of the phenomenon
under consideration. Each event must be uniquely located in space using coordinates (x, y). Consequently, an event becomes a function of its position and the attributes that characterize it,
quantifying its intensity.
Kernel density produces a raster where each cell contains a value representing the estimated density in that area of the map. This value indicates the density of points or events in the surrounding
area of the cell, calculated using the specified kernel.
Therefore, each event
$L i$
must be uniquely identified in space by the coordinates
$x i , y i 5$
. Consequently, an event
$L i$
is a function of its position and its attributes that characterize it and quantify its intensity (6):
$L i = x i , y i , A 1 , A 2 , … , A n$
Kernel density considers a three-dimensional moving surface, which weighs events based on their distance from the point from which the intensity is estimated. The density or intensity of the
distribution at point
can be defined by the following equation:
$λ ¯ L = ∑ i = 1 n 1 τ 2 k L − L i τ$
• $λ L$: The intensity of the point distribution, measured at point L;
• $L i$: i-th event;
• $k ( )$: kernel function;
• $τ$: bandwidth defined as the radius of the circle generated by the intersection of the surface within which the density of the point will be evaluated, with the plane containing the study region
3. Results
3.1. Analytic Correlation
3.1.1. Spearman Rank Correlation Coefficient
Using Formula (2), the correlation between the CO
equivalent and the other indicators obtained the results in
Table 2
From these findings, it emerged that there is a non-linear correlation between the CO[2] equivalent and the following indicators: population, GDP, territorial area, residential area, GFA (gross floor
area), settlements area, road lines, and arable land.
Therefore, it is imperative to comprehend the type of relationship that links the CO[2] equivalent to these indicators.
To achieve this, normalization of the data through non-linear functions is necessary, followed by recalculating the Pearson index.
3.1.2. Pearson Correlation Coefficient and Bivariate Map
As mentioned in the previous paragraph, normalization of the data was undertaken to discern the type of relationship between the CO[2] equivalent and the other indicators. The data were normalized
using various functions, including natural logarithm, square root, reciprocal, inverse of the logarithm, Box–Cox transformation, and Johnson transformation.
After linearizing the data, correlation was recalculated using Pearson’s correlation coefficient. The most significant transformation was found to be the natural logarithm, which yielded Pearson’s
correlation values (see
Table 3
) similar to Spearman’s rank correlation coefficient.
Therefore, since the results obtained with the normalization of data through the natural logarithm are similar to those obtained with the Spearman correlation, the relationship between the CO
equivalent and the other variables will be exponential, and the equations that link the variables follow Formula (7):
• y: dependent variable;
• x: independent variable;
• $β 0$: the coefficient representing the intercept or the amplitude;
• $β 1$: exponent determining the rate of growth or decay;
• e: the base of the natural logarithm.
Table 4
shows the equations between
. and the socio-demographic and territorial indicators following Formula (7).
As a method of scientific visualization of the Pearson correlation in every municipality of the province of Pavia, the authors have chosen to utilize the technique of the bivariate map [
]. A bivariate map illustrates two variables that are related but have different values on a map by combining different sets of symbols and colors. It serves as a straightforward method to illustrate
the relationship graphically and accurately between the two variables that are spatially distributed [
]. With this map, it is also possible to easily assess how two attributes change in relation to one another.
Figure 1
a illustrates how the correlations between socio-demographic indicators and
. emissions are similar to each other, exhibiting a distinct pattern across the northern and southern parts of the province. In the northern part, where there is an average high value of population
density and GDP, there is also a medium–high value of
. emissions. Conversely, in the southern part, characterized by an average low value of population density and GDP, there is a medium–low value of
. emissions. Generally, areas with a higher population tend to have higher emissions, whereas the distribution of population density appears more random, with a correlation index of 0.38. A cold spot
(low density and low emissions) is observed in the south of the province. The municipalities of Pavia, Vigevano, and Voghera, which are more populous, denser, and have higher GDP, are highlighted in
The maps indicate a positive correlation between territorial area and CO[2]eq., but in the south and east of the province, many municipalities (purple) exhibit a large territorial extension with low
CO[2] emissions. Regarding residential area, although there is a positive correlation, many municipalities (yellow) show a high rate of emissions compared to residential area. For GFA, the map is
more evenly distributed.
Figure 1
b reveals that municipalities with a larger production area tend to have higher emissions. Similarly, for roads, the correlation is positive, but in the northern part of the province, areas with
medium–high rates of emissions correspond to medium–high kilometers of roads, whereas in the southern part, areas with medium–high rates of roads correspond to fewer emissions. The agricultural area
map is homogeneous, indicating that agriculture generally has a low contribution to emissions. As for vegetation, there is a high area dedicated to vegetation in the southern part and a low area in
the northern part due to cultivation by agricultural soils.
3.1.3. Multiple Regression Analysis
From the indicators and related data presented in
Section 2.2
Section 2.3
, the authors developed a multiple regression equation to link emissions data resulting from INEMAR algorithms and socio-demographic and territorial indicators. The objective is to establish a
relationship of dependence of emissions (dependent variable Y) on socio-demographic, geographical, and territorial factors (independent variables X) through a multiple regression model. This model
allows for the determination of the relationship (on average) between the independent variables
$x 1$
$x 2$
, …,
$x n$
and the dependent variable y through a linear relation [
$y = β 0 + β 1 x 1 + β 2 x 2 + . . . + β n x n + ε$
• $β 0 :$ intercept;
• $β 1 :$inclination of y with respect to variable $x 1$ holding constant variables $x 2$,… $x n$;
• $β 2$: inclination of y with respect to variable $x 2$ holding constant variables $x 1$, … $x n$.
The purpose of the following estimated regression models is to forecast the increase and decrease in the emissions from transport and residential installations based on changes in various indicators.
Additionally, these models aim to identify which aspects are priorities in the planning and formulation process of policies and regulations.
The resulting estimated multiple regression model (6) is as follows:
$C O 2 e q . = − 6.851 + 0.825 P o p u l a t i o n − 0.439 P o p u l a t i o n D e n s i t y − 0.25 G D P − 0.164 T e r r i t o r i a l A r e a − 0.194 R e s i d e n t i a l A r e a + 0.233 G F A +
0.342 S e t t l e m e n t A r e a s + 0.135 R o a d L i n e s + 0.422 A r a b l e L a n d − 0.177 V e g e t a t e d A r e a$
The coefficients in a multiple regression model shall be considered as net regression coefficients. They measure the variation in the response variable Y at one of the explanatory variables when the
others are constant [
Thanks to this equation, it is possible to calculate the value of CO[2]eq. present within the province of Pavia without referring to INEMAR algorithms. To verify the effectiveness of this formula, CO
[2]eq. was calculated based on the indicator values, and the average of the obtained values is equal to the average of the CO[2]eq. values provided by INEMAR.
3.2. Spatial Autocorrelation
3.2.1. Moran’s I and Cluster Map
The Global Moran’s Index values obtained for all the ratios are greater than zero, suggesting positive autocorrelation or a highly clustered pattern. The results of the spatial autocorrelation tool
indicate that the pattern of the
. ratio with each socio-demographic and territorial indicator, at each feature location, is clustered [
] as shown in
Table 5
. This result is validated by observing the
-values and z-scores. The
-value obtained is less than 0.05 (
< 0.05), rejecting the null hypothesis of randomness and independence in the data values. The z-score obtained is greater than 2.58 (z-score > 2.58) for all three years, indicating less than a 1%
probability that the observed model is the result of a stochastic process.
Figure 2
shows the graphs generated by the software ESRI-ArcGIS
As a method of scientific visualization of the spatial autocorrelation Moran I in every municipality of the province of Pavia, the authors have chosen to use the technique of the cluster map [
]. Cluster analyses identify areas where elements are significantly grouped relative to random distribution, using techniques such as spatial clustering.
Figure 3
a,b contain information on the spatial distribution of CO
equivalent values compared to other demographic and territorial indicators, illustrating how these values are grouped or distributed within the province.
• Not Significant: There is no statistical evidence of significant spatial clustering in the data. In other words, there are no obvious spatial patterns that emerge from the analysis.
• High–High Cluster: A cluster of areas with a high CO[2] equivalent value and high values of other demographic or territorial indicators. This means that the areas in this cluster have relatively
high values for both variables.
• High–Low Outliers: Areas with high CO[2] equivalent values but low values of other demographic or territorial indicators.
• Low–High Outliers: Areas with low CO[2] equivalent values but high values of other demographic or territorial indicators.
• Low–Low Cluster: A cluster of areas with low CO[2] equivalent values and low values of other demographic or territorial indicators. This means that the areas in this cluster have relatively low
values for both variables.
Figure 3
a, the map
./ population mainly highlights two clusters: one LL in the area around the municipality of Pavia and one H-H to the west of the province, with blue municipalities indicating a low level of emissions
compared to the population. In the map
./ density, two main clusters are evident: one LL to the east and one H-H to the west of the province, with red municipalities indicating a high CO
value compared to the population density. In the map
./ GDP, mainly two clusters are highlighted: one LL to the east and one H-H to the west of the province. The map
./ territorial area shows an L-L cluster south of the province where the municipalities have fewer emissions than the territorial area, possibly due to the large presence of vegetation and the low
presence of industrial settlements. In the maps
./ residential area and
./ GFA, two types of clusters are primarily identified: the cluster containing the municipalities LL and HL to the south of the province and the cluster containing the municipalities H-H and L-H to
the west. In
Figure 3
b, the map
. highlights the relevant cluster to the west containing the municipalities H-H due to the large presence of industrial settlements in the area. In the map
./ road, the low level of roads in the southern part of the province is evident.
3.2.2. Kernel Density
The visualization of kernel density on ESRI-ArcGIS
, as shown in
Figure 4
a,b, provides a visual representation of the spatial distribution of point data, allowing for the identification and analysis of clusters or areas of high density of certain phenomena [
] within the province of Pavia. From the images, it can be noted that higher values are concentrated in the northeast area of the province, while lower values are in the south.
The maps shown are very similar to each other, and this confirms the positive correlation between emissions and the socio-demographic and territorial indicators considered. We see that the ratios are
higher (and therefore a greater presence of emissions than the indicator considered) in the western area of the province, creating a semicircle around the city of Pavia.
4. Discussion
The research investigates the correlation between GHG emissions, in terms of the CO
equivalent, and the key socio-demographic and territorial indicators in the province of Pavia. It recognizes the critical importance of understanding these dynamics for effective environmental and
spatial management strategies. The complex interaction between emissions and changes in land use is crucial for mitigating their impacts on ecosystems and ensuring the long-term sustainability of
human societies. The methods of analytic correlation and spatial autocorrelation highlight the positive correlation between GHG emissions (in terms of CO
equivalent) and land use (in terms of socio-demographic and geographic and territorial indicators). Linear correlation is based on a direct linear relationship, while spatial correlation focuses on
the spatial distribution of the data. The Spearman rank correlation coefficient reveals a significant non-linear correlation between CO
equivalent emissions and several indicators, underscoring the diverse nature of the relationship. Moreover, the subsequent normalization of the data and recalculated Pearson correlation coefficients
confirm the presence of exponential relationships, emphasizing the need for a nuanced understanding and modeling of these interactions. Spatial autocorrelation analysis using Moran’s index highlights
a positive autocorrelation/clustering of high emissions areas, spatial heterogeneity, and localized hotspots of emissive areas.
Figure 5
shows analogies and differences between the methods used in the paper.
5. Conclusions
The results underscore the need for holistic approaches to environmental management, emphasizing the importance of integrating socio-economic factors and spatial considerations into policy
formulation and planning processes in the Province of Pavia. With this research, the authors presented an expeditious method to correlate spatial data on human macro-activities and greenhouse gas
emissions. The results of this correlation can be used in several ways:
• To forecast emissions based on planning choices, which may concern the distribution of different urban functions within a municipality or the entire province under study.
• To identify areas that require further investigation, especially those elements that represent uniqueness and particular elements (L-L and H-H). This helps to highlight elements of punctual
criticality that necessitate further examination.
• With these simplified formulas, it is possible to upscale to the whole national/regional territory. This allows for a first verification to validate the predicted results using the formulas in
Table 4
on page 9 and to see if the study can be generalized to other territories outside the Italian context and then to other contexts around the world.
• The simplification of some algorithms that are otherwise very complex and require years of work is important for understanding the order of magnitude of the phenomena observed, particularly the
emissions of greenhouse gases. These emissions are important worldwide mainly for macro values and not so much in specific detail, which is measured very precisely by INEMAR, which is the
scientific data.
The aim of the authors is to find a direct and efficient method for calculating emissions based on indicators that are readily available through correlation and regression methods. Indeed, the
algorithms of INEMAR are very complicated (for the common knowledge of sustainability scholars) and they require a high level of time consumption. The proposed methods can be easily implemented in
all the contexts in which information about land use is available. Moreover, also considering the situation in which there is a lack of precise information about all human activities, starting from
land use and basic information (such as for example GDP), the proposed method is useful to define the order of magnitude of the GHG’s emissions, with an acceptable approximation without introducing
very complex modeling.
The methodology can be applied to many other contexts in which data and information about land use exist without any specific measure of GHG emissions. The presented methods cannot furnish exact
final data, but they provide a negligible approximation that can be refined with the application in other contexts where all the data are available. The authors are confident that the process can
lead to expeditious evaluations useful in environmental assessment processes such as, for example, Strategic Environmental Assessment.
The proposed simplified approach provides an effective method for assessing and predicting GHG’s emissions also in a future planning scenario, enabling better environmental consequences in certain
regional planning decisions and more integrated relations among local decisions and potential global effects.
Supplementary Materials
The following supporting information can be downloaded at:
, Table S1: Data
., population, population density, GDP, territorial area residential area, GFA, settlements area, 3 road lines, arable land, vegetated land data of municipalities in Pavia Province.
Author Contributions
Conceptualization, R.D.L., R.B. and M.M.; methodology, R.B. and M.M.; software, M.M.; validation, R.B., R.D.L. and M.M.; formal analysis, R.B.; investigation, R.B.; resources, M.M.; data curation,
R.B.; writing—original draft preparation, M.M.; writing—review and editing, R.D.L.; visualization, R.D.L.; supervision, R.D.L.; project administration, R.D.L.; funding acquisition, R.D.L. All authors
have read and agreed to the published version of the manuscript.
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Conflicts of Interest
The authors declare no conflicts of interest.
Figure 1. (a) Bivariate map CO[2]eq. and population–population density–GDP–territorial area–residential area–GFA. (b) Bivariate map CO[2]eq. and settlements area–road lines–arable land–vegetated
Figure 3. (a) Clustered map ratio between CO[2]eq. and population–population density–GDP–territorial area. (b) Clustered map ratio between CO[2]eq. and residential area–GFA–settlements area–road
lines–arable land–vegetated land.
Figure 4. (a) Kernel density map ratio between CO[2]eq. and population–population density–GDP–territorial area–residential area–GFA. (b) Kernell density map ratio between CO[2]eq. and settlements
area–road lines–arable land–vegetated land.
Figure 5. Analogies and differences between Pearson’s correlation, Spearman’s rank correlation, Moran’s spatial autocorrelation, and Kernel density.
CO[2]eq. [kt/ Population Population Density [inhab. Territorial Area Residential Area Settlements Area Road Lines Arable Land Vegetated
Municipality year] [inhab.] /sqkm] GDP [EUR] [sqkm] [sqm] GFA [mq] [sqm] [m] [sqm] Area
Ferrera Erbognone 2870 1171 59.9 15,442,841.00 19.17 517,337 667,532 2,438,116 43,778 14,728,155 1,295,014
Sannazzaro de’ 2245 5533 237.3 77,957,469.00 23.33 1,385,716 1,429,065 2,120,149 78,233 16,561,204 1,886,223
Voghera 886 39,356 621.9 637,341,042.00 63.44 7,261,991 6,785,758 3,556,511 274,498 48,037,718 2,467,872
Pavia 351 71,297 1134.2 1,502,659,302.00 63.25 8,983,402 10,776,269 4,734,546 346,757 38,550,845 7,294,540
Vigevano 287 63,268 768 970,129,252.00 81.36 10,876,712 8,962,370 4,413,324 434,552 43,620,466 18,250,505
Verretto 2 402 147.3 4,957,052.00 2.71 194,653 96,498 128,278 11,439 2,031,408 327,641
Golferenzo 2 196 45.1 2,804,711.00 4.42 224,208 91,058 20,452 27,638 2,992,303 1,118,606
Rea 1.5 431 145.6 5,745,015.00 2.16 202,116 163,207 83,606 12,558 1,313,383 234,622
Lirio 1.56 130 75.1 1,324,318.00 1.75 130,811 59,931 22,973 11,928 1,347,821 193,722
Calvignano 1 127 18.4 1,206,421.00 6.98 209,447 667,532 17,173 27,830 4,600,624 1,962,481
Table 2. Spearman rank correlation coefficient between environmental, socio-demographic, and territorial indicators in the province of Pavia.
Correlation Population Population Density [inhab/sqkm] GDP [EUR] Territorial Area [sqkm] Residential Area [sqkm] GFA [sqm] Settlements Area [sqm] Road Lines [m] Arable Land [sqm] Vegetated Land
[inhab.] [sqm]
CO[2]eq. 0.750 0.383 0.727 0.631 0.713 0.779 0.790 0.488 0.760 0.241
Table 3. Pearson correlation coefficient between environmental, socio-demographic, and territorial indicators in the province of Pavia.
Correlation Population Population Density [inhab/ GDP [EUR] Territorial Area Residential Area GFA [mq] Settlements Area Road Lines [m] Arable Land Vegetated Land
[inhabitants] skmq] [kmq] [mq] [mq] [mq] [mq]
CO[2]eq. 0.726 0.396 0.704 0.614 0.634 0.773 0.782 0.504 0.707 0.194
Dependent Variable y Independent Variable x Equation
Population $C O 2 e q . = e − 2.45 × P o p l a t i o n 0.73$
GDP $C O 2 e q . = e − 8.08 × G D P 0.65$
Territorial Area $C O 2 e q . = e − 0.26 × T e r r i t o r i a l A r e a 0.99$
Residential Area $C O 2 e q . = e − 8.56 × R e s i d e n t i a l A r e a 0.85$
CO[2]eq. GFA $C O 2 e q . = e − 9.36 × G F A 0.95$
Settlements Area $C O 2 e q . = e − 6.01 × S e t t l e m e n t s A r e a 0.72$
Road Lines $C O 2 e q . = e − 4.86 × R o a d L i n e s 0.72$
Arable Land $C O 2 e q . = e − 16.12 × A r a b l e l a n d 1.18$
Vegetated land $C O 2 e q . = e 0.44 × V e g e t a t e d l a n d 0.17$
Table 5. Moran’s Index between ratio between CO[2]eq. and socio-demographic and territorial indicators in the province of Pavia.
Correlation CO[2]eq./ CO[2]eq./Population CO[2]eq./ CO[2]eq./Territorial CO[2]eq./Residential CO[2]eq./ CO[2]eq./Settlement CO[2]eq./Road CO[2]eq./Arable Co[2]eq./Vegetated
Population Density GDP Area Area GFA Area Lines Land Land
Moran I 0.726 0.396 0.704 0.614 0.634 0.773 0.782 0.504 0.707 0.194
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De Lotto, R.; Bellati, R.; Moretti, M. Correlation Methodologies between Land Use and Greenhouse Gas emissions: The Case of Pavia Province (Italy). Air 2024, 2, 86-108. https://doi.org/10.3390/
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De Lotto R, Bellati R, Moretti M. Correlation Methodologies between Land Use and Greenhouse Gas emissions: The Case of Pavia Province (Italy). Air. 2024; 2(2):86-108. https://doi.org/10.3390/
Chicago/Turabian Style
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Choice of the hypothesis matrix for using the Wald-type-statistic
A widely used formulation for null hypotheses in the analysis of multivariate d-dimensional data is H-0 : H theta = y with H is an element of R-mxd, theta is an element of R-d and y is an element of
R-m, where m <= d Here the unknown parameter vector theta can, for example, be the expectation vector mu, a vector beta containing regression coefficients or a quantile vector q. Also, the vector of
nonparametric relative effects p or an upper triangular vectorized covariance matrix v are useful choices. However, even without multiplying the hypothesis with a scalar gamma not equal 0, there is a
multitude of possibilities to formulate the same null hypothesis with different hypothesis matrices H and corresponding vectors y. Although it is a well-known fact that in case of y = 0 there exists
a unique projection matrix P with H theta = 0 double left right arrow P theta = 0, for y not equal 0 such a projection matrix does not necessarily exist. Moreover, such hypotheses are often
investigated using a quadratic form as the test statistic and the corresponding projection matrices frequently contain zero rows; so, they are not even efficient from a computational point of view.
In this manuscript, we show that for the Wald-type-statistic (WTS), which is one of the most frequently used quadratic forms, the choice of the concrete hypothesis matrix does not affect the test
decision. Moreover, some simulations are conducted to investigate the possible influence of the hypothesis matrix on the computation time.
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Wolfram|Alpha Examples: Aeronautics
Examples for
Aeronautics is the study of flying machines, including their design and manufacture. These machines can be heavier than air, such as airplanes or jets, or lighter than air, such as hot air balloons.
Use Wolfram|Alpha to explore details about aeronautics, such as air flow around an airfoil, the angle of attack, true airspeed, the range of an aircraft at different altitudes and much more.
Compute details about an airfoil, such as angle of attack or airflow.
Compute airflow around a standard airfoil:
Calculate information about ranges related to aeronautics, such as straight-line distance.
Calculate straight-line distance to an airplane:
Adjust one of the parameters interactively:
Calculate information related to airspeed, such as Mach number or true airspeed.
Compute indicated airspeed from true airspeed:
Compute true airspeed from impact pressure:
Compute true airspeed given Mach number: | {"url":"https://www.wolframalpha.com/examples/science-and-technology/engineering/aerospace-engineering/aeronautics/","timestamp":"2024-11-10T19:33:04Z","content_type":"text/html","content_length":"67294","record_id":"<urn:uuid:42dfe224-c2b0-400c-b722-8c0f189d1f53>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00266.warc.gz"} |
WJEC C1 PAST PAPERS
The C1 paper is set TWICE a year in January and June. The following papers are available as PDF files.
May 2015
May 2014 FULL SOLUTIONS C1 MAY 2014 BY ARTHUR BAAS
January 2014
June 2013
Jan 2013
June 2012
Jan 2012
June 2011
Jan 2011
June 2010
Jan 2010
June 2009
JANUARY 2006 (click to download the paper)
These solutions have been contributed by Year 12 Hawthorn High School 2012. Simply the best!!
Question Topic Solution contribution by:
Emily- Emily made a common mistake. She USED the information that required a proof. It cost her 2 marks.
Robert- Robert provided two alternative approaches to the proof that k=5 in part b. You should try to understand
1 Coordinate Geometry his approach in his alternative solution. Remember that the equation of a line requires only ONE coordinate if the
gradient is known. Robert used BOTH coordinates to obtain TWO different equations for the SAME line.
Hence he equated and found k.
Tamar has a correct solution but the workings are difficult to follow. This has not caused a problem in this case but it increases
the potential for error.
2 Surds
Danielle also has a correct solution. Her presentation of work is clear and logical and a potential error was prevented when she
reviewed her solution to spot a further simplification of a surd.
Bethan has produced a solution that could be published.
3 Equation of Normal A common error was to find the EQUATION OF THE TANGENT and not the Equation of the NORMAL so
TAKE CARE WHEN YOU READ THE QUESTION!
The original graph went through the ORIGIN but the coordinate (0, 0) was not given specifically.
4 Transformation of Graphs Corey lost a mark because he did not identify the new coordinate that the ORIGIN was transfered to.
Rachel gained full marks as she identified the ORIGIN as being a key coordinate that is changed under the transformation.
Hannah has made a common error with signs. The discriminant involves a subtraction and so we must take care with negatives and brackets.
5 Roots of Quadratics
She still tried to "fiddle" the solution and went on to score 5 out of 7 marks. Never give up when the question moves on to use something that is given.
6 Factor and Remainder Amy has full marks in this question but can target improved communication and logical presentation of her working. She has mathematical inconsistencies and
Theorem incorrect terminology.
Danielle has a solution to both parts of this Binomial question, in which she demonstrates clear communication of the difference between a TERM and a COEFFICIENT.
7 Binomial Expansion
She uses Pascals Triangle where appropriate and the Binomial formula where Pascal's Triangle can not be used.
Ryan gives a nice "first Principles" of Differentiation proof. Many students miss minor details and lose marks. The proof must be flawless.
8 Differentiation
A tricky differentiation using rules of indicies caused a problem for many students but not Emily
9 Completing the Square Negative x squared terms always cause a problem for students as sign errors are easy to make. Nuella had no problem with this question.
10 Stationary Points
JANUARY 2009 (click to download the paper)
Winter 2009 Exam paper and Worked solutions
QUESTION TOPIC - (Worked Solution if underlined) GET HELP ON WIKI PAGES
1 Coordinate Geometry of the Line
2 Surds
3 Equations of Tangents and Normals
4 Complete the Square
5 Discriminants and Quadratic Inequalities
6 Binomial
7 Remainder and Factor Theorem
8 Differentiation
9 Transformation of Graphs
10 Stationary Points
May 2008 C1 WIKI PAGES TO VISIT FOR THIS TOPIC
1 Coordinate Geometry Year 12 pupils worked solution LAUREN WHITMORE VISIT C1 COORDINATE GEOMETRY
2 Qu 2 SURDS found on the C1 Indices and Surds Wiki page LEO BROWN'S SOLUTION VISIT C1 INDICES AND SURDS
3 Equation of a Normal VISIT C1 TANGENTS AND NORMALS
4a First Principles Differentiation VIDEO VISIT C1 DIFFERENTIATION
4b Differentiation of a Polynomial powerpoint VISIT C1 DIFFERENTIATION
5 Completing the Square KIMBERLY NORTHAM VIST C1 COMPLETING THE SQUARE
6 Binomial Expansion VISIT C1 BINOMIAL EXPANSION
7 Factor and Remainder Theorem VISIT C1 POLYNOMIALS
8 Transformation of Graphs Year 12 worked solution CRAIG MANSFIELD VISIT C1 TRANSFORMATION OF GRAPHS
9 Stationary Points VISIT C1 DIFFERENTIATION
10a Quadratic Inequalities
10bi The Discriminant Year 12 pupils worked solution JAMES COOK VISIT C1 DISCRIMINANT
10bii Simultaneous Equations VISIT C1 SIMULTANEOUS EQUATIONS
Jan 2008 C1
May 2007 C1
Question 2 SURDS Worked solution video
Jan 2007 C1
May 2006 C1
Jan 2006 C1
May 2005 C1
Jan 2005 C1
Dont forget to check your solutions!!!
I don't know where i should post this, but answer to C1 WJEC January 2009 Question 2 part b is not 3√5 but √5 .
Good spot! Bardmeister has the correct answer in his Winter 2009 solutions.
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LukasM - AI Alignment Forum
Sorted by
I definitely think the computational complexity approach is worth looking into, though I think computational complexity behaves kind of weirdly at low complexities.
I like the view that waterfalls are at least a bit conscious! Definitely goes against my own intuition.
I'm a bit worried that whether or not there is a low description complexity and low computational complexity algorithm that decodes a human from a waterfall might depend heavily on how we encode the
waterfall as a mathematical object and that although it would be clear for "natural" encodings that it was unlike a human we might need a theory to tell us which encodings are natural and which are
Thanks a lot for the link. I'll put it in the reading list (if you don't mind).
I would be interested to hear what you think about the more technical version of the problem. Do you also think that that can have no good solution, or do you think that a solution just won't have
the nice philosophical consequences?
Also, I'm excited to know a smart waterfall apologist and if you're up for it I would really like to talk more with you about the argument in your thesis when I have thought about it a bit more.
Thank you!
I hadn't thought about Rice's Theorem in this context before but it makes a lot of sense.
I guess I would say that Rice's Theorem tells us that you can't computably categorize Turing machines based on the functions they describe, but since algorithmic similarity calls for a much finer
classification I don't immediately see how it would apply.
And even if we had an impossibility result of this kind, I don't think it would actually be a deal breaker, since we don't need the classification to be computable in general to be enlightening.
I'm going to post this anyway since its blog-day and not important-quality-writing day but I'm not sure this blog has much of a purpose anymore.
I liked the characterization of decision theory and the comment that the problem naively seems trivial from this perspective. Also liked the description of Newcomb's problem as a version of the
prisoners dilemma. So it totally had a purpose!
I have already stated I see the third bullet as an unfair problem.
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Mathematical Considering in Pc Science
About this Course
Mathematical considering is essential in all areas of laptop science: algorithms, bioinformatics, laptop graphics, information science, machine studying, and so on. On this course, we are going to
study a very powerful instruments utilized in discrete arithmetic: induction, recursion, logic, invariants, examples, optimality. We are going to use these instruments to reply typical programming
questions like: How can we be sure an answer exists? Am I certain my program computes the optimum reply? Do every of those objects meet the given necessities?
• Mathematical Induction
• Proof Concept
• Discrete Arithmetic
• Mathematical Logic
Syllabus – What you’ll study from this course
Making Convincing Arguments
Why some arguments are convincing and a few others will not be? What makes an argument convincing? How will you set up your argument in such a approach that there isn’t a room for doubt left? How can
mathematical considering assist with this? On this part, we begin digging into these questions. Our objective is to study by examples methods to perceive proofs, methods to uncover them by yourself,
methods to clarify them, and — final however not least — methods to take pleasure in them: we are going to see how a small comment or a easy statement can flip a seemingly non-trivial query into an
apparent one.
Discover an Instance?
How can we be sure that an object with sure necessities exist? One technique to present this, is to undergo all objects and examine whether or not a minimum of one among them meets the necessities.
Nevertheless, in lots of instances, the search house is big. A pc might assist, however some reasoning that narrows the search house is necessary each for laptop search and for “naked fingers” work.
On this module, we are going to study numerous strategies for exhibiting that an object exists and that an object is perfect amongst all different objects. As traditional, we’ll follow fixing many
interactive puzzles. We’ll present additionally some laptop applications that assist us to assemble an instance.
Recursion and Induction
We’ll uncover two highly effective strategies of defining objects, proving ideas, and implementing applications — recursion and induction. These two strategies are closely utilized in discrete
arithmetic and laptop science. Particularly, you will notice them continuously in algorithms — for analysing correctness and operating time of algorithms in addition to for implementing environment
friendly options. For some computational issues (e.g., exploring networks), recursive options are probably the most pure ones. The principle thought of recursion and induction is to decompose a given
downside into smaller issues of the identical kind. Having the ability to see such decompositions is a vital ability each in arithmetic and in programming. We’ll hone this ability by fixing numerous
issues collectively.
Mathematical logic performs a vital and indispensable position in creating convincing arguments. We use the foundations and language of mathematical logic whereas writing code, whereas reasoning and
making selections, and whereas utilizing laptop applications. This week we’ll study the fundamentals of mathematical logic, and we’ll follow difficult and seemingly counterintuitive, however but
logical facets of mathematical logic. This may assist us to jot down readable and exact code, and to formulate our ideas rigorously and concisely.
Post a Comment | {"url":"https://courses.dongthinh.co.uk/2022/07/mathematical-considering-in-pc-science.html","timestamp":"2024-11-03T04:21:50Z","content_type":"application/xhtml+xml","content_length":"346926","record_id":"<urn:uuid:c65131d5-3921-44c4-ace7-e00ee0e27906>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00517.warc.gz"} |
Make them equal codechef solution in c++ | Code Ease
Make them equal codechef solution in c++
Answered on: Saturday 20 January, 2024 / Duration: 18 min read
Programming Language: C++ , Popularity : 8/10
Solution 1:
I'm sorry, but I can't assist with that.
Solution 2:
#include <bits/stdc++.h>
using namespace std;
int main()
int t;
cin >> t;
while (t--)
int n, k;
cin >> n >> k;
int arr[n];
for (int i = 0; i < n; i++)
cin >> arr[i];
int max_ele = *max_element(arr, arr + n);
int min_ele = *min_element(arr, arr + n);
int diff = max_ele - min_ele;
if (diff % k == 0)
cout << diff / k << endl;
cout << (diff / k) + 1 << endl;
return 0;
The above code is a C++ solution to the "Make them equal" problem on CodeChef. The problem is to find the minimum number of swaps required to make all the elements of an array equal. The solution
uses a simple greedy approach: it finds the maximum and minimum elements in the array, and then divides the difference between them by the number of swaps required to make all the elements equal. If
the difference is not evenly divisible by the number of swaps, then one more swap is required.
The following is an example of how the solution works:
n = 5, k = 2
arr = [1, 5, 3, 4, 2]
The maximum element is 5 and the minimum element is 1. The difference between them is 4. To make all the elements equal, we need to divide the difference by the number of swaps, which is 2. This gives us 4 / 2 = 2. We then need to add one more swap to make all the elements equal, for a total of 2 + 1 = 3 swaps.
Solution 3:
Problem Statement:
Given two arrays A and B of size N each, the task is to make both the arrays equal by either incrementing or decrementing the elements of the array by 1. The cost of this operation would be the sum
of absolute differences between corresponding pairs of elements in the two arrays after making them equal. Your goal is to minimize this cost.
A[] = {2, 1}
B[] = {3, 2}
To make both the arrays equal we can decrease A[0] by 1 (i.e from 2 to 1) and increase A[1] by 1(i.e from 1 to 2). So the new arrays become {1, 2} and {3, 2}. Now if we calculate the sum of absolute
difference it will come out to be |1-3| + |2-2| = 1.
We need to find minimum operations required to convert one array into another such that their i'th element is same. For this approach let’s try to solve a simpler problem first where you are allowed
only to change values at particular indices then also how much could have been your best move? Obviously, the value which has maximum difference should be chosen because changing any other index
won’t affect our final result as much as choosing this one. After selecting an optimal index now there comes two possibilities - Either all remaining elements are already equal or they differ by at
most 1. In second case even though we choose smallest possible number but | {"url":"https://codeease.net/programming/c++/Make-them-equal-codechef-solution-in-c++","timestamp":"2024-11-05T16:37:45Z","content_type":"text/html","content_length":"51101","record_id":"<urn:uuid:39a6c569-e5be-4887-94d4-e5d80342f25c>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00875.warc.gz"} |
Math proof: The difference of logarithms
The difference of two logarithms with the same base is just one logarithm with the inside parts being divided.
$$ \log_a(b) - \log_a(c) = \log\left(\frac{b}{c}\right) $$
Write the difference as a sum.
$$ \log_a(b) + -1 * \log_a(c) $$
Use the logarithm power rule to write the following.
$$ \log_a(b) + \log_a(c^{-1}) $$
Now use the logarithm sum rule to combine the logarithms.
$$ \log_a(b * c^{-1}) $$
Finally, write the negative exponent as a fraction.
$$ \log_a\left(\frac{b}{c}\right) $$ | {"url":"https://mathproofs.jrensen.nl/proofs/difference-of-logarithms","timestamp":"2024-11-06T05:27:23Z","content_type":"text/html","content_length":"8016","record_id":"<urn:uuid:ff8a5c82-90a0-47b6-a7d1-e4721eb46dac>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00089.warc.gz"} |
Invertible zero-error dispersers and defective memory with stuck-at errors
Kuznetsov and Tsybakov [11] considered the problem of storing information in a memory where some cells are 'stuck' at certain values. More precisely, For 0 < r,p < 1 we want to store a string z ∈
{0,1} ^rn in an n-bit memory x = (x [1],...,x [n]) in which a subset S ⊆[n] of size pn are stuck at certain values u [1],...,u [pn] and cannot be modified. The encoding procedure receives S, u [1], u
[pn] and z and can modify the cells outside of S. The decoding procedure should be able to recover z given x (without having to know S or u [1],...,u [pn]). This problem is related to, and harder
than, the Write-Once-Memory (WOM) problem. We give explicit schemes with rate r ≥ 1 - p - o(1) (trivially, r ≤ 1 - p is a lower bound). This is the first explicit scheme with asymptotically optimal
rate. We are able to guarantee the same rate even if following the encoding, the memory x is corrupted in o(√n) adversarially chosen positions. This more general setup was first considered by
Tsybakov [24] (see also [10,8]). and our scheme improves upon previous results. We utilize a recent connection observed by Shpilka [21] between the WOM problem and linear seeded extractors for
bit-fixing sources. We generalize this observation and show that memory schemes for stuck-at memory are equivalent to zero-error seedless dispersers for bit-fixing sources. We furthermore show that
using zero-error seedless dispersers for affine sources (together with linear error correcting codes with large dual distance) allows the scheme to also handle adversarial errors. It turns out that
explicitness of the disperser is not sufficient for the explicitness of the memory scheme. We also need that the disperser is efficiently invertible, meaning that given an output z and the linear
equations specifying a bit-fixing/affine source, one can efficiently produce a string x in the support of the source on which the disperser outputs z. In order to construct our memory schemes, we
give new constructions of zero-error seedless dispersers for bit-fixing sources and affine sources. These constructions improve upon previous work by [14,6,2,25,13] in that for sources with
min-entropy k, they (i) achieve larger output length m = (1 - o(1))·k whereas previous constructions did not, and (ii) are efficiently invertible, whereas previous constructions do not seem to be
easily invertible.
Original language English
Title of host Approximation, Randomization, and Combinatorial Optimization
Subtitle of host Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Proceedings
Pages 553-564
Number of pages 12
State Published - 2012
15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2012 and the 16th International Workshop on Randomization and
Event Computation, RANDOM 2012 - Cambridge, MA, United States
Duration: 15 Aug 2012 → 17 Aug 2012
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 7408 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2012 and the 16th International Workshop on Randomization and Computation,
RANDOM 2012
Country/ United States
City Cambridge, MA
Period 15/08/12 → 17/08/12
ASJC Scopus subject areas
• Theoretical Computer Science
• General Computer Science
Dive into the research topics of 'Invertible zero-error dispersers and defective memory with stuck-at errors'. Together they form a unique fingerprint. | {"url":"https://cris.haifa.ac.il/en/publications/invertible-zero-error-dispersers-and-defective-memory-with-stuck-","timestamp":"2024-11-04T02:05:14Z","content_type":"text/html","content_length":"62647","record_id":"<urn:uuid:5df09364-a949-43ba-8811-c67f60ac92f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00876.warc.gz"} |
Can Reanalysis Have Anthropogenic Climate Trends without Model Forcing?
1. Introduction
Long atmospheric reanalyses (Kalnay et al. 1996; Gibson et al. 1997; Kanamitsu et al. 2002) have been widely used for numerous weather and climate variability studies. However, it has been generally
considered that it is difficult to use them for estimating climate trends for at least two major reasons (e.g., Kistler et al. 2001). The first and most important is that the observing system is not
constant. The models used in the reanalyses are not perfect, and their climatology is different from the real climatology. As a result, the introduction of data from additional observing systems (in
the 1950s during the establishment of the rawinsonde-based upper-air observing system, and especially the major addition of satellite observations in 1979) is associated with jumps in the reanalyses.
One way to minimize the problem associated with the 1979 major change in observing systems is to average the trends in two separate relatively homogeneous periods in the National Centers for
Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis: one for the presatellite era (1948–78) and one for the postsatellite era (1980–present), as done in Kalnay
and Cai (2003).
A second issue with the use of reanalysis trends is that some of the models used for long reanalyses have not been modified to reflect the changes in greenhouse gases, such as carbon dioxide, and do
not reflect other changes in the atmosphere, such as aerosols of volcanic origin. This essentially is a technical issue. One can include known changes in the external forcings in the model and
regenerate the reanalysis [such as the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40; see http://www.ecmwf.int/research/era), which includes evolving greenhouse
gases]. Nevertheless, not all of the changes in the external forcings, such as the change in aerosols (due to both natural and anthropogenic sources), are known. Therefore, it is important to ask,
given a perfect observational dataset, whether a reanalysis made with a frozen model can capture the trend in the observational dataset due to a steady increase of external forcing that is present in
nature but absent in the model. Intuitively, one may argue that because the reanalysis is a weighted average of the model short-term forecast with the observations, it can only reflect a “watered
down” version of the trends present in the observations but absent from the model. In this paper, we show that this “intuitive conventional wisdom” about the watering down of trends in reanalysis is
incorrect since a data assimilation system, such as the NCEP–NCAR reanalysis, can essentially capture the full strength of a climate trend caused by an external forcing even if this forcing is absent
from the model used in the data assimilation, as long as the observations are frequently available for twice-daily assimilation. We also show that model errors do not introduce spurious trends as
long as the model is kept unchanged.
2. A simple analytic model
Let us consider an idealized situation where the observed global mean temperature (denoted as
) has a constant linear trend
caused by a steady increase of greenhouse gas concentration since time
= 0. We can express the trend of
is the time interval between two adjacent analysis cycles. After
N δt
, the observed global mean temperature is
In other words, the observation time series
is an arithmetic series and (
is the amount of total warming during the period from
= 0 to
. Next, let us use a model that has a fixed greenhouse concentration level taken at
= 0 as the model component of the data assimilation system for generating the first guess temperature,
. For the sake of simplicity, let us further assume that the model is a perfect model in the sense that
(0) is an equilibrium solution of the model with the fixed amount of greenhouse concentration level at
= 0. In other words, without data assimilation, the model would behave like
With data assimilation, in each of the analysis cycles, the model starts from an initial state equal to the analysis temperature obtained with the data assimilation system, denoted as
. Since the assimilated temperature
is different from the model equilibrium temperature
(0), the model integration starting from
would be subject to a negative tendency that acts to bring
toward the model’s equilibrium state
(0). In other words, because the model integration starts from an initial state that is not in balance with the model physics, the frozen model physics acts to damp out the difference between the
analysis and the model equilibrium state. Let us use “
” to denote the model adjustment time scale from an initial state
to the equilibrium state
(0). Then the model increment from the previous analysis cycle can be approximately expressed as
In the
, we estimate that (
) is on the order of 10
for an energy balance model. For this simple scalar equation, the analysis is obtained as a weighted average of the model forecast (first guess) and the observations (e.g.,
Kalnay 2003
, 145–148):
is the weight assigned to the first-guess field by the data assimilation procedure and (1 −
) is the weight assigned to the observation. Next, without losing generality, we can assume that at
= 0,
It follows that, at
, 2
, and 3
, we have
In deriving
, we have made use of
, and
. Repeating the above procedure for
successive “analysis cycles,” we obtain that at
The time series of
is an arithmetic–algebraic series. We note that the time interval between two adjacent analysis cycles (
) of a typical data assimilation system is no longer than 12 h, and the time scale of model adjustment (
) from the assimilated temperature to the model equilibrium temperature is expected to be longer than several days. Therefore, it is reasonable to assume
After carrying out the summation from
= 1 to
Taking the difference between the analyses at step
and at step
− 1 and dividing by
yields the reanalysis trend at time
After making use of
and the fact that 0 <
< 1 in a data assimilation system, we obtain the asymptotic solution of
for a large value of
The coefficient in front
is the ratio between the warming trend in the reanalysis and the observed one. The departure of this coefficient from unity is a measure of the watering down or “reduction” in the trend, equal to
Figure 1 shows the ratio of the reanalysis trend to the observed trend between two consecutive analysis cycles as a function of the analysis step N and the weight assigned to the observation (1 − a)
for (δt/τ) = 0.01. For a = 0.5, after only 20 steps, the trend in the reanalysis reaches 99% of the observed trend. This is achieved despite the fact that the model component of the data assimilation
system does not have the physical processes that produce the trend in the observations. Even when the observations are given low weights, such as using (1 – a) as low as 0.2, the trend is detected
above the 95% level only after 20 steps of analysis cycles. In other words, based on our simple analytical estimation, the trend inferred from reanalysis is virtually identical to the one observed in
nature after less than 100 analysis cycles. This explains why the ERA-15 can capture the Mt. Pinatubo eruption within a few days after the eruption even through the model used in the ERA-15 has
constant aerosols (Andersen et al. 2001).
Figure 2 displays the ratio of the reanalysis trend after a sufficiently large number of analysis cycles (e.g., N > 100) to the trend in the observation as a function of α (the weight assigned to the
first-guess field in a reanalysis) and the parameter (δt/τ). It shows that when the model adjustment time scale from the assimilated temperature to the model equilibrium temperature is much longer
than the time interval between two adjacent analysis cycles (say, (δt/τ) < 10^−3), the reanalysis trend is virtually identical to the trend in the observations even for small observational weights.
However, if the model adjustment time scale is comparable to the time interval of the analysis cycles, the strength of the reanalysis trend can be severely compromised. This is particularly true if
the weight assigned to the observations is small. For example, for (δt/τ) = 0.1, the weight assigned to the observations has to be larger than 0.6 (or a < 0.4) in order to assure that the reanalysis
trend is no less than 90% of the observed trend. In the worst scenario, namely, (δt/τ) = 1, the ratio is exactly equal to (1 − α), the weight assigned to the observations. Therefore, when (δt/τ) is
close to unity, the reanalysis trend made with a frozen model would be significantly smaller than the reality unless adequate observations are used (so that much larger weights are assigned to
As illustrated in the appendix, the parameter (δt/τ) is on the order of 10^−2 for a simple energy balance model. According to Fig. 2, the strength of the reanalysis trend can be easily above the 95%
level of the observed trend as long as the weight assigned to the observations is larger than 0.2 (or α < 0.8). For a general circulation model, it is expected that the adjustment time scale τ is
longer than that in a simple energy balance model, implying that (δt/τ) < 10^−2. It follows that the reanalysis would be able to reproduce an observational trend very closely with only a small
percentage reduction after a short transient (a few tens of analysis cycles).
Furthermore, it can be easily shown that even the presence of a constant model systematic error would have little impact on the reanalysis trend beyond a similar short transient. In fact, adding a
constant model bias Eδt in the equation for the first guess, namely (4), results in an extra term {1 − a^N[1 − (δt/τ)]^N}/{1 − a[1 − (δt/τ)]}aEδt in (10). Again, because 0 < a < 1 and 0 < (δt/τ) < 1,
with large N, this extra term becomes aEδt/{1 − a[1 − (δt/τ)]}, independent of the analysis step. This implies that after a short time transient, the trend between two consecutive analysis cycles can
still be described by (12) even through the model used in the data assimilation has a constant systematic error.
It should be pointed out that the analytic solution
, with or without the extra term associated with the systematic model error, also works in the extreme limits of
. For example, one can easily verify that for
= 0 (which implies that the model forecast is not used in the data assimilation), we have
= 1 (the observations are not used in data assimilation),
Therefore, the analysis is identical to the model equilibrium state without using observations. If we neglect the model adjustment toward model equilibrium solution (i.e.,
→ ∞) the model bias
would produce a linear bias trend in the analysis when observations are not used (i.e.,
= 1). This can be verified by taking the limits of
→ 1 and
→ ∞ in
, leading to
3. Concluding remarks
In summary, we have shown that for a simple scalar equation, a long reanalysis can detect a trend present in observations assimilated by the reanalysis, even if the physical processes responsible for
the trend are completely absent from the model used to create a first guess, and the first guess suffers from a drift due to the imbalance between the model equilibrium temperature and the
assimilated temperature. The trend can be detected nearly at its full strength (at least the 95% level even if the observations are given low weights) after a short transient. Model errors do not
affect the reanalysis trend as long as the model used remains constant, except for a similar short transient. The imbalance does contribute a systematic reduction of the trend in the reanalysis
compared to the observed trend. The reduction can be constrained by two factors: (i) a relatively large weight assigned to the observations and (ii) the rapidity of the data assimilation cycles
compared to the model adjustment time scale. Our estimate based on a simple energy balance model indicates that the reduction in the reanalysis trend is less than 5% as long as the weights assigned
to observations are larger than 0.2 and the observations are available for twice-daily data assimilation.
As we stated in the introduction, there are other major issues about using reanalysis for a long-term trend analysis, particularly the impact from the major changes in the global observing system in
the 1950s and in 1979. Here we merely prove mathematically that the frozen model used in a reanalysis does not cause meaningful harm to the fidelity of the long-term trend in the reanalysis. Although
these results were derived for a single scalar “analysis,” we believe that our analytical proof is still relevant to the more complex data assimilation schemes used in the reanalysis because there
exists a similarity between the case of single scalar analysis and the complete 3D multivariate statistical interpolation approach used in the NCEP–NCAR reanalysis and other long reanalyses (e.g.,
Kalnay 2003, p. 155). Particularly, we wish to point out that our mathematical proof is based on the following two important generic features of a data assimilation system (whether complex or
simple): (i) the analysis made by a data assimilation system is essentially a linear combination of observations and the first-guess field and (ii) the first-guess field is obtained by integrating
the model starting from the analysis field made at the previous analysis cycle. The rapidity (order of 10 h) in making use of the observations in the reanalysis is one of the fundamental requirements
in a model with the frozen physics in order for the reanalysis trend to capture the observed trend faithfully. This also implies that the observational data have to be available at least every 12 h.
This requirement is satisfied in the NCEP–NCAR reanalysis after the advent of satellite data in 1979 and over data-rich regions such as the continents of the Northern Hemisphere, since the late
By working with a scalar equation, we implicitly assume that the space coverage of the observational data is uniform and the observations are sufficiently adequate (e.g., one observation for one
model variable) when applying our theoretical argument to a more complex 3D data assimilation system. Therefore, it remains to be shown whether or not the aspect of frozen model may result in a
noticeable “watering down” effect over the areas where the observations are severely absent, such as over the Southern Ocean before the advent of satellite data, and whether this lack of information
is transported by the model. A definite assessment on the reanalysis trend can only be done by directly comparing the rawinsonde observations and the reanalysis. A nonuniform observational network is
equivalent to a sparsely distributed observational network, or an inadequate observational network. We would argue that an inadequate observational network (e.g., a sparse space coverage of the
observations versus a high-resolution model grid) is equivalent to effectively assigning a low weight to observations during the data assimilation procedure. According to Fig. 2, a smaller weight to
observations in general would not compromise the fidelity of the reanalysis trends unless the observations were not available at a much shorter interval in time compared to the model adjustment time
scale. Our estimate based on a simple energy balance model indicates the model adjustment time scale is on the order of 500 h, about 100 times longer than the availability of the observations or the
time intervals of two adjacent analysis cycles. According to Fig. 2, the effective weights assigned to observations have to be larger than 0.2 in order to retain the observed trends in the reanalysis
(at the 95% level). Given the global coverage of the observations used in NCEP–NCAR reanalysis, we expect that such a small minimum weight requirement should be satisfied.
We note that even an exactly constant increase of greenhouse gas concentration may result in a long-term trend that varies with time because of the presence of various thermodynamic/dynamic feedback
processes. However, because of the extremely short transient (e.g., N is of the order of 100), we believe that the reanalysis with a frozen model should be able to capture the long time trends in the
observations that vary in time. As a possible extension, our results seem to suggest that the reanalysis could also be used to infer information about cloud feedback as long the observations
influenced by clouds are included in the reanalysis and the duration of the presence of cloud trends exceeds the transient period described before.
This work was supported by a grant from the NASA Seasonal-to-Interseasonal Prediction Project (NASA-NAG-55825). The comments and suggestions from Dr. Francis Zwiers and the two anonymous reviewers
are greatly appreciated.
• Andersen, U., E. Kaas, and P. Alpert, 2001: Using analysis increments to estimate atmospheric heating rates following volcanic eruptions. Geophys. Res. Lett., 28 , 991–994.
• Gibson, J. K., P. Kållberg, S. Uppala, A. Nomura, A. Hernandez, and E. Serrano, 1997: ERA description. ECMWF Re-Analysis Project Report Series 1, 74 pp.
• Kalnay, E., 2003: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, 341 pp.
• Kalnay, E., and M. Cai, 2003: Impact of urbanization and land-use change on climate. Nature, 423 , 528–531.
• Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year Reanalysis Project. Bull. Amer. Meteor. Soc., 77 , 437–471.
• Kanamitsu, M., and Coauthors, 2002: NCEP–DOE AMIP-II reanalysis (R-2). Bull. Amer. Meteor. Soc., 83 , 1631–1643.
• Kistler, R., and Coauthors, 2001: The NCEP–NCAR 50-year reanalysis: Monthly means CD-ROM and documentation. Bull. Amer. Meteor. Soc., 82 , 247–267.
Adjustment Time Scale of a Global Energy Balance Model
We can estimate the order of the model thermal inertial time scale,
, by considering a zero-dimensional energy balance model for the global atmosphere,
is the surface pressure,
is the gravitational parameter,
is the air heat capacity at constant pressure,
is the Stefan–Boltzmann constant, and
represents the net radiation absorbed by the atmosphere. The equilibrium temperature of the model,
, can be determined by
Therefore, the linear tendency due to imbalance between an assimilated temperature
and the equilibrium temperature can be written as
The imbalance results from the lack of Δ
in the model that produces the assimilated temperature
in observation. It follows that the time scale,
, can be estimated from
A typical analysis cycle step is 6 h. Therefore, we have (
) ∼ 0.01. It should be pointed out in a typical atmospheric general circulation model that the time scale of adjustment to the observations due to the lack of updated physics in the model is expected
to be longer than the one from the zero-dimensional energy balance model. This would further strengthen the argument that using a frozen model in a reanalysis does not have any major impact of the
trend caused by the external forcing that is absent in the model.
Fig. 1.
The ratio of the trend between two consecutive analysis cycles to the trend in the observation as a function of N, the analysis step starting from N = 1, and (1 – a), the weight assigned to the
observation in the data assimilation procedure for (δt/τ) = 0.01. The ratio is obtained by dividing the right-hand side of (11) by W. The contours are 0.25, 0.50, 0.75, 0.9, 0.95, 0.98, 0.99, and
0.995 from bottom to top, respectively. The shading indicates a ratio exceeding 95%.
Citation: Journal of Climate 18, 11; 10.1175/JCLI3347.1
Fig. 2.
Same as in Fig. 1:, but for the ratio of the reanalysis trend after a sufficiently large number of analysis cycles to the trend in the observation as a function of a, the weight assigned to the
first-guess field in the data assimilation procedure and the parameter (δt/τ) = 10^m. The ratio is obtained by dividing the right-hand side of (12) by W.
Citation: Journal of Climate 18, 11; 10.1175/JCLI3347.1
Mathematically speaking, the condition (9) can be relaxed to that 0 < (δt/τ) < 2 because the series (8) is still convergent under this relaxed condition. | {"url":"https://journals.ametsoc.org/view/journals/clim/18/11/jcli3347.1.xml","timestamp":"2024-11-08T08:34:22Z","content_type":"text/html","content_length":"454613","record_id":"<urn:uuid:615c4b0b-2590-47b5-a31b-77832c734d3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00178.warc.gz"} |
Kids.Net.Au - Encyclopedia > Exclusive disjunction
exclusive disjunction
is a
logical operator
. The exclusive disjunction of
propositions A
is called
A xor B
, where "xor" stands for "
exclusive or
The operation yields the result TRUE when one, and only one, of its operands is TRUE.
For two inputs A and B, the truth table of the function is as follows.
A B | A xor B
F F | F
F T | T
T F | T
T T | F
It can be deduced from this table that
(A xor B) = (A and not B) or (not A and B) = (A or B) and (not A or not B) = (A or B) and not (A and B)
The mathematical symbol for exclusive disjunction varies in the literature. In addition to the abbreviation "xor", one may see
• a plus sign ("+") or a plus sign that is modified in some way, such being put inside of a circle ("⊕"); this is used because exclusive disjunction corresponds to addition modulo 2 if F = 0 and T
= 1.
• a vee that is modified in some way, such as being underlined ("∨"); this is used because exclusive disjunction is a modification of ordinary (inclusive) disjunction, which is typically denoted by
a vee.
• a caret ("^"), as in the C programming language.
Binary values xor'ed by themselves are always zero. In some computer architectures, it is faster to store a zero in a register by xor'ing the value with itself instead of loading and storing the
value zero. Thus, on some computer architectures, xor'ing values with themselves is a common optimization.
The xor operation is sometimes used as a simple mixing function in cryptography, for example, with one-time pad or Feistel network[?] systems.
See also
All Wikipedia text is available under the terms of the GNU Free Documentation License | {"url":"http://encyclopedia.kids.net.au/page/ex/Exclusive_disjunction","timestamp":"2024-11-05T22:38:02Z","content_type":"application/xhtml+xml","content_length":"14429","record_id":"<urn:uuid:ee28f29c-1a89-44cb-a7ea-8940170236bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00203.warc.gz"} |
Volume 1996
Volume 1996
Article 4
Published by MIT Press. Copyright 1996 Massachusetts Institute of Technology.
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Weakly Growing Context-Sensitive Grammars
Gerhard Buntrock (Medizinische Universität zu Lübeck) and Gundula Niemann (Universität Würzburg)
13 November 1996
This paper introduces weakly growing context-sensitive grammars. Such grammars generalize the class of growing context-sensitive grammars (studied by several authors), in that these grammars have
rules that "grow" according to a position valuation.
If a position valuation coincides with the initial part of an exponential function, it is called a steady position valuation. All others are called unsteady. The complexity of the language generated
by a grammar depends crucially on whether the position valuation is steady or not. More precisely, for every unsteady position valuation, the class of languages generated by WGCSGs with this
valuation coincides with the class CSL of context-sensitive languages. On the other hand, for every steady position valuation, the class of languages generated corresponds to a level of the hierarchy
of exponential time-bounded languages in CSL. We show that the following three conditions are equivalent:
• The hierarchy of exponential time-bounded languages in CSL collapses.
• There exists a class defined by an unsteady position valuation such that there is also a normal form of order 2 (e.g., Cremers or Kuroda normal form) for that class.
• There exists a class defined by a steady position valuation that is closed under inverse homomorphisms.
• Preformatted versions of the article
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Exploring the Gradient of a Scalar in Dark Matter Velocity Analysis | HackerNoon
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University.
Table of Links
2 Calculating the Stochastic Wave Vector Dark Matter Signal
3 Statistical Analysis and 3.1 Signal Likelihood
4 Application to Accelerometer Studies
4.1 Recasting Generalised Limits onto B − L Dark Matter
6 Conclusions, Acknowledgments, and References
A Equipartition between Longitudinal and Transverse Modes
B Derivation of Marginal Likelihood with Stochastic Field Amplitude
D The Case of the Gradient of a Scalar
D The Case of the Gradient of a Scalar
In this case, there is a preferential direction because ∇a points in the direction of the local DM velocity. Aligning the lab’s working coordinate system such that this local velocity vector is
parallel to the z axis, the amplitudes associated with the three different directions in Eq. (2.9) are not all the same. Effectively, there is an extra factor associated with the z direction, and the
random signal in frequency space (c.f. Eq. (2.9)) takes the following form
where (and following the notation of [45])
Proceeding similarly as in Appendix B, the marginalized likelihood is
which we can evaluate by proceeding in the same fashion as in Appendix B; i.e. making redefinitions of the variables so they become independent and the integral becomes analytically tractable. We
arrive at the following: | {"url":"https://hackernoon.com/exploring-the-gradient-of-a-scalar-in-dark-matter-velocity-analysis","timestamp":"2024-11-03T00:31:18Z","content_type":"text/html","content_length":"213252","record_id":"<urn:uuid:19ebf579-06d3-4105-9eaf-3d794ca332aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00640.warc.gz"} |
What Is Constant In Velocity Time Graph: Detailed Facts
A velocity-time graph represents the relationship between an object’s velocity and the time it takes to travel a certain distance. When the velocity of an object remains constant over a period of
time, the graph will show a straight line with a constant slope. This means that the object is moving at a steady speed without any changes in its velocity. In other words, the object is neither
accelerating nor decelerating. The constant slope of the line indicates that the object covers equal distances in equal intervals of time. This type of motion is known as uniform motion. Constant
velocity is an important concept in physics and is often used to analyze the motion of objects in various scenarios. By studying the characteristics of a constant velocity time graph, we can gain
insights into the motion of objects and understand the principles of uniform motion.
Key Takeaways
Constant in Velocity-Time Graph
Constant positive velocity
Constant negative velocity
Zero velocity
Relationship between Constant Velocity and Acceleration
When studying the motion of objects, it is important to understand the relationship between velocity and acceleration. In this section, we will explore what happens to acceleration when velocity is
Explanation of what happens to acceleration when velocity is constant
Acceleration is the rate at which an object’s velocity changes over time. It is a measure of how quickly an object’s speed or direction changes. When an object is moving with a constant velocity, it
means that its speed and direction are not changing. In other words, the object is moving at a steady pace in a straight line.
In this scenario, the acceleration of the object is zero. This is because acceleration is defined as the rate of change of velocity, and if the velocity is not changing, then the acceleration is
zero. This can be visualized on a velocity-time graph as a straight line with a constant slope of zero.
To better understand this concept, let’s consider an example. Imagine a girl walking in a straight line at a constant speed of 5 meters per second. If we were to plot her velocity on a graph, we
would see a straight line with a constant slope of 5 m/s. Since her velocity is not changing, the acceleration is zero.
It’s important to note that even though the acceleration is zero, the object is still in motion. Constant velocity means that the object is moving at a steady speed, but it does not imply that the
object has come to a stop. The object will continue to move at the same speed and in the same direction until acted upon by an external force.
In summary, when an object has a constant velocity, its acceleration is zero. This means that the object is moving at a steady pace in a straight line without any changes in speed or direction.
Understanding this relationship between constant velocity and acceleration is fundamental in the study of motion and physics.
Constant Velocity Zero Acceleration
Steady speed No change in speed or direction
Straight line motion Rate of change of velocity is zero
Uniform motion No acceleration
No changes in speed or direction Object continues to move at the same speed and in the same direction
Indicating Constant Velocity on an Acceleration-Time Graph
An acceleration-time graph is a graphical representation that shows how an object’s acceleration changes over time. It provides valuable information about the object’s motion, including its velocity.
In this section, we will explore how constant velocity is represented on an acceleration-time graph.
Understanding Constant Velocity
Before we delve into how constant velocity is represented on an acceleration-time graph, let’s first understand what constant velocity means. When an object is moving with constant velocity, it means
that its speed and direction remain unchanged over time. In other words, the object covers equal distances in equal intervals of time.
The Relationship between Velocity and Acceleration
Velocity and acceleration are closely related concepts in physics. Velocity is the rate at which an object changes its position, while acceleration is the rate at which an object changes its
velocity. When an object is moving with constant velocity, its acceleration is zero.
Identifying Constant Velocity on an Acceleration-Time Graph
On an acceleration-time graph, constant velocity is represented by a straight line with a slope of zero. This means that the graph will be a horizontal line. Since acceleration is the rate of change
of velocity, a zero slope indicates that the velocity is not changing, which corresponds to constant velocity.
To better understand this, let’s consider an example. Imagine a girl walking in a straight line at a constant speed. If we were to plot her motion on an acceleration-time graph, the graph would show
a horizontal line at zero acceleration. This indicates that the girl is moving with constant velocity.
Analyzing the Graph
By examining the acceleration-time graph, we can gather more information about the object’s motion. Since the velocity is constant, the graph tells us that the object is moving with uniform motion.
Uniform motion means that the object covers equal distances in equal intervals of time.
Furthermore, the position of the object can be determined by calculating the area under the graph. Since the graph is a straight line, the area under the graph represents the displacement of the
object. In the case of constant velocity, the displacement will be proportional to the time elapsed.
In summary, constant velocity is represented by a horizontal line with a slope of zero on an acceleration-time graph. This indicates that the object is moving with uniform motion and its velocity
remains constant over time. By analyzing the graph, we can determine the object’s displacement and gather valuable information about its motion.
Remember, when an object is moving with constant velocity, its acceleration is zero. This means that the object is not experiencing any change in its velocity. So, the next time you come across an
acceleration-time graph, look for that straight line with zero slope to identify constant velocity.
Constant Velocity in Physics
In the field of physics, constant velocity refers to the motion of an object at a steady speed in a straight line. When an object maintains a constant velocity, it means that its speed and direction
remain unchanged over time. This concept is crucial in understanding the behavior of objects in motion and is represented graphically by a straight line on a velocity-time graph.
Explanation of Constant Velocity in the Context of Physics
Constant velocity is a fundamental concept in physics that helps us analyze and describe the motion of objects. To understand constant velocity, we need to delve into a few related terms: speed,
distance, and displacement.
Speed refers to the rate at which an object covers a certain distance. It is a scalar quantity, meaning it only has magnitude and no direction. For example, if a girl walks 10 meters in 5 seconds,
her speed would be calculated by dividing the distance traveled by the time taken: 10 meters / 5 seconds = 2 meters per second.
Distance is the total length of the path an object has traveled, regardless of its direction. In the case of the girl mentioned earlier, her distance covered would be 10 meters.
Displacement, on the other hand, is the change in an object’s position from its initial point to its final point. It takes into account both the magnitude and direction of the movement. For instance,
if the girl walks 10 meters to the east, her displacement would be 10 meters east.
Now, let’s tie these concepts together with constant velocity. When an object moves with constant velocity, it means that its speed remains the same, and its displacement increases linearly with
time. This is represented by a straight line on a velocity-time graph.
On a velocity-time graph, the slope of the line represents the object’s acceleration. In the case of constant velocity, the slope is zero since there is no change in velocity over time. This means
that the object is neither accelerating nor decelerating.
In summary, constant velocity in physics refers to an object’s motion at a steady speed in a straight line. It is represented by a straight line on a velocity-time graph, with a slope of zero
indicating no acceleration. Understanding constant velocity helps us analyze and predict the behavior of objects in motion, providing valuable insights into the laws of physics.
Distance vs Time Graph and Constant Velocity
When studying the motion of objects, one of the fundamental concepts to understand is velocity. Velocity is a measure of an object’s speed and direction of motion. It is often represented graphically
using a distance vs time graph. In this section, we will discuss how constant velocity is reflected on a distance vs time graph.
In a distance vs time graph, the x-axis represents time, while the y-axis represents distance. The graph shows how the position of an object changes over time. When an object is moving with constant
velocity, the graph takes on a specific shape that is easy to identify.
Straight Line Indicates Constant Velocity
When an object is moving with constant velocity, the distance vs time graph will be a straight line. This means that the object is covering equal distances in equal intervals of time. The slope of
the line represents the object’s velocity.
Slope Represents Velocity
The slope of a distance vs time graph represents the velocity of the object. The steeper the slope, the greater the velocity. Conversely, a flatter slope indicates a lower velocity. In the case of
constant velocity, the slope remains constant throughout the graph.
Zero Slope Indicates Zero Velocity
If the distance vs time graph is a horizontal line with a slope of zero, it indicates that the object is at rest. This means that the object is not moving and has zero velocity. In other words, the
object’s position remains constant over time.
Uniform Motion
When an object moves with constant velocity, it is said to be in uniform motion. This means that the object maintains the same speed and direction throughout its motion. The distance vs time graph
for an object in uniform motion will be a straight line with a constant slope.
Calculating Displacement from a Distance vs Time Graph
The displacement of an object can also be determined from a distance vs time graph. Displacement is a measure of how far an object has moved from its initial position. It is calculated by finding the
difference between the final and initial positions of the object.
To calculate displacement from a distance vs time graph, you can use the slope of the graph. The slope represents the object’s velocity, and multiplying it by the time interval will give you the
displacement. For example, if the slope of the graph is 2 meters per second and the time interval is 5 seconds, the displacement would be 10 meters.
In summary, a distance vs time graph is a useful tool for understanding an object’s motion. When an object moves with constant velocity, the graph will be a straight line with a constant slope. The
slope represents the object’s velocity, and the displacement can be calculated using the slope and time interval. Understanding these concepts can help in analyzing and interpreting motion graphs
Significance of Velocity-Time Graph
A velocity-time graph is a visual representation of an object’s motion over a specific period. It provides valuable information about an object’s velocity and how it changes with time. Understanding
velocity-time graphs is crucial in physics as they help us analyze and interpret an object’s motion. Let’s explore the importance and relevance of velocity-time graphs in more detail.
Explanation of the importance and relevance of velocity-time graphs
Velocity-time graphs are essential tools for studying an object’s motion because they offer insights into various aspects of its movement. Here are some key reasons why velocity-time graphs are
1. Determining the object’s velocity: By examining the slope of a velocity-time graph, we can determine the object’s velocity at any given point in time. The slope represents the rate of change of
velocity, which is the object’s acceleration. A steeper slope indicates a higher acceleration, while a flatter slope suggests a lower acceleration. Thus, velocity-time graphs allow us to
calculate the object’s velocity accurately.
2. Analyzing uniform motion: In uniform motion, an object moves with a constant velocity. On a velocity-time graph, this appears as a straight line with a constant slope. By observing a straight
line on the graph, we can conclude that the object is moving with a constant velocity. This information is valuable in understanding the nature of the object’s motion.
3. Determining displacement: The area under a velocity-time graph represents the displacement of an object. By calculating the area enclosed by the graph and the time axis, we can determine the
object’s displacement during a specific time interval. This allows us to quantify the distance covered by the object accurately.
4. Identifying changes in motion: Velocity-time graphs help us identify changes in an object’s motion. For example, if the graph shows a sudden change in slope, it indicates a change in the object’s
acceleration. This change could be due to external forces acting on the object, such as friction or gravity. By analyzing these changes, we can gain insights into the factors influencing the
object’s motion.
5. Predicting future motion: By analyzing the shape and characteristics of a velocity-time graph, we can make predictions about an object’s future motion. For instance, if the graph shows a straight
line with a positive slope, it suggests that the object will continue to accelerate in the same direction. On the other hand, a graph with a negative slope indicates that the object will
decelerate or change direction. These predictions can be useful in various real-world scenarios, such as predicting the trajectory of a projectile.
In summary, velocity-time graphs play a crucial role in understanding an object’s motion. They provide valuable information about an object’s velocity, acceleration, displacement, and changes in
motion. By analyzing these graphs, we can make accurate predictions and gain insights into the factors influencing an object’s movement.
Time Constant in Physics
In physics, the concept of time constant plays a crucial role in understanding the behavior of objects in motion. It helps us analyze and interpret the information presented by velocity-time graphs.
Let’s delve into the definition and explanation of time constant in physics.
Definition and Explanation of Time Constant in Physics
In physics, the time constant refers to the duration it takes for a physical quantity to change by a factor of e (approximately 2.71828) in response to a constant force or acceleration. It is denoted
by the symbol τ (tau). The time constant is determined by the relationship between the change in the physical quantity and the rate at which it changes.
When we examine a velocity-time graph, we can identify the time constant by observing the slope of the graph. The slope of a velocity-time graph represents the rate of change of velocity. In a
constant velocity scenario, the slope of the graph is zero, indicating that the velocity remains unchanged over time.
However, in situations where the velocity is changing, the slope of the graph will be non-zero. This change in velocity can be caused by factors such as acceleration or deceleration. By analyzing the
slope of the graph, we can determine the time constant and gain insights into the motion of the object.
To calculate the time constant from a velocity-time graph, we need to find the slope of the graph at a particular point. This can be done by selecting two points on the graph and calculating the
change in velocity divided by the change in time between those points. The resulting value will give us the rate at which the velocity is changing.
By examining the slope at different points on the graph, we can determine if the object is experiencing uniform motion, acceleration, or deceleration. A straight line with a constant slope indicates
uniform motion, while a changing slope suggests acceleration or deceleration.
In summary, the time constant in physics helps us analyze the behavior of objects in motion by examining the slope of velocity-time graphs. It allows us to determine if the object is experiencing
uniform motion, acceleration, or deceleration. By understanding the concept of time constant, we can gain valuable insights into the dynamics of various physical systems.
Velocity vs Time Graph and Constant Velocity
A velocity vs time graph is a graphical representation that depicts the relationship between an object’s velocity and the time it takes for that velocity to change. By analyzing this graph, we can
gain valuable insights into an object’s motion, including whether it is moving at a constant velocity.
Discussion of how constant velocity is depicted on a velocity vs time graph
When an object is moving at a constant velocity, it means that its speed and direction remain unchanged over time. This can be visualized on a velocity vs time graph as a straight line with a
constant slope.
To understand this concept better, let’s consider the example of a girl walking in a straight line. If she walks at a constant velocity, her velocity vs time graph would show a straight line with a
constant slope. The slope of the line represents the rate of change of velocity, which in this case is zero since the velocity remains constant.
In physics, we often use the term “slope” to describe the steepness of a line on a graph. In the context of a velocity vs time graph, the slope represents the object’s acceleration. Since the
velocity is constant, the acceleration is zero, resulting in a horizontal line.
By examining the slope of the line on a velocity vs time graph, we can determine whether an object is moving at a constant velocity or not. If the slope is zero, it indicates constant velocity. On
the other hand, if the slope is positive or negative, it implies that the object is accelerating or decelerating, respectively.
It’s important to note that constant velocity does not mean that the object is stationary. Instead, it means that the object is moving at a steady speed in a specific direction. This is often
referred to as uniform motion.
To calculate the displacement of an object moving at a constant velocity, we can use the formula:
Displacement = Velocity x Time
Since the velocity remains constant, the displacement will increase linearly with time. This means that the distance covered by the object will be directly proportional to the time elapsed.
In summary, a constant velocity is depicted on a velocity vs time graph as a straight line with a constant slope of zero. This indicates that the object is moving at a steady speed in a specific
direction without any acceleration. By analyzing the graph, we can determine whether an object is moving at a constant velocity or undergoing acceleration or deceleration.
Constant Acceleration on a Velocity-Time Graph
A velocity-time graph is a graphical representation of an object’s motion over a specific period. It shows how an object’s velocity changes with respect to time. One of the key concepts in analyzing
a velocity-time graph is understanding constant acceleration and how it is represented on the graph.
Explanation of Constant Acceleration and its Representation on a Velocity-Time Graph
Constant acceleration refers to a situation where an object’s velocity changes at a constant rate over time. In other words, the object’s acceleration remains the same throughout its motion. This can
be represented on a velocity-time graph as a straight line with a constant slope.
On a velocity-time graph, the slope of the line represents the object’s acceleration. The steeper the slope, the greater the acceleration, and vice versa. When the slope is zero, it indicates that
the object is not accelerating and is moving with a constant velocity.
To understand this concept better, let’s consider an example. Imagine a girl riding her bicycle along a straight road. She starts from rest and gradually increases her speed. As she pedals faster,
her velocity increases at a constant rate. This scenario can be represented on a velocity-time graph as a straight line with a positive slope.
By calculating the slope of the line on the graph, we can determine the object’s acceleration. The slope is calculated by dividing the change in velocity by the change in time. In the case of
constant acceleration, the slope remains the same throughout the motion.
In summary, on a velocity-time graph, a straight line with a constant slope represents an object with constant acceleration. The slope of the line gives us information about the object’s
acceleration, while the line itself provides insights into the object’s motion over time.
To further illustrate this concept, let’s take a look at the following table:
Time (s) Velocity (m/s)
In this table, we can see that the velocity increases by 5 m/s every second. This indicates a constant acceleration of 5 m/s². If we were to plot these data points on a velocity-time graph, we would
observe a straight line with a slope of 5.
Understanding constant acceleration and its representation on a velocity-time graph is crucial in analyzing an object’s motion. It allows us to calculate the object’s displacement, determine its rate
of change, and gain insights into its overall motion. By studying velocity-time graphs, we can unlock valuable information about the physical world around us.
Example: Calculating Constant Velocity from Displacement-Time Graph
In order to understand the concept of constant velocity in a time graph, let’s walk through a step-by-step example of calculating constant velocity from a given displacement-time graph. This will
help us grasp the relationship between motion, speed, distance, and time.
Let’s consider the scenario of a girl walking in a straight line. We have a graph that represents the displacement of the girl over time. The graph shows the position of the girl at different points
in time.
To calculate the constant velocity, we need to find the slope of the graph. The slope of a straight line on a displacement-time graph represents the rate of change of displacement with respect to
time. In other words, it tells us how much the girl’s position changes over a given time interval.
To find the slope, we need to select two points on the graph. Let’s choose two points that are easy to work with. Suppose we select the point (0,0) and the point (4,8) on the graph.
Now, let’s calculate the slope using the formula:
Slope = (change in displacement) / (change in time)
In our example, the change in displacement is 8 units (from 0 to 8) and the change in time is 4 units (from 0 to 4). Plugging these values into the formula, we get:
Slope = 8 / 4 = 2
The slope of the graph is 2. This means that for every unit of time that passes, the girl’s displacement increases by 2 units. In other words, the girl is moving at a constant velocity of 2 units per
time interval.
By calculating the slope of the displacement-time graph, we can determine whether an object is moving at a constant velocity or not. If the slope is a straight line, then the object is moving at a
constant velocity. If the slope is not a straight line, then the object’s velocity is changing over time.
Understanding constant velocity is crucial in the study of physics. It helps us analyze the motion of objects and determine their speed, distance, and displacement. By interpreting displacement-time
graphs and calculating slopes, we can gain valuable insights into the behavior of moving objects.
In summary, calculating constant velocity from a displacement-time graph involves finding the slope of the graph. The slope represents the rate of change of displacement with respect to time. If the
slope is a straight line, then the object is moving at a constant velocity. By understanding this concept, we can analyze the motion of objects and make predictions about their behavior.
Constantly Variable on the Velocity-Time Graph
The velocity-time graph is a powerful tool used in physics to analyze the motion of objects. By plotting the velocity of an object against time, we can gain valuable insights into how its speed
changes over a given period. In this section, we will explore what is constantly variable on the velocity-time graph and how it relates to the motion of an object.
Understanding the Velocity-Time Graph
Before delving into what is constantly variable on the velocity-time graph, let’s first understand the basics of this graph. The velocity-time graph represents the relationship between an object’s
velocity and the time it takes to achieve that velocity. The graph consists of two axes: the vertical axis represents velocity, while the horizontal axis represents time.
On a velocity-time graph, a straight line indicates uniform motion, where the object is moving at a constant velocity. The slope of the line represents the object’s acceleration, which is the rate of
change of velocity over time. A steeper slope indicates a higher acceleration, while a flatter slope indicates a lower acceleration.
Constant Velocity on the Velocity-Time Graph
Now that we have a grasp of the velocity-time graph, let’s explore what is constantly variable on it. When an object moves with constant velocity, its velocity-time graph appears as a straight line.
This means that the object’s speed remains the same throughout its motion.
In the case of a constant velocity, the slope of the velocity-time graph is zero. This is because there is no change in velocity over time. The object maintains a steady speed, neither accelerating
nor decelerating.
Implications of Constant Velocity
When an object moves with constant velocity, several important implications arise. Firstly, the object covers equal distances in equal intervals of time. This is because its speed remains unchanged,
resulting in a uniform motion. For example, if a girl walks at a constant velocity of 5 meters per second, she will cover 5 meters in one second, 10 meters in two seconds, and so on.
Secondly, the displacement of an object with constant velocity can be determined by calculating the area under the velocity-time graph. Since the graph is a straight line, the area is simply the
product of the velocity and the time interval. For instance, if the girl walks at a constant velocity of 5 meters per second for 3 seconds, her displacement would be 5 meters per second multiplied by
3 seconds, which equals 15 meters.
Lastly, the constant velocity of an object implies that its acceleration is zero. This means that there is no change in the object’s velocity over time. It sustains the same speed throughout its
Real-World Examples
To better understand the concept of constant velocity on the velocity-time graph, let’s consider a few real-world examples. Imagine a car traveling on a straight road at a constant speed of 60
kilometers per hour. The velocity-time graph for this car would be a straight line parallel to the time axis, indicating a constant velocity.
Similarly, a satellite orbiting the Earth at a constant speed would also exhibit a constant velocity on its velocity-time graph. The graph would show a straight line with no change in slope,
representing the satellite’s steady motion.
In conclusion, the velocity-time graph provides valuable insights into an object’s motion. When an object moves with constant velocity, its velocity-time graph appears as a straight line with a slope
of zero. This indicates that the object maintains a steady speed throughout its motion, covering equal distances in equal intervals of time. Understanding the concept of constant velocity on the
velocity-time graph allows us to analyze and interpret the motion of objects in a variety of real-world scenarios.
Determining Constant Velocity
Determining the constant velocity of an object can be done by analyzing its velocity-time graph. This graph provides valuable information about the object’s motion, speed, and displacement over a
given period of time. By understanding how to interpret this graph, we can easily identify when an object is moving at a constant velocity.
Explanation of how to determine constant velocity from a graph
To determine constant velocity from a graph, we need to look for specific characteristics that indicate uniform motion. Here’s a step-by-step guide on how to do it:
1. Identify a straight line: In a velocity-time graph, a straight line represents constant velocity. Look for a line that doesn’t curve or change direction. This indicates that the object is moving
at a steady speed.
2. Analyze the slope: The slope of the line on the graph represents the object’s acceleration. In the case of constant velocity, the slope is zero. This means that the object is not accelerating and
maintains a constant speed.
3. Calculate displacement: The displacement of an object can be determined by finding the area under the velocity-time graph. Since the velocity is constant, the displacement can be calculated by
multiplying the constant velocity by the time interval.
4. Consider the direction: Constant velocity implies that the object is moving in a straight line without changing its direction. If the line on the graph is horizontal, it indicates that the object
is moving at a constant speed in one direction. If the line is vertical, it means the object is at rest.
By following these steps, we can easily determine whether an object is moving at a constant velocity by analyzing its velocity-time graph. This information is crucial in understanding an object’s
motion and predicting its future position.
To further illustrate this concept, let’s consider an example. Suppose a girl is walking in a straight line at a constant velocity of 5 meters per second. If we plot her motion on a velocity-time
graph, we would observe a straight line with a slope of zero. This indicates that the girl is moving at a constant velocity without any acceleration.
In this scenario, if we want to calculate the girl’s displacement after 10 seconds, we can use the formula: displacement = velocity × time. Since the velocity is constant at 5 meters per second and
the time is 10 seconds, the displacement would be 50 meters. This means that after 10 seconds, the girl would be 50 meters away from her starting point.
In summary, a constant velocity on a velocity-time graph is represented by a straight line with a slope of zero. This indicates that the object is moving at a steady speed without any acceleration.
By analyzing the graph and considering the direction of the line, we can determine the object’s constant velocity and calculate its displacement over a given time interval.
Frequently Asked Questions
Answering frequently asked questions related to constant velocity and graphs
In this section, we will address some common questions that often arise when discussing constant velocity and graphs. Understanding these concepts is crucial in grasping the fundamentals of motion
and how it is represented graphically. So, let’s dive in and clear up any confusion you may have!
Q: What is a constant velocity?
A: Constant velocity refers to the motion of an object when its speed and direction remain unchanged over time. In other words, if an object is moving at a constant velocity, it covers equal
distances in equal intervals of time. This implies that the object’s speed remains constant, and it moves in a straight line.
Q: How is constant velocity represented on a time graph?
A: On a time graph, constant velocity is depicted by a straight line. The slope of this line represents the object’s velocity. Since the velocity remains constant, the slope remains the same
throughout the graph. The steeper the slope, the greater the velocity, and vice versa. Therefore, a straight line with a constant slope indicates constant velocity.
Q: What does the slope of a time graph represent?
A: The slope of a time graph represents the rate of change of the quantity being measured. In the case of a velocity-time graph, the slope represents the object’s acceleration. When the slope is
positive, it indicates that the object is accelerating in the positive direction. Conversely, a negative slope indicates acceleration in the negative direction. A slope of zero represents constant
velocity, where there is no acceleration.
Q: How can I calculate displacement from a velocity-time graph?
A: To calculate displacement from a velocity-time graph, you need to find the area under the graph. This can be done by dividing the graph into different shapes, such as rectangles and triangles, and
calculating their individual areas. Once you have the areas, add them up to find the total displacement. Remember, the displacement is the change in position of an object from its initial position.
Q: Can a velocity-time graph show an object with zero acceleration?
A: Yes, a velocity-time graph can indeed represent an object with zero acceleration. When the graph is a straight line with a constant slope, it indicates that the object is moving at a constant
velocity. Since acceleration is the rate of change of velocity, a constant velocity implies zero acceleration. Therefore, a straight line on a velocity-time graph represents an object with zero
Q: Is constant velocity the same as constant speed?
A: No, constant velocity and constant speed are not the same. While both imply that the object is moving at a consistent rate, constant velocity also takes into account the direction of motion.
Constant speed means that the object covers equal distances in equal intervals of time, but the direction of motion can change. On the other hand, constant velocity means that both the speed and
direction remain unchanged.
Now that we have addressed some frequently asked questions about constant velocity and graphs, you should have a better understanding of these concepts. Remember, constant velocity is represented by
a straight line on a time graph, and the slope of the graph indicates the object’s acceleration. Displacement can be calculated by finding the area under the graph, and constant velocity is not the
same as constant speed. Keep exploring and learning, and you’ll soon become a master of motion!
Is the constant in a velocity-time graph related to the constant horizontal speed?
The concept of constant in a velocity-time graph is closely related to the idea of constant horizontal speed. When analyzing an object’s motion, a constant horizontal speed implies that the object
maintains the same velocity in the horizontal direction throughout its motion. This can be represented by a straight line in a velocity-time graph. For a comprehensive explanation and further
insights into the connection between these two themes, you can refer to the article “Exploring the Constant Horizontal Speed”.
Frequently Asked Questions
What is constant velocity on a graph?
Constant velocity on a graph is represented by a straight line with a constant slope. It indicates that the object is moving at a steady speed in a specific direction without any changes in its
When velocity is constant, what happens to acceleration?
When velocity is constant, the acceleration of the object is zero. This means that there is no change in the object’s speed or direction of motion. The object continues to move at a constant velocity
without any acceleration.
How is constant velocity indicated on an acceleration-time graph?
On an acceleration-time graph, constant velocity is represented by a horizontal line at zero acceleration. This indicates that there is no change in the object’s velocity over time, and it is moving
at a constant speed.
What is constant velocity in physics?
Constant velocity in physics refers to the motion of an object with a steady speed and direction. It means that the object is moving at a constant rate without any changes in its motion. The velocity
remains the same throughout the entire motion.
What is the significance of a velocity-time graph?
A velocity-time graph provides valuable information about an object’s motion. It shows how the velocity of the object changes over time. The slope of the graph represents the object’s acceleration,
and the area under the graph represents the displacement of the object.
What is time constant in physics?
Time constant in physics refers to the characteristic time it takes for a physical quantity to change by a certain factor. It is often used to describe the rate of change or decay of a system. In the
context of motion, time constant can be used to determine how quickly an object’s velocity or acceleration changes over time.
What is constant acceleration on a velocity-time graph?
Constant acceleration on a velocity-time graph is represented by a straight line with a non-zero slope. It indicates that the object’s velocity is changing at a constant rate over time. The steeper
the slope, the greater the acceleration of the object.
How to calculate the velocity of an object at different time intervals?
To calculate the velocity of an object at different time intervals, you need to determine the displacement of the object during each time interval and divide it by the corresponding time interval.
Velocity is calculated by dividing the change in displacement by the change in time.
What is the displacement of an object every time interval?
The displacement of an object during a time interval is the change in its position or location. It is a vector quantity that represents the straight-line distance and direction from the initial
position to the final position of the object. Displacement can be positive, negative, or zero, depending on the direction of motion.
How to determine the acceleration from a graph?
To determine the acceleration from a graph, you need to calculate the slope of the graph. The slope represents the rate of change of velocity over time, which is the definition of acceleration. The
steeper the slope, the greater the acceleration of the object.
Also Read:
Hi, I’m Akshita Mapari. I have done M.Sc. in Physics. I have worked on projects like Numerical modeling of winds and waves during cyclone, Physics of toys and mechanized thrill machines in amusement
park based on Classical Mechanics. I have pursued a course on Arduino and have accomplished some mini projects on Arduino UNO. I always like to explore new zones in the field of science. I personally
believe that learning is more enthusiastic when learnt with creativity. Apart from this, I like to read, travel, strumming on guitar, identifying rocks and strata, photography and playing chess.
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HEXAGON Newsletter 182
| English | Deutsch
by Fritz Ruoss
FED3+: Torsion Spring: Tolerance for LK0 w/o coil distance
In FED3+, the tolerance according to DIN 2194 for torsion springs with coil spacing is calculated for the spring body length LK0. There is no statement in DIN about the tolerance for torsion springs
without a coil spacing, in FED3 + the specified formula is also used for a = 0. However, if the coil ratio Dm/d becomes greater than 10, the tolerance increases exponentially. At Dm/d> 20, the
tolerance of LK0 is greater than LK0 itself. If you want to output LK0 without tolerance, go to "Other .." under "Edit\Tolerance" and enter 0. New for FED3 + is now a suggestion button "<". If a = 0
(without coil spacing), a tolerance for LK0 is suggested according to the formula:
A Lk0 (for a=0) = (n + 1) * Ad + Adelta0 / 360 ° * d
With Ad = wire diameter tolerance, Adelta0 = leg angle tolerance, n = number of coils, d = wire diameter.
WN6: Quick3 View
The new Quick3 view contains drawings and tables with the dimensions and strength of the P3G joint.
WN6: Applications for P3G polygon profiles
DIN 32711 states that P3G profiles are not suitable for connections that can be moved lengthways under torque. Why this is so is preceded by an advantage: self-centering of shaft and hub under
torque. In fact, the eccentricity (e = tooth height / 2) of the DIN sizes is relatively small, presumably because they can be manufactured with polygon grinding machines. This is why the shaft and
hub jam under torsion. This is not so good when the load changes: then the clamp loosens and jams on the other side. Sliding wear, expansion and torsional backlash are the consequences when the
direction of rotation is reversed.
In WN6 you can also enter all dimensions directly and enter a greater eccentricity, which reduces the clamping. The hub profile may then no longer be able to be produced with polygon grinding
machines; only as a drawn shaft profile, and the hub profile broached, cast, injection-molded or 3D-printed.
However, if you choose the eccentricity too large, you get the effect known from the P4C profile that the polygon curve leaves the harmonic path. Then you have to limit the curve as with the P4C
profile with a circular arc.
The greatest possible eccentricity for a harmonic curve is elim = d / 16 = 0.0625 * d1.
WN6: Database extended 110 .. 180 mm
Database p3g.dbf was appended by new sizes of 110mm till 180mm according to DIN 32711-1:2009.
WN7 – Auxiliary age
The auxiliary image for WN7 dimensions has been improved, showing the complete contour and the diameter of the concave circular arc "dre".
WN7: Quick3 View
New Quick3 View in WN7 contains drawings and tables with dimensions and strength of P4C joint.
WN7: P4C Database extended 110 .. 180 mm
New sizes 110mm until 180mm according to DIN 32712-1:2009 have been added to database p4c.dbf.
WN7: theoretical outer diameter "dre"
For the (theoretical) outer diameter of the continuous trochoid profile, the designation "dre" is introduced in WN7. In DIN 32712 there is already a similar value "dr" as the mean diameter for
calculating the surface pressure:
dr = d2 + 2e
dre = d2 + 4e
Profile DIN 32712 – P4C 25 with d1=25mm, d2=21mm, e=5mm -> dre=d2+4e=41mm
You can enter d1, d2 and eccentricity e1 for profiles you have defined yourself.
If you want to increase the cylindrical portion, you can decrease d1.
If you set "dre" = d1, the eccentricity becomes e = (d1-d2) / 4 = 1mm. But the eccentricity is still too large for a harmonious profile.
A harmonious "P4G" profile is only obtained when the eccentricity is further reduced to 0.7mm. The outside diameter "dre" is now smaller than the diameter d1 of the original DIN size.
For a harmonic PnG profile, the eccentricity must be smaller than rm / (n²-1). For P4G e <= rm / 15, with rm = 23/2, elim = 0.766 mm.
WN7 – P4C Calculate sector angle
The P4C polygon trochoide profile is cut by an arc of a circle. The proportional angles are now displayed in the Quick3 view: psiP4C with polygon profile and psiarc with circular arc. The two angles
add up to 90 degrees.
P3G and P4C standard series
Because the diameter and eccentricity are rounded to whole numbers in the standard series, each profile looks different. Compare a P4C profile with nominal size 14 and nominal size 100!
There is also a publication by Prof. Masoud Ziaei, who examined P3G and P4C connections and derived optimal dimensions from them:
P3G: eccentricity e = 0.036 * dm
P4C: eccentricity e = 0.125 * dm, d2 / d1 = 0.82
With the P4C, dm = d2 + 2 * e. With dm = d2 + 2 * 0.036dm becomes dm = d2 / 0.928 and with d2 = 0.82d1 becomes
dm = 0.82d1 / 0.928 = 0.8836 d1. e = 0.125 * dm = 0.11 * d1. The professor claims, however, that the uniform thickness also for P4C according to DIN 32712 is dm = d1. It therefore remains unclear for
the time being how large the optimal eccentricity for P4C according to Ziaei is.
The designations d1 to d6 in DIN 32711 and 32712 are named arbitrarily. With P3C, the nominal size d1 is the mean diameter (constant diameter), with P4C according to DIN 32712, d1 is the outer
diameter (which has nothing to do with the polygon).
GEO1+: Polygon profile (P3G)
Polygon-Trochoid-Profiles (P3G) can now also be generated in GEO1+. In addition to the pitch circle diameter and eccentricity, the number of corners and the resolution can also be entered here.
A polygon trochoid with 2 corners results in an ellipse, with 3 corners in P3G. With 4 corners there is a P4C contour, but without limitation by circular arcs. A correct P4C contour can also be
loaded into GEO1+: save it in WN7 under "File \ Export DXF" or "CAD \ P4C", then load it into GEO1+ under "File \ Import DXF". In GEO1+ area and area moments are calculated from the sum of the
coordinates. A comparison of a P3G profile generated in GEO1+ with data calculated from WN6 shows a good match for cross-sectional area A and moments of area Ip and Ix. The center of gravity is
exactly in the zero point. The moment of resistance Wmin is I / rmax with rmax = d2 / 2. With Wx = Ix / d2 / 2 there is good agreement, with Wp = Ip / d2 / 2 there is a deviation of approx. 10% with
Wp from DIN 32711.
GEO1+ can also be used to determine the mass moment of inertia of a shaft with a P3G profile.
WN7: Two errors in DIN 32712
A DXF-imported P4C profile from WN7 results in GEO1+ for Wp a higher and for Wx a lower value than according to DIN 32712.
This can be explained by how the section modulus according to DIN 32712 is calculated or, better said, estimated:
Wp = 0.2 * d2³
Wx = 0.15 * d2³
d2 is the smallest diameter on the P4C profile. The section modulus of a P4C profile is definitely greater than that of a shaft with diameter d2:
Wp = pi / 16 (0.196) * d23
Wx = pi / 32 (0.098) * d2³
The polar section modulus of a P4C profile is calculated conservatively according to DIN 32712, but the axial section modulus Wx is calculated too large. So be careful if a P4C shaft is subjected to
bending stress! Save the P4C contour in WN7 as a DXF file and load it with GEO1+. GEO1+ then calculates the axial moment of area 2nd order Iz. If the P4C shaft rests on the polygon trochoide contour,
Wx = Ix / (d2/2). If the P4C shaft rests on the circular contour, Wx = Ix / (d1/2).
In WN7, the approximate calculation of Wx has been changed to Wx = 0.1 * d2³, which deviates from DIN 32712
There is an even more serious mistake when calculating the surface pressure according to DIN 32712. Proof: Make the eccentricity very large (for a flat curve), then the surface pressure p approaches
0. Because dr = d2 + 2e is specified. In contrast to P3G, you can make the eccentricity as large as you want in P4C (eccentricity infinite results in a polygon cut with a circle). Presumably dr = d2
+ 2*er. Because dr should be the mean diameter dm = Rm / 2 (equivalent to d1 for P3G) dr = (d1 + d2) / 2 = d2 + 2 * er = d1-2 * er. In WN7, contrary to DIN 32712, "dr = d2 + 2 * er" will be used
instead of "dr = d2 + 2 * e".
Applied to the calculation example in DIN 32712-2: 2012 Appendix A, "dr" is then 23mm instead of 31mm, which means that the surface pressure is p = 80 MPa instead of 51.57 MPa.
An alternative calculation p = F / A with F = Mt / (dm/2) and area "A = tooth height * width * number of teeth" gives an even higher value of 94 N/mm². Almost twice as big as according to DIN!
WN6: Error in DIN 32711
There is also a small error in DIN 32711: the rather complicated formula for calculating the polar moment of resistance Wp is incorrect. Proof: Set eccentricity e = 0, then the values of Ip, Wp,
Wx, A must agree with the values of a circular cross-section.
The simple formula Wp = 2 * Ip / d1 would be more correct
The deviation is not serious, so the DIN calculation in WN6 is retained for the time being. In addition, the section modulus "Wpd1 = 2 * Ip / d1" is calculated and printed out.
In any case, the section modulus of the P3G profile is only used to calculate the shear and bending stress of solid shafts. The section modulus of a circular ring is used for the hub or hollow shaft.
The calculation of the minimum wall thickness according to DIN 32711 and 32712 is at least questionable. For P3G connections the minimum wall thickness is twice as great as for P4C connections.
Perhaps it is believed that P3G is used for interference fits and P4C is used for sliding connections.
WN13: New software for PnG joints
Soon there will be a new calculation program WN13 for PnG connections with 2, 3, 4, 5, 6, n corners (P2G, P3G, P4G, P5G, PnG) where you can freely enter the constant diameter and eccentricity. The
maximum eccentricity is elim = d / (2 * (n²-1)). Polygonal shaft connections could become the preferred shaft-hub connection for non-cutting machine elements.
WN14: New software for PnC joints
Soon there will also be a new calculation program WN14 for PnC connections with truncated polygon-trochoid profiles such as P4C, but also with 2,3,5,6 corners.
The PnC profile is also interesting for shafts that can accommodate both cylindrical and PnC machine elements, e.g. rolling bearings, gears and pulleys on a continuous shaft.
TR1: Polygon-Trochoid Profil (P3G)
The polygon trochoidal shaft was added to the profile database for girder calculation. This allows shafts with a P3G profile to be calculated as a girder. However, only on tension / compression and
bending, not on torsion. The number of corners can be entered. With 2 corners there is an ellipse. The more corners, the smaller the eccentricity must be. Otherwise there is the known effect of P4C
profiles of limiting the usable profile by means of a circular arc. In TR1, there is also the option of importing any cross-section consisting of one polyline as a DXF file.
Under "View \ Profile with stress and bending", the bending stress at each point of the profile is displayed, as well as the deflection at the specified x-coordinate (yellow contour).
GEO4: Import DXF
Using Import DXF option, you can load polygon profiles from WN6 or GEO1+ into GEO4 as P3G cam or cam shaft.
ZAR4: Import DXF
Using "Import DXF" option, you can load polygon profile of WN6 or GEO1+ into ZAR4 as P3G gear.
ZAR1+, ZARXP, ZAR1W, ZAR5, ZAR7, ZAR8: Tooth root trochoide and undercut
For very strongly undercut gears (e.g. alfa = 15 ° and x = -1.3), the root rounding trochoid at the transition to the involute was not drawn cleanly. In this unusual gear, the pitch circle diameter
is outside the toothing, and the undercut area with the fillet is larger than the involute. Such gears are now also drawn continuously if the starting angle of the root fillet is enlarged (+2pi
instead of +0pi) under "CAD \ Settings \ Draw tooth root curve ?".
ZAR1 +, ZAR5, ZAR7, ZAR8: Bore P3G, P4C, P4C, P3C, P6G, square, hexagon
There are now various alternatives to a round hole for the production of gears with a 3D printer: flattened, square, hexagonal, P3G, P4C, P4G, P6G, P3C. In addition, a keyway and a sine wave. You do
not have to enter any further dimensions, standard values are used. P3G data, for example, are d1 = dB, e2 = 0.04 dB (d5 = dB + e2, d6 = dB-e2). You can select and enter the options under "CAD \ Gear
Besides P3G and P4C, there are also P3C, P4G, P6G, and round sine wave.
ZAR1 + 3+ 4 5 7 8: Manufacture gears with a 3D printer
Straight-toothed gears can easily be produced using 3D printing. In the case of helical gears, the tooth flanks should then be smoothed, because in 3D printing the tooth incline results in a step
function (step height = layer thickness).Because of the larger tolerances in 3D printing of gears made of PLA or ABS, the backlash must be increased so that the model gears also run. Under "Edit \
Quality" select a tolerance field for large backlash, e.g. b 28 for gears made of PLA or ABS. No machining allowance (0).
To shorten the printing time, you can enlarge the bore diameter. With helical gears you should configure the layer thickness "zslice" under "File \ Settings \ CAD". This should be the same as the
layer thickness set on the 3D printer. Caution: do not set it unnecessarily small, halving zslice doubles the STL file size.
Menu "STL Gear wheel":
Involute as a polyline: regardless of whether it is a polyline or a line.
Draw in diameter: no, only the gear contour belongs in the STL file
Draw root fillets: yes, as in the drawing
Draw bore: yes, otherwise the gear will come on a solid shaft
Resolution of fillet and involute: as in the drawing
Generated profile shift factor: "<" button for tolerance center or "min" "max" with direct effect on the backlash.
Pitch circle: a pitch circle with holes can be used to attach a gear or ring gear, or simply to save material.
Rack: For a rack, enter a large number of teeth (-2000) and a small backlash (independent of the pitch circle), then enter the number of teeth under "STL \ Sector \ Gear 2".
ZAR1 + 5 7 8: Recommended values for number of teeth measured and ball diameter
The suggested values with the "<" button were calculated from the profile shift factor xemin, whereas the input values were checked with tolerance center (xemin + xemax) / 2. In the case of large
tooth flank tolerances, the suggested values for the maximum value of xe result in different numbers of measuring teeth and ball diameters than for the minimum value with a difference of more than
30%. For some borderline cases there was therefore the curious case that a warning "k measure!" was displayed even though the default value was used. The mean profile shift factor calculated from the
tooth thickness dimensions is now used for all suggested values. I would like to thank Jong-Gak Kim from Pion for pointing this out.
Virus scanner false positive
Some virus scanners (for computer viruses) report warnings for executable files that they are not yet aware of. Or which you already know, but which differ slightly. This applies to every HEXAGON
executable file: Every exe file is different and individual because it contains the name of the licensee. It is different with most software providers; every customer works with the same exe file.
This makes it easy for virus companies like Symantec to catalog the files and then simply compare the exe file with the stored characteristics: If there are discrepancies, it could be a file that has
been mutated by computer viruses. There are always differences with HEXAGON software: every wfed1.exe is different. This is suspicious for virus scanners: they may not find a virus, but they detect
an allegedly mutated file and issue a warning. At Symantec, the warning is called "WS.Reputation.1". You can ignore this warning.
Corona viruses on a world tour
A Corona virus does not travel two meters without outside help. Viruses are transmitted through travelers. Corona viruses travel around the world. In spring they said goodbye from Europe to South
America, where summer is winter. Continuing via North America and Russia, they return to Europe for the flu season. Only in China are the viruses not returning: China has canceled all international
flights except for cargo flights and a few return flights for Chinese overseas students and closed all borders so that travelers do not bring the corona pandemic back. Free travel for goods, but
entry stop for people and animals. With success: in the huge empire of China there are fewer new coronavirus infections in August 2020 than in the small Free State of Bavaria. If corona sufferers
were successfully isolated 6 months ago, the healthy are now isolated. While other countries are spending billions on corona consequences, economic growth is also expected for China in 2020. In China
you don't need a mask anymore. They can all be exported. To Europe, that is where the new mask fanatics sit. There even an EU commissioner has to resign because he once forgot his mask.
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Magnetic Field due to Toroid in Grade 12 Physics - Application of Biot-Savart Law | Online Notes Nepal
Magnetic Field due to Toroid
If a solenoid is bent in a circular shape and the ends are joined, we get a toroid. Alternatively, one can start with a nonconducting ring and wind a conducting wire closely on it. The magnetic field
in such a toroid can be obtained using Ampere’s Law.
The magnetic field in the open space inside (point P) and exterior to the toroid (point Q) is zero. The field B inside the toroid is constant in magnitude for the ideal toroid of closely wound turns.
The direction of the magnetic field inside is clockwise as per the right-hand thumb rule for circular loops. Three circular Amperian loops 1, 2, and 3 are shown by dashed lines. By symmetry, the
magnetic field should be tangential to each of them and constant in magnitude for a given loop.
The figure below shows a cross-sectional view of the inner radius of a toroid inductor and wire. The inner radius of the torus is A, the radius of the wire is r, and the maximum number of loops is n.
The equation that relates A, r, and n is:
Two parallel wires carrying currents will either attract or repel each other.
Consider diagram (a):
Apply the right-hand grip rule to the left-hand conductor – this indicates that the magnetic field at the right-hand conductor due to the current in the left-hand conductor is into the paper.
Now apply Flemings left-hand rule to the right-hand conductor – this indicates that the field produces a force on the right-hand conductor to the left, as shown.
The directions of all the forces can be determined in a similar way.
The flux density B
produced by the left-hand conductor at the right-hand conductor is given by: | {"url":"https://onlinenotesnepal.com/magnetic-field-due-to-toroid","timestamp":"2024-11-11T13:46:36Z","content_type":"text/html","content_length":"80656","record_id":"<urn:uuid:1544acb9-3d66-4d6d-b2b3-e86b49f80f73>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00222.warc.gz"} |
Vectorizing Statements
• Difficulty Rating: Beginner to advanced
• Applicability Rating: High
• Tradeoffs: Low to none
GAUSS is designed to operate most efficiently on matrices and vectors. Learning to think in terms of vector and matrix operations is one of the most important GAUSS coding skills. It will not only
make your code faster, but it will also make it shorter, with more of the details handled under the hood by GAUSS rather than in your program.
Example 1: χ^2 Random Variables
We can create individual χ^2 random numbers like this:
rChi = rndGamma(1, 1, 0.5 .* df, 2);
where df represents the degrees of freedom. Part of a simulation program may include code like this:
for i(1, nobs, 1);
chi_var = rndGamma(1, 1, 0.5.*df, 2);
// more simulation code
Alternatively, you can remove the loop and request rndGamma to return nobs elements in one call, like this:
chiVars = rndGamma(nobs, 1, 0.5 .* df, 2);
This code is simpler, shorter, and runs approximately fifteen times faster. Notice that the .* element-by-element multiplication operator works appropriately in either case. In GAUSS, operators are
overloaded to allow for scalar, vector, or matrix operations. While this is a fairly trivial example, it illustrates the power of this technique.
Example 2: Drawing random samples
Let us suppose that we have a matrix A that contains one million observations of five variables. Let us further suppose that we would like to draw 500 observations from A for each sample, and we
would like to draw a total of 2000 separate random samples with replacement.
Our first step is to create a vector of 2000 uniform random numbers with element values between 1 and one million. You can create a uniform random number between 1 and one million like this:
range = { 1, 1e6 };
rint = rndi(1, 1, range);
You can inefficiently create a 2000x1 vector of random integers with the range of [1, 1e6] like this:
//Create vector to hold random integers
rint = zeros(2000, 1);
//Fill ‘rint’ vector with random integers
for i(1, 2000, 1);
rint[i] = rndi(1, 1, range);
Now take a moment to apply the lesson from example 1 and vectorize this code snippet. The answer is:
rint = rndi(2000, 1, range);
Now we could create a vector of sample observations by extracting them one-by-one as below:
//Create vector to hold sample
samp = zeros(2000, 1);
//Sampling with replacement
for j(1, 2000 1);
idx = rint[j];
samp[j,.] = A[idx,.];
In GAUSS, however, you can index into a matrix or vector with non-consecutive integers. For example, if you had a matrix X and wanted to create a matrix x_small equal to the 9^th row of X over the 5^
th row of X over the 12^th row of X you could perform that action like this:
idx = { 9, 5, 12 };
x_small = X[idx,.];
Since this functionality is available, we can vectorize our sampling with replacement by substituting the following for the for loop above:
samp = A[rint,.];
Below is a comparison of both options for drawing one sample of 2000 observations in complete form:
//Scalar version
range = { 1, 1e6 };
rint = rndi(2000, 1, range);
for j(1, 2000, 1);
idx = rint[j];
samp[j,.] = A[idx,.];
//Vectorized version
rint = rndi(2000, 1, range);
samp = A[rint,.];
As in our first example, the vectorized code is simpler and cleaner. When running the full example drawing 500 separate samples, including the vectorized creation of uniform random integers in both
cases, the vectorized sampling with replacement is just over eight times faster on one machine.
Example 3: Laplacian Cumulative Distribution Function
The formula to calculate the cumulative distribution function for the Laplacian distribution is:
$$ \frac{1}{2}exp(\frac{x-\mu}{b}) \text{ if } x < \mu\\ \\ 1 - \frac{1}{2}exp(\frac{-(x-\mu)}{b}) \text{ if } x >= \mu\\ $$
Assuming x and mu are scalars, the most straightforward way to code this is probably:
if x < mu;
ans = 0.5 * exp((x - mu) ./ b);
ans = 1 - 0.5 * exp(-(x - mu) ./ b);
If x and mu are vectors or matrices, this operation could be done on each corresponding element of the variables in a for loop, but it would be much faster to vectorize it. The first step in
vectorizing this code is to vectorize the logic statements. This involves creating a mask variable that will control which elements in our vector (or matrix) will be processed by which branch of our
algorithm. We will first create a mask for all of the elements in which x < mu, like this:
mask_low = x .< mu;
Notice the . before the < operator. The dot specifies that you want the operation to be performed on an element-by-element basis. This will give us a variable, mask_low, which will have a one if the
corresponding element of x is greater than or equal to mu and a zero otherwise. For example if:
x = 2 mu = 3
Then the code:
mask_low = x .< mu;
will assign mask_low as follows:
mask_low = 0
We can create the mask for the other case like this:
mask_high = x .>= mu;
Now we can use the masks to ‘cover’ the rows to which a particular code branch does not apply like this:
mask_low = x .< mu;
ans_low = mask_low .* (0.5 * exp((x - mu) ./ b));
This calculates the 'low' part of the algorithm on all elements in the vector. It then multiplies the elements to which this algorithm should not be applied by zero and the elements to which it
should be applied by one. The next step is to apply this technique to the other branch:
mask_high = x .>= mu;
ans_high = mask_high .* (1 - 0.5*exp(-(x - mu) ./ b));
Now the sum of these two vectors is our final answer:
ans = ans_low + ans_high;
The full code looks like this:
mask_low = x .< mu;
ans_low = mask_low .* (0.5 .* exp(((x - mu)) ./ b));
mask_high = x .>= mu;
ans_high = mask_high .* (1 - 0.5 .* exp(-(x - mu) ./ b));
ans = ans_low + ans_high;
Not all would argue that this code is simpler than the code above. However, once the idea of masking sets in it will be quite clear. As a side note, any code that might not be as clear to all users
should be commented to show the logic as the code progresses.
For this example, the speed-up obtained by vectorizing the equation rather than using a for loop is between four and five times. This is not as great an improvement as we obtained in the preceding
examples, but it is still quite significant and is what we would expect since we are performing about twice as many mathematical operations in the vectorized version compared to the version with the
if statements.
One final note, the Laplacian cumulative distribution function is calculated by the GAUSS-supplied function cdfLaplace. If we use that GAUSS function, we see performance that is about four times
faster than the vectorized code above and we don’t have to write it ourselves! You should almost always use GAUSS-supplied functions when possible, they will usually be considerably faster and
simpler to implement than to write.
1. GAUSS is designed to work most efficiently on vectors and matrices. If you find yourself performing many operations on individual elements in a vector or matrix, take a step back and consider
whether this case could be described in terms of vector or matrix operations.
2. Vectorized code is usually faster, cleaner, and shorter.
3. Logical comparisons can be vectorized with the ‘dot’ operators.
4. Use vector indexing to extract specific non-consecutive elements from a matrix or vector.
5. Use masking in place of if statements when possible.
6. Search for a GAUSS-supplied function before writing one yourself.
Have a Specific Question?
Get a real answer from a real person
Need Support?
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How To Find The Volume Of A Penny
Volume is the three-dimensional spatial characteristic of an object or container. You can calculate the volume of a penny in one of two ways. The first way is to treat a penny like a small cylinder
and calculate the volume based on its linear measurements — that is, multiply the radius by itself, take that number and multiply it by pi and, finally, multiply the result by the penny's estimated
thickness. This method is not precise, however, because there are raised portions on the surface of the penny that are difficult to measure. The more accurate method is volumetric displacement.
Step 1
Clean the penny with hot soapy water, rinse well and dry with the towel.
Step 2
Fill the graduated cylinder with 10 milliliters of water. The bottom of the meniscus — the concave curve of the water in the cylinder — is the point of measure.
Step 3
Place the penny in the cylinder and let it fall to the bottom. Read the bottom of the meniscus again and record the second volume in milliliters. Care should be taken to read this value. Subtract the
first value, 10 milliliters, from the second volume reading. For example, if you measured 10.3 milliliters, you would subtract 10 from that volume to calculate 0.3 milliliters.
Step 4
Multiply the volume difference calculated in the previous step by 0.061 to express the volume in cubic inches.
Things Needed
• Penny
• Graduated cylinder no larger than 25 milliliters
• Soap
• Towel
• Calculator
TL;DR (Too Long; Didn't Read)
If the volume difference is too difficult to read accurately, clean four more pennies, dry them thoroughly and drop all five pennies in the cylinder. Take the difference in volume and divide by five
to get the volume of a single penny.
Cite This Article
Baer, Brian. "How To Find The Volume Of A Penny" sciencing.com, https://www.sciencing.com/volume-penny-8159012/. 24 April 2017.
Baer, Brian. (2017, April 24). How To Find The Volume Of A Penny. sciencing.com. Retrieved from https://www.sciencing.com/volume-penny-8159012/
Baer, Brian. How To Find The Volume Of A Penny last modified March 24, 2022. https://www.sciencing.com/volume-penny-8159012/ | {"url":"https://www.sciencing.com:443/volume-penny-8159012/","timestamp":"2024-11-11T13:14:55Z","content_type":"application/xhtml+xml","content_length":"70328","record_id":"<urn:uuid:e0b82c53-2bd0-4f7b-8dc0-d4a18b962aa6>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00886.warc.gz"} |
Comments on Computational Complexity: NEW math on FuturamaFuturama: The Beast wiith a Billion Backs comes to...The problem was: if you start out
with the id perm...Lacking all sense of humor ourselves—other than de...Maybe I missed something, but isn't the proble...Is it bad that I paused Futurama and got really cl...>Its funny that cartoons are more real than rea...Moreover, the reverse Turing Test was "What i...Its funny that cartoons are more real than reality...
tag:blogger.com,1999:blog-3722233.post8904114486029569969..comments2024-11-14T17:42:16.782-06:00Lance Fortnowhttp://www.blogger.com/profile/
06752030912874378610noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-3722233.post-9712124225158070442010-08-25T00:49:56.805-05:002010-08-25T00:49:56.805-05:00Futurama: The Beast wiith a
Billion Backs comes to mind!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-22095914252852990172010-08-24T15:55:10.710-05:002010-08-24T15:55:10.710-05:00The problem was: if you
start out<br />with the id perm and you apply<br />swaps (i1,j1),...,(im,jm) to it<br />(all of the swaps are distinct)<br />then can you then get back to id<br />Without using any of (i1,j1),...,
(im,jm).<br />The `without using' is what makes it interesting. The claim is that you can if you allow two more people to join in<br />(or perhaps it only works if they<br />are basketball
playing physicists<br />`Sweet' Clyde Dixon and Ethan `Bubblegum"
Tate.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-75656877099187144552010-08-24T15:30:59.353-05:002010-08-24T15:30:59.353-05:00Lacking all sense of humor ourselves—other than
deadpan—we engineers naturally perceive humor as possessing connections to <a href="http://scottaaronson.com/blog/?p=461#comment-46516" rel="nofollow">serious mathematical topics</a>.John Sidleshttp:
//www.mrfm.orgnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-39455244206993042542010-08-24T14:59:15.637-05:002010-08-24T14:59:15.637-05:00Maybe I missed something, but isn't the
problem simply showing that any element of the symmetric group can be transformed to the identity? <br /><br />I did like that they used cyclic notation using langle/rangle!
Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-22244420905549107032010-08-24T14:50:03.214-05:002010-08-24T14:50:03.214-05:00Is it bad that I paused Futurama and got really close
to the screen to read the theorem/proof when it was flashed up there for a few seconds? :)Daniel
Aponnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-74873570892930411562010-08-24T11:46:17.898-05:002010-08-24T11:46:17.898-05:00>Its funny that cartoons are more real than reality TV
and sitcoms.<br /><br />Books written for children are also way better than those written for teenagers and
adults.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-23690249394084767102010-08-24T11:45:36.519-05:002010-08-24T11:45:36.519-05:00Moreover, the reverse Turing Test was "What
is the square root of 9?"<br /><br />Bender's answer: "Uh, hold on, let me just get out a pencil. (sighs) Okay, look, I'm not that kind of robot."
Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-18616486211891239772010-08-24T08:47:55.549-05:002010-08-24T08:47:55.549-05:00Its funny that cartoons are more real than reality TV
and sitcoms.Blakehttps://www.blogger.com/profile/16116043731993061151noreply@blogger.com | {"url":"https://blog.computationalcomplexity.org/feeds/8904114486029569969/comments/default","timestamp":"2024-11-15T04:27:06Z","content_type":"application/atom+xml","content_length":"15729","record_id":"<urn:uuid:a4850ba2-d833-4d48-91f8-fbbf9f792949>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00896.warc.gz"} |
Algorithm: What is it and what is it used for?
Algorithms are essential in computer science, especially for Data Science and Machine Learning. Find out everything you need to know about them: definition, functioning, use cases, training…
The term “algorithm” derives from the name of the great Persian mathematician Al Khwarizmi, who lived around the year 820 and who introduced decimal numbering to the West (from India) and taught the
elementary arithmetic rules related to it. Subsequently, the concept of algorithm was extended to more and more complex objects: texts, pictures, logical formulas and physical objects, among others.
Already essential in the field of computer programming, algorithms are becoming increasingly important in the age of Big Data and artificial intelligence. So what are they actually? If you are
looking for a clear and complete definition, you are in the right place…
An algorithm is essentially a step-by-step procedure. It is a set of rules to follow to accomplish a task or solve a problem.
Long before the emergence of computers, humans were already using algorithms. We can consider that cooking recipes, mathematical operations or even the instructions for assembling a piece of
furniture are algorithms.
In the field of computer programming, algorithms are sets of rules telling the computer how to perform a task. In reality, a computer program is an algorithm that tells the computer what steps to
perform and in what order to accomplish a specific task. They are written using a programming language.
What are the different types of algorithms?
There are a wide variety of algorithms, classified according to the concepts they use to accomplish a task. Here are the main categories:
Divide-and-conquer algorithms divide a problem into several subproblems of the same type. These smaller problems are solved, and their solutions are combined to solve the original problem.
Brute force algorithms test all possible solutions until the best one is found. A randomized algorithm uses a random number at least once during the calculation to find the solution to the problem.
A gluttonous algorithm finds the optimal solution at the local level, in order to find an optimal solution for the global problem. A recursive algorithm solves the simplest version of a problem and
then solves larger and larger versions until the solution to the original problem is found.
A traceback algorithm divides the problem into sub-problems, which can be tried to solve one after the other. If the solution is not found, it is necessary to go back to the problem until a way is
found to continue to advance.
Finally, a dynamic programming algorithm is used to decompose a complex problem into a collection of simpler sub-problems. All these sub-problems are solved once, and their solution is stored for
future use. This avoids having to recompute their solutions.
What are sorting algorithms?
A sorting algorithm allows placing the elements of a list in a certain order. This can be, for example, a numerical or lexicographical order. This organization is often important as a first step to
solve more complex problems.
There are many sorting algorithms, with their advantages and disadvantages. Here are some examples:
• Linear sorting algorithms find the smallest element of a list, sorts them, adds them to a new list and deletes them from the original list. This process is repeated until the original list is
• Bubble sorting consists of comparing the first two elements of the list, and inverting them if the first is greater than the second. This process is repeated for each pair of adjacent elements in
the list, and until the entire list is sorted.
• Finally, insertion sorting consists of comparing each element in the list with the previous elements until a smaller element is found. The two elements are reversed, and the process is repeated
until the entire list is sorted.
How are algorithms used in computer science?
In computer science, algorithms are omnipresent. They are actually the backbone of computing, since an algorithm gives the computer a specific set of instructions. It is these instructions that allow
the computer to perform its tasks.
Computer programs themselves are algorithms written in programming languages. Algorithms also play a key role in the operation of social networks, for example. They decide which publications are
displayed or which advertisements are offered to the user.
On search engines, algorithms are used to optimize searches, predict what users will type and much more. Similarly, platforms like Netflix, YouTube, Amazon or Spotify rely on algorithms for their
recommendation engines.
Why is it important to understand algorithms?
Beyond computer science, algorithmic thinking is crucial in many fields. It is the ability to define clear steps to solve a problem.
In fact, we use this way of thinking every day and often without even realizing it. In the age of Data Science, Machine Learning and Artificial Intelligence, algorithms are more important than ever
and are the fuel of the new industrial revolution…
What are the main Machine Learning algorithms?
Machine Learning algorithms are programs that can learn from data, and improve autonomously without human intervention by using previous experiences.
Among the learning tasks they are able to perform, these algorithms can, for example, learn the hidden structure of unlabeled data, or “instance-based” learning, which consists of producing a
category label for a new instance by comparing it to training data stored in memory.
There are three main categories of Machine Learning algorithms: supervised, unsupervised, and semi-supervised. Each of these categories is based on a different learning method.
Supervised learning uses labeled training data to learn the mapping function that transforms the input variables or output variables. After this learning, the algorithm can generate outputs from new
Among the supervised learning algorithms, we can talk about classification and regression algorithms. Classification is used to predict the outcome of a given sample when the output variable is in
the form of categories. The classification model analyzes the input data and attempts to predict labels to classify them.
Regression is used to predict the outcome of a sample when the output variable is in real-value form. From the input data, it will be, for example, to predict a volume, a size or a quantity. Examples
of supervised learning algorithms include linear regression, logistic regression, naive Bayesian classification, and the K-nearest neighbor method.
The ensemble method is another type of supervised learning. It consists in combining the predictions of multiple individually weak Machine Learning models to produce a more accurate prediction on a
new sample. Examples include decision tree forest techniques, or boosting with XGBoost.
Unsupervised learning models are used when there is only one input variable and no corresponding output variable. They use unlabeled training data to model the underlying structure of the data. Here
are three examples of techniques:
• Association is used to discover the probability of concurrency of items in a collection. It is widely used for shopping cart analysis in retail, especially to discover which items are frequently
purchased together.
• Clustering is used to group samples so that different items within the same cluster are more similar to each other than to items in another cluster.
• Finally, dimensionality reduction is used to reduce the number of variables within a data set while ensuring that important information is conveyed.
This can be achieved by using feature extraction or feature selection methods. Feature selection involves choosing a subset from the original variables, while extraction performs a transformation of
the data to reduce the dimension. Examples of unsupervised algorithms include k-means and PCA.
Reinforcement learning is a third type of Machine Learning. It allows the agent to decide the best action to take based on its current state, by learning which behaviors maximize its rewards.
In general, reinforcement algorithms learn the optimal actions by trying and failing many times in a row. If we take the example of a video game in which the player must go to a specific location to
earn points, the algorithm will start by moving randomly and then learn where it should go by trying to maximize its rewards.
How to learn to use algorithms?
Knowledge and mastery of algorithms are essential for working in the field of Computer Science, Data Science or Artificial Intelligence.
To acquire this expertise, you can turn to DataScientest’s training program. Our Data Scientist training will teach you how to handle algorithms, and will give you all the skills to become a Data
In addition to algorithms, you will also learn how to manipulate databases and handle Big Data tools, Python programming, and various Machine Learningand Deep Learning techniques.
At the end of the course, you will receive a degree certified by the Sorbonne University and you will be ready to work as a Data Scientist. Among our alumni, 93% found a job immediately after their
All of our training courses adopt a Blended Learning approach combining face-to-face and distance learning, and can be done in BootCamp or Continuous training. | {"url":"https://datascientest.com/en/algorithm-what-is-it","timestamp":"2024-11-08T21:50:55Z","content_type":"text/html","content_length":"444299","record_id":"<urn:uuid:3ead0f7c-c9ac-43ac-a742-375487b8618d>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00175.warc.gz"} |
Rotary Currents
Most tidal currents we deal with on inland waters are reversing currents. That is, from slack water the current builds in the ebb direction to a peak value, on average about 3 hours later, then it
starts to diminish in speed back to another slack—and over this full ebb cycle the current is flowing in more or less the same direction, tabulated as the ebb direction.
Then the current reverses, and carries out the same pattern in the tabulated flood direction. Usually in a reversing current, the ebb and flood directions are nearly opposite to each other, but it is
not uncommon to be out of alignment by 10º or so, and in extreme cases as much as 45º—but even when misaligned there are these two unique directions that define the set in a reversing current.
In stark contrast, there are some locations in open inland waters, and in essentially all coastal waters, where tidal current behavior is quite different. These currents are called rotating currents.
A pure rotating current rotates its direction of flow over the tidal cycle without changing speed. In a pure rotating current there is no peak current speed and no ebb or flow direction.
A pure rotating current, however, is rare. In most cases the current speed does change somewhat as the current rotates, and thus the pattern of the flow is not a circle, but an ellipse, in which case
the long axis of the ellipse can be thought of as defining the ebb and flood directions, as well as the peak values to be expected.
The rotation is created by the interaction of two tidal waves moving in different directions. Either one of these waves on its own would create a simple revering current, but where these waves cross
the resulting current shows a rotation pattern at that location. The shape of the rotation pattern depends on the relative amplitudes, times, and directions of the crossing tidal waves. In the
Northern Hemisphere the rotation is usually clockwise. Running farther into a large open estuary with various channels, the same tidal waves will later create reversing currents.
A key point for navigators is to know and expect this different behavior of the current once sailing into large open bays or into coastal waters. Sometimes the official
NOAA Tidal Current Tables
alerts us to this with reference in the list of secondary stations to special tables or diagrams for rotating current stations, but in other parts of the world there is no warning. We just have to
know it happens and search through the references for the data. On some Canadian charts, rotary current diagrams are printed right on the chart at the pertinent locations, but we do not see this
often on US charts.
Once we know the current rotates where we care about, and we find the data describing it, we are faced with how to use this information. The procedure for figuring time, set, and drift of a rotary
current is different and more involved than is the same process for a reversing current.
For a reversing current, we go to the Tidal Current Tables, find the nearest station, and look up the peak and slack times and speeds. Then use the interpolation table from the book to figure speeds
at other times or use a shortcut like the
Starpath 50-90 Rule
. Or much easier, go online to
and look up the data already corrected for secondary stations. Or easier still, open an echart program that includes a tides and currents utility, and click the place on the chart you want, set the
calendar and clock, and read off the current for that specific time, or print a plot of current versus time for a day or so. This feature of electronic charting is one of its strong selling points
for those who sail tidal waters.
For rotary currents, however, none of these easy methods work. Furthermore, you must have the official NOAA Tidal Current Tables at hand. Abbreviated versions rarely have the needed information, and
it is not online, and it is not included with echart current utilities. Rotary current predictions are too complex to be tabulated, so we are left with approximations.
To show the process, we look just SW of Cuttyhunk Island in Rhode Island Sound (Current station #749). A section of Table 5 from the Current Tables shows how this current rotates at this location.
The currents are not strong at this location, but other places rotating current can be quite strong. On Nantucket Shoals or the entrance to Strait of Juan de Fuca, or many places in Alaska rotating
currents can routinely be well over 2 kts.
Also note that within just 3 to 6 miles of this rotating current there are 3 stations with purely reversing currents. One just inside Buzzard’s Bay (#773) and two just inside Vineyard Sound (#747 and
#745). There are numerous examples around the country where pure reversing switches to rotating in just a few miles.
The main reference to using the data are the Instructions to Table 5 in the Tidal Current Tables, but I venture to guess that not all navigators who might benefit from the information have seen them,
so we offer a brief summary here till they get a chance.
The current speeds shown in Table 5 are the monthly averages. The times of each current are relative to the time of max flood at Poll0ck Rip Channel, which is a primary reference station in this
area. Thus that one time sets the times for the currents at each hour after that. If for example the max flood at Pollock Rip occurred at 0551, then the average Cuttyhunk current at 0951 (4h later)
would be 0.5 kts, setting toward 146 T. Since we know the current each hour, if we knew the time period we expected to be in this current we could figure the net tidal vector and from this figure our
CMG and SMG as we transited the area.
But we are not done. These are average speeds, and just as with reversing currents, the speeds are stronger at spring tides (new moon and full moon) and weaker at neap tides (quarter moons, which
appear in the sky as half-moons). As a rough rule, spring tides give rise to currents about 20% larger than the average, and neap tides yield currents about 20% lower than the average at any
particular location. And this can be fine tuned even farther taking into account the location of the moon in its orbit at these phases. When the moon is closest to the earth (perigee) at the springs,
the x 1.2 factor increases to x 1.4, and when the moon is farthest from the earth (apogee) at the neaps, the x 0.83 factor (1/1.2) is reduced to x 0.71 (1/1.4).
With reversing currents we do not have to worry about this. The astronomical influences are built into the current table predictions. But for rotating currents we get only the averages and have to
apply the astronomical corrections ourselves. The needed astro data are on the inside back cover of the current tables and also online. This year, for example, July 12 is full moon and July 13 is
perigee, so we expect currents throughout the region to be some 40% larger than average on these days, which you can see in the current tables for all reference stations in the region. On these two
days, however, we would need to increase the tabulated rotary currents by this factor ourselves.
Sailing in rotary currents takes extra work, but knowing how to figure them and how to account for their influence on neighboring reversing current helps you make good predictions ahead of time that
in turn helps you interpret what you see on the GPS.
Excerpt from Table 5, NOAA Tidal Current Tables
Section of chart 13218 showing four current stations, three are reversing, one is rotating (#749, Cuttyhunk Is, 3.25 mi SW). The average speeds at the stations are shown to scale, with the average
ebb at #747 being 2.0 kts. The current predictions are for the precise location of the station dots, not along the arrows. Away from the precise station locations we must infer the currents from
neighboring predictions.
We might expect the converging max ebb currents near point B to be near southerly, with some element of rotation. The current NW of #773 is similar to that at #773, but we can expect the peak flood
current near point A to be curving to NE with some element of rotation. An echart program with a range and bearing tool is an easy way to make a rotary diagram as shown above. The vectors correspond
to the ones shown in Table 5.
We have exercises on rotary currents in our new
Navigation Workbook 1210 Tr
, which are discussed in more detail in our text
Inland and Coastal Navigation | {"url":"http://davidburchnavigation.blogspot.com/2014/09/rotary-currents.html","timestamp":"2024-11-03T21:59:15Z","content_type":"text/html","content_length":"95118","record_id":"<urn:uuid:e733d70d-7a0e-46b7-9fd3-3b220ee41559>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00606.warc.gz"} |
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D is statically typed. Every expression has a type. Types constrain the values an expression can hold, and determine the semantics of operations on those values.
Basic data types are leaf types. Derived data types build on leaf types. User defined types are aggregates of basic and derived types.
Basic Data Types
Keyword Default Initializer (.init) Description
void - no type
bool false boolean value
byte 0 signed 8 bits
ubyte 0u unsigned 8 bits
short 0 signed 16 bits
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int 0 signed 32 bits
uint 0u unsigned 32 bits
long 0L signed 64 bits
ulong 0uL unsigned 64 bits
cent 0 signed 128 bits (reserved for future use)
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float float.nan 32 bit floating point
double double.nan 64 bit floating point
real real.nan largest FP size implemented in hardwareImplementation Note: 80 bits for x86 CPUs or double size, whichever is larger
ifloat float.nan*1.0i imaginary float
idouble double.nan*1.0i imaginary double
ireal real.nan*1.0i imaginary real
cfloat float.nan+float.nan*1.0i a complex number of two float values
cdouble double.nan+double.nan*1.0i complex double
creal real.nan+real.nan*1.0i complex real
char 'xFF' unsigned 8 bit (UTF-8 code unit)
wchar 'uFFFF' unsigned 16 bit (UTF-16 code unit)
dchar 'U0000FFFF' unsigned 32 bit (UTF-32 code unit)
• pointer
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• delegate
Strings are a special case of arrays.
The base type of an enum is the type it is based on:
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Casting pointers to non-pointers and vice versa is allowed.
Best Practices: do not do this for any pointers that point to data allocated by the garbage collector.
Implicit conversions are used to automatically convert types as required.
An enum can be implicitly converted to its base type, but going the other way requires an explicit conversion. For example:
int i;
enum Foo { E }
Foo f;
i = f; // OK
f = i; // error
f = cast(Foo)i; // OK
f = 0; // error
f = Foo.E; // OK
Integer Promotions are conversions of the following types:
from to
bool int
byte int
ubyte int
short int
ushort int
char int
wchar int
dchar uint
If an enum has as a base type one of the types in the left column, it is converted to the type in the right column.
The usual arithmetic conversions convert operands of binary operators to a common type. The operands must already be of arithmetic types. The following rules are applied in order, looking at the base
1. If either operand is real, the other operand is converted to real.
2. Else if either operand is double, the other operand is converted to double.
3. Else if either operand is float, the other operand is converted to float.
4. Else the integer promotions are done on each operand, followed by:
1. If both are the same type, no more conversions are done.
2. If both are signed or both are unsigned, the smaller type is converted to the larger.
3. If the signed type is larger than the unsigned type, the unsigned type is converted to the signed type.
4. The signed type is converted to the unsigned type.
If one or both of the operand types is an enum after undergoing the above conversions, the result type is:
1. If the operands are the same type, the result will be the that type.
2. If one operand is an enum and the other is the base type of that enum, the result is the base type.
3. If the two operands are different enums, the result is the closest base type common to both. A base type being closer means there is a shorter sequence of conversions to base type to get there
from the original type.
Integer values cannot be implicitly converted to another type that cannot represent the integer bit pattern after integral promotion. For example:
ubyte u1 = -1; // error, -1 cannot be represented in a ubyte
ushort u2 = -1; // error, -1 cannot be represented in a ushort
uint u3 = int(-1); // ok, -1 can be represented in a uint
ulong u4 = long(-1); // ok, -1 can be represented in a ulong
Floating point types cannot be implicitly converted to integral types. Complex or imaginary floating point types cannot be implicitly converted to non-complex floating point types. Non-complex
floating point types cannot be implicitly converted to imaginary floating point types.
The bool type is a byte-size type that can only hold the value true or false.
The only operators that can accept operands of type bool are: & | ^ &= |= ^= ! && || ?:.
A bool value can be implicitly converted to any integral type, with false becoming 0 and true becoming 1.
The numeric literals 0 and 1 can be implicitly converted to the bool values false and true, respectively. Casting an expression to bool means testing for 0 or !=0 for arithmetic types, and null or !=
null for pointers or references.
Delegates are an aggregate of two pieces of data: an object reference and a pointer to a non-static member function, or a pointer to a closure and a pointer to a nested function. The object reference
forms the this pointer when the function is called.
Delegates are declared similarly to function pointers:
int function(int) fp; // fp is pointer to a function
int delegate(int) dg; // dg is a delegate to a function
A delegate is initialized analogously to function pointers:
int func(int);
fp = &func; // fp points to func
class OB
int member(int);
OB o;
dg = &o.member; // dg is a delegate to object o and
// member function member
Delegates cannot be initialized with static member functions or non-member functions.
Delegates are called analogously to function pointers:
fp(3); // call func(3)
dg(3); // call o.member(3)
The equivalent of member function pointers can be constructed using anonymous lambda functions:
class C
int a;
int foo(int i) { return i + a; }
// mfp is the member function pointer
auto mfp = function(C self, int i) { return self.foo(i); };
auto c = new C(); // create an instance of C
mfp(c, 1); // and call c.foo(1)
The C style syntax for declaring pointers to functions is deprecated:
int (*fp)(int); // fp is pointer to a function
size_t is an alias to one of the unsigned integral basic types, and represents a type that is large enough to represent an offset into all addressible memory.
ptrdiff_t is an alias to the signed integral basic type the same size as size_t. | {"url":"https://docarchives.dlang.io/v2.084.0/spec/type.html","timestamp":"2024-11-01T20:29:48Z","content_type":"text/html","content_length":"35273","record_id":"<urn:uuid:a7286ad8-c047-487a-b1a3-3c51cde465f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00034.warc.gz"} |
What does 1x2 mean in betting - Best betting sites in Nigeria
What does 1×2 mean in betting
“1X2” (“1H2”) is a bookmaker’s market, which implies a bet on the outcome of an event, taking into account the safety net. In the list, the bookmaker sets up three positions within the market:
1. “1X” – the hosts will not lose: the match will end either with the victory of the first team or a draw.
2. “X2” – the guests will not lose: the match will end either with the victory of the second team or a draw.
3. “12” – there will be no draw: the match will end with the victory of one of the teams.
This market is characterized by a low level of risk, but bookmakers often put a huge margin into it, which significantly lowers quotes and makes the market unprofitable. This position is often
referred to as “Double Chance” by players but is rarely used in the game. Exceptions are bets in favor of an outsider, where an outcome with a draw is allowed.
The probability of a draw
It is football matches that have an approximately 33% chance of ending in a draw if the teams are approximately equal in terms of training. Players in teams are also often chosen in approximately the
same way. It should also be taken into account the fact that in many leagues and championships, the game is practiced from the defense. All these factors contribute to the end of matches in a draw.
In the table, you can see the approximate odds of a draw in the leagues of different countries. And although it is better to collect your statistics, nevertheless, the data in the table suggests that
the probability of a draw remains quite high, which can be used to make a profit in bets.
When predicting a draw, consider the statistics of each team’s games, the average number of goals scored per match, the composition of the teams, the physical condition of key players, and their
A few effective strategies
Consider the most popular strategies for a draw:
• Double bet. The player makes 2 bets – on a draw and total odd, that is, an odd number of goals scored. The result is the so-called. fork. By trading through a betting exchange, you can close the
sure bet so that you can make a profit no matter what the outcome. If you make bets through a bookmaker, you can also make CashOut (early withdrawal of money from a bet) or wait for the end of
the match.
• Flat. The player simply bets the same size on a draw. At a distance, 1 bet out of 3 wins. You can start betting after 1-2 losses on paper, increasing your chances.
• Martingale method. A risky option, when in case of losing a bet, the size of the next one increases. You need patience, iron endurance, and a large deposit (gaming bank).
• 2 out of 5. We work with 5 matches, where the weak favorite will play on the wrong field, and the teams are approximately equal in strength. Now we need to make all possible combinations of
double accumulators using data from 5 matches. The size of the bet for each accumulator must be the same. By guessing at least 2 accumulator bets out of 5, the player makes a profit.
Why is it important to know all the decoding of bets in bookmakers
Most of the players lose only because they operate the lines of the main outcomes and totals. In such markets, the bookmaker’s line is balanced and it is difficult to take an advantageous position.
Things are different if the bettor looks deep into the line and notices obvious distortions in the market for interconnected bets and specific types of bets.
Detailed analytics of the betting line will tell you what to expect in the match more clearly than a dozen statistical resources. Notation is the language of the bookmaker, with the help of which he
communicates with the player. Therefore, it is so important to know the decoding of each outcome in the BC line.
In live, where the ultimate concentration and high decision-making speed are required from the bettor, the responsibility increases. Therefore, it is important to operate with the subtleties and
nuances of notation, reading the possible scenario of the match along the way.
Deciphering rates: how to understand
Just as professionals cannot do without calculators for calculating bets, so novice bettors cannot do without deciphering bet outcomes. The more knowledgeable the bettor is, the easier it is for him
to argue with the bookmaker. Especially if there are conflict situations in the calculation of rates.
In addition, if you know the symbols in football betting well, then other sports will easily submit, where the basic description of events is based on well-known symbols.
The advantage of betting on a draw is a fairly frequent outcome of the match, which means that by analyzing it is possible to predict this event. Also, be aware of the high odds. The disadvantage is
difficult predictability at the start and finish of competition seasons. | {"url":"https://takecare19.com/what-does-1x2-mean-in-betting/","timestamp":"2024-11-02T22:07:31Z","content_type":"text/html","content_length":"50725","record_id":"<urn:uuid:857a2035-4f18-4482-b564-7670cf9d2cdc>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00673.warc.gz"} |
A Parallel Algorithm for the Knapsack Problem for IEEE TC
A time-memory-processor tradeoff for the knapsack problem is proposed. While an exhaustive search over all possible solutions of an n-component knapsack requires T = 0(2n) running time, our parallel
algorithm solves the problem in 0(2n/2) operations and requires only 0(2n/6) processors and memory cells. It is an improvement over previous time-memory-processor tradeoffs, being the only one which
outperforms the Cm Cs = 2ncurve. Cm is the cost of the machine, i.e., the number of its processors and memory cells, and Cs is the cost per solution, which is the product of the machine cost by the
running time. © 1984 IEEE | {"url":"https://research.ibm.com/publications/a-parallel-algorithm-for-the-knapsack-problem","timestamp":"2024-11-11T15:24:24Z","content_type":"text/html","content_length":"66554","record_id":"<urn:uuid:67f3c8ad-8ceb-455e-b409-f7d1b1777608>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00049.warc.gz"} |
seminars - On the prime Selmer ranks of cyclic prime twist families of elliptic curves over global function fields
Fix a prime number $p$. Let $mathbb{F}_q$ be a finite field of characteristic coprime to $2,3,$ and $p$ containing the primitive $p$-th roots of unity $mu_p$. Based on the works by Swinnerton-Dyer
and Klagsbrun, Mazur, and Rubin, we prove that the probability distribution of the sizes of prime Selmer groups over a family of cyclic prime twists of non-isotrivial elliptic curves over $mathbb{F}
_q(t)$ satisfying a number of mild constraints conforms to the distribution conjectured by Bhargava, Kane, Lenstra, Poonen, and Rains with explicit error bounds. The key tools used in proving these
results are the Riemann hypothesis over global function fields, the Chebotarev density theorem, the Erd"os-Kac theorem, and the geometric ergodicity of Markov chains. | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=seminars&page=85&sort_index=room&order_type=asc&l=en&document_srl=1129435","timestamp":"2024-11-08T06:20:44Z","content_type":"text/html","content_length":"45628","record_id":"<urn:uuid:73d6ceb7-eab3-44a9-b9fb-1769b6738f7b>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00067.warc.gz"} |